Dissipative measure valued solutions for general conservation laws

TL;DR

This paper develops a unified framework for dissipative measure-valued solutions in general conservation laws, establishing measure-valued-strong uniqueness for various complex systems including new examples like magnetohydrodynamics.

Contribution

It introduces a comprehensive framework for measure-valued solutions applicable to diverse systems, extending the measure-valued-strong uniqueness principle to new equations.

Findings

Unified framework for measure-valued solutions

Extension of uniqueness principle to new systems

Examples include magnetohydrodynamics and shallow water equations

Abstract

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued -- strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system et al. and there are also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems, and most importantly, we give examples of equations, for which the aspect of measure-valued -- strong uniqueness has not been considered before, like incompressible magentohydrodynamics and shallow water magnetohydrodynamics.