*Preprint*

**Inserted:** 27 nov 2010

**Year:** 2010

**Notes:**

This paper is meant to appear in the proceedings of the PDE Seminar of Ecole Polytechnique, where I gave a talk about gradient flow in Wasserstein spaces and applications to crowds. In particular, I detail(ed) the approach by ``vertical'' perturbations for optimality conditions that is complementary to the classical one. Here I only show easy examples where such an approach quickly gives an existence result without any convexity assumptions, and the intention is to undergo a deeper analysis later on, in a possible forthcoming paper

**Abstract:**

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-SavarÃ©, 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 ``additive'' form. 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).

**Keywords:**
Optimal transport, minimizing movements, Congestion, Necessary conditions for optimality

**Download:**