Calculus of Variations and Geometric Measure Theory

M. Colombo - G. Crippa - E. Marconi - L. V. Spinolo

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

created by marconi on 09 Jun 2022



Inserted: 9 jun 2022
Last Updated: 9 jun 2022

Year: 2022


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.