Weak-strong uniqueness for measure-valued solutions

TL;DR

This paper establishes the weak-strong uniqueness property for measure-valued solutions of the incompressible Euler equations, confirming their consistency with classical solutions when they exist, and extends this result to conservation laws.

Contribution

It proves the weak-strong uniqueness for measure-valued solutions of the Euler equations and conservation laws, clarifying their relation to classical solutions.

Findings

Measure-valued solutions coincide with classical solutions when the latter exist.

Weak-strong uniqueness holds for DiPerna's measure-valued solutions.

Global existence for measure-valued solutions is established for L2 initial data.

Abstract

We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by R.DiPerna and A.Majda, and in particular global existence to any L2 initial data was proven. Whether measure-valued solutions agree with classical solutions if the latter exist has apparently remained open. We also show that DiPerna's measure-valued solutions to systems of conservation laws have the weak-strong uniqueness property.