Almost global wellposedness of the 2-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.

On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary

“…See also Wu [40] for the proof of the second part of this proposition using these results and the Tb Theorem.…”

Two dimensional water waves in holomorphic coordinates II: global solutions

“…The local in time existence and uniqueness of solutions was proved in [16,22,23], both for finite and infinite depth. Later, Wu [24] proved almost global existence for small localized data. Very recently, global results for small localized data were independently obtained by Alazard-Delort [3] and by Ionescu-Pusateri [14].…”

“…(iii) almost global well-posedness for small localized data, refining and simplifying Wu's approach in [24].…”

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.…”