Critical Thresholds in a Relaxation Model for Traffic Flows

In this paper, we consider a hyperbolic relaxation system arising from a dynamic continuum traffic flow model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in this model. Thus the usual sub-characteristic condition only holds marginally. In spite of this obstacle, we prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation system. We identify five upper thresholds for finite time singularity in solutions and three lower thresholds for global existence of smooth solutions. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope. Our results show that the shorter the drivers’ responding time to the traffic, the larger the set of initial conditions leading to global smooth solutions which correctly predicts the empirical findings for traffic flows.