Vol. 15, No. 1, 2022

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Existence and stability of unidirectional flocks in hydrodynamic Euler alignment systems

Daniel Lear and Roman Shvydkoy

Vol. 15 (2022), No. 1, 175–196

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 multidimensional 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 a traveling wave with rapidly aligned velocity field.

In the context of multiscale models introduced by Shvydkoy and Tadmor (Multiscale Model. Simul. 19:2 (2021), 1115–1141) 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.

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flocking, alignment, Cucker–Smale, Mikado solutions, Euler alignment
Mathematical Subject Classification 2010
Primary: 92D25
Secondary: 35Q35, 76N10
Received: 24 November 2019
Revised: 24 May 2020
Accepted: 15 September 2020
Published: 16 March 2022
Daniel Lear
Department of Mathematics, Statistics, and Computer Science
University of Illinois
Chicago, IL
United States
Roman Shvydkoy
Department of Mathematics, Statistics, and Computer Science
University of Illinois
Chicago, IL
United States