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Description

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid subject to self-gravitational force and neglecting surface tension in two space dimensions. We show that for smooth data which are size $\epsilon$ perturbations of an equilibrium state, the solution exists and remains smooth for time of at least $O(\epsilon^{-2})$. This should be compared with the lifespan $O(\epsilon^{-1})$ provided by local well-posedness. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate system has no quadratic nonlinear terms. This is a joint work with Lydia Bieri, Shuang Miao and Sohrab Shahshahani.