A Volterra equation approach to the local limit of nonlocal traffic models

Abstract

We consider a class of nonlocal conservation laws modeling traffic flow, given by ${\partial }_t{u}_{\epsilon }+{\partial }_x(V(u_{\epsilon }\ast {\gamma }_{\epsilon })u_{\epsilon })=0$ with ${\gamma }_{\epsilon }(\cdot ):={\epsilon }^{-1}\gamma (\cdot /\epsilon )$ for a suitable convex convolution kernel $\gamma$. Since the work of Colombo et al. (Arch. Ration. Mech. Anal., 2023), thanks to uniform ${\mathrm{L}}^{\infty }$- and TV-estimates, it is known that $w_{\epsilon }:=u_{\epsilon }\ast {\gamma }_{\epsilon }$ converges to the entropy solution of the local scalar conservation law ${\partial }_{t}u+{\partial }_x(V(u)u)=0$ as $\epsilon \searrow 0$. However, the convergence of $\{u_{\epsilon }{\}}_{\epsilon >0}$ itself has not been fully addressed so far. In this direction, a known result applies specifically to the case of an exponential kernel, where the identity $\epsilon {\partial }_x{w}_{\epsilon }=w_{\epsilon }-u_{\epsilon }$ is fundamental. In this work, we address this gap in the literature and prove that $\{u_{\epsilon }{\}}_{\epsilon >0}$ converges to the same limit $u$ under the mild additional assumption that the initial datum belongs to ${\mathrm{L}}^1(\mathbb{R})$. Our analysis exploits, through a Fourier approach, the stability properties of the more general Volterra-type equation ${\partial }_x{w}_{\epsilon }={\gamma }_{\epsilon }^{\prime }\ast u_{\epsilon }-{\gamma }_{\epsilon }(0)u_{\epsilon }$, thereby deducing the convergence of $u_{\epsilon }$ from that of $w_{\epsilon }$.

6 pages