Turning point principle for stability of viscous gaseous stars

TL;DR

This paper demonstrates that the stability transition of viscous gaseous stars modeled by Navier-Stokes-Poisson equations aligns with the inviscid case, confirming the turning point principle and establishing a link between linear and nonlinear stability.

Contribution

It extends the turning point principle to viscous gaseous stars and proves the equivalence of linear and nonlinear stability for the Navier-Stokes-Poisson system.

Findings

The number of unstable modes matches between viscous and inviscid models.

Stability transitions occur at extrema of the mass-radius curve.

Linear stability implies nonlinear asymptotic stability.

Abstract

We consider stability of non-rotating viscous gaseous stars modeled by the Navier-Stokes-Poisson system. Under general assumptions on the equations of states, we proved that the number of unstable modes for the linearized Navier-Stokes-Poisson system equals that of the linearized Euler-Poisson system modeling inviscid gaseous stars. In particular, the turning point principle holds true for non-rotating stars with or without viscosity. That is, the transition of stability only occurs at the extrema of the total mass and the number of unstable modes is determined by the mass-radius curve. For the proof, we establish an infinite dimensional Kelvin-Tait-Chetaev theorem for a class of linear second order PDEs with dissipation. Moreover, we prove that linear stability implies nonlinear asymptotic stability and linear instability implies nonlinear instability for Navier-Stokes-Poisson system.