Long-time existence for multi-dimensional periodic water waves

TL;DR

This paper proves a long-time existence result for multi-dimensional periodic water waves with small initial data, overcoming difficulties posed by weak small divisors and resonances using structural and normal form techniques.

Contribution

It provides the first extended lifespan result for quasilinear equations with non-trivial resonances on a multi-dimensional torus.

Findings

Solutions exist up to times of order ^{-5/3+} for small initial data.

Utilizes Hamiltonian structure and time-reversibility to handle resonances.

Employs normal form transformations and sharp small divisor bounds.

Abstract

We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of $\varepsilon^{-5/3+}$, for almost all values of the gravity and surface tension parameters. Besides the quasilinear nature of the equations, the main difficulty is to handle the weak small divisors bounds for quadratic and cubic interactions, growing with the size of the largest frequency. To overcome this difficulty we use (1) the (Hamiltonian) structure of the equations which gives additional smoothing close to the resonant hypersurfaces, (2) another structural property, connected to time-reversibility, that allows us to handle "trivial" cubic resonances, (3) sharp small divisors lower bounds on three and four-waves modulation functions based on counting arguments, and (4) partial normal form transformations and symmetrization arguments in the Fourier space. Our theorem appears to be the first extended lifespan result for quasilinear equations with non-trivial resonances on a multi-dimensional torus.