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Global well-posedness for the three dimensional Muskat problem in the critical Sobolev space

Francisco Gancedo, Omar Lazar

Abstract

We prove that the 3D stable Muskat problem is globally well-posed in the critical Sobolev space $\dot H^2 \cap \dot W^{1,\infty}$ provided that the semi-norm $\Vert f_0 \Vert_{\dot H^{2}}$ is small enough. Consequently, this allows the Lipschitz semi-norm to be arbitrarily large. The proof is based on a new formulation of the 3D Muskat problem that allows to capture the hidden oscillatory nature of the problem. The latter formulation allows to prove the $\dot H^{2}$ {\emph{a priori}} estimates. In the literature, all the known global existence results for the 3D Muskat problem are for small slopes (less than 1). This is the first arbitrary large slope theorem for the 3D stable Muskat problem.

Global well-posedness for the three dimensional Muskat problem in the critical Sobolev space

Abstract

We prove that the 3D stable Muskat problem is globally well-posed in the critical Sobolev space provided that the semi-norm is small enough. Consequently, this allows the Lipschitz semi-norm to be arbitrarily large. The proof is based on a new formulation of the 3D Muskat problem that allows to capture the hidden oscillatory nature of the problem. The latter formulation allows to prove the {\emph{a priori}} estimates. In the literature, all the known global existence results for the 3D Muskat problem are for small slopes (less than 1). This is the first arbitrary large slope theorem for the 3D stable Muskat problem.

Paper Structure

This paper contains 37 sections, 16 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. A new formulation of the 3D Muskat problem
  4. Functional setting and notations
  5. A priori estimates in $\dot H^2$
  6. Estimates of the less singular term: $\mathcal{T}=\sum_{i=1}^{5} \mathcal{T}_i$
  7. Estimate of $\mathcal{T}_1$
  8. Estimate of $\mathcal{T}_2$
  9. Estimate of $\mathcal{T}_{3}$
  10. Estimate of $\mathcal{T}_{4}$
  11. Estimate of $\mathcal{T}_{5}$
  12. Symmetrization and useful identities
  13. Estimates of the most singular term : $\mathcal{S}=\sum_{i=1}^4 S_i$
  14. Algebraic decomposition of the most singular term : $\mathcal{S}$
  15. Estimate of $\mathcal{S}_{1}$
  16. ...and 22 more sections

Key Result

Theorem \oldthetheorem

Let $\mathcal{F}(x)=C({1+x^{2}})^{-3/2}$, where $C>0$ is a fixed constant. For any initial data $f_{0} \in \dot H^{2} \cap \dot W^{1,\infty}$ with $\Vert f_{0} \Vert_{\dot H^{2}}< \mathcal{F}(\Vert f_{0}\Vert_{\dot W^{1,\infty}})$ small enough, then, there exists a unique global solution $f$ to the

Theorems & Definitions (20)

  • Theorem \oldthetheorem
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition \oldthetheorem
  • Lemma 6.1
  • Lemma 7.1
  • Lemma 7.2
  • Lemma 7.3
  • Lemma 8.1
  • ...and 10 more