A Uniqueness Condition for Conservation Laws with Discontinuous Gradient-Dependent Flux

TL;DR

This paper establishes a simple condition ensuring the uniqueness of entropy solutions for scalar conservation laws with discontinuous, gradient-dependent flux, extending previous results on vanishing viscosity and front tracking approximations.

Contribution

It introduces a new condition that guarantees the uniqueness of weak entropy solutions in conservation laws with discontinuous flux functions.

Findings

The condition ensures all entropy solutions coincide with the semigroup trajectory.

It guarantees the uniqueness of solutions in the considered class.

The approach simplifies the analysis of conservation laws with discontinuous flux.

Abstract

The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. In the stable case where $f(u)<g(u)$ for all $u\in R$, it was proved in [1] that the limits of vanishing viscosity approximations form a contractive semigroup w.r.t. the $L^1$ distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In the present paper we introduce a simple condition that guarantees that every weak, entropy admissible solution of a Cauchy problem coincides with the corresponding semigroup trajectory, and hence is unique.