Unidirectional flocks in hydrodynamic Euler Alignment system II: Singular models

Abstract

In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels $\ddot{I}•(x):=|x{|}^{-(n+\hat{I}\pm )}$ for $\hat{I}\pm \in (0,2)$. The solutions describe unidirectional parallel motion of agents governing multi-dimensional collective behavior of flocks. Here, we consider the range $1<\hat{I}\pm <2$ 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.

18 pages