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On the Cauchy problem for water gravity waves T. Alazard, N. Burq, C. Zuily

Summary: On the Cauchy problem for water gravity waves
T. Alazard, N. Burq, C. Zuily
The purpose of this article is to clarify the Cauchy theory of the gravity water
waves equations (without surface tension) as well in terms of regularity indexes
for the initial conditions as for the smoothness of the bottom of the domain
(namely no regularity assumption is assumed on the bottom). In terms of
Sobolev embeddings, the initial surfaces we consider turn out to be only of C3/2
class and consequently have unbounded curvature. We also take benefit from
our low regularity result and an elementary (though seemingly yet unknown)
observation to solve a question raised by Boussinesq on the water-wave system
in a canal.
Sharp results for the microlocal analysis of the Dirichlet-Neumann operator
in rough domains are proved. After suitable para-linearizations, we show that
the water waves system can be arranged into an explicit symmetric system
of quasilinear waves equation type. As an illustration of this reduction, we
show that in fact following the analysis by Bahouri-Chemin and Tataru for
quasi-linear wave equations, using Strichartz estimates, the regularity threshold
can be further lowered, which allows to solve the water waves system for non
lipschitz initial velocity fields.


Source: Alazard, Thomas - Département de Mathématiques, Université de Paris-Sud 11


Collections: Mathematics