Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems

TL;DR

This paper introduces new classes of solutions for hydrodynamic Euler alignment systems that describe unidirectional flocking behavior, demonstrating their stability, convergence, and potential for complex multiscale formations.

Contribution

It develops the existence, stability, and flocking theory for unidirectional solutions, extending understanding of collective motion in multi-dimensional systems.

Findings

Solutions are globally well-posed in multi-dimensional settings.

Long-term convergence to traveling wave solutions with aligned velocities.

Formation of multiscale Mikado structures resembling real flocking behavior.

Abstract

In this note we reveal new classes of solutions to hydrodynamic Euler alignment systems governing collective behavior of flocks. The solutions describe unidirectional parallel motion of agents, and are globally well-posed in multi-dimensional settings subject to a threshold condition similar to the one dimensional case. We develop the flocking and stability theory of these solutions and show long time convergence to traveling wave with rapidly aligned velocity field. In the context of multi-scale models introduced in \cite{ST-multi} our solutions can be superimposed into Mikado formations -- clusters of unidirectional flocks pointing in various directions. Such formations exhibit multiscale alignment phenomena and resemble realistic behavior of interacting large flocks.