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Measure-theoretic analysis and nonlinear conservation laws

By Gui-qiang Chen, Monica Torres, William P. Ziemer, To Leon and Simon Friendship

Abstract

Abstract: We discuss some recent developments and trends of applying measure-theoretic analysis to the study of nonlinear conservation laws. We focus particularly on entropy solutions without bounded variation and Cauchy fluxes on oriented surfaces which are used to formulate the balance law. Our analysis employs the Gauss-Green formula and normal traces for divergence-measure fields, Young measures and compensated compactness, blow-up and scaling techniques, entropy methods, and related measuretheoretic techniques. Nonlinear conservation laws include multidimensional scalar conservation laws, strictly hyperbolic systems of conservation laws, isentropic Euler equations, two-dimensional sonic-subsonic flows, and degenerate parabolic-hyperbolic equations. Some open problems and trends on the topics are addressed and an extensive list of references is also provided. Key Words and Phrases. Measure-theoretic analysis, geometric measures, conservation laws, Cauchy fluxes, balance laws, entropy solutions, divergence-measure fields, sets of finite perimeter, Gauss-Green formula, normal traces, oriented surfaces, compactness, Young measures, compensated compactness, blow-up, scaling, axioms for continuum thermodynamics, field equations

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.352.4058
Provided by: CiteSeerX
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