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A model for the formation and evolution of traffic jams. (English) Zbl 1153.90003

The authors establish and analyse a traffic flow model which describes the formation and dynamics of traffic jams. It consists of a pressureless gas dynamics system under a maximal constraint on the density and is derived through a singular limit of the Aw-Rascle model, where – in contrast to the model mentioned before – the pseudo-pressure \(p(n)\) tends to infinity as the density \(n\) of cars approaches the maximal density \(n^*\). In this model the velocity at which perturbations of traffic in front propagate backwards tend to infinity when \(n\to n^*\) more congestion will imply shorter reaction time of drivers yielding increasing propagation velocity.
Assuming additionally that the velocity offset (the difference between the velocity on an empty road and the actual velocity) is infinitesimally small as long as traffic is not congested but becomes large immediately when the traffic reaches a congestion state, the authors draw the conclusion that in the limit this leads to the pressureless gas dynamics model.
From this analysis, they deduce the particular dynamical behavior of clusters (or traffic jams), defined as intervals where the density limit is reached. An existence result for a generic class of initial data is proved by means of an approximation of the solution by a sequence of clusters. Finally, numerical simulations are produced.

MSC:

90B20 Traffic problems in operations research
68U20 Simulation (MSC2010)
70K70 Systems with slow and fast motions for nonlinear problems in mechanics
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