Sobolev Estimates for Two Dimensional Gravity Water Waves

The authors' goal in this volume is to apply a normal forms method to estimate the Sobolev norms of the solutions of the water waves equation. They construct a paradifferential change of unknown, without derivatives losses, which eliminates the part of the quadratic terms that bring non zero contributions in a Sobolev energy inequality. The authors' approach is purely Eulerian: they work on the Craig-Sulem-Zakharov formulation of the water waves equation. In addition to these Sobolev estimates, the authors also prove $L^2$-estimates for the $\\partial_{{x}}^{{^\\alpha}}Z^{{\\beta}}$-derivatives of the solutions of the water waves equation, where $Z$ is the Klainerman vector field $t\\partial_{{t}}+2x\\partial_{{x}}$. These estimates are used in one of the book's references. In that reference, the authors prove a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data, and they obtain an asymptotic description in physical coordinates of the solution, which shows that modified scattering holds. The proof of this global in time existence result relies on the simultaneous bootstrap of some Holder and Sobolev a priori estimtes for the action of iterated Klainerman vector fields on the solutions of the water waves equation. The present volume contains the proof of the Sobolev part of that bootstrap.