Long-time existence for multi-dimensional periodic water waves

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 $\epsilon$, smooth solutions exist up to times of the order of ${\epsilon }^{-5/3+}$, 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.

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References in corpus (3)

• Time quasi-periodic gravity water waves in finite depth
• Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions
• Recent advances on the global regularity for irrotational water waves

Cited by in corpus (5)

• Long-time existence for semi-linear beam equations on irrational tori
• Long-time stability of the quantum hydrodynamic system on irrational tori
• Enhanced existence time of solutions to evolution equations of Whitham type
• Long time solutions for quasi-linear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori
• Justification of Peregrine soliton from full water waves