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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 domainwhich is initially an $O(\epsilon_0)$ perturbation of a flat interface. Weassume that the initial vorticity is of size $O(\epsilon_1)$ and prove aregularity 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 thevorticity equation; this needs to be performed in the full three dimensionaldomain and is necessary to effectively remove the irrotational components fromthe quadratic stretching terms and uniformly control the vorticity. Anotherdifficulty is to obtain sharp decay for the irrotational component of thevelocity and the interface; to do this we perform a dispersive analysis on theboundary equations, which are forced by a singular contribution from therotational component of the velocity. As a corollary of our result, when$\epsilon_1$ goes to zero we recover the celebrated global regularity resultsof Wu (Invent. Math. 2012) and Germain, Masmoudi and Shatah (Ann. of Math.2013) in the irrotational case.

Long time regularity for 3d gravity waves with vorticity