Global solutions for the gravity water waves equation in dimension 3

2012/01/31 by Pierre Germain, Nader Masmoudi, Jilal Shatah · 32 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.4007/annals.2012.175.2.6

Abstract

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 . 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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