Abstract and Applied Analysis

Volume 2013 (2013), Article ID 947291, 5 pages

http://dx.doi.org/10.1155/2013/947291

Research Article

Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows

Yong Zhou,¹ Jishan Fan,² and Gen Nakamura³

¹Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

²Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China

³Department of Mathematics, Inha University, Incheon 402-751, Republic of Korea

Received 6 November 2012; Accepted 14 February 2013

Academic Editor: Giovanni P. Galdi

Copyright © 2013 Yong Zhou 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.

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocity and small . We also give a regularity criterion of the problem with the Dirichlet boundary condition , on .

1. Introduction and Main Results

Let be a bounded domain with smooth boundary , and is the unit outward normal vector on . We consider the global strong solution to the density-dependent incompressible liquid crystal flow [1–4] as follows:

in with initial and boundary conditions

where denotes the density, the velocity, the unit vector field that represents the macroscopic molecular orientations, and the pressure. The symbol denotes a matrix whose th entry is , and it is easy to find that .

When is a given constant unit vector, then (1), (2), and (3) represent the well-known density-dependent Navier-Stokes system, which has received many studies; see [5–7] and references therein.

When and , Xu and Zhang [8] proved global existence of weak solutions to the problem if , and

When and (6) is replaced by

Lin et al. [9] proved the global existence of weak solutions to the system (1)–(5) and (8), which are smooth away from at most finitely many singular times, and they also prove a regularity criterion

When and the term in (4) is replaced by , then the problem has been studied in [10–15].

Very recently, Wen and Ding [16] proved the global existence and uniqueness of strong solutions to the problem (1)–(6) with small and and the local strong solutions with large initial data when is a smooth bounded domain.

Fan et al. [17] studied the regularity criterion of the Cauchy problem (1)–(5) when .

We will prove the following.

Theorem 1. Let , for some , , and with , and in . If

with an absolute constant in (22), then the problem (1)–(6) has a unique global-in-time strong solution satisfying

Remark 2. When , Theorem 1 is also correct, thus improving the result in [18], where and are assumed to be small.

Next, we consider (1)–(4) with as follows:

We will prove the following.

Theorem 3. Let and with and in and for some . If satisfies

then the strong solution can be extended beyond .

Remark 4. In [9], the authors prove the regularity criterion (9) for the problem (12)–(16), and our condition (17) is weaker than (9). Moreover, (17) is scaling invariant for (12)–(14).

2. Proof of Theorem 1

This section is devoted to the proof of Theorem 1. Since the local-in-time well-posedness has been proved in [16], we only need to establish a priori estimates. Also, by the local well-posedness result in [16], we note that is absolutely continuous on for any given .

By the maximum principle, it follows from (1) and (2) that

Testing (3) by and using (1) and (2), we see that

Testing (4) by , using , we find that

Summing up (19) and (20) and integrating over , we get

Since on , we have the following Gagliardo-Nirenberg inequality:

By (20) and the Ladyzhenskaya inequality in 2D, we derive

On the other hand, since , we have

If the initial data , then there exists such that for any ,

We denote by the maximal time such that (25) holds on . Therefore, by (23), (24), and (25), it follows that for any ,

which gives

which implies that if the initial data satisfies

Let be a maximal existence time for the solution . Then, (18), (21), and (27) ensure that by continuity argument.

Testing (3) by , using (1), (18), (21), (22), , and the Gagliardo-Nirenberg inequalities, we obtain

On the other hand, (3) can be rewritten as

By the -theory of Stokes system, we have

which yields

Inserting (32) into (29), we deduce that

Applying to (4), testing by , using , (21) and (22), and the Gagliardo-Nirenberg inequalities, we have

Here, we have used the Gagliardo-Nirenberg inequalities

Combining (33) and (34) and using the Gronwall inequality, we have

Now, by the similar calculations as those in [17], we arrive at

This completes the proof.

3. Proof of Theorem 3

This section is devoted to the proof of Theorem 3. By the results in [9], we only need to prove (9).

Similar to (21), we still have

We will use the following Gagliardo-Nirenberg inequalities:

Testing (14) by , using , (40), (41), and (42), we have

which gives (9).

This completes the proof.

Acknowledgments

The authors would like to thank the referees for careful reading and helpful suggestions. This work is partially supported by the Zhejiang Innovation Project (Grant no. T200905), the ZJNSF (Grant no. R6090109), and the NSFC (Grant no. 11171154).

References

S. Chandrasekhar, Liquid Crystalsed, Cambridge University Press, 2nd edition, 1992.

J. L. Ericksen, “Hydrostatic theory of liquid crystals,” Archive for Rational Mechanics and Analysis, vol. 9, pp. 371–378, 1962.

F. M. Leslie, “Some constitutive equations for liquid crystals,” Archive for Rational Mechanics and Analysis, vol. 28, no. 4, pp. 265–283, 1968.

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.

R. Danchin, “Density-dependent incompressible fluids in bounded domains,” Journal of Mathematical Fluid Mechanics, vol. 8, no. 3, pp. 333–381, 2006.

H. Kim, “A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations,” SIAM Journal on Mathematical Analysis, vol. 37, no. 5, pp. 1417–1434, 2006.

J. Fan and T. Ozawa, “Regularity criteria for the 3D density-dependent Boussinesq equations,” Nonlinearity, vol. 22, no. 3, pp. 553–568, 2009.

X. Xu and Z. Zhang, “Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows,” Journal of Differential Equations, vol. 252, no. 2, pp. 1169–1181, 2012.

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.

F.-H. Lin and C. Liu, “Partial regularity of the dynamic system modeling the flow of liquid crystals,” Discrete and Continuous Dynamical Systems, vol. 2, no. 1, pp. 1–22, 1996.

F.-H. Lin and C. Liu, “Existence of solutions for the Ericksen-Leslie system,” Archive for Rational Mechanics and Analysis, vol. 154, no. 2, pp. 135–156, 2000.

F. Lin and C. Liu, “Static and dynamic theories of liquid crystals,” Journal of Partial Differential Equations, vol. 14, no. 4, pp. 289–330, 2001.

J. Fan and B. Guo, “Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ,” Science in China. Series 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. Series A, vol. 25, no. 3, pp. 859–867, 2009.

Y. Zhou and J. Fan, “A regularity criterion for the nematic liquid crystal flows,” Journal of Inequalities and Applications, vol. 2010, Article ID 589697, 9 pages, 2010.

H. Wen and S. Ding, “Solutions of incompressible hydrodynamic flow of liquid crystals,” Nonlinear Analysis: Real World Applications, vol. 12, no. 3, pp. 1510–1531, 2011.

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.

J. Li, “Global strong and weak solutions to nematic liquid crystal flow in two dimensions,” http://arxiv.org/abs/1211.0131