Turning point principle for the stability of viscous gaseous stars

We investigate the stability of non-rotating viscous gaseous stars, which are modeled by the Navier-Stokes-Poisson system. Based on general hypotheses concerning the equation of states, we demonstrate that the count of unstable modes in the linearized Navier-Stokes-Poisson system matches that of the linearized Euler-Poisson system modeling inviscid gaseous stars. In particular, the turning point principle holds for the non-rotating stars with viscosity. This principle asserts that the stability of these stars is determined by the mass-radius curve parameterized by the center density. The transition of stability only occurs at the extrema of the total mass. To substantiate our claims, we formulate an infinite-dimensional Kelvin-Tait-Chetaev Theorem for a class of abstract second-order linear equations with dissipation. Moreover, we prove that linear stability implies nonlinear asymptotic stability and linear instability implies nonlinear instability for the Navier-Stokes-Poisson system under spherically symmetric perturbations.