ISRN Mathematical Analysis

VolumeΒ 2012Β (2012), Article IDΒ 248473, 7 pages

http://dx.doi.org/10.5402/2012/248473

## Continuation Criterion for the 2D Liquid Crystal Flows

^{1}Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China^{2}Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received 16 December 2011; Accepted 17 January 2012

Academic Editors: A.Β Montes-Rodriguez and T.-P.Β Tsai

Copyright Β© 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 consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion for the case with wave maps, and we prove the existence of a global-in-time strong solutions for the case with biharmonic wave maps.

#### 1. Introduction

First, we consider the following simplified liquid crystal flows in two space dimensions [1]: where is the velocity, is the pressure, and represents the macroscopic average of the liquid crystal orientation field with values in the unit circle.

The first two equations (1.1) and (1.2) are the well-known Navier-Stokes system with the Lorentz force . The last equation (1.3) is the well-known wave maps when .

It is a simple matter to show that the system (1.1)β(1.4) has a unique local-in-time smooth solution when with in . The aim of this paper is to study the regularity criterion of smooth solutions to the problem (1.1)β(1.4). We will prove the following.

Theorem 1.1. *Let with in and let be a smooth solution of (1.1)β(1.4) on some interval with . Assume that
**
Then the solution can be extended beyond .*

is the homogeneous Besov space. We have ; see Triebel [2].

In the proof of Theorem 1.1, we will use the logarithmic Sobolev inequalities [3β6]: for , and the Gagliardo-Nirenberg inequalities: with , and , and the product estimate due to Kato-Ponce [7]: with and .

Motivated by the problem (1.1)β(1.4), we consider the following liquid crystal flows: The last two equations (1.13) and (1.14) are the biharmonic wave maps. It is also a simple matter to show that the problem (1.11)β(1.15) has at least one local-in-time strong solution. The aim of this paper is to prove the global-in-time regularity. We obtain the following.

Theorem 1.2. *Let with in . Then there exists at least a global-in-time smooth solution:
**
for any .*

*Remark 1.3. *We are unable to prove the uniqueness of strong solutions in Theorem 1.2.

#### 2. Proof of Theorem 1.1

We only need to prove a priori estimates.

Testing (1.1) by , using (1.2), we see that

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

Summing up (2.1) and (2.2), we get from which we get

Applying to (1.1), testing by , using (1.2) and (1.10), we derive where

Taking to (1.3), testing by , we have

By using (1.10), (2.4), and (1.9), can be bounded as follows:

By using (1.10), can be bounded as

Combining (2.5), (2.7), (2.8), and (2.9) and using (1.6), (1.7), (1.8), and the Gronwall lemma, we arrive at

This completes the proof.

#### 3. Proof of Theorem 1.2

For simplicity, we only present a priori estimates.

First, we still have (2.1).

Testing (1.13) by , using , we have

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

which yields

Applying to (1.11), testing by , using (1.2) and (1.10), we deduce that where

Applying to (1.13), we have

Since we easily see that

Testing (3.6) by , using (3.8), we obtain

By the same calculations as those in [8], we have

By using (1.10), can be bounded as

Combining (3.4), (3.9), (3.10), and (3.11) and using (1.6) and the Gronwall lemma, we conclude that

This completes the proof.

#### Acknowledgment

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

#### References

- F. Lin and C. Liu, βStatic and dynamic theories of liquid crystals,β
*Journal of Partial Differential Equations*, vol. 14, no. 4, pp. 289β330, 2001. View at Google Scholar - H. Triebel,
*Theory of Function Spaces. II*, vol. 84, Birkhäuser, Basel, Switzerland, 1992. View at Publisher Β· View at Google Scholar - H. Brézis and T. Gallouet, βNonlinear Schrödinger evolution equations,β
*Nonlinear Analysis*, vol. 4, no. 4, pp. 677β681, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Brézis and S. Wainger, βA note on limiting cases of Sobolev embeddings and convolution inequalities,β
*Communications in Partial Differential Equations*, vol. 5, no. 7, pp. 773β789, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Ozawa, βOn critical cases of Sobolev's inequalities,β
*Journal of Functional Analysis*, vol. 127, no. 2, pp. 259β269, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Kozono, T. Ogawa, and Y. Taniuchi, βThe critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,β
*Mathematische Zeitschrift*, vol. 242, no. 2, pp. 251β278, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Kato and G. Ponce, βCommutator estimates and the Euler and Navier-Stokes equations,β
*Communications on Pure and Applied Mathematics*, vol. 41, no. 7, pp. 891β907, 1988. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Fan and T. Ozawa, βOn regularity criterion for the 2D wave maps and the 4D biharmonic wave maps,β in
*Current Advances in Nonlinear Analysis and Related Topics*, vol. 32 of*GAKUTO International Series Mathematical Sciences and Applications*, pp. 69β83, Gakkōtosho, Tokyo, Japan, 2010. View at Google Scholar Β· View at Zentralblatt MATH