ATTENTION : exceptionnellement un LUNDI

Speaker: José A. Carrillo

Title: Nonlocal Aggregation-Diffusion Equations: fast diffusion and partial concentration

Abstract:

In the first part I will give an overview of nonlocal aggregation-diffusion equations and the state of the art on results for homogeneous kernels and nonlinear diffusions in the degenerate case. I will explain part of the qualitative behavior of the solutions, numerical explorations and ideas on the proofs of some results. I will also discuss the applications they have in mathematical biology and their connections.

The seminar part will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This part of the talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernandez, R. Frank, D. Gomez-Castro, F. Hoffmann, M. Lewin, and J. L, Vazquez.