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arXiv · 1905.05424

Quadratic life span of periodic gravity-capillary water waves

Massimiliano Berti·Roberto Feola·Luca Franzoi
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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 $ε$ in a sufficiently smooth Sobolev space lead to a solution that remains in an $ε$-ball of the same Sobolev space up to times of order $ε^{-2}$. We exploit that the $3$-waves resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.

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Massimiliano Berti, Roberto Feola, Luca Franzoi. 2019-05-14. Quadratic life span of periodic gravity-capillary water waves. https://arxiv.org/abs/1905.05424

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