On the Hughes Model and Numerical Aspects
  We study a crowd model proposed by R. Hughes and we describe a numerical approach to solve it. The Hughes model comprises a Fokker-Planck equation coupled with an eikonal equation with Dirichlet or Neumann data. First, we establish a priori estimates for the solutions. Second, we study radial solutions and identify a shock formation mechanism. Third, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. Finally, we propose a new numerical method and consider two numerical examples. 
Diogo A. Gomes, Roberto M. Velho
math.AP math.OC