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Global solutions for the gravity water waves equation in dimension 3. (English. Abridged French version) Zbl 1177.35168

Summary: 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 \(L^\infty\). To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B45 A priori estimates in context of PDEs
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