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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.
Böhme et al. (Tue,) studied this question.