On formation of singularity of spherically symmetric nonbarotropic flows

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 for the breakdown of the classical solution. For smooth initial data away from vacuum, we prove that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the density concentrates or vanishes or the 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.