On the motion of the free boundary of a self-gravitating incompressible fluid

The motion of the boundary of an incompressible fluid body subject to its self-gravitatioal force can be described by the free boundary problem of the Euler-Poisson system. The structure of the equation has similarities with that of the water wave problem, but an important difference is that the constant gravity in water waves is replaced by a nonlinear self-gravitaty. In this talk, we present some recent results on the well-posedness of this problem and give a lower bound on the lifespan of smooth solutions in two dimensions. In particular, we show that the Taylor sign condition always holds leading to local well-posedness, and for smooth data of size $\epsilon$ a unique smooth solution exists for time greater than or equal to $O(1/{\epsilon}^2)$. This is achieved by constructing an appropriate quantity and a coordinate transformation such that the new quantity in the new coordinate system satisfies an equation without quadratic nonlinearities. This is joint work with L. Bieri, S. Miao, and S. Wu.