A counterexample to well-posedness of entropy solutions to the compressible Euler system

TL;DR

This paper demonstrates that for the isentropic compressible Euler equations in multiple dimensions, entropy solutions can lack uniqueness even from smooth initial conditions, challenging previous assumptions about well-posedness.

Contribution

It provides a counterexample showing failure of uniqueness of entropy solutions in multiple dimensions, extending the understanding of solution behavior in compressible fluid dynamics.

Findings

Entropy solutions are not unique in multiple dimensions.

Failure of well-posedness occurs even from smooth initial data.

The methods adapt De Lellis-Székelyhidi techniques to this context.

Abstract

We deal with entropy solutions to the Cauchy problem for the isentropic compressible Euler equations in the space-periodic case. In more than one space dimension, the methods developed by De Lellis-Sz\'ekelyhidi enable us to show failure of uniqueness on a finite time-interval for entropy solutions starting from any continuously differentiable initial density and suitably constructed bounded initial linear momenta.