'The Hughes' model for pedestrian flow
Seminar Room 1, Newton Institute
AbstractI will present a recent result in collaboration with Peter A Markowich, Jan F Pletschmann, and Marie T Wolfram (DAMTP, University of Cambridge) about the mathematical theory of the Hughes' model for the flow of pedestrians (cf. Hughes 2002), which consists of a continuity equation with logistic mobility coupled with an eikonal equation for the potential describing the common sense of the task.
We consider an approximation of such model in which the eikonal equation is replaced by an elliptic approximation.
For such an approximated model we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kruzkov, in which the boundary conditions are posed following the approach by Bardos et al. We use BV estimates on the density and stability estimates on the potential in order to prove uniqueness.
Moreover, we analyze the evolution of characteristics for the original Hughes' model in one space dimension and study the behaviour of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations.