7R6. Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. - PG LeFloch (Center de Math Appliquees and CNRS, Ecole Polytechnique, Palaiseau, 91128, France). Birkhauser Verlag AG, Basel, Switzerland. 2002. 294 pp. Softcover. ISBN 3-7643-6687-7. \$34.95.

Reviewed by J Novotny (Inst of Thermomech, Dolejskova 5, Prague, 182 00, Czech Republic).

The book presents a self-contained modern mathematical theory of hyperbolic systems of nonlinear partial differential equations of first order in divergence form, which are also called hyperbolic systems of conservation laws. These equations arise in many areas of continuum physics (compressible fluid dynamics, phase transition dynamics, nonlinear elastodynamics…), where fundamental balance laws are formulated for mass, momentum, total energy of fluid, or solid continuum.

Solutions to these systems may lead to singularities (shock waves) appearing even when smooth initial data are given. As established, weak solutions are not unique unless some entropy condition is imposed.

The text contains existence, uniqueness, and continuous dependence of classical (compressive) entropy solutions on initial data. The latest results of the author and his collaborators on uniqueness of entropy solutions with bounded variations and continuous dependence are included.

Part one of the book describes scalar conservation laws and part two, systems of conservation laws. The Riemann problem, classical and nonclassical Riemann solvers are studied. Also the developing theory of nonclassical (under compressive) entropy solutions is presented. Existence theory for the Cauchy problem for classical entropy solutions, for both convex and general flux, and nonclassical entropy solutions are studied in detail. Continuous dependence of the solutions in $L1$ norm is proved.

The study of nonclassical shock waves is based on the concept of a kinetic relation introduced by the author for general hyperbolic systems and derived from singular limits of hyperbolic conservation laws with balanced diffusion and dispersion terms.

Basic courses of functional analysis and modern methods for partial differential equations are necessary for studying of this book. No preliminary knowledge of continuum physics is required, however, basic knowledge is useful for better understanding.

The book contains a number of pertinent figures completing well the theoretical explanations. The book does not contain a subject index, nevertheless it contains bibliographical notes to each chapter and a large bibliography.

Up to now, no book clearly presented the most important principles of classical and modern theory of hyperbolic conservation laws together with recent developments in this field. This book, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, can be considered as a concise and comprehensive monograph and at the same time a textbook for graduate students. The book should be particularly suitable for graduate students, courses for PhD students, and also for researchers working in the fields of modern theory and numerical analysis of nonlinear hyperbolic partial differential equations, and in theoretical continuum physics. It is suitable especially for young researchers, who want to become familiar with the basic principles, the current state of knowledge, and the latest, most important results in the mathematical theory of hyperbolic conservation laws.

This book is recommended for purchase by university libraries, departments of mathematics and physics, and seriously interested individuals.