Séminaire d'analyse (archives)

Dans cet exposé, je présenterai des résultats concernant le problème de Cauchy pour les ondes non linéaires avec données aléatoires et/ou une force stochastique en dimension deux. Après avoir expliqué la construction de la mesure de Gibbs associée au Hamiltonien de l'équation et la nécessité de renormaliser, je présenterai un schéma de preuve du caractère bien posé dans le cas particulier du tore. Enfin, j'expliquerai comment contourner l'approche classique par analyse de Fourier de la construction des objets stochastiques principaux afin d'étendre ces résultats à une surface compacte sans bords plus générale.

We consider general classes of nonlinear Schr\"odinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\varepsilon$, for generic initial data of size $\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\varepsilon^{M+1}$. This implies that for such initial data $u(0)$ we control the Sobolev norm of the solution $u(t)$ for time of order $\varepsilon^{-M}$. Furthermore this property is locally stable: if $v(0)$ is sufficiently close to $u(0)$ (of order $\varepsilon^{3/2}$) then the solution $v(t)$ is also controled for time of order $\varepsilon^{-M}$. (Joint work with Erwan Faou and Joackim Bernier)

We consider the problem of inversion of weighted Radon transforms. This problem arises in different tomographies and, in particular, in emission tomographies. We present old and very recent results on this problem. This talk is based, in particular, on recent works [Goncharov, Novikov, 2016, 2018], [Goncharov, 2017].

We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. The key ingredients of the proof are sparse domination and probabilistic representation of the Riesz vector. This type of proof has the significant advantage that it allows for a much stronger conclusion, giving us a new dimensionless weighted $L^p$ estimate.

I will describe an approach to studying the Kakeya maximal function in high dimensions via the Guth--Katz polynomial partitioning method. Although the approach does not currently produce better bounds than the record set by Katz--Tao, it is rather flexible, provides a lot of interesting structural information and gives rise to some interesting algebraic/geometric problems.

Motivated by some classical results of Meyer, Pichorides and Zygmund, we present a variant of Yano's extrapolation theorem for analytic Hardy spaces over the torus. Some related questions will also be discussed.

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This prove a conjecture of Zakharov-Dyachenko based on the formal Birkhoff integrability of the waver waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order $\epsilon^{−3}$.

First I will describe a new pseudodifferential calculus for (pseudo-)Riemannian spaces, which in our opinion (my, D.Siemssen's and A.Latosiński's) is the most appropriate way to study operators on such a manifold. I will briefly describe its applications to computations of the asymptotics the heat kernel and Green's operator on RIemannian manifolds. Then I will discuss analogous applications to Lorentzian manifolds, relevant for QFT on curved spaces. I will mention an intriguing question of the self-adjointness of the Klein-Gordon operator. I will describe the construction of the (distinguished) Feynman propagator on asymptotically static spacetimes. I will show how our pseudodifferential calculus can be used to compute the full asymptotics around the diagonal of various inverses and bisolutions of the Klein-Gordon operator.

14h, Peter Topalov : On the group of almost periodic diffeomorphisms and its exponential map

We define the group of almost periodic diffeomorphisms on the Euclidean plane $\mathbb{R}^n$. We then study the properties of its Riemannian and Lie group exponential map and provide applications to fluid dynamics.

15h, Alexei Iantchenko : Semiclassical inverse problems for elastic surface waves in isotropic media

We carry out a semiclassical analysis of surface waves in Earth which is stratified near its boundary at some scale comparable to the wave length.

Propagation of such waves is governed by effective Hamiltonians which are non-homogeneous principal symbols of some pseudodifferential operators. Each Hamiltonian is identified with an eigenvalue in the discreet spectrum of a locally 1D Schr{\"o}dinger-like operator on the one hand, and generates a flow identified with surface wave bicharacteristics in the two-dimensional boundary on the other hand.

The eigenvalues exist under certain assumptions reflecting that wave speeds near the boundary are smaller than in the deep interior. This assumption is naturally satisfied by the structure of Earth's crust and mantle.

Using these Hamiltonians, we obtain pseudodifferential surface wave equations. In case of isotropic medium the equations decouple into Rayleigh and Love waves. In both cases we perform a comprehensive analysis of the recovery of the S-wavespeed from the semiclassical spectrum.

The approach follows the ideas of Colin de Verdière on acoustic surface waves and is joint work with Maarten V. de Hoop, Jian Zhai, Rice University, and Gen Nakamura, Hokkaido University