In the spirit of the macroscopic crowd motion models with hard congestion (i.e., a strong density
constraint
)
introduced by Maury et al. some years ago, we analyze a variant of the same models
where diffusion of the agents is also taken into account. From the modeling point of
view, this means that individuals try to follow a given spontaneous velocity, but are
subject to a Brownian diffusion, and have to adapt to a density constraint
which introduces a pressure term affecting the movement. From the point
of view of PDEs, this corresponds to a modified Fokker–Planck equation,
with an additional gradient of a pressure (only living in the saturated zone
) in
the drift. We prove existence and some estimates, based on optimal transport
techniques.
Keywords
diffusive crowd motion model, Fokker–Planck equation,
density constraint, optimal transportation
