On entropy weak solutions of Hughes' model for pedestrian motion (Q353369)

The authors study a generalization of Hughes model for pedestrian flow, described by a conservation law for the number of pedestrians coupled with an eikonal equation giving the direction of the flow. This model reduces to the conservation law \(\rho_t+F(t,x,\rho)_x=0\), \(t>0\), \(x\in (-1,1)\), with the flux \(F(t,x,\rho)\) discontinuous on a curve \(x=\xi(t)\), which depends non-locally on \(\rho\) through the solution of the eikonal equation \(|\phi_x|=c(\rho)\). The notion of entropy weak solutions to the initial-boundary problem for this equation is introduced and some qualitative properties of entropy solutions are derived (including the maximum principle). The authors describe the structure of the solutions for Riemann-type initial data and study the optimization of evacuation time depending on the cost function \(c(\rho)\). All results are illustrated by numerical simulations.