Crowd dynamics and conservation laws with nonlocal constraints and capacity drop

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce a nonlocal constraint to restrict the flux at the exit to a maximum value p(ξ), where ξ is the weighted averaged instantaneous density of the crowd in an upstream vicinity of the exit. Choosing a non-increasing constraint function p(·), we are able to model the capacity drop phenomenon at the exit. Existence and stability results for the Cauchy problem with Lipschitz constraint function p(·) are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. In view of the construction of explicit examples (one is provided), we discuss the Riemann problem with discretized piecewise constant constraint p(·). We illustrate the fact that nonlocality induces loss of self-similarity for the Riemann solver; moreover, discretization of p(·) may induce non-uniqueness and instability of solutions.