Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Bogdan-Vasile Matioc, Georg Prokert

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.

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

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.

Paper Structure

This paper contains 9 sections, 20 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. An auxiliary fixed-time problem
  5. The $L_2$-resolvent of the hydrodynamic double-layer potential operator
  6. The resolvent of the hydrodynamic double-layer potential operator in higher order Sobolev spaces
  7. The contour integral formulation
  8. Linearization, localization, and proof of the main result
  9. The hydrodynamic double-layer potential near $\Gamma$

Key Result

Theorem 1.1

Let $s\in(3/2,2)$ be given. Then, the following statements hold true:

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3: Spectral properties of $\mathbb{D}(f)$ and $\mathbb{D}(f)^\ast$
  • proof
  • ...and 29 more