Global solutions for the gravity water waves system in 2d

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 \cite{WuAG}. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case \cite{GMS2,Wu3DWW}. 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.Comment: final version to be publishe

Global regularity for 2d water waves with surface tension

We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method, based on energy estimates and dispersive analysis, which allows us to deal with strong singularities arising from time resonances in the applications of the normal form method and nonlinear scattering. As a result, we are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.Comment: 100 pages. References update