Abstract:
In this talk, we present some recent results on nonlocal regularizations of (scalar) hyperbolic conservation laws, where the flux function depends on the solution through the convolution with a given kernel. These models are widely used to describe vehicular traffic, where each car adjusts its velocity based on a weighted average of the traffic density ahead.
First, we establish the existence, uniqueness, and maximum principle for solutions of the nonlocal problem under mild assumptions on the kernel and flux function. We then investigate the convergence of the solution to the entropy admissible one of the corresponding local conservation law when the nonlocality is shrunk to a local evaluation (i.e., when the rescaled kernel tends to a Dirac delta distribution). For convex kernels (and, in particular, for kernels of exponential type), we analyze this singular limit first for initial data of bounded variation, and then, using Oleinik-type estimates, for merely bounded ones. We then introduce a suitable Godunov-type numerical scheme for the nonlocal problem and study its asymptotic compatibility: we prove its convergence, with a rate, as the mesh size and the nonlocal parameter tend to zero, to the entropy solution of the local scalar conservation law. Finally, we present a recent breakthrough: for some classes of non-convex kernels, we employ tools from the theory of compensated compactness to establish the nonlocal-to-local convergence.
This talk is based on several papers written in recent years in collaboration with the following coauthors: M. Colombo, G. M. Coclite, J.-M. Coron, G. Crippa, K. Huang, A. Keimer, E. Marconi, L. Pflug, L. Spinolo, and E. Zuazua.
Bio:
Nicola De Nitti earned his Ph.D. in Mathematics, summa cum laude, from FAU Erlangen-Nürnberg in July 2023 under the supervision of Enrique Zuazua. He was a postdoctoral researcher in Maria Colombo's group at EPFL from September 2023 to March 2025. Since April 2025, he has been a tenure-track assistant professor in mathematical analysis at the University of Pisa.
