Spherically Symmetric Isentropic Compressible Flows with Density-Dependent Viscosity Coefficients

Abstract

We prove the existence of global weak solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients when the initial data are large and spherically symmetric by constructing suitable aproximate solutions. We focus on the case where those coefficients vanish on vacuum. The solutions are obtained as limits of solutions in annular regions between two balls, and the equations hold in the sense of distribution in the entire space-time domain. In particular, we prove the existence of spherically symmetric solutions to the Saint–Venant model for shallow water.