# Global Regularity for the 2D Micropolar Fluid Flows with Mixed Partial Dissipation and Angular Viscosity

Authors: Zujin Zhang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper establishes the global existence and uniqueness of classical solutions to the 2D micropolar fluid  flows with mixed partial dissipation and angular viscosity.


---

## Body

## 1. Introduction

In this paper, we investigate the Cauchy problem for the viscous incompressible micropolar fluid flows. In three-dimensional case it can be expressed as(1)
v
t
-
(
ν
+
κ
)
Δ
v
-
2
κ
∇
×
w
+
∇
π
+
(
v
·
∇
)
v
=
0
,
w
t
-
γ
Δ
w
-
(
α
+
β
)
∇
∇
·
w
+
4
κ
w
-
2
κ
∇
×
v
+
(
v
·
∇
)
w
=
0
,
∇
·
v
=
0
,
v
(
0
)
=
v
0
,
w
(
0
)
=
w
0
.
Here, v
=
(
v
1
,
v
2
,
v
3
) is the divergence-free fluid velocity field, π is a scalar pressure, w
=
(
w
1
,
w
2
,
w
3
) is the microrotation field (angular velocity of the rotation of the particles of the fluid), and the constant ν
≥
0 is the Newtonian kinetic viscosity, κ
>
0 is the dynamics microrotation viscosity, and α
,
β
,
γ
≥
0 are the angular viscosities (see, e.g., [1, 2]).The micropolar fluid equations (1) enable us to consider some physical phenomena that cannot be treated by the classical Navier-Stokes equations (w
=
0 in (1)), such as the motion of animal blood, liquid crystals, and dilute aqueous polymer solutions. Physically, (1)1 represents the conservation of linear momentum, (1)2 reflects the conservation of angular momentum, and (1)3 is the incompressibility of the fluid, specifying the conservation of mass.Besides their physical applications, the micropolar fluid equations (1) are also mathematically important. The existence of weak and strong solutions was established by Galdi and Rionero [3] and Yamaguchi [4], respectively.In this paper, we study the global regularity problem of the2D micropolar fluid equations. Assuming that the velocity component in the z-direction is zero and the axes of rotation of particles are parallel to the z-axis, that is,
(2)
v
=
(
v
1
,
v
2
,
0
)
,
w
=
(
0,0
,
w
3
)
,
we obtain by gathering (2) into (1)
(3)
v
t
-
(
ν
+
κ
)
Δ
v
-
2
κ
∇
×
w
+
∇
π
+
(
v
·
∇
)
v
=
0
,
w
t
-
γ
Δ
w
+
4
κ
w
-
2
κ
∇
×
v
+
(
v
·
∇
)
w
=
0
,
∇
·
v
=
0
,
v
(
0
)
=
v
0
,
w
(
0
)
=
w
0
,
where v
=
(
v
1
,
v
2
) is a vector and w is a scalar. Here and in what follows, we use the notations
(4)
∇
×
v
=
∂
x
v
2
-
∂
y
v
1
,
∇
×
w
=
(
∂
y
w
,
-
∂
x
w
)
.
The global regularity of (3) with full viscosity has been established by Łukaszewicz [2] (see also [5] for more explicit result). The purpose of this paper is to investigate the global regularity of the 2D micropolar fluid flows with mixed partial dissipation and angular viscosity. To be precise, we will consider the following system:
(5)
v
t
-
(
ν
+
κ
)
v
x
x
-
2
κ
∇
×
w
+
∇
π
+
(
v
·
∇
)
v
=
0
,
w
t
-
γ
w
y
y
+
4
κ
w
-
2
κ
∇
×
v
+
(
v
·
∇
)
w
=
0
,
∇
·
v
=
0
,
v
(
0
)
=
v
0
,
w
(
0
)
=
w
0
.Our study is partially motivated by the global well-posedness of the2D MHD equations with partial viscosities (see [6, 7], for instance), that of the 2D Boussinesq equations with partial viscosity (see, e.g., [8, 9]), and that of the 2D micropolar fluid equations with zero angular viscosity [10].The main result of this paper now reads.Theorem 1.
Supposeν
>
0, κ
>
0, (
v
0
,
w
0
)
∈
H
2
(
ℝ
2
) with ∇
·
v
0
=
0. Then (5) with initial data (
v
0
,
w
0
) possesses a unique global classical solution (
v
,
w
). In addition, for any T
>
0, (
v
,
w
) satisfies
(6)
(
v
,
w
)
∈
L
∞
(
0
,
T
;
H
2
)
,
ω
x
∈
L
2
(
0
,
T
;
H
1
)
,
w
y
∈
L
2
(
0
,
T
;
H
2
)
,
where ω
=
∇
×
v is the vorticity.Remark 2.
Using the same method in this paper, we may also establish the global regularity for the following system:(7)
v
t
-
(
ν
+
κ
)
v
y
y
-
2
κ
∇
×
w
+
∇
π
+
(
v
·
∇
)
v
=
0
,
w
t
-
γ
w
x
x
+
4
κ
w
-
2
κ
∇
×
v
+
(
v
·
∇
)
w
=
0
,
∇
·
v
=
0
,
v
(
0
)
=
v
0
,
w
(
0
)
=
w
0
.The rest of this paper is organized as follows. In Section2, we recall an elementary lemma from [7]. Section 3 is devoted to establishing the a priori bounds for ∥
ω
∥
2 and ∥
∇
w
∥
2, while the bounds for ∥
∇
ω
∥
2 and ∥
∇
2
w
∥
2 are provided in Section 4. With the a priori estimates in Sections 3 and 4, we may conclude the proof of Theorem 1 as in [7]. Throughout this paper, the L
2-norm of a function f is denoted by ∥
f
∥
2.


## 2. An Elementary Lemma

We recall in this section the following elementary lemma from [7].Lemma 3.
Assume thatf, g, g
y, h, and h
x all belong to L
2
(
ℝ
2
). Then,
(8)
∬
‍
|
f
g
h
|
d
x
d
y
≤
2
∥
f
∥
2
∥
g
∥
2
1
/
2
∥
g
y
∥
2
1
/
2
∥
h
∥
2
1
/
2
∥
h
x
∥
2
1
/
2
.Proof.
We provide a proof of (8) simpler than that of [7].
Applying Hölder inequality,(9)
F
2
(
x
)
=
∫
‍
2
F
(
x
)
F
′
(
x
)
d
x
≤
2
(
∫
‍
|
F
(
x
)
|
2
d
x
)
1
/
2
(
∫
‍
|
F
x
(
x
)
|
2
d
x
)
1
/
2
.
Thus,
(10)
sup
⁡
x
∈
ℝ
|
F
(
x
)
|
≤
2
(
∫
‍
|
F
(
x
)
|
2
d
x
)
1
/
4
(
∫
‍
|
F
x
(
x
)
|
2
d
x
)
1
/
4
.
Consequently,
(11)
∬
‍
|
f
g
h
|
d
x
d
y
≤
∫
‍
[
(
∫
‍
|
f
|
2
d
x
)
1
/
2
(
∫
‍
|
g
|
2
d
x
)
1
/
2
sup
⁡
x
∈
ℝ
|
h
|
]
d
y
≤
2
∫
‍
[
(
∫
‍
|
f
|
2
d
x
)
1
/
2
(
∫
‍
|
g
|
2
d
x
)
1
/
2
h
h
h
h
h
h
h
h
h
h
×
(
∫
‍
|
h
|
2
d
x
)
1
/
4
(
∫
‍
|
h
x
|
2
d
x
)
1
/
4
]
d
y
≤
2
∥
f
∥
2
[
sup
⁡
y
∈
ℝ
(
∫
‍
|
g
|
2
d
x
)
1
/
2
]
∥
h
∥
2
1
/
2
∥
h
x
∥
2
1
/
2
≤
2
∥
f
∥
2
(
∫
‍
sup
⁡
y
∈
ℝ
|
g
|
2
d
x
)
1
/
2
∥
h
∥
2
1
/
2
∥
h
x
∥
2
1
/
2
≤
2
∥
f
∥
2
[
∫
‍
(
∫
‍
|
g
|
2
d
y
)
1
/
2
(
∫
‍
|
g
y
|
2
d
y
)
1
/
2
d
x
]
1
/
2
×
∥
h
∥
2
1
/
2
∥
h
x
∥
2
1
/
2
≤
2
∥
f
∥
2
∥
g
∥
2
1
/
2
∥
g
y
∥
2
1
/
2
∥
h
∥
2
1
/
2
∥
h
x
∥
2
1
/
2
.


## 3. A Priori Bounds for∥
ω
∥
2 and ∥
∇
w
∥
2

In this section, we establish the a priori bounds for∥
ω
∥
2 and ∥
∇
w
∥
2. First, we have the following energy estimates.Proposition 4.
Assume(
v
,
w
) solves (5) on [
0
,
T
]. Then,
(12)
∥
v
(
t
)
∥
2
2
+
∥
w
(
t
)
∥
2
2
+
(
ν
+
κ
)
∫
0
t
‍
∥
v
x
(
τ
)
∥
2
2
d
τ
+
γ
∫
0
t
‍
∥
w
y
(
τ
)
∥
2
2
d
τ
≤
C
∥
(
v
0
,
w
0
)
∥
2
2
.
Here C is a constant depending only on ν, κ, γ, and T.Proof.
Taking the inner product of (5)1 with v and (5)2 with w in L
2
(
ℝ
3
), respectively, we deduce
(13)
1
2
d
d
t
∥
(
v
,
w
)
∥
2
2
+
(
ν
+
κ
)
∥
v
x
∥
2
2
+
γ
∥
w
y
∥
2
2
+
4
κ
∥
w
∥
2
2
=
4
κ
∫
‍
(
∇
×
w
)
·
v
d
x
d
y
≡
I
,
where we use the following facts (the first one being well-known in the mathematical theory of fluid dynamics, and its proof is provided in the appendix):
(14)
∇
·
v
=
0
⟹
∫
‍
[
(
v
·
∇
)
v
]
·
Δ
v
d
x
d
y
=
0
,
∫
‍
(
∇
×
w
)
·
v
d
x
d
y
=
∫
‍
w
·
(
∇
×
v
)
d
x
d
y
,
∫
‍
∇
π
·
v
d
x
d
y
=
-
∫
‍
π
(
∇
·
v
)
d
x
d
y
=
0
.
Now, I can be dominated as
(15)
I
=
4
κ
∫
‍
(
∇
×
w
)
·
v
d
x
d
y
=
4
κ
∫
‍
(
w
y
v
1
-
w
x
v
2
)
d
x
d
y
=
4
κ
∫
‍
(
w
y
v
1
+
w
∂
x
v
2
)
d
x
d
y
≤
γ
2
∥
w
y
∥
2
2
+
C
∥
v
∥
2
2
+
ν
+
κ
2
∥
v
x
∥
2
2
+
C
∥
w
∥
2
2
.
Substituting (15) into (13), we obtain (12) by invoking Gronwall inequality.Remark 5.
Due to the partial dissipation and angular viscosity, we are not able to establish the uniform boundedness of∥
(
v
(
t
)
,
w
(
t
)
)
∥
2 on [
0
,
∞
) but rather the exponential growth:
(16)
∥
(
v
(
t
)
,
w
(
t
)
)
∥
2
≤
e
C
t
∥
(
v
0
,
w
0
)
∥
2
.Now, we are in a position to derive the bounds for∥
ω
∥
2 and ∥
∇
w
∥
2.Proposition 6.
Assume as in Proposition4. Then the vorticity ω
=
∇
×
v and w satisfy
(17)
∥
(
ω
(
t
)
,
∇
w
(
t
)
)
∥
2
2
+
2
(
ν
+
κ
)
∫
0
t
‍
∥
ω
x
(
τ
)
∥
2
2
d
τ
+
2
γ
∫
0
t
‍
∥
∇
w
y
(
τ
)
∥
2
2
d
τ
≤
2
∥
(
ω
0
,
∇
w
0
)
∥
2
2
.Proof.
Taking the curl of (5)1, we find
(18)
ω
t
-
(
ν
+
κ
)
ω
x
x
-
2
κ
∇
×
(
∇
×
w
)
+
(
v
·
∇
)
ω
=
0
.
Then, taking the inner product of (18) with ω and (5)2 with -
Δ
w in L
2
(
ℝ
2
), respectively, we obtain
(19)
1
2
d
d
t
∥
(
ω
,
∇
w
)
∥
2
2
+
(
ν
+
κ
)
∥
ω
x
∥
2
2
+
γ
∥
∇
w
y
∥
2
2
+
4
κ
∥
∇
w
∥
2
2
=
2
κ
∫
‍
∇
×
(
∇
×
w
)
·
ω
d
x
d
y
+
2
κ
∫
‍
∇
×
v
·
Δ
w
d
x
d
y
=
-
2
κ
∫
‍
Δ
w
·
ω
d
x
d
y
+
2
κ
∫
‍
∇
×
v
·
Δ
w
d
x
d
y
=
0
.
Applying Gronwall inequality, we may complete the proof of Proposition 6.


## 4. A Priori Bounds for∥
∇
ω
∥
2 and ∥
∇
2
w
∥
2

This section is devoted to deriving the a priori bounds for∥
∇
ω
∥
2 and ∥
∇
2
w
∥
2.Proposition 7.
Assume as in Proposition6. Then,
(20)
∥
(
∇
ω
(
t
)
,
Δ
w
(
t
)
)
∥
2
2
+
(
ν
+
κ
)
∫
0
t
‍
∥
∇
ω
x
(
τ
)
∥
2
2
d
τ
+
γ
∫
0
t
‍
∥
Δ
w
y
(
τ
)
∥
2
2
d
τ
≤
C
∥
(
∇
ω
0
,
Δ
w
0
)
∥
2
2
.
Here C is a constant depending only on ν, κ, γ, and T.Proof.
Taking the inner product of (18) with -
Δ
ω and (5)2 with Δ
2
w in L
2
(
ℝ
3
), respectively, we find
(21)
1
2
d
d
t
∥
∇
ω
∥
2
2
+
(
ν
+
κ
)
∥
∇
ω
x
∥
2
2
=
-
2
κ
∫
‍
∇
×
(
∇
×
w
)
·
Δ
ω
d
x
d
y
+
∫
‍
[
(
v
·
∇
)
ω
]
·
Δ
ω
d
x
d
y
,
1
2
d
d
t
∥
Δ
w
∥
2
2
+
γ
∥
Δ
w
y
∥
2
2
=
-
4
κ
∫
‍
w
·
Δ
2
w
d
x
d
y
+
2
κ
∫
‍
ω
·
Δ
2
w
d
x
d
y
-
∫
‍
[
(
v
·
∇
)
w
]
·
Δ
2
w
d
x
d
y
.
Gathering the above equations together, noticing that
(22)
∇
×
(
∇
×
w
)
=
-
Δ
w
,
∇
·
v
=
0
⟹
∫
‍
[
(
v
·
∇
)
ω
]
·
Δ
ω
d
x
d
y
=
-
∫
‍
[
(
∇
v
·
∇
)
ω
]
·
∇
ω
d
x
d
y
,
we see
(23)
1
2
d
d
t
∥
(
∇
ω
,
Δ
w
)
∥
2
2
+
(
ν
+
κ
)
∥
∇
ω
x
∥
2
2
+
γ
∥
Δ
w
y
∥
2
2
=
-
4
κ
∫
‍
|
Δ
w
|
2
d
x
d
y
+
4
κ
∫
‍
Δ
ω
·
Δ
w
d
x
d
y
+
∫
‍
[
(
∇
v
·
∇
)
ω
]
·
∇
ω
d
x
d
y
-
∫
‍
Δ
[
(
v
·
∇
)
w
]
·
Δ
w
d
x
d
y
.
Expanding the right-hand side of (23) gives
(24)
1
2
d
d
t
∥
(
∇
ω
,
Δ
w
)
∥
2
2
+
(
ν
+
κ
)
∥
∇
ω
x
∥
2
2
+
γ
∥
Δ
w
y
∥
2
2
+
4
κ
∥
Δ
w
∥
2
2
≤
4
κ
∫
‍
ω
x
x
·
Δ
w
d
x
d
y
+
4
κ
∫
‍
ω
y
y
·
Δ
w
d
x
d
y
+
∫
‍
∂
x
v
1
·
ω
x
·
ω
x
d
x
d
y
+
∫
‍
∂
x
v
2
·
ω
y
·
ω
x
d
x
d
y
+
∫
‍
∂
y
v
1
·
ω
x
·
ω
y
d
x
d
y
+
∫
‍
∂
y
v
2
·
ω
y
·
ω
y
d
x
d
y
+
C
∫
‍
|
Δ
v
|
·
|
∇
w
|
·
|
Δ
w
|
d
x
d
y
+
C
∫
‍
|
∇
v
|
·
|
∇
2
w
|
·
|
Δ
w
|
d
x
d
y
≡
∑
i
=
1
8
‍
K
i
.
For K
1, applying Hölder inequality yields
(25)
K
1
=
4
κ
∫
‍
ω
x
x
·
Δ
w
d
x
d
y
≤
∫
‍
ν
+
κ
12
|
∇
ω
x
|
2
+
C
|
Δ
w
|
2
d
x
d
y
.
For K
2, integrating by parts gives
(26)
K
2
=
4
κ
∫
‍
ω
y
y
·
Δ
w
d
x
d
y
=
-
4
κ
∫
‍
ω
y
·
Δ
w
y
d
x
d
y
≤
∫
‍
C
|
∇
ω
|
2
+
γ
6
|
Δ
w
y
|
2
d
x
d
y
.
For K
3, we apply Lemma 3 to deduce
(27)
K
3
=
∫
‍
∂
x
v
1
·
ω
x
·
ω
x
d
x
d
y
≤
C
∥
∂
x
v
1
∥
2
1
/
2
∥
∂
x
y
v
1
∥
2
1
/
2
∥
ω
x
∥
2
1
/
2
∥
ω
x
x
∥
2
1
/
2
∥
ω
x
∥
2
≤
C
∥
ω
∥
2
1
/
2
∥
ω
x
∥
2
1
/
2
∥
ω
x
∥
2
1
/
2
∥
ω
x
x
∥
2
1
/
2
∥
ω
x
∥
2
≤
C
∥
ω
x
∥
2
2
∥
∇
ω
x
∥
2
1
/
2
(
By Proposition
6
,
ω
∈
L
∞
(
0
,
T
;
L
2
)
)
≤
ν
+
κ
12
∥
∇
ω
x
∥
2
2
+
C
∥
ω
x
∥
2
2
/
3
∥
∇
ω
∥
2
2
.
Similarly, we have
(28)
K
4
=
∫
‍
∂
x
v
2
·
ω
y
·
ω
x
d
x
d
y
≤
C
∥
∂
y
v
2
∥
2
1
/
2
∥
∂
x
y
v
2
∥
2
1
/
2
∥
ω
y
∥
2
1
/
2
∥
ω
y
x
∥
2
1
/
2
∥
ω
x
∥
2
≤
C
∥
ω
∥
2
1
/
2
∥
ω
x
∥
2
1
/
2
∥
ω
y
∥
2
1
/
2
∥
∇
ω
x
∥
2
1
/
2
∥
ω
x
∥
2
≤
C
∥
ω
x
∥
2
3
/
2
∥
ω
x
∥
2
1
/
2
∥
∇
ω
x
∥
2
1
/
2
≤
ν
+
κ
12
∥
∇
ω
x
∥
2
2
+
C
∥
ω
x
∥
2
2
/
3
∥
∇
ω
∥
2
2
;
K
5
=
∫
‍
∂
y
v
1
·
ω
x
·
ω
y
d
x
d
y
≤
C
∥
∂
y
v
1
∥
2
∥
ω
x
∥
2
1
/
2
∥
ω
x
y
∥
2
1
/
2
∥
ω
y
∥
2
1
/
2
∥
ω
y
x
∥
2
1
/
2
≤
C
∥
ω
∥
2
∥
ω
x
∥
2
1
/
2
∥
ω
y
∥
2
1
/
2
∥
∇
ω
x
∥
2
≤
ν
+
κ
12
∥
∇
ω
x
∥
2
2
+
C
∥
∇
ω
∥
2
2
;
K
6
=
∫
‍
∂
y
v
2
·
ω
y
·
ω
y
d
x
d
y
=
-
∫
‍
∂
x
v
1
·
ω
y
·
ω
y
d
x
d
y
=
2
∫
‍
v
1
·
ω
y
·
ω
y
x
d
x
d
y
≤
C
∥
v
1
∥
2
1
/
2
∥
∂
y
v
1
∥
2
1
/
2
∥
ω
y
∥
2
1
/
2
∥
ω
y
x
∥
2
1
/
2
∥
ω
y
x
∥
2
≤
C
∥
ω
y
∥
2
1
/
2
∥
∇
ω
x
∥
2
3
/
2
≤
ν
+
κ
12
∥
∇
ω
x
∥
2
2
+
C
∥
∇
ω
∥
2
2
.
Now, for K
7, K
8, we use Lemma 3 and Young inequality to see
(29)
K
7
=
C
∫
‍
|
Δ
v
|
·
|
∇
w
|
·
|
Δ
w
|
d
x
d
y
≤
C
∥
Δ
v
∥
2
1
/
2
∥
Δ
v
x
∥
2
1
/
2
∥
∇
w
∥
2
∥
Δ
w
∥
2
1
/
2
∥
Δ
w
y
∥
2
1
/
2
≤
C
∥
∇
ω
∥
2
1
/
2
∥
∇
ω
x
∥
2
1
/
2
∥
∇
w
∥
2
∥
Δ
w
∥
2
1
/
2
∥
Δ
w
y
∥
2
1
/
2
≤
ν
+
κ
12
∥
∇
ω
x
∥
2
2
+
γ
6
∥
Δ
w
y
∥
2
2
+
C
(
∥
∇
w
∥
2
2
+
∥
Δ
w
∥
2
2
)
;
(30)
K
8
=
C
∫
‍
|
∇
v
|
·
|
∇
2
w
|
·
|
Δ
w
|
d
x
d
y
≤
C
∥
∇
v
∥
2
1
/
2
∥
∇
v
x
∥
2
1
/
2
∥
∇
2
w
∥
2
1
/
2
∥
∇
2
w
y
∥
2
1
/
2
∥
Δ
w
∥
2
≤
C
∥
ω
∥
2
1
/
2
∥
ω
x
∥
2
1
/
2
∥
Δ
w
∥
2
1
/
2
∥
Δ
w
y
∥
2
1
/
2
∥
Δ
w
∥
2
≤
γ
6
∥
Δ
w
y
∥
2
2
+
C
∥
ω
x
∥
2
2
/
3
∥
Δ
w
∥
2
2
.
Gathering (25)–(30) into (24) yields
(31)
d
d
t
∥
(
∇
ω
,
Δ
w
)
∥
2
2
+
(
ν
+
κ
)
∥
∇
ω
x
∥
2
2
+
γ
∥
Δ
w
y
∥
2
2
≤
C
(
1
+
∥
ω
x
∥
2
2
/
3
)
∥
(
∇
ω
,
Δ
w
)
∥
2
2
.
According to Proposition 6, we may invoke Gronwall inequality to deduce (20).



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*Source: 709746-2014-05-12.xml*