Self-Similar Solutions to the Compressible Euler Equations and their Instabilities

TL;DR

This paper constructs and analyzes self-similar solutions to the compressible Euler equations, revealing their instability and the potential for shock formation prior to singularity development.

Contribution

It provides the first construction of smooth, non-vacuum self-similar solutions and characterizes their spectral stability, highlighting their inherent instabilities.

Findings

Existence of smooth, non-vacuum self-similar solutions

Spectral characterization of radial perturbations

Numerical evidence of shock formation before singularity

Abstract

This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evidence that one of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable.