Conservation Laws in the Modeling of Moving Crowds

which depends Lipschitz continuously from the data and, by [12, Theorem 2.6], also from v and ~v. According to (2), at time t the pedestrian at x moves along a prescribed trajectory, an integral curve of ~v, with a speed v(ρ) that depends on ρ evaluated at point x and time t. On the contrary, Section 2 is devoted to (1) with the speed of the individual at x depending on an average of the density ρ in a neighborhood of x. The resulting model has a rich analytical structure, the solutions being also differentiable with respect to the data and to the speed law. In Section 3 the direction chosen by the pedestrian at x depends from an average of the density gradient ∇ρ around x, while his/her speed depends from ρ evaluated at x. The resulting solutions display qualitative properties usually seen in context