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Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces

Helmut Abels, Bogdan-Vasile Matioc

Abstract

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in(1,\infty)$. The Sobolev space $W^s_p(\mathbb{R})$ with $s=1+1/p$ is a critical space for this problem. We prove, for $s\in (1+1/p,2),$ that the Rayleigh-Taylor condition identifies an open subset of $W^s_p(\mathbb{R})$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces

Abstract

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an -setting with . The Sobolev space with is a critical space for this problem. We prove, for that the Rayleigh-Taylor condition identifies an open subset of within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

Paper Structure

This paper contains 5 sections, 24 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On the resolvent set of $\mathbb{A}(f)$
  4. The abstract evolution problem
  5. Preparatory results used in Section \ref{['Sec:4']}

Key Result

Theorem 1

Let $a:\mathbb{R}\to\mathbb{R}$ be continuously differentiable with bounded and Hölder-continuous first derivative. For $f\in{\rm C}^\infty_0(\mathbb{R})$ let Given $p\in (1,\infty)$, the operator $T_a$ has an extension $T_a\in \mathcal{L}(L_p(\mathbb{R}))$ and it holds that The constant $C_p$ depends only on $p$.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1: The Hörmander condition
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 39 more