Singularities of solutions to compressible Euler equations with vacuum

TL;DR

This paper investigates finite time singularity formation in solutions to the compressible Euler equations with vacuum, showing that certain smooth initial conditions lead to blow-up of the velocity and sound speed norms.

Contribution

It provides new results on singularity formation for symmetric initial data in compressible Euler equations with vacuum, without requiring compact support.

Findings

Finite time blow-up of the $H^3$ norm of velocity and sound speed

Singularity formation occurs under symmetric initial conditions with vanishing sound speed at the origin

Results apply to both two and three space dimensions

Abstract

Presented are two results on the formation of finite time singularities of solutions to the compressible Euler equations in two and three space dimensions for isentropic, polytropic, ideal fluid flows. The initial velocity is assumed to be symmetric and the initial sound speed is required to vanish at the origin. They are smooth in Sobolev space $H^3$, but not required to have a compact support. It is shown that the $H^3$ norm of the velocity field and the sound speed will blow up in a finite time.