Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

Bogdan–Vasile Matioc,Georg Prokert

doi:10.1017/prm.2020.82

Abstract

We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. 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-layer potential and abstract results on nonlinear parabolic evolution equations.

Highlights

One of the standard methods in the analysis of moving boundary problems is the reformulation of these problems as evolution equations in function spaces to which methods of Functional Analysis can be applied, depending on the character of the problem under investigation
The difficulty of this typically consists in the fact that the resulting evolution equations are nonlocal and strongly nonlinear
For moving boundary problems with domains of general shape this approach typically involves the transformation of the moving domain to a fixed reference domain by an unknown, time-dependent diffeomorphism, and the use of solution operators for boundary value problems with variable coefficients on this reference domain

Summary

Introduction

One of the standard methods in the analysis of moving boundary problems is the reformulation of these problems as evolution equations in function spaces to which methods of Functional Analysis can be applied, depending on the character of the problem under investigation. It is this analogy that enables us to study the moving boundary problem of two-phase Stokes flow driven by capillarity (at least in 2D and with equal viscosity in both phases) along the same lines as for the Muskat problem This has first been exploited in [4] to obtain an existence result for all positive times, with initial data that are small in a space of Fourier transforms of bounded measures. The structure of the paper is as follows: In § 2 we consider the underlying twophase boundary value problem for the Stokes equations (1.1)− with fixed interface, and show that it is solved by the so-called hydrodynamic single-layer potential We prove this by investigating its behaviour near and on the interface (recovering results from [8] in our slightly different setting) and show that it vanishes in the far-field limit. Throughout the paper, some longer proofs are deferred to appendices

The fixed time problem

Uniqueness

A class of singular integral operators

Formulation of the evolution equation and the main result