Global solutions for the gravity water waves equation in dimension 3

We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L 2 related norms, with dispersive estimates, which give decay in L 8 . To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.

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