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

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

“…Note that the same formula has also been derived by Alazard-Delort [3] and IfrimTataru [19]. We comment that the modified scattering property is only a by-product of the L 1 decay estimate.…”

“…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.…”

“…We mentioned that all initial data are assumed to be small at the level of L 2 P H 1=2 in all previous results [3,19,21]. That is to say, kjrj 1=2 k L 2 is not only finite but also small.…”

“…We only briefly introduce and explain the main ideas behind this subsection. More details can be found in Appendix C. Interested readers can refer to [3] for more elaborated details.…”

“…As the system (1.3) lacks symmetries, one fails at the beginning of the energy estimate due to the quasilinear nature. Thanks to the work of Alazard-Métivier [4], the works of Alazard-Burq-Zuily [1,2], and the work of the Alazard-Delort [3], their paralinearization method helps us to see the good structures inside the system (1.3) and to find the good substitution variables U 1 and U 2 . The system of equations satisfied by U 1 and U 2 has requisite symmetries to avoid losing derivatives when doing the energy estimate.…”