Fields Institute Seminar
INLN, 06560 Valbonne, France
Existence of Standing gravity waves in deep water
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, symmetric with respect to
the vertical axis, 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 that there are infinitely
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 order, despite of these infinitely many resonances. Numerical
studies (Bryant-Stiassnie 1994) computed hundreds of terms in the series,
in starting with suitable combinations of two or three eigenmodes.
We are now able to construct infinitely many of such formal expansions
in powers of the amplitude, with a leading order containing suitable
combinations of any finite number of eigenmodes.
The problem of existence of such solutions, corresponding to the above
formal expansions, was open since Stokes (1847). For a finite depth
layer the standing wave problem was recently solved by Plotnikov and
Toland (2001). In such a problem there is not the above degeneracy.
In the present case, we use a formulation of Zakharov leading to a nonlocal
second order PDE. The problem combines several serious difficulties:
infinitely many resonances, highest order derivatives in the nonlinear
terms than in the linear ones. This leads to the need of an appropriate
version of the Nash-Moser implicit function theorem. The major difficulty
is to invert the linearized operator near a non zero point, leading
to a second order hyperbolic PDE with periodic coefficients, nonlocal
in space. Successive changes of variables allow to reformulate this
inversion as a small divisor problem. We show the existence of the standing
waves for a set of values of the amplitude for which 0 is a Lebesgue
point (hence containing at least an infinite sequence of values of the
parameter tending to a critical value).
C.Amick, J.Toland. Proc. Roy. Soc. Lond. A 411 (1987), 123-137.
G.Iooss. J.Math. Fluid Mech. 4 (2002) 155-185.
P.Plotnikov, J.Toland. Arch. Rat. Mech. Anal.159 (2001) 1-83.
G.Iooss, J.Toland, P.Plotnikov. On the standing wave problem with infinite
depth (in preparation).
Yong Jung Kim, The Fields Institute
Asymptotic convergence in fast diffusion
Abstract: In this talk a technique based on the dynamics of Newtonian
potentials is introduced and the optimal convergence to the Barenblatt
solution is obtained for the fast diffusion. Possible application of
this technique to other cases such as porous medium equations or p-laplacian
will be discussed.