[Solutions globales pour les équations des ondes de surface en dimension 3]
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 , avec des estimations dispersives, qui donnent la décroissance dans . 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 related norms, with dispersive estimates, which give decay in . 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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@article{CRMATH_2009__347_15-16_897_0,
author = {Germain, P. and Masmoudi, Nader and Shatah, Jalal},
title = {Global solutions for the gravity water waves equation in dimension 3},
journal = {Comptes Rendus. Math\'ematique},
pages = {897--902},
publisher = {Elsevier},
volume = {347},
number = {15-16},
year = {2009},
doi = {10.1016/j.crma.2009.05.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.05.005/}
}
TY - JOUR AU - Germain, P. AU - Masmoudi, Nader AU - Shatah, Jalal TI - Global solutions for the gravity water waves equation in dimension 3 JO - Comptes Rendus. Mathématique PY - 2009 SP - 897 EP - 902 VL - 347 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.05.005/ DO - 10.1016/j.crma.2009.05.005 LA - en ID - CRMATH_2009__347_15-16_897_0 ER -
%0 Journal Article %A Germain, P. %A Masmoudi, Nader %A Shatah, Jalal %T Global solutions for the gravity water waves equation in dimension 3 %J Comptes Rendus. Mathématique %D 2009 %P 897-902 %V 347 %N 15-16 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.05.005/ %R 10.1016/j.crma.2009.05.005 %G en %F CRMATH_2009__347_15-16_897_0
Germain, P.; Masmoudi, Nader; Shatah, Jalal. Global solutions for the gravity water waves equation in dimension 3. Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 897-902. doi : 10.1016/j.crma.2009.05.005. http://www.numdam.org/articles/10.1016/j.crma.2009.05.005/
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