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Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

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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 \({\mathbf{L^{1}_{loc}}}\)) to the unique Kružkov entropy solution of the conservation law. The initial data are taken in \({\mathbf{L}^\infty}\), nonnegative, and with compact support, hence we are able to handle densities with a vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, for example in the Lighthill-Whitham-Richards model for traffic flow) with a possible degenerate slope near the vacuum state. The proof of the result is based on discrete \({\mathbf{BV}}\) estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect \({\mathbf{L}^\infty \mapsto \mathbf{BV}}\) for nonlinear scalar conservation laws is intrinsic to the discrete model.

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Authors and Affiliations

  1. Department of Information Engineering, ComputerScience, and Mathematics, University of L’Aquila, Via Vetoio 1, Coppito, 67100, L’Aquila, Italy

    M. Di Francesco

  2. ICM, University of Warsaw, ul. Prosta 69, 00-838, Warsaw, Poland

    M.D. Rosini

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  1. M. Di Francesco
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Correspondence to M. Di Francesco.

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Communicated by A. Bressan

M. Di Francesco is supported by the Marie Curie CIG (Career Integration Grant) DifNonLoc - Diffusive Partial Differential Equations with Nonlocal Interaction in Biology and Social Sciences and by the Ministerio de Ciencia e Innovación, grant MTM2011-27739-C04-02. M.D. Rosini is supported by ICM. Projekt został sfinansowany ze środków Narodowego Centrum Nauki przyznanych na podstawie decyzji nr: DEC-2011/01/B/ST1/03965.

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Di Francesco, M., Rosini, M. Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit. Arch Rational Mech Anal 217, 831–871 (2015). https://doi.org/10.1007/s00205-015-0843-4

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