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ON THE WATER WAVES EQUATIONS WITH SURFACE T. ALAZARD, N. BURQ, AND C. ZUILY
 

Summary: ON THE WATER WAVES EQUATIONS WITH SURFACE
TENSION
T. ALAZARD, N. BURQ, AND C. ZUILY
Abstract. The purpose of this article is to clarify the Cauchy theory
of the water waves equations 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). Our main
result is that, following the approach developed in [1], after suitable par-
alinearizations, the system can be arranged into an explicit symmetric
system of Schr¨odinger type. We then show that the smoothing effect
for the (one dimensional) surface tension water waves is in fact a rather
direct consequence of this reduction, and following this approach, we are
able to obtain a sharp result in terms of regularity of the indexes of the
initial data, and weights in the estimates.
Contents
1. Introduction 1
2. The Dirichlet-Neumann operator 5
3. Paralinearization 12
4. Symmetrization 26
5. A priori estimates 36

  

Source: Alazard, Thomas - Département de Mathématiques, Université de Paris-Sud 11
Frey, Pascal - Laboratoire Jacques-Louis Lions, Université Pierre-et-Marie-Curie, Paris 6

 

Collections: Mathematics