Ecole d'hiver « Dynamics and PDE's »,

Saint-Etienne de Tinée, 2-7 Fevrier 2014

Mini-cours :

Enno Lenzman : "New Tools for Nonlocal Elliptic Problems 1"

Enno Lenzman : "New Tools for Nonlocal Elliptic Problems 2"

Uniqueness (modulo symmetries) for optimizers of nonlocal elliptic

problems plays a central role in the classification, stability and blowup

analysis of nonlinear dispersive equations with nonlocal dispersion (e.g.

generalized Benjamin-Ono equation).

In this mini-course, I will start with a brief review of the seminal work

initiated by Amick, Toland, Lieb, and Weinstein in this area. Followed by

this, I will discuss my recent joint work with Frank and Silvestre, which

brings in a set of new tools to settle uniqueness problems for linear and

nonlinear elliptic problems with nonlocal operators (e.g. the fractional

Laplacian). More specifically, I plan to cover the following topics:

1) Topological bounds by harmonic extensions

2) Hamiltonian estimates and local monotonicity formulae for radial

nonlocal problems

3) Nonlinear continuation techniques

The level of the course is intended to be broadly accessible by

non-experts.

Vadim Kaloshin : « The Arnold diffusion via normally hyperbolic invariant cylinders. »

The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff,

says that a typical Hamiltonian system on a typical energy surface

has a dense orbit. This question is wide open. In the early sixties,

Arnold constructed a nearly integrable Hamiltonian system presenting

instabilities and he conjectured that such instabilities existed in typical

nearly integrable Hamiltonian systems.

Key ideas of a recent proof of Arnol'd conjecture for two and a half

degrees will be explained. The lectures are based on a joint work with

Bernard-Zhang, and another with Zhang. The approach is based on

constructing a net of normally hyperbolic invariant cylinders and a version

of Mather variational method.

Raphaël Krikorian : « Quasiperiodic Schrödinger operators: spectral and dynamical aspects »

The aim of this mini-course is to study the spectral properties of 1D Schrödinger operators with quasiperiodic potentials. In this regard, an important tool is to study the dynamics of the related so-called Schrödinger cocycles. This approach proved to be very useful these last two decades and led recently to some spectacular results obtained in particular by Artur Avila.

The plan of the course should be the following:

I will recall some basic facts on the spectral theory of dynamically defined Schrödinger operators and introduce the fundamental notion of density of states. I will then define the dynamical notions of quasiperiodic cocycle, rotation number, Lyapunov exponent, hyperbolicity (uniform or not) and describe some links between these dynamical concepts and their spectral counterparts.I will explain how the reducibility theory of quasiperiodic cocycles (based on KAM theory) is important in understanding the absolutely continuous part of the spectrum and how non-uniform hyperbolicity is related to the pure point part of the spectrum. I will put some special emphasis on the almost-Mathieu operator.

Yvan Martel : "On multi-solitons for the quartic generalizedKorteweg-de Vries equation."

Exposés :

Pietro Baldi : "Gravity Capillary Standing Water Waves."

We consider the water waves problem in Zakharov Hamiltonian formulation

for an infinitely deep 2-dimensional ocean with gravity and capillarity.

We construct small amplitude, standing solutions of Sobolev regularity

(where "standing" means periodic in time and space, and not travelling).

The result holds for values of the surface tension coefficient in a set of full Lebesgue measure, and for values of the time-frequency in a set of asymptotically full Lebesgue measure (because of small divisors).

The main ingredient of the proof are:

Nash-Moser scheme, bifurcation analysis, and pseudo-differential calculus.

This is a joint work with T.Alazard.

Massimiliano Berti: "KAM for quasi-linear KdV".

We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear autonomous Hamiltonian and reversible perturbations of KdV. This is a joint work with P. Baldi and R. Montalto.

Guan Huang : "Averaging for nonlinear PDEs"

Thomas Kappeler:"On normalized differentials on hyperelliptic curves of infinite genus."

With a view towards applications to the focusing NLS equation

we develop a new approach for constructing normalized holomorphic

differentials on hyperelliptic curves of infinite genus and obtain uniform

asymptotic estimates for their zeroes. This is joint work with Peter Topalov.

Eric
Lombardi :"Homoclinic
orbits with many loops near a 0^{2}
iω resonant fixed point of Hamiltonian systems."

In this talk we study the dynamics near
the equilibrium point of a family of Hamiltonian systems in the
neighborhood of a 0^{2} iω resonance. The existence of
a family of periodic orbits surrounding the equilibrium is well-known
and we show here the existence of homoclinic connections with several
loops for every periodic orbit close to the origin, except the origin
itself. The same problem was studied before for reversible non
Hamiltonian vector fields, and the splitting of the homoclinic orbits
lead to exponentially small terms which prevent the existence of
homoclinic connections with one loop to exponentially small periodic
orbits. The same phenomenon occurs here but we get round this
difficulty thanks to geometric arguments specific to Hamiltonian
systems and by studying homoclinic

orbits with many loops. This a joined work with Patrick Bernard and Tiphaine Jezequel.

Peter De Maesschalck : "Transitory canard cycles"

Geneviève Raugel : « Dynamique de l'équation de Klein-Gordon amortie en dimension trois d'espace- Dynamics of the damped Klein-Gordon equation in the three-dimensional case »

Joint work with N. Burq and W. Schlag.

In this talk, we consider the damped focusing Klein-Gordon equation in the radial case in $R^3$. After having given some classical results such as the existence of ground-states, the global dynamics below the ground state energy, etc.., we quickly recall the nice recent results of K. Nakanishi and W. Schlag for the undamped Klein-Gordon equation in the radial case. Using invariant manifold theory together with variational arguments, these authors have described the global dynamics for initial data with energy slightly above the ground state energy. Applying similar techniques and also the existence of foliations, we extend their results to the damped equation. In this case, we describe the dynamics for initial data with energy above the ground state energy and all the way below the energy of all non-zero bound states.