Local well-posedness for the motion of a compressible, self-gravitating liquid with free surface boundary

TL;DR

This paper proves local well-posedness for the free boundary problem of a self-gravitating compressible liquid, using smoothed Euler equations and advanced energy and elliptic estimates.

Contribution

It introduces a novel approach by solving a tangentially-smoothed Euler system in Lagrangian coordinates with uniform energy bounds.

Findings

Established local well-posedness for the problem.

Developed uniform energy estimates as smoothing parameter approaches zero.

Utilized optimal elliptic estimates related to boundary regularity.

Abstract

We establish the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity. We do this by solving a tangentially-smoothed version of Euler's equations in Lagrangian coordinates which satisfies uniform energy estimates as the smoothing parameter goes to zero. The main technical tools are delicate energy estimates and optimal elliptic estimates in terms of boundary regularity, for the Dirichlet problem and Green's function.