On strong local alignment in the kinetic Cucker-Smale model

TL;DR

This paper rigorously justifies the limit of the Motsch-Tadmor flocking model as the interaction radius shrinks, establishing the strong local alignment in the kinetic Cucker-Smale model through advanced mathematical analysis.

Contribution

It provides a rigorous mathematical proof that the kinetic Cucker-Smale model with strong local alignment is the limit of the Motsch-Tadmor model as the interaction radius approaches zero.

Findings

Established the limit of the Motsch-Tadmor model to the strong local alignment model.

Used velocity averaging lemmas and $L^p$ estimates for the analysis.

Confirmed the formal motivation of the strong local alignment term.

Abstract

In two recent papers the authors study the existence of weak solutions and the hydrodynamic limit of kinetic flocking equations with strong local alignment. The introduction of a strong local alignment term to model flocking behavior was formally motivated in these papers as a limiting case of an alignment term proposed by Motsch and Tadmor. In this paper, we rigorously justify this limit, and show that the considered equation is indeed a limit of the Motsch-Tadmor model when the radius of interaction goes to zero. The analysis involves velocity averaging lemmas and several $L^p$ estimates.