On admissibility criteria for weak solutions of the Euler equations

TL;DR

This paper investigates the limitations of current admissibility criteria in uniquely identifying weak solutions of the Euler equations, revealing non-uniqueness even under common energy conditions for certain initial data.

Contribution

It demonstrates that several admissibility criteria do not guarantee uniqueness of weak solutions for the Euler equations and extends this non-uniqueness to the p-system of gas dynamics.

Findings

Admissibility criteria do not ensure uniqueness for some initial data.

Non-uniqueness of solutions in multiple dimensions.

Counterexamples for the p-system of gas dynamics.

Abstract

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.