Maxime Laborde, Université Paris Cité : Wasserstein gradient flow of optimal transport problems : Application to city dynamics

Résumé : In 1998, Jordan, Kinderlehrer and Otto introduced gradient flows in the Wasserstein space to prove existence and uniqueness of parabolic equations under very weak assumptions on the initial condition. In this talk, we show that this method provides a good framework to study well-posedness and the long time behavior of systems of parabolic equations coupled via transport problems. First, we are interested in gradient flow of functionals involving Optimal Transport problem. A simple example consists in solving two equations coupled through the solution to the very degenerate Monge-Ampère equation which may appear as an evolution model for cities. Then, we are going to add noise in the transport between populations which leads to study Wasserstein gradient flow of the Entropic Optimal Transport problem that means to solve systems of PDEs coupled through a Schrödinger system.