On Traffic Flow with Nonlocal Flux: A Relaxation Representation

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 $\rho$ 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 $\varepsilon \rightarrow 0$, the limit yields a weak solution to the corresponding conservation law $\rho _t + ( \rho v(\rho ))_x=0$. In the case where the velocity $v(\rho )= a-b\rho$ is affine, using the Hardy–Littlewood rearrangement inequality we prove that the limit is the unique entropy-admissible solution to the scalar conservation law.