Global well-posedness of arbitrarily large Lipschitz solutions for the Muskat problem with surface tension

TL;DR

This paper establishes the global existence and uniqueness of solutions for the 2D Muskat problem with surface tension, even when the initial interface slope is arbitrarily large, provided the data is small in a critical space.

Contribution

It proves global well-posedness for the 2D Muskat problem with surface tension for large Lipschitz initial data, extending previous results to arbitrarily large slopes.

Findings

Unique global solutions exist for large Lipschitz initial data.

Solutions are well-posed for initial data small in a critical space.

The interface slope can be arbitrarily large without losing well-posedness.

Abstract

We prove a global well-posedness result for the 2D Muskat problem with surface tension. Given any regular enough initial data which is small in some critical space but possibly large in Lipschitz, we prove that the associated Cauchy problem has a unique global solution. Our result allows for the slope of the interface between the two fluids to be arbitrarily large.