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| Tuesday, 6 Feb. | 15:00 | Isabelle Gallagher | Profile decompositions and the Navier-Stokes Equations |
| The aim of this talk is to present the Profile Decomposition Theorem of P. Gerard (1999) and to apply it to sequences of solutions of the Navier-Stokes equations. The Profile Decomposition Theorem is a precise analysis of the defect of compactness in Sobolev injections, in terms of translations and dilations of fixed functions. We use that result to prove that the profile decompostion of a bounded sequence of initial data for the Navier-Stokes equations can be transported (for a short time) by the nonlinear evolution. We finally present some open questions which could possibly be dealt with using such methods. | |||
| 16:30 | Remi Carles | Concentration at one point for semiclassical NLS equations | |
| We consider semi-classical nonlinear Schroedinger equations in one space dimension whose data oscillate so that rays focus at the origin at time 1. We underscore critical scales for the infulence of the nonlinear term. For subcritical indices, we study how the present problem is related to scattering theory. Finally, for supercritical cases, formal computations suggest that new strongly nonlinear phenomena occur. Which phenomena? | |||
| Wednesday, 7 Feb. | 15:00 | Sahbi Keraani | Semiclassical limit for a class of Nonlinear Schroedinger equations with potential |
| We improve the recent result of J.C. Bronski and R.L. Jerrard on the semi-classical limit of the nonlinear focusing Schroedinger equation with potential, for initial data of the form Q((x-a)/h)*exp(ixv/h), where Q is the ground state of the associated unscaled problem. We give a sharp description of the asymptotic behavior of this family of solutions. This allows us, in particular, to completely describe the dynamics of the associated Wigner measure. | |||
| Thursday, 8 Feb. | 15:00 | Clotilde Fermanian | Wigner measures and eigenvalue crossings |
| Propagation of Wigner measures for a given system suggets the following question: how do Wigner measures proagate through energy level crossings? We discuss a 2x2 system which displays such a crossing. We introduce two-scale Wigner measures to describe how the usual Wigner transforms concentrate on trajectories that push through the crossing points. Then we derive explicit formulae for the branching of such measures. These formulae are generalizations of the Landau-Zener formulae. | |||
| 16:30 | Patrick Gerard | NLS on compact surfaces | |
| We discuss generalizations of Strichartz estimates and dispersion inequalities for the Schroedinger equation on a compact Riemannian manifold. As a consequence, we obtain global existence and uniqueness of strong solutions to any nonlinear defocusing Schroedinger equation with polynomial nonlinearity in two space dimensions. | |||
| Friday, 9 Feb. | 15:00 | Alex Gottlieb | Resampling from jackknife pseudovalues and the propagation of chaos |
| Given N iid samples you want to estimate some feature of the distribution from which they are sampled, so you calculate a statistic from those N samples and hope that it gives a good estimate. To assess its accuracy, you would like to perform the same calculation on many other data sets of the same kind, to simulate the distribution of the sample statistic. But you only one data set! To surmount this problem, you may (sometimes) regard subsets of the given data set as "new" data sets. This technique is called the "jackknife." Although the "pseudovalues" that result are highly dependent, they have a sort of asymptotic independence called "molecular chaos." | |||
| 16:30 | Luc Miller | Geometric conditions for controllability and boundary Wigner measures | |
| Bardos, Lebeau and Rauch have given necessary and sufficient conditions for exact controllability of the wave equation from the boundary in terms of the rays of geometrical optics (BLR 88, 92). We recall how to prove this type of result using Wigner measures (after Lebeau 96, Burq 97, Burq-Gerard 97, Burq-Lebeau 99) and present work in progress on applying this method to the context of transparent obstacles (i.e. transmission problem) where new phenomena arise: interference at refraction points and continuous radiation along gliding rays. | |||
| Tuesday, 13 Feb. | 11:30 | Alexis Vasseur | Classical and and quantum transport in random media |
| In this joint work with F. Poupaud, we consider a semi-classical limit in random media. More precisely, we derive a linear Boltzmann equation from the Schroedinger equation with stochastic potential in the so-called weak coupling limit and prove the convergence. The potential depends on time and wether the associated time scale is the same as the quantum scale or is larger, accordingly, we obtain at the limit either a diffusive or an elastic collision term. | |||
| Wednesday 14 Feb. | 11:30 | Nicolas Burq | Smoothing effects for Schroedinger operators and resonances |
| We study in this talk the link between the smoothing effects for for Schroedinger operators and the estimates for the boundary value of the resolvent of the operator which are obtained for non-trapping geometries. | |||
| 15:00 | Hailiang Li | Global existence and dispersion limit of quantum hydrodynamics | |
| We consider first the global existence and large-time behavior of classical selection of quantum hydrodynamics on a bounded domain. Then we discuss the dispersion limit before singularity. |