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The multiphase Muskat problem with equal viscosities in two dimensions

Jonas Bierler, Bogdan-Vasile Matioc

Abstract

We study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with equal viscosities in a vertical homogeneous porous medium identified with $\mathbb{R}^2$ under the effect of gravity. We first formulate the governing equations as a strongly coupled evolution problem for the functions that parameterize the sharp interfaces between the fluids. Afterwards we prove that the problem is of parabolic type and establish its well-posedness together with two parabolic smoothing properties. For solutions that are not global we exclude, in a certain regime, that the interfaces come into contact along a curve segment.

The multiphase Muskat problem with equal viscosities in two dimensions

Abstract

We study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with equal viscosities in a vertical homogeneous porous medium identified with under the effect of gravity. We first formulate the governing equations as a strongly coupled evolution problem for the functions that parameterize the sharp interfaces between the fluids. Afterwards we prove that the problem is of parabolic type and establish its well-posedness together with two parabolic smoothing properties. For solutions that are not global we exclude, in a certain regime, that the interfaces come into contact along a curve segment.

Paper Structure

This paper contains 4 sections, 16 theorems, 179 equations.

Table of Contents

  1. Introduction and the main results
  2. Mapping properties
  3. The generator property and the proof of the main results
  4. An extension of Privalov's theorem

Key Result

Theorem 1.1

Let $r\in(3/2,2)$, $c_\infty>0$, and $f,\,h\in H^r({\mathbb R})$ with $c_\infty+f>h$ be given. Then the boundary value problem has a unique solutionThe pressures $(p_1,p_2,p_3)$ are unique only up to the same additive constant.$(v_1,v_2,v_3,p_1,p_2,p_3)$ with Moreover, setting $v\coloneqq{} v_1{\bf 1}_{\Omega_1}+v_2{\bf 1}_{\Omega_2}+v_3{\bf 1}_{\Omega_3}$, it holds for $z\coloneqq{}(x,y)\in{\ma

Theorems & Definitions (32)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 22 more