Analysis Seminar

We study interacting particles behaving according to a reaction-diffusion equation with nonlinear diffusion and nonlocal attractive interaction. This class of partial differential equations is a generalization of the Patlak-Keller-Segel model for bacterial chemotaxis, and has a nice gradient flow structure with respect to the Wasserstein-2 distance that allows us to make links to variations of well-known functional inequalities. Depending on the nonlinearity of the diffusion, the choice of interaction potential and the space dimensionality, we obtain different regimes. This talk will give an overview of recent advances in the fair-competition regime, when attractive and repulsive forces are in balance, as well as the diffusion-dominated and attraction-dominated regimes.