# Quadratic life span of periodic gravity-capillary water waves

## TL;DR

This paper rigorously reduces the gravity-capillary water waves equations to Birkhoff normal form, demonstrating that small initial data solutions persist for long times despite potential resonances and chaotic dynamics.

## Contribution

It provides a Birkhoff normal form reduction for the system and proves long-time existence of solutions for small initial data, accounting for possible resonances.

## Key findings

- Normal form reduction up to cubic degree.
- Long-time stability for small initial data.
- Finite 3-wave resonances and chaotic dynamics possible.

## 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 $\epsilon$ in a sufficiently smooth Sobolev space lead to a solution that remains in an $\epsilon$-ball of the same Sobolev space up to times of order $\epsilon^{-2}$. We exploit that the $3$-waves resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.