On the uniqueness of solutions to hyperbolic systems of conservation laws

TL;DR

This paper proves the uniqueness of dissipative weak solutions to hyperbolic conservation laws under certain Besov space regularity and one-sided bounds, extending the relative entropy method to broader settings.

Contribution

It establishes a universal regularity condition for uniqueness of solutions across various hyperbolic systems, including elasticity and Euler equations.

Findings

Uniqueness holds for solutions in Besov spaces with exponent > 1/2.

The method applies to systems like elasticity, shallow water MHD, and Euler.

Constructs a non-Lipschitz solution satisfying the uniqueness criteria.

Abstract

For general hyperbolic systems of conservation laws we show that dissipative weak solutions belonging to an appropriate Besov space $B^{\alpha,\infty}_q$ and satisfying a one-sided bound condition are unique within the class of dissipative solutions. The exponent $\alpha>1/2$ is universal independently of the nature of the nonlinearity and the Besov regularity need only be imposed in space when the system is expressed in appropriate variables. The proof utilises a commutator estimate which allows for an extension of the relative entropy method to the required regularity setting. The systems of elasticity, shallow water magnetohydrodynamics, and isentropic Euler are investigated, recovering recent results for the latter. Moreover, the article explores a triangular system motivated by studies in chromatography and constructs an explicit solution which fails to be Lipschitz, yet satisfies the conditions of the presented uniqueness result.