Ecole d'hiver « Dynamics and PDE's »,

Saint-Etienne de Tinée, 3-8 Fevrier 2013

Mini-cours :

Paulo Cordaro : "Analytic and Gevrey regularity for solutions to Hörmander operators"

We shall consider real linear PDE defined by sums of squares of real-analytic vector fields plus first order terms satisfying a finite order condition (Hörmander operators) and study the analytic and Gevrey regularity for their solutions. We will start from the celebrate example of Baouendi-Goulaouic (1972) and will describe many of the known examples and models that appeared since then in order to motivate the introduction of a conjecture stated by F.Treves, which will be studied in detail. We shall also describe the phenomena that appear when hyperfunction solutions are considered as well as the effect of first order terms in the Gevrey regularity.

Pierre Raphael : "Blow for the generalized KdV equation near the ground state: a case study."

I
will consider the L^{2}
critical generalized KdV equation. This equation admits a family of
solitons or solitary waves which are expected to be the smallest
nonlinear waves. I will give a complete description of the flow near
this explicit nonlinear wave. For a suitable class of perturbations,
only three regimes can occur:

-(Blow up): the solution blows up in finite time in a universal way;

- (Soliton): the solution is global in time and behaves like a solitary wave;

- (Exit); the solution leaves a neighborhood of the solitary wave (and is then conjectured to scatter).

I will present the main formal arguments behind the proof of this result, as well as the fundamental new objects behind the proof, and in particular the notion of minimal blow up element. These results correspond to a series of 3 joint papers with Yvan Martel (Ecole Polytechnique) and Frank Merle (Cergy Pontoise and IHES).

Marcel Guardia : "Growth of Sobolev norms for the cubic defocusing NLS"

The study of solutions of Hamiltonian PDEs undergoing growth of Sobolev norms as time tends to infinity has drawn considerable attention in recent years. The importance of growth of Sobolev norms is due to the fact that it implies that the solution transfers energy to higher modes as time evolves.

Consider the cubic defocusing nonlinear Schrödinger equation with periodic boundary conditions and fix s>1. Colliander, Keel, Staffilani, Tao and Takaoka (2010) proved the existence of solutions whose s-Sobolev norm grows in time by any given factor R. One of the key steps of their approach is the study of a finite dimensional toy model which is a good first order for some solutions of the cubic defocusing NLS.

In these lectures we will show how, using Dynamical Systems techniques, one can refine the study of this toy model to obtain solutions with s-Sobolev norm growing in polynomial time in R. I will also explain how these improvements allow us to show that the growth of Sobolev norms can also be attained by solutions of the cubic defocusing NLS with a convolution potential.

Part of the results explained in these lectures are joint work with Vadim Kaloshin.

Michela Procesi :" Normal form for the NLS"

I will discuss the structure of the normal form for the NLS on a torus.

It is well known that in general this normal form is non-integrable and has a rich structure. I will analyze this structure by using ideas and techniques coming from the theory of Cayley graphs and prove that for appropriate choices of the initial data the normal form is an integrable and non-degenerate quadratic form. Then I will show how this form may be integrated explicitly via a non-perturbative change of variables related to the Cayley graph structure. Finally one can reduce this integrable hamiltonian to its canonical form using the theory of classification of

symplectic matrices. In the case of the cubic NLS, due to non-degeneracy

we may reduce to the standard quadratic (and for appropriate regions even

elliptic) diagonal form.

Wilhem Schlag : "Long-term dynamics of dispersive equations."

These lectures will provide an introduction

to some aspects of nonlinear dispersive equations. We will consider Hamiltonian

equations, mostly the cubic Klein-Gordon equation for the sake of simplicity.

By means of the defocusing equation, which has a positive definite energy, we will

illustrate the method of concentration-compactness in order to obtain scattering for

large times.

The focusing equation admits both blowup in finite time as well as dispersion

for small data. In addition, there are soliton-type solutions, amongst which

the ground state plays a special role. Near the energy of the ground state

we will be able to exhibit a center-stable manifold which is related to the

hyperbolic dynamics of the linearized flow around the ground state.

For data from either side of the manifold one obtains a finite-time blowup

vs. scattering dichotomy.

Exposés :

Pietro Baldi : "KAM for quasi-linear and fully nonlinear forced KdV."

(Joint work with Massimiliano Berti and Riccardo Montalto.)

We prove the existence of quasi-periodic, small amplitude solutions for

quasi-linear and fully nonlinear forced perturbations of KdV equations.

For Hamiltonian or reversible nonlinearities we also obtain the linear stability of the solutions.

The proofs are based on

(i) a Nash-Moser iterative scheme in Sobolev scales.

(ii) A regularization procedure, which conjugates the linearized operator to a differential operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and (pseudo-)differential operators.

(iii) A reducibility KAM scheme, which completes the reduction to constant coefficients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues.

Nicolas Burq : "Control for Schrödinger equation on tori."

The purpose of this talk is to present some recent results on the control of Schrödinger equations on tori, in the presence of a rough potential. This is a joint work with J. Bourgain (IAS Princeton) and M. Zworski (U-California, Berkeley).

Bassam Fayad : Lois limites d'actions de Z^d et applications aux approximations diophantiennes."

Gérard Iooss : "Few small divisor problems in Fluid Mechanics."

* What is a small divisor problem?

* Failures of Lyapunov-Schmidt method.

* 3D traveling periodic gravity waves.

* 2D standing gravity waves on an infinitely deep fluid layer.

* Faraday problem and quasipatterns.

Sergei Kuksin : "KAM-theory for PDE with one and many space-variables."

In my talk I will discuss the space-multidimensional "KAM for PDE" theory, its differences with the space-onedimensional case and with the Craig-Wayne-Bourgain approach. I will explain what is done in corresponding works of Bourgain, Eliasson-Kuksin, Berti-Bolle and Grebert-Eliasson-Kuksin.

Gilles Lebeau : "Hypoelliptic random walks."

Laurent Michel : "Tunnel effect for random walk."

We study natural random walks on Euclidean space endowed with a probability density. We investigate the convergence speed of such random walk towards its stationary distribution.

The answer is given by estimates of the spectral gap of the associated Markov operator that we study in a semiclassical regime. We show that this gap is of order exp(-S/h), where h>0 is the semiclassical parameter in the problem. The computation, of the constant S, involves study of tunnel effect in the spirit of works of Helffer-Sjöstrand and Helffer-Klein-Nier on Witten laplacian. One of the main ingredient of the proof, is to exhibit a super-symmetric structure for our operator.