Eric A. Carlen (Rutgers) : Classical and Quantum Master Equations for N-particle Systems and their Kinetic Limits.
Résumé : We present recent results as well as background on models for systems of N particles undergoing random binary collisions, focusing on propagation of chaos and the rate of convergence to equilibrium. These questions arise from the work of Mark Kac and his investigation into the probabilistic structure underlying the Boltzmann equation. Recently, the quantum mechanical variation on Kac’s question has begun to be investigated. There are novel difficulties due to the fact that in quantum mechanics, conditional probability is not always well-defined. Nonetheless, a substantial quantum analog of of the Kac program can be carried out, and this leads to an interesting and novel class of quantum kinetic equations.
Bernard Helffer (Paris-Sud 11 and Nantes) : On non self-adjoint spectral problems occurring in superconductivity, kinetic theory and control theory.
Résumé : In this course we would like to discuss spectral properties of some non self-adjoint operators appearing in the analysis of the long time behavior of the solutions of the time dependent Ginzburg Landau system (due to Eliashberg-Gorkov) and to consider in particular the global stability of the stationary normal solutions. Similar questions arise for the Fokker-Planck operator or in questions in Control theory. The recent theorems have been obtained in collaboration with Y. Almog, X. Pan, R. Henry, K. Beauchard and L. Robbiano.
Clément Mouhot (Cambridge) : Some quantitative factorisation semigroup methods for PDEs
Résumé : In this course we would like to give an overview of the various problems of linear and nonlinear stability in kinetic theory —and beyond— that have been studied recently by various authors with the help of an abstract theory for deriving quantitative estimates on localisation of spectrum and semigroup growth when changing functional spaces. We have developed the latter theory in several works with Mischler and other collaborators.