Large time existence for $3D$ water-waves and asymptotics

Borys Alvarez-Samaniego (1) , David Lannes (1)

1 IMB - Institut de Mathématiques de Bordeaux

We rigorously justify in $3D$ the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves 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 the 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.

Mots clés: water-waves; Boussinesq; KP; Shallow-water; Green-Naghdi equations; Serre equations; Dirichlet-Neumann operator

Domaines: Equations aux dérivées partielles [math.AP]

Borys Alvarez-Samaniego, David Lannes. Large time existence for $3D$ water-waves and asymptotics. 2007. ⟨hal-00128402v2⟩