Singular limit for a class of nonlocal conservation laws via compensated compactness

TL;DR

This paper proves the convergence of solutions of nonlocal traffic flow models to local conservation laws using compensated compactness, without relying on total variation bounds, for a class of kernels including non-convex ones.

Contribution

It introduces a novel approach to establish compactness and convergence for nonlocal conservation laws without total variation bounds, covering non-convex kernels and general initial data.

Findings

Established strong L^1_loc convergence of solutions as kernels concentrate to delta functions.

Proved convergence for piecewise constant and strictly monotone kernels.

Resolved a long-standing open problem on nonlocal-to-local limits for non-convex kernels.

Abstract

We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t u_\varepsilon + \partial_x(V(u_\varepsilon \ast \gamma_\varepsilon) u_\varepsilon) = 0$, with a rescaled convolution kernel $\gamma_\varepsilon(\cdot) := \varepsilon^{-1}\gamma(\cdot/\varepsilon)$. We establish the strong $\mathrm L^1_{\mathrm{loc}}$-convergence of weak solutions $u_\varepsilon$ toward the entropy-admissible solution of the corresponding local conservation law as the kernel $\gamma_\varepsilon$ concentrates to a Dirac delta distribution when $\varepsilon \searrow 0$. In contrast to previous literature, we obtain compactness of the family $\{u_\varepsilon \ast \gamma_\varepsilon\}_{\varepsilon>0}$ without relying on total variation bounds or Ole\u{\i}nik-type estimates. Instead, we establish $\mathrm L^2$-type bounds on its entropy production and use the theory of compensated compactness, assuming that the initial datum merely belongs to $\mathrm L^1\cap \mathrm L^\infty$. Our results are twofold. First, we establish the nonlocal-to-local limit for the piecewise constant kernel $\gamma(\cdot) := {1}_{[-1,0]}(\cdot)$ combined with the affine velocity function from Greenshields' traffic model. Second, we prove the limit for strictly monotone kernels along with decreasing velocity functions. These results settle a long-standing open problem concerning the nonlocal-to-local convergence for non-convex kernels.