We prove that the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary are well-posed in Sobolev space. The methodology consists of a fixed-point argument using a tangential smoothing operator, followed by energy estimates

In this paper, we prove a priori estimates in Lagrangian coordinates for the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary. The estimates show that on a finite time interval we control five derivatives of the fluid velocity and five and a half derivatives of the coordinates of the moving domain.