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_\varepsilon + \partial_x(V(u_\varepsilon \ast γ_\varepsilon)\, u_\varepsilon) = 0 $ with $ γ_\varepsilon(\cdot) := \varepsilon^{-1}γ(\cdot/\varepsilon) $ for a suitable convex convolution kernel $γ$. 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_\varepsilon := u_\varepsilon \ast γ_\varepsilon $ converges to the entropy solution of the local scalar conservation law $ \partial_t u + \partial_x(V(u)\, u) = 0 $ as $\varepsilon \searrow 0$. However, the convergence of $ \{u_\varepsilon\}_{\varepsilon > 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 $ \varepsilon \partial_x w_\varepsilon = w_\varepsilon - u_\varepsilon $ is fundamental. In this work, we address this gap in the literature and prove that $ \{u_\varepsilon\}_{\varepsilon > 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 $\partial_xw_\varepsilon=γ'_\varepsilon\ast u_\varepsilon-γ_\varepsilon(0)u_\varepsilon$, thereby deducing the convergence of $u_\varepsilon$ from that of $w_\varepsilon$.