Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence

We prove an almost global existence result for space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth, a full measure set of surface tensions, and any small and smooth enough initial datum. The proof demands a novel approach—that we call paradifferential Hamiltonian Birkhoff normal form for quasi-linear PDEs—in presence of resonant wave interactions: the normal form is not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) breaks the Hamiltonian structure. A major achievement of this paper is to correct (possibly) unbounded paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. Thanks to a deep cancellation, our symplectic correctors are smoothing perturbations of the identity. Thus we are able to preserve both the paradifferential structure and the Hamiltonian nature of the equations. Such Darboux procedure is written in an abstract functional setting applicable also in other contexts.