On Formation of Singularity of Spherically Symmetric Nonbarotropic Flows

TL;DR

This paper investigates conditions under which smooth, spherically symmetric solutions to the heat-conductive compressible Navier-Stokes-Fourier system break down in finite time, focusing on density concentration, vanishing, or unbounded velocity.

Contribution

It establishes a precise criterion for the finite-time breakdown of classical solutions in spherically symmetric compressible flows with heat conduction.

Findings

Classical solutions break down if density concentrates or vanishes or velocity becomes unbounded.

Vacuum formation around the center can lead to solution singularity.

The breakdown criterion is both necessary and sufficient for smooth solutions.

Abstract

We study an initial boundary value problem on a ball for the heat-conductive system of compressible Navier-Stokes-Fourier equations, in particular, a criterion of breakdown of the classical solution. For smooth initial data away from vacuum, it is proved that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the {\bf density} {\it concentrates} or {\it vanishes} or the {\bf velocity} becomes unbounded around the center. One possible situation is that a vacuum ball appears around the center and the density may concentrate on the boundary of the vacuum ball simultaneously.