On the Cauchy problem for the Muskat equation. II: Critical initial data

Abstract

We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in $L^2$. Moreover, we prove that the solution exists globally in time under a smallness assumption.

This is a sequel to our previous paper arXiv:2009.04343