# Solution of conservation laws via convergence space completion by

Solution of conservation laws via convergence space completion by Dennis Ferdinand Agbebaku Submitted in partial fulfilment of requirements for the degree of Magister Scientiae in the Department of Mathematics and Applied Mathematics in the Faculty of Natural and Agricultural Sciences University of Pretoria Pretoria August 2011 © University of Pretoria Declaration I, the undersigned, hereby declare that the thesis submitted herewith for the degree Master of Science to the University of Pretoria contains my own, independent work and has not previously been submitted by me or any other person for any degree at this or any other University. Name: Dennis Ferdinand Agbebaku Date: August 2011 iii To the memory of my late aunty Mrs Roseline Segun Okhuiegbe, and my late step-mother Mrs Philomina Agbebaku, both of whom passed on during the course of this work. Memory is the one thing death cannot destroy. iv Title Solution of conservation laws via convergence space completion Name Dennis Ferdinand Agbebaku Supervisor Dr JH van der Walt Co-supervisor Prof R. Anguelov Department Mathematics and Applied Mathematics Degree Magister Scientiae Summary In this thesis we consider generalized solutions of scalar conservation laws. In this regard, the Order Completion Method for systems of nonlinear PDEs is modiﬁed in a suitable way. In particular, with a given Cauchy problem for scalar conservation law, we associate an injective mapping T : M −→ N , where M and N are suitable spaces of suﬃciently smooth functions, independent of the given conservation law, so that the initial value problem may be expressed as one equation Tu = h (1) for a suitable h ∈ N . Uniform convergence structures are introduced on the spaces M and N in such a way that the mapping T is a uniformly continuous embedding. Thus there exists a unique, injective uniformly continuous mapping T ♯ : M♯ −→ N ♯ , where M♯ and N ♯ denote the completions of M and N , respectively, that extends T. Thus we arrive at a generalized version of the equation (1), namely, T ♯ u♯ = h (2) where the unknown function u is supposed to belong to M. Any solution of (2), if it exists, is interpreted as a generalized solution of (1). Note that due to the injectivity of T ♯ , the equation (2) has at most one solution. Furthermore, the space M of generalized functions may be identiﬁed in a natural way with a set of Hausdorﬀ continuous interval valued functions. Therefore the solution of (2) has a solution, which agrees with the well known entropy solution. Acknowledgements I thank God for the grace and strength He gave me to complete this thesis. To Him alone be all glory. I would also like to express my sincerely appreciate to every person and organization that have made some contributions, directly or indirectly, to the successful completion of this thesis. First of all, I thank the University of Pretoria for their full ﬁnancial support throughout the two-year period of this research work, under the UP M.Sc pilot bursary programme. My profound gratitude goes to my supervisors, Prof Roumen Anguelov and Dr Jan Harm van der Walt, who introduce me to convergence spaces and their applications to PDEs. I found their mathematical ingenuity second to none. Their patience and careful guidance throughout this research work have given some strength to my mathematical research understanding. I sincerely appreciate their eﬀort, the attention they gave me, and the many useful discussions I had with each of them during the course of writing this thesis. I also thank them for reading the preliminary drafts of this thesis and making useful suggestions. I am grateful for their supervision and recommendation for the UP M.Sc pilot bursary. I also appreciate my virtuous wife Mrs Grace Agbebaku for her patience, endurance and prayers during the period of my study. My beloved children: Victor, Virtue and Virginia have been very patient too, enduring and strong during my absence, the lord bless you and keep you. I like to thank Prof Jean Lubuma- the Head of department, Mrs Yvonne Mcdermot-the Head of academic administration and the entire members of staﬀ of the Department of Mathematics and Applied Mathematics University of Pretoria for their kind assistance. My association with them was very proﬁtable to this research work. I would also like to thank Prof (Mrs) Okeke -the immediate Past Dean, Prof M.O Oyesanya -the current Dean of the faculty of Physical Sciences University of Nigeria Nsukka, the Head of Department of Mathematics -Dr G. C. E. Mbah and Prof M. O Osilike for their support and recommendations to the University for me to pursue this programme. I also thank my employer - University of Nigeria Nsukka- for granting me study leave to enable me pursue this study programme. I like to thank Pastor John Amoni- for his prayers and encouragement, the Akpor family, the Kruger family, the Agypong family, Bro Jide, Bro Baridam, vi the Ogunniyi family, and the entire members of the Deeper Christian life Ministry South Africa, not mentioned for space, for their prayers, concern, care and ﬁnancial support during my studies. To my My Nigerian friends Kola, Bode, Mr Oke, Adaku to mention a few, I want to say your presence has been a source of strength and encouragement. Thank you all. My colleagues in the Department of Mathematics and Applied Mathematics University of Pretoria, namely Mr Zakaria Ali , Mr Yibetal Aldane to mention a few were helpful and I appreciate your support. Contents Declaration ii Dedication iii Summary iv Acknowledgement v Contents 1 1 Introduction 1.1 Nonlinear Hyperbolic Conservation Laws . . . . . . . . . . . . . . 1.1.1 Introduction to Scalar conservation laws . . . . . . . . . . 1.1.2 Examples of Conservation laws . . . . . . . . . . . . . . . 1.1.3 Solutions to Scalar Conservation Laws . . . . . . . . . . . 1.1.4 Solutions of Scalar Conservation Laws via Vanishing Viscosity 1.1.5 Compensated Compactness Methods for Nonlinear Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Convergence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Uniform Convergence Structure . . . . . . . . . . . . . . . 1.2.2 Convergence vector spaces . . . . . . . . . . . . . . . . . . 1.3 Hausdorﬀ Continuous Functions . . . . . . . . . . . . . . . . . . . 1.4 The Order Completion Method . . . . . . . . . . . . . . . . . . . 1.4.1 Main Ideas of Convergence space Completion . . . . . . . 1.5 Summary of the Main Results . . . . . . . . . . . . . . . . . . . . 35 37 47 52 56 60 64 68 2 Hausdorﬀ Continuous Solution of Scalar Conservation laws 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Convergence Vector Spaces for Conservation Laws . . . . . . . 2.3 Approximation results . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Requirements for u0 . . . . . . . . . . . . . . . . . . . 2.4 Existence and uniqueness results . . . . . . . . . . . . . . . . . 70 70 72 86 91 94 . . . . . . . . . . 3 6 6 7 9 25 2 CONTENTS 3 Concluding Remarks 3.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Topics for further research . . . . . . . . . . . . . . . . . . . . . . Bibliography 99 99 99 101 Chapter 1 Introduction The mathematical models for real-world problems occurring in Physics, Chemistry, Economics, Engineering and Biology are usually expressed in the form of partial diﬀerential equations (PDEs) with associated initial and/or boundary values. We consider only initial value problems, consisting of a PDE T (x, t, D)u(x, t) = h(x, t), x ∈ Ω, t > 0 and an initial condition u(x, 0) = u0 (x), x ∈ Ω where Ω ⊆ Rn is open and h a suitable function. The fundamental mathematical question, concerns the well-posedness of the problem. Recall [58] that the given initial value problem of a PDE is well-posed if the problem has a solution, if the solution is unique and if the solution depends continuously on the data given in the problem. Each of the three issues involve in the concept of well-posedness is nontrivial in its own right. It is well known that an initial value problem may fail to have a classical solution on the whole domain of deﬁnition of the equation. Indeed, a nonlinear analytic PDE will, according to the well known Cauchy-Kowalevskaia Theorem [71], admit an analytic solution which is deﬁned on a neighborhood of any noncharacteristic hyper surface on which analytic initial data is speciﬁed. However, outside this neighborhood the solution may fail to exist. In particular, the solution will typically exhibit singularities outside the mentioned neighborhood of analyticity. Thus, the solutions cannot be guaranteed to exists on the whole domain of deﬁnition of the given PDE. In fact, even a linear equation without initial conditions may fail to have a solution, as shown by the following example due to Lewy [83]. Example 1.1. Consider the linear operator A(u) = ux + iuy − 2i(x + iy)ut (x, y, t) ∈ R3 . There exist C ∞ -smooth functions h for which the equation A(u) = h has no 4 Introduction solution in D′ -distributions in any neighborhood of any point in R3 , see [83] for details. 2 In view of the local nature of solutions of an initial value problems, in general, it is clear that there is an interest in solutions to PDEs that may fail to be classical on the whole domain of deﬁnition of the respective PDE. Such generalized or weak solutions to PDEs are obtained as elements of suitable spaces of generalized functions, that is, objects which retain certain essential features of the usual real or complex valued functions. Many mathematicians specializing in nonlinear partial diﬀerential equations (PDEs) believe that there is no general and type independent theory for the existence and basic regularity properties of generalized solutions of PDEs. In fact, the ﬁrst chapter of the book [14] by V. I. Arnold starts with the following statement: “In contrast to ordinary diﬀerential equations, there is no uniﬁed theory of partial diﬀerential equations. Some equations have their own theories, while others have no theory at all. The reason for this complexity is a more complicated geometry ...” This seeming inability of mathematical theories to deal with PDEs in a uniﬁed way may be attributed to the inherent limitations of the customary, linear topological theories for the solution of PDEs themselves, rather than to any fundamental conceptual obstacles. In this regard we may note that the spaces of generalized functions that are typically used in the study of solutions of linear and nonlinear PDEs cannot deal with suﬃciently large classes of singularities. Indeed, due to the celebrated Sobolev Embedding Theorem [112], none of the Sobolev spaces can deal with the most simple singular functions, such as the Heaviside step function 0 if x < 0 H(x) = 1 if x ≥ 0 Furthermore, Colombeau generalized functions [50] as well as distributions, cannot handle an analytic function with an essential singularity at one single point, such as f (z) = e1/z . The great Picard Theorem states that such a function will assume every complex number, with possibly one exception, as a value in every neighborhood of the singular point. It will therefore violate the polynomial growth conditions that are imposed on the Colombeau generalized functions near singularities. However, there are two recent theories that provide general and type independent results regarding the existence and basic regularity properties of large classes of PDEs, namely, the Order Completion Method(OCM) [93] and the generalized 5 method of Steepest Descent in suitably constructed Hilbert spaces, introduced by Neuberger [89, 90, 91, 92]. The Order Completion Method is based on the Dedekind order completion of suitable spaces of (piecewise) smooth functions, and applies to what may be considered all continuous nonlinear PDEs. Furthermore, the solution so obtained satisﬁes a blanket regularity property. In particular, the solutions may be assimilated with Hausdorﬀ continuous interval valued functions [10]. The recent reformulation and enrichment of the OCM in terms of suitable uniform convergence spaces and their completions has greatly improved the regularity properties as well as the understanding of the structure of solutions, [118, 119, 120, 121]. The underlying ideas upon which the method of Steepest Descent is based does not depend on the particular form of the PDE involved, and is therefore type independent. However, the relevant techniques involve several highly technical aspects which have, as of yet, not been resolved for a class of equations comparable to that to which the OCM applies. However, the numerical computation of solutions, based on this theory, has advanced beyond the proven scope of the underlying analytical techniques, see for instance [92]. In this Thesis we study a class of ﬁrst order PDEs that may serve as mathematical descriptions of physical conservation laws, such as the laws of gas dynamics and the laws of electromagnetism. In particular, we apply the Order Completion Method, as formulated in the context of Convergence spaces as well as uniform convergence spaces completion [119, 120, 121] to the ﬁrst order nonlinear Cauchy problem of conservation laws. Furthermore, we show how the Convergence Space Completion Method can be applied to solve the initial value problem of the Burgers equation. We construct the entropy solution of the Burgers equation and show how it can be assimilated with the space of H-continuous functions. In the rest of Chapter 1, some of the concepts and theories that are used in obtaining our results are discussed. The existence and uniqueness of weak solutions of the Cauchy problem of conservation laws is discussed in Section 1.1. Some of the admissibility conditions for singling out a unique solution are considered, namely, the Lax, Oleinik and entropy conditions. Other techniques, namely, the vanishing viscosity method and the compensated compactness technique, for obtaining the entropy solution of a conservation law are also discussed. Some existence and uniqueness results for solutions of conservation laws obtained by Hopf, Lax, Oleinik and Kruzhkov, are also discussed. An introduction to spaces of Hausdorﬀ continuous functions is presented in Section 1.3. Section 1.2 is an introduction to the theory of convergence spaces. Section 1.4 addresses the main ideas underlying the Order Completion Method. Chapter 1 ends with a summary of the main results in this thesis. 6 Introduction 1.1 Nonlinear Hyperbolic Conservation Laws 1.1.1 Introduction to Scalar conservation laws A conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. In particular, any change in such a conserved quantity can only occur as a result of an “inﬂux” or an “outﬂow” of this quantity into or out of the system respectively. The exact mathematical model for a single conservation law in one spatial dimension is given by the ﬁrst order PDE ut + (f (u))x = 0. (1.1) Here u is the conserved quantity while f is the ﬂux. Integrating equation (1.1) over some interval [a, b] leads to ∫ ∫ b d b u(x, t)dx = ut (x, t)dx dt a a ∫ b =− f (u(x, t))x dx a = f (u(x, a)) − f (u(x, b)) = [ in ﬂow at a] − [ out ﬂow at b] In other words, the quantity u is neither created nor destroyed. In particular, the total amount of u contained in the interval [a, b] can only change due to the ﬂow of u across the two endpoints. In general, if u = (u1 , · · · uk ) is a vector of conserved quantities, depending on time t and n independent variables x1 , · · · , xn , then the ﬂux of u out of any bounded region Ω ⊆ Rn is given by ∫ F(u) · ndS. ∂Ω Here F : Rk −→ Mk×n = {A : A is a matrix of order k × n} is the ﬂux, n denotes the outward unit normal to ∂Ω and dS the surface element on ∂Ω. Since any change in u in such a domain Ω over time can only be due to the ‘in ﬂow’ or ‘out ﬂow’ of u into or out of Ω, it follows that ∫ ∫ d udx = − F(u) · ndS. (1.2) dt Ω ∂Ω Note that the integral on the right of (1.2) measures the ﬂow out of Ω, hence the minus sign. Assuming that F, u and ∂Ω are suﬃciently smooth, we may apply the Divergence Theorem to equation (1.2) so that ∫ ∫ d udx = − ∇ · F(u)dx. dt Ω Ω Nonlinear Hyperbolic Conservation Laws 7 Taking the derivatives with respect to t under the integral sign we obtain ∫ [ut + ∇ · F(u)]dx = 0. Ω the Mean Value Theorem implies the diﬀerential form of conservation laws, which is given by ut + ∇ · F(u) = 0. The study presented in this thesis is concerned mainly with the Cauchy problem for strictly hyperbolic systems in one spatial dimension. That is, ut + (F(u))x = 0 in R × (0, ∞) u(x, 0) = u0 (x) x ∈ R, (1.3) (1.4) where u = (u1 , · · · , uk ), F : Rk −→ Rk and u0 = (u10 , · · · , uk0 ) is the initial value of u. If A(u) = Ju F(u) is the k × k Jacobian matrix of the function F at the point u, then the system (1.3) can be written in the form ut + A(u)ux = 0. (1.5) Deﬁnition 1.2. We say that a system of conservation laws is strictly hyperbolic if the matrix A(u) has k real, distinct eigenvalues, say λ1 (u) < · · · < λk (u). (1.6) for every u. 1.1.2 Examples of Conservation laws As mentioned, systems of conservation laws such as (1.3) may serve as mathematical models for certain real-world phenomena. In particular, such equations appear as precise mathematical descriptions of physical conservation laws. In this section we mention a few examples of conservation laws that arise in applications. Example 1.3 (Traﬃc Flow). Let u(x, t) denote the density of cars on a highway at point x at time t. For example, u may be the number of cars per kilometer. Assume that u is continuous and that the speed s of cars depends only on their density, that is, s = s(u). We also assume that the speed s of the cars decreases ds as the density u increases, that is du < 0. Given any two points a and b on the highway, the number of cars between a and b varies according to the law ∫ ∫ b d b u(x, t)dx = − [s(u)u]x dx. (1.7) dt a a Since (1.7) holds for all a, b ∈ R this leads to the conservation law ut + [s(u)u]x = 0 8 Introduction Here the ﬂux is given by F (u) = s(u)u. In practice the ﬂux F is often taken to be a2 F (u) = a1 (ln( ))u, 0 < u < a2 , u for suitable constants a1 and a2 . Example 1.4 (The p-system). The p - system is a simple model for isentropic (constant entropy) gas dynamics. If v is the speciﬁc volume and u the velocity of the gas, then the equations are vt − ux = 0 ut + (p(v))x = 0 The ﬂux p is given as p(v) = kv −λ , k ≥ 0, λ ≥ 1 where k and λ are constants. In applications λ is chosen such that λ ∈ [1, 3] for most gases; in particular λ = 75 for air. In the region v > 0, the system is strictly hyperbolic. Indeed ( ) 0 −1 A = JF = p′ (v) 0 √ has real distinct eigenvalues λ = ± −p′ (v). 2 Example 1.5 (Gas dynamics). The Euler equations for the dynamics of a compressible, non-viscous gas is given by vt − ux = 0 ut + px = 0 (conservation of mass) (conservation of momentum) u2 (ν + )t + (pu)x = 0 (conservation of energy). 2 −1 Here v = ρ , where ρ is the density and v is the speciﬁc volume. The velocity in the gas is u, while ν is the internal energy and p the pressure. The system is closed by an additional equation p = p(ν, v) called the equation of state, which depend on the particular gas under consideration. 2 Example 1.6 (Electromagnetism). Let E be the electric intensity, D the electric induction, H the magnetic intensity, B the magnetic induction, I the electric current and q the heat ﬂux in an electromagnetic system. The conservation laws of electromagnetism are ∂t B + ∇ × E = 0 ∇·B =0 ∂t D − ∇ × H + I = 0 ∂t E + ∇ · (E × H + q) = 0 (Faraday′ s law) (Ampere′ s law) (conservation of energy). 2 Nonlinear Hyperbolic Conservation Laws 1.1.3 9 Solutions to Scalar Conservation Laws In this section we are concerned with initial value problems for scalar conservation laws in one spatial dimension ut + (f (u))x = 0 in R × (0, ∞) u(x, 0) = u0 (x) x ∈ R. (1.8) (1.9) Here u : R × [0, ∞) −→ R is the unknown conserved quantity, f ∈ C ∞ (R) is the ﬂux and u0 : R −→ R is the initial value. When solving the Cauchy problem (1.8) - (1.9), one is typically confronted with the following diﬃculties: Even in the case of a C ∞ - smooth initial condition u0 , the initial value problem (1.8) - (1.9) may not have a classical solution on the whole domain of deﬁnition of the equation (1.8). Indeed, solutions of (1.8) - (1.9) may develop discontinuities after a ﬁnite time. Classical Solutions A classical solution of the Cauchy problem (1.8) - (1.9) is a continuously diﬀerentiable function satisfying equations (1.8) - (1.9). One can obtain the classical solution of equation (1.8) - (1.9) by the method of characteristics. To do this, let the ﬂux function f be given, and assume that equation (1.8) is genuinely nonlinear. That is, f ′ (u) ̸= constant for all u, which further implies f ′′ (u) > 0 for all u. (1.10) If u ∈ C 1 (R × [0, ∞)) is a solution of the Cauchy problem, then we deﬁne the characteristic curves in R × [0, ∞) as the level curves of u. That is, for any y ∈ R the characteristic curve through the point (y, 0) consists of the set of points where u(x, t) = u(y, 0) = u0 (y). At every point (x, t) on the characteristic curve through (y, 0), (1.8) and (1.9) imply that ∇u(x, t) · ⟨f ′ (u0 (y)), 1⟩ = 0. Therefore ⟨1, −f ′ (u0 (y))⟩ is tangent to the curve at every point. Thus the characteristic through (y, 0) is a straight line with equation x(t) = y + tf ′ (u0 (y)). Since u(x, t) = u0 (y) for every point (x, t) on the curve, we may express the solution of (1.8) - (1.9) implicitly as u = u0 (x − tf ′ (u)). The Implicit Function Theorem may now be used to solve for u. The classical solution of (1.8) - (1.9) found above is unique, but may fail to exist for all t > 0 as the following theorem shows 10 Introduction Theorem 1.7. [109, Proposition 2.1.1] Assume that u0 ∈ C 1 (R), together with its derivative, is bounded on R. Set { +∞ if f ′ (u0 ) is an increasing f unction (1.11) T∗ = d ′ f (u0 ))−1 otherwise. −(inf dx Then (1.8) - (1.9) has a unique solution u ∈ C 1 (R × (0, T ∗ )). For T > T ∗ , (1.8) - (1.9) has no classical solutions on R × [0, T ). We now give an example to illustrate the nonexistence of classical solution for some time t > 0. Example 1.8. Consider the initial value problem for Burger’s equation u2 ut + ( )x = 0 in R × (0, ∞) 2 u(x, 0) = u0 (x) x ∈ R. (1.12) (1.13) Using the method of characteristics discussed above we see that for a C 1 -smooth function u0 , a classical solution u is given by the implicit equation u(x, t) = u0 (x − tu(x, t)), t ≥ 0, x ∈ R. (1.14) By the Implicit Function Theorem, we can obtain u(y, s) from (1.14) for y and s in suitable neighborhoods of x and t respectively, whenever 1 + tu′0 (x − tu(x, t)) ̸= 0. (1.15) If u′0 (x) ≥ 0 for all x ∈ R, then condition (1.15) is clearly satisﬁed for all (x, t), so that the Cauchy problem (1.12) - (1.13) has a unique solution on R × (0, ∞). However, if u′0 (x0 ) < 0 for some x0 ∈ R then for certain values of t > 0 the condition (1.15) may fail. Therefore, violation of condition (1.15) implies that the classical solution u fails to exist for the respective values of t and x. If we take if x ≤ 0 1 u0 (x) = 1 − x if 0 ≤ x ≤ 1 (1.16) 0 if x ≥ 1. then the unique classical solution of (1.12) - (1.13) is given by if x < t 1 1−x if t ≤ x ≤ 1, t<1 u(x, t) = 1−t 0 if x ≥ 1. Clearly, the classical solution of (1.12) - (1.13), with u0 as in (1.16), breaks down at t = 1. It should be noted that the breakdown of the solution u(x, t) at t = 1 for initial data u0 given in (1.16) is not due to the lack of smoothness of u0 , but to the fact that u′0 (x) = −1 < 0 for x ∈ [0, 1]. 2 In view of the nonexistence, in general, of global classical solutions, one is forced to consider suitable generalized solutions of (1.8) - (1.9). Nonlinear Hyperbolic Conservation Laws 11 Weak solutions and non-uniqueness One well known and much studied generalized formulation of (1.8) - (1.9) is the weak form of the initial value problem. Let us assume temporarily that u is a classical solution of (1.8) - (1.9). The idea is to multiply equation (1.8) with a smooth function ϕ and integrate by parts. More precisely, let ϕ be a test function, that is, ϕ : R × [0, ∞) → R (1.17) has compact support and is C ∞ - smooth. We denote the set of all such test functions by C0∞ (R × [0, ∞)). Multiply equation (1.8) by ϕ and integrate by parts to get ∫ ∞∫ ∞ 0= (ut + (f (u))x )ϕdxdt 0 −∞ ∫ ∞∫ ∞ ∫ ∞∫ ∞ ∫ ∞ =− uϕt dxdt − f (u)ϕx dxdt − uϕ|t=0 dx. 0 −∞ 0 −∞ −∞ In view of the initial condition (1.9), we obtain ∫ ∞∫ ∞ ∫ ∞ uϕt + f (u)ϕx dxdt + u0 ϕ|t=0 dx = 0. 0 −∞ (1.18) −∞ In contradistinction with equations (1.8) - (1.9), equation (1.18) does not involve any derivative of u, thus equation (1.18) makes sense not only for smooth functions, but also for bounded and measurable functions u and u0 . We thus arrive at the following deﬁnition of a weak solution of (1.8) -(1.9). Deﬁnition 1.9. We say that u ∈ L∞ (R×(0, ∞)) is a weak solution of (1.8)-(1.9) if the equation (1.18) holds for each test function ϕ ∈ C0∞ (R × [0, ∞)). If u ∈ C 1 (R × (0, ∞)) is a weak solution of (1.8) - (1.9) then u satisﬁes (1.8) (1.9). That is, a C 1 -smooth weak solution is a classical solution of equation (1.8) - (1.9). Thus the concept of weak solution of (1.8) - (1.9) is a generalization of the classical notion of solution. Remark 1.10. Equation (1.8) can also be written in the form: ut + a(u)ux = 0, with a(u) = f ′ (u). (1.19) At the level of classical solutions, equations (1.8) and (1.19) are equivalent. That is, u ∈ C 1 (R × [0, ∞)) is a solution of (1.8) if and only if u is a solution of (1.19). However, if u has a discontinuity, then the left hand side of equation (1.19) may contain a product of a discontinuous function a(u) with the distributional derivative ux . Such a product is typically not well deﬁned, see for instance [103]. Working with the equation in the form of (1.8) avoids this diﬃculty when dealing with weak solutions as deﬁned in Deﬁnition 1.9. 12 Introduction We give some examples to illustrate the non-uniqueness of solution to the initial value problem (1.8) - (1.9). Examples 1.11. (i) Consider the initial value problem of the Burgers equation (1.12) - (1.13). If we take initial data to be (1.16) then it can be shown that the function { 1 if x < 1+t 2 u1 (x, t) = 0 if x > 1+t 2 is a weak solution to the initial value problem (1.8), (1.16). (ii) Again consider the initial value problem (1.12) - (1.13) with initial data { 0 if x < 0 u0 (x) = (1.20) 1 if x > 0. { The function u1 (x, t) = 0 if x < 1 if x > is a weak solution to the initial value function 0 x u2 (x, t) = t 1 t 2 t 2 problem (1.8), (1.20). However, the if x < 0 if 0 < x < t if x > t is a solution to the initial value problem (1.8), (1.20), but cannot be classiﬁed as a weak solution according to Deﬁnition 1.9. (iii) A more spectacular example of the loss of uniqueness of solution is the following. Consider the Burgers equation (1.12) with the initial data { −1 if x < 0 u0 (x) = (1.21) 1 if x > 0. For every α ∈ [1, ∞) the function −1 −α uα (x, t) = +α +1 if x < (1−α) 2 t if (1−α) 2 t< x <0 if 0 < x < (α−1) 2 t (α−1) if 2 t < x (1.22) is a solution of (1.12) - (1.13) with u0 as in (1.21). It can be shown that only the solution for which α = 1 satisﬁes the deﬁnition of a weak solution. 2 One diﬃculty that arises in the study of weak solutions of (1.8) - (1.9) is related to the uniqueness of such solutions. In contradistinction with classical solution of (1.8) - (1.9), weak solutions are not unique as shown in the following, Nonlinear Hyperbolic Conservation Laws 13 Example 1.12. Consider the initial value problem of the Burgers equation (1.12) - (1.13) with initial condition { 0 if x < 0 u0 (x) = (1.23) 1 if x ≥ 0. For every α ∈ (0, 1), the function uα deﬁned as αt 0 if x < 2 , α if αt ≤ x < (1+α)t uα (x, t) = 2 2 1 if x ≥ (1+α)t , 2 is a weak solution of the initial value problem. The underlying physical laws that are modeled as mathematical conservation laws are deterministic in nature. That is, the future state of a system that evolves according to (1.8) is completely determined by the initial condition (1.9) of the system. From this point of view, the non uniqueness of weak solutions of (1.8) - (1.9), as demonstrated in Example 1.12, is unacceptable. In particular, in the context of physical systems that may be modeled through (1.8) - (1.9), the non uniqueness of weak solutions of the Cauchy problem may be interpreted as follows: The state of the system at time t > 0 is not completely determined by the weak formulation of (1.8) - (1.9) alone. Therefore further additional conditions, motivated by physical consideration, must be imposed on the weak solutions of (1.8) - (1.9) in order to obtain the unique solution that describes the evolution of the underlying physical system. In this regard, let u be a weak solution of (1.8) - (1.9). Assume that u has continuous ﬁrst order partial derivatives everywhere in the open set Ω ⊆ R×[0, ∞) except on a smooth curve C in Ω with equation x = x(t). 6 Curve of discontinuity C x = ρt Ωl Ωr Figure 1.1 That is, limits of u from left and from right of curve C exist. Hence u has a jump discontinuity across C. Let Ωl and Ωr be the parts of Ω on the left and on the right of curve C respectively, see Figure 1.1. Furthermore, since u is smooth on either side of the curve C, it is smooth in Ωl 14 Introduction and Ωr . Because u is a weak solution of (1.8) - (1.9), we have ∫ ∫ uϕt + (f (u))ϕx dxdt = 0, (1.24) Ω for all ϕ ∈ C0∞ (Ω). Thus, if suppϕ ⊂ Ωr , then ∫ ∫ ∫ ∫ 0= uϕt + (f (u))ϕx dxdt = − [ut + (f (u))x ]ϕdxdt. Ω (1.25) Ωr which implies ut + (f (u))x = 0 in Ωr . (1.26) ut + (f (u))x = 0 in Ωl . (1.27) Similarly, From (1.24) we get ∫ ∫ uϕt + f (u)ϕx dxdt 0= ∫Ω ∫ = ∫ ∫ uϕt + f (u)ϕx dxdt + Ωl uϕt + f (u)ϕx dxdt. (1.28) Ωr Now using the fact that u is C 1 -smooth in Ωr and Green’s Theorem, we ﬁnd that ∫ ∫ ∫ ∫ (uϕt + f (u)ϕx )dxdt = [(uϕ)t + (f (u)ϕ)x ]dxdt Ωr Ωr ∫ = (−uϕ)dx + (f (u)ϕ)dt ∂Ωr ∫ = ∫ (−uϕ)dx + (f (u)ϕ)dt + (−uϕ)dx + (f (u)ϕ)dt C ∂Ω Since ϕ = 0 on ∂Ω, we have ∫ ∫ ∫ (uϕt + f (u)ϕx )dxdt = (−ur ϕ)dx + (f (ur )ϕ)dt. Ωr C where ur the right limit of u on the curve C. Similarly, ∫ ∫ ∫ (uϕt + f (u)ϕx )dxdt = − (−ul ϕ)dx + (f (ul )ϕ)dt Ωl (1.29) C (1.30) Nonlinear Hyperbolic Conservation Laws 15 where ul is the left limit of u on the curve C. Substituting equations (1.29) and (1.30) into equation (1.28) we have ∫ 0 = (−ul + ur )ϕdx + (f (ul ) − f (ur ))ϕdt ∫C ϕ[−(ul − ur )dx + (f (ul ) − f (ur ))dt] = (1.31) C which further implies [−(ul − ur )dx + (f (ul ) − f (ur ))dt] = 0. This implies dx = (f (ul ) − f (ur )) dt in Ω along the curve C, which may be expressed as (ul − ur ) (f (ul ) − f (ur )) = ẋ(ul − ur ). (1.32) ρ[[u]] = [[f (u)]], (1.33) We write this as where [[u]] = ul − ur is the jump in u across the curve C, [[f (u)]] = f (ul ) − f (ur ) is the jump in f (u) and ρ = dx dt is the speed of curve C. Relation (1.33) is known as the jump condition. Equation (1.33) is popularly known as Rankine Hugoniot condition. Remark 1.13. We remark here that if u is a piecewise smooth solution to the initial value problem (1.8) - (1.9), then u satisﬁes the jump condition if and only if it is a weak solution. In other words, every weak solution to the initial value problem (1.8) - (1.9) satisﬁes the jump condition. Conversely, every piecewise smooth solution to the initial value problem that satisﬁes the jump condition is a weak solution to the initial value problem (1.8) - (1.9). This follows from the above derivation of the jump condition. However, if u is a weak solution which is bounded and measurable, then ul and ur in condition (1.32) have to be interpreted as ul (x, t) = lim inf u(y, t) y−→x, ur (x, t) = lim supu(y, t). y−→x, Examples 1.14. (i) Applying the jump condition to the Burgers’ equation (1.12) where f (u) = 1 2 dx 2 u , we ﬁnd that the speed of propagation of the discontinuities is dt = ρ = 1 2 (ul + ur ). 16 Introduction (ii) Again, applying the jump condition to the solutions uα of Example 1.12, along the lines of we see that by the jump condition, ρ = α2 and ρ = (1+α) 2 (1+α)t discontinuity x = αt respectively for each α ∈ (0, 1). Thus, 2 and x = 2 the jump condition alone is not suﬃcient to determine the unique, physically relevant solution of the Cauchy problem (1.8) - (1.9). 2 Admissibility conditions and the Entropy Condition From Example 1.12, it is clear that the set of weak solutions of a given initial value problem (1.8)-(1.9) may include various solutions which are not physically relevant. In order to single out a unique solution that is physically and/or mathematically relevant, suitable additional requirements, which we shall call admissibility conditions, are imposed on such solutions, see for instance [52, 79]. These admissibility conditions, such as entropy conditions, are typically motivated by some physical considerations. In the literature, various admissibility conditions have been introduced. In this section, we recall some of these conditions.The main results in which these admissibility conditions are employed to single out the unique, physically relevant solution to the Cauchy problem (1.8) -(1.9) are also discussed. Admissibility condition 1 (The Oleinik inequality) Oleinik [95] introduced the Lipschitz condition, with respect to x, for ﬁxed t for a genuinely nonlinear single conservation law (1.8) given by u(x + a, t) − u(x, t) E ≤ . a t a > 0, t > 0. (1.34) Here E = inf1f ′′ is independent of x, t, and a. Using the Lax-Friedreich ﬁnite diﬀerence scheme, Oleinik showed that if f is convex, which implies that f ′′ > 0, then there exists precisely one weak solution of the Cauchy problem (1.8) - (1.9) satisfying (1.34). Note that the weak solutions of (1.8) - (1.9) that satisﬁes (1.34) will, for any ﬁxed t > 0 have x-diﬀerence quotient bounded from above. As t tends to 0, the upper bound for the x-diﬀerence quotients may tend to plus inﬁnity. The Oleinik inequality (1.34) was motivated by the fact that if u′0 ≥ 0, a classical solution u of (1.8) - (1.9) exists with ux = So that ux ≤ u′0 1 + tf ′′ (u0 )u′0 . 1 K ≤ , t > 0 and K > 0 a constant tf ′′ (u0 ) t Nonlinear Hyperbolic Conservation Laws 17 which is an limiting version of the Oleinik inequality (1.34). It is therefore reasonable for a solution of the Cauchy problem (1.8) - (1.9) to satisfy the inequality (1.34). The basic idea of the ﬁnite diﬀerence scheme in PDE is to replace derivatives with appropriate ﬁnite diﬀerences. The main result, concerning solutions satisfying (1.34), which is also found in [111], is given below. Theorem 1.15. [111, Theorem 16. 1] Let u0 ∈ L∞ (R), and let f ∈ C 2 (R) with f ′′ (u) > 0 on {u : |u| ≤ ∥u0 ∥∞ }. Let M = ∥u0 ∥L∞ , µ = inf{f ′′ (u) : |u| ≤ ∥u0 ∥∞ } and A = sup{|f ′ (u)| : |u| ≤ ∥u0 ∥∞ }. Then there exists exactly one weak solution u of (1.8)-(1.9) satisfying the following: (a) There exists a constant E > 0, depending only on M, µ and A, such that for every a > 0, t > 0, and x ∈ R, the inequality u(x + a, t) − u(x, t) E < . a t (1.35) holds. (b) |u(x, t)| ≤ M, ∀ (x, t) ∈ R × [0, ∞). (c) u is stable and depends continuously on u0 in the following sense: If u0 , v0 ∈ L∞ (R)∩L1 (R) with ∥v0 ∥∞ ≤ ∥u0 ∥∞ , and v is the solution of (1.8) with initial data v0 satisfying (1.35), then for every x1 , x2 ∈ R, with x1 < x2 , and every t > 0, ∫ x2 −At ∫ x2 |u(x, t) − v(x, t)|dx ≤ |u0 (x) − v0 (x)|dx. (1.36) x1 x1 −At Remark 1.16. (i) An immediate consequence of (1.35) is that for any t > 0, the solution u(·, t) is of locally bounded total variation, that is u ∈ BVloc , which means the total variation of u is bounded in every compact subset of R × [0, ∞). To see this, let us deﬁne a function v(x, t) = u(x, t) − E x. t Then if a > 0 (1.35) implies E a < 0. t That is, v is a decreasing function with respect to x and thus has locally bounded total variation with respect to x. Hence u is of locally bounded total variation since a linear function is also of locally bounded total variation. Thus even though u0 is only L∞ , the solution u(·, t) is fairly regular. In fact, we can conclude that it has at most a countable number of jump discontinuities, and it is diﬀerentiable almost everywhere. v(x + a, t) − v(x, t) = u(x + a, t) − u(x, t) − 18 Introduction (ii) Theorem 1.15 is limited to single conservation laws in one spatial dimension. An analogue of the Oleinik inequality (1.35) has not been found for systems of conservation laws. (iii) The Oleinik inequality (1.35) implies that ul > ur as we move across a curve of discontinuity. To see this, note that the function v(x, t) = u(x, t) − Et x is bounded in a domain (x1 , x2 ) × [0, ∞) containing the curve of discontinuity. Then v has left and right limits with respect to x at each point since it is decreasing with respect to x as noted in (i). Consequently u(x, t) has left and right limits at each point. For any point c on the line of discontinuity we have E E ur − ul = lim+ u(x, t) − lim+ x − lim− u(x, t) + lim− x x−→c x−→c x−→c t x−→c t = lim+ v(x, t) − lim− v(x, t) < 0, x−→c x−→c which implies ul > ur as we move across a curve of discontinuity. Admissibility condition 2 (The Lax inequality) The inequality f ′ (ul ) > ρ > f ′ (ur ) for all t > 0. (1.37) was introduced by Lax [78]. The inequality (1.37) implies that the characteristics starting on either sides of the curve of discontinuity should intersect each other on the curve, see ﬁgure 1.2. At this point of intersection, u has two values which is impossible, so that there is a jump discontinuity at that point. Indeed, if u′0 < 0, there are two points y1 , y2 ∈ R such that y1 < y2 and u1 = u0 (y1 ) > u0 (y2 ) = u2 . If (1.37) holds then f ′ (u0 (y1 )) > f ′ (u0 (y2 )) so that the characteristics drawn from 2 −y1 points (y1 , 0) and (y2 , 0) intersect at the point when t = f ′ (u0 (y1y))−f ′ (u (y )) with u 0 2 having values u(y1 ) and u(y2 ) at that point. The Lax inequality can be obtained from the Jump condition. To see this, let f be a convex function. Then f ′′ > 0 which implies that f ′ is increasing. Thus if ul > ur , then f ′ (ul ) > f ′ (ur ). By the Mean Value Theorem there exists ζ ∈ [ur , ul ] such that f ′ (ζ) = f (ul ) − f (ur ) = ρ. ul − ur Since f ′ is increasing we have that f ′ (ul ) > f ′ (ζ) > f ′ (ur ), which leads to the Lax inequality (1.37). However, not all weak solutions of equations (1.8) - (1.9) satisfying the jump condition (1.33) will also satisfy the Nonlinear Hyperbolic Conservation Laws 19 Lax condition (1.37). For example, the lines of discontinuity in the solutions obtained in Example 1.12 that are shown to satisfy the jump condition do not satisfy the Lax inequality (1.37). If all the discontinuities of a weak solution satisfy condition (1.37), then no characteristics drawn backward will intersects the curve of discontinuity, see ﬁgure 1.2. A discontinuity satisfying both the jump condition (1.33) and the Lax inequality (1.37) is called a shock. A weak solution having only shocks as discontinuity is called shock wave solution. Lax showed that there is exactly one shock wave solution u of the equations (1.8) if the initial condition is such that the left initial state is greater than the right initial state, that is ul > ur . If we consider the Burger’s equation and take the initial condition to be { 1 if x < 0, u0 (x) = 0 if x > 0, then the shock wave solution is expressed as { 1 if x < ρt, u(x, t) = 0 if x > ρt. 2) and the Lax condition f ′ (u1 ) > ρ > f ′ (u2 ) Clearly, the jump condition ρ = (u1 +u 2 are both satisﬁed if and only if ρ = 21 . 6 Curve of discontinuity C x = ρt R Ωl Ωr Figure 1.2 A more general example is the Riemann’s problem illustrated below. Example 1.17 (The Riemann’s Problem). The Riemann’s problem is the Cauchy problem ut + (f (u))x = 0 in R × (0, ∞) { ul if x < 0 u(x, 0) = u0 (x) = ur if x > 0. Here, ul , ur ∈ R are the left and right initial states of u. Note that ul ̸= ur . If 20 Introduction ul > ur the shock wave solution to the Riemann’s problem is { u(x, t) := ul if x < ρt, ur if x > ρt. 2 The main result by Lax is summarized in the following Theorem 1.18. Let f : R −→ R be a C 2 - smooth and convex function. If u0 ∈ L1 (R) then there exists a weak solution u of the Cauchy problem (1.8) (1.9) which satisﬁes (1.37). The solution u is deﬁned as ( x − y0 u(x, t) = b t ) for each t > 0 and a.e. x ∈ R (1.38) where y0 = y0 (x, t) is the the value of y that minimizes K(x, y, t) = U0 (y) + tG( x−y ). t Here the function b(s) is deﬁned as b(s) = (f ′ (s))−1 and G(s) is deﬁned as the solution of dG(s) = b(s), ds G(c) = 0, with c = f ′ (0), and ∫ U0 (y) = y −∞ u0 (s)ds. The discontinuity of the solution u constructed in Theorem 1.18 are shocks, which means u satisﬁes the Lax inequality (1.37). In addition, u has the semi group property which means that if u(x, t1 ) is taken as a new initial value, the solution u(x, t2 ) at t2 > t1 corresponding to the initial condition u(x, t1 ) equals u(x, t1 + t2 ). Remark 1.19. (i) For ﬁxed t > 0, the function y0 (x, t) is an increasing function of x, see [79, Lemma 3.3]. (ii)The shock wave solution constructed in Theorem 1.18 satisﬁes the Oleinik inequality (1.34). Indeed, since b and y(x, t) are increasing functions, then for Nonlinear Hyperbolic Conservation Laws x2 > x1 we have that ( ) ( ) ( ) x1 − y0 (x1 , t) u(x1 , t) = b t x1 − y0 (x2 , t) ≥b t x2 − y0 (x2 , t) ≥b t = u(x2 , t) − α 21 −α x2 − x1 t x2 − x1 ; t which implies α u(x2 , t) − u(x1 , t) ≤ , x2 − x1 t here α > 0 is a Lipschitz constant for the function b. A generalized form of the Lax condition (1.37) was given by Oleinik [96]. For 0 ≤ α ≤ 1, f (αur + (1 − α)ul ) ≤ αf (ur ) + (1 − α)f (ul ) f (αur + (1 − α)ul ) ≥ αf (ur ) + (1 − α)f (ul ) if ul > ur , if ul < ur . (1.39) (1.40) The inequality (1.39) implies that f is convex. Geometrically this means that the graph of f over the interval [ur , ul ] lies below the chord of f drawn from the point (ul , f (ul )) to the point(ur , f (ur )). On the other hand, the inequality (1.40) implies that f is concave, which means that the graph of f over the interval [ul , ur ] lies above the chord of f drawn from the point (ul , f (ul )) to the point(ur , f (ur )). We now discuss the relationship between the Lax inequality (1.37) and the generalized Oleinik inequality (1.39). To start with, the convexity of f implies that the inequality (1.39) is equivalent to f (u∗ ) − f (ul ) f (ur ) − f (u∗ ) ≥ . u∗ − ul ur − u∗ (1.41) for every u∗ = αur + (1 − α)ul , with 0 < α < 1. Combining the inequality (1.41) with the Mean Value Theorem, we have that there exists ζ ∈ [ur , ul ] such that f ′ (ζ) = f (ul ) − f (ur ) =ρ ul − ur and f (u∗ ) − f (ul ) f (ur ) − f (u∗ ) ′ ≥ f (ζ) ≥ . u∗ − ul ur − u∗ (1.42) 22 Introduction Now Taking limits as u∗ −→ ul and u∗ −→ ur in (1.42), we have f ′ (ul ) ≥ ρ ≥ f ′ (ur ), which is the Lax inequality. Thus, the generalized inequality (1.39) implies the Lax inequality. On the other hand, if the ﬂux function f, is a convex function, which imply f ′′ > 0, then the Lax inequality (1.37) would implies the inequality (1.39). Essentially, all the conditions considered so far require that the ﬂux function f be convex or concave. Krushkov [74] introduced a more general admissibility condition for a ﬂux function f that is not necessarily convex or concave. One advantage of the Kruzkov condition is that it is also applicable to systems of conservation laws in more than one dimension, whereas the Oleinik condition is limited to scalar conservation laws in one dimension. Although the Lax inequality is applicable to systems of conservation laws, it still requires the convexity of the ﬂux function f , moreover the Lax inequality is limited to systems in one spatial dimension. Kruzkov’s admissibility condition is given below. Admissibility condition 3 (The Entropy condition) The admissibility condition discussed in this section was ﬁrst introduced by Kruzhkov [74]. It is formulated in terms of entropy/entropy ﬂux pairs. A pair (Φ, Ψ) of real C ∞ - smooth functions is called an entropy/entropy ﬂux pair for the conservation law (1.8) if Ψ′ (z) = Φ′ (z)f ′ (z), for all z ∈ R. (1.43) The function Ψ is called an entropy ﬂux function for the entropy function Φ. For every convex function Φ we can ﬁnd a corresponding entropy ﬂux function Ψ given by ∫ z Ψ(z) = Φ′ (z)f ′ (z), z ∈ R. (1.44) z0 For each entropy/entropy ﬂux pair (Φ, Ψ) one may formulate an admissibility condition, known as entropy condition given by ∫∫ Φ(u)ϕt + Ψ(u)ϕx dxdt ≥ 0, Ω ⊆ R × [0, ∞) Ω (1.45) ∀ϕ ∈ C0∞ (R × (0, ∞)), ϕ ≥ 0, supp ϕ ⊂ Ω. Deﬁnition 1.20 (Entropy solution). The function u ∈ C([0, ∞), L1 (R))∩L∞ (R× (0, ∞)) is called an entropy solution of the Cauchy problem (1.8) - (1.9) if it satisﬁes the entropy condition (1.45) for each entropy/entropy-ﬂux pair (Φ, Ψ), and u(., t) → u0 in L1 (R) as t → 0. The main result on the existence and uniqueness of entropy solution is given below. Nonlinear Hyperbolic Conservation Laws 23 Theorem 1.21. [109, Theorem 2.3.5] For every function u0 ∈ L∞ (R), there ∩ exists one and only one entropy solution u ∈ L∞ (R × [0, T )) C([0, T ); L1loc (R)) of (1.8) -(1.9). The entropy solution u satisﬁes the maximum principle ∥u∥L∞ (R×[0,T )) = ∥u0 ∥L∞ (R) . Remark 1.22. Theorem 1.21 is valid in several spatial dimension, [75], [109]. By Theorem 1.21, one can construct a semi group operator S(t), associated with the entropy solution u(x, t) with respect to the initial data u0 and time t > 0 written as, St u0 (x) = u(x, t). The semi group S : D ×[0, ∞) −→ D with D ⊂ L1 (R) a closed domain containing all functions with bounded total variation, has the following properties [30, 102]: (i) S0 u = u, St+s u = St Ss u, (ii) St is uniformly Lipschitz continuous w.r.t time and initial data: There exists L, L′ > 0 such that ∥St u0 − Ss v0 ∥ ≤ L∥u0 − v0 ∥ + L′ |t − s|. In his proof, Kruzhkov considered a family of entropy-entropy ﬂux pairs (Φk , Ψk )k∈N , but Panov [98] has shown that it is not necessary to consider the whole family of entropy/entropy ﬂux pair. A single pair of entropy/entropy ﬂux pair (Φ, Ψ) is suﬃcient to characterize entropy solutions of (1.8) - (1.9). We now show that the entropy solution u of the Cauchy problem (1.8) - (1.9) is a weak solution, see [59]. In this regard, assume u is C 1 - smooth in the left subregion Ωl and right subregion Ωr of some region Ω ⊆ (R × [0, ∞)) divided by a smooth curve C. Let u also satisfy the entropy condition. If we take Φ(u) = ±u and Ψ(u) = F (u) in (1.45) we see that ut + f (u)x = 0 in Ωl , Ωr . Integrating (1.45) by parts we get ∫ ∫ ∫ ∫ Φ(u)ϕt + Ψ(u)ϕx dxdt + Φ(u)ϕt + Ψ(u)ϕx dxdt ≥ 0 Ωl Ωr from where we deduce ∫ ϕ[(Φ(ul ) − Φ(ur ))n2 + (Ψ(ul ) − Ψ(ur ))n1 ]dS ≥ 0 (1.46) C where n = (n1 , n2 ) is the unit normal to C pointing from Ωl to Ωr . Suppose that the curve C is given by x = s(t) for some smooth function s : [0, ∞) −→ R. Then ṡ) . Consequently (1.46) becomes n = (n1 , n2 ) = √(1,− 1+ṡ2 ṡ(Φ(ur ) − Φ(ul )) ≥ Ψ(ur ) − Ψ(ul ) along C, (1.47) 24 Introduction which leads to the jump condition ṡ[[u]] = [[f (u)]]. (1.48) Thus the entropy condition (1.45) satisﬁes the jump condition and thus a weak solution. Suppose ul > ur . Fix u∗ such that ul > u∗ > ur and deﬁne the entropy/entropy ﬂux pair as { (z − u∗ ) ifz − u∗ > 0 Φ(z) = 0 otherwise ∫ and z Ψ(z) = sgn+ (v − u)f ′ (v)dv. ul Then and Φ(ur ) − Φ(ul ) = u∗ − ul Ψ(ur ) − Ψ(ul ) = f (u∗ ) − f (ul ). Consequently (1.47) implies ṡ(u∗ − ul ) ≥ f (u∗ ) − f (ul ) which, when combined with (1.48), gives f (u∗ ) − f (ul ) . ṡ ≤ u∗ − ul (1.49) Similarly, if ur > ul and ur > u∗ > ul then f (ur ) − f (u∗ ) ṡ ≥ . ur − u∗ (1.50) Conditions (1.49) and (1.50) gives condition (1.41). Thus the entropy condition (1.45) implies the Lax condition. Remark 1.23. (i) Note that any entropy solution of (1.8) - (1.9) is also a weak solution of (1.8). This follows if we set Φ(z) = z, z ∈ R, in which case Ψ = f. (ii) If u ∈ C 1 (Ω) is a classical solution of the initial value problem, then Φ′ (u)(ut + (f (u))x ) = 0 for any convex function Φ. This further implies 0 = Φ′ (u)ut + Φ′ (u)f ′ (u)ux = Φ′ (u)ut + Ψ′ (u)ux , with Ψ any entropy ﬂux associated with Φ. This veriﬁes that a classical solution is also an entropy solution. The theory of hyperbolic conservation laws has developed in a number of directions. One major approach consists of considering weak solutions in suitable Nonlinear Hyperbolic Conservation Laws 25 spaces of functions with bounded variation (BV functions). The problem, and a very diﬃcult one, is to prove that various approximating schemes such as the vanishing viscosity methods, the Glimm scheme, wave front tracking etc, converge to the entropy solution, see [3, 16, 26, 27, 64] for details. The BV approach consists of proving convergence of these schemes under assumption on the initial condition u0 related to its total variation. Typically, one assumes that the total variation satisﬁes a smallness condition, see [22]. Another approach is to construct weak solutions through weak convergence and compensated compactness arguments, see for instance [87, 114, 130] and Section 1.1.5. In the next section we discuss the vanishing viscosity method for conservation laws. 1.1.4 Solutions of Scalar Conservation Laws via Vanishing Viscosity The role of the entropy condition in conservation laws is to distinguish between the physically relevant weak solution and other, possibly irrelevant weak solutions. One method for obtaining and analyzing entropy solutions to hyperbolic conservation laws is to modify the given conservation law by adding a small perturbation term to the right-hand side of the equation, for example, εuxx , with ε << 1, to obtain from (1.8) a regularized equation uεt + (f (uε ))x − εuxx = 0 (1.51) The motivation that is often given for the study of the Cauchy problem (1.8) (1.9) through the regularized problem ut + (f (u))x = εuxx in R × (0, ∞), u(x, 0) = u0 (x) x ∈ R. ε>0 (1.52) (1.53) is that a physically and mathematically correct solution of (1.8) - (1.9) should arise as the limit of the solution uε of (1.52) - (1.53), as the parameter ε tends to zero. This method is generally known as the vanishing viscosity method [21, 23, 52, 111, 109, 113, 128, 129]. In this regard, we may recall that the model for thermoelastic materials under adiabatic conditions is a ﬁrst order system of hyperbolic PDEs, while that for thermoviscoelastic, heat-conducting materials is a second order PDE, containing a diﬀusive term, see [52]. Every material has a degree of viscous response and conducts heat. Classifying a material as an elastic nonconductor of heat simply means that viscosity and heat conductivity are negligible, but not totally absent. The consequence of this is that the theory of adiabatic thermoelasticity may be physically meaningful only as a limiting case of thermoviscoelasticity, with viscosity and heat conductivity tending to zero see [52, 111]. In the same way hyperbolic conservation laws are considered as a limiting case of the parabolic equation (1.52). 26 Introduction We note here that solutions of nonlinear PDEs are in general highly unstable with respect to small perturbations of the equation. Thus in spite of the physical intuition underlying such viscosity methods, the rigorous mathematical analysis of the limiting behavior of solutions of equations like (1.52) - (1.53) as ε tends to 0 is highly non trivial. It is well known that for any ε > 0, and for bounded and measurable initial data, there exists a unique classical solution uε of the parabolic equation (1.52) (1.53), see [56, 95, 130]. This unique solution uε of equation (1.52) - (1.53) is called a viscosity solution of (1.52) - (1.53). The following general theorem guarantees the existence of a sequence of solutions to the parabolic problem (1.52) - (1.53). Theorem 1.24. [130, Theorem 1.0.2] (i) For any ﬁxed ε > 0, the Cauchy problem (1.52) - (1.53) with u0 ∈ L∞ has a local classical solution uε ∈ C ∞ (R × (0, τ )) for a small time τ, which depends only on the L∞ norm of the initial data u0 . (ii) If the solution uε has an a priori L∞ bound ∥uε (·, t)∥L∞ ≤ M (ε, T ) for t ∈ [0, T ], then the solution exists on R × [0, T ]. (iii) The solution uε satisﬁes: lim uε (x, t) = 0, |x|−→∞ if lim u0 (x) = 0. |x|−→∞ Following the standard theory for parabolic equations, the global existence of a solution can easily be obtained by applying the contraction mapping principle to an integral representation of the solution. Whenever there is a local solution with a priori L∞ bound, the domain of existence of solution can be extended, step by step, to any further time T since the step-time depends only on the L∞ norm of the initial condition. Details of the proof can be found in [77, 111]. The two fundamental questions concerning the solution uε of (1.52) - (1.53) are the following. (i) In what sense does the sequence of functions uε converge to a limit function u as ε tends to 0? (ii) Given that uε converges to some u in a speciﬁed way, in what sense can we interpret u as a solution of the Cauchy problem (1.8) - (1.9)? In particular, if uε is the unique classical solution of (1.52)-(1.53) and uε converges to some function u as ε tends to 0, is u an entropy solution of the Cauchy problem (1.8) - (1.9)? A partial answer to the above questions is given in the following Theorem, see [52, 58, 75]. Theorem 1.25. [52, Theorem 6.3.1] Suppose uε is the solution of (1.52), (1.53), and assume that for some sequence {εn } with εn → 0 as n → ∞, we have that uεn is norm bounded in L∞ and uεn (x, t) → u(x, t) as εn −→ 0 in R for almost all Nonlinear Hyperbolic Conservation Laws 27 (x, t) ∈ R × [0, ∞). In other words, uεn → u boundedly a.e on R × [0, ∞). Then u is an entropy solution of (1.8)-(1.9) on R × [0, ∞). Remark 1.26. (i) Since the weak solutions of (1.8)-(1.9) are in L∞ , and are typically not continuous, it may happen that as the smooth function uε approaches u the functions uεx and uεxx become unbounded, in a neighborhood of a point of discontinuity of u. Thus establishing the convergence uε → u is a highly nontrivial issue. (ii)If uε converges to u in the weak sense only; the sequence F (uε ) will converge in the weak sense but not to F (u). In this regard, we have the following Theorem 1.27 ([78], [80]). If the sequence of functions un converges in the weak sense to a limit u, then f (un ) converges in the weak sense to f (u) if and only if un → u strongly in L1 . The theory of scalar conservation laws via vanishing viscosity was initiated by E. Hopf in his 1950 paper [69]. In that paper, Hopf considered the viscous Burgers equation ( ε 2) (u ) ε ut + = εuεxx in R × (0, ∞) (1.54) 2 x uε (x, 0) = u0 (x) R × {t = 0}, (1.55) and showed that the solution to the Cauchy problem (1.54) - (1.55) can be expressed via an explicit formula ∫ ∞ x−y − 1 K(x,y,t) e 2ε dy uε (x, t) = −∞ (1.56) ∫ ∞ t − 1 K(x,y,t) 2ε e dy −∞ where (x − y)2 K(x, y, t) = + 2t The result by Hopf is stated below. ∫ y u0 (s)ds. (1.57) 0 Theorem 1.28. [69, Hopf E.] Suppose u0 ∈ L1loc (R) is such that ∫ x u0 (ξ)dξ = o(x2 ) for |x| large. (1.58) 0 Then there exists a classical solution of equation (1.54)-(1.55) given by (1.56). The solution uε satisﬁes the initial condition: For all a ∈ R, ∫ x ∫ a ε u (ξ, t)dξ −→ u0 (ξ)dξ as x −→ a, t −→ 0. (1.59) 0 0 If, in addition, u0 (x) is continuous at x = a then uε (x, t) −→ u0 (a) as x −→ a, t −→ 0. (1.60) A solution of (1.54) - (1.55) which is C 2 -smooth in the interval 0 < t < T and satisﬁes (1.59) for each value of a necessarily coincides with (1.56) in the interval. 28 Introduction In his proof Hopf’s technique was to ﬁrst transform the equation (1.54) - (1.55) into the heat equation zt − εzxx = 0, z(x, 0) = z0 (x) = e− 2ε 1 ∫x 0 (1.61) (u0 (ν))dν (1.62) using the transformation equation 1 −( 2ε z=e ∫ uε dx) with inverse uε = −2ε(log z)x = −2ε( zx ). z The solution of (1.61) - (1.62) is then obtained as ∫ ∞ (x−y)2 1 uε (y, 0)e− 4εt . z(x, t) = √ 4πεt −∞ (1.63) (1.64) Substituting the expression (1.64) for z(x, t) in (1.63) one obtains the formula (1.56) as the unique solution of equations (1.54) - (1.55). The condition (1.58) on the initial value is necessary to guarantee the convergence of the deﬁnite integral (1.64), thus also those in the expression (1.56) for the solution of (1.54) - (1.55). The function K(x, y, t) has the following properties which are used in the sequel (P1) K(x, y, t) is a continuous function of y with x, t being ﬁxed. (P2) For any ﬁxed value of x and t, the function K(x, y, t) attains its minimum value at one or several values of y. Furthermore, the set {y : K(x, y, t) = minK(x, z, t)} z∈R (1.65) is a compact set. (P3) K(x, ymin (x, t), t) = K(x, ymax (x, t), t) = m(x, t) is a continuous function of x, t in the half plane t > 0, where ymin and ymax denote the minimum and maximum values of y, respectively, for which K(x, y, t) attains its minimum. Note that the assumption (1.58) is essential for obtaining property (P2). Indeed, using (1.58) one can see that K 1 = >0 2t |y|−→∞ y 2 lim which implies that the set (1.65) is bounded. The fact that it is closed follows from the continuity of K. It follows from (P2) that the set (1.65) is bounded in R. Hence the functions ymin (x, t) = min{y : K(x, y, t) = minK(x, z, t)} z∈R Nonlinear Hyperbolic Conservation Laws 29 and ymax (x, t) = max{y : K(x, y, t) = minK(x, z, t)}. z∈R are well deﬁned. The functions ymin and ymax have the following properties, [69, Lemma 1 and Lemma 3 ]. (Y1) If x1 < x2 then ymax (x1 , t) ≤ ymin (x2 , t). (Y2) ymin (x− , t) = ymin (x, t), ymax (x+ , t) = ymax (x, t), where ymin (x− , t) is the left limit of ymin and ymax (x+ , t) the right limit of ymax , for ﬁxed t. (Y3) lim ymin (x, t) = +∞, x−→+∞ lim ymax (x, t) = −∞. x−→−∞ (Y4) As functions of x and t, ymin and ymax are lower semi-continuous and upper semi-continuous respectively. Therefore both functions are continuous at all points (x, t) where ymin (x, t) = ymax (x, t). Let us recall [4] the deﬁnition of lower semi-continuous and upper semi-continuous functions. In this regard, let R∗ = R∪{±∞} be the set of extended real numbers. A function u : Ω −→ R∗ is called lower semi-continuous at point (α, θ) ∈ Ω if for every m < u(α, θ) there exist η > 0 such that |x − α| < η, |t − θ| < η, =⇒ m < u(x, t). A function u : Ω −→ R∗ is called upper semi-continuous at point (α, θ) ∈ Ω if for every m > u(α, θ) there exist η > 0 such that |x − α| < η, |t − θ| < η, =⇒ m > u(x, t). A function u : Ω −→ R∗ is called lower(upper) semi-continuous in Ω if it is lower(upper) semi-continuous at every point of Ω. It is clear, from property (Y1), that ymin and ymax are monotone functions in x. Since a monotone function has only a denumerable number of discontinuities one can conclude that, for any t > 0, ymin (x, t) = ymax (x, t) for all x except at some denumerable set of values of x where ymin < ymax . That is, for t > 0 the set {x : ymin (x, t) < ymax (x, t)} is countable. (1.66) The convergence Theorem for the solution uε of (1.54) - (1.55) is stated as follows Theorem 1.29. [69, Theorem 3] Let uε (x, t) be the solution of (1.54) - (1.55) with u0 satisfying (1.58). Then for all x and t > 0, x − ymax (x, t) ≤ lim inf uε (α, θ) ≤ t α→x θ→t ε→0 lim sup uε (α, θ) ≤ α→x θ→t ε→0 x − ymin (x, t) . t 30 Introduction In particular, lim uε (α, θ) = α→x θ→t ε→0 x − ymax (x, t) x − ymin (x, t) = t t holds at every point (x, t), t > 0, at which ymax (x, t) = ymin (x, t). Deﬁne the function u as u(x, t) := lim uε (α, θ) α→x θ→t ε→0 (1.67) at every point (x, t), t > 0 where this limit exists. By Theorem 1.29, the limit (1.67) will exist at every point where ymin (x, t) = ymax (x, t). At these points, u is well deﬁned and continuous. Furthermore, for each t > 0, u has a denumerably many discontinuities, as we noted above. Therefore we can conclude that the set of discontinuities has measure zero. Thus the convergence of uε to u as given by (1.67) is almost everywhere. We deﬁne u and u on R × [0, ∞), by u(x, t) = lim inf uε (α, θ) α→x θ→t ε→0 (1.68) = sup{inf{uε (α, θ) : |α − x| < η, |θ − t| < η, ε < η} : η > 0} and u(x, t) = lim sup uε (α, θ) α→x θ→t ε→0 (1.69) = inf{sup{uε (α, θ) : |α − x| < η, |θ − t| < η, ε < η} : η > 0} Note that u(x, t) ≤ u(x, t) (x, t) ∈ R × [0, ∞). and u(x, t) = u(x, t) = u(x, t) whenever u(x, t) is deﬁned for (x, t) ∈ R × [0, ∞). The function u(x, t) deﬁned by (1.67) is a weak solution to the Cauchy problem of the inviscid Burgers equation (1.12) - (1.13). To see this, let us note ﬁrst that Nonlinear Hyperbolic Conservation Laws 31 a solution uε of (1.54) - (1.55) satisﬁes the equation ∫ ∞ ∫ ∞∫ ∞ (uε )2 ε ϕx }dxdt + u0 ϕ|t=0 dx {u ϕt + 2 −∞ 0 −∞ ∫ ∞∫ ∞ +ε uε ϕxx dxdt = 0, 0 (1.70) −∞ C0∞ (Ω). for each test function ϕ ∈ From the convergence Theorem 1.29 it follows that every point (x, t) has a neighborhood in which the solutions uε of (1.54) (1.55) are uniformly bounded as ε tends to 0. As a result of this, one can pass through the limit ε −→ 0 in (1.70) with ϕ being ﬁxed. Thus we have ∫ ∞∫ ∞ ∫ ∞ u2 {uϕt + ϕx }dxdt + u0 ϕ|t=0 dx = 0, 2 −∞ 0 −∞ which shows that u(x, t) is a weak solution of the Cauchy problem for the inviscid Burgers equation (1.12). Furthermore, the limit function u(x, t) deﬁned by (1.67) is an entropy solution of the equation (1.12) - (1.13). To see this, let (Φ, Ψ) be any given pair of entropy/entropy-ﬂux pair for the equation (1.8). Multiply equation (1.54) by Φ′ (uε ) uεt Φ′ (uε ) + uε uεx Φ′ (uε ) = εuεxx Φ′ (uε ) = ε((Φ(uε ))xx − Φ′′ (uε )(uε )2 ). Using (1.43) we get (Φ(uε ))t + (Ψ(uε ))x = ε((Φ(uε ))xx − Φ′′ (uε )(uε )2 ). (1.71) Now multiply equation (1.71) by ϕ ∈ C0∞ (R × (0, ∞)), ϕ ≥ 0 and integrate over R × [0, ∞). ∫ ∞ ∫ ∞ (Φ(uε )ϕt + Φ(uε )ϕx )dxdt −∞ 0 ∫ ∞∫ ∞ ∫ ∞∫ ∞ ε = ε Φ(u )ϕxx dxdt − ε Φ′′ (uε )(uε )2 ϕdxdt. 0 −∞ 0 ∫−∞ ∞ ∫ ∞ ≥ ε Φ(uε )ϕxx dxdt (1.72) −∞ 0 since Φ′′ (uε )(uε )2 ϕ ≥ 0. Again from the convergence Theorem 1.29 it follows that every point (x, t) has a neighborhood in which the solutions uε of (1.54) - (1.55) are uniformly bounded as ε tends to 0. Moreover, the function Φ(u) is convex and thus continuous. As such one can pass to the limit as ε −→ 0 in (1.72) with ϕ being ﬁxed. Thus we have ∫ ∞∫ ∞ (Φ(u)ϕt + Φ(u)ϕx )dxdt ≥ 0. −∞ 0 32 Introduction which shows that u(x, t) is an entropy solution of the Cauchy problem for the inviscid Burgers equation (1.12). Lax [78] obtained a result similar to that of Hopf by showing that the weak solution ( ) x − y0 u(x, t) = b for each t > 0 and a.e. x ∈ R t obtained in Theorem 1.18 can be written as the limit of the solution uε of the Burgers equations(1.12) - (1.13). To see this, consider the equation ut + (f (u))x = 1 u2 uxx , f (u) = . 2n 2 (1.73) with the initial condition un (x, 0) = u0 (x) Then similar to Hopf’s result, we see that the function ∫ ∞ x−y −nK(x,y,t) b( )e dy un = −∞∫ ∞ t −nK(x,y,t) dy −∞ e (1.74) (1.75) is a solution to the equation (1.73) - (1.74) and we deﬁne u(x, t) = lim un . n→∞ As it was in case of Hopf’s result, the convergence of un to u as n tends to ∞ is almost everywhere, see [79]. Likewise, deﬁne ∫∞ x−y −nK dy −∞ f (b( t ))e ∫∞ fn = . (1.76) −nK dy e −∞ Then f = lim fn , a.e. n→∞ Here, x−y ), t the function b(s) is deﬁned as b(s) = (f ′ (s))−1 , G(s) is deﬁned as the solution of K(x, y, t) = U0 (y) + tG( dG(s) = b(s), ds G(c) = 0, with f ′ (0) = c, ∫ and U0 (y) = y u0 (s)ds. 0 If we denote by Vn the function Vn = log ∫ ∞ −∞ e−nK(x,y,t) dy Nonlinear Hyperbolic Conservation Laws then un = − 33 1 ∂ Vn n ∂x and 1∂ Vn n ∂t provided that f (b(z)) = zb(z) − G(z). It then follows that fn (x, t) = − (un )t + (fn )x = 0. (1.77) Multiply equation (1.77) by a test function ϕ ∈ C0∞ (Ω) and integrate to get ∫ ∫ un ϕt + fn ϕx = 0, Ω letting n → ∞ we obtain the limit relation ∫ ∫ uϕt + f (u)ϕx = 0, Ω This shows that u is a weak solution of equations (1.8) - (1.9), see [78, Theorem 2.1]. For an arbitrary function f, there are no explicit formula for the solution to the viscous equation. However, Oleinik [94] proved that for a general convex or concave function, the solutions of the parabolic problem (1.52) - (1.53) tends to a weak solution of (1.8). A simpler proof was given by Ladyzhenskaya in [76]. As mentioned in Section 1.1.3, see in particular Theorem 1.15, Oleinik [95, 96] showed that there exists a unique solution of (1.8)-(1.9) that satisﬁes the admissibility condition (1.34), provided that the ﬂux function f is convex. This solution is constructed as a limit of solutions uε of equation (1.52) -(1.53) obtained through a ﬁnite diﬀerence scheme introduced by Lax in [80], see also [81, 82]. It was subsequently shown that u is in fact the unique solution of (1.8) - (1.9) satisfying (1.34), see [111, Theorem 16.11]. Kruzhkov [74], [75] introduced a new method to apply the vanishing viscosity method to a larger class of equations. For initial data u0 ∈ L∞ , he proved existence and uniqueness of the classical solution uε (x, t) of (1.52)-(1.53). Using a family of entropy-entropy ﬂux pairs (Φk , Ψk )k∈R where Φk (u) = |u − k| and Ψk (u) := sgn(u − k)(f (u) − f (k)), he showed that the solution uε (x, t) of equations (1.52) - (1.53) converges as ε tends to 0 almost everywhere to a weak solution u(x, t) of the Cauchy problem (1.8) -(1.9). Theorem 1.30. [75, Kruzhkov]] Let u0 ∈ L∞ (R). Then the solution uε (x, t) of problem (1.52) - (1.53) converges as ε −→ 0 almost everywhere in R × [0, T ) to a function u(x, t) which is a weak solution of the problem (1.8) - (1.9). 34 Introduction In the proof of the above theorem, a priori bounds (independent of ε) were obtained for the solutions uε (x, t) which ensures the compactness of the family of functions {uε (x, t) : t > 0} with respect to the L1 - norm. This in turn guarantees the existence of a subsequence uεn of uε that converges almost everywhere to the weak solution u(x, t). Thus a weak solution of the Cauchy problem (1.8) - (1.9) is constructed as the limit of solution uε of the parabolic problem (1.52) - (1.53). The following theorem shows that the weak solution constructed above is an entropy solution. Theorem 1.31. Let u0 ∈ L∞ (R). If uε (x, t) converges to a function u(x, t) almost everywhere as ε −→ 0 in R × [0, T ). Then the solution u is the entropy solution of the Cauchy problem (1.8) - (1.9). The properties of the solution u(x, t) of problem (1.8) - (1.9) is addressed in the following result, see also [74, 75]. Theorem 1.32. [109, Proposition 2.3.6] Let u0 , v0 ∈ L∞ and u and v be the entropy solutions of (1.8) -(1.9) associated with u0 and v0 respectively. Let M = sup{|f ′ (s)| : s ∈ [inf(u0 (x), v0 (x)), sup(u0 (x), v0 (x))]}. Then the following properties are satisﬁed: (P1) For all t > 0 and every interval [a, b], we have ∫ b ∫ b+M t |v(x, t) − u(x, t)|dx ≤ |v0 (x) − u0 (x)|dx. a a+M t (P2) If u0 and v0 coincide on [x0 − δ, x0 + δ] for some δ >, 0 then u and v coincide on the triangle {(x, t) : |x − x0 | + M t < δ}. (P3) If u0 − v0 ∈ L1 (R), then u(t) − v(t) ∈ L1 (R) ∀ t > 0, where u(t) := u(·, t) and v(t) := v(·, t). Moreover, ∥v(t) − u(t)∥L1 (R) ≤ ∥v0 − u0 ∥L1 (R) , and ∫ R ∫ (v(x, t) − u(x, t))dx = R (v0 (x) − u0 (x))dx. (P4) If u0 ∈ L1 (R), then u(t) ∈ L1 (R), for all t > 0, and ∫ ∫ ∥u(t)∥L1 (R) ≤ ∥u0 ∥L1 (R) , u(x, t)dx = u0 (x)dx. R R (P5) If u0 (x) ≤ v0 (x) for almost all x ∈ R, then u(x, t) ≤ v(x, t) for almost all (x, t) ∈ R × [0, ∞). (P6) If u0 has bounded total variation, then u(t) has bounded total variation for all t > 0 and T V (u(t)) ≤ T V (u0 ). Nonlinear Hyperbolic Conservation Laws 35 Remark 1.33. The proof of the above Theorem 1.32 is based on the fact that the semigroup operator St of (1.8) is a contraction in L1 (R) ∩ L∞ (R) with respect to the L1 -norm. This fact is expressed in property (P3), which implies that if u0 ∈ BV, then u ∈ BV for all t > 0 as stated in property (P6). Property (P4) is a consequence of property (P3), and it leads to property (P5). 1.1.5 Compensated Compactness Methods for Nonlinear Conservation Laws As mentioned, the vanishing viscosity method involves the construction of physically meaningful solution of the Cauchy problem (1.8) - (1.9) as the limit of the solutions of the parabolic problem (1.52) - (1.53). The strategy usually adopted in the literature is to obtain a priori bounds on solutions uε , of (1.52) - (1.53), that is, to show that ∥uε ∥L∞ ≤ C, with C a constant independent of ε. Such an estimate is then used to show that uε converges to some function u, in an appropriate sense, as ε −→ 0. The ﬁnal step is usually to show, using a suitable compactness argument, that the limit function u is the entropy solution to the Cauchy problem (1.8) - (1.9). From the forgoing discussion it is clear that an essential part of the vanishing viscosity method is the study of the compactness of the set {uε (x, t) : ε > 0} of solutions of the viscous problem (1.52) - (1.53) with respect to the L1 topology. That is, the possibility of obtaining the strong convergence of a subsequence uεn , where εn −→ 0 as n −→ ∞. These compactness arguments are closely related to the decay of entropy solution for large time, see for instance [41, 42, 43, 44, 45, 46] and [79]. The issue of compactness of the set {uε : ε > 0} has been addressed mainly in the following ways: (i) The compactness approach is based on a priori BV bounds, which has proven to be more useful in the study of scalar conservation laws or systems of conservation laws, see Section 1.1.4, [21] and [52]. A wide range of numerical examples on the applications of this compactness approach to ﬁnite diﬀerence approximations exist, see for instance [65], [66], [78], [94]. However, the approach is essentially limited to systems in one spatial dimension, except for the Kruzkov’s multidimensional BV-based existence result which rely on the translation invariance of the underlying solution operator. (ii) Compensated Compactness, developed by Tartar [114], [116] and Murat [87], [88], is based on certain L2 -type, H −1 −compact entropy production bounds, which replaces the BV bounds in the BV compactness method, see [47, 49, 130]. So far existence results based on compensated compactness arguments are limited to a system of two conservation laws in two spatial dimension. 36 Introduction In the light of the importance of compactness arguments in the study of nonlinear conservation laws, we discuss brieﬂy some of the main point related to such compactness methods and their applications to nonlinear conservation laws. BV Compactness If a collection of functions {uε (x, t) : ε > 0} satisﬁes ∥uε ∥L∞ ≤ C1 and T V {∥uε ∥} ≤ C2 for all ε > 0, where C1 and C2 are constants independent of ε, then by Helly’s Compactness Theorem [28, Theorem 2.3] there exists a sequence εk −→ 0 such that the sequence {uεk } converges almost everywhere to some u ∈ L∞ . The use of BV compactness framework in proving existence and uniqueness of solution of (1.8) - (1.9) can be found in [21], [64], [95] and [124], see [51] for other applications. Compactness in L1 Suppose that {uε : ε > 0} satisﬁes (i) ∥uε ∥L1 ≤ C, C > 0 a constant independent of ε. (ii) {uε (x, t) : ε > 0} is equicontinuous in L1loc (R × [0, ∞)), that is, for any compact subset Ω ⊆ R × [0, ∞), ∥uε (x + ∆x, t + ∆t) − uε (x, t)∥L1 −→ 0 uniformly on Ω as ∆x, ∆t −→ 0. Then there exists a sequence εk −→ 0 such that the sequence uεk −→ u in L1loc (R × [0, ∞). The above compactness in L1 was applied by Kruzhkov to prove existence and uniqueness of entropy solution to the conservation laws [75]. Remark 1.34. (i) Existence of C ∞ solution uε follows from the boundedness of the initial data and Theorem 1.24. The proof of the compactness of the entropy bound is similar to the case where solutions are in L∞ , which was brieﬂy shown above. (ii) We remark here that the concept of generalized solutions of hyperbolic systems of conservation laws is a straight forward generalization of scalar conservation laws discussed above, we therefore omit it and refer the reader [2, 13, 29, 30, 31, 34, 35, 36, 37, 55, 60, 67, 110, 115] for details. Convergence Spaces 1.2 37 Convergence Spaces The Hausdorﬀ-Kuratowski-Bourbaki concept of general topology has proved to be very useful in analysis .One such useful application is the powerful methods of linear functional analysis initiated by Banach [18] within the setting of metric spaces. However, several deﬁciencies of the Hausdorﬀ-Kuratowski-Bourbaki topology emerged in the middle of the twentieth century. The most serious of these deﬁciencies is the fact that there is in general no natural topological structure for function spaces. Recall that if X, Y and Z are sets, then the exponential law Z X×Y ≃ (Z X )Y (1.78) holds. This means there is a canonical one-to-one mapping between the spaces of functions f : X × Y −→ Z (1.79) g : Y −→ Z X = {h : X −→ Z} (1.80) and That is, with any function (1.79) one can associate the function f˜ : Y ∋ y 7−→ f (·, y) ∈ Z X (1.81) deﬁned through f˜(y) : X ∋ x 7−→ f (x, y) ∈ Z. Conversely, with the function (1.80) one may associate the function g̃ : X × Y −→ Z deﬁned by g̃(x, y) = g(y)(x) ∈ Z. For topological spaces X, Y and Z, the exponential law can be written as C(X × Y, Z) ≃ C(Y, C(X, Z)). (1.82) Consider a continuous function f : X × Y −→ Z, (1.83) with the mapping (1.83) we associate the mapping Ff : Y ∋ y 7−→ Ff (y) ∈ C(X, Z) (1.84) Ff (y) : X ∋ x 7−→ f (x, y) ∈ Z. (1.85) deﬁned by 38 Introduction Conversely, with a continuous function F : Y −→ C(X, Z) (1.86) fF : X × Y −→ Z (1.87) associate the mapping deﬁned as fF : X × Y ∋ (x, y) 7−→ (F (y))(x) ∈ Z. Let C(X, Y ) be equipped with the compact open topology, which has a subbasis { } K ⊆ Xcompact S(K, U ) U ⊆ Y open where S(K, U ) = {f ∈ C(X, Y ) : f (K) ⊆ U }. If X is locally compact and Hausdorﬀ, then the mapping (1.87) associated with the mapping (1.86) is continuous whenever the mapping (1.86) is continuous . Hence, whenever X is locally compact and Hausdorﬀ , the mappings (1.83) (1.87) deﬁne a bijection χ : C(X × Y, Z) f 7−→ Ff ∈ C(Y, C(X, Z). (1.88) Moreover, if Y and Z are also locally compact then the mapping (1.88) is a homeomorphism. Thus (1.82) holds for locally compact spaces X, Y and Z and the compact open topology on the relevant spaces of continuous functions. However, if the assumptions of local compactness on any of the spaces X, Y or Z are relaxed, then either the mapping (1.87) fails to be continuous, or the mapping (1.88) will no longer be a homeomorphism. Thus, unless all the spaces X, Y and Z are locally compact, there is no topology on C(X, Y ) so that the above construction holds, see for instance [25, 86, 122]. Another failure of the Hausdorﬀ-Kuratowski-Bourbaki concept of topology, from the perspective of applications to analysis, concerns the issue of generality. In this regards, we may mention that there are several natural and important notions of convergence that cannot be associated with the Hausdorﬀ-KuratowskiBourbaki topology. For instance, the point-wise almost everywhere convergence is not topological. To see this, we recall the following example, see [97, 101, 123]. Example 1.35. Let M (R) denote the set of real Lebesgue measurable functions on R. Consider the sequence (um n ), where { m 1 m−1 n ≤x ≤ n um (x) = n 0 otherwise. Convergence Spaces 39 For any m, , n ∈ N and ε > 0 we have m−1 m , ]. n n m lim mes(Am n (ε)) = 0 So that (un ) converges to 0 in measure. However, m Am n (ε) = {x ∈ R : un (x) ≥ ε} ⊆ [ Thus m,n−→∞ m (un )does not converge to 0 almost everywhere. Indeed, for all a ∈ R, a > 0 and N ∈ N, there exists m, n ≥ N such that um n (a) = 1. Now suppose that there is some topology τae on M (R) so that the sequence that converges with respect to τae are precisely those that converge almost everywhere. Since (um n ) does not converge to 0 almost everywhere, it follows that there is a τae neighborhood V of 0 such that ∀ k∈N: ∃ mk , nk ≥ k : k um nk ̸∈ V. In this way, we obtain a subsequence (unmkk ) of (unm ) so that ∀ k∈N unmkk ̸∈ V. (1.89) Thus no subsequence of (unmkk ) converges almost everywhere to 0. However, a well known result [68, Theorem 11.26] states that every sequence which converges in measure has a subsequence which converges almost everywhere to the same limit. Therefore (unmkk ) has a subsequence which converges almost everywhere to 0, which is a contradiction. Thus there is no topology which induces convergence almost everywhere on the set of measurable functions. 2 One solution of the above mentioned limitations of Hausdorﬀ-KuratowskiBourbaki topology is provided by the theory of convergence spaces, see [20, 25, 123], which is a more general notion of topology. Our main focus in this section is to introduce some of the fundamental concepts related to convergence spaces. A convergence space is a set together with a designated collection of ﬁlters. Recall that a ﬁlter F on a set X is a nonempty collection of subsets of X such that (i) The empty set does not belong to F. (ii) For all F ∈ F and for all G ⊆ X, if G ⊇ F, then G ∈ F (iii) If F, G ∈ F , then F ∩ G ∈ F. A subset B ⊆ F is a ﬁlter basis of F if each set in F contains a set in B. The ﬁlter F is said to be generated by B. We then write F = [B]. If A ⊆ X, the ﬁlter generated A is written as [A]. That is [A] = {F ⊆ X : F ⊇ A} . In particular for x ∈ X, [x] is the ﬁlter generated by {x}. The ﬁlter [x] is called the principal ultraﬁlter generated by x. Recall that a ﬁlter G on X is called an 40 Introduction ultraﬁlter if G ̸⊂ F for all ﬁlter F on X. The intersection of two ﬁlters F and G on X is deﬁned as F ∩ G = [{F ∪ G : F ∈ F , G ∈ G}] If F is a ﬁlter on X, and G is a ﬁlter on Y, then the product of the ﬁlters F and G is a ﬁlter on X × Y which is deﬁned as F × G = [{F × G : F ∈ F, G ∈ G}] If ﬁlters F and G on X are such that G ⊆ F , then we say that F is ﬁner than G, or alternatively G is coarser than F. If F and G are ﬁlters on X, the ﬁlter F ∨ G may not exist. However, if A ∩ B ̸= ∅ for all A ∈ F and all B ∈ G then F ∨ G = [{A ∩ B : A ∈ F, B ∈ G}] exists. If f : X −→ Y is a mapping then we deﬁne the image ﬁlter of F under f as f (F) = [{f (F ) : F ∈ F}]. If (xn ) is a sequence in X, then we deﬁne the Frechét ﬁlter associated with (xn ) as ⟨(xn )⟩ = [{{xn : n ≥ k} : k ∈ N}]. Recall [20, 123] that a given topological space (X, τ ) may be completely described by specifying the convergence associated with the topology τ. In particular, a ﬁlter F on X converges to x ∈ X if and only if F ⊇ Vτ (x), where Vτ (x) denotes the τ -neighborhood ﬁlter at x ∈ X. For each x ∈ X we may denote the set of all ﬁlters converging to x with respect to τ by λτ (x). That is, λτ (x) = {F a ﬁlter on X : F ⊇ Vτ (x)} A sequence (xn ) converges to x ∈ X with respect to the topology τ if ∀ V ∈ Vτ (x) : ∃ NV ∈ N : x n ∈ V ∀ n ≥ NV (1.90) (1.91) This implies that ⟨(xn )⟩ ⊇ Vτ (x). Conversely, if ⟨(xn )⟩ ⊇ Vτ (x), then (1.91) must hold. Thus the deﬁnition of ﬁlter convergence in a topological space is a straight forward generalization of the corresponding notion of convergence for sequences in a topological space. A convergence structure on a set X is a generalization of the topological convergence (1.90) and is deﬁned as follows Deﬁnition 1.36. Let X be a nonempty set. A convergence structure on X is the mapping λ from X to the power set of the set of all ﬁlters on X that satisﬁes the following for all x ∈ X : Convergence Spaces 41 (i) [x] ∈ λ(x) (ii) If F, G ∈ λ(x), then F ∩ G ∈ λ(x). (iii) If F ∈ λ(x), then G ∈ λ(x), for all ﬁlters G ⊇ F. The pair (X, λ) is called a convergence space. Whenever F ∈ λ(x) we say F converges to x and write “F −→ x”. Remark 1.37. Let λ and µ be two convergence structures on the same set X. Then λ is ﬁner than µ (or µ is coarser than λ) if for every x ∈ X, λ(x) ⊆ µ(x). That is, λ has fewer convergent ﬁlters than µ. As mentioned, convergence spaces are more general than topological spaces. However the concepts of continuity, embedding, homeomorphisms, open set and closure of a set generalize to the more general context of convergence spaces. In this regard, let X and Y be convergence spaces with convergence structures λX and λY respectively. A mapping f : X −→ Y is said to be continuous at a point x ∈ X if f (F) = [{f (F ) : F ∈ F}] −→ f (x) whenever F −→ x ∈ X. The mapping f is continuous if it is continuous at every point of X. Furthermore, f is called a homeomorphism if it is a bijection with both f and f −1 are continuous. It is an embedding if it is a homeomorphism onto its co-domain. Clearly the topological convergence (1.90) satisﬁes the conditions of Deﬁnition 1.36. Examples of non-topological convergence structures include the following. Example 1.38 (Almost every where convergence structure). Let X be the set of real-valued measurable functions on a measure space (Ω, A, µ). Let a convergence structure λae be deﬁne on X as follows: a ﬁlter F converges to f in (X, λae ) if F converges to f almost everywhere in Ω. Then λae is a convergence structure. In particular, a sequence (un ) in X converges almost everywhere to u ∈ X if and only if ⟨(un )⟩ converges to u with respect to λae . As we have shown in Example 1.35, almost everywhere convergence is not topological, see also [20]. Example 1.39 (Order convergence structure). [12] Let X be an Archimedean vector lattice [24, 84, 117]. A ﬁlter F on X converges to u in X with respect to the order convergence structure λ0 if and only if ∃ (αn ), (βn ) ⊂ X : (i) αn ≤ αn+1 ≤ βn+1 ≤ βn , n ∈ N (ii) sup{αn : n ∈ N} = u = inf{βn : n ∈ N} (iii) [{[αn , βn ] : n ∈ N}] ⊆ F . A sequence (un ) in X converges to u ∈ X with respect to λ0 if and only if (un ) 42 Introduction order converges to u. That is, ∃ (αn ), (βn ) ⊂ X : (i) αn ≤ αn+1 ≤ un ≤ βn+1 ≤ βn , n ∈ N (ii) sup{αn : n ∈ N} = u = inf{βn : n ∈ N}. The order convergence structure is not topological. To see this, consider the Archimedean vector lattice C(R), and the sequence (un ) ⊂ C(R) given by 1 − n|x − qn | if |x − qn | < n1 un (x) = (1.92) 1 0 if |x − qn | ≥ n where {qn | n ∈ N} = [0, 1] ∩ Q. The complement of any subset of Q ∩ [0, 1] is dense in [0, 1]. The sequence (un ) does not order converge to 0. For any N0 ∈ N we have βN0 (x) = sup{un (x) : n ≥ N0 } = 1, x ∈ [0, 1) This means that a sequence (βn ) ⊆ C(R) such that un ≤ βn for all n ∈ N cannot decrease to 0. Thus if there is a topology τ on C 0 (R) that induces order convergence, then there is some τ -neighborhood V of 0 and a subsequence (unk ) of (un ) which is always outside of V. Let (qnk ) denote the sequence of rational numbers associated with the subsequence (unk ) according to (1.92). Since the sequence (qnk ) is bounded, there exist a subsequence (qnki ) of (qnk ) that converges to some q ∈ [0, 1]. Let (unki ) be the sequence associated with the sequence of rational numbers (qnki ). Then ∀ε>0: ∃ Nε ∈ N : unki (x) = 0, whenever|x − qn | > ε and nki > Nε . For each j ∈ N set εj = µnki 1 j and let the sequence (µnki ) ⊆ C 0 (R) be deﬁned as if |x − q| ≥ 2εj 0 1 if |x − q| ≤ εj = |x−q| + 2 if εj < |x − q| < 2εj εj (1.93) whenever Nεj < nki < Nεj+1 . The sequence (µnki ) decreases to 0, and 0 ≤ unki ≤ µnki , This means that the sequence (unki ) order converges to 0. Therefore it must eventually be in V, a contradiction. Thus the topology τ cannot exist. Example 1.40 (Continuous convergence structure). [20] Let X and Y be convergence spaces, C(X, Y ) the space of all continuous functions from X to Y and ωX,Y : C(X, Y ) × X −→ Y the evaluation mapping. That is, ωX,Y (f, x) = f (x) for all f ∈ C(X, Y ) and all x ∈ X. A ﬁlter H converges to f ∈ C(X, Y ) with respect to the continuous Convergence Spaces 43 convergence structure λc if and only if ωX,Y (H × F) −→ f (x) for all x ∈ X and all F −→ x ∈ X. The universal property of the continuous convergence structure is states as follows: Let X, Y, Z be convergence spaces. Then the mapping h : Z −→ C(X, Y ) is continuous if and only if the associated mapping h̃ : Z × X −→ Y (z, x) deﬁned by h̃(z, x) = h(z)(x) is continuous. For more examples and a detailed exposition on convergence spaces see [20, 38, 40, 48, 57, 73]. One method for constructing new convergence spaces from given ones is to make use of initial and ﬁnal convergence structures. Subspaces, product spaces, projective limits, quotient spaces and inductive limits are examples of initial or ﬁnal convergence structure. Let X be a set, (Xi )i∈I a collection of convergence spaces and, for each i ∈ I, fi : X −→ Xi a mapping. A ﬁlter F on X converges to x in the initial convergence structure λX with respect to the family of mapping (fi )i∈I if and only if fi (F) −→ fi (x) in Xi for all i ∈ I. To see that λX is a convergence structure on X, note the following (i) For each i ∈ I we have fi ([x]) = [{fi ({x}) : {x} ∈ [x]}] = [{fi (x) : {x} ∈ [x]}] = [fi (x)]Xi −→ fi (x) which shows that [x] ∈ λX (x). (ii) Let F, G ∈ λX (x). Then for each i ∈ I, we have fi (F ∩ G) = [{fi (F ∪ G) : F ∈ F, G ∈ G}] = [{fi (F ) ∪ fi (G) : F ∈ F, G ∈ G}] = fi (F) ∩ fi (G) −→ fi (x) thus F ∩ G ∈ λX (x). (iii) Let F ∈ λX (x) and F ⊆ G. Then fi (G) = [{fi (G) : G ∈ G}] ⊇ [{fi (F ) : F ∈ F}] = fi (F) −→ fi (x) thus for each i ∈ I, G ∈ λX (x). 44 Introduction The initial convergence structure λX with respect to the family of mapping (fi )i∈I is the coarsest convergence structure on X making each of the mapping fi : X −→ Xi continuous. That is, for any other convergence structure λ on X such that each fi is continuous we have λ(x) ⊆ λX (x), x ∈ X. Example 1.41. Let (Xi )i∈I be a family of convergence spaces, and let X be the Cartesian product of the family (Xi ). That is ∏ X= Xi . i∈I The product convergence structure on X is the initial convergence structure with respect to the projection mapping πi : X −→ Xi , i ∈ I deﬁned as πi ((xj )j∈I ) = xi ∈ Xi . A ﬁlter F on X converges to x = (xi ), in X if and only if , for each i ∈ I πi (F) −→ πi (x) ∈ Xi That is ∀ i∈I ∃ ∏ Fi ∈ λXi (xi ) : i∈I Fi ⊆ F . Here ∏ [{ Fi = i∈I ∏ i∈I }] Fi ∈ Fi i ∈ I Fi Fi = Xi for all but ﬁnitely many i ∈ I denotes the Tychonoﬀ product of the family of ﬁlters (Fi )i∈I . 2 Example 1.42. Let X be a convergence space and M a subset of X. The subspace convergence structure λM on M is the initial convergence structure with respect to the inclusion mapping iM : M −→ X given by iM (x) = x ∈ X, x ∈ M. A ﬁlter F on M converges to x in M if and only if [{ }] ∃ F ∈F : [F]X = G ⊆ X F ⊆G converges to x in X. Convergence Spaces 45 Let X be a set, (Xi )i∈I a collection of convergence spaces and, for each i ∈ I, fi : Xi −→ X a mapping. A ﬁlter F on X converges to a point x in the ﬁnal convergence structure with respect to the family of mapping (fi )i∈I if and only if F = [x] or ∃ indices i1 · · · ik ∈ I : ∃ point xn ∈ Xin , n = 1, · · · , k : ∃ ﬁlters Fn ∈ λXin (xn ), n = 1 · · · k : 1)fi (xn ) = x, i = 1 · · · , k 2)fi1 (F1 ) ∩ · · · ∩ fik (Fk ) ⊆ F (1.94) Example 1.43. [20] Let X be a convergence space, Y a set and q : X −→ Y a surjective mapping. The quotient convergence structure λq on Y is the ﬁnal convergence structure with respect to the mapping q. A ﬁlter F on Y converges to y ∈ Y if and only if ∃ points x1 , · · · , xk ∈ X : ∃ ﬁlters F1 , · · · , Fk on X : 1) Fi ∈ λX (xi ), i = 1, · · · , k : 2) q(xi ) = y, i = 1, · · · , k : 3) q(F1 ) ∩ · · · ∩ q(Fk ) ⊆ F . (1.95) If X and Y are convergence spaces, and q : X −→ Y a surjection so that Y carries the quotient convergence structure with respect to q, then q is called a convergence quotient mapping. 2 The ﬁnal convergence structure is the ﬁnest convergence structure making all the mapping (fi )i∈I continuous. That is for any other convergence structure λ in X such that the mapping fi is continuous we have λX (x) ⊆ λ(x) Let X be a convergence space. For any x ∈ X a set V ⊆ X is a neighborhood of x if V belongs to every ﬁlter that converges to x. That is, ) ( ∀ F ∈ λX (x) V ∈ VλX (x) ⇐⇒ V ∈F where VλX (x) denotes the neighborhood ﬁlter at x. A set V ⊆ X is open if and only if it is a neighborhood of each of its elements. The concept of adherence in the context of convergence spaces is the generalization of the closure of a subset A of a topological space X. In a topological space X, the closure of a set A ⊆ X consists of A, together with all cluster points of A. That is, { } ∀ V ∈ Vτ (x) clτ (A) = x ∈ X . V ∩ A ̸= ∅ 46 Introduction Therefore for each x ∈ cl(A), the ﬁlter F = [{V ∩ A : V ∈ Vτ (x)}] converges to x and A ∈ F. Conversely, if there is a ﬁlter F ∈ λτ (x) such that A ∈ F, then it follows from (1.90) that A intersects every neighborhood of x so that x ∈ cl(A). This means that the closure of a set A ⊆ X is he set of of all points x ∈ X such that A belongs to some ﬁlter F that converges to x with respect to τ. In a convergence space the adherence of a set A ⊆ X is the set } { ∃ F ∈ λX (x) : . aλX (A) = x ∈ X A∈F That is, x ∈ aλX (A) if there is a ﬁlter that converges to x and contains A. Where there is no confusion we shall simply denote the adherence of A by a(A). The set A ⊆ X is closed if a(A) = A. Many of the familiar properties of the closure operator of a topological space also hold for the adherence operator in a convergence space. Some of these properties are stated in the following. Proposition 1.44. [20] Let X be a convergence space. Then the following hold: (i) a(A) ⊆ a(B) if A ⊆ B for all A, B ⊂ X (ii) a(∅) = ∅ (iii) A ⊆ a(A) for all A ⊆ X (iv) a(A ∪ B) = a(A) ∪ a(B) for all A, B ⊆ X. (v) f (a(A)) ⊆ a(f (A)) for all A ⊆ X, and f : X −→ Y continuous. In a non-topological convergence space, the adherence operator is, in general, not idempotent. That is, for some A ⊂ X a(A) ̸= a(a(A)). If a convergence space X is such that ∀ x∈X Vλx (x) ∈ λX (x) then the convergence space is called pre-topological and the convergence structure λX is called a pre-topology. Every topological space is pre-topological but the converse is not true, see for instance [20]. Indeed, one of the characterization of topological convergence spaces is the following Proposition 1.45. [20] A convergence space X is topological if and only if X is pre-topological and the adherence operator is idempotent. The notions of Hausdorﬀ, T1 and regular spaces in convergence spaces coincide with the usual ones in the case of a topological space. A convergence space X is called a Hausdorﬀ space if every convergent ﬁlter converges to a unique limit. It Convergence Spaces 47 is called a T1 space if every ﬁnite subset of X is closed , and it is called a regular space if F ∈ λX (x) =⇒ a(F) = [{a(F ) : F ∈ F}] ∈ λX (x) Note that a Hausdorﬀ space is a T1 space. To see this, let X be a Hausdorﬀ space and let A ⊆ X be a ﬁnite set. The set A is a ﬁnite union of singleton sets. Therefore it suﬃces to show that the singleton set {y}, for y ∈ X, is closed. If x ∈ a({y}) then there exists a ﬁlter F −→ x and {y} ∈ F. This implies that F ⊆ [y]. Therefore [y] −→ x. Since X is Hausdorﬀ it follows that x = y. Thus a({y}) = {y}. Conversely, a regular T1 space is Hausdorﬀ. This is because if X is regular and T1 , and a ﬁlter F converges to x and y then a(F) also converges to x, and to y. Then x, y ∈ a(F ) for all F ∈ F. So that a(F) = [{a(F ) : F ∈ F}] ⊆ [x]. Hence [x] converges to y. This implies that y ∈ a({x}). But X is T1 , hence x = y. Thus X is Hausdorﬀ. Subspaces, product and projective limits of T1 spaces, Hausdorﬀ spaces and regular spaces are also T1 , Hausdorﬀ and regular, respectively, as shown in [20, Proposition 1.4.2]. 1.2.1 Uniform Convergence Structure In this section we discuss some of the basic aspects of the theory of uniform convergence spaces which is a generalization of the theory of uniform spaces. Recall [20] that a uniformity on a set X is a ﬁlter U on X × X such that the following conditions are satisﬁed. (i) ∆ ⊆ U for each U ∈ U. (ii) If U ∈ U, then U −1 ∈ U. (iii) For each U ∈ U there are some V ∈ U such that V ◦ V ⊆ U. Here ∆ = {(x, x) : x ∈ X} denotes the diagonal in X × X. If U and V are subsets of X × X then U −1 = {(x, y) ∈ X × X : (y, x) ∈ U } and the composition of U and V is deﬁned as { } ∃ z∈X: U ◦ V = (x, y) ∈ X × X . (x, z) ∈ V and (z, y) ∈ U A uniformity UX on X induces a topology on X in the following way: A set A ⊆ X is open in X if ∀ x∈A: ∃ U ∈ UX : U [x] ⊆ A 48 Introduction where U [x] = {y ∈ X|(x, y) ∈ U }. A ﬁlter F on X is a Cauchy ﬁlter if and only if UX ⊆ F × F. A uniform space is complete if and only if every Cauchy ﬁlter on X converges to some point x ∈ X. We recall [70] the deﬁnition of uniform continuity of function deﬁned on a uniform space. Let X and Y be uniform spaces. A mapping f : X −→ Y is uniformly continuous if and only if ∀ U ∈ UY (f −1 × f −1 )(U ) ∈ UX . The mapping f is uniform embedding if it is injective and its inverse f −1 is uniformly continuous on the subspace f (X) of Y. Furthermore, f is a uniform isomorphism, if it is a uniform embedding which is surjective. The main result due to Weil [126], in connection with completeness of uniform spaces assert that for any Hausdorﬀ uniform space X, one can ﬁnd a complete Hausdorﬀ uniform space X ♯ and a uniform embedding iX : X −→ X ♯ such that iX (X) is dense in X ♯ . Moreover, for any complete, Hausdorﬀ uniform space Y and any uniformly continuous mapping f : X −→ Y there is a uniformly continuous mapping f ♯ : X ♯ −→ Y such that the diagram f X - Y f♯ iX R (1.96) X♯ commutes. Remark 1.46. Note that not every topology τX on a set X is induced by a uniformity UX . In fact, it was shown in [126] that a given topology τX on X is induced by a uniformity UX if and only if the topology τX is completely regular. Hence the class of uniform spaces is rather small in comparison to the class of all topological spaces. Deﬁnition 1.47. Let X be a set. A family JX of ﬁlters on X × X is called a uniform convergence structure if the following holds: (i) [x] × [x] ∈ JX for every x ∈ X (ii) If U ∈ JX and U ⊆ V, then V ∈ JX Convergence Spaces 49 (iii) If U, V ∈ JX , then U ∩ V ∈ JX . (iv) If U ∈ JX , then U −1 ∈ JX . (v) If U, V ∈ JX , then U ◦ V ∈ JX whenever U ◦ V exists. The pair (X, JX ) is called a uniform convergence space. If U and V are ﬁlters on X × X then U −1 is deﬁned as U −1 = [{U −1 : U ∈ U}]. If U ◦ V ̸= ∅ for all U ∈ U and V ∈ V then the ﬁlter U ◦ V exists and it is deﬁned as U ◦ V = [{U ◦ V : U ∈ U , V ∈ V}]. Uniform convergence spaces generalizes the concept of a uniform space in the sense that every uniformity UX on X give rise to a unique uniform convergence structure JUX deﬁned through U ∈ JUX =⇒ UX ⊆ U . Every uniform convergence structure JX on X induces a convergence structure λJX on X deﬁned by ∀ x∈X ∀ F a ﬁlter on X F ∈ λJX (x) ⇐⇒ F × [x] ∈ JX The convergence structure λJX is called the induced convergence structure. The induced convergence structure need neither be topological nor completely regular, but rather satisﬁes more general separation properties, see [20]. Every reciprocal convergence structure λX is induced by a uniform convergence structure. Recall that a convergence structure is called reciprocal if ∀ x, y ∈ X λX (x) = λX (y) or λX (x) ∩ λX (y) = ∅ (1.97) Note that if a convergence space is Hausdorﬀ then it is reciprocal but the converse is not true. Given a reciprocal convergence structure λX on X, the associated uniform convergence structure JλX on X × X, deﬁned by ∃ x1 · · · xk ∈ X ∃ F1 · · · Fk ﬁlters on X : (1.98) U ∈ JλX ⇐⇒ (1) Fi ∈ λX (xi ) for i = 1 · · · k (2) (F1 × F1 ) ∩ · · · ∩ (Fk × Fk ) ⊆ U is a uniform convergence structure that induces a convergence structure λX . In particular, every Hausdorﬀ convergence structure is induced by the associated uniform convergence structure (1.98). A Hausdorﬀ uniform convergence space is characterized by the following [20, Proposition2.1.10] 50 Introduction Proposition 1.48. A uniform convergence space (X, JX ) is a Hausdorﬀ uniform convergence space if and only if ∀ U ∈ JX ∀ x, y ∈ X, x ̸= y : ∃ U ∈U : (x, y) ̸∈ U. As mentioned in Section 1.2, new convergence spaces can be constructed from existing ones using the initial and ﬁnal convergence structure. This is also true of uniform convergence spaces. The initial uniform convergence structure is constructed as follows: Let X be a set and (Xi , Ji )i∈I a family of convergence spaces. For each i ∈ I let fi : X −→ Xi be a mapping. The initial uniform convergence structure J on X × X with respect to the mapping fi is deﬁned as ( ∀ i∈I U ∈ J ⇐⇒ (1.99) (fi × fi )(U) ∈ Ji . The initial uniform convergence structure J induces the initial convergence structure λJ , see [20, Proposition 2.2.2]. Subspaces and product uniform convergence spaces are typical example of initial uniform convergence structure. Let X be a set and (Xi , Ji )i∈I a family of convergence spaces. For each i ∈ I let fi : Xi −→ X be a mapping. The ﬁnal uniform convergence structure J on X × X with respect to the mapping fi is deﬁned as ∃ U 1 · · · Un ∈ J0 : U ∈ J ⇐⇒ ∃ x1 · · · xk ∈ X : (1.100) U1 ∩ · · · ∩ Un ∩ ([x1 ] × [x1 ]) ∩ · · · ∩ ([xk ] × [xk ]) ⊆ U . where J0 is a family of ﬁlters V on X × X deﬁned by ∃ i1 · · · in ∈ I : V ∈ J0 ⇐⇒ ∃ Vk ∈ Jik : (fi1 × fi1 )(V1 ) ◦ · · · ◦ (fi1 × fi1 )(Vk ) ⊆ V. Quotient uniform convergence structure is an example of ﬁnal uniform convergence structure. We remark here that the ﬁnal uniform convergence structure does not induce the ﬁnal convergence structure, refer to [20, 62] for more details. The concepts of uniform continuity, Cauchy ﬁlters, completeness and completion extend to uniform convergence spaces in a natural way. In this regard let X and Y be uniform convergence spaces. A mapping f : X −→ Y is uniformly continuous if ∀ U ∈ JX (f × f )(U) ∈ JY . A uniformly continuous mapping f is called a uniformly continuous embedding if it is injective and f −1 is uniformly continuous on the subspace f (X) of Y. A Convergence Spaces 51 uniformly continuous embedding is a uniformly continuous isomorphism if it is also surjective. A ﬁlter F on X is called a Cauchy ﬁlter if F × F ∈ JX . Some important properties of Cauchy ﬁlters are stated in the following, see [20, Proposition 2.3.2 - 2.2.3]. Proposition 1.49. Let (X, JX ) be a uniform convergence space. Then the following hold: (i) Each convergent ﬁlter is a Cauchy ﬁlter. (ii) If F is a Cauchy ﬁlter and F ⊆ G then G is a Cauchy ﬁlter. (iii) Let F be a Cauchy ﬁlter and let F ⊆ G. If G −→ x ∈ X, then F −→ x. (iv) If F and G are Cauchy ﬁlters and F ∨ G exists then F ∩ G is a Cauchy ﬁlter. (v) If F is a Cauchy ﬁlter, G a ﬁlter on X such that F × G ∈ JX then G is a Cauchy ﬁlter. (vi) If (Y, λY ) is a uniform convergence space, f : X −→ Y is a uniformly continuous mapping, and F is a Cauchy ﬁlter on X then f (F) is a Cauchy ﬁlter on Y. A uniform convergence space X is said to be complete if every Cauchy ﬁlter converges to a point in X. Proposition 1.50. Let (X, JX ) be a complete uniform convergence space. Then the following hold: (i) Each closed subspace of a complete uniform convergence space is complete. (ii) If (X, JX ) is Hausdorﬀ, then a subspace of (X, JX ) is complete if and only if it is closed. (iii) The product of complete uniform convergence spaces is complete. Example 1.51. The associated uniform convergence space of a reciprocal convergence space is complete. 2 The Weil concept of completion of uniform spaces has been extended to the more general setting of uniform convergence spaces, see [127]. Indeed, if X is a Hausdorﬀ uniform convergence space, then there exists a complete, Hausdorﬀ uniform convergence space X ♯ and a uniformly continuous embedding iX : X −→ X ♯ such that iX (X) is dense in X ♯ . Moreover, the completion X ♯ of X satisﬁes the universal property: If Y is a complete Hausdorﬀ uniform convergence space and 52 Introduction f : X −→ Y is uniformly continuous, then there exists a uniformly continuous mapping f ♯ : X ♯ −→ Y such that the diagram X iX f Y f♯ (1.101) ? X♯ commutes. X ♯ is called the Wyler completion of X. This completion is unique up to uniformly continuous isomorphism. In general, the Wyler completion of uniform convergence spaces does not preserve subspaces. Indeed, the completion of a subspace of a uniform convergence space X will, in general, not be a subspace of the completion X ♯ , [121]. The following Theorems gives the characterization of the completion of a subspace X of a uniform convergence space Y. Theorem 1.52. Let X be a subspace of the uniform convergence space Y. Let i : X −→ Y be the inclusion mapping. Then there exists an injective, uniformly continuous mapping i♯ : X ♯ −→ Y ♯ , which extends the mapping i. In particular, i♯ (X ♯ ) = aY ♯ (iY (X)), where aY ♯ denote the adherence operator in X ♯ and iY denote the uniformly continuous embedding associated with the completion Y ♯ . Furthermore, the uniform convergence structure on Y ♯ is the smallest complete, Hausdorﬀ uniform continuous structure on aY ♯ (X), with respect to inclusion so that X is contained in X ♯ as a dense subspace. Theorem 1.53. Let X and Y be uniformly convergence spaces, and φ : X −→ Y a uniformly continuous embedding. Then there exists an injective uniformly continuous mapping φ♯ : X ♯ −→ Y ♯ , where X ♯ and Y ♯ are the completions of X and Y respectively, which extends φ. 1.2.2 Convergence vector spaces Let V be a vector space over the scalar ﬁeld K of real or complex numbers. A convergence structure λV on V is called a vector space convergence structure if the vector space operations + : (V, λV ) × (V, λV ) −→ (V, λV ) and · : K × (V, λV ) −→ (V, λV ) are continuous. In this case V is called a convergence vector space. Convergence Spaces 53 Examples 1.54. 1. Every topological vector space is a convergence vector space. Recall [99, 106] that a vector space V over the scalar ﬁeld K of real or complex numbers is called a topological vector space if V is endowed with a topology τV such that + : (V, τV ) × (V, τV ) −→ (V, τV ) and · : K × (V, τV ) −→ (V, τV ) are (jointly) continuous. 2. Let X be a convergence space and V a convergence vector space. Then Cc (X, V ) is a convergence vector space. In particular Cc (X) = Cc (X, R) is a convergence vector space. 3. Let X and Y be convergence vector spaces. Then Lc (X, Y ), which is the set L(X, Y ) of all continuous linear mapping between X and Y endowed with the subspace convergence structure from Cc (X, Y ), is a convergence vector space. In particular, Lc (X) = Lc (X, R) is a convergence vector space. The space Lc (X) is the continuous dual space of the convergence vector space X. It is the canonical dual in the setting of convergence vector spaces. 2 The following Lemma gives some properties of convergence vector spaces which are well-known in the topological case, see [20]. Lemma 1.55. Let V be a convergence vector space. Then the following statements hold. (i) For each a ∈ V the translation mapping Ta : V ∋ x 7→ a + x ∈ V is a homeomorphism. (ii) For all x ∈ V F ∈ λV (x) ⇐⇒ F − x ∈ λV (0) (iii) If W is another convergence vector space then a linear mapping f : V −→ W is continuous if and only if it is continuous at 0. A standard procedure for constructing a vector space convergence structure or for showing that a given convergence structure is a vector space convergence structure, is given by the following proposition [20]. Proposition 1.56. Let V be a vector space over K and let V(0) be the zero neighborhood ﬁlter on K. Let S be a family of ﬁlters on V satisfying the the following conditions: 54 Introduction (i) If F ∈ S and G ∈ S then F ∩ G ∈ S. (ii) If F ∈ S then G ∈ S for all ﬁlters G ⊇ F . (iii) If F ∈ S and G ∈ S then F + G ∈ S. (iv) If F ∈ S then V(0)F ∈ S. (v) If F ∈ S then αF ∈ S for all α ∈ K (vi) For all x ∈ V, V(0)x ∈ S. Then the mapping λV from V to the power set of all the set of ﬁlters on V deﬁned by F ∈ λV (x) ⇐⇒ F − x ∈ S. is a vector space convergence structure on V. As mentioned, a convergence space X is topological if and only if it is pretopological and the adherence operator is idempotent. However, for a convergence vector space to be topological it is suﬃcient for V to be pre-topological. That is, a convergence vector space is topological if and only if it is pre-topological. Also, a convergence vector space is Hausdorﬀ if and only if the set {0} is closed, see [20]. A convergence vector space is equipped with a natural uniform convergence structure, called the induced uniform convergence structure, which is denoted as JV . In this regard, let V be a convergence vector space, and let U be a ﬁlter on V × V. Then ∃ F a ﬁlter on V : U ∈ JV ⇐⇒ (1)F −→ 0 (1.102) (2)∆(F) ⊆ U . Here ∆(F) = [{∆(F ) : F ∈ F}] and for any set F ⊆ V ∆(F ) = {(x, y) ∈ V × V : x − y ∈ F }. (1.103) Lemma 1.57. Let V be a convergence vector space. Then for all A, B ⊆ V and for all ﬁlters F, G on V we have (i) ∆(A ∩ B) = ∆(A) ∩ ∆(B). (ii) ∆(A ∪ B) = ∆(A) ∪ ∆(B). (iii) ∆(F ∩ G) = ∆(F ) ∩ ∆(G). (iv) If U is a ﬁlter on V × V, then the ﬁlter [{A ⊆ V : ∆(A) ∈ U}] is an ultraﬁlter if U is an ultraﬁlter. (v) For any x ∈ V, F × [x] ⊇ ∆(G) =⇒ F ⊇ G + x Convergence Spaces 55 (vi) ∆(F + G) ⊆ ∆(F) ◦ ∆(G). Here F + G = [{F + G : F ∈ F, G ∈ G}] = [{{x + y : x ∈ F, y ∈ G} : F ∈ F , G ∈ G}] (vii) a(∆(F)) ⊇ ∆(a(F)). The convergence structure induced by the uniform convergence structure JV agrees with the vector space convergence structure λV , that is, λJV = λV . If V and W are convergence vector space and a linear mapping f : V −→ W is continuous then f is uniformly continuous, see for instance [20, Proposition 2.5.3]. Note that the induced uniform convergence structure of a reciprocal convergence vector space is not in general the associated uniform convergence structure and hence need not be complete. In a convergence vector space Cauchy ﬁlters are characterized as follows: A ﬁlter F on V is a Cauchy ﬁlter if and only if F − F converges to 0. The Wyler completion of unform convergence spaces does not preserve algebraic structures. If V is a convergence vector space carrying its induced uniform convergence structure, then its completion V ♯ is naturally a convergence vector space. However the uniform structure does not induce a vector convergence structure and so one has to consider its “convergence vector space modiﬁcation” which has all the desired properties of a convergence vector space completion, see [20, 61], see also [63, 100]. If V is a Hausdorﬀ convergence vector space, it is not possible in general to embed it into a complete convergence vector space, since V may contain unbounded Cauchy ﬁlters. However it is possible to modify its completion V ♯ so that it contains V as a dense subspace, satisﬁes the universal extension property for linear mappings, and every bounded Cauchy ﬁlter converges. See for instance [62]. However if every Cauchy ﬁlter F in V is bounded, that is, there is some F ∈ F so that V(0)F −→ 0, then there is a complete, Hausdorﬀ convergence vector space V ♯ and a linear embedding iV : V −→ V ♯ . such that iV is dense in V ♯ . Furthermore, for every complete Hausdorﬀ convergence vector space W and every continuous linear mapping f : V −→ W there exists a continuous linear mapping f ♯ : V ♯ −→ W so that the diagram f V W iV f♯ (1.104) ? V♯ commutes. Below are some important examples of complete convergence vector spaces [20]. 56 Introduction Examples 1.58. (i) If X is any convergence space, then Cc (X) is a complete convergence vector space. (ii) If V is a convergence vector space, then Lc (V ) is a complete convergence vector space. 1.3 Hausdorﬀ Continuous Functions In this section we discuss Hausdorﬀ continuous (H-continuous) extended real interval valued functions deﬁned on a metric space X, see [5, 7, 8, 9, 108, 123]. Interval valued functions are traditionally associated with validated computing, where they naturally appear as error bounds for numerical and theoretical computations, see for instance [1, 72]. Sendov [107], see also [6], introduced the concept of H-continuous functions in connection with Hausdorﬀ approximations of real functions of real a variable. We now recall the basic notations and concepts involve in H-continuous functions. In this regard, let IR denote the set of all closed real intervals [a, a] = {x ∈ R : a ≤ x ≤ a}. That is, IR = {a = [a, a] : a, a ∈ R}, and let IR∗ denote the set of all extended, closed real intervals. That is, IR∗ = {a = [a, a] : a, a ∈ R∗ }, where R∗ = R ∪ {±∞} denote the extended real line with the usual ordering. Clearly, IR ⊂ IR∗ . Given an interval a = [a, a] ∈ IR∗ , the number w(a) = a − a is called the width of a, and |a| = max{|a|, |a|} is called the modulus of a. An interval a is a proper interval if w(a) > 0 and a point interval, if w(a) = 0. If we identify a ∈ R∗ with the point interval [a, a] ∈ IR∗ , then R∗ ⊂ IR∗ . On IR∗ we deﬁne the partial order through a ≤ b ⇐⇒ a ≤ b and a ≤ b. Let X be a metric space. Denote by A(X) the set of all interval valued function deﬁned on X. That is, A(X) = {u : X −→ IR∗ }. Since R∗ ⊂ IR∗ we have that A(X) ⊂ A(X), where A(X) = {u : X −→ R∗ }. On A(X) we deﬁne the pointwise partial order through ( ∀ x∈X u ≤ v ⇐⇒ u(x) ≤ v(x) (1.105) Hausdorﬀ Continuous Functions 57 Note that if u ∈ A(X) then for all x ∈ X the value of u at x is the interval [u(x), u(x)]. Hence the function u can be written as u = [u, u] where u, u ∈ A(X) and u ≤ u. The concept of a H-continuous function is formulated in terms of extended Baire operators. The extended Baire operators are deﬁned as follows: Let D ⊆ X be dense. For u ∈ A(X) and η > 0 we denote by I(η, D, u) the function I(η, D, u)(x) = inf{u(y)|y ∈ Bη (x) ∩ D}, x ∈ X, and S(η, D, u) the function S(η, D, u)(x) = sup{u(y)|y ∈ Bη (x) ∩ D}, x ∈ X. The function I(D, ·) : A(X) −→ A(X) is deﬁned by I(D, u)(x) = supI(η, D, u)(x), η>0 x∈X (1.106) and the function S(D, ·) : A(X) −→ A(X) is deﬁned by S(D, u)(x) = inf S(η, D, u)(x), η>0 x ∈ X. (1.107) In fact, since I(η, D, u)(x) < I(δ, D, u)(x), x ∈ X and S(η, D, u)(x) > S(δ, D, u)(x), x ∈ X whenever η < δ, it follows that I(D, u)(x) = limI(η, D, u)(x) x ∈ X η→0 and S(D, u)(x) = limS(η, D, u)(x) x ∈ X η→0 The operators I(D, ·) and S(D, ·) are called Lower and Upper extended Baire operators respectively. The operator F (D, ·) : A(X) −→ A(X) deﬁned by F (D, u)(x) = [I(D, u)(x), S(D, u)(x)], x∈X (1.108) is called the Graph Completion Operator. In the case when D = X the set D is omitted from the argument and we write I(u)(x) = I(X, u)(x), S(u)(x) = S(X, u)(x), F (u)(x) = F (X, u)(x) The operators (1.106), (1.107) and (1.108) satisfy the following properties. (C1 ) I(u) ≤ u ≤ S(u), u ∈ A(X) (C2 ) I, S, F and their compositions are idempotent. That is, for all u ∈ A(X), 58 Introduction (i) I(I(u)) = I(u) (ii) S(S(u)) = S(u) (iii) F (F (u)) = F (u) (iv) (I ◦ S)((I ◦ S)(u)) = (I ◦ S)(u) (C3 ) I, S, F and their compositions are monotone. That is, for all u, v ∈ A(X), (i) I(u) ≤ I(v) (ii) S(u) ≤ S(v) u ≤ v =⇒ (iii) F (u) ≤ F (v) (iv) (I ◦ S)(u) ≤ (I ◦ S)(v) The operator F is monotone with respect to inclusion, that is u(x) ⊆ v(x), x ∈ X =⇒ F (u)(x) ⊆ F (v)(x), x ∈ X. Furthermore, it is easy to see that, for u ∈ A(X), the functions I(u) and S(u) are lower and upper semi-continuous functions, respectively, on X. We now deﬁne the set of Hausdorﬀ continuous function. Deﬁnition 1.59. A function u ∈ A(X) is called H-continuous if for every function v ∈ A(X) which satisﬁes the inclusion v(x) ⊆ u(x), x ∈ X, we have F (v)(x) = u(x), x ∈ X. Denote by H(X) ⊆ A(X) the set of all H-continuous functions on X. Clearly all continuous real valued functions are H-continuous, that is C(X) ⊆ H(X). Indeed, if u is continuous then u is both upper and lower semi-continuous and hence F (u) = [I(u), S(u)] = [u, u] = u Furthermore, let v ∈ A(X) be such that v(x) ⊆ u(x). Then v(x) = u(x), x ∈ X and hence F (v)(x) = F (u)(x) = u(x), x ∈ X which shows that u is H-continuous. The set H(X) inherits the partial order (1.105). Equipped with this partial order, the set H(X) is a complete lattice. That is, ∀ A ⊆ H(X) ∃ u0 , v0 ∈ H(X) : (i) u0 = sup A (ii) v0 = inf A (1.109) The supremum and inﬁmum in (1.109) may be describe as follows: If ϕ : X ∋ x 7→ sup{u(x) : u ∈ A} ∈ R∗ and ψ : X ∋ x 7→ inf{u(x) : u ∈ A} ∈ R∗ then u0 = F (I(S(ϕ))), v0 = F (S(I(ψ))) Below are some examples of H-continuous functions which are not continuous. Hausdorﬀ Continuous Functions 59 Examples 1.60. (i) Let X = R. The function 1 if x > 0 u(x) = [−1, 1] if x = 0 −1 if x < 0 is H-continuous. (ii) Let X = {(x, t) : t ≥ 0} ⊆ R2 . For (x, t) ∈ X the function 1 if t ∈ [0, 1), x < t − 1 x if t ∈ [0, 1), x ∈ [t − 1, 0] t−1 if t ∈ [0, 1) x > 0 0 u(x, t) = 1 if t ≥ 1, x < t−1 2 t−1 [0, 1] if t ≥ 1, x = 2 0 if t ≥ 1, x > t−1 2 is H-continuous. This function arises as a shockwave solution of the nonlinear conservation law. The lower and upper Baire operators can be written in terms of u and u. Indeed, it is clear that I(u) = I(u) and S(u) = S(u) Hence F (u) = [I(u), S(u)] Therefore we have that, ( ) { u = I(u), u is lower semi − continuous, F (u) = u ⇐⇒ ⇐⇒ u = S(u) u is upper semi − continuous H-continuous functions are characterized as follows: Theorem 1.61. [4] Let u = [u, u] ∈ A(X). The following conditions are equivalent: (a) The function u is H-continuous. (b) F (u) = F (u) = u 60 Introduction (c) S(u) = u, I(u) = u. H-continuous functions may be constructed as follows: Theorem 1.62. Let u ∈ A(X). The functions F (S(I(u))) and F (I(S(u))) are H-continuous. The set H(X) of H-continuous functions contains the following three important subsets. The set { } ∀ x∈X: Hf t (X) = u ∈ H(X) , (1.110) u(x) ∈ IR of all ﬁnite H-continuous functions, the set { } ∃ Γ ⊂ X closed nowhere dense : Hnf (X) = u ∈ H(X) , x ∈ X\Γ =⇒ u(x) ∈ IR of nearly ﬁnite H-continuous functions, and the set } { ∃ [a, a] ∈ IR : , Hb (X) = u ∈ Hf t (X) u(x) ⊆ [a, a], x ∈ X (1.111) (1.112) of bounded H-continuous functions. Since the functions in C(X) assume values which are ﬁnite real numbers, we have the following inclusions: C(X) ⊆ Hf t (X) ⊆ H(X) and Cb (X) ⊆ Hb (X) ⊆ Hf t (X) ⊆ H(X). Here Cb (X) denotes the space of all bounded continuous functions. It has been shown, see [4], that the set Hf t (X) is Dedekind order complete and thus contains the Dedekind order completion of C(X) if X is an arbitrary topological space. If X is a metric space then the space Hf t (X) is the Dedekind order completion of C(X). 1.4 The Order Completion Method In this section we discuss the Order Completion Method (OCM) for nonlinear PDEs. The OCM is a type independent theory for the existence and basic regularity of solutions to nonlinear PDEs, based on the order completion of partially ordered sets of functions. This theory yields the existence and uniqueness of generalized solutions to arbitrary continuous nonlinear PDEs. The Order Completion Method 61 Let us consider a nonlinear PDE of order at most m of the form T (x, D)u(x) = h(x), x ∈ Ω. (1.113) Here Ω ⊆ Rn is open, and h ∈ C 0 (Ω). The nonlinear operator T (x, D) is deﬁned in terms of a jointly continuous function F : Ω × Rm −→ R by setting T (x, D)u(x) = F (x, u(x), · · · , Dα u(x), · · · ), |α| ≤ m, x ∈ Ω, (1.114) for any u ∈ C m (Ω). We assume that the PDE (1.113) satisﬁes h(x) ∈ int{F (x, ζ)|ζ ∈ Rm }, x ∈ Ω. (1.115) Under this condition, the following fundamental approximation result holds [120]. Theorem 1.63. Suppose that (1.115) holds. Then for all ε > 0 there exists Γε ⊂ Ω closed and nowhere dense and uε ∈ C m (Ω\Γε ) such that h(x) − ε < T (x, D)uε (x) ≤ h(x), x ∈ Ω\Γε . The OCM consists of using the Theorem 1.63, interpreted in appropriate function spaces, to construct solutions of the PDE (1.113). We summarize this construction below, see [10, 11, 93, 103, 104] for a detailed exposition. In this regard, m consider the space Cnd (Ω) deﬁned as follows: For any integer 0 ≤ m < ∞, set ∃ Γ ⊂ Ω closed, nowhere dense : m Cnd (Ω) = u ∈ A(Ω) 1) u : Ω\Γ −→ R (1.116) m 2) u ∈ C (Ω\Γ) m Clearly, C m (Ω) ⊆ Cnd (Ω), 0 ≤ m ≤ ∞. Since the mapping F that deﬁnes T (x, D) through (1.114) is continuous, it follows that if u ∈ C m (Ω\Γ) with Γ ⊂ Ω closed nowhere dense, then T (x, D)u ∈ C 0 (Ω\Γ). That is, with the operator T (x, D) we can associate a mapping m 0 T (x, D) : Cnd (Ω) −→ Cnd (Ω). (1.117) 0 0 On Cnd (Ω) we deﬁne an equivalence relation as follows: For any u, v ∈ Cnd (Ω), we have ∃ Γ ⊂ Ω closed nowhere dense : (1.118) u ∼ v ⇐⇒ 1) u, v ∈ C 0 (Ω\Γ) 2) u(x) = v(x), x ∈ Ω\Γ 0 the quotient space Cnd (Ω)/ ∼ is denoted by M0 (Ω). We also introduce an equivm m alence relation on Cnd (Ω) in the following way: For any u, v ∈ Cnd (Ω), u ∼T v ⇐⇒ T u ∼ T v. (1.119) 62 Introduction m The space Mm T (Ω) is deﬁned as the quotient space Cnd (Ω)/ ∼T . The mapping (1.117) induces an injective mapping 0 Tb : Mm T (Ω) −→ M (Ω) in a canonical way, so that the diagram T m Cnd (Ω) - q1 0 (Ω) Cnd q2 ? Mm T (Ω) Tb - (1.120) (1.121) ? M0 (Ω) commutes, with q1 and q2 canonical quotient mappings associated with the equivalence relations (1.118) and (1.119) respectively. The mapping Tb is deﬁned as m follows: If U ∈ Mm T (Ω) is the ∼T - equivalence class generated by u ∈ Cnd (Ω), then Tb(U ) is the ∼T - equivalence class generated by T u. On the space M0 (Ω), deﬁne a partial order as follows: For any H, G ∈ M0 (Ω), ∃ h ∈ H, g ∈ G, Γ ⊂ Ω closed nowhere dense : H ≤ G ⇐⇒ (1) h, g ∈ C 0 (Ω\Γ) (1.122) (2) h ≤ g on Ω\Γ b On the space Mm T (Ω) deﬁne a partial order ≤T through the mapping T as follows: For any U, V ∈ Mm T (Ω) U ≤T V ⇐⇒ TbU ≤ TbV in M0 (Ω). (1.123) With respect to the partial orders (1.122) and (1.123) on M0 (Ω) and Mm T (Ω), respectively, the mapping Tb is an order isomorphic embedding [93]. That is, Tb is injective and ∀ U, V ∈ Mm T (Ω) : U ≤T V ⇐⇒ TbU ≤ Tb According to the McNeille completion Theorem [85], see also [93, Appendix], there exists a unique Dedekind complete partially ordered sets (M0 (Ω)♯ , ≤) and ♯ (Mm T (Ω) , ≤T ), and order isomorphic embeddings m ♯ iMmT (Ω) : Mm T (Ω) −→ MT (Ω) and iM0 (Ω) : M0 (Ω) −→ M0 (Ω)♯ so that the following universal property is satisﬁed: For every order isomorphic embedding 0 S : Mm T (Ω) −→ M (Ω) The Order Completion Method 63 0 there exists a unique order isomorphic embedding S ♯ : Mm T (Ω) −→ M (Ω) so that the diagram S Mm M0 (Ω) T (Ω) iMmT (Ω) iM0 (Ω) S♯ ? ♯ Mm T (Ω) - (1.124) ? M0 (Ω)♯ commutes. In particular, there exists a unique order isomorphic embedding Tb♯ : Mm (Ω)♯ −→ M0 (Ω)♯ , T which is an extension of the mapping Tb so that the diagram Tb Mm (Ω) M0 (Ω) T iMmT (Ω) iM0 (Ω) Tb♯ ? ♯ Mm T (Ω) - (1.125) ? M0 (Ω)♯ commutes. In this way we arrive at an extension of the nonlinear PDE (1.113). ♯ In this regard, any solution U ♯ ∈ Mm T (Ω) of the equation Tb♯ U ♯ = f is considered a generalized solution of (1.113). The main existence and uniqueness result for solutions of the PDE (1.113) is stated below. Theorem 1.64. [93] If the PDE (1.113) satisﬁes the condition (1.115) then there ♯ exists a unique solution U ♯ ∈ Mm T (Ω) such that Tb♯ U ♯ = f As shown in [4], this generalized solution to the PDE (1.113) may be assimilated with usual Hausdorﬀ continuous functions in Hnf (Ω). Indeed, the Dedekind order completion M0 (Ω)♯ of M0 (Ω) is order isomorphic with the space Hnf of nearly ﬁnite H-continuous functions on Ω. Thus, since 0 ♯ Tb♯ : Mm T (Ω) −→ M (Ω) is an order isomorphic embedding, one may obtain an order isomorphic embedding Tb♯ : Mm T (Ω) −→ Hnf (Ω) 0 ♯ so that Mm T (Ω) is order isomorphic with a subspace of Hnf (Ω). 64 Introduction 1.4.1 Main Ideas of Convergence space Completion One major deﬁciency of the OCM, as formulated in Section 1.4, is that the spaces of generalized functions containing solutions of a PDE (1.113) may to a large extent depend on the particular nonlinear operator T (x, D). Furthermore, there is no concept of generalized partial derivative for generalized functions. Recently, [119], [120], [121] these issues were resolved by introducing suitable uniform convergence spaces. Here we recall brieﬂy the main ideas underlying this new approach. To illustrate the convergence space completion method we introduce normal lower and upper semi-continuous functions which are deﬁned through, u ∈ A(Ω) is normal lower semi − continuous at x0 ∈ Ω ⇔ I(S(u(x0 ))) = u(x0 ), u ∈ A(Ω) is normal upper semi − continuous at x0 ∈ Ω ⇔ S(I(u(x0 ))) = u(x0 ). A function is normal lower or normal upper semi-continuous on Ω if it is normal lower or normal upper semi-continuous at every point x0 ∈ Ω, [4], [54]. Every continuous function is both normal upper semi continuous and normal lower semi continuous. Deﬁnition 1.65. A normal lower semi-continuous function is called nearly ﬁnite whenever the set {x ∈ Ω : u(x) ∈ R} is open and dense in Ω. We denote by, N L(Ω), the set of all nearly ﬁnite normal lower semi-continuous functions on Ω. That is, { } (1)(I ◦ S)u(x) = u(x) N L(Ω) = u ∈ A(Ω) (2) {x ∈ Ω : u(x) ∈ R} is open and dense in Ω Note that every continuous, real valued function is nearly ﬁnite normal lower semi-continuous. Thus we have that C(Ω) ⊆ N L(Ω). Now consider the space m MLm (Ω) = {u ∈ N L(Ω) : u ∈ Cnd (Ω)}. (1.126) The space MLm (Ω) is a sublattice of N L(Ω). In particular, the space 0 ML0 (Ω) = {u ∈ N L(Ω) : u ∈ Cnd (Ω)}, is σ-order dense in N L(Ω). This means for each u ∈ N L(Ω) ∃ (λn ), (µn ) ⊂ ML0 (Ω) : (i) λn ≤ λn+1 ≤ µn+1 ≤ µn nN, (ii) sup{λn : n ∈ N} = u = inf{µn : n ∈ N}. (1.127) On the space ML0 (Ω) we deﬁne a uniform convergence structure as follows: The Order Completion Method 65 Deﬁnition 1.66. Let Λ consists of all nonempty order intervals in ML0 (Ω). Let J0 denote the family of ﬁlters on ML0 (Ω) × ML0 (Ω) deﬁned as follows: ∃ k∈N: ∀ j = 1, · · · , k : j ∃ Λj = {In } ⊆ Λ : ∃ uj ∈ N L(Ω) : U ∈ J0 ⇐⇒ (1.128) j j (i) I ⊆ I , n ∈ N n n+1 (ii) lim inf {Inj } = uj = lim sup{Inj } n−→∞ n−→∞ (iii) ([Λ1 ] × [Λ1 ]) ∩ · · · ∩ ([Λk ] × [Λk ]) ⊆ U . The uniform convergence structure J0 is uniformly Hausdorﬀ, ﬁrst countable and induces the convergence structure λJ0 on ML0 (Ω) given by ∃ (λn ), (µn ) ⊂ ML0 (Ω) : (i) λn ≤ λn+1 ≤ µn+1 ≤ µn nN, F ∈ λJ0 ⇐⇒ (1.129) (ii) sup{λn : n ∈ N} = u = inf{µn : n ∈ N} (iii) [{[λn , µn ] : n ∈ N}] ⊂ F We now consider the PDE (1.113). With the operator T (x, D) one may associate a mapping T : MLm (Ω) −→ ML0 (Ω) (1.130) deﬁned by T (u)(x) = (I ◦ S)(F (x, u, · · · , Dα u, · · · ))(x), x∈Ω (1.131) where Dα u = (I ◦ S)(Dα u). On the space MLm (Ω) consider the equivalence relation ∼T induced by T through ∀ u, v ∈ MLm (Ω) : u ∼T v ⇐⇒ T u = T v (1.132) m Denote by MLm T (Ω) the quotient space MLT (Ω)\ ∼T . With the mapping (1.130) one may associate in a canonical way an injective mapping 0 Tb : MLm (1.133) T (Ω) −→ ML (Ω) such that the diagram MLm (Ω) T - qT ? MLm T (Ω) ML0 (Ω) id Tb - ? ML0 (Ω) (1.134) 66 Introduction commutes. Here, qT denotes the canonical quotient map associated with the equivalence relation (1.132) and id is the identity map on ML0 (Ω). On the space MLm T (Ω) we consider the initial uniform convergence structure m JT with respect to the mapping Tb: For any ﬁlter U ∈ MLm T (Ω) × MLT (Ω) U ∈ JT ⇐⇒ (Tb × Tb)(U) ∈ J0 (1.135) Since the mapping Tb is injective, it follows that the space MLm T (Ω) is uniformly m 0 isomorphic to the subspace Tb(MLT (Ω)) of ML (Ω), see [120]. Thus the mapping Tb is a uniformly continuous embedding. The Wyler completion of the space (ML0 (Ω), J0 ) is the space N L(Ω) equipped with the uniform convergence structure J0♯ deﬁned as follows, see [120]. Deﬁnition 1.67. Let Λ consists of all nonempty order intervals in ML0 (Ω). Let J0♯ denote the family of ﬁlters on N L(Ω) × N L(Ω) deﬁned as follows ∃ k∈N: ∀ i = 1, · · · , k ∃ Λi = {Ini : n ∈ N} ⊆ Λ : ∃ ui ∈ N L(Ω) : ♯ i U ∈ J0 ⇐⇒ (1.136) (i) In+1 ⊆ Ini n ∈ N (ii) lim inf {Ini } = ui = lim sup{Ini } n→∞ n→∞ k ∩ (iii) (([Λi ] × [Λi ]) ∩ ([ui ] × [ui ])) ⊆ U. i=k The completion of the space MLm T (Ω) is denoted by N LT (Ω), and is realized as a subspace of N L(Ω). In particular, the mapping Tb extends uniquely to an injective uniformly continuous mapping Tb♯ : N LT (Ω) −→ N L(Ω). This is summarized in the following commutative diagram. Tb m MLT (Ω) ML0 (Ω) ϕ ? N LT (Ω) ψ Tb♯ - (1.137) ? N L(Ω) Here ϕ and ψ are the canonical uniformly continuous embeddings associated with the completions N LT (Ω) and N L(Ω), respectively. A ﬁrst existence and uniqueness result for the generalized solutions of the PDE (1.113) is given below. The Order Completion Method 67 Theorem 1.68. For every f ∈ C 0 (Ω) satisfying (1.115), there exists a unique U ♯ ∈ N LT (Ω) such that Tb♯ U ♯ = f. Theorem 1.68 is essentially a reformulation of Theorem 1.64 in the context of uniform convergence spaces.Thus the mentioned deﬁciencies of the OCM also applies to Theorem 1.68. However, by introducing a parallel construction of spaces of generalized functions, which is independent of the particular nonlinear operator T we may resolve these diﬃculties. In this regard, we introduce on MLm (Ω) the initial uniform convergence structure Jm with respect to the partial derivatives Dα : MLm (Ω) −→ ML0 (Ω). (1.138) That is, ( U ∈ Jm ⇐⇒ ∀ α ∈ Nn , |α| ≤ m : (Dα × Dα )(U) ∈ J0 thus each of the mappings (1.138) is uniformly continuous so that the mapping D : MLm (Ω) −→ ML0 (Ω). is a uniformly continuous embedding, therefore [121] D extends uniquely to an injective, uniformly continuous mapping D♯ : N Lm (Ω) −→ N L(Ω)µ . (1.139) where N Lm (Ω) denotes the completion of MLm (Ω). This gives a ﬁrst and basic regularity result: The generalized functions in N Ln (Ω) may be represented, through their generalized partial derivatives, as normal lower semi-continuous functions. Indeed, the mapping (1.139) may be represented as D♯ (u) = ((Dα )♯ )|α|≤m where (Dα )♯ denotes the extension of Dα to N Lm (Ω). Theorem 1.69. The mapping T : MLm (Ω) −→ ML0 (Ω) deﬁned in (1.130) - (1.131) is uniformly continuous. In view of Theorem 1.69 the mapping T extends to a unique uniformly continuous mapping T ♯ : N Lm (Ω) −→ N L(Ω) 68 Introduction so that the diagram MLm (Ω) T - ML0 (Ω) φ ψ ? m N L (Ω) T♯ - (1.140) ? N L(Ω) commutes. Here φ and ψ are the uniformly continuous embeddings associated with the completion N Lm (Ω) and N L(Ω), respectively. The main existence result for the solutions of (1.113) in N L(Ω) is the following Theorem 1.70. If for each x ∈ Ω there is some ζ ∈ Rm and neighbourhoods V and W of x and ζ so that F (x, ζ) = f (x) and F : V × W :−→ R is open, then there exists u♯ ∈ N Lm (Ω) such that T ♯ u♯ = f The relationship Between Theorem 1.68 and Theorem 1.70 is summarized as follows: If F (x, ·) : Rm −→ R is open and surjective for each x ∈ Ω, then the PDE (1.113) admits generalized m ♯ 0 solutions U ♯ ∈ N Lm T (Ω) and u ∈ N L (Ω) for every f ∈ C (Ω) and U ♯ = {u♯ ∈ N Lm (Ω)|T ♯ u♯ = f }. Thus the generalized solution in N Lm T (Ω) may be viewed as the set of all solutions m in N L (Ω). 1.5 Summary of the Main Results In chapter two of this thesis we present the main results obtained. The Order Completion Method, in particular the formulation of this Theory in terms of uniform convergence spaces presented in Section 1.4.1 is modiﬁed for single conservation laws in one spatial dimension. The following points are addressed. • Suitable convergence vector spaces are introduced for the formulation of question of existence and uniqueness of generalized solution of the mentioned conservation law. The completion of this space is described in terms of the set of ﬁnite H-continuous functions. Summary of the Main Results 69 • The issue of existence of generalized solution of conservation laws is formulated in an operator theoretic context. It is shown that each conservation law, with a given initial condition, admits at most one generalized solution. • Existence of a generalized solution for the Burgers equation is demonstrated. It is also shown that this solution is the entropy solution of the Burgers equation. Chapter 2 Hausdorﬀ Continuous Solution of Scalar Conservation laws 2.1 Introduction In this chapter we study the solutions of the initial value problem ut + (f (u))x = 0, in R × (0, ∞) u(x, 0) = u0 (x), x ∈ R (2.1) (2.2) in the context of Order Completion Method, and in particular the formulation and extension of the theory introduced in [118], [120] and [122], see also Sections 1.4 and 1.4.1 in the introduction. In particular, the general theory developed in [120] is adapted so as to deliver the entropy solution of (2.1) - (2.2). In this regard, we introduce suitable convergence vector spaces M and N . With the initial value problem (2.1)-(2.2) a mapping T : M −→ N (2.3) is associated so that (2.1)-(2.2) may be written as one single equation Tu = h for a suitable h ∈ N . The vector space convergence structure on M and N are constructed in such a way that the mapping (2.3) is uniformly continuous. In this way we obtain a canonical uniformly continuous extension T ♯ : M♯ −→ N ♯ of (2.3) to the completions M♯ and N ♯ of M and N , respectively. Any solution u♯ ∈ M♯ of the equation T ♯ u♯ = h (2.4) is interpreted as a generalized solution of the initial value problem (2.1) -(2.2). The main result presented in this chapter concerns existence, uniqueness and regularity of the solutions of (2.4). In this regard, we prove the following: Introduction 71 (A) Equation (2.4) has at most one solution u♯ ∈ M♯ . (B) M♯ may be identiﬁed with a set of H-continuous functions, thus the solution of (2.4) is H-continuous, if it exists. (C) There exist a solution u♯ ∈ M♯ for the initial value problem (2.1)-(2.2) with f (u) = (u)2 . 2 This solution can be identiﬁed with the entropy solution of the Burgers equation. The main novelty of the approach developed here is that the theory of entropy solution of scalar conservation laws is developed in an operator - theoretic setting. In this regard, we may recall, see for instance [105], that weak solutions methods for the solutions of linear and nonlinear PDEs involve an ad hoc extension of a partial diﬀerential operator associated with a given PDE. Given topological vector spaces X and Y of suﬃciently smooth functions, and a partial diﬀerential operator T : X −→ Y, (2.5) a Cauchy sequence (un ) in X is constructed so that the sequence (T un ) converges to some h ∈ Y. The sequence (un ) being a Cauchy sequence in X, converges to some u♯ in the completion X ♯ of X. Now, based on the convergence un −→ u♯ , T un −→ h of one single sequence (un ), u♯ is declared to be a generalized solution of the PDE T u = h. This amounts to an ad hoc extension of the mapping (2.5) to a mapping T ♯ : X ∪ {u♯ } −→ Y. In the case of a linear PDE this approach turns out to be well founded, due to the automatic continuity of certain linear mappings on topological vector spaces. However, in the case of non-linear PDEs such methods can, and often do, lead to non-linear stability paradoxes, see for instance [105, Chapter1, Section 8]. The result obtained in this chapter places the theory of entropy solutions of conservation laws on a ﬁrm operator - theoretic footing. The rest of this Chapter is organized as follows. In Section 2.2 we discuss the convergence vector spaces used in our result. The approximation result needed for the existence of a solution is discussed in Section 2.3. Existence and uniqueness results for the Burgers equation are presented in Section 2.4. Hausdorﬀ Continuous Solution of Scalar Conservation laws 72 2.2 Convergence Vector Spaces for Conservation Laws As mentioned, the novelty in the approach in this section to the conservation law (2.1) - (2.2) is based mostly on the diﬀerent way of constructing the operator equation T u = h associated with the conservation law. It is essential for this development that the classical solution of the problem (2.1) - (2.2) is unique whenever it exists. We present below a precise formulation of the uniqueness result. In this regard, we now deﬁne the following convergence vector spaces. Let M = {u ∈ C 1 (R × (0, ∞)) ∩ C 0 (R × [0, ∞)) : u(·, 0) ∈ U0 } (2.6) N = C 0 (R × (0, ∞)) × U0 (2.7) and where U0 is a set of initial conditions. In the literature U0 is deﬁned in diﬀerent ways. Here we take U0 = {h ∈ C 0 (R) : lim h(x), lim h(x) exist} x−→∞ x−→−∞ (2.8) The following result is an extended formulation of Theorem 6.2 in [28]. Theorem 2.1. Let f be Liptschitz on compacta. For u, v ∈ M set ϕ = ut + (f (u))x ψ = vt + (f (v))x . (2.9) (2.10) Then there exists L such that ∫ b ∫ b+Lt |v(x, t) − u(x, t)| dx ≤ |u(x, 0) − v(x, 0)| dx a a−Lt ∫ t ∫ b+Lt |ϕ − ψ| dxdt. + 0 (2.11) a−Lt Proof. Since lim u(x, 0), lim u(x, 0), lim v(x, 0) and lim v(x, 0) exist, then x−→∞ x−→−∞ x−→∞ x−→−∞ u, v are both bounded. Let u(x, t), v(x, t) ∈ [−d, d], x ∈ R. Using the fact that f is Liptschitz on compacta there exists L such that |f (w) − f (w′ )| ≤ L |w − w′ | for w, w′ ∈ [−d, d]. (2.12) From (2.9)-(2.10) we have that ψ − ϕ = (v − u)t + (f (v) − f (u))x . Multiply equation (2.13) by the function 1 if v − u > 0 sgn(v − u) = −1 if v − u < 0 0 if u = v (2.13) Convergence Vector Spaces for Conservation Laws 73 to get (ψ − ϕ)sgn(v − u) = |v − u|t + ((f (v) − f (u))x )sgn(v − u), which can be written as |v − u|t + ((f (v) − f (u))sgn(v − u))x = (ψ − ϕ)sgn(v − u). (2.14) Now integrate (2.14) over the trapezium { } 0 ≤ t ≤ τ; D = (x, t) ∈ R × [0, ∞) , a − L(τ − t) ≤ x ≤ b + L(τ − t) (2.15) for arbitrary ﬁxed τ > 0. Then ∫∫ ∫∫ (|v − u|t ) dxdt + (((f (v) − f (u))sgn(v − u))x ) dxdt ∫D∫ D (ψ − ϕ)sgn(v − u)dxdt. = (2.16) D t 6 (a, τ ) L1 (b, τ ) L3 L4 L2 a − Lτ a b b + Lτ x Figure 2.1 Apply Green’s Theorem to the left hand side of equation (2.16). Note that, see Figure 2.1, on L1 L2 L3 L4 : t = τ, dt = 0 : t = 0, dt = 0 : x = a + L(t − τ ), : x = b − L(t − τ ), dx = Ldt dx = −Ldt. Hausdorﬀ Continuous Solution of Scalar Conservation laws 74 Therefore ∫∫ ∫∫ (((f (v) − f (u))sgn(v − u))x ) dxdt (|v − u|t ) dxdt + ∫D∫ D (|v − u|t − (−((f (v) − f (u))sgn(v − u))x )) dxdt = D I |v − u| dx − ((f (v) − f (u))sgn(v − u))dt = L1 +L2 +L3 +L4 b ∫ ∫ = ∫a τ − ∫0 a−Lτ τ (((f (v((C3 (t), t))) − f (u((C3 (t), t))))sgn(v − u)))dt τ (L |v(C4 (t), t) − u(C4 (t), t)|)dt + ∫0 |v0 (x) − u0 (x)| dx |v(x, τ ) − u(x, τ )| dx − (L |v(C3 (t), t) − u(C3 (t), t)|)dt + ∫0 b+Lτ τ (((f (v(C4 (t), t)) − f (u(C4 (t), t)))sgn(v − u)))dt − 0 where C3 (t) = a + L(t − τ ), That is, ∫∫ ∫∫ (|v − u|t ) dxdt + D b ∫ (((f (v) − f (u))sgn(v − u))x ) dxdt D a ∫0 τ (L |v(C3 (t), t) − u(C3 (t), t)|)dt (((f (v((C3 (t), t))) − f (u((C3 (t), t))))sgn(v − u)))dt (L |v(C4 (t), t) − u(C4 (t), t)|)dt + − |v0 (x) − u0 (x)| dx τ ∫0 τ ∫0 b+Lτ a−Lτ + − ∫ |v(x, τ ) − u(x, τ )| dx − = ∫ C4 (t) = b − L(t − τ ). τ (((f (v(C4 (t), t)) − f (u(C4 (t), t)))sgn(v − u)))dt 0 Using the inequality (2.12) we see that L |v(C3 (t), t) − u(C3 (t), t)| − ((f (v((C3 (t), t))) − f (u((C3 (t), t))))sgn(v − u)) ≥ 0 Convergence Vector Spaces for Conservation Laws 75 and L |v(C4 (t), t) − u(C4 (t), t)| − ((f (v(C4 (t), t)) − f (u(C4 (t), t)))sgn(v − u)) ≥ 0 Therefore ∫∫ ∫∫ (|v − u|t ) dxdt + D b D ∫ ≥ (((f (v) − f (u))sgn(v − u))x ) dxdt ∫ b+Lτ |v(x, τ ) − u(x, τ )| dx − a 0 v (x) − u0 (x) dx, a−Lτ which further implies that ∫∫ (ψ − ϕ)sgn(v − u)dxdt ∫D∫ (|v − u|t + ((f (v) − f (u))sgn(v − u))x ) dxdt = D b ∫ ∫ b+Lτ |v(x, τ ) − u(x, τ )| dx − ≥ a 0 v (x) − u0 (x) dx. a−Lτ Thus we obtain the inequality ∫ b |v(x, τ ) − u(x, τ )| dx a ∫ b+Lτ ∫∫ 0 0 v (x) − u (x) dx + ≤ (ψ − ϕ)sgn(v − u)dxdt a−Lτ D ∫ b+Lτ ∫∫ 0 0 v (x) − u (x) dx + |ψ − ϕ| dxdt ≤ a−Lτ D as required. Consider the operator T : M −→ N (2.17) deﬁned by ( Tu = ut + (f (u))x u(·, 0) ) ( = T1 u T2 u ) (2.18) The mentioned uniqueness of a classical solution of (2.1) -(2.2) is extended in the following way. Lemma 2.2. The operator T is injective 76 Hausdorﬀ Continuous Solution of Scalar Conservation laws Proof. The injectivity of the operator T follows from Theorem 2.1. Indeed, let Tu = Tv for some u, v ∈ M. Then for any t > 0, a, b ∈ R, a < b we have ∫ b ∫ b+Lt |v(x, t) − u(x, t)| dx ≤ |T2 u − T2 v| dx a a−Lt ∫ t ∫ b+Lt + |T1 u − T1 v| dxdt 0 a−Lt = 0. By the continuity of u and v this implies u = v. Convergence Structures on M and N On the considered spaces M and N we deﬁne the respective convergence structures as follows: On M we consider the following convergence structure which we denote as λ1 . Given a ﬁlter F on M, we have ∃ (αn ), (βn ) ⊆ C 0 (R × [0, ∞)) : (i) αn ≤ αn+1 ≤ u ≤ βn+1 ≤ βn , n ∈ N ∫ b F ∈ λ1 (u) ⇐⇒ (2.19) (βn (x, t) − αn (x, t))dx −→ 0 (ii) a for t ≥ 0, a, b ∈ R, a ≤ b (iii) [{[αn , βn ] : n ∈ N}] ⊆ F. Here the interval [αn , βn ] is considered in M with respect to the usual point-wise order, that is, [αn , βn ] = {v ∈ M : αn (x, t) ≤ v(x, t) ≤ βn (x, t), x ∈ R, t ∈ [0, ∞)} Proposition 2.3. The convergence structure λ1 is a Hausdorﬀ vector space convergence structure. Proof. We ﬁrst show that λ1 is a convergence structure on M by showing that λ1 satisﬁes the deﬁnition of a convergence structure given in (1.36). (i) Consider u ∈ M. In (2.19) set αn = βn = u for all n ∈ N. We see that conditions (2.19)(i) and (ii) are satisﬁed, and [{[αn , βn ] : n ∈ N}] = [u]. Therefore [u] ∈ λ1 . (1) (1) (ii) Let F, G ∈ λ1 (u) be ﬁlters on M. Then there exist sequences (αn ), (βn ) (2) (2) and (αn ), (βn ) on C 0 (R, [0, ∞)), converging to the same limit, which can be associated with ﬁlters F and G respectively according to (2.19). Denote Convergence Vector Spaces for Conservation Laws (1) (1) (2) 77 (2) αn = inf{αn , αn }, βn = sup{βn , βn }, n ∈ N. Clearly, αn ≤ αn+1 ≤ u ≤ βn+1 ≤ βn . Moreover, we have ∀ a, b ∈ R, a ≤ b ∫ b βn (x, t) − αn (x, t)dx −→ 0. a Furthermore, we have that [αn1 , βn1 ] ⊆ [αn , βn ] and [αn2 , βn2 ] ⊆ [αn , βn ]. Therefore, [αn1 , βn1 ] ∪ [αn2 , βn2 ] ⊆ [αn , βn ], which implies [{[αn , βn ] : n ∈ N}] ⊆ F ∩ G. (iii) Let F ∈ λ1 (u). Let G be a ﬁlter ﬁner than F. Then there exist sequences (αn ), (βn ) on C 0 (R, [0, ∞)) satisfying (2.19)(i), (ii) and [{[αn , βn ] : n ∈ N}] ⊆ F ⊆ G. Hence G ∈ λ1 (u). It follows from (i) - (iii) above that λ1 is a convergence structure. Next, we show that addition and scalar multiplication are continuous. In this regard, let F −→ u and G −→ v with respect to λ1 . Then there exist sequences (αn1 ), (βn1 ) ⊂ C 0 (R × [0, ∞)) that can be associated with the ﬁlter F according to (2.19) and sequences (αn2 ), (βn2 ) ⊂ C 0 (R × [0, ∞)) that can be associated with the ﬁlter G according to (2.19). Therefore, we have (a) from (2.19)(i) we have 1 2 1 2 αn1 + αn2 ≤ αn+1 + αn+1 ≤ u + v ≤ βn+1 + βn+1 ≤ βn1 + βn2 . (b) From (2.19)(ii) we have ∫ b (βn1 (x, t) + βn2 (x, t) − αn1 (x, t) − αn2 (x, t))dx ∫a b ∫ b 1 1 = (βn (x, t) − αn (x, t)) + (βn2 (x, t) − αn2 (x, t))dx −→ 0. a a (c) From (2.19)(iii) we have that ∀ n∈N ∃ F ∈F ∃ G∈G: F ⊆ [αn1 , βn1 ] and G ⊆ [αn2 , βn2 ] Hausdorﬀ Continuous Solution of Scalar Conservation laws 78 Hence, F + G ⊆ [αn1 , βn1 ] + [αn2 , βn2 ] ⊆ [αn1 + αn2 , βn1 + βn2 − αn1 (x, t)] so that [{[αn1 + αn2 , βn1 + βn2 − αn1 (x, t)] : n ∈ N}] ⊆ F + G. It thus follows from (a) - (c) above that F + G ∈ λ1 (u + v), which shows that addition is continuous. To show that scalar multiplication is continuous, let F ∈ λ1 (u) and let (αn ), (βn ) on C 0 (R, [0, ∞)) be sequences associated with F according to (2.19). Then for any constant c ∈ R, c ≥ 0 we have (a) cαn ≤ cαn+1 ≤ cu ≤ cβn+1 ≤ cβn (b) ∀ t ≥ 0, a, b ∈ R, a ≤ b ∫ b ∫ b (cβn (x, t) − cαn (x, t))dx = c(βn (x, t) − αn (x, t))dx −→ 0 a a (c) ∀ n ∈ N ∃ F ∈ F such that cF ⊆ [cαn , cβn ]. Which implies [{[cαn , cβn ] : n ∈ N}] ⊆ cF. The case c < 0 is treated in a similar way. Thus, cF ∈ λ1 (cu) which implies that scalar multiplication is continuous. We now show that λ1 is Hausdorﬀ. To do this we need to show that the set {0} is closed. Let u ∈ a({0}). Then ∃ F −→ u : {0} ∈ F Since F −→ u, it follows that there exist sequences (αn ), (βn ) ⊆ C 0 (R × [0, ∞)) satisfying (i) (ii) αn ≤ αn+1 ≤ u ≤ βn+1 ≤ βn , n ∈ N ∫ b (βn (x, t) − αn (x, t))dx −→ 0 a (2.20) ∀ t ≥ 0, a, b ∈ R, a ≤ b (iii) [{[αn , βn ] : n ∈ N}] ⊆ F . But F ⊆ [0], which implies, from (2.20)(iii), that [{[αn , βn ] : n ∈ N}] ⊆ F ⊆ [0]. Which means, ∀ n∈N ∃ A∋0: A ⊆ [αn , βn ]. This implies αn ≤ 0 ≤ βn . Taking limit as n −→ ∞ we have u = 0. Hence a({0}) = {0}, which implies that {0} is closed. This completes the proof. Convergence Vector Spaces for Conservation Laws 79 Let us recall that given a ﬁlter F on M and u ∈ M, F converges to u with respect to the subspace convergence structure induced on M by the order convergence structure, see Example 1.42, whenever ∃ sequences (αn ), (βn ) ⊂ C 0 (R × [0, ∞)) : (i) αn ≤ αn+1 ≤ βn+1 ≤ βn (ii) sup{αn : n ∈ N} = u = inf{βn : n ∈ N} (iii) [{[αn , βn ] : n ∈ N}] ⊆ F. (2.21) where the inﬁmum and the supremum are both taken in C 0 (R × [0, ∞)). Denote this induced convergence structure on M by λs . The convergence structure λ1 on M is closely related to λs . In this regard, we have the following. Lemma 2.4. Let F converge to u with respect to λ1 . Then F converge to u with respect to the convergence structure λs . Proof. Let (αn ) and (βn ) be sequences associated with F according to (2.19). Conditions (2.21)(i)and (2.21)(iii) follows from (2.19)(i) and (2.19)(iii), respectively. It follows from (2.19)(i) that u ∈ M ⊂ C 0 (R × [0, ∞)) is an upper bound of {αn : n ∈ N} and a lower bound of {βn : n ∈ N} in C 0 (R × [0, ∞)). We need to show that it is the least upper bound for {αn : n ∈ N} and the greatest lower bound for {βn : n ∈ N}. Let v be an upper bound for {αn : n ∈ N} and w be a lower bound of {βn : n ∈ N} in C 0 (R × [0, ∞)) such that v ≤ u ≤ w. Then for any a, b ∈ R, a ≤ b, and t ≥ 0 we have ∫ b ∫ b (u(x, t) − v(x, t))dx ≤ (βn (x, t) − αn (x, t))dx −→ 0 as n −→ 0. a Therefore a ∫ b (u(x, t) − v(x, t))dx = 0. a Using the continuity of u−v and the fact that u(x, t)−v(x, t) ≥ 0 for all x ∈ [a, b] and t ≥ 0 we obtain u = v. Similarly, ∫ b (w(x, t) − u(x, t))dx = 0 a which implies w = v. Using the fact that sup{αn : n ∈ N} ≤ u, inf{βn : n ∈ N} ≥ u and v, w are arbitrary we obtain sup{αn : n ∈ N} = u = inf{βn : n ∈ N}. This completes the proof. Note that the convergence structure λ1 is ﬁner than the convergence structure λs . Indeed, it follows from Lemma 2.4 that λ1 (u) ⊂ λs (u). The convergence vector space M is equipped with the induced uniform convergence structure JM deﬁned as follows, see (1.102): Let U be a ﬁlter on M × M. Hausdorﬀ Continuous Solution of Scalar Conservation laws 80 Then U ∈ JM ⇐⇒ ∃ F a ﬁlter on M : (1)F ∈ λ1 (0) (2)∆(F) ⊆ U (2.22) Lemma 2.5. The operator T : M −→ N is uniformly continuous with respect to the vector space convergence structures deﬁned on N as follows: ( π1 (F) −→ u weakly in L1 F −→ (u, h) ⇐⇒ π2 (F) −→ h in L1loc . Here π1 is the projection on C 0 (R × (0, ∞)) and π2 is the projection on U0 . Proof. We need to show that ∀ F −→ 0 in M ∃ G −→ 0 in N : (T × T )(∆(F)) ⊇ ∆(G). (2.23) Let F −→ 0 in M and let (αn ) and (βn ) be sequences associated with F according to (2.19). It is suﬃcient to prove that (T × T )(∆([{[αn , βn ] : n ∈ N}] ⊇ ∆(G) for some ﬁlter G in N such that G −→ 0 in N . Let ϕ ∈ C0∞ (R × [0, ∞)) be a test function. Let d = max{ − min α1 (x, t), max β1 (x, t)}. Then using (x,t)∈ supp ϕ (x,t)∈ supp ϕ (2.19)(i) we have that αn (x, t) ∈ [−d, d], βn (x, t) ∈ [−d, d], for (x, t) ∈ supp ϕ. Let −d ≤ u, v ≤ d. Then there exists Lϕ such that |f (u(x, t)) − f (v(x, t))| ≤ Lϕ |u(x, t) − v(x, t)| for (x, t) ∈ supp ϕ (2.24) where Lϕ is the lipschitz constant of f on the compact interval [−d, d]. For any n ∈ N deﬁne ∞ ∞ ∫ ∫ π (g)(x, t)ϕdxdt 1 −∞ 0 ∫ ∫ ≤ (|ϕ | + L |ϕ |) (β − α ) dxdt, t ϕ x n n Ω . (2.25) Gn = g ∈ N ∞ for all ϕ ∈ C (R × [0, ∞)) ϕ 0 ∫b ∫b π2 (g)(x)dx ≤ (βn (x, 0) − αn (x, 0))dx a a for any a, b ∈ R, a ≤ b Let G = [{Gn : n ∈ N}]. (2.26) Convergence Vector Spaces for Conservation Laws 81 From (2.25) we see that the ﬁlter G −→ 0 in N . It remains to show that the inclusion in (2.23) holds. Equivalently, we need to show that ∀ G∈G ∃ F ∈F : (T × T )(∆(F )) ⊆ ∆(G). For n ∈ N let (u, v) ∈ ∆([αn , βn ]), that is u − v ∈ [αn , βn ]. Then for every test function ϕ and real intervals [a, b] we have ∫ ∫ ∫∫ (T1 v − T1 u)ϕdxdt = ((v − u)t + (f (v) − f (u))x )ϕdxdt Ω supp ϕ ∫∫ = (v − u)ϕt + (f (v) − f (u))ϕx dxdt supp ϕ ∫∫ ≤ (|v − u| |ϕt | + |f (v) − f (u)| |ϕx |)dxdt. supp ϕ ∫∫ (|ϕt | + L |ϕx |) |v − u| dxdt. ≤ supp ϕ ∫∫ (|ϕt | + L |ϕx |)(βn − αn )dxdt ≤ supp ϕ ∫∫ (|ϕt | + L |ϕx |)(βn − αn )dxdt. = Ω where Ω = R × [0, ∞), and ∫ b ∫ b (T2 v − T2 u)dx = (v(x, 0) − u(x, 0))dx a a ∫ b ≤ (|v(x, 0) − u(x, 0)|)dx ∫a b ≤ (|v(x, 0) − u(x, 0)|)dx a ∫ b ≤ (βn (x, 0) − αn (x, 0))dx. a Therefore, (T × T )(u, v) = (T u, T v) ∈ {(p, q) : p − q ∈ Gn } = ∆(Gn ), which implies that (T × T )∆([αn , βn ]) ⊆ ∆(Gn ). Hence (T × T )(∆([{[αn , βn ] : n ∈ N}])) ⊇ ∆([{gn : n ∈ N}]) = ∆(G). Hausdorﬀ Continuous Solution of Scalar Conservation laws 82 This completes the proof. Corollary 2.6. Let F be a Cauchy ﬁlter on M. Then ( π1 (T (F)) is weakly L1 Cauchy π2 (T (F))is Cauchy in L1loc . On the space N we consider the ﬁnal uniform convergence structure JN ,T which is deﬁned as follows: ∃ V ∈ JM : U ∈ JN ,T ⇐⇒ ∃ ϕ1 · · · ϕk ∈ N : (2.27) (T × T )(V) ∩ ([ϕ1 ] × [ϕ1 ]) ∩ · · · ∩ ([ϕk ] × [ϕk ]) ⊆ U. Proposition 2.7. The uniform convergence structure JN ,T is Hausdorﬀ. Proof. We need to show that ∀ ϕ, ψ ∈ N , ϕ ̸= ψ : ∀ U ∈ JN ,T : ∃ U ∈U : (ϕ, ψ) ̸∈ U. Let ϕ, ψ ∈ N be such that ϕ ̸= ψ. Set U = (T × T )(V) ∩ ([ϕ1 ] × [ϕ1 ] ∩ · · · ∩ [ϕk ] × [ϕk ]) with basis U = (T × T )(V ) ∪ ((ϕ1 , ϕ1 ), · · · , (ϕk , ϕk )), V ∈ V. Suppose (ϕ, ψ) ∈ U, then (ϕ, ψ) ̸∈ ((ϕ1 , ϕ1 ), · · · , (ϕk , ϕk )) which implies (ϕ, ψ) ∈ (T × T )(V ), V ∈ V. Since T is injective it follows that (T −1 ϕ, T −1 ψ) ∈ V, V ∈ V. Thus T −1 ϕ = T −1 ψ since JM is Hausdorﬀ so that ϕ = ψ, which is a contradiction. Hence (ϕ, ψ) ̸∈ U, for some U ∈ U. This completes the proof. Note that the ﬁnal uniform convergence structure JN ,T is the ﬁnest uniform convergence structure on N making T uniformly continuous, see [20]. Thus we have the following Corollary 2.8. If F is a Cauchy ﬁlter on N with respect to JN ,T , then π1 (F) is weakly Cauchy in L1 and π2 (F) is Cauchy in L1loc . Proof. The result follows from Corollary 2.6. We now apply the completion process. In this regard, the Wyler completion of M is constructed in the following way. Denote by C[M] the set of all Cauchy ﬁlters on M, and deﬁne an equivalence relation on C[M] through F ∼C G ⇔ F ∩ G ∈ C[M]. (2.28) Let us denote by M♯ the quotient space C[M]/ ∼C . For F ∈ C[M], denote the equivalence class generated by F with respect to (2.28) by [F]. One may identify Convergence Vector Spaces for Conservation Laws 83 M with a subset of M♯ by associating each u ∈ M with λ1 (u) ⊂ C[M]. From the deﬁnition of a convergence structure given in (1.36) it is clear that λ1 (u) is indeed a ∼C -equivalence class. Furthermore, since λ1 is Hausdorﬀ, the mapping iM : M ∋ u 7→ λ1 (u) ∈ M♯ is injective. Thus we may consider the convergence space M as a subset of M♯ The Wyler completion of M is the set M♯ , equipped with the following vector space convergence structure, see for instance [100]: ( ∃ F1 · · · Fn ∈ [F] : ♯ ∩ ∩ ∩ ∩ ∩ G ∈ λ1 ([F]) ⇔ (2.29) iM (F1 ) · · · iM (Fn ) [F1 ] · · · [Fn ] ⊆ G Similarly, let C[N ] be the set of all Cauchy ﬁlters on N , and deﬁne an equivalence relation on C[N ] through F ∼C G ⇔ F ∩ G ∈ C[N ]. (2.30) The Wyler completion of (N , JN ,T ) is the set N ♯ equipped with the uniform convergence structure JN♯ ,T which is deﬁned as follows ∃ V ∈ JN ,T : ∃ Cauchy ﬁlters F1 · · · Fk ∈ C[N ]\λJN ,T : ∩ U ∈ JN♯ ,T ⇐⇒ (2.31) (iN × iN )(V) [(iN (F1 ) × [F1 ]) ∩ ([F1 ] × iN (F1 )) ∩ · · · ∩(iN (Fk ) × [Fk ]) ∩ (Fk × iN (Fk ))] ⊆ U, where λJN ,T denotes the convergence structure induced by the ﬁnal convergence structure JN ,T . In general, λJN ,T is not a ﬁnal convergence structure, see [20] and Section 1.2.1. Since T : M −→ N is uniformly continuous there exists a unique uniformly continuous mapping T ♯ : M♯ −→ N ♯ such that the diagram T - N M iM iN ? M♯ T♯ - (2.32) ? N♯ commutes, where iM and iN are the uniformly continuous embeddings associated with the completion M♯ and N ♯ , respectively. Furthermore, since T is injective, it follows by the deﬁnition of JN ,T that T is a uniformly continuous embedding. That is, T −1 is uniformly continuous on T (M) ⊂ N . Therefore the mapping T ♯ is injective as well. We now give a concrete description of the completion M♯ of M as a subset of the space of ﬁnite Hausdorﬀ continuous function H. In this regard, the following characterization of Cauchy ﬁlters is essential. Hausdorﬀ Continuous Solution of Scalar Conservation laws 84 Proposition 2.9. A ﬁlter F on M is a Cauchy ﬁlter with respect to the vector space convergence structure λ1 if and only if ∃ (αn ), (βn ) ⊆ C 0 (R × [0, ∞)) : (i) αn ≤ αn+1 ≤ βn+1 ≤ βn ∫b (2.33) (ii) a (βn (x, t) − αn (x, t))dx −→ 0 ∀ t > 0, a, b ∈ R, a ≤ b (iii) [{[αn , βn ] : n ∈ N}] ⊆ F. (1) (1) Proof. Let (2.33) hold. Then set αn = αn − βn and βn = βn − αn on C 0 (R × (1) (1) [0, ∞)). Therefore the sequences αn and βn satisfy the following. (1) (1) (1) (1) (i) αn ≤ αn+1 ≤ 0 ≤ βn+1 ≤ βn . This follows from (2.33)(i). ∫ b (1) (1) (ii) (βn (x, t) − αn (x, t))dx −→ 0. This is because a ∫ b ∫ (1) (βn (x, t) − (1) αn (x, t))dx = a b ∫a (βn − αn − αn + βn )dx b (βn − αn )dx −→ 0. =2 a (1) (1) (iii) [{[αn , βn ] : n ∈ N}] ⊆ F − F . To see this, observe that from (2.33)(iii) we have ∀ n∈N ∃ F ∈F : F ⊆ [αn , βn ]. It follows that F − F ⊆ [αn , βn ] − [αn , βn ] ⊆ [αn − βn , βn − αn ] = [αn(1) , βn(1) ]. Thus, [{[αn(1) , βn(1) ] : n ∈ N}] ⊆ F − F, which implies F − F converges to zero. Hence F is Cauchy with respect to λ1 . Conversely, let F on M be a Cauchy ﬁlter with respect to λ1 . Then F − F ∈ λ1 (0). Let αn , βn ⊆ C 0 (R × [0, ∞)) be sequences associated with F − F according to (2.19). It follows from (2.19)(iii) that ∀ n∈N ∃ F ∈F : F − F ⊆ [αn , βn ]. Choose any v ∈ F. Then F ⊆ F − F + v. Since the ultraﬁlter [v] ∈ λ1 (v), it follows that there exists sequences αn1 , βn1 ⊆ C 0 (R × [0, ∞)) such that [{[αn1 , βn1 ] : Convergence Vector Spaces for Conservation Laws 85 n ∈ N}] ⊆ [v]. Therefore, F ⊆ F − F + v ⊆ [αn , βn ] + [αn1 , βn1 ] ⊆ [αn + αn1 , βn + βn1 ], which implies [{[αn + αn1 , βn + βn1 ] : n ∈ N}] ⊆ F . Denote α̃n = αn +αn1 and β̃n = βn +βn1 . clearly, α̃n ≤ α̃n+1 ≤ β̃n+1 ≤ β̃n . Moreover, for all a, b ∈ R, a ≤ b we have ∫b ∫b (β̃n − α̃n )dx = a (βn + βn1 − αn − αn1 )dx a ∫b ∫b (βn − αn ) + = a (βn1 − αn1 )dx −→ 0. a Hence there exists sequences (α̃n ), (β̃n ) ⊆ C 0 (R × [0, ∞)) satisfying (2.19). This completes the proof. Consider some p ∈ M♯ . Then there exists a Cauchy ﬁlter G on M such that G −→ p in M♯ . Then G is Cauchy with respect to λs as well. Therefore there exists u ∈ H such that G −→ u in H with respect to the order convergence structure on H. We deﬁne the mapping η : M♯ −→ H via η(p) = u (2.34) Theorem 2.10. The map η is well deﬁned, that is, if G, V are Cauchy ﬁlters in M and G, V −→ p in M♯ . Then G and V converge to the same limit u in H. Proof. Let G, V −→ p in M♯ . Then G ∩ V −→ p in M♯ . But G ∩ V is a Cauchy ﬁlter with respect to λ1 . Therefore it converges in H. Let G ∩ V −→ w in H, then G ⊇ G ∩ V implies that G −→ w. Similarly, V −→ w in H. The proof is complete. Theorem 2.11. The map η is injective. Proof. Let η(p) = η(q) = u for some p, q ∈ M♯ . There exist Cauchy ﬁlters G1 , G2 on M such that G1 −→ p, G2 −→ q in M♯ and G1 , G2 −→ u ∈ H. Let (i) (i) (αn ), (βn ) be the sequences associated with Gi , i = 1, 2 in terms of (2.33). Let (1) (2) αn = inf{αn , αn } in C 0 (R × [0, ∞)), that is, αn is the point-wise minimum of (1) (2) (1) (2) αn and αn . Similarly, βn = sup{βn , βn }. Clearly, αn , βn ∈ C 0 (R × [0, ∞)) and the sequences (αn ), (βn ) are monotone increasing and decreasing respectively. It is also easy to see that [{[αn , βn ] : n ∈ N}] ⊆ G1 ∩ G2 . In order to associate the sequences (αn ) and (βn ) with G1 ∩ G2 in terms of (2.33) we need to show that the property (2.33)(ii) is satisﬁed. From the deﬁnition of the order convergence Hausdorﬀ Continuous Solution of Scalar Conservation laws 86 (i) (i) structure we have αn (x, t) ≤ u(x, t) ≤ βn (x, t) i = 1, 2. Using the fact that max{x, y} ≤ x + y for x ≥ 0, y ≥ 0 we obtain βn (x, t) − αn (x, t) = max{βn(1) (x, t), βn(2) (x, t)} − u(x, t) + u(x, t) − min{αn(1) (x, t), αn(2) (x, t)} = max{βn(1) (x, t) − u(x, t), βn(2) (x, t) − u(x, t)} + max{u(x, t) − αn(1) (x, t), u(x, t) − αn(2) (x, t)} ≤ βn(1) (x, t) − u(x, t) + βn(2) (x, t) − u(x, t) + u(x, t) − αn(1) (x, t) + u(x, t) − αn(2) (x, t) = βn(1) (x, t) − αn(1) (x, t) + βn(2) (x, t) − αn(2) (x, t). Then we have ∫ a b (βn (x, t) − αn (x, t))dx ∫ b ≤ (βn(1) (x, t) − αn(1) (x, t))dx a ∫ b + (βn(2) (x, t) − αn(2) (x, t))dx −→ 0. a Therefore G1 ∩ G2 is a Cauchy ﬁlter with respect to λ1 in M. This means that G1 ∩ G2 converges in M♯ . Let G1 ∩ G2 −→ w in M♯ . Then G1 and G2 being ﬁner that G1 ∩ G2 also converges to w. Hence p = w = q. 2.3 Approximation results In this section we consider the Cauchy problem for the viscous Burgers equation of the form 1 ( δ,ε )2 δ,ε in R × (0, ∞) vtδ,ε + v x = εvxx (2.35) 2 v δ (x, 0) = u0 (x) − 2δ, δ > 0 in R × {t = 0} (2.36) which is the Cauchy problem of the viscous Burgers equation with a vertical shift by 2δ in the initial condition. Using the auxiliary problem (2.35) - (2.36) and techniques for problems of monotonic type, [125], we show how the entropy solution of the inviscid Burgers equation [39, 53, 69] 1 ut + (u)2x = 0 2 with the initial condition in R × (0, ∞) (2.37) u(x, 0) = u0 (x) in R × {t = 0}. (2.38) Approximation results 87 is approximated from below. Applying Hopf’s technique [69], see also Section 1.1.4, to equations (2.35) (2.36), we have a solution similar to (1.56) where K(x, y, t) is replaced with K δ (x, y, t), deﬁned below in (2.40). Theorem 1.28 may be stated as follows Theorem 2.12. Suppose u0 ∈ L1loc (R) is such that (1.58) holds. Then there exists a unique classical solution of equation (2.35)-(2.36) given by ∫ ∞ x−y − 1 K δ (x,y,t) 2ε dy t e (2.39) v δ,ε (x, t) = −∞ ∫ ∞ − 1 K δ (x,y,t) 2ε dy e −∞ where (x − y)2 K (x, y, t) = + 2t ∫ y u0 (s)ds − 2δy. (2.40) u0 (ξ)dξ − 2δa as x −→ a, t −→ 0, (2.41) δ 0 with the following properties: (i) For all a ∈ R, ∫ x ∫ v δ,ε (ξ, t)dξ −→ 0 a 0 (ii) If u0 (x) is continuous at x = a then v δ,ε (x, t) −→ u0 (a) − 2δ as x −→ a, t −→ 0. (2.42) Furthermore, a solution of (2.35) - (2.36) which is C 2 -smooth in an interval 0 < t < T and satisﬁes (2.41) for each value of a ∈ R necessarily coincides with (2.39) in this interval. The function K δ (x, y, t) satisﬁes properties (P1) - (P3) given in Section 1.1.4. Using this fact we now introduce the functions δ ymin = min{y : K δ (x, y, t) = minK δ (x, z, t)} z∈R and δ ymax = max{y : K δ (x, y, t) = minK δ (x, z, t)} z∈R Observe that K δ (x, y, t) = K(x, y, t) − 2δy for x , t ﬁxed, Lemma 2.13. For each δ > 0 we have δ ymin (x, t) = ymin (x + 2δt, t) δ ymax (x, t) = ymax (x + 2δt, t). (2.43) Hausdorﬀ Continuous Solution of Scalar Conservation laws 88 Proof. From (1.57) we get K(x + 2δt, y, t) − 2δx − 2δ 2 t ∫ y (x + 2δt − y)2 = + u0 (s)ds − 2δx − 2δ 2 t 2t 0 ∫ y 2 (x − y) + 4(x − y)δt + 4δ 2 t2 + u0 (s)ds − 2δx − 2δ 2 t = 2t ∫ y0 2 (x − y) = + 2δx − 2δy + 2δ 2 t + u0 (s)ds − 2δx − 2δ 2 t 2t 0 ∫ y 2 (x − y) = + u0 (s)ds − 2δy 2t 0 δ = K (x, y, t) This implies that for ﬁxed x and t, the functions K δ (x, y, t) and K(x, y, t) diﬀer by a constant. Therefore they have the same set of minimizers, which implies the statement of the Lemma. δ δ The functions ymin and ymax have the properties (Y1) - (Y4) given in Section δ δ 1.1.4, see also [69]. Hence, the functions ymin and ymax are monotone functions in δ δ x. Moreover, for every t ≥ 0, ymin (x, t) = ymax (x, t) for all x ∈ R with the possible δ δ exception of a denumerable set of values of x where ymin (x, t) < ymax (x, t). It follows from property (Y1) that δ x − ymin (x, t) x − ymax (x + δt) ≤ t t From Theorem 1.29 we have that for all x and t > 0, δ x − ymax (x, t) ≤ lim inf v δ,ε (α, θ) ≤ t α→x θ→t ε→0 (2.44) δ x − ymin (x, t) lim sup v (α, θ) ≤ (2.45) t α→x θ→t ε→0 δ,ε and, in particular, that v δ (x, t) = lim v δ,ε (α, θ) = α→x θ→t ε→0 δ δ x − ymax (x, t) x − ymin (x, t) = t t (2.46) δ δ holds at every point (x, t) where ymax (x, t) = ymin (x, t). The following Lemma shows that the functions u and u deﬁned in (1.68) and (1.69) are lower and upper semi-continuous respectively. Lemma 2.14. The functions u and u deﬁned in (1.68) and (1.69) are lower semi-continuous and upper semi-continuous respectively. Approximation results 89 Proof. Let u > m for some m ∈ R and let µ be such that u > m + µ. Since u(x, t) = sup{inf{uε (α, θ) : |α − x| < η, |θ − t| < η, ε < η} : η > 0}, it follows that ∃ η>0: inf{uε (α, θ) : |α − x| < η, |θ − t| < η, ε < η} > m + µ. Therefore, uε (α, θ) > m + µ if |α − x| < η, |θ − t| < η, ε < η. Let x̃ ∈ (x − η, x + η) and t̃ ∈ (t − η, t + η). Then u(x̃, t̃) = lim inf uε (α, θ) ≥ m + µ > m. α → x̃ θ → t̃ ε→0 Since the last inequality holds for all x̃ ∈ (x − η, x + η) and t̃ ∈ (t − η, t + η) it shows that u is lower semi- continuous. The proof of upper semi-continuity of u is done in a similar way. It follows from Lemma 2.14 that the functions v δ (x, t) = lim inf v δ,ε (α, θ) α→x θ→t ε→0 (2.47) v δ (x, t) = lim sup v δ,ε (α, θ) α→x θ→t ε→0 (2.48) and are lower semi-continuous and upper semi-continuous respectively. Lemma 2.15. The functions v δ deﬁned by (2.48) and u deﬁned by (1.69) satisfy the following inequality v δ (x, t) ≤ u(x + δt) − δ x ∈ R, t ≥ 0. Hausdorﬀ Continuous Solution of Scalar Conservation laws 90 Proof. From the inequality (2.45), it follows that δ x − ymin (x, t) x − ymin (x + 2δt, t) v (x, t) ≤ = t t x − ymax (x + δt, t) x + δt − ymax (x + δt, t) = −δ ≤ t t ≤ lim inf uε (z + δτ, τ ) − δ z→x τ →t ε→0 δ = u(x + δt, t) − δ as required. Consider the viscous problem 1 ( δ,ρ,ε )2 δ,ρ,ε w = εwxx R × (0, ∞) x 2 wδ,ρ,ε (x, 0) = I(ρ, u0 )(x) − 2δ R × {t = 0} wtδ,ρ,ε + (2.49) (2.50) Lemma 2.16. Let wδ,ρ,ε denote the solution of the Cauchy problem (2.49) - (2.50). Then [ ρ] δ,ρ w (x, t) ≤ u(x, t) − δ, x ∈ R, t ∈ 0, . (2.51) δ where wδ,ρ (x, t) = lim sup wδ,ρ,ε (α, θ) as it is in (1.69). α→x θ→t ε→0 Proof. Consider the viscous problem (2.35) - (2.36) with solution v δ,ε and the Cauchy problem 1 δ,σ,ε ztδ,σ,ε + (z δ,σ,ε )2x = εzxx 2 δ,σ,ε z (x, 0) = u0 (x + σ) − 2δ = v δ,ε (x + σ, 0) From (2.50) we have that wδ,ρ,ε (x, 0) = I(ρ, uε )(x, 0) − 2δ ≤ uε (x + σ, 0) − 2δ = u0 (x + σ) − 2δ ∀ |σ| ≤ ρ = z δ,σ,ε (x, 0) = v δ,ε (x + σ, 0). It follows from [125, Chapter IV 25II] that wδ,ρ,ε (x, t) ≤ v δ,ε (x + σ, t) ∀ t > 0, x ∈ R, |σ| ≤ ρ. (2.52) (2.53) Approximation results 91 Therefore, wδ,ρ (x, t) = lim sup wδ,ρ,ε (α, θ) ≤ ε→0 α→x θ→t lim sup v δ,ε (α + σ, θ) = v δ (x + σ, t), ε→0 α→x θ→t that is, By Lemma 2.15 wδ,ρ (x, t) ≤ v δ (x + σ, t), ∀ |σ| < ρ ≤ u(x + δt + σ, t) − δ. Now, for ﬁxed x and t take σ = −δt. Then wδ (x, t) ≤ u(x, t) − δ as required. 2.3.1 Requirements for u0 Lemma 2.17. Assume that lim u0 (x) x−→∞ and lim u0 (x) x−→−∞ exist. Then condition (1.58) is satisﬁed. Proof. Let lim u0 (x) = β then x−→∞ ∃ M : |u0 (x) − β| < 1 for x > M lim x−→∞ | ∫x 0 ∫x ∫M u0 (s)ds| | 0 u0 (s)ds| + | M u0 (s)ds| ≤ lim x−→∞ x2 x2 ∫M | u0 (s)ds| (|β| + 1)(x − M ) ≤ lim 0 + lim x−→∞ x−→∞ x2 x2 =0 The following Lemmas are consequences of condition (2.54) on u0 . Lemma 2.18. Suppose conditions (2.54) holds. Then for every ε > 0 lim uε (x, t) = lim u0 (x) x→+∞ x → +∞ t → t̃ and lim uε (x, t) = lim u0 (x) x→−∞ x → −∞ t → t̃ (2.54) Hausdorﬀ Continuous Solution of Scalar Conservation laws 92 where uε is the solution of the viscous Burger’s equation (1.54) (1.55). Proof. Let N > 0. For any ε < 1 and x > N we have ∫ N ∫ N ( ) 2 ∫y 2 1 (x−y) 1 1 (x−y) + − u (s)ds 0 − 2ε K(x,y,t) − 2ε 4t 2ε 4t 0 dy u (y)e = dy u (y)e e 0 0 −∞ −∞ ∫ N ( ) 2 ∫y 2 1 (x−y) 1 (x−y) − + u (s)ds 0 0 ≤ max e− 2ε 4t |u0 (y)|e 2ε 4t dy y∈(−∞,N ] −∞ ∫ N ( ) 2 ∫y 2 1 (N −y) 1 (x−N ) − + u (s)ds 0 0 |u0 (y)|e 2 4t dy (2.55) ≤ e− 2ε 4T −∞ Taking the limit as x −→ +∞ we have that the expression on the right of (2.55) converges to zero, so that ∫ N 1 lim u0 (y)e− 2ε K(x,y,t) dy = 0 x−→+∞ −∞ which implies ∫ N u0 (y)e− 2ε K(x,y,t) dy = 0 1 lim x−→+∞ Similarly, −∞ ∫ lim x−→+∞ N 1 K(x,y,t) − 2ε e −∞ (2.56) dy = 0 which implies ∫ N lim x−→+∞ e− 2ε K(x,y,t) dy = 0 1 (2.57) −∞ Now consider the solution uε of the viscous Burgers equation (1.54) - (1.55) which is given as ∫∞ 1 − 2ε K(x,y,t) dy −∞ u0 (y)e ε u (x, t) = ∫ ∞ − 1 K(x,y,t) 2ε dy −∞ e ∫N ∫∞ 1 1 K(x,y,t) − 2ε K(x,y,t) − 2ε u (y)e dy + dy 0 −∞ N u0 (x)e = ∫ N − 1 K(x,y,t) ∫ N − 1 K(x,y,t) 2ε dy + ∞ e 2ε dy −∞ e Let σ > 0 and N be such that β − σ < u0 (x) < β + σ whenever |x| > N, where β = lim u0 (x). Then x→+∞ ∫ ∞ − 1 K(x,y,t) 1 K(x,y,t) − 2ε 2ε dy + (β + σ) dy u (y)e 0 N e uε (x, t) ≤ −∞ ∫ N ∫ 1 1 K(x,y,t) dy + N e− 2ε K(x,y,t) dy − 2ε e −∞ ∞ ∫N Approximation results 93 Using (2.56) and (2.57) we have that lim sup uε (x, t) ≤ x → +∞ t → t̃ lim sup x → +∞ t → t̃ [∫ N −∞ lim sup x → +∞ t → t̃ Therefore u0 (y)e− 2ε K(x,y,t) dy + (β + σ) 1 [∫ N e− 2ε K(x,y,t) dy + −∞ [ 1 ∫∞ ∫N ∞ ∫∞ N e− 2ε K(x,y,t) dy 1 e− 2ε K(x,y,t) dy 1 1 K(x,y,t) − 2ε ] ] . ] dy lim sup (β + σ) N e x → +∞ t → t̃ [∫ ] = β + σ. lim sup uε (x, t) = N − 1 K(x,y,t) 2ε lim sup dy x → +∞ ∞ e x → +∞ t → t̃ t → t̃ Similarly, [ ∫∞ 1 − 2ε K(x,y,t) ] lim sup (β − σ) N e dy x → +∞ t → t̃ [∫ ] lim sup uε (x, t) ≥ = β − σ. N − 1 K(x,y,t) 2ε lim sup dy x → +∞ ∞ e x → +∞ t → t̃ t → t̃ Since σ is arbitrary, we have lim sup uε (x, t) = lim inf uε (x, t) = β = lim u0 (x) x→+∞ x → +∞ x → +∞ t → t̃ t → t̃ which implies lim uε (x, t) = lim u0 (x) x→+∞ x → +∞ t → t̃ as required. The proof of the second part of the Lemma is similar. As an easy consequence of Lemma 2.18, we obtain Corollary 2.19. For any t̃ ≥ 0 we have lim u(x, t) = lim sup u(x, t) = lim inf u(x, t) = β = lim u0 (x) x→+∞ x → +∞ x → +∞ x → +∞ t → t̃ t → t̃ t → t̃ Hausdorﬀ Continuous Solution of Scalar Conservation laws 94 and lim u(x, t) = lim sup u(x, t) = lim inf u(x, t) = β = lim u0 (x) x→−∞ x → −∞ x → −∞ x → −∞ t → t̃ t → t̃ t → t̃ where u and u are deﬁned by (1.68) and (1.68) respectively. 2.4 Existence and uniqueness results In this section we prove existence result for solution of the equation T ♯ u♯ = 0 in the case of the Burgers equation. More precisely,(for every ) uo ∈ U0 we construct 0 a Cauchy sequence (wk ) in M such that T wk −→ in N . The approximau0 tion results in the previous session are utilized for this purpose. Let δk = 41k and ρk = 21k , k ∈ N. For every k ∈ N consider the problem (2.49) - (2.50) with δ = δk and ρ = ρk . Using Lemma 2.16 we obtain the following inequality wδk ,ρk (x, t) ≤ wδk+1 ,ρk+1 (x, t) − (δk − δk+1 ), [ ] 2k+1 x ∈ R, t ∈ 0, . 3 (2.58) Indeed, we have wδk ,ρk (x, 0) = I(ρk , u0 )(x) − 2δk = I(ρk − ρk+1 , wδk+1 ,ρk+1 (·, 0))(x) − 2(δk − δk+1 ). Hence the inequality (2.58) follows from Lemma 2.16 with u = wδk+1 ,ρk+1 , δ = δk − δk+1 and ρ = ρk − ρk+1 . The upper bound for the time interval is obtained as follows 1 1 1 − 2k+1 (2 − 1) 2k+1 ρ ρk − ρk+1 2k 2k+1 = = 1 = = . 1 1 δ δk − δk+1 3 − (4 − 1) 4k 4k+1 4k+1 The construction of the Cauchy sequence is based on the following Lemma 2.20. For every k there exists εk such that wδ2k ,ρ2k ,εk satisﬁes wδ2k−1 ,ρ2k−1 (x, t) ≤ wδ2k ,ρ2k ,εk (x, t) ≤ wδ2k+1 ,ρ2k+1 (x, t), (2.59) 4k for x ∈ R, and t ∈ [0, ]. 3 Proof. Assume the opposite, that is, there exists k > 0 such that for every ε > 0 k there exists (xε , tε ) with tε ∈ [0, 43 ] such that one of the inequalities in (2.59) is violated. Since tε is in a compact interval, there exists a sequence (εn ) such that Existence and uniqueness results 95 k (tεn ) converges. Let tεn −→ t̃ ∈ [0, 43 ]. At least one of the inequalities in (2.59) is violated for a subsequence of (εn ). To avoid too many notations we denote this subsequence by (εn ). Assume the second inequality is violated. The other case is dealt with in a similar way. Now let us consider the sequence (xεn ). Case 1. The sequence (xεn ) has an accumulation point x̃ ∈ R. Then there is a subsequence converging to x̃. Without loss of generality we may assume that xεn −→ x̃. Then wδ2k ,ρ2k (x̃, t̃) ≥ lim supwδ2k ,ρ2k ,εk (xεn , tεn ) n−→∞ δ2k+1 ,ρ2k+1 ≥w (x̃, t̃) which contradicts (2.58). Case 2. The sequence (xεn ) is unbounded. Then it has a subsequence diverging to +∞ or −∞. Let us denote this subsequence again by (xεn ) and let it converge to +∞ (the case of −∞ is treated similarly). Then using Corollary 2.19 we have lim wδ2k ,ρ2k ,εn (xεn , tεn ) ≥ lim wδ2k+1 ,ρ2k+1 (xεn , tεn ) n−→∞ n−→∞ = lim I(ρ2k+1 , u0 )(x) − 2δ2k+1 n−→∞ = lim u0 (x) − 2δ2k+1 . n−→∞ (2.60) On the other hand, by Lemma 2.18 lim wδ2k ,ρ2k ,εn (xεn , tεn ) = lim I(ρ2k , u0 )(x) − 2δ2k n−→∞ n−→∞ = lim u0 (x) − 2δ2k . n−→∞ (2.61) The relations (2.60) and (2.61) lead to lim u0 (x) − 2δ2k ≥ lim u0 (x) − 2δ2k+1 n−→∞ n−→∞ which is impossible since δ2k > δ2k+1 . The contradictions obtained in Case 1 and Case 2 proves the statement of the lemma. Now we construct an increasing sequence (αn ) in M as follows. Set αk (x, t) = wδ2k ,ρ2k ,εk (x,t) (x, t) for x ∈ R, t ∈ [0, 4k−1 ]. Then αk is extended for t ∈ [4k−1 , ∞) in such a way that αk ∈ C 1 (R × [0, ∞)), αk−1 (x, t) ≤ αk (x, t) < inf wδ2p ,ρ2p ,εp (x, t) p>k (2.62) Note that for every (x, t) the sequence (wδ2p ,ρ2p ,εp (x, t)) is eventually monotone increasing so that the inﬁmum in the inequality (2.62) is ﬁnite. The inequality αk−1 (x, t) ≤ αk (x, t) is obtained from (2.59) for x ∈ R and t ∈ [0, 4k−1 ] and from (2.62) for x ∈ R and t ∈ [4k−1 , ∞). It is also easy to see from Lemma 2.16 that αk (x, t) ≤ u(x, t), x ∈ R, t ≥ 0. Hausdorﬀ Continuous Solution of Scalar Conservation laws 96 Lemma 2.21. At any point (x, t) we have αk (x, t) ≥ u(x − 3δ2k−1 t − ρ2k−1 , t) − 3δ2k−1 − 2ρ2k−1 t for suﬃciently large k. Proof. Let the point (x, t), t > 0 be ﬁxed. Let k be so large that 4k−1 > t. Then αk (x, t) = wδ2k ,ρ2k ,εk (x, t) ≥ wδ2k−1 ,ρ2k−1 (x, t) ≥ wδ2k−1 ,ρ2k−1 (x, t) 2k−1 x − ymax (x, t) ≥ t (2.63) where 2k−1 ymax (x, t) = max{y : K 2k−1 (x, y, t) = minK 2k−1 (x, z, t)} z∈R ∫ y (x − y)2 K (x, y, t) = + I(ρ2k−1 , u0 )(s)ds − 2δ2k−1 y. 2t 0 2k−1 2k−1 Then ymax (x, t) is a solution to ∂K∂y = 0. That is, and 2k−1 2k−1 ymax (x, t) − x 2k−1 + I(ρ2k−1 , u0 )(ymax (x, t)) − 2δ2k−1 = 0. t Then there exists γ = γ(x, t), |γ| ≤ ρ2k−1 such that 2k−1 ymax (x, t) + γ(x, t) − (x + 2δ2k−1 t + γ(x, t)) 2k−1 + u0 (ymax (x, t) + γ(x, t)) = 0 t or equivalently ∂K 2k−1 (x + 2δ2k−1 t + γ(x, t), ymax (x, t) + γ(x, t), t) = 0 ∂y Therefore, 2k−1 ymax (x + 2δ2k−1 t + γ(x, t), t) ≥ ymax (x, t) + γ(x, t) Furthermore, using the monotonicity of ymax we have 2k−1 ymax (x, t) ≤ ymax (x + 2δ2k−1 t + γ(x, t), t) − γ(x, t) ≤ ymax (x + 2δ2k−1 t + ρ2k−1 , t) + ρ2k−1 . From (2.63) we obtain x − ymax (x + 2δ2k−1 t + ρ2k−1 , t) ρ2k−1 αk (x, t) ≥ − t t x − ymin (x + 3δ2k−1 t + ρ2k−1 , t) ρ2k−1 > − t t x + 3δ2k−1 t − ymin (x + 3δ2k−1 t + ρ2k−1 , t) 2ρ2k−1 = − − 3δ2k−1 t t 2ρ2k−1 ≥ u(x + 3δ2k−1 t + ρ2k−1 , t) − − 3δ2k−1 . t Existence and uniqueness results 97 This completes the proof. In a similar way one constructs a decreasing sequence (βk ) such that at any point (x, t) we have u(x, t) ≤ βk (x, t) ≤ u(x − 3δ2k−1 t − ρ2k−1 , t) + 3δ2k−1 + 2ρ2k−1 . t Clearly, αk ≤ wδ2k ,ρ2k ,εk ≤ βk . In order to prove that (wδ2k ,ρ2k ,εk ) is a Cauchy sequence in M, it remains to show that (2.33)(ii) holds. Let t > 0 and a, b ∈ R, such that a ≤ b. For all suﬃciently large k ∫ b (βk (x, t) − αk (x, t))dx a ∫ b 4ρ2k−1 ]dx ≤ [u(x − 3δ2k−1 t − ρ2k−1 , t) − u(x + 3δ2k−1 t + ρ2k−1 , t) + 6δ2k−1 + t a ∫ b+3δ2k−1 t+ρ2k−1 ∫ b−3δ2k−1 t−ρ2k−1 = u(x, t)dx − u(x, t)dx + (6δ2k−1 t + ρ2k−1 )(b − a) a+3δ2k−1 t+ρ2k−1 b+3δ2k−1 t+ρ2k−1 ∫ ∫ a−3δ2k−1 t−ρ2k−1 a+3δ2k−1 t+ρ2k−1 u(x, t)dx + = b−3δ2k−1 t−ρ2k−1 u(x, t)dx + (6δ2k−1 t + ρ2k−1 )(b − a). a−3δ2k−1 t−ρ2k−1 The last expression tends to 0 as k −→ ∞. Hence ∫ b (βk (x, t) − αk (x, t))dx −→ 0 as k −→ ∞. a Thus the Frechét ﬁlter ⟨(wδ2k ,ρ2 k,εk )⟩ associated with the sequence (wδ2k ,ρ2 k,εk ) deﬁnes an element p of M♯ . Moreover, in the topology of N we have T1 wδ2k ,ρ2 k,εk −→ 0 and T2 wδ2k ,ρ2 k,εk −→ u0 . ( Therefore T ♯p = 0 u0 ) . In this way we have proved the following Theorem 2.22. For any u0 ∈ U0 there exists a unique p ∈ M♯ such that ) ( 0 . T ♯p = u0 This means that the initial value problem for the Burgers equation has a unique solution in M♯ . 98 Hausdorﬀ Continuous Solution of Scalar Conservation laws It is easy to see that (wδ2k ρ2k εk ) order converges to the unique H-continuous function u = [u, u]. Hence we have η(p) = u. Therefore the Burger’s equation has an H-continuous solution which corresponds to the well-known entropy solution. In particular, u = u almost everywhere, and any real valued function v such that v(x, t) ∈ u(x, t) for all (x, t) ∈ R × [0, ∞) satisﬁes the entropy condition for the Burger’s equation. Chapter 3 Concluding Remarks 3.1 Main results We considered the Cauchy problem of nonlinear conservation law with smooth ﬂux function and continuous initial condition in the context of Convergence Space Completion Method. In particular, the Convergence Space Completion Method was applied to the nonlinear operator equation derived from the Cauchy problem for nonlinear scalar conservation law. In this regard, suitable uniform convergence spaces were introduced. The completions of these uniform convergence spaces were obtained through the Wyler completion process. In addition, a uniformly continuous and injective mapping was obtained as an extension of the nonlinear operator derived from the Cauchy problem. It was shown that the extended operator equation has at most one generalized solution which can be identiﬁed with the entropy solution in the case of the Burgers equation. Thus we obtained an existence and uniqueness result for the operator equation of the Burgers equation. The uniqueness of solution follows from the injectivity of the extended operator. It was further shown that the space of generalized solutions can be identiﬁed with the space of Hausdorﬀ continuous functions, thus the unique solution of the Burgers equation so obtained is identiﬁed with a Hausdorﬀ continuous function. This provides a further regularity property for the generalized solution of the Burgers equation. 3.2 Topics for further research In this work we have applied the Order Completion Method, which is a general and type independent theory for existence and regularity of generalized solutions for large class of systems of nonlinear PDEs, to obtained the entropy solution of Burgers equation. The application of the Order Completion Method to the case of a more general ﬂux function is very important and should be considered. 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