Crowd movements: gradient flow in Wasserstein spaces under density constraints.

Crowd movements: gradient flow in Wasserstein spaces under density constraints.

A crowd is located in a room and everybody wants to exit, thus moving towards, say, a single small exit. Yet, the density r of the crowd cannot exceed a fixed bound (r £ 1). The actual velocity that every agent may realize is hence obtained in the following way: take the speed they would like (the unit vector pointing to the door), and project the whole velocity field in L²(r) on the cone of feasible velocities, i.e. those who have a positive divergence on the set where the density already saturates the constraint.

This evolution is actually a gradient flow in the space W₂ of probability densities for the functional which associates to any r satisfying the density constraint the mean value òD dr, where D is the distance function to the exit, and +¥ to any probability violating the same constraint.

This gradient flow approach allows to give existence results for a problem that had previously been studied in a microscopical way (with people represented by small disks, and the density constraint replaced by a non-overlap condition).

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