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

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.