Gradient flows in Wasserstein spaces and applications to crowd movement

Santambrogio, Filippo¹

¹ Laboratoire de Mathématiques d’Orsay Faculté des Sciences Université Paris-Sud XI 91405 Orsay cedex France

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 27, 16 p.

Résumé

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${ℝ}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step are obtained by looking at perturbation of the form ${\rho }_{\epsilon }=(1-\epsilon )\rho +\epsilon \tilde{\rho }$ instead of ${\rho }_{\epsilon }={(id+\epsilon \xi )}_{\#}\rho$. The ideas to make this approach rigorous are presented in the case of a Fokker-Planck equation, possibly with an interaction term, and then the paper is concluded by a section, where this method is applied to the original problem of crowd motion (referring to a recent paper in collaboration with B. Maury and A. Roudneff-Chupin for the details).