Long time regularity for 3d gravity waves with vorticity

We consider the Cauchy problem for the full free boundary Euler equations in $3$d with an initial small velocity of size $O(\epsilon_0)$, in a moving domain which is initially an $O(\epsilon_0)$ perturbation of a flat interface. We assume that the initial vorticity is of size $O(\epsilon_1)$ and prove a regularity result up to times of the order $\epsilon_1^{-1+}$, independent of $\epsilon_0$. A key part of our proof is a normal form type argument for the vorticity equation; this needs to be performed in the full three dimensional domain and is necessary to effectively remove the irrotational components from the quadratic stretching terms and uniformly control the vorticity. Another difficulty is to obtain sharp decay for the irrotational component of the velocity and the interface; to do this we perform a dispersive analysis on the boundary equations, which are forced by a singular contribution from the rotational component of the velocity. As a corollary of our result, when $\epsilon_1$ goes to zero we recover the celebrated global regularity results of Wu (Invent. Math. 2012) and Germain, Masmoudi and Shatah (Ann. of Math. 2013) in the irrotational case.