  We study the two-phase Stokes flow driven by surface tension with two fluids
of equal viscosity, separated by an asymptotically flat interface with graph
geometry. The flow is assumed to be two-dimensional with the fluids filling the
entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces
up to critical regularity. The main technical tools are an analysis of
nonlinear singular integral operators arising from the hydrodynamic
single-layer potential and abstract results on nonlinear parabolic evolution
equations.
