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.
Table of Contents:
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.
Review :
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. a ] the original text has been reorganized so as to streamline the exposition, enrich the collection of examples, and improve the notation. a ] The bibliography has been considerably expanded a ] ." (Evgeniy Panov, Zentralblatt MATH, Vol. 1078, 2006)
"This comprehensive book is about rigorous mathematical theory of balance and conservation laws a ] . The statements of theorems are carefully and precisely written. The proofs are canonical and illuminating a ] . 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 a ] ." (Katarina Jegdic, SIAM Review, Vol. 48 (3), 2006)
From the reviews of the second edition:
"The second edition of the famous book Grundlehren der Mathematischen Wissenchaften 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)
