### 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.

Original language | English (US) |
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Pages (from-to) | 691-754 |

Number of pages | 64 |

Journal | Annals of Mathematics |

Volume | 175 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2012 |

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Germain, P., Masmoudi, N., & Shatah, J. (2012). Global solutions for the gravity water waves equation in dimension 3.

*Annals of Mathematics*,*175*(2), 691-754. https://doi.org/10.4007/annals.2012.175.2.6