Global solutions and asymptotic behavior for two dimensional gravity water waves (Q2788605)

The authors study the two-dimensional motion of an incompressible, irrotational, inviscid fluid in a domain with free surface and of infinite depth when gravity is accounted for and surface tension is negligible (gravity waves). They obtain a global existence result for the Cauchy problem of the associated Craig-Sulem-Zakharov system, assuming that initial data are sufficiently small with sufficiently strong decay at infinity. An asymptotic description of the solution as time goes to infinity is given.NEWLINENEWLINEThe proof of the global existence result relies on a bootstrap argument of \(L^2\) and \(L^\infty\) estimates for iterated Klainerman vector fields acting on the unknowns. The \(L^2\) estimates are given in the parallel paper [the author, Sobolev estimates for two dimensional gravity water waves. Paris: Société Mathématique de France (SMF) (2015; Zbl 1360.35002)].