Global well-posedness for the one-phase Muskat problem in 3D

TL;DR

This paper proves global well-posedness and exponential decay of solutions for the three-dimensional one-phase Muskat problem with Lipschitz initial data, extending results previously known only in two dimensions.

Contribution

It develops a new approach for 3D Muskat problem using Dahlberg-Kenig regularity and layer potential techniques, addressing challenges unique to three dimensions.

Findings

Unique global-in-time solutions exist for Lipschitz initial data.

Hölder norms of solutions decay exponentially over time.

Extends 2D results to 3D with new analytical tools.

Abstract

This paper is concerned with the long time dynamics of the free boundary of a Darcy fluid in three space dimensions, also known as the one-phase Muskat problem. The dynamics of the free boundary is governed by a nonlocal fully nonlinear parabolic partial differential equation. It is proven that for any periodic Lipschitz graph given as initial data, the problem has a unique global-in-time solution which satisfies the equation in the strong sense. Moreover, all H\"older norms of the solution decay exponentially in time. These results have been previously established in two space dimensions. This paper addresses new challenges to extend the results to the more difficult three dimensional setting. The approach developed is critical in three space dimensions and crucially relies on Dahlberg-Kenig's $W^{1, 2+\varepsilon}$ optimal regularity for layer potentials together with delicate structures of the Dirichlet-to-Neumann operator and layer potentials in Lipschitz domains of $\mathbb{T}^2\times \mathbb{R}$.