### Abstract

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: w_{ε}(s) = ε^{- 1}e^{-}^{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 ρ_{t}+ (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 language | English (US) |
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Pages (from-to) | 1213-1236 |

Number of pages | 24 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 237 |

Issue number | 3 |

DOIs | |

State | Accepted/In press - Jan 1 2020 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

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## Cite this

*Archive for Rational Mechanics and Analysis*,

*237*(3), 1213-1236. https://doi.org/10.1007/s00205-020-01529-z