Existence and Nonlinear Stability of Rotating Star Solutions of the Compressible Euler-Poisson Equations

TL;DR

This paper establishes the existence and nonlinear stability of rotating star solutions to the 3D compressible Euler-Poisson equations, expanding the understanding of their steady states and stability under perturbations.

Contribution

It proves the existence of rotating star solutions via a variational approach and demonstrates their nonlinear and local-in-time stability under specific perturbations.

Findings

Existence of rotating star solutions with prescribed angular momentum and mass.

Nonlinear dynamical stability of these solutions.

Uniform a-priori estimates for entropy-weak solutions.

Abstract

We prove existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler-Poisson (EP) equations in 3 spatial dimensions, with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty-Beals paper. We prove the nonlinear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove local in time stability of $W^{1, \infty}(\RR^3)$ solutions where the perturbations are entropy-weak solutions of the EP equations. Finally, we give a uniform (in time) a-priori estimate for entropy-weak solutions of the EP equations.