Quadratic life span of periodic gravity-capillary water waves

Abstract

We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of 3-waves resonances for general values of gravity, surface tension and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton-ripples). Nevertheless we prove that for all the values of gravity, surface tension and depth, initial data that are of size $\epsilon$ in a sufficiently smooth Sobolev space lead to a solution that remains in an $\epsilon$-ball of the same Sobolev space up to times of order ${\epsilon }^{-2}$. We exploit that the $3$-waves resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.