2014/04/30 by Marco Di Francesco, Massimiliano D. Rosini · 1 citation
Engineering · Mathematics · #Conservation law #Degenerate energy levels #Entropy (arrow of time) #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows #Geometry #Limit (mathematics) #Mathematical analysis #Mathematics #Monotone polygon #Monotonic function #Navier-Stokes equation solutions #Nonlinear system #Physics #Quantum mechanics #Scalar (mathematics) #math.AP
paper · pdf · doi:10.1007/s00205-015-0843-4
arxiv created 2015/01/18 · openalex publication_date 2015/01/29 · arxiv updated 2015/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 L1loc) to the unique Kruzkov entropy solution of the conservation law. The initial data are taken in L1∩ L^∞, 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 L1∩ L^∞ ↦ BV for nonlinear scalar conservation laws is intrinsic of the discrete model.