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

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 $p=\infty$.