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

TL;DR

This paper establishes local well-posedness and global existence results for the Muskat equation with critical initial data in Lipschitz spaces, advancing understanding of its well-posedness under specific regularity conditions.

Contribution

It proves local well-posedness for the Muskat equation with critical Lipschitz initial data and global existence under smallness assumptions, extending previous results.

Findings

Local well-posedness in critical Lipschitz space

Global existence for small initial data

Advancement in understanding Muskat equation regularity

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.