Large time existence for 3D water-waves and asymptotics

Abstract

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.Comment: Revised version of arXiv:math.AP/0702015 (notations simplified and remarks added) To appear in Inventione

About global existence and asymptotic behavior for two dimensional gravity water waves

“…Moreover, one may prove as well tame estimates, that complement similar results due to Craig, Schanz and Sulem (see [18] and [42,Chapter 11], and [8,28]), and establish bounds for the approximation of G(η)ψ (resp. F (η)ψ) by its Taylor expansion at order two G ≤2 (η)ψ (resp.…”

Variable depth KdV equations and generalizations to more nonlinear regimes

“…Making assumptions on the size of ε, β, α, and μ one is led to derive (simpler) asymptotic models from (1.3). In the shallow-water scaling (μ 1), one can derive (when no smallness assumption is made on ε, β and α) the so-called Green-Naghdi equations (see [14,25] for a derivation and [2] for a rigorous justification). For one-dimensional surfaces and over uneven bottoms these equations couple the free surface elevation ζ to the vertically averaged horizontal component of the velocity,…”

“…At this point, a classical method is to choose an asymptotic regime, in which we look for approximate models and hence for approximate solutions. More recently Alvarez-Samaniego and Lannes [2] rigorously justified the relevance of the main asymptotical models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations (GN), Serre approximation, full-dispersion model and deep-water equations. Some of these models capture the existence of solitary water waves and the associated phenomenon of soliton manifestation [17].…”

“…It is known (see [2]) that the Green-Naghdi equations give, under the scaling (1.9) with β = O(ε) a correct approximation of the exact solutions of the full water waves equations (with a precision O(μ 2 t) and…”

Experimental study of the particle paths in solitary water waves

“…Korteweg & de Vries [3] derived a theory for irrotational solitary shallow water waves of small amplitude. Recently, Alvarez-Samaniego & Lannes [4] discussed the various regimes suitable for the modelling of water waves. In contrast to the case of periodic travelling waves, for solitary waves, it is known that the method of linearization is not appropriate, even for waves of small amplitude [5,6].…”