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.