Error Estimation in the Mean-Field Limit of Kinetic Flocking Models with Local Alignments

TL;DR

This paper develops a particle system with moderate interactions to accurately approximate kinetic flocking models with singular forces and local alignments, establishing error bounds for the mean-field limit.

Contribution

It introduces a novel particle system with regularized singular forces and local alignment approximations, providing rigorous error estimates for the mean-field convergence.

Findings

Existence of weak solutions for flocking equations.

Error estimates for the mean-field approximation.

Effective particle system with logarithmic parameter scaling.

Abstract

In this paper, we present an innovative particle system characterized by moderate interactions, designed to accurately approximate kinetic flocking models that incorporate singular interaction forces and local alignment mechanisms. We establish the existence of weak solutions to the corresponding flocking equations and provide an error estimate for the mean-field limit. This is achieved through the regularization of singular forces and a nonlocal approximation strategy for local alignments. We show that, by selecting the regularization and localization parameters logarithmically with respect to the number of particles, the particle system effectively approximates the mean-field equation.