[Une réduction paradifférentielle du système des vagues de gravité-capillarité à basse régularité et applications]
We consider in this article the system of gravity-capillary waves in all dimensions and under the Zakharov/Craig-Sulem formulation. Using a paradifferential approach introduced by Alazard-Burq-Zuily, we symmetrize this system into a quasilinear dispersive equation whose principal part is of order . The main novelty, compared to earlier studies, is that this reduction is performed at the Sobolev regularity of quasilinear pdes: with , being the dimension of the free surface.
From this reduction, we deduce a blow-up criterion involving solely the Lipschitz norm of the velocity trace and the -norm of the free surface. Moreover, we obtain an a priori estimate in the -norm and the contraction of the solution map in the -norm using the control of a Strichartz norm. These results have been applied in establishing a local well-posedness theory for non-Lipschitz initial velocity in our companion paper [24].
Dans cet article, nous étudions le système des vagues de gravité-capillarité en toutes dimensions, dans la formulation de Zakharov, Craig et Sulem. À l’aide d’une approche paradifférentielle introduite par Alazard, Burq et Zuily, nous symétrisons ce système en une équation dispersive quasilinéaire dont le terme principal est d’ordre . La principale nouveauté par rapport aux études précédentes est que cette réduction est effectuée au niveau de régularité des EDPs quasilinéaires : avec , étant la dimension de la surface libre. À partir de cette réduction, nous déduisons un critère d’explosion n’impliquant que la norme Lipschitz de la trace de la vitesse et la norme de la surface libre. En outre, nous obtenons une estimation a priori de la norme et la contraction de l’application solution dans la norme , en utilisant le contrôle d’une norme de Strichartz. Ces résultats ont été utilisés pour développer une théorie de Cauchy locale pour des vitesses initiales non Lipschitz, dans le papier compagnon [24].
Révisé le :
Accepté le :
Publié le :
DOI : 10.24033/bsmf.2750
Keywords: Gravity-capillary waves, paradifferential reduction, blow-up criterion, a priori estimate, contraction of the solution map.
Mots-clés : Vagues de gravité-capillarité, réduction paradifférentielle, critère d’explosion, estimations a priori, contraction de l’application solution.
de Poyferré, Thibault 1 ; Nguyen, Quang-Huy 2
@article{BSMF_2017__145_4_643_0,
author = {de Poyferr\'e, Thibault and Nguyen, Quang-Huy},
title = {A {Paradifferential} {Reduction} for the {Gravity-capillary} {Waves} {System} at {Low} {Regularity} and {Applications}},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {643--710},
year = {2017},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {145},
number = {4},
doi = {10.24033/bsmf.2750},
mrnumber = {3770970},
zbl = {1397.35237},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2750/}
}
TY - JOUR AU - de Poyferré, Thibault AU - Nguyen, Quang-Huy TI - A Paradifferential Reduction for the Gravity-capillary Waves System at Low Regularity and Applications JO - Bulletin de la Société Mathématique de France PY - 2017 SP - 643 EP - 710 VL - 145 IS - 4 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2750/ DO - 10.24033/bsmf.2750 LA - en ID - BSMF_2017__145_4_643_0 ER -
%0 Journal Article %A de Poyferré, Thibault %A Nguyen, Quang-Huy %T A Paradifferential Reduction for the Gravity-capillary Waves System at Low Regularity and Applications %J Bulletin de la Société Mathématique de France %D 2017 %P 643-710 %V 145 %N 4 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2750/ %R 10.24033/bsmf.2750 %G en %F BSMF_2017__145_4_643_0
de Poyferré, Thibault; Nguyen, Quang-Huy. A Paradifferential Reduction for the Gravity-capillary Waves System at Low Regularity and Applications. Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 4, pp. 643-710. doi: 10.24033/bsmf.2750
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