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 γ 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 C1/2-Hölder 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. Those uniform convergence
are of fundamental importance in the study of vacuum free boundary problems
which are missing in the previous results for global weak solutions. Moreover, the results
obtained in this paper apply to much broader cases of viscosities than those in Fang and
Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions
when the adiabatic exponent γ lies in the most physically relevant range. Finally, this
paper extends the previous local-in-time theory for strong solutions to a global-in-time
one.