Hydrodynamic limit of the kinetic Cucker-Smale flocking model

TL;DR

This paper rigorously analyzes the hydrodynamic limit of a kinetic Cucker-Smale flocking model with added noise and confinement, showing convergence to an Euler-type flocking system using entropy methods.

Contribution

It provides a rigorous derivation of the Euler-type flocking system as the singular limit of a kinetic model with strong local alignment and noise.

Findings

Convergence of the kinetic model to an Euler-type system.

Inclusion of strong local alignment and noise effects.

Use of entropy methods for proof.

Abstract

The hydrodynamic limit of a kinetic Cucker-Smale model is investigated. In addition to the free-transport of individuals and the Cucker-Smale alignment operator, the model under consideration includes a strong local alignment term. This term was recently derived as the singular limit of an alignment operator due to Motsch and Tadmor. The model is enhanced with the addition of noise and a confinement potential. The objective of this work is the rigorous investigation of the singular limit corresponding to strong noise and strong local alignment. The proof relies on the relative entropy method and entropy inequalities which yield the appropriate convergence results. The resulting limiting system is an Euler-type flocking system.