Global solutions for the gravity water waves equation in dimension 3

Abstract

We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L 2 related norms, with dispersive estimates, which give decay in L ∞ . To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.

Global Infinite Energy Solutions for the 2D Gravity Water Waves System

“…On the long-time behavior side, we start with the breakthrough work of Wu [34], where she proved almost global existence for the 2D gravity water waves system for small initial data; then Germain-Masmoudi-Shatah [14] and Wu [35] proved global existence for the 3D gravity water waves system for small initial data. When the surface tension effect is considered but the gravity effect is neglected (the so-called capillary waves system), Germain-Masmoudi-Shatah [15] proved global existence of the 3D capillary waves system for small initial data.…”

On the free surface Navier-Stokes equation in the inviscid limit

“…More recent results have been obtained by using other approaches [11,33]. Note that much more can be said when u is assumed in addition to be irrotationnal (we obtain the famous water-waves system), we refer for example to [38], [25], [39], [13] and the talk by Nicolas Burq. Nevertheless, note that irrotational solutions are not interesting for our problem since in the context of the Navier-Stokes equation, vorticity on the boundary is automatically created.…”

Justification of the Nonlinear Schrödinger Equation for the Evolution of Gravity Driven 2D Surface Water Waves in a Canal of Finite Depth

“…For a third formulation of the water wave problem in which the top surface is parametrized by arc length a local existence and uniqueness theorem has been shown in [3][4][5]. Recently, various almost global and global existence results have been proven in [1,[12][13][14][15][16]18,40,41]. The existence and uniqueness theorems for the water wave problem can be distinguished according to whether or not one considers the two dimensional or three dimensional problem, finite or infinite depth, with or without surface tension, regularity of the initial conditions and the coordinates which have been chosen to formulate the problem.…”

“…Recently, the nature and effects of resonances in the water wave problem have also been examined for the 2D water wave problem by Wu [40] and for the three dimensional water wave problem in by Germain et al [12] and with an alternative method by Wu [41] in establishing (almost) global existence results in case of infinite depth, that is, ω 2 = |k|. However, due to the different goal in [12] the normal-form transform does not have to be inverted and the loss of regularity occurs in such a way that the local existence method of the untransformed system still can be used.…”

“…However, due to the different goal in [12] the normal-form transform does not have to be inverted and the loss of regularity occurs in such a way that the local existence method of the untransformed system still can be used.…”