Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes

Abstract

We consider a scalar conservation law with a discontinuous flux function. The fluxes are non-convex, have multiple points of extrema and can have arbitrary intersections. We propose an entropy formulation based on interface connections and associated jump conditions at the interface. We show that the entropy solutions with respect to each choice of interface connection exist and form a contractive semi-group in L¹. Existence is shown by proving convergence of a Godunov type scheme by a suitable modification of the singular mapping approach. This extends the results of [3] to the general case of non-convex flux geometries.