Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit

TL;DR

This paper rigorously derives the unique entropy solution of a scalar nonlinear conservation law as the large particle limit of a follow-the-leader microscopic model, applicable to traffic flow and similar systems.

Contribution

It provides a rigorous proof connecting microscopic follow-the-leader models to macroscopic scalar conservation laws, including cases with vacuum and degenerate velocities.

Findings

Empirical measures from the particle system converge to the entropy solution.

The discrete model exhibits intrinsic BV regularization effects.

The approach applies to general velocity functions, including classical traffic models.

Abstract

We prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in $L^1_{loc}$) to the unique Kruzkov entropy solution of the conservation law. The initial data are taken in $L^1\cap L^\infty$, nonnegative, and with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, e.g. in the Lighthill-Whitham-Richards model for traffic flow) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect $L^1\cap L^\infty \mapsto BV$ for nonlinear scalar conservation laws is intrinsic of the discrete model.