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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 234809, 4 pages

http://dx.doi.org/10.1155/2014/234809

## Regularity Criterion for the Nematic Liquid Crystal Flows in Terms of Velocity

^{1}Department of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China^{2}School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China

Received 7 February 2014; Accepted 15 May 2014; Published 30 June 2014

Academic Editor: Gaohang Yu

Copyright © 2014 Ruiying Wei et al. 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 regularity criterion for the 3D nematic liquid crystal flows in the framework of anisotropic Lebesgue space. More precisely, we proved some sufficient conditions in terms of velocity or the fractional derivative of velocity in one direction.

#### 1. Introduction

This paper is devoted to the regularity criterion for the three-dimensional nematic liquid crystal flows: with initial data where is the velocity field, represents the macroscopic average of the nematic liquid crystal orientation field, and is the scalar pressure. The symbol denotes a matrix whose th entry is given by for ; here . Since the sizes of the viscosity constants do not play important roles in our proof, for simplicity, we assume all these positive constants to be one.

The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [1–4]; the model (1) is a simplified system of Ericksen-Leslie model which was first introduced by Lin in [5], and one of the most significant works is given by Lin and Liu [6]; more precisely, they established global existence for weak solutions and classical solutions. Recently, Liu et al. in [7] established the regularity criterion for (1) as follows: One may refer to some interesting and important regularity criteria of nematic liquid crystal flows studied by many authors (see, e.g., [8–13] and the references therein). When is constant, the system (1) becomes the well-known Navier-Stokes equations. The regularity of solutions to the 3D NS equations has been widely investigated during the past fifty years; see, for example, [14–22] and so on. The aim of this paper is to establish a new regularity criterion by providing sufficient condition in terms of velocity or the fractional derivative of velocity in one direction in the framework of anisotropic Lebesgue space.

Throughout the paper, the norm of the Lebesgue spaces is denoted by and denoted the directional derivatives of a function by , the symbol , , , and belongs to the permutation group . Denote

Theorem 1. *Let with the initial data , and let the pair be the weak solution to the liquid crystal flows (1)-(2) on for some . If satisfies
**
then can be extended beyond .*

Theorem 2. *Let with the initial data , and let the pair be the weak solution to the liquid crystal flows (1)-(2) on for some . If satisfies
**
then can be extended beyond .*

Corollary 3. *Under the assumption of Theorem 2, if we fix , then the sufficient condition is that
*

*Remark 4. *Comparing with the corresponding results in [7], it is obvious that the conclusion of Corollary 3 is an improvement version of Theorem 1.1 in [7] in some sense.

*2. The Proof of Theorems 1 and 2*

*In this section, we will prove Theorems 1 and 2. For convenience, we first recall the following three-dimensional Sobolev and Ladyzhenskaya inequalities in the whole space (see, e.g., [23–25]).*

*Lemma 5. Let , , and , . There hold that
*

*Proof of Theorem 1. *Suppose that is the maximal interval of the existence of the local smooth solution. If , then there is nothing to prove; on the other side, for , our strategy is to show that
under the assumption (5). As a result, the interval cannot be a maximal interval of existence, which leads to a contradiction.

We multiply (1)_{1} by and integrate over and, similarly, multiply (1)_{2} by and integrate over and then by adding two results above and using the fact that , we obtain
Here we used the facts that div and ; here denotes the usual inner product of , which implies

Besides, we multiply (1)_{2} by and integrate over and get
which implies

Multiplying the first equation of (1) by and integrating over . Similarly, by taking on both sides of the second equation of (1), by multiplying the resulting equation by , by integrating over , and then by adding two results above and taking the divergence-free condition div into account, we obtain

In the following, we establish the bounds of , for the first term ; thanks to Lemma 5 and using Young’s inequality, we have
For the second term , similar to estimate of , we have
For the term , using Hölder’s inequality, Young’s inequality, and (13), one has
Substituting the above estimates (15)–(17) into (14), we obtain
Integrating (18) from 0 to , using Hölder’s inequality and Young’s inequality, one has

Finally, applying Gronwall’s inequality and using condition (5), then can be extended beyond . This completes the proof of Theorem 1.

*Proof of Theorem 2. *When , combining Theorem 1 and using the following imbedding theorem, one can get the conclusion that
When , our strategy is to show that
is a sufficient condition. We can verify that integral term satisfies the conditions of Theorem 1 with . Applying Lemma 5, Hölder’s inequality, and the interpolation theorem, one can conclude that, for ,
where with and we have used the fact that implies . Using Hölder’s inequality, one has
where .

According to the fact that and , we have
This together with Theorem 1 gives the desired result of Theorem 2.

* Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments*

*This work is partially supported by the NNSFC (Grant no. 11271381), China, 973 Program (Grant no. 2011CB808002), Guangdong Provincial Culture of Seedling of China (no. 2013LYM0081), Guangdong Provincial NSF of China (no. S2012010010069), the Shaoguan Science and Technology Foundation (no. 313140546), and Science Foundation of Shaoguan University.*

*References*

*References*

- J. L. Ericksen, “Conservation laws for liquid crystals,”
*Transactions of the Society of Rheology: Journal of Rheology*, vol. 5, no. 1, pp. 23–34, 1961. View at Publisher · View at Google Scholar · View at MathSciNet - J. L. Ericksen, “Continuum theory of nematic liquid crystals,”
*Res Mechanica*, vol. 21, no. 4, pp. 381–392, 1987. View at Google Scholar · View at Scopus - J. L. Ericksen, “Liquid crystals with variable degree of orientation,”
*Archive for Rational Mechanics and Analysis*, vol. 113, no. 2, pp. 97–120, 1990. View at Publisher · View at Google Scholar · View at MathSciNet - F. Leslie,
*Theory of Flow Phenomenum in Liquid Crystals*, vol. 4, Springer, New York, NY, USA, 1979. - F.-H. Lin, “Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena,”
*Communications on Pure and Applied Mathematics*, vol. 42, no. 6, pp. 789–814, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Liu, J. Zhao, and S. Cui, “A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity,”
*Journal of Mathematical Physics*, vol. 52, no. 3, Article ID 033102, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. Fan and B. Guo, “Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ${R}^{3}$,”
*Science in China A: Mathematics*, vol. 51, no. 10, pp. 1787–1797, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Liu and J. Zhao, “Logarithmically improved blow-up criteria for the nematic liquid crystal flows,”
*Nonlinear Analysis: Real World Applications*, vol. 16, pp. 178–190, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - T. Huang and C. Wang, “Blow up criterion for nematic liquid crystal flows,”
*Communications in Partial Differential Equations*, vol. 37, no. 5, pp. 875–884, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Lin, J. Lin, and C. Wang, “Liquid crystal flows in two dimensions,”
*Archive for Rational Mechanics and Analysis*, vol. 197, no. 1, pp. 297–336, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, X. Wang, and Z.-A. Yao, “Remarks on regularity criteria for the weak solutions of liquid crystals,”
*Journal of Evolution Equations*, vol. 12, no. 4, pp. 801–812, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, T. Tang, and L. Liu, “An Osgood type regularity criterion for the liquid crystal flows,”
*Nonlinear Differential Equations and Applications*, vol. 21, no. 2, pp. 253–262, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - G. P. Galdi,
*An Introduction to the Mathematical Theory of the Navier-Stokes Equations*, vol. 1, Springer, New York, NY, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - G. P. Galdi,
*An Introduction to the Mathematical Theory of the Navier-Stokes Equations*, vol. 2, Springer, New York, NY, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - P. Penel and M. Pokorný, “Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity,”
*Applications of Mathematics*, vol. 49, no. 5, pp. 483–493, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Cao and E. S. Titi, “Regularity criteria for the three-dimensional Navier-Stokes equations,”
*Indiana University Mathematics Journal*, vol. 57, no. 6, pp. 2643–2661, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Gala, “Remarks on regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure,”
*Applicable Analysis*, vol. 92, no. 1, pp. 96–103, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Jia and Y. Zhou, “Remarks on regularity criteria for the Navier-Stokes equations via one velocity component,”
*Nonlinear Analysis: Real World Applications*, vol. 15, pp. 239–245, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Zhou and M. Pokorný, “On the regularity of the solutions of the Navier-Stokes equations via one velocity component,”
*Nonlinearity*, vol. 23, no. 5, pp. 1097–1107, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, “A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component,”
*Communications on Pure and Applied Analysis*, vol. 12, no. 1, pp. 117–124, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, Z.-A. Yao, M. Lu, and L. Ni, “Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations,”
*Journal of Mathematical Physics*, vol. 52, no. 5, Article ID 053103, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. A. Adams,
*Sobolev Spaces*, Academic Press, New York, NY, USA, 1975. - X. Zheng, “A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component,”
*Journal of Differential Equations*, vol. 256, no. 1, pp. 283–309, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - O. A. Ladyzhenskaya,
*Mathematical Theory of Viscous Incompressible Flow*, Gordon and Breach, New York, NY, USA, 2nd edition, 1969, English translation. View at MathSciNet

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