Structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness

TL;DR

This paper studies the structure of solutions to multidimensional conservation laws with discontinuous flux, demonstrating the existence of traces and a generalized Kato inequality, which are used to establish solution uniqueness.

Contribution

It introduces a framework for analyzing entropy solutions with discontinuous flux, proving trace existence and a generalized Kato inequality for the first time.

Findings

Entropy solutions admit traces on flux discontinuities.

A generalized Kato inequality is established for solution pairs.

Applications demonstrate improved uniqueness results.

Abstract

We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given.