In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels $\phi (x) := {\lvert x \rvert}^{- (n + \alpha)}$ for $\alpha \in (0,2)$. The solutions describe unidirectional parallel motion of agents governing multi-dimensional collective behavior of flocks. Here, we consider the range $1\lt \alpha \lt 2$ and establish the global regularity of smooth solutions, together with a full description of their long-time dynamics. Specifically, we develop the flocking theory of these solutions and show long-time convergence to traveling wave with rapidly aligned velocity field.

中文翻译：

---

流体动力学欧拉对准系统中的单向群II：奇异模型

在本说明中，我们继续研究具有强奇异通信内核$ \ phi（x）的流体动力学Euler对准系统的单向解：= $ {\ lvert x \ rvert} ^ {-（n + \ alpha）} $ for $ \ alpha \ in（0,2）$。这些解决方案描述了控制群体的多维集体行为的代理的单向平行运动。在这里，我们考虑$ 1 \ lt \ alpha \ lt 2 $的范围，并建立光滑解的全局正则性，以及它们长期动态的完整描述。具体而言，我们开发了这些解决方案的植绒理论，并显示了速度场快速对齐的行波的长时间收敛性。