New entropy conditions for scalar conservation laws with discontinuous flux

TL;DR

This paper introduces new entropy conditions for scalar conservation laws with discontinuous flux, ensuring existence and uniqueness of solutions without requiring nonlinearity, even with multiple flux crossings.

Contribution

It proposes Kruzhkov type entropy conditions that handle discontinuous fluxes, allowing multiple crossings and removing the need for genuine nonlinearity assumptions.

Findings

Proves existence of entropy solutions under new conditions

Establishes uniqueness of solutions with discontinuous flux

Allows multiple flux crossings without additional assumptions

Abstract

We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under assumptions on the flux which provide the maximum principle. In particular, we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions.