On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations

TL;DR

This paper demonstrates the non-uniqueness of continuous entropy solutions to the 3D isentropic compressible Euler equations, constructing infinitely many smooth solutions without shocks and analyzing their regularity.

Contribution

It introduces a method to construct infinitely many continuous entropy solutions with specific regularity properties, challenging uniqueness results.

Findings

Existence of infinitely many smooth entropy solutions without shocks.

Constructed solutions have density smoothness and momentum $eta$-H"older continuity for $eta<1/7$.

Provided a continuous entropy solution satisfying the entropy inequality strictly.

Abstract

We consider the Cauchy problem for the isentropic compressible Euler equations in a three-dimensional periodic domain under general pressure laws. For any smooth initial density away from the vacuum, we construct infinitely many entropy solutions with no presence of shock. In particular, the constructed density is smooth and the momentum is $\alpha$-H\"older continuous for $\alpha<1/7$. Also, we provide a continuous entropy solution satisfying the entropy inequality strictly.