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The Muskat problem with surface tension and equal viscosities in subcritical $L_p$-Sobolev spaces

Anca-Voichita Matioc, Bogdan-Vasile Matioc

Abstract

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $W^s_p(\mathbb{R})$, where ${p\in(1,2]}$ and ${s\in(1+1/p,2)}$. This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $W^{\overline{s}-2}_p(\mathbb{R})$, where ${\overline{s}\in(1+1/p,s)}$. Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

The Muskat problem with surface tension and equal viscosities in subcritical $L_p$-Sobolev spaces

Abstract

In this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces , where and . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in , where . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

Paper Structure

This paper contains 5 sections, 18 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Some singular integral operators
  3. A functional analytic framework for the Muskat problem
  4. Freezing the kernels of singular operators in Sobolev spaces with negative exponent
  5. Estimates for some pointwise multipliers

Key Result

Theorem 1.1

Let $p\in(1,2]$ and $1+1/p<\overline s<s<2$. Then, the Muskat problem P possesses for each $f_0\in W^s_p({\mathbb R})$ a unique maximal solution $f:=f(\,\cdot\,; f_0)$ such that with $T^+=T^+(f_0)\in(0,\infty]$ denoting the maximal time of existence. Moreover, the following properties hold true:

Theorems & Definitions (37)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 27 more