Global well-posedness for the 2D Muskat problem with slope less than 1

Abstract

We prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the product of the maximal and minimal slopes is strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon}$. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Nazarov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation.