Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 589697, 9 pages
doi:10.1155/2010/589697
Research Article
A Regularity Criterion for the Nematic Liquid
Crystal Flows
Yong Zhou1 and Jishan Fan2, 3
1
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
Department of Applied Mathematics, Nanjing Forestry University, Nanjing, Jiangsu 210037, China
3
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
2
Correspondence should be addressed to Yong Zhou, yzhoumath@zjnu.edu.cn
Received 25 September 2009; Accepted 16 April 2010
Academic Editor: Michel C. Chipot
Copyright q 2010 Y. Zhou and J. Fan. 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.
A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is
established.
1. Introduction
We consider the following hydrodynamical systems modeling the flow of nematic liquid
crystal materials 1, 2:
ut u · ∇u ∇π − μΔu −λ∇ · ∇d ∇d Δd − fd ⊗ d ,
dt u · ∇d − d · ∇u γ Δd − fd ,
div u 0,
v, d|t0 v0 , d0 1.1
1.2
1.3
in R3 .
1.4
ux, t ∈ R3 is the velocity field of the flow. dx, t ∈ R3 is the averaged macroscopic/
continuum molecular orientations vector in R3 . πx, t is a scalar function representing the
pressure including both the hydrostatic part and the induced elastic part from the orientation
field. μ is a positive viscosity constant. The constant λ represents the competition between
kinetic energy and potential energy. The constant γ is the microscopic elastic relaxation time
Deborah number for the molecular orientation field. fd 1/2 |d|2 − 1d. For simplicity,
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Journal of Inequalities and Applications
we will take μ λ γ 1. The 3 × 3 matrix is defined by ∇ ∇dij ∂i d · ∂j d. ⊗ is the
usual Kronecker multiplication, for example, a ⊗ bij ai bj for a, b ∈ R3 .
Very recently, results for the local existence of classical solutions for the problems 1.1–
1.4 were presented in 3. The aim of this paper is to establish a regularity criterion for it.
We will prove the following.
Theorem 1.1. Let u0 , d0 ∈ H 2 × H 3 with div u0 0 in R3 . Suppose that a local smooth solution
u, d satisfies
T
0
∇utrLp
dt < ∞,
1 lne ∇utLp with
2 3
2, 2 ≤ p ≤ 3.
r p
1.5
Then u, d can be extended beyond T .
Remark 1.2. Equation 1.5 can be regarded as a logarithmically improved regularity criterion
of the form ∇u ∈ Lr 0, T ; Lp R3 with 2/r 3/p 2. Condition 1.5 only involves the
velocity field u, which plays a dominant role in regularity theorem. Similar phenomenon
already appeared in the studies of MHD equations see 4–6 for details.
Remark 1.3. When λ 0 in 1.1, then 1.1 and 1.2 are the well-known Navier-Stokes
equations. Similar conditions to 1.5 have been established in 7–10. But previous methods
can not be used here.
Remark 1.4. A natural region for p in 1.5 should be 3/2 ≤ p ≤ ∞, but we only can prove it
for 2 ≤ p ≤ 3 here. We are unable to establish any other regularity criterion in terms of u or π.
2. Proof of Theorem 1.1
Since we deal with the regularity conditions of the local smooth solutions, we only need to
establish the needed a priori estimates. We mainly will follow the method introduced in 9.
First, it has been proved in 3 that
1 d
2 dt
R3
2
|u|2 x, t |∇d|2 x, t |d|2 − 1 x, t dx
R3
2
2
2.1
|∇u| x, t |Δd − fd| x, t dx 0.
Hence
uL∞ 0,T ;L2 uL2 0,T ;H 1 ≤ C.
2.2
Journal of Inequalities and Applications
3
Multiplying 1.3 by d, integration by parts yields
1 d
2 dt
R3
≤
|d|2 x, tdx R3
1
2
|∇d|2 x, t |d|4 x, t dx
R3
|d|2 x, t d · ∇u · dx, t dx
R3
|d|4 x, tdx 2.3
1
|d|2 x, t |∇u|2 x, t dx.
2
R3
Thanks to 2.1, 2.2, and the Gronwall inequality, we get
dL∞ 0,T ;H 1 dL2 0,T ;H 2 ≤ C.
2.4
Let u u1 , u2 , u3 T and d d1 , d2 , d3 T , then the ith i 1, 2, 3 component of u satisfies
3
∂t ui u · ∇ui ∂i π − Δui − ∂j
j1
∂i dk ∂j dk Δdi − |d|2 − 1 di dj
.
2.5
k
Multiplying 2.5 by −Δui , after integration by parts, summing over i, and using 1.2, we
find that
1 d
2 dt
2
R3
|∇u| x, tdx −
i,j,k
−
R3
−
R3
i,j
|Δu|2 x, tdx
∂k uj · ∂j ui · ∂k ui dx −
i,k
R3
R3
R3
i,k
∂i dk · ∇Δdk · ∇ui dx i,j
∂j
|d|2 − 1 di dj · Δui dx
: I1 I2 I3 I4 I5 .
Δdk · ∂i ∇dk · ∇ui dx
R3
∂j dj Δdi · Δui dx
2.6
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Journal of Inequalities and Applications
Applying Δ on 1.3, multiplying it by Δd, and using 1.2, we have
1 d
2 dt
R3
|Δd|2 x, tdx R3
i,k
R3
|∇Δd|2 x, t Δfd · Δdx, t dx
∂i dk · ∇Δdk · ∇ui dx −
R3
i,j,k
R3
i,j
−2
i,j
R3
dj Δdi · ∂j Δui dx −
R3
i,j
∂i ∂j dk · ∂j ∇dk · ∇ui dx
Δdj Δdi · ∂j ui dx
2.7
∇dj · ∂j ui · ∇Δdi dx
: I6 I7 I8 I9 I10 .
Combining 2.6 and 2.7 together, noting that I3 I6 0, I4 I8 0, we deduce that
1 d
2 dt
R3
2
2
|∇u| x, t |Δd| x, t dx R3
R3
|Δu|2 x, tdx
2.8
2
|∇Δd| x, t Δfd · Δdx, t dx I1 I2 I5 I7 I9 I10 .
We do estimates for Ii i 1, 2, 5, 7, 9, 10 as follows:
I1 ≤ C∇uLp ∇u2L2p/p−1
21−3/2p
≤ C∇uLp ∇uL2
3/p
2.9
ΔuL2
2p/2p−3
≤ Δu2L2 C∇uLp
∇u2L2 ,
for any > 0.
Here we have used the following Gagliardo-Nirenberg inequality:
1−3/2p
∇uL2p/p−1 ≤ C∇uL2
3/2p
ΔuL2 .
2.10
Similarly, by using 2.10, we have
I2 I7 I9 ≤ C∇uLp Δd2L2p/p−1
21−3/2p
≤ C∇uLp ΔdL2
2p/2p−3
≤ ∇Δd2L2 C∇uLp
3/p
2.11
∇ΔdL2
Δd2L2 ,
for any > 0.
Journal of Inequalities and Applications
5
I5 is simply bounded as follows:
I5 ≤ C
R3
|d| |d|3 |∇d| · |Δu|dx
≤ C dL6 ∇dL3 d3L6 ∇dL∞ ΔuL2
≤ C∇dL3 ∇dL∞ ΔuL2
1/2
1/4
3/4
ΔuL2
≤ C ∇d1/2
Δd
∇d
∇Δd
2
2
2
2
L
L
L
L
2.12
≤ Δu2L2 CΔdL2 C∇Δd3/2
L2
≤ Δu2L2 CΔd2L2 ∇Δd2L2 C,
for any > 0.
When p 2 or 3, I10 can be estimated easily and hence omitted here. If 2 < p < 3, we
do estimates as follows:
I10 ≤ C∇uLp ∇dL2p/p−2 ∇ΔdL2
2−3/p
3/p
≤ C∇uLp · ΔdL2
· ∇ΔdL2
2p/2p−3
≤ ∇Δd2L2 C∇uLp
2.13
· Δd2L2 ,
for any > 0. Here we have used the Gagliardo-Nirenberg inequality:
2−3/p
∇dL2p/p−2 ≤ CΔdL2
3/p−1
∇ΔdL2
.
2.14
Finally, we omit the trivial term
R3
Δfd · Δd dx −
R3
i
∂i fd · ∂i Δd dx.
2.15
Now, putting the above estimates for Ii s into 2.8 and taking small enough, we obtain
d
dt
R3
2
2
|∇u| |Δd| dx 2p/2p−3
≤ C∇uLp
R3
|Δu|2 |∇Δd|2 dx
∇u2L2 Δd2L2 CΔd2L2 C
2p/2p−3
1 ∇u2L2 Δd2L2 .
≤ C 1 ∇uLp
2.16
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Journal of Inequalities and Applications
Due to the integrability of 1.5, we conclude that for any small constant > 0, there exists a
time T∗ < T such that
T
T∗
2p/2p−3
1 ∇utLp
dt ≤ .
1 lne ∇utLp 2.17
Easily, from 2.16 and 2.17 it follows that
d
1 ∇u2L2 Δd2L2
dt
1 ∇uLp
≤C
lne ΔuL2 ∇ΔdL2 1 ∇u2L2 Δd2L2 ,
1 lne ∇uLp 2p/2p−3
2.18
which implies that for t ∈ T∗ , T ,
C
∇ut2L2
Δdt2L2
≤ C 1 supΔu·L2 sup∇Δd·L2
T∗ ,t
.
2.19
T∗ ,t
We are going to do the estimate for Δu and ∇Δd. To this end, we introduce the following
commutator estimates due to the work of Kato and Ponce 11:
α α−1 Λ fg − fΛα g p ≤ C g
Λ
L
Lq1
∇f p Λα f p g q ,
L1
L2
L2
α
Λ fg p ≤ C f p1 Λα g q1 Λα f p2 g q2 ,
L
L
L
L
L
2.20
2.21
where Λα −Δα/2 , for α > 1, and 1/p 1/p1 1/q1 1/p2 1/q2 .
Applying Δ to 2.5 and multiplying it by Δui , after integration by parts, and summing
over i yield
1 d
2 dt
2
R3
|Δu| x, tdx R3
|∇Δu|2 x, tdx
≤ Δu · ∇u − u · ∇ · Δu · Δu dx R3
i,j
i,j
i,j
R3
R
R3
∂j Δ |d|2 − 1 di dj · Δui dx i,j
R3
∂j Δ ∂i d · ∂j d · Δui dx
dj Δ2 di · ∂j Δui dx
∇dj · ∇Δdi · ∂j Δui dx
Δdi · Δdj · ∂j Δui dx 2
3
3
: J1 J2 J3 J4 J5 J6 .
i,j
R
2.22
Journal of Inequalities and Applications
7
Applying Λ3 to 1.3, multiplying it by Λ3 d, we deduce that
1 d
2 dt
2
R3
|Λ3 d| x, tdx 2
R3
|Λ4 d| x, tdx
Λ3 u · ∇d − u · ∇Λ3 d · Λ3 d dx
≤ R3
R3
−
Λ3 fd · Λ3 d dx −
i,j
i,j
R3
R3
dj Δ2 di · ∂j Δui dx
∂j ui Δdj · Δ di dx − 2
2
i,j
R3
2.23
∇dj · ∇∂j ui · Δ2 di dx
: J7 J8 J9 J10 J11 .
Summing up 2.22 and 2.23, using J4 J9 0, we have
1 d
2 dt
R3
2
2
|Δu| x, t |Λ d| x, t dx 3
R3
2
|∇Δu|2 x, t |Λ4 d| x, t dx
2.24
≤ J1 J2 J3 J5 J6 J7 J8 J10 J11 .
Now we estimate each term Ji as follows.
By using 2.20, we estimate J1 as
J1 ≤ C∇uL3 Δu2L3 ≤ C∇u3/4
∇Δu1/4 · ∇u1/2
∇Δu3/2
L2
L2
L2
≤ ∇Δu2L2 C∇u10
L2 ,
2.25
for any > 0;
here we used the following Gagliardo-Nirenberg inequalities:
∇uL3 ≤ C∇u3/4
∇Δu1/4
,
L2
L2
ΔuL3 ≤ C∇u1/4
∇Δu3/4
.
L2
L2
2.26
Using 2.21, we estimate J2 as
J2 ≤ C∇dL∞ Λ4 dL2 ΔuL2
5/4
≤ CΔd3/4
Λ4 dL2 · ∇u1/2
∇Δu1/2
L2
L2
L2
2.27
2
≤ ∇Δu2L2 Λ4 dL2 C∇u4L2 Δd6L2 ,
for any > 0. Here we have used the following Gagliardo-Nirenberg inequalities:
1/4
∇dL∞ ≤ CΔd3/4
Λ4 dL2 ,
L2
ΔuL2 ≤ C∇u1/2
∇Δu1/2
.
L2
L2
2.28
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Journal of Inequalities and Applications
J3 only involves lower derivatives of d and is easy to handle, so we omit it here:
J5 ≤ CΔd2L4 ∇ΔuL2
3/4
2.29
≤ CΔd5/4
Λ4 dL2 ∇ΔuL2
L2
2
≤ ∇Δu2L2 Λ4 dL2 CΔd10
L2 ,
for any > 0. Here we have used
3/8
ΔdL4 ≤ CΔd5/8
Λ4 dL2 ,
L2
J6 ≤ C∇dL6 ∇ΔdL3 ∇ΔuL2
2.30
3/4
≤ CΔdL2 · Δd1/4
Λ4 dL2 ∇ΔuL2
L2
2
≤ ∇Δu2L2 Λ4 dL2 CΔd10
L2 ,
for any > 0. Where we have used the following inequality
3/4
2.31
∇ΔdL3 ≤ CΔd1/4
Λ4 dL2 .
L2
By using 2.20, we estimate J7 as follows:
2
J7 ≤ C∇uL2 Λ3 dL4 CΛ3 uL2 ∇dL4 Λ3 dL4
7/4
7/8
≤ C∇uL2 Δd1/4
Λ4 dL2 CΛ3 uL2 ∇dL4 Δd1/8
Λ4 dL2
L2
L2
2.32
2
2
≤ Λ3 uL2 Λ4 dL2 CΔd2L2 ∇u8L2 CΔd2L2 ∇d16
L4 ,
for any > 0. Here we have used
7/8
Λ3 dL4 ≤ CΔd1/8
Λ4 dL2 .
L2
2.33
The term J8 is trivial, and we omit it here:
J10 ≤ CΔdL∞ ∇uL2 Λ4 dL2
7/4
≤ C∇uL2 · Δd1/4
· Λ4 dL2
L2
2.34
2
≤ Λ4 dL2 C∇u8L2 Δd2L2 ,
for any > 0. Where we have used the following inequality:
3/4
ΔdL∞ ≤ CΔd1/4
Λ4 dL2 .
L2
2.35
Journal of Inequalities and Applications
9
Finally, using 2.26, J11 can be bounded as follows:
J11 ≤ C∇dL6 ΔuL3 Λ4 dL2
3/4
≤ CΔdL2 · ∇u1/4
· Λ3 uL2 Λ4 dL2
L2
2
2.36
2
≤ Λ3 uL2 Λ4 dL2 C∇u2L2 Δd8L2 ,
for any > 0. Now, inserting the above estimates for Ji s into 2.24, using 2.19, and taking be small enough, we get
uL∞ 0,T ;H 2 uL2 0,T ;H 3 ≤ C,
dL∞ 0,T ;H 3 dL2 0,T ;H 4 ≤ C.
2.37
This completes the proof.
Acknowledgments
The authors thank the referee for his/her careful reading and helpful suggestions. This work
is partially supported by Zhejiang Innovation Project Grant no. T200905, NSF of Zhejiang
Grant no. R6090109, and NSF of China Grant no. 10971197.
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