Long-term regularity of 2D gravity water waves

The two dimensional gravity water wave problem concerns the motion of an incompressible fluid occupying half the 2D space and flowing under its own gravity. In this paper we study long-term regularity of solutions evolving from small but non-localized initial data. Our main result is that if the $H^s$ norm of the initial data is $\epsilon$, where $s\gg 1$ and $\epsilon \ll 1$, then the equation is wellposed at least for a time proportional to ${\epsilon }^{-4}$, improving on the ${\epsilon }^{-3}$ lifespan obtained in \cite{BeFePu,Wu2DL}. We also study period water waves and show a lifespan bridging the gap between non-periodic waves and waves with a period of 1.