Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem by A

Publisher Description

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 longstanding open problems of uniqueness and stability of entropy weak solutions. This monograph is the first to present a comprehensive account of these new, fundamental advances, mainly obtained by theauthor together with several collaborators. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. The book is addressed to graduate students as well as researchers. Both the elementary and the more advanced material are carefully explained, helping the reader's visual intuition with over 70 figures. A set of problems, with varying difficulty, is given at the end of each chapter. These exercises are designed to verify and expand a student'sunderstanding of the concepts and techniques previously discussed. For researchers, this book will provide an indispensable reference for the state of the art, in the field of hyperbolic systems ofconservation laws. The last chapter contains a large, up to date list of references, preceded by extensive bibliographical notes.

Author Biography

Alberto Bressan is at SISSA, Trieste.

Table of Contents

1. Introduction 2. Mathematical preliminaries 3. Semilinear and quasilinear systems 4. Discontinuous solutions 5. The Riemann problem 6. The single conservation law 7. The Cauchy problem for systems 8. Stability 9. Uniqueness 10. Qualitative properties 11. Bibliographical notes Index

Review

An excellent and self-contained treatment of the mathematical theory of hyperbolic systems of conservation laws ... written in a clear and self-contained way and will be of great value for graduate students and specialists in the field. * EMS *

Long Description

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 monograph is the first to present a comprehensive account of these new, fundamental advances, mainly obtained by the author together with several collaborators. It also includes a detailed analysis of the stability
and convergence of the front tracking algorithm. The book is addressed to graduate students as well as researchers. Both the elementary and the more advanced material are carefully explained, helping the reader's visual intuition with over 70 figures. A set of problems, with varying difficulty, is given at the end of each chapter. These exercises are designed to verify and expand a student's understanding of the concepts and techniques previously discussed. For researchers, this book
will provide an indispensable reference for the state of the art, in the field of hyperbolic systems of conservation laws. The last chapter contains a large, up to date list of references, preceded by extensive bibliographical notes.

Review Quote

"Hyperbolic systems of conservation laws has become such a mature topic in the past decade that several researchers decided to write a book on the subject recently. . This book is entirely devoted to the achievements of the author's long-standing program. The author's results have been the most impressive obtained about conservation laws in the past decade. . .Bressan's style is fluid, and several exercises help doctoral students to digest each chapter. . .Presumably, this book will be often cited in research papers on hyperbolic conservation laws. It is recommended to researchers and graduate students who specialize in this field."--Mathematical Review

Feature

Systematic presentation of the fundamental, new results on uniqueness and stability of solutions
Only book containing detailed description and analysis of front-tracking algorithm for construction of approximate solutions.
75 figures explaining elementary and advanced material.
Problems set to each chapter.
Indispensable reference for researchers in the field, with extensive bibliographical notes

Details