Prochainement

Pas d'événement prévu ce mois

Passés

Vendredi 15 mai 15:00-16:00 Thibault Lefeuvre  (Université Paris-Saclay (Orsay))
Microlocal regularity of solutions to cohomological equations

Plus d'infos...

Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : On a closed manifold endowed with a uniformly hyperbolic flow — or Anosov, in the literature —, a certain number of dynamical/geometrical problems (structural stability, marked length spectrum rigidity, study of transparent connections, ...) involve a class of equations called cohomological equations. Usually, one can construct ``by hand" a Hölder continuous solution to these equations but proving smoothness is harder. I will explain how one can relate the study of these equations to microlocal analysis. The key estimate to prove is a radial source estimate in Hölder-Zygmund spaces (and more generally, in Besov spaces), which is a kind of propagation of singularities in phase space. This was first used in the context of hyperbolic flows by Dyatlov-Zworski in Sobolev spaces. However, their proof is based on a positive commutator argument and the sharp Gärding inequality and does not seem to generalize to Hölder-Zygmund spaces. This is an ongoing project with Yannick Guedes Bonthonneau.

Microlocal regularity of solutions to cohomological equations  Version PDF

Vendredi 15 mai 14:00-15:00 Emmanuel Grenier  (ENS Lyon)
TBA

Plus d'infos...

Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : TBS

TBA  Version PDF

Jeudi 7 mai 11:07-14:00 David Lannes  (Université de Bordeaux)
Dispersive perturbations of hyperbolic initial boundary value problems

Plus d'infos...

Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : We will show in this talk how some models for the description of the interactions of waves with floating structures can be formulated as hyperbolic initial boundary value problems or (depending on the model chosen for the propagation of the waves), dispersive perturbations of such problems. After recalling some classical results on hyperbolic initial boundary value problems (in particular on the nature of the admissible boundary conditions), we will explain how the presence of a dispersive perturbation in the equations drastically changes the nature of the equations. These different behaviors raise several questions, one of which being nature of the dispersionless limit. We will show that the presence of dispersive boundary layers make this limit singular, and explain how to control them on an example motivated by a model for wave-structure interactions.

Dispersive perturbations of hyperbolic initial boundary value problems  Version PDF

Jeudi 7 mai 11:09-12:09 France Hoffmann  (California Institute of Technology)
Kalman-Wasserstein Gradient Flows

Plus d'infos...

Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : We study a class of interacting particle systems that may be used for optimization. By considering the mean-field limit one obtains a nonlinear Fokker-Planck equation. This equation exhibits a gradient structure in probability space, based on a modified Wasserstein distance which reflects particle correlations : the Kalman-Wasserstein metric. This setting gives rise to a methodology for calibrating and quantifying uncertainty for parameters appearing in complex computer models which are expensive to run, and cannot readily be differentiated. This is achieved by connecting the interacting particle system to ensemble Kalman methods for inverse problems. This is joint work with Alfredo Garbuno-Inigo (Caltech), Wuchen Li (UCLA) and Andrew Stuart (Caltech).

Kalman-Wasserstein Gradient Flows  Version PDF

Jeudi 30 avril 15:00-16:30 Isabelle Gallagher 
From Newton to Boltzmann, fluctuations and large deviations

Plus d'infos...

Lieu : url https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : I will report on a recent work, joint with Th. Bodineau, L. Saint-Raymond and S. Simonella, in which we develop a rigorous theory of macroscopic fluctuations for a hard sphere gas outside thermal equilibrium, in the Boltzmann-Grad limit : in particular we study deviations from the Boltzmann equation (describing the asymptotic dynamics of the empirical density) and provide, for short kinetic times, both a central limit theorem and large deviation bounds.

Notes de dernières minutes : Attention, exceptionnellement, le séminaire aura lieu le jeudi 30 avril au lieu du vendredi 1er mai

From Newton to Boltzmann, fluctuations and large deviations  Version PDF

Jeudi 30 avril 14:00-15:00 Felix Otto   (Max-Planck-Gesellschaft Leipzig)
Stochastic homogenization

Plus d'infos...

Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : In engineering applications, heterogeneous media are often described in statistical terms.
This partial knowledge is sufficient to determine the effective, i. e. large-scale behavior.
This effective behavior may be inferred from the Representative Volume Element (RVE) method.
I report on last years’ progress on the quantitative understanding of what is called
stochastic homogenization of elliptic partial differential equations :
optimal error estimates of the RVE method and the homogenization error,
and the leading-order characterization of fluctuations.
Methods connect to elliptic regularity theory, and in fact lead to a fresh look upon
this classical area, and to concentration of measure arguments.
We put a special emphasis on annealed Calderón-Zygmund estimates.

Notes de dernières minutes : Attention exceptionnellement le séminaire aura lieu le jeudi 30 avril au lieu du vendredi 1er mai

Stochastic homogenization  Version PDF

Vendredi 24 avril 15:00-16:30 Cyril Letrouit  (Ecole Normale Supérieure (Paris))
Subelliptic wave equations are never observable

Plus d'infos...

Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : In this talk, we explain a result we obtained recently, concerning the wave equation with a sub-Riemannian (i.e. subelliptic) Laplacian. Given a manifold $M$, a measurable subset $\omega\subset M$, a time $T_0$ and a subelliptic Laplacian $\Delta$ on $M$, we say that the wave equation with Laplacian $\Delta$ is observable on $\omega$ in time $T_0$ if any solution $u$ of $\partial_tt^2u-\Delta u=0$ with fixed initial energy satisfies $\int_0^T_0\int_\omega |u|^2dxdt\geq C$ for some constant $C>0$ independent on $u$.
It is known since the work of Bardos-Lebeau-Rauch that the observability of the elliptic wave equation, i.e. with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the geometric control condition (GCC), which stipulates that any geodesic ray meets $\omega$ within time $T_0$. We show that in the subelliptic case, as soon as $M\backslash \omega$ has non-empty interior and $\Delta$ is ``subelliptic but not elliptic", GCC is never verified, which implies that subelliptic wave equations are never observable. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the distance on $M$ associated to $\Delta$) spending a long time in $M\setminus \omega$.

Subelliptic wave equations are never observable  Version PDF

Vendredi 24 avril 14:00-16:00 C. Mouhot  (University of Cambridge)
Unified approach to fluid approximation of linear kinetic equations with heavy tails

Plus d'infos...

Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : The rigorous fluid approximation of linear kinetic equations was first obtained in the late 70s when the equilibrium distribution decays faster than polynomials. In this case the limit is a diffusion equation. In the case of heavy tail equilibrium distribution (with infinite variance), the first rigorous derivation was obtained in 2011 in a joint paper with Mellet and Mischler, in the case of scattering operators. The limit shows then anomalous diffusion ;
it is a governed by a fractional diffusion equation. Lebeau and Puel proved last year the first similar result for Fokker-Planck operator, in dimension 1 and assuming that the equilibrium distribution has finite mass. Fournier and Tardif gave an alternative probabilistic proof, more general (covering any dimension and infinite-mass equilibrium distribution) but non-constructive. We present a unified elementary approach, fully quantitative, that covers all previous cases as well as new ones. This is a joint work with Emeric Bouin (Université Paris-Dauphine).

Unified approach to fluid approximation of linear kinetic equations with heavy tails  Version PDF