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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 947291, 5 pages
Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows
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.
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 and , Xu and Zhang  proved global existence of weak solutions to the problem if , and
When and (6) is replaced by
Very recently, Wen and Ding  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.
We will prove the following.
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 .
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 , we only need to establish a priori estimates. Also, by the local well-posedness result in , we note that is absolutely continuous on for any given .
Testing (4) by , using , we find that
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 ,
On the other hand, (3) can be rewritten as
By the -theory of Stokes system, we have which yields
Here, we have used the Gagliardo-Nirenberg inequalities
Now, by the similar calculations as those in , we arrive at
This completes the proof.
3. Proof of Theorem 3
Similar to (21), we still have
We will use the following Gagliardo-Nirenberg inequalities:
This completes the proof.
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).
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