Hydrodynamic limit of a coupled Cucker-Smale system with strong and weak internal variable relaxation

TL;DR

This paper derives the hydrodynamic limit of a multiscale agent system with alignment and internal variables, revealing different macroscopic behaviors under strong and weak relaxation regimes.

Contribution

It introduces a novel fluid-particle interaction model and rigorously analyzes the hydrodynamic limits for both relaxation regimes in a coupled kinetic-Euler system.

Findings

Hydrodynamic limits are established for both strong and weak internal variable relaxation.

Inertial effects vanish under strong relaxation, but internal variable dynamics persist under weak relaxation.

The analysis applies to both Lipschitz and weakly singular influence functions.

Abstract

In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. It consists of a kinetic equation coupled with an Euler-type equation inspired by the thermomechanical Cucker--Smale (TCS) model. We propose a novel drag force for the fluid-particle interaction reminiscent of Stokes' law. Whilst the macroscopic species is regarded as a self-organized background fluid that affects the kinetic species, the latter is assumed sparse and does not affect the macroscopic dynamics. We propose two hyperbolic scalings, in terms of a strong and weak relaxation regime of the internal variable towards the background population. Under each regime, we prove the rigorous hydrodynamic limit towards a coupled system composed of two Euler-type equations. Inertial effects of momentum and internal variable in the kinetic species disappear for strong relaxation, whereas a nontrivial dynamics for the internal variable appears for weak relaxation. Our analysis covers both the case of Lipschitz and weakly singular influence functions