Expanding Solutions to Free Boundary 3D Spherically Symmetric Compressible Navier-Stokes-Poisson Equations near the Lane-Emden Stars

Abstract

We consider the gravitational Navier-Stokes-Poisson equations with the equation of state $P(\rho)=K\rho^{\gamma}$, where $\gamma\in(\frac{6}{5},\frac{4}{3}]$, which models the viscous polytropic gaseous stars. We prove the existence of global weak solutions to the equations with constant viscosity and radially symmetric initial data. For $\gamma=\frac{4}{3}$, we require the initial data having mass less than the mass of the Lane-Emden stars; for $\gamma\in(\frac{6}{5},\frac{4}{3})$, we require that the initial data belong to an invariant set where initial initial data can be taken near the Lane-Emden stars. For $\gamma\in(\frac{6}{5},\frac{4}{3})$, we show that the invariant set contains some initial data that are not allowed in previous literature. We also prove the support of any strong solution expands to infinity for the Navier-Stokes-Poisson equations with constant viscosity and a class of density-dependent viscosity, which indicates the strong instability of Lane-Emden solutions for the Navier-Stokes-Poisson equations.