On unified theory for scalar conservation laws with fluxes and sources discontinuous with respect to the unknown

TL;DR

This paper introduces a kinetic formulation for multi-dimensional scalar conservation laws with discontinuous fluxes and sources, establishing existence, uniqueness, and equivalence of solutions under minimal regularity assumptions.

Contribution

It develops a new kinetic approach for problems with discontinuous fluxes and sources, extending the theory to non-monotone source terms and non-smooth flux functions.

Findings

Existence of entropy measure-valued solutions

Equivalence of entropy weak and kinetic solutions

Uniqueness under Hölder continuity of flux at zero

Abstract

We deal with the Cauchy problem for multi-dimensional scalar conservation laws, where the fluxes and the source terms can be discontinuous functions of the unknown. The main novelty of the paper is the introduction of a~kinetic formulation for the considered problem. To handle the discontinuities we work in the framework of re-parametrization of the flux and the source functions, which was previously used for Kru\v{z}kov entropy solutions. Within this approach we obtain a fairly complete picture: existence of entropy measure valued solutions, entropy weak solutions and their equivalence to the kinetic solution. The results of existence and uniqueness follow under the assumption of H\"{o}lder continuity at zero of the flux. The source term, what is another novelty for the studies on problems with discontinuous flux, is only assumed to be one-side Lipschitz, not necessarily monotone function.