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Turning point principle for stability of viscous gaseous stars

Ming Cheng, Zhiwu Lin, Yucong Wang

TL;DR

This work establishes a turning-point framework for the stability of non-rotating viscous gaseous stars by linking the viscous Navier-Stokes-Poisson system to its inviscid Euler-Poisson limit. A central achievement is an infinite-dimensional Kelvin-Tait-Chetaev theorem showing that the number of unstable modes of the dissipative problem equals the number of negative modes of the associated self-adjoint operator, which in turn yields a turning-point principle: stability changes occur only at extrema of the mass along the mass-radius curve. Under sharp linear stability ($n^-(L_\mu)=0$ and $dM_\mu/d\mu\neq0$), the authors prove nonlinear asymptotic stability for spherically symmetric perturbations; conversely, when $n^-(L_\mu)>0$ and $dM_\mu/d\mu\neq0$, nonlinear instability is established via a nonlinear bootstrap argument. The results hold for general equations of state (P1)-(P2), extend known polytropic white-dwarf cases, and clarify the impact of viscosity on stability through a unified, coordinate-robust energy framework that leverages Lagrangian formulations and spectral data from the mass-radius curve.

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.

Turning point principle for stability of viscous gaseous stars

TL;DR

This work establishes a turning-point framework for the stability of non-rotating viscous gaseous stars by linking the viscous Navier-Stokes-Poisson system to its inviscid Euler-Poisson limit. A central achievement is an infinite-dimensional Kelvin-Tait-Chetaev theorem showing that the number of unstable modes of the dissipative problem equals the number of negative modes of the associated self-adjoint operator, which in turn yields a turning-point principle: stability changes occur only at extrema of the mass along the mass-radius curve. Under sharp linear stability ( and ), the authors prove nonlinear asymptotic stability for spherically symmetric perturbations; conversely, when and , nonlinear instability is established via a nonlinear bootstrap argument. The results hold for general equations of state (P1)-(P2), extend known polytropic white-dwarf cases, and clarify the impact of viscosity on stability through a unified, coordinate-robust energy framework that leverages Lagrangian formulations and spectral data from the mass-radius curve.

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.
Paper Structure (15 sections, 28 theorems, 309 equations)

This paper contains 15 sections, 28 theorems, 309 equations.

Table of Contents

  1. Introduction
  2. Kelvin-Tait-Chetaev Theorem in infinite dimension
  3. The turning point principle for non-rotating viscous stars.
  4. Nonlinear stability under the spherically symmetric perturbation
  5. Formulation in Lagrangian coordinates
  6. Linear stable estimates
  7. Nonlinear estimates for stability
  8. The positive quadratic form
  9. Lower-order estimates
  10. High-order estimates
  11. Local existence
  12. Nonlinear instability under the spherically symmetric perturbation
  13. The linear unstable estimates
  14. Formulation in Lagrangian mass coordinates
  15. Nonlinear estimates for instability

Key Result

Theorem 1.1

When $\frac{dM_\mu}{d\mu} \neq0$, we have $n_{EP}^{u}\left( \mu\right)$$=n_{NSP}^{u}\left( \mu\right) .$

Theorems & Definitions (51)

  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 41 more