## Large time existence for 3D water waves and asymptotics.(English)Zbl 1131.76012

Summary: We rigorously justify in three dimensions the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a nondimensionalized version of water wave equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep water, from small to large surface and bottom variations, and from fully to weakly transverse waves.
The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.

### MSC:

 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics 35Q35 PDEs in connection with fluid mechanics

### Keywords:

asymptotic models; well-posedness; energy bounds
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