Stability of quasi-entropy solutions of non-local scalar conservation laws

TL;DR

This paper establishes the stability, existence, and uniqueness of entropy solutions for non-local scalar conservation laws, providing a framework to analyze numerical schemes and convergence rates for traffic flow models.

Contribution

It introduces a general stability theorem for non-local conservation laws, including approximate solutions from numerical schemes, and derives convergence rates for particle methods.

Findings

Proved stability of entropy solutions under perturbations.

Established conditional existence and uniqueness for non-local flux problems.

Derived convergence rates for particle methods in traffic models.

Abstract

We prove the stability of entropy solutions of nonlinear conservation laws with respect to perturbations of the initial datum, the space-time dependent flux and the entropy inequalities. Such a general stability theorem is motivated by the study of problems in which the flux $P[u](t,x,u)$ depends possibly non-locally on the solution itself. For these problems we show the conditional existence and uniqueness of entropy solutions. Moreover, the relaxation of the entropy inequality allows to treat approximate solutions arising from various numerical schemes. This can be used to derive the rate of convergence of the recent particle method introduced in [Radici-Stra 2021] to solve a one-dimensional model of traffic with congestion, as well as recover already known rates for some other approximation methods.