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_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.