Expanding solutions to the compressible Navier-Stokes equations with degenerate viscosities in spherical symmetry: global existence and inviscid limit

Abstract

This paper is devoted to studying the strong solutions to the vacuum free boundary problem for the isentropic compressible Navier-Stokes equations with density-dependent viscosities under spherical symmetry, which models the motions of isentropic compressible viscous flows surrounded by vacuum. We construct a class of expanding global solutions when the initial data is a small perturbation of the expanding affine solutions for the adiabatic exponent $γ>1$ and the viscosity coefficients proportional to $ρ^γ$, with $ρ$ being the fluid density. In addition, when the viscosity coefficients tend to zero, the perturbed solutions are proved to converge to the solutions to the compressible Euler equations with an explicit converging rate of viscosity coefficients.