A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients

Abstract

Local solutions of the multidimensional Navier-Stokes equations for isentropic compressible flow are constructed with spherically symmetric initial data between a solid core and a free boundary connected to a surrounding vacuum state. The viscosity coefficients $λ, μ$ are proportional to $ρ^θ}$, $0<θ<γ$, where $ρ$ is the density and $γ>1$ is the physical constant of polytropic fluid. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.