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.
Berti, M., Feola, R., & Franzoi, L. (2021). Quadratic Life Span of Periodic Gravity-capillary Water Waves. WATER WAVES, 3(1), 85-115 [10.1007/s42286-020-00036-8].
Titolo: | Quadratic Life Span of Periodic Gravity-capillary Water Waves | |
Autori: | ||
Data di pubblicazione: | 2021 | |
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Citazione: | Berti, M., Feola, R., & Franzoi, L. (2021). Quadratic Life Span of Periodic Gravity-capillary Water Waves. WATER WAVES, 3(1), 85-115 [10.1007/s42286-020-00036-8]. | |
Handle: | http://hdl.handle.net/11590/397000 | |
Appare nelle tipologie: | 1.1 Articolo in rivista |