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 et al. (Commun Partial Differ Equ 44(12):1285–1298, 2019), we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space $$B^{\alpha ,\infty }_{p}$$ 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, $$B^{\alpha ,\infty }_{3}$$ for $$\alpha >1/3$$ . Our proof relies on commutator estimates for Besov functions and the relative entropy method.