Presents an exposition of the mathematical theory of hyperbolic system laws, intended for readers familiar with the qualitative theory or partial differential equations. This edition contains a chapter recounting developments on the vanishing viscosity method.

This masterly 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. 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 1st edition. In addition, a substantial portion of 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 updated and expanded, now comprising over one thousand titles.

Balance Laws.- Introductioin 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 Law.- Genuinely Nonlinear Systems of Two Conservation Laws.- The Random Choice Method.- The Front Tracking Method and Standard Riemann Semigroups.- Construction of BV Volutions by the Vanishing Viscosity Method.- Compensated Compactness.- Bibliography.- Author Index.- Subject Index.