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.

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 ρ ε =(1-ε)ρ+ερ ˜ instead of ρ ε =(id+εξ) # ρ. 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).

Santambrogio, Filippo 1

1 Laboratoire de Mathématiques d’Orsay Faculté des Sciences Université Paris-Sud XI 91405 Orsay cedex France 
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Santambrogio, Filippo. Gradient flows in Wasserstein spaces and applications to crowd movement. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 27, 16 p. http://archive.numdam.org/item/SEDP_2009-2010____A27_0/

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