An existence result for conservation laws having BV spatial flux heterogeneities - without concavity

Abstract

We prove existence for an initial value problem featuring a conservation law whose flux has a discontinuous spatial dependence. For the type of flux considered here, where the spatial dependence occurs in a very general form, a uniqueness result is known, but the existence question was open until the recent work of Piccoli and Tournus (2018) [22]. Piccoli and Tournus proved existence using approximate solutions generated by the wave front tracking algorithm. A concavity assumption plays a simplifying role in their analysis. The main contribution of the present paper is an extension of this existence theorem in the absence of the concavity assumption. We accomplish this via finite difference approximations.