Nonradial stability of self-similarly expanding Goldreich-Weber stars

Goldreich-Weber solutions constitute a finite-parameter of expanding and collapsing solutions to the mass-critical Euler-Poisson system. A subclass of this family corresponds to compactly supported density profiles suitably modulated by the dynamic radius of the star that expands at the self-similar rate $\lambda(t)_{t\to\infty}\sim t^{2/3}$. We prove that any given self-similarly expanding Goldreich-Weber star is codimension-4 stable in the class of irrotational perturbations. The codimension-4 condition is optimal and reflects the presence of 4 unstable directions of the linearised operator in self-similar coordinates, which are induced by the conservation of the energy and the momentum. This result can be viewed as the codimension-1 nonlinear stability of the"manifold"of self-similarly expanding GW-stars in the class of irrotational perturbations.