Fast diffusion, mean field drifts and reverse HLS inequalities

(Cogne, 3-7 June 2019)

This course is devoted to drift-diffusion equations with linear and non-linear diffusions and mean field drifts of various types. A central feature is the identification of an asymptotic or stationary solution and the use of a free energy (or relative entropy). By linearizing the free energy, we obtain a functional framework in which we can analyze the spectrum of the linearized operator. The corresponding spectral gap determines the optimal asymptotic rate of convergence, but can also be used to establish the optimal constant in the underlying functional inequality which relates the free energy and the relative Fisher informations. We shall develop these ideas in the pure diffusive case (fast diffusions, with and without weights and Bakry-Emery methods: Lecture 1) and on two examples of drift-diffusion equations (with linear diffusion and mean-field drifts: a flocking model in Lecture 2 and the Keller-Segel model in Lecture 3), before exposing the reverse Hardy-Littlewood-Sobolev (HLS) inequalities and the corresponding entropies, a problem in which many questions are still open.

CEREMADE - Université Paris- Dauphine
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