  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
$\mathbb{T}^2\times \mathbb{R}$.
