I will present an almost global in time existence result of small amplitude 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 and any surface tension belonging to a full measure set. The proof demands a Hamiltonian paradifferential Birkhoff normal form reduction for quasi-linear PDEs in presence of resonant wave interactions: the normal form may be not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. A major difficulty is that usual paradifferential calculus used to prove local well posedness (as the celebrated Alinhac good unknown) does not preserve the Hamiltonian structure. A major novelty of this paper is to develop an algorithmic perturbative procedure à la Darboux to correct usual paradifferential transformations to symplectic maps, up to an arbitrary degree of homogeneity. The symplectic correctors turn out to be smoothing perturbations of the identity, and therefore only slightly modify the paradifferential structure of the equations. The Darboux procedure which recovers the nonlinear Hamiltonian structure is written in an abstract functional setting, in order to be applicable also in other contexts. This is a joint work with Maspero and Murgante.
The talk will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487
Note the unusual start time.