A characterization on breakdown of smooth spherically symmetric solutions of the isentropic system of compressible Navier--Stokes equations

Abstract

We study an initial boundary value problem on a ball for the isentropic system of compressible Navier--Stokes 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 concentration of mass forms around the center in Lagrangian coordinate system. In other words, in Euler coordinate system, either the density concentrates or vanishes around the center. For the latter case, 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.