On reduced inertial PDE models for Cucker-Smale flocking dynamics

TL;DR

This paper introduces a new reduced PDE model for Cucker-Smale flocking that depends only on particle position, providing a direct link to particle dynamics and offering analytical and numerical insights.

Contribution

The paper presents a novel reduced inertial PDE model for flocking that is not derived from the mean-field limit but directly reduces empirical densities at the particle level.

Findings

The reduced PDE satisfies a natural flocking definition.

Quantification of the discrepancy between PDE and particle system in specific cases.

Numerical simulations support the theoretical analysis.

Abstract

In particle systems, flocking refers to the phenomenon where particles' individual velocities eventually align. The Cucker-Smale model is a well-known mathematical framework that describes this behavior. Many continuous descriptions of the Cucker-Smale model use PDEs with both particle position and velocity as independent variables, thus providing a full description of the particles mean-field limit (MFL) dynamics. In this paper, we introduce a novel reduced inertial PDE model consisting of two equations that depend solely on particle position. In contrast to other reduced models, ours is not derived from the MFL, but directly includes the model reduction at the level of the empirical densities, thus allowing for a straightforward connection to the underlying particle dynamics. We present a thorough analytical investigation of our reduced model, showing that: firstly, our reduced PDE satisfies a natural and interpretable continuous definition of flocking; secondly, in specific cases, we can fully quantify the discrepancy between PDE solution and particle system. Our theoretical results are supported by numerical simulations.