Nonlocal regularizations of hyperbolic conservation laws
数学专题报告
报告题目(Title)：Nonlocal regularizations of hyperbolic conservation laws
报告人(Speaker)：Nicola De Nitti 助理教授（Università di Pisa）
地点(Place)：后主楼1223
时间(Time)：2025年12月10日（周三）16:30-17:30
邀请人(Inviter)：许孝精
报告摘要
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. 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 for initial data with possibly unbounded variation.
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.
