The method of compensated compactness as a technique for studying hyperbolic conservation laws is of fundamental importance in many branches of applied mathematics. Until now, however, most accounts of this method have been confined to research papers. Offering the first comprehensive treatment, Hyperbolic Conservation Laws and the Compensated Compactness Method gathers together into a single volume the essential ideas and developments.
The authors begin with the fundamental theorems, then consider the Cauchy problem of the scalar equation, build a framework for L8 estimates of viscosity solutions, and introduce the Invariant Region Theory. The study then turns to methods for symmetric systems of two equations and two equations with quadratic flux, and the extension of these methods to the Le Roux system. After examining the system of polytropic gas dynamics (g-law), the authors first study two special systems of one-dimensional Euler equations, then consider the general Euler equations for one-dimensional compressible fluid flow, and extend that method to systems of elasticity in L8 space. Weak solutions for the elasticity system are introduced and an application to adiabatic gas flow through porous media is considered. The final four chapters explore applications of the compensated compactness method to the relaxation problem.
With its careful account of the underlying ideas, development of applications in key areas, an inclusion of the author's own contributions to the field, this monograph will prove a welcome addition to the literature and to your library.
Table of Contents
THEORY OF COMPENSATED COMPACTNESS
ELEMENTARY IN HYPERBOLIC SYSTEM
A SYMMETRY SYSTEM
A SYSTEM OF QUADRATIC FLUX
LE ROUX SYSTEM
SYSTEM OF POLYTROPIC GAS DYNAMICS
TWO SYSTEMS OF EULER-EQUATIONS
GENERAL SYSTEM OF EULER-EQUATIONS
EXTENDED SYSTEMS OF ELASTICITY
L[p] CASE FOR SYSTEMS OF ELASTICITY
ELEMENTARY IN RELAXATION
RELAXATION WITH DIFFUSION
SYSTEMS WITH STIFF RELAXATION
RELAXATION FOR 3 x 3 SYSTEMS
"The book is carefully written and will be appreciated both by PhD students and experts in the field as one of a few books gathering the knowledge … [previously] dispersed among research papers."
- EMS Newsletter, Dec. 2004