Global Well-Posedness for the One-Phase Muskat Problem in 3D

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 Hölder 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 \(\mathbb {T}^2\times \mathbb {R}\).