On entropy solutions of scalar conservation laws with discontinuous flux

TL;DR

This paper develops a framework for entropy solutions to scalar conservation laws with discontinuous flux, establishing existence, stability, and conditions for uniqueness, and analyzing limits of periodic solutions.

Contribution

It introduces a new notion of entropy solutions for conservation laws with discontinuous flux and proves key properties including existence, stability, and partial uniqueness.

Findings

Existence of largest and smallest entropy solutions.

Stability and monotonicity of these solutions.

Uniqueness holds for periodic initial data, but can fail generally.

Abstract

We introduce the notion of entropy solutions (e.s.) to a conservation law with an arbitrary jump continuous flux vector and prove existence of the largest and the smallest e.s. to the Cauchy problem. The monotonicity and stability properties of these solutions are also established. In the case of a periodic initial function we derive the uniqueness of e.s. Generally, the uniqueness property can be violated, which is confirmed by an example. Finally, we proved that in the case of single space variable a weak limit of a sequence of spatially periodic e.s. is an e.s. as well.