Nonradial stability of expanding Goldreich-Weber stars

TL;DR

This paper proves the nonlinear stability of certain expanding Goldreich-Weber star solutions to the Euler-Poisson system, highlighting differences between linear and self-similar expansion regimes and identifying unstable directions.

Contribution

It establishes the nonlinear stability of linearly expanding stars and codimension-4 stability of self-similarly expanding stars against irrotational perturbations, revealing the structure of unstable modes.

Findings

Linearly expanding stars are nonlinearly stable.

Self-similarly expanding stars are codimension-4 stable.

Unstable directions are linked to conservation laws.

Abstract

Goldreich-Weber solutions constitute a finite-parameter of expanding and collapsing solutions to the mass-critical Euler-Poisson system. Two subclasses of this family correspond to compactly supported density profiles suitably modulated by the dynamic radius of the star that expands at the self-similar rate $\lambda(t)_{t\to\infty}\sim t^{\frac23}$ and linear rate $\lambda(t)_{t\to\infty}\sim t$ respectively. We prove two results: any linearly expanding Goldreich-Weber star is nonlinearly stable, while any given self-similarly expanding Goldreich-Weber star is codimension-4 nonlinearly stable against irrotational perturbations. The codimension-4 condition in the latter result is optimal and reflects the presence of 4 unstable directions in the linearised dynamics in self-similar coordinates, which are induced by the conservation of the energy and the momentum. This result can be viewed as a codimension-1 nonlinear stability of the moduli space of self-similarly expanding Goldreich-Weber stars against irrotational perturbations.