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Archive for Rational Mechanics and Analysis

, Volume 217, Issue 3, pp 831–871 | Cite as

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

  • M. Di Francesco
  • M.D. Rosini
Article

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.

Keywords

Weak Solution Entropy Solution Entropy Inequality Lebesgue Dominate Convergence Theorem Rigorous Derivation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Information Engineering, ComputerScience, and MathematicsUniversity of L’AquilaL’AquilaItaly
  2. 2.ICM, University of WarsawWarsawPoland

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