The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock

TL;DR

This paper demonstrates that the Riemann problem for the multidimensional isentropic gas dynamics system is generally ill-posed with multiple entropy solutions, except in the smooth case, completing previous research on solution uniqueness.

Contribution

It extends prior work by proving non-uniqueness of entropy solutions for all initial states, except smooth solutions, in the multidimensional isentropic Euler equations.

Findings

Existence of infinitely many entropy solutions for certain initial states.

Uniqueness holds only for smooth solutions.

Completes the classification of solution uniqueness for the problem.

Abstract

In this paper we consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. The latter consists only of two constant states, where one state lies on the lower and the other state on the upper half plane. The aim is to investigate if there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. For some initial states this question has been answered by E. Feireisl and O. Kreml, and also G.-Q. Chen and J. Chen, where there exists a unique entropy solution. For other initial states E. Chiodaroli, O. Kreml and C. De Lellis showed that there are infinitely many entropy solutions. For still other initial states the question on uniqueness remained open and this will be the content of this paper. This paper can be seen as a completion of the aforementioned papers by showing that the solution is non-unique in all cases (except if the solution is smooth).