Le système de Serre-Green-Naghdi (SGN) est un modèle fortement non-linéaire et faiblement dispersif pour la propagation des vagues. Il possède une structure Hamiltonienne, et le problème de Cauchy est bien posé dans des espaces de Sobolev d'indice suffisamment élevé.
The simplex Hilbert transform is a singular integral form related to the multilinear Hilbert transform and Carleson’s operator. Boundedness of the simplex Hilbert transform is is one of the major open problems in harmonic analysis. In this talk we discuss bounds for a superposition of simplex Hilbert transforms in low dimensions, and related singular integral forms. Joint work with Joris Roos.
Dans cet exposé nous nous proposerons de revisiter la théorie de Harris-Meyn-Tweedie sur les semi-groupes de Markov, et nous nous intéresserons plus particulièrement au cas sous-géométrique, c’est-à-dire, lorsque le retour vers l’équilibre n’est pas à vitesse exponentielle. Nous commencerons par établir l’existence d’un état d’équilibre et un taux de convergence vers celui-ci pour une classe d’opérateurs stochastiques à l’aide d’arguments simples, déterministes et constructifs.
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.
