Hyperbolic Conservation Laws in Continuum Physics

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Overview

This exposition of the mathematical theory of hyperbolic system laws brings out the intimate connection with continuum thermodynamics, emphasizing issues in which the analysis may reveal something about the physics and, in return, the underlying physical structure may direct and drive the analysis. The reader should have a certain mathematical sophistication and be familiar with the rudiments of the qualitative theory of partial differential equations, whereas the required notions from continuum physics are introduced from scratch.
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Editorial Reviews

From the Publisher
From the reviews of the second edition:

"The second edition of the famous book Grundlehren der Mathematischen Wissenschaften 325 is devoted to the mathematical theory of hyperbolic conservation and balance laws. The author is known as one of the leading experts in the field. His masterly written book is, surely, the most complete exposition in the subject of conservations laws. … the original text has been reorganized so as to streamline the exposition, enrich the collection of examples, and improve the notation. … The bibliography has been considerably expanded … ." (Evgeniy Panov, Zentralblatt MATH, Vol. 1078, 2006)

"This comprehensive book is about rigorous mathematical theory of balance and conservation laws … . The statements of theorems are carefully and precisely written. The proofs are canonical and illuminating … . This book is sure to convince every reader that working in this area is challenging, enlightening, and joyful. I heartily recommend this book to anyone who wants to learn about the foundations of the theory of balance and conservation laws and their generic relations to continuum physics … ." (Katarina Jegdic, SIAM Review, Vol. 48 (3), 2006)

From the reviews of the third edition:

“Covers the existence, uniqueness, continuous dependence, and qualitative behavior of entropy solutions for, successively, scalar conservation laws, systems of two conservation laws, and general strictly hyperbolic systems. … This book is now widely recognized as the ‘Bible’ on hyperbolic conservation laws, and has contributed to the training of many students while providing a huge amount of invaluable information (including a 100 page bibliography) for researchers working in this field or related areas.” (Philippe G. LeFloch, Mathematical Reviews, Issue 2011 i)

“The third edition of the famous book by C. M. Dafermos … devoted to the mathematical theory of hyperbolic conservation and balance laws. This masterly written book is, surely, the most entire exposition in the subject of conservation laws.”­­­ (Evgeniy Panov, Zentralblatt MATH, Vol. 1196, 2010)

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Product Details

Table of Contents

I Balance laws 1
II Introduction to continuum physics 25
III Hyperbolic systems of balance laws 51
IV The Cauchy problem 67
V Entropy and the stability of classical solutions 87
VI The L[superscript 1] theory for Scalar conservation laws 123
VII Hyperbolic systems of balance laws in one-space dimension 173
VIII Admissible shocks 205
IX Admissible wave fans and the Riemann problem 239
X Generalized characteristics 295
XI Genuinely nonlinear Scalar conservation laws 301
XII Genuinely nonlinear systems of two conservation laws 343
XIII The random choice method 405
XIV The front tracking method and standard Riemann semigroups 443
XV Construction of BV solutions by the vanishing viscosity method 483
XVI Compensated compactness 511
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