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

Bogdan-Vasile Matioc, Georg Prokert

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.

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

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.

Paper Structure

This paper contains 9 sections, 22 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. The fixed time problem
  3. The evolution problem
  4. A class of singular integral operators
  5. Formulation of the evolution equation and the main result
  6. The hydrodynamic potential near $\Gamma$
  7. The hydrodynamic potential in the far-field limit
  8. Smoothness of some nonlinear operators
  9. Some auxiliary results related to localization

Key Result

Theorem 2.1

Given $f\in H^3(\mathbb{R})$, Problem fixtimeeq has the unique solution $(v^\pm,q^\pm)$ given by defvq or, equivalently, repvq.

Theorems & Definitions (45)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 35 more