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This paper is devoted to the proof of a global existence result for the waterwaves equation with smooth, small, and decaying at infinity Cauchy data. Weobtain moreover an asymptotic description in physical coordinates of thesolution, which shows that modified scattering holds.  The proof is based on a bootstrap argument involving $L^2$ and $L^\infty$estimates. The $L^2$ bounds are proved in a companion paper of this article.They rely on a normal forms paradifferential method allowing one to obtainenergy estimates on the Eulerian formulation of the water waves equation. Wegive here the proof of the uniform bounds, interpreting the equation in asemi-classical way, and combining Klainerman vector fields with the descriptionof the solution in terms of semi-classical lagrangian distributions. This,together with the $L^2$ estimates of the companion paper, allows us to deduceour main global existence result.

Global solutions and asymptotic behavior for two dimensional gravity water waves