  In this work, we study the well-posedness of a system of partial differential
equations that model the dynamics of a two-dimensional Stokes bubble immersed
in two-dimensional ambient Stokes fluid of the same viscosity that extends to
infinity under the effect of surface tension. We assume that the two fluids are
immiscible and incompressible and that there is no interfacial jump in the
fluid velocity. For this PDE system, a circular fluid bubble is a steady-state
solution. Given an initial contour for the fluid bubble which is sufficiently
close to a circle, we show that there exists a unique, global-in-time solution.
This unique solution decays to a circle exponentially fast, which means that
circular fluid bubbles are stable steady-state solutions. We also obtain a
result concerning the regularity of the unique solution, that although the
initial perturbation around a circular contour is assumed to be of low
regularity, any later perturbation becomes real analytic, hence smooth.
