Existence and stability of unidirectional flocks in hydrodynamic Euler Alignment systems

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 [16] 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.

Sticky particle Cucker-Smale dynamics and the entropic selection principle for the 1D Euler-alignment system

“…A discrete one-sided Lipschitz condition. We now digress briefly to discuss the quantities (43) e i,j (t) := ψ j (t) − ψ i (t)…”

“…). The right-hand side vanishes if the velocity is unidirectional, i.e., u(x, t) = u(x, t)h with a fixed direction h ∈ R d ; in this case, the same threshold as in the 1D setting holds [43]. However, for general u, the term (∇ x • u) 2 − trace((∇ x u) 2 ) does not vanish and is difficult to control (but c.f.…”

“…However, for general u, the term (∇ x • u) 2 − trace((∇ x u) 2 ) does not vanish and is difficult to control (but c.f. [43] for the 'almost unidirectional' case). Partial results are available in 2D [55,29], for radial solutions [57], and recently in higher dimensions [54].…”