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 (Invent Math 177(1):45–135, 2009). We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case (Germain et al., Ann Math 175(2):691–754, 2012; Wu, Invent Math 184(1):125–220, 2011). 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.
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