Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 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 Nakamura
3
1
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
3
Department of Mathematics, Inha University, Incheon 402-751, Republic of Korea
Correspondence should be addressed to Yong Zhou; yzhoumath@zjnu.edu.cn
Received 6 November 2012; Accepted 14 February 2013
Academic Editor: Giovanni P. Galdi
Copyright © 2013 Yong Zhou et al. Tis 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.
Te initial-boundary value problem for the density-dependent fow 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
0
and small ∇
0
. We also give a regularity criterion ∇ ∈
(0,;
(Ω)) ((2/) + (2/) = 1, 2 < ≤ ∞) of the problem
with the Dirichlet boundary condition =0, =
0
on Ω.
1. Introduction and Main Results
Let Ω⊆ R
2
be a bounded domain with smooth boundary
Ω, and is the unit outward normal vector on Ω. We
consider the global strong solution to the density-dependent
incompressible liquid crystal fow [1–4] as follows:
div = 0, (1)
+ div () = 0, (2)
() + div (⊗)+∇−Δ=−∇⋅ (∇⊙∇), (3)
+⋅∇−Δ=|∇|
2
, (4)
in (0,∞)×Ω with initial and boundary conditions
(,,) (⋅, 0) = (
0
,
0
,
0
) in Ω, (5)
= 0,
=0 on Ω, (6)
where denotes the density, the velocity, the unit vector
feld that represents the macroscopic molecular orientations,
and the pressure. Te symbol ∇ ⊙ ∇ denotes a matrix
whose (,)th entry is
, and it is easy to fnd 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 ≡1 and Ω := R
2
, Xu and Zhang [8] proved
global existence of weak solutions to the problem if
0
∈
2
,∇
0
∈
2
,|
0
|=1, and
exp (216(
0
2
2 +
1
16
)
2
)
∇
0
2
2 <
1
16
. (7)
When ≡1 and (6) is replaced by
= 0, =
0
on Ω. (8)
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 fnitely many singular times, and they also prove a
regularity criterion
∈
2
(0,;
2
(Ω)). (9)
When =1 and the term |∇|
2
in (4) is replaced by (1 −
||
2
), then the problem has been studied in [10–15].
Very recently, Wen and Ding [16] proved the global exist-
ence and uniqueness of strong solutions to the problem (1)–
(6) with small
0
and ∇
0
and the local strong solutions with
large initial data when Ω⊆ R
2
is a smooth bounded domain.