Hyperbolic Conservation Laws in Continuum Physics

Front Cover
Springer, May 26, 2016 - Mathematics - 826 pages

This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of
(a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics;
(b) specialists in continuum mechanics who may need analytical tools;
(c) experts in numerical analysis who wish to learn the underlying mathematical theory; and
(d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws.

This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles.

From the reviews of the 3rd edition:

"This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH

"A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews


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I Balance Laws
II Introduction to Continuum Physics
III Hyperbolic Systems of Balance Laws
IV The Cauchy Problem
V Entropy and the Stability of Classical Solutions
VI The L1 Theory for Scalar Conservation Laws
VII Hyperbolic Systems of Balance Laws in OneSpace Dimension
VIII Admissible Shocks
XII Genuinely Nonlinear Systems of Two Conservation Laws
XIII The Random Choice Method
XIV The Front Tracking Method and Standard Riemann Semigroups
XV Construction of BV Solutions by the Vanishing Viscosity Method
XVI BV Solutions for Systems of Balance Laws
XVII Compensated Compactness
XVIII Steady and Selfsimilar Solutions in MultiSpace Dimensions

IX Admissible Wave Fans and the Riemann Problem
X Generalized Characteristics
XI Scalar Conservation Laws in One Space Dimension
Author Index
Subject Index

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About the author (2016)

Professor Dafermos received a Diploma in Civil Engineering from the National Technical University of Athens (1964) and a Ph.D. in Mechanics from the Johns Hopkins University (1967). He has served as Assistant Professor at Cornell University (1968-1971),and as Associate Professor (1971-1975) and Professor (1975- present) in the Division of Applied Mathematics at Brown University. In addition, Professor Dafermos has served as Director of the Lefschetz Center of Dynamical Systems (1988-1993, 2006-2007), as Chairman of the Society for Natural Philosophy (1977-1978) and as Secretary of the International Society for the Interaction of Mathematics and Mechanics. Since 1984, he has been the Alumni-Alumnae University Professor at Brown.
In addition to several honorary degrees, he has received the SIAM W.T. and Idalia Reid Prize (2000), the Cataldo e Angiola Agostinelli Prize of the Accademia Nazionale dei Lincei (2011), the Galileo Medal of the City of Padua (2012), and the Prize of the International Society for the Interaction of Mechanics and Mathematics (2014). He was elected a Fellow of SIAM (2009) and a Fellow of the AMS (2013). In 2016 he received the Wiener Prize, awarded jointly by the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM).