Global solutions for two-phase Hele-Shaw bubble for a near-circular initial shape

Using a vortex sheet method we prove global existence of a near circular initial bubble in a Hele-Shaw cell with surface tension and generally finite nonzero viscosity ratio between fluids inside and outside the bubble. The circular shape is shown to be asymptotically stable for all sufficiently smooth small perturbation. The initial condition in this case, while smooth, need not be analytic.