On Traffic Flow with Nonlocal Flux: a Relaxation Representation

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: $w_\varepsilon(s)=\varepsilon ^{-1} e^{-s/\varepsilon}$. By a transformation of coordinates, the problem can be reformulated as a $2\times 2$ hyperbolic system with relaxation. Uniform BV bounds on the solution are thus obtained, independent of the scaling parameter $\varepsilon$. Letting $\varepsilon\to 0$, 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.

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