ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 935045, 10 pagesdoi:10.5402/2012/935045

Research Article

Regularity Criterion for the 3D Nematic Liquid Crystal Flows

Jishan Fan¹ and Tohru Ozawa²

¹Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China²Department 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.

Abstract.

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.

1. Introduction

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 [5] 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 [6] and Fan-Ozawa [7] 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.

Theorem 1.1. Let with in . Let be a smooth solution to the problem (1.1)–(1.4) on . If satisfies for some with , then the solution can be extended beyond .

When the penalization parameter , (1.1)–(1.4) reduce to

When , then (1.9) is the well-known harmonic heat flow equation onto a sphere.

Fan-Gao-Guo [8] proved the following blow-up criteria:

We will prove the folowing theorem

Theorem 1.2. Let with in . Let be a smooth solution to the problem (1.7)–(1.10) on . If the following condition is satisfied: for some with , then the solution can be extended beyond .

2. Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. Since it is well-known that there are and a unique smooth solution to the problem (1.1)–(1.4) in , we only need to show a priori estimates.

Testing (1.1) by and using (1.3), we see that

Testing (1.2) by and using (1.3), we find that

Summing up (2.1) and (2.2), we infer that

Testing (1.2) by and using (1.3), we deduce that which yields

Next, we prove the following estimate:

Without loss of generality, we assume that . Multiplying (1.2) by , we get with and . Then (2.6) follows immediately from by the maximum principle.

Testing (1.1) by and using (1.3), we see that

Applying to (1.2), testing by , and using (1.3), we find that

Summing up (2.8) and (2.9), we get

By using (2.6), is simply bounded as

By using the inequalities [9]

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

Now, we derive a bound on defined by (2.18). To this end, we will use the following commutator and product estimates due to Kato-Ponce [10]: with and .

Applying to (1.1), testing by , and using (1.3), (2.20), (2.21) and (2.19), we obtain Here we have used the following Gagliardo-Nirenberg inequalities:

Taking to (1.2), testing by , and using (1.3), (2.20), (2.23), and (2.6), we have

Summing up (2.22) and (2.24) and taking small enough, we arrive at

This completes the proof.

3. Proof of Theorem 1.2

In this section, we will prove Theorem 1.2. Since it is easy to prove that there are and a unique smooth solution to the problem (1.7)–(1.10) in , we only need to prove a priori estimates.

First, as in the previous section, we still have (2.1).

Testing (1.9) by , using and , we see that

Summing up (2.1) and (3.1), we find 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

Applying Gronwall’s inequality to (3.7) in the interval , one has (2.19).

As in the previous section, we still have (2.22).

Similarly to (2.24), we obtain is bounded as that in (2.24); then can be bounded as that in (2.24).

Combining (2.22) and (3.9) and taking small enough, we conclude that This completes the proof.

Acknowledgment

The paper is supported by NSFC (no. 11171154).

References

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.