On Traffic Flow with Nonlocal Flux: A Relaxation Representation

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35 Scopus citations


We consider a conservation law model of traffic flow, where the velocity of each car depends on a weighted average of the traffic density ρ ahead. The averaging kernel is of exponential type: wε(s) = ε- 1e-s/ε. By a transformation of coordinates, the problem can be reformulated as a 2 × 2 hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter ε. Letting ε→ 0 , the limit yields a weak solution to the corresponding conservation law ρt+ (ρv(ρ)) x= 0. In the case where the velocity v(ρ) = a- bρ is affine, using the Hardy–Littlewood rearrangement inequality we prove that the limit is the unique entropy-admissible solution to the scalar conservation law.

Original languageEnglish (US)
Pages (from-to)1213-1236
Number of pages24
JournalArchive for Rational Mechanics and Analysis
Issue number3
StatePublished - Sep 1 2020

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering


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