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 Nakamura3
1
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
3
Department of Mathematics, Inha University, Incheon 402-751, Republic of Korea
2
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. 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 𝑢0 and small ∇𝑑0 . We also give a regularity criterion ∇𝑑 ∈ 𝐿𝑝 (0, 𝑇; 𝐿𝑞 (Ω)) ((2/𝑞) + (2/𝑝) = 1, 2 < 𝑞 ≤ ∞) of the problem
with the Dirichlet boundary condition 𝑢 = 0, 𝑑 = 𝑑0 on 𝜕Ω.
1. Introduction and Main Results
Let Ω ⊆ R2 be a bounded domain with smooth boundary
𝜕Ω, and 𝜈 is the unit outward normal vector on 𝜕Ω. We
consider the global strong solution to the density-dependent
incompressible liquid crystal flow [1–4] as follows:
div 𝑢 = 0,
(1)
𝜕𝑡 𝜌 + div (𝜌𝑢) = 0,
When 𝑑 is a given constant unit vector, then (1), (2),
and (3) represent the well-known density-dependent NavierStokes system, which has received many studies; see [5–7] and
references therein.
When 𝜌 ≡ 1 and Ω := R2 , Xu and Zhang [8] proved
global existence of weak solutions to the problem if 𝑢0 ∈
𝐿2 , ∇𝑑0 ∈ 𝐿2 , |𝑑0 | = 1, and
1 2 󵄩
1
󵄩2
󵄩 󵄩2
exp (216(󵄩󵄩󵄩𝑢0 󵄩󵄩󵄩𝐿2 + ) ) 󵄩󵄩󵄩∇𝑑0 󵄩󵄩󵄩𝐿2 < .
16
16
When 𝜌 ≡ 1 and (6) is replaced by
(2)
𝜕𝑡 (𝜌𝑢) + div (𝜌𝑢 ⊗ 𝑢) + ∇𝜋 − Δ𝑢 = −∇ ⋅ (∇𝑑 ⊙ ∇𝑑) , (3)
𝜕𝑡 𝑑 + 𝑢 ⋅ ∇𝑑 − Δ𝑑 = |∇𝑑|2 𝑑,
(4)
in (0, ∞) × Ω with initial and boundary conditions
(𝜌, 𝑢, 𝑑) (⋅, 0) = (𝜌0 , 𝑢0 , 𝑑0 )
𝑢 = 0,
𝜕𝜈 𝑑 = 0
in Ω,
on 𝜕Ω,
(5)
(6)
where 𝜌 denotes the density, 𝑢 the velocity, 𝑑 the unit vector
field that represents the macroscopic molecular orientations,
and 𝜋 the pressure. The symbol ∇𝑑 ⊙ ∇𝑑 denotes a matrix
whose (𝑖, 𝑗)th entry is 𝜕𝑖 𝑑𝜕𝑗 𝑑, and it is easy to find that ∇𝑑 ⊙
∇𝑑 = ∇𝑑𝑇 ∇𝑑.
𝑢 = 0,
𝑑 = 𝑑0
on 𝜕Ω.
(7)
(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 finitely many singular times, and they also prove a
regularity criterion
𝑑 ∈ 𝐿2 (0, 𝑇; 𝐻2 (Ω)) .
(9)
When 𝜌 = 1 and the term |∇𝑑|2 in (4) is replaced by (1 −
|𝑑| )𝑑, 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 𝑢0 and ∇𝑑0 and the local strong solutions with
large initial data when Ω ⊆ R2 is a smooth bounded domain.
2
2
Abstract and Applied Analysis
Testing (3) by 𝑢 and using (1) and (2), we see that
Fan et al. [17] studied the regularity criterion of the
Cauchy problem (1)–(5) when Ω := R2 .
We will prove the following.
1 𝑑
∫ 𝜌𝑢2 𝑑𝑥 + ∫ |∇𝑢|2 𝑑𝑥 = − ∫ (𝑢 ⋅ ∇) 𝑑 ⋅ Δ𝑑 𝑑𝑥.
2 𝑑𝑡
1,𝑟
Theorem 1. Let 0 < 𝑚 ≤ 𝜌0 ≤ 𝑀 < ∞, 𝜌0 ∈ 𝑊 for some
𝑟 ∈ (2, ∞), 𝑢0 ∈ 𝐻01 ∩ 𝐻2 , and 𝑑0 ∈ 𝐻3 with div 𝑢0 =0, and
|𝑑0 | = 1 in Ω. If
2
𝐶2 󵄩
1
1
󵄩󵄩
󵄩2
󵄩2
, (10)
󵄩󵄩∇𝑑0 󵄩󵄩󵄩𝐿2 exp [216 0 (󵄩󵄩󵄩√𝜌0 𝑢0 󵄩󵄩󵄩𝐿2 + 2 ) ] ≤
𝑚
8𝐶0
8𝐶02
with an absolute constant 𝐶0 in (22), then the problem (1)–(6)
has a unique global-in-time strong solution (𝜌, 𝑢, 𝑑) satisfying
󵄩󵄩 󵄩󵄩
󵄩󵄩𝜌󵄩󵄩𝐿∞ (0,𝑇;𝑊1,𝑟 ) ≤ 𝐶,
󵄩󵄩 󵄩󵄩
󵄩󵄩𝜌𝑡 󵄩󵄩𝐿∞ (0,𝑇;𝐿𝑟 ) ≤ 𝐶,
‖𝑢‖𝐿∞ (0,𝑇;𝐻2 )∩𝐿2 (0,𝑇;𝑊2,𝑠 ) ≤ 𝐶,
𝑓𝑜𝑟𝑠𝑜𝑚𝑒 𝑠 > 2,
(11)
Testing (4) by −Δ𝑑 − |∇𝑑|2 𝑑, using |𝑑| = 1, we find that
1 𝑑
󵄨
󵄨2
∫ |∇𝑑|2 𝑑𝑥 + ∫ 󵄨󵄨󵄨󵄨Δ𝑑 + |∇𝑑|2 𝑑󵄨󵄨󵄨󵄨 𝑑𝑥 = ∫ (𝑢 ⋅ ∇) 𝑑 ⋅ Δ𝑑 𝑑𝑥.
2 𝑑𝑡
(20)
Summing up (19) and (20) and integrating over (0, 𝑇), we
get
𝑇
󵄨
󵄨
∫ (𝜌𝑢2 + |∇𝑑|2 ) 𝑑𝑥 + 2 ∫ ∫ (|∇𝑢|2 + 󵄨󵄨󵄨󵄨Δ𝑑 + |∇𝑑|2 𝑑󵄨󵄨󵄨󵄨) 𝑑𝑥 𝑑𝑡
0
‖𝑑‖𝐿∞ (0,𝑇;𝐻3 ) ≤ 𝐶.
󵄨2
󵄨
≤ ∫ (𝜌0 𝑢02 + 󵄨󵄨󵄨∇𝑑0 󵄨󵄨󵄨 ) 𝑑𝑥.
2
Remark 2. When Ω := R , Theorem 1 is also correct, thus
improving the result in [18], where 𝑢0 and ∇𝑑0 are assumed
to be small.
Next, we consider (1)–(4) with 𝜌 ≡ 1 as follows:
div 𝑢 = 0,
(12)
𝜕𝑡 𝑢 + 𝑢 ⋅ ∇𝑢 + ∇𝜋 − Δ𝑢 = −∇ ⋅ (∇𝑑 ⊙ ∇𝑑) ,
(13)
𝜕𝑡 𝑑 + 𝑢 ⋅ ∇𝑑 − Δ𝑑 = |∇𝑑|2 𝑑,
(14)
𝑢 = 0,
𝑑 = 𝑑0
on 𝜕Ω,
(𝑢, 𝑑) (⋅, 0) = (𝑢0 , 𝑑0 )
in Ω.
(15)
(16)
We will prove the following.
Theorem 3. Let 𝑢0 ∈ 𝐿2 and 𝑑0 ∈ 𝐻1 with div 𝑢0 = 0 and
|𝑑0 | = 1 in Ω and 𝑑0 ∈ 𝐶2,𝛽 (𝜕Ω) for some 𝛽 ∈ (0, 1). If 𝑑
satisfies
∇𝑑 ∈ 𝐿𝑞 (0, 𝑇; 𝐿𝑝 ) ,
2 2
+ = 1,
𝑞 𝑝
2 < 𝑝 ≤ ∞,
(17)
then the strong solution (𝑢, 𝑑) can be extended beyond 𝑇 > 0.
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 ∇𝑑 is absolutely
continuous on [0, 𝑇] for any given 𝑇 > 0.
By the maximum principle, it follows from (1) and (2) that
0 < 𝑚 ≤ 𝜌 ≤ 𝑀 < ∞.
(18)
(19)
(21)
Since 𝜕𝜈 𝑑 = 0 on (0, ∞) × 𝜕Ω, we have the following
Gagliardo-Nirenberg inequality:
‖∇𝑑‖2𝐿4 ≤ 𝐶0 ‖∇𝑑‖𝐿2 ‖Δ𝑑‖𝐿2 .
(22)
By (20) and the Ladyzhenskaya inequality in 2D, we
derive
1 𝑑
󵄨
󵄨2
∫ |∇𝑑|2 𝑑𝑥 + ∫ 󵄨󵄨󵄨󵄨Δ𝑑 + |∇𝑑|2 𝑑󵄨󵄨󵄨󵄨 𝑑𝑥
2 𝑑𝑡
≤ ‖𝑢‖𝐿4 ‖∇𝑑‖𝐿4 ‖Δ𝑑‖𝐿2
≤ √2‖𝑢‖1/2
⋅ √𝐶0 ‖∇𝑑‖1/2
‖∇𝑢‖1/2
‖Δ𝑑‖3/2
𝐿2
𝐿2
𝐿2
𝐿2
≤
‖Δ𝑑‖2𝐿2
+ 216𝐶02 ‖𝑢‖2𝐿2 ‖∇𝑢‖2𝐿2 ‖∇𝑑‖2𝐿2
8
≤
𝐶2 󵄩
‖Δ𝑑‖2𝐿2
󵄩2 󵄩
󵄩2
+ 216 0 (󵄩󵄩󵄩√𝜌0 𝑢0 󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩∇𝑑0 󵄩󵄩󵄩𝐿2 ) ‖∇𝑢‖2𝐿2 ‖∇𝑑‖2𝐿2 .
8
𝑚
(23)
On the other hand, since (𝑎 + 𝑏)2 ≥ (𝑎2 /2) − 𝑏2 , we have
‖Δ𝑑‖2𝐿2
󵄨2
󵄨
− ‖∇𝑑‖4𝐿4
∫ 󵄨󵄨󵄨󵄨Δ𝑑 + |∇𝑑|2 𝑑󵄨󵄨󵄨󵄨 𝑑𝑥 ≥
2
‖Δ𝑑‖2𝐿2
≥
− 𝐶02 ‖∇𝑑‖2𝐿2 ‖Δ𝑑‖2𝐿2 .
2
(24)
If the initial data ‖∇𝑑0 ‖2𝐿2 < (1/𝐶02 )(1/8), then there exists
𝑇1 > 0 such that for any 𝑡 ∈ [0, 𝑇1 ],
‖∇𝑑 (𝑡)‖2𝐿2 ≤
1 1
⋅ .
𝐶02 4
(25)
Abstract and Applied Analysis
3
We denote by 𝑇1∗ the maximal time such that (25) holds
on [0, 𝑇1∗ ]. Therefore, by (23), (24), and (25), it follows that for
any 𝑡 ∈ [0, 𝑇1∗ ],
𝑑
1
∫ |∇𝑑|2 𝑑𝑥 + ‖Δ𝑑‖2𝐿2
𝑑𝑡
4
≤ 432
≤ 432
𝐶02
󵄩2 󵄩
󵄩2
󵄩
(󵄩󵄩√𝜌 𝑢 󵄩󵄩 2 + 󵄩󵄩∇𝑑 󵄩󵄩 2 ) ‖∇𝑢‖2𝐿2 ‖∇𝑑‖2𝐿2
𝑚 󵄩 0 0 󵄩𝐿 󵄩 0 󵄩𝐿
(26)
𝐶02 󵄩󵄩
1
󵄩2
(󵄩󵄩√𝜌0 𝑢0 󵄩󵄩󵄩𝐿2 + 2 ) ‖∇𝑢‖2𝐿2 ‖∇𝑑‖2𝐿2 ,
𝑚
8𝐶0
󵄩 󵄩
‖Δ𝑢‖𝐿2 ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿2
󵄩
󵄩
≤ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 + 𝐶‖𝑢‖𝐿4 ‖∇𝑢‖𝐿4 + 𝐶‖∇𝑑‖𝐿4 ‖Δ𝑑‖𝐿4
󵄩
󵄩
≤ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 + 𝐶‖∇𝑢‖𝐿2 ‖Δ𝑢‖1/2
+ 𝐶‖∇𝑢‖𝐿2
𝐿2
+ 𝐶‖Δ𝑑‖𝐿2 ‖∇Δ𝑑‖1/2
+ 𝐶‖Δ𝑑‖𝐿2 ,
𝐿2
󵄩
󵄩
‖Δ𝑢‖𝐿2 ≤ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 + 𝐶‖∇𝑢‖2𝐿2 + 𝐶
1 𝑡
∫ ‖Δ𝑑(𝜏)‖2𝐿2 𝑑𝜏
4 0
+ 𝐶‖Δ𝑑‖𝐿2 ‖∇Δ𝑑‖1/2
+ 𝐶‖Δ𝑑‖𝐿2 .
𝐿2
2
×∫
0
‖∇𝑢‖2𝐿2 𝑑𝜏]
(27)
2
2
𝐶 󵄩
1
󵄩
󵄩2
󵄩2
≤ 󵄩󵄩󵄩∇𝑑0 󵄩󵄩󵄩𝐿2 exp [216 0 (󵄩󵄩󵄩√𝜌0 𝑢0 󵄩󵄩󵄩𝐿2 + 2 ) ]
𝑚
8𝐶0
1 𝑑
∫ |∇𝑢|2 𝑑𝑥 + ∫ 𝜌𝑢𝑡2 𝑑𝑥
2 𝑑𝑡
󵄩3/2
󵄩
󵄩
󵄩
≤ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 ‖∇𝑢‖𝐿2 + 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 (‖∇𝑢‖2𝐿2 + ‖∇𝑢‖𝐿2 )
󵄩
󵄩
󵄩
󵄩
+ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 ‖Δ𝑑‖𝐿2
+ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 ‖Δ𝑑‖𝐿2 ‖∇Δ𝑑‖1/2
𝐿2
1󵄩
1
󵄩2
≤ 󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 + 𝐶‖∇𝑢‖4𝐿2 + 𝐶 + ‖∇Δ𝑑‖2𝐿2 + 𝐶‖Δ𝑑‖4𝐿2 .
8
8
(33)
1
≤
,
8𝐶02
which implies that 𝑇1∗ = 𝑇 if the initial data satisfies
2
2
𝐶 󵄩
1
1
󵄩󵄩
󵄩2
󵄩2
. (28)
󵄩󵄩∇𝑑0 󵄩󵄩󵄩𝐿2 exp [216 0 (󵄩󵄩󵄩√𝜌0 𝑢0 󵄩󵄩󵄩𝐿2 + 2 ) ] ≤
𝑚
8𝐶0
8𝐶02
Let 𝑇∗ be a maximal existence time for the solution
(𝜌, 𝑢, 𝑑). Then, (18), (21), and (27) ensure that 𝑇∗ = ∞ by
continuity argument.
Testing (3) by 𝑢𝑡 , using (1), (18), (21), (22), |𝑑| = 1, and the
Gagliardo-Nirenberg inequalities, we obtain
Applying Δ to (4), testing by Δ𝑑, using |𝑑| = 1, (21) and
(22), and the Gagliardo-Nirenberg inequalities, we have
1 𝑑
∫ |Δ𝑑|2 𝑑𝑥 + ∫ |∇Δ𝑑|2 𝑑𝑥
2 𝑑𝑡
󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨󵄨∇ (|∇𝑑|2 𝑑)󵄨󵄨󵄨󵄨 |∇Δ𝑑| 𝑑𝑥 + ∫ |∇ (𝑢 ⋅ ∇𝑑)| |∇Δ𝑑| 𝑑𝑥
≤ 𝐶 (‖∇𝑑‖3𝐿6 + ‖∇𝑑‖𝐿4 ‖Δ𝑑‖𝐿4 + ‖𝑢‖𝐿4 ‖Δ𝑑‖𝐿4
+‖∇𝑢‖𝐿2 ‖∇𝑑‖𝐿∞ ) ‖∇Δ𝑑‖𝐿2
1 𝑑
∫ |∇𝑢|2 𝑑𝑥 + ∫ 𝜌𝑢𝑡2 𝑑𝑥
2 𝑑𝑡
≤ 𝐶 (‖∇𝑑‖𝐿2 ‖Δ𝑑‖2𝐿2 + ‖Δ𝑑‖𝐿2 ‖∇Δ𝑑‖1/2
+ ‖Δ𝑑‖𝐿2
𝐿2
= − ∫ 𝜌𝑢 ⋅ ∇𝑢 ⋅ 𝑢𝑡 𝑑𝑥 − ∫ 𝑢𝑡 ⋅ ∇𝑑 ⋅ Δ𝑑 𝑑𝑥
+ ‖∇𝑢‖1/2
‖Δ𝑑‖1/2
‖∇Δ𝑑‖1/2
𝐿2
𝐿2
𝐿2
󵄩
󵄩
≤ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 (‖𝑢‖𝐿4 ‖∇𝑢‖𝐿4 + ‖∇𝑑‖𝐿4 ‖Δ𝑑‖𝐿4 )
+ ‖∇𝑢‖1/2
+ ‖∇𝑢‖𝐿2
‖Δ𝑑‖1/2
𝐿2
𝐿2
× ‖∇𝑑‖1/2
) ‖∇Δ𝑑‖𝐿2
‖∇Δ𝑑‖1/2
𝐿2
𝐿2
󵄩
󵄩
+ ‖𝑢‖1/2
)
≤ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 [‖𝑢‖1/2
‖∇𝑢‖𝐿2 (‖Δ𝑢‖1/2
𝐿2
𝐿2
𝐿2
+ ‖𝑑‖1/2
)]
+ ‖∇𝑑‖1/2
‖Δ𝑑‖𝐿2 (‖∇Δ𝑑‖1/2
𝐿2
𝐿2
𝐿2
󵄩
󵄩
+ ‖∇𝑢‖𝐿2 + ‖Δ𝑑‖𝐿2
≤ 𝐶󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩𝐿2 (‖∇𝑢‖𝐿2 ‖Δ𝑢‖1/2
𝐿2
× ‖∇Δ𝑑‖1/2
+ ‖Δ𝑑‖𝐿2 ) .
𝐿2
1
≤ ‖∇Δ𝑑‖2𝐿2 + 𝐶‖Δ𝑑‖4𝐿2 + 𝐶 + 𝐶‖∇𝑢‖4𝐿2 .
8
(34)
Here, we have used the Gagliardo-Nirenberg inequalities
(29)
‖∇𝑑‖3𝐿6 ≤ 𝐶‖∇𝑑‖𝐿2 ‖Δ𝑑‖2𝐿2 ,
‖∇𝑑‖2𝐿∞ ≤ ‖∇𝑑‖𝐿2 ‖∇Δ𝑑‖𝐿2 ,
On the other hand, (3) can be rewritten as
−Δ𝑢 + ∇𝜋 = 𝑓 := −𝜌𝑢𝑡 − 𝜌𝑢 ⋅ ∇𝑢 − ∇ ⋅ (∇𝑑 ⊙ ∇𝑑) .
(32)
Inserting (32) into (29), we deduce that
𝐶 󵄩
1
󵄩2
󵄩2
󵄩
≤ 󵄩󵄩󵄩∇𝑑0 󵄩󵄩󵄩𝐿2 exp [432 0 (󵄩󵄩󵄩√𝜌0 𝑢0 󵄩󵄩󵄩𝐿2 + 2 )
𝑚
8𝐶0
𝑇1∗
(31)
which yields
which gives
‖∇𝑑 (𝑡)‖2𝐿2 +
By the 𝐻2 -theory of Stokes system, we have
(30)
‖Δ𝑑‖2𝐿4 ≤ 𝐶‖Δ𝑑‖𝐿2 ‖∇Δ𝑑‖𝐿2 + 𝐶‖Δ𝑑‖𝐿2 .
(35)
4
Abstract and Applied Analysis
Combining (33) and (34) and using the Gronwall inequality, we have
‖𝑢‖𝐿∞ (0,𝑇;𝐻1 ) + ‖𝑢‖𝐿2 (0,𝑇;𝐻2 ) ≤ 𝐶,
󵄩󵄩󵄩√𝜌𝑢𝑡 󵄩󵄩󵄩 2
󵄩
󵄩𝐿 (0,𝑇;𝐿2 ) ≤ 𝐶,
(36)
‖𝑑‖𝐿∞ (0,𝑇;𝐻2 ) + ‖𝑑‖𝐿2 (0,𝑇;𝐻3 ) ≤ 𝐶.
(38)
(37)
Now, by the similar calculations as those in [17], we arrive
at
󵄩󵄩
󵄩
󵄩󵄩(𝑢𝑡 , ∇𝑑𝑡 )󵄩󵄩󵄩𝐿∞ (0,𝑇;𝐿2 )∩𝐿2 (0,𝑇;𝐻1 ) ≤ 𝐶,
‖(𝑢, ∇𝑑)‖𝐿∞ (0,𝑇;𝐻2 ) ≤ 𝐶,
‖𝑢‖𝐿2 (0,𝑇;𝑊2,𝑠 ) ≤ 𝐶 for some 𝑠 > 2,
󵄩󵄩 󵄩󵄩
󵄩󵄩 󵄩󵄩
󵄩󵄩𝜌𝑡 󵄩󵄩𝐿∞ (0,𝑇;𝐿𝑟 ) ≤ 𝐶.
󵄩󵄩𝜌󵄩󵄩𝐿∞ (0,𝑇;𝑊1,𝑟 ) ≤ 𝐶,
(39)
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
𝑇
󵄨
󵄨
∫ (𝑢2 + |∇𝑑|2 ) 𝑑𝑥 + 2 ∫ ∫ (|∇𝑢|2 + 󵄨󵄨󵄨󵄨Δ𝑑 + |∇𝑑|2 𝑑󵄨󵄨󵄨󵄨) 𝑑𝑥 𝑑𝑡
0
≤
∫ (𝑢02
󵄨2
󵄨
+ 󵄨󵄨󵄨∇𝑑0 󵄨󵄨󵄨 ) 𝑑𝑥.
(40)
We will use the following Gagliardo-Nirenberg inequalities:
1−(2/𝑝)
‖𝑢‖𝐿2𝑝/(𝑝−2) ≤ 𝐶‖𝑢‖𝐿2
1−(2/𝑝)
‖∇𝑑‖𝐿2𝑝/(𝑝−2) ≤ 𝐶‖∇𝑑‖𝐿2
2/𝑝
‖∇𝑢‖𝐿2 ,
2/𝑝
‖Δ𝑑‖𝐿2 + 𝐶‖∇𝑑‖𝐿2 .
(41)
(42)
Testing (14) by −Δ𝑑, using |𝑑| = 1, (40), (41), and (42), we
have
1 𝑑
∫ |∇𝑑|2 𝑑𝑥 + ∫ |Δ𝑑|2 𝑑𝑥
2 𝑑𝑡
= ∫ (𝑢 ⋅ ∇𝑑 − |∇𝑑|2 𝑑) Δ𝑑 𝑑𝑥
≤ (‖𝑢‖𝐿2𝑝/(𝑝−2) ‖∇𝑑‖𝐿𝑝 + ‖∇𝑑‖𝐿𝑝 ‖∇𝑑‖𝐿2𝑝/(𝑝−2) ) ‖Δ𝑑‖𝐿2
1−(2/𝑝)
≤ 𝐶‖∇𝑑‖𝐿𝑝 (‖𝑢‖𝐿2
2/𝑝
‖∇𝑢‖𝐿2 + ‖∇𝑑‖𝐿2
1−(2/𝑝)
+ ‖∇𝑑‖𝐿2
2/𝑝
2/𝑝
‖Δ𝑑‖𝐿2 ) ‖Δ𝑑‖𝐿2
2/𝑝
≤ 𝐶‖∇𝑑‖𝐿𝑝 (‖∇𝑢‖𝐿2 + 1 + ‖Δ𝑑‖𝐿2 ) ‖Δ𝑑‖𝐿2
1
4/𝑝
4/𝑝
≤ ‖Δ𝑑‖2𝐿2 + 𝐶‖∇𝑑‖2𝐿𝑝 (‖∇𝑢‖𝐿2 + 1 + ‖Δ𝑑‖𝐿2 )
4
1
2𝑝/(𝑝−2)
≤ ‖Δ𝑑‖2𝐿2 + ‖∇𝑢‖2𝐿2 + 𝐶‖∇𝑑‖𝐿𝑝
+ 𝐶,
2
which gives (9).
This completes the proof.
(43)
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).
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