Global ill-posedness of the isentropic system of gas dynamics

TL;DR

This paper demonstrates that the isentropic compressible Euler system in two dimensions with quadratic pressure law is globally ill-posed by showing the existence of infinitely many weak solutions for certain initial data, including Lipschitz continuous ones.

Contribution

It establishes the global ill-posedness of the 2D isentropic Euler system for specific initial conditions, including classical Riemann data and Lipschitz initial data.

Findings

Existence of infinitely many admissible weak solutions for certain Riemann data.

Construction of Lipschitz initial data leading to multiple solutions.

Ill-posedness of the system in the class of bounded weak solutions.

Abstract

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law $p({\rho}) = {\rho}^2$ and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions.