Critical Thresholds in Relaxation Systems with Resonance of Characteristic Speeds

In this paper, we consider hyperbolic relaxation systems arising from dynamic continuum traffic flow models including the well-known Payne and Whitham (PW) model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in many physical scenarios in traffic flow, for which the usual subcharacteristic condition only marginally holds. 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 systems. We identify a lower threshold for finite time singularity in solutions and an upper threshold for the global existence of the smooth solution for two traffic models with different choices of traffic pressure and equilibrium velocity. The obtained thresholds are represented in terms of the initial slopes of the Riemann invariants and the initial density. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope.