E-Book Overview
Systems of conservation laws arise naturally in physics and chemistry. To understand them and their consequences (shock waves, finite velocity wave propagation) properly in mathematical terms requires, however, knowledge of a broad range of topics. This book sets up the foundations of the modern theory of conservation laws describing the physical models and mathematical methods, leading to the Glimm scheme. Building on this the author then takes the reader to the current state of knowledge in the subject. The maximum principle is considered from the viewpoint of numerical schemes and also in terms of viscous approximation. Small waves are studied using geometrical optics methods. Finally, the initial-boundary problem is considered in depth. Throughout, the presentation is reasonably self contained, with large numbers of exercises and full discussion of all the ideas. This makes it an ideal text for graduate courses in the area of partial differential equations.
E-Book Content
Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves DENIS SERRE Translated by I . N. SNEDDON CAMBRIDGE UNIVERSITY PRESS Systems of Conservation Laws 1 Systems of conservation laws arise naturally in several areas of physics and chemistry. To understand them and their consequences (shock waves, finite velocity wave propagation) properly in mathematical terms requires, however, knowledge of a broad range of topics. This book sets up the foundations of the modern theory of conservation laws describing the physical models and mathematical methods, leading to the Glimm scheme. Building on this the author then takes the reader to the current state of knowledge in the subject. In particular, he studies in detail viscous approximations, paying special attention to viscous profiles of shock waves. The maximum principle is considered from the viewpoint of numerical schemes and also in terms of viscous approximation, whose convergence is studied using the technique of compensated compactness. Small waves are studied using geometrical optics methods. Finally, the initial–boundary problem is considered in depth. Throughout, the presentation is reasonably self-contained, with large numbers of exercises and full discussion of all the ideas. This will make it ideal as a text for graduate courses in the area of partial differential equations. Denis Serre is Professor of Mathematics at the Ecole Normale Sup´erieure de Lyon and was a Member of the Institut Universitaire de France (1992–7). This page intentionally left blank Systems of Conservation Laws 1 Hyperbolicity, Entropies, Shock Waves DENIS SERRE Translated by I. N. SNEDDON PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING) FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 IRP 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia http://www.cambridge.org Originally published in French by Diderot as Systèmes de lois de conservation I: hyperbolicité, entropies, ondes de choc and © 1996 Diderot First published in English by Cambridge University Press 1999 as Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves English translation © Cambridge University Press 1999 This edition © Cambridge University Press (Virtual Publishing) 2003 First published in printed format 1999 A catalogue record for the original printed book is available from the British Library and from the Library of Congress Original ISBN 0 521 58233 4 hardback ISBN 0 511 00900 3 virtual (netLibrary Edition) To Paul and Fanny This page intentionally left blank Contents Acknowledgments Introduction page xi xiii 1 Some models 1.1 Gas dynamics in eulerian variables 1.2 G