Global solution of spherically symmetric compressible Navier–Stokes equations with bounded density and density-dependent viscosity

Abstract

We consider the compressible Navier–Stokes equations with viscosities $\mu \left(\rho \right)=\rho ,\lambda \left(\rho \right)=0$ in bounded domains when the initial data are spherically symmetric, which covers the Saint-Venant model for the motion of shallow water. First, based on the exploitation of the one-dimensional feature of symmetric solutions, we prove the global existence of weak solutions with initial vacuum, where the upper bound of the density is obtained. Then, with more conditions imposed on the nonvacuum initial data, we obtain the global weak solution which is a strong one away from the symmetry center. The analysis allows for the possibility that a vacuum state emerges at the symmetry center; in particular, we give the uniform bound of the radius of the vacuum domain.