Shocks Make the Riemann Problem for the Full Euler System in Multiple Space Dimensions Ill-posed

TL;DR

This paper demonstrates that the Riemann problem for the full Euler system in multiple dimensions is ill-posed, with multiple admissible solutions existing when the self-similar solution contains even a single shock.

Contribution

It extends previous results by proving ill-posedness for the full Euler system with just one shock, advancing understanding of solution non-uniqueness in multi-dimensional gas dynamics.

Findings

Existence of infinitely many admissible weak solutions with one shock

Ill-posedness of the Riemann problem in multiple dimensions for the full Euler system

Extension of prior results from simpler systems to the full Euler system

Abstract

The question of (non-)uniqueness of one-dimensional self-similar solutions to the Riemann problem for hyperbolic systems of gas dynamics in sets of multi-dimensional admissible weak solutions was addressed in recent years in several papers culminating in [17] with the proof that the Riemann problem for the isentropic Euler system with a power law pressure is ill-posed if the one-dimensional self-similar solution contains a shock. Natural question then arises whether the same holds also for a more involved system of equations, the full Euler system. After the first step in this direction was made in [1], where the ill-posedness was proved in the case of two shocks appearing in the self-similar solution, we prove in this paper that the presence of just one shock in the self-similar solution implies the same outcome, i.e. the existence of infinitely many admissible weak solutions to the multi-dimensional problem.