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We prove that the unique entropy solution to a scalar nonlinear conservationlaw with strictly monotone velocity and nonnegative initial condition can berigorously obtained as the large particle limit of a microscopicfollow-the-leader type model, which is interpreted as the discrete Lagrangianapproximation of the nonlinear scalar conservation law. More precisely, weprove that the empirical measure (respectively the discretised density)obtained from the follow-the-leader system converges in the 1-Wassersteintopology (respectively in $L^1_{loc}$) to the unique Kruzkov entropy solutionof 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 densitieswith vacuum. Our result holds for a reasonably general class of velocity maps(including all the relevant examples in the applications, e.g. in theLighthill-Whitham-Richards model for traffic flow) with possible degenerateslope near the vacuum state. The proof of the result is based on discrete BVestimates and on a discrete version of the one-sided Oleinik-type condition. Inparticular, we prove that the regularizing effect $L^1\cap L^\infty \mapsto BV$for nonlinear scalar conservation laws is intrinsic of the discrete model.

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