CMX Lunch

We study interacting particles behaving according to a reaction-diffusion equation with nonlinear diffusion and nonlocal attractive interaction. This class of partial differential equations has a very nice gradient flow structure that allows us to make links to homogeneous functionals and variations of well-known functional inequalities. However, the convexity properties of these functionals are not known, and we make use of optimal transport techniques to draw connections between the minimizers of the energy and the equilibria of the equation. Depending on the nonlinearity of the diffusion, the choice of interaction potential and the space dimensionality, we obtain different regimes. Our goal is to understand better the asymptotic behavior of solutions in each of these regimes, starting with the fair-competition regime where attractive and repulsive forces are in balance. No prior knowledge of PDE theory is required as I will give a quick introduction to the notions involved. This is joint work with José A. Carrillo and Vincent Calvez.