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

TL;DR

This paper proves well-posedness and analyzes stability of a two-phase quasistationary Stokes flow with surface tension and gravity, modeling immiscible fluids separated by a sharp interface, using potential theory and nonlinear parabolic analysis.

Contribution

It establishes well-posedness in subcritical Sobolev spaces and investigates the stability of flat interface equilibria considering physical parameters.

Findings

Well-posedness in all subcritical spaces $ ext{H}^r( ext{S})$, $r ext{ in }(3/2,2)$.

Stability analysis of flat equilibria depending on fluid properties.

Formulation of the free boundary problem as a nonlinear nonlocal parabolic PDE.

Abstract

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 $\mathrm{H}^r(\mathbb{S})$, $r\in(3/2,2)$. Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.