# Reduced fluid models for self-propelled particles interacting through alignment

## TL;DR

This paper derives simplified fluid models for self-propelled particles with alignment interactions by analyzing the asymptotic behavior of kinetic models under large alignment frequency, revealing a reduction to measures on a velocity sphere.

## Contribution

It introduces a novel asymptotic analysis framework for kinetic models with alignment, leading to macroscopic fluid equations supported on a velocity sphere.

## Key findings

- Derivation of macroscopic fluid models from kinetic equations.
- Identification of measures supported on a velocity sphere.
- Application of averaging techniques from magnetic confinement studies.

## Abstract

The asymptotic analysis of kinetic models describing the behavior of particles interacting through alignment is performed. We will analyze the asymptotic regime corresponding to large alignment frequency where the alignment effects are dominated by the self propulsion and friction forces. The former hypothesis leads to a macroscopic fluid model due to the fast averaging in velocity, while the second one imposes a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity space. The analysis relies on averaging techniques successfully used in the magnetic confinement of charged particles. The limiting particle distribution is supported on a sphere, and therefore we are forced to work with measures in velocity. As for the Euler-type equations, the fluid model comes by integrating the kinetic equation against the collision invariants and its generalizations in the velocity space. The main difficulty is their identification for the averaged alignment kernel in our functional setting of measures in velocity.