Free boundary problem for the equation of spherically symmetric motion of viscous gas

We study the spherically symmetric motion of viscous barotropic gas surrounding a ball. The basic equation is the compressible Navier-Stokes equation. We are interested in the density distribution which contacts with vacuum at a finite radius. This is a free boundary problem. After rewriting the equation in the Lagrangean coordinate, we construct approximate solutions by discretizing the mass variable. Passing to a limit, we find a global weak solution.