Séminaire d'analyse (archives)

In this talk I will introduce the Anderson-Gross-Pitaevskii equation, a nonlinear Schrödinger equation with both a spatial white noise potential and a smooth confining potential. Due to the spatial roughness of the white noise in dimension 2, a renormalization procedure is needed. I will briefly present this procedure which relies on a good integrability estimate on the kernel of an unbounded operator. Then, I will present a paracontroled approach to the confining Anderson operator in order to obtain Strichartz estimates. With Strichartz estimates at hand, I will present my local wellposedness result in two steps. First the low-regularity wellposedness through the usual fix point argument, and then unconditional local wellposedness in the energy space using propagation of regularity.

Depuis les travaux de Lebeau et Robbiano (1995), il est connu que l'équation de la chaleur est contrôlable depuis des sous-ensembles qui satisfont une inégalité spectrale : les normes L² sur le sous-ensemble et sur la variété doivent être équivalentes pour des fonctions à support spectral borné, avec une constante qui croît au plus exponentiellement en le seuil de fréquence.

Après avoir introduit ces notions, je parlerai d'une collaboration en cours avec Alix Deleporte (Université Paris-Saclay) et Jean Lagacé (King's College London), dans laquelle nous caractérisons les ensembles satisfaisant cette inégalité spectrale sur n'importe quelle variété à courbure sectionnelle bornée. La condition nécessaire et suffisante, dite d'épaisseur, est une condition d'équidistribution de la mesure de l'ensemble dans toute la variété. Si le temps le permet, j'expliquerai comment l'inégalité spectrale et la notion d'épaisseur interagissent avec la géométrie du problème, et quelles conséquences de l'hypothèse de courbure bornée sont pertinentes dans l'étude. Les outils employés combinent théorie du contrôle, analyse géométrique et analyse harmonique.

En s'appuyant sur les travaux de Garg, Gurvits, Oliveira et Wigderson, on démontre que les données géométriques de Brascamp-Lieb sont, dans un certain sens, omniprésentes au sein des données admissibles. Cela répond à une question soulevée par Bennett et Tao dans leurs travaux récents sur l'inégalité adjointe de Brascamp-Lieb.

In this talk, I will present several recent results, in collaboration with Nicolas Burq, on the control theory of Schrödinger propagators on tori. Our goal is to address the following conjecture: on a torus of arbitrary dimension, Schrödinger propagators with bounded potentials are observable, and therefore controllable, from arbitrary space-time domains of positive Lebesgue measure.

Using a scheme that combines (1) approximation of rough functions by continuous functions, (2) the cluster structure of lattice points near paraboloids, and (3) mathematical induction on the dimension, we reduce the conjecture to certain integrability bounds for linear Schrödinger waves. These bounds are weaker than Bourgain’s conjectured periodic Strichartz estimates but remain nontrivial. In particular, our criteria imply the observability conjecture for the one-dimensional torus.

Applications of our results include Cantor--Lebesgue type theorems and uniform nonvanishing estimates for quantum limits.

Consider a particle randomly moving in a bounded (planar) domain starting at any given point within. Assume it bounces against the boundary and consider $\Sigma$, a small part of that boundary. What is the expected time we need to wait before the particle hits $\Sigma$ ? This question is known as the narrow escape problem. We can also consider the related question : what is the probability that the particle hits $\Sigma$ before another given subset of the boundary $\Gamma$ ? In this talk, I will address these questions and give quantitative answers in the asymptotic regime where the lengths of the windows tend to 0. To tackle the problem, I will prove a Feynman-Kac formula, linking the stochastic process studied to a deterministic PDE which has the form of a Poisson equation with mixed boundary conditions. Then, constructing appropriate quasimodes to this PDE, we are able to derive sharp asymptotics for the expected time and probabilities.

The full regularity of harmonic maps from a given surface into an arbitrary Riemannian manifold has been proved by Hélein in 1991. This is not true anymore when the domain has dimension strictly greater than 2, Rivière constructed an example of harmonic map from a 3-dimensional domain which is everywhere discontinuous in 1995. There are many possible generalizations of these maps to the higher dimensional case in order to recover the regularity of some "optimal" maps. For most of these generalizations, the optimal regularity in full generality is still open. In this talk, we will discuss some recent progress obtained for n-harmonic maps. This is a joint work with Armin Schikorra.

Quantum mechanics is well approximated by classical physics when Planck's constant is considered small, i.e., in the semi-classical limit. Typically, one can study an observable associated with a particle, such as its momentum or its position, and show that its dynamics is given by classical dynamics at first order, with corrections of the order of Planck's constant. In this talk, I will present more precisely the concept of semi-classical limits, the standard mathematical results known for non-relativistic quantum mechanics, and my work that concerns the semi-classical limit in the context of relativistic quantum mechanics. Concretely, I will show how to adapt the modulated energy method to the Klein-Gordon and Klein-Gordon-Maxwell equations and how to recover relativistic mechanics (instead of classical mechanics) at the semi-classical limit

We consider a system of N Brownian particles interacting through a long-range smooth potential. It is known that "propagation of chaos" holds in the mean-field scaling. Assume indeed that the initial distribution of the particles is chaotic, i.e. that the particles are independent and identically distributed. Then, for any given time, and as N becomes large, the distribution of particles remains chaotic. Moreover, the distribution of a typical particle is given by the solution of a Vlasov-Fokker-Planck equation.

In this talk, we will investigate the creation of chaos phenomenon. Starting from an initial distribution of particles which is only exchangeable, we prove that in some weak norm, propagation of chaos holds up to an error stemming from initial correlations, exponentially damped over time. This is a joint work with Armand Bernou and Mitia Duerinckx.