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

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.