M. Di Francesco

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

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

Deterministic particle approximation of scalar conservation laws

{\mathbf{L^{1}_{loc}}}

Condensation phenomena in nonlinear drift equations

Lloc1) to the unique Kružkov entropy solution of the conservation law. The initial data are taken in

Deterministic particle approximation of the Hughes model in one space dimension

{\mathbf{L}^\infty}

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

L∞, 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