I will present some recent progress in an ongoing project with S. Buschenhenke (Kiel), I. Ikromov (Samarkand) and D. Müller (Kiel) where we obtain the range of $p$ for which the maximal operator associated to hypersurfaces in $R^3$ is bounded on $L^p$. We will see, with a particular example, how, when the so-called height is less than $2$, it is not what determines the $p$ range.
In this talk, we will first review some classical results on the so-called ’spectral inequalities’, which yield a sharp quantification of the unique continuation of the spectral family associated with the Laplace-Beltrami operator in a compact manifold. In a second part, we will discuss how to obtain the spectral inequalities associated to the Schrodinger operator -\Delta_x + V(x), in \mathbb{R}^d, in any dimension $d\geq 1$, where V=V(x) is a real analytic potential. In particular, we can handle some long-range potentials. This is a joint work with Prof G.
Classical and flag paraproducts arise naturally in the study of nonlinear PDEs. While multi-parameter paraproduct has been studied thoroughly, known estimates for flag paraproducts only involve single parameter. We will state $L^p$ estimates for a particular case of the bi-parameter flag paraproduct on some restricted function spaces. We will discuss the key ingredient of the proof - a stopping-time argument which combines information from subspaces to obtain estimates on the entire space.
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.
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.
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 $\varepsilon^{M+1}$.
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.
