Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Three-phase Muskat problem: uniform lifespan with respect to the width of the strip between interfaces

Ángel Castro, Liangchen Zou

Abstract

We consider the three-phase Muskat problem with different densities and the same viscosities. The lifespan of the solutions with respect to the width of the strip between interfaces is studied. Indeed, the interfaces are parameterized by the graph of two functions $f(x,t)$ and $g(x,t)$ and we impose that $||f(\cdot,0)-g(\cdot,0)||_{L^\infty}\leq Cσ$ and $\inf_x |f(\cdot,0)-g(\cdot,0)|\geq cσ.$ It is shown, under stronger assumption on $f(x,0)$ and $g(x,0)$, local existence independent of the parameter $σ$ (with $σ$ small enough). In order to prove such a result, we need to work in analytic spaces.

Three-phase Muskat problem: uniform lifespan with respect to the width of the strip between interfaces

Abstract

We consider the three-phase Muskat problem with different densities and the same viscosities. The lifespan of the solutions with respect to the width of the strip between interfaces is studied. Indeed, the interfaces are parameterized by the graph of two functions and and we impose that and It is shown, under stronger assumption on and , local existence independent of the parameter (with small enough). In order to prove such a result, we need to work in analytic spaces.

Paper Structure

This paper contains 20 sections, 20 theorems, 540 equations.

Table of Contents

  1. Introduction
  2. A brief summary of previous results on two-phase stable case.
  3. Almost-sharp fronts for SQG
  4. Organization of the paper.
  5. Functional setting and main Theorem
  6. Equations for multi-phase Muskat problem.
  7. Preliminaries
  8. Statement of the main result
  9. A priori energy estimate
  10. Decomposition of the most singular terms
  11. Symmetric part and parabolicity
  12. Lower order terms
  13. Anti-symmetric part
  14. Transport terms
  15. Commutators-Safe terms
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $f_0$ and $g_0$ be analytic functions in $H^k_{\gamma_0}$ (see Section preliminaries), $\rho_0<\rho_1<\rho_2$, $0<\sigma<1$, and $\mu_1$, $\mu_2$ be given by (mu). Then there exist small constants $\epsilon_0$, $\epsilon_1>0$, independent of $\sigma$, such that if there exists a unique analytic (in space) solution $(f(x,t), g(x,t))$ to f-g in an interval $t\in [0, T]$, with $T>0$ independent

Theorems & Definitions (36)

  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3: Parabolicity and choice of the unknown
  • Remark 2.4
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['summary3.1']}
  • Lemma 3.2
  • ...and 26 more