Communications in Mathematical Sciences

Volume 19 (2021)

Number 3

Unidirectional flocks in hydrodynamic Euler alignment system II: singular models

Pages: 807 – 828

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n3.a11

Authors

Daniel Lear (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Roman Shvydkoy (Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Il., U.S.A.)

Abstract

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

Keywords

flocking, emergence, fractional dissipation, Cucker–Smale, Euler alignment

2010 Mathematics Subject Classification

35Q35, 76N10, 92D25

Full Text (PDF format)

The research of R.S. is supported by NSF grant DMS-1813351.

Received 9 June 2020

Accepted 2 November 2020

Published 5 May 2021