Uniqueness of dissipative solutions to the complete Euler system

2019/05/16 by Ghoshal, Shyam Sundar, Jana, Animesh
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.06919

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^α,∞_p with α>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^α,∞₃ for α>1/3. Our proof relies on commutator estimates for Besov functions and the relative entropy method.

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