Global existence for the confined Muskat problem

TL;DR

This paper proves the global existence of Lipschitz continuous solutions for the stable confined Muskat problem with small initial data, showing bounded amplitude and slope over time, extending previous infinite-depth results.

Contribution

It establishes global existence for the confined Muskat problem with small initial data, allowing for bounded slope growth, thus generalizing prior infinite-depth findings.

Findings

Amplitude remains uniformly bounded over time

Slope remains bounded, even if it grows

Results extend to finite depth scenarios

Abstract

In this paper we show global existence of Lipschitz continuous solution for the stable Muskat problem with finite depth (confined) and initial data satisfying some smallness conditions relating the amplitude, the slope and the depth. The cornerstone of the argument is that, for these \emph{small} initial data, both the amplitude and the slope remain uniformly bounded for all positive times. We notice that, for some of these solutions, the slope can grow but it remains bounded. This is very different from the infinite deep case, where the slope of the solutions satisfy a maximum principle. Our work generalizes a previous result where the depth is infinite.