Well-posedness and stability for the two-phase periodic quasistationary Stokes flow

The two-phase horizontally periodic quasistationary Stokes flow in ${\mathbb{R}}^2$, describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, which is parameterized as the graph of a function $f=f(t)$, is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function $f$. Based on abstract parabolic theory, it is proven that the problem is well-posed in all subcritical spaces ${\mathrm{H}}^r(\mathbb{S})$, $r\in (3\mathrm{/}2,2)$. Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.