- About this Journal ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 935045, 10 pages
Regularity Criterion for the 3D Nematic Liquid Crystal Flows
1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan
Received 17 January 2012; Accepted 14 February 2012
Academic Editors: A. Carpio, Y. Liu, and G. A. Seregin
Copyright © 2012 Jishan Fan and Tohru Ozawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.
The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [1–4]. However, since the equations are too complicated, we consider the first simplified Ericksen-Leslie system: which include the velocity vector , the scalar pressure being and the direction vector . with , a positive constant. , and hence .
Lin-Liu  proved that the system (1.1)–(1.4) has a unique smooth solution globally in 2 space dimensions and locally in 3 dimensions. They also proved the global existence of weak solutions. However, the regularity of solutions to the system is still open. Fan-Guo  and Fan-Ozawa  showed the following regularity criteria: where denotes the homogeneous Besov space.
The first aim of this paper is to prove a new regularity criterion as follows.
When , then (1.9) is the well-known harmonic heat flow equation onto a sphere.
Fan-Gao-Guo  proved the following blow-up criteria:
We will prove the folowing theorem
2. Proof of Theorem 1.1
Next, we prove the following estimate:
By using (2.6), is simply bounded as
By using the inequalities 
can be bounded as follows:
We bound and as follows:
Here we used the Gagliardo-Nirenberg inequality Inserting the above estimates into (2.10), we derive
Due to (1.6), one concludes that for any small constant , there exists such that
For any , we set
Applying Gronwall’s inequality to (2.16) in the interval , one has
This completes the proof.
3. Proof of Theorem 1.2
First, as in the previous section, we still have (2.1).
Testing (1.9) by , using and , we see that
Similarly to (2.10), we have Here , and are the same as that in (2.10) and can be bounded as in the previous section. The corresponding last term is written and bounded as Here we have used the following inequality [11, 12]: and the Gagliardo-Nirenberg inequality
Substituting the above estimates into (3.3), we obtain
Due to (1.12), one concludes that for any small constant , there exists such that
As in the previous section, we still have (2.22).
The paper is supported by NSFC (no. 11171154).
- J. L. Ericksen, “Conservation laws for liquid crystals,” Transactions of The Society of Rheology, vol. 5, pp. 23–34, 1961.
- J. Ericksen, “Continuum theory of nematic liquid crystals,” Res Mechanica, vol. 21, pp. 381–392, 1987.
- J. L. Ericksen, “Liquid crystals with variable degree of orientation,” Archive for Rational Mechanics and Analysis, vol. 113, no. 2, pp. 97–120, 1990.
- F. Leslie, Theory of Flow Phenomenum in Liquid Crystals, vol. 4, Springer, New York, NY, USA, 1979.
- F.-H. Lin and C. Liu, “Nonparabolic dissipative systems modeling the flow of liquid crystals,” Communications on Pure and Applied Mathematics, vol. 48, no. 5, pp. 501–537, 1995.
- J. Fan and B. Guo, “Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ,” Science in China A, vol. 51, no. 10, pp. 1787–1797, 2008.
- J. Fan and T. Ozawa, “Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals,” Discrete and Continuous Dynamical Systems A, vol. 25, no. 3, pp. 859–867, 2009.
- J. Fan, H. Gao, and B. Guo, “Regularity criteria for the Navier-Stokes-Landau-Lifshitz system,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 29–37, 2010.
- H. Kozono and Y. Shimada, “Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations,” Mathematische Nachrichten, vol. 276, pp. 63–74, 2004.
- T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.
- S. Machihara and T. Ozawa, “Interpolation inequalities in Besov spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 5, pp. 1553–1556, 2003.
- Y. Meyer, “Oscillating patterns in some nonlinear evolution equations,” in Mathematical Foundation of Turbulent Viscous Flows, vol. 1871 of Lecture Notes in Mathematics, pp. 101–187, Springer, Berlin, Germany, 2006.