Measure-theoretic analysis and nonlinear conservation laws

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 compact ness, blow-up and scaling techniques, entropy methods, and related measure-theoretic 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.