Hyperbolic Systems of Conservation Laws The One-Dimensional Cuachy Problem

by Bressan, Alberto

• ISBN13: 9780198507000
• ISBN10: 0198507003
• Format: Hardcover
• Copyright: 2000-12-21
• Publisher: Oxford University Press

Summary

This book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves. This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and the semigroup approach. These techniques provide a solution to the long standing open problems of uniqueness and stability of entropy weak solutions. This volume is the first to present a comprehensive account of these new, fundamental advances. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. A set of problems, with varying difficulty is given at the end of each chapter to verify and expand understanding of the concepts and techniques previously discussed. For researchers, this book will provide an indispensable reference to the state of the art in the field of hyperbolic systems of conservation laws.

Table of Contents

Introduction	1	(5)
Mathematical preliminaries	6	(34)
Implicit function theorems	7	(1)
Some linear algebra	7	(2)
Distributions	9	(2)
Functions with bounded variation	11	(6)
BV functions of two variables	17	(5)
Fixed points of contractive maps	22	(1)
Differentiability of Lipschitz-continuous functions	23	(2)
Approximation of Lipschitz flows	25	(2)
Taylor estimates	27	(5)
Weak convergence of measures	32	(2)
Basic ODE theory	34	(6)
Problems	38	(2)
Semilinear and quasilinear systems	40	(33)
Explicit solutions	41	(1)
A single semilinear equation	42	(2)
The scalar quasilinear equation	44	(2)
Semilinear systems: broad solutions	46	(7)
Continuity and continuous dependence of broad solutions	53	(4)
Classical solutions	57	(5)
Lipschitz solutions	62	(2)
Quasilinear systems	64	(9)
Problems	71	(2)
Discontinuous solutions	73	(17)
Weak solutions	73	(3)
Rankine--Hugoniot conditions	76	(4)
Coordinate transformations	80	(1)
Admissibility conditions	81	(4)
The scalar case	85	(5)
Problems	88	(2)
The Riemann problem	90	(18)
Centred rarefaction waves	91	(1)
Shocks	92	(7)
General solution of the Riemann problem	99	(3)
An example	102	(1)
Isentropic gas dynamics	103	(5)
Problems	107	(1)
The single conservation law	108	(15)
Piecewise constant approximations	109	(4)
Global existence of BV solutions	113	(1)
Uniqueness	114	(4)
A contractive semigroup	118	(5)
Problems	120	(3)
The Cauchy problem for systems	123	(30)
Wave-front tracking approximations	124	(3)
A front tracking algorithm	127	(6)
Global existence of front tracking approximations	133	(10)
Existence of solutions	143	(3)
Further interaction estimates	146	(3)
Approximate conservation of waves	149	(1)
A positively invariant domain	150	(3)
Problems	152	(1)
Stability	153	(17)
Stability of front tracking approximations	153	(7)
Proof of the main estimates	160	(8)
Proof of L1 stability	168	(2)
Problems	169	(1)
Uniqueness	170	(30)
Uniqueness of the semigroup	171	(7)
A characterization of semigroup trajectories	178	(6)
Uniqueness of entropy weak solutions	184	(16)
Problems	196	(4)
Qualitative properties	200	(29)
Wave measures	201	(9)
Decay of positive waves	210	(8)
Global structure of solutions	218	(11)
Problems	228	(1)
Bibliographical notes	229	(10)
References	239	(10)
Index	249