Well-posedness of the Euler system of gas dynamics

TL;DR

This paper introduces a new two-step selection criterion for dissipative measure-valued solutions of the Euler gas dynamics system, aiming to identify physically relevant solutions with vanishing energy defect over time.

Contribution

It proposes a novel selection process based on entropy production and energy maximization, distinguishing between weak solutions and turbulent measure-valued solutions.

Findings

Selected solutions depend measurably on initial data

Turbulent solutions' energy defect vanishes over time

The method identifies physically relevant solutions

Abstract

We propose a new two-step selection criterion applicable to the dissipative measure--valued solutions of the Euler system of gas dynamics. The process consists of a successive maximisation of the entropy production rate and the total energy defect, i.e. maximisation of the turbulent energy. If the selected solution is a weak solution of the Euler system, then it is identified in the first step. Solutions selected in the second step are truly measure--valued maximising the energy defect. Accordingly, they are called turbulent solutions. The energy defect of turbulent solutions vanishes with growing time. The selected solutions depend in a Borel--measurable way on the initial data. In particular, they are almost continuously dependent on the initial data.