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2002 | Buch

Fundamental Concepts and Examples Kapitel lesen Erstes Kapitel lesen

Hyperbolic Systems of Conservation Laws

The Theory of Classical and Nonclassical Shock Waves

verfasst von: Philippe G. LeFloch

Verlag: Birkhäuser Basel

Buchreihe : Lectures in Mathematics ETH Zürich

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

This set of lecture notes was written for a Nachdiplom-Vorlesungen course given at the Forschungsinstitut fUr Mathematik (FIM), ETH Zurich, during the Fall Semester 2000. I would like to thank the faculty of the Mathematics Department, and especially Rolf Jeltsch and Michael Struwe, for giving me such a great opportunity to deliver the lectures in a very stimulating environment. Part of this material was also taught earlier as an advanced graduate course at the Ecole Poly technique (Palaiseau) during the years 1995-99, at the Instituto Superior Tecnico (Lisbon) in the Spring 1998, and at the University of Wisconsin (Madison) in the Fall 1998. This project started in the Summer 1995 when I gave a series of lectures at the Tata Institute of Fundamental Research (Bangalore). One main objective in this course is to provide a self-contained presentation of the well-posedness theory for nonlinear hyperbolic systems of first-order partial differential equations in divergence form, also called hyperbolic systems of con­ servation laws. Such equations arise in many areas of continuum physics when fundamental balance laws are formulated (for the mass, momentum, total energy . . . of a fluid or solid material) and small-scale mechanisms can be neglected (which are induced by viscosity, capillarity, heat conduction, Hall effect . . . ). Solutions to hyper­ bolic conservation laws exhibit singularities (shock waves), which appear in finite time even from smooth initial data.

Inhaltsverzeichnis

Frontmatter

Fundamental Concepts and Examples

Chapter I. Fundamental Concepts and Examples
Abstract
In this first chapter, we present some basic definitions and concepts which will be of constant use in this course. We also discuss the main difficulties of the theory and briefly indicate the main results to be established in the forthcoming chapters.
Philippe G. LeFloch

Scalar Conservation Laws

Frontmatter
Chapter II. The Riemann Problem
Abstract
In this chapter, we study the Riemann problem for scalar conservation laws. In Section 1 we discuss several formulations of the entropy condition. Then, in Section 2 we construct the classical entropy solution satisfying, by definition, all of the entropy inequalities; see Theorems 2.1 to 2.4. Next in Section 3, imposing only that solutions satisfy a single entropy inequality, we show that undercompressive shock waves are also admissible and we determine a one-parameter family of solutions to the Riemann problem; see Theorem 3.5. Finally in Sections 4 and 5, we construct nonclassical entropy solutions which, by definition, satisfy a single entropy inequality together with a kinetic relation; see Theorem 4.1 for concave-convex flux-functions and Theorem 5.4 for convex-concave flux-functions.
Philippe G. LeFloch
Chapter III. Diffusive-Dispersive Traveling Waves
Abstract
In this chapter we study a large class of diffusive-dispersive equations associated with scalar conservation laws. We investigate the existence of traveling wave solutions which, as was pointed out earlier (Theorem I-5.4), converge to shock wave solutions of (1.1) as the diffusion and the dispersion tend to zero. The corresponding shock set can be determined and compared with the one obtained in Chapter II by applying entropy inequalities. The present chapter demonstrates the relevance of the construction given in Chapter II. We confirm here that classical shock waves are independent of the small-scale mechanisms, while nonclassical shock waves require the kinetic relation determined by the given diffusive-dispersive operator. In Section 1 we consider the effect of the diffusion only; see Theorem 1.2. In Section 2 we determine the kinetic relation explicitly for the conservation law with cubic flux and linear diffusion-dispersion terms; see Theorem 2.3. The main result in this chapter for general flux-functions are stated in Section 3; see Theorem 3.3. The proofs of the results given in Section 3 are postponed to Sections 4 and 5.
Philippe G. LeFloch
Chapter IV. Existence Theory for the Cauchy Problem
Abstract
This chapter is devoted to the general existence theory for scalar conservation laws in the setting of functions with bounded variation. We begin, in Section 1, with an existence result for the Cauchy problem when the flux-function is convex. We exhibit a solution given by an explicit formula (Theorem 1.1) and prove the uniqueness of this solution (Theorem 1.3). The approach developed in Section 1 is of particular interest as it reveals important features of classical entropy solutions. However, it does not extend to non-convex fluxes or nonclassical solutions, and an entirely different strategy based on Riemann solvers and wave front tracking is developed in the following sections. In Sections 2 and 3, we discuss the existence of classical and of nonclassical entropy solutions to the Cauchy problem, respectively; see Theorems 2.1 and 3.2 respectively. Finally in Section 4, we derive refined estimates for the total variation of solutions (Theorems 4.1 to 4.3) which represent a preliminary step toward the forthcoming discussion of the Cauchy problem for systems (in Chapters VII and VIII).
Philippe G. LeFloch
Chapter V. Continuous Dependence of Solutions
Abstract
In the present chapter, we investigate the continuous dependence of solutions to scalar conservation laws. In Section 2, we study a class of hyperbolic equations with discontinuous coefficient, and we establish a general stability result in L 1 when the coefficient does not contain rarefaction-shocks; see Definition 1.1 and the main result in Theorem 1.7. Next, in Section 2 we apply this setting to conservation laws with convex flux; see Theorems 2.2 and 2.3. The proofs in Section 2 are based on the key observation that no rarefaction-shock (in the sense of Section 1) can arise from comparing two entropy solutions. In Section 3, we derive a sharp estimate in a weighted L 1 norm, which provides a quantitative bound on the decrease of the L 1 norm; see Theorem 3.1. Finally, in Section 4 we state the generalization to nonclassical solutions.
Philippe G. LeFloch

Systems of Conservation Laws

Frontmatter
Chapter VI. The Riemann Problem
Abstract
In this first chapter on systems we explicitly construct the classical and the nonclassical entropy solutions to the Riemann problem associated with a strictly hyperbolic system of conservation laws. The initial data consist of single jump discontinuities of sufficiently small strength. As was already observed with scalar conservation laws (Chapter II), solutions can be obtained by combining shock waves and rarefaction waves together. Motivated by the applications (Sections I-3 and I-4) we are primarily interested in systems endowed with a strictly convex entropy pair and in solutions satisfying a single entropy inequality.
Philippe G. LeFloch
Chapter VII. Classical Entropy Solutions of the Cauchy Problem
Abstract
In this chapter we establish the existence of a classical entropy solution to the Cauchy problem associated with a strictly hyperbolic system of conservation laws when the initial data have small total variation. We cover here the general class of systems whose each characteristic field is either genuinely nonlinear or concave-convex. With minor changes, the results in this chapter extend to linearly degenerate and convex-concave fields. In Section 1 we discuss fundamental properties of (exact and approximate) classical entropy solutions to the Riemann problem, studied earlier in Sections VI-1 and VI-2. The key property is given by the interactions estimates in Theorem 1.1: at each interaction, the wave strengths may increase by an amount which is bounded by the product of the strengths of the two incoming waves. In Section 2 we describe the approximation scheme which generalizes the one given in Section IV-2 for scalar conservation laws, and we state the main existence result; see Theorem 2.1. Technical aspects of the proof are postponed to Section 3. Finally, in Section 4 we briefly discuss pointwise regularity properties of the solutions.
Philippe G. LeFloch
Chapter VIII. Nonclassical Entropy Solutions of the Cauchy Problem
Abstract
In this chapter we give a general existence result for nonclassical entropy solutions to the Cauchy problem associated with a system of conservation laws whose characteristic fields are genuinely nonlinear or concave-convex. (The result can be extended to linearly degenerate and convex-concave fields as well.) The proof is based on a generalization of the algorithm described in Chapter VII. Here, we use the nonclassical Riemann solver based on a given kinetic function for each concave-convex as was described in Section VI-3. Motivated by the examples arising in the applications (see Chapter III) we can assume that the kinetic functions satisfy the following threshold condition: any shock wave with strength less than some critical value is classical. In Section 1 we introduce a generalized total variation functional which is non-increasing for nonclassical solutions (Theorem 1.4) and whose decay rate can be estimated (Theorem 1.5). In Section 2 we introduce a generalized interaction potential and we extend Theorem IV-4.3 to nonclassical solutions; see Theorem 2.1. Section 3 and 4 are concerned with the existence and regularity theory for systems; see Theorems 3.1 and 4.2 respectively.
Philippe G. LeFloch
Chapter IX. Continuous Dependence of Solutions
Abstract
In this chapter, we investigate the L 1 continuous dependence of solutions for systems of conservation laws. We restrict attention to solutions generated in the limit of piecewise approximate solutions and we refer to Chapter X for a discussion of the uniqueness of general solutions with bounded variation. In Section 1 we outline a general strategy based on a L 1 stability result for a class of linear hyperbolic systems with discontinuous coefficients. The main result in Theorem 1.5 shows that the sole source of instability would be the presence of rarefaction-shocks. In Section 2 we apply the setting to systems with genuinely nonlinear characteristic fields; see Theorem 2.3. One key observation here is that rarefaction-shocks never arise from comparing two classical entropy solutions to systems of conservation laws. In Section 3 we provide a sharp version of the continuous dependence estimate which shows that the L 1 distance between two solutions is “strictly decreasing”; see Theorem 3.2. Finally, in Section 4 we state the generalization to nonclassical entropy solutions.
Philippe G. LeFloch
Chapter X. Uniqueness of Entropy Solutions
Abstract
In this chapter, we establish a general uniqueness theorem for nonlinear hyperbolic systems. Solutions are sought in the space of functions with bounded variation, slightly restricted by the so-called tame variation condition (Definition 1.1). The results of existence and continuous dependence established in previous chapters covered solutions obtained as limits of piecewise constant approximate solutions with uniformly bounded total variation (in Chapters IV and V for scalar conservation laws and in Chapters VII to IX for systems). Our purpose now is to cover general functions with bounded variation and to establish a general uniqueness theory for hyperbolic systems of conservation laws.
Philippe G. LeFloch
Backmatter
Metadaten
Titel
Hyperbolic Systems of Conservation Laws
verfasst von
Philippe G. LeFloch
Copyright-Jahr
2002
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8150-0
Print ISBN
978-3-7643-6687-2
DOI
https://doi.org/10.1007/978-3-0348-8150-0