This masterly exposition of the mathematical theory of hyperbolic system for conservation laws brings out the intimate connection with continuum thermodynamics, by emphasising 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 is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of the qualitative theory of partial differential equations, whereas the required notions from continuum physics are introduced from scratch. 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.

The 2nd edition contains a new chapter recounting the exciting recent developments on the vanishing viscosity method. Numerous new sections have been incorporated in preexisting chapters, to introduce newly derived results or present older material, omitted in the first edition, whose relevance and importance has been underscored by current trends in research. In addition, a substantal portion of the original text has been revamped so as to streamline the exposition, enrich the collection of examples and improve the notation. The bibliography has been updated and expanded as well, now comprising over one thousand titles.

Résumé
The seeds of continuum physics were planted with the works of the natural philo- phers of the eighteenth century, most notably Euler; by the mid-nineteenth century, the trees were fully grown and ready to yield fruit. It was in this environment that the study of gas dynamics gave birth to the theory of quasilinear hyperbolic systems in divergence form, commonly called hyperbolic conservation laws; and these two subjects have been traveling hand in hand over the past one hundred and ?fty years. This book aims at presenting the theory of hyperbolic conservation laws from the standpoint of its genetic relation to continuum physics. Even though research is still marching at a brisk pace, both ?elds have attained by now the degree of maturity that would warrant the writing of such an exposition. Intherealmofcontinuumphysics,materialbodiesarerealizedascontinuous- dia, and so-called extensive quantities, such as mass, momentum and energy, are monitored through the ?elds of their densities, which are related by balance laws and constitutive equations. A self-contained, though skeletal, introduction to this branch of classical physics is presented in Chapter II. The reader may ?esh it out with the help of a specialized text on the subject.

Contenu
Balance Laws.- to Continuum Physics.- Hyperbolic Systems of Balance Laws.- The Cauchy Problem.- Entropy and the Stability of Classical Solutions.- The L1 Theory for Scalar Conservation Laws.- Hyperbolic Systems of Balance Laws in One-Space Dimension.- Admissible Shocks.- Admissible Wave Fans and the Riemann Problem.- Generalized Characteristics..- Genuinely Nonlinear Scalar Conservation Laws.- Genuinely Nonlinear Systems of Two Conservation Laws.- The Random Choice Method.- The Front Tracking Method and Standard Riemann Semigroups.- Construction of BV Solutions by the Vanishing Viscosity Method.- Compensated Compactness.