Le séminaire a lieu le jeudi
17 juillet 2003 de 15h00 à 16h00
dans la salle de réunion
du PHC (Observatoire de Nice).
Standing waves in deep water
Gérard Iooss
INLN (Nice)
This is a joint work with J.Toland and P.Plotnikov.
We consider the classical problem of the two-dimensional
potential flow of time and space periodic gravity waves in an infinitely
deep layer of perfect fluid, with no surface
tension at the free surface. It is well known from linear
theory that there are infinitely many eigenmodes for any rational value
of the unique dimensionless parameter (one says
It was proved only in 1987 by Amick and Toland that an
expansion in power series of the amplitude of a single eigenmode can be
computed at all orders, despite these infinitely
many resonances. This formal result has been now extended
to all possibly bifurcating multi-modal standing waves (G.I. 2002).
The standing wave problem for a finite depth layer was
recently solved by Plotnikov and Toland (2001). In this problem the above
mentioned infinitely many resonances are not
available and that is why the infinite depth problem
is more difficult. We use a formulation of Zakharov leading to a nonlocal
second order PDE. We start with an approximate
solution built as the formal solution above, for insuring
the invertibility of the infinite dimensional bifurcation equation, and
we use the Nash-Moser implicit function theorem.
The major difficulty is to invert the linearized operator
near a non zero point, where we use averaging techniques coming from hyperbolic
equations theory, applied to a suitable
reformulation of the linearized problem, and which keeps
the invertibility of the infinite dimensional bifurcation part. We show
the existence of the standing waves for a
Lebesgue set of values of the parameter (hence containing
at least an infinite sequence of values of the parameter tending to a critical
value).
References:
C.Amick, J.Toland. Proc. Roy. Soc. Lond. A 411 (1987),
123-137.
G.Iooss. J.Math. Fluid Mech. 4 (2002) 155-185.
G.iooss, P.Plotnikov, J.Toland. paper in preparation
(2003)
P.Plotnikov, J.Toland. Arch. Rat. Mech. Anal.159 (2001)
1-83.