Global solutions for the gravity water waves system in 2d

Abstract

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu [Wu09]. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case [GMS12a,Wu11]. In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2d.

Two dimensional water waves in holomorphic coordinates II: global solutions

“…Here we improve the result in (iii) to a global statement, drastically improving and simplifying the earlier results of Alazard-Delort [3] and by Ionescu-Pusateri [14].…”

“…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]. Extensive work was also done on the same problem in three or higher space dimensions, and also on related problems with surface tension, vorticity, finite bottom, etc.…”

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

“…Until very recently, we had several results. Global existence of the 2D gravity water waves for small initial data was first proved by Ionescu-Pusateri [21], and a similar result was proved independently by Alazard-Delort [3] in Eulerian coordinates. More recently Hunter-Ifrim-Tataru [16] used holomorphic coordinates to give a different proof of the almost global existence result; then Ifrim-Tataru [19] extended it to global existence in holomorphic coordinates.…”

“…For a nonneutral Schwartz physical velocity v, the restricted velocity potential behaves like the antiderivative of v, which belongs to L 1 : Remark 1.4. The property of the modified scattering in the infinite energy setting is the same as the small energy setting in [21,23]. We first describe the modified scattering property and then give a comment.…”

“…The Z-normed space in (4.3) is similar but much weaker than the one used in [21]. Theˇused there is 1 100 .…”