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 three-wave 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 leads to a solution that remains in an ε-ball of the same Sobolev space up times of order ε−2. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.

Quadratic Life Span of Periodic Gravity-capillary Water Waves / Berti, M.; Feola, R.; Franzoi, L.. - In: WATER WAVES. - ISSN 2523-367X. - 3:1(2021), pp. 85-115. [10.1007/s42286-020-00036-8]

Quadratic Life Span of Periodic Gravity-capillary Water Waves

Berti, M.;Franzoi, L.
2021

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 three-wave 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 leads to a solution that remains in an ε-ball of the same Sobolev space up times of order ε−2. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.
3
1
85
115
10.1007/s42286-020-00036-8
https://arxiv.org/abs/1905.05424
Berti, M.; Feola, R.; Franzoi, L.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/116809
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