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 a characterization result for generating $${\mathcal {A}}$$ A -free Young measures in terms of potential operators including uniform $$L^{\infty }$$ L ∞ -bounds. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing that are 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 $$L^p$$ L p -bounds for $$1<p<\infty $$ 1 < p < ∞ instead of $$p=\infty $$ p = ∞ .