Comptes Rendus
no. 15-16
Partial Differential Equations
Global solutions for the gravity water waves equation in dimension 3
[Solutions globales pour les équations des ondes de surface en dimension 3]

Présenté par : Gilles Lebeau

P. Germain1 ; Nader Masmoudi1 ; Jalal Shatah1
1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012-1185, USA
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 897-902.

Nous montrons l'existence de solutions globales pour les équations des ondes de surface en dimension 3 avec gravité seulement, dans le cas de petites données initiales. La preuve combine des estimations d'énergie, qui donnent le contrôle de normes de type L2, avec des estimations dispersives, qui donnent la décroissance dans L∞. Ces estimations dispersives sont obtenues grâce à une analyse dans l'espace de Fourier, qui repose sur l'étude des résonances en temps et en espace.

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 L2 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.05.005
P. Germain 1 ; Nader Masmoudi 1 ; Jalal Shatah 1

1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012-1185, USA 
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P. Germain; Nader Masmoudi; Jalal Shatah. Global solutions for the gravity water waves equation in dimension 3. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 897-902. doi : 10.1016/j.crma.2009.05.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.005/

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