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. While 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.

中文翻译：

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具有强和弱内变松弛的耦合Cucker-Smale系统的流体动力学极限

在本文中，我们提出了一个多尺度系统的流体动力学极限，该系统描述了具有对齐相互作用和内部变量影响的两个代理群体的动力学。它由一个动力学方程和一个受热机械 Cucker-Smale (TCS) 模型启发的欧拉型方程组成。我们提出了一种新的阻力，用于流体-粒子相互作用，让人联想到斯托克斯定律。虽然宏观物种被认为是影响动力学物种的自组织背景流体，但后者被假定为稀疏且不影响宏观动力学。我们提出了两个双曲线尺度，就内部变量对背景群体的强和弱弛豫机制而言。在每种制度下，我们证明了对由两个欧拉型方程组成的耦合系统的严格流体动力学极限。对于强弛豫，动量和内部变量的惯性效应消失，而对于弱弛豫，内部变量的非平凡动力学出现。我们的分析涵盖了 Lipschitz 的情况和弱奇异影响函数。