The seeds of Continuum Physics were planted with the works of the natural philosophers 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 enviยญ ronment that the study of gas dynamics gave birth to the theory of quasilinear hyperbolic systems in divergence form, commonly called "hyperbolic conservaยญ tion laws"; and these two subjects have been traveling hand-in-hand over the past one hundred and fifty years. This book aims at presenting the theory of hyperยญ bolic conservation laws from the standpoint of its genetic relation to Continuum Physics. Even though research is still marching at a brisk pace, both fields have attained by now the degree of maturity that would warrant the writing of such an exposition. In the realm of Continuum Physics, material bodies are realized as continuous media, and so-called "extensive quantities", such as mass, momentum and energy, are monitored through the fields 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 flesh it out with the help of a specialized text on the subject
CONTENT
I. Balance Laws -- II. Introduction to Continuum Physics -- III. Hyperbolic Systems of Balance Laws -- IV. The Initial-Value Problem: Admissibility of Solutions -- V. Entropy and the Stability of Classical Solutions -- VI. The L1 Theory of the Scalar Conservation Law -- VII. Hyperbolic Systems of Balance Laws in One-Space Dimension -- VIII. Admissible Shocks -- IX. Admissible Wave Fans and the Riemann Problem -- X. Generalized Characteristics -- XI. Genuinely Nonlinear Scalar Conservation Laws -- XII. Genuinely Nonlinear Systems of Two Conservation Laws -- XIII. The Random Choice Method -- XIV. The Front Tracking Method and Standard Riemann Semigroups -- XV. Compensated Compactness -- Author Index
