  We study the two-phase Stokes flow driven by surface tension for two fluids
of different viscosities, separated by an asymptotically flat interface
representable as graph of a differentiable function. 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 and double layer
potential, spectral results on the corresponding integral operators, and
abstract results on nonlinear parabolic evolution equations.
