On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary

TL;DR

This paper studies the stability and long-term behavior of a self-gravitating incompressible fluid with a free boundary, proving solutions persist longer than expected for small perturbations by removing quadratic nonlinearities.

Contribution

It introduces a nonlinear transformation that eliminates quadratic terms in the Euler-Poisson system, extending the lifespan of solutions for small perturbations beyond standard local well-posedness.

Findings

Solutions exist for at least cε^{-2} time for small perturbations

The Taylor sign condition always holds in this setting

Local well-posedness is established for the system

Abstract

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler-Poission system in moving bounded simply connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data which are small perturbations of size $\epsilon$ of these static states, measured in appropriate Sobolev spaces, the solution exists and remains of size $\epsilon$ on a time interval of length at least $c\epsilon^{-2},$ where $c$ is a constant independent of $\epsilon.$ This should be compared with the lifespan $O(\epsilon^{-1})$ provided by local well-posdness. The key ingredient of our proof is finding a nonlinear transformation which removes quadratic terms from the nonlinearity. An important difference with the related gravity water waves problem is that unlike the constant gravity for water waves, the self-gravity in the Euler-Poisson system is nonlinear. As a first step in our analysis we also show that the Taylor sign condition always holds and establish local well-posedness for this system.