Maximal dissipation and well-posedness for the compressible Euler system

TL;DR

This paper explores the well-posedness of the compressible Euler system using the principle of maximal dissipation, demonstrating non-uniqueness through convex integration and proposing maximal dissipation as a potential criterion for uniqueness.

Contribution

It adapts the maximal dissipation principle to weak solutions of the Euler system and uses convex integration to show non-uniqueness, suggesting a new criterion for solution selection.

Findings

Convex integration produces oscillatory solutions violating maximal dissipation.

Counterexamples demonstrate non-uniqueness of weak solutions.

Maximal dissipation may serve as a criterion for selecting unique solutions.

Abstract

We discuss the problem of well-posedness of the compressible (barotropic) Euler system in the framework of weak solutions. The principle of maximal dissipation introduced by C.M. Dafermos is adapted and combined with the concept of admissible weak solutions. We use the method of convex integration in the spirit of the recent work of C.DeLellis and L.Szekelyhidi to show various counterexamples to well-posedness. On the other hand, we conjecture that the principle of maximal dissipation should be retained as a possible criterion of uniqueness as it is violated by the oscillatory solutions obtained in the process of convex integration.