Global Solutions for Small Data to the Hele-shaw Problem

We analyze an equation governing the motion of an interface between two fluids in a pressure field. In two dimensions, the interface is described by a conformal mapping which is analytic in the exterior of the unit disc. This mapping obeys a nonlocal nonlinear equation. When there is no pumping at infinity, there is conservation of area and contraction of the length of the interface. We prove global in time existence for small analytic perturbations of the circle as well as nonlinear asymptotic stability of the steady circular solution. The same method yields well-posedness of the Cauchy problem in the presence of pumping.