Por Claude Bardos (Laboratoire Jacques-Louis Lions, Paris).

In this talk we will discuss a system of PDEs that describes the evolution of a density of electrons under a mean-field approximation. It is simple enough to lead to some basic mathematical problems which received over the recent years complete or partial answers. For instance:

1. Short and long time existence of solutions.
2. Derivation of the solution of N interacting particles as N tends to inifity. On the other hand it is rich enough to lead to the formulation of qualitative issues appearing in real physical problem like the stability of plasma in Tokomak or laser confined plasma. In particular

1. Landau damping for small analytic perturbations of a stable homogeneous profile.
2. Local in time instabilities (in particular under the Penrose instability condition) and so on.

Following an ongoing project, I intend to elaborate on this last point. I will start with the quasilinear approximation: a non linear diffusion equation which shares some similarity (and difference in the range of applications ) with the Balescu Lenard or the Fokker Planck equations which will be considered as connection between the Penrose unstable solutions, the Penrose stable (O. Penrose. Phys. Fluids, 3:258 265, 1960) and at the end of the day the Mouhot Villani version of Landau Damping.

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