| | |
Summary: Paralinearization of the Dirichlet to Neumann
operator, and regularity of three-dimensional
water waves
Thomas Alazard Guy M´etivier
Abstract
This paper is concerned with a priori C
regularity for three-
dimensional doubly periodic travelling gravity waves whose fundamen-
tal domain is a symmetric diamond. The existence of such waves was
a long standing open problem solved recently by Iooss and Plotnikov.
The main difficulty is that, unlike conventional free boundary prob-
lems, the reduced boundary system is not elliptic for three-dimensional
pure gravity waves, which leads to small divisors problems. Our main
result asserts that sufficiently smooth diamond waves which satisfy a
diophantine condition are automatically C
. In particular, we prove
that the solutions defined by Iooss and Plotnikov are C
. Two no-
table technical aspects are that (i) no smallness condition is required
and (ii) we obtain an exact paralinearization formula for the Dirichlet
|
