Which measure-valued solutions of the monoatomic gas equations are generated by weak solutions?

TL;DR

This paper investigates conditions under which measure-valued solutions of the isentropic Euler equations for monoatomic gases can be approximated by weak solutions, highlighting key differences from the incompressible case and providing new characterization results.

Contribution

It establishes sufficient conditions for generating weak solutions from measure-valued solutions and extends existing characterizations to an $L^{ abla}$-setting, especially for solutions with two Dirac measures.

Findings

Not all measure-valued solutions are generated by weak solutions in the compressible case.

Provides an $L^{ abla}$-variant of Fonseca and Müller's characterization for $ ext{A}$-free Young measures.

Discusses necessary conditions and the gap caused by only having uniform $L^p$ bounds for $1<p< abla$.

Abstract

Contrary to the incompressible case not every measure-valued solution of the compressible Euler equations can be generated by weak solutions or a vanishing viscosity sequence. In the present paper we give sufficient conditions on an admissible measure-valued solution of the isentropic Euler system to be generated by weak solutions. As one of the crucial steps we prove an $L^{\infty}$-variant of Fonseca and M\"uller's characterization result for generating $\mathcal{A}$-free Young measures in terms of potential operators. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing also necessary conditions for generating a measure-valued solution by weak solutions or a vanishing viscosity sequence and will point out that the resulting gap mainly results from obtaining only uniform $L^p$-bounds for $1<p<\infty$ instead of $p=\infty$.