On the Two-Dimensional Muskat Problem with Monotone Large Initial Data

TL;DR

This paper proves the global existence of weak solutions for the two-dimensional Muskat problem with large, monotone initial data, using a maximum principle and well-posedness results.

Contribution

It introduces new techniques to establish global solutions for large initial data in the Muskat problem, extending previous results.

Findings

Global existence of weak solutions for large initial data

Development of a new maximum principle for the derivative

Well-posedness results with specific asymptotics

Abstract

We consider the evolution of two incompressible, immiscible fluids with different densities in porous media, known as the Muskat problem [21], which in two dimensions is analogous to the Hele-Shaw cell [26]. We establish, for a class of large and monotone initial data, the global existence of weak solutions. The proof is based on a local well-posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function.