Hyperbolic Conservation Laws in Continuum Physics

1. Front Matter

   Pages I-XIX

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2. Balance Laws

   Pages 1-23

3. Introduction to Continuum Physics

   Pages 25-49

4. Hyperbolic Systems of Balance Laws

   Pages 51-66

5. The Cauchy Problem

   Pages 67-86

6. Entropy and the Stability of Classical Solutions

   Pages 87-122

7. The L¹ Theory for Scalar Conservation Laws

   Pages 123-172

8. Hyperbolic Systems of Balance Laws in One-Space Dimension

   Pages 173-204

9. Admissible Shocks

   Pages 205-238

10. Admissible Wave Fans and the Riemann Problem

    Pages 239-293

11. Generalized Characteristics.

    Pages 295-300

12. Genuinely Nonlinear Scalar Conservation Laws

    Pages 301-342

13. Genuinely Nonlinear Systems of Two Conservation Laws

    Pages 343-404

14. The Random Choice Method

    Pages 405-441

15. The Front Tracking Method and Standard Riemann Semigroups

    Pages 443-536

16. Construction of BV Solutions by the Vanishing Viscosity Method

    Pages 483-509

17. Compensated Compactness

    Pages 511-536

18. Back Matter

    Pages 537-626

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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.

• Entropy
• hyperbolic conservation laws
• partial differential equation
• partial differential equations
• thermodynamics

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)

• Division of Applied Mathematics, Brown University, Providence, USA

  Constantine M. Dafermos

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