A paradifferential reduction for the gravity-capillary waves system at low regularity and applicationsReportar como inadecuado




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1 ENS - Ecole Normale Supérieure 2 LM-Orsay - Laboratoire de Mathématiques d-Orsay

Abstract : 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 $3-2$. The main novelty, compared to earlier studies, is that this reduction is performed at the Sobolev regularity of quasilinear pdes: $H^sR^d$ with $s>3-2+d-2$, $d$ 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 $C^{5-2+}$-norm of the free surface. Moreover, we obtain an a priori estimate in the $H^s$-norm and the contraction of the solution map in the $H^{s-3-2}$-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.

Keywords : gravity-capillary waves a priori estimates blow-up criteria paradifferential calculus





Autor: Thibault De Poyferré - Quang-Huy Nguyen -

Fuente: https://hal.archives-ouvertes.fr/



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