Title | Quadratic Life Span of Periodic Gravity-capillary Water Waves |

Publication Type | Journal Article |

Year of Publication | 2021 |

Authors | Berti, M, Feola, R, Franzoi, L |

Volume | 3 |

Issue | 1 |

Pagination | 85 - 115 |

Date Published | 2021/04/01 |

ISBN Number | 2523-3688 |

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 $$ \varepsilon $$in a sufficiently smooth Sobolev space leads to a solution that remains in an $$ \varepsilon $$-ball of the same Sobolev space up times of order $$ \varepsilon ^{-2}$$. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form. |

URL | https://doi.org/10.1007/s42286-020-00036-8 |

Short Title | Water Waves |

## Quadratic Life Span of Periodic Gravity-capillary Water Waves

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