Hyperbolic Conservation Laws in Continuum Physics
Description
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.
Reviews / Votes
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. . 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)
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Content
1.1 Formulation of the Balance Law 1.2 Reduction to Field Equations 1.3 Change of Coordinates 1.4 Systems of Balance Laws 1.5 Companion Systems of Balance Laws 1.6 Weak and Shock Fronts 1.7 Survey of the Theory of BV Functions 1.8 BV Solutions of Systems of Balance Laws 1.9 Rapid Oscillations and the Stabilizing Effect of Companion Banalce Laws 1.10 Notes 2. Introduction to Continuum Physics 2.1 Bodies and Motions 2.2 Balance Laws in Continuum Physics 2.3 The Balance Laws of Continuum Thermomechanics 2.4 Material Frame Indifference 2.5 Thermoelasticity 2.6 Thermoviscoelasticity 2.7 Notes 3. Hyperbolic Systems of Balance Laws
3.1 Hyperbolicity 3.2 Entropy-Entropy Flux Pairs 3.3 Examples of Hyperbolic Systems of Balance Laws 3.4 Notes 4. The Initial-Value Problem: Admissibility of Solutions 4.1 The Initial-Value Problem 4.2 The Burgers Equation and Nonuniqueness of Weak Solutions 4.3 Entropies and Admissible Solutions 4.4 The Vanishing Viscosity Approach 4.5 Initial-Boundary Value Problems 4.6 Notes 5. Entropy and the Stability of Classical Solutions
5.1 Convex Entropy and the Existence of Classical Solutions 5.2 Convex Entropy and the Stability of Classical Solutions 5.3 Partially Convex Entropies and Involutions 5.4 Notes 6. The L1 Theory of Scalar Balance Laws
7. Hyperbolic Systems of Balance Laws in One Space Dimension 8. Admissible Shocks
9. Admissible Wave Fans and the Riemann Problem
10. Generalized Characteristics
11. Genuinely Nonlinear Scalar Conservation Law
12. Genuinely Nonlinear Systems of Two Conservation Laws
13. The Random Choice Method
14. The Method of Wave Front Tracking and the Standard Riemann Semigroup
15. The Method of Compensated Compactness
Bibliography.