Global Well-posedness for the Three Dimensional Muskat Problem in the Critical Sobolev Space View Full Text


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Article Info

DATE

2022-09-01

AUTHORS

Francisco Gancedo, Omar Lazar

ABSTRACT

We prove that the three dimensional stable Muskat problem is globally well-posed in the critical Sobolev space H˙2∩W˙1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}^2 \cap \dot{W}^{1,\infty }$$\end{document} provided that the semi-norm ‖f0‖H˙2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f_0 \Vert _{\dot{H}^{2}}$$\end{document} is small enough. Consequently, this allows the Lipschitz semi-norm to be arbitrarily large. The proof is based on a new formulation of the three dimensional Muskat problem that allows for the capture at the hidden oscillatory nature of the problem. The latter formulation allows to prove the H˙2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}^{2}$$\end{document}a priori estimates. In the literature, all the known global existence results for the three dimensional Muskat problem are for small slopes (less than 1). This is the first arbitrary large slope theorem for the three dimensional stable Muskat problem. More... »

PAGES

141-207

References to SciGraph publications

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  • 2013-04-06. Breakdown of Smoothness for the Muskat Problem in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
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  • 2008-08-08. A Maximum Principle for the Muskat Problem for Fluids with Different Densities in COMMUNICATIONS IN MATHEMATICAL PHYSICS
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  • 2020-03-29. Paralinearization of the Muskat Equation and Application to the Cauchy Problem in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 2021-04-03. On the Cauchy Problem for the Muskat Equation. II: Critical Initial Data in ANNALS OF PDE
  • 2011. Fourier Analysis and Nonlinear Partial Differential Equations in NONE
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