Deterministic Particle Models for Scalar Conservation Laws and Related Models

Abstract:
Deterministic follow-the-leader Lagrangian particle schemes are well known to be a very efficient tool to approximate traffic flow and pedestrian flow PDE models. I will recall a set of recent results in collaboration with M. D. Rosini (Ferrara), S. Fagioli (L’Aquila), E. Radici (L’Aquila), G. Stivaletta (L’Aquila), and G. Russo (Catania) on the rigorous formulation via many particle limits in this setting for a relatively wide set of problems including Cauchy problems and Initial-Boundary-Value problems for scalar conservation laws, second order models for traffic flow, the Hughes model for pedestrian movements, nonlocal transport equations with nonlinear mobility and external potentials. These results are based on BV estimates for the approximating piece-wise constant density. A relevant issue in this procedure is ability to detecting the L^infinity - BV smoothing effect of a scalar conservation law under a suitable convexity assumption for the flux. In all cases, the scheme allows to solve the target PDE problem in the classical entropy sense.