Global Regularity for the Fractional Euler Alignment System
Abstract
© 2017 Springer-Verlag GmbH Germany We study a pressureless Euler system with a non-linear
density-dependent alignment term, originating in the Cucker–Smale swarming models.
The alignment term is dissipative in the sense that it tends to equilibrate the velocities.
Its density dependence is natural: the alignment rate increases in the areas of high
density due to species discomfort. The diffusive term has the order of a fractional
Laplacian (Formula presented.). The corresponding Burgers equation with a linear dissipation
of this type develops shocks in a finite time. We show that the alignment nonlinearity
enhances the dissipation, and the solutions are globally regular for all (Formula
presented.). To the best of our knowledge, this is the first example of such regularization
due to the non-local nonlinear modulation of dissipation.
Type
Journal articlePermalink
https://hdl.handle.net/10161/15910Published Version (Please cite this version)
10.1007/s00205-017-1184-2Publication Info
Do, T; Kiselev, A; Ryzhik, L; & Tan, C (2017). Global Regularity for the Fractional Euler Alignment System. Archive for Rational Mechanics and Analysis. pp. 1-37. 10.1007/s00205-017-1184-2. Retrieved from https://hdl.handle.net/10161/15910.This is constructed from limited available data and may be imprecise. To cite this
article, please review & use the official citation provided by the journal.
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Show full item recordScholars@Duke
Alexander A. Kiselev
William T. Laprade Distinguished Professor of Mathematics
My current research interests focus on mathematical fluid mechanics and mathematical
biology.In the past, I have also worked on reaction-diffusion equations and spectral
theory of Schredinger operators.

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