June 25, 2018

Fields Institute Seminar

Gerard Iooss,
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 many resonances).

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.