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 φ(x):=|x|-(n+α) for α∈(0,2). The solutions describe unidirectional parallel motion of agents governing multi-dimensional collective behavior of flocks. Here, we consider the range 1<α<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.
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