自适应通信强度的欧拉对齐系统的微观、介观与宏观描述
Microscopic, mesoscopic, and macroscopic descriptions of the Euler alignment system with adaptive communication strength
AI总结：
本文为自适应通信强度的欧拉对齐系统建立微观、介观与宏观描述，通过平均场极限和极限区域推导严格联系，并证明动力学方程的Maxwellian弛豫，数值验证其定性类似Motsch-Tadmor模型。
摘要：
本文是我们先前关于$\st$-模型的联合工作的延续（见[《具有自适应通信强度的欧拉对齐系统的适定性与长时间行为》，已接收于Abel研讨会论文集，亦见arXiv:2310.00269, 2023]）。$\st$-模型由第一作者在[《环境平均》，EMS Surv. Math. Sci., 11 (2024), no. 2, 277413]中引入，是一种对齐模型，其对齐力强度$\st$沿平均速度场传输。强度的传输设计使得其允许一个$e$-量，$e = \partial_x u + \st$，该量在1D中控制正则性，类似于经典的Cucker-Smale情形。$\st$-模型的实用性在于其具有多样性，能够定性表现得像Motsch-Tadmor模型，而后者尚无全局正则性理论。\n\n本文旨在通过阐述和证明$\st$-模型所源自的微观和介观描述，将其置于更坚实的物理基础上。微观系统的一个显著特征是它是一个离散-连续系统：粒子的位置和速度是离散对象，而强度是一个主动的连续标量函数。我们通过平均场极限建立了从微观到介观描述的严格推导，并在单动能和Maxwellian极限区域中建立了从介观到宏观描述的严格推导。我们还通过建立当速度平均由Favre过滤给出时1D中向Maxwellian的弛豫，解决了动力学Fokker-Planck-Alignment方程的长时间行为。作为我们先前工作中已呈现的数值结果的补充，我们通过粒子模拟提供了额外的数值证据，表明$\st$-模型定性表现得像Motsch-Tadmor模型。
英文摘要：
This is a continuation of our previous joint work on the $\st$-model in[\textit{Well-posedness and long time behavior of the Euler Alignment System with adaptive communication strength}, accepted at the Abel Symposium Proceedings, also arXiv:2310.00269, 2023]. The $\st$-model, introduced by the first author in [\textit{Environmental averaging}. EMS Surv. Math. Sci., 11 (2024), no. 2, 277413],is an alignment model with the property that the strength of the alignment force, $\st$, is transported along an averaged velocity field. The transport of the strength is designed so that it admits an $e$-quantity, $e = \partial_x u + \st$, which controls regularity in 1D similarly to the classical Cucker-Smale case. The utility of the $\st$-model is that it has the versatility to behave qualitatively like the Motsch-Tadmor model, for which global regularity theory is not known. This paper aims to put the $\st$-model on firmer physical grounds by formulating and justifying the microscopic and mesoscopic descriptions from which it arises. A distinctive feature of the microscopic system is that it is a discrete-continuous system: the position and velocity of the particles are discrete objects, while the strength is an active continuum scalar function. We establish a rigorous passage from the microscopic to the mesoscopic description via the Mean Field Limit and from the mesoscopic to the macroscopic description in the monokinetic and Maxwellian limiting regimes. We also address the long-time behavior of the kinetic Fokker-Planck-Alignment equation by establishing the relaxation to the Maxwellian in 1D when the velocity averaging is given by the Favre filtration. As a supplement to the numerical results already presented in our previous work, we provide additional numerical evidence, via a particle simulation, that the $\st$-model behaves qualitatively like the Motsch-Tadmor model.
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