On the Cauchy problem for the Muskat equation with non-Lipschitz initial data

TL;DR

This paper investigates the Muskat equation's Cauchy problem with initial data in a critical Sobolev space, establishing the first local and global well-posedness results for non-Lipschitz free surface initial conditions.

Contribution

It provides novel well-posedness results for the Muskat equation with initial data lacking Lipschitz regularity, expanding understanding of the equation's behavior.

Findings

First local well-posedness result for non-Lipschitz initial data

Global well-posedness established under critical Sobolev conditions

Extension of well-posedness theory to rough initial surfaces

Abstract

This article is devoted to the study of the Cauchy problem for the Muskat equation. We consider initial data belonging to the critical Sobolev space of functions with three-half derivative in $L^2$, up to a fractional logarithmic correction. As a corollary, we obtain the first local and global well-posedness results for initial free surface which are not Lipschitz.