Séminaire d'Analyse

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Séminaire d'Analyse Institut Henri-Poincaré - Fondation Sciences Mathématiques de Paris. Organisateurs : Jean-Yves Chemin, Sergiu Klainerman, Cédric Villani

Part. 1/2 -Séminaire d'analyse IHP-Fondation Sciences Mathématiques de Paris -Conférence du lundi 28 mars 2011 - Global dynamics above the ground state for the energy critical focussing NLW par Joachim Krieger (EPFL)Résumé :I'll discuss recent joint work with Kenji Nakanishi and Wilhelm Schlag on the existence of certain open data sets with energy strictly above that of the ground state for which one controls the global in time dynamics, in the context of the energy critical focussing nonlinear wave equation. This work builds in part on recent work by Duyckaerts, Kenig and Merle.

Part. 2/2 -Séminaire d'analyse IHP-Fondation Sciences Mathématiques de Paris -Conférence du lundi 28 mars 2011 - Global dynamics above the ground state for the energy critical focussing NLW par Joachim Krieger (EPFL)Résumé :I'll discuss recent joint work with Kenji Nakanishi and Wilhelm Schlag on the existence of certain open data sets with energy strictly above that of the ground state for which one controls the global in time dynamics, in the context of the energy critical focussing nonlinear wave equation. This work builds in part on recent work by Duyckaerts, Kenig and Merle.

Part. 1/2Séminaire d'analyse IHP-Fondation Sciences Mathématiques de Paris -Conférence du lundi 28 mars 2011 -Blow up for some energy critical geometrical models par Pierre Raphaël (Institut de Mathématiques de Toulouse (IMT), CNRS - Université Paul Sabatier UMR5219) -Résumé :I will consider energy critical geometric models which appear in various physical situations (cristal physics, ferromagnetism), and which are respectively the parabolic, wave and Schrodinger analogue of the stationary harmonic map problem: the harmonic heat flow, the wave map problem and the Schrodinger map, for maps from (Rtimes R^2) to S^2. For these three problems, I will describe a natural class of smooth solutions which lead to finite time blow up dynamics at some universal blow up speed. This regime is stable for wave maps and the heat flow within the classs of corotational symmetry, and a completely new phenomenon occurs for the Schrodinger map: the remaining rotation fredoom in equivariant symmetry stabilizes the system, and leads to a codimension one blow up phenomenon. This is joint work with Frank Merle, Igor Rodnianski and Remi Schweyer.

Partie 1/2 -Hyperbolic dispersive estimates, topological pressure, and applicationspar Maciej Zworski (Berkeley University)Séminaire d'analyse IHP-Fondation Sciences Mathématiques de ParisConférence du lundi 7 février 2011Résumé :Following the work of Anantharaman and Nonnenmacher, Nonnenmacher and the speaker developed estimates for semiclassical propagators for open chaotic systems: under conditions on the topological pressure of the classical system one obtains exponential decay in time. This gives resonance free strips, resolvent estimates, local smoothing estimates. In related work of Wunsch and the speaker similar estimates are obtained for normally hyperbolic trapped sets. Dyatlov applied these to quasinormal modes for black holes which gives exponential decay of linear waves in Kerr-deSitter background.

Partie 2/2 -Hyperbolic dispersive estimates, topological pressure, and applicationspar Maciej Zworski (Berkeley University)Séminaire d'analyse IHP-Fondation Sciences Mathématiques de ParisConférence du lundi 7 février 2011Résumé :Following the work of Anantharaman and Nonnenmacher, Nonnenmacher and the speaker developed estimates for semiclassical propagators for open chaotic systems: under conditions on the topological pressure of the classical system one obtains exponential decay in time. This gives resonance free strips, resolvent estimates, local smoothing estimates. In related work of Wunsch and the speaker similar estimates are obtained for normally hyperbolic trapped sets. Dyatlov applied these to quasinormal modes for black holes which gives exponential decay of linear waves in Kerr-deSitter background.

Partie 1/2 -On the uniqueness of stationary black holes in vacuumpar Alex Ionescu (Princeton University)Séminaire d'analyse IHP-Fondation Sciences Mathématiques de Parisorganisé par Jean-Yves Chemin, Sergiu Klainerman et Cédric VillaniConférence du 31 janvier 2011- Résumé :I will discuss first the concept of unique continuation and present some classical theorems in the subject. I will then discuss some recent work, joint with Sergiu Klainerman and Spyros Alexakis, on the uniqueness properties of the Kerr solutions in the class of regular stationary solutions of the Einstein vacuum equations.

partie 1/2Optimal transport and rearrangement tools for some hamiltonian PDEs with dissipation par Yann Brenier (CNRS Nice et UPMC) est le quatrième rendez-vous du Séminaire d'analyse IHP-FSMP, organisé par Jean-Yves Chemin, Sergiu Klainerman et Cédric Villani.Résumé :There are important hamiltonian PDEs that may produce singularities in finite time.Sometimes, their solutions can be extended beyond singularities to the expense of some dissipation mechanism. A classical example is the so-called inviscid Burgers equation with the concepts of shock waves and entropy solutions.A very ambitious goal would be to address the 3D Euler equations of incompressible fluid mechanics in a similar way, following Kolmogorov analysis of turbulence.In this lecture, we present a somewhat simpler example where such a strategy can be followed using tools borrowed from rearrangement and optimal transport theories. This is related to the Vlasov-Poisson system with mono-kinetic distributions, and the related problem of "reconstructing the early universe", in cosmology, following Zeldovich, Peebles, and, more recently, Uriel Frisch.

partie 2/2Optimal transport and rearrangement tools for some hamiltonian PDEs with dissipation par Yann Brenier (CNRS Nice et UPMC) est le quatrième rendez-vous du Séminaire d'analyse IHP-FSMP, organisé par Jean-Yves Chemin, Sergiu Klainerman et Cédric Villani.Résumé :There are important hamiltonian PDEs that may produce singularities in finite time.Sometimes, their solutions can be extended beyond singularities to the expense of some dissipation mechanism. A classical example is the so-called inviscid Burgers equation with the concepts of shock waves and entropy solutions.A very ambitious goal would be to address the 3D Euler equations of incompressible fluid mechanics in a similar way, following Kolmogorov analysis of turbulence.In this lecture, we present a somewhat simpler example where such a strategy can be followed using tools borrowed from rearrangement and optimal transport theories. This is related to the Vlasov-Poisson system with mono-kinetic distributions, and the related problem of "reconstructing the early universe", in cosmology, following Zeldovich, Peebles, and, more recently, Uriel Frisch.

Partie 1/2The Dixmier unitarizability problem for group representations par Gilles Pisier (Texas A&M et UPMC - Institut de Mathématiques de Jussieu UMR 7586 CNRS-UPMC).Résumé :A uniformly bounded representation of a group on Hilbert space is called unitarizable if it is similar to a unitary one. A group G is called unitarizable if every uniformly bounded representation on it is unitarizable. In 1950, Dixmier (as well as Day independently) proved that amenable implies unitarizable and then asked whether the converse holds. We will review the history of this problem, describe several partial results and discuss recent progress due to Ozawa and Monod based on the Gaboriau-Lyons result that the free group F2 ”randomly” embeds in any non-amenable group.