Uniqueness of dissipative solutions to the complete Euler system

Abstract

Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure-valued solutions. Motivated from [Feireisl, Ghoshal and Jana, Commun. Partial Differ. Equ., 2019], we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space $B_p^{\alpha ,\infty }$ with $\alpha >1/2$. We prove that the Besov solution satisfying the above-mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one-sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, $B_3^{\alpha ,\infty }$ for $\alpha >1/3$. Our proof relies on commutator estimates for Besov functions and the relative entropy method.