On reduced inertial PDE models for Cucker–Smale flocking dynamics

In particle systems, flocking refers to the phenomenon where particles’ individual velocities eventually align. The Cucker–Smale (CS) model is a well-known mathematical framework that describes this behaviour. Many continuous descriptions of the CS 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.