ON SCALAR HYPERBOLIC CONSERVATION LAWS WITH A DISCONTINUOUS FLUX

Abstract

We study the Cauchy problem for scalar hyperbolic conservation laws with a flux that can have jump discontinuities. We introduce new concepts of entropy weak and measure-valued solution that are consistent with the standard ones if the flux is continuous. Having various definitions of solutions to the problem, we then answer the question what kind of properties the flux should possess in order to establish the existence and/or uniqueness of solution of a particular type. In any space dimension we establish the existence of measure-valued entropy solution for a flux having countable jump discontinuities. Under the additional assumption on the Hölder continuity of the flux at zero, we prove the uniqueness of entropy measure-valued solution, and as a consequence, we establish the existence and uniqueness of weak entropy solution. If we restrict ourselves to one spatial dimension, we prove the existence of weak solution to the problem where the flux has merely monotone jumps; in such a setting we do not require any continuity of the flux at zero.