Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Critical thresholds in a relaxation system with resonance of characteristic speeds

Pages: 511 - 521, Volume 24, Issue 2, June 2009      doi:10.3934/dcds.2009.24.511

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Tong Li - Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, United States (email)
Hailiang Liu - Department of Mathematics, Iowa State University, Ames, IA 50011, United States (email)

Abstract: We study critical threshold phenomena in a dynamic continuum traffic flow model known as the Payne and Whitham (PW) model. This model is a quasi-linear hyperbolic relaxation system, and when equilibrium velocity is specifically associated with pressure, the equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system. For a scenario of physical interest we identify a lower threshold for finite time singularity in solutions and an upper threshold for the global existence of the smooth solution. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope.

Keywords:  Critical thresholds, singularity formation, quasi-linear relaxation model, global regularity, traffic flow.
Mathematics Subject Classification:  Primary: 35B30, 35B40; Secondary: 90B20.

Received: November 2007;      Revised: August 2008;      Available Online: March 2009.