On Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem

TL;DR

This paper establishes the nonlinear asymptotic stability of Lane-Emden solutions for viscous gaseous stars within a specific adiabatic constant range, demonstrating global existence, regularity, and convergence of solutions to the equilibrium state.

Contribution

It proves the stability and detailed asymptotic behavior of solutions to the viscous star problem for the first time within the specified parameter range.

Findings

Global-in-time strong solutions exist for small perturbations.

Solutions converge to Lane-Emden solutions with explicit rates.

The vacuum boundary behavior is precisely characterized.

Abstract

This paper proves the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant $\gamma$ lies in the stability range $(4/3, 2)$. It is shown that for small perturbations of a Lane-Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary provided that $\gamma$ lies in $(4/3, 2)$. The key is to establish the global-in-time regularity uniformly up to the vacuum boundary, which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of the Lane-Emden solution with detailed convergence rates, and detailed large time behaviors of solutions near the vacuum boundary.