On Strong Local Alignment in the Kinetic Cucker-smale Model

In the recent papers [4, 5] 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 [6]. In this paper, we rigorously justify this limit, and show that the equation considered in [4, 5] 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 Lp estimates.