Section: Application Domains

Congested Crowd motion

The volume preserving property appears naturally in this context where motion is constrained by the density of player.

• Optimal Mass Transportation and MFG theories can be an extremely powerful tool to attack some of these problems arising from spatial economics or to design new ones. For instance, various urban/traffic planning models have been proposed by Buttazzo, Santambrogio, Carlier ([10]   [40]   [32] ) in recent years.

• Many models from PDEs and fluid mechanics have been used to give a description of people or vehicles moving in a congested environment. These models have to be classified according to the dimension (1D model are mostly used for cars on traffic networks, while 2D models are most suitable for pedestrians), to the congestion effects (“soft” congestion standing for the phenomenon where high densities slow down the movement, “hard” congestion for the sudden effects when contacts occur, or a certain threshold is attained), and to the possible rationality of the agents Maury et al [69] recently developed a theory for 2D hard congestion models without rationality, first in a discrete and then in a continuous framework. This model produces a PDE that is difficult to attack with usual PDE methods, but has been successfully studied via Optimal Mass Transportation techniques again related to the JKO gradient flow paradigm.