Numerical Study of Non-uniqueness for 2D Compressible Isentropic Euler Equations

TL;DR

This paper numerically investigates the non-uniqueness of solutions in 2D compressible Euler equations with singular initial vorticity, revealing potential issues with well-posedness.

Contribution

It introduces a numerical approach to demonstrate multiple solutions for specific initial conditions, highlighting fundamental non-uniqueness in the equations.

Findings

Numerical evidence of multiple solutions with spiraling vorticity singularities.

Identification of obstructions to well-posedness in 2D compressible Euler systems.

Use of positivity-preserving discontinuous Galerkin method for simulations.

Abstract

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.