Global well-posedness for the 2D stable Muskat problem in $H^{3/2}$

Abstract

We prove a global existence result of a unique strong solution in ${\dot{H}}^{5/2}\cap {\dot{H}}^{3/2}$ with small ${\dot{H}}^{3/2}$ semi-norm for the 2D Muskat problem, hence allowing the interface to have arbitrary large finite slopes and finite energy (thanks to the $L^2$ maximum principle). The proof is based on the use of a new formulation of the Muskat equation that involves oscillatory terms. Then, a careful use of interpolation inequalities in homogeneneous Besov spaces allows us to close the {\emph{a priori}} estimates.

33 pages. Some typos corrected, to appear in Annales scientifiques de l'École normale supérieure