19-34 M. Berti, R. Feola, L. Franzoi

Quadratic life span of periodic gravity-capillary water waves (496K, PDF) May 15, 19

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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.

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