Macroscopic traffic flow models Macroscopic traffic flow models describe the traffic flow as a liquid that is characterized by macroscopic quantities such as flow, density, and means speed of vehicles. Since connected and automated vehicles use sensors and can communicate their presence and state to other vehicles, they can "nudge" following vehicles and increase their speed. In [C5], [J5], we focused on proving the existence and uniqueness of classical solutions for a non-local conservation law on a ring-road with possible nudging (or ‘look behind’) terms. More specifically, we considered traffic models described by partial differential equations of the form
The first-order traffic model discussed above presents many limitations such as unrealistic vehicle accelerations or decelerations and predicts that the exit flow from a congested area is equal to the capacity flow. This does not apply to real traffic networks (capacity drop). These limitations motivate us to study the effects of non-local terms and in particular the effects of nudging for second-order models. Thus, in the papers [J7] and [J9] the following systems of partial differential equations were proposed that describe the macroscopic flow of vehicle traffic
and
Finally, in paper [J7] we proved that for model (2) without viscosity terms, if the density is sufficiently small, then the macroscopic model has a solution that exponentially approximates the equilibrium velocity (in the sup norm) while the density converges exponentially to a traveling wave.