Long-term regularity of 3D gravity water waves

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 $|x|^α$, for any $α> 0$. In the periodic case, if the initial data lives on an $R$ by $R$ torus, and $ε$ close to the constant solution, then the life span of the solution is at least $R/(ε^2(\log R)^2)$.

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