Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary

In this paper, we obtain the global weak solution to the 3D spherically symmetric compressible isentropic Navier-Stokes equations with arbitrarily large, vacuum data and free boundary when the shear viscosity it is a positive constant and the bulk viscosity lambda(rho) = rho(beta) with beta > 0. The analysis of the upper and lower bound of the density is based on some well-chosen functionals. In addition, the free boundary can be shown to expand outward at an algebraic rate in time.