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

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$\ddot{\mathrm{o}}$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.

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References in corpus (3)

• Non-linear stability of gaseous stars
• Existence and Nonlinear Stability of Rotating Star Solutions of the Compressible Euler-Poisson Equations
• Nonlinear Dynamical Stability of Newtonian Rotating White Dwarfs and Supermassive Stars