Global wellposedness of the 3-D full water wave problem

Abstract

Abstract. We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate 1/t.

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.…”

Bilinear Dispersive Estimates Via Space Time Resonances, Dimensions Two and Three

“…It was then successfully applied by the same authors to the three-dimensional waterwaves equation [10,11], see also Wu [20]. Very recently, related ideas were brought to bear on the more difficult one-dimensional water-waves equation by Ionescu and Pusateri [14] and Alazard and Delort [1,2].…”

“…Our interest lies in stability of the equilibrium state given by (h, ψ) = (0, 0). It was established in [10] [20] in the case g > 0, σ = 0, and in [11] in the case g = 0, σ > 0; to be more specific, it is proved in these papers that, for data sufficiently close to (0, 0) in a sufficiently strong topology, the resulting solution is global, and scatters in an L 2 -type space as t → ∞.…”

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

“…This difficulty already arises for local existence results, and was solved initially by Nalimov [37] and Wu [43,44]. For long time existence problems, Wu [46] uses arguments combining the Eulerian and Lagrangian formulations of the system. Our approach in [5] is purely Eulerian.…”

“…In her breakthrough result [45], Wu proved that the maximal time of existence T ε is larger or equal to e c/ε for d = 1. Then Germain-Masmoudi-Shatah [22] and Wu [46] have shown that the Cauchy problem for three-dimensional waves is globally in time well-posed for ε small enough (with linear scattering in Germain-Masmoudi-Shatah and no assumption about the decay to 0 at spatial infinity of |D x | 1 2 ψ in Wu). Germain-Masmoudi-Shatah recently proved global existence for pure capillary waves in dimension d = 2 in [21].…”