Existence and Asymptotic Behavior of Measure-Valued Solutions for Degenerate Conservation Laws

Abstract

We consider two classes of typical degenerate hyperbolic systems of conservation laws to provide a general approach for solving the existence and large-time asymptotic behavior of measure-valued solutions for initial-boundary value problems. Some existence theorems of the measure-valued solutions are established. The convergence of large time-averages of the measure-valued solutions to a Dirac mass, concentrated at the input state on the boundary, is proved for almost each fixed space variable. Although the measure-valued solutions of the initial-boundary problems may not be unique in general, our results indicate that the asymptotic equilibrium of these measure-valued solutions is unique.