On existence, stability and many-particle approximation of solutions of 1D Hughes' model with linear costs

This paper deals with the one-dimensional formulation of Hughes’ model for pedestrian flows in the setting of entropy solutions. In this model, the mass conservation equation for the pedestrian density authorizes non-classical shocks at the location of the so-called turning curve. We consider linear (more precisely, affine) cost functions, whose slopes $\alpha\geq0$ correspond to different crowd behaviours. We prove for the first time an existence result in the framework of entropy solutions, for general data. Differently from the partial existence results available in the literature, our existence result allows for the possible presence of non-classical shocks. The proofs are based on a sharply formulated many-particle approximation scheme, with careful treatment of interactions of particles with the turning curve. First, we rigorously establish the well-posedness of this many-particle scheme. Then we develop a local compactness argument that permits to circumvent the lack of available BV bounds in a vicinity of the turning curve, while proving consistency of the approximation scheme with the entropy formulation.