Séminaire d'analyse (archives)

The Degasperis-Procesi equation (DP) is a spatial one-dimensional model for nonlinear shallow waters phenomena and it is one of the few known Hamiltonian PDEs which is completely integrable, namely it possesses infinitely many constants of motion. Moreover this equation is quasi-linear, namely the nonlinear terms contain derivatives of the same order of the linear part. In this talk I will show a recent result of existence and stability of small amplitude quasi-periodic solutions for Hamiltonian perturbations of the DP equation on the circle. This result is based on a combination of Nash-Moser / KAM schemes and pseudo differential calculus techniques. There are several issues in dealing with this problem:

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 $\varepsilon^{-M}$.

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].

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.