Hamiltonian PDEs in fluid dynamics: local and long time existence results

The most important result which we obtained in the thesis concern about long time existence of water waves. In particular we prove 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, any surface tension belonging to a full measure set and for any small and smooth enough initial datum.