文档介绍：Lectures in Mathematics – ETH Z¨urich, Birkh¨auser Verlag, Basel (2002).
HYPERBOLIC SYSTEMS OF CONSERVATION LAWS
The theory of classical and nonclassical shock waves
Philippe G. LeFloch
Author address:
Centre de Math´ematiques Appliqu´ees & Centre National de la Recherche
Scientifique, UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France.
E-mail: lefl******@.
To my wife Claire
To Olivier, Aline, Bruno
Contents
Preface ix
Chapter I. Fundamental concepts and examples
1. Hyperbolicity, genuine nonlinearity, and entropies 1
2. Shock formation and weak solutions 8
3. Singular limits and the entropy inequality 13
4. Examples of diffusive-dispersive models 17
5. ic relations and traveling waves 22
Part 1. SCALAR CONSERVATION LAWS
Chapter II. The Riemann problem
1. Entropy conditions 29
2. Classical Riemann solver 31
3. Entropy dissipation function 36
4. Nonclassical Riemann solver for concave-convex flux 41
5. Nonclassical Riemann solver for convex-concave flux 47
Chapter III. Diffusive-dispersive traveling waves
1. Diffusive traveling waves 51
2. ic functions for the cubic flux 53
3. ic functions for general flux 59
4. Traveling waves for a given speed 68
5. Traveling waves for a given diffusion-dispersion ratio 77
Chapter IV. Existence theory for the Cauchy problem
1. Classical entropy solutions for convex flux 81
2. Classical entropy solutions for general flux 88
3. Nonclassical entropy solutions 93
4. Refined estimates 112
Chapter V. Continuous dependence of solutions
1. A class of linear hyperbolic equations 118
2. L1 continuous dependence estimate 126
3. Sharp version of the continuous dependence estimate 133
4. Generalizations 135
viii CONTENTS
Part 2. SYSTEMS OF CONSERVATION LAWS
Chapter VI. The Riemann problem
1. Shock and rarefaction waves 139
2. Classical Riemann solver 148
3. Entropy dissipation and wave sets 156
4. ic relation and nonclassical Riemann solver 163
Chapter VII. C