Spherically Symmetric Barotropic Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

This paper is concerned with the exterior problem and the initial boundary value problem for the spherically symmetric barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data. For the exterior problem and the initial boundary value problem, we prove that there exists a unique global piecewise regular solution for piecewise regular initial density with arbitrarily large jump discontinuity. Moreover, we show that the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially as time tends to infinity.