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].

Quadratic Life Span of Periodic Gravity-capillary Water Waves

Berti M.;Feola R.;
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.
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].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/397000
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