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We also give the detailed proof of some results announced in the paper [Andreianov, Donadello, Rosini,

*Crowd dynamics and conservation laws with nonlocal constraints and capacity drop*], which is devoted to existence and stability for a more general class of Cauchy problems subject to Lipschitz continuous non--local point constraints.

In this paper we present a macroscopic phase transition model with a local point constraint on the flow. Its motivation is, for instance, the modelling of the evolution of vehicular traffic along a road with pointlike inhomogeneities characterized by limited capacity, such as speed bumps, traffic lights, construction sites, toll booths, etc. The model accounts for two different phases, according to whether the traffic is low or heavy. Away from the inhomogeneities of the road the traffic is described by a first order model in the free-flow phase and by a second order model in the congested phase. To model the effects of the inhomogeneities we propose two Riemann solvers satisfying the point constraints on the flow.

In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of the unknown function at the node of the graph and, moreover, the diffusivity can be degenerate. Our main result concerns a necessary and sufficient algebraic condition for the existence of traveling waves in the graph. We also study in great detail some examples corresponding to quadratic and logarithmic flux functions, for different diffusivities, to which our results apply.

We consider the follow-the-leader approximation of the Aw-Rascle-Zhang (ARZ) model for traffic flow in a multi population formulation. We prove rigorous convergence to weak solutions of the ARZ system in the many particle limit in presence of vacuum. The result is based on uniform ${\mathbf{BV}}$ estimates on the discrete particle velocity. We complement our result with numerical simulations of the particle method compared with some exact solutions to the Riemann problem of the ARZ system.

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