Nonlocal traffic models with general kernels: singular limit, entropy admissibility, and convergence rate

TL;DR

This paper studies the convergence of nonlocal traffic flow models with general kernels to local conservation laws, establishing conditions for convergence, entropy admissibility, and convergence rates, with implications for traffic modeling accuracy.

Contribution

It extends convergence results to more general kernels in traffic models, introducing natural assumptions and a criterion for entropy admissibility.

Findings

Established convergence under natural assumptions and convexity of kernels.

Provided a criterion for entropy admissibility of the limit solution.

Demonstrated the necessity of convexity with a counter-example.

Abstract

Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counter-example showing that the convexity assumption is necessary for our main compactness estimate.