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A simple model to handle the flow of people in emergency evacuationsituations is considered: at every point x, the velocity U(x) that individualsat x would like to realize is given. Yet, the incompressibility constraintprevents this velocity field to be realized and the actual velocity is theprojection of the desired one onto the set of admissible velocities. Instead oflooking at a microscopic setting (where individuals are represented by rigiddiscs), here the macroscopic approach is investigated, where the unknwon is theevolution of the density . If a gradient structure is given, say U is theopposite of the gradient of D where D is, for instance, the distance to theexit door, the problem is presented as a Gradient Flow in the Wasserstein spaceof probability measures. The functional which gives the Gradient Flow isneither finitely valued (since it takes into account the constraints on thedensity), nor geodesically convex, which requires for an ad-hoc study of theconvergence of a discrete scheme.

A MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE