Maximal dissipation and well-posedness of the Euler system of gas dynamics

TL;DR

This paper establishes that dissipative solutions satisfying maximal dissipation are weak solutions and introduces a simple selection procedure for unique solutions of the Euler system of gas dynamics.

Contribution

It proves the equivalence of dissipative solutions with maximal dissipation to admissible weak solutions and proposes a new, refined criterion for unique solution selection.

Findings

Dissipative solutions with maximal dissipation are necessarily admissible weak solutions.

A simple, at most two-step, selection procedure for unique solutions is proposed.

A refined criterion guarantees uniqueness for any finite energy initial data.

Abstract

We show that any dissipative (measure-valued) solution of the compressible Euler system that complies with Dafermos' criterion of maximal dissipation is necessarily an admissible weak solution. In addition, we propose a simple, at most two step, selection procedure to identify a unique semigroup solution in the class of dissipative solutions to the Euler system. Finally, we introduce a refined version of Dafermos' criterion yielding a unique solution of the problem for any finite energy initial data.