# Global m-Equivariant Solutions of Nematic Liquid Crystal Flows in Dimension Two

Global m-Equivariant Solutions of Nematic Liquid Crystal Flows in Dimension Two In this article we construct a global solution of the simplified Ericksen-Leslie system. We show that the velocity of the solution can be decomposed into the sum of three parts. The main flow is governed by the Oseen vortex with the same circulation Reynolds number as the initial fluid. The secondary flow has finite kinetic energy and decay in the speed (1 + t)−2 as $${t \rightarrow \infty}$$ t → ∞ . The third part is a minor flow whose kinetic energy decays faster than the secondary flow. As for the orientation variable, our solution has a phase function which diverges logarithmically to $${\infty}$$ ∞ as $${t \rightarrow \infty}$$ t → ∞ . This indicates that the orientation variable will keep rotating around the z-axis while $${t \rightarrow \infty}$$ t → ∞ . This phenomenon results from a non-trivial coupling between the orientation variable and a fluid with a non-zero circulation Reynolds number. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Rational Mechanics and Analysis Springer Journals

# Global m-Equivariant Solutions of Nematic Liquid Crystal Flows in Dimension Two

, Volume 226 (2) – Jun 24, 2017
42 pages

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Physics; Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics
ISSN
0003-9527
eISSN
1432-0673
D.O.I.
10.1007/s00205-017-1144-x
Publisher site
See Article on Publisher Site

### Abstract

In this article we construct a global solution of the simplified Ericksen-Leslie system. We show that the velocity of the solution can be decomposed into the sum of three parts. The main flow is governed by the Oseen vortex with the same circulation Reynolds number as the initial fluid. The secondary flow has finite kinetic energy and decay in the speed (1 + t)−2 as $${t \rightarrow \infty}$$ t → ∞ . The third part is a minor flow whose kinetic energy decays faster than the secondary flow. As for the orientation variable, our solution has a phase function which diverges logarithmically to $${\infty}$$ ∞ as $${t \rightarrow \infty}$$ t → ∞ . This indicates that the orientation variable will keep rotating around the z-axis while $${t \rightarrow \infty}$$ t → ∞ . This phenomenon results from a non-trivial coupling between the orientation variable and a fluid with a non-zero circulation Reynolds number.

### Journal

Archive for Rational Mechanics and AnalysisSpringer Journals

Published: Jun 24, 2017

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