Capillarity driven Stokes flow: the one-phase problem as small viscosity limit

TL;DR

This paper analyzes the quasistationary Stokes flow driven by surface tension in a 2D fluid, reformulating it as a nonlinear evolution problem, proving well-posedness, and connecting it to the small viscosity limit of two-phase flows.

Contribution

It introduces a nonlinear parabolic formulation for the free boundary Stokes flow and establishes well-posedness and smoothing properties, linking it to the small viscosity limit of two-phase flows.

Findings

Proved well-posedness in Sobolev spaces up to critical regularity.

Established parabolic smoothing properties of solutions.

Identified the problem as the small viscosity limit of two-phase Stokes flow.

Abstract

We consider the quasistationary Stokes flow that describes the motion of a two-dimensional fluid body under the influence of surface tension effects in an unbounded, infinite-bottom geometry. We reformulate the problem as a fully nonlinear parabolic evolution problem for the function that parameterizes the boundary of the fluid with the nonlinearities expressed in terms of singular integrals. We prove well-posedness of the problem in the subcritical Sobolev spaces $H^s(\mathbb{R})$ up to critical regularity, and establish parabolic smoothing properties for the solutions. Moreover, we identify the problem as the singular limit of the two-phase quasistationary Stokes flow when the viscosity of one of the fluids vanishes.