# Qualitative properties of solutions to partial differential equationsl

“volumeV” — 2009/8/3 — 0:35 — page i — #1 Jindřich Nečas Center for Mathematical Modeling Lecture notes Volume 5 Qualitative properties of solutions to partial differential equations Volume edited by E. Feireisl, P. Kaplický and J. Málek Dedicated to the memory of Professor Tetsuro Miyakawa “volumeV” — 2009/8/3 — 0:35 — page ii — #2 “volumeV” — 2009/8/3 — 0:35 — page iii — #3 Jindřich Nečas Center for Mathematical Modeling Lecture notes Volume 5 Editorial board Michal Beneš Pavel Drábek Eduard Feireisl Miloslav Feistauer Josef Málek Jan Malý Šárka Nečasová Jiřı́ Neustupa Antonı́n Novotný Kumbakonam R. Rajagopal Hans-Georg Roos Tomáš Roubı́ček Daniel Ševčovič Vladimı́r Šverák Managing editors Petr Kaplický Vı́t Průša “volumeV” — 2009/8/3 — 0:35 — page iv — #4 “volumeV” — 2009/8/3 — 0:35 — page v — #5 Jindřich Nečas Center for Mathematical Modeling Lecture notes Qualitative properties of solutions to partial differential equations Dedicated to the memory of Professor Tetsuro Miyakawa Dorin Bucur Alessandro Morando Laboratoire de Mathématiques CNRS UMR 5127 Université de Savoie, Campus Scientiﬁque 73376 Le-Bourget-Du-Lac France Dipartimento di Matematica Università di Brescia Via Valotti 9 25133 Brescia Italy Grzegorz Karch Instytut Matematyczny Uniwersytet Wroclawski pl. Grunwaldzki 2/4 50-384 Wroclaw Poland Roger Lewandowski UMR 6625 Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France Paolo Secchi Dipartimento di Matematica Università di Brescia Via Valotti 9 25133 Brescia Italy Andro Mikelić Université de Lyon Lyon, F-69003 France Université Lyon 1, Institut Camille Jordan, UMR CNRS Bât. Braconnier 43, Bd du onze novembre 1918 69622 Villeurbanne Cedex France Paola Trebeschi Dipartimento di Matematica Università di Brescia Via Valotti 9 25133 Brescia Italy Volume edited by E. Feireisl, P. Kaplický and J. Málek “volumeV” — 2009/8/3 — 0:35 — page vi — #6 2000 Mathematics Subject Classification. 35-06, 35B99 Key words and phrases. partial differential equations, qualitative properties, geometric perturbation, rough domains, anomalous diffusion, hyperbolic systems Abstract. The text provides a record of lectures given by the visitors of the Jindřich Nečas Center for Mathematical Modeling in academic years 2006-2009. The lecture notes are focused on qualitative properties of solutions to evolutionary equations. All rights reserved, no part of this publication may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior written permission of the publisher. c Jindřich Nečas Center for Mathematical Modeling, 2009 c MATFYZPRESS Publishing House of the Faculty of Mathematics and Physics Charles University in Prague, 2009 ISBN 978-80-7378-088-3 “volumeV” — 2009/8/3 — 0:35 — page vii — #7 Preface The aim of the present volume is to acquaint the interested reader with various qualitative properties of solutions to evolutionary equations. The topics written by leading experts in their respective ﬁelds are not necessarily related. A part of the volume consists of lecture notes of the international summer school EVEQ 2008, held in Prague, 16–20 June 2008. The contributions are presented in alphabetical order according to the name of the ﬁrst author. The article by Dorin Bucur documents a series of lectures delivered by the author at the Nečas Center for Mathematical Modeling in 2006 and 2007. Its aim is to study the behavior of solutions to certain partial diﬀerential equations posed on domains with the rough (rapidly oscillating) boundaries. Grzegorz Karch in his EVEQ lecture addresses a new topic, namely evolutionary equations with anomalous diffusion. The contribution of Roger Lewandowski is devoted to problems related to turbulence associated with ﬂuid motions. The paper of Andro Mikelić is closely related to that of Dorin Bucur. It addresses the problem of eﬀective boundary conditions on domains with rough boundaries. The ﬁnal contribution to the volume is written by another EVEQ lecturer Paolo Secchi and his collaborators Alessandro Morando and Paola Trebeschi. Here, they present general results concerning regularity of solutions to hyperbolic systems with characteristic boundary. We ﬁrmly believe that the fascinating variety of rather diﬀerent topics covered in this volume will contribute to inspiring and motivating research studies in the future. This volume is dedicated to the memory of Tetsuro Miyakawa. He visited the Nečas Center spending two months in Prague in fall 2008 as a senior lecturer. He gave a series of lectures on “On the existence and asymptotic behavior of dissipative 2D quasi-geostrophic ﬂows” and we felt that the extended form of his lecture notes should be included in this volume. However, his sudden death makes this impossible. We are very thankful to Yoshiyuki Kagei for a commemorative note with the complete list of research papers of Tetsuro Miyakawa. Prague, 30 June 2009 Eduard Feireisl Petr Kaplický Josef Málek vii “volumeV” — 2009/8/3 — 0:35 — page viii — #8 “volumeV” — 2009/8/3 — 0:35 — page ix — #9 Tetsuro Miyakawa (1948–2009) Tetsuro Miyakawa was born on March 10, 1948, in a small city in the middlenorth part of Japan. He suddenly passed away on February 11, 2009. He made major contributions to the ﬁeld of mathematical analysis of the incompressible Navier–Stokes equation. He analyzed this equation by his sophisticated technique with great insight and established signiﬁcant results. He developed an Lp semigroup approach to the Navier-Stokes equation, which has become a fundamental framework in the analysis of this ﬁeld. He introduced various function spaces suited to the analysis of the Navier–Stokes equation. One of his main contributions is found in the theory of weak solutions of the Navier–Stokes ﬂows in exterior domains to which he devoted much of his energies in the prime of his life. His recent works concern with space-time asymptotic behavior of solutions in unbounded domains. It seems to me that his last interest was still in the decay properties of weak solutions in exterior domains. He was very kind, especially to young people. Fukuoka, 30 June 2009 Yoshiyuki Kagei Bibliography 1. Inoue, A., Miyakawa, T., and Yoshida, K. : Some properties of solutions for semilinear heat equations with time lag, J. Diﬀerential Equations 24 (1977), 383–396. ix “volumeV” — 2009/8/3 — 0:35 — page x — #10 x TETSURO MIYAKAWA (1948–2009) 2. Inoue, A., Miyakawa, T. : On the existence of solutions for linearized Euler’s equation, Proc. Japan Acad. 55A (1979), 282–285. 3. Miyakawa, T. : The Lp approach to the Navier–Stokes equations with the Neumann boundary condition, Hiroshima Math. J. 10 (1980), 517–537. 4. Miyakawa, T. : On the initial value problem for the Navier–Stokes equations in Lp spaces, Hiroshima Math. J. 11 (1981), 9–20. 5. Miyakawa, T. : On nonstationary solutions of the Navier–Stokes equations in an exterior domain, Hiroshima Math. J. 12 (1982), 115–140. 6. Miyakawa, T. and Teramoto, Y. : Existence and periodicity of weak solutions of the Navier–Stokes equations in a time dependent domain, Hiroshima Math. J. 12 (1982), 513–528. 7. Giga, Y. and Miyakawa, T. : A kinetic construction of global solutions of ﬁrst order quasilinear equations, Duke Math. J. 50 (1983), 505–515. 8. Miyakawa, T. : A kinetic approximation of entropy solutions of ﬁrst order quasilinear equations, “Recent Topics in Nonlinear PDE, Hiroshima”, Ed. by M. Mimura and T. Nishida, Lecture Notes in Num. Appl. Anal. 6 (1983), 93–105, North-Holland Publ. Co., Amsterdam. 9. Miyakawa, T. : Construction of solutions of a semilinear parabolic equation with the aid of the linear Boltzmann equation, Hiroshima Math. J. 14 (1984), 299–310. 10. Giga, Y. and Miyakawa, T. : Solutions in Lr of the Navier–Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267–281. 11. Giga, Y., Miyakawa, T. and Oharu, S. : A kinetic approach to general ﬁrst order quasilinear equations, Trans. Amer. Math. Soc. 287 (1985), 723–743. 12. Kajikiya, R. and Miyakawa, T. : On L2 decay of weak solutions of the Navier– Stokes equations in Rn , Math. Z. 192 (1986), 135–148. 13. Borchers, W. and Miyakawa, T. : L2 decay for the Navier–Stokes ﬂow in halfspaces, Math. Ann. 282 (1988), 139–155. 14. Giga, Y., Miyakawa, T. and Osada, H. : Two-dimensional Navier–Stokes ﬂow with measures as initial vorticity, Arch. Rational Mech. Anal. 104 (1988), 223–250. 15. Miyakawa, T. and Sohr, H. : On energy inequality, smoothness and large time behavior in L2 for weak solutions of the Navier–Stokes equations in exterior domains, Math. Z. 199 (1988), 455–478. 16. Giga, Y. and Miyakawa, T. : Navier–Stokes ﬂow in R3 with measures as initial vorticity and Morrey spaces, Comm. Partial Diﬀerential Equations 14 (1989), 577–618. 17. Borchers, W. and Miyakawa, T. : Algebraic L2 decay for Navier–Stokes ﬂows in exterior domains, Acta Math. 165 (1990), 189–227. 18. Miyakawa, T. : On Morrey spaces of measures: basic properties and potential estimates, Hiroshima Math. J. 20 (1990), 213–222. 19. Borchers, W. and Miyakawa, T. : On large time behavior of the total kinetic energy for weak solutions of the Navier–Stokes equations in unbounded domains, The Navier–Stokes Equations, Theory and Numerical Methods, Ed. by J. Heywood, K. Masuda, R. Rautmann and V. A. Solonnikov, Lecture Notes in Math. 1431, Springer-Verlag, Berlin, 1990. “volumeV” — 2009/8/3 — 0:35 — page xi — #11 TETSURO MIYAKAWA (1948–2009) xi 20. Borchers, W. and Miyakawa, T. : Algebraic L2 decay for Navier–Stokes ﬂows in exterior domains, II, Hiroshima Math. J. 21 (1991), 621–640. 21. Miyakawa, T. and Yamada, M. : Planar Navier–Stokes ﬂows in a bounded domain with measures as initial vorticities, Hiroshima Math. J. 22 (1992), 401–420. 22. Borchers, W. and Miyakawa, T. : L2 -decay for Navier–Stokes ﬂows in unbounded domains, with applications to exterior stationary ﬂows, Arch. Rational Mech. Anal. 118 (1992), 273–295. 23. Chen, Z.-M., Kagei, Y. and Miyakawa, T. : Remarks on stability of purely conductive steady states to the exterior Boussinesq problem, Adv. Math. Sci. Appl. 1 (1992), 411–430. 24. Borchers, W. and Miyakawa, T. : On some coercive estimates for the Stokes problem in unbounded domains, The Navier–Stokes Equations II, Theory and Numerical Methods, Ed. by J. Heywood, K. Masuda, R. Rautmann and V. A. Solonnikov, Lecture Notes in Math. 1530, Springer-Verlag, Berlin, 1992. 25. Miyakawa, T. : The Helmholtz decomposition of vector ﬁelds in some unbounded domains, Math. J. Toyama Univ. 17 (1994), 115–149. 26. Borchers, W. and Miyakawa, T. : On stability of exterior stationary Navier– Stokes ﬂows, Acta Math. 174 (1995), 311–382. 27. Miyakawa, T. : On uniqueness of steady Navier–Stokes ﬂows in an exterior domain, Adv. Math. Sci. Appl. 5 (1995), 411–420. 28. Miyakawa, T. : Hardy spaces of solenoidal vector ﬁelds, with applications to the Navier–Stokes equations, Kyushu J. Math. 50 (1996), 1–64. 29. Miyakawa, T. : Remarks on decay properties of exterior stationary Navier– Stokes ﬂows. Proceedings of the Mathematical Society of Japan International Research Institute in Nonlinear Waves, Sapporo, July, 1995, ; Gakko-Tosho Publ., Tokyo, 1997. 30. Miyakawa, T. : On L1 stability of stationary Navier–Stokes ﬂows in Rn , J. Math. Sci. Univ. Tokyo 4 (1997), 67–119. 31. Chen, Z.-M. and Miyakawa, T. : Decay properties of weak solutions to a perturbed Navier–Stokes system in Rn , Adv. Math. Sci. Appl. 7 (1997), 741–770. 32. Miyakawa, T. : Application of Hardy space techniques to the time-decay problem for incompressible Navier–Stokes ﬂows in Rn , Funkcial. Ekvac. 41 (1998), 383– 434. 33. Miyakawa, T. : On stationary incompressible Navier–Stokes ﬂows with fast decay and the vanishing ﬂux condition, Progress in Nonlinear Diﬀerential Equations and Their Applications 35, Ed. by J. Escher and G. Simonett, Birkhäuser, Basel–Berlin–Boston, 1999, pp. 535–552. 34. Miyakawa, T. : On space-time decay properties of nonstationary incompressible Navier–Stokes ﬂows in Rn , Funkcial. Ekvac. 43 (2000), 541–557. 35. Fujigaki, Y. and Miyakawa, T. : Asymptotic proﬁles of nonstationary incompressible Navier–Stokes ﬂows in the whole space, SIAM J. Math. Anal. 33 (2001), 523–544. 36. Fujigaki, Y. and Miyakawa, T. : Asymptotic proﬁles of nonstationary incompressible Navier–Stokes ﬂows in the half-space, Methods Appl. Anal. 8 (2001), 121–158. “volumeV” — 2009/8/3 — 0:35 — page xii — #12 xii TETSURO MIYAKAWA (1948–2009) 37. Miyakawa, T. and Schonbek, M.E. : On optimal decay rates for weak solutions to the Navier–Stokes equations in Rn , Mathematica Bohemica 126 (2001), 443– 455. 38. Miyakawa, T. : Asymptotic proﬁles of nonstationary incompressible Navier– Stokes ﬂows in Rn and Rn+ , The Navier–Stokes equations: theory and numerical methods, Ed. by R. Salvi, Proceedings of the International Conference at Varenna, Lecture Notes in Pure and Appl. Math. 223, Marcel Dekker Inc., New York, 2002, pp. 205–219. 39. Miyakawa, T. : Notes on space-time decay properties of nonstationary incompressible Navier–Stokes ﬂows in Rn , Funkcial Ekvac. 45 (2002), 271–289. 40. Miyakawa, T. : On upper and lower bounds of rates of decay for nonstationary Navier–Stokes ﬂows in the whole space, Hiroshima Math. J. 32 (2002), 431–462. 41. Fujigaki, Y. and Miyakawa, T. : On solutions with fast decay of nonstationary Navier–Stokes system in the half-space, Nonlinear Problems in Mathematical Physics and Related Topics, dedicated to O. A. Ladyzhenskaya, Ed. by V. A, Solonnikov et al, Int. Math. Ser. (N. Y.), 1, Kluwer/Plenum, New York, 2002. 42. He, C. and Miyakawa, T. : On L1 -summability and asymptotic proﬁles for smooth solutions to Navier–Stokes equations in a 3D exterior domain, Math. Z. 245 (2003), 387–417. 43. Miyakawa, T. : d’Alembert’s paradox and the integrability of pressure for nonstationary incompressible Euler ﬂows in a two-dimensional exterior domain, Kyushu J. Math. 60 (2006), 345–361 44. He, C. and Miyakawa, T. : Nonstationary Navier–Stokes ﬂows in a two-dimensional exterior domain with rotational symmetries, Indiana Univ. Math. J. 55 (2006), 1483–1555. 45. He, C. and Miyakawa, T. : On two-dimensional Navier–Stokes ﬂows with rotational symmetries, Funkcial. Ekvac. 49 (2006), 163–192. 46. Tun, May Thi and Miyakawa, T. : A proof of the Helmholtz decomposition of vector ﬁelds over the half-space, Adv. Math. Sci. Appl. 18 (2008), 199–217. 47. He, C. and Miyakawa, T. : On weighted-norm estimates for nonstationary incompressible Navier–Stokes ﬂows in a 3D exterior domain, J. Diﬀerential Equations 246 (2009), 2355–2386. “volumeV” — 2009/8/3 — 0:35 — page xiii — #13 Contents Preface vii Tetsuro Miyakawa (1948–2009) ix Part 1. The rugosity eﬀect Dorin Bucur 1 Chapter 1. Some classical examples 1. Introduction 2. The example of Cioranescu and Murat: a strange term coming from somewhere else 3. Babuška’s paradox 4. The Courant–Hilbert example for the Neumann–Laplacian spectrum 5. The rugosity eﬀect 5 5 5 6 8 8 Chapter 2. Variational analysis of the rugosity eﬀect 1. Scalar elliptic equations with Dirichlet boundary conditions 2. The rugosity eﬀect in ﬂuid dynamics 11 11 16 Bibliography 23 Part 2. Nonlinear evolution equations with anomalous diﬀusion Grzegorz Karch 25 Chapter 1. Lévy operator 1. Probabilistic motivations – Wiener and Lévy processes 2. Convolution semigroup of measures and Lévy operator 3. Fractional Laplacian 4. Maximum principle 5. Integration by parts and the Lévy operator 29 29 31 35 36 39 Chapter 2. Fractal Burgers equation 1. Statement of the problem 2. Viscous conservation laws and rarefaction waves 3. Existence o solutions 4. Decay estimates 5. Convergence toward rarefaction waves for α ∈ (1, 2) 6. Self-similar solution for α = 1 7. Linear asymptotics for 0 < α < 1 45 45 46 47 48 49 50 51 xiii “volumeV” — 2009/8/3 — 0:35 — page xiv — #14 xiv CONTENTS 8. Probabilistic summary 52 Chapter 3. Fractal Hamilton–Jacobi–KPZ equations 1. Kardar, Parisi & Zhang and Lévy operators 2. Assumptions and preliminary results 3. Large time asymptotics – the deposition case 4. Large time asymptotics – the evaporation case 53 53 54 56 58 Chapter 4. Other equations with Lévy operator 1. Lévy conservation laws 2. Nonlocal equation in dislocation dynamics 59 59 61 Bibliography 65 Part 3. On a continuous deconvolution equation Roger Lewandowski 69 Chapter 1. Introduction and main facts 1. General orientation 2. Towards the models 3. Approximate deconvolution models 4. The deconvolution equation and outline of the remainder 73 73 74 75 76 Chapter 2. Mathematical tools 1. General background 2. Basic Helmholtz ﬁltration 79 79 80 Chapter 3. From discrete to continuous deconvolution operator 1. The van Cittert algorithm 2. The continuous deconvolution equation 3. Various properties of the deconvolution equation 4. An additional convergence result 83 83 84 85 86 Chapter 4. Application to the Navier–Stokes equations 1. Dissipative solutions to the Navier–Stokes equations 2. The deconvolution model 89 89 91 Bibliography 101 Part 4. Rough boundaries and wall laws Andro Mikelić 103 Chapter 1. Rough boundaries and wall laws 1. Introduction 2. Wall law for Poisson’s equation with the homogeneous Dirichlet condition at the rough boundary 3. Wall laws for the Stokes and Navier–Stokes equations 4. Rough boundaries and drag minimization 107 107 Bibliography 131 108 120 129 “volumeV” — 2009/8/3 — 0:35 — page xv — #15 CONTENTS xv Part 5. Hyperbolic problems with characteristic boundary Paolo Secchi, Alessandro Morando, Paola Trebeschi 135 Chapter 1. Introduction 1. Characteristic IBVP’s of symmetric hyperbolic systems 2. Known results 3. Characteristic free boundary problems 139 139 142 143 Chapter 2. Compressible vortex sheets 1. The nonlinear equations in a ﬁxed domain 2. The L2 energy estimate for the linearized problem 3. Proof of the L2 -energy estimate 4. Tame estimate in Sobolev norms 5. The Nash–Moser iterative scheme 149 152 154 156 158 160 Chapter 3. An example of loss of normal regularity 1. A toy model 2. Two for one 3. Modiﬁed toy model 167 167 169 171 Chapter 4. Regularity for characteristic symmetric IBVP’s 1. Problem of regularity and main result 2. Function spaces 3. The scheme of the proof of Theorem 4.1 175 175 178 180 Bibliography 191 Appendix A. The Projector P 195 Appendix B. Kreiss-Lopatinskiı̆ condition 197 Appendix C. Structural assumptions for well-posedness 199 “volumeV” — 2009/8/3 — 0:35 — page xvi — #16 “volumeV” — 2009/8/3 — 0:35 — page 1 — #17 Part 1 The rugosity effect Dorin Bucur “volumeV” — 2009/8/3 — 0:35 — page 2 — #18 2000 Mathematics Subject Classification. 35B40, 49Q10 Key words and phrases. geometric perturbation, partial diﬀerential equations, boundary behaviour, rough domains Abstract. This paper surveys the series of lectures given by the author at the Nečas Center for Mathematical Modelling in 2006 and 2007. The main purpose is the study of the boundary behaviour of solutions of some partial differential equations in domains with rough boundaries. Several classical examples are recalled: the strange term “coming from somewhere else” of Cioranescu–Murat, Babuška’s paradox, the Courant–Hilbert example and the rugosity effect in fluid dynamics. Some classical and recent results on the shape stability of partial differential equations with Dirichlet boundary conditions are presented. In particular we describe different ways to deal with the rugosity effect in fluid dynamics or contact mechanics. Acknowledgement. The visit of D.B. was supported by the Nečas Center for Mathematical Modelling. “volumeV” — 2009/8/3 — 0:35 — page 3 — #19 Contents Chapter 1. Some classical examples 1. Introduction 2. The example of Cioranescu and Murat: a strange term coming from somewhere else 3. Babuška’s paradox 4. The Courant–Hilbert example for the Neumann–Laplacian spectrum 5. The rugosity eﬀect 5 5 5 6 8 8 Chapter 2. Variational analysis of the rugosity eﬀect 1. Scalar elliptic equations with Dirichlet boundary conditions 2. The rugosity eﬀect in ﬂuid dynamics 2.1. The vector case: in a scalar setting... 2.2. The Stokes equation 11 11 16 16 16 Bibliography 23 3 “volumeV” — 2009/8/3 — 0:35 — page 4 — #20 “volumeV” — 2009/8/3 — 0:35 — page 5 — #21 CHAPTER 1 Some classical examples 1. Introduction The behaviour of the solutions of partial diﬀerential equations or the spectrum of some diﬀerential operators as a consequence of geometric domains perturbations is a classical question which has both theoretical and numerical issues. It is natural to expect that if Ωε is a “nice” perturbation of a smooth open set Ω, then the solution of some partial diﬀerential equation deﬁned on Ωε converges to the solution of the same equation on Ω. While this is indeed a reasonable guess corresponding to the reality, there are many “simple” situations where dramatic changes can be produced by “small” geometric perturbations. We recall some classical examples of such geometric perturbations and give the main tools for handling the particular case of Dirichlet boundary conditions and of the rugosity eﬀect. We underline the fact that the Dirichlet boundary conditions are much easier to deal with than Neumann or Robin boundary conditions (see [7, 20, 25]). The rugosity eﬀect can be seen as sort of eﬀect of partial Dirichlet boundary conditions for vector valued solutions, which interact with the geometric perturbation. In the sequel, we show how small geometric perturbations can produce huge eﬀects on the solution of the partial diﬀerential equations, or on the spectrum of some diﬀerential operators. The word small is not clear and may have signiﬁcantly diﬀerent interpretations. Overall, the perturbations are certainly small in terms of Lebesgue measure but they have also some other features which at a ﬁrst sight may lead to the false intuition that the perturbations would leave the behaviour of the partial diﬀerential equation unchanged. 2. The example of Cioranescu and Murat: a strange term coming from somewhere else We consider an open set Ω contained in the unit square S in R2 and f ∈ L2 (S). For every n ∈ N we introduce Cn = n [ B (i/n,j/n),rn , i,j=0 Ωn = Ω \ Cn , 2 where rn = e−cn , c > 0 being a ﬁxed positive constant. If we denote by un the weak solution of −∆un = f in Ωn un ∈ H01 (Ωn ). 5 (1.1) “volumeV” — 2009/8/3 — 0:35 — page 6 — #22 6 1. SOME CLASSICAL EXAMPLES 2 rn = e−cn one can prove that un ⇀u weakly in H01 (S), where u solves −∆u + cu = f in Ω u ∈ H01 (Ω). (1.2) We refer the reader to [14] for a detailed proof of the passage to the limit as n → ∞. The proof is elementary and comes from a direct computation as follows: one introduces the functions zn ∈ H 1 (S): 0 on Cn ln p(x − i/n)2 + (y − j/n)2 + cn2 zn = on B (i/n,j/n),1/2n \ Cn cn2 − ln(2n) Sn 1 on S \ i,j=0 B (i/n,j/n),1/2n . Then, for every ϕ ∈ C0∞ (Ω), one can take zn ϕ as test function in equation (1.1). The passage to the limit for n → ∞ can be performed completely to arrive to the weak form of (1.2). The explanation of the fact that a union of small perforations of measure less 2 than πn2 e−2cn rapidly converging to zero can produce a huge eﬀect on the equation can be completely understood in terms of Γ-convergence (see [19]). The eﬀect is observed by the presence of the “strange term” cu in the limit equation. For a complete description of this phenomenon in relationship with optimal design problems we refer to the recent book [7]. 3. Babuška’s paradox We consider the sequence (Pn )n of regular polygons with n edges, inscribed in the unit circle in R2 . As n → ∞, it is reasonable to expect that the solutions of (some) partial diﬀerential equations set on Pn would converge to the solution on the disc. This is indeed the case for some partial diﬀerential equations of second order, like the Laplace equation with homogeneous Dirichlet or Neumann boundary conditions (with a ﬁxed admissible right hand side, see [7]). Nevertheless, as Babuška noticed (see [2] and also [29]) this is not anymore the case for a fourth order equation of bi-laplacian type as equilibrium problems in the bending of simply supported Kirchhoﬀ-Love plates (see for a detailed explanation [2] and also [29], [21]). “volumeV” — 2009/8/3 — 0:35 — page 7 — #23 4. THE COURANT–HILBERT EXAMPLE FOR THE NEUMANN–LAPLACIAN SPECTRUM 7 Precisely, we consider the constant force f = 1 and 0 ≤ σ < 21 . For every bounded Lipschitz open set Ω ⊆ R2 , the solution of the following minimization problem: min{u ∈ H 2 (Ω) ∩ H01 (Ω) : Z Ω 1 |∆u|2 + (1 − σ)(u2xy − uxx uyy ) − udx} 2 is denoted uΩ . Then uΩ is a formal weak solution of the following partial diﬀerential equation ∆2 u = 1 in Ω (1.3) u = ∆u − (1 − σ)k nu = 0 on ∂Ω k being the curvature of the boundary. It turns out that if Ω has a polygonal shape, as Pn does, then the term Z (1 − σ)(u2xy − uxx uyy )dx Ω vanishes identically in the energy functional above (see [24, Lemma 2.2.2]). So that, the solution uPn is also solution of the minimization problem Z 1 2 1 min{u ∈ H (Pn ) ∩ H0 (Pn ) : |∆u|2 − udx}, 2 Pn and formal weak solution of ∆2 u = 1 in Pn u = ∆u = 0 on ∂Pn (1.4) When n → ∞, one can notice that uPn converges in L∞ to the solution of (1.4) on The sequence (Pn )n of regular polygons “converges” to the disc the disc, which is diﬀerent from the solution of (1.3) on the disc. This means, that the approximation of the disc by the sequence of regular polygons (Pn )n for equation (1.3) does not hold! The implications of this non-stability result for equation (1.3) in numerical analysis are obvious. “volumeV” — 2009/8/3 — 0:35 — page 8 — #24 8 1. SOME CLASSICAL EXAMPLES The parameters ε, η, µ vanish with diﬀerent speeds 4. The Courant–Hilbert example for the Neumann–Laplacian spectrum One considers the Neumann–Laplacian eigenvalues associated to the following Lipschitz domain, which depends on the small parameters ε, η, µ > 0. Precisely, the values of η, µ will be chosen dependently on ε. By abuse of notation, let us denote Ωε the perturbed domain and by Ω the limit square. Since Ωε is Lipschitz, the spectrum of the Neumann Laplacian consists only of eigenvalues satisfying formally −∆u = λk (Ωε )u in Ωε (1.5) ∂u ∂n = 0 on ∂Ωε for some function u ∈ H 1 (Ω), u 6≡ 0. The eigenvalues can be ordered, counting their multiplicities 0 = λ1 (Ωε ) < λ2 (Ωε ) ≤ ... Using the continuous dependence of the eigenvalues for smooth domain perturbations (see [7, 15]) or, alternatively, the deﬁnition of the eigenvalues with the Rayleigh quotient, for every c ∈ (0, λ2 (Ω)) and ε small enough, one can choose µ = ε and η ∈ (0, ε) such that λ2 (Ωε ) = c. Consequently, when ε → 0, the ﬁrst nonzero eigenvalue of the NeumannLaplacian on Ωε will converge to c, which is diﬀerent from the ﬁrst nonzero eigenvalue associated to Ω. The conclusion is that a “small” geometric perturbation of the square Ω leads to an uncontrollable behaviour of the Neumann-Laplacian spectrum (see [7] for details). 5. The rugosity eﬀect For simplicity, the Stokes equation with perfect slip boundary conditions (on a piece of the boundary) is considered in the 2D-rectangle Ω = (0, L) × (0, 1). Roughly speaking, the rugosity eﬀect is the following: a geometric perturbation of the boundary at a microscopic scale may transform perfect slip boundary conditions in total adherence. We refer the reader to [13] for a description of this phenomenon if the perturbation of the boundary has a periodic structure: x Γε = {(x, 1 + εϕ( )) : x ∈ (0, L)}, ε where ϕ ∈ C 2 [0, L], ϕ(0) = ϕ(L), is extended by periodicity on R. “volumeV” — 2009/8/3 — 0:35 — page 9 — #25 5. THE RUGOSITY EFFECT 9 Example of periodic rugosity. The amplitude ε of the perturbation vanishes. This phenomenon occurs (in 2D) as soon as some rugosity is present (i.e. ∇ϕ 6≡ 0) in particular the boundary Γε is not ﬂat. This means for the periodic case above that ϕ 6≡ ϕ(0)! It is a consequence of the oscillating normal in relationship with the non-penetration condition satisﬁed by the solutions uε · nε = 0 on Γε , where nε is the normal vector on the oscillating boundary. Recent results in [9, 10, 11] give more hints on how arbitrary rugosity acts on the solution of a Stokes (or Navier-Stokes) equation, precisely by “driving” the ﬂow on the boundary and by introducing some friction matrix. In the next chapter we give some explanations of the rugosity eﬀect, from the variational point of view. In particular one may use the results on the geometric perturbations for scalar elliptic equations with Dirichlet boundary conditions, since the perfect slip boundary conditions for vector valued PDEs can be seen as sort of partial Dirichlet boundary conditions for vector PDEs. The inﬂuence of the rugosity in the presence of complete adherence is a diﬀerent problem, and we refer the reader to [26]. In this case, the complete adherence is preserved in the limit, the challenge being to ﬁnd better approximations of the solutions associated to the rough boundaries in a smooth domain where the complete adherence is replaced by a wall law (see also [4]). “volumeV” — 2009/8/3 — 0:35 — page 10 — #26 “volumeV” — 2009/8/3 — 0:35 — page 11 — #27 CHAPTER 2 Variational analysis of the rugosity effect 1. Scalar elliptic equations with Dirichlet boundary conditions Let D ⊆ RN be a bounded open set, f ∈ H −1 (D) (one can consider f ∈ L2 (D) for simplicity) and Ωε be a geometrical perturbation of Ω ⊆ D. We consider the Dirichlet problem for the Laplacian on the moving domain −∆uε = f in Ωε (2.1) uε ∈ H01 (Ωε ). The question we deal with is whether the convergence uε → u holds, and in which norm? The following abstract result can be found in [7]. It gives a ﬁrst elementary approach to study whether or not the solution of the Dirichlet problem (2.1) is stable for an arbitrary geometric perturbation. The main drawback of this (abstract) result is that for particular geometric perturbations of non-smooth sets it does not give a clear answer whether or not the solution is stable. Theorem 2.1. Assertions (1) to (4) below are equivalent: (1) For every f ∈ H −1 (D), uε → u in H01 (D)-strong; (2) For f = 1, uε → u in H01 (D)-strong; (3) H01 (Ωε ) converges in the sense of Mosco to H01 (Ω), i.e. M1) For all φ ∈ H01 (Ω) there exists a sequence φε ∈ H01 (Ωε ) such that φε converges strongly in H01 (D) to φ. M2) For every sequence φεk ∈ H01 (Ωεk ) weakly convergent in H01 (D) to a function φ, φ ∈ H01 (Ω). (4) If Fε : L2 (D) → R ∪ {+∞}, R 2 if u ∈ H01 (Ωε ) D |∇u| dx Fε (u) = +∞ otherwise then Fε Γ-converges in L2 (D) to F , i.e. • ∀φε → φ in L2 (D) then F (φ) ≤ lim inf Fε (φε ) ε→0 • ∀φ ∈ L2 (D) there exists φε → φ in L2 (D) s.t. F (φ) ≥ lim sup Fε (φε ) ε→0 Remark 2.2. From the previous theorem, it appears clearly that the solution of the equations with the right hand side equal to 1 plays a crucial role. For simplicity, let us denote wε the solutions for f ≡ 1. Assume now that (Ωε )ε is a sequence of 11 “volumeV” — 2009/8/3 — 0:35 — page 12 — #28 12 2. VARIATIONAL ANALYSIS OF THE RUGOSITY EFFECT arbitrary open subsets of D and that for some f ∈ H −1 (D) uε ⇀u and wε ⇀w weakly in H01 (D). Here the limit set Ω is not given, so we wonder whether u and w are solutions on some set Ω? If such set exists, its identification would not be complicated since by the maximum principle one should have Ω = {x : w(x) > 0}. This set may be quasi-open, in general. In practice, from the example of Cioranescu and Murat, one can notice that the set Ω may not exists because of the new term which appears: the strange term. In fact, one can formalise the emerging of this strange term (which in general will be a positive Borel measure, maybe infinite but absolute continuous with respect to capacity), and give a full interpretation through Γ-convergence arguments. Let ϕ ∈ C0∞ (D) and take wε ϕ as test function in (2.1) on Ωε . Then (we integrate over D for simplicity) Z R f wε ϕdx = D ∇uε ∇(wε ϕ)dx Z Z D ∇uε ∇wε ϕdx ∇uε ∇ϕwε dx + = ZD ZD = ∇uε ∇ϕwε dx − uε ∇wε ∇ϕdx − h∆wε , ϕuε iH −1 ×H01 Z ZD ZD uε ϕdx. uε ∇wε ∇ϕdx + ∇uε ∇ϕwε dx − = D D D Let ε → 0 and use Z Z ∇u∇wϕdx + h∆w, uϕiH −1 (D)×H01 (D) . u∇w∇ϕdx = − D D Consequently, Z D ∇u∇(ϕw)dx + h∆w + 1, uϕiH −1 ×H01 = Z f ϕwdx. (2.2) D But ν = ∆w + 1 ≥ 0 in D′ (D) is a non-negative Radon measure belonging to H −1 (D). In fact, the positivity can be easily proven for smooth sets, and then use the weak convergence in H −1 (D): ∆wε + 1 ⇀ ∆w + 1. We formally write Z Z Z f ϕwdx, (2.3) uϕwdµ = ∇u∇(ϕw)dx + D D D where µ is the Borel measure defined by Z +∞ if cap(B ∩ {w = 0}) > 0 1 µ(B) = dν if cap(B ∩ {w = 0}) = 0. B w (2.4) Using the density of {wϕ : ϕ ∈ C0∞ (D)} in H01 (D) ∩ L2 (D, µ), it turns out that u solves in a weak sense the following problem −∆u + uµ = f in D (2.5) u ∈ H01 (D) ∩ L2 (D, µ). i.e. ∀ϕ ∈ H01 (D) ∩ L2 (D, µ) Z D ∇u∇ϕdx + Z D uϕdµ = Z D f ϕdx. “volumeV” — 2009/8/3 — 0:35 — page 13 — #29 1. SCALAR ELLIPTIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS 13 In the case of the example of Cioranescu-Murat, the measure µ equals cL⌊Ω and +∞ on S \ Ω, where L is the Lebesgue measure. This phenomenon, called relaxation, plays a crucial role in optimal design problems (see [7]). It can be formalised as follows, in terms of Γ-convergence of the energy functionals (point (4) in Theorem 2.1). Theorem 2.3. Let (Ωε )ε be an arbitrary sequence of open subsets of D. There exists a sub-sequence (still denoted using the same index) and a functional F : L2 (D) → R ∪ {+∞} such that Fε Γ-converges in L2 (D) to F . Moreover, F can be represented as Z Z F (u) = D |∇u|2 dx + u2 dµ D where µ is a positive Borel measure, absolutely continuous with respect to capacity. Remark 2.4. A way to prove this theorem (see Theorem 2.12 in the next paragraph for the vector case), is to prove in a first step the compactness result (which is of topological nature) and in a second step to use representation theorems in order to find the form of the Γ-limit functional. Remark 2.5. The measure µ above, is precisely the measure computed with the help of the solutions wε for the right hand side f ≡ 1. It is quite easy to notice that for every f ∈ H −1 (D) we have that Γ-converges to Fε (·) − 2hf, ·iH −1 (D)×H01 (D) F (·) − 2hf, ·iH −1 (D)×H01 (D) . As the Γ-convergence implies the convergence of the minimizers of the functionals, one gets the strong convergence L2 (D) (and weak H01 (D)) of uε to the solution of (2.5) for every admissible right hand side f . Notice the very important fact, that the measure µ is independent on f , being only an effect of the geometric perturbation. Remark 2.6. When the measure is known? The measure can be computed explicitly for very few geometric perturbations, often with periodic character. There are formulas giving in general the value of the measure in terms of the limits of local capacities of Ωcε ∩ B for a well chosen family of balls [16, 17]). Remark 2.7. Also notice, that for some particular geometric perturbations, e.g. when one of the assertions of Theorem 2.1 holds, the relaxation process does not occur, and so the measure µ coming from Theorem 2.3 corresponds to a (quasi)open set Ω, i.e. µ(A) = 0 if cap(A ∩ Ωc ) = 0 and µ(A) = +∞ if cap(A ∩ Ωc ) > 0. Remark 2.8. When classical stability holds? That means that in the limit no relaxation occurs and the (quasi)-open set Ω can be identified. Below are some situations when the geometric limit is identified. • Increasing sequences of domains: this case is very easy, the geometric limit is the union of the open sets (direct use of Theorem 2.1). • Decreasing sequences of domains: this problem is not so simple. Yet, what is the limit domain? The intersection of a decreasing sequence of open sets is not, in general, an open set. One may suspect that the interior of the intersection is the right limit, but the answer is not always affirmative. “volumeV” — 2009/8/3 — 0:35 — page 14 — #30 14 2. VARIATIONAL ANALYSIS OF THE RUGOSITY EFFECT Keldysh gave the answer to this problem in 1962 [27], and introduced a new regularity concept, called stability (see [7] for an interpretation through Γconvergence). • Perturbations satisfying some geometric constraints: if the domains satisfy a uniform geometric constraint forcing the boundary to avoid oscillations, or new holes to appear, than no relaxation occurs, and the limit set Ω can be identified by some geometric convergence, precisely in the Hausdorff complementary topology (see [7]). Here is an example of a domain satisfying a pointwise cone condition: there exists a non trivial cone C (of dimension N or N − 1 ) such that for every point x ∈ ∂Ωε there exists a cone congruent to C with vertex at x and lying in Ωcε . If every Ωε satisfy this condition with the cone C, then no relaxation occurs, and the geometric limit can be identified. In R2 a 2D cone is a triangle and a 1D cone is a segment. This condition is related to a uniformity property of the Wiener criterion (see Theorem 2.9 below). Pointwise cone condition • Perturbations satisfying some topological constraints: in two dimensions of the space provided the number of the connected components of the complements Ωcε is uniformly bounded (roughly speaking there is a uniformly bounded number of holes) the relaxation process does not hold and the limit can be identified in the Hausdorff complementary topology. This result is due to Šverák [28] and opened the way of intensive use of potential theory in understanding the behaviour of the solutions uε near the oscillating boundaries. In fact, in any other dimension of the space the topological constraint is not relevant. The “equivalent” constraint is a density property in terms of capacity (see [7]). The use of capacity estimates in terms of the Wiener criterion allows us to handle the local oscillations of the solutions (see [23], [7]). For the convenience of the reader we recall the deﬁnition of the capacity: let E ⊆ D be two sets in RN , such that D is open. The capacity of E in D is Z cap(E, D) = inf{ |∇u|2 + |u|2 dx, u ∈ UE,D } D “volumeV” — 2009/8/3 — 0:35 — page 15 — #31 1. SCALAR ELLIPTIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS 15 where UE,D stand for the class of all functions u ∈ H01 (D) such that u ≥ 1 a.e. in an open set containing E. We recall the following result from [12] (see also [7]). Theorem 2.9. Assume that Ωε converges in the Hausdorff complementary topology to some open set Ω and that there exists a function g : (0, 1] × (0, 1] → (0, +∞) such that lim g(r, R) = +∞ r→0 and for every ε > 0, x ∈ ∂Ωε , 0 < r < R < 1 we have Z R cap(Ωεc ∩ Bx,t , Bx,2t ) dt ≥ g(r, R). cap(Bx,t , Bx,2t ) t r Then uε → u in H01 (D). Remark 2.10. Notice that this theorem involves a quantitative estimate of the complement of Ωε near the boundary and not its smoothness. A particular situation when this theorem can be applied, is the so called capacity density condition. For some positive constant c and for t ∈ (0, r) independent on ε, the stronger estimate cap(Ωcε ∩ Bx,t , Bx,2t ) ≥c cap(Bx,t , Bx,2t ) holds for every x ∈ ∂Ωε . The uniform minoration of the local capacity of the complement If ω is a smooth open subset of an (N − 1) dimensional manifold, such that {0} ⊆ ω ⊆ B(0, 21 ) and F = ∪α∈ZN Tα (ω), then all the sets (εF )ε satisfy uniformly a capacity density condition. Here Tα (ω) is the translation of ω by the vector α. Remark 2.11. Recent advances on the stability question involve convergence N of solutions in L∞ . Indeed, for right hand sides f ∈ L 2 +ε (D), the solutions uε belong to L∞ (Ωε ) so that a natural question is to seek if uε converges to u in L∞ (D). This problem is not anymore of variational type and relies on the study of the oscillations near the boundaries related to some geometric information. A characterization of the stability is given in [6]. We refer the reader to [1, 3, 20] for more results concerning this question. “volumeV” — 2009/8/3 — 0:35 — page 16 — #32 16 2. VARIATIONAL ANALYSIS OF THE RUGOSITY EFFECT 2. The rugosity eﬀect in ﬂuid dynamics 2.1. The vector case: in a scalar setting... The rugosity eﬀect can be seen as the inﬂuence of partial Dirichlet boundary conditions on the behaviour of the solutions of vector valued PDEs. In order to make the relationship with the scalar case, we give below an example of scalar equation with partial Dirichlet boundary conditions. Here the word “partial” is understood in a geometric sense: there are small regions with perfect support of a membrane (homogeneous Dirichlet boundary conditions) and small regions with free membrane boundary conditions. We consider a rectangle Ω ⊆ R2 , f ∈ L2 (Ω) and a sequence of closed sets Γε ⊆ ∂Ω (for example located on the upper edge Γ of Ω). We consider the Laplace equation with mixed Dirichlet and Neumann homogeneous boundary conditions. −∆uε = f in Ω uε = 0 on Γε (2.6) ∂uε = 0 on Γ \ Γε ∂n uε = 0 on ∂Ω \ Γ When ε → 0, for a subsequence one has uε → u weakly in H 1 (D) and the limit u solves the same equation on Ω but with Robin boundary conditions on the upper edge! There exists a positive measure µ such that u solves in a weak sense −∆u = f in Ω ∂u (2.7) + µu = 0 on Γ ∂n u = 0 on ∂Ω \ Γ This result ﬁts precisely into the theory of the ﬁrst section of this chapter. Indeed, one can formally reﬂect Ω and uε with respect to Γ, in Ωr and ur , respectively and obtain that uε together with its reﬂection, is solution of the Laplace equation with Dirichlet boundary conditions on Γε ∪ ∂(Ω ∪ Ωr ) in Ω ∪ Ωr . In this way, the Neumann b.c. can be ignored and all results of the previous section apply, thus the presence of the measure µ in the limit process. 2.2. The Stokes equation. For simplicity, we consider the following situation Ω = (0, 1)N ⊆ RN , N ≥ 2. Let us denote T = (0, 1)N −1 and a sequence of functions ϕε : T → R such that ϕε ∈ W 1,∞ (T ), kϕε k∞ ≤ ε and k∇ϕε k∞ ≤ M , for some M > 0 independent on ε. If x = (x1 , .., xN ) ∈ RN , by x̂ we denote x̂ = (x1 , .., xN −1 ). Then , we introduce the perturbed domains Ωε = {x ∈ RN : x̂ ∈ T, 0 < xN < 1 + ϕε (x̂)}, And denote Γε = {x ∈ RN : x̂ ∈ T, xN = 1 + ϕε (x̂)}. “volumeV” — 2009/8/3 — 0:35 — page 17 — #33 2. THE RUGOSITY EFFECT IN FLUID DYNAMICS 17 Let f ∈ L2loc (RN ). We consider the Stokes equation on Ωε with perfect slip boundary conditions on Γε and total adherence boundary conditions on ∂Ωε \ Γε . −div D[uε ] + ∇pε = f in Ωε div uε = 0 in Ωε uε · nε = 0 on Γε (2.8) (D[u ] · n ) = 0 on Γ ε ε tan ε uε = 0 on ∂Ωε \ Γε It is easy to notice that the solutions uε ∈ H 1 (Ωε ) are uniformly bounded, as a consequence of the uniform Korn inequality in the equi-Lipschitz domains Ωε . For a subsequence (still denoted using the same index) we have that 1Ωε uε and L2 (Rn ) → 1Ω u, (2.9) L2 (Rn ) (2.10) 1Ωε ∇uε ⇀ 1Ω ∇u. The question is: what is the equation satisfied by u? It is not complicated to observe that u satisﬁes in a weak sense the equation (by multiplication with test functions with free divergence in H01 (Ω)) and −div D[u] + ∇p = f in Ω div u = 0 in Ω, in the sense of distributions. As well, on the part of ∂Ω which is not oscillating, namely ∂Ω \ Γ, one gets immediately u = 0. Several approaches are available in the literature in order to understand the behaviour of the solution on the upper boundary. Below there is an intuitive justiﬁcation of the rugosity phenomenon in R2 . Let us consider the function ϕ(x) = |x − 21 | deﬁned on [0, 1] and extended by periodicity on R. Moreover, the upper boundaries Γε of the two dimensional sets are given by the functions ϕε (x) = εϕ( xε ). If we denote n1 and n2 the two normals at the boundaries, for every solution uε we have uε · n1 = 0 on Lε and uε · n2 = 0 on Rε (Lε stands for the segments of Γε which correspond to the locally increasing part of ϕε and Rε to the complement). At this point, we use the vanishing information for the scalar H 1 -functions uε · n1 and uε · n2 . As pointed in the previous paragraph, both Lε and Rε satisfy a capacity density condition and converge in the Hausdorﬀ metric to the segment Γ = [0, 1] × {1}. Consequently u · n1 = 0 and u · n2 = 0 on Γ. “volumeV” — 2009/8/3 — 0:35 — page 18 — #34 18 2. VARIATIONAL ANALYSIS OF THE RUGOSITY EFFECT As n1 and n2 are linearly independent, we conclude with u = 0 on Γ. For general rugosity it is more diﬃcult to follow the normals. Below we brieﬂy describe four methods. Example of “arbitrary” rugosity. The amplitude ε of the perturbation vanishes. Method 1: use of Young measures. In order to handle the oscillations of the boundaries, a very eﬃcient way to describe the limit(s) of ∇φε is the use of Young measures. We refer the reader to [22] for an introduction to Young measures. The passage to the limit of the impermeability condition uε · (∇φε , −1) = 0 may give a substantial information provided that the support of the Young measures associated to the sequence (∇φε )ε is large enough. We refer the reader to [9] for a description of this method. Here are some examples where the rugosity eﬀect is produced under mild assumptions (see [9]). • periodic boundaries of the form ϕε (x′ ) = εϕ( xε ) for some Lipschitz function deﬁned on T ; • crystalline boundaries; • riblets; • etc. Method 2: use of capacity estimates. This method relies on the previous paragraph on scalar functions. One may mimic the intuitive example above but, as normals vary, should work with cones of normals instead of discrete normals. For example, let us ﬁx a vector n and denote by C(n) a cone of axis n and opening ω. “volumeV” — 2009/8/3 — 0:35 — page 19 — #35 2. THE RUGOSITY EFFECT IN FLUID DYNAMICS 19 Then, if for some point x we have uε (x) · nε (x) = 0 and assume that nε (x) ∈ C(n). We get |uε (x) · n| ≤ |n − nε (x)||uε (x)|. In order to make the idea clear, let us assume that uε are moreover uniformly bounded in L∞ , i.e. for some M > 0 and for every ε we have |uε |∞ ≤ M . Consequently, for the point x we have |uε (x) · n| ≤ M c(ω), where c(ω) depends only on the opening of the cone and vanishes for ω → 0. In particular, this means that (|uε · n| − M c(ω))+ vanishes at x, and in general on the region where the normals nε (x) are deﬁned and belong to the cone C(n). Consequently, for the scalar sequence of functions (|uε · n| − M c(ω))+ we can fully use the scalar setting for Dirichlet Laplacian by estimating precisely in capacity the size of the region where the normals nε (x) belong to C(n). If this region satisfy a density capacity condition (which is likely to be the case for periodic boundaries and well chosen n) then in the limit we get (|u · n| − M c(ω))+ = 0 on Γ. Making ω → 0, we get u · n = 0 on Γ. In order to give a general frame, let us consider V ∈ W 1,∞ (RN , RN ). As in the scalar case, one can construct a measure supported on Γ which is associated to V and counts the energy eﬀect of the asymptotical rugosity of ∂Ωε when ε → 0, into the direction of the ﬁeld V . The fact that the ﬁeld V is ﬁxed a priori allows, roughly speaking, to use the previous results for scalar functions by considering the family of scalar functions (vε · V )ε . Typically, the argument above for V = n can be used. Nevertheless, in order to give a general framework and avoid unnecessary hypotheses as uniform boundedness in L∞ , one can formally consider energy functionals of the form Fε : L2 (RN ) → R ∪ {+∞}, R |∇(u · V )|2 dx if u ∈ H 1 (Ωε ), u · nε = 0 on Γε , u = 0 on ∂Ωε \ Γε RN Fε (u) = +∞ otherwise and to investigate their inferior Γ-limit. We consider the family MV of positive Borel measures, absolutely continuous with respect to the capacity, such that for every sequence vεk ∈ H 1 (Ωεk , RN ), vεk · nεk = 0 on Γεk vεk = 0 on ∂Ωεk \Γεk and such that vεk → v in the sense of relations (2.9)-(2.10), then Z Z Z |∇(v · V )|2 dx + (v · V )2 dµ ≤ lim inf |∇(vεk · V )|2 dx. D D k→∞ D The equality vεk · nεk = 0 is understood pointwise where the normal exists and for a quasi continuous representative of v. Since at least the zero measure can be considered above, MV 6= ∅ so that µV = sup{µ : µ ∈ MV } is well deﬁned. The measure µV is supported on Γ and takes into account precisely the rugosity eﬀect on ∂Ω in the direction of the ﬁeld V from an energetic point of view. If, as in the scalar case, one can prove that µ = ∞Γ , then we get u · V = 0 on Γ, so that “volumeV” — 2009/8/3 — 0:35 — page 20 — #36 20 2. VARIATIONAL ANALYSIS OF THE RUGOSITY EFFECT the ﬂow is orthogonal to V on Γ. This argument works properly in several cases when computations can be carried out, e.g. the periodical case. Method 3: uniform estimates. Let us denote Uε = (0, 1)N −1 × {1 − 2ε}. Provided some uniformity on the rugosities ϕε , one can prove the existence of a constant C > 0, independent on ε such that for every solution of the Stokes equation (2.8), we have Z Z Uε |uε |2 dσ ≤ Cε Ωε |∇uε |2 dx. Of course, if such an estimate holds and since the solutions (uε )ε have uniformly bounded energy, then as ε → 0 one gets u = 0 on Γ. We refer to [8, 13] for estimates of this kind in the periodic case, and to [5] for improvements of the periodic case, if the Lipschitz hypothesis is removed. Method 4: representation by Γ-convergence. In order to ﬁnd the general form of the limit problem, in [11] it is used an approach based on Γ-convergence. Theorem 2.12. Let ε → 0 and let f ∈ L2loc (RN , RN ) be given. Let {uε }ε>0 be the family of (weak) solutions to the Stokes equation (2.8) in Ωε . Then, at least for a suitable subsequence we have 1Ωε uε → 1Ω u (strongly) in L2 (RN , RN ), 1Ωε ∇uε → 1Ω ∇u weakly in L2 (RN , RN ×N ), and there exists a suitable trio {µ, A, V} independent of the driving force f such that • µ is a capacitary measure concentrated on Γ • {V}x∈Γ is a family of vector subspaces in RN −1 • A is a positive symmetric matrix function A defined on Γ and u is a solution in Ω of the Stokes equation with friction-driven b.c. −div D[u] + ∇p = f in Ω div u = 0 in Ω u(x) ∈ V (x) for q.e. x ∈ Γ (2.11) h i D[u] · n + µAu · v = 0 for any v ∈ V (x), x ∈ Γ u(x) = 0 for q.e. x ∈ ∂Ω \ Γ. The sense in which u solves the equation (2.11) is the following: u is solution of the minimization of Z Z Z 1 1 |D[v]|2 + |v|2 dx + vT Avdµ − f · vdx, (2.12) J (v) := 2 Ω 2 ∂Ω Ω on n o v ∈ H 1 (Ω, RN ) div v = 0 in Ω, v(x) ∈ V (x) for q. e. x ∈ Γ, v = 0 on ∂Ω \ Γ . Proof. The main steps of the proof are the following: Step 1. introduce energy functionals involving the boundary constraint: uε ·nε = 0 and remove incompressibility condition; Step 2. use representation results of the Γ-limit for vector valued functionals (see [18] and also [16, 17] for scalar or vector equations for Dirichlet boundary conditions); “volumeV” — 2009/8/3 — 0:35 — page 21 — #37 2. THE RUGOSITY EFFECT IN FLUID DYNAMICS 21 Step 3. prove that the measure is concentrated on the surface; Step 4. use a diagonal argument in order to handle the incompressibility condition. This theorem gives the general form of the limit problem, but in any particular situation, speciﬁc computations should be carried out in order to identify the trio {µ, A, V}. “volumeV” — 2009/8/3 — 0:35 — page 22 — #38 “volumeV” — 2009/8/3 — 0:35 — page 23 — #39 Bibliography [1] W. ARENDT, D. DANERS, Uniform convergence for elliptic problems on varying domains, Math. Nachr. 280 (2007), no. 1-2, 28–49. [2] I. BABUŠKA, Stabilität des Definitionsgebietes mit Rücksicht auf grundlegende Probleme der Theorie der partiellen Differentialgleichungen auch im Zusammenhang mit der Elastizitätstheorie. I, II. (Russian) Czechoslovak Math. J. 11 (86) 1961 76–105, 165–203. [3] M. BIEGERT, D. DANERS, Local and global uniform convergence for elliptic problems on varying domains. J. Diﬀerential Equations 223 (2006), no. 1, 1–32. [4] D. BRESCH, V. MILISIC, Vers des lois de parois multi-échelle implicites. C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 833–838. [5] J. BŘEZINA, Asymptotic properties of solutions to the equations of incompressible fluid mechanics Preprint 1/2009 Necas Center for Math. Model. (accepted to Journal of Mathematical Fluid Mechanics) [6] D. BUCUR, Characterization of the shape stability for nonlinear elliptic problems. J. Diﬀerential Equations 226 (2006), no. 1, 99–117. [7] D. BUCUR, G. BUTTAZZO, Variational Methods in Shape Optimization Problems. Progress in Nonlinear Diﬀerential Equations 65, Birkhäuser Verlag, Basel (2005). [8] D. BUCUR, E. FEIREISL, The incompressible limit of the full Navier-StokesFourier system on domains with rough boundaries E. FEIREISL, Nonlinear Analysis: Real World Applications (to appear) 2009. [9] D. BUCUR, E. FEIREISL, Š. NEČASOVÁ, and J. WOLF, On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries. J. Differential Equations, 244:2890–2908, 2008. [10] D. BUCUR, E. FEIREISL, Š. NEČASOVÁ, On the asymptotic limit of ﬂows past a ribbed boundary J. Math. Fluid Mech. , 2008. To appear. [11] , D. BUCUR, E. FEIREISL, Š. NEČASOVÁ, Boundary behavior of viscous ﬂuids: Inﬂuence of wall roughness and friction-driven boundary conditions Preprint Université de Savoie , 2008. [12] D. BUCUR, J. P. ZOLÉSIO, Wiener’s criterion and shape continuity for the Dirichlet problem. Boll. Un. Mat. Ital., B 11 (4) (1997), 757–771. [13] J. CASADO-DIÁZ, E. FERNÁNDEZ-CARA, J. SIMON, Why viscous ﬂuids adhere to rugose walls: A mathematical explanation. J. Differential Equations, 189:526–537, 2003. [14] D. CIORANESCU, F. MURAT, Un terme étrange venu d’ailleurs. Nonlinear partial diﬀerential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), pp. 98–138, 389–390, Res. Notes in Math., 60, 23 “volumeV” — 2009/8/3 — 0:35 — page 24 — #40 24 Bibliography Pitman, Boston, Mass.-London (1982). [15] R. COURANT, D. HILBERT, Methods of mathematical physics. Vol. I. and 2. Interscience Publishers, Inc., New York, 1953 and 1962. [16] G. DAL MASO, Γ-convergence and µ-capacities. Ann. Scuola Norm. Sup. Pisa, 14 (1988), 423–464. [17] G. DAL MASO, A. DE FRANCESCHI, Limits of nonlinear Dirichlet problems in varying domains. Manuscripta Math., 61 (1988), 251–268. [18] G. DAL MASO, A. DE FRANCESCHI, E. VITALI, Integral representation for a class of C 1 -convex functionals. J. Math. Pures Appl. (9) 73 (1994), no. 1, 1–46. [19] G. DAL MASO, U. MOSCO, Wiener criteria and energy decay for relaxed Dirichlet problems. Arch. Rational Mech. Anal., 95 (1986), 345–387. [20] D. DANERS, Domain perturbation for linear and semi-linear boundary value problems, Handbook of diﬀernetial equations (to appear). [21] C. DAVINI, Γ-convergence of external approximations in boundary value problems involving the bi-Laplacian. Proceedings of the 9th International Congress on Computational and Applied Mathematics (Leuven, 2000). J. Comput. Appl. Math. 140 (2002), no. 1-2, 185–208. [22] L.C. EVANS, Weak convergence methods for nonlinear partial differential equations. CBMS Regional Conference Series in Mathematics, 74. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. [23] J. FREHSE, Capacity methods in the theory of partial differential equations. Jahresber. Deutsch. Math.-Verein. 84 (1) (1982), 1-44. [24] P. GRISVARD, Singularities in boundary value problems. Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 22. Masson, Paris; Springer-Verlag, Berlin, 1992. [25] A. HENROT, M. PIERRE, Variation et optimisation de formes. Une analyse géométrique. Mathématiques & Applications (Berlin) 48. Berlin: Springer, 2005. [26] W. JAEGER, A. MIKELIĆ, On the roughness-induced eﬀective boundary conditions for an incompressible viscous ﬂow. J. Differential Equations, 170:96– 122, 2001. [27] M.V. KELDYSH, On the Solvability and Stability of the Dirichlet Problem. Amer. Math. Soc. Translations, 51-2 (1966), 1–73. [28] V. ŠVERÁK, On optimal shape design. J. Math. Pures Appl., 72 (1993), 537– 551. [29] G. SWEERS, A survey on boundary conditions for the biharmonic, Complex Variables and Elliptic Equations (to appear). “volumeV” — 2009/8/3 — 0:35 — page 25 — #41 Part 2 Nonlinear evolution equations with anomalous diffusion Grzegorz Karch “volumeV” — 2009/8/3 — 0:35 — page 26 — #42 2000 Mathematics Subject Classification. 35K55, 35B40, 35Q53, 60J60, 60J60 Key words and phrases. Lévy process, Lévy operator, fractal Burgers equation, Fractal Hamilton–Jacobi–KPZ equations, large time asymptotics of solutions Abstract. This is the review article on nonlinear pseudodifferential equations involving Lévy semigroup generators–used in physical models where the diffusive behavior is affected by hopping and trapping phenomena. In first chapter, properties of Lévy generators are discussed. Results on the large time asymptotics of solutions to the fractal Burgers equation are presented in Chapter 2. A generalization of the Kardar–Parisi–Zhang equation modeling the ballistic rain of particles onto the surface is discussed in Chapter 3. In the last chapter, some other classes of nonlinear evolution equations with Lévy operators are briefly described. These are the lectures notes presented by the author at EVEQ 2008—International Summer School on Evolution Equations Prague, Czech Republic, June 16-20, 2008. Acknowledgement. The author gratefully thanks Nečas Center for Mathematical Modeling, the Faculty of Mathematics and Physics of the Charles University, and Institute of Mathematics of the Czech Academy of Sciences for the warm hospitality and for the support. The preparation of this paper was also supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389 and by the MNiSW grant N201 022 32 / 09 02. “volumeV” — 2009/8/3 — 0:35 — page 27 — #43 Contents Chapter 1. Lévy operator 1. Probabilistic motivations – Wiener and Lévy processes 2. Convolution semigroup of measures and Lévy operator 3. Fractional Laplacian 4. Maximum principle 5. Integration by parts and the Lévy operator 29 29 31 35 36 39 Chapter 2. Fractal Burgers equation 1. Statement of the problem 2. Viscous conservation laws and rarefaction waves 3. Existence o solutions 4. Decay estimates 5. Convergence toward rarefaction waves for α ∈ (1, 2) 6. Self-similar solution for α = 1 7. Linear asymptotics for 0 < α < 1 8. Probabilistic summary 45 45 46 47 48 49 50 51 52 Chapter 3. Fractal Hamilton–Jacobi–KPZ equations 1. Kardar, Parisi & Zhang and Lévy operators 2. Assumptions and preliminary results 3. Large time asymptotics – the deposition case 4. Large time asymptotics – the evaporation case 53 53 54 56 58 Chapter 4. Other equations with Lévy operator 1. Lévy conservation laws 2. Nonlocal equation in dislocation dynamics 59 59 61 Bibliography 65 27 “volumeV” — 2009/8/3 — 0:35 — page 28 — #44 “volumeV” — 2009/8/3 — 0:35 — page 29 — #45 CHAPTER 1 Lévy operator 1. Probabilistic motivations – Wiener and Lévy processes In 1827 the Scottish botanist Robert Brown observed that pollen grains suspended in liquid performed an irregular motion, caused by the random collisions with the molecules of the liquid, see Figure 1. The hits occur a large number of times in any small interval of time, independently of each other and the eﬀect of a particular hit is small compared to the total eﬀect. The physical theory of this motion (and the probabilistic derivation of the heat equation, see (1.2)) was set up by Einstein in 1905. All those facts suggest that this motion is random, and has the following properties: (i) it has independent increments; (ii) increments are Gaussian random variables; (iii) the motion is continuous. Property (i) means that the displacements of a pollen particle over disjoint time intervals are independent random variables. Property (ii) is not surprising in view of the central-limit theorem. To describe this motion mathematically, we recall ﬁrst that a random variable X : Ω → R is called Gaussian with mean m and variance σ 2 (and one uses the Figure 1. Starting at the origin trajectory of a Brownian motion. 29 “volumeV” — 2009/8/3 — 0:35 — page 30 — #46 30 1. LÉVY OPERATOR notation X ∼ N (m, σ 2 )) if, for every Borel set A ⊂ R Z (x − m)2 1 √ exp − P ({ω ∈ Ω : X(ω) ∈ A}) = dx. 2σ 2 2πσ A A random variable X = (X1 , ..., Xn ) : Ω → Rn is called Gaussian if all linear combinations of the random variables Xk , k = 1, ..., n, are Gaussian. Norbert Wiener proposed to model the Brownian motion by a continuous time stochastic process {W (t)}t≥0 (see Deﬁnition 1.1, below). Here, W (t, ω) is a random variables for each t ≥ 0 which is interpreted as the position at time t of the pollen grain ω. Definition 1.1. The stochastic process {W (t)}t≥0 is called the Wiener process, if it fulﬁls the following conditions • W (0) = 0 with probability equal to one: P ({ω ∈ Ω : W (0, ω) = 0}) = 1, • W (t) has independent increments: for every sequence 0 ≤ t0 < t1 < · · · < tn , the random variables W (t0 ), W (t1 ) − W (t0 ), . . . , W (tn ) − W (tn−1 ) are independent, • trajectories of W are continuous with probability equal to one • for all 0 ≤ s ≤ t, we have Wt − Ws ∼ N (0, t − s). It is possible to prove that such processes exist and probabilists have studied systematically their properties, see the book by Revuz and Yor [55]. Now, for every x ∈ Rn and every function u0 ∈ C(Rn ) ∩ L∞ (Rn ) we deﬁne the average Z u(x, t) = E(u0 (x + W (t))) = Rn u0 (x + y) N (0, t)(dy), (1.1) where “E” denotes the mathematical expectation and N (0, t)(dy) = (2πt)−n/2 e−|x| 2 /(2t) dy is the probability measure on Rn called the centered Gaussian measure. Here, the process x + W (t) denotes the Wiener process (or Brownian motion) started at x. By a direct calculation, it is possible to check that the function u = u(x, t) from (1.1) is the solution of the initial value problem for the heat equation 1 ∆u for x ∈ Rn , t > 0, 2 u(x, 0) = u0 (x). ut = (1.2) In other words, in (1.1), we obtained a solution of the heat equation starting a Wiener process at each point x ∈ Rn and computing the average (the mathematical expectation) of all trajectories started at x. However, there are several examples from ﬂuid mechanics, solid state physics, polymer chemistry, and mathematical ﬁnance leading to non-Gaussian processes where the trajectories are no longer continuous (they have jumps as shown on Figure 1). Such phenomena appear to be well modeled by Lévy processes (named after the French mathematician Paul Lévy), where the assumption on the continuity of trajectories from the deﬁnition of a Wiener process is replaced by the more general notion of continuity in probability. “volumeV” — 2009/8/3 — 0:35 — page 31 — #47 2. CONVOLUTION SEMIGROUP OF MEASURES AND LÉVY OPERATOR 31 Figure 2. Two pictures of the same trajectory of a pure jump Lévy process. On the right hand side, points of jumps of this trajectory were connected by straight lines in order to make the motion more visible. Definition 1.2. The stochastic process {X(t)}t≥0 on the probability space (Ω, F, P ) is called the Lévy process if it fulﬁls the following conditions: • • • • X(0) = 0 with probability equal to one, X(t) has independent increments, the probability distribution of X(s + t) − X(s) is independent of s, the process X(t) is continuous in probability, namely, lims→t P (|Xs − Xt | > ε) = 0. Note that the mathematical assumption on the continuity in probability admits Lévy processes having trajectories with jumps (see Remark 1.13). We refer the reader to the review articles by Applebaum [6] and Woyczyński [59] for several applications of Lévy processes as well as to the book by Bertoin [12] for mathematical results. Now, with a given Lévy process X(t), we associate the family of probability measures µt on Rn deﬁned by the formula Z µt (dy) ≡ P ({ω ∈ Ω : X(t, ω) ∈ A}) A for every Borel set A ⊂ Rn . Next, similarly as in the case of a Wiener process, for every u0 ∈ C(Rn ) ∩ L∞ (Rn ) and x ∈ Rn , we deﬁne the function Z u0 (x + y) µt (dy) (1.3) u(x, t) = E(u0 (x + X(t))) = Rn where x + X(t) is a Lévy process started at x. In the next section, using purely non-probabilistic language, we shall identify the initial value problem satisﬁed by the function u = u(x, t). 2. Convolution semigroup of measures and Lévy operator As it was explained in the previous section, the chaotic motion described by the Wiener process or, more generally, by the Lévy process can be described (in a “volumeV” — 2009/8/3 — 0:35 — page 32 — #48 32 1. LÉVY OPERATOR purely analytic way) by the family of probability measures {µt }t≥0 on Rn with the properties stated in the following deﬁnition. Definition 1.3. The family of nonnegative Borel measures {µt }t≥0 on Rn is called the convolution semigroup if (1) µt (Rn ) = 1 for all t ≥ 0; (2) µs ∗ µt = µt+s for s, t ≥ 0 and µ0 = δ0 (the Dirac delta) (3) µt → δ0 vaguely as t → 0, namely, Z ϕ(y) µt (dy) → ϕ(0) as t → 0 Rn for every test function ϕ ∈ Cc (Rn ) (smooth and compactly supported). Obviously, we deal with probability measures by condition (1). Item (2) is the analytic way to say that the increments of the corresponding stochastic process are independent. The continuity in probability of the process is encoded in (3). The following theorem results directly from Deﬁnition 1.3. Theorem 1.4. Let {µt }t≥0 be a convolution semigroup of measure on Rn . There exists a function a : Rn → C such that the equality µ bt (ξ) = (2π)−n/2 e−ta(ξ) n holds for all ξ ∈ R and t ≥ 0. Proof. Recall that the Fourier transform of a measures is deﬁned as Z µ bt (ξ) = (2π)−n/2 e−ixξ µt (dx). Rn n Now, for ﬁxed ξ ∈ R we consider the mapping φξ : [0, ∞) 7→ C deﬁned by Z φξ (t) = (2π)n/2 µ bt (ξ) = e−ixξ µt (dx). Rn By condition (ii) from Deﬁnition 1.3, we obtain φξ (s + t) = φξ (t)φξ (s), (1.4) because the Fourier transform changes a convolution into a product. Moreover, the convergence stated in (3) of Deﬁnition 1.3 implies limt→0 φξ (t) = 1. The functional equation (1.4) has the well-known unique (continuous at zero) solution. Hence, for every ξ ∈ Rn there is a unique complex number a(ξ) such that φξ (t) = e−ta(ξ) for all t ≥ 0. Definition 1.5. The function a = a(ξ) obtained in Theorem 1.4 is called the symbol of the convolution semigroup of measures {µt }t≥0 . Now, we are in a position to deﬁne the pseudodiﬀerential operator, which plays the main role in these lecture notes. Definition 1.6. Lévy operator L is the pseudodiﬀerential operator with the symbol a = a(ξ) corresponding to a certain convolution semigroup of measures. In c other words, Lv(ξ) = a(ξ)b v (ξ). Let us now explain the connection between the convolution semigroup and the corresponding initial value problem with Lévy operator. This is the ﬁrst step toward studying evolution equations with Lévy operators. “volumeV” — 2009/8/3 — 0:35 — page 33 — #49 2. CONVOLUTION SEMIGROUP OF MEASURES AND LÉVY OPERATOR 33 Theorem 1.7. Denote by a = a(ξ) the symbol of the convolution semigroup {µt }t≥0 in Rn . For every sufficiently regular (bounded) function u0 = u0 (x) the convolution Z u(x, t) = Rn u0 (x − y) µt (dy). (1.5) is the solution of the initial value problem x ∈ Rn , t ≥ 0 ut = −Lu, u(x, 0) = u0 (x). (1.6) (1.7) Proof. If we compute the Fourier transform of the function in (1.5) and of the equation (1.6), we see that, for every ξ ∈ Rn , the function u b(ξ, t) = (2π)−n/2 e−a(ξ) u b0 (ξ) (cf. Theorem 1.4) is the solution of the ordinary diﬀerential equation u bt (ξ, t) = −a(ξ)b u(ξ, t) supplemented with the initial datum u b0 (ξ). The initial value problem (1.6)-(1.7) describes so-called anomalous diffusion. Remark 1.8. Notice that the convolution stated in (1.5) differs from the convolutions in (1.1) and in (1.3). Obviously, both expressions are equivalent because it suffices to replace any probability measure µt (dy) by µt (−dy). In this work, we prefer to use the standard notation from (1.5). Remark 1.9. Using a more sophisticated language, one can say that the operator −L generates a strongly continuous semigroup e−tL of linear operators on L2 (R) given by (1.5). This is the sub-Markovian semigroup, namely, 0≤v≤1 implies 0 ≤ e−tL v ≤ 1 almost everywhere (see e.g. [37, Chapter 4] for more details). Example 1.10. There is the well-known connection between the Cauchy problem for the heat equation ut = ∆u, u(x, 0) = u0 (x) x ∈ Rn , t ≥ 0 (1.8) and the following convolution semigroup (“dy” means the Lebesgue measure) µt (dy) = (4πt)−n/2 e−|y| 2 /(4t) dy for all t > 0. Indeed, the solution of the initial value problem (1.8) (for not too bad initial conditions) has the form Z 2 u0 (x − y)(4πt)−n/2 e−|y| /(4t) dy. u(x, t) = Rn 2 In this case, we have the equality µbt (ξ) = (2π)−n/2 e−t|ξ| from which we immediately obtain the symbol a(ξ) = |ξ|2 of this convolution semigroup and the corresponding Lévy operator L = −∆. “volumeV” — 2009/8/3 — 0:35 — page 34 — #50 34 1. LÉVY OPERATOR Example 1.11. Now, let us show that, for every fixed b ∈ Rn , the first order differential operator L = b · ∇ is the Lévy operator with the symbol a(ξ) = ib · ξ. Indeed, in this case, we should consider the initial value problem for the transport equation ut + b · ∇u = 0, u(x, 0) = u0 (x) (1.9) with the well-known solution u(x, t) = u0 (x − bt). Note that this solution takes the form from (1.5) for the convolution semigroup of measures µt (dx) = δtb (the Dirac delta at tb). It is possible to characterize all Lévy operators. Theorem 1.12 (Lévy–Khinchin formula). Assume that a : Rn → C is the symbol of a certain convolution semigroup of measures on Rn . Then there exist • a constant c ≥ 0, • a vector b ∈ Rn , • a symmetric positive semidefinite quadratic form q on Rn q(ξ) = n X ajk ξj ξk , j,k=1 • a nonnegative Borel measure Π on Rn satisfying Π({0}) = 0 and Z min(1, |η|2 ) Π(dη) < ∞ (1.10) Rn such that the following representation is valid Z 1 − e−iηξ − iηξ1I{|η|<1} (η) Π(dη). a(ξ) = ib · ξ + q(ξ) + (1.11) Rn Moreover, this representation is unique. In other words, taking into account Theorem 1.4, we may reformulate the Lévy–Khinchin Theorem 1.12 as follows: the Fourier transform of any convolution semigroup {µt }t≥0 of measures on Rn is of the form µ bt (ξ) = (2π)−n/2 e−ta(ξ) where the symbol a = a(ξ) is given by (1.11). One should also remember the reverse implication: for every c, b, q, Π as in Theorem 1.12, the function a = a(ξ) in (1.11) is the symbol of certain convolution semigroup of measures (see [37, Thm. 3.7.8]) hence the corresponding pseudodiﬀerential operator is a Lévy operator. Here, we skip the long proof of Theorem (1.12) and we refer the reader to [37, Ch. 3.7] for an analytic reasoning (based on properties of the Fourier transform of a measure) which leads to representation (1.11). However, to understand deeper this representation, one should look at probabilistic arguments which lead to Theorem 1.12. We sketch and discuss them in Remark 1.13, below. Now, let us emphasize that, since every Lévy operator is deﬁned by the Fourier c transform as Lu(ξ) = a(ξ)b u(ξ), using the explicit form of the symbol a = a(ξ) given in (1.11) and inverting the Fourier transform we obtain the most general form of “volumeV” — 2009/8/3 — 0:35 — page 35 — #51 3. FRACTIONAL LAPLACIAN 35 the Lévy operator: Lu(x) =b · ∇u(x) − − Z Rn n X ajk j,k=1 ∂2u ∂xj ∂xk u(x − η) − u(x) − η · ∇u(x)1I{|η|<1} (η) Π(dη). (1.12) The ﬁrst term on the right-hand side of (1.12) corresponds to the transport operator recalled in Example 1.11. Note that the matrix (ajk )nj,k=1 is assumed to be nonnegative-deﬁnite; if it is not degenerate, a linear change of the variables transforms the second term in (1.12) into the usual Laplacian −∆ on Rn which corresponds to the Brownian part of the diﬀusion modeled by L. The integral on the right-hand side of (1.12) is called the pure jump part of the Lévy operator and the Lévy measure Π describes the statistical properties of jumps of the corresponding Lévy process. Remark 1.13. In the study of evolution equations with Lévy operator, it is useful to keep in mind probabilistic arguments which lead to the Lévy–Khinchin formula (1.12). The probabilistic proof of Theorem 1.12 consists in showing that any Lévy process {X(t)}t≥0 (cf. Definition 1.2) can be expressed as the sum of three independent Lévy processes X(t) = X (1) (t) + X (2) (t) + X (3) (t), where • X (1) is a linear transform of a Brownian motion with drift • X (2) is a compound Poisson process having only jumps of size at least 1, • X (3) is a pure-jump martingale only with jumps of size less than 1. Moreover, this decomposition is unique. Note that the process X (1) has continuous trajectories almost surely, and is expressed by the first and the second term on the right-hand side of (1.12). Now, we should decompose the integral term in (1.12) into two parts: the integral describing large jumps |η| ≥ 1 modeled by Poisson process X (2) and to the integral corresponding to the pure-jump martingale X (3) for small jumps |η| < 1. Details of this proof, which can be understood by non-probabilists, can be found in the first chapter of the excellent book by J. Bertoin [12]. 3. Fractional Laplacian Let us now present the most important example of the Lévy operator which will often appear in these lectures. Choosing, in formula (1.12), b = 0, ajk = 0 for all j, k ∈ {1, ..., n}, and the following Lévy measure Π(dη) = C(α) |η|n+α with α ∈ (0, 2) (1.13) and with a certain explicit constant C = C(α) > 0 we obtain the so-called α-stable anomalous diﬀusion operator L = (−∆)α/2 with the symbol a(ξ) = |ξ|α for 0 < α ≤ 2. (1.14) “volumeV” — 2009/8/3 — 0:35 — page 36 — #52 36 1. LÉVY OPERATOR Using the symmetry of the Lévy measure, we can rewrite (1.12) in this particular case as Z u(x − η) − u(x) dη. (1.15) (−∆)α/2 u(x) = −C(α) lim ε→0 |η|≥ε |η|n+α Calculations based only on the properties of the Fourier transform which shows the equivalence of deﬁnitions (1.14) and (1.15) can be also found e.g. in [27, Thm. 1]. The corresponding convolution semigroup of measures has a density µt (dx) = pα (x, t) dx for all t > 0, where the function pα (x, t) can be computed via the Fourier α transform pbα (ξ, t) = e−t|ξ| (c.f. Theorem 1.4). In particular, pα (x, t) = t−n/α Pα (xt−1/α ), (1.16) α where Pα is the inverse Fourier transform of e−|ξ| (see [37, Ch. 3] for more details). It is well known that for every α ∈ (0, 2) the function Pα is smooth, nonnegative, and satisﬁes the estimates 0 < Pα (x) ≤ C(1 + |x|)−(α+n) and |∇Pα (x)| ≤ C(1 + |x|)−(α+n+1) (1.17) for a constant C and all x ∈ Rn . Moreover, Pα (x) = c0 |x|−(α+n) + O |x|−(2α+n) , and ∇Pα (x) = −c1 x|x|−(α+n+2) + O |x|−(2α+n+1) , where c0 = α2α−1 π −(n+2)/2 sin(απ/2)Γ and as |x| → ∞, as |x| → ∞, (1.18) (1.19) α + n α Γ , 2 2 α + n + 2 α Γ . 2 2 We refer to [21] for a proof of the formula (1.18) with the explicit constant c0 . The optimality of the estimate of the lower order term in (1.18) is due Kolokoltsov [46, Eq. (2.13)], where higher order expansions of Pα are also computed. The proof of the asymptotic expression (1.19) and the value of c1 can be deduced from (1.18) using an identity by Bogdan and Jakubowski [22, Eq. (11)]. c1 = 2πα2α−1 π −(n+4)/2 sin(απ/2)Γ 4. Maximum principle In this section and in the next one, we recall properties of Lévy operators which are useful in the study of nonlinear equations. We begin with the maximum principle which is well known in the case of elliptic and parabolic problems. Here, we present results for Lévy operators, but they can be formulated in a much more general case of generators of Feller semigroups, see [37, Sec. 4.5]. Definition 1.14. We say that the operator (A, D(A)) satisﬁes the positive maximum principle if for any ϕ ∈ D(A) the fact that 0 ≤ ϕ(x0 ) = supx∈Rn ϕ(x) for some x0 ∈ Rn implies Aϕ(x0 ) ≤ 0. Remark 1.15. Obviously, the operators Aϕ = ϕ′′ and , more generally, Aϕ = ∆ϕ satisfy the positive maximum principle. “volumeV” — 2009/8/3 — 0:35 — page 37 — #53 4. MAXIMUM PRINCIPLE 37 Theorem 1.16. Denote by L the Lévy diffusion operator. Then A = −L satisfies the positive maximum principle. Proof. We present two diﬀerent arguments which are based on diﬀerent properties of the Lévy operator L. Let ϕ ∈ D(L) and assume that 0 ≤ ϕ(x0 ) = sup ϕ(x) for some x∈Rn x0 ∈ Rn . First argument. Using the Lévy–Khinchin representation (1.12) we obtain that the following quantity −Lϕ(x0 ) = −b · ∇ϕ(x0 ) + + Z Rn n X ajk j,k=1 ∂ 2 ϕ(x0 ) ∂xj ∂xk ϕ(x0 − η) − ϕ(x0 ) − n X j=1 ηj ∂ϕ(x0 ) 1I{|η|<1} (η) Π(dη) ∂xj is nonpositive because the ﬁrst term on the right-hand side is equal to zero since x0 is the point of the maximum of ϕ, the second term is nonpositive due to the property of the matrix {ajk }nj,k=1 (see Theorem 1.12), and the integrand of the third term is nonpositive because ϕ(x0 − η) ≤ ϕ(x0 ) for all η ∈ Rn . Second argument. Recall that, by Theorem 1.7, the solution of the problem x ∈ Rn , t ≥ 0, ut = −Lu, u(x, 0) = ϕ(x) is given by u(x, t) = Z Rn ϕ(x − y)µt (dy). Hence, by the deﬁnition of the derivative ∂t , we have −Lϕ(x0 ) = lim+ t→0 u(x0 , t) − ϕ(x0 ) . t R Now, the right-hand side is nonpositive for any t > 0 because Rn µt (dy) = 1 and because Z ϕ(x0 − y) − ϕ(x0 ) µt (dy) ≤ 0 u(x0 , t) − ϕ(x0 ) = Rn by the deﬁnition of x0 and since the measures µt are nonnegative. Next, we prove an analogous result for bounded functions which not necessarily attain their points of global maximum. Here, we follow an argument from [27, Thm. 2]. Lemma 1.17. Let ϕ ∈ Cb2 (Rn ). Assume that the sequence {xn }n≥1 ⊂ Rn satisfies ϕ(xn ) → supx∈Rn ϕ(x). Then lim ∇ϕ(xn ) = 0 n→∞ and lim sup −Lϕ(xn ) ≤ 0. n→∞ “volumeV” — 2009/8/3 — 0:35 — page 38 — #54 38 1. LÉVY OPERATOR Proof. By the assumption, the matrix D2 ϕ has bounded coeﬃcients, hence there exists C > 0 such that sup ϕ(x) ≥ ϕ(xn + z) ≥ ϕ(xn ) + ∇ϕ(xn ) · z − C|z|2 . (1.20) x∈Rn Since the sequence ∇ϕ(xn ) is bounded, passing to the subsequence, we can assume that ∇ϕ(xn ) → p. Consequently, passing to the limit in (1.20) we obtain the inequality 0 ≥ p · z − C|z|2 for every z ∈ Rn . Choosing z = tp and letting t → 0+ , we get p = 0. Now, we prove that lim supn→∞ −Lϕ(xn ) ≤ 0. Note ﬁrst that, by the deﬁnition of the sequence {xn }n , we have ϕ(xn + z) − ϕ(xn ) ≤ sup ϕ − ϕ(xn ) → 0 x∈Rn as n → ∞. Hence lim supn→∞ ϕ(xn + z) − ϕ(xn ) ≤ 0 and equivalently for ∇ϕ(xn ) → 0, lim sup ϕ(xn + z) − ϕ(xn ) − ∇ϕ(xn ) · z ≤ 0. n→∞ Finally, it suﬃces to use the Fatou lemma in the expression Z Lϕ(xn ) = ϕ(xn − z) − u(xn ) − z · ∇ϕ(xn )1I{|z|<1} (z) Π(dz), Rn because the Lévy measure Π is nonnegative. We are in a position to prove the main comparison principle for equations with Lévy operators. Theorem 1.18. Assume that u ∈ Cb (Rn ×[0, T ])∩Cb2(Rn ×[ε, T ]) is the solution of the equation ut = −Lu + v(x, t) · ∇u, (1.21) where L is the Lévy operator represented by (1.12) and v = v(x, t) is a given and sufficiently regular function with values in Rn . Then u(x, 0) ≤ 0 u(x, t) ≤ 0 implies for all, x ∈ Rn , t ∈ [0, T ]. Proof. We extract this proof from [27, Proof of Prop. 2]. It is an easy exercise using assumptions imposed on u to show that the function Φ(t) = sup u(x, t) x∈Rn is well-deﬁned and continuous. Our goal is to show that Φ is locally Lipschitz and Φ′ (t) ≤ 0 almost everywhere. To show the Lipschitz continuity of Φ, for every ε > 0 we chose xε such that sup u(x, t) = u(xε , t) + ε. x∈Rn Now, we ﬁx t, s ∈ I, where I ⊂ (0, T ) is a bounded and closed interval and we suppose (without loss of generality) that Φ(t) ≥ Φ(s). Using the deﬁnition of Φ “volumeV” — 2009/8/3 — 0:35 — page 39 — #55 5. INTEGRATION BY PARTS AND THE LÉVY OPERATOR 39 and regularity of u we obtain 0 ≤ Φ(t) − Φ(s) = sup u(x, t) − sup u(x, s) x∈Rn x∈Rn ≤ ε + u(xε , t) − u(xε , s) ≤ ε + sup |u(x, t) − u(x, s)| x∈Rn ≤ ε + |t − s| sup x∈Rn ,t∈I |∇t u(x, t)|. Since ε > 0 and t, s ∈ I are arbitrary, we immediately obtain that the function Φ is locally Lipschitz hence, by the Rademacher theorem, diﬀerentiable almost everywhere, as well. Let us now diﬀerentiate Φ(t) = supx∈Rn u(x, t) with respect to t > 0. By the Taylor expansion, for 0 < s < t, we have u(x, t) ≤ u(x, t − s) + sut (x, t) + Cs2 . Hence, using equation (1.21), we obtain u(x, t) ≤ sup u(x, t − s) + s − Lu(x, t) + v(x, t)∇u(x, t) + Cs2 . (1.22) x Substituting in (1.22) x = xn , where u(xn , t) → supx u(x, t) as n → ∞, passing to the limit using Lemma 1.17, we obtain the inequality sup u(x, t) ≤ sup u(x, t − s) + Cs2 x x which can be transformed into Φ(t) − Φ(s) ≤ Cs. s For s ց 0, we obtain Φ′ (t) ≤ 0 in those t, where Φ is diﬀerentiable. 5. Integration by parts and the Lévy operator We have seen in the previous section that any pseudodiﬀerential operator given by the Lévy-Khinchin formula (1.12) satisﬁes the maximum principle typical for the Laplace operator. Now, we present other properties of Lévy operators which will allow us to show energy-type estimates for solutions of some evolution equations. It is worth to emphasize that equalities and inequalities, proved in the case of Laplacian integrating by parts, can be generalized for any Lévy operator by using suitable convex inequalities. Let us illustrate this phenomenon by proving the Kato inequality. Theorem 1.19 (Kato inequality for Laplacian). For every ϕ ∈ Cc∞ (Rn ), Z (−∆ϕ) sgn ϕ dx ≥ 0. Rn Proof. Let us begin with the following smooth approximation of the sign function d p s . gε (s) = ε + s2 = √ ds ε + s2 “volumeV” — 2009/8/3 — 0:35 — page 40 — #56 40 1. LÉVY OPERATOR Note that gε′ (s) ≥ 0 and gε (s) → sgn s as ε → 0. Now, we integrate by parts to obtain Z Z |∇ϕ|2 gε′ (ϕ) dx ≥ 0. (−∆ϕ) gε (ϕ) dx = Rn Rn To complete the proof, it suﬃces to pass to the limit ε → 0 on the left-hand side using the Lebesgue dominated convergence theorem. Theorem 1.20 (Kato inequality for Lévy operator). For every ϕ ∈ Cc∞ (Rn ) and for every Lévy operator represented by (1.12), we have Z (Lϕ) sgn ϕ dx ≥ 0. Rn Proof. According to Deﬁnitions 1.6 and 1.3 we denote by {µt }t≥0 the convolution semigroup of measures corresponding to the Lévy operator L. Recall that Z −tL u0 (x − y) µt (dx) (1.23) e u0 (x) ≡ u(x, t) = Rn is the solution of the initial value problem (1.6)-(1.7). In particular, we have Lϕ = lim+ t→0 ϕ − e−tL ϕ . t Consequently, it suﬃces to show that, for every t > 0, we have the inequality Z (ϕ − e−tL ϕ) sgn ϕ dx ≥ 0 Rn which is equivalent to Z Rn |ϕ| dx ≥ Z (e−tL ϕ) sgn ϕ dx. (1.24) Rn We complete the proof of inequality (1.24) by using the formula (1.23), the Fubini theorem, and the fact that µt is the probability measure for every t ≥ 0 as follows Z Z Z Z −tL t ≤ (e ϕ) sgn ϕ dx |ϕ(x − y)| µ (dy) dx = |ϕ| dx. Rn Rn Rn Rn Let us present the next result which looks like an integration by parts for any Lévy operator. Theorem 1.21 (Strook–Varopoulos inequality). Assume that L is a Lévy operator. For every p ∈ (1, ∞) and ϕ ∈ Cc∞ (Rn ) such that ϕ ≥ 0 we have Z Z p−1 4 2 (Lϕ) ϕp−1 dx. (1.25) (Lϕp/2 ) ϕp/2 dx ≤ p n n R R Remark 1.22. Note that, for L = b · ∇ with any fixed b ∈ Rn , both sides of the Strook–Varopoulos inequality (1.25) are equal to 0. On the other hand, if L = −∆, “volumeV” — 2009/8/3 — 0:35 — page 41 — #57 5. INTEGRATION BY PARTS AND THE LÉVY OPERATOR 41 we integrate by parts to obtain the equality Z Z |∇ϕ|2 ϕp−2 dx (−∆ϕ) ϕp−1 dx = (p − 1) n n R ZR = (p − 1) |∇ϕ ϕp/2−1 |2 dx Rn Z p−1 =4 2 |∇ϕp/2 |2 dx. p Rn Sketch of proof of Theorem 1.21. Inequality (1.25) was proved by Strook [56] and Varopoulos [57]. We also refer the reader to the review article by Liskevich and A. Semenov [48] (the preprint is available on the V.A. Liskevisch webpage) for the proof of this inequality in the case of much more general Markov semigroups. Here, we emphasize the main steps of the proof of (1.25), only. Step 1. Let α > 0 and β > 0 be such that α + β = 2. Then the following inequality (xα − y α )(xβ − y β ) ≥ αβ(x − y)2 holds true for all x ≥ 0 and y ≥ 0. Step 2. As before, we use the relation Z Z 1 (f − e−tL f ) g dx, (Lf ) g dx = lim t→0+ t Rn Rn valid for all f, g ∈ D(L). Step 3. We use inequality from Step 1 and formula (1.5) (remember that µt is a probability measure) to show Z Z (f − e−tL f ) f dx (f α − e−tL f α ) f β dx ≥ αβ Rn Rn for every f ∈ D(L) such that f ≥ 0 and for α + β = 2. Step 4. Finally, we substitute in the inequality form Step 3 2 2 p−1 f = ϕp/2 , α = , β = 2 − , αβ = 4 2 , p p p and, after dividing by t, we pass to the limit t → 0+ to conclude the proof. Remark 1.23 (General Strook–Varopoulos inequality). The Kato inequality combined with the Strook–Varopoulos inequality give the following estimate 4(p − 1) hL|ϕ|p/2 , |ϕ|p/2 i ≤ hLϕ, |ϕ|p−1 sgn ϕi p2 (1.26) for every ϕ ∈ D(L). This inequality is more suitable for studying sign changing solutions. Theorem 1.24 (Convexity inequality, see e.g. [25, 27, 40]). Let u ∈ Cb2 (Rn ) and g ∈ C 2 (R) be a convex function. Then Lg(u) ≤ g ′ (u)Lu. (1.27) Remark 1.25. Note that, in the one dimensional case, for L = −∂x2 we have −(g(u))xx = −g ′′ (u)u2x − g ′ (u)uxx ≤ −g ′ (u)uxx since g ′′ ≥ 0. “volumeV” — 2009/8/3 — 0:35 — page 42 — #58 42 1. LÉVY OPERATOR Proof of Theorem 1.24. The convexity of the function g leads to the inequality g(u(x − η)) − g(u(x)) ≥ g ′ (u(x))[u(x − η) − u(x)], which can by immediately reformulate as follows g(u(x − η)) − g(u(x)) − η · ∇g(u(x)) ≥ g ′ (u(x))[u(x − η) − u(x) − η · ∇u(x)] for any η ∈ Rn . To complete the proof, it suﬃces to apply the Lévy-Khinchin form of any Lévy operator given in (1.12). Now, we state an important application of the convexity inequality (1.27). Corollary 1.26. Let g ∈ C 2 (R) be a convex function. Assume g(u) ∈ D(L) and Lg(u) ∈ L1 (Rn ). Then Z Z g ′ (u(x))Lu(x) dx. Lg(u(x)) dx ≤ 0 = Rn Rn Proof. Denoting v(x) = g(u(x)) and using properties of the (inverse) Fourier transform we obtain Z Z (a b v )ˇ(x) dx = (2π)n/2 a(0)b v (0) = 0, Lv(x) dx = Rn Rn because a(0) = 0 (cf. (1.11)). Now, it suﬃces to apply inequality (1.27). Corollary 1.27. Any Lévy diffusion operator L satisfies Z Rn p (Lu) (u − k)+ dx ≥ 0 for each 1 < p < ∞ and all constants k ≥ 0. Remark 1.28. Note that the general Strook–Varopoulos inequality C(p)hL|ϕ|p/2 , |ϕ|p/2 i ≤ hLϕ, |ϕ|p−1 sgn ϕi can be obtained immediately from the convexity inequality (1.27), applied with the convex function g(ϕ) = |ϕ|p/2 for p > 2. Here, however, we have got the nonoptimal constant 2 4(p − 1) C(p) = ≤ for every p > 2. p p2 We conclude this section by the proof of a particular case of the Gagliardo– Nirenberg inequality. The proof of the following theorem uses an argument from the celebrated work by Nash [53] where, on page 935, the author emphasized that this argument was shown to him by E.M. Stein. Theorem 1.29 (Nash inequality). Let 0 < α. There exists a constant CN > 0 such that 2(1+α) ≤ CN kΛα/2 wk22 kwk2α (1.28) kwk2 1 for all functions w = w(x) satisfying w ∈ L1 (R) and Λα/2 w ∈ L2 (R). “volumeV” — 2009/8/3 — 0:35 — page 43 — #59 5. INTEGRATION BY PARTS AND THE LÉVY OPERATOR 43 Proof. For every R > 0, we decompose the L2 -norm of the Fourier transform of w as follows Z 2 2 |w(ξ)| b dξ kwk2 = C R Z Z 2 dξ + CR−α ≤ Ckwk b 2∞ |ξ|α |w(ξ)| b dξ |ξ|≤R |ξ|>R ≤ CRkwk21 + CR−α kΛα/2 wk22 . 1/(1+α) we obtain (1.28). Choosing R = kΛα/2 wk22 /kwk21 “volumeV” — 2009/8/3 — 0:35 — page 44 — #60 “volumeV” — 2009/8/3 — 0:35 — page 45 — #61 CHAPTER 2 Fractal Burgers equation 1. Statement of the problem To see properties of a Lévy operator “in action”, we present recent results on the asymptotic behavior of solutions of the Cauchy problem for the nonlocal conservation law ut + Λα u + uux = 0, u(0, x) = u0 (x) α 2 x ∈ R, t > 0, (2.1) (2.2) 2 α/2 where Λ = (−∂ /∂x ) is the Lévy operator deﬁned via the Fourier transform α \ α (Λ v)(ξ) = |ξ| vb(ξ), see Section 3. Remark 2.1. Following [13], we will call equation (2.1) the fractal Burgers equation. There are two reasons for using here the word “fractal”. We want to emphasize the fractal nature of the symmetric α-stable stochastic process which corresponds to the operator Λα . In this sense, the usual viscous Burgers equation (i.e. (2.1) with α = 2) should be also called the fractal Burgers equation. Moreover, we would like to distinguish our equation (2.1) from the fractional Burgers equation with the fractional derivative with respect to time t. Equations of this type appear in the study of growing interfaces in the presence of self-similar hopping surface diﬀusion [49]. Moreover, in their recent papers, Jourdain, Méléard, and Woyczynski [38, 39] gave probabilistic motivations to study equations with the anomalous diﬀusion, when Laplacian (corresponding to the Wiener process) is replaced by a more general pseudodiﬀerential operator generating the Lévy process. In particular, the authors of [38] studied problem (2.1)-(2.2), where the initial condition u0 is assumed to be a nonconstant function with bounded variation on R. In other words, a.e. on R, Z x u0 (x) = c + m(dy) = c + H ∗ m(x) (2.3) −∞ with c ∈ R, m being a ﬁnite signed measure on R, and H(y) denoting the unit step function 1I{y≥0} . Observe that the gradient v(x, t) = ux (x, t) satisﬁes vt + Λα v + (vH ∗ v)x = 0, v(·, 0) = m. (2.4) If m is a probability measure on R, the equation (2.4) is a nonlinear Fokker-Planck equation. In the case of an arbitrary ﬁnite signed measure, the authors of [38] associate (2.4) with a suitable nonlinear martingale problem. Next, they study the convergence of systems of particles with jumps as the number of particles tends to +∞. As a consequence, the weighted empirical cumulative distribution functions 45 “volumeV” — 2009/8/3 — 0:35 — page 46 — #62 46 2. FRACTAL BURGERS EQUATION of the particles converge to the solution of the martingale problem connected to (2.4). This phenomena is called the propagation of chaos for problem (2.1)–(2.2) and we refer the reader to [38] for more details and additional references. Motivated by the results from [38], we study problem (2.1)–(2.2) under the crucial assumption α ∈ (1, 2) and with the initial condition of the form (2.3). In our main result, we assume that u0 is a function satisfying u0 − u− ∈ L1 ((−∞, 0)) and u0 − u+ ∈ L1 ((0, +∞)) with u− < u+ , R where u− = c and u+ − u− = R m(dx). (2.5) Remark 2.2. For c = 0 and a probability measure m, the function u0 is called the probability cumulative distribution function. 2. Viscous conservation laws and rarefaction waves It is well known [31, 35, 50] that the asymptotic proﬁle as t → ∞ of solutions of the viscous Burgers equation ut − uxx + uux = 0 (2.6) (i.e. equation (2.1) with α = 2) supplemented with an initial datum satisfying (2.5) is given by the so-called rarefaction wave. This is the continuous function x/t ≤ u− , u− , R R u− ≤ x/t ≤ u+ , w (x, t) = W (x/t) = x/t , (2.7) u+ , x/t ≥ u+ , which is the unique entropy solution of following Riemann problem wtR + wR wxR = 0, wR (x, 0) = w0R (x) = (2.8) u− , u+ , x < 0, x > 0. (2.9) Below, we use the smooth approximations of rarefaction waves, namely, the solutions of the following Cauchy problem wt − wxx + wwx = 0, u− , w(x, 0) = w0 (x) = u+ , (2.10) x < 0, . x > 0. (2.11) Lemma 2.3. Let u− < u+ . Problem (2.10)–(2.11) has the unique, smooth, global-in-time solution w(x, t) satisfying i) u− < w(t, x) < u+ and wx (t, x) > 0 for all (x, t) ∈ R × (0, ∞); ii) for every p ∈ [1, ∞], there exists a constant C = C(p, u− , u+ ) > 0 such that kwx (t)kp ≤ Ct−1+1/p , kwxx (t)kp ≤ Ct−3/2+1/(2p) and kw(t) − wR (t)kp ≤ Ct−(1−1/p)/2 , for all t > 0, where wR (x, t) is the rarefaction wave (2.7). “volumeV” — 2009/8/3 — 0:35 — page 47 — #63 3. EXISTENCE O SOLUTIONS 47 All results stated in Lemma 2.3 are deduced from the explicit formula for solutions of (2.10)–(2.11) and detailed calculations can be found in [31] with some additional improvements contained in [44, Section 3]. Finally, let us necessarily recall the fundamental paper of Il’in and Oleinik [35] who showed the convergence toward rarefaction waves of solutions of the nonlinear equation ut − uxx + f (u)x = 0 under strict convexity assumption imposed on f . That idea was next extended in several diﬀerent directions and we refer the reader, e.g., to [31, 50, 51, 54] for an overview of know results and additional references. 3. Existence o solutions The basic questions on the existence and the uniqueness of solutions of problem (2.1)–(2.2) with α ∈ (1, 2) have been answered in the papers [26, 27]. Theorem 2.4. ([26, Thm. 1.1], [27, Thm. 7]) Let α ∈ (1, 2) and u0 ∈ L∞ (R). There exists the unique solution u = u(x, t) to problem (2.1)–(2.2) in the following sense: for all T > 0, u ∈ Cb ((0, T ) × R) and, f or all a ∈ (0, T ), u ∈ Cb∞ ((a, T ) × R), u satisf ies (2.1) on (0, T ) × R, u(t, ·) → u0 in L∞ (R) weak − ∗ as t → 0. Moreover, the following inequality holds true ku(t)k∞ ≤ ku0 k∞ for all t > 0. (2.12) ∞ Remark 2.5. Notice that L (R) is not a separable Banach space. Hence, the statement u(t, ·) → u0 in L∞ (R) weak–∗ means that, for every ϕ ∈ L1 (R), we have R R u(x, t) − u0 (x) ϕ(x) dx → 0 as t → 0. The proof of Theorem 2.4 is based on the Banach ﬁxed point argument applied to the integral formulation of the Cauchy problem (2.1)–(2.2) Z t u(t) = Sα (t)u0 − Sα (t − τ )u(τ )ux (τ ) dτ, (2.13) 0 where Sα (t)u0 = pα (t) ∗ u0 (x). (2.14) α Here, the fundamental solution pα (x, t) of the linear equation ∂t v + Λ v = 0 can α be computed via the Fourier transform pbα (ξ, t) = e−t|ξ| and its properties are discussed in Section 3. Hence, by the Young inequality for the convolution, we obtain the estimates kSα (t)vkp ≤ Ct−(1−1/p)/α kvk1 , k(Sα (t)v)x kp ≤ Ct −(1−1/p)/α−1/α kvk1 (2.15) (2.16) for every p ∈ [1, ∞] and all t > 0. Moreover, we can replace v in (2.15) and in (2.16) by any signed measure m. In that case, kvk1 should be replaced by kmk. Note also that inequality (2.12) is the immediate consequence of Theorem 1.18. Now, let us deal with α ∈ (0, 1]. It was shown in [2] (see also [45]) that solutions of the initial value problem (2.1)–(2.2) can become discontinuous in ﬁnite time if 0 < α < 1. Hence, in order to deal with discontinuous solutions, the notion of “volumeV” — 2009/8/3 — 0:35 — page 48 — #64 48 2. FRACTAL BURGERS EQUATION entropy solutions in the sense of Kruzhkov was extended by Alibaud [1] to nonlocal problem (2.1)–(2.2). Theorem 2.6 ([1]). Assume that 0 < α ≤ 1 and u0 ∈ L∞ (R). There exists the unique entropy solution u = u(x, t) to the Cauchy problem (2.1)–(2.2). This solution u belongs to C([0, ∞); L1loc (R)) and satisfies u(0) = u0 . Moreover, we have the following maximum principle: essinf u0 ≤ u ≤ esssup u0 . The occurrence of discontinuities in ﬁnite time of entropy solutions of (2.1)– (2.2) with α = 1 seems to be not clear. Regularity results have recently been obtained [24, 45, 52] for a large class of initial conditions, that does unfortunately not include general L∞ initial data. Nevertheless, Theorem 2.6 provides the existence and the uniqueness of global-in-time the entropy solution even for the critical case α = 1. 4. Decay estimates Due to possible singularities of solutions of (2.1)–(2.2) with α ∈ (0, 1), from now on, we study solutions of the Cauchy problem for the regularized fractal Burgers equation with ε > 0 if α ∈ (0, 1] and ε = 0 for α ∈ (1, 2) uεt + Λα uε − εuεxx + uε uεx = 0, ε u (x, 0) = u0 (x). x ∈ R, t > 0, (2.17) (2.18) The procedure now is, roughly speaking, to make the asymptotic study of uε with stability estimates uniform in ε. Next, we pass to the limit ε → 0 using the theory developed in [3] in order to obtain for solutions of (2.1)–(2.2). Most of the results of this section are inspired from [41] and, when it is the case, the reader is referred to precise proofs in that paper. One can show (as in Theorem 2.4) that problem (2.17)–(2.18) admits the unique global-in-time smooth solution that satisﬁes the maximum principle. If, moreover, the initial datum u0 can be written in the form (2.3) for a constant c ∈ R and a signed ﬁnite measure m on R, the solution uε = uε (x, t) of problem (2.17)–(2.18) satisﬁes uεx ∈ C((0, T ]; Lp (R)) for each 1 ≤ p ≤ ∞ and every T > 0. Here, for the proofs of those properites, one should follow [41, Thm. 2.2]. Main properties of uεx (x, t) are contained in the following theorem. Theorem 2.7. Assume that 0 < α ≤ 2, ε > 0, and u0 is of the form (2.3) with c ∈ R and a finite nonnegative measure m(dx) on R. Denote by uε = uε (x, t) the unique solution of problem (2.17)–(2.18). Then (i) uεx (x, t) ≥ 0 for all x ∈ R and t > 0, (i) for every p ∈ [1, ∞] there exists C = C(p) > 0 independent of ε such that n o kuεx (t)kp ≤ C(p) min t−(1/α)(1−1/p) kmk, t−(1−1/p) kmk1/p (2.19) for all t > 0 Sketch of proof. To prove this result, it suﬃces to modify slightly the argument from [41, Thm. 2.3] as follows. We write the equation for v = uεx vt + Λα v − εvxx + (uε uεx )x = 0 (2.20) “volumeV” — 2009/8/3 — 0:35 — page 49 — #65 5. CONVERGENCE TOWARD RAREFACTION WAVES FOR α ∈ (1, 2) 49 and we note that, due to the Kato inequality (c.f. Theorems 1.19 and 1.20), we have the “good” sign of the following quantities Z Z −ε vxx (x, t)ϕ(v(x)) dx ≥ 0 and Λα v(x, t)ϕ(v(x)) dx ≥ 0 R R for any nondecreasing function ϕ. Hence, to prove Theorem 2.7 it suﬃces to rewrite all inequalities from [41, Proof of Thm. 2.3] skipping each term containing ε. Here, we recall that argument proving inequality (2.19) for p = 2, only. For v = uεx ≥ 0, we multiply equation (2.20) by v and integrate over R: Z Z Z 1 1 d kvk22 + ε (vx )2 dx + vΛα v dx + v 3 dx = 0. (2.21) 2 dt 2 R R R Note that second, third, and forth term of identity (2.21) are nonnegative. Let us use the third term and skip the other two. Applying Nash inequality (1.28) to estimate the third term of (2.21) we obtain d 2(1+α) −1 kv(t)k22 + 2CN kmk−2α kv(t)k2 ≤ 0, dt which, after integration, leads to kv(t)k2 ≤ C1 kmkt−1/(2α) with C1 = (CN /2α)1/(2α) . This is the ﬁrst decay estimate on the right-hand side of (2.19) with p = 2. To show R 3 the second inequality in (2.19), one should proceed analogously using the term R v dx. The idea of the proof of (2.19) for p 6= 2 is similar and uses Strook-Varopoulos inequality (1.25) combined with Nash inequality (1.28), see [41] for more details. To show Theorem 2.7.i, one should apply either the comparison principle from Theorem 1.18 (see [27]) or an energy argument based on Corollary 1.27 (see [41, Thm. 2.3]) . In the study of the large time asymptotics to (2.1)–(2.2), we also need the following asymptotic stability result. Theorem 2.8. Let α ∈ (0, 2). Assume that uε and f uε are two solutions of the regularized problem (2.17)–(2.18) with initial conditions u0 and u e0 of the form (2.3), the both of with finite signed measures m and m, e respectively. Suppose, moreover, that the measure m e is nonnegative and u0 − u e0 ∈ L1 (R). Then, for every p ∈ [1, ∞] there exists a constant C = C(p) > 0 independent of ε such that for all t>0 kuε (t) − f uε (t)kp ≤ Ct−(1−1/p)/α ku0 − u e0 k1 . Proof. Here, it suﬃces to copy calculations from [41, Proofs of Thm. 2.2 and Lemma 3.1] skipping each term containing ε as it was explained in the proof of Theorem 2.7. 5. Convergence toward rarefaction waves for α ∈ (1, 2) Now, we are in a position to state the result for the large time asymptotics of solutions of (2.1)–(2.2) with α ∈ (1, 2). Here, we use estimates from the previous section assuming that ε = 0. “volumeV” — 2009/8/3 — 0:35 — page 50 — #66 50 2. FRACTAL BURGERS EQUATION Theorem 2.9 ([41]). Let α ∈ (1, 2) and the initial datum u0 be of the form (2.3) with c ∈ R and m being a finite measure on R (not necessarily nonnegative). i 3−α We assume, moreover, the (2.5) holds true. For every p ∈ α−1 , ∞ there exists C > 0 independent of t such that ku(t) − wR (t)kp ≤ Ct−[α−1−(3−α)/p]/2 log(2 + t) for all t > 0. Proof. In view of Lemma 2.3, we my replace the rarefaction wave wR (x, t) by its smooth approximation w = w(x, t). Next, using the Gagliardo-Nirenberg inequality we have a ku(t) − w(t)kp ≤ C kux (t)k∞ + kwx (t)k∞ ku(t) − w(t)k1−a p0 for every 1 < p0 < p < ∞. Since both quantities kux(t)k∞ and kwx (t)k∞ decay in time by (2.19), the proof is completed by applying Lemma 2.10, stated below. Lemma 2.10. For p0 = (3 − α)/(α − 1), the following estimate is valid ku(t) − w(t)kp0 ≤ C log(2 + t). Proof. The function v = u − w satisﬁes 1 vt + Λα v + [v 2 + 2vw]x = −Λα w + wxx . 2 We multiply this equation by |v|p−2 v and we integrate over R to obtain Z Z Z 1 1 d p α p−2 |v| dx + (Λ v)(|v| v) dx + [v 2 + 2vw]x |v|p−2 v dx p dt 2 Z = (−Λα w + wxx )(|v|p−2 v) dx. The second and the third term on the left hand side are nonnegative, hence we skip them. Using the Hölder inequality on the right-hand side we obtain the following diﬀerential inequality d kv(t)kpp ≤ p (kΛα w(t)kp + kwxx (t)kp ) kv(t)kp−1 , p dt which, after integration, leads to Z t kΛα w(τ )kp + kwxx (τ )kp dτ. kv(t)kp ≤ kv(t0 )kp + t0 Now, we use the decay properties of the smooth approximation of rarefaction waves from Lemma 2.3 to complete the proof (see [41, Lemma 3.3] for more details). 6. Self-similar solution for α = 1 Using the uniqueness result from [1] (see Theorem 2.6 above) combined with a standard scaling technique, one can show that equation (2.1) with α = 1 has self-similar solutions. “volumeV” — 2009/8/3 — 0:35 — page 51 — #67 7. LINEAR ASYMPTOTICS FOR 0 < α < 1 51 Theorem 2.11. Assume α = 1. The unique entropy solution U = U (x, t) of the initial value problem (2.1)–(2.2) with the initial condition u− , x < 0, U0 (x) = (2.22) u+ , x > 0, is self-similar, i.e. it has the form U (x, t) = U (x/t, 1) for all x ∈ R and all t > 0. Note that the function U0 from (2.22) is of the form (2.3) for the measure m := (u+ − u− )δ0 (here, δ0 denotes the Dirac delta at 0). In [3], we show that the self-similar proﬁle U (x, 1) from Theorem 2.11 enjoys the following properties: • Regularity: U (·, 1) is Lipschitz-continuous. • Monotonicity: U (·, 1) is non-decreasing with limx→±∞ U (x, 1) = u± . • Symmetry: For all y ∈ R, we have the equality u− + u+ . U (c + y, 1) = c − U (c − y, 1) , where c := 2 • Convexity: U (·, 1) is convex (resp. concave) on (−∞, c] (resp. on [c, +∞)). −u− • Decay at inﬁnity: We have Ux (x, 1) ∼ u+2π |x|−2 , as |x| → ∞. 2 This self-similar solution U = U (x, t) describes the large time asymptotics of other solutions of (2.1)–(2.2) with α = 1. Theorem 2.12. Let α = 1. Let u = u(x, t) be the entropy solution of problem (2.1)–(2.2) corresponding to the initial condition of the form (2.3) satisfying (2.5). Denote by U = U (x, t) the self-similar solution from Theorem 2.11. For every p ∈ [1, ∞] there exists a constant C = C(p) > 0 such that ku(t) − U (t)kp ≤ Ct−(1−1/p) ku0 − U0 k1 for all t > 0. Proof. This result is the immediate consequence of Theorem 2.8 by passing to the limit ε → 0. We refer the reader to [3] for more details concerning self-similar solutions of equation (2.1). 7. Linear asymptotics for 0 < α < 1 In the case where α < 1, the Duhamel principle (2.13) combined with the decay estimates (2.19) allow us to show that the nonlinearity in (2.1) is negligible in the asymptotic expansion of solutions. Theorem 2.13 ([3]). Let 0 < α < 1 and u = u(x, t) be the entropy solution of (2.1)–(2.2) corresponding to the initial condition of the form (2.3) satisfying (2.5). Denote by Sα (t)u0 the solution of the linear initial value problem ut + Λα u = 0, 1 u(x, 0) = u0 (x). For every p ∈ 1−α , ∞ there exists C = C(p) > 0 such that ku(t) − Sα (t)u0 kp ≤ Cku0 k∞ kmkt1−(1/α)(1−1/p) for all t > 0. (2.23) “volumeV” — 2009/8/3 — 0:35 — page 52 — #68 52 2. FRACTAL BURGERS EQUATION It follows from the proof of Theorem 2.13 that inequality (2.23) is valid for 1 , ∞ , only. every p ∈ [1, ∞]. Its right-hand-side, however, decays for p ∈ 1−α Actually, the asymptotic term Sα (t)u0 in (2.23) can be written in a self-similar way. Corollary 2.14. Under the assumptions of Theorem 2.13, we have Z as t → ∞, c + Hα (t) m(dx) − u(t) → 0 p R Rx where Hα (x, t) := −∞ pα (y, t) dy and pα (x, t) is the fundamental solution of the linR ear equation ut + Λα u = 0. Moreover, if we assume in addition that R |x||m|(dx) < ∞, then we have the following rate of convergence Z c + Hα (t) m(dx) − u(t) p R Z ≤ C ku0 k∞ kmk + |x||m|(dx) t1−(1/α)(1−1/p) , (2.24) for some constant C = C(p). R Notice that c + Hα (x, t) m(dx) is nothing else than the solution of problem ut + Λα u = 0, u(x, 0) = U0 (x) (U0 being deﬁned in (2.22)). It is well-known that this solution is eﬀectively self-similar with the scaling Hα (x, t) = Hα xt−1/α , 1 , see also the homogeneity property (1.16). 8. Probabilistic summary Let us summarize our results on large time behavior of solutions of the initial value problem (2.1)–(2.2). In the case α > 1, the diﬀusive term in (2.1) is negligible in the asymptotic expansion of solutions (see Theorem 2.9), whereas in the case α < 1, the nonlinear convection term does not appear in the asymptotics of solutions (cf. Theorem 2.13). In the case α = 1, both terms have to be taken into account (cf. Theorem 2.12). To conclude, let us emphasize the probabilistic meaning of these results. We have already mentioned that the solution u of (2.1)–(2.2) supplemented with the initial datum of the form (2.3) with c = 0 and with a probability measure m on R is the cumulative distribution function for every t ≥ 0. This family of probabilities deﬁned by problem (2.1)–(2.2) converges, as t → ∞, toward • the uniform distribution on the interval [0, t] if 1 < α ≤ 2 (see Theorem 2.9); • the one parameter family of new laws constructed in Theorem 2.11 if α = 1 (see Theorem 2.12); • the symmetric α-stable laws pα (t) if 0 < α < 1 (cf. Theorem 2.13 and Corollary 2.14). “volumeV” — 2009/8/3 — 0:35 — page 53 — #69 CHAPTER 3 Fractal Hamilton–Jacobi–KPZ equations 1. Kardar, Parisi & Zhang and Lévy operators The well-known Kardar–Parisi–Zhang (KPZ) equation λ |∇h|2 2 was derived in [43] as a model for growing random interfaces. Recall that the interface is parameterized here by the transformation Σ(t) = (x, y, z = h(x, y, t)), so that h = h(x, y, t) is the surface elevation function, ν > 0 is identiﬁed in [43] as a “surface tension” or “high diﬀusion coeﬃcient”, ∆ and ∇ stand, respectively, for the usual Laplacian and gradient diﬀerential operators in spatial variables, and λ ∈ R scales the intensity of the ballistic rain of particles onto the surface. An alternative, ﬁrst-principles derivation of the KPZ equation (cf. [49], for more detailed information and additional references) makes three points: • The Laplacian term can be interpreted as a result of the surface transport of adsorbed particles caused by the standard Brownian diﬀusion; • In several experimental situations a hopping mechanism of surface transport is present which necessitates augmentation of the Laplacian by a nonlocal term modeled by a Lévy stochastic process; • The quadratic nonlinearity is a result of truncation of a series expansion of a more general, physically justiﬁed, nonlinear even function. ht = ν∆h + These observations lead us to consider in this paper a nonlinear nonlocal equation of the form ut = −Lu + λ|∇u|q , (3.1) where the Lévy diﬀusion operator deﬁned in (1.12). In this chapter, we assume (for the sake of the simplicity of the exposition) that there is no transport term in the Lévy operator (1.12), namely b = 0. Recall also that if the matrix (ajk )nj,k=1 in (1.12) is not degenerate, a linear change of the variables transforms the corresponding term in (1.12) into the usual Laplacian −∆ on Rn . Relaxing the assumptions that led to the quadratic expression in the classical KPZ equation, the nonlinear term in (3.1) has the form q/2 , λ|∇u|q = λ |∂x1 u|2 + ... + |∂xn u|2 where q is a constant parameter. To study the interaction of the “strength” of the nonlocal Lévy diﬀusion parameterized by the Lévy measure Π, with the “strength” of the nonlinear term, parameterized by λ and q, we consider in (3.1) the whole range, 1 < q < ∞, of the nonlinearity exponent. 53 “volumeV” — 2009/8/3 — 0:35 — page 54 — #70 54 3. FRACTAL HAMILTON–JACOBI–KPZ EQUATIONS Finally, as far as the intensity parameter λ ∈ R is concerned, we distinguish two cases: • The deposition case: Here, λ > 0 characterizes the intensity of the ballistic deposition of particles on the evolving interface, • The evaporation case: Here, λR < 0, and the model displays a time-decay of the total “mass” M (t) = Rn u(x, t) dx of the solution (cf. equation (3.12), below). Equation (3.1) will be supplemented with the nonnegative initial datum, u(x, 0) = u0 (x), (3.2) 1,∞ n 1 and our standing assumptions are that u0 ∈ W (R ), and u0 − K ∈ L (Rn ), for some constant K ∈ R; as usual, W , with some superscripts, stands for various Sobolev spaces. Remark 3.1. The long-time behavior of solutions of the viscous HamiltonJacobi equation ut = ∆u+λ|∇u|q , with λ ∈ R, and q > 0, has been studied by many authors, see e.g. [5, 9, 10, 11, 30, 47], and the references therein. The dynamics of solutions of this equation is governed by two competing effects, one resulting from the diffusive term ∆u, and the other corresponding to the “hyperbolic” nonlinearity |∇u|q . The above-cited papers aimed at explaining how the interplay of these two effects influences the large-time behavior of solutions depending on the values of q and the initial data. Below, we are going to present results from [42] where we follow strategy from Remark 3.1, as well. Hence, in [42], we want to understand the interaction of the diﬀusive nonlocal Lévy operator (1.12) with the power-type nonlinearity. Our results can be viewed as extensions of some of the above-quoted work. However, their physical context is quite diﬀerent and, to prove them, new mathematical tools have to be developed. 2. Assumptions and preliminary results The basic assumption throughout the paper is that the Lévy operator L is a “perturbation” of the fractional Laplacian (−∆)α/2 (see Section 3) or, more precisely, that it satisﬁes the following condition: • The symbol a of the operator L can be written in the form a(ξ) = ℓ|ξ|α + k(ξ), (3.3) where ℓ > 0, α ∈ (0, 2]. and the pseudodiﬀerential operator K, corresponding to the symbol k, generates a strongly continuous semigroup of operators on Lp (Rn ), 1 ≤ p ≤ ∞, with norms uniformly bounded in t. Observe that, without loss of generality (rescaling the spatial variable x), we can assume that the scaling constant ℓ in (3.3) is equal to 1. Also, note that the above assumptions on the operator K are satisﬁed if the Fourier transform of the function e−tk(ξ) is in L1 (Rn ), for every t > 0, and its L1 -norm is uniformly bounded in t. The study of the large time behavior of solutions of the nonlinear problem (3.1)-(3.2) will necessitate the following supplementary asymptotic condition on the Lévy operator L: “volumeV” — 2009/8/3 — 0:35 — page 55 — #71 2. ASSUMPTIONS AND PRELIMINARY RESULTS 55 • The symbol k = k(ξ) appearing in (3.3) satisﬁes the condition k(ξ) = 0. ξ→0 |ξ|α (3.4) lim The assumptions (3.3) and (3.4) are fulﬁlled, e.g., by multifractal diffusion operators k X aj (−∆)αj /2 , L = −a0 ∆ + j=1 with a0 ≥ 0, aj > 0, 1 < αj < 2, and α = min1≤j≤k αj , but, more generally, one can consider here L = (−∆)α/2 + K, where K is a generator of another Lévy semigroup. Nonlinear conservation laws with such nonlocal operators were studied in [16, 17, 18]. In view of the assumption (3.3) imposed on its symbol a(ξ), the semigroup e−tL satisﬁes the following decay estimates analogous to those in (2.15)–(2.16) (cf. [18, Sec. 2], for details): ke−tL vkp k∇e −tL vkp ≤ Ct−n(1−1/p)/α kvk1 , ≤ Ct −n(1−1/p)/α−1/α kvk1 , (3.5) (3.6) for each p ∈ [1, ∞], all t > 0, and a constant C depending only on p and n. The sub-Markovian property of e−tL implies that, for every p ∈ [1, ∞], ke−tL vkp ≤ kvkp . (3.7) Moreover, for each p ∈ [1, ∞], we have k∇e−tL vkp ≤ Ct−1/α kvkp . (3.8) Let us also note that under the assumption (3.4), the large time behavior of e−tL is described by the fundamental solution pα (x, t), deﬁned in (1.16), of the linear equation ut + (−∆)α/2 u = 0, see [42, Lemma 6.1] for more details. We are now in a position to present our results concerning the nonlinear problem (3.1)-(3.2) starting with the fundamental problems of the existence, the uniqueness, and the regularity of solutions. Note that at this stage no restrictions are imposed on the sign of the parameter λ and the initial datum u0 . Consequently, all results of Theorem 3.2 are valid for both the deposition and the evaporation cases. Theorem 3.2. Assume that the symbol a = a(ξ) of the Lévy operator L satisfies condition (3.3) with an α ∈ (1, 2]. Then, for every u0 ∈ W 1,∞ (Rn ), λ ∈ R, and T > 0, problem (3.1)-(3.2) has the unique solution u in the space X = C([0, T ), W 1,∞ (Rn )). If, additionally, there exists a constant K ∈ R such that u0 − K ∈ L1 (Rn ), then u − K ∈ C([0, T ], L1 (Rn )), and sup t1/α k∇u(t)k1 < ∞. (3.9) 0<t≤T Moreover, for all t ≥ 0, ku(t)k∞ ≤ ku0 k∞ , and k∇u(t)k∞ ≤ k∇u0 k∞ , (3.10) “volumeV” — 2009/8/3 — 0:35 — page 56 — #72 56 3. FRACTAL HAMILTON–JACOBI–KPZ EQUATIONS and the following comparison principle is valid: for any two initial data satisfying condition, for all x ∈ Rn , u0 (x) ≤ ũ0 (x), the corresponding solutions satisfy the inequality u(x, t) ≤ ũ(x, t), for all x ∈ Rn , and t ≥ 0. Remark 3.3. Note that if u is a solution of (3.1) then so is u − K, for any constant K ∈ R. Hence, without loss of generality, in what follows we will assume that K = 0. In a recent publication, Droniou and Imbert [27] study a nonlinear-nonlocal viscous Hamilton-Jacobi equation of the form ut + (−∆)α/2 u + F (t, x, u, ∇u) = 0. For α ∈ (0, 2), and under very general assumptions on the nonlinearity, they construct a unique, global-in-time viscosity solution for initial data from W 1,∞ (Rn ), and emphasize (cf. [27, Remark 5]) that an analogous result can be obtained in the case of more general nonlocal operators, including those given by formula (1.12). That unique solution also satisﬁes the maximum principle (cf. Theorem 1.18) which implies inequalities (3.10), and the comparison principle contained in Theorem 3.2. Finally, the L1 -property of solutions stated in (3.9) (under the additional assumption u0 − K ∈ L1 (Rn )) is proved in [42, Section 3]. Here we only mention that the reasoning used in the construction of solutions of (3.1)-(3.2) involves the integral (mild) equation Z t u(t) = e−tL u0 + λ e−(t−τ )L |∇u(τ )|q dτ, (3.11) 0 estimates (3.10), and the Banach ﬁxed point argument. 3. Large time asymptotics – the deposition case Once the global-in-time solution u is constructed, it is natural to ask questions about its behavior as t → ∞. From now onwards, equation (3.1) will be supplemented with the nonnegative integrable initial datum (3.2). In view of Theorem 3.2, the standing assumption u0 ∈ W 1,∞ (Rn ) ∩ L1 (Rn ) allows as to deﬁne the “mass” of the solution of (3.1)-(3.2) by the formula Z u(x, t) dx M (t) = ku(t)k1 = Rn (3.12) Z tZ Z |∇u(x, s)|q dxds u0 (x) dx + λ = Rn 0 Rn To show last equality, note that since, for every t ≥ 0, µt in the representation (1.5) is a probability measure it follows from the Fubini theorem, and from the representation (1.5), that Z Z Z Z −tL t e u0 (x) dx = u0 (y) dy, u0 (x − y) µ (dy)dx = Rn Rn and, similarly, Z Z Rn t 0 Rn Rn e−(t−τ )L|∇u(x, τ )|q dτ dx = Z tZ 0 Rn |∇u(x, τ )|q dxdτ. “volumeV” — 2009/8/3 — 0:35 — page 57 — #73 3. LARGE TIME ASYMPTOTICS – THE DEPOSITION CASE 57 Hence, identity (3.12) is immediately obtained from equation (3.11) by integrating it with respect to x. The large-time behavior of M (t) is one of the principal objects of presented in this chapter. It turns out that in the deposition case, i.e., for λ > 0, the function M (t) is nondecreasing in t (cf. equation (3.12)) and, for suﬃciently small q, escapes to +∞, as t → ∞. Theorem 3.4. Let λ > 0, 1 < q ≤ n+α n+1 , and suppose that the symbol a of the Lévy operator L satisfies conditions (3.3) and (3.4) with α ∈ (1, 2]. If u = u(x, t) is a solution of (3.1) with an initial datum satisfying conditions 0 ≤ u0 ∈ L1 (Rn ) ∩ W 1,∞ (Rn ), and u0 ≡ \ 0, then limt→∞ M (t) = +∞. When q is greater that the critical exponent (n + α)/(n + 1), we are able to show that, for suﬃciently small initial data, the mass M (t) is uniformly bounded in time. Theorem 3.5. Let λ > 0, q > n+α n+1 , and suppose that the symbol a of the Lévy operator L satisfies conditions (3.3) and (3.4) with α ∈ (1, 2]. If, either ku0 k1 or k∇u0 k∞ is sufficiently small, then limt→∞ M (t) = M∞ < ∞. The smallness assumption from Theorem 3.5 can be easily formulated if we limit ourselves to L = (−∆)α/2 . In this case, it suﬃces to assume that the (q(n+1)−α−n)/(α−1) quantity ku0 k1 k∇u0 k∞ is smaller than a given (and small) number independent of u0 . This fact, for α = 2, is in perfect agreement with the assumption imposed in [47]. To see this result, note that, for every R > 0, the equation ut = −(−∆)α/2 u + λ|∇u|q is invariant under rescaling uR (x, t) = Rb u(Rx, Rα t), −1/(1+b) with b = (α − q)/(q − 1). Choosing R = k∇u0 k∞ we immediately obtain 1+b k∇u0,R k∞ = R k∇u0 k∞ = 1. Hence, the conclusion follows from the smallness assumption imposed on ku0,R k1 in Theorem 3.5 and from the identity ku0,R k1 = (q(n+1)−α−n)/(α−1) ku0 k1 Rb−n = ku0 k1 k∇u0 k∞ . If the Lévy operator L has a non-degenerate Brownian part, and if q ≥ 2, we can strengthen the assertion of Theorem 3.5 and show that the mass of every solution (not necessary small) is bounded as t → ∞. Theorem 3.6. Let λ > 0, q ≥ 2, and suppose that the Lévy diffusion operator L has a non-degenerate Brownian part. Then, each nonnegative solution Rof (3.1)-(3.2) with an initial datum u0 ∈ W 1,∞ (Rn ) ∩ L1 (Rn ) has the mass M (t) = Rn u(x, t) dx increasing to a finite limit M∞ , as t → ∞. The smallness assumption imposed in Theorem 3.5 seems to be necessary. Indeed, for L = −∆, it is known that if λ > 0, and (n + 2)/(n + 1) < q < 2, then there exists a solution of (3.1)-(3.2) such that limt→∞ M (t) = +∞ (cf. [11] and [9, Thm. 2.4]). Moreover, if ku0 k1 and k∇u0 k∞ are “large”, then the large-time behavior of the corresponding solution is dominated by the nonlinear term ([9]), and one can expect that M∞ = ∞. We conjecture that analogous results hold true at least for the α-stable operator (fractional Laplacian) L = (−∆)α/2 , and for q satisfying the inequality (n + α)/(n + 1) < q < α. We also conjecture that the critical exponent q = 2, for L = −∆, should be replaced by q = α if L has a nontrivial α-stable part. In this case, for q ≥ α, we expect that, as t → ∞, the mass of any nonnegative “volumeV” — 2009/8/3 — 0:35 — page 58 — #74 58 3. FRACTAL HAMILTON–JACOBI–KPZ EQUATIONS solution converges to a ﬁnite limit, just like in Theorem 3.6. Our expectation is that the proof of this conjecture can be based on a reasoning similar to that contained in the proof of Theorem 3.6. However, at this time, we were unable to obtain those estimates in a more general case. 4. Large time asymptotics – the evaporation case In the evaporation case, λ < 0, the mass M (t) is a nonincreasing function of t (see equation (3.12)), and the question, answered in the next two theorems, is when it decays to 0 and when it decays to a positive constant. Theorem 3.7. Let λ < 0, 1 ≤ q ≤ n+α n+1 , and suppose that the symbol a of the Lévy operator L satisfies conditions (3.3) and (3.4). If u is a nonnegative solution of (3.1)-(3.2) with an initial datum satisfying 0 ≤ u0 ∈ W 1,∞ (Rn ) ∩ L1 (Rn ), then limt→∞ M (t) = 0. Again, when q is greater than the critical exponent, the diﬀusion eﬀects prevails for large times and, as t → ∞, the mass M (t) converges to a positive limit. Theorem 3.8. Let λ < 0, q > n+α n+1 , and suppose that the symbol a of the Lévy operator L satisfies condition (3.3). If u is a nonnegative solution of (3.1)-(3.2) with an initial datum satisfying 0 ≤ u0 ∈ W 1,∞ (Rn )∩L1 (Rn ), then limt→∞ M (t) = M∞ > 0. The proof of Theorem 3.8 is based on the decay estimates of k∇u(t)kp proven in [42, Thm. 3.8]. However, as was the case for λ > 0, we can signiﬁcantly simplify that reasoning for Lévy operators L with nondegenerate Brownian part, and q ≥ 2; see [42, Remark 5.3]. Our ﬁnal result shows that when the mass M (t) tends to a ﬁnite limit M∞ , as t → ∞, the solutions of problem (3.1)-(3.2) display a self-similar asymptotics dictated by the fundamental solution of the linear equation ut + (−∆)α/2 u = 0 which is given by the formula Z α 1 −n/α −1/α pα (x, t) = t pα (xt , 1) = eixξ e−t|ξ| dξ, (3.13) /2 (2π) Rn see Section 3. More precisely, we have Theorem 3.9. Let u = u(x, t) be a solution of problem (3.1)-(3.2) with u0 ∈ L1 (Rn ) ∩ W 1,∞ (Rn ), and suppose that the symbol a of the Lévy operator L satisfies conditions (3.3) and (3.4). If limt→∞ M (t) = M∞ exists and is finite, then lim ku(t) − M∞ pα (t)k1 = 0. t→∞ (3.14) If, additionally, ku(t)kp ≤ Ct−n(1−1/p)/α , (3.15) for some p ∈ (1, ∞], all t > 0, and a constant C independent of t, then, for every r ∈ [1, p), lim tn(1−1/r)/α ku(t) − M∞ pα (t)kr = 0. (3.16) t→∞ Remark 3.10. Note that, in the case M∞ = 0, the results of Theorem 3.9 only give that, as t → ∞, ku(t)kr decays to 0 faster than t−n(1−1/r)/α . “volumeV” — 2009/8/3 — 0:35 — page 59 — #75 CHAPTER 4 Other equations with Lévy operator 1. Lévy conservation laws In this section, we present asymptotic results for the Cauchy problem for nonlinear pseudodiﬀerential equations of the form ut + Lu + ∇N u = 0, u(x, 0) = u0 (x), n n (4.1) + where u = u(x, t), x ∈ R , t ≥ 0, u : R × R → R, −L is a (linear) generator of a symmetric positive semigroup e−tL on L1 (Rn ), with the symbol deﬁned by the Lévy–Khintchine formula (1.12). The solutions of the Cauchy problem (4.4) have to be understood in some weak sense and several options are here available and have been studied in the papers quoted below. For the sake of this presentation let us just say that as the mild solution of (4.4) we mean a solution of the integral equation Z t −tL u(t) = e u0 − ∇ · e−(t−τ )L (N u)(τ ) dτ, (4.2) 0 motivated by the classical Duhamel formula. Such equations are used in physical models where the diﬀusive behavior is aﬀected by hopping, trapping and other nonlocal, but possibly self-similar, phenomena (see, e.g., [7, 8, 23, 29, 58, 59]). Recently, the questions of existence, uniqueness, regularity, and temporal asymptotics have been studied for certain special cases of equation (4.4), in particular, the fractal Burgers equation (see, [14]), ut + (−∆)α/2 u + c · ∇(u|u|r−1 ) = 0, c ∈ Rn , (4.3) and the one-dimensional multifractal conservation laws (see [16]), ut + Lu + f (u)x = 0, (4.4) with the multifractal operator L = −a0 ∆ + k X aj (−∆)αj /2 , (4.5) j=1 0 < αj < 2, aj > 0, j = 0, 1, . . . , k, where (−∆)α/2 , 0 < α < 2, is the fractional Laplacian deﬁned is deﬁned in Section 3. All these equations are generalizations of the classical Burgers equation ut − uxx + (u2 )x = 0. 59 (4.6) “volumeV” — 2009/8/3 — 0:35 — page 60 — #76 60 4. OTHER EQUATIONS WITH LÉVY OPERATOR Let us brieﬂy sketch our results from [16, 17, 18] in the particular case of the Cauchy problem ut + (−∆)α/2 u + b · ∇ (u|u|q ) = 0, u(x, 0) = u0 (x). (4.7) Intuitively speaking, our results from [16, 17] have shown that, for q suﬃciently large, the ﬁrst order asymptotics (as t → ∞) for solutions of (4.7) is essentially linear. Theorem 4.1 (Linear asymptotics). Assume that α ∈ (1, 2) and q > 1. Let u be the solution of the Cauchy problem (4.7). Suppose that the initial datum satisfies Z u0 (x) dx = M u0 ∈ L1 (Rn ) and Rn for some fixed M ∈ R. If q > (α − 1)/n, then then tn(1−1/p)/α ku(t) − M pα (t)kp → 0 as t → ∞, for each p ∈ [1, ∞], where pα (x, t) is defined in (1.16). By contrast with the results in Theorem 4.1, let us note, that the ﬁrst order asymptotics of solutions of the Cauchy problem for the Burgers equation (4.6) is described by the relation where t(1−1/p)/2 ku(t) − UM (t)kp → 0, 1 UM (x, t) = t−1/2 exp(−x2 /4t) K(M ) + 2 as Z √ x/ t t → ∞, exp(−ξ 2 /4) dξ 0 !−1 is the, so-called, source solution such that u(x, 0) = M δ0 . It is easy to verify that this solution is self-similar, i.e., UM (x, t) = t−1/2 U (xt−1/2 , 1). Thus, the long time behavior of solutions of the classical Burgers equation is genuinely nonlinear, i.e., it is not determined by the asymptotics of the linear heat equation. As it turns out that genuinely nonlinear behavior of the Burgers equation is due to the precisely matched balancing inﬂuence of the regularizing Laplacian diﬀusion operator and the gradient-steepening quadratic inertial nonlinearity. The next result ﬁnds such a matching critical nonlinearity exponent for the nonlocal multifractal Burgers equation. Theorem 4.2 (Nonlinear asymptotics). Assume that α ∈ (1, 2) and q > 0. Let u be the solution of the Cauchy problem (4.7). Suppose that Z u0 (x) dx = M. u0 ∈ L1 (Rn ) and Rn If q = (α − 1)/n, then tn(1−1/p)/α ku(t) − UM (t)kp → 0 as t → ∞, for each p ∈ [1, ∞], where UM (x, t) = t−n/α UM (xt−1/α , 1) is the unique self-similar solution of the equation ut + (−∆)α/2 u + b · ∇(u|u|(α−1)/n ) = 0 with the initial datum M δ0 . “volumeV” — 2009/8/3 — 0:35 — page 61 — #77 2. NONLOCAL EQUATION IN DISLOCATION DYNAMICS 61 2. Nonlocal equation in dislocation dynamics Dislocations are line defects in crystals whose typical length is ∼ 10−6 m and their thickness is ∼ 10−9 m. When the material is submitted to shear stresses, these lines can move in the crystallographic planes and this dynamics can be observed using electron microscopy. The elementary mechanisms at the origin of the deformation of monocrystals are rather well understood, however, many questions concerning the plastic behavior of materials containing a high density of defects are still open. Hence, in recent years, new physical theories describing the collective behavior of dislocations have been developed and numerical simulations of dislocations have been performed. We refer the reader to the recent publications [4, 36] for the comprehensive references about modeling of dislocation dynamics. One possible (simpliﬁed) model of the dislocation dynamics is given by the system of ODEs X V ′ (yi − yj ) for i = 1, ..., N, (4.8) ẏi = F − V0′ (yi ) − j∈{1,...,N }\{i} where F is a given constant force, V0 is a given potential and V is a potential of two-body interactions. One can think of yi as the position of dislocation straight lines. In this model, dislocations can be of two types, + or −, depending on the sign of their Burgers vector (see the book by Hirth and Lothe [34] for a physical deﬁnition of the Burgers vector). Self-similar solutions (i.e. solutions of the form yi (t) = g(t)Yi with constant Yi ) of system (4.8) with the particular potential V ′ (z) = z1 as well as their role in the asymptotic behavior of other solutions of (4.8) were studied by Head in [32]. More recently, Forcadel et al. showed in [28, Th. 8.1] that, under suitable assumptions on V0 and V in (4.8), the rescaled “cumulative distribution function” ! N X 1 t ρε (x, t) = ε − + (4.9) H x − εyi 2 i=1 ε (where H is the Heaviside function) satisﬁes (as a discontinuous viscosity solution) the following nonlocal eikonal equation ε ρ (·, t) x ε ε +M (x) |ρεx (x, t)| (4.10) ρt (x, t) = c ε ε for (x, t) ∈ R × (0, +∞), with c(y) = V0′ (y) − F . Here, M ε is the nonlocal operator deﬁned by Z ε M (U )(x) = J(z) E U (x + εz) − U (x) dz, (4.11) R where J(z) = V ′′ (z) on R \ {0} and E is the modiﬁcation of the integer part: E(r) = k + 1/2 if k ≤ r < k + 1. Note that the nonlocal operator M ε describes the interactions between dislocation lines, hence, interactions are completely characterized by the kernel J. Next, under the assumption that the kernel J is a suﬃciently smooth, even, nonnegative function with the following behavior at inﬁnity 1 J(z) = 2 for all |z| ≥ R0 (4.12) |z| “volumeV” — 2009/8/3 — 0:35 — page 62 — #78 62 4. OTHER EQUATIONS WITH LÉVY OPERATOR and for some R0 > 0, the rescaled cumulative distribution function ρε , deﬁned in (4.9), is proved to converge (cf. [28, Th. 2.5]) towards the unique solution of the corresponding initial value problem for nonlinear diﬀusion equation e ut = H(−Λu, ux), (4.13) e is a continuous function and Λ is a Lévy operator of where the Hamiltonian H order 1. It is deﬁned for any function U ∈ Cb2 (R) and for r > 0 by the formula Z 1 U (x + z) − U (x) − zU ′ (x)1{|z|≤r} dz (4.14) −ΛU (x) = C(1) |z|2 R with a constant C(1) > 0. Finally, in the particular case of c ≡ 0 in (2.6), we have e H(L, p) = L|p| (cf. [28, Th. 2.6]) which allows us to rewrite equation (4.13) in the form ut + |ux |Λu = 0. (4.15) One can show that the deﬁnition of Λ is independent of r > 0, hence, we 1/2 ﬁx r = 1. In fact, for suitably chosen C(1), Λ = Λ1 = −∂ 2 /∂x2 is the \ 1 pseudodiﬀerential operator deﬁned in the Fourier variables by (Λ w)(ξ) = |ξ|w(ξ) b (cf. formula (4.21) below). In this particular case, equation (4.15) is an integrated form of a model studied by Head [33] for the self-dynamics of a dislocation density represented by ux . Indeed, denoting v = ux we may rewrite equation (4.15) as vt + (|v|Hv)x = 0, (4.16) where H is the Hilbert transform deﬁned by [ (Hv)(ξ) = −i sgn(ξ) vb(ξ). Let us recall two well known properties of this transform Z v(y) 1 dy and Λ1 v = Hvx . Hv(x) = P.V. π R x−y (4.17) (4.18) Motivated by physics described above, in [15], we study the following initial value problem for the nonlinear and nonlocal equation involving u = u(x, t) ut = −|ux | Λα u on R × (0, +∞), u(x, 0) = u0 (x) for x ∈ R. (4.19) (4.20) where the assumptions on the initial datum u0 will be precised later. Here, for α ∈ α/2 (0, 2), Λα = −∂ 2 /∂x2 is the pseudodiﬀerential operator discussed in Section 3. Recall that the operator Λα has the Lévy–Khintchine integral representation for every α ∈ (0, 2) Z dz w(x + z) − w(x) − zw′ (x)1{|z|≤1} −Λα w(x) = C(α) , (4.21) |z|1+α R where C(α) > 0 is a constant. This formula (discussed in Chapter 1 for functions w in the Schwartz space) allows us to extend the deﬁnition of Λα to functions which are bounded and suﬃciently smooth, however, not necessarily decaying at inﬁnity. As we have already explained (cf. equation (4.15)), in the particular case α = 1, equation (4.19) is a mean ﬁeld model that has been derived rigorously in [28] as the limit of a system of particles in interactions (cf. (4.8)) with forces V ′ (z) = z1 . Here, “volumeV” — 2009/8/3 — 0:35 — page 63 — #79 2. NONLOCAL EQUATION IN DISLOCATION DYNAMICS 63 the density ux means the positive density |ux | of dislocations of type of the sign of ux . Moreover, the occurrence of the absolute value |ux | in the equation allows the vanishing of dislocation particles of the opposite sign. In the work [15], we study the general case α ∈ (0, 2) that could be seen as a mean ﬁeld model of particles modeled by system (4.8) with repulsive interactions V ′ (z) = z1α . Here, we would like also to keep in mind that (4.19) is the simplest nonlinear anomalous diﬀusion model (described by the Lévy operator Λα ) which degenerates for ux = 0. In work [15], we construct explicitly the self-similar solution of (4.19)-(4.20) and we prove its asymptotic stability. Moreover, we show the existence and the uniqueness of viscosity solutions of (4.19)-(4.20) as well as decay estimates using properties of the Lévy operator Λα presented in Chapter 1. “volumeV” — 2009/8/3 — 0:35 — page 64 — #80 64 4. OTHER EQUATIONS WITH LÉVY OPERATOR Figure 1. Wroclaw. The view of the Grunwaldzki Bridge From: http://wikitravel.org/en/Wroclaw Wroclaw in Polish, formally known as Breslau in German, is a large undiscovered gem of a city in southwestern Poland in the historic region of Silesia. It boasts fascinating architecture, many rivers and bridges, and a lively and metropolitan cultural scene. It is a city with a troubled past, having seen much violence and devastation, and was almost completely destroyed during the end of the Second World War. However, it has been brilliantly restored and can now be counted amongst the highlights of Poland, and all of Central Europe. As Poland rushes headlong into further integration with the rest of Europe, now is the time to visit before the tourist hordes (and high prices) arrive. Read Norman Davies’ and Roger Moorhouse’s Microcosm: Portrait of a Central European City to understand the complicated history of the town. “volumeV” — 2009/8/3 — 0:35 — page 65 — #81 Bibliography [1] N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ. 7 (2007), 145–175. [2] N. Alibaud, J. Droniou, J. Vovelle, Occurrence and non-appearance of shock in fractal Burgers equations, J. Hyperbolic Diﬀer. Equ 4 (2007), 479–499. [3] N. Alibaud, C. Imbert, G. Karch, Asymptotic properties of entropy solutions to fractal Burgers equation, (2009), preprint. [4] O. Alvarez, P. Hoch, Y. Le Bouar, and R. Monneau, Dislocation dynamics: short time existence and uniqueness of the solution, Arch. Rat. 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Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl. 81 (2002), 343–378. [12] J. Bertoin, Lévy Processes, Cambridge University Press, 1996. [13] P. Biler, T. Funaki, W.A. Woyczynski, Fractal Burgers equations, J. Diﬀ. Eq. 148 (1998), 9–46. [14] P. Biler, T. Funaki, W.A. Woyczynski, Interacting particle approximation for nonlocal quadratic evolution problems, Prob. Math. Stat. 19 (1999), 267–286. [15] P. Biler, G. Karch, R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions,(2009), arXiv:0812.4979v1 [math.AP]. [16] P. Biler, G. Karch, & W.A. Woyczyński, Asymptotics for multifractal conservation laws, Studia Math. 135 (1999), 231–252. 65 “volumeV” — 2009/8/3 — 0:35 — page 66 — #82 66 Bibliography [17] P. Biler, G. Karch, & W.A. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators, Studia Math. 148 (2001), 171–192. [18] P. Biler, G. Karch, & W.A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. I.H. Poincaré – Analyse non linéaire, 18 (2001), 613–637. [19] P. Biler, W.A. Woyczynski, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math. 59 (1998), 845–869. [20] P. Biler, G. Karch, W. A. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators, Studia Math. 148 (2001), 171–192. [21] R. M. Blumenthal, R. K. Getoor, Some theorems on stable processes, Trans. Math. Soc. 95 N.2, 263–273 (1960). [22] K. Bogdan, T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (2007), 179–198. [23] A. Carpinteri, F. Mainardi , Eds., Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, 1997. [24] C. H. Chan, M. Czubak, Regularity of solutions for the critical N -dimensional Burgers equation, (2008), 1–31. arXiv:0810.3055v3 [math.AP] [25] A. Córdoba and D. Córdoba, A maximum principle applied to quasigeostrophic equations, Comm. Math. Phys. 249 (2004), 511–528. [26] J. Droniou, T. Gallouët, J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ. 3 (2002), 499–521. [27] J. Droniou and C. Imbert, Fractal first order partial differential equations, Arch. Rat. Mech. Anal. 182 (2006), 299–331. [28] N. Forcadel, C. Imbert, R. Monneau, Homogenization of the dislocation dynamics and of some particle systems with two-body interactions, Discrete Contin. Dyn. Syst. Ser. A 23 (2009), 785–826. [29] P. Garbaczewski, Lévy processes and relativistic quantum dynamics, in Chaos – The Interplay Between Stochastic and Deterministic Behaviour, P. Garbaczewski, M. Wolf and A. Weron, Eds., Springer-Verlag, 1996. [30] B. Gilding, M. Guedda and R. Kersner, The Cauchy problem for ut = ∆u + |∇u|q , J. Math. Anal. Appl. 284 (2003), 733–755. [31] Y. Hattori and K. Nishihara, A note on the stability of the rarefaction wave of the Burgers equation, Japan J. Indust. Appl. Math. 8 (1991), 85–96. [32] A. K. Head, Dislocation group dynamics I. Similarity solutions od the n-body problem, Phil. Magazine 26 (1972), 43–53. [33] A. K. Head, Dislocation group dynamics III. Similarity solutions of the continuum approximation, Phil. Magazine 26 (1972), 65–72. [34] J. R. Hirth, L. Lothe, Theory of Dislocations, Second Ed., Krieger, Malabar, Florida, 1992. [35] A. M. Il’in and O. A. Oleinik, Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. (N.S.) 51 (93) (1960), 191–216. [36] C. Imbert, R. Monneau, E. Rouy, Homogenization of first order equations, with (u/ε)-periodic Hamiltonians. Part II: application to dislocations dynamics, Comm. Partial Diﬀ. Eq. 33 (2008), 479–516. “volumeV” — 2009/8/3 — 0:35 — page 67 — #83 Bibliography 67 [37] N. Jacob, Pseudo-diﬀerential operators and Markov processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London, 2001. [38] B. Jourdain, S. Méléard, and W. Woyczyński, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli 11 (2005), 689–714. [39] B. Jourdain, S. Méléard, and W. Woyczyński, A probabilistic approach for nonlinear equations involving the fractional Laplacian and singular operator, Potential Analysis 23 (2005), 55–81. [40] N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys. 255 (2005), 161–181. [41] G. Karch, C. Miao and X. Xu, On convergence of solutions of fractal burgers equation toward rarefaction waves, SIAM J. Math. Anal. 39 (2008), 1536– 1549. [42] G. Karch & W.A. Woyczynski, Fractal Hamilton-Jacobi-KPZ equations, Trans. Amer. Math. Soc. 360 (2008), 2423–2442. [43] M. Kardar, G. Parisi, Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889–892. [44] S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math. 58 (2004), 211–250. [45] A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation, (2008), 1–35. arXiv:0804.3549v1 [math.AP] [46] V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc. 80 (2000), 725–768. [47] Ph. Laurençot and Ph. Souplet, On the growth of mass for a viscous HamiltonJacobi equation, J. Anal. Math. 89 (2003), 367–383. [48] V. A. Liskevich, Yu. A. Semenov, Some problems on Markov semigroups, Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras, 163–217, Math. Top., 11, Akademie Verlag, Berlin, 1996. [49] J.A. Mann and W.A. Woyczyński, Growing fractal interfaces in the presence of self-similar hopping surface diffusion, Phys. A 291 (2001), 159–183. [50] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 3 (1986), 1–13. [51] A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys. 144 (1992), 325–335. [52] C. Miao, G. Wu, Global well-posedness of the critical Burgers equation in critical Besov spaces, (2008), 1–21. arXiv:0805.3465v3 [math.AP] [53] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. [54] M. Nishikawa and K. Nishihara, Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions, Trans. Amer. Math. Soc. 352 (1999), 1203–1215. [55] D. Revuz, M. Yor, Continuous martingales and Brownian motion, second edition, Springer-Verlag 1994. [56] D.W. Stroock, An Introduction to the Theory of Large Deviations, SpringerVerlag, New York, 1984. “volumeV” — 2009/8/3 — 0:35 — page 68 — #84 68 Bibliography [57] N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), 240–260. [58] W.A. Woyczyński, Burgers-KPZ Turbulence – Göttingen Lectures, Lecture Notes in Mathematics 1700, Springer-Verlag 1998. [59] W.A. Woyczyński, Lévy processes in the physical sciences, in Lévy Processes — Theory and Applications, T. Mikosch, O. Barndorﬀ-Nielsen and S. Resnick, Eds., Birkhäuser, Boston 2000, 31pp. “volumeV” — 2009/8/3 — 0:35 — page 69 — #85 Part 3 On a continuous deconvolution equation Roger Lewandowski “volumeV” — 2009/8/3 — 0:35 — page 70 — #86 2000 Mathematics Subject Classification. 76Fxx, 35Q30 Key words and phrases. turbulence, deconvolution models, existence of solution Abstract. We introduce in this paper the notion of “Continuous Deconvolution Equation” in a 3D periodic case. We first show how to derive this new equation from the Van Cittert algorithm. Next we show many mathematical properties of the solution to this equation. Finally, we show how to use it to introduce a new turbulence model for high Reynolds number flows. Acknowledgement. Roger Lewandowski warmly thanks the Nečas Center for Mathematical Modeling in the Faculty of Mathematics and Physics of Charles University in Prague (Czech Republic), for the hospitality in November 2008 and in May 2009. Special thanks are also addressed to Josef Málek for many stimulating conversations on Navier–Stokes equations. The deconvolution equation (1.10) has been discovered following a talk between Roger Lewandowski and Edriss Titi during the stay of R. Lewandowski in Weizmann Institute (Israel) in September 2007. It has been fixed by both during the stay of E. Titi in IRMAR, Rennes (France) in June 2008. R. Lewandowski also warmly thanks Weizmann Institute for the hospitality as well as E. Titi for many stimulating discussions. Finally, the work of Roger Lewandowski is partially supported by the ANR project 08FA300-01. “volumeV” — 2009/8/3 — 0:35 — page 71 — #87 Contents Chapter 1. Introduction and main facts 1. General orientation 2. Towards the models 3. Approximate deconvolution models 4. The deconvolution equation and outline of the remainder 73 73 74 75 76 Chapter 2. Mathematical tools 1. General background 2. Basic Helmholtz ﬁltration 79 79 80 Chapter 3. From discrete to continuous deconvolution operator 1. The van Cittert algorithm 2. The continuous deconvolution equation 3. Various properties of the deconvolution equation 4. An additional convergence result 83 83 84 85 86 Chapter 4. Application to the Navier–Stokes equations 1. Dissipative solutions to the Navier–Stokes equations 2. The deconvolution model 2.1. A priori estimates 2.2. Compactness property 89 89 91 93 94 Bibliography 101 71 “volumeV” — 2009/8/3 — 0:35 — page 72 — #88 “volumeV” — 2009/8/3 — 0:35 — page 73 — #89 CHAPTER 1 Introduction and main facts 1. General orientation It is well known since Kolmogorov’s work [25], that to simulate an incompressible 3D turbulent ﬂow using the Navier–Stokes equations, ∂t u + (u · ∇) u − ν∆u + ∇p = f , ∇ · u = 0, (1.1) u(0, x) = u0 , requires about N b = Re9/4 points in a numerical grid (details are available in [19] or [35]). Here, Re = U L/ν denotes the Reynolds number (see a rigorous deﬁnition in 4.1.1, below in the text). For realistic ﬂows, such as those involved in mechanical engineering or in geophysics, Re is of order 108 -1010 , sometimes much more. Therefore, the number of points N b necessary for the simulation is huge and the amount of memory computational algorithms need distinctively exceeds memory size of most powerful modern computers. This is why one needs “turbulent models” in order to reduce the appropriate number of grid points, and to simulate at least averages of turbulent ﬂows. There are two main families of turbulent models: statistical models, such as the well known k-ε model (see in [31] and [35]), and Large Eddy Simulations models (see in [9] and [36]), known as “LES models”. This paper deals with LES models family. The idea behind LES is to simulate the “large scale” of the ﬂow, trying to keep energy information on the “small” scales. Eddy viscosities are mostly involved in those models. Many models also emerged without eddy viscosity, such as Bardina’s models [3] or related (see in [32], [28], [27] [26]), as well as the family of α-models and related (see for instance in [17], [21], [24], [13], [12]). All of them are still considered as LES models. They mainly aim to regularize the nonlinear term (u · ∇) u in the Navier–Stokes equations. This idea takes inspiration in the work of Jean Leray in 1934 [30]. At this time computers did not exist and people were not thinking about numerical simulations of ﬂows past aircraft wings or about numerical simulations in weather forecast. They were mostly trying to ﬁnd analytical solutions to the 3D Navier–Stokes equations in cases of laminar ﬂows or where geometrical symmetries occur as well as where special 2D approximations were legitimate, the general case remaining out of reach. Such calculus is well explained in the famous book by G. Batchelor [4]. Therefore the question is whether the Navier–Stokes equations in the general case 73 “volumeV” — 2009/8/3 — 0:35 — page 74 — #90 74 1. INTRODUCTION AND MAIN FACTS have a solution or do not have a solution even if it is not possible to give analytical formula for these solutions. Jean Leray showed the existence of what we call today “a dissipative weak solution” to the Navier–Stokes equation in the whole space R3 (see deﬁnition 4.2 below in the text). To do this, he ﬁrst constructed approximated smooth solutions to the Navier–Stokes equations. Secondly, using some compactness arguments, he considered the limit of a subsequence, showing that this limit is a dissipative weak solution, called formerly “turbulent solution”. By “dissipative” solution we mean a distributional solution satisfying the energy inequality (see (4.6) below in the text). We still do not know if there is a unique dissipative solution in the general case, and also if it does or does not develop singularities in ﬁnite time. The question of singularities for particular dissipative solutions called “suitable weak solutions”, is studied in the very famous paper by Caﬀarelli–Kohn–Nirenberg [11]. 2. Towards the models To build approximated smooth solutions, J. Leray got the idea to replace the nonlinear term (u · ∇) u by ((u ⋆ ρε ) · ∇) u, where (ρε )ε>0 is a sequence of molliﬁers: doing like this, he introduced the ﬁrst LES models without knowing it, a long time before Smagorinsky published his ﬁrst paper in 1953 [37], Smagorinsky being often considered as a main pioneer of LES. This idea of smoothing the nonlinear term can be generalized in many other cases, such as in the periodic case we consider in this paper. In this case, one can regularize the Navier–Stokes equations by using the so called “Helmholtz equation”. Let u be an incompressible periodic ﬁeld u (∇ · u = 0), the mean value of which, m(u) (see (2.2) below in the text), being equal to zero. Notice that in the rest of the paper, all ﬁelds we consider will have a zero mean value for compatibility reasons. We do not mention it every time so far no risk of confusion occurs. Such a ﬁeld u being given, let us consider the Stokes problem Au = −α2 ∆u + u + ∇π = u, ∇ · u = 0. (1.2) The parameter α is the “small parameter”. It is generally agreed that α must be taken about the numerical grid size in numerical simulations, even if this claim is sometimes subject to caution. The Leray-α model is the one where the nonlinear term in the Navier–Stokes equations is regularized by taking (u · ∇) u in place of (u · ∇) u. The Bardina’s model of order zero is the one where one replaces the nonlinear term by (u · ∇) u. The solutions of these approximated Navier–Stokes equations are supposed to give approximations of mean values of pressure and velocity ﬁelds. To see this, let us take average of (1.1). We get the following “right” equation for u, ∂t u + (u · ∇) u − ν∆u + ∇p = f , ∇ · u = 0, u(0, x) = u0 , (1.3) “volumeV” — 2009/8/3 — 0:35 — page 75 — #91 3. APPROXIMATE DECONVOLUTION MODELS that we can rewrite as ∂t u + Bα (u, u) − ν∆u + ∇p = f + Bα (u, u) − (u · ∇) u, ∇ · u = 0, u(0, x) = u0 , 75 (1.4) where Bα (u, u) is a nonlinear term depending on α and that is “regular enough”. In the model, Bα (u, u) must replace (u · ∇) u, and Rα = Bα (u, u) − (u · ∇) u is a residual stress that we neglect for more or less legitimate physical or numerical reasons. Then the principle of the model consists in simulating ﬂows by computing an approximation of u and p, denoted by uα and pα , being a solution to ∂t uα + Bα (uα , uα ) − ν∆uα + ∇pα = f , (1.5) ∇ · uα = 0, uα (0, x) = u0 . Such a model is relevant if: • Bα correctly ﬁlters high frequencies and describes with accuracy low frequencies. • System (1.5) has a unique “smooth enough” solution when u0 ∈ L2loc 2 ). By “smooth enough” we mean u ∈ L∞ ([0, T ], (therefore u0 ∈ Hloc 1 3 2 2 3 1 (Hloc ) ) ∩ L ([0, T ], (Hloc ) ), p ∈ L2 ([0, T ], Hloc ), in any time interval [0, T ]. • The unique solution (uα , pα ) to (1.5) satisﬁes an energy balance like (4.15) (and not only an energy inequality like (4.6), see below in the text), for α ﬁxed. • There is a subsequence of the sequence (uα , pα )α>0 which converges (in a certain sense) to a dissipative weak solution to (1.1) when α goes to zero. We must say that there are many Bα such that the last three points on the list above are satisﬁed. But in order to use these equations to simulate realistic ﬂows we must check the ﬁrst point. Unfortunately there is no rigorous deﬁnition that can make this point precise, see also linked notion of “cut frequency”. 3. Approximate deconvolution models In 1999 and later, Adams and Stolz ([1], [39], [38], [2] ) were considering “the Bardina’s model of order zero” where Bα (u, u) = ∇ · (u ⊗ u) = (u · ∇)u. In order to improve reconstruction of the right ﬁeld in numerical simulations, they got the idea to apply a “deconvolution operator DN ”. To do it, they introduced a parameter N of deconvolution, using the discrete “van Cittert algorithm” (see in [10]), w0 = u, (1.6) wN +1 = wN + (u − A−1 wN ), where the operator A is deﬁned in (1.2). The deconvolution operator is deﬁned by HN (u) = wN = DN (u). It is ﬁxed such that for given incompressible ﬁeld u, HN (u) = DN (u) goes to u in a certain sense (see Section 1 in Chapter 3 below). Therefore, the model consists in replacing the nonlinear term by Bα,N (u, u) = ∇ · (DN (u) ⊗ DN (u)), “volumeV” — 2009/8/3 — 0:35 — page 76 — #92 76 1. INTRODUCTION AND MAIN FACTS that gives model ∂t uα,N + ∇ · (DN (uα,N ) ⊗ DN (uα,N )) − ν∆uα,N + ∇pα,N = DN (f ), ∇ · uα,N = 0, uα,N (0, x) = DN (u0 ) = HN (u0 ). (1.7) This model is called an “Approximate deconvolution model”. Existence, regularity and uniqueness of a solution to this model for general deconvolution order N , were proved by Dunca–Epshteyn in 2006 [16]. The case N = 0 was already studied in detail before in [26], [27], [32]. Questions of accuracy and error estimates were also studied in [28] for general order of deconvolution N . The exciting point in model (1.7) is that it formally “converges” to the right averaged Navier–Stokes equations (1.3) when N goes to inﬁnity and α is ﬁxed. A suitable choice of the deconvolution order N combined with a suitable choice of α, gives hope that we can approach with a good accuracy the right average of the real ﬁeld, deﬁned by the Navier–Stokes equations (expecting uniqueness of the dissipative solution). Therefore, we had to investigate the problem of the convergence of (uα,N , pα,N )N ∈N to a solution of the mean Navier–Stokes equations (1.3) when N goes to inﬁnity. This problem is very tough, and we got very recently ideas how to solve it [8]. Earlier, in [29], we got an idea to introduce a simpliﬁed deconvolution model, where the nonlinear term is (HN (u) · ∇) u ∂t uα,N + (HN (uα,N ) · ∇) uα,N − ν∆uα,N + ∇pα,N = HN (f ), ∇ · uα,N = 0, (1.8) uα,N (0, x) = HN (u0 ). In [29] we proved existence, uniqueness and regularity of a solution (uα,N , pα,N ) to (1.8), and also that a subsequence of the sequence (uα,N , pα,N )N ∈N converges, in a certain sense, to a dissipative weak solution of the Navier–Stokes equations for a ﬁxed α, when N goes to inﬁnity. 4. The deconvolution equation and outline of the remainder All the models we have shown above have been well studied in the periodic case. This calls for the question of adapting them in cases of realistic boundary conditions. We have considered an ocean forced by the atmosphere, under the rigid lid hypothesis with a mean ﬂux condition on the surface (see in [31]). As we started working on this question, it appeared soon that we were not able to do the job for the Adams–Stolz deconvolution model (1.7), often known as ADM model. Indeed, if we keep the natural boundary condition on the surface, we cannot write an identity like Z Ω ∇ · (DN (u) ⊗ DN (u)) · u = 0, though it is the key to get the L2 ([0, T ], (H 2 )3 ) ∩ L∞ ([0, T ], (H 1 )3 ) estimate in the periodic case. Therefore even modeling of the boundary condition remains an open problem in task to derive an ADM model which ﬁts with the physics and has good mathematical properties. “volumeV” — 2009/8/3 — 0:35 — page 77 — #93 4. THE DECONVOLUTION EQUATION AND OUTLINE OF THE REMAINDER 77 Facing the diﬃculty in the question of boundary conditions in model (1.7), we turned to another deconvolution model we have in hand, the model (1.8), although we take ADM model (1.7) for the best one in this class of models. Indeed, (1.7) really approaches the averaged Navier–Stokes equations for high deconvolution’s order making it a right LES model, at least formally, when model (1.8) approaches the right Navier–Stokes equations, fading the role of α, a fact we cannot physically interpret, although it shows a good numerical behavior (see in [5]). We next thought that ﬁxing the van Cittert algorithm with realistic boundary conditions would be easy. Unfortunately, we had troubles when rewriting it in the form (1.6), precisely because of the boundary conditions. This is why we decided to replace the van Cittert algorithm by a continuous variational problem. Our key observation is that this algorithm can be written in the form ∆wN +1 − ∆wN 2 + wN +1 + ∇πN +1 = u, (1.9) −α δτ with δτ = 1. This is precisely the ﬁnite diﬀerence equation corresponding to the continuous equation ∂w + w + ∇π = u, −α2 ∆ ∂τ (1.10) ∇ · w = 0, w(0, x) = u. We set Hτ (u) = w(τ, x). The parameter τ is a non dimensional parameter. We call it “deconvolution parameter”. Equation (1.10) is called the “deconvolution equation”. The corresponding LES model becomes ∂t uα,τ + (Hτ (uα,τ ) · ∇) uα,τ − ν∆uα,τ + ∇pα,τ = Hτ (f ), ∇ · uα,τ = 0, (1.11) uα,τ (0, x) = Hτ (u0 ). This model appears ﬁrst in [7] and [6], in the case of the ocean. It also constitutes a part of the PhD thesis of A.-C. Bennis [5], who made very good numerical tests in 2D cases with the software FreeFem++ [23], showing that this model deserves further numerical investigations in realistic 3D situations, compared with in situ data, a work which remains to be done. The goal of the rest of this paper is to study in detail the deconvolution equation and the related model (1.11) in the 3D periodic case. For pedagogical reasons and for the simplicity, we study the deconvolution equation in the scalar case. By virtue of periodicity, we can express the solution of this equation in terms of Fourier’s series. The same analysis holds for incompressible 3D ﬁelds. We next show existence and uniqueness of a solution (uα,τ , pα,τ ) to problem (1.11) for α and τ ﬁxed, the solution being “regular enough”. We ﬁnish the paper by showing that there exists a sequence τn which goes to inﬁnity when n goes to inﬁnity, and such that the sequence (uα,τn , pα,τn )n∈N converges to a dissipative weak solution to the Navier–Stokes equations when n goes to inﬁnity, always when α is ﬁxed. “volumeV” — 2009/8/3 — 0:35 — page 78 — #94 78 1. INTRODUCTION AND MAIN FACTS The rest of the paper is organized as follows. We start by giving some mathematical tools such as function spaces we are working with and the Helmholtz equation. We next turn to the study of the continuous deconvolution equation. As we have already said, for the sake of simplicity and clarity we will show results in the scalar case, so far the generalization to incompressible ﬁelds is straightforward. In a last section, we will study the model (1.11) and prove the announced results. “volumeV” — 2009/8/3 — 0:35 — page 79 — #95 CHAPTER 2 Mathematical tools 1. General background Let L ∈ R⋆+ , Ω = [0, L]3 ⊂ R3 . By (e1 , e2 , e3 ) we denote the orthonormal basis in R3 , x = (x1 , x2 , x3 ) ∈ R3 denotes a point in R3 . Let us ﬁrst start with some basic deﬁnitions. Definition 2.1. (1) A function u : R3 → C is said to be Ω-periodic if and only if for all x ∈ R3 , for all (p, q, r) ∈ Z3 one has u(x + L(p e1 + q e2 + r e3 )) = u(x). (2) Dper denotes all functions Ω-periodic of class C ∞ . (3) We put T3 = 2πZ3 /L. Let T3 be the torus deﬁned by T3 = R3 /T3 . (4) When p ∈ [1, ∞[, by Lp (T3 ) we denote the function space deﬁned by Lp (T3 ) = {u : R3 → C, u ∈ Lploc (R3 ), u is Ω−periodic}, endowed with the p1 R norm ||u||0,p = L13 T3 |u(x)|p dx . When p = 2, L2 (T3 ) is a Hermitian space with the Hermitian product Z 1 u(x)v(x)dx. (u, v) = 3 (2.1) L T3 R (5) Let u ∈ L1 (T3 ). We put m(u) = Ω u(x)dx. s (6) Let s ∈ R+ . By Hper,0 (R3 ) we denote the space s s Hper,0 (R3 ) = {u : R3 → C, u ∈ Hloc (R3 ), u is Ω − periodic, m(u) = 0}. (2.2) s The space Hper,0 (R3 ) is endowed by the induced topology of the classical space H s (T3 ). (7) For k = (k1 , k2 , k3 ) ∈ T3 , we put |k|2 = k12 + k22 + k32 , |k|∞ = supi |ki |, In = {k ∈ T3 ; |k|∞ ≤ n}. (8) We say that a Ω-periodic function P is a trigonometric polynomial if there P exists n ∈ N and coeﬃcients ak , k ∈ In , and such that P = k∈In ak ei k·x . The degree of P is the greatest q such that there is a k with |k|∞ = q and ak 6= 0. (9) By Vn we denote the ﬁnite dimensional space of all trigonometric polynomials of degree less than n with mean value equal to zero, X uk ei k·x , u0 = 0}, Vn = {u = k∈In and IPn the orthogonal projection from L2 (T3 ) onto its closed subspace Vn . (10) Let us put I3 = T3⋆ = (2πZ3 /L) \ {0}. 79 “volumeV” — 2009/8/3 — 0:35 — page 80 — #96 80 2. MATHEMATICAL TOOLS A real number s being given, we consider the function space IHs deﬁned by IHs = ( 3 u : R → C, u = We put ||u||s,2 = X k∈I3 X uk e i k·x , u0 = 0, k∈T3 |k|2s |uk |2 X k∈T3 ! 21 , (u, v)s = 2s 2 ) |k| |uk | < ∞ . X k∈I3 |k|2s uk v k . (2.3) (2.4) In the formula above, v k stands for the complex conjugate of vk . The following can be proved (see in [33]) • For all s ≥ 0, the space IHs is a Hermitian space, isomorphic to space s Hper,0 (R3 ). • One always has (IHs )′ = IH−s , Definition 2.2. Let s ≥ 0 and IHR s be a closed subset of IHs made of all real valued functions u ∈ IHs , IHR s = {u ∈ IHs , ∀x ∈ T3 , u(x) = u(x)}. (2.5) 2. Basic Helmholtz ﬁltration Let α > 0, s ≥ 0, u ∈ IHs and let u ∈ IHs+2 be the unique solution to the equation −α2 ∆u + u = u. (2.6) We are aware that u could be confused with the complex conjugate of u instead of the solution of the Helmholtz equation (2.6). Unfortunately, this is also the usual notation used by many authors working on the topic. This is why we decided to keep the notations like that, expecting that no confusion will occur. We also shall denote by A the operator IHs+2 −→ IHs , A: (2.7) w −→ −α2 ∆w + w. Therefore, one has u = A−1 u. P It is easily checked that if u = k∈T3 uk ei k·x , then X uk u= ei k·x . 1 + α2 |k|2 (2.8) (2.9) k∈T3 Formula (2.9) yields easily estimates 1 (2.10) ||u||s,2 , ||u − u||s,2 ≤ α||u||s+1,2 . α2 We will sometimes use notation uα instead of u, if we need to recall the dependence on parameter α. ||u||s+2,2 ≤ Theorem 2.3. Assume u ∈ IHs . Then the sequence (uα )α>0 converges strongly to u in the space IHs . “volumeV” — 2009/8/3 — 0:35 — page 81 — #97 2. BASIC HELMHOLTZ FILTRATION 81 Proof. By deﬁnition, one has X α2 |k|2 2 2 ||u − u||s,2 = |k|2s |uk |2 . 1 + α2 |k|2 k∈I3 Let ε > 0. As u ∈ IHs , there exists N be such that X ε |k|2s |uk |2 ≤ , 2 k∈I3 \IN 2 2 2 2 and since α |k| /(1 + α |k| ) ≤ 1, X α2 |k|2 2 ε IN = |k|2s |uk |2 ≤ . 1 + α2 |k|2 2 k∈I3 \IN On the other hand, because the set IN is ﬁnite, X α2 |k|2 2 |k|2s |uk |2 = 0. lim 2 |k|2 α→0 1 + α ⋆ k∈IN Therefore, there exists α0 > 0 be such that for each α ∈ ]0, α0 [ one has X α2 |k|2 2 ε JN = |k|2s |uk |2 ≤ . 2 |k|2 1 + α 2 ⋆ k∈IN u||2s,2 As ||u − = IN + JN , then for all α ∈ ]0, α0 [, one has ||u − u||2s,2 ≤ ε ending the proof like that. “volumeV” — 2009/8/3 — 0:35 — page 82 — #98 “volumeV” — 2009/8/3 — 0:35 — page 83 — #99 CHAPTER 3 From discrete to continuous deconvolution operator 1. The van Cittert algorithm Let us consider the operator DN = N X (I − A−1 )n . n=0 We introduce the operator HN (u) = DN (u). A straightforward calculation gives ! X X i k·x = uk e HN k∈I3 1− k∈I3 (3.1) α2 |k|2 1 + α2 |k|2 N +1 ! uk ei k·x . (3.2) One can prove the following (see in [29]): • Let s ∈ R, u ∈ IHs . Then HN (u) ∈ IHs+2 and ||HN (u)||s+2,2 ≤ C(N, α)||u||s,2 , where C(N, α) blows up when α goes to zero and/or N goes to inﬁnity. This is due to the fact N +1 ! α2 |k|2 N +1 1− ≈ 2 2 as |k|∞ → ∞. 2 2 1 + α |k| α |k| • The operator HN maps continuously IHs into IHs and ||HN ||L(IHs ) = 1. • Let u ∈ IHs . Then the sequence (HN (u))N ∈N converges strongly to u in IHs when N goes to inﬁnity. Let us put w0 = u, uN = HN (u). We now show how one can compute each wN thanks to the van Cittert algorithm (see also in [10]), starting from the deﬁnition wN = N X (I − A−1 )n u. (3.3) n=0 When A−1 acts on both sides in (3.3), one gets A−1 wN = N X n=0 A−1 (I − A−1 )n u = − N X (I − A−1 )n+1 u + n=0 83 N X (I − A−1 )n u n=0 = −wN +1 + u + wN . (3.4) “volumeV” — 2009/8/3 — 0:35 — page 84 — #100 84 3. FROM DISCRETE TO CONTINUOUS DECONVOLUTION OPERATOR In summary, the van Cittert algorithm is the following: w0 = u, wN +1 = wN + (u − A−1 wN ). (3.5) 2. The continuous deconvolution equation Applying A on both sides of (3.5) yields AwN +1 − AwN + wN = Au = u. Using the deﬁnition of A, Aw = −α2 ∆w + w, one deduces the equality −α2 (∆wN +1 − ∆wN ) + wN +1 = u. (3.6) Here is the analogy. Let δτ > 0 be a real number and consider the equation ∆wN +1 − ∆wN + wN +1 = u. (3.7) −α2 δτ We notice the following facts: • equation (3.6) is a special case of equation (3.7) when δτ = 1, • equation (3.7) is a ﬁnite diﬀerence scheme that corresponds to the equation satisﬁed by the variable w = w(τ, x), τ > 0, ∂w + w = u, −α2 ∆ (3.8) ∂τ w(0, x) = u(x), with the zero mean condition m(w) = 0 so far u also satisﬁes m(u) = 0 as well as m(u) = 0. We call equation (3.8) the continuous deconvolution equation. The parameter τ is dimensionless. We call it the deconvolution parameter. Before doing anything, we ﬁrst make change of variable v(τ, x) = w(τ, x)−u(x). The variable v is solution to the equation ∂v + v = 0, −α2 ∆ (3.9) ∂τ v(0, x) = u(x) − u(x), with periodic boundary conditions. We also keep in mind that we impose all variables to have a zero mean value over a cell, a fact we shall not recall every time. In the rest of this section, we will study in detail the solution of problem (3.9) and thus problem (3.8) that we will solve completely. To do this, we will express the solution in terms of Fourier series. We search for a solution v(τ, x) as X v(τ, x) = vk (τ )ei k·x , (3.10) k∈I3 with initial condition, with obvious notation, vk (0) = − α2 |k|2 uk = (u − u)k . 1 + α2 |k|2 (3.11) “volumeV” — 2009/8/3 — 0:35 — page 85 — #101 3. VARIOUS PROPERTIES OF THE DECONVOLUTION EQUATION 85 We deduce that each mode at frequency k satisﬁes the diﬀerential equation ( dvk + vk = 0, α2 |k|2 (3.12) dτ vk (0) = (u − u)k . We deduce that τ − vk (τ ) = (u − u)k e α2 |k|2 . Therefore, the general solution to problem (3.8) is X α2 |k|2 − τ +i k·x w(τ, x) = u(x) − , uk e α2 |k|2 2 2 1 + α |k| (3.13) (3.14) k∈I3 where u= X uk ei k·x . k∈I3 3. Various properties of the deconvolution equation We now prove general properties satisﬁed by the solution of the deconvolution equation, using either equation (3.8) itself, or formula (3.14). In the following, we put Hτ (u) = Hτ (u)(τ, x) = w(τ, x), (3.15) where v(τ, x) is the solution to equation (3.8). Lemma 3.1. Let s ∈ R, u ∈ IHs . Then for all τ ≥ 0, Hτ (u) ∈ IHs and ||Hτ (u)||s,2 ≤ 2||u||s,2 . (3.16) Proof. Since one has for every τ ≥ 0 and every k ∈ I3 α2 |k|2 − τ 0≤ e α2 |k|2 ≤ 1, 2 2 1 + α |k| the result is a direct consequence of (3.14). Lemma 3.2. Let α > 0 be fixed, s ∈ R and u ∈ IHs . Then (Hτ (u))τ >0 converges strongly to u in IHs , when τ → ∞. Proof. One has u − Hτ (u) = which yields ||u − Hτ (u)||2s,2 = X k∈I3 X α2 |k|2 − τ +i k·x uk e α2 |k|2 , 2 2 1 + α |k| k∈I3 2s |k| α2 |k|2 1 + α2 |k|2 2 |uk |2 e − α22τ |k|2 2τ ≤ e− α2 ||u||s,2 . Therefore, limτ →0 ||u − Hτ (u)||s,2 = 0, and the proof is ﬁnished. Lemma 3.3. Let α > 0 and τ ≥ 0 be fixed, s ∈ R and u ∈ IHs . Then Hτ (u) ∈ IHs+2 and one has C(L)(1 + τ ) ||Hτ (u)||s+2,2 ≤ ||u||s,2 , (3.17) α2 where C(L) is a constant which only depends on the box size L. “volumeV” — 2009/8/3 — 0:35 — page 86 — #102 86 3. FROM DISCRETE TO CONTINUOUS DECONVOLUTION OPERATOR Proof. Let us write equation (3.8) in the form −α2 ∆ ∂Hτ (u) = u − Hτ (u). ∂τ Since we already know that u − Hτ (u) ∈ IHs , we deduce from the standard elliptic theory that ∂Hτ (u) C(L) 3C(L) ∂Hτ (u) ≤ ||u − Hτ (u)||s,2 ≤ ||u||s,2 ∈ IHs+2 , ∂τ ∂τ s+2,2 α2 α2 (3.18) We now write Z τ ∂Hτ ′ (u) ′ Hτ (u) = u + dτ , ∂τ ′ 0 The result is a consequence of (3.18) combined with (2.10). 4. An additional convergence result We ﬁnish this section devoted to the continuous deconvolution equation by a convergence result. Indeed, when one studies existence result for some variational problem such as the Navier–Stokes equations and related, one usually must prove some compactness or continuity result. In all cases, there is one moment when one faces the question of studying a sequence (un )n∈N of approximated solutions which converges to some u in a certain sense, and one must identify the equation satisﬁed by u. The problem we are working with uses the operator u → Hτ (u). Among many compactness results that we potentially can prove, we will restrict ourself to one we will use in the next section. We are studying evolution problems. Therefore the functions (and later the ﬁelds) we consider are time dependent, that means u = u(t, x) for x ∈ T3 and t belongs to a time interval [0, T ]. Let s ≥ 0; the space L2 ([0, T ], IHs ) can easily be described to be a set of all functions u : T3 → C that can be decomposed as Fourier series (see in [33]) Z T X X |k|2s uk (t)ei k·x be such that ||u||2L2 ([0,T ],IHs ) = u= |uk (t)|2 dt < ∞. k∈I3 k∈I3 0 Lemma 3.4. Let α > 0 and τ > 0 be fixed. Let (un )n∈N be a sequence in L2 ([0, T ], IHs ) which converges strongly to u in the space L2 ([0, T ], IHs ). Therefore, (Hτ (un ))n∈N converges to Hτ (u) strongly in L2 ([0, T ], IHs ) when n → ∞. Proof. We use formula (3.14) to estimate Hτ (un ) − Hτ (u). Therefore one has, with obvious notations X α2 |k|2 − τ +i k·x . (3.19) (uk,n − uk )e α2 |k|2 Hτ (un ) − Hτ (u) = un − u + 1 + α2 |k|2 k∈I3 This yields the estimate ||Hτ (un ) − Hτ (u)||L2 ([0,T ],IHs ) ≤ 2||un − u||L2 ([0,T ],IHs ) , (3.20) “volumeV” — 2009/8/3 — 0:35 — page 87 — #103 4. AN ADDITIONAL CONVERGENCE RESULT because α2 |k|2 − τ e α2 |k|2 ≤ 1. 2 2 1 + α |k| The result is then a direct consequence of (3.20). 87 “volumeV” — 2009/8/3 — 0:35 — page 88 — #104 “volumeV” — 2009/8/3 — 0:35 — page 89 — #105 CHAPTER 4 Application to the Navier–Stokes equations 1. Dissipative solutions to the Navier–Stokes equations Let us start by writing again the Navier–Stokes equations: ∂t u + (u · ∇) u − ν∆u + ∇p = f , ∇ · u = 0, m(u) = 0, m(p) = 0, u(0, x) = u0 (4.1) Here, u stands for the velocity and p for the pressure, and they are both the unknowns. Since the ﬁelds are real valued and periodic, one can consider them as ﬁelds from T3 to R3 for the velocity, from T3 to R for the pressure. The right hand side f is a datum of the problem as well as the kinematic viscosity ν > 0. Recall that Z Z p(t, x)dx. u(t, x)dx, m(p) = m(u) = Ω Ω Recall that for ﬁelds satisfying ∇ · u = 0, one always has (u · ∇)u = ∇ · (u ⊗ u). We shall use sometimes this identity when we need it, without special warnings. Let us recall some facts and notation. Definition 4.1. (1) The Reynolds number Re is deﬁned as Re = U L/ν, where L is the box size, U is a typical velocity scale, for instance 1/2 Z Z 1 1 T 2 |u(t, x)| dx dt, U = lim T →∞ T 0 L3 Ω where lim stands for the generalized Banach limit (see in [14], [15] and [18]). (2) Let s ≥ 0. We set 3 IHs = {u ∈ (IHR s) , ∇ · u = 0}. The space IHs is a closed subset of (IHs )3 and contains real valued vector ﬁelds, see (2.2), endowed with the Hermitian product, for u = (u1 , u2 , u3 ), v = (v1 , v2 , v3 ), (u, v)s = (u1 , v1 )s + (u1 , v1 )s + (u1 , v1 )s (see (2.4)). We still denote ||u||s,2 = (||u1 ||2s,2 + ||u2 ||2s,2 + ||u3 ||2s,2 )1/2 . ′ (3) We put W −1,p (T3 ) = (W 1,p (T3 ))′ for 1/p + 1/p′ = 1, p ≥ 1. We also put IH−s = (IHs )′ for s ≥ 0. (4) The usual case we keep in mind for the data in the Navier–Stokes equations, is the case u0 ∈ IH0 and f ∈ L2 ([0, T ], (H 1 (T3 )3 )′ ), noting that (H 1 (T3 )3 )′ ⊂ IH−1 . 89 “volumeV” — 2009/8/3 — 0:35 — page 90 — #106 90 4. APPLICATION TO THE NAVIER–STOKES EQUATIONS Definition 4.2. We say that (u, p) is a dissipative solution to the Navier– Stokes equations (4.1) in time interval [0, T ] if: (1) The following holds: u ∈ L2 ([0, T ], IH1 ) ∩ L∞ ([0, T ], IH0 ), (4.2) p∈L (4.3) 5/3 ([0, T ] × T3 ), ∂t u ∈ L5/3 ([0, T ], (W −1,5/3 (T3 ))3 ). (2) limt→0 ||u(t, ·) − u0 (·)||0,2 = 0 (3) ∀ v ∈ L5/2 ([0, T ], W 1,5/2 (T3 )3 ) one has for all t ∈ [0, T ], Z tZ Z tZ ∇u : ∇v dxdt′ u ⊗ u : ∇v + ν (∂t u, v) − 0 0 T3 − T3 Z tZ 0 T3 p (∇ · v) = (4.4) Z t (f , v), (4.5) 0 where (·, ·) stands here for the duality product between W 1,5/2 (T3 )3 and W −1,5/3 (T3 )3 , noting that (H 1 (T3 )3 )′ ⊂ W −1,5/3 (T3 )3 . (4) The energy inequality holds, for all t ∈ [0, T ], Z tZ Z 1 |∇u(t′ , x)|2 dxdt |u(t, x)|2 + ν 2 T3 0 T3 Z t Z 1 2 |u0 (x)| dx + (f , u)dt′ , (4.6) ≤ 2 T3 0 where (·, ·) stands here for the duality product between IH1 and IH−1 , noting that (H 1 (T3 )3 )′ ⊂ IH−1 . Remark 4.3. This definition makes sense once u0 ∈ IH0 and f ∈ L2 ([0, T ], (H (T3 )3 )′ ), giving a sense to the integrals on the right hand side of (4.5) and (4.6). Moreover, by interpolation we see that the regularity conditions in point (1) in the definition above, make sure that u ∈ L10/3 ([0, T ] × T3 )3 . When combining this fact with the regularity for ∂t u, we see that all integrals on the right hand side of (4.5) are well defined. 1 Remark 4.4. The condition imposed on the pressure, p ∈ L5/3 ([0, T ] × T3 ), is directly satisfied when we already have the estimate u ∈ L∞ ([0, T ], IH0 ) ∩ L2 ([0, T ], IH1 ). Indeed, when one takes the divergence of the momentum equation formally using ∇ · u = 0 (included in the definition of the function space IHs in 4.1.2), we get the following equation for the pressure −∆p = ∇ · (∇ · (u ⊗ u)). (4.7) ∞ Now, by using Hölder’s inequality, it is easy to check that u ∈ L ([0, T ], IH0 ) ∩ L2 ([0, T ], IH1 ) implies u ∈ L10/3 ([0, T ] × T3 )3 . Therefore, ∇ · (∇ · (u ⊗ u)) ∈ L5/3 ([0, T ], W −2,5/3 ) and by the standard elliptic theory it follows p ∈ L5/3 ([0, T ] × T3 ). Let us recall a result due to J. Leray [30]. “volumeV” — 2009/8/3 — 0:35 — page 91 — #107 2. THE DECONVOLUTION MODEL 91 Theorem 4.5. Assume that u0 ∈ IH0 and f ∈ L2 ([0, T ], (H 1 (T3 )3 )′ ). Then the Navier–Stokes equations (4.1) have a dissipative solution. We still do not know whether • this solution is unique, • if it develops singularities in ﬁnite time, even if u0 and f are smooth. 2. The deconvolution model The deconvolution equation for incompressible ﬁelds takes the form −α2 ∆ ∂w + w + ∇π = u, ∂τ ∇ · w = 0, m(w) = 0, m(π) = 0, w(0, x) = u, (4.8) where u is such that m(u) = ∇ · u = 0, and u is the solution of the Stokes problem Au = −α2 ∆u + u + ∇ξ = u, ∇ · u = 0, (4.9) m(u) = 0, m(ξ) = 0. In the equations above, π and ξ are necessary Lagrange multipliers, involved because of the zero divergence constraint. In the following we set Hτ (u)(t, x) = w(τ, t, x), where w(τ, t, x) is the solution for the deconvolution parameter τ at a ﬁxed time t. Of course H0 (u) = u. A straightforward adaptation of the results of Section 3 in Chapter 3 combined with classical results related to the Stokes problem (see [22]) yield that Lagrange multipliers π and ξ are both equal to zero, and that the following facts are satisﬁed: (1) Let u ∈ L∞ ([0, T ], IH0 ). Then for all τ ≥ 0, Hτ (u) ∈ L∞ ([0, T ], IH2 ) and one has sup ||Hτ (u)||2,2 ≤ C sup ||u||0,2 , (4.10) t≥0 t≥0 where the constant C depends on τ and blows up when τ goes to inﬁnity. Thanks to Sobolev injection theorem, we deduce from (4.10) that in addition Hτ (u) ∈ L∞ ([0, T ] × T3 )3 , ||Hτ (u)||L∞ ([0,T ]×T3 )3 ≤ C(τ, α, sup ||u||0,2 ). (4.11) t≥0 (2) Let u ∈ L2 ([0, T ], IH1 ). Then the following estimate holds: Z T Z T 2τ −α 2 2 ||u(t, ·) − Hτ (u)(t, ·)||1,2 dt ≤ e ||u(t, ·)||21,2 dt. 0 (4.12) 0 In particular, the sequence (Hτ (u))τ >0 goes strongly to u in the space L2 ([0, T ], IH1 ) when τ goes to inﬁnity and α > 0 is ﬁxed. “volumeV” — 2009/8/3 — 0:35 — page 92 — #108 92 4. APPLICATION TO THE NAVIER–STOKES EQUATIONS Let us consider the problem ∂t uα,τ + (Hτ (uα,τ ) · ∇) uα,τ − ν∆uα,τ + ∇pα,τ = Hτ (f ), ∇ · uα,τ = 0, m(uα,τ ) = 0, m(pα,τ ) = 0, uα,τ (0, x) = Hτ (u0 ), (4.13) with periodic boundary conditions. Definition 4.6. We say that (uα,τ , pα,τ ) is a weak solution to Problem (4.13) if the following properties are satisﬁed: (1) uα,τ ∈ L∞ ([0, T ], IH1 ) ∩ L2 ([0, T ], IH1 ), ∂t uα,τ ∈ (L2 ([0, T ] × T3 ))3 , p ∈ L2 ([0, T ], IH1 ), (2) limt→0 ||uα,τ (t, ·) − Hτ (u0 )||0,2 = 0, (3) ∀ v ∈ L2 ([0, T ], (H 1 (T3 )3 )), Z T 0 Z +ν T3 Z 0 ∂t uα,τ · v + T Z T3 Z 0 T Z T3 ∇uα,τ : ∇v + (Hτ (uα,τ ) · ∇)uα,τ · v Z 0 T Z T3 ∇p · v = Z T 0 Z T3 Hτ (f ) · v, (4) the following energy balance holds for all t ∈ [0, T ], Z tZ Z 1 |∇uα,τ (t′ , x)|2 dxdt′ |uα,τ (t, x)|2 dx + ν 2 T3 0 T3 Z tZ Z 1 2 Hτ (f ) · uα,τ dxdt′ . |Hτ (u0 )(x)| dx + = 2 T3 0 T3 (4.14) (4.15) We now prove the following two results. Theorem 4.7. Assume that u0 ∈ IH0 and f ∈ L2 ([0, T ], (H 1 (T3 )3 )′ ). Then Problem (4.13) admits a unique weak solution (uα,τ , pα,τ ). Theorem 4.8. Assume that u0 ∈ IH0 and f ∈ L2 ([0, T ], (H 1 (T3 )3 )′ ). Then there exists a sequence (τn )n∈N which goes to infinity when n goes to infinity and such that the sequence (uα,τn , pα,τn )n∈N goes to a dissipative weak solution of the Navier–Stokes equations. Proof of Theorem 4.7. A complete proof of Theorem 4.7 would use the Galerkin method. We construct approximations as solutions of variational problems in the ﬁnite dimensional spaces Vn , thanks to the Cauchy–Lipchitz theorem. Afterwards we derive estimates in order to ﬁnally pass to the limit. To make the paper easy and not too diﬃcult, we bypass the construction of approximations in ﬁnite dimensional spaces, a procedure we have already completed for similar models (see for instance in [29]). The general Galerkin method is well explained in the famous book by J. L. Lions published in 1969 [34]. Therefore, we concentrate our eﬀort on two main points that make the result true: • a priori estimates, • the compactness property and how to pass to the limit in the equations. “volumeV” — 2009/8/3 — 0:35 — page 93 — #109 2. THE DECONVOLUTION MODEL 93 2.1. A priori estimates. For the simplicity, we write (u, p) instead of (uα,τ , pα,τ ). We perform computations assuming that (u, p) are enough regular to validate the integrations by parts we do. We also keep in mind that the boundary terms compensate each other in the integrations by parts, thanks to the periodicity. Therefore no boundary terms occur in these computations. As usual, we take u as a test function in (4.13), and we integrate by parts on T3 and on the time interval [0, t] for some t ∈ [0, T ], using ∇ · u = 0 as well as ∇ · (Hτ (u)) = 0. We get in particular Z (Hτ (u) · ∇) u · u = 0, T3 and therefore Z Z Z Z 1 1 2 2 2 |∇u| = Hτ (f ) · u. |u| + ν |Hτ (u0 )| + 2 {t}×T3 2 T3 [0,t]×T3 [0,t]×T3 (4.16) As u0 ∈ IH0 , Hτ (u0 ) ∈ IH2 , and recall that ||Hτ (u0 )||0,2 ≤ 2||u0 ||0,2 . Similarly, Z Hτ (f ) · u ≤ C||f ||−1,2 ||u||1,2 , [0,t]×T3 where again C do not depend on τ and α. Here and in the rest, we still denote the norm on (H 1 (T3 )3 )′ by || · ||−1,2 . Therefore, (4.16) yields Z sup |u|2 ≤ C(||u0 ||0,2 , ||f ||−1,2 ), (4.17) t∈[0,T ] Z {t}×T3 [0,t]×T3 |∇u|2 ≤ C(||u0 ||0,2 , ||f ||−1,2 , ν). (4.18) Next, we use fact (1) (Hτ (u) ∈ L∞ ([0, T ] × T3)3 ) and estimate (4.11) together with (4.17). This yields in particular A = (Hτ (u)·∇)u ∈ L2 ([0, T ]×T3)3 , ||A||L2 ([0,T ]×T3 )3 ≤ C(τ, α, ||u0 ||0,2 , ||f ||−1,2 ). (4.19) Let us now take ∂t u as a test function in equation (4.13), and we integrate on [0, t] × T3 , using ∇ · (∂t u) = 0. Therefore we get Z Z 1 2 |∂t u| + |∇u|2 2 [0,t]×T3 {t}×T3 Z Z Z 1 2 Hτ (f ) · ∂t u (4.20) A · ∂t u + |∇Hτ (u0 )| + = 2 T3 [0,t]×T3 [0,t]×T3 Since Hτ (u0 ) ∈ IH2 and Hτ (f ) ∈ L2 ([0, T ], H 1(T3 )3 ), using (4.19) combined with Cauchy–Schwarz and Young inequalities, we deduce from (4.20) Z |∂t u|2 ≤ C(τ, α, ||u0 ||0,2 , ||f ||−1,2 ), (4.21) [0,t]×T3 Z sup |∇u|2 ≤ C(τ, α, ||u0 ||0,2 , ||f ||−1,2 ). (4.22) t∈[0,T ] {t}×T3 In other words ∂t u ∈ L2 ([0, T ] × T3 )3 and u ∈ L∞ ([0, T ], IH1 ). In fact, one easily veriﬁes that ∂t u ∈ L2 ([0, T ], IH0 ). “volumeV” — 2009/8/3 — 0:35 — page 94 — #110 94 4. APPLICATION TO THE NAVIER–STOKES EQUATIONS We now get a bound for u in the space L2 ([0, T ], IH2 ). For it, let us consider a ﬁxed t ∈ [0, T ] and let us write the Navier–Stokes equations (4.13) in the form of Stokes problem −ν∆u + ∇p = Hτ (f ) − A − ∂t u, ∇ · u = 0, (4.23) m(u) = 0, m(p) = 0. Classical results on the Stokes problem yield the estimate ||u||2IH2 + ||p||2H 1 (T3 ) ≤ C1 (ν)||Hτ (f ) − A − ∂t u||2L2 (T3 ) ≤ C2 (ν)(||Hτ (f )||2L2 (T3 ) + ||A||2L2 (T3 ) + ||∂t u||2L2 (T3 ) ) (4.24) We now integrate (4.24) with respect to time. We get ||u||L2 ([0,T ],IH2 ) + ||p||L2 ([0,T ],H 1 (T3 )) ≤ C(ν, τ, α, ||u0 ||0,2 , ||f ||−1,2 ), (4.25) where we have used the regularizing eﬀect of Hτ and estimates (4.19) and (4.21). In summary, we get: (1) u is in L2 ([0, T ], IH1 )∩L∞ ([0, T ], IH0 ) and therefore p ∈ L5/3 ([0, T ], IL5/3 ) and ∂t u ∈ L5/3 ([0, T ], W −1,5/3 (T3 )3 ). The bounds only depend on the data ν, ||u0 ||0,2 and ||f ||−1,2 . (2) u ∈ L2 ([0, T ], IH2 ) ∩ L∞ ([0, T ], IH1 ) and p ∈ L2 ([0, T ], H 1 (T3 )). The bounds depend on the data ν, ||u0 ||0,2 and ||f ||−1,2 as well as on the deconvolution parameter τ and the ﬁltration parameter α. In particular these bounds blow up when τ goes to inﬁnity and/or α goes to zero. (3) ∂t u ∈ L2 ([0, T ], IH0 ). The bounds depend on the data ν, ||u0 ||0,2 and ||f ||−1,2 as well as on the deconvolution parameter τ and the ﬁltration parameter α. 2.2. Compactness property. Let us now consider a sequence (un , pn )n∈N of “smooth” solutions to problem (4.13). We aim to show that we can extract from this sequence a subsequence which converges in a certain sense to a solution of problem (4.13), when n goes to inﬁnity. Fact (2) makes sure that we can extract a subsequence, still denoted (un , pn )n∈N , such that un −→ u weakly in L2 ([0, T ], IH2 ), ∞ un −→ u weakly-star in L ([0, T ], IH1 ), pn −→ p 2 1 weakly in L ([0, T ], H (T3 )). (4.26) (4.27) (4.28) Let us now ﬁnd a strong compactness property. We have the following IH2 ⊂ IH1 ⊂ IH0 , the injections being continuous, compact and dense. We know that (∂t un )n∈N is bounded in L2 ([0, T ], IH0 ) while (un )n∈N is bounded in L2 ([0, T ], IH2 ) We deduce from Aubin–Lions lemma (see in [34]) that un −→ u strongly in L2 ([0, T ], IH1 ). (4.29) ∂t un −→ ∂t u weakly in L2 ([0, T ] × T3 )3 . (4.30) Finally, it is easily checked that we can extract an other subsequence such that “volumeV” — 2009/8/3 — 0:35 — page 95 — #111 2. THE DECONVOLUTION MODEL 95 Notice that the limit (u, p) satisﬁes points (1), (2) and (3) on the list above. It remains to show that (u, p) is a solution to problem (4.13). Let us start with the initial data, writing Z t un (t) = Hτ (u0 ) + ∂t un dt′ . 0 2 It is easy to pass to limit here in L ([0, T ] × T3 )3 , to get for free relation Z t u(t) = Hτ (u0 ) + ∂t u dt′ , 0 that tells us that u ∈ C 0 ([0, T ], IH0 ) and that u(0, x) = Hτ (u0 (x)). In fact we have a much better result since u ∈ C 0 ([0, T ], IH1 ). The proof is left to the reader. Let us now pass to the limit in the momentum equation. Let v ∈ L2 ([0, T ], IH1 ) be a test vector ﬁeld. One obviously has—when n goes to inﬁnity— Z TZ Z TZ ∂t u · v, ∂t un · v −→ Z 0 Z 0 T Z 0 T3 T3 T Z ∇un : ∇v −→ T3 pn (∇ · u) −→ 0 Z T 0 Z T 0 where we have used the identity Z Z p (∇ · u) = − T3 T3 Z Z T3 T3 ∇u : ∇v, T3 p (∇ · u), (4.31) ∇p · v. It remains to treat the term (Hτ (un )·∇)un which constitutes the novelty. This is why we focus our attention on it. Let us remark that (∇un )n∈N goes strongly to ∇u in the space L2 ([0, T ] × T3 )9 . On the other hand, applying Lemma 3.4, we get that (Hτ (un ))n∈N converges to Hτ (u) in L2 ([0, T ] × T3 )9 when n goes to inﬁnity. Therefore the sequence ((Hτ (un ) · ∇)un )n∈N goes strongly to (Hτ (u) · ∇)u in L1 ([0, T ] × T3 )3 . Finally, since the sequence ((Hτ (un ) · ∇)un )n∈N is bounded in L2 ([0, T ] × T3)3 , it converges weakly, up to a subsequence, to some g in L2 ([0, T ] × T3 )3 . The result above and uniqueness of the limit allows us to claim that g = (Hτ (u) · ∇)u. Consequently Z TZ Z TZ (Hτ (u) · ∇)u · v. (Hτ (un ) · ∇)un · v −→ 0 T3 0 T3 In summary, (u, p) satisﬁes: (1) u ∈ L2 ([0, T ], IH2 ) ∩ L∞ ([0, T ], IH1 ), p ∈ L2 ([0, T ], H 1 (T3 )), (2) limt→0 ||u(t, ·) − Hτ (u0 )||0,2 = 0, (3) ∀v ∈ L2 ([0, T ], IH1 ) : Z Z ∇p · v [∂t u · v + (Hτ (u) · ∇)u · v + ν∇u · ∇v] + [0,T ]×T3 [0,T ]×T3 Z Hτ (f ) · v. (4.32) = [0,T ]×T3 “volumeV” — 2009/8/3 — 0:35 — page 96 — #112 96 4. APPLICATION TO THE NAVIER–STOKES EQUATIONS Uniqueness is proven exactly like in [29], and we skip the details. Moreover, taking u as a test vector ﬁeld, which is a legitimate operation, and integrating in space and time using ∇ · u = 0 yields the energy equality Z Z Z Z 1 1 |∇u|2 = f · u. |u|2 + ν |u0 |2 + 2 {t}×T3 2 T3 [0,T ]×T3 [0,T ]×T3 Therefore, (u, p) is a smooth weak solution to problem (4.13), which concludes the proof of Theorem 4.7. Proof of Theorem 4.8. We ﬁnish the paper by proving the convergence result when τ goes to inﬁnity. We note that for solution (uτ , pτ ) the grid parameter α is ﬁxed. In this case, we only can use estimates (4.17) and (4.18). We also use estimate (4.12). Let us ﬁrst write the equation for the pressure: −∆pτ = ∇ · (∇ · (Hτ (uτ ) ⊗ uτ )). (4.33) ||pτ ||L5/3 ([0,T ]×T3 ) ≤ C. (4.34) ∂t uτ = −∇ · (Hτ (uτ ) ⊗ uτ ) + ν∆uτ − ∇pτ + Hτ (f ), (4.35) ||∂t uτ ||L5/3 ([0,T ],W −1,5/3 (T3 )3 ) ≤ C. (4.36) This yields, by interpolation combining (4.17), (4.18) and (4.12), existence of a constant C = C(ν, ||u0 ||0,2 , ||f ||−1,2 ) such that When writing we obtain the existence of a constant C = C(ν, ||u0 ||0,2 , ||f ||−1,2 ) such that We are now well prepared to pass to the limit. Thanks to all these bounds, there exists (τn )n∈N which goes to inﬁnity when n goes to inﬁnity and such that there exists u ∈ L2 ([0, T ], IH1 ) ∩ L∞ ([0, T ], IH0 ) and p ∈ L5/3 ([0, T ] × T3 ) such that uτn −→ u weakly in L2 ([0, T ], IH1 ), ∞ (4.37) uτn −→ u weakly star in L ([0, T ], IH0 ), (4.38) pτn −→ p (4.39) weakly in L 5/3 ([0, T ] × T3 ), when n goes to inﬁnity. We must prove that (u, p) is a dissipative weak solution to the Navier–Stokes equations. Let us start with the compactness result derived from Aubin–Lions lemma. We have H 1 (T3 ) ⊂ L10/3 (T3 ) ⊂ W −1,5/3 (T3 ), the injections being dense and continuous, the ﬁrst one being compact (since 10/3 < 6, 6 being the critical exponent in the 3D case). Therefore, applying again Aubin– Lions lemma using the bound on (uτn )n∈N in L2 ([0, T ], IH1 ) ⊂ L2 ([0, T ], H 1 (T3 )3 ) and the bound on (∂t uτn )n∈N in L5/3 ([0, T ], W −1,5/3 (T3 )3 ), we see that (uτn )n∈N is compact in L5/3 ([0, T ], L10/3 (T3 )3 ). Then we have in particular uτn −→ u strongly in L5/3 ([0, T ] × T3 )3 . (4.40) ∀q < 10/3 : uτn −→ u strongly in Lq ([0, T ] × T3 )3 . (4.41) Using Egorov’s theorem combined with Lebesgue inverse theorem, we deduce from (4.40) combined with the bound in L10/3 that “volumeV” — 2009/8/3 — 0:35 — page 97 — #113 2. THE DECONVOLUTION MODEL 97 Let us again consider (∂t uτn )n∈N . The bound (4.36) authorizes us to extract a subsequence (still using the same notation) and such that ∂t uτn −→ g weakly in L5/3 ([0, T ], W −1,5/3 )3 . (4.42) ∞ We must prove that g = ∂t u. Let ϕ be a C ﬁeld deﬁned on [0, T ] × T3 and such that ϕ(0, x) = ϕ(T, x) = 0. Then one has Z Z uτn · ∂t ϕ. ∂t uτn · ϕ = − [0,T ]×T3 [0,T ]×T3 Passing to the limit in this equality using (4.42) yields Z Z u · ∂t ϕ, g·ϕ =− [0,T ]×T3 [0,T ]×T3 which tells us that g = u in the distributional sense, and also in Lp sense by uniqueness of the limit. From now, v ∈ L5/2 ([0, T ], W 1,5/2 (T3 )3 ) is a ﬁxed test vector ﬁeld. We have the obvious following convergences when n goes to inﬁnity, Z ∂t uτn · v −→ (∂t u · v), Q Z Z ∇u : ∇v, ∇uτn : ∇v −→ Q Q Z Z (4.43) p (∇ · v), pτn (∇ · v) −→ Q Q Z Z f · v, Hτn (f ) · v −→ Q Q where Q = [0, T ]×T3 for the simplicity, (·, ·) stands for the duality product between L5/2 ([0, T ], W 1,5/2 (T3 )3 ) and L5/3 ([0, T ], W −1,5/3(T3 )3 ), and where we also have used Lemma 3.4. We now have to deal with the nonlinear term. We ﬁrst notice that (Hτn (un ) ⊗ uτn )n∈N is bounded in L5/3 (Q)9 . Thus—up to a subsequence—it converges weakly in L5/3 (Q)9 to a guy named h for the time being. That means Z Z h : ∇v. (4.44) Hτn (un ) ⊗ uτn : ∇v −→ Q Q The challenge is to prove that h = u ⊗ u. We already know that uτn converges to u strongly in L10/3−ε (Q) (ε > 0 and as usual “small”). Let us study the sequence Hτn (un ). It obviously converges to u but we must specify in which space and in which topology. We shall work in a L2 space type (2 < 10/3). We can write Hτn (un ) − u = Hτn (un − u) + Hτn (u) − u. Thanks to (3.16), we have for any ﬁxed time t, ||Hτn (un − u)(t, ·)||20,2 ≤ 2||(un − u)(t, ·)||20,2 , an inequality that we integrate on the time interval [0, T ]. This ensures that the sequence (Hτn (un − u))n∈N converges to zero in L2 (Q)3 when n goes to inﬁnity. “volumeV” — 2009/8/3 — 0:35 — page 98 — #114 98 4. APPLICATION TO THE NAVIER–STOKES EQUATIONS Applying Lemma 3.4, we deduce that the sequence (Hτn (u) − u)n∈N converges to zero in L2 (Q)3 when n goes to inﬁnity. In summary, we obtain the convergence of (Hτn (un ) ⊗ uτn )n∈N to u ⊗ u in L1 (Q)3 , making sure that h = u ⊗ u and also thanks to (4.44), Z Z u ⊗ u : ∇v. (4.45) Hτn (un ) ⊗ uτn : ∇v −→ Q Q In conclusion, (u, p) satisﬁes (4.5). Point 1 in deﬁnition (4.2) is already checked. To conclude our proof, it remains to prove points 2 (initial data) and 4 (energy inequality). We start with the energy inequality. We already know that (uτn , pτn ) satisﬁes the energy equality (4.15). Let 0 ≤ t1 < t2 ≤ T , and integrate (4.15) on the time interval [t1 , t2 ]. We get Z Z Z t2 Z t Z 1 t2 2 |uτn (t, x)| dxdt + ν |∇uτn (t′ , x)|2 dxdt′ dt 2 t1 T3 t1 0 T3 Z t2 Z t Z Z t2 − t1 = |Hτn (u0 )(x)|2 dx + Hτn (f ) · uτn dxdt′ dt. (4.46) 2 t1 T3 0 T3 Because (Hτn (f ))n∈N converges strongly to f in L2 ([0, T ], (H 1 (T3 )3 )′ ) while (uτn )n∈N converges weakly to u in L2 ([0, T ], IH1 ), the standard arguments yield Z t2 Z t Z Z t2 ′ (f , u)dt. (4.47) Hτn (f ) · uτn dxdt dt −→ t1 0 t1 T3 Analogous arguments also tell Z Z t2 − t1 t2 − t1 |Hτn (u0 )(x)|2 dx −→ |u0 (x)|2 dx. 2 2 T3 T3 As we know that (uτn )n∈N goes to u strongly in L2 (Q)3 , we have Z Z Z Z 1 t2 1 t2 2 |uτn (t, x)| dxdt −→ |u(t, x)|2 dxdt. 2 t1 T3 2 t1 T3 (4.48) (4.49) Finally, by the standard arguments in analysis (see for instance in [31]), the weak convergence of (uτn )n∈N to u in L2 ([0, T ], IH1 ) yields Z t2 Z t Z Z t2 Z t Z |∇u(t′ , x)|2 dxdt′ dt ≤ lim inf |∇uτn (t′ , x)|2 dxdt′ dt. (4.50) t1 0 T3 n∈N t1 0 T3 When one combines (4.46) together with (4.47),(4.48), (4.49) and (4.49), we obtain Z Z Z t2 Z t Z 1 t2 |u(t, x)|2 dxdt + ν |∇u(t′ , x)|2 dxdt′ dt 2 t1 T3 t1 0 T3 Z t2 Z t2 − t1 2 ≤ (f , u)dt, (4.51) |u0 (x)| dx + 2 t1 T3 an inequality which holds for every t1 , t2 such that 0 ≤ t1 < t2 ≤ T . We deduce that u satisﬁes the energy inequality (4.15). To ﬁnish the proof, we have to study the initial data. Let us ﬁrst notice that u(t, ·)t>0 is bounded in L2 (T3 )3 . Therefore, we can ﬁnd a sequence (tn )n∈N which “volumeV” — 2009/8/3 — 0:35 — page 99 — #115 2. THE DECONVOLUTION MODEL 99 converges to 0 and a ﬁeld k ∈ L2 (T3 )3 such that u(tn , ·)n∈N converges weakly in L2 (T3 )3 to k. The ﬁrst task is to prove that k = u0 . We start from the equality Z t (4.52) ∂t uτn dt′ , uτn (t, ·) = Hτn (u0 ) + 0 −1,5/3 an equality that we consider in the space W (T3 )3 . Using a straightforward variant of Lemma 3.4 and the convergence results proved above, we can pass to the limit in (4.52), to get in W −1,5/3 (T3 )3 , Z t ∂t u dt′ . (4.53) u(t, ·) = u0 + 0 Because ∂t u ∈ L5/3 ([0, T ], W −1,5/3 (T3 )3 ) ⊂ L1 ([0, T ], W −1,5/3 (T3 )3 ), this last equality says that u(0, ·) = u0 at least in W −1,5/3 (T3 )3 , and consequently in L2 (T3 )3 . Therefore we have k = u0 . Since the limit is unique, we deduce that the whole sequence u(t, ·)t>0 converges weakly in IH0 to u0 when t goes to zero. Moreover, one has ||u||0,2 ≤ lim inf ||u(t, ·)||0,2 . (4.54) t→0 On the other hand, when one lets t go to zero in the energy inequality, we get lim sup ||u(t, ·)||0,2 ≤ ||u0 ||0,2 . (4.55) t→0 We deduce that lim ||u(t, ·)||0,2 = ||u0 ||0,2 , t→0 which combined with the weak convergence yields lim ||u0 − u(t, ·)||0,2 = 0. t→0 (4.56) This concludes the question concerning initial data and also the proof of Theorem 4.8. Remark 4.9. Without too much effort, one can prove that the approximated velocity in model (4.13) lies in the space C([0, T ], IH1 ). Concerning the Navier– Stokes equation, it is well known that the trajectories are continuous in L2 (T3 )3 with respect to its weak topology. Nevertheless, one may wonder about the strong continuity of the trajectory at t = 0 that we have proved here. This approach indeed seems not to be usual in the folklore of the Navier–Stokes equations. However, it fits with the famous local regularity result due to Fujita–Kato [20]. “volumeV” — 2009/8/3 — 0:35 — page 100 — #116 “volumeV” — 2009/8/3 — 0:35 — page 101 — #117 Bibliography [1] N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, 2001. [2] , A subgrid-scale deconvolution approach for shock capturing, Journal of Computational Physics, 178 (2002), pp. 391–426. [3] J. Bardina, J. Ferziger, and W. Reynolds, Improved subgrid scale models for large eddy simulation, AIAA paper, 80 (1980), p. 1357. [4] G. Batchelor, The theory of homogeneous turbulence, Cambridge University Press, 1953. [5] A. C. Bennis, Étude de quelques modèles de turbulence pour l’océanographie, Thèse de l’Université Rennes 1, n◦ 3762 (2008). [6] A. C. Bennis, R. Lewandowski, and E. Titi, A generalized Leraydeconvolution model of turbulence, In Progress, (2009). , Simulations de l’écoulement turbulent marin avec un modèle de [7] déconvolution, Comptes-Rendus de l’Académie des Sciences de Paris, under press, arXiv:0809.2655, (2009). [8] L. Berselli and R. Lewandowski, Convergence of ADM models to NavierStokes Equations, In progress, (2009). [9] L. C. Berselli, T. Iliescu, and W. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Springer, Berlin, 2005. [10] M. Bertero and B. Boccacci, Introduction to Inverse Problems in Imaging, IOP Publishing Ltd, 1998. [11] L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. on Pure and applied maths, 35 (1982), pp. 771–831. [12] S. Chen, C. Foias, D. Holm, E. Olson, E. S. Titi, and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, D133 (1999), pp. 49–65. [13] V. V. Chepyzhov, E. S. Titi, and M. I. Vishik, On the convergence of the leray-alpha model to the trajectory attractor of the 3D Navier-Stokes system, Matematicheskii Sbornik, 12 (2007), pp. 3–36. [14] C. Doering and C. Foias, Energy dissipation in body-forced turbulence, J. Fluid Mech., 467 (2002), pp. 289–306. [15] C. Doering and J. Gibbon, Applied analysis of the Navier-Stokes equations, Cambridge University Press, 1995. [16] A. Dunca and Y. Epshteyn, On the stolz-adams de-convolution model for the large eddy simulation of turbulent flows, SIAM J. Math. Anal., 37 (2006), pp. 1890–1902. 101 “volumeV” — 2009/8/3 — 0:35 — page 102 — #118 102 Bibliography [17] C. Foias, D. D. Holm, and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 152 (2001), pp. 505–519. [18] C. Foias, O. Manley, R. Rosa, and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, 2001. [19] U. Frisch, Turbulences, Cambridge, 1995. [20] H. Fujita and T. Kato, On the Navier-Stokes initial value problem, Arch. Rat. Mach. Anal., 16 (1964), pp. 269–315. [21] B. J. Geurts and D. D. Holm, Leray and LANS-alpha modeling of turbulent mixing, Journal of Turbulence, 00 (2005), pp. 1–42. [22] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, 1986. [23] F. Hecht, O. Pironneau, A. L. Hyaric, and K. Ohtsua, Freefem++ manual v2.21, http://www.freefem.org/ﬀ++/, (2006). [24] A. A. Ilyin, E. M. Lunasin, and E. S. Titi, A modified leray-alpha subgridscale model of turbulence, Nonlinearity, 19 (2006), pp. 879–897. [25] A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers, Dokl. Akad. Nauk SSSR, 30 (1941), pp. 301–305. [26] W. Layton and R. Lewandowski, A simple and stable scale similarity model for large eddy simulation: energy balance and existence of weak solutions, Applied Math. letters, 16 (2003), pp. 1205–1209. , On a well posed turbulence model, Continuous Dynamical Systems series [27] B, 6 (2006), pp. 111–128. [28] , Residual stress of approximate deconvolution large eddy simulation models of turbulence, Journal of Turbulence, 7 (2006), pp. 1–21. , A high accuracy Leray-deconvolution model of turbulence and its limiting [29] behavior, Analysis and Applications, 6 (2008), pp. 23–49. [30] J. Leray, Sur les mouvements d’une liquide visqueux emplissant l’espace, Acta Math., 63 (1934), pp. 193–248. [31] R. Lewandowski, Analyse Mathématique et Océanographie, collection RMA, Masson, Paris, 1997. [32] , Vorticities in a LES model for 3D periodic turbulent flows, Journ. Math. Fluid. Mech., 8 (2006), pp. 398–42. , Approximations to the Navier-Stokes Equations, In progress, 2009. [33] [34] J. L. Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires, Gauthiers-Villard, 1969. [35] B. Mohammadi and O. Pironneau, Analysis of the k-Epsilon model, collection RMA, Masson, 1994. [36] P. Sagaut, Large eddy simulation for Incompressible flows, Springer, Berlin, 2001. [37] J. Smagorinsky, The dynamical influence of large-scale heat sources and sinks on the quasi-stationary mean motions of the atmosphere, Quarterly Journal of the Royal Meteorological Society, 79 (1953), pp. 342–366. [38] S. Stolz and N. A. Adams, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids, 10 (1999), pp. 1699–1701. [39] S. Stolz, N. A. Adams, and L. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows, Physics of ﬂuids, 13 (2001), pp. 997–1015. “volumeV” — 2009/8/3 — 0:35 — page 103 — #119 Part 4 Rough boundaries and wall laws Andro Mikelić “volumeV” — 2009/8/3 — 0:35 — page 104 — #120 2000 Mathematics Subject Classification. 35B27, 35Q30, 35B20 Key words and phrases. rough boundary, boundary layers, wall laws Abstract. We consider Laplace and Stokes operators in domains with rough boundaries and search for an effective boundary condition. The method of homogenization, coupled with the boundary layers, is used to obtain it. In the case of the homogeneous Dirichlet condition at the rough boundary, the effective law is Navier’s slip condition, used in the computations of viscous flows in complex geometries. The corresponding effective coefficient is determined by upscaling. It is given by solving an appropriate boundary layer problem. Finally we address application to the drag reduction. In this review article we will explain how those results are obtained, give precise references for technical details and present open problems. Acknowledgement. This research was partially supported by the Groupement MOMAS (Modélisation Mathématique et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) (PACEN/CNRS, ANDRA, BRGM, CEA, EDF, IRSN). The working group on wall laws was taking place during the visit of A.M. to the Nečas center for mathematical modeling, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Prague, Czech Republic, in March/April 2008. The author is grateful to the Nečas center for mathematical modeling for the hospitality during his stay in March/April 2008. Special thanks go to E. Feireisl and Š. Nečasová from the Center. “volumeV” — 2009/8/3 — 0:35 — page 105 — #121 Contents Chapter 1. Rough boundaries and wall laws 107 1. Introduction 107 2. Wall law for Poisson’s equation with the homogeneous Dirichlet condition at the rough boundary 108 2.1. The geometry and statement of the model problem 109 2.2. Laplace’s boundary layer 112 2.3. Rigorous derivation of the wall law 115 2.4. Some further questions: almost periodic rough boundaries and curved rough boundaries 118 3. Wall laws for the Stokes and Navier–Stokes equations 120 3.1. Navier’s boundary layer 121 3.2. Justiﬁcation of the Navier slip condition for the laminar 3D Couette ﬂow 124 3.3. Wall laws for ﬂuids obeying Fourier’s boundary conditions at the rough boundary 129 4. Rough boundaries and drag minimization 129 Bibliography 131 105 “volumeV” — 2009/8/3 — 0:35 — page 106 — #122 “volumeV” — 2009/8/3 — 0:35 — page 107 — #123 CHAPTER 1 Rough boundaries and wall laws 1. Introduction Boundary value problems involving rough boundaries arise in many applications, like ﬂows on surfaces with ﬁne longitudinal ribs, rough periodic surface diﬀraction, cracks for elastic bodies in such situations etc. An important class of problems is modeling reinforcement by thin layers with oscillating thickness (see e.g. Buttazzo and Kohn [25] and references therein). Reinforcement is described by an important contrast in the coeﬃcients and in the Γ-limit a Robin type boundary condition, with coeﬃcients of order 1, is obtained. Its value is calculated using ﬁnite cell auxiliary problems. Next we can mention homogenization of elliptic problems with the Neumann boundary conditions in domain with rapidly oscillating locally periodic boundaries, depending on small parameter. For more details we refer to [27]. The main goal of this review is to discuss the eﬀective boundary conditions for the Laplace equation and the Stokes system with homogeneous Dirichlet condition at the rough boundary. In ﬂuid mechanics the widely accepted boundary condition for viscous ﬂows is the no-slip condition, expressing that ﬂuid velocity is zero at an immobile solid boundary. It is only justiﬁed where the molecular viscosity is concerned. Since the ﬂuid cannot penetrate the solid, its normal velocity is equal to zero. This is the condition of non-penetration. To the contrary, the absence of slip is not very intuitive. For the Newtonian ﬂuids, it was established experimentally and contested even by Navier himself (see [44]). He claimed that the slip velocity should be proportional to the shear stress. The kinetic-theory calculations have conﬁrmed Navier’s boundary condition, but they give the slip length proportional to the mean free path divided by the continuum length (see [47]). For practical purposes it means a zero slip length, justifying the use of the no-slip condition. In many cases of practical signiﬁcance the boundary is rough. An example is complex boundaries in the geophysical ﬂuid dynamics. Compared with the characteristic size of a computational domain, such boundaries could be considered as rough. Other examples involve sea bottoms of random roughness and artiﬁcial bodies with periodic distribution of small bumps. A numerical simulation of the ﬂow problems in the presence of a rough boundary is very diﬃcult since it requires many mesh nodes and handling of many data. For computational purposes, an artiﬁcial smooth boundary, close to the original one, is taken and the equations are solved in the new domain. This way the rough boundary is avoided, but the boundary conditions at the artiﬁcial boundary are not given by the physical principles. It is clear that the non-penetration condition v · n = 0 should be kept, but there 107 “volumeV” — 2009/8/3 — 0:35 — page 108 — #124 108 1. ROUGH BOUNDARIES AND WALL LAWS are no reasons to keep the full no-slip condition. Usually it is supposed that the shear stress is a non-linear function F of the tangential velocity. F is determined empirically and its form varies for diﬀerent problems. Such relations are called the wall laws and classical Navier’s condition is one example. Another well-known example is modeling of the turbulent boundary layer close to the rough surface by a logarithmic velocity profile r r τw 1 y τw + C + (ks+ ) (1.1) vτ = ln ρ κ µ ρ where vτ is the tangential velocity, y is the vertical coordinate and τw the shear stress. ρ denotes the density and µ the viscosity. κ ≈ 0.41 is the von Kármán’s constant and C + is a function of the ratio ks+ of the roughness height ks and the thin wall sublayer thickness δv = vµτ . For more details we refer to the book of Schlichting [49]. Justifying the logarithmic velocity proﬁle in the overlap layer is mathematically out of reach for the moment. Nevertheless, after recent results [35] and [37] we are able to justify the Navier’s condition for the laminar incompressible viscous ﬂows over periodic rough boundaries. In [37] the Navier law was obtained for the Couette turbulent boundary layer. We note generalization to random rough boundaries in [15]. In the text which follows, we are going to give a review of rigorous results on Navier’s condition. Somewhat related problem is the homogenization of the Poisson equation in a domain with a periodic oscillating boundary and we start by discussing that situation. 2. Wall law for Poisson’s equation with the homogeneous Dirichlet condition at the rough boundary In our knowledge, mathematically rigorous investigations of the eﬀective wall laws started with the paper by Achdou and Pironneau [1]. They considered Poisson’s equation in a ring with many small holes close to the exterior boundary. They create an oscillating perforated annular layer close to the outer boundary. The amplitude and the period of the oscillations are of order ε and the homogeneous Dirichlet condition is imposed on the solution. In the paper by Achdou and Pironneau [1] the homogenized problem was derived. The rough boundary was replaced by a smooth artiﬁcial one and the corresponding wall law was the Robin boundary condition, saying that the eﬀective solution u was proportional to the characteristic roughness ε times its normal derivative. The proportionality constant was calculated using an auxiliary problem for Laplace’s operator in a ﬁnite cell. Nevertheless, in [1] the conductivity of the thin layer close to the boundary is not small and, contrary to [25], the homogenized boundary condition contains an ε. Consequently, it is not clear that using the ﬁnite cell for the auxiliary problem gives the the H 1 -error estimate from [1]. Despite this slight criticism, the reference [1] is a pioneering work since it was ﬁrst to point out that a) keeping homogeneous Dirichlet boundary condition gives an approximation; b) the wall law is a correction of the previous approximation and c) the wall laws are valid for curved rough boundaries. “volumeV” — 2009/8/3 — 0:35 — page 109 — #125 2. WALL LAW FOR POISSON’S EQUATION 109 The readable error estimate for the wall laws, in the case of Poisson’s equation and the ﬂat rough boundary is in the paper by Allaire and Amar [4]. They considered a rectangular domain having one face which was a periodic repetition of εΓg and the same boundary value problem as in [1] except periodic lateral conditions. Then they introduced the following auxiliary boundary layer problem in the inﬁnite strip Γg ×]0, +∞[ : Find a harmonic function ψ, ∇ψ ∈ L2 , periodic in y ′ = (y1 , . . . , yn−1 ) and having a value on Γg equal to its parametric form. The classical theory (see e.g. [46] or [39]) gives existence of a unique solution which decays exponentially to a constant d. The conclusion of [4] was that the homogenized solution ūε obeyed the wall law ∂ ūε on the artiﬁcial boundary and gave an interior H 1 -approximation of ūε = εd ∂x n order ε3/2 . We note the diﬀerence in determination of the proportionality constant in the wall law between papers [1] and [4]. It should be pointed out that there is a similarity between the homogenization of Poisson’s equation in partially perforated domain and obtaining wall laws for the same equations in presence of rough boundaries. In [30] an eﬀective Robin condition, analogous to one from [1] and [4] was obtained for the artiﬁcial boundary in the case of partially perforated domain. Other important work on Laplace’s operator came from the team around Y. Amirat and J. Simon. They were interested in the question if presence of the roughness diminishes the hydrodynamical drag. We will be back to this question in Section 4. In [7] and [8] they undertook study on the Couette ﬂow over a rough plate. For the special case of longitudinal grooves, the problem is reduced to the Laplace operator. This research for the case of Laplace operator and for complicated roughness was continued in the doctoral thesis [28] and articles [11], [12] and [21]. Even if the homogeneous Dirichlet condition at the rough boundary is meaningful mostly for ﬂow problems, it makes sense to study the case of Poisson equation. Following Bechert and Bartenwerfer [17] we can interpret it as simpliﬁed Stokes system for longitudinal ribs at the outer boundary. Mathematically, it is much easier to treat Laplace’s operator than technically complicated Stokes system. We start with a simple problem, which would serve us to present the main ideas. Figure 1. Domain Ωε with the rough boundary B ε . 2.1. The geometry and statement of the model problem. We consider the Poisson equation in a domain Ωε = P ∪ Σ ∪ Rε consisting of the parallelepiped P = (0, L1 )×(0, L2 )×(0, L3), the interface Σ = (0,L1 )×(0, L2)×{0} and the layer of roughness Rε = ∪{k∈Z2 } ε (Y + (k1 b1 , k2 b2 , −b3 )) ∩ ((0, L1 ) × (0, L2 ) × (−εb3 , 0)). The canonical cell of roughness Y ⊂ (0, b1 ) × (0, b2 ) × (0, b3 ) is deﬁned in Subsection 2.2. Let Υ = ∂Y \ Σ. For simplicity we suppose that L1 /(εb1 ) and L2 /(εb2 ) “volumeV” — 2009/8/3 — 0:35 — page 110 — #126 110 1. ROUGH BOUNDARIES AND WALL LAWS are integers. Let I = {k ∈ Z2 : 0 ≤ k1 ≤ L1 /b1 ; 0 ≤ k2 ≤ L2 /b2 }. Then, the rough boundary is B ε = ∪{k∈I} ε Υ + (k1 b1 , k2 b2 , −b3 ) . It consists of a large number of periodically distributed humps of characteristic length and amplitude ε, small compared with a characteristic length of the macroscopic domain. Finally, let Σ2 = (0, L1 ) × (0, L2 ) × {L3 }. We suppose that f ∈ C ∞ (Ωε ), periodic in (x1 , x2 ) with period (L1 , L2 ), and consider the following problem: −∆v ε = f vε = 0 vε in Ωε , (1.2) on B ε ∪ Σ2 , (1.3) is periodic in (x1 , x2 ) with period (L1 , L2 ). (1.4) Obviously problem (1.2)–(1.4) admits a unique solution in H(Ωε ), where H(Ωε ) = {ϕ ∈ H 1 (Ωε ) : ϕ = 0 on B ε ∪ Σ2 , ϕ is periodic in x′ = (x1 , x2 ) with period (L1 , L2 )}. (1.5) By elliptic regularity, v ε ∈ C ∞ (Ωε ). Every element of H(Ωε ) is extended by zero to (0, L1 ) × (0, L2 ) × (−b3 , 0) \ Rε . STEP 1: Zero order approximation We consider the problem −∆u0 = f u0 = 0 u0 in P , (1.6) on Σ ∪ Σ2 , (1.7) is periodic in (x1 , x2 ) with period (L1 , L2 ). (1.8) Obviously problem (1.6)–(1.8) admits a unique solution in H(P ) and, after extension by zero to (0, L1 ) × (0, L2 ) × (−b3 , 0), it is also an element of H(Ωε ). Obviously v ε ⇀ u0 , weakly in H(P ). We wish to have an error estimate. First we need estimates of the L2 -norms of the function in a domain and at a boundary using the L2 -norm of the gradient. Here the geometrical structure is used in essential way. We have: Proposition 1.1. Let ϕ ∈ H(Ωε ). Then we have kϕkL2 (Σ) ≤ Cε1/2 k∇x ϕkL2 (Ωε \P )3 , kϕkL2 (Ωε \P ) ≤ Cεk∇x ϕkL2 (Ωε \P )3 . (1.9) (1.10) This result is well-known and we give its proof only for the comfort of the reader. Proof. Let ϕ̃(y) = ϕ(εy), y ∈ Y +(k1 , k2 , −b3 ). Then ϕ̃ ∈ H 1 (Y +(k1 , k2 , −b3 )), ∀k, and ϕ = 0 on Υ + (k1 , k2 , −b3 ). Therefore by the trace theorem and the Poincaré’s inequality Z Z 2 | ∇y ϕ̃ |2 dy. | ϕ̃(ỹ, 0) | dỹ ≤ C {y3 =0}∩Ȳ +(k1 ,k2 ) Y +(k1 ,k2 ,−b3 ) “volumeV” — 2009/8/3 — 0:35 — page 111 — #127 2. WALL LAW FOR POISSON’S EQUATION 111 Change of variables and summation over k gives Z 1/2 Z 1/2 | ϕ(x̃, 0) |2 dx̃ ≤ Cε1/2 | ∇x ϕ(x) |2 dx Rε Σ and (1.9) is proved. Inequality (1.10) is well-known (see e.g. Sanchez-Palencia [48]). Next we introduce w = v ε − u0 . Then we have ( 0, in P , −∆w = f, in Rε , and w ∈ H(Ωε ) satisﬁes the variational equation Z Z Z ∂u0 f ϕ dx, ∇w∇ϕ dx = ϕ dS + − Rε Ωε Σ ∂x3 (1.11) ∀ϕ ∈ H(Ωε ). After testing (1.12) by ϕ = w, and using Proposition 1.1 we get Z Z Z ∂u0 √ |∇w|2 dx ≤ f w dx + w dS ≤ C ε||w||L2 (Rε ) . Ωε Rε Σ ∂x3 We conclude that √ ||∇(v ε − u0 )||L2 (Ωε ) ≤ C ε. (1.12) (1.13) (1.14) Could we get some more precise error estimates? Answer is positive. First, after recalling that the total variation of ∇w is given by Z Z 1 ε 3 ε |∇w| dx = sup w div s dx : s ∈ C0 (Ω ; R ), |s(x)| ≤ 1, ∀x ∈ Ω , Ωε Ωε we conclude that ||v ε − u0 ||BV (Ωε ) ≤ Cε. (1.15) Next, we need the notion of the very weak solution of the Poisson equation: Definition 1.2. Function B ∈ L2 (P ) is called a very weak solution of the problem −∆B = G ∈ H −1 (P ) 2 B = ξ ∈ L (Σ ∪ Σ2 ) B if − Z P B∆ϕ dx − Z Σ2 in P on Σ ∪ Σ2 (1.16) is periodic in (x1 , x2 ) with period (L1 , L2 ). ∂ϕ ξ dS + ∂x3 Z Σ ∂ϕ ξ dS = ∂x3 Z P Gϕ dx, ∀ϕ ∈ H(P ) ∩ C 2 (P̄ ). We recall the following result on very weak solutions to Poisson equation, which is easily proved using transposition: Lemma 1.3. The problem (1.16) has a unique very weak solution such that ||B||L2 (Σ) ≤ C ||ξ||L2 (Σ∪Σ2 ) + ||G||H −1 (P ) , (1.17) ||B||L2 (P ) ≤ C ||ξ||L2 (Σ∪Σ2 ) + ||G||H −1 (P ) . “volumeV” — 2009/8/3 — 0:35 — page 112 — #128 112 1. ROUGH BOUNDARIES AND WALL LAWS Direct consequence of Lemma 1.3 is the estimate ||v ε − u0 ||L2 (Σ) ≤ Cε, ||v ε − u0 ||L2 (P ) ≤ Cε. (1.18) Now we see that if we want to have a better estimate, an additional correction is needed. 2.2. Laplace’s boundary layer. The eﬀects of roughness occur in a thin layer surrounding the rough boundary. In this subsection we construct the 3D boundary layer, which will be used in taking into account the eﬀects of roughness. We start by prescribing the geometry of the layer. Let bj , j = 1, 2, 3 be three positive constants. Let Z = (0, b1 ) × (0, b2 ) × (0, b3 ) and let Υ be a Lipschitz surface y3 = Υ(y1 , y2 ), taking values between 0 and b3 . We suppose that the rough surface ∪k∈Z2 Υ+(k1 b1 , k2 b2 , 0) is also a Lipschitz surface. We introduce the canonical cell of roughness (the canonical hump) by Y = {y ∈ Z : b3 > y3 > max {0, Υ(y1 , y2 )}}. The crucial role is played by an auxiliary problem. It reads as follows: Find β that solves in Z + ∪ (Y − b3~e3 ) −∆y β = 0 [β]S (·, 0) = 0 ∂β (·, 0) = 1 ∂y3 S β=0 (1.19) (1.20) on (Υ − b3~e3 ), ′ β is y = (y1 , y2 )-periodic, (1.21) (1.22) where S = (0, b1 ) × (0, b2 ) × {0}, Z + = (0, b1 ) × (0, b2 ) × (0, +∞), and Zbl = Z + ∪ S ∪ (Y − b3~e3 ). Figure 2. Boundary layer containing the canonical roughness. Let V = z ∈ L2loc (Zbl ) : ∇y z ∈ L2 (Zbl )3 ; z = 0 on (Υ − b3~e3 ); and z is y ′ = (y1 , y2 )-periodic}. Then, by Lax–Milgram lemma, there is a unique β ∈ V satisfying Z Z ∇β∇ϕ dy = − ϕ dy1 dy2 , ∀ϕ ∈ V. (1.23) Zbl S By the elliptic theory, any variational solution β to (1.19)–(1.22) satisﬁes β ∈ V ∩ C ∞ (Z + ∪ (Y − b3~e3 )) . “volumeV” — 2009/8/3 — 0:35 — page 113 — #129 2. WALL LAW FOR POISSON’S EQUATION 113 Lemma 1.4. For every y3 > 0 we have Z Z Z b1 Z b2 β(y1 , y2 , y3 ) dy1 dy2 = C bl = β dy1 dy2 = − 0 0 S Zbl |∇β(y)|2 dy < 0. (1.24) Next, let a > 0 and let β a be the solution for (1.19)–(1.22) with S replaced by Sa = (0, b1 ) × (0, b2 ) × {a} and Z + by Za+ = (0, b1 ) × (0, b2 ) × (a, +∞). Then we have Z Z b2 b1 β a (y1 , y2 , a) dy1 = C bl − ab1 b2 . C a,bl = 0 0 (1.25) Proof. Integration of the equation (1.19) over the section, gives for any y3 > a Z b1 Z b2 d2 β a (y1 , y2 , y3 ) dy1 dy2 = 0 on (a, +∞). (1.26) dy32 0 0 Rb Rb Since β a ∈ V , we conclude that 0 1 0 2 β a (y1 , y2 , y3 ) dy1 dy2 is constant on (a, +∞). Then the variational equation (1.23) yields (1.24). Next we have Z b1 Z b2 a,bl β a (y1 , y2 , c) dy1 dy2 , ∀c ≥ a. C = 0 0 Let 0 ≤ c1 < a < c2 . Integration of the equation (1.19) over (c1 , c2 ) gives Z b1 Z b2 a ∂β a ∂β (y1 , y2 , c2 ) − (y1 , y2 , a + 0) ∂y3 ∂y3 0 0 ∂β a ∂β a + (y1 , y2 , a − 0) − (y1 , y2 , c1 ) dy1 dy2 = 0. ∂y3 ∂y3 Hence from (1.20) and (1.26) we get Z b1 Z b2 d β a (y1 , y2 , y3 ) dy1 dy2 = −b1 b2 , dy3 0 0 and Z b1 0 Z 0 b2 β a (y1 , y2 , y3 ) dy1 dy2 = (a − y3 )b1 b2 + C a,bl , for c1 < y3 < a for 0 ≤ y3 ≤ a. The variational equation for β a − β reads Z b1 Z b2 Z (ϕ(y1 , y2 , a) − ϕ(y1 , y2 , 0)) dy1 dy2 , ∇(β a − β)∇ϕ dy = − 0 Zbl 0 (1.27) ∀ϕ ∈ V. Testing with ϕ = β a − β and using (1.27) yields Z b1 Z b2 Z (β a (y1 , y2 , a) − β a (y1 , y2 , 0)) dy1 dy2 = ab1 b2 . |∇(β a − β)|2 dy = − 0 Zbl From the other hand Z Z |∇(β a − β)|2 dy = Zbl Zbl and formula (1.25) is proved. 0 |∇β a |2 dy + Z Zbl |∇β|2 dy − 2 Z Zbl ∇β a ∇β dy = C bl − C a,bl “volumeV” — 2009/8/3 — 0:35 — page 114 — #130 114 1. ROUGH BOUNDARIES AND WALL LAWS Next we search to establish the exponential decay. For the Laplace operator the result is known for long time. General reference for the decay of solutions to boundary layer problems corresponding to the operator − div(A∇u), with bounded and positively deﬁnite matrix A is [46], where a Saint Venant type estimate was proved. A very readable direct proof for similar setting and covering our situation, is in [4] and in [6]. Nevertheless one of the ﬁrst known proofs for the case of second order elliptic operators in divergence form is in [39]. Here we will present the main steps of that approach from late seventies. This early result is based on the following Tartar’s lemma: Lemma 1.5. (Tartar’s lemma) Let V and V0 be two real Hilbert spaces such that V0 ⊂ V with continuous injection. Let a be a continuous bilinear form on V × V0 and M a surjective continuous linear map between V and V0 . We assume that a(u, M u) ≥ α||u||2V , α > 0, ∀u ∈ V (1.28) and f ∈ V0′ . Then there exists a unique u ∈ V such that a(u, v) =< f, v >V0′ ,V0 , ∀v ∈ V0 . (1.29) Proof. For the proof see [39]. We note that this is a variant of Lax–Milgram lemma. Now we suppose that A = A(y) is a matrix such that A(y)ξ · ξ ≥ CA |ξ|2 , a.e. and 1 ||Aij ||∞ ≤ C̄A ; g ∈ Hper (S); eδ0 y3 f ∈ L2 (Z + ) for some δ0 > 0, (1.30) and consider the problem − divy (A(y)∇y β) = f β=g in Z + , (1.31) on S, (1.32) ′ β is y = (y1 , y2 )-periodic. (1.33) We have the following result Proposition 1.6. Under conditions (1.30) the problem (1.31)–(1.33) admits a unique solution such that for some δ ∈ (0, δ0 ) we have Z ∞ Z b1 Z b2 e2δy3 |∇y β|2 dy < +∞, 0 Z 0 ∞ Z 0 b1 Z 0 b2 0 0 2 Z b1 Z b2 1 β(t, y3 ) dt dy < +∞. e2δy3 β − b1 b2 0 0 (1.34) Proof. We just repeat the main steps from the proof from [39]. It relies on Tartar’s lemma. We introduce the spaces V and V0 by 1 V = {z ∈ L2loc ((0, +∞); Hper ((0, b1 ) × (0, b2 ))) : eδy3 ∇z ∈ L2 (Z + ) and z|S = 0}, V0 = {z ∈ V : eδy3 z ∈ L2 (Z + )}. “volumeV” — 2009/8/3 — 0:35 — page 115 — #131 2. WALL LAW FOR POISSON’S EQUATION the associated bilinear form is Z A∇u∇(e2δy3 v) dy, a(u, v) = Z+ and the linear form is < f, v >V0′ ,V0 = Z 115 u ∈ V, v ∈ V0 , (1.35) v ∈ V0 . (1.36) e2δy3 f v dy, Z+ Obviously, the linear form is continuous for δ ≤ δ0 . Same property holds for the bilinear form a. In the next step we introduce the operator M by setting Z y3 Z b1 Z b2 2δ M u(y) = u(y) − e−2δ(y3 −t) u(y1 , y2 , t) dy1 dy2 dt. (1.37) b1 b2 0 0 0 1 Using Poincaré’s inequality in Hper ((0, b1 ) × (0, b2 )) we get eδy3 M u ∈ L2 (Z + ) and M u ∈ V0 for δ < δ0 . (1.38) We note that R y M is surjective since the equation M u = v, v ∈ V0 , admits a solution u = v + 2δ 0 3 < v >(0,b1 )×(0,b2 ) (t) dt ∈ V. Concerning ellipticity, a direct calculation yields a(u, M u) ≥ (α − 2δCP ||A||∞ )||eδy3 ∇u||L2 (Z + ) , (1.39) 1 where Cp is the constant in Poincaré’s inequality in Hper ((0, b1 )×(0, b2 )). Therefore, o n 1 α for δ < min δ0 , 2Cp ||A||∞ ) we have the ellipticity and the Proposition is proved. Next, by reﬁning the result of Proposition 1.6 we get the pointwise exponential decay, as in [46]. 2.3. Rigorous derivation of the wall law. After constructing the boundary layer, we are ready for passing to the next order STEP 2: Next order correction From the proof of (1.13) we see that the main contribution comes from the term corresponding to the artiﬁcial interface Σ. Therefore one should eliminate the R 0 ϕ dS. The correction is given through a new unknown ubl,ε and we term Σ ∂u ∂x3 bl,ε ε search for u ∈ H(Ω ) such that Z Z ∂u0 ∇ubl,ε ∇ϕ dx = 0, ∀ϕ ∈ H(Ωε ). (1.40) ϕ dS + Σ ∂x3 Ωε Since the geometry is periodic this problem can be written as (Z X ∂u0 ϕ|x3 =0 dS Υ+(εk1 b1 ,εk2 b2 ) ∂x3 x3 =0 {k∈Z2 : (εk1 ,εk2 )∈(0,L1 )×(0,L2 )} ) Z + εZbl +(εk1 b1 ,εk2 b2 ,0) ∇ubl,ε ∇ϕ dx = 0. (1.41) “volumeV” — 2009/8/3 — 0:35 — page 116 — #132 116 For 1. ROUGH BOUNDARIES AND WALL LAWS ∂u0 ∂x3 Σ ∂u0 ∂x3 constant, by uniqueness, the solution to (1.41) would read ubl,ε = εβ xε , where β is the solution for (1.23). In general this is not the case, but Σ this is the candidate for a good approximation. Also, the boundary layer function β does not satisfy the homogeneous Dirichlet boundary condition at Σ2 . In order to have correct boundary condition we introduce an auxiliary function v by −∆v = 0 in P, ∂u0 on Σ, v= ∂x3 (1.42) Σ v=0 on Σ2 , v is (y1 , y2 )-periodic. Therefore we search for ubl,ε in the form C bl C bl x ∂u0 − + ubl,ε = ε β H(x3 ) v(x)H(x ) − wε , 3 ε b1 b2 ∂x3 Σ b1 b2 where C bl < 0 is a uniquely determined constant such that eδy3 β(y) − 2 + (1.43) C bl b1 b2 ∈ L (Z ) (the boundary layer tail). By Proposition 1.6 we know that such constant exists and is uniquely determined. Next by direct calculation, as in [35], we get 0 • div ∇ β xε ∂u is bounded by Cε3/2 in H −1 . ∂x3 Σ • Jump of the normal derivative of εv at Σ leads also to a term which is bounded by Cε3/2 in H −1 . • Corresponding terms in Rε are even smaller. bl C bl 0 Then after testing by wε = v ε −u0 +ε β xε − bC1 b2 H(x3 ) ∂u ∂x3 + b1 b2 v(x)H(x3 ) , Σ we get that ||∇wε ||L2 (Ωε ) ≤ Cε3/2 , (1.44) ||wε ||L2 (Σ) + ||wε ||L2 (Ω) ≤ Cε2 . STEP 3: Derivation of the wall law Having obtained a good approximation for the solution of the original problem, we get the wall law. We start by a formal derivation: At the interface Σ we have ∂β xε ∂u0 ∂u0 ∂v ε = − + O(ε) ∂x3 ∂x3 ∂x3 ∂x3 and vε u0 ∂u0 x + O(ε). β = − ε ε ∂x3 ε Z b1 Z b2 β(y1 , y2 , 0) dy1 dy2 and that the After averaging, and using that C bl = 0 0 mean of the normal derivative is zero, we obtain the familiar form of the wall law ueﬀ = −ε C bl ∂ueﬀ b1 b2 ∂x3 on Σ, (1.45) “volumeV” — 2009/8/3 — 0:35 — page 117 — #133 2. WALL LAW FOR POISSON’S EQUATION 117 where ueﬀ is the average over the impurities and C bl < 0 is deﬁned by (1.24). The higher order terms are neglected. Let us now give a rigorous justiﬁcation of the wall law (1.45). First we introduce the eﬀective problem: −∆ueﬀ = f ueﬀ = −ε in P eﬀ bl bl eﬀ C ∂u C ∂u =ε b1 b2 ∂x3 b1 b2 ∂n on Σ, ueﬀ = 0 on Σ2 , ueﬀ is (y1 , y2 )-periodic. (1.46) How close is ueﬀ to v ε ? In the diﬀerence C bl C bl ∂u0 x ε eﬀ − H(x3 ) + v(x)H(x3 ) − ueﬀ , v − u = wε + u0 − ε β ε b1 b2 ∂x3 Σ b1 b2 bl 0 the error estimate (1.44) implies that wε is negligible. Next ε β xε − bC1 b2 ∂u ∂x3 Σ is O(ε3/2 ) in L2 (P ) and O(ε2 ) in L1 (P ). Therefore it is enough to consider the bl function zε = u0 − ε bC1 b2 v(x) − ueﬀ . What do we know about this function? bl First, we have ∆ u0 − ε bC1 b2 v(x) − ueﬀ = 0 in P . Then on the lateral boundaries and on Σ2 it satisﬁes homogeneous boundary conditions. Finally on Σ we have bl 2 C C bl ∂zε ∂v 2 +ε . zε = −ε b1 b2 ∂x3 b1 b2 ∂x3 Hence zε solves the variational equation Z Z Z b1 b2 εC bl ∂v ∇zε ∇ϕ dx − ϕ dS, ∀ϕ ∈ H(P ). (1.47) z ϕ dS = − ε bl εC b b ∂x 1 2 3 P Σ Σ Testing (1.47) by ϕ = zε yields ||∇zε ||L2 (P ) ≤ Cε3/2 , ||zε ||L2 (Σ) ≤ Cε2 and ||zε ||L2 (P ) ≤ Cε2 . (1.48) Using (1.44), (1.48) and estimates for the boundary layer β we conclude that ||v ε − ueﬀ ||L2 (P ) ≤ Cε3/2 , 3/2 1 (P ) ≤ Cε ||v ε − ueﬀ ||Hloc , ε eﬀ (1.49) 2 ||v − u ||L1 (P ) ≤ Cε . Note that the approximation on Σ is not good. In fact the boundary layer is concentrated around Σ and there is a price to pay for neglecting it. STEP 4: Invariance of the wall law It remains to prove that translation of the artiﬁcial boundary of order O(ε) does not change our eﬀective solution. We have established in Lemma 1.4 the formula (1.25), showing how the boundary tail changes with translation of the artiﬁcial interface for a. Next using the smoothness of ueﬀ we ﬁnd out that ueﬀ (·, x3 − aε) satisﬁes the wall law at x3 = a with error O(ε2 ). Now if f does not depend on x3 , we see that the translation of the artificial boundary at O(ε) changes the result at order O(ε2 ). Things are more complicated if f depends on x3 . “volumeV” — 2009/8/3 — 0:35 — page 118 — #134 118 1. ROUGH BOUNDARIES AND WALL LAWS 2.4. Some further questions: almost periodic rough boundaries and curved rough boundaries. In the above sections the roughness was periodic. This corresponds to uniformly distributed rough elements. This is acceptable for industrially produced surfaces. Natural rough surfaces contain random irregularly distributed roughness elements. In applications it is important to derive wall laws for random surfaces. The natural question to be raised is if our construction still works in that case. In estimates we were using Poincaré’s inequality and clearly one should impose that our roughness layer does not become of large size with positive probability. But the real diﬃculty is linked to construction of boundary layers without periodicity assumption. In this direction there is a recent progress for ﬂow problems (see e.g. [15]), but still there are open questions. Let us discuss the question of decay at inﬁnity of boundary layers which is crucial for our estimates. We will follow the results by Amar et al from [5]. For sake of simplicity, we shall work in R2 . Our equation will be posed in the half space Π = {(x, y) ∈ R2 : y > 0}, whose boundary ∂Π is the real axis {(x, y) ∈ R2 : y = 0}. Let h : R → R be a smooth function, which is almost-periodic in the sense of Bohr (simply, almost-periodic), which means that for every δ > 0, there exists a strictly positive number ℓδ > 0 such that for every real interval of length ℓδ there exists a number τδ satisfying supx∈R |h(x + τδ ) − h(x)| ≤ δ. A well known reference on almost-periodic functions is the book [19]. For any almost-periodic function h, the asymptotic average 1 M [h] = lim T →+∞ 2T Z T h(x) dx −T is well deﬁned. Furthermore we can associate with h its generalized Fourier series, given by h(x) ∼ X λ∈R h̃(λ)eiλx , 1 T →+∞ 2T h̃(λ) = lim Z T h(x)e−iλx dx. −T The number h̃(λ) is the Fourier coeﬃcient of h associated to the frequency λ. It is well known that there exists at most a countable set of frequencies for which the Fourier coeﬃcients are diﬀerent from zero. Also the Parseval identity holds. Now, in analogy with the periodic case and with almost-periodic data on ∂Π, we expect to ﬁnd solutions to Laplace equation that are almost-periodic in the x variable and decay to a certain constant, say d, as y tends to inﬁnity. In the periodic case d was equal to the average of h. In the almost-periodic case, d is given by the asymptotic average M [h], that we may ﬁx to be zero without loss of generality. In analogy with the periodic case, we introduce the following space of weakly decaying “volumeV” — 2009/8/3 — 0:35 — page 119 — #135 2. WALL LAW FOR POISSON’S EQUATION functions L2ap (Π) = 2 Z ||ψ|| = 0 ( 119 ψ : x → ψ(x, y) is almost-periodic ∀y ≥ 0, +∞ " 1 lim T →+∞ 2T Z T 2 # ψ (x, y) dx −T dy = Z +∞ M ψ 0 2 ) (y) dy < +∞ (1.50) As noted in [5], a trouble with L2ap (Π) is that it is not complete. This is a known disadvantage of Besicovitch’s spaces. Next we study our boundary layer problem. For a given smooth almost-periodic function h it reads ∆ψ = 0 in Π, ψ(x, 0) = h(x) on ∂Π, (1.51) M [h] = 0. It is well known that the unique smooth bounded solution for (1.51) is given by Z yh(t) 1 dt. (1.52) ψ(x, y) = π R (x − t)2 + y 2 Then we have the following result Theorem 1.7. (see [5]) Let ψ be the unique bounded solution of (1.51). Then, for every fixed y > 0, the function x → ψ(x, y) is an almost-periodic function. Moreover, for any given γ0 > 0, the following equivalence condition holds: ||ψeγy || < +∞ for every 0 < γ < γ0 if and only if h̃(λ) = 0 for every |λ| < γ0 . Further analysis in [5] lead to the conclusion that the necessary and suﬃcient condition for the exponential decay is that the frequencies λ of h are far from zero. It is worthwhile to point out that, in the purely periodic case, the frequencies are always far from zero and hence the exponential decay of the solution is in accordance with previous theorem. On the contrary, in the general almost-periodic case, the exponential decay property fails if the frequencies of h accumulate at zero. Diﬃculties are illustrated through the following explicit example from [5]: P+∞ 1 x Let h(x) = sin 2 3 n=1 n n . Then the the series converges uniformly, the function h is well deﬁned, almost-periodic and satisﬁes M [h] = 0. With this h, the problem (1.51) has a unique bounded solution +∞ x X 2 1 sin ψ(x, y) = e−y/n , with ||ψ|| = +∞. 2 3 n n n=1 In this case not only that we do not have an exponential decay, but ψ is even not in the space L2ap (Π). We can only conclude that a reasonable theory would be possible in a correct setting and with well-prepared data. Next diﬃculty is linked with the fact that in nature one has to handle curved rough boundaries. In the pioneering paper [1] the roughness was linked to a curved circular boundary. This work continued mainly with formal multiscale expansions and numerical simulations for ﬂow problems (see [2], [3], [43] and references therein). “volumeV” — 2009/8/3 — 0:35 — page 120 — #136 120 1. ROUGH BOUNDARIES AND WALL LAWS Nevertheless, there is a recent article [41] by Madureira and Valentin, with analysis of the curvature inﬂuence on 2D eﬀective wall laws. Their geometry is essentially annular and it was possible to describe the rough surface using just angular variable. Their boundary layer problems are posed in an open angle and the connection with known results is to be established. Also their Laplace’s operator in polar coordinates systematically misses a term. The paper gives ideas but not really the complete construction of the approximation. Furthermore, we note that the two-dimensional case is very special because it allows for a global isometric parametrization of the boundary, while in the multidimensional case even the correct formulation of the problem setting is not obvious. Derivation of the approximations and eﬀective boundary conditions for solutions of the Poisson equation on a domain in Rn whose boundary diﬀers from the smooth boundary of a domain Rn by rapid oscillations of size ε, was considered in [45]. More precisely, the Poisson equation was supposed in a bounded or unbounded domain Ω of Rn , n ≥ 2, with smooth compact boundary Γ = ∂Ω, being an (n − 1)-dimensional Riemannian manifold. Using the unit outer normal ν to Γ, the tubular neighborhood of Γ was deﬁned by the mapping T : (x, t) → x + tν(x), deﬁned on Γ × (−δ, δ). Then, using a function γ ε from Γ to R such that |γ ε (x)| ≤ εM < δ/2 on Γ, and that γ ε is locally ε-periodic through an atlas of charts, it was possible to deﬁne a rough boundary Γε = T (x, γ ε (x)); x ∈ Γ. For this fairly general geometric situation it was possible to accomplish the steps 1 to 3 from the above construction, for the ﬂat rough boundary. The wall law (1.45) was obtained again. Nevertheless, it was found that the coeﬃcient C bl depends on position. The position was present as a parameter in the boundary layer construction. The construction from [45] is to be extended to systems, most notably to the Stokes system. 3. Wall laws for the Stokes and Navier–Stokes equations In the text which follows we will try to give a brief resume of the results concerning the wall laws for the incompressible Stokes and Navier–Stokes equations. Also we will recall the basic steps of the construction of the boundary layer corrections, following the approach from [31]. Flow problems over rough surfaces were considered by O. Pironneau and collaborators in [43], [2] and [3]. The paper [43] considers the ﬂow over a rough surface and the ﬂow over a wavy sea surface. It discusses a number of problems and announces a rigorous result for an approximation of the Stokes ﬂow. Similarly, in the paper [2] numerical calculations are presented and rigorous results in [3] are announced. Finally, in the paper [3] the stationary incompressible ﬂow at high Reynolds number Re ∼ 1ε over a periodic rough boundary, with the roughness period ε, is considered. An asymptotic expansion is constructed and, with the help of boundary layer correctors deﬁned in a semi-inﬁnite cell, eﬀective wall laws are obtained. A numerical validation is presented, but there are no mathematically rigorous convergence results. The error estimate for the approximation, announced in [2], was not proved in [3]. We mention also the article [14]. In this section we are going to present a sketch of the justiﬁcation of the Navier slip law by the technique developed in [30] for Laplace’s operator and then in “volumeV” — 2009/8/3 — 0:35 — page 121 — #137 3. WALL LAWS FOR THE STOKES AND NAVIER–STOKES EQUATIONS 121 [31] for the Stokes system. The result for a 2D laminar stationary incompressible viscous ﬂow over a rough boundary is in [35]. It presents a generalization of the analogous results on the justiﬁcation of the law by Beavers and Joseph [16] for a tangential viscous ﬂow over a porous bed, obtained in [32], [33], [34] and [36]. For a review we refer to [42] and [38]. In the subsections which follow we consider a 3D Couette ﬂow over a rough boundary. In Subsection 3.1 we introduce the corresponding boundary layer problem and in Subsection 3.2 we present the main steps in obtaining the Navier slip condition from [37]. 3.1. Navier’s boundary layer. As observed in hydrodynamics, the phenomena relevant to the boundary occur in a thin layer surrounding it. We are not interested in the boundary layers corresponding to the inviscid limit of the Navier–Stokes equations, but we undertake to construct the viscous boundary layer describing effects of the roughness. There is a similarity with boundary layers describing eﬀects of interfaces between a perforated and a non-perforated domain. The corresponding theory for the Stokes system is in [31] and, in a more pedagogical way, in [42]. In this subsection we are going to present a sketch of construction of the main boundary layer, used for determining the coeﬃcient in Navier’s condition. It is natural to call it the Navier’s boundary layer. In [35] the 2D boundary layer was constructed and the 3D case was studied in [37] . We suppose the layer geometry from the beginning of the subsection 2.2. Following the construction from [35], the crucial role is played by an auxiliary problem. It reads as follows: For a given constant vector λ ∈ R2 , ﬁnd {β λ , ω λ } that solve −∆y β λ + ∇y ω λ = 0 in Z + ∪ (Y − b3~e3 ) λ divy β = 0 λ β S (·, 0) = 0 λ λ {∇y β − ω I}~e3 S (·, 0) = λ βλ = 0 λ λ in Zbl (1.54) on S (1.55) on S (1.56) on (Υ − b3~e3 ) (1.57) is y = (y1 , y2 )-periodic, (1.58) ′ {β , ω } (1.53) where S = (0, b1 ) × (0, b2 ) × {0}, Z + = (0, b1 ) × (0, b2 ) × (0, +∞), and Zbl = Z + ∪ S ∪ (Y − b3~e3 ). Let V = {z ∈ L2loc (Zbl )3 : ∇y z ∈ L2 (Zbl )9 ; z = 0 on (Υ − b3~e3 ); divy z = 0 in Zbl and z is y ′ = (y1 , y2 )-periodic}. Then, by the Lax–Milgram lemma, there is a unique β λ ∈ V satisfying Z Z ∀ϕ ∈ V. (1.59) ∇β λ ∇ϕ dy = − ϕλ dy1 dy2 , Zbl S Using De Rham’s theorem we obtain a function ω λ ∈ L2loc (Zbl ), unique up to a constant and satisfying (1.53). By the elliptic theory, {β λ , ω λ } ∈ V ∩ C ∞ (Z + ∪ (Y − b3~e3 ))3 × C ∞ (Z + ∪ (Y − b3~e3 )), for any solution to (1.53)–(1.58). In the neighborhood of S we have β λ − (λ1 , λ2 , 0)(y3 − y32 /2)e−y3 H(y3 ) ∈ W 2,q and ω λ ∈ W 1,q , ∀q ∈ [1, ∞). Then we have “volumeV” — 2009/8/3 — 0:35 — page 122 — #138 122 1. ROUGH BOUNDARIES AND WALL LAWS Lemma 1.8. ([31], [32], [42]). For any positive a, a1 and a2 , a1 > a2 , the solution {β λ , ω λ } satisfies Z b1 Z b2 β2λ (y1 , y2 , a) dy1 dy2 = 0, Z 0 b1 b1 0 Cλbl = 0 b2 ω λ (y1 , y2 , a1 ) dy1 dy2 = Z 2 X b2 0 Z b1 βjλ (y1 , y2 , a1 ) dy1 dy2 = Cλj,bl λj = j=1 Z S Z b1 0 β λ λ dy1 dy2 = − Z b2 ω λ (y1 , y2 , a2 ) dy1 dy2 , 0 0 0 0 Z Z Z Z b2 0 Zbl βjλ (y1 , y2 , a2 ) dy1 dy2 , j = 1, 2, |∇β λ (y)|2 dy < 0. (1.60) Lemma Let λ ∈ R2 and let {β λ , ω λ } be the solution for (1.53)–(1.58) R 1.9. P2 P2 λ satisfying S ω dy1 dy2 = 0. Then β λ = j=1 β j λj and ω λ = j=1 ω j λj , where R {β j , ω j } ∈ V × L2loc (Zbl ), S ω j dy1 dy2 = 0, is the solution for (1.53)–(1.58) with λ = ~ej , j = 1, 2. Lemma 1.10. Let a > 0 and let β a,λ be the solution for (1.53)–(1.58) with S replaced by Sa = (0, b1 ) × (0, b2 ) × {a} and Z + by Za+ = (0, b1 ) × (0, b2 ) × (a, +∞). Then we have Z b1 Z b2 β a,λ (y1 , y2 , a)λ dy1 = Cλbl − a | λ |2 b1 b2 (1.61) Cλa,bl = 0 0 Proof. It goes along the same lines as Lemma 2 from [35] and we omit it. Lemma 1.11. (see [37]) Let {β j , ω j } be as in Lemma 1.8 and let Mij = R j S βi dy1 dy2 be the Navier matrix. Then the matrix M is symmetric negatively definite. 1 b1 b2 Lemma 1.12. (see [37]) Let Y have the mirror symmetry with respect to yj , where j is 1 or 2. The the matrix M is diagonal. Lemma 1.13. (see [37]) Let us suppose that the shape of the boundary doesn’t depend on y2 . Then for λ = ~e2 the system (1.53)–(1.58) has the solution β 2 = (0, β22 (y1 , y3 ), 0) and ω 2 = 0, where β22 is determined by − ∂ 2 β22 ∂ 2 β22 − =0 2 ∂y1 ∂y 2 2 3 β2 (·, 0) = 0 2 ∂β2 (·, 0) = 1 ∂y3 β22 = 0 β22 in (0, b1 ) × (0, +∞) ∪ (Y ∩ {y2 = 0} − b3~e3 ) (1.62) on (0, b1 ) × {0} (1.63) on (0, b1 ) × {0} (1.64) on (Υ ∩ {y2 = 0} − b3~e3 ), (1.65) is y1 -periodic, (1.66) “volumeV” — 2009/8/3 — 0:35 — page 123 — #139 3. WALL LAWS FOR THE STOKES AND NAVIER–STOKES EQUATIONS 123 Furthermore, for λ = ~e1 , the system (1.53)–(1.58) has the solution β 1 = (β11 (y1 , y3 ), 0, β31 (y1 , y3 )) and ω 1 = ω(y1 , y3 ) satisfying − ∂βj1 ∂βj1 ∂ω − + =0 ∂y12 ∂y32 ∂yj ∂β11 ∂β31 + =0 ∂y1 ∂y3 1 βj (·, 0) = 0 in (0, b1 ) × (0, +∞) ∪ (Y ∩ {y2 = 0} − b3~e3 ), (1.67) in Zbl ∩ {y2 = 0} (1.68) j = 1 and j = 3 on (0, b1 ) × {0}, j = 1 and j = 3 [ω] = 0 1 ∂β1 (·, 0) = 1 on (0, b1 ) × {0}, ∂y3 1 ∂β3 (·, 0) = 1 ∂y3 on (Υ ∩ {y2 = 0} − b3~e3 ), β11 = β31 = 0 {β11 , β31 , ω} Finally, (1.69) (1.70) (1.71) is y1 -periodic. (1.72) Z 1 b1 1 β1 (y1 , 0) dy1 b1 0 = M21 = 0 Z 1 b1 2 = β2 (y1 , 0) dy1 b1 0 M11 = M12 M22 and | M11 |≤| M22 |. (1.73) Lemma 1.14. Let {β j , ω j }, j = 1 and j = 3, be as in Lemma 1.8. Then we have α j | D curly β (y) | ≤ Ce o n −2πy3 min b1 ,b1 1 2 , | β j (y) − (M1j , M2j , 0) | ≤ C(δ)e−δy3 , α j | D β (y) | ≤ C(δ)e | ω j (y) | ≤ Ce −δy3 y3 > 0, α ∈ N2 ∪ (0, 0), ( y3 > 0, o n ( , n o −2πy3 min b1 ,b1 1 2 , ∀δ < 2π min 1 1 b1 , b2 2 y3 > 0, α ∈ N n , ∀δ < 2π min y3 > 0. 1 1 b1 , b2 (1.74) o Proof. As in [32] we take the curl of the equation (1.53) and obtain the j following problem for ξm = curl β j m , m = 1, 2, 3 j ∆ξm =0 in Z + j ξm ∈ W 1−1/q,q (S) ∀q < +∞ j ξm (1.75) ′ is periodic in y = (y1 , y2 ) j Now Tartar’s lemma from [39] (see Lemma 1.5) implies an exponential decay of ∇ξm j j to zero and of ξm . Since ξm ∈ L2 (Z + ), this constant equals to zero. Furthermore, “volumeV” — 2009/8/3 — 0:35 — page 124 — #140 124 1. ROUGH BOUNDARIES AND WALL LAWS having established an exponential decay, we are in situation to apply the separation of variables. Then explicit calculations, analogous to those in [36], give the ﬁrst estimate in (1.74). In the next step we use the following identity, holding for the divergence free ﬁelds: −∆β j = curl curl β j = curl ξ j and the same arguing as above leads to the second and the third estimate. After taking the divergence of the equation (1.53) we ﬁnd out that the pressure is harmonic in Z + . Since the averages of the pressure over the sections {y3 = a} are zero, we obtain the last estimate in (1.75). Corollary 1.15. The system (1.53)–(1.58) defines a boundary layer. 3.2. Justiﬁcation of the Navier slip condition for the laminar 3D Couette ﬂow. A mathematically rigorous justiﬁcation of the Navier slip condition for the 2D Poiseuille ﬂow over a rough boundary is in [35]. Rough boundary was the periodic repetition of a basic cell of roughness, with characteristic heights and lengths of the impurities equal to a small parameter ε. Then the ﬂow domain was decomposed to a rough layer and its complement. The no-slip condition was imposed on the rough boundary and there were inﬂow and outﬂow boundaries, not interacting with the humps. The ﬂow was governed by a given constant pressure drop. The mathematical model were the stationary Navier–Stokes equations. In [35] the ﬂow under moderate Reynolds numbers was considered and the following results were proved: a) A non-linear stability result with respect to small perturbations of the smooth boundary with a rough one; b) An approximation result of order ε3/2 ; c) Navier’s slip condition was justiﬁed. In this review we are going to present analogous results for a 3D Couette ﬂow from [37]. We consider a viscous incompressible ﬂuid ﬂow in a domain Ωε deﬁned in Subsection 2.1. ~ = (U1 , U2 , 0), the Then, for a ﬁxed ε > 0 and a given constant velocity U Couette ﬂow is described by the following system −ν∆vε + (vε ∇)vε + ∇pε = 0 in Ωε , (1.76) ε ε in Ω , (1.77) ε ε on B , (1.78) on Σ2 = (0, L1 ) × (0, L2 ) × {L3 } (1.79) div v = 0 v =0 ~ vε = U ε ε {v , p } is periodic in (x1 , x2 ) with period (L1 , L2 ) (1.80) Rε ε where ν > 0 is the kinematic viscosity and Ω p dx = 0. Let us note that a similar problem was considered in [9], but in an inﬁnite strip with a rough boundary. In [9] the authors were looking for solutions periodic in (x1 , x2 ), with the period ε(b1 , b2 ). “volumeV” — 2009/8/3 — 0:35 — page 125 — #141 3. WALL LAWS FOR THE STOKES AND NAVIER–STOKES EQUATIONS 125 Since we need not only existence for a given ε, but also the a priori estimates independent of ε, we give a non-linear stability result with respect to rough perturbations of the boundary, leading to uniform a priori estimates. First, we observe that the Couette ﬂow in P , satisfying the no-slip conditions at Σ, is given by v0 = U2 x3 U1 x3 ~ x3 , ~e1 + ~e2 = U L3 L3 L3 p0 = 0. (1.81) q Let |U | = U12 + U22 . Then it is easy to see that v0 is the unique solution for the Couette ﬂow in P if |U |L3 < 2ν, i.e. if the Reynolds number is moderate. We extend the velocity ﬁeld to Ωε \ P by zero. The idea is to construct the solution to (1.76)–(1.80) as a small perturbation to the Couette ﬂow (1.81). Before the existence result, we prove an auxiliary lemma: Lemma 1.16. ([35]). Let ϕ ∈ H 1 (Ωε \ P ) be such that ϕ = 0 on B ε . Then we have kϕkL2 (Ωε \P ) ≤ Cεk∇ϕkL2 (Ωε \P )3 , kϕkL2 (Σ) ≤ Cε 1/2 k∇ϕkL2 (Ωε \P )3 . (1.82) (1.83) Now we are in position to prove the desired non-linear stability result: Theorem 1.17. ([37]). Let |U |L3 ≤ ν. Then there exists a constant C0 = 3/4 L3 ν 3/4 the problem (1.76)–(1.80) C0 (b1 , b2 , b3 , L1 , L2 ) such that for ε ≤ C0 |U| Rε has a unique solution {vε , pε } ∈ H 2 (Ωε )3 × H 1 (Ωε ), Ω pε dx = 0, satisfying √ |U | . k∇(vε − v0 )kL2 (Ωε )9 ≤ C ε L3 (1.84) Moreover, kvε kL2 (Σ)3 √ |U | kvε kL2 (Ωε \P )3 ≤ Cε ε , L3 |U | + kvε − v0 kL2 (P )3 ≤ Cε , L3 |U | √ ε, kpε − p0 kL2 (P ) ≤ C L3 (1.85) (1.86) (1.87) where C = C(b1 , b2 , b3 , L1 , L2 ). Therefore, we have obtained the uniform a priori estimates for {vε , pε }. Moreover, we have found that Couette’s ﬂow in P is an O(ε) L2 -approximation for vε . Following the approach from [35], the Navier slip condition should correspond to taking into the account the next order corrections for the velocity. Then formally “volumeV” — 2009/8/3 — 0:35 — page 126 — #142 126 1. ROUGH BOUNDARIES AND WALL LAWS we get vε = v0 − 2 ε X j x Uj β − (Mj1 , Mj2 , 0)H(x3 ) L3 j=1 ε − 2 x3 ε X (Mj1 , Mj2 , 0)H(x3 ) + O(ε2 ) Uj 1 − L3 j=1 L3 where v0 is the Couette velocity in P and the last term corresponds to the counterﬂow generated by the motion of Σ. Then on the interface Σ 2 ∂vε j Uj 1 X ∂βji Ui = − + O(ε) ∂x3 L3 L3 i=1 ∂y3 and 2 x 1 ε 1 X Ui βji + O(ε). v j =− ε L3 i=1 ε After averaging we obtain the familiar form of the Navier slip condition ueﬀ j = −ε 2 X ∂ueﬀ Mji i ∂x3 i=1 on Σ, (1.88) where ueﬀ is the average over the impurities and the matrix M is deﬁned in Lemma 1.11. The higher order terms are neglected. Now let us make this formal asymptotic expansion rigorous. It is clear that in P the ﬂow continues to be governed by the Navier–Stokes system. The presence of the irregularities would only contribute to the eﬀective boundary conditions at the lateral boundary. The R leading contribution for the estimate (1.84) were the interface integral terms Σ ϕj . Following the approach from [35], we eliminate it by using the boundary layer-type functions x x β j,ε (x) = εβ j and ω j,ε (x) = ω j , x ∈ Ωε , j = 1, 2, (1.89) ε ε where {β j , ω j } is deﬁned in Lemma 1.8. We have, for all q ≥ 1 and j = 1, 2, 1 j,ε kβ − ε(M1j , M2j , 0)kLq (P )3 + kω j,ε kLq (P ) + k∇β j,ε kLq (Ω)9 = Cε1/q ε (1.90) and −∆β j,ε + ∇ω j,ε = 0 j,ε div β = 0 j,ε β Σ (·, 0) = 0 j,ε j,ε {∇β − ω I}e3 Σ (·, 0) = ej in Ωε \ Σ, (1.91) ε in Ω , (1.92) on Σ, (1.93) on Σ. (1.94) As in [35] stabilization of β j,ε towards a nonzero constant velocity ε M1j , M2j , 0 , at the upper boundary, generates a counterﬂow. It is given by the 3D Couette ﬂow x3 i d = 1 − L3 ~ei and g i = 0. “volumeV” — 2009/8/3 — 0:35 — page 127 — #143 3. WALL LAWS FOR THE STOKES AND NAVIER–STOKES EQUATIONS 127 Now, we would like to prove that the following quantities are o(ε) for the velocity and O(ε) for the pressure: 2 + X 1 x x + ~, ~ − ε Uj β j ( ) + ε 3 M U x3 U U ε (x) = vε − (1.95) L3 ε L3 j=1 P ε = pε + 2 ν X Uj ω j,ε . L3 j=1 (1.96) Then we have the following result: Theorem 1.18. ([37]). Let U ε be given by (1.95) and P ε by (1.96). Then U ε ∈ H 1 (Ωε )3 , U ε = 0 on Σ, it is periodic in (x1 , x2 ), exponentially small on Σ2 and div U ε = 0 in Ωε . Furthermore, ∀ϕ satisfying the same boundary conditions, we have the following estimate Z Z Z Z 2 X ~ x+ U ∂U ε 3 ε ε ν P div ϕ + ∇U ∇ϕ − Uj U3ε ϕ ϕ+ ∂xj L3 Ωε L3 j=1 Ωε Ωε Ωε Z ε |U |2 ε 0 0 (v − v )∇ (v − v )ϕ ≤ Cε3/2 k∇ϕkL2 (Ωε )9 + . (1.97) L3 Ωε let Corollary 1.19. ([37]). Let U ε (x) and P ε be defined by (1.95)–(1.96) and ε≤ ν 6/7 min |U | ν 1/7 3/7 , C(b1 , b2 , b3 , L1 , L2 )L3 |U |1/7 . 4(|M | + kβkL∞ ) (1.98) Then vε , constructed in Theorem 1.17, is a unique solution to (1.76)–(1.80) and |U |2 , νL3 |U |2 . ≤ Cε2 νL3 k∇U ε kL2 (Ωε )9 + P ε kL2 (P ) ≤ Cε3/2 kU ε kL2 (P )3 + kU ε kL2 (Σ)3 (1.99) (1.100) The estimates (1.99)–(1.100) allow to justify Navier’s slip condition. Remark 1.20. It is possible to add further correctors and then our problem would contain an exponentially decreasing forcing term. This is in accordance with [9] for the Navier–Stokes system and with [7], [8] and [13] for the Stokes system. For the case of rough boundaries with different characteristic heights and lengths we refer to the doctoral dissertation of I. Cotoi [28]. The estimate (1.98) is of the same order in ε as the H 1 -estimate in [4], obtained for the Laplace operator. The advantage of our approach is that we are going to obtain the Navier slip condition with a negatively definite matricial coefficient. Now we introduce the eﬀective Couette–Navier ﬂow through the following boundary value problem: “volumeV” — 2009/8/3 — 0:35 — page 128 — #144 128 1. ROUGH BOUNDARIES AND WALL LAWS Find a velocity ﬁeld ueﬀ and a pressure ﬁeld peﬀ such that −ν∆ueﬀ + (ueﬀ ∇)ueﬀ + ∇peﬀ = 0 in P , (1.101) in P , (1.102) ueﬀ = (U1 , U2 , 0) on Σ2 , (1.103) ueﬀ 3 on Σ, (1.104) on Σ, j = 1, 2, (1.105) eﬀ div u =0 =0 ueﬀ j = −ε 2 X Mji i=1 ∂ueﬀ i ∂x3 is periodic in (x1 , x2 ) {ueﬀ , peﬀ } with period (L1 , L2 ) If |U |L3 ≤ ν, the problem (1.101)–(1.106) has a unique solution −1 ε ~ , x ∈ P, ~ + x3 − 1 I− M U ueﬀ = (ũeﬀ , 0), ũeﬀ = U L3 L3 p eﬀ (1.106) (1.107) x ∈ P. = 0, Let us estimate the error made when replacing {vε , pε , Mε } by {ueﬀ , peﬀ , Meﬀ }. Theorem 1.21. ([37]). Under the assumptions of Theorem 1.17 we have k∇(vε − ueﬀ )kL1 (P )9 ≤ Cε, √ |U | εkvε − ueﬀ kL2 (P )3 + kvε − ueﬀ kL1 (P )3 ≤ Cε2 . L3 (1.108) (1.109) Our next step is to calculate the tangential drag force or the skin friction Z 1 ∂vε j ε Ft,j = (x1 , x2 , 0) dx1 dx2 , j = 1, 2. (1.110) ν L1 L2 Σ ∂x3 Theorem 1.22. ([37]). Let the skin friction Ftε be defined by (1.110). Then we have 2 ε ~ + ε MU Ft − ν 1 U ~ ≤ Cε2 |U | 1 + ν . (1.111) L3 L3 νL3 L3 |U | Corollary 1.23. . Let Fteﬀ = ν L13 I − eﬀ ε L3 M −1 ~ be the tangential drag U force corresponding to the effective velocity u . Then we have 2 ν eﬀ ε 2 |U | 1+ |Ft − Ft | ≤ Cε νL3 L3 |U | (1.112) Remark 1.24. We see that the presence of the periodic roughness diminishes the tangential drag. The contribution is linear in ε, and consequently rather small. It coincides with the conclusion from [8] that for laminar flows there is no palpable drag reduction. Nevertheless, we are going to see in the next subsection that the calculations from the laminar case could be useful for turbulent Couette flow. “volumeV” — 2009/8/3 — 0:35 — page 129 — #145 4. ROUGH BOUNDARIES AND DRAG MINIMIZATION 129 3.3. Wall laws for ﬂuids obeying Fourier’s boundary conditions at the rough boundary. In number of situations, the adherence conditions, that are used to describe ﬂuid behavior when moderate pressure and low surface stresses are involved, are no longer valid. Physical considerations lead to slip boundary conditions. These conditions are of particular interest in the study of polymers, blood ﬂow, and ﬂow through ﬁlters. We mention also the near wall models from turbulence theories. These conditions are of Fourier’s type and in number of recent publications, authors undertook the homogenization of Stokes and Navier–Stokes equations in such setting. An early reference is [10], but it was the work of Simon et al [26] which attracted lot of interest. This is a fast developing research area and we mention only the articles [22] and [23]. In most cases the eﬀective boundary condition is the no-slip condition. Consequently, the boundary layers do not enter into the wall law and the eﬀective models are valid for much larger class of the rough boundaries than the wall law derived in the previous section. Finally, we mention that there is a work on roughness induced wall laws for geostrophic ﬂows. For more information see the article [20] and references therein. 4. Rough boundaries and drag minimization Drag reduction for planes, ships and cars reduces signiﬁcantly the spending of the energy, and consequently the cost for all type of land, sea and air transportation. Drag-reduction adaptations were important for the survival of Avians and Nektons, since their eﬃciency or speed, or both, have improved. Essentially, there are three forms of drag. The largest drag component is pressure or form drag. It is particularly troublesome when ﬂow separation occurs. The two remaining drag components are skin-friction drag and drag due to lift. Skinfriction drag is the result of the no-slip condition on the surface. Those components are present for both laminar (low Reynolds number) or turbulent (high Reynolds number) ﬂows. There are several drag-reduction methods and here we discuss only the use of drag-reducing surfaces. For an overview of other techniques we refer to Bushnell, Moore [24]. The inspiration comes from morphological observations. It is known that the skin of fast sharks is covered with tiny scales having little longitudinal ribs on their surface (shark dermal denticles). These are tiny ridges, closely spaced (less than 100 µm apart and still less in height). We note that the considered sharks have a length of approximately 2 m and swim at Reynolds numbers Re ≈ 3 · 107 (see e.g. Vogel [50]). Such grooves are similar to ones used on the yacht “Stars and Strips” in America’s Cup ﬁnals and seem to reduce the skin-friction for O(10%) (see [24]). In the applications, the main interest is in the turbulent case. Mathematical modeling of the turbulent ﬂows in the presence of solid walls is still out of reach. However the turbulent boundary layers on surfaces with ﬁne roughness contain a viscous sublayer. It was found that the viscous sublayer exhibits a streaky structure. Those “low-speed streaks” are believed to be produced by slowly rotating longitudinal vortices. For a streaky structure, with a preferred lateral wavelength, a turbulent shear stress reduction was observed. “volumeV” — 2009/8/3 — 0:35 — page 130 — #146 130 1. ROUGH BOUNDARIES AND WALL LAWS The experimental facts were theoretically explained in the papers by Bechert and Bartenwerfer [17] and Luchini, Manzo and Pozzi [40] (see also [18] and references in mentioned articles). In [37] the theory developed in the laminar situation was applied to the turbulent ﬂow. It is known that the turbulent Couette ﬂow has a 2-layer structure. There is a large core layer where the molecular momentum transfer can be neglected and a thin wall layer (or sublayer) where both turbulent and molecular momentum transfer are important. The ﬂow in the wall layer is governed by the turbulent viscous shear stress τw , supposed to depend only on time. In connection with τw q authors use the friction velocity v = τρw , where ρ is the density. Then the wall layer thickness is δv = νv , we suppose that our riblets remain all the time in the pure viscous sublayer and try to apply the analysis from the Subsection 3.2. The with L3 = δv and velocity q corresponding equations are (1.76)–(1.80) q τw τw v = ρ = (v1 , v2 , 0) at x3 = δv . Since δv ρ = ν < 2ν, our results from Subsection 3.2 are applicable and we get 2 ε F − ν v + ε M v ≤ C ε|U | . (1.113) t δv δv δv q Since δv = ν τρw , we see that the eﬀects of roughness are signiﬁcant. √ √ For the shark skin ε/δv = 0.1, L3 = δv = 10−3 = ν and |U | = ν = 10−3 . The uniqueness condition from Corollary 1.19 applies if ε ≤ Cν 9/4 . Since ε ≈ 10−4 and ν 9/14 ≈ 1.389 · 10−4 . We see that our theory is applicable to the swimming of Nektons. For more details we refer to [36]. Furthermore, let us suppose the geometry of the rough boundary from [17] and [40]. Then M is diagonal and the origins of the cross and longitudinal ﬂows are at the characteristic walls coordinates (see [49]) y + = δεv M11 and y + = δεv M22 , respectively. Hence the proposition is to model the ﬂow in the viscous sublayer in the presence of the rough boundary by the Couette–Navier proﬁle (1.106) instead of the simple Couette proﬁle in the smooth case. We note that these observations were implemented numerically into a shape optimization procedure in [29]. The numerically obtained drag reduction conﬁrmed the theoretical predictions from [36]. “volumeV” — 2009/8/3 — 0:35 — page 131 — #147 Bibliography [1] Y. Achdou, O. Pironneau, Domain decomposition and wall laws, C. R. Acad. Sci. Paris, Série I, 320 (1995), p. 541–547. [2] Y. Achdou, O. Pironneau, F. Valentin, Shape control versus boundary control, eds F.Murat et al , Equations aux dérivées partielles et applications. Articles dédiés à J.L.Lions, Elsevier, Paris, 1998, p. 1–18. [3] Y. Achdou, O. Pironneau, F. Valentin, Effective Boundary Conditions for Laminar Flows over Periodic Rough Boundaries, J. Comp. Phys., 147 (1998), p. 187–218. [4] G. Allaire, M. Amar, Boundary layer tails in periodic homogenization, ESAIM: Control, Optimisation and Calculus of Variations 4(1999), p. 209– 243. [5] M. Amar, M. Tarallo, S. Terracini, On the exponential decay for boundary layer problems. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1139– 1144. [6] M. Amar, A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions. Discrete Contin. Dynam. Systems 6 (2000), no. 3, 537–556. [7] Y. Amirat, J. Simon, Influence de la rugosité en hydrodynamique laminaire, C. R. Acad. Sci. Paris, Série I, 323 (1996), p. 313–318. [8] Y. Amirat, J. Simon, Riblet and Drag Minimization, in Cox, S (ed) et al., Optimization methods in PDEs, Contemp. Math, 209, p. 9–17, American Math. Soc., Providence, 1997. [9] Y. Amirat, D. Bresch, J. Lemoine, J. Simon, Effect of rugosity on a flow governed by Navier-Stokes equations, Quart. Appl. Math. 59 (2001), no. 4, 769–785. [10] Y. Amirat, B. Climent, E. Fernández-Cara, J. Simon, The Stokes equations with Fourier boundary conditions on a wall with asperities, Math. Methods Appl. Sci. 24 (2001), no. 5, 255–276. [11] Y. Amirat, O. Bodart, Boundary layer correctors for the solution of Laplace equation in a domain with oscillating boundary, Z. Anal. Anwendungen 20 (2001), no. 4, 929–940. [12] Y. Amirat, O. Bodart, U. De Maio, A. Gaudiello, Asymptotic approximation of the solution of the Laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal. 35 (2004), no. 6, 1598–1616. [13] Y. Amirat, O. Bodart, U. De Maio, A. Gaudiello, Asymptotic approximation of the solution of Stokes equations in a domain with highly oscillating boundary, Ann. Univ. Ferrara Sez. VII Sci. Mat. 53 (2007), no. 2, 135–148. 131 “volumeV” — 2009/8/3 — 0:35 — page 132 — #148 132 Bibliography [14] G. Barrenechea, P. Le Tallec, F. Valentin, New wall laws for the unsteady incompressible Navier-Stokes equations on rough domains. M2AN Math. Model. Numer. Anal. 36 (2002), no. 2, 177–203. [15] A. Basson, D. Gérard-Varet, Wall laws for fluid flows at a boundary with random roughness. Comm. Pure Appl. Math. 61 (2008), no. 7, 941–987. [16] G. S. Beavers, D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech. 30(1967), p. 197–207. [17] D. W. Bechert, M. Bartenwerfer, The viscous flow on surfaces with longitudinal ribs , J. Fluid Mech. 206(1989), p. 105–129. [18] D. W. Bechert, M. Bruse, W. Hage, J. G. T. van der Hoeven, G. Hoppe, Experiments on drag reducing surfaces and their optimization with an adjustable geometry, preprint, spring 1997. [19] AS. Besicovitch, Almost periodic functions, Dover Publications, Cambridge Univ. Press, 1954. [20] D. Bresch, D. Gérard-Varet, Roughness-induced effects on the quasigeostrophic model. Comm. Math. Phys. 253 (2005), no. 1, 81–119. [21] D. Bresch, V. Milisic, Vers des lois de parois multi-échelle implicites, C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 833–838. [22] D. Bucur, E. Feireisl, Š. Nečasová, On the asymptotic limit of flows past a ribbed boundary. J. Math. Fluid Mech. 10 (2008), no. 4, 554–568. [23] D. Bucur, E. Feireisl, Š. Nečasová, J. Wolf, On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries. J. Diﬀerential Equations 244 (2008), no. 11, 2890–2908. [24] D.M. Bushnell, K.J. Moore, Drag reduction in nature, Ann. Rev. Fluid Mech. 23(1991), p. 65–79. [25] G. Buttazzo, R.V. Kohn, Reinforcement by a Thin Layer with Oscillating Thickness, Appl. Math. Optim. 16(1987), p. 247-261. [26] J. Casado-Dı́az, E. Fernández-Cara, J. Simon, Why viscous fluids adhere to rugose walls: a mathematical explanation, J. Diﬀerential Equations 189 (2003), no. 2, 526–537. [27] G.A. Chechkin, A. Friedman, A. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary, J. Math. Anal. Appl. 231 (1999), no. 1, 213–234. [28] I. Cotoi, Etude asymptotique de l’écoulement d’un fluide visqueux incompressible entre une plaque lisse et une paroi rugueuse, doctoral dissertation, Université Blaise Pascal, Clermont-Ferrand, January 2000. [29] E. Friedmann, The Optimal Shape of Riblets in the Viscous Sublayer, J. Math. Fluid Mech., DOI 10.1007/s00021-008-0284-z, 2008. [30] W. Jäger, A. Mikelić, Homogenization of the Laplace equation in a partially perforated domain , prépublication no. 157, Equipe d’Analyse Numérique Lyon-St-Etienne, September 1993, published in “Homogenization, In Memory of Serguei Kozlov” , eds. V. Berdichevsky, V. Jikov and G. Papanicolaou, p. 259–284, Word Scientiﬁc, Singapore, 1999. [31] W. Jäger, A. Mikelić, On the Boundary Conditions at the Contact Interface between a Porous Medium and a Free Fluid, Annali della Scuola Normale Superiore di Pisa, Classe Fisiche e Matematiche - Serie IV 23 (1996), Fasc. 3, “volumeV” — 2009/8/3 — 0:35 — page 133 — #149 Bibliography 133 p. 403–465. [32] W. Jäger, A. Mikelić, On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness, Communications on Pure and Applied Mathematics 51 (1998), p. 1073–1121. [33] W. Jäger, A. Mikelić, On the boundary conditions at the contact interface between two porous media, in Partial diﬀerential equations, Theory and numerical solution, eds. W. Jäger, J. Nečas, O. John, K. Najzar, et J. Stará, Chapman and Hall/CRC Research Notes in Mathematics no 406, 1999. pp. 175–186. [34] W. Jäger, A. Mikelić, On the interface boundary conditions by Beavers, Joseph and Saffman, SIAM J. Appl. Math. , 60(2000), p. 1111-1127. [35] W. Jäger, A. Mikelić, On the roughness-induced effective boundary conditions for a viscous flow, J. of Diﬀerential Equations, 170(2001), p. 96–122. [36] W. Jäger, A. Mikelić, N. Neuß, Asymptotic analysis of the laminar viscous flow over a porous bed, SIAM J. on Scientiﬁc and Statistical Computing, 22(2001), p. 2006–2028. [37] W. Jäger, A. Mikelić, Couette Flows over a Rough Boundary and Drag Reduction, Communications in Mathematical Physics , Vol. 232 (2003), p. 429–455. [38] W. Jäger, A. Mikelić, Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization, accepted for publication in Transport in Porous Media, 2009. DOI : 10.1007/s11242-009-9354-9 [39] J. L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Control, Gordon and Breach, New York, 1981. [40] P. Luchini, F. Manzo, A. Pozzi, Resistance of a grooved surface to parallel flow and cross-flow, J. Fluid Mech. 228(1991), p. 87–109. [41] A. Madureira, F. Valentin, Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1450–1473. [42] A. Mikelić, Homogenization theory and applications to filtration through porous media, chapter in Filtration in Porous Media and Industrial Applications, by M. Espedal, A.Fasano and A. Mikelić, Lecture Notes in Mathematics Vol. 1734, Springer-Verlag, 2000, p. 127–214. [43] B. Mohammadi, O. Pironneau, F. Valentin, Rough Boundaries and Wall Laws, Int. J. Numer. Meth. Fluids, 27 (1998), p. 169–177. [44] C. L. M. H. Navier, Sur les lois de l’équilibre et du mouvement des corps élastiques, Mem. Acad. R. Sci. Inst. France, 369 (1827). [45] N. Neuss, M. Neuss-Radu, A. Mikelić, Effective Laws for the Poisson Equation on Domains with Curved Oscillating Boundaries, Appl. Anal., Vol. 85 (2006), no. 5, 479–502. [46] O.A. Oleinik, G.A. Iosif’jan, On the behavior at infinity of solutions of second order elliptic equations in domains with noncompact boundary, Math. USSR Sbornik 40(1981), p. 527–548. [47] R. L. Panton, Incompressible Flow, John Wiley and Sons, New York, 1984. [48] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer Lecture Notes in Physics 127, Springer-Verlag, Berlin, 1980. [49] H. Schlichting, K. Gersten, Boundary-Layer Theory, 8th Revised and Enlarged Edition, Springer-Verlag, Berlin, 2000. “volumeV” — 2009/8/3 — 0:35 — page 134 — #150 134 Bibliography [50] S. Vogel, Life in Moving Fluids, 2nd ed., Princeton University Press, Princeton, 1994. “volumeV” — 2009/8/3 — 0:35 — page 135 — #151 Part 5 Hyperbolic problems with characteristic boundary Paolo Secchi, Alessandro Morando, Paola Trebeschi “volumeV” — 2009/8/3 — 0:35 — page 136 — #152 2000 Mathematics Subject Classification. Primary 35L40, 35L50, 35Q35, 76N10, 76E17 Key words and phrases. symmetric hyperbolic systems, mixed initial-boundary value problem, free boundary problem, characteristic boundary, anisotropic Sobolev spaces, vortex sheet, Euler equations, MHD equations, free boundary Abstract. In this lecture notes we consider mixed initial-boundary value problems with characteristic boundary for symmetric hyperbolic systems. First, we recall the main results of the regularity theory. Among the applications, we describe some free boundary problems for the equations of motion of inviscid compressible flows in Fluid Dynamics and ideal MHD. These are problems where the free boundary is a characteristic hypersurface and the Lopatinskiı̆ condition for the associated linearized equations holds only in weak form. In particular, we describe the result obtained in some joint papers by J.F. Coulombel and P. Secchi about the stability and existence of 2D compressible vortex sheets. Then we present a general result about the regularity of solutions to characteristic initial-boundary value problems for symmetric hyperbolic systems. We assume the existence of the strong L2 −solution, satisfying a suitable energy estimate, without assuming any structural assumption sufficient for existence, such as the fact that the boundary conditions are maximally dissipative or satisfy the Kreiss-Lopatinskiı̆ condition. We show that this is enough in order to get the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces H∗m , provided the data are sufficiently smooth. Acknowledgement. This article is an extended version of a series of lectures given by the first author at “EVEQ 2008: International Summer School on Evolutionary Equations” in Prague, June 2008. P. Secchi expresses his deep gratitude to the organizers for the kind invitation and hospitality in Prague. “volumeV” — 2009/8/3 — 0:35 — page 137 — #153 Contents Chapter 1. Introduction 1. Characteristic IBVP’s of symmetric hyperbolic systems 2. Known results 3. Characteristic free boundary problems 3.1. Compressible vortex sheets 3.2. Strong discontinuities for ideal MHD 139 139 142 143 145 145 Chapter 2. Compressible vortex sheets 1. The nonlinear equations in a ﬁxed domain 2. The L2 energy estimate for the linearized problem 3. Proof of the L2 -energy estimate 4. Tame estimate in Sobolev norms 5. The Nash–Moser iterative scheme 5.1. Preliminary steps 5.2. Description of the iterative scheme 149 152 154 156 158 160 160 162 Chapter 3. An example of loss of normal regularity 1. A toy model 2. Two for one 3. Modiﬁed toy model 167 167 169 171 Chapter 4. Regularity for characteristic symmetric IBVP’s 1. Problem of regularity and main result 2. Function spaces 3. The scheme of the proof of Theorem 4.1 3.1. The homogeneous IBVP, tangential regularity 3.1.1. Regularity of the stationary problem (4.28) 3.2. The nonhomogeneous IBVP, case m = 1 3.3. The nonhomogeneous IBVP, proof for m ≥ 2 3.4. Purely tangential regularity 3.5. Tangential and one normal derivatives 3.6. Normal derivatives 175 175 178 180 181 183 186 188 188 189 190 Bibliography 191 Appendix A. The Projector P 195 Appendix B. Kreiss-Lopatinskiı̆ condition 137 197 “volumeV” — 2009/8/3 — 0:35 — page 138 — #154 138 CONTENTS Appendix C. Structural assumptions for well-posedness 199 “volumeV” — 2009/8/3 — 0:35 — page 139 — #155 CHAPTER 1 Introduction 1. Characteristic IBVP’s of symmetric hyperbolic systems For a given integer n ≥ 2, let Ω be an open bounded connected subset of Rn , and let ∂Ω denote its boundary. For T > 0 we set QT = Ω×]0, T [ and ΣT = ∂Ω×]0, T [. We are interested in the following initial-boundary value problem (shortly written IBVP) Lu = F in QT , Mu = G on ΣT , (1.1) u|t=0 = f in Ω, where L is a ﬁrst order linear partial diﬀerential operator L = A0 (x, t, u)∂t + n X Ai (x, t, u)∂i + B(x, t, u), (1.2) i=1 ∂ ∂ and ∂i := ∂x , i = 1, . . . , n. ∂t := ∂t i The coeﬃcients Ai , B, for i = 0, . . . , n, are real N × N matrix-valued functions, deﬁned on QT . The unknown u = u(x, t), and the data F = F (x, t), G = G(x, t), f = f (x) are vector-valued functions with N components, deﬁned on QT , ΣT and Ω respectively. M = M (x, t) is a given real d × N matrix-valued function; M is supposed to have maximal constant rank d. Let ν(x) = (ν1 (x), . . . , νn (x)) be the unit outward normal to ∂Ω at a point x; then n X Ai (x, t)νi (x) Aν (x, t) = i=1 is called the boundary matrix. Definition 1.1. L is symmetric hyperbolic if the matrix A0 is deﬁnite positive and symmetric on QT , and the matrices Ai , for i = 1, . . . , n, are also symmetric. Definition 1.2. The boundary is said characteristic if the boundary matrix Aν is singular on ΣT . The boundary is characteristic of constant multiplicity if the boundary matrix Aν is singular on ΣT and rank Aν (x, t) is constant for all (x, t) ∈ ΣT . The boundary is uniformly characteristic if the boundary matrix Aν is singular on ΣT and rank Aν (x, t) is constant in a neighborhood of ΣT . The assumption that the boundary is characteristic of constant multiplicity yields that the number of negative eigenvalues (counted with multiplicity) of Aν is constant on the connected components of ΣT . 139 “volumeV” — 2009/8/3 — 0:35 — page 140 — #156 140 1. INTRODUCTION The case when the boundary matrix Aν is singular on ΣT and rank Aν (x, t) is not constant on ΣT is said nonuniformly characteristic. This case is also physically quite interesting, but only partial results are known, see [40, 47, 56, 57]. In the present notes we will not discuss this problem. Example 1.3. Let us consider the Euler equations for inviscid compressible fluids ∂t ρ + ∇ · (ρ v) = 0 , (1.3) ∂t (ρ v) + ∇ · (ρ v ⊗ v) + ∇p = 0 , ∂t (ρe + 12 ρ|v|2 ) + ∇ · ρv(e + 12 |v|2 ) + vp = 0 . Here ρ denotes the density, S the entropy, v the velocity field, p = p(ρ, S) the pressure (such that p′ρ > 0), and e = e(ρ, S) the internal energy. The ”Gibbs relation” T dS = de + p dV 1 the specific volume) yields ρ ∂e ∂e ∂e p=− = ρ2 , T = . ∂V S ∂ρ S ∂S ρ (with T the absolute temperature, and V = Therefore (1.3) is a closed system for the vector of unknowns (ρ, v, S). For smooth solutions, system (1.3) can be rewritten as ρ p ρ (∂t p + v · ∇p) + ∇ · v = 0, (1.4) ρ{∂t v + (v · ∇)v} + ∇p = 0, ∂t S + v · ∇S = 0. This is a quasi-linear symmetric hyperbolic system since it can be written in the form (ρp /ρ)(∂t + v · ∇) ∇· 0 p v = 0. ∇ ρ(∂t + v · ∇)I3 0 S ∂t + v · ∇ 0 0T The boundary matrix is If v · ν = 0, then (ρp /ρ)v · ν ν Aν = 0 νT ρv · νI3 0T 0 0 . v·ν ker Aν = {U ′ = (p′ , v ′ , S ′ ) : p′ = 0, v ′ · ν = 0}, and rank Aν = 2. Example 1.4. Let us consider the equations of ideal Magneto-Hydrodynamics (MHD) for the motion of an electrically conducting fluid, where ”ideal” means that “volumeV” — 2009/8/3 — 0:35 — page 141 — #157 1. CHARACTERISTIC IBVP’S OF SYMMETRIC HYPERBOLIC SYSTEMS 141 the effect of viscosity and electrical resistivity is neglected. The equations read ∂t ρ + ∇ · (ρ v) = 0 , ∂t (ρ v) + ∇ · (ρ v ⊗ v − H ⊗ H) + ∇(p + 21 |H|2 ) = 0 , ∂ H − ∇ × (v × H) = 0 , t (1.5) ρe + 12 (ρ|v|2 + |H|2 ) ∂ t +∇ · ρv(e + 21 |v|2 ) + vp + H × (v × H) = 0 , ∇ ·H = 0. Here ρ denotes the density, S the entropy, v the velocity field, H the magnetic field, p = p(ρ, S) the pressure (such that p′ρ > 0), and e = e(ρ, S) the internal energy. The constraint ∇ · H = 0 may be considered as a restriction on the initial data. For smooth solutions, system (1.5) is written in equivalent form as a quasilinear symmetric hyperbolic system: p ρp /ρ 0T 0T 0 0 ρI 0 0 v 3 3 ∂ + 0 03 I3 0 t H S 0 0T 0T 1 (ρp /ρ)v · ∇ ∇ 0 0 ∇· ρv · ∇I3 H∇ · −H · ∇I3 0T 0T ∇(·) · H − H · ∇I3 v · ∇I3 0T 0 p v 0 = 0. 0 H S v·∇ A different symmetrization can be obtained by the introduction of the total pressure q = p + |H|2 /2: ρ p ρ {(∂t + v · ∇)q − H · (∂t + (v · ∇))H} + ∇ · v = 0, ρ(∂t + (v · ∇))v + ∇q − (H · ∇)H = 0, (1.6) (∂t + (v · ∇))H − (H · ∇)v− ρp − ρ H{(∂t + v · ∇)q − H · (∂t + (v · ∇))H} = 0, ∂t S + v · ∇S = 0. This system can be rewritten as ρp /ρ 0T −(ρp /ρ)H T 0 ρI3 03 −(ρp /ρ)H 03 a0 0 0T 0T where (ρp /ρ)v · ∇ ∇· ∇ ρv · ∇I3 −(ρp /ρ)Hv · ∇ −H · ∇I3 0 0T 0 q v 0 ∂ + 0 t H S 1 −(ρp /ρ)H T v · ∇ 0 q v −H · ∇I3 0 = 0, a0 v · ∇ 0 H S v·∇ 0T a0 = I3 + (ρp /ρ)H ⊗ H. “volumeV” — 2009/8/3 — 0:35 — page 142 — #158 142 1. INTRODUCTION The boundary matrix is: (ρp /ρ)v · ν ν Aν = −(ρp /ρ)Hv · ν 0 νT ρv · νI3 −H · νI3 0T −(ρp /ρ)H T v · ν −H · νI3 a0 v · ν 0T 0 0 . 0 v·ν The rank of the boundary matrix depends accordingly on the conditions satisfied by (ρ, v, H, S) at the boundary. (i) If v · ν = 0, H · ν = 0, then ker Aν = {U ′ = (q ′ , v ′ , H ′ , S ′ ) : q ′ = 0, v ′ · ν = 0}, rank Aν = 2. (ii) If H · ν = 0 and v · ν 6= 0, v · ν 6= |H| √ ρ ± c(ρ), then ker Aν = {0}. This yields that the boundary matrix is invertible; in this case the boundary is noncharacteristic. (iii) If v · ν = 0 and H · ν 6= 0, then ker Aν = {v ′ = 0, νq ′ − H · νH ′ = 0}, rank Aν = 6. 2. Known results It is well-known that full regularity (existence in usual Sobolev spaces H m (Ω)) of solutions to characteristic IBVP’s for symmetric hyperbolic systems can’t be expected, in general, because of the possible loss of normal regularity at ∂Ω. This fact has been ﬁrst noticed by Tsuji [71], see also Majda–Osher [30]. Ohno-Shirota [42] have proved that a mixed problem for the linearized MHD equations is ill-posed in H m (Ω) for m ≥ 2. Generally speaking, one normal derivative (w.r.t. ∂Ω) is controlled by two tangential derivatives, see [9]. The loss of normal regularity and the relation between normal and tangential derivatives will be shown in Chapter 3 with a very simple example. The natural function space is the weighted anisotropic Sobolev space H∗m (Ω) := {u ∈ L2 (Ω) : Z α ∂xk1 u ∈ L2 (Ω), |α| + 2k ≤ m}, where Z1 = x1 ∂x1 and Zj = ∂xj for j = 2, . . . , n, if Ω = {x1 > 0} (a more rigorous deﬁnition will be given in Section 2). This function space has been ﬁrst introduced by Chen Shuxing [9] and Yanagisawa Matsumura [73] for the study of ideal MHD equations. Most of the theory has been developed for symmetric hyperbolic systems and maximal non-negative boundary conditions: Definition 1.5. The boundary space ker M is said maximally non-negative for Aν if, for every (t, x) ∈ ΣT , (Aν (t, x)u, u) ≥ 0 for all u ∈ ker M (t, x), and ker M (t, x) is not properly contained in any other subspace having this property. “volumeV” — 2009/8/3 — 0:35 — page 143 — #159 3. CHARACTERISTIC FREE BOUNDARY PROBLEMS 143 Linear L2 theory with maximal non-negative boundary conditions and characteristic boundaries with constant multiplicity has been developed by Rauch [46], where the tangential regularity of solutions is also proved. Existence and regularity theory in H∗m (Ω) has been treated by Guès [21], Ohno, Shizuta and Yanagisawa [44], Secchi [51, 53, 55, 58], Shizuta [62]. Resolution of the MHD equations in H∗m (Ω) may be found in the already cited paper [73] and in Secchi [52, 59]. Applications to general relativity are in [20, 66], see also [49]. An extension to nonhomogeneous strictly dissipative boundary conditions has been considered by Casella, Secchi and Trebeschi in [7, 60]. For problems with a nonuniformly characteristic boundary we refer again to [40, 47, 56, 57]. Remark 1.6. There is a very important exception to the phenomenon of the loss of normal regularity at ∂Ω. This is given by the IBVP for the Euler compressible equations under the slip boundary condition v · ν = 0, see Example 1.3. The latter is a maximal non-negative boundary condition and the boundary matrix is singular at ∂Ω with constant rank Aν = 2. That IBVP for the Euler equations can be solved in the usual Sobolev spaces H m (Ω), i.e. solutions have full regularity with respect to the normal direction to the boundary, see [3, 50]. The reason is due to the vorticity equation, which represents an additional conservation law that can be used in order to estimate those normal derivatives that one cannot obtain by the inversion of the noncharacteristic part of the boundary matrix. A similar remark holds for compressible vortex sheets, see Chapter 2, where solutions have full regularity with respect to the normal direction to the boundary, but a loss of regularity with respect to the initial data. So far for characteristic boundaries of constant multiplicity and maximal nonnegative boundary conditions. For more general boundary conditions, some results have been proven for symmetrizable hyperbolic systems under suitable structural assumptions, that we brieﬂy describe in Appendix C. Instead of maximal non-negative boundary conditions, the theory deals with uniform Kreiss–Lopatinskiı̆ conditions (UKL) (that we introduce in Appendix B). Moreover the boundary is assumed to be uniformly characteristic. The general theory has received major contributions by Majda and Osher [30], Ohkubo [41], Benzoni and Serre [4]. In the same framework we may also quote the papers about elasticity by Morando and Serre [35, 36]. 3. Characteristic free boundary problems In general, the global existence of regular solutions of quasi-linear hyperbolic systems can’t be expected because the formation of singularities in ﬁnite time may occur. The breakdown of the smoothness property may come from the appearance of discontinuities in the solution, i.e. shock waves which develop no matter how smooth the initial data are, see [64]. In contrast to the 1D case, in higher space dimensions there is no general existence theorem for solutions which allows discontinuities. A fundamental part in the study of quasi-linear hyperbolic equations is the Riemann problem, i.e. the initial value problem where initial data are piecewise constant with a jump in between. “volumeV” — 2009/8/3 — 0:35 — page 144 — #160 144 1. INTRODUCTION This initial discontinuity generates elementary waves of three kinds: centered rarefaction waves, shock waves and contact discontinuities. In general, the solution of the Riemann problem is expected to develop singularities or fronts of the above kind for all the characteristic ﬁelds. Since the general case is too diﬃcult, we will restrict the problem to the case with only one single wave front separating two smooth states. This is a free boundary problem because the single wave front separating the two smooth states on either sides is part of the unknowns. The ﬁrst attempt to extend the theory to several space variables is due to Majda [28, 29], who showed the short-time existence and stability of a single shock wave. See also Blokhin-Trakhinin [5] and the references therein for a diﬀerent approach. A general presentation of Majda’s result with some improvements may be found in Métivier [32]. See [19] for the uniform stability of weak shocks when the shock strength tends to zero. The existence of rarefaction waves was then showed by Alinhac [1], the existence of sound waves by Métivier [31]. While rarefaction waves are continuous solutions with only a singularity at the initial time given by the initial jump, shocks and contact discontinuities are solutions with a discontinuity which persists in time; it is therefore useful to point out the diﬀerences between these two cases. Let us brieﬂy recall the main deﬁnitions. Consider a general N × N system of conservation laws in Rn n X ∂xj fj (U ) = 0, ∂t U + j=1 ∞ N N where fj ∈ C (R ; R ). Given U ∈ RN , denote Aj (U ) := fj′ (U ), A(U, ν) := n X j=1 νj Aj (U ) ∀ν ∈ Rn ; let λk (U, ν) be the (real) eigenvalues (characteristic ﬁelds) of the matrix A(U, ν), λ1 (U, ν) ≤ · · · ≤ λN (U, ν). Let us denote by rk (U, ν) the right eigenvectors of A(U, ν). Consider a planar discontinuity at (t, x) with front Σ := {ν · (x − x) = σ(t − t)}, where σ is the velocity of the front. Denote by U ± the values of U at (t, x) from each side of Σ. We have the following deﬁnition introduced by Lax [26]. Deﬁnition. U ± is a shock if there exists k ∈ {1, . . . , N } such that ∇λk · rk 6= 0 ∀U , ∀ν (λk is said genuinely nonlinear) and λk−1 (U − , ν) < σ < λk (U − , ν), λk (U + , ν) < σ < λk+1 (U + , ν). The ﬁrst inequality on the left (resp. the last on the right) is ignored when k = 1 (resp. k = N ). U ± is a contact discontinuity if there exists k ∈ {1, . . . , N } such that ∇λk ·rk ≡ 0 ∀U , ∀ν (λk is said linearly degenerate) and λk (U + , ν) ≤ σ = λk (U − , ν) “volumeV” — 2009/8/3 — 0:35 — page 145 — #161 3. CHARACTERISTIC FREE BOUNDARY PROBLEMS 145 or λk (U + , ν) = σ ≤ λk (U − , ν). In case of shocks, the deﬁnition shows that the velocity of the front is always diﬀerent from the characteristic ﬁelds. It follows that the shock front is a noncharacteristic interface. On the contrary, the contact discontinuity is a characteristic interface, because of the possible equalities. Another crucial diﬀerence between shocks and contact discontinuities is that in the ﬁrst case one has the uniform stability, which is the extension of the uniform Kreiss–Lopatinskiı̆ condition for standard mixed problems. In case of contact discontinuities the Kreiss–Lopatinskiı̆ condition holds only weakly, and not uniformly. This fact has consequences for the apriori energy estimate of solutions. In the following we describe some characteristic free boundary value problems for piecewise smooth solutions: 2D vortex sheets for compressible Euler equations (which will be considered in more detail in Chapter 2) and the strong discontinuities of ideal MHD. Other characteristic interfaces are the rarefaction waves and the sound waves, see [1, 31]. 3.1. Compressible vortex sheets. 2D vortex sheets are piecewise smooth solutions for the compressible Euler equations for barotropic ﬂuids: ( ∂t ρ + ∇x · (ρ u) = 0 , (1.7) ∂t (ρ u) + ∇x · (ρ u ⊗ u) + ∇x p(ρ) = 0 , where t ≥ 0, x ∈ R2 . At the unknown discontinuity front Σ = {x1 = ϕ(x2 , t)} it is required that ∂t ϕ = v ± · ν, [p] = 0, where [p] = p+ − p− denotes the jump across Σ, and ν is a normal vector to Σ. Stability and existence of solutions to the above problem have been proven by Coulombel-Secchi [16]. We will consider this problem in more detail in Chapter 2. For a comparison with the following analysis about strong discontinuities for ideal MHD, it may be useful to notice that here the mass ﬂux j = j ± := ρ± (v ± ·ν−∂t ϕ) = 0 at Σ. 3.2. Strong discontinuities for ideal MHD. Consider a solution (ρ, v, H, S) of ideal MHD equations (1.5) in R3 , with a single front of discontinuity Σ = {x1 = ϕ(x2 , x3 , t)}. This is a piecewise smooth function which solves (1.5) on either side of the front and, in order to be a weak solution, satisﬁes the Rankine–Hugoniot jump conditions at Σ, taking the form [j] = 0, [HN ] = 0, + j[vτ ] = HN [Hτ ], j[e + 12 |v|2 + |H|2 2ρ ] j[vN ] + [q]|N |2 = 0, + j[Hτ /ρ] = HN [vτ ] + [qvN − HN (v · H)] = 0, (1.8) “volumeV” — 2009/8/3 — 0:35 — page 146 — #162 146 1. INTRODUCTION where N = (1, −∂x2 ϕ, −∂x3 ϕ) is a normal vector to Σ, and we have set vN = v · N, HN = H · N, vτ = v − vN N, Hτ = H − HN N, j := ρ(vN − ∂t ϕ) (mass ﬂux), q := p + 12 |H|2 (total pressure). The Rankine–Hugoniot conditions (1.8) may be satisﬁed in diﬀerent ways. This leads to diﬀerent kinds of strong discontinuities classiﬁed as follows, see [25]: Definition 1.7. (i) The discontinuity front Σ is a MHD shock if j ± 6= 0, [ρ] 6= 0; j ± 6= 0, [ρ] = 0; (ii) Σ is called an Alfvén or rotational discontinuity (Alfvén shock) if (iii) Σ is a contact discontinuity if j ± = 0, ± HN 6= 0; (iv) Σ is a current-vortex sheets (also called tangential discontinuities) if j ± = 0, ± HN = 0. Except for MHD shocks, which are noncharacteristic interfaces, all the above free boundaries are characteristic surfaces. Accordingly to the above classiﬁcation, the Rankine–Hugoniot conditions (1.8) are satisﬁed as follows: (1) If Σ is an Alfvén discontinuities then: H [p] = 0, [S] = 0, [HN ] = 0, [|H|2 ] = 0, [v − √ ] = 0, ρ (1.9) p + ± ± ± + j = j = ρ (vN − ∂t ϕ) = HN ρ 6= 0. As the modulus and the normal component of H are continuous across the front, in general H may only change its direction. For this reason Alfvén discontinuities are also called rotational discontinuities. Moreover, since the mass ﬂux j is diﬀerent from 0, Alfvén discontinuities are sometimes called Alfvén shocks. Consider the problem obtained by linearizing equations (1.5) and (1.9) around a piecewise constant solution of (1.9). This problem may be formulated as a nonstandard boundary value problem, which is well-posed if the analogue of the Kreiss– Lopatinskiı̆ condition is satisﬁed (see Appendix B for the deﬁnition). It has been shown by Syrovatskii [65] for incompressible MHD, and Ilin Trakhinin [24] for compressible MHD, that such planar Alfvén discontinuities are never uniformly stable, that is the uniform Lopatinskiı̆ condition is always violated. In fact, planar Alfvén discontinuities are either violently unstable or weakly stable. Violent instability means that the Kreiss–Lopatinskiı̆ condition is violated, so that there exist exponentially exploding modes of instability. This instability corresponds to ill-posedness in the sense of Hadamard. Weak stability means that the Kreiss- Lopatinskiı̆ condition is satisﬁed (there are no growing modes) but not uniformly. In this case the solution can become unstable, but the instability is much slower to develop than in the case of violent instability. “volumeV” — 2009/8/3 — 0:35 — page 147 — #163 3. CHARACTERISTIC FREE BOUNDARY PROBLEMS 147 Another remarkable fact about Alfvén discontinuities is that the symbol of the operator associated to the function ϕ describing the unknown front Σ, that is obtained from (1.9), is not elliptic. This leads to an additional loss of regularity of the front. As already mentioned, Alfvén discontinuities are characteristic interfaces. (2) If Σ is a contact discontinuity then: [v] = 0, [H] = 0, [p] = 0. (1.10) (We may have [ρ] 6= 0, [S] 6= 0.) The boundary conditions (1.10) are maximally non-negative (but not strictly dissipative). Using this fact, some a priori estimates for the solution of (1.5), (1.10) have been proven by Blokhin-Trakhinin [5], by the energy method. As for Alfvén discontinuities, the symbol associated to the front in (1.10) is not elliptic. Again, the front of contact discontinuities is characteristic, Example 1.4 (iii). (3) If Σ is a current-vortex sheets then: ± ∂t ϕ = vN , [q] = 0, ± HN = 0. (1.11) (We may have [vτ ] 6= 0, [Hτ ] 6= 0, [ρ] 6= 0, [S] 6= 0.) The analysis of the linearized equations around a piecewise constant solution of (1.11) shows that planar current-vortex sheets are never uniformly stable (i.e. the uniform Lopatinskiı̆ condition is always violated). They are either weakly stable or violently unstable (Hadamard ill-posedness). The symbol associated to the front is elliptic. Again the front is characteristic, Example 1.4 (i). The stability and existence of current-vortex sheets has been studied by Trakhinin. In [68] Trakhinin has considered the linearized equations with variable coeﬃcients obtained from a new symmetrization of the MHD equations. Under the assumption H + × H − 6= 0 on Σ, and a smallness condition on [vτ ] 6= 0, he has shown that the boundary conditions (1.11) are maximally non-negative (but not strictly dissipative). Using this fact he has proved by the energy method an a priori estimate in space H∗1 (Ω), without loss of regularity w.r.t. the initial data (but not w.r.t. the coeﬃcients). For the stability in the incompressible case see [39, 69]. In [70] Trakhinin has proved the existence of current-vortex sheets. First he has extended the a priori estimate of [68] and proved a tame estimate in anisotropic Sobolev spaces H∗m (Ω). Then the existence of the solution to the nonlinear problem has been shown by adapting a Nash–Moser iteration. This strategy is explained with more details in Chapter 2 on vortex sheets. “volumeV” — 2009/8/3 — 0:35 — page 148 — #164 “volumeV” — 2009/8/3 — 0:35 — page 149 — #165 CHAPTER 2 Compressible vortex sheets Let us consider Euler equations of isentropic gas dynamics in the whole space R2 . Denoting by u the velocity of the ﬂuid and ρ the density, the equations read: ( ∂t ρ + ∇ · (ρu) = 0 , (2.1) ∂t (ρu) + ∇ · (ρu ⊗ u) + ∇ p = 0 , where p = p(ρ) is the pressure law, a C ∞ function of ρ, deﬁned on ]0, +∞[, with p′ (ρ) > 0 for all ρ. The speed of sound c(ρ) in the ﬂuid is then deﬁned by the relation p c(ρ) := p′ (ρ) . Let Σ := {x2 = ϕ(t, x1 )} be a smooth interface and (ρ, u) a smooth function on either side of Σ: ( (ρ+ , u+ ) if x2 > ϕ(t, x1 ) (ρ, u) := (ρ− , u− ) if x2 < ϕ(t, x1 ). Definition 2.1. (ρ, u) is a weak solution of (2.1) if and only if it is a classical solution on both sides of Σ and it satisﬁes the Rankine–Hugoniot conditions at Σ: ∂t ϕ [ρ] − [ρu · ν] = 0 , ∂t ϕ [ρu] − [(ρu · ν)u] − [p]ν = 0 , (2.2) where ν := (−∂x1 ϕ, 1) is a (space) normal vector to Σ. As usual, [q] = q + − q − denotes the jump of a quantity q across the interface Σ. Following Lax [26], we shall say that (ρ, u) is a contact discontinuity if (2.2) is satisﬁed in the following way: ∂t ϕ = u+ · ν = u− · ν , p+ = p− . Because p is monotone, the previous equalities read ∂t ϕ = u+ · ν = u− · ν , ρ+ = ρ− . (2.3) Since the density and the normal velocity are continuous across the interface Σ, the only jump experimented by the solution is on the tangential velocity. (Here, normal and tangential mean normal and tangential with respect to Σ). For this reason, a contact discontinuity is a vortex sheet and we shall make no distinction in the terminology we use. 149 “volumeV” — 2009/8/3 — 0:35 — page 150 — #166 150 2. COMPRESSIBLE VORTEX SHEETS Therefore the problem is to show the existence of a weak solution to (2.1), (2.3). Observe that the interface Σ, or equivalently the function ϕ, is part of the unknowns of the problem; we thus deal with a free boundary problem. Due to (2.3), the free boundary is characteristic with respect to both left and right sides. Some information comes from the study of the linearized equations near a piecewise constant vortex sheet. The stability of linearized equations for planar and rectilinear compressible vortex sheets around a piecewise constant solution has been analysed some time ago by Miles [18, 34], using tools of complex analysis. For a vortex sheet in Rn , n ≥ 2, the situation may be summarized as follows (see [61]): for n ≥ 3, the problem is always violently unstable; for n = 2, there exists a critical value for the jump of the tangential velocity such that : √ if |[u · τ ]| < 2√2c(ρ) the problem is violently unstable (subsonic case), if |[u · τ ]| > 2 2c(ρ) the problem is weakly stable (supersonic case), (2.4) p where c(ρ) := p′ (ρ) is the sound speed and τ is a tangential unit vector to Σ. As will be seen below, the problem given by the linearized equations obtained from (2.1) and the transmission boundary conditions at the interface (2.3) may be formulated as a nonstandard boundary value problem, which is well-posed if the analogue of the Kreiss–Lopatinskiı̆ condition is satisﬁed (see Appendix B). In (2.4), violent instability means that the Kreiss–Lopatinskiı̆ condition is violated, so that there exist exponentially exploding modes of instability. This instability corresponds to ill-posedness in the sense of Hadamard. Weak stability means that the Kreiss–Lopatinskiı̆ condition is satisﬁed (there are no growing modes), but not uniformly. In this case the solution can become unstable, but the instability is much slower to develop than in the case of violent instability. In the instability case no a priori energy estimate for the solution is possible, because of the ill-posedness. In the weak stability case, an L2 − L2 energy estimate (for the solution with respect to the data) is not expectable, because the Kreiss– Lopatinskiı̆ condition doesn’t hold uniformly. However, it is reasonable to look for an energy estimate with loss of derivatives with respect to the data. This is diﬀerent from the case of shocks, where the Kreiss–Lopatinskiı̆ condition holds uniformly, so that an energy estimate without loss of derivatives may be proved (see Majda [28, 29]). The above result in 2D formally agrees with the theory of incompressible vortex sheets. In fact, in the incompressible limit the speed of sound tends to inﬁnity, and the above result yields that two-dimensional incompressible vortex sheets are always unstable (the Kelvin–Helmhotz instability). Recalling that in the theory for incompressible vortex sheets, solutions are shown to exist in the class of analytic functions, one may look for analytic solutions also in the compressible instability case; here the existence of a local in time analytic solution for the nonlinear problem may be obtained by applying Harabetian’s result [22]. “volumeV” — 2009/8/3 — 0:35 — page 151 — #167 2. COMPRESSIBLE VORTEX SHEETS 151 √ In the transition case |[u · τ ]| = 2 2c(ρ) the problem is also weakly stable, as shown in [14], in a weaker sense than in the supersonic case. The complete analysis of linear stability of contact discontinuities for the nonisentropic Euler equations is carried out in [13], for both cases n = 2 and n = 3. From now on we consider the 2D supersonic weakly stable regime. In the paper [15], written with J.F. Coulombel, we show that supersonic constant vortex sheets are linearly stable, in the sense that the linearized system (around a piecewise constant solution) obeys an L2 -energy estimate. The failure of the uniform Kreiss–Lopatinskiı̆ condition yields an energy estimate with the loss of one tangential derivative from the source terms to the solution. Moreover, since the problem is characteristic, the estimate we prove exhibits a loss of control on the trace of the solution. We also consider the linearized equations around a perturbation of a constant vortex sheet, and we show that these linearized equations with variable coeﬃcients obey the same energy estimate with loss of one derivative w.r.t. the source terms. In a second paper [16] written with J.F. Coulombel, we consider the nonlinear problem and prove the existence of supersonic compressible vortex sheets solutions. To prove our result we ﬁrst extend the energy estimate of solutions to the linearized equations to Sobolev norms, by application of the L2 -estimate to tangential derivatives and combination with an a priori estimate for normal derivatives obtained by the energy method from a vorticity-type equation, see Remark 1.6. Here solutions have full regularity with respect to the normal direction to the boundary; therefore they can be estimated in usual Sobolev spaces H m instead of anisotropic Sobolev spaces H∗m , in spite of the characteristic boundary. The failure of the uniform Kreiss–Lopatinskiı̆ condition yields another type of loss of regularity, i.e. the loss of derivatives from the source terms to the solution. The new estimate extended to Sobolev norms shows the loss of one derivative with respect to the source terms, and the loss of three derivatives with respect to the coeﬃcients. The loss is ﬁxed, and we can thus solve the nonlinear problem by a Nash–Moser iteration scheme. Recall that the Nash–Moser procedure was already used to construct other types of waves for multidimensional systems of conservation laws, see, e.g., [1, 19]. However, our Nash–Moser procedure is not completely standard, since the tame estimate for the linearized equations will be obtained under certain nonlinear constraints on the state about which we linearize. We thus need to make sure that these constraints are satisﬁed at each iteration step. The rest of the present chapter is devoted to the presentation of these results. In [16] we also show how a similar analysis yields the existence of weakly stable shock waves in isentropic gas dynamics, and the existence of weakly stable liquid/vapor phase transitions. In [17] we prove that suﬃciently smooth 2-D compressible vortex sheets are unique. Similar arguments to those of [15] have been considered by Morando and Trebeschi [38] in the analysis of the linearized stability of 2D vortex sheets for the nonisentropic Euler equations. Adapting the proof of [16], Trakhinin [70] has shown the existence of currentvortex sheets in MHD, see Section 3.2. “volumeV” — 2009/8/3 — 0:35 — page 152 — #168 152 2. COMPRESSIBLE VORTEX SHEETS 1. The nonlinear equations in a ﬁxed domain The interface Σ := {x2 = ϕ(t, x1 )} is an unknown of the problem. We ﬁrst straighten the unknown front in order to work in a ﬁxed domain. Let us introduce the change of variables (τ, y1 , y2 ) → (t, x1 , x2 ), (t, x1 ) = (τ, y1 ), x2 = Φ(τ, y1 , y2 ), where Φ : {(τ, y1 , y2 ) : y2 > 0} → R, is a smooth function such that ∂y2 Φ(τ, y1 , y2 ) ≥ κ > 0 , Φ(τ, y1 , 0) = ϕ(t, x1 ) . We deﬁne the new unknowns + (ρ+ ♯ , u♯ )(τ, y1 , y2 ) := (ρ, u)(τ, y1 , Φ(τ, y1 , y2 )) , − (ρ− ♯ , u♯ )(τ, y1 , y2 ) := (ρ, u)(τ, y1 , Φ(τ, y1 , −y2 )) . ± The functions ρ± ♯ , u♯ are smooth on the ﬁxed domain {y2 > 0}. For convenience, we drop the ♯ index and only keep the + and − exponents. Then, we again write (t, x1 , x2 ) instead of (τ, y1 , y2 ). Let us denote u± = (v ± , u± ). The existence of compressible vortex sheets amounts to proving the existence of smooth solutions to the following ﬁrst order system: ρ± ∂ ∂t ρ± + v ± ∂x1 ρ± + (u± − ∂t Φ± − v ± ∂x1 Φ± ) ∂xx2 Φ± +ρ± ∂x1 v ± + ρ± ∂x2 u± ∂x2 Φ± − ρ± ∂x1 Φ± ∂x2 Φ± 2 ∂x2 v ± = 0 , v± ∂ ∂t v ± + v ± ∂x1 v ± + (u± − ∂t Φ± − v ± ∂x1 Φ± ) ∂xx2 Φ± ′ ± + p ρ(ρ± ) ∂x1 ρ± − ± p′ (ρ± ) ∂x1 Φ ρ± ∂x2 Φ± 2 ∂x2 ρ± = 0 , (2.5) u± ∂ ∂t u± + v ± ∂x1 u± + (u± − ∂t Φ± − v ± ∂x1 Φ± ) ∂xx2 Φ± ′ ± + p ρ(ρ± ± ) ∂x2 ρ ∂x2 Φ± 2 = 0, in the ﬁxed domain {x2 > 0}, where Φ± (t, x1 , x2 ) := Φ(t, x1 , ±x2 ) , both deﬁned on the half-space {x2 > 0}. The equations are not suﬃcient to determine the unknowns U ± := (ρ± , v ± , u± ) and Φ± . In fact, the change of variables is only requested to map Σ to {x2 = 0} and is arbitrary outside Σ. In order to simplify the transformed equations of motion we may prescribe that Φ± solve the eikonal equations ∂t Φ± + v ± ∂x1 Φ± − u± = 0 (2.6) in the domain {x2 > 0}. This choice has the advantage that the boundary matrix of the system for U ± has constant rank in the whole space domain {x2 ≥ 0}, and not only at the boundary. “volumeV” — 2009/8/3 — 0:35 — page 153 — #169 1. THE NONLINEAR EQUATIONS IN A FIXED DOMAIN 153 The equations for U ± are only coupled through the boundary conditions Φ + = Φ− = ϕ , (v + − v − ) ∂x1 ϕ − (u+ − u− ) = 0 , ∂t ϕ + v + ∂x1 ϕ − u+ = 0 , ρ+ − ρ− = 0 , (2.7) on the ﬁxed boundary {x2 = 0}, which are obtained from (2.3). We will also consider the initial conditions ± ± (ρ± , v ± , u± )|t=0 = (ρ± 0 , v0 , u0 )(x1 , x2 ) , R2+ ϕ|t=0 = ϕ0 (x1 ) , (2.8) in the space domain = {x1 ∈ ρ , x2 > 0}. Thus, compressible vortex sheet solutions should solve (2.5), (2.6), (2.7), (2.8). There exist many simple solutions of (2.5), (2.6), (2.7) that correspond (for the Euler equations (2.1) in the original variables) to stationary rectilinear vortex sheets: ( if x2 > 0, (ρ, v, 0) , (ρ, u) = (ρ, −v, 0) , if x2 < 0, where ρ, v ∈ R, ρ > 0. Up to Galilean transformations, every rectilinear vortex sheet has this form. In the straightened variables, this stationary vortex sheet corresponds to the following smooth (stationary) solution to (2.5), (2.6), (2.7): ρ ± ± U ≡ ±v , Φ (t, x) ≡ ±x2 , ϕ ≡ 0 . (2.9) 0 In this paper, we shall assume v > 0, but the opposite case can be dealt with in the same way. The following theorem is our main result: for the nonlinear problem (2.5), (2.6), (2.7), (2.8) of supersonic compressible vortex sheets we prove the existence of solutions close enough to the piecewise constant solution (2.9). Theorem 2.2. [15, 16] Let T > 0, and let µ ∈ N, with µ ≥ 6. Assume that the stationary solution defined by (2.9) satisfies the “supersonic”condition: √ v > 2 c(ρ) . (2.10) Assume that the initial data (U0± , ϕ0 ) have the form U0± = U ± + U̇0± , with U̇0± ∈ H 2µ+3/2 (R2+ ), ϕ0 ∈ H 2µ+2 (R), and that they satisfy sufficient compatibility conditions. Assume also that (U̇0± , ϕ0 ) have a compact support. Then, there exists δ > 0 such that, if kU̇0± kH 2µ+3/2 (R2+ ) + kϕ0 kH 2µ+2 (R) ≤ δ, then there ± exists a solution U ± = U + U̇ ± , Φ± = ±x2 + Φ̇± , ϕ of (2.5), (2.6), (2.7), (2.8), on the time interval [0, T ]. This solution satisfies (U̇ ± , Φ̇± ) ∈ H µ (]0, T [×R2+ ), and ϕ ∈ H µ+1 (]0, T [×R). For the compatibility conditions as for all the other details we refer the reader to [15, 16]. The rest of the chapter is organized as follows: in section 2 we introduce the linearized equations around a perturbation of the piecewise constant solution (2.9) “volumeV” — 2009/8/3 — 0:35 — page 154 — #170 154 2. COMPRESSIBLE VORTEX SHEETS and state the basic a priori L2 estimate while in section 3 we describe the main steps of its proof. In 4 we give a tame a priori estimate in Sobolev spaces for the solution of the linearized problem. In section 5, we reduce the nonlinear problem (2.5), (2.6), (2.7), (2.8), to another nonlinear system with zero initial data; then we describe the Nash–Moser iteration scheme that will be used to solve this reduced problem. 2. The L2 energy estimate for the linearized problem We introduce the linearized equations around a perturbation of the piecewise constant solution (2.9). More precisely, let us consider the functions Ur,l = U ± + U̇r,l (t, x1 , x2 ), Φr,l = ±x2 + Φ̇r,l (t, x1 , x2 ), where U̇r,l ∈ W 2,∞ (Ω) , Φ̇r,l ∈ W 3,∞ (Ω) , (Ur,l , Φr,l ) satisfy (2.6), (2.7), and the perturbations U̇r,l and Φ̇r,l have compact support. Let us consider the linearized equations around Ur,l , Φr,l with solutions denoted by U± , Ψ± . The equations take a simpler form by the introduction of the new unknowns (cfr. [1]) U̇+ := U+ − Ψ+ ∂x Ur , ∂x2 Φr 2 U̇− := U− − Ψ− ∂x Ul . ∂x2 Φl 2 (2.11) Then the equations are diagonalized and transformed to an equivalent form with constant (singular) boundary matrix. Denote the new unknowns by W ± . The linearized equations are then equivalent to Nr W + := Ar0 ∂t W + + Ar1 ∂x1 W + + I2 ∂x2 W + + Ar0 Cr W + = F + , (2.12) Nl W − := Al0 ∂t W − + Al1 ∂x1 W − + I2 ∂x2 W − + Al0 Cl W − = F − , r,l r,l with suitable matrices Ar,l = Cr,l (Ur,l , Φr,l ), and boundary j = Aj (Ur,l , Φr,l ), C matrix I2 := diag (0, 1, 1). We have 2,∞ Ar,l (Ω), j ∈ W Cr,l ∈ W 1,∞ (Ω). In view of the results in [1, 19], in (2.12) we have dropped the zero order terms in Ψ+ , Ψ− . The linearized boundary conditions are Ψ+ |x =0 = Ψ− |x =0 = ψ , 2 2 b ∇ψ + M U|x2 =0 = g , with suitable matrices b = b(Ur,l ), M = M(∇ϕ) and where U = (U+ , U− )T , ∇ψ = (∂t ψ, ∂x1 ψ)T and g = (g1 , g2 , g3 )T . Introducing W ± the linearized boundary “volumeV” — 2009/8/3 — 0:35 — page 155 — #171 2. THE L2 ENERGY ESTIMATE FOR THE LINEARIZED PROBLEM 155 conditions become equivalent to Ψ+ = Ψ− = ψ , f | = g. B(W nc , ψ) := b ∇ψ + b̌ ψ + MW x2 =0 (2.13) Here W = (W + , W − )T , and b(Ur,l ) ∈ W 2,∞ (R2 ) , b̌(∂x2 Ur,l , ∇ϕ, ∂x2 Φr,l ) ∈ W 1,∞ (R2 ) , f r,l , ∇ϕ, ∇Φr,l ) ∈ W 2,∞ (R2 ). M(U Observe that the boundary conditions involve both ψ and W . Moreover, the matrix M only acts on the noncharacteristic part W nc := (W2+ , W3+ , W2− , W3− ) of the vector W . Our ﬁrst goal is to obtain an L2 a priori estimate of the solution to the linearized problem (2.12),(2.13). Let us deﬁne Ω := {(t, x1 , x2 ) ∈ R3 s.t. x2 > 0} = R2 × R+ . The boundary ∂Ω = {x2 = 0} is identiﬁed to R2 . Deﬁne also Hγs = Hγs (R2 ) := {u ∈ D′ (R2 ) s.t. exp(−γt)u ∈ H s (R2 )} , equipped with the norm kukHγs := k exp(−γt)ukH s (R2 ) . Deﬁne similarly the space Hγk (Ω). The space L2 (R+ ; Hγs (R2 )) is equipped with the norm Z +∞ 2 |||v|||L2 (H s ) := kv(·, x2 )k2Hγs (R2 ) dx2 . γ 0 2 In the sequel, the variable in R is (t, x1 ) while x2 is the variable in R+ . Our ﬁrst result is the following (here we denote N := (Nr , Nl )). Theorem 2.3. [15] Assume that the particular solution defined by (2.9) satisfies (2.10), that (Ur,l , Φr,l ) satisfy (2.6), (2.7), and that the perturbation (U̇r,l , Φ̇r,l ) is sufficiently small in W 2,∞ (Ω) × W 3,∞ (Ω) and has compact support. Then, for all γ ≥ 1 large enough and for all (W, ψ) ∈ Hγ2 (Ω) × Hγ2 (R2 ), the following estimate holds: 2 γ |||W |||L2γ (Ω) + kW nc |x2 =0 k2L2 + kψk2H 1 γ ≤C 1 γ3 2 |||N W |||L2 (Hγ1 ) + 1 γ2 kB(W γ nc , ψ)k2H 1 γ (2.14) . Observe that there is the loss of one (tangential) derivative for W with respect to the source terms, but no loss of derivatives for the front function ψ (as in Majda’s work [28, 29] on shock waves). Since the problem is characteristic, only the trace of the noncharacteristic part of the solution may be controlled at the boundary. The loss of control regards the tangential velocity. “volumeV” — 2009/8/3 — 0:35 — page 156 — #172 156 2. COMPRESSIBLE VORTEX SHEETS 3. Proof of the L2 -energy estimate We describe the main steps of the proof of the above Theorem 2.3. (1) Paralinearization of the equations. Using the paradiﬀerential calculus of Bony [6] and Meyer [33], we substitute in the equations the paradiﬀerential operators (w.r.to the tangential variables (t, x1 )) and obtain a system of ordinary diﬀerential equations with derivatives in x2 and symbols instead of derivatives in (t, x1 ). This step essentially reduces to the constant coeﬃcient case. (2) Elimination of the front. The projected boundary condition onto a suitable subspace of the frequency space gives an elliptic equation of order one for the front ψ. This property is a key point in our work since it allows to eliminate the unknown front and to consider a standard boundary value problem with a symbolic boundary condition (this ellipticity property is also crucial in Majda’s analysis on shock waves [28, 29]). One obtains an estimate of the form kψk2H 1 ≤ C γ12 kB(W nc , ψ)k2H 1 + kW nc |x2 =0 k2L2 γ γ γ +error terms , with no loss of regularity with respect to the source terms. In view of (2.14), it is enough to estimate W . (3) Problem with reduced boundary conditions. The projection of the boundary condition onto the orthogonal subspace gives a boundary condition involving only W nc , i.e. without involving ψ. Thus we are left with the (paradiﬀerential version of the) linear problem for W Nr W + = Ar0 ∂t W + + Ar1 ∂x1 W + + I2 ∂x2 W + + Ar0 Cr W + = F + , x2 > 0 , Nl W − = Al0 ∂t W − + Al1 ∂x1 W − + I2 ∂x2 W − + Al0 Cl W − = F − , x2 > 0 , f W| = Πg, ΠM x2 =0 x2 = 0 , (2.15) where Π denotes the suitable projection operator. For this problem the boundary is characteristic with constant multiplicity, as in the analysis of Majda and Osher [30]. Diﬀerently from [30], our problem satisﬁes a Kreiss–Lopatinskiı̆ condition in the weak sense and not uniformly. In fact, the Lopatinskiı̆ determinant associated to the boundary condition vanishes at some points. Recalling that the uniform Kreiss–Lopatinskiı̆ condition is a necessary and suﬃcient condition for the L2 estimate with no loss of derivatives, the failure of the uniform Kreiss–Lopatinskiı̆ condition yields necessarily a loss of derivatives with respect to the source terms. The proof of the main energy estimate is based on the construction of a degenerate Kreiss’ symmetrizer. We add the techniques of Majda and Osher [30] for the analysis of characteristic boundaries to Coulombel’s technique [10, 11] for the analysis of the singularities near the frequencies where the Lopatinskiı̆ condition fails. “volumeV” — 2009/8/3 — 0:35 — page 157 — #173 3. PROOF OF THE L2 -ENERGY ESTIMATE 157 In order to explain the main ideas, let us consider for simplicity the linearization around the piecewise constant solution (2.9). c=W c (δ, η) is the Then, instead of (2.15), we have a problem of the form ( W Fourier transform in (t, x1 )) c + A2 (τ A0 + iηA1 )W nc d β(τ, η)W c dW dx2 = 0, =b h, x2 > 0 , x2 = 0 , (2.16) where τ = δ + iη and where A0 , A1 , A2 are matrices with constant coeﬃcients. Because of the characteristic boundary, the two ﬁrst equations do not involve differentiation with respect to the normal variable x2 : c + − ic2 η W c + + ic2 η W c+ (τ + ivr η) W 1 2 3 − − 2 2 c − ic η W c + ic η W c− (τ + ivl η) W 1 2 3 = 0, = 0. c + and W c − that we plug into the For Re τ > 0, we obtain an expression for W 1 1 other equations in (2.16). This operation yields a system of O.D.E. of the form: nc d dW dx2 nc , d = A(τ, η) W nc b d β(τ, η)W (0) = h , x2 > 0, x2 = 0. By microlocalization, the analysis is performed locally in the neighborhood of points (τ, η) with Re τ ≥ 0. In points with Re τ > 0 the matrix A(τ, η) is regular and the Lopatinskiı̆ determinant doesn’t vanish; therefore in the neighborhood of those points we can construct a classical Kreiss’ symmetrizer. This symmetrizer would yield an L2 estimate with no loss of derivatives. When Re τ = 0 we ﬁnd points of the following type: 1) Points where A(τ, η) is diagonalizable and the Lopatinskiı̆ condition is satisﬁed. In these points the analysis is the same as for the interior points with Re τ > 0. Therefore we can construct a classical Kreiss’ symmetrizer. This symmetrizer would yield an L2 estimate with no loss of derivatives. 2) Points where A(τ, η) is diagonalizable and the Lopatinskiı̆ condition breaks down. The points where the Lopatinskiı̆ determinant vanishes correspond to critical speeds which are exactly the speeds of the kink modes in [2]. Since the Lopatinskiı̆ determinant has simple roots, it behaves like γ =Re τ uniformly in a neighborhood of the points. Using this fact and the diagonalizability of A(τ, η) we construct a degenerate Kreiss’ symmetrizer; this yields an L2 estimate with loss of one derivative. 3) Points where A(τ, η) is not diagonalizable. In those points, the Lopatinskiı̆ condition is satisﬁed. Diﬀerently from the other cases we construct a suitable non-diagonal symmetrizer. This case doesn’t yield a loss of derivatives. 4) Poles of A(τ, η). At those points, the Lopatinskiı̆ condition is satisﬁed. The matrix A(τ, η) is not smoothly diagonalizable. Consequently, Majda and Osher [30] construction of a symmetrizer in this case involves a singularity in the “volumeV” — 2009/8/3 — 0:35 — page 158 — #174 158 2. COMPRESSIBLE VORTEX SHEETS symmetrizer. We avoid this singularity and construct a smooth symmetrizer by working on the original system (2.15). In the end, we consider a partition of unity to patch things together and we get the degenerate Kreiss’ symmetrizer used in order to derive the energy estimate. 4. Tame estimate in Sobolev norms Our second result concerns the well-posedness in Sobolev norm. In view of the future application to the initial-boundary value problem, we consider functions deﬁned up to time T . Let us set ΩT := {(t, x1 , x2 ) ∈ R3 s.t. − ∞ < t < T, x2 > 0} , ωT := {(t, x1 , x2 ) ∈ R3 s.t. − ∞ < t < T, x2 = 0} . (2.17) Theorem 2.4. [16] Let T > 0 and m ∈ N. Assume that (i) the particular ± ± solution U defined by (2.9) satisfies (2.10), (ii) (U + U̇r,l , ±x2 + Φ̇r,l ) satisfies (2.6) and (2.7), (iii) the perturbation (U̇r,l , Φ̇r,l ) ∈ Hγm+3 (ΩT ) has compact support and is sufficiently small in H 6 (ΩT ). Then there exist some constants C > 0 and γ ≥ 1 such that, if (F± , g) ∈ H m+1 (ΩT ) × H m+1 (ωT ) vanish in the past (i.e. for t < 0), then there exists a unique solution (W ± , ψ) ∈ H m (ΩT ) × H m+1 (ωT ) to (2.12), (2.13) that vanishes in the past. Moreover the following estimate holds: kW kHγm (ΩT ) + k m kW|nc x2 =0 Hγ (ωT ) + kψkHγm+1 (ωT ) ≤ C kF kHγm+1 (ΩT ) +kgkHγm+1 (ωT ) + kF kHγ4 (ΩT ) + kgkHγ4 (ωT ) k(U̇r,l , Φ̇r,l )kHγm+3 (ΩT ) . (2.18) Observe that there is the loss of one derivative for W with respect to the source terms, and the loss of three derivatives with respect to the coeﬃcients. Again we have no loss of derivatives for the front function ψ (as in Majda’s work [28, 29] on shock waves). For the forthcoming analysis of the nonlinear problem by a Nash–Moser procedure it’s important to observe that (2.18) is a ”tame estimate” (roughly speaking: linear in high norms which are multiplied by low norms). Proof. We describe the main steps of the proof of the above theorem. (1) Estimate of tangential derivatives. The tangential derivatives ∂th ∂xk1 W and the front function ψ are estimated by diﬀerentiation along tangential directions of the equations and application of the L2 energy estimate given in Theorem 2.3. We “volumeV” — 2009/8/3 — 0:35 — page 159 — #175 4. TAME ESTIMATE IN SOBOLEV NORMS 159 obtain √ γ |||W |||L2 (H m ) + kW nc |x2 =0 kHγm (ωT ) + kψkHγm+1 (ωT ) γ ≤ C γ |||F |||L2 (Hγm+1 ) + kgkHγm+1 (ωT ) + kW kW 1,∞ (ΩT ) |||(U̇r,l , ∇Φ̇r,l |||Hγm+2 (ΩT ) + nc + kW|x =0 kL∞ (ωT ) + kψkW 1,∞ (ωT ) k(U̇r,l , ∂x2 U̇r,l , ∇Φ̇r,l )|x2 =0 kH m+1 (ωT ) , 2 (2.19) where ||| · |||L2 (Hγm ) denotes the norm of L2 (R+ ; Hγm (ωT )). 2) Estimate of the linearized vorticity. Consider the original non linear equations. On both sides of the interface the solution is smooth and satisﬁes ρ(∂t u + (u · ∇)u) + ∇ p(ρ) = 0 . Hence the vorticity ξ := ∂x1 u − ∂x2 v satisﬁes on both sides ∂t ξ + u · ∇ξ + ξ∇ · u = 0 . Recalling that the interface is a streamline and that there is continuity of the normal velocity across the interface, this suggests the possibility of estimates of the vorticity on either part of the front. This leads to introduce the ”linearized vorticity” 1 ξ̇± := ∂x1 u̇± − ∂x1 Φr,l ∂x2 u̇± + ∂x2 v̇± . ∂x2 Φr,l Then ∂t ξ˙+ + vr ∂x1 ξ˙+ = ∂x1 F2+ − ∂t ξ˙− + vl ∂x1 ξ̇− = ∂x1 F2− − + 1 ∂x2 Φr (∂x1 Φr ∂x2 F2 + ∂x2 F1+ ) + Λr1 · ∂x1 U̇+ + Λr2 · ∂x2 U̇+ , − 1 ∂x2 Φl (∂x1 Φl ∂x2 F2 + ∂x2 F1− ) + Λl1 · ∂x1 U̇− + Λl2 · ∂x2 U̇− , (2.20) r,l ± ∞ 2 where Λ1,2 are C functions of (U̇r,l , ∇U̇r,l , ∇Φ̇r,l , ∇ Φ̇r,l ) and where F1,2 are C ∞ functions of Ur,l , ∇Φr,l and depend linearly on F ± and W ± . A standard energy argument may be applied to (2.20). In fact we may observe that, if we take any derivative ∂ α of (2.20), multiply by ∂ α ξ ± and integrate over ωT , then the usual integrations by parts give no boundary terms. We obtain the estimate γkξ̇± kHγm−1 (ΩT ) ≤ C kF kHγm (ΩT ) + kF kL∞ (ΩT ) k∇Φ̇r,l kHγm (ΩT ) +kW kHγm (ΩT ) + kW kW 1,∞ (ΩT ) kU̇r,l kH m+1 (ΩT ) + k∇Φ̇r,l kH m (ΩT ) . (2.21) 3) Estimate of normal derivatives. We have ∂x2 W1± = = 1 ˙± ) − ∂x Φr,l (∂x Tr,l W ± )3 − (∂x Tr,l W ± )2 , u̇ − ξ Φ (∂ ∂ ± r,l x x 2 2 1 1 2 h∂x1 Φr,l i2 “volumeV” — 2009/8/3 — 0:35 — page 160 — #176 160 2. COMPRESSIBLE VORTEX SHEETS where Tr,l = T (Ur,l , Φr,l ) denotes a suitable invertible matrix such that W ± = −1 Tr,l U̇± (recall that U̇± is the unknown deﬁned in (2.11)). The above equality shows that we may estimate ∂x2 W1± by the previous steps. The estimate of normal derivatives ∂x2 W nc of the noncharacteristic part of the solution follows directly from the equations: r,l r,l r,l ± ± W± , I2 ∂x2 W ± = F ± − Ar,l 0 ∂t W − A1 ∂x1 W − A0 C since I2 := diag (0, 1, 1), W nc := (W2+ , W3+ , W2− , W3− ). We obtain for k = 1, . . . , m |||∂xk2 W |||L2 (Hγm−k ) ≤ C kF kHγm−1 (ΩT ) + kξ˙± kHγm−1 (ΩT ) + kξ̇± kL∞ (ΩT ) k∇Φ̇r,l kHγm−1 (ΩT ) (2.22) +kW kL∞ (ΩT ) k(U̇r,l , ∇Φ̇r,l )kHγm (ΩT ) + |||W |||L2 (Hγm ) + kW kHγm−1 (ΩT ) . By a combination of (2.19), (2.21) and (2.22) we ﬁnally obtain (2.18). The existence of the solution of the linear problem (2.12), (2.13) is a consequence of the well-posedness result of [12]. 5. The Nash–Moser iterative scheme 5.1. Preliminary steps. We reduce the nonlinear problem (2.5), (2.6), (2.7), (2.8), to a new problem with solution vanishing in the past. We proceed as follows. ± 1) Given initial data U0± = U + U̇0± , U̇0± ∈ H 2µ+3/2 (R2+ ), and ϕ0 ∈ H 2µ+2 (R), ± U̇0 and ϕ0 with compact support and small enough, there exist an approximate ”solution” U a , Φa , ϕa , such that U a − U = U̇ a ∈ H 2µ+2 (Ω), Φa± ∓ x2 = Φ̇a± ∈ H 2µ+3 (Ω), ϕa ∈ H 2µ+5/2 (ω), and such that ∂tj L(U a , Φa )|t=0 = 0 , a a a for j = 0, . . . , 2µ , a ∂t Φ + v ∂x1 Φ − u = 0 , a− ϕ = Φa+ |x2 =0 = Φ|x2 =0 a , ϕa ) = 0 . B(U|x 2 =0 a , (2.23) (2.24) (2.25) (2.26) The functions U̇ a , Φ̇a± , ϕa satisfy a suitable a priori estimate and may be taken with compact supports. 2) We write the equations (2.5), (2.7) for U = (U + , U − ), Φ = (Φ+ , Φ− ) in the form L(U, Φ) = 0 , B(U|x2 =0 , ϕ) = 0 , and introduce ( f a := −L(U a , Φa ) , t > 0 , f a := 0 , t < 0. Because U̇ a ∈ H 2µ+2 (Ω) and Φ̇a ∈ H 2µ+3 (Ω), (2.23) yields f a ∈ H 2µ+1 (Ω). “volumeV” — 2009/8/3 — 0:35 — page 161 — #177 5. THE NASH–MOSER ITERATIVE SCHEME 161 + 3) For all real number T > 0, we let Ω+ T , and ωT denote the sets ωT+ :=]0, T [ ×ρ , + + Ω+ T :=]0, T [ ×ρ× ]0, +∞[= ωT × R . Given the approximate solution (U a , Φa ) and the function f a , then (U, Φ) = (U a , Φa )+ + − (V, Ψ) is a solution on Ω+ T of (2.5), (2.6), (2.7), (2.8), if V = (V , V ), Ψ = + − (Ψ , Ψ ) satisfy the following system: L(V, Ψ) = f a , in ΩT , E(V, Ψ) := ∂t Ψ + (v a + v) ∂x1 Ψ − u + v ∂x1 Φa = 0 , in ΩT , Ψ+ |x on ωT , 2 =0 = Ψ− |x 2 =0 =: ψ , B(V|x2 =0 , ψ) = 0 , on ωT , (V, Ψ) = 0 , for t < 0 , (2.27) where L(V, Ψ) := L(U a + V, Φa + Ψ) − L(U a , Φa ) , B(V|x2 =0 , ψ) := B(U|ax 2 =0 + V|x2 =0 , ϕa + ψ) . (2.28) We note that (V, Ψ) = 0 satisfy (2.27) for t < 0, because f a = 0 for t < 0, and B(U|ax =0 , ϕa ) = 0 for all t ∈ R. Therefore the initial nonlinear problem on Ω+ T is 2 now substituted by a problem on ΩT . The initial data (2.8) are absorbed into the equations by the introduction of the approximate solution (U a , Φa , ϕa ), and the problem has to be solved in the class of functions vanishing in the past (i.e., for t < 0), which is exactly the class of functions in which we have a well-posedness result for the linearized problem, see Theorem 2.4. 4) We solve problem (2.27) by a Nash–Moser type iteration. This method requires a family of smoothing operators. For T > 0, s ≥ 0, and γ ≥ 1, we let Fγs (ΩT ) := u ∈ Hγs (ΩT ) , u = 0 for t < 0 . The deﬁnition of Fγs (ωT ) is entirely similar. Proposition 2.5. Let T > 0, γ ≥ 1, and let M ∈ N, with M ≥ 4. There exists a family {Sθ }θ≥1 of operators \ Sθ : Fγ3 (ΩT ) × Fγ3 (ΩT ) −→ Fγβ (ΩT ) × Fγβ (ΩT ) , β≥3 and a constant C > 0 (depending on M ), such that ∀ α, β ∈ {1, . . . , M } , kSθ U kHγβ (ΩT ) ≤ C θ(β−α)+ kU kHγα (ΩT ) , kSθ U − U kHγβ (ΩT ) ≤ C θβ−α kU kHγα (ΩT ) , 1≤β≤α≤M, d ∀ α, β ∈ {1, . . . , M } . k Sθ U kHγβ (ΩT ) ≤ C θβ−α−1 kU kHγα (ΩT ) , dθ Moreover, (i) if U = (u+ , u− ) satisfies u+ = u− on ωT , then Sθ u+ = Sθ u− on ωT , (ii) the following estimate holds: k(Sθ u+ − Sθ u− )|x2 =0 kHγβ (ωT ) ≤ C θ(β+1−α)+ k(u+ − u− )|x2 =0 kHγα (ωT ) , ∀ α, β ∈ {1, . . . , M } . “volumeV” — 2009/8/3 — 0:35 — page 162 — #178 162 2. COMPRESSIBLE VORTEX SHEETS There is another family of operators, still denoted Sθ , that acts on functions that are defined on the boundary ωT , and that enjoy the above properties with the norms k · kHγα (ωT ) . In our case it appears to be convenient the choice M := 2µ + 3. 5.2. Description of the iterative scheme. The iterative scheme starts from V0 = 0, Ψ0 = 0, ψ0 = 0. Assume that Vk , Ψk , ψk are already given for k = 0, . . . , n and verify (Vk , Ψk , ψk ) = 0 , for t < 0, − = ψ , on ωT . = (Ψ ) (Ψ+ ) k | | k x2 =0 k x2 =0 Given θ0 ≥ 1, let us set θn := (θ02 + n)1/2 , and consider the smoothing operators Sθn . Let us set Vn+1 = Vn + δVn , Ψn+1 = Ψn + δΨn , ψn+1 = ψn + δψn . (2.29) We introduce the decomposition (L is deﬁned in (2.28)) L(Vn+1 , Ψn+1 ) − L(Vn , Ψn ) = L(U a + Vn+1 , Φa + Ψn+1 ) − L(U a + Vn , Φa + Ψn ) = L′ (U a + Vn , Φa + Ψn )(δVn , δΨn ) + e′n = L′ (U a + Sθn Vn , Φa + Sθn Ψn )(δVn , δΨn ) + e′n + e′′n , where e′n denotes the usual “quadratic”error of Newton’s scheme, and e′′n the “substitution”error, due to the regularization of the state where the operator is calculated. Thanks to the properties of the smoothing operators, we have (Sθn Ψ+ n )|x2 =0 = ♯ − (Sθn Ψn )|x2 =0 and we denote ψn the common trace of these two functions. With this notation, we have B((Vn+1 )|x2 =0 , ψn+1 ) − B((Vn )|x2 =0 , ψn ) = B′ (U a + Vn )|x2 =0 , ϕa + ψn ((δVn )|x2 =0 , δψn ) + ẽ′n = B′ (U a + Sθn Vn )|x2 =0 , ϕa + ψn♯ ((δVn )|x2 =0 , δψn ) + ẽ′n + ẽ′′n , where ẽ′n denotes the “quadratic”error, and ẽ′′n the “substitution”error. The inversion of the operator (L′ , B′ ) requires the linearization around a state satisfying the constraints (2.6), (2.7). We thus need to introduce a smooth modiﬁed state, denoted Vn+1/2 , Ψn+1/2 , ψn+1/2 , that satisﬁes the above mentioned constraints, see [16] for details of the construction. Accordingly, we introduce the decompositions L(Vn+1 , Ψn+1 )−L(Vn , Ψn ) = L′ (U a +Vn+1/2 , Φa +Ψn+1/2 )(δVn , δΨn )+e′n +e′′n +e′′′ n , and B((Vn+1 )|x2 =0 , ψn+1 ) − B((Vn )|x2 =0 , ψn ) = B′ (U a + Vn+1/2 )|x2 =0 , ϕa + ψn+1/2 ((δVn )|x2 =0 , δψn ) + ẽ′n + ẽ′′n + ẽ′′′ n , ′′′ where e′′′ n , ẽn denote the second “substitution”errors. The ﬁnal step is the introduction of the “good unknown”: ∂x2 (U a + Vn+1/2 ) . (2.30) δ V̇n := δVn − δΨn ∂x2 (Φa + Ψn+1/2 ) “volumeV” — 2009/8/3 — 0:35 — page 163 — #179 5. THE NASH–MOSER ITERATIVE SCHEME 163 This leads to L(Vn+1 , Ψn+1 ) − L(Vn , Ψn ) = L′e (U a + Vn+1/2 , Φa + Ψn+1/2 )δ V̇n n o δΨn ∂x2 L(U a + Vn+1/2 , Φa + Ψn+1/2 ) , + e′n + e′′n + e′′′ n + a ∂x2 (Φ + Ψn+1/2 ) (2.31) and B((Vn+1 )|x2 =0 , ψn+1 ) − B((Vn )|x2 =0 , ψn ) = B′e ((U a + Vn+1/2 )|x2 =0 , ϕa + ψn+1/2 )((δ V̇n )|x2 =0 , δψn ) + ẽ′n + ẽ′′n + ẽ′′′ n , (2.32) Here L′e δ V̇ denotes the “eﬀective” linear operator obtained by linearizing LδV , substituting the good unknown δ V̇ in place of the unknown δV and neglecting the zero order term in δΨ, see (2.12). Similarly, B′e is the operator obtained from linearization of the boundary conditions and the introduction of the good unknown. For the sake of brevity we set n o 1 a a Dn+1/2 := L(U + V , Φ + Ψ ) , ∂ x n+1/2 n+1/2 ∂x2 (Φa + Ψn+1/2 ) 2 B′n+1/2 := B′e (U a + Vn+1/2 )|x2 =0 , ϕa + ψn+1/2 . Let us also set en := e′n + e′′n + e′′′ n + Dn+1/2 δΨn , ẽn := ẽ′n + ẽ′′n + ẽ′′′ n. The iteration proceeds as follows. Given V0 := 0 , f0 := Sθ0 f a , V1 , . . . , Vn , f1 , . . . , fn−1 , e0 , . . . , en−1 , Ψ0 := 0 , g0 := 0 , Ψ1 , . . . , Ψn , g1 , . . . , gn−1 , ẽ0 , . . . , ẽn−1 , we ﬁrst compute for n ≥ 1 En := n−1 X ek , ψ0 := 0 , E0 := 0 , Ẽ0 := 0 , ψ1 , . . . , ψn , Ẽn := k=0 n−1 X ẽk . k=0 These are the accumulated errors at the step n. Then we compute fn , and gn from the equations: n n X X (2.33) gk + Sθn Ẽn = 0 , fk + Sθn En = Sθn f a , k=0 k=0 and we solve the linear problem L′e (U a + Vn+1/2 , Φa + Ψn+1/2 ) δ V̇n = fn in ΩT , B′n+1/2 ((δ V̇n )|x2 =0 , δψn ) = gn on ωT , δ V̇n = 0, for t < 0 , δψn = 0 (2.34) “volumeV” — 2009/8/3 — 0:35 — page 164 — #180 164 2. COMPRESSIBLE VORTEX SHEETS − ﬁnding (δ V̇n , δψn ). Now we need to construct δΨn = (δΨ+ n , δΨn ) that satisﬁes ± (δΨn )|x2 =0 = δψn . Using the explicit expression of the boundary conditions in (2.34), we ﬁrst note that δψn solves the equation: + ∂t δψn + (v a+ + vn+1/2 )|x2 =0 ∂x1 δψn ( ) + )|x2 =0 ∂x2 (v a+ + vn+1/2 ∂x2 (ua+ + u+ n+1/2 )|x2 =0 a + ∂x1 (ϕ + ψn+1/2 ) − δψn ∂x2 (Φa+ + Ψ+ ∂x2 (Φa+ + Ψ+ n+1/2 )|x2 =0 n+1/2 )|x2 =0 + ∂x1 (ϕa + ψn+1/2 ) (δ v̇n+ )|x2 =0 − (δ u̇+ n )|x2 =0 = gn,2 , (2.35) and the equation − ∂t δψn + (v a− + vn+1/2 )|x2 =0 ∂x1 δψn ( ) − )|x2 =0 ∂x2 (v a− + vn+1/2 ∂x2 (ua− + u− n+1/2 )|x2 =0 a + ∂x1 (ϕ + ψn+1/2 ) − δψn ∂x2 (Φa− + Ψ− ∂x2 (Φa− + Ψ− n+1/2 )|x2 =0 n+1/2 )|x2 =0 + ∂x1 (ϕa + ψn+1/2 ) (δ v̇n− )|x2 =0 − (δ u̇− n )|x2 =0 = gn,2 − gn,1 . (2.36) − We shall thus deﬁne δΨ+ n , δΨn as the solutions to the following equations: + a+ ∂t δΨ+ + vn+1/2 ) ∂x1 δΨ+ n n + (v ( ) + + a+ a+ + v ) ∂ + u ) ∂ x x 2 (v 2 (u n+1/2 n+1/2 + + ∂x1 (Φa+ + Ψn+1/2 ) δΨ+ − n ∂x2 (Φa+ + Ψ+ ∂x2 (Φa+ + Ψ+ n+1/2 ) n+1/2 ) + + + + ∂x1 (Φa+ + Ψ+ n+1/2 ) δ v̇n − δ u̇n = RT gn,2 + hn , (2.37) and − a− ∂t δΨ− + vn+1/2 ) ∂x1 δΨ− n n + (v ) ( − ∂x2 (ua− + u− ) ∂x2 (v a− + vn+1/2 n+1/2 ) − a− δΨ− − + ∂x1 (Φ + Ψn+1/2 ) n a− + Ψ− ) (Φ ) ∂x2 (Φa− + Ψ− ∂ x 2 n+1/2 n+1/2 − − − + ∂x1 (Φa− + Ψ− n+1/2 ) δ v̇n − δ u̇n = RT (gn,2 − gn,1 ) + hn . (2.38) In (2.37), and (2.38), the source terms h± n have to be chosen suitably. First we require that h± n vanish on the boundary ωT , and in the past, so that the unique smooth solutions to (2.37) and (2.38) will vanish in the past, and will satisfy the ± continuity condition (δΨ± n )|x2 =0 = δψn . In order to compute the source terms hn , we use a decomposition that is similar to (2.31) for the operator E (deﬁned in (2.27)). We have: E(Vn+1 , Ψn+1 ) − E(Vn , Ψn ) = E ′ (Vn+1/2 , Ψn+1/2 )(δVn , δΨn ) + ê′n + ê′′n + ê′′′ n , (2.39) where ê′n is the “quadratic”error, ê′′n is the ﬁrst “substitution”error, and ê′′n is the second “substitution”error. We denote ên := ê′n + ê′′n + ê′′′ n , Ên := n−1 X k=0 êk . “volumeV” — 2009/8/3 — 0:35 — page 165 — #181 5. THE NASH–MOSER ITERATIVE SCHEME 165 Using the good unknown (2.30), and omitting the ± superscripts, we compute E ′ (Vn+1/2 , Ψn+1/2 )(δVn , δΨn ) = ∂t δΨn + (v a + vn+1/2 ) ∂x1 δΨn ∂x2 (ua + un+1/2 ) ∂x2 (v a + vn+1/2 ) a δΨn − + ∂x1 (Φ + Ψn+1/2 ) ∂x2 (Φa + Ψn+1/2 ) ∂x2 (Φa + Ψn+1/2 ) + ∂x1 (Φa + Ψn+1/2 ) δ v̇n − δ u̇n . Consequently, (2.37) and (2.39) yield + + + + + E(Vn+1 , Ψ+ n+1 ) − E(Vn , Ψn ) = RT gn,2 + hn + ên . Summing these relations, and using E(V0+ , Ψ+ 0 ) = 0, we get + E(Vn+1 , Ψ+ n+1 ) = RT = RT n X k=0 n X + h+ gk,2 + k + Ên+1 k=0 + E((Vn+1 )|x2 =0 , ψn+1 ) − Ẽn+1,2 + n X + h+ k + Ên+1 , k=0 where in the last equality, we have summed (2.32) and used the relation + )|x2 =0 , ψn+1 ) , B((Vn+1 )|x2 =0 , ψn+1 ) = E((Vn+1 2 which simply shows that the second line of the boundary operator B coincides with E at the boundary, see the deﬁnitions in (2.27), (2.28). The previous relations lead to the following deﬁnition of the source term h+ n: n X k=0 + h+ k + Sθn Ên − RT Ẽn,2 = 0 . The deﬁnition of h− n is entirely similar: n X k=0 − h− k + Sθn Ên − RT Ẽn,2 + RT Ẽn,1 = 0 . Once δΨn is computed, the function δVn is obtained from (2.30), and the functions Vn+1 , Ψn+1 , ψn+1 are obtained from (2.29). Finally, we compute en , ên , ẽn from L(Vn+1 , Ψn+1 ) − L(Vn , Ψn ) = fn + en , + + + + + E(Vn+1 , Ψ+ n+1 ) − E(Vn , Ψn ) = RT gn,2 + hn + ên , − − − − − E(Vn+1 , Ψ− n+1 ) − E(Vn , Ψn ) = RT (gn,2 − gn,1 ) + hn + ên , (2.40) B((Vn+1 )|x2 =0 , ψn+1 ) − B((Vn )|x2 =0 , ψn ) = gn + ẽn . To compute V1 , Ψ1 , ψ1 we only consider steps (2.34), (2.37), (2.38), (2.40) for n = 0. “volumeV” — 2009/8/3 — 0:35 — page 166 — #182 166 2. COMPRESSIBLE VORTEX SHEETS Adding (2.40) from 0 to N , and combining with (2.33) gives L(VN +1 , ΨN +1 ) − f a = (SθN − I)f a + (I − SθN )EN + eN , + + E(VN++1 , Ψ+ N +1 ) = RT E((VN +1 )|x2 =0 , ψN +1 ) + (I − SθN )(ÊN − RT ẼN,2 ) +ê+ N − RT ẽN,2 , − − E(VN−+1 , Ψ− N +1 ) = RT E((VN +1 )|x2 =0 , ψN +1 ) + (I − SθN )(ÊN − RT (ẼN,2 − ẼN,1 )) +ê− N − RT (ẽN,2 − ẽN,1 ) , B (VN +1 )|x2 =0 , ψN +1 = (I − SθN )ẼN + ẽN . Because SθN → I as N → +∞, and since we expect (eN , ên , ẽN ) → 0, we will formally obtain the solution of the problem (2.27) from L(VN +1 , ΨN +1 ) → f a , B((VN +1 )|x2 =0 , ψN +1 ) → 0, and E(VN +1 , ΨN +1 ) → 0. The rigorous proof of convergence follows from a priori estimates of Vk , Ψk , ψk proved by induction for every k. In the limit we obtain a solution (V, Ψ) on ΩT of (2.27), vanishing in the past, which yields that (U, Φ) = (U a , Φa ) + (V, Ψ) is a solution on Ω+ T of (2.5), (2.6), (2.7), (2.8). This concludes the proof of Theorem 2.2. “volumeV” — 2009/8/3 — 0:35 — page 167 — #183 CHAPTER 3 An example of loss of normal regularity 1. A toy model In Ω = R2+ = {x > 0} let us consider the linear IBVP ut + ux + vy = 0 v + u = 0 t y u|x=0 = 0 (u, v)|t=0 = (u0 , v0 ), (3.1) In matrix form the diﬀerential equations can be written as u u u 1 0 0 1 + + = 0. ∂t ∂ ∂ v 0 0 x v 1 0 y v Clearly the system is symmetric hyperbolic and the boundary is (uniformly) characteristic. It is also immediate to verify that the boundary condition is maximally nonnegative. We look for a priori estimates of the solution. Assume that (u0 , v0 ) ∈ H 1 (Ω) with ||(u0 , v0 )||H 1 (Ω) ≤ K. (I) We multiply the ﬁrst equation by u, the second one by v, integrate over (0, t) × Ω and obtain (|| · || stands for || · ||L2 (Ω) ) ||u(t, )||2 + ||v(t, )||2 = ||u0 ||2 + ||v0 ||2 ∀t > 0. It follows that ||u(t, )|| + ||v(t, )|| ≤ C(K) ∀t > 0. (II) Consider the tangential derivatives (uy , vy ). By taking the y−derivative of the problem we see that (uy , vy ) solves the same problem as (u, v), with initial data (u0y , v0y ). In particular it satisﬁes the same boundary condition as (u, v). It follows that ||uy (t, )||2 + ||vy (t, )||2 = ||u0y ||2 + ||v0y ||2 ∀t > 0. Thus ||uy (t, )|| + ||vy (t, )|| ≤ C(K) ∀t > 0. (III) By taking the t−derivative of the equations we see that (ut , vt ) is also a solution. This yields ||ut (t, )||2 + ||vt (t, )||2 = ||ut (0, )||2 + ||vt (0, )||2 = ||u0x + v0y ||2 + ||u0y ||2 , ||ut (t, )|| + ||vt (t, )|| ≤ C(K) 167 ∀t > 0. “volumeV” — 2009/8/3 — 0:35 — page 168 — #184 168 3. AN EXAMPLE OF LOSS OF NORMAL REGULARITY (IV ) From ux = −ut − vy we may estimate the normal derivative ux : ||ux (t, )|| ≤ ||ut (t, )|| + ||vy (t, )|| ≤ C(K) ∀t > 0. Let P be the orthogonal projection onto ker Aν (x, t)⊥ . Then u u P = v 0 (this is called the noncharacteristic component of (u, v)T ). (V ) Now we want to estimate the normal derivative vx . We have u 0 (I − P ) = v v (3.2) (3.3) (called the characteristic component of (u, v)T ). Take the x−derivative of the second equation in (3.1): vtx + uxy = 0. (3.4) Take also the y−derivative of the ﬁrst equation in (3.1): uty + uxy + vyy = 0. R R Multiply (3.4) by vx and integrate over Ω. Then ( = Ω dxdy) R R 1 d 2 2 dt ||vx || = − uxy vx = (uty + vyy )vx = d dt = d dt = d dt = d dt R uy vx − R R uy vx + uy vx + Z R R uy vtx + uy uxy − 1 2 uy vx − R 1 2 | R R (u2y )x − Z |x=0 vyy vx vy vxy 1 2 R (vy2 )x u2y (t, 0, y)dy + {z } =0 1 2 Z |x=0 vy2 (t, 0, y)dy. From vty = −uyy , we have R R R 1 d 2 2 dt |x=0 vy dy = |x=0 vy vty dy = − |x=0 vy uyy dy = 0. Then Z |x=0 vy2 (t, 0, y)dy = We then obtain d 2 dt ||vx || Z |x=0 d = 2 dt R 2 v0y (y)dy = constant in time. uy vx + R v 2 (y)dy. |x=0 0y Integration in time between 0 and t > 0 gives R R R 2 (y)dy. ||vx (t, )||2 = ||v0x ||2 + 2 uy vx − 2 u0y v0x + t |x=0 v0y “volumeV” — 2009/8/3 — 0:35 — page 169 — #185 2. TWO FOR ONE 169 By the Young’s inequality we ﬁnally obtain R 1 2 2 2 t |x=0 v0y (y)dy − C1 (K) ≤ ||vx (t, )|| ≤ ≤ 2t This shows that R 2 |x=0 v0y (y)dy + C2 (K), t > 0. vx (t, ) ∈ L2 (Ω) for t > 0 if and only if v0 ∈ H 1 (∂Ω). By the trace theorem, v0 ∈ H 1 (Ω) only gives v0|∂Ω ∈ H 1/2 (∂Ω). Therefore (u0 , v0 ) ∈ H 1 (Ω) 6⇒ (u(t, ), v(t, )) ∈ H 1 (Ω) for t > 0. 2. Two for one We consider the problem of determining a function space X characterized by the property of persistence of regularity, that is such that (u0 , v0 ) ∈ X ⇒ (u(t, ), v(t, )) ∈ X, We assume (u0 , v0 ) ∈ H 2 (Ω) ∀t > 0 . with ||(u0 , v0 )||H 2 (Ω) ≤ K2 . After the above analysis, we don’t expect to obtain (u(t, ), v(t, )) ∈ H 2 (Ω). Calculations as above give ∂th ∂yk u(t, ), ∂th ∂yk v(t, ) ∈ L2 (Ω), t > 0, h + k ≤ 2, with norms bounded by C(K2 ). By the t and y diﬀerentiation of the ﬁrst equation in (3.1) we readily obtain utx = −utt − vty ∈ L2 (Ω), 2 ||utx (t, )|| ≤ C(K2 ), uxy = −uty − vyy ∈ L2 (Ω), ||uxy (t, )|| ≤ C(K2 ), 1 t > 0, t > 0. v0 ∈ H (Ω) yields v0|∂Ω ∈ H (∂Ω), so that by the above analysis vx (t, ) ∈ L2 (Ω), ||vx (t, )|| ≤ C(K2 ), 0 < t < T, for any T < +∞. We look for an estimate of the mixed derivative vxy . Here we start from ( utyy + uxyy + vyyy = 0, vtxy + uxyy = 0. Multiply the second equation by vxy and integrate over Ω. Then R R 1 d 2 2 dt ||vxy || = − uxyy vxy = (utyy + vyyy )vxy = d dt = d dt = d dt R R Z uyy vxy − uyy vxy + R R uyy vxy − uyy vtxy − uyy uxyy − 1 2 | Z |x=0 R 1 2 vyy vxyy R 2 (vyy )x u2yy (t, 0, y)dy + {z =0 } 1 2 Z |x=0 2 vyy (t, 0, y)dy. “volumeV” — 2009/8/3 — 0:35 — page 170 — #186 170 3. AN EXAMPLE OF LOSS OF NORMAL REGULARITY Since vtyy = −uyyy , we have R R R 1 d 2 2 dt |x=0 vyy dy = |x=0 vyy vtyy dy = − |x=0 vyy uyyy dy = 0, again by the boundary condition on u. It follows that Z Z 2 2 v0yy (y)dy = constant in time. vyy (t, 0, y)dy = |x=0 |x=0 We then obtain d 2 dt ||vxy || d = 2 dt R uyy vxy + R 2 |x=0 v0yy (y)dy. Integrating in time between 0 and t > 0 yields R 1 2 2 2 t |x=0 v0yy (y)dy − C1 (K2 ) ≤ ||vxy (t, )|| ≤ ≤ 2t R 2 |x=0 v0yy (y)dy + C2 (K2 ), t > 0. It follows that, if v0 ∈ H 2 (Ω), but v0|∂Ω 6∈ H 2 (∂Ω), then vxy (t, ) 6∈ L2 (Ω). Since uxx = −utx − vxy and utx (t, ) ∈ L2 (Ω), then uxx (t, ) 6∈ L2 (Ω). A fortiori we also have vxx (t, ) 6∈ L2 (Ω). The ﬁrst two cases that we have considered suggest to deﬁne the following functions spaces. Given m ≥ 1, let us deﬁne the anisotropic Sobolev spaces K∗m (Ω) = {u ∈ L2 (Ω)|∂xk ∂yh u ∈ L2 (Ω) for 2k + h ≤ m}, m K∗∗ (Ω) = {u ∈ L2 (Ω)|∂xk ∂yh u ∈ L2 (Ω) for 2k + h ≤ m + 1, h ≤ m}. 1 0 Observe that K∗∗ (Ω) = H 1 (Ω). When m = 0 we set K∗0 = K∗∗ = L2 . Deriving the m above a priori estimates we had assumed (u0 , v0 ) ∈ H (Ω), m = 1, 2, but not all the derivatives had been used. We go back and check which particular derivatives have to be L2 in order to get the estimates. We summarize as follows 1 1 (I) If u0 ∈ K∗∗ (Ω), v0 ∈ K∗1 (Ω), then u(t, ) ∈ K∗∗ (Ω), v(t, ) ∈ K∗1 (Ω). If 1 1 v0|∂Ω 6∈ H (∂Ω) then v(t, ) 6∈ K∗∗ (Ω). 2−h 2−h (II) If ∂th u(0, ) ∈ K∗∗ (Ω), ∂th v(0, ) ∈ K∗2−h (Ω), then ∂th u(t, ) ∈ K∗∗ (Ω), 2−h 2 2 6 K∗∗ (Ω). ∈ K∗ (Ω), h = 0, 1, 2. If v0|∂Ω 6∈ H (∂Ω) then v(t, ) ∈ ∂th v(t, ) (III) In order to check if this is the correct choice when m is odd we also 3−h 3−h prove: if ∂th u(0, ) ∈ K∗∗ (Ω), ∂th v(0, ) ∈ K∗3−h (Ω), then ∂th u(t, ) ∈ K∗∗ (Ω), 3−h 2 3 h ∂t v(t, ) ∈ K∗ (Ω), h = 0, . . . , 3. If v0x|∂Ω ∈ H (∂Ω), but v0|∂Ω 6∈ H (∂Ω), then vxx (t, ) 6∈ L2 (Ω) and thus v may loose two normal derivatives even if the data are in H 3 (Ω). Under the same assumption one also shows vxyy (t, ) 6∈ L2 (Ω), uxxy (t, ) 6∈ 3 L2 (Ω); it follows that v(t, ) 6∈ K∗∗ (Ω). These are not yet the best choices for the function spaces appropriate for the general problem. A better insight is obtained with a little modiﬁcation of the model problem. “volumeV” — 2009/8/3 — 0:35 — page 171 — #187 3. MODIFIED TOY MODEL 171 3. Modiﬁed toy model Let σ ∈ C ∞ (R+ ) be a monotone increasing function such that σ(x) = x in a neighborhood of the origin and σ(x) = 1 for any x large enough. In Ω = R2+ = {x > 0} we consider the linear initial-boundary value problem ut + ux + σvx + vy = 0 v + σu + σv + u = 0 t x x y (3.5) u = 0 |x=0 (u, v)|t=0 = (u0 , v0 ). Dropping the terms with σ we get (3.1). This is a symmetric hyperbolic system with variable coeﬃcients. The boundary matrix is singular at the boundary with constant rank 1, thus the boundary is characteristic; again the boundary condition is maximally nonnegative. Let us assume (u0 , v0 ) ∈ H 2 (Ω). It is understood that not all the derivatives of the initial data will be used and we will have to take care of that. We multiply the ﬁrst equation by u, the second one by v, and integrate over Ω. After integrating by parts, using the boundary condition and σ(0) = 0, we obtain Z Z 1 1 d (||u||2 + ||v||2 ) = σ ′ uv + σ′ v2 . (3.6) 2 dt 2 By the Gronwall lemma it follows that ||u(t, )||2 + ||v(t, )||2 ≤ eCt (||u0 ||2 + ||v0 ||2 ), t > 0. (3.7) By taking the y− and t−derivative of the problem we see that (uy , vy ) and (ut , vt ) are solution of the diﬀerential equations and boundary condition in (3.5) (with suitable initial data, of course). From the previous calculation it follows that d 2 dt (||uy || + ||vy ||2 ) ≤ C(||uy ||2 + ||vy ||2 ), d 2 dt (||ut || + ||vt ||2 ) ≤ C(||ut ||2 + ||vt ||2 ). (3.8) Now we consider the normal derivatives. From the ﬁrst equation we have ux = −ut − σvx − vy , which shows the necessity to estimate at ﬁrst σvx . We apply the weighted diﬀerential operator σ∂x to both equations in (3.5) and obtain the system (σux )t + (σux )x + σ(σvx )x + (σvx )y = σ ′ ux (σv ) + σ(σu ) + σ(σv ) + (σu ) = 0 x t x x x x x y (3.9) σu = 0 x|x=0 (σux , σvx )|t=0 = (σux0 , σvx0 ). The boundary condition follows from σ(0) = 0. We multiply the two equations respectively by σux , σvx and integrate over Ω. The result is similar to (3.6); we only have to write σux instead of u, σvx instead of v, and take care of the forcing term in (3.9). We obtain Z Z Z 1 1 d (||σux ||2 + ||σvx ||2 ) = σ ′ (σux )(σvx ) + σ ′ (σvx )2 + σ ′ ux (σux ). 2 dt 2 “volumeV” — 2009/8/3 — 0:35 — page 172 — #188 172 3. AN EXAMPLE OF LOSS OF NORMAL REGULARITY This yields 1 d (||σux ||2 + ||σvx ||2 ) ≤ C(||σux ||2 + ||σvx ||2 + ||ux || ||σux ||). (3.10) 2 dt The estimate may be closed by using the ﬁrst equation of (3.5) which gives ||ux || ≤ ||ut || + ||σvx || + ||vy ||. (3.11) Let us denote by ∂⋆ any of the derivatives ∂t , σ∂x , ∂y , and set also ∂⋆α = ∂tα0 (σ∂x )α1 ∂yα2 for α = (α0 , α1 , α2 ). We substitute (3.11) in (3.10), add (3.8), apply the Gronwall lemma and ﬁnally obtain ||∂⋆ u(t, )|| + ||∂⋆ v(t, )|| + ||ux (t, )|| (3.12) ≤ CeCt (||∂⋆ u(0, )|| + ||∂⋆ v(0, )||), t > 0. Let us consider the second order derivatives. As (uy , vy ) and (ut , vt ) solve problem (3.5) with suitable initial data, the estimate (3.12) holds also with (uy , vy ) and (ut , vt ) replacing (u, v). Substituting ux = −(ut + σvx + vy ) in the right-hand side of (3.9) we can also obtain an estimate for ((σ∂x )2 u, (σ∂x )2 v). Hence we ﬁnd P α α |α|=2 (||∂⋆ u(t, )|| + ||∂⋆ v(t, )||) + ||∂⋆ ux (t, )|| (3.13) P ≤ CeCt |α|=2 (||∂⋆α u(0, )|| + ||∂⋆α v(0, )||), t > 0. Let us denote ∂∗α = (σ∂x )α1 ∂yα2 for α = (α1 , α2 ). Given m ≥ 1, we deﬁne the anisotropic Sobolev spaces H∗m (Ω) = {u ∈ L2 (Ω)|∂∗α ∂xk u ∈ L2 (Ω) for |α| + 2k ≤ m}, m H∗∗ (Ω) = {u ∈ L2 (Ω)|∂∗α ∂xk u ∈ L2 (Ω) for |α| + 2k ≤ m + 1, |α| ≤ m}. 0 1 = L2 . Notice that (Ω) = H 1 (Ω). When m = 0 we set H∗0 = H∗∗ Observe that H∗∗ m in H∗ (Ω) there is one normal derivative ∂x every two tangential derivatives ∂∗ . In m the space H∗∗ (Ω) every normal derivative admits one more tangential derivative m than in H∗ (Ω). m From (3.13) we see that H∗∗ (Ω) is a good space for u in the sense that we have 2 u(t, ) ∈ H∗∗ (Ω) with ∂t u(t, ) ∈ H∗1 (Ω). Notice that the normal regularity of u follows from the ﬁrst equation in (3.5), that we can write as ux = −(ut +σvx +vy ). More generally, recalling that P denotes ⊥ the orthogonal projection onto ker Aν (x, t) , the normal regularity of the noncharu acteristic component P follows by inverting in the equations the nonsingular v part of the boundary matrix Aν (x, t), in a neighborhood of the boundary. On the other hand, the good choice for v is the space H∗m (Ω). In fact, the second order tangential derivatives are already estimated in (3.13). It rests to estimate the ﬁrst order normal derivative vx . We diﬀerentiate the second equation in (3.5) w.r.t. x and obtain the transporttype equation ∂t vx + σ∂x vx + σ ′ vx = −(σ ′ ux + σ∂x ux + uxy ). (3.14) “volumeV” — 2009/8/3 — 0:35 — page 173 — #189 3. MODIFIED TOY MODEL 173 We notice that no boundary condition is needed for (3.14) because σ(0) = 0. We also observe that the right-hand side has already been estimated. By multiplying (3.14) by vx and integrating over Ω, plus an integration by parts, we get an estimate for vx in L2 (Ω). Thus we have obtained v(t, ) ∈ H∗2 (Ω) with ∂t v(t, ) ∈ H∗1 (Ω). More generally, applying the projection I − P to the diﬀerential equations in (3.5) gives a transport-type equation for the normal derivatives of the characteristic u component (I − P ) , with vanishing boundary matrix (no need of a boundary v condition) and right-hand side estimated at previous step. Then, an energy argument gives the a priori estimate. A similar strategy will be employed in Section 3.3. The above analysis gives an answer to the problem set at the beginning of Section 2 of determining a function space X characterized by the property of persistence of regularity, that is such that (u0 , v0 ) ∈ X ⇒ (u(t, ), v(t, )) ∈ X, ∀t > 0 . The function space X is characterized as follows: (i) If (u0 , v0 ) ∈ H∗m (Ω)×H∗m (Ω) are such that (∂tk u(0, ), ∂tk v(0, )) ∈ H∗m−k (Ω)× m−k H∗ (Ω), for k = 1, . . . , m, then (u(t, ), v(t, )) ∈ H∗m (Ω) × H∗m (Ω) with (∂tk u(t, ), ∂tk v(t, )) ∈ H∗m−k (Ω) × H∗m−k (Ω), for k = 1, . . . , m, ∀t > 0 . Alternatively one may require (using the equations one can prove that (i) and (ii) are equivalent): m−k m (Ω), ∂tk v(0, ) ∈ (Ω), v0 ∈ H∗m (Ω) are such that ∂tk u(0, ) ∈ H∗∗ (ii) If u0 ∈ H∗∗ m−k m m H∗ (Ω), for k = 1, . . . , m, then u(t, ) ∈ H∗∗ (Ω), v(t, ) ∈ H∗ (Ω) with ∂tk u(t, ) ∈ m−k (Ω), ∂tk v(t, ) ∈ H∗m−k (Ω), for k = 1, . . . , m, ∀t > 0 . H∗∗ “volumeV” — 2009/8/3 — 0:35 — page 174 — #190 “volumeV” — 2009/8/3 — 0:35 — page 175 — #191 CHAPTER 4 Regularity for characteristic symmetric IBVP’s 1. Problem of regularity and main result We consider an initial-boundary value problem for a linear Friedrichs symmetrizable system, with characteristic boundary of constant multiplicity. It is wellknown that for solutions of symmetric or symmetrizable hyperbolic systems with characteristic boundary the full regularity (i.e. solvability in the usual Sobolev spaces H m ) cannot be expected generally because of the possible loss of derivatives in the normal direction to the boundary, see [42, 71] and Chapter 3. The natural space is the anisotropic Sobolev space H∗m , which comes from the observation that the one order gain of normal diﬀerentiation should be compensated by two order loss of tangential diﬀerentiation (cf. [9]). The theory has been developed mostly for characteristic boundaries of constant multiplicity (see Deﬁnition 1.2 or the deﬁnition in assumption (B)) and maximally non-negative boundary conditions, see Deﬁnition 1.5 and [9, 21, 44, 51, 53, 54, 55, 62]. However, there are important characteristic problems of physical interest where boundary conditions are not maximally non-negative. Under the more general Kreiss–Lopatinskiı̆ condition (KL), see Appendix B, the theory has been developed for problems satisfying the uniform KL condition with uniformly characteristic boundaries (when the boundary matrix has constant rank in a neighborhood of the boundary), see [4, 30] and references therein. In this chapter we are interested in the problem of the regularity. We assume the existence of the strong L2 −solution, satisfying a suitable energy estimate, without assuming any structural assumption suﬃcient for existence, such as the fact that the boundary conditions are maximally dissipative or satisfy the Kreiss–Lopatinskiı̆ condition. We show that this is enough in order to get the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces H∗m , provided the data are suﬃciently smooth. Obviously, the present results contain in particular what has been previously obtained for maximally nonnegative boundary conditions. Let Ω be an open bounded subset of Rn (for a ﬁxed integer n ≥ 2), lying locally on one side of its smooth, connected boundary Γ := ∂Ω. For any real T > 0, we set QT := Ω×]0, T [ and ΣT := Γ×]0, T [; we also deﬁne Q∞ := Ω × [0, +∞[, Σ∞ := ∂Ω × [0, +∞[, Q := Ω × R and Σ := ∂Ω × R. We are interested in the following IBVP Lu = F, in QT (4.1) M u = G, on ΣT u|t=0 = f, in Ω, (4.2) (4.3) 175 “volumeV” — 2009/8/3 — 0:35 — page 176 — #192 176 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S where L is the ﬁrst order linear partial diﬀerential operator n X L = ∂t + Ai (x, t)∂i + B(x, t), (4.4) i=1 ∂ ∂ ∂t := ∂t , ∂i := ∂x , i = 1, . . . , n and Ai (x, t), B(x, t) are N × N real matrix-valued i functions of (x, t), for a given integer size N ≥ 1, deﬁned over Q∞ . The unknown u = u(x, t) and the data F = F (x, t), f = f (x) are real vector-valued functions with N components, deﬁned on QT and Ω respectively. In the boundary conditions (4.2), M is a smooth d × N matrix-valued function of (x, t), deﬁned on Σ∞ , with maximal constant rank d. The boundary datum G = G(x, t) is a d−vector valued function, deﬁned on ΣT . Let us denote by ν(x) := (ν1 (x), . . . , νn (x)) the unit outward normal to Γ at the point x ∈ Γ; then n X Ai (x, t)νi (x) , (x, t) ∈ Σ∞ , (4.5) Aν (x, t) = i=1 is the boundary matrix. Let P (x, t) be the orthogonal projection onto the orthogonal complement of ker Aν (x, t), denoted ker Aν (x, t)⊥ ; it is deﬁned by Z 1 P (x, t) = (λ − Aν (x, t))−1 dλ , (x, t) ∈ Σ∞ , (4.6) 2πi C(x,t) where C(x, t) is a closed rectiﬁable Jordan curve with positive orientation in the complex plane, enclosing all and only all non-zero eigenvalues of Aν (x, t). Denoting again by P an arbitrary smooth extension on Q∞ of the above projection, P u and (I − P )u are called respectively the noncharacteristic and the characteristic components of the vector ﬁeld u = u(x, t). Examples of projector P for problems of physical interest are given in Appendix A. See also (3.2), (3.3) in Section 1. We study the problem (4.1)-(4.3) under the following assumptions: (A) The operator L is Friedrichs symmetrizable, namely for all (x, t) ∈ Q∞ there exists a symmetric positive deﬁnite matrix S0 (x, t) such that the matrices S0 (x, t)Ai (x, t), for i = 1, · · · , n, are also real symmetric; this n P Ai (x, t)ξi is diagoimplies, in particular, that the symbol A(x, t, ξ) = i=1 nalizable with real eigenvalues, whenever (x, t, ξ) ∈ Q∞ × Rn . (B) The boundary is characteristic, with constant multiplicity, namely the boundary matrix Aν is singular on Σ∞ and has constant rank 0 < r := rank Aν (x, t) < N for all (x, t) ∈ Σ∞ ; this assumption, together with the symmetrizability of L and that Γ is connected, yields that the number of negative eigenvalues of Aν (the so-called incoming modes) remains constant on Σ∞ . (C) ker Aν (x, t) ⊆ ker M (x, t), for all (x, t) ∈ Σ∞ ; moreover d = rank M (x, t) must equal the number of negative eigenvalues of Aν (x, t). (D) The orthogonal projection P (x, t) onto ker Aν (x, t)⊥ , (x, t) ∈ Σ∞ , can be extended as a matrix-valued C ∞ function over Q∞ . Concerning the solvability of the IBVP (4.1)-(4.3), we state the following wellposedness assumption: “volumeV” — 2009/8/3 — 0:35 — page 177 — #193 1. PROBLEM OF REGULARITY AND MAIN RESULT 177 (E) Existence of the L2 −weak solution. Assume that S0 , Ai ∈ Lip(Q∞ ), for i = 1, . . . , n. For all T > 0 and all matrices B ∈ L∞ (QT ), there exist constants γ0 ≥ 1 and C0 > 0 such that for all F ∈ L2 (QT ), G ∈ L2 (ΣT ), f ∈ L2 (Ω) there exists a unique solution u ∈ L2 (QT ) of (4.1)-(4.3), with data (F, G, f ), satisfying the following properties: i. u ∈ C([0, T ]; L2 (Ω)); ii. P u| ΣT ∈ L2 (ΣT ); iii. for all γ ≥ γ0 and 0 < τ ≤ T the solution u enjoys the following a priory estimate Rτ e−2γτ ||u(τ )||2L2 (Ω) + γ 0 e−2γt ||u(t)||2L2 (Ω) dt + Rτ 0 e−2γt ||P u|∂Ω (t)||2L2 (∂Ω) dt (4.7) Rτ ≤ C0 ||f ||2L2 (Ω) + 0 e−2γt ( γ1 ||F (t)||2L2 (Ω) + ||G(t)||2L2 (∂Ω) ) dt . When the IBVP (4.1)-(4.3) admits an a priori estimate of type (4.7), with F = Lu, G = M u, for all τ > 0 and all suﬃciently smooth functions u, one says that the problem is strongly L2 well posed, see e.g. [4]. A necessary condition for (4.7) is the validity of the uniform Kreiss–Lopatinskiı̆ condition (UKL) (see Appendix B, an estimate of type (4.7) has been obtained by Rauch [45]). On the other hand, UKL is not suﬃcient for the well posedness and other structural assumptions have to be taken into account, see Appendix C and [4]. Finally, we require the following technical assumption that for C ∞ approximations of problem (4.1)-(4.3) one still has the existence of L2 −solutions. This stability property holds true for maximally nonnegative boundary conditions and for uniform KL conditions. (F) Given matrices (S0 , Ai , B) ∈ CT (H∗σ ) × CT (H∗σ ) × CT (H∗σ−2 ), where σ ≥ (k) (k) [(n + 1)/2] + 4, enjoying properties (A)-(E), let (S0 , Ai , B (k) ) be C ∞ matrix-valued functions converging to (S0 , Ai , B) in CT (H∗σ ) × CT (H∗σ ) × CT (H∗σ−2 ) as k → ∞, and satisfying properties (A)-(D). Then, for k suﬃciently large, property (E) holds also for the approximating problems (k) (k) with coeﬃcients (S0 , Ai , B (k) ). The solution of (4.1)-(4.3), considered in the statements (E), (F), must be intended in the sense of Rauch [46]. This means that for all v ∈ H 1 (QT ) such that v| ΣT ∈ (Aν (ker M ))⊥ and v(T, ·) = 0 in Ω, there holds: Z Z T Z T Z ∗ hAν g, vi dσx dt + hf, v(0)i dx, hu(t), L v(t)i dt = hF (t), v(t)i dt − 0 0 ΣT Ω where L∗ is the adjoint operator of L and g is a function deﬁned on ΣT such that M g = G. Notice also that for such a weak solution to (4.1)-(4.3), the boundary condition (4.2) makes sense. Indeed, in [46, Theorem 1] it is shown that for any u ∈ L2 (QT ), with Lu ∈ L2 (QT ), the trace of Aν u on ΣT exists in H −1/2 (ΣT ). Moreover, for a given boundary matrix M (x, t) satisfying assumption (C), there exists another matrix M0 (x, t) such that M (x, t) = M0 (x, t)Aν (x, t) for all (x, t) ∈ “volumeV” — 2009/8/3 — 0:35 — page 178 — #194 178 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S Σ∞ . Therefore, for L2 −solutions of (4.1) one has M u = G on ΣT ⇐⇒ M0 Aν u|ΣT = G on ΣT . (4.8) In order to study the regularity of solutions to the IBVP (4.1)-(4.3), the data F , G, f need to satisfy some compatibility conditions. The compatibility conditions are deﬁned in the usual way (see [48]). Given the IBVP (4.1)-(4.3), we recursively deﬁne f (h) by formally taking h − 1 time derivatives of Lu = F , solving for ∂th u and evaluating it at t = 0; for h = 0 we set f (0) := f . The compatibility condition of order k ≥ 0 for the IBVP reads as p X p (∂tp−h M )| t=0 f (h) = ∂th G| t=0 , on Γ , p = 0, . . . , k . (4.9) h h=0 In the framework of the preceding assumptions, we are able to prove the following theorem. Theorem 4.1. [37] Let m ∈ N and s = max{m, (n + 1)/2 + 5}. Assume that S0 , Ai ∈ CT (H∗s ), for i = 1, . . . , n, and B ∈ CT (H∗s−1 ) (or B ∈ CT (H∗s ) if m = s). Assume also that problem (4.1)-(4.3) obeys the assumptions (A)-(F). Then for all F ∈ H∗m (QT ), G ∈ H m (ΣT ), f ∈ H∗m (Ω), with f (h) ∈ H∗m−h (Ω) for h = 1, . . . , m, satisfying the compatibility condition (4.9) of order m − 1, the unique solution u to (4.1)–(4.3), with data (F, G, f ), belongs to CT (H∗m ) and P u| ΣT ∈ H m (ΣT ). Moreover u satisfies the a priori estimate ||u||CT (H∗m ) + ||P u| ΣT ||H m (ΣT ) ≤ Cm |||f |||m,∗ + ||F ||H∗m (QT ) + ||G||H m (ΣT ) , (4.10) with a constant Cm > 0 depending only on Ai , B. The function spaces involved in the statement above (cf. also the assumption (F)), and the norms appearing in the energy estimate (4.10) are introduced in the next section. 2. Function spaces For every integer m ≥ 1, H m (Ω), H m (QT ) denote the usual Sobolev spaces of order m, over Ω and QT respectively. In order to deﬁne the anisotropic Sobolev spaces, ﬁrst we need to introduce the diﬀerential operators in tangential direction. Throughout the paper, for every j = 1, 2, . . . , n the diﬀerential operator Zj is deﬁned by Z1 := x1 ∂1 , Zj := ∂j , for j = 2, . . . , n . Then, for every multi-index α = (α1 , . . . , αn ) ∈ Nn , the tangential diﬀerential operator Z α of order |α| = α1 + · · · + αn is deﬁned by setting Z α := Z1α1 . . . Znαn (we also write, with the standard multi-index notation, ∂ α = ∂1α1 . . . ∂nαn ). We denote by Rn+ the n−dimensional positive half-space Rn+ := {x = (x1 , x′ ) ∈ Rn : x1 > 0 x′ := (x2 , . . . , xn ) ∈ Rn−1 }. For every positive integer m, the tangential (or “volumeV” — 2009/8/3 — 0:35 — page 179 — #195 2. FUNCTION SPACES 179 m conormal) Sobolev space Htan (Rn+ ) and the anisotropic Sobolev space H∗m (Rn+ ) are deﬁned respectively by: m Htan (Rn+ ) := {w ∈ L2 (Rn+ ) : Z α w ∈ L2 (Rn+ ) , |α| ≤ m} , H∗m (Rn+ ) := {w ∈ L2 (Rn+ ) : Z α ∂1k w ∈ L2 (Rn+ ) , |α| + 2k ≤ m} , (4.11) (4.12) and equipped respectively with norms X ||Z α w||2L2 (Rn+ ) , ||w||2Htan m (Rn ) := + |α|≤m X ||Z α ∂1k w||2L2 (Rn+ ) . ||w||2H∗m (Rn+ ) := (4.13) (4.14) |α|+2k≤m To extend the deﬁnition of the above spaces to an open bounded subset Ω of Rn (fulﬁlling the assumptions made at the beginning of the previous section), we proceed as follows. First, we take an open covering {Uj }lj=0 of Ω such that Uj ∩ Ω, j = 1, . . . , l, are diﬀeomorphic to B+ := {x1 ≥ 0, |x| < 1}, with Γ corresponding to ∂B+ := {x1 = 0, |x| < 1}, and U0 ⊂⊂ Ω. Next we choose a smooth partition of unity {ψj }lj=0 subordinate to the covering {Uj }lj=0 . We say that a distribution m u belongs to Htan (Ω), if and only if ψ0 u ∈ H m (Rn ) and, for all j = 1, . . . , l, m n m ψj u ∈ Htan (R+ ), in local coordinates in Uj . The space Htan (Ω) is provided with the norm l X 2 (4.15) ||ψj u||2Htan ||u||2Htan m (Rn ) . m (Ω) := ||ψ0 u||H m (Rn ) + + j=1 The anisotropic Sobolev space H∗m (Ω) is deﬁned in a completely similar way as the set of distributions u in Ω such that ψ0 u ∈ H m (Rn ) and ψj u ∈ H∗m (Rn+ ), in local coordinates in Uj , for all j = 1, . . . , l; it is provided with the norm ||u||2H∗m (Ω) := ||ψ0 u||2H m (Rn ) + l X j=1 ||ψj u||2H∗m (Rn ) . + (4.16) m The deﬁnitions of Htan (Ω) and H∗m (Ω) do not depend on the choice of the coorl dinate patches {Uj }j=0 and the corresponding partition of unity {ψj }lj=0 , and the norms arising from diﬀerent choices of Uj , ψj are equivalent. For an extensive study of the anisotropic Sobolev spaces, we refer the reader to [37, 43, 44, 53, 58, 63]; here we just remark that the continuous imbeddings p m (Ω) , H∗m (Ω) ֒→ H∗p (Ω) , Htan (Ω) ֒→ Htan m m m H (Ω) ֒→ H∗ (Ω) ֒→ Htan (Ω) , ∀m ≥ 1 , m [m/2] 1 (Ω) (Ω) , H∗1 (Ω) = Htan H∗ (Ω) ֒→ H ∀m ≥ p ≥ 1 , (4.17) 0 hold true. For the sake of convenience, we also set H∗0 (Ω) = Htan (Ω) = L2 (Ω). The m m spaces Htan (Ω), H∗ (Ω), endowed with their norms (4.15), (4.16), become Hilbert m (QT ) and H∗m (QT ). spaces. Analogously, we deﬁne the spaces Htan m Let C ([0, T ]; X) denote the set of all m-times continuously diﬀerentiable functions over [0, T ], taking values in a Banach space X. We deﬁne the spaces m CT (Htan ) := m \ j=0 m−j C j ([0, T ]; Htan (Ω)) , CT (H∗m ) := m \ j=0 C j ([0, T ]; H∗m−j (Ω)) , “volumeV” — 2009/8/3 — 0:35 — page 180 — #196 180 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S equipped respectively with the norms ||u||2CT (H m tan ) := ||u||2CT (H m ) := ∗ For the initial datum f we set m P j=0 m P j=0 supt∈[0,T ] ||∂tj u(t)||2H m−j (Ω) , tan supt∈[0,T ] ||∂tj u(t)||2H m−j (Ω) . |||f |||2m,∗ := (4.18) ∗ m X j=0 ||f (j) ||2H m−j (Ω) . ∗ 3. The scheme of the proof of Theorem 4.1 The proof of Theorem 4.1 is made of several steps. In order to simplify the forthcoming analysis, hereafter we only consider the case when the operator L has smooth coeﬃcients. For the general case of coeﬃcients with the ﬁnite regularity prescribed in Theorem 4.1, we refer the reader to [37]; this case is treated by a reduction to the smooth coeﬃcients case, based upon the stability assumption (F). Thus, from now on, we assume that S0 , Ai , B are given functions in C ∞ (Q∞ ). Just for simplicity, we even assume that the coeﬃcients Ai of L are symmetric matrices (in this case the matrix S0 reduces to IN , the identity matrix of size N ); the case of a symmetrizable operator can be easily reduced to this one, just by the application of the symmetrizer S0 to system (4.1) (see [37] for details). Below, we introduce the new unknown uγ (x, t) := e−γt u(x, t) and the new data Fγ (x, t) := e−γt F (x, t), Gγ (x, t) = e−γt G(x, t). Then problem (4.1)-(4.3) becomes equivalent to (γ + L)uγ = Fγ in QT , M uγ = Gγ , on ΣT , (4.19) uγ | t=0 = f , in Ω . Let us now summarize the main steps of the proof of Theorem 4.1. 1. We ﬁrstly consider the homogeneous IBVP (γ + L)uγ = Fγ in QT , M uγ = Gγ on ΣT , uγ |t=0 = 0 in Ω . (4.20) We study (4.20), by reducing it to a stationary boundary value problem (see (4.26)), for which we deduce the tangential regularity. From the tangential regularity of this stationary problem, we deduce the tangential regularity of the homogeneous problem (4.20) (see the next Theorem 4.2). 2. We study the general problem (4.19). The anisotropic regularity, stated in Theorem 4.1, is obtained in two steps. 2.i Firstly, from the tangential regularity of problem (4.20) we deduce the anisotropic regularity of (4.19) at the order m = 1. 2.ii Eventually, we obtain the anisotropic regularity of (4.19), at any order m > 1, by an induction argument. “volumeV” — 2009/8/3 — 0:35 — page 181 — #197 3. THE SCHEME OF THE PROOF OF THEOREM 4.1 181 3.1. The homogeneous IBVP, tangential regularity. In this section, we concentrate on the study of the tangential regularity of solutions to the IBVP (4.19), where the initial datum f is identically zero, and the compatibility conditions are fulﬁlled in a more restrictive form than the one given in (4.9). More precisely, we consider the homogeneous IBVP (4.20) where, for a given integer m ≥ 1, we assume that the data Fγ , Gγ satisfy the following conditions: ∂th Fγ | t=0 = 0 , ∂th Gγ | t=0 = 0 , h = 0, . . . , m − 1 . (4.21) One can prove that conditions (4.21) imply the compatibility conditions (4.9) of order m − 1, in the case f = 0. Theorem 4.2. Assume that Ai , B, for i = 1, . . . , n, are in C ∞ (Q∞ ), and that problem (4.20) satisfies assumptions (A)-(E); then for all T > 0 and m ∈ N there exist constants Cm > 0 and γm , with γm ≥ γm−1 , such that for all γ ≥ γm , for all m Fγ ∈ Htan (QT ) and all Gγ ∈ H m (ΣT ) satisfying (4.21) the unique solution uγ to m (4.20) belongs to Htan (QT ), the trace of P uγ on ΣT belongs to H m (ΣT ) and the a priori estimate 1 2 2 2 γ||uγ ||2Htan ||F || + ||G || (4.22) m (Q ) +||P uγ| Σ ||H m (Σ ) ≤ Cm m m γ Htan (QT ) γ H (ΣT ) T T T γ is fulfilled. The ﬁrst step to prove Theorem 4.2 is reducing the original mixed evolution problem (4.20) to a stationary boundary value problem, where the time is allowed to span the whole real line and it is treated then as an additional tangential variable. To make this reduction, we extend the data Fγ , Gγ and the unknown uγ of (4.20) to all positive and negative times, by following methods similar to those of [4, Ch.9]. In the sequel, for the sake of simplicity, we remove the subscript γ from the unknown uγ and the data Fγ , Gγ . Because of (4.21), we extend F and G through ] − ∞, 0], by setting them equal to zero for all negative times; then for times t > T , we extend them by “reﬂection”, following Lions–Magenes [27, Theorem 2.2]. Let us denote by F̆ and Ğ the resulting m extensions of F and G respectively; by construction, F̆ ∈ Htan (Q) and Ğ ∈ H m (Σ). As we did for the data, the solution u to (4.20) is extended to all negative times, by setting it equal to zero. To extend u also for times t > T , we exploit the assumption (E). More precisely, for every T ′ > T we consider the mixed problem (γ + L)u = F̆| ]0,T ′ [ in QT ′ , M u = Ğ| ]0,T ′ [ , on ΣT ′ , u| t=0 = 0 , in Ω . (4.23) Assumption (E) yields that (4.23) admits a unique solution uT ′ ∈ C([0, T ′ ]; L2 (Ω)), such that P uT ′ ∈ L2 (ΣT ′ ) and the energy estimate ||uT ′ (T ′ )||2L2 (Ω) + γ||uT ′ ||2L2 (Q ′ ) + ||P uT ′ | ΣT ′ ||2L2 (Σ ′ ) T T ≤ C ′ γ1 ||F̆| ]0,T ′ [ ||2L2 (Q ′ ) ) + ||Ğ| ]0,T ′ [ ||2L2 (Σ ′ ) T (4.24) T is satisﬁed for all γ ≥ γ ′ and some constants γ ′ ≥ 1 and C ′ > 0 depending only on T ′ (and the norms ||Ai ||Lip(QT ′ ) , ||B||L∞ (QT ′ ) ). From the uniqueness of the L2 −solution, we infer that for arbitrary T ′′ > T ′ ≥ T “volumeV” — 2009/8/3 — 0:35 — page 182 — #198 182 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S we have uT ′′ = uT ′ (uT := u) over ]0, T ′[. Therefore, we may extend u beyond T , by setting it equal to the unique solution of (4.23) over ]0, T ′ [ for all T ′ > T . Thus we deﬁne ( uT ′ (t) , ∀ t ∈]0, T ′ [ , ∀ T ′ > T , ŭ(t) := (4.25) 0, ∀t < 0. Since ŭ, F̆ , Ğ are all identically zero for negative times, we can take arbitrary smooth extensions of the coeﬃcients of the diﬀerential operator L and the boundary operator M (originally deﬁned on Q∞ and Σ∞ ) on Q and Σ respectively, with the only care to preserve rankAν = r and rankM = d and kerAν ⊂ kerM for all t < 0. These extensions, that we ﬁx once and for all, are denoted again by Ai , B, M . Moreover, we denote by L the corresponding extension on Q of the diﬀerential operator (4.4). By construction, we have that ŭ solves the boundary value problem (BVP) (γ + L)u = F̆ in Q , M u = Ğ , on Σ . (4.26) Using the estimate (4.24), for all T ′ > T , and noticing that the extended data F̆ , Ğ, as well as the solution ŭ, vanish identically for large t > 0, we derive that ŭ enjoys the following estimate 1 2 2 2 2 γ||ŭ||L2 (Q) + ||P ŭ| Σ ||L2 (Σ) ≤ C̆ (4.27) ||F̆ ||L2 (Q) + ||Ğ||L2 (Σ) , γ for all γ ≥ γ̆, and suitable constants γ̆ ≥ 1, C̆ > 0. For the sake of simplicity, in the sequel we remove the superscript from the unknown ŭ and the data F̆ , Ğ of (4.26). The next step is to move from the BVP (4.26) to a similar BVP posed in the (n + 1)−dimensional positive half-space Rn+1 := {(x1 , x′ , t) : x1 > 0, (x′ , t) ∈ Rn }. + n+1 To make this reduction into a problem in R+ , we follow a standard localization procedure of the problem (4.26) near the boundary of the spatial domain Ω; this is done by taking a covering {Uj }lj=0 of Ω and a partition of unity {ψj }lj=0 subordinate to this covering, as in Section 2. Assuming that each patch Uj , j = 1, . . . , l, is suﬃciently small, we can write the resulting localized problem in the form (γ + L)u = F in Rn+1 , + M u = G , on Rn . (4.28) As a consequence of the localization, the data F and G of the problem (4.28) are m m n functions in Htan (Rn+1 + ) and H (R ) respectively; without loss of generality, we may also assume that the forcing term F and the solution u are supported in the set B+ × [0, +∞[, and the boundary datum G is supported in ∂B+ × [0, +∞[. In (4.28)1 , L is now a diﬀerential operator in Rn+1 of the form L = ∂t + n X Ai (x, t)∂i + B(x, t) , (4.29) i=1 where the coeﬃcients Ai , B are matrix-valued functions of (x, t) belonging to n+1 ∞ C(0) (Rn+1 of (matrix-valued) func+ ), namely the space of the restrictions onto R+ tions in C0∞ (Rn+1 ). Let us remark that the boundary matrix of (4.28) is now “volumeV” — 2009/8/3 — 0:35 — page 183 — #199 3. THE SCHEME OF THE PROOF OF THEOREM 4.1 183 −A1 | {x1 =0} , because the outward unit vector to the boundary is ν = (−1, 0, . . . , 0). It is a crucial step that the previously described localization process can be performed in such a way that A1 has the following block structure I,I A1 AI,II 1 , (x, t) ∈ Rn+1 , (4.30) A1 (x, t) = II,II + A AII,I 1 1 I,II II,I II,II where AI,I are respectively r × r, r × (N − r), (N − r) × r, 1 , A1 , A1 , A1 (N − r) × (N − r) sub-matrices. Moreover, AI,I 1 (x, t) is invertible over the support of u(x, t) and we have AI,II = 0, 1 AII,I = 0, 1 AII,II = 0, 1 in {x1 = 0} × Rnx′ ,t . (4.31) In view of assumption (C), we may even assume that the matrix M in the boundary condition (4.28)2 is just M = (Id , 0), where Id is the identity matrix of size d. According to (4.30), let us decompose the unknown u as u = (uI , uII ); then we have P u = (uI , 0). Following the arguments of [8], one can prove that a local counterpart of the global estimate (4.27), associated to the stationary problem (4.26), can be attached to the local problem (4.28). More precisely, there exist constants C0 > 0 and γ0 ≥ 1 such n+1 2 that for all ϕ ∈ L2 (Rn+1 + ), supported in B+ × [0, +∞[, such that Lϕ ∈ L (R+ ) and γ ≥ γ0 there holds γ||ϕ||2L2 (Rn+1 ) + ||ϕI| {x1 =0} ||2L2 (Rn ) + ≤ C0 ( γ1 ||(γ + L)ϕ||2L2 (Rn+1 ) + ||M ϕ| {x1 =0} ||2L2 (Rn ) ) . (4.32) + 3.1.1. Regularity of the stationary problem (4.28). The analysis performed in the previous section shows that the tangential regularity of the homogeneous IBVP (4.20) can be deduced from the study of the regularity of the stationary BVP (4.28). For this stationary problem, we are able to show that if the data F and G belong m m n 2 to Htan (Rn+1 + ) and H (R ) respectively, and the L a priori estimate (4.32) is m fulﬁlled, then the L2 −solution of the problem (4.28) belongs to Htan (Rn+1 + ), the I m n trace of its noncharacteristic part u belongs to H (R ) and the estimate of order m γ||u||2H m (Rn+1 )+||uI| {x1 =0} ||2H m (Rn ) ≤ Cm γ1 ||F ||2H m (Rn+1 ) +||G||2H m (Rn ) (4.33) tan + tan + is satisﬁed with some constants Cm > 0, γm ≥ 1 and for all γ ≥ γm . Then we recover the tangential regularity of the solution u to problem (4.26), posed on Q = Ω × R, and we ﬁnd an associated estimate of order m analogous to (4.33). Recalling that the solution u to (4.26) is the extension of the solution uγ of the homogeneous IBVP (4.20), from the tangential regularity of u we can now derive the m tangential regularity of uγ , namely that uγ ∈ Htan (QT ) and P uγ | ΣT ∈ H m (ΣT ). To get the energy estimate (4.22), we observe that the extended data F̆ and Ğ are deﬁned in such a way that m (Q) ≤ C||Fγ ||H m (Q ) , ||F̆ ||Htan T tan ||Ğ||H m (Σ) ≤ C||Gγ ||H m (ΣT ) , with positive constant C independent of Fγ , Gγ and γ. “volumeV” — 2009/8/3 — 0:35 — page 184 — #200 184 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S In order to prove the announced tangential regularity of the BVP (4.28), we adapt the classical technique of Friedrichs’ molliﬁers to our setting. More precisely, following Nishitani and Takayama [40], we introduce a “tangential” molliﬁer Jε well suited to the tangential Sobolev spaces. Let χ be a function in C0∞ (Rn+1 ). For all n+1 2 0 < ε < 1, we set χε (y) := ε−(n+1) χ(y/ε). We deﬁne Jε : L2 (Rn+1 + ) → L (R+ ) by Z (4.34) Jε w(x) := w(x1 e−y1 , x′ − y ′ )e−y1 /2 χε (y)dy , Rn+1 which diﬀers from the one introduced in Rauch [46] by the factor e−y1 /2 . Using Jε we follow the same lines in Tartakoﬀ [67], Nishitani and Takayama [40] to get regularity of the weak solution u. Starting from a classical characterization of the ordinary Sobolev spaces given in [23, Theorem 2.4.1], the following characterization of tangential Sobolev spaces m Htan (Rn+1 + ) by means of Jε can be proved. Proposition 4.3. Assume that χ ∈ C0∞ (Rn+1 + ) satisfies the following conditions: χ b(ξ) = O(|ξ|p ) as ξ → 0, f or some p ∈ N; χ b(tξ) = 0 , f or all t ∈ R , implies ξ = 0. (4.35) (4.36) m Then for all m ∈ N with m < p, we have that u ∈ Htan (Rn+1 + ) if and only if m−1 a. u ∈ Htan (Rn+1 + ); R1 2 b. 0 ||Jε u||L2 (Rn+1 ) ε−2m 1 + + δ2 ε2 −1 dε ε is uniformly bounded for 0 < δ ≤ 1. m−1 In view of Proposition 4.3, showing that the solution u ∈ Htan (Rn+1 + ) of (4.28) m actually belongs to Htan (Rn+1 + ) amounts to provide a uniform bound, with respect to δ, for the integral quantity appearing in [b.], computed for the mollified solution Jε u. To get this bound, the scheme is the following: 1. We notice that Jε u solves the following BVP (γ + L)Jε u = Jε F + [L, Jε ]u , in Rn+1 , + M Jε u = Gε , on Rn , (4.37) where [L, Jε ] is the commutator between the operators L and Jε , and Gε is a suitable boundary datum that can be computed from the original datum G and the function χǫ involved in (4.34) (see [37]). 2. Since the BVP (4.37) is the same as (4.28), with data Jε F + [L, Jε ]u and Gε , the L2 estimate (4.32) applied to (4.37) gives that the L2 −norm of Jε u can be estimated by γ||Jε u||2L2 (Rn+1 ) + ||Jε uI| {x1 =0} ||2L2 (Rn ) + ≤ C0 γ1 ||Jε F + [L, Jε ]u||2L2 (Rn+1 ) + ||Gε ||2L2 (Rn ) . (4.38) + 3. From the preceding estimate, we immediately derive, for the integral quantity in [b.] and the analogous integral quantity associated to the trace of “volumeV” — 2009/8/3 — 0:35 — page 185 — #201 3. THE SCHEME OF THE PROOF OF THEOREM 4.1 185 noncharacteristic part of the solution, the following bound −1 R1 2 dε γ 0 ||Jε u||2L2 (Rn+1 ) ε−2m 1 + εδ2 ε + −1 R1 2 dε + 0 ||Jε uI|{x1 =0} ||2L2 (Rn ) ε−2m 1 + εδ2 ε −1 R 1 1 dε δ2 2 −2m ≤ C0 ( γ 0 ||Jε F ||L2 (Rn+1 ) ε 1 + ε2 ε + −1 R 1 1 dε δ2 2 −2m + γ 0 ||[L, Jε ]u||L2 (Rn+1 ) ε 1 + ε2 ε + −1 R1 δ2 dε 2 −2m + 0 ||Gε ||L2 (Rn ) ε 1 + ε2 ε ). (4.39) m m n Since F ∈ Htan (Rn+1 + ) and G ∈ H (R ), the ﬁrst and the last integrals in the right-hand side of (4.39) can be estimated by ||F ||2H m (Rn+1 ) and tan + ||G||2H m (Rn ) respectively. It remains to provide a uniform estimate for the middle integral involving the commutator [L, Jε ]u. For this term we get the following estimate −1 δ2 dε 2 −2m 1 + ||[L, J ]u|| ε n+1 ε 2 2 ε ε 0 L (R+ ) −1 R1 dε δ2 −2m 2 1 + ε2 ≤ C 0 ||Jε u||L2 (Rn+1 ) ε ε R1 (4.40) + +Cγ 2 ||u||2H m−1 (Rn+1 ) + C||F ||2H m tan n+1 ) tan (R+ + . The estimate (4.40) is obtained by treating separately the diﬀerent contributions to the commutator [L, Jε ] associated to the diﬀerent terms in the expression (4.29) of L (see [37] for details). The terms of the commutator involving the tangential derivatives [Ai ∂i , Jε ], for i = 2, . . . , n (note that [∂t , Jε ] = 0), and the zero-th order term [B, Jε ] are estimated by using [40, Lemma 9.2]. The term [A1 ∂1 , Jε ], involving the normal derivative ∂1 , needs a more careful analysis; to estimate it, it is essential to make use of the structure (4.30), (4.31) of the boundary matrix in (4.28). Actually, by inverting AI,I in (4.28)1 , we can write ∂1 uI as the sum of space-time 1 tangential derivatives by ∂1 uI = ΛZu + R , where n X II −1 , Aj Zj u)I + AI,II ΛZu = −(AI,I (∂t uI + 1 ∂1 u 1 ) j=2 −1 (F − γu − Bu)I . R = (AI,I 1 ) Here, we use the fact that, if a matrix A vanishes on {x1 = 0}, we can write A∂1 u = HZ1 u, where H is a suitable matrix; this trick transforms some normal derivatives into tangential derivatives. Combining the inequalities (4.39) and (4.40), and arguing by ﬁnite induction on m to estimate ||u||H m−1 (Rn+1 ) in the right-hand side of (4.40), tan + “volumeV” — 2009/8/3 — 0:35 — page 186 — #202 186 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S we get the desired uniform bounds of the integrals −1 R1 dε δ2 2 −2m 1 + ||J u|| ε n+1 ε ε2 ε , 0 L2 (R+ ) −1 R1 2 dε ||Jε uI|{x1 =0} ||2L2 (Rn ) ε−2m 1 + δε2 ε , 0 appearing in the left-hand side of (4.39). From this, in view of Proposition m I 4.3 and [23, Theorem 2.4.1], we conclude that u ∈ Htan (Rn+1 + ) and u ∈ m n H (R ). The a priori estimate (4.33) is deduced from (4.39), by following the same arguments. 3.2. The nonhomogeneous IBVP, case m = 1. For nonhomogeneous IBVP, we mean the problem (4.1)-(4.3) where the initial datum f is diﬀerent from zero. As announced before, we ﬁrstly prove the statement of Theorem 4.1 for m = 1, namely we prove that, under the assumptions (A)-(F), for all F ∈ H∗1 (QT ), G ∈ H 1 (ΣT ) and f ∈ H∗1 (Ω), with f (1) ∈ L2 (Ω), satisfying the compatibility condition M|t=0 f|∂Ω = G|t=0 , the unique solution u to (4.1)–(4.3), with data (F, G, f ), belongs to CT (H∗1 ) and P u| ΣT ∈ H 1 (ΣT ); moreover, there exists a constant C1 > 0 such that u satisﬁes the a priori estimate ||u||CT (H∗1 ) +||P u| ΣT ||H 1 (ΣT ) ≤ C1 |||f |||1,∗ +||F ||H∗1 (QT ) +||G||H 1 (ΣT ) . (4.41) To this end, we approximate the data with regularized functions satisfying one more compatibility condition. In this regard we get the following result, for the proof of which we refer to [37] and the references therein. Lemma 4.4. Assume that problem (4.1)-(4.3) obeys the assumptions (A)-(E). Let F ∈ H∗1 (QT ), G ∈ H 1 (ΣT ), f ∈ H∗1 (Ω), with f (1) ∈ L2 (Ω), such that M|t=0 f|∂Ω = G|t=0 . Then there exist Fk ∈ H 3 (QT ), Gk ∈ H 3 (ΣT ), fk ∈ H 3 (Ω), (1) such that M|t=0 fk = Gk|t=0 , ∂t M|t=0 fk + M|t=0 fk = ∂t Gk|t=0 on ∂Ω, and such (1) that Fk → F in H∗1 (QT ), Gk → G in H 1 (ΣT ), fk → f in H∗1 (Ω), fk → f (1) in L2 (Ω), as k → +∞. Given the functions Fk , Gk , fk as in Lemma 4.4, we ﬁrst calculate through (1) (2) equation Lu = Fk , u|t=0 = fk , the initial time derivatives fk ∈ H 2 (Ω), fk ∈ 1 3 H (Ω). Then we take a function wk ∈ H (QT ) such that wk|t=0 = fk , (1) ∂t wk|t=0 = fk , (2) 2 ∂tt wk|t=0 = fk . Notice that this yields (Lwk )|t=0 = Fk|t=0 , ∂t (Lwk )|t=0 = ∂t Fk|t=0 . (4.42) Now we look for a solution uk of problem (4.1)-(4.3), with data Fk , Gk , fk , of the form uk = vk + wk , where vk is solution to Lvk = Fk − Lwk , M vk = Gk − M wk , vk|t=0 = 0, in QT on ΣT in Ω. (4.43) “volumeV” — 2009/8/3 — 0:35 — page 187 — #203 3. THE SCHEME OF THE PROOF OF THEOREM 4.1 187 Let us denote again ukγ = e−γt uk , vkγ = e−γt vk and so on. Then (4.43) is equivalent to (γ + L)vkγ = Fkγ − (γ + L)wkγ , in QT (4.44) M vkγ = Gkγ − M wkγ , on ΣT vkγ|t=0 = 0, in Ω. We easily verify that (4.42) yields (Fkγ − (γ + L)wkγ )|t=0 = 0 , ∂t (Fkγ − (γ + L)wkγ )|t=0 = 0 (1) and M|t=0 fk|∂Ω = Gk|t=0 , ∂t M|t=0 fk|∂Ω + M|t=0 fk|∂Ω = ∂t Gk|t=0 yield (Gkγ − M wkγ )|t=0 = 0, ∂t (Gkγ − M wkγ )|t=0 = 0. Thus the data of problem (4.44) obey conditions (4.21) for h = 0, 1; then we may 2 apply to (4.44) Theorem 4.2 for γ large enough and ﬁnd vk ∈ Htan (QT ), with 2 2 P vk|ΣT ∈ H (ΣT ). Accordingly, we infer that uk ∈ Htan (QT ) ֒→ CT (H∗1 ) and P uk|ΣT ∈ H 2 (ΣT ). Moreover uk ∈ L2 (QT ) solves Luk = Fk , M uk = Gk , uk|t=0 = fk , in QT on ΣT in Ω. (4.45) Arguing as in the previous section, we take a covering {Uj }lj=0 of Ω and a related partition of unity {ψj }lj=0 , and we reduce problem (4.45) into a corresponding problem posed in the positive half-space Rn+ , with new data Fk ∈ H 3 (Rn+ ×]0, T [), Gk ∈ H 3 (Rn−1 ×]0, T [), fk ∈ H 3 (Rn+ ), and boundary matrix M = (Id , 0). We also write the similar problem solved by the ﬁrst order derivatives Zuk = 1 (Z1 uk , . . . , Zn+1 uk ) ∈ Htan (QT ) = H∗1 (QT ) (where Zn+1 = ∂t ). Here a crucial remark regards the commutators of L and the tangential operators Zi , see [37, 46]: there exist matrices Γβ , Γ0 , Ψ such that P i = 1, . . . , n + 1. [L, Zi ] = − |β|=1 Γβ Z β + Γ0 + ΨL, (4.46) Therefore the commutators contain only tangential derivatives, and no normal derivative. Since assumption (E) is satisﬁed, applying the a priori estimate (4.7) to a difference of solutions uh − uk of those problems satisﬁed by the ﬁrst order derivatives readily gives ||uk − uh ||CT (H∗1 ) + ||P (uk − uh )|ΣT ||H 1 (ΣT ) ≤ C |||fk − fh |||1,∗ + ||Fk − Fh ||H∗1 (QT ) + ||Gk − Gh ||H 1 (ΣT ) . From Lemma 4.4, we infer that {uk } is a Cauchy sequence in CT (H∗1 ) and {P uk| ΣT } is a Cauchy sequence in H 1 (ΣT ). Therefore there exists a function in CT (H∗1 ) which is the limit of {uk }. Passing to the limit in (4.45) as k → ∞, we see that this function is a solution to (4.1)-(4.3). The uniqueness of the L2 −solution yields u ∈ CT (H∗1 ) and P u| ΣT ∈ H 1 (ΣT ). Applying the a priori estimate (4.7) to the solution uk of (4.45) and its ﬁrst order derivatives, and passing to the limit ﬁnally gives (4.41). This completes the proof of Theorem 4.1 for m = 1 in the case of C ∞ coeﬃcients. As we already say, here we do not deal with the case of less regular coeﬃcients, for which the reader is referred to [37, Sect. 5]. “volumeV” — 2009/8/3 — 0:35 — page 188 — #204 188 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S 3.3. The nonhomogeneous IBVP, proof for m ≥ 2. The proof proceeds by ﬁnite induction on m. Assume that Theorem 4.1 holds up to m − 1. Let f ∈ H∗m (Ω), F ∈ H∗m (QT ), G ∈ H m (ΣT ), with f (k) ∈ H∗m−k (Ω), k = 1, · · · , m, and assume also that the compatibility conditions (4.9) hold up to order m − 1. By the inductive hypothesis there exists a unique solution u of the problem (4.1)-(4.3) such that u ∈ CT (H∗m−1 ). In order to show that u ∈ CT (H∗m ), we have to increase the regularity of u by order one, that is by one more tangential derivative and, if m is even, also by one more normal derivative. This can be done as in [51, 53], with the small change of the elimination of the auxiliary system (introduced in [51, 53]) as in [7, 55]. At every step we can estimate some derivatives of u through equations where in the right-hand side we can put other derivatives of u that have already been estimated at previous steps. The reason why the main idea in [51] works, even though here we do not have maximally nonnegative boundary conditions, is that for the increase of regularity we consider the system (4.50) of equations for purely tangential derivatives of the type of (4.1)-(4.3), where we can use the inductive assumption, and other systems (4.52), (4.53) of equations for mixed tangential and normal derivatives where the boundary matrix vanishes identically, so that no boundary condition is needed and we can apply an energy method, under the assumption of the symmetrizable system. Without entering in too many details we brieﬂy describe the diﬀerent steps of the proof, for the reader’s convenience. It can be useful to compare this strategy with Section 3. As before, we take a covering {Uj }lj=0 of Ω and a partition of unity {ψj }lj=0 subordinate to this covering. Assuming that each patch Uj , j = 1, . . . , l, is suﬃciently small we can write the resulting localized problem in the form Lu = F, M u = G, u|t=0 = f, in Rn+ ×]0, T [, on {x1 = 0} × Rxn−1 ×]0, T [, ′ in Rn+ . (4.47) with L as in (4.29), and M = (Id , 0). The boundary matrix −Aj1 has the block form as in (4.30), (4.31). According to (4.30), let us decompose the unknown u as u = (uI , uII ); then we have P u = (uI , 0). Hereafter we will denote Z = (Z1 , . . . , Zn+1 ), Zn+1 = ∂t . 3.4. Purely tangential regularity. Let us start byconsidering all the tanI ∂ u 1 gential derivatives Z α u, |α| = m−1. We decompose ∂1 u = . By inverting ∂1 uII I AI,I 1 in (4.47)1 , we can write ∂1 u as the sum of tangential derivatives by ∂1 uI = ΛZu + R where (4.48) n X II −1 , Aj Zj u)I + AI,II (An+1 Zn+1 u + ΛZu = (AI,I 1 ∂1 u 1 ) j=2 −1 R = (AI,I (Bu − F )I . 1 ) Here and below, everywhere it is needed, we use the fact that, if a matrix A vanishes on {x1 = 0}, we can write A∂1 u = HZ1 u, where H is a suitable matrix “volumeV” — 2009/8/3 — 0:35 — page 189 — #205 3. THE SCHEME OF THE PROOF OF THEOREM 4.1 189 such that ||H||H∗s−2 (Ω) ≤ c||A||H∗s (Ω) , see [37, App. B]; this trick transforms some normal derivatives into tangential derivatives. We obtain Λ ∈ CT (H∗s−2 ). Applying the operator Z α to (4.47), with α = (α′ , αn+1 ), α′ = (α1 , · · · , αn ), and substituting (4.48) gives equation (5.3) in [51], that is α L(Z u) + X (ZAn+1 Zn+1 + + |γ|=|α|−1 0 γ ZAj Zj )Z u + j=2 αn+1 ΛZ(Z1α1 −1 Z2α2 · · · Zn+1 u) |γ|=|α|−1 −α1 A1 X n X α ZA1 |γ|=|α|−1 n+1 ZA1 Z − α1 A1 Z1α1 −1 Z2α2 · · · Zn+1 γ X 0 ∂1 uII ΛZ(Z γ u) 0 = Fα , (4.49) with Fα ∈ H∗1 (QT ), see [51] for its explicit expression. Equation (4.49) takes the form (L + B)Z α u = Fα with B ∈ CT (H∗s−3 ). Then we consider the problem satisﬁed by the vector of all tangential derivatives Z α u of order |α| = m − 1. From (4.49) this problem takes the form (L + B)Z α u = F MZ α u = Z α G Z α u|t=0 = f˜ where L= L .. in Rn+ ×]0, T [, on {x1 = 0} × Rxn−1 ×]0, T [, ′ in Rn+ , . L , M= M .. (4.50) . M , B ∈ CT (H∗s−3 ) is a suitable linear operator and F is the vector of all right-hand ′ sides Fα . The initial datum f˜ is the vector of functions Z α f (αn+1 ) . We have F ∈ H∗1 (QT ), f˜ ∈ H∗1 , Z α G ∈ H 1 (ΣT ). Moreover the data satisfy the compatibility conditions of order 0. We infer that the solution of (4.50) satisﬁes Z α u ∈ CT (H∗1 ), for all |α| = m − 1. 3.5. Tangential and one normal derivatives. We apply to the part II of (4.47)1 the operator Z β ∂1 , with |β| = m − 2. We obtain equation (28) in [7], that is n X X ZAj ∂j )Z γ (ZA0 ∂t + (L + ∂1 A1 )Z β + (4.51) j=1 |γ|=|β|−1 βn+1 II,II β1 −1 β2 II ∂1 u = G, −β1 A1 ∂1 Z1 Z2 · · · Zn+1 where the exact expression of G may be found in [7]. It is important to observe that G contains only tangential derivatives of order at most m. Hence, we can estimate it by using the previous step and infer G ∈ L2 (QT ). Using (4.48) again, we write (4.51) as ˜ β ∂1 uII = G, (L̃ + C)Z (4.52) “volumeV” — 2009/8/3 — 0:35 — page 190 — #206 190 4. REGULARITY FOR CHARACTERISTIC SYMMETRIC IBVP’S where L̃ = L̃ .. . L̃ + and where C˜ ∈ CT (H∗s−2 ) is a suitable linear with L̃ = operator. Here a crucial point is that (4.52) is a transport-type equation, because the boundary matrix of L̃ vanishes at {x1 = 0}. Thus we do not need any boundary condition. We infer that equation (4.52) has a unique solution Z β ∂1 uII ∈ CT (L2 ) := C([0, T ]; L2 (Rn+ )), for all |β| = m − 2. Using (4.48) again, we deduce Z β ∂1 u ∈ CT (L2 ), for all |β| = m − 2. ∂t AII,II 0 Pn II,II ∂j j=1 Aj 3.6. Normal derivatives. The last step is again by induction, as in [51], page 867, (ii). For convenience of the reader, we provide a brief sketch of the proof. Suppose that for some ﬁxed k, with 1 ≤ k < [m/2], it has already been shown that Z α ∂1h u belongs to CT (L2 ), for any h and α such that h = 1, · · · , k, |α|+2h ≤ m. From (4.48) it immediately follows that Z α ∂1k+1 uI ∈ CT (L2 ). It rests to prove that Z α ∂1k+1 uII ∈ CT (L2 ). We apply operator Z α ∂1k+1 , |α| + 2k = m − 2, to the part II of (4.47)1 and obtain an equation similar to (4.52) of the form (4.53) (L̃ + C˜k )Z α ∂ k+1 uII = Gk , 1 where C˜k ∈ CT (H∗s−3 ) is a suitable linear operator. The right-hand side Gk contains derivatives of u of order m (in H∗m , i.e. counting 1 for each tangential derivative and 2 for normal derivatives), but contains only normal derivatives that have already been estimated. We infer Gk ∈ L2 (QT ). Again it is crucial that the boundary matrix of L̃ vanishes at {x1 = 0}. We infer that the solution Z α ∂1k+1 uII is in CT (L2 ) for all α, k with |α| + 2k = m − 2. By repeating this procedure we obtain the result for any k ≤ [m/2], hence u ∈ CT (H∗m ). 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Systems of conservation laws. II: Geometric structures, oscillations and mixed problems. (Systèmes de lois de conservation. II: Structures géométriques, oscillations et problèmes mixtes.). Fondations. Paris: Diderot Editeur. iii, 306 p. FF 195.00 , 1996. [62] Y. Shizuta. On the ﬁnal form of the regularity theorem for solutions to the characteristic initial boundary value problem for symmetric hyperbolic systems. Proc. Japan Acad. Ser. A Math. Sci., 76(4):47–50, 2000. [63] Y. Shizuta, K. Yabuta. The trace theorems in anisotropic Sobolev spaces and their applications to the characteristic initial boundary value problem for symmetric hyperbolic systems. Math. Models Methods Appl. Sci., 5(8):1079– 1092, 1995. [64] T. C. Sideris. Formation of singularities in solutions to nonlinear hyperbolic equations. Arch. Ration. Mech. Anal., 86:369–381, 1984. [65] S.I. Syrovatskii. Magnetohydrodynamics. Uspekhi Fizicheskikh Nauk, 62:247– 303, 1957. [66] B. Szilágyi, J. Winicour. Well-posed initial-boundary evolution in general relativity. Phys. Rev. D, 68(4):041501, 2003. [67] D. Tartakoﬀ. Regularity of solutions to boundary value problems for ﬁrst order systems. Indiana Univ. Math. J., 21:1113–1129, 1972. [68] Y. Trakhinin. Existence of compressible current-vortex sheets: Variable coefﬁcients linear analysis. Arch. Ration. Mech. Anal., 177(3):331–366, 2005. [69] Y. Trakhinin. On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem. Math. Methods Appl. Sci., 28(8):917–945, 2005. [70] Y. Trakhinin. The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics. Arch. Ration. Mech. Anal., 191(2):245–310, 2009. [71] M. Tsuji. Regularity of solutions of hyperbolic mixed problems with characteristic boundary. Proc. Japan Acad., 48:719–724, 1972. [72] T. Yanagisawa. The initial boundary value problem for the equations of ideal magneto- hydrodynamics. Hokkaido Math. J., 16:295–314, 1987. [73] T. Yanagisawa, A. Matsumura. The ﬁxed boundary value problems for the equations of ideal magnetohydrodynamics with a perfectly conducting wall condition. Comm. Math. Phys., 136(1):119–140, 1991. “volumeV” — 2009/8/3 — 0:35 — page 195 — #211 APPENDIX A The Projector P In this Appendix we see in various examples of physical interest what is the form of the orthogonal projection P onto (ker Aν )⊥ , and which is the price of the hypothesis P ∈ C ∞ , taken in assumption (D) of Theorem 4.1, Section 1. Example A.1. Consider the Euler equations (1.4). The boundary matrix is: (ρp /ρ)v · ν νT 0 ν ρv · νI3 0 . Aν = T 0 0 v·ν If v · ν = 0, then ker Aν = {U ′ = (p′ , v ′ , S ′ ) : p′ = 0, v ′ · ν = 0}, The projection onto (ker Aν )⊥ is: 1 P = 0 0 0T ν⊗ν 0T 0 0 , 0 and P has the regularity of ν: if ∂Ω ∈ C ∞ then P ∈ C ∞ . is: Example A.2. Consider the ideal MHD equations (1.6). The boundary matrix (ρp /ρ)v · ν νT −(ρp /ρ)H T v · ν 0 ν ρv · νI3 −H · νI3 0 . Aν = −(ρp /ρ)Hv · ν −H · νI3 a0 v · ν 0 0 0T 0T v·ν (i) If v · ν = 0, H · ν = 0, then the projection P onto (ker Aν )⊥ is: 1 0T 0T 0 0 ν ⊗ ν 0 3 0 , P = 0 03 03 0 0 0T 0T 0 and P has the regularity of ν: if ∂Ω ∈ C ∞ then P ∈ C ∞ . √ ± c(ρ), then (ii) If H · ν = 0 and v · ν 6= 0, v · ν 6= |H| ρ ker Aν = {0}, (Noncharacteristic boundary) (iii) If v · ν = 0 and H · ν 6= 0, then P = Id ∈ C ∞ . ker Aν = {v ′ = 0, νq ′ − H · νH ′ = 0}, (ker Aν )⊥ = {H · ν q ′ + H ′ · ν = 0, S ′ = 0} 195 “volumeV” — 2009/8/3 — 0:35 — page 196 — #212 196 A. THE PROJECTOR P The projection onto (ker Aν )⊥ is: Λ 0 P = −Λ(H · ν)ν 0 0T I3 03 0T −Λ(H · ν)ν T 03 I3 − Λν ⊗ ν 0T 0 0 . 0 0 where Λ := [1 + (H · ν)2 ]−1 . P has the (finite) regularity of H · ν (for ∂Ω ∈ C ∞ ), while we would need at least P ∈ C m+1 (this is probably the least that we can ask instead of assumption (D) of Theorem 4.1, i.e. in place of P ∈ C ∞ ). Therefore our method does not seem to be applicable in this case. It is interesting to notice that, in spite of that, this problem may have full regularity (solvability in H m ), see Yanagisawa [72]. “volumeV” — 2009/8/3 — 0:35 — page 197 — #213 APPENDIX B Kreiss-Lopatinskiı̆ condition For the sake of simplicity, instead of an initial-boundary value problem, we consider the boundary value problem (BVP) ( Lu = F , in {x1 > 0} , (B.1) M u = G , on {x1 = 0} . where L := ∂t + n X Aj ∂xj j=1 is a hyperbolic operator (with eigenvalues of constant multiplicity); moreover Aj ∈ MN ×N , j = 1, . . . , n, with constant entries. For simplicity we assume that the boundary is noncharacteristic, i.e. det A1 6= 0. We also assume that M ∈ Md×N with constant entries, rank(M ) = d where d denotes the number of positive eigenvalues of the matrix A1 . Let u = u(x1 , x′ , t) (x′ = (x2 , . . . , xn )) be a solution to (B.1) for F = 0 and G = 0. Let u b = u b(x1 , η, τ ) be Fourier-Laplace transform of u w.r.t. x′ and t respectively (η and τ dual variables of x′ and t respectively). Then u b solves the ODE problem ( db u u , x1 > 0 , dx1 = A(η, τ )b (B.2) Mu b(0) = 0 , ! n P −1 Aj ηj . Let E − (η, τ ) denote the stable subτ In + i where A(η, τ ) := −(A1 ) j=2 space of (B.2). Definition B.1. Problem (B.1) satisﬁes the Kreiss-Lopatinskiı̆ condition (KL) if: kerM ∩ E − (η, τ ) = {0}, ∀(η, τ ) ∈ Rn−1 × C, ℜτ > 0. An equivalent formulation is the following. Proposition B.2. The Kreiss-Lopatinskiı̆ condition holds if and only if ∀(η, τ ) ∈ Rn−1 × C, ℜτ > 0, ∃C = C(η, τ ) > 0 : |A1 V | ≤ C|M V | ∀V ∈ E − (η, τ ). When the constant in the above estimate is independent of (η, τ ) we have the so-called uniform Kreiss-Lopatinskiı̆ condition: 197 “volumeV” — 2009/8/3 — 0:35 — page 198 — #214 198 B. KREISS-LOPATINSKIĬ CONDITION Definition B.3. Problem (B.1) satisﬁes the Uniform Kreiss-Lopatinskiı̆ condition (UKL) if: ∃C > 0 : ∀(η, τ ) ∈ Rn−1 × C, ℜτ > 0 : |A1 V | ≤ C|M V | ∀V ∈ E − (η, τ ). An useful tool for checking whether (KL) or (UKL) holds is given by the Lopatinskiı̆ determinant. For all (η, τ ) ∈ Rn−1 × C, ℜτ > 0, let {X1 (η, τ ), . . . , Xd (η, τ )} be an orthonormal basis of E − (η, τ ) (dim E − (η, τ ) = rank M = d). The assumption that the eigenvalues have constant multiplicity yields that Xj (η, τ ), j = 1, . . . , d, and E − (η, τ ) can be extended to all (η, τ ) 6= (0, 0) with ℜτ = 0. Definition B.4. The Lopatinskiı̆ determinant is the determinant deﬁned by ∆(η, τ ) := det [M (X1 (η, τ ), . . . , Xd (η, τ ))] ∀(η, τ ) ∈ Rn−1 × C, ℜτ ≥ 0. Proposition B.5. The Kreiss-Lopatinskiı̆ condition holds if and only if ∆(η, τ ) 6= 0 , ∀ℜτ > 0, ∀η ∈ Rn−1 . ∆(η, τ ) 6= 0 , ∀ℜτ ≥ 0, ∀η ∈ Rn−1 . The Uniform Kreiss-Lopatinskiı̆ condition holds if and only if Below we summarize the relation between the Kreiss-Lopatinskiı̆ condition and the well posedness of (B.1). 1. det A1 6= 0 (i.e. noncharacteristic boundary) - (KL) does NOT hold ⇒ (B.1) is ill posed in Hadamard’s sense; - (UKL) ⇔ L2 −strong well posedness of (B.1); - (KL) holds but NOT (UKL) ⇒ Weak well posedness of (B.1) (energy estimate with possible loss of regularity?). 2. det A1 = 0 (i.e. characteristic boundary) - (KL) does NOT hold ⇒ (B.1) is ill posed in Hadamard’s sense; - (UKL) + structural assumptions (see Appendix C) on L ⇒ L2 −strong well posedness of (B.1). “volumeV” — 2009/8/3 — 0:35 — page 199 — #215 APPENDIX C Structural assumptions for well-posedness For more general boundary conditions than those maximally non-negative, the well posedness has been proven for symmetrizable hyperbolic systems under suitable structural assumptions. Instead of maximally non-negative boundary conditions, the theory deals with uniform Kreiss-Lopatinskiı̆ conditions (UKL), see Appendix B. Moreover the boundary is assumed to be uniformly characteristic, see Deﬁnition 1.2. The general theory has received major contributions by Majda and Osher [30], Ohkubo [41], Benzoni and Serre [4]. In the same framework we may also quote the papers about elasticity by Morando and Serre [35, 36]. We brieﬂy recall these results. (I) Majda and Osher [30] prove the well-posedness of (1.1) under the following assumptions: - the operator L is symmetric hyperbolic, with variable coeﬃcients, - the boundary is uniformly characteristic, - the uniform Kreiss-Lopatinskiı̆ condition (UKL) holds, - several technical assumptions on L and M , among which the symbol of L is such that: n X a1 (η) a2,1 (η)T Aj ηj = , (C.1) A(η) := a2,1 (η) a2 (η) j=2 where a2 (η) has only simple eigenvalues for |η| = 1. In (C.1) the block decomposition is as in (4.30), where a1 (η) takes the place of the invertible part AI,I 1 (x, t). The above assumptions are satisﬁed in many interesting cases: strictly hyperbolic systems, MHD equations, Maxwell’s equations, linearized shallow water equations. They are not satisﬁed by the 3D isotropic elasticity, where a2 (η) = 03 . (II) Benzoni-Gavage and Serre [4] prove the well-posedness of (1.1) under the following assumptions: - L is symmetric hyperbolic, with constant coeﬃcients, M is constant, - the boundary is uniformly characteristic, and ker Aν ⊂ ker M , - the uniform Kreiss-Lopatinskiı̆ condition (UKL) holds, - Instead of (C.1), one has a1 (η) a2,1 (η)T A(η) = (C.2) a2,1 (η) 0 with a1 (η) = 0. This is the case of Maxwell’s equations and linearized acoustics. Unfortunately the above assumption (C.2) is not satisﬁed by isotropic elasticity, where a1 (η) 6= 0. 199 “volumeV” — 2009/8/3 — 0:35 — page 200 — #216 200 C. STRUCTURAL ASSUMPTIONS FOR WELL-POSEDNESS (III) The well posedness of linear isotropic elasticity in 2D and 3D has been shown by Morando and Serre [35, 36] by the construction ad hoc of a symbolic Kreiss symmetrizer. “volumeV” — 2009/8/3 — 0:35 — page 201 — #217 (Eds.) E. Feireisl, P. Kaplický and J. Málek Qualitative properties of solutions to partial diﬀerential equations Published by MATFYZPRESS Publishing House of the Faculty of Mathematics and Physics Charles University, Prague Sokolovská 83, CZ – 186 75 Praha 8 as the 282 publication The volume was typeset by the authors using LATEX Printed by Reproduction center UK MFF Sokolovská 83, CZ – 186 75 Praha 8 First edition Praha 2009 ISBN 978-80-7378-088-3 “volumeV” — 2009/8/3 — 0:35 — page 202 — #218 “volumeV” — 2009/8/3 — 0:35 — page 203 — #219 Jindřich Nečas Jindřich Nečas was born in Prague on December 14th, 1929. He studied mathematics at the Faculty of Natural Sciences at the Charles University from 1948 to 1952. After a brief stint as a member of the Faculty of Civil Engineering at the Czech Technical University, he joined the Czechoslovak Academy of Sciences where he served as the Head of the Department of Partial Diﬀerential Equations. He held joint appointments at the Czechoslovak Academy of Sciences and the Charles University from 1967 and became a full time member of the Faculty of Mathematics and Physics at the Charles University in 1977. He spent the rest of his life there, a signiﬁcant portion of it as the Head of the Department of Mathematical Analysis and the Department of Mathematical Modeling. His initial interest in continuum mechanics led naturally to his abiding passion to various aspects of the applications of mathematics. He can be rightfully considered as the father of modern methods in partial diﬀerential equations in the Czech Republic, both through his contributions and through those of his numerous students. He has made signiﬁcant contributions to both linear and non-linear theories of partial diﬀerential equations. That which immediately strikes a person conversant with his contributions is their breadth without the depth being compromised in the least bit. He made seminal contributions to the study of Rellich identities and inequalities, proved an inﬁnite dimensional version of Sard’s Theorem for analytic functionals, established important results of the type of Fredholm alternative, and most importantly established a signiﬁcant body of work concerning the regularity of partial diﬀerential equations that had a bearing on both elliptic and parabolic equations. At the same time, Nečas also made important contributions to rigorous studies in mechanics. Notice must be made of his work, with his collaborators, on the linearized elastic and inelastic response of solids, the challenging ﬁeld of contact mechanics, a variety of aspects of the Navier–Stokes theory that includes regularity issues as well as important results concerning transonic ﬂows, and ﬁnally non-linear ﬂuid theories that include ﬂuids with shear-rate dependent viscosities, multi-polar ﬂuids, and ﬁnally incompressible ﬂuids with pressure dependent viscosities. Nečas was a proliﬁc writer. He authored or co-authored eight books. Special mention must be made of his book “Les méthodes directes en théorie des équations elliptiques” which has already had tremendous impact on the progress of the subject and will have a lasting inﬂuence in the ﬁeld. He has written a hundred and forty seven papers in archival journals as well as numerous papers in the proceedings of conferences all of which have had a signiﬁcant impact in various areas of applications of mathematics and mechanics. Jindřich Nečas passed away on December 5th , 2002. However, the legacy that Nečas has left behind will be cherished by generations of mathematicians in the Czech Republic in particular, and the world of mathematical analysts in general. “volumeV” — 2009/8/3 — 0:35 — page 204 — #220 Jindřich Nečas Center for Mathematical Modeling The Nečas Center for Mathematical Modeling is a collaborative eﬀort between the Faculty of Mathematics and Physics of the Charles University, the Institute of Mathematics of the Academy of Sciences of the Czech Republic and the Faculty of Nuclear Sciences and Physical Engineering of the Czech Technical University. The goal of the Center is to provide a place for interaction between mathematicians, physicists, and engineers with a view towards achieving a better understanding of, and to develop a better mathematical representation of the world that we live in. The Center provides a forum for experts from diﬀerent parts of the world to interact and exchange ideas with Czech scientists. The main focus of the Center is in the following areas, though not restricted only to them: non-linear theoretical, numerical and computer analysis of problems in the physics of continua; thermodynamics of compressible and incompressible ﬂuids and solids; the mathematics of interacting continua; analysis of the equations governing biochemical reactions; modeling of the non-linear response of materials. The Jindřich Nečas Center conducts workshops, house post-doctoral scholars for periods up to one year and senior scientists for durations up to one term. The Center is expected to become world renowned in its intended ﬁeld of interest. ISBN 978-80-7378-088-3

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