**Winter
school « dynamics and pde », Saint-Etienne de
tinée 2015**

**Mini-Cours:**

*Raphael Cote

"Energy partition for the wave equation and applications to the profile decomposition for wave maps"

We investigate how the energy of a solution to the radial linear wave

equation spreads out for large times. We obtain explicit formulas

involving the initial data, and see for example that the energy

concentrates around the light cone. We are also able to compute the

amount of energy which remains outside the light cone, which actually

depends crucially on the parity of the space dimension.

These results play an important role in the proof of the profile

decomposition for nonlinear wave type equations, that is the description

of the long time behaviour of solutions. We will in particular discuss

the proofs in the case of equivariant wave maps.

These are joint works with Carlos Kenig, Andrew Lawrie and Wilhlem Schlag.

*Boris Dubrovin

"Hamiltonian partial differential equations and Painleve' transcendents 1"

"Hamiltonian partial differential equations and Painleve' transcendents 2"

In the first part I will explain basic notions of geometry of weakly dispersive Hamiltonian partial differential equations (PDEs) including a perturbative approach to integrability. There are various types of phase transitions in solutions to such PDEs. Conjecturally the critical behaviour of a generic solution can be described by certain Painleve' transcendents. I will explain motivations for such Universality Conjecture and formulate rigorous results and open problems.

*Isabelle Gallagher

"From molecular to fluid dynamics"

"Hilbert's 6th problem consists in justifying rigourously the mechanism leading from classical, molecular dynamics to fluid motion. Using Boltzmann's equation as an intermediate description, we shall explain in this mini-course how this is possible in the simplified, linear settings of the heat and Stokes equations. The main difficulty is the justification of Boltzmann's equation starting from Newton's laws, on diffractive times. We shall therefore concentrate in the presentation of Lanford's proof of convergence to the Boltzmann equation, insisting on the particularities of the linear and linearized case (leading respectively to the heat and Stokes equations). This corresponds to joint works with Thierry Bodineau and Laure Saint-Raymond."

*Patrick Gérard

"Quasiperiodic and turbulent solutions of the cubic Szegö equation"

We will show how the precise description of quasiperiodic solutions to

the cubic Szegö equation allows to establish the genericity of solutions

with large time unbounded high Sobolev norms.

The proof relies on a nonlinear Fourier transform inherited from the Lax pair structure

for this equation. This is a jointwork with Sandrine Grellier.

*Jiangong You

"Reducibility and its applications in spectral theory"

I will talk about the reducibility of quasi-periodic linear systems,

and its applications in spectral theory including counter-examples to Kotani-Last

conjecture, dry Ten Martini problem and phase transition of Almost Mathieu Operators

based on joint works with Avila, Hou and Zhou.

**Exposés**

*Massimiliano Berti

" KAM for PDEs"

I will discuss new existence results of quasi-periodic solutions for PDEs, like for wave equations in higher space

dimension and for water-wawes.

*Nicolas Burq

"Concentration of quasimodes around submanifolds"

"The purpose of this talk is to present some new results on the speed of concentration on submanifolds of quasi-modes (or eigenfunctions) for the Laplace operator. We deduce some estimates on the stabilization of wave equations. This is a joint work with C. Zuily"

*Zaher Hani

"Energy dynamics across scales and wave turbulence in Hamiltonian PDE"

Broadly speaking, we will be interested in the mathematical study of a central physical problem, namely that of energy transfer in Hamiltonian systems. More precisely, suppose energy is initially injected only in a fraction of all the possible degrees of freedom of the system, how will this energy be redistributed as time passes by? This problematic arises in several phenomena from heat transfer to wave dynamics in oceans and fluid plasma. We will focus our attention on dispersive Hamiltonian systems, and see how this problem translates into deep analytical questions about the long-time behavior of solutions to the corresponding nonlinear PDE. We will survey some attempts to understand these questions from a deterministic and statistical viewpoint, and report on recent progress in both directions.

*Georgi Popov

"Asymptotic quantization with an exponentially small error. "

The aim of this talk is to provide a construction of different types of quasi-modes with an exponentially small discrepancy for semi-classical formally selfadjoint differential operators with (analytic) Gevrey coefficients in an (analytic) Gevrey smooth manifold of dimension greater or equal to two. There are three types of quasi-modes we are interested in which are distinguished by their micro-supports - quasi-modes associated with a given Kronecker invariant torus of the classical Hamiltonian with a Diophantine vector of rotation, quasi-modes associated with a large family of such tori in any dimension and the so called Shnirelman quasi-modes associated with gaps between invariant tori when the dimension of the manifold is two.

The construction is based on Birkhoff and Quantum Birkhoff normal forms in Gevrey classes and on the calculus of Fourier Integral Operators in Gevrey classes.

An analogue of the effective stability in the semi-classical limit is obtained as well.

*Michela Procesi

"Growth of Soblev norms for the analyitc non-linear Schrodinger equation on the torus."

I will discuss some recent results in collaboration with Emanuele

Haus and Marcel Guardia.

I will consider the analytic non-linear Schrodinger equation on a

two-dimensional torus and exhibit orbits whose Sobolev norms grow with

time. The main point is to make use of an accurate combinatorial analysis

in order to reduce to a sufficiently simple toy model, similar in many ways

to the one used for the case of the cubic NLS.

*Jérémie Szeftel

"Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space"

abstract: We study time-like hypersurfaces with vanishing mean curvature in the 3+1 dimensional Minkowski space, which are the hyperbolic counterparts to the minimal surface equation. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid. More precisely, we prove in a certain class of symmetry the existence, in the neighborhood of the catenoid initial data, of a co-dimension 1 Lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid. This is joint work with Roland Donninger, Joachim Krieger and Willie Wong.

*Jiquiang Zheng

"Scattering theory for nonlinear Schrödinger equations with inverse-square potential"

In this talk, we study the long-time behavior of solutions to nonlinear

Schr\"o-dinger equations with some critical rough potential of

$a|x|^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with $P_a=-\Delta+a|x|^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schr\"odinger equation with inverse square potential in energy space

$H^1(\R^n)$. Following a concentration-compactness approach, we also

obtain such scattering theory for the defocusing energy-critical nonlinear Schr\"odinger equation with inverse square potential in $\dot H^1(\R^n)$-space.