Entropy Admissibility of the Limit Solution for a Nonlocal Model of Traffic Flow

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: $w_{\epsilon }(s)={\epsilon }^{-1}{e}^{-s/\epsilon }$. For any decreasing velocity function $v$, we prove that, as $\epsilon \to 0$, the limit of solutions to the nonlocal equation coincides with the unique entropy-admissible solution to the scalar conservation law ${\rho }_t+(\rho v(\rho ){)}_x=0$.