M. Berti, R. Feola, L. Franzoi
Quadratic life span
of periodic gravity-capillary water waves
(496K, PDF)
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 times of order \epsilon^{-2}. We exploit that the 3-waves resonances are finitely
many, and the Hamiltonian nature of the Birkhoff normal form.