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Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2165–2175 | Cite as

Global Existence and Stability of Nearly Aligned Flocks

  • Roman ShvydkoyEmail author
Article

Abstract

We study regularity of a hydrodynamic singular model of collective behavior introduced in Shvydkoy and Tadmor (Trans Math Appl 1(1):tnx001, 2017). In this note we address the question of global well-posedness in multi-dimensional settings. It is shown that any initial data \((u,\rho )\) with small velocity variations \(|u(x) - u(y)| < \varepsilon \) relative to its higher order norms, gives rise to a unique global regular solution which aligns and flocks exponentially fast. Moreover, we prove that the limiting flocks are stable.

Keywords

Flocking Alignment Cucker–Smale Fractional Laplacian 

Mathematics Subject Classification

92D25 35Q35 76N10 

Notes

Acknowledgements

Research was supported in part by NSF grant DMS 1515705, and the College of LAS at UIC. The author thanks Eitan Tadmor for stimulating discussions

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer Science, M/C 249University of IllinoisChicagoUSA

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