Non-uniqueness of admissible weak solution to the Riemann problem for the full Euler system in 2D

TL;DR

This paper demonstrates that for the full 2D Euler system, the Riemann problem admits infinitely many entropy admissible weak solutions when initial data lead to a 1D solution with two shocks and a contact discontinuity, indicating non-uniqueness.

Contribution

It extends the non-uniqueness results from isentropic to full Euler equations in 2D for Riemann problems with complex wave structures.

Findings

Existence of infinitely many solutions for certain 2D Riemann problems.

Non-uniqueness persists in the full Euler system.

Results highlight challenges in well-posedness of multi-dimensional conservation laws.

Abstract

The question of well- and ill-posedness of entropy admissible solutions to the multi-dimensional systems of conservation laws has been studied recently in the case of isentropic Euler equations. In this context special initial data were considered, namely the 1D Riemann problem which is extended trivially to a second space dimension. It was shown that there exist infinitely many bounded entropy admissible weak solutions to such a 2D Riemann problem for isentropic Euler equations, if the initial data give rise to a 1D self-similar solution containing a shock. In this work we study such a 2D Riemann problem for the full Euler system in two space dimensions and prove the existence of infinitely many bounded entropy admissible weak solutions in the case that the Riemann initial data give rise to the 1D self-similar solution consisting of two shocks and possibly a contact discontinuity.