Qualitative properties of solutions to partial differential equationsl

Qualitative properties of solutions to partial differential equationsl
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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
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Part 1
The rugosity effect
Dorin Bucur
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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.
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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
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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)
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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]).
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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]).
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“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
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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.
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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.
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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
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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.
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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)}.
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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.
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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.
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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
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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);
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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}.
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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
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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).
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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
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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
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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.
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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
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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.
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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 = −∆.
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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
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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)
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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.
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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→∞
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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
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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
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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 = −∆,
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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.
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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).
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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
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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
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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).
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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
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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)
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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.
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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.
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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)
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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).
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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
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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:
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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)
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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.
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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
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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 .
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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)
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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 .
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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|
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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,
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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.
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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
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Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras, 163–217, Math. Top., 11, Akademie Verlag, Berlin, 1996.
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Eds., Birkhäuser, Boston 2000, 31pp.
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Part 3
On a continuous deconvolution
equation
Roger Lewandowski
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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.
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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
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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
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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)
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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)),
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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.
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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.
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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.
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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
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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 .
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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.
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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)
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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)
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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.
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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)
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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
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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
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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].
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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.
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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.
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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 ).
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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
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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
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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
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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.
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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
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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].
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[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
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[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.
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Part 4
Rough boundaries and wall laws
Andro Mikelić
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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.
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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
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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
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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.
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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 )
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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 )
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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 ) .
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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 )) .
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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
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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 + )}.
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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)
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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)
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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 .
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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
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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).
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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
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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
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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)
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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,
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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 ).
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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
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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.
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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:
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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.
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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.
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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].
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Part 5
Hyperbolic problems with
characteristic boundary
Paolo Secchi, Alessandro Morando, Paola Trebeschi
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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.
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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
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138
CONTENTS
Appendix C. Structural assumptions for well-posedness
199
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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
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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
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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.
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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.
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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.
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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)
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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)
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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.
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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.
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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
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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].
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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.
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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.
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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)
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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
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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.
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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.
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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
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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
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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
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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).
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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 } .
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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 )
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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)
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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 .
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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.
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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.
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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.
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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
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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.
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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.
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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
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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)
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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∗∗
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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
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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:
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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) ∈
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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
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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)) ,
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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.
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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
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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
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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.
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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
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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
+
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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)
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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].
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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)
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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 ).
The a priori estimate (4.10) follows from (4.41) (namely estimate (4.10) in case
m = 1) applied to the solution of (4.50), plus standard L2 energy estimates for
equations (4.52) and (4.53), and the direct estimate of the normal derivative of
u by tangential derivatives via (4.48). All products of functions are estimated in
spaces H∗m by using the properties of these spaces given in [37, App. B]. We refer
the reader to [7, 37, 51, 53] for all details.
This concludes the proof of Theorem 4.1.
“volumeV” — 2009/8/3 — 0:35 — page 191 — #207
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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
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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
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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
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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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