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

The two-phase horizontally periodic quasistationary Stokes flow in R², 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 Hʳ(S), r∈(3/2,2). Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.