Small inertia limit for coupled kinetic swarming models

TL;DR

This paper rigorously analyzes the small-inertia limit in multi-species kinetic models of swarming, establishing convergence to macroscopic equations under various interaction assumptions.

Contribution

It provides a rigorous connection between kinetic and macroscopic models for swarming by analyzing the small-inertia limit with different regularity conditions on interaction kernels.

Findings

Proves existence of solutions to the kinetic system.

Establishes uniform estimates with respect to inertia.

Shows convergence to macroscopic models as inertia tends to zero.

Abstract

We investigate various versions of multi-dimensional systems involving many species, modeling aggregation phenomena through nonlocal interaction terms. We establish a rigorous connection between kinetic and macroscopic descriptions by considering the small-inertia limit at the kinetic level. The results are proven either under smoothness assumptions on all interaction kernels or under singular assumptions for \emph{self-interaction} potentials. Utilizing different techniques in the two cases, we demonstrate the existence of a solution to the kinetic system, provide uniform estimates with respect to the inertia parameter, and show convergence towards the corresponding macroscopic system as the inertia approaches zero.