Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 676353, 6 pages
http://dx.doi.org/10.1155/2013/676353
Research Article
A Regularity Criterion for the Magneto-Micropolar Fluid
−1
Equations in 𝐵̇ ∞,∞
Zhihao Tang,1 Gang Wang,2 and Haiwa Guan3
1
Department of Fundamentals, Henan Polytechnic Institute, Nanyang, Henan 473009, China
Shandong Transport Vocational College, Weifang, Shandong 261206, China
3
Department of Public Teaching, Wenzhou Vocational College of Science and Technology, Wenzhou, Zhejiang 325000, China
2
Correspondence should be addressed to Haiwa Guan; haiwaguan1234@126.com
Received 16 January 2013; Accepted 2 March 2013
Academic Editor: Fuyi Xu
Copyright © 2013 Zhihao Tang 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 paper is dedicated to study of the Cauchy problem for the magneto-micropolar fluid equations in three-dimensional spaces. A
new logarithmically improved regularity criterion for the magneto-micropolar fluid equations is established in terms of the pressure
−1
in the homogeneous Besov space 𝐵̇ ∞,∞ .
1. Introduction
This paper concerns with the regularity of weak solutions to
the magneto-micropolar fluid equations in three dimensions
as
𝜕𝑡 V − (𝜇 + 𝜒) ΔV + V ⋅ ∇V − 𝑏 ⋅ ∇𝑏 + ∇ (𝑝 + 𝑏2 )
− 𝜒∇ × 𝜔 = 0,
𝜕𝑡 𝜔 − 𝛾Δ𝜔 − 𝜅∇ div 𝜔 + 2𝜒𝜔 + V ⋅ ∇𝜔 − 𝜒∇ × V = 0,
𝜕𝑡 𝑏 − ]Δ𝑏 + V ⋅ ∇𝑏 − 𝑏 ⋅ ∇V = 0,
(1)
div V = div 𝑏 = 0,
V (0, 𝑥) = V0 (𝑥) ,
𝜔 (0, 𝑥) = 𝜔0 (𝑥) ,
𝑏 (0, 𝑥) = 𝑏0 (𝑥) ,
where V(𝑡, 𝑥) = (V1 (𝑡, 𝑥), V2 (𝑡, 𝑥), V3 (𝑡, 𝑥)) ∈ R3 denotes the
velocity of the fluid at a point 𝑥 ∈ R3 , 𝑡 ∈ [0, 𝑇), 𝜔(𝑡, 𝑥) ∈
R3 , 𝑏(𝑡, 𝑥) ∈ R3 , and 𝑝(𝑡, 𝑥) ∈ R denote, respectively, the
microrotational velocity, the magnetic field, and the hydrostatic pressure. 𝜇, 𝜒, 𝜅, 𝛾, ] are positive numbers associated
to properties of the material: 𝜇 is the kinematic viscosity, 𝜒 is
the vortex viscosity, 𝜅 and 𝛾 are spin viscosities, and 1/] is the
magnetic Reynold. 𝑢0 , 𝜔0 , 𝑏0 are initial data for the velocity,
the angular velocity, and the magnetic field with properties
div 𝑢0 = 0 and div 𝑏0 = 0. For more detailed background,
we refer the readers to [1–3].
As we know, the problem of global regularity or finite
time singularity for the weak solutions of the magneto-micropolar fluid equations model with large initial data still
remains unsolved since (1) includes the 3D Navier-Stokes
equations. It is of interest that the regularity of the weak solutions is under preassumption of certain growth conditions.
There are a lot of lectures to study the regularity of weak
solutions of the magneto-micropolar fluid equations (see, [4–
6]). The purpose of this paper is to establish a new logarithmically improved regularity criterion for the micropolar fluid
−1
equations in terms of the pressure in Besov space 𝐵̇ ∞,∞ . Now
we state the main results as follows.
Theorem 1. Let (V0 (𝑥), 𝜔0 (𝑥), 𝑏0 (𝑥)) ∈ 𝐻1 (R3 ). Let 𝑇 > 0 and
(V, 𝜔, 𝑏) be a weak solution to the system (1). If the pressure filed
𝑃 satisfies the following condition:
∫
𝑇
0
‖𝑃 (𝑡, ⋅)‖2𝐵̇ −1
∞,∞
1 + ln (𝑒 + ‖𝑃 (𝑡, ⋅)‖𝐵̇ −1 )
𝑑𝑡 < ∞,
∞,∞
then the weak solution (V, 𝜔, 𝑏) is regular on [0, 𝑇].
(2)
2
Discrete Dynamics in Nature and Society
−1
Remark 2. Since the space 𝐵̇ ∞,∞ is wider than Ṁ 2,3 , so our
result resolves the limit case 𝑟 = 1 in [7], which greatly
improves the result in [7].
Lemma 5 (see [11]). Let 𝑓 ∈ 𝑊1,𝑠 (R3 ) (𝑠 ≥ 1), and 𝑟 ≥ 1,
then there exists a positive constant 𝐶 independent of 𝑓 such
that
−1
Remark 3. Since the space 𝐵̇ ∞,∞ is wider than 𝐿3/𝑟,∞ , hence
our result extends and improves the recent results given by
[4].
󵄩 󵄩1−𝛼 󵄩 󵄩𝛼
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑓󵄩󵄩𝐿𝛾 ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿2 󵄩󵄩󵄩∇𝑓󵄩󵄩󵄩𝐿2 ,
where
2. Preliminaries and Lemmas
𝛼=
Throughout this paper, we introduce some function spaces,
notations, and important inequalities.
Let 𝑒𝑡Δ denote the heat semigroup defined by
𝑡Δ
𝑒 𝑓 = 𝐾𝑡 ∗ 𝑓,
𝐾𝑡 = (4𝜋𝑡)
−3/2
|𝑥|2
exp (−
)
4𝑡
(3)
for 𝑡 > 0 and 𝑥 ∈ R3 , where ∗ denotes the convolution of
functions defined on R3 .
We now recall the definition of the homogeneous Besov
−𝛼
space with negative indices 𝐵̇ ∞,∞ on R𝑛 and the homoge𝛼
neous Sobolev space 𝐻̇ 𝑞 of exponent 𝛼 > 0. It is known (p.
−𝛼
if and only if
192 of [8]) that 𝑓 ∈ S󸀠 (R3 ) belongs to 𝐵̇
∞,∞
𝑒𝑡Δ ∈ 𝐿∞ for all 𝑡 > 0 and 𝑡𝛼/2 ‖𝑒𝑡Δ ‖∞ ∈ 𝐿∞ (0, ∞; 𝐿∞ ). The
−𝛼
norm of 𝐵̇ ∞,∞ is defined, up to equivalence, by
󵄩 󵄩
󵄩󵄩 󵄩󵄩 −𝛼
= sup (𝑡𝛼/2 󵄩󵄩󵄩󵄩𝑒𝑡Δ 󵄩󵄩󵄩󵄩∞ ) .
󵄩󵄩𝑓󵄩󵄩𝐵̇
∞,∞
𝑡>0
(4)
𝛼
We introduce now the homogeneous Sobolev space 𝐻̇ 𝑞 (R3 ),
which is defined by the set of functions 𝑓 ∈ 𝐿𝑟 (R3 ), 1/𝑟 =
(1/𝑞) − (𝑠/3) such that (−Δ)𝑠/2 𝑓 ∈ 𝐿𝑞 (R3 ). This space is
endowed with the norm
󵄩󵄩 󵄩󵄩 𝛼 󵄩󵄩󵄩
𝑠/2 󵄩
󵄩󵄩𝑓󵄩󵄩𝐻̇ 𝑞 = 󵄩󵄩(−Δ) 𝑓󵄩󵄩󵄩󵄩𝐿𝑞 ,
(5)
𝛼
𝛼
and when 𝑞 = 2, we just let 𝐻̇ 2 (R3 ) = 𝐻̇ (R3 ). Additionally,
we have the following inclusion relations (see, e.g., [9]):
1/2
−1
𝐻̇ (R3 ) ⊂ 𝐿3 (R3 ) ⊂ 𝐿3,∞ (R3 ) ⊂ 𝐵̇ ∞,∞ (R3 ) ,
1/2
−1
𝐻̇ (R3 ) ⊂ 𝐿3 (R3 ) ⊂ Ṁ 2,3 (R3 ) ⊂ 𝐵̇ ∞,∞ (R3 )
(6)
󵄩
󵄩󵄩 󵄩󵄩
𝑠/2 󵄩𝑝/𝑞 󵄩 󵄩1−(𝑝/𝑞)
󵄩󵄩𝑓󵄩󵄩𝐿𝑞 ≤ 𝐶󵄩󵄩󵄩󵄩(−Δ) 𝑓󵄩󵄩󵄩󵄩𝐿𝑝 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐵̇ −𝛼 .
∞,∞
(10)
For given initial data (V0 , 𝜔0 , 𝑏0 ) ∈ 𝐻1 (R3 ), the weak solution
is the same as the local strong solution (V, 𝜔, 𝑏) in a local
interval (0, 𝑇) as in the discussion of Navier-Stokes equations.
For the uniqueness and existence of local strong solution,
we refer to [1]. Thus, it proves that Theorem 1 is reduced
to establish a priori estimates uniformly in (0, 𝑇) for strong
solutions. With the use of the a priori estimates, the local
strong solution (V, 𝜔, 𝑏) can be continuously extended to 𝑡 =
𝑇 by a standard process to obtain global regularity of the weak
solution. Therefore, we assume that the solution (V, 𝜔, 𝑏) is
sufficiently smooth on (0, 𝑇).
Proof of Theorem 1. We show that Theorem 1 holds under
condition (1). To prove the theorem, we need the 𝐿4 -estimate.
For this purpose, taking the inner product of the first equation of (1) with |𝑢|2 𝑢 and integrating by parts, it can be deduced that
1 𝑑
‖𝑢‖4 4 + (𝜇 + 𝜒) ‖|∇𝑢| |𝑢|‖2𝐿2
4 𝑑𝑡 𝐿
1
󵄩2
󵄩
(𝜇 + 𝜒) 󵄩󵄩󵄩󵄩∇|𝑢|2 󵄩󵄩󵄩󵄩𝐿2
2
+
(11)
2
2
≤ 2 ∫ |𝑃| |𝑢| |∇𝑢| 𝑑𝑥 + 3𝜒 ∫ |𝑤| |𝑢| |∇𝑢| 𝑑𝑥
R3
R3
󵄨
󵄨
− ∫ |𝑏| 󵄨󵄨󵄨󵄨∇ (|𝑢|2 𝑢)󵄨󵄨󵄨󵄨 |𝑏| 𝑑𝑥,
R3
where we used the following relations by the divergence-free
condition div 𝑢 = 0:
∫ 𝑢 ⋅ ∇𝑢 ⋅ |𝑢|2 𝑢𝑑𝑥 =
R3
1
∫ 𝑢 ⋅ ∇|𝑢|4 𝑑𝑥 = 0,
2 R3
∫ Δ𝑢 ⋅ |𝑢|2 𝑢𝑑𝑥 = − ∫ |∇𝑢|2 |𝑢|2 𝑑𝑥 −
R3
R3
1
󵄨
󵄨2
∫ 󵄨󵄨󵄨∇|𝑢|2 󵄨󵄨󵄨󵄨 𝑑𝑥,
2 R3 󵄨
∫ ∇ × 𝜔 ⋅ |𝑢|2 𝑢𝑑𝑥
(7)
In particular, for 𝑠 = 1, 𝑝 = 2, and 𝑞 = 4, we get 𝛼 = 1 and
󵄩󵄩 󵄩󵄩
󵄩 󵄩1/2 󵄩 󵄩1/2
󵄩󵄩𝑓󵄩󵄩𝐿4 ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐻̇ 1 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐵̇ −1 .
∞,∞
(1/𝑟) − (1/𝛾)
.
(1/3) − (1/𝑠) − (1/𝑟)
3. Proof of Theorem 1
with continuous injection.
Lemma 4 (see [10]). Let 1 < 𝑝 < 𝑞 < ∞ and 𝑠 = 𝛼((𝑞/𝑝) −
1) > 0. Then there exists a constant 𝐶 depending only on 𝛼, 𝑝,
𝛼
−𝛼
and 𝑞 such that for all 𝑓 ∈ 𝐻̇ 𝑝 (R3 ) ∩ 𝐵̇ ∞,∞ (R3 ),
(9)
(8)
R3
= − ∫ |𝑢|2 𝜔 ⋅ ∇ × 𝑢𝑑𝑥 − ∫ 𝜔 ⋅ ∇|𝑢|2 × 𝑢𝑑𝑥,
R3
|∇ × 𝑢| ≤ |∇𝑢| ,
R3
|∇ |𝑢|| ≤ |∇𝑢| .
(12)
Discrete Dynamics in Nature and Society
3
Similarly, taking the inner product of the second equation of
(1) with |𝜔|2 𝜔 and integrating by parts, it can be inferred that
𝛾󵄩
1 𝑑
󵄩2
‖𝜔‖4𝐿4 + 𝛾‖|∇𝜔| |𝜔|‖2𝐿2 + 󵄩󵄩󵄩󵄩∇|𝜔|2 󵄩󵄩󵄩󵄩𝐿2 + 𝑘‖ div 𝜔‖2𝐿2
4 𝑑𝑡
2
(13)
+ 2𝜒‖𝜔‖4𝐿4 ≤ 3𝜒 ∫ |𝑢| |𝜔|2 |∇𝜔| 𝑑𝑥.
R3
Using an argument similar to that used in deriving the estimate (11)–(13), it can be obtained for the third equation of (1)
that
1 𝑑
‖𝑏‖4 4 + ‖|∇𝑏| |𝑏|‖2𝐿2 + 2‖∇ |𝑏| |𝑏|‖2𝐿2
4 𝑑𝑡 𝐿
󵄨
󵄨
≤ ∫ |𝑏| 󵄨󵄨󵄨󵄨∇ (|𝑏|2 𝑏)󵄨󵄨󵄨󵄨 |𝑢| 𝑑𝑥.
3
R
With the help of (8) and (19), by the Hölder inequality and
the Young inequality, we deduce that
1
∫ |𝑃|2 |V|2 𝑑𝑥
2 R3
1
≤ ‖𝑃‖2𝐿4 ‖V‖2𝐿4 ≤ 𝐶‖∇𝑃‖𝐿2 ‖𝑃‖𝐵̇ −1 ‖V‖2𝐿4
∞,∞
2
≤ 𝐶 (‖|V| |∇V|‖𝐿2 + ‖|𝑏| |∇𝑏|‖𝐿2 ) ‖𝑃‖𝐵̇ −1 ‖V‖2𝐿4
∞,∞
1/2
= (‖|V| |∇V|‖𝐿2 + ‖|𝑏| |∇𝑏|‖𝐿2 ) (𝐶‖𝑃‖2𝐵̇ −1 ‖V‖4𝐿4 )
∞,∞
(14)
≤
1
(‖|V| |∇V|‖2𝐿2 + ‖|𝑏| |∇𝑏|‖2𝐿2 ) + 𝐶‖𝑃‖2𝐵̇ −1 ‖V‖4𝐿4 .
∞,∞
4
(20)
Adding up (11), (13), and (14), then we obtain
So the term 𝐼1 can be estimated as
1 𝑑
(‖𝑢‖4𝐿4 + ‖𝜔‖4𝐿4 + ‖𝑏‖4𝐿4 ) + (𝜇 + 𝜒) ‖|∇𝑢| |𝑢|‖2𝐿2
4 𝑑𝑡
+
𝛾󵄩
1
󵄩2
󵄩2
󵄩
(𝜇 + 𝜒) 󵄩󵄩󵄩󵄩∇|𝑢|2 󵄩󵄩󵄩󵄩𝐿2 + 𝛾‖|∇𝜔| |𝜔|‖2𝐿2 + 󵄩󵄩󵄩󵄩∇|𝜔|2 󵄩󵄩󵄩󵄩𝐿2
2
2
+ 𝑘‖ div 𝜔‖2𝐿2 + 2𝜒‖𝜔‖4𝐿4 + ‖|∇𝑏| |𝑏|‖2𝐿2 + 2‖∇ |𝑏| |𝑏|‖2𝐿2
2
𝐼1 ≤
1
1
󵄨
󵄨2
∫ 󵄨󵄨󵄨∇|V|2 󵄨󵄨󵄨󵄨 𝑑𝑥 + (‖|V| |∇V|‖2𝐿2 + ‖|𝑏| |∇𝑏|‖2𝐿2 )
2 R3 󵄨
4
+ 𝐶‖𝑃‖2𝐵̇ −1 ‖V‖4𝐿4 .
∞,∞
(21)
2
≤ 2 ∫ |𝑃| |𝑢| |∇𝑢| 𝑑𝑥 + 3𝜒 ∫ |𝑤| |𝑢| |∇𝑢| 𝑑𝑥
R3
R3
󵄨
󵄨
+ 3𝜒 ∫ |𝑢| |𝜔|2 |∇𝜔| 𝑑𝑥 − ∫ |𝑏| 󵄨󵄨󵄨󵄨∇ (|𝑢|2 𝑢)󵄨󵄨󵄨󵄨 |𝑏| 𝑑𝑥
3
3
R
R
󵄨
󵄨
𝐼4 ≤ ∫ |𝑏|2 |𝑢| 󵄨󵄨󵄨󵄨∇|𝑢|2 󵄨󵄨󵄨󵄨 𝑑𝑥.
󵄨
󵄨
+ ∫ |𝑏| 󵄨󵄨󵄨󵄨∇ (|𝑏|2 𝑏)󵄨󵄨󵄨󵄨 |𝑢| 𝑑𝑥
R3
R3
≜ 𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 + 𝐼5 .
(15)
Applying the Hölder inequality and the Young inequality for
𝐼2 , it follows that
𝐼2 ≤
𝜒+𝜇
‖|∇𝑢| |𝑢|‖2𝐿2 + 𝐶 (‖𝑢‖4𝐿4 + ‖𝜔‖4𝐿4 ) .
2
(16)
Arguing similarly to above, it can be derived for 𝐼3 that
𝐼3 ≤
𝛾
‖|∇𝜔| |𝜔|‖2𝐿2 + 𝐶 (‖𝑢‖4𝐿4 + ‖𝜔‖4𝐿4 ) .
2
(17)
Considering the term 𝐼1 , by virtue of the Cauchy inequality,
we have
𝐼1 ≤
1
1
󵄨
󵄨2
∫ 󵄨󵄨󵄨󵄨∇|V|2 󵄨󵄨󵄨󵄨 𝑑𝑥 + ∫ |𝑃|2 |V|2 𝑑𝑥.
3
2 R
2 R3
(19)
(22)
Since 𝑢 ∈ 𝐿2 (R3 ) ∩ 𝐿6 (R3 ) and using Cauchy inequality, generalized Hölder inequality, Gagliardo-Nirenberg inequality,
and Sobolev imbedding theorem, we obtain
𝜒 + 𝜇 󵄩󵄩
󵄩 󵄩
󵄩2
󵄩
󵄩2
󵄩
󵄩
󵄩󵄩∇|𝑢|2 󵄩󵄩󵄩 2
𝐼4 ≤ 𝐶󵄩󵄩󵄩󵄩|𝑏|2 |𝑢|󵄩󵄩󵄩󵄩𝐿2 󵄩󵄩󵄩󵄩∇|𝑢|2 󵄩󵄩󵄩󵄩𝐿2 ≤ 𝐶󵄩󵄩󵄩󵄩|𝑏|2 |𝑢|󵄩󵄩󵄩󵄩𝐿2 +
󵄩𝐿
4 󵄩
𝜒 + 𝜇 󵄩󵄩
󵄩2
󵄩 󵄩2
󵄩󵄩∇|𝑢|2 󵄩󵄩󵄩 2
≤ 𝐶󵄩󵄩󵄩󵄩|𝑏|2 󵄩󵄩󵄩󵄩𝐿6 ‖𝑢‖2𝐿3 +
󵄩𝐿
4 󵄩
𝜒 + 𝜇 󵄩󵄩
󵄩2
󵄩2
󵄩
󵄩󵄩∇|𝑢|2 󵄩󵄩󵄩 2
≤ 𝐶󵄩󵄩󵄩󵄩∇|𝑏|2 󵄩󵄩󵄩󵄩𝐿2 ‖𝑢‖𝐿2 ‖∇𝑢‖𝐿2 +
󵄩𝐿
4 󵄩
𝜒 + 𝜇 󵄩󵄩
󵄩2
󵄩󵄩∇|𝑢|2 󵄩󵄩󵄩 2 .
≤ 𝐶‖|𝑏| ∇ |𝑏|‖2𝐿2 +
󵄩𝐿
4 󵄩
(18)
Let us bound the integral (1/2) ∫R3 |𝑃|2 |V|2 𝑑𝑥. Applying the
divergence operator div to the first equation of (1), one formally has 𝑃 = ∑3𝑖,𝑗=1 𝑅𝑖 𝑅𝑗 (𝑢𝑖 𝑢𝑗 − 𝑏𝑖 𝑏𝑗 ), where 𝑅𝑗 denotes the
𝑗th Riesz operator. By the Calderon-Zygmund inequality, we
have
‖∇𝑃‖𝐿2 ≤ 𝐶 (‖|V| |∇V|‖𝐿2 + ‖|𝑏| |∇𝑏|‖𝐿2 ) .
Next we have the following estimate for the term 𝐼4 :
(23)
The last term of (15) can be treated in the same way as
1󵄩
󵄩
󵄨
󵄩2
󵄨
󵄩
𝐼5 ≤ 𝐶 ∫ |𝑏|2 |𝑢| 󵄨󵄨󵄨󵄨∇|𝑏|2 󵄨󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶󵄩󵄩󵄩󵄩|𝑏|2 |𝑢|󵄩󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩󵄩∇|𝑏|2 󵄩󵄩󵄩󵄩𝐿2
8
R3
≤ 𝐶‖|𝑏| ∇ |𝑏|‖2𝐿2 .
(24)
4
Discrete Dynamics in Nature and Society
Inserting the estimates (15) and (21) into (14), it follows that
𝑑
(‖V‖44 + ‖𝜔‖44 + ‖𝑏‖44 )
𝑑𝑡
≤
𝐶‖𝑃‖2𝐵̇ −1 ‖V‖4𝐿4
∞,∞
+
𝐶 (‖V‖44
+
‖𝜔‖44
1 𝑑
‖∇V‖2𝐿2 + ‖ΔV‖2𝐿2
2 𝑑𝑡
‖𝑏‖44 )
+
second equation of (1) by (−Δ𝜔), and the third equation of (1)
by (−Δ𝑏), by integration by parts over R3 , we get
≤ 𝐶‖𝑃‖2𝐵̇ −1 (‖V‖44 + ‖𝜔‖44 ) + 𝐶 (‖V‖44 + ‖𝜔‖44 + ‖𝑏‖44 )
= ∫ (V ⋅ ∇) V ⋅ ΔV 𝑑𝑥
R3
∞,∞
≤ 𝐶 (1 +
+ ∫ (𝑏 ⋅ ∇) 𝑏 ⋅ ΔV 𝑑𝑥 − ∫ curl 𝜔ΔV 𝑑𝑥
‖𝑃 (𝑡, ⋅)‖2𝐵̇ −1
∞,∞
1 + ln (𝑒 + ‖𝑃 (𝑡, ⋅)‖𝐵̇ −1 )
R3
)
≤ ‖V‖𝐿4 ‖∇V‖𝐿4 ‖ΔV‖𝐿2
∞,∞
× [1 + ln (𝑒 + ‖𝑃 (𝑡, ⋅)‖𝐵̇ −1 )] (‖V‖44 + ‖𝜔‖44 + ‖𝑏‖44 )
∞,∞
≤ 𝐶 (1 +
‖𝑃 (𝑡, ⋅)‖2𝐵̇ −1
∞,∞
1 + ln (𝑒 + ‖𝑃 (𝑡, ⋅)‖𝐵̇ −1 )
× [1 + ln (𝑒 +
≤ 𝐶 (1 +
+
‖𝑃 (𝑡, ⋅)‖2𝐵̇ −1
∞,∞
1 + ln (𝑒 + ‖𝑃 (𝑡, ⋅)‖𝐵̇ −1 )
‖𝜔‖44
+
∞,∞
1 + ln (𝑒 + ‖𝑃 (𝑡, ⋅)‖𝐵̇ −1 )
)
1
‖ΔV‖2𝐿2 + 𝐶‖∇𝜔‖2𝐿2
16
2
16
2
2
+ 𝐶‖𝑏‖16
𝐿4 ‖𝑏‖𝐿2 + 𝐶‖V‖𝐿4 ‖V‖𝐿2 + 𝐶‖𝜔‖𝐿2 ,
1 𝑑
‖∇𝜔‖2𝐿2 + ‖Δ𝜔‖2𝐿2 + ‖∇ div 𝜔‖2𝐿2 + 2‖∇𝜔‖2𝐿2
2 𝑑𝑡
= ∫ (V ⋅ ∇) 𝜔 ⋅ Δ𝜔 𝑑𝑥 − ∫ curl VΔ𝜔 𝑑𝑥
)
≤ ‖V‖𝐿4 ‖∇𝜔‖𝐿4 ‖Δ𝜔‖𝐿2 + ‖∇V‖𝐿2 ‖Δ𝜔‖𝐿2
∞,∞
× [1 + ln (𝑒 + 𝑦 (𝑡))] (‖V‖44 + ‖𝜔‖44 + ‖𝑏‖44 ) ,
(25)
where 𝑦(𝑡) is defined by
󵄩
󵄩2 󵄩
󵄩2 󵄩
󵄩2
𝑦 (𝑡) =: sup (󵄩󵄩󵄩󵄩Λ3 V󵄩󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩󵄩Λ3 𝜔󵄩󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩󵄩Λ3 𝑏󵄩󵄩󵄩󵄩𝐿2 ) .
𝑇 ≤𝑠≤𝑡
(26)
≤ ‖V‖𝐿4 ‖𝜔‖1/8
‖Δ𝜔‖7/8
‖Δ𝜔‖𝐿2
𝐿2
𝐿2
+ 𝐶‖V‖1/2
‖ΔV‖1/2
‖Δ𝜔‖𝐿2
𝐿2
𝐿2
1
1
2
2
≤ ‖ΔV‖2𝐿2 + ‖Δ𝜔‖2𝐿2 + 𝐶‖V‖16
𝐿4 ‖𝜔‖𝐿2 + 𝐶‖V‖𝐿2 ,
8
8
0
Applying Gronwall’s inequality on (25) for the interval [𝑇0 , 𝑡],
one has
sup (‖V‖44 + ‖𝜔‖44 + ‖𝑏‖44 ) ≤ 𝐶0 exp (𝐶𝜀 (1 + ln (𝑒 + 𝑦 (𝑡))))
𝑇0 ≤𝑠≤𝑡
≤ 𝐶0 exp (2𝐶𝜀 ln (𝑒 + 𝑦 (𝑡)))
= ∫ (V ⋅ ∇𝑏) Δ𝑏 𝑑𝑥 − ∫ (𝑏 ⋅ ∇V) Δ𝑏 𝑑𝑥
2𝐶𝜀
(27)
provided that
∫
𝑇0
∞,∞
1 + ln (𝑒 + ‖𝑃 (𝑡, ⋅)‖𝐵̇ −1 )
𝑑𝑠 < 𝜀 ≪ 1,
R3
≤ ‖V‖𝐿4 ‖∇𝑏‖𝐿4 ‖Δ𝑏‖𝐿2 + ‖𝑏‖𝐿4 ‖∇V‖𝐿4 ‖Δ𝑏‖𝐿2
≤ 𝐶0 (𝑒 + 𝑦 (𝑡))
‖𝑃 (𝑡, ⋅)‖2𝐵̇ −1
(30)
1 𝑑
‖∇𝑏‖2𝐿2 + ‖Δ𝑏‖2𝐿2
2 𝑑𝑡
R3
𝑡
(29)
1
1
1
≤ ‖ΔV‖2𝐿2 + ‖Δ𝜔‖2𝐿2 + ‖Δ𝑏‖2𝐿2
8
8
8
× [1 + ln (𝑒 + ‖V (𝑡, ⋅)‖2𝐿6 )] (‖V‖44 + ‖𝜔‖44 + ‖𝑏‖44 )
≤ 𝐶 (1 +
≤
‖ΔV‖7/8
‖ΔV‖𝐿2
‖V‖𝐿4 ‖V‖1/8
𝐿2
𝐿2
+
‖𝑏‖44 )
∞,∞
‖𝑃 (𝑡, ⋅)‖2𝐵̇ −1
+ ‖𝑏‖𝐿4 ‖∇𝑏‖𝐿4 ‖ΔV‖𝐿2 + ‖∇𝜔‖𝐿2 ‖ΔV‖𝐿2
+ ‖𝑏‖𝐿4 ‖𝑏‖1/8
‖Δ𝑏‖7/8
‖ΔV‖𝐿2
𝐿2
𝐿2
)
∞,∞
‖𝑃 (𝑡, ⋅)‖𝐿3 )] (‖V‖44
R3
(28)
≤ ‖V‖𝐿4 ‖𝑏‖1/8
‖Δ𝑏‖7/8
‖ΔV‖𝐿2
𝐿2
𝐿2
+ ‖𝑏‖𝐿4 ‖V‖1/8
‖ΔV‖7/8
‖Δ𝑏‖𝐿2
𝐿2
𝐿2
1
1
2
16
2
≤ ‖ΔV‖2𝐿2 + ‖Δ𝑏‖2𝐿2 + 𝐶‖V‖16
𝐿4 ‖𝑏‖𝐿2 + 𝐶‖𝑏‖𝐿4 ‖V‖𝐿2 ,
8
8
(31)
∞,∞
where 𝐶0 is a positive constant depending on 𝑇0 .
Next we will estimate the 𝐿2 -norm of ∇V, ∇𝜔, and ∇𝑏. We
multiply both sides of the first equation of (1) by (−ΔV), the
where we have used the Gagliardo-Nirenberg inequality:
󵄩󵄩 󵄩󵄩
󵄩 󵄩1/8 󵄩 󵄩7/8
󵄩󵄩∇𝑓󵄩󵄩𝐿4 ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿2 󵄩󵄩󵄩Δ𝑓󵄩󵄩󵄩𝐿2 .
(32)
Discrete Dynamics in Nature and Society
5
Combining (29), (30), and (31) and using the definition of the
weak solution, we deduce that
where we used (33) with 𝑠 = 3, 𝑝 = 3/2, 𝑝1 = 𝑞1 = 𝑝2 =
𝑞2 = 3 and the following inequalities:
󵄩󵄩 3 󵄩󵄩1/4
󵄩Λ V󵄩󵄩 2 ,
‖∇V‖𝐿3 ≤ 𝐶‖∇V‖3/4
𝐿2 󵄩
󵄩
󵄩𝐿
‖∇V‖2𝐿2 + ‖∇𝜔‖2𝐿2 + ‖∇𝑏‖2𝐿2
6𝐶𝜀
≤ 𝐶𝐶0 (𝑒 + 𝑦 (𝑡))
(𝑡 − 𝑇0 )
󵄩2 󵄩
󵄩2
󵄩
+ 󵄩󵄩󵄩∇V (⋅, 𝑇0 )󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩∇𝜔 (⋅, 𝑇0 )󵄩󵄩󵄩𝐿2 .
If we use the existing estimate (31) for 𝑇0 < 𝑡 < 𝑇, (36) reduces
to
Finally we go to the estimate for 𝐻3 -norm of V, 𝜔, and 𝑏. In the
following calculations, we will use the following commutator
estimate due to Kato and Ponce [12]:
󵄩󵄩 𝑠
𝑠 󵄩
󵄩󵄩Λ (𝑓𝑔) − 𝑓Λ 𝑔󵄩󵄩󵄩𝐿𝑝
󵄩
󵄩 󵄩 󵄩
󵄩
󵄩 󵄩 󵄩
≤ (󵄩󵄩󵄩∇𝑓󵄩󵄩󵄩𝐿𝑝1 󵄩󵄩󵄩󵄩Λ𝑠−1 𝑔󵄩󵄩󵄩󵄩𝐿𝑞1 + 󵄩󵄩󵄩Λ𝑠 𝑓󵄩󵄩󵄩𝐿𝑝2 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿𝑞2 ) ,
(34)
1 𝑑
󵄨
󵄨2 󵄨
󵄨2 󵄨
󵄨2
∫ 󵄨󵄨󵄨Λ3 V󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨Λ3 𝜔󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨Λ3 𝑏󵄨󵄨󵄨󵄨 𝑑𝑥
2 𝑑𝑡 R3 󵄨
𝐴3 + 𝐴5 + 𝐴6 + 𝐴7 ≤
1 󵄩󵄩 4 󵄩󵄩2 󵄩󵄩 4 󵄩󵄩2 󵄩󵄩 4 󵄩󵄩2
(󵄩󵄩Λ V󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩Λ 𝜔󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩Λ 𝑏󵄩󵄩󵄩𝐿2 )
6 󵄩
+ 𝐶𝐶0 (𝑒 + 𝑦 (𝑡))
.
(39)
For 𝐴 2 and 𝐴 4 , we have
𝐴2 + 𝐴4 ≤
1 󵄩󵄩 4 󵄩󵄩2 󵄩󵄩 4 󵄩󵄩2
󵄩
󵄩2 󵄩
󵄩2
(󵄩󵄩Λ V󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩Λ 𝜔󵄩󵄩󵄩𝐿2 ) + 𝐶 (󵄩󵄩󵄩󵄩Λ3 V󵄩󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩󵄩Λ3 𝜔󵄩󵄩󵄩󵄩𝐿2 )
6 󵄩
1 󵄩󵄩 4 󵄩󵄩2 󵄩󵄩 4 󵄩󵄩2
(󵄩󵄩Λ V󵄩󵄩󵄩𝐿2 + 󵄩󵄩󵄩Λ 𝜔󵄩󵄩󵄩𝐿2 ) + 𝐶𝐶0 (𝑒 + 𝑦 (𝑡)) .
6 󵄩
(40)
Inserting the above estimates (38)–(40) into (35), we obtain
󵄨
󵄨
󵄨2
󵄨2
+ ∫ 󵄨󵄨󵄨󵄨Λ3 div V󵄨󵄨󵄨󵄨 𝑑𝑥 + 2 ∫ 󵄨󵄨󵄨󵄨Λ3 𝜔󵄨󵄨󵄨󵄨 𝑑𝑥
3
3
R
R
𝑑
󵄨
󵄨2 󵄨
󵄨2
∫ 󵄨󵄨󵄨Λ3 V󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨Λ3 𝜔󵄨󵄨󵄨󵄨 𝑑𝑥
𝑑𝑡 R3 󵄨
3
(3/4)+(39/2)𝐶𝜀
= − ∫ [Λ (V ⋅ ∇V) − V ⋅ ∇Λ V] Λ V 𝑑𝑥
≤ 𝐶𝐶0 (𝑒 + 𝑦 (𝑡))
R3
+ ∫ Λ3 curl 𝜔 ⋅ Λ3 V 𝑑𝑥
R3
− ∫ [Λ3 (V ⋅ ∇𝜔) − V ⋅ ∇Λ3 𝜔] Λ3 𝜔 𝑑𝑥
(38)
Using (37) again, we have
≤
󵄨
󵄨
󵄨
󵄨2
󵄨2
󵄨2
+ ∫ 󵄨󵄨󵄨󵄨Λ4 V󵄨󵄨󵄨󵄨 𝑑𝑥 + ∫ 󵄨󵄨󵄨󵄨Λ4 𝜔󵄨󵄨󵄨󵄨 𝑑𝑥 + ∫ 󵄨󵄨󵄨󵄨Λ4 𝑏󵄨󵄨󵄨󵄨 𝑑𝑥
R3
R3
R3
3
1󵄩
󵄩2
(3/4)+(39/2)𝐶𝜀
.
𝐴 1 ≤ 󵄩󵄩󵄩󵄩Λ4 V󵄩󵄩󵄩󵄩𝐿2 + 𝐶𝐶0 (𝑒 + 𝑦 (𝑡))
6
(3/4)+(39/2)𝐶𝜀
with 𝑠 > 0, Λ𝑠 = (−Δ)𝑠/2 and (1/𝑝) = (1/𝑝1 ) + (1/𝑞1 ) =
(1/𝑝2 ) + (1/𝑞2 ). Taking the operation Λ3 on both sides of (1),
then multiplying them by Λ3 V, Λ3 𝜔, and Λ3 𝑏, and integrating
by parts over R3 , we have
3
(37)
󵄩󵄩 4 󵄩󵄩5/6
󵄩󵄩 3 󵄩󵄩
󵄩Λ V󵄩󵄩 2 .
󵄩󵄩Λ V󵄩󵄩 3 ≤ 𝐶‖∇V‖1/6
𝐿2 󵄩
󵄩
󵄩𝐿
󵄩𝐿
󵄩
(33)
(35)
(41)
+ 𝐶𝐶0 (𝑒 + 𝑦 (𝑡)) .
Gronwall’s inequality implies the boundness of 𝐻3 -norm of
V, 𝜔, and 𝑏 provided that 39𝐶𝜀 < (1/2), which can be
achieved by the absolute continuous property of integral (2).
This completes the proof of Theorem 1.
R3
Acknowledgment
+ ∫ Λ3 curl V ⋅ Λ3 𝜔 𝑑𝑥
R3
3
3
The authors thank Professor Xiaohong Fan for his profitable
discussion and suggestions.
3
+ ∫ [Λ (𝑏 ⋅ ∇𝑏) − 𝑏 ⋅ ∇Λ 𝑏] Λ V 𝑑𝑥
R3
References
− ∫ [Λ3 (V ⋅ ∇𝑏) − V ⋅ ∇Λ3 𝑏] Λ3 𝑏 𝑑𝑥
R3
+ ∫ [Λ3 (𝑏 ⋅ ∇V) − 𝑏 ⋅ ∇Λ3 V] Λ3 𝑏 𝑑𝑥
R3
≡ 𝐴 1 + 𝐴 2 + 𝐴 3 + 𝐴 4 + 𝐴 5 + 𝐴 6 + 𝐴 7.
Hence 𝐴 1 can be estimated as
󵄩
󵄩󵄩 3 󵄩󵄩1/4 󵄩󵄩 4 󵄩󵄩5/3
󵄩2
󵄩Λ V󵄩󵄩 2 󵄩󵄩Λ V󵄩󵄩 2
𝐴 1 ≤ 𝐶‖∇V‖𝐿3 󵄩󵄩󵄩󵄩Λ3 V󵄩󵄩󵄩󵄩𝐿3 ≤ 𝐶‖∇V‖13/2
𝐿2 󵄩
󵄩
󵄩𝐿 󵄩
󵄩𝐿
1󵄩
󵄩󵄩 3 󵄩󵄩3/2
󵄩2
≤ 󵄩󵄩󵄩󵄩Λ4 V󵄩󵄩󵄩󵄩𝐿2 + 𝐶‖∇V‖13/2
2 󵄩
󵄩󵄩Λ V󵄩󵄩󵄩𝐿2 ,
𝐿
6
(36)
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
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Stochastic Analysis
Abstract and
Applied Analysis
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Hindawi Publishing Corporation
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Optimization
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014