Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

TL;DR

This paper analyzes two-phase Stokes flow driven by surface tension between fluids of different viscosities in 2D, proving well-posedness and smoothing properties using advanced integral operator analysis.

Contribution

It introduces a rigorous mathematical framework for two-phase Stokes flow with different viscosities, including well-posedness and regularity results in Sobolev spaces.

Findings

Proved well-posedness of the flow model.

Established parabolic smoothing effects.

Analyzed nonlinear singular integral operators.

Abstract

We study the two-phase Stokes flow driven by surface tension for two fluids of different viscosities, separated by an asymptotically flat interface representable as graph of a differentiable function. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single and double layer potential, spectral results on the corresponding integral operators, and abstract results on nonlinear parabolic evolution equations.