# Preprint 2014-007

# Rigorous derivation of the Lighthill–Whitham–Richards model from the follow the-leader model as many particle limit

## Marco Di Francesco and Massimiliano D. Rosini

**Abstract:**
We prove that the unique entropy solution to the macroscopic
Lighthill–Witham–Richards model for traffic flow
can be rigorously obtained as the large particle limit
of the microscopic follow-the-leader model,
which is interpreted as the discrete Lagrangian approximation of the former.
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 $L^1_{\mathrm{loc}}$)
to the unique entropy solution of the Lighthill–Witham–Richards equation
in the Kruzkov sense.
The initial data are taken in $L^\infty$ 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)
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 $L^\infty\to\mathrm{BV}$
regularizing effect is
intrinsic of the discrete model.