  We consider the quasistationary Stokes flow that describes the motion of a
two-dimensional fluid body under the influence of surface tension effects in an
unbounded, infinite-bottom geometry. We reformulate the problem as a fully
nonlinear parabolic evolution problem for the function that parameterizes the
boundary of the fluid with the nonlinearities expressed in terms of singular
integrals. We prove well-posedness of the problem in the subcritical Sobolev
spaces $H^s(\mathbb{R})$ up to critical regularity, and establish parabolic
smoothing properties for the solutions. Moreover, we identify the problem as
the singular limit of the two-phase quasistationary Stokes flow when the
viscosity of one of the fluids vanishes.
