  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 $B^{\alpha,\infty}_{p}$ 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.
