  We study a fundamental model in fluid mechanics--the 3D gravity water wave
equation, in which an incompressible fluid occupying half the 3D space flows
under its own gravity. In this paper we show long-term regularity of solutions
whose initial data is small but not localized. Our results include: almost
global wellposedness for unweighted Sobolev initial data and global
wellposedness for weighted Sobolev initial data with weight $|x|^\alpha$, for
any $\alpha > 0$. In the periodic case, if the initial data lives on an $R$ by
$R$ torus, and $\epsilon$ close to the constant solution, then the life span of
the solution is at least $R/(\epsilon^2(\log R)^2)$.
