Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities

TL;DR

This paper proves the global-in-time stability and detailed asymptotic behavior of viscous gaseous star models with density-dependent viscosities, extending previous results to degenerate viscosity cases at vacuum boundaries.

Contribution

It establishes the asymptotic stability of Lane-Emden solutions for viscous gaseous stars with density-dependent viscosities, including the case where viscosities degenerate at vacuum boundaries.

Findings

Global existence of strong solutions with regularity up to vacuum boundary

Convergence of solutions to Lane-Emden solutions over time

Detailed behavior of solutions near vacuum boundary

Abstract

The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent $\gamma$ lies in the stability regime $(4/3, 2)$, by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. The results obtained in this paper extend those in \cite{LXZ} of the authors for the constant viscosities to the case of density dependent viscosities which are degenerate at vacuum states.