# On a System of Equations of a Non-Newtonian Micropolar Fluid

Authors: G. M. de Araújo; M. A. F. de Araújo; E. F. L. Lucena
Journal: Journal of Applied Mathematics
 (2015)

---

## Abstract

We investigate a problem for a model of a non-Newtonian micropolar fluid coupled system. The problem has been considered in a bounded, smooth domain ofR
3 with Dirichlet boundary conditions. The operator stress tensor is given by τ
(
e
(
u
)
)
=
[
(
ν
+
ν
0
M
(
|
e
(
u
)
|
2
)
)
e
(
u
)
]. To prove the existence of weak solutions we use the method of Faedo-Galerkin and compactness arguments. Uniqueness and periodicity of solutions are also considered.


---

## Body

## 1. Introduction

LetΩ be a bounded domain in R
d with smooth boundary ∂
Ω, and let T
>
0. We denote by Q
T the time space cylinder I
×
Ω, with lateral boundary Σ
=
I
×
∂
Ω, where I
=
(
0
,
T
) is a time interval. The unsteady flows of incompressible fluids in a boundary domain Ω
⊂
R
d, d
>
1, are described by the system of equations
(1)
ρ
∂
u
∂
t
-
∇
·
τ
e
u
+
ρ
u
·
∇
u
=
-
∇
p
+
ρ
f
in
Q
T
,
∇
·
u
=
0
in
Q
T
,
u
=
0
on
⁡
Σ
T
,
u
0
=
u
0
in
Ω
,
where u
=
(
u
1
,
u
2
,
…
,
u
d
) is the velocity, p represents the pressure, ρ is a positive constant determining the density of a material, f
=
(
f
1
,
f
2
,
…
,
f
d
) stands for the given external body forces, τ
:
R
sym
d
2
→
R
sym
d
2 denotes the extra stress tensor, e
:
R
d
→
R
sym
d
2 denotes the symmetric part of the velocity gradient; that is,
(2)
e
u
=
1
2
∇
u
+
∇
u
T
,
whose components are defined as in [1] by
(3)
2
e
i
j
u
=
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
,
i
,
j
=
1,2
,
…
,
d
and R
sym
d
2 represents the set of all symmetric d
×
d matrices; that is,
(4)
R
sym
d
2
=
D
∈
R
d
2
;
D
i
j
=
D
j
i
,
i
,
j
=
1,2
,
…
,
d
.
Note, for example, that when τ
(
e
(
u
)
) is of the form
(5)
τ
(
e
(
u
)
)
=
μ
0
(
1
+
e
u
p
-
2
)
e
(
u
)
,
with p
=
2, problem (1) turns into the Navier-Stokes system, which is a model for Newtonian fluids. In the expression (5), |
e
(
u
)
| denotes the usual Euclidean matrix norm. We observe that (5) can be written in the form
(6)
τ
(
e
(
u
)
)
=
μ
0
M
e
u
2
e
(
u
)
,
where M
:
R
0
+
→
R
0
+, M
∈
C
0
(
0
,
∞
) is the generalized viscosity function. Fluids constituted by (6) are sometimes named fluids with shear-dependent viscosity. Models belonging to this class of non-Newtonian fluid mechanics are frequently used in several fields of chemistry, glaciology, biology, and geology, as discussed by Malek et al. [2].The first mathematical investigations of problem (1) was done by Ladyzhenskaya in 1963, where she proposed to study system (1) with (5) and p
=
4. Combining monotone operator theory and compactness arguments, she proved the existence of weak solution to model (1), if p
≥
1
+
2
d
/
d
+
2, and their uniqueness if p
≥
d
+
2
/
2. See also Lions [3] for another proof of the same results. More results are known about problem (1) obtained in a series of papers, including those of Malek et al. [2], Malek et al. [4], Frehse and Málek [5], Malek et al. [1], and other mathematicians.The equations below describe the motion of Newtonian micropolar fluids:(7)
∂
u
∂
t
-
(
ν
+
ν
r
)
Δ
u
+
(
u
·
∇
)
u
+
∇
p
=
2
ν
r
∇
×
w
+
f
in
Q
T
,
∂
w
∂
t
-
(
c
a
+
c
d
)
Δ
w
+
(
u
·
∇
)
w
-
(
c
o
+
c
d
-
c
a
)
∇
(
∇
·
w
)
+
λ
w
=
2
ν
r
∇
×
u
+
g
in
Q
T
,
∇
·
u
=
0
in
Q
T
,
u
=
0
on
⁡
Σ
T
,
w
=
0
on
⁡
Σ
T
,
u
0
=
u
0
in
Ω
,
w
0
=
w
0
in
Ω
,
where u
(
x
,
t
)
,
w
(
x
,
t
)
∈
R
3 and p
(
x
,
t
)
∈
R, denoting for (
x
,
t
)
∈
Q, respectively, the unknown velocity, the microrotational velocity, and the hydrostatic pressure of the fluid and λ is a positive constant. The positive constants ν and ν
r are, respectively, theNewtonian andmicrorotational viscosity. The positive constants c
0, c
a, and c
d are calledcoefficients of angular viscosities and satisfy c
0
+
c
d
>
c
a.The main difference with respect to modeled fluids by the Navier-Stokes is that the rotation of the particles is taken into account. The above approach was introduced by Eringen [6]. The nonlinear coupled system (7) can be used to model the behavior of liquid crystals, polymeric fluids, and blood under some circumstances (see, e.g., [7]). These systems have been mainly analyzed in the book of Lukaszewicz [8].The problem that we study in this work consists in supposing that in system (7) the fluid is of the type (5). More precisely, we investigate the mixed problem: let Ω be a bounded domain in R
3 with smooth boundary ∂
Ω, and let T
>
0. We denote by Q
T the time space cylinder I
×
Ω, with lateral boundary Σ
=
I
×
∂
Ω, where I
=
(
0
,
T
) is a time interval. We find that u
,
w
:
Q
T
→
R
3 and p
:
Q
T
→
R solving the following system of equations:
(8)
u
′
-
∇
·
τ
e
u
+
u
·
∇
u
+
∇
p
=
∇
×
w
+
f
in
Q
T
,
w
′
-
ν
1
∇
·
e
w
+
u
·
∇
w
+
λ
1
w
=
λ
2
∇
×
u
+
g
in
Q
T
,
∇
·
u
=
0
in
Q
T
,
u
=
0
on
⁡
Σ
T
,
w
=
0
on
⁡
Σ
T
,
u
0
=
u
0
in
Ω
,
w
0
=
w
0
in
Ω
,
where the extra stress tensor is given by τ
(
e
(
u
)
)
=
(
ν
+
ν
0
M
(
|
e
(
u
)
|
2
)
)
e
(
u
), e
(
u
) as in (2) and (3), ν
0, ν
1, λ
1, and λ
2 are positives constants, u
=
(
u
1
,
u
2
,
u
3
), and ∇
×
u is given by
(9)
∇
×
u
=
∂
u
3
∂
x
2
-
∂
u
2
∂
x
3
,
∂
u
1
∂
x
3
-
∂
u
3
∂
x
1
,
∂
u
2
∂
x
1
-
∂
u
1
∂
x
2
;
the same holds for ∇
×
w. Let us consider M
:
(
0
,
∞
)
→
(
0
,
∞
) satisfying the hypothesis
(10)
M
∈
C
1
0
,
∞
,
M
>
M
0
>
0
,
M
′
>
0
,
(11)
c
1
e
u
2
≤
M
(
e
u
2
)
≤
c
2
e
u
2
,
where M
0, c
1, and c
2 are positive constants. We observe that if M is a constant function, then problem (8) reduces to problem (7).


## 2. Notation and Main Results

In order to solve problem (8) we need some notations about Sobolev spaces. We use standard notation of L
p
(
Ω
), W
m
,
p
(
Ω
), and C
p
(
Ω
) for functions that are defined on Ω and range in R and the notation of L
p
(
Ω
), W
m
,
p
(
Ω
), and C
p
(
Ω
) for functions that range in R
d. We also work with the spaces L
p
(
I
;
W
m
,
p
(
Ω
)
) or L
p
(
Q
T
).By〈
·
,
·
〉 we will represent the duality pairing between X and X
′, with  X
′ being the topological dual of the space X. We Also define the followings spaces:
(12)
V
=
φ
∈
D
Ω
;
∇
·
φ
=
0
.
V
p
=
V
p
(
Ω
) is the closure of V in the space W
1
,
p
(
Ω
), p
∈
(
1
,
∞
). In particular, V
=
V
2. The norm of gradient in V
p is given by
(13)
∇
u
p
≡
∫
Ω
‍
∇
u
x
p
d
x
1
/
p
.
The inner product and norm in V is given, respectively, by
(14)
u
,
v
=
∑
i
,
j
=
1
3
‍
∫
Ω
‍
∂
u
i
∂
x
j
x
∂
v
i
∂
x
j
x
d
x
,
u
2
=
∑
i
,
j
=
1
3
‍
∫
Ω
‍
∂
u
i
∂
x
j
(
x
)
2
d
x
.
H
=
H
(
Ω
) is the closure of V in the space L
2
(
Ω
), with inner product and norm defined, respectively, by
(15)
u
,
v
=
∑
i
=
1
3
‍
∫
Ω
‍
u
i
x
v
i
x
d
x
,
u
2
=
∑
i
=
1
3
‍
∫
Ω
‍
u
i
x
2
d
x
.Remark 1.
H
0
1
(
Ω
) and L
2
(
Ω
) are Hilbert’s spaces. We note that H
0
1
(
Ω
)
↪
c
L
2
(
Ω
)
↪
H
-
1
(
Ω
), where the first embeddings are compact.We introduce the following bilinear and the trilinear forms, as well as the convention of summation of indices, that is,α
i
β
j instead of ∑
i
,
j
=
1
d
‍
α
i
β
j:
(16)
a
u
,
v
=
∫
Ω
‍
∂
u
i
∂
x
j
x
∂
v
i
∂
x
j
x
d
x
=
u
,
v
,
∀
u
,
v
∈
V
,
(17)
b
u
,
v
,
w
=
∫
Ω
‍
u
i
x
∂
v
j
∂
x
i
x
w
j
x
d
x
,
∀
u
,
v
,
w
∈
V
.
We note that (see Lions [3])
(18)
b
u
,
v
,
w
=
-
b
u
,
w
,
v
,
∀
u
∈
V
,
∀
v
,
w
∈
H
0
1
(
Ω
)
.
We also introduce the notations
(19)
A
u
=
-
Δ
u
,
B
u
v
=
u
·
∇
v
,
∀
u
,
v
∈
V
,
(20)
K
u
=
-
∇
·
M
e
u
2
e
u
,
∀
u
∈
V
.
According to this, we have
(21)
A
u
,
v
=
a
u
,
v
,
∀
u
,
v
∈
V
,
(22)
B
u
v
,
w
=
b
u
,
v
,
w
,
∀
u
∈
V
,
∀
v
,
w
∈
H
0
1
(
Ω
)
,
(23)
K
u
,
v
=
∫
Ω
‍
M
e
u
2
e
i
j
u
e
i
j
v
d
x
,
∀
u
,
v
∈
V
.Remark 2.
We observe thatM
′
>
0 implies for all u
1
,
u
2
∈
V that
(24)
K
u
1
-
K
u
2
,
u
1
-
u
2
=
∫
Ω
‍
M
(
e
u
1
2
)
e
i
j
(
u
1
)
-
M
(
e
u
2
2
)
e
i
j
(
u
2
)
×
e
i
j
(
u
1
)
-
e
i
j
(
u
2
)
d
x
≥
0
.
Therefore K
:
V
→
V
′ is a monotonous operator.Definition 3.
Letu
0
∈
H
,
w
0
∈
L
2
(
Ω
), f
∈
L
4
/
3
(
I
,
V
′
), and g
∈
L
2
(
I
;
H
-
1
(
Ω
)
). A weak solution to (8) is a pair of functions {
u
,
w
}, such that
(25)
u
∈
L
∞
(
I
;
H
)
∩
L
4
(
I
;
V
4
)
∩
L
4
(
I
;
V
)
,
w
∈
L
∞
(
I
;
L
2
(
Ω
)
)
∩
L
2
(
I
;
H
0
1
(
Ω
)
)
satisfying the following identity:
(26)
∫
0
T
‍
u
′
t
,
φ
d
t
+
ν
∫
0
T
‍
a
(
u
(
t
)
,
φ
)
d
t
+
∫
0
T
‍
b
(
u
(
t
)
,
u
(
t
)
,
φ
)
d
t
+
ν
0
∫
0
T
‍
∫
Ω
‍
M
e
u
t
2
e
i
j
u
t
e
i
j
φ
d
x
d
t
=
∫
0
T
‍
(
∇
×
w
(
t
)
,
φ
)
d
t
+
∫
0
T
‍
f
t
,
φ
d
t
,
∀
φ
∈
D
(
I
;
V
)
,
∫
0
T
‍
w
′
t
,
ϕ
d
t
+
ν
1
∫
0
T
‍
a
w
t
,
ϕ
d
t
+
ν
1
∫
0
T
‍
(
∇
·
w
(
t
)
,
∇
·
ϕ
)
d
t
+
∫
0
T
‍
b
(
u
(
t
)
,
w
(
t
)
,
ϕ
)
d
t
+
λ
1
∫
0
T
‍
(
w
(
t
)
,
ϕ
)
d
t
=
λ
2
∫
0
T
‍
(
∇
×
u
(
t
)
,
ϕ
)
d
t
+
∫
0
T
‍
g
t
,
ϕ
d
t
,
∀
ϕ
∈
D
(
I
;
D
(
Ω
)
)
,
u
0
=
u
0
,
w
(
0
)
=
w
0
.Lemma 4 (Korn’s inequality).
Let1
<
p
<
∞. Then, there exists a constant K
p
=
K
p
(
Ω
), such that the inequality
(27)
K
p
v
W
1
,
p
(
Ω
)
≤
e
(
v
)
L
p
(
Ω
)
is fulfilled for all v satisfying either v
∈
W
0
1
,
p
(
Ω
), where Ω
⊂
R
d is open and bounded with ∂
Ω
⊂
C
1.Proof.
The proof of this lemma can be found in [1], page 169.Lemma 5.
Letp
≥
2, τ
:
R
s
y
m
d
2
→
R
s
y
m
d
2, and Φ
:
R
0
+
→
R
0
+ and the assumptions below are satisfied for all B
,
D
∈
R
s
y
m
d
2 and i
,
j
,
k
,
l
=
1
,
…
,
d:
(28)
∂
i
j
Φ
(
D
2
)
=
τ
i
j
(
D
)
,
Φ
(
0
)
=
∂
i
j
Φ
(
0
)
=
0
,
∂
i
j
∂
k
l
Φ
(
D
2
)
B
i
j
B
k
l
≥
C
1
1
+
D
p
-
2
B
2
,
∂
i
j
∂
k
l
Φ
(
D
2
)
≤
C
2
1
+
D
p
-
2
.
Then, there exist positive constants C
i, i
=
3,4
,
5, such that
(29)
C
3
(
1
+
D
p
-
2
)
D
2
≤
Φ
(
D
2
)
≤
C
4
1
+
D
p
,
(30)
τ
B
-
τ
D
·
(
B
-
D
)
≥
C
5
B
-
D
2
.Proof.
The proof of this lemma can be found in [4], page 263.Lemma 6 (Vitali).
LetΩ be a bounded domain in R
n and f
m
:
Ω
→
R integrable for every m
∈
N. Assume that (1)
lim
⁡
m
→
∞
f
m
(
x
) exists and is finite for almost all x
∈
Ω;
(2)
for everyε
>
0 there exists δ
>
0 such that
(31)
sup
⁡
m
∈
N
∫
H
‍
f
m
x
d
x
<
ε
,
∀
H
∈
Ω
,
H
<
δ
;then
(32)
lim
⁡
⁡
m
→
∞
∫
Ω
‍
f
m
(
x
)
d
x
=
∫
Ω
‍
lim
⁡
m
→
∞
f
m
(
x
)
d
x
.Proof.
The proof of this lemma can be found in [9], page 63.Lemma 7.
Considerd
≥
3 and s
,
r
∈
R, with s
>
2, r
>
d, verifying 2
/
s
+
d
/
r
=
1. If u
∈
L
r
(
Ω
), then
(33)
b
u
,
v
,
w
≤
c
u
L
r
(
Ω
)
v
w
2
/
s
w
d
/
r
for all v
,
w
∈
V, where c
≥
0 is a constant independent of u, v, and w.Proof.
The proof of this lemma can be found in [3], page 84.Theorem 8.
Ifd
≤
3, u
0
∈
H, w
0
∈
L
2
(
Ω
), f
∈
L
4
/
3
(
I
;
V
′
), and g
∈
L
2
(
I
;
H
-
1
(
Ω
)
), then there exist a weak solution to problem (8).Theorem 9.
Under the assumptions of Theorem8 with d
=
2, problem (8) has a unique weak solution.Theorem 10 (periodic solutions).
Under the assumptions of Theorem8 there exist a pair of functions (
u
,
w
) such that
(34)
u
∈
L
∞
(
I
;
H
)
∩
L
4
(
I
;
V
4
)
∩
L
4
(
I
;
V
)
,
w
∈
L
∞
(
I
;
L
2
(
Ω
)
)
∩
L
2
(
I
;
H
0
1
(
Ω
)
)
,
u
′
(
t
)
,
φ
+
ν
a
(
u
(
t
)
,
φ
)
+
ν
0
(
K
u
(
t
)
,
φ
)
+
(
B
u
u
(
t
)
,
φ
)
=
(
∇
×
w
(
t
)
,
φ
)
+
(
f
(
t
)
,
φ
)
,
w
′
(
t
)
,
ϕ
+
ν
1
a
(
w
(
t
)
,
ϕ
)
+
ν
1
(
∇
·
w
(
t
)
,
∇
·
ϕ
)
+
(
B
u
w
(
t
)
,
ϕ
)
+
λ
1
(
w
(
t
)
,
ϕ
)
=
λ
2
∇
×
u
t
,
ϕ
+
g
t
,
ϕ
,
u
0
=
u
T
,
w
0
=
w
T
,
∀
φ
∈
V
,
∀
ϕ
∈
H
0
1
Ω
i
n
t
h
e
s
e
n
s
e
o
f
D
′
0
,
T
.Theorem 11.
Assuming thatd
≤
3, u
0
∈
V
∩
V
4, w
0
∈
H
0
1
(
Ω
), f
∈
L
4
/
3
(
I
;
V
′
), and g
∈
L
2
I
;
H
-
1
(
Ω
) there exist a unique weak solution to problem (8) such that
(35)
u
′
∈
L
2
(
I
;
H
)
,
u
∈
L
∞
(
I
;
V
4
)
,
u
∈
L
∞
(
I
;
V
)
,
w
∈
L
∞
I
;
H
0
1
(
Ω
)
,
w
′
∈
L
2
I
;
L
2
Ω
.


## 3. Proofs of the Results

Proof of Theorem8.
We will show the existence of a weak solution to system (8) employing the Galerkin approximations. For that purpose we consider (
φ
ν
)
ν
∈
N
⊂
V, a basis of eigenvectors of the Stokes operator and (
ϕ
μ
)
μ
∈
N
⊂
H
0
1
(
Ω
) a basis of eigenvectors of Lamé. We represent by V
m
=
[
φ
1
,
…
,
φ
m
]
⊂
V the subspace generated by {
φ
1
,
…
,
φ
m
} and W
m
=
[
ϕ
1
,
…
,
ϕ
m
]
⊂
H
0
1
(
Ω
) the subspace generated by {
ϕ
1
,
…
,
ϕ
m
}. Let us also consider the pair (
u
m
,
w
m
), such that
(36)
u
m
x
,
t
=
∑
r
=
1
m
‍
g
r
m
t
φ
r
x
,
w
m
(
x
,
t
)
=
∑
r
=
1
m
‍
h
r
m
(
t
)
ϕ
r
(
x
)
are the solution of the approximate problem
(37)
u
m
′
t
,
φ
r
+
ν
(
A
u
m
(
t
)
,
φ
r
)
+
ν
0
(
K
u
m
(
t
)
,
φ
r
)
+
B
u
m
u
m
t
,
φ
r
=
∇
×
w
m
t
,
φ
r
+
f
t
,
φ
r
,
r
=
1
,
…
,
m
,
w
m
′
t
,
ϕ
r
-
ν
1
(
∇
·
e
(
w
)
,
ϕ
r
)
+
B
u
m
w
m
t
,
ϕ
r
+
λ
1
(
w
m
(
t
)
,
ϕ
r
)
=
λ
2
(
∇
×
u
m
(
t
)
,
ϕ
r
)
+
(
g
(
t
)
,
ϕ
r
)
,
r
=
1
,
…
,
m
,
u
m
0
=
u
0
m
⟶
u
0
,
strongly
in
H
,
w
m
0
=
w
0
m
⟶
w
0
,
strongly
in
L
2
(
Ω
)
.
The system of ordinary differential equations (37) has a local solution on an interval [
0
,
t
m
[, 0
<
t
m
<
T. The first estimate permits us to extend this solution to the whole interval [
0
,
T
].
First Estimate. We sometimes omit the parameter t. Multiplying both sides of (37)1 by g
r
m and (37)2 by h
r
m, next adding from r
=
1 to r
=
m, we obtain
(38)
1
2
d
d
t
u
m
t
2
+
ν
u
m
(
t
)
2
+
ν
0
∫
Ω
‍
M
(
e
u
m
t
2
)
e
i
j
u
m
t
2
d
x
≤
w
m
t
u
m
(
t
)
+
f
(
t
)
V
′
u
m
(
t
)
,
(39)
1
2
d
d
t
w
m
t
2
+
ν
1
w
m
(
t
)
2
+
ν
1
∇
·
w
m
t
2
+
λ
1
w
m
t
2
≤
λ
2
u
m
t
w
m
t
+
g
(
t
)
H
-
1
(
Ω
)
w
m
(
t
)
,
because b
(
u
,
u
,
u
)
=
b
(
u
,
w
,
w
)
=
0, for all u
∈
V, for all w
∈
H
0
1
(
Ω
) (see Lions [3]), |
∇
×
u
m
|
=
|
∇
u
m
|
=
u
m and (
∇
×
w
m
,
u
m
)
=
(
w
m
,
∇
×
u
m
) (see Lukaszewicz [8]). Now using Young’s inequality we obtain from (38) and (39), respectively:
(40)
1
2
d
d
t
u
m
t
2
+
ν
u
m
(
t
)
2
+
ν
0
c
1
K
2
u
m
(
t
)
V
4
4
+
ν
2
u
m
(
t
)
4
≤
ν
4
u
m
(
t
)
2
+
c
ν
w
m
t
2
+
ν
2
2
u
m
(
t
)
4
+
c
ν
2
f
(
t
)
V
′
4
/
3
,
(41)
1
2
d
d
t
w
m
t
2
+
ν
1
w
m
(
t
)
2
≤
ν
4
u
m
(
t
)
2
+
c
ν
w
m
t
2
+
ν
1
2
w
m
(
t
)
2
+
c
ν
1
g
(
t
)
H
-
1
(
Ω
)
2
.
From (27) (Korn’s inequality) and (11) we can get
(42)
ν
0
2
∫
Ω
‍
M
(
e
u
m
2
)
e
i
j
u
m
2
d
x
≥
ν
0
c
1
2
e
(
u
m
)
L
4
(
Ω
)
4
≥
ν
0
c
1
K
2
u
m
V
4
4
,
ν
0
2
∫
Ω
‍
M
(
e
u
m
2
)
e
i
j
u
m
2
d
x
≥
ν
0
c
1
2
e
(
u
m
)
L
4
(
Ω
)
4
≥
ν
2
u
m
4
.
Adding inequalities (40) and (41) and integrating from 0 to t, with 0
≤
t
≤
T, we conclude
(43)
(
u
m
t
2
+
w
m
t
2
)
+
ν
0
c
1
K
∫
0
t
‍
u
m
(
s
)
V
4
4
d
s
+
ν
2
∫
0
t
‍
u
m
(
s
)
4
d
s
+
ν
1
∫
0
t
‍
w
m
(
s
)
2
d
s
≤
C
+
C
∫
0
t
‍
u
m
s
2
+
w
m
s
2
d
s
.
By using Gronwall’s inequality, we can write
(44)
u
m
t
2
+
w
m
t
2
≤
C
.
Therefore, it follows from (43) that
(45)
u
m
is
bounded
in
L
∞
(
I
;
H
)
,
(46)
(
u
m
)
is
bounded
in
L
4
(
I
;
V
4
)
,
(47)
(
u
m
)
is
bounded
in
L
4
(
I
;
V
)
,
(48)
(
w
m
)
is
bounded
in
L
∞
(
I
;
L
2
(
Ω
)
)
,
(49)
(
w
m
)
is
bounded
in
L
2
(
I
;
H
0
1
(
Ω
)
)
.
Second Estimate. We consider P
m
:
V
→
V
m as the orthogonal projections from V to V
m:
(50)
P
m
u
=
∑
j
=
1
m
‍
u
,
φ
j
φ
j
,
∀
u
(
t
)
∈
V
.
We also consider the adjoint operator to P
m which is P
m
*
:
V
′
→
V
′. We note that P
m
*
u
m
′
=
u
m
′. By the choice of the special basis (
φ
ν
), we obtain
(51)
P
m
L
(
V
,
V
)
≤
1
,
P
m
*
L
(
V
′
,
V
′
)
≤
1
.
It follows from (37)1, (21), (22), and (23) that
(52)
u
m
′
=
-
ν
P
m
*
A
u
m
-
ν
0
P
m
*
K
u
m
-
P
m
*
B
u
m
+
P
m
*
∇
×
w
m
+
P
m
*
f
.
We have |
〈
A
u
m
,
v
〉
|
≤
u
m
v, for all u
m
(
t
)
,
v
(
t
)
∈
V. Therefore (47) implies
(53)
A
u
m
is
bounded
in
L
4
(
I
;
V
′
)
↪
L
4
/
3
(
I
;
V
′
)
.
Let u
m
(
t
)
,
v
(
t
)
∈
V. From (20), Hölder’s inequality, and (11) we take
(54)
K
u
m
,
v
≤
M
(
e
u
m
2
)
e
(
u
m
)
,
∇
v
≤
c
e
u
m
3
v
≤
c
∇
u
m
3
v
≤
u
m
3
v
.
Therefore, from (47), we obtain
(55)
K
u
m
is
bounded
in
L
4
/
3
(
I
;
V
′
)
.
From d
≤
3 we derive H
0
1
(
Ω
)
↪
L
4
(
Ω
). Using (17) and Hölder’s inequality we conclude
(56)
B
u
m
u
m
,
v
≤
u
m
L
4
(
Ω
)
u
m
v
L
4
(
Ω
)
≤
c
u
m
2
v
for all u
m
(
t
)
,
v
(
t
)
∈
V. Therefore, from (47)(57)
B
u
m
u
m
is
bounded
in
L
2
(
I
;
V
′
)
↪
L
4
/
3
(
I
;
V
′
)
.
On the other hand, let w
m
(
t
)
∈
H
0
1
(
Ω
), and v
(
t
)
∈
V
(58)
∇
×
w
m
,
v
=
w
m
,
∇
×
v
≤
w
m
v
≤
c
w
m
v
(see Lukaszewicz [8], pp. 116). It follows from (49) that
(59)
∇
×
w
m
is
bounded
in
L
2
(
I
;
V
′
)
↪
L
4
/
3
(
I
;
V
′
)
.
It follows from (53)–(59), (51), and hypothesis about f that
(60)
u
m
′
is
bounded
in
L
4
/
3
(
I
;
V
′
)
.
Analogously letR
m
:
H
0
1
(
Ω
)
→
W
m be the orthogonal projections
(61)
R
m
w
=
∑
j
=
1
m
‍
w
,
ϕ
j
ϕ
j
,
∀
w
∈
H
0
1
(
Ω
)
.
We also consider the adjoint operator to R
m, which is R
m
*
:
H
-
1
(
Ω
)
→
H
-
1
(
Ω
). We have R
m
*
w
m
′
=
w
m
′ and by the choice of the special basis (
ϕ
ν
), we can get
(62)
R
m
L
(
H
0
1
(
Ω
)
,
H
0
1
(
Ω
)
)
≤
1
,
R
m
*
L
(
H
-
1
(
Ω
)
,
H
-
1
(
Ω
)
)
≤
1
.
From (37)2, (21), (22), and (23)(63)
w
m
′
=
-
ν
1
R
m
*
∇
·
e
w
m
-
R
m
*
B
w
m
-
λ
1
R
m
*
w
m
+
λ
2
R
m
*
∇
×
u
m
+
R
m
*
g
.
We note that |
〈
∇
·
e
(
w
m
)
,
v
〉
|
=
|
〈
∇
w
m
,
∇
v
〉
|
≤
w
m
v, for all w
m
(
t
)
,
v
(
t
)
∈
H
0
1
(
Ω
). Thus, (49) implies
(64)
∇
·
e
(
w
m
)
is
bounded
in
L
2
I
;
H
-
1
(
Ω
)
.
Analogously and by using the embedding H
0
1
(
Ω
)
↪
L
2
(
Ω
) we obtain
(65)
(
w
m
)
is
bounded
in
L
2
I
;
H
-
1
(
Ω
)
.
On the other hand |
〈
∇
×
u
m
,
v
〉
|
≤
u
m
v, for all u
m
(
t
)
,
v
(
t
)
∈
H
0
1
(
Ω
). Now, by using (47), we have
(66)
(
∇
×
u
m
)
is
bounded
in
L
2
I
;
H
-
1
(
Ω
)
.
Finally assuming that d
=
2, it follows from (18), (22), and Hölder’s inequality that
(67)
B
u
m
w
m
,
v
=
b
u
m
,
w
m
,
v
=
b
u
m
,
v
,
w
m
≤
c
u
m
L
4
(
Ω
)
v
w
m
L
4
(
Ω
)
for all u
m
(
t
)
,
w
m
(
t
)
,
v
(
t
)
∈
H
0
1
(
Ω
). Therefore, B
w
m
H
-
1
Ω
2
≤
c
|
u
m
|
u
m
|
w
m
|
w
m.
Because ifd
=
2, then, u
L
4
(
Ω
)
2
≤
c
|
u
|
u (see Lions [3]). Now using Young’s inequality we get
(68)
B
u
m
w
m
H
-
1
(
Ω
)
2
≤
c
u
m
2
u
m
2
+
c
w
m
2
w
m
2
.
Therefore, (47)–(49) and (45) permit us to obtain
(69)
(
B
u
m
w
m
)
is
bounded
in
L
2
I
;
H
-
1
(
Ω
)
,
d
=
2
.
Analogously and assuming that d
=
3 we obtain from Lemma 7
(70)
B
u
m
w
m
H
-
1
(
Ω
)
≤
u
m
L
6
(
Ω
)
w
m
1
/
2
w
m
1
/
2
.
H
0
1
(
Ω
)
↪
L
6
(
Ω
), because d
=
3. Thus,
(71)
B
u
m
w
m
H
-
1
Ω
2
≤
c
u
m
2
w
m
w
m
≤
c
u
m
4
+
c
w
m
2
w
m
2
.
Therefore, (47)–(49) imply that
(72)
(
B
u
m
w
m
)
is
bounded
in
L
2
I
;
H
-
1
(
Ω
)
,
d
=
3
.
It follows from (64)–(72) and hypothesis about g that
(73)
(
w
m
′
)
is
bounded
in
L
2
I
;
H
-
1
(
Ω
)
.
We note that (45)–(49), (60), (73), and the Aubin-Lions lemma imply that there exist subsequences of (
u
m
) and (
w
m
), still denoted by (
u
m
) and (
w
m
), such that
(74)
u
m
⟶
u
strongly
in
L
2
I
;
H
,
a
.
e
.
in
Q
T
,
(75)
u
m
⟶
u
weak
star
in
L
∞
(
I
;
H
)
,
(76)
u
m
⟶
u
weakly
in
L
4
(
I
;
V
)
,
(77)
u
m
′
⟶
u
′
weakly
in
L
4
/
3
(
I
;
V
′
)
,
(78)
w
m
⟶
w
strongly
in
L
2
I
;
L
2
Ω
,
a
.
e
.
in
Q
T
,
(79)
w
m
⟶
w
weak
star
in
L
∞
(
I
;
L
2
(
Ω
)
)
,
(80)
w
m
⟶
w
weakly
in
L
2
(
I
;
H
0
1
(
Ω
)
)
,
(81)
w
m
′
⟶
w
′
weakly
in
L
2
(
I
;
H
-
1
(
Ω
)
)
,
(82)
K
u
m
⟶
χ
weakly
in
L
4
/
3
(
I
;
V
′
)
.
We note that (46) and (60) imply that u
∈
C
0
(
I
;
H
). Similarly, (49) and (73) imply that w
∈
C
0
(
I
;
L
2
(
Ω
)
). Thus, it does make sense to consider u
(
0
)
=
u
0 and w
(
0
)
=
w
0.
In order to prove that(83)
∫
0
T
‍
b
(
u
m
,
u
m
,
φ
)
⟶
∫
0
T
‍
b
(
u
,
u
,
φ
)
,
∀
φ
∈
D
(
I
;
V
)
,
∫
0
T
‍
b
(
u
m
,
w
m
,
ϕ
)
⟶
∫
0
T
‍
b
(
u
,
w
,
ϕ
)
,
∀
ϕ
∈
D
I
;
D
(
Ω
)
,
we use (74) and (78) (see [1], pp. 210). Now, we note that
(84)
∫
Q
T
‍
e
i
j
(
w
m
)
e
i
j
(
ϕ
)
d
x
d
t
⟶
∫
Q
T
‍
e
i
j
(
w
)
e
i
j
(
ϕ
)
d
x
d
t
,
or equivalently
(85)
∫
0
T
‍
∇
·
e
(
w
m
)
,
ϕ
d
t
⟶
∫
0
T
‍
∇
·
e
(
w
)
,
ϕ
d
t
results from (80). The other terms of (37)2 are obtained in the usual manner. In order to prove that
(86)
∫
Q
T
M
(
e
u
m
2
)
e
i
j
(
u
m
)
e
i
j
(
φ
)
d
x
d
t
⟶
∫
Q
T
M
(
e
u
2
)
e
i
j
(
u
)
e
i
j
(
φ
)
d
x
d
t
,
we use the fact ∇
u
m
→
∇
u a.e. in Q
T, (see [5] pp. 565-566). Therefore,
(87)
∇
u
m
2
⟶
∇
u
2
a
.
e
.
in
Q
T
;
that is,
(88)
e
u
m
2
⟶
e
u
2
a
.
e
.
in
Q
T
.
Since M
∈
C
1
(
0
,
∞
) we obtain from (88)(89)
M
e
u
m
2
⟶
M
e
u
2
a
.
e
.
in
Q
T
.
Thus,
(90)
M
(
e
u
m
2
)
e
i
j
(
u
m
)
e
i
j
(
φ
)
⟶
M
(
e
u
2
)
e
i
j
(
u
)
e
i
j
(
φ
)
a.e. in Q
T and for all φ
∈
D
(
I
;
D
(
Ω
)
). Using (46) and (11) we obtain
(91)
∫
Q
T
‍
M
e
u
m
2
e
i
j
(
u
m
)
e
i
j
(
φ
)
d
x
d
t
≤
C
∫
Q
T
‍
e
u
m
3
e
i
j
φ
d
x
d
t
≤
C
∫
Q
T
‍
e
u
m
3
d
x
d
t
≤
C
∫
Q
T
‍
∇
u
m
3
d
x
d
t
≤
C
.
It follows that
(92)
M
e
u
m
2
e
i
j
(
u
m
)
e
i
j
(
φ
)
∈
L
1
(
Q
T
)
.
Moreover, if H
⊂
Q
T is a measurable set, we have from (11), (46), and Hölder’s inequality that
(93)
∫
H
‍
M
(
e
u
m
2
)
e
i
j
(
u
m
)
e
i
j
(
φ
)
d
x
d
t
≤
c
∫
H
‍
e
u
m
3
e
φ
d
x
d
t
≤
c
∫
Q
T
‍
e
u
m
4
d
x
d
t
3
/
4
∫
H
‍
e
φ
4
d
x
d
t
1
/
4
≤
c
∫
Q
T
‍
∇
u
m
4
d
x
d
t
3
/
4
H
1
/
4
≤
c
H
1
/
4
.
Therefore,
(94)
sup
⁡
m
∈
N
∫
H
‍
M
(
e
u
m
2
)
e
i
j
(
u
m
)
e
i
j
(
φ
)
d
x
d
t
≤
c
H
1
/
4
.
Assuming that |
H
| is sufficiently small, we obtain
(95)
sup
⁡
m
∈
N
∫
H
‍
M
(
e
u
m
2
)
e
i
j
(
u
m
)
e
i
j
(
φ
)
d
x
d
t
≤
ε
,
for all ε
∈
R. Now using (90), (92), (95), and Vitali’s lemma we can derive (86). Therefore, we can write χ
=
K
u in L
4
/
3
(
I
;
H
-
1
(
Ω
)
). The convergences (74)–(82) and (85) and (86) allow us to pass the limit on system (37), with φ
r and q
r being fixed to obtain
(96)
u
′
+
ν
A
u
+
ν
0
K
u
+
B
u
u
=
∇
×
w
+
f
in
L
4
/
3
(
I
;
V
′
)
,
w
′
+
ν
1
∇
·
e
(
w
)
+
B
u
w
+
λ
1
w
=
λ
2
∇
×
u
+
g
in
L
2
I
;
H
-
1
(
Ω
)
.
This concludes the proof of Theorem8.Proof of Theorem9.
Let(
u
1
,
w
1
) and (
u
2
,
w
2
) be weak solutions to Problem (8). Then,
(97)
u
1
,
u
2
∈
L
∞
(
I
;
H
)
∩
L
4
(
I
;
V
)
∩
L
4
(
I
,
V
4
)
,
w
1
,
w
2
∈
L
∞
I
;
L
2
(
Ω
)
∩
L
∞
I
;
H
0
1
(
Ω
)
.
Consider u
~
=
u
1
-
u
2 and w
~
=
w
1
-
w
2. Then, (
u
~
,
w
~
) satisfies
(98)
u
~
′
+
ν
A
u
~
+
ν
0
(
K
u
1
-
K
u
2
)
+
(
B
u
1
u
1
-
B
u
2
u
2
)
=
∇
×
w
~
,
w
~
′
+
ν
1
A
w
~
+
ν
1
∇
(
∇
·
w
~
)
+
(
B
u
1
w
1
-
B
u
2
w
2
)
+
λ
1
w
~
=
λ
2
∇
×
u
~
,
u
~
(
0
)
=
w
~
(
0
)
=
0
,
where the first equality has been considered in L
4
/
3
(
I
;
V
′
) and the second in L
2
H
-
1
(
Ω
). We take the duality in (98)1 and (98)2 with u
~ and w
~, respectively, to obtain
(99)
1
2
d
d
t
u
~
2
+
ν
u
~
2
+
ν
0
K
u
1
-
K
u
2
,
u
~
+
B
u
1
u
1
-
B
u
2
u
2
,
u
~
=
∇
×
w
~
,
u
~
,
1
2
d
d
t
w
~
2
+
ν
1
w
~
2
+
ν
1
∇
·
w
~
2
+
B
u
1
w
1
-
B
u
2
w
2
,
w
~
+
λ
1
w
~
2
=
λ
2
∇
×
u
~
,
w
~
,
u
~
(
0
)
=
w
~
(
0
)
=
0
.
We note that
(100)
B
u
1
u
1
t
-
B
u
2
u
2
t
,
u
~
t
=
b
(
u
~
(
t
)
,
u
1
(
t
)
,
u
~
(
t
)
)
,
B
u
1
w
1
t
-
B
u
2
w
2
t
,
w
~
t
=
b
(
u
~
(
t
)
,
w
2
(
t
)
,
w
~
(
t
)
)
.
From the monotonicity of K we have 〈
K
u
1
-
K
u
2
,
u
~
〉
≥
0. Thus,
(101)
1
2
d
d
t
u
~
2
+
ν
u
~
2
+
b
(
u
~
(
t
)
,
u
1
(
t
)
,
u
~
(
t
)
)
≤
∇
×
w
~
,
u
~
,
1
2
d
d
t
w
~
2
+
ν
1
w
~
2
+
λ
1
w
~
2
+
b
u
~
t
,
w
2
t
,
w
~
t
≤
λ
2
∇
×
u
~
,
w
~
.
Adding the inequalities above, we derive
(102)
1
2
d
d
t
u
~
2
+
w
~
2
+
ν
u
~
2
+
ν
1
w
~
2
≤
w
~
u
~
+
λ
2
u
~
w
~
+
b
u
~
,
u
1
,
u
~
+
b
u
~
,
w
2
,
w
~
.
In other words,
(103)
1
2
d
d
t
u
~
2
+
w
~
2
+
ν
u
~
2
+
ν
1
w
~
2
≤
ν
4
u
~
2
+
ν
4
u
~
2
+
c
ν
w
~
2
+
b
u
~
,
u
1
,
u
~
+
b
u
~
,
w
2
,
w
~
.
Considering d
=
2, we get H
0
1
(
Ω
)
↪
L
4
(
Ω
). Moreover (see Lions [3])
(104)
u
L
4
(
Ω
)
≤
c
u
1
/
2
u
1
/
2
.
Thus, using (17), Hölder’s inequality, (104), and Young’s inequality we take
(105)
b
u
~
,
u
1
,
u
~
+
b
u
~
,
w
2
,
w
~
≤
u
~
L
4
Ω
2
u
1
+
u
~
L
4
Ω
w
2
w
~
L
4
Ω
≤
c
u
1
u
~
u
~
+
w
2
u
~
1
/
2
u
~
1
/
2
w
~
1
/
2
w
~
1
/
2
≤
ν
4
u
~
2
+
c
ν
u
1
2
u
~
2
+
ν
2
u
~
2
ν
1
w
~
+
c
u
~
w
~
w
2
2
≤
ν
4
u
~
2
+
c
ν
u
1
2
u
~
2
+
ν
4
u
~
2
+
ν
1
w
~
2
+
c
w
2
2
u
~
2
+
c
w
2
2
w
~
2
.
It follows from (103) that we can write
(106)
1
2
d
d
t
u
~
2
+
w
~
2
≤
c
1
+
u
1
2
+
w
2
2
u
~
2
+
w
~
2
.
Integrating from 0 to t we obtain
(107)
u
~
t
2
+
w
~
t
2
≤
c
∫
0
t
‍
1
+
u
1
s
2
+
w
2
s
2
(
u
~
s
2
+
w
~
s
2
)
d
s
.
Applying Gronwall’s inequality in (107), we deduce by using (47) and (49) that
(108)
u
1
t
=
u
2
t
,
w
1
t
=
w
2
t
,
∀
t
∈
0
,
T
.
Theorem 9 has been proved.Proof of Theorem10.
Under the assumptions and notations defined in the proof of the Theorem8 we know that the system (37) has a solution whatever initial value (
u
m
(
0
)
,
w
m
(
0
)
)
∈
V
m
×
W
m. To proveTheorem 10, we first show that there exist an approximate solution for (37), such that
(109)
u
m
(
0
)
,
w
m
(
0
)
=
u
m
(
T
)
,
w
m
(
T
)
.
For this purpose, let us take φ
=
u
m and ϕ
=
w
m in (37) to obtain
(110)
1
2
d
d
t
u
m
2
+
ν
u
m
2
+
ν
0
∫
Ω
M
e
u
m
2
e
i
j
u
m
2
d
x
≤
w
m
u
m
+
f
V
′
u
m
,
(111)
1
2
d
d
t
w
m
2
+
ν
1
∫
Ω
e
i
j
w
m
2
d
x
+
λ
1
w
m
2
≤
λ
2
u
m
w
m
+
g
w
m
,
because b
(
u
,
u
,
u
)
=
0 (see Lions [3]), |
∇
×
u
m
|
=
|
∇
u
m
|
=
u
m, and (
∇
×
w
m
,
u
m
)
=
(
w
m
,
∇
×
u
m
) (see Lukaszewicz [8]). Using (11) and (10) and (27) (Korn’s inequality), we obtain from (110) and (111), respectively:
(112)
1
2
d
d
t
u
m
2
+
ν
u
m
2
+
ν
0
M
0
e
u
m
2
≤
w
m
u
m
+
f
V
′
u
m
,
(113)
1
2
d
d
t
w
m
2
+
ν
1
K
w
m
2
≤
λ
2
u
m
w
m
+
g
w
m
.
After usual computations, we can derive
(114)
d
d
t
u
m
2
+
w
m
2
+
ν
u
m
2
+
ν
1
K
w
m
2
≤
C
u
m
2
+
w
m
2
+
f
V
′
2
+
g
2
.
Considering the embeddings V
↪
H and H
0
1
(
Ω
)
↪
L
2
(
Ω
), there exists a constant c
2 such that
(115)
d
d
t
u
m
2
+
w
m
2
+
c
2
u
m
2
+
c
2
w
m
2
≤
C
(
u
m
2
+
w
m
2
+
f
V
′
2
+
g
2
)
.
Multiplying by e
c
2
t and integrating on [
0
,
t
), we obtain
(116)
u
m
t
2
+
w
m
t
2
≤
e
-
c
2
t
u
m
0
2
+
w
m
0
2
+
C
+
C
∫
0
t
‍
u
m
s
2
+
w
m
s
2
d
s
.
By Gronwall’s inequality, we can write
(117)
u
m
t
2
+
w
m
t
2
≤
e
-
c
2
t
u
m
0
2
+
w
m
0
2
+
C
,
for all t
∈
[
0
,
T
]. Let θ
(
t
)
=
e
-
c
2
t; we have 0
<
θ
(
t
)
<
1. Thus,
(118)
u
m
T
2
+
w
m
T
2
≤
θ
u
m
0
2
+
w
m
0
2
+
C
,
where θ
=
θ
(
t
) is a positive constant, such that 0
<
1
-
θ
<
1. Therefore, C
<
C
/
1
-
θ. Taking R
>
0, such that C
/
1
-
θ
<
R
2, we obtain C
<
(
1
-
θ
)
R
2. Choosing the initial data (
u
m
(
0
)
,
w
m
(
0
)
)
∈
V
m
×
W
m, such that
(119)
u
m
0
2
<
R
2
2
,
w
m
0
2
<
R
2
2
.
We obtain from (118)(120)
u
m
T
2
+
w
m
T
2
≤
θ
(
u
m
0
2
+
w
m
0
2
)
+
C
<
θ
R
2
+
(
1
-
θ
)
R
2
=
R
2
.
Therefore, |
u
m
(
0
)
|
2
+
|
w
m
(
0
)
|
2
<
R
2 implies that |
u
m
(
T
)
|
2
+
|
w
m
(
T
)
|
2
<
R
2.
Now we defineσ
:
B
R
(
0
)
∩
V
m
×
W
m
→
B
R
(
0
)
∩
V
m
×
W
m, such that
(121)
σ
u
m
(
0
)
,
w
m
(
0
)
=
u
m
(
T
)
,
w
m
(
T
)
,
where B
R
(
0
)
=
{
(
u
,
w
)
∈
H
×
L
2
(
Ω
)
;
|
u
|
2
+
|
w
|
2
<
R
}. We note that σ is a continuous function because the solution of the (37) depends continuously of the initial data. We also note that (118) implies σ
(
B
R
(
0
)
)
⊂
B
R
(
0
). Therefore, it follows from Brower fixed-point theorem that σ has a fixed point:
(122)
(
u
0
m
,
w
0
m
)
∈
B
R
(
0
)
⊂
V
m
×
W
m
.
In other words, σ
(
u
0
m
,
w
0
m
)
=
(
u
0
m
,
w
0
m
). Taking the initial data (
u
0
m
,
w
0
m
) in (37), that is, (
u
m
(
0
)
,
w
m
(
0
)
)
=
(
u
0
m
,
w
0
m
), we obtain
(123)
(
u
m
(
0
)
,
w
m
(
0
)
)
=
(
u
m
(
T
)
,
w
m
(
T
)
)
.
Therefore, (37) has a periodic solution. Next, we obtain estimates to (37) with the initial data (
u
0
m
,
w
0
m
) as in the proof of Theorem 8. We obtain
(124)
u
m
⟶
u
weakly
in
L
4
(
I
;
V
4
)
,
(125)
u
m
′
⟶
u
′
weakly
in
L
4
/
3
(
I
;
V
′
)
,
(126)
w
m
⟶
w
weakly
in
L
2
(
I
;
H
0
1
(
Ω
)
)
,
(127)
w
m
′
→
w
′
weakly
in
L
2
(
I
;
H
-
1
(
Ω
)
)
,
where u is the solution to problem (8) in the sense of Definition 3. The convergences (124) and (125) allow us to derive
(128)
∫
0
T
‍
d
d
t
u
m
(
s
)
,
v
θ
(
s
)
d
s
⟶
∫
0
T
‍
d
d
t
u
(
s
)
,
v
θ
(
s
)
d
s
,
for all v
∈
V and θ
∈
D
(
0
,
T
), with θ
(
T
)
=
0. In other words,
(129)
u
m
0
,
v
⟶
u
0
,
v
,
∀
v
∈
V
.
The same argument with θ
∈
D
(
0
,
T
) and θ
(
0
)
=
0 allows us to derive
(130)
(
u
m
(
T
)
,
v
)
⟶
(
u
(
T
)
,
v
)
,
∀
v
∈
V
.
It follows from (129) and (130) that u
(
0
)
=
u
(
T
). Analogously, from (126) and (127), we obtain
(131)
(
w
m
(
0
)
,
v
¯
)
⟶
(
w
(
0
)
,
v
¯
)
,
∀
v
¯
∈
L
2
(
Ω
)
,
(
w
m
(
T
)
,
v
¯
)
⟶
(
w
(
T
)
,
v
¯
)
,
∀
v
¯
∈
L
2
(
Ω
)
.
Therefore, w
(
0
)
=
w
(
T
). Theorem 10 has been proved.Proof of Theorem11.
Let us considerτ
:
R
sym
d
2
→
R
sym
d
2 and the corresponding potential Φ
:
R
0
+
→
R
0
+ such that
(132)
τ
D
=
M
D
2
D
,
∀
D
∈
R
sym
d
2
,
Φ
D
2
=
∫
0
D
2
‍
M
s
d
s
,
∀
D
∈
R
sym
d
2
.
It is possible to verify (see Malek et al. [1]) that (132) satisfy the assumptions (28) of Lemma 5.
To obtain some estimate foru
m
′, we make φ
r
=
u
m
′ in (37)1, to obtain applying Schwarz’s inequality
(133)
u
m
′
2
+
ν
∫
Ω
‍
e
i
j
(
u
m
)
e
i
j
(
u
m
′
)
d
x
+
ν
0
∫
Ω
‍
τ
i
j
(
e
(
u
m
)
)
e
i
j
u
m
′
d
x
≤
∫
Ω
‍
u
m
i
∂
u
m
j
∂
x
i
u
m
j
′
d
x
+
u
m
′
∇
×
w
m
+
f
u
m
′
.
Remark 12. We note that (see [4])
(134)
∫
Ω
‍
τ
i
j
e
v
e
i
j
v
′
d
x
=
d
d
t
∫
Ω
‍
Φ
e
v
2
d
x
.
Applying Young’s inequality, (49) and (134), we obtain from (141) that
(135)
1
2
u
m
′
2
+
ν
2
d
d
t
e
u
m
2
+
ν
0
d
d
t
∫
Ω
‍
Φ
(
e
u
m
2
)
d
x
≤
c
+
c
∫
Ω
‍
u
m
2
∇
u
m
2
d
x
.
We observe that (46) implies u
m
(
t
)
∈
V
4
↪
L
∞
(
Ω
) because d
≤
3. On the other hand, (47) implies ∇
u
m
(
t
)
∈
L
2
(
Ω
). Thus, applying the Hölder’s inequality in (135), integrating on (
0
,
t
), with t
≤
T, and after applying Korn’s inequality, we obtain
(136)
∫
0
t
‍
u
m
′
s
2
d
s
+
ν
1
K
u
m
2
+
2
ν
0
∫
Ω
‍
Φ
(
e
u
m
2
)
d
x
≤
c
+
2
ν
0
∫
Ω
‍
Φ
(
e
u
m
0
2
)
d
x
+
ν
1
K
u
m
0
2
+
c
∫
0
t
‍
u
m
s
L
∞
(
Ω
)
2
u
m
(
s
)
2
d
s
.
Observing that (46) implies u
m
(
t
)
L
∞
(
Ω
)
2
∈
L
2
(
0
,
T
) and also u
m
(
t
)
2
∈
L
2
(
0
,
T
), because of (47), so we can obtain by using Hölder’s inequality in (136)(137)
∫
0
t
‍
u
m
′
s
2
d
s
+
ν
1
K
u
m
2
+
2
ν
0
∫
Ω
‍
Φ
(
e
u
m
2
)
d
x
≤
c
because (29) and u
0
∈
V
∩
V
4. Now we note that (29) and Korn’s inequality imply
(138)
∫
Ω
‍
Φ
(
e
u
m
2
)
d
x
≥
C
3
e
(
u
m
)
L
4
(
Ω
)
4
≥
C
3
K
u
m
V
4
4
.
It follows from (137) that
(139)
∫
0
t
‍
u
m
′
s
2
d
s
+
ν
1
K
u
m
2
+
C
3
K
u
m
V
4
4
≤
c
.
Inequality (139) permits us to obtain the following estimates:
(140)
(
u
m
′
)
is
bounded
in
L
2
(
I
;
H
)
,
(
u
m
)
is
bounded
in
L
∞
(
I
;
V
)
,
(
u
m
)
is
bounded
in
L
∞
(
I
;
V
4
)
.
Now we will obtain some estimate forw
m
′. For that purpose, we make ϕ
r
=
w
m
′ in (37)2 to obtain
(141)
w
m
′
2
+
ν
1
∫
Ω
‍
e
i
j
(
w
m
)
e
i
j
(
w
m
′
)
d
x
+
λ
1
2
d
d
t
w
m
2
≤
∫
Ω
‍
u
m
i
∂
w
m
j
∂
x
i
w
m
j
′
d
x
+
λ
2
u
m
w
m
′
+
g
w
m
′
.
Applying Young’s inequality and using (47) we derive
(142)
1
2
w
m
′
2
+
ν
1
2
d
d
t
e
w
m
2
+
λ
1
2
d
d
t
w
m
2
≤
c
+
c
∫
Ω
‍
u
m
2
∇
w
m
2
d
x
.
We observe that (46) implies u
m
(
t
)
∈
V
4
↪
L
∞
(
Ω
) because d
≤
3. On the other hand, (49) implies ∇
w
m
(
t
)
∈
L
2
(
Ω
). Thus, applying Hölder’s inequality in (142), integrating on (
0
,
t
), with t
≤
T, and after applying Korn’s inequality, we obtain
(143)
∫
0
t
‍
w
m
′
s
2
d
s
+
ν
1
K
w
m
2
≤
c
+
c
∫
0
t
‍
u
m
s
L
∞
(
Ω
)
2
w
m
(
s
)
2
d
s
because w
0
∈
H
0
1
(
Ω
). Furthermore (46) implies u
m
(
t
)
L
∞
(
Ω
)
2
∈
L
2
(
0
,
T
) and (49) implies u
m
(
t
)
∈
L
1
(
0
,
T
). Thus, by using Gronwall’s inequality in (143), we conclude
(144)
(
w
m
′
)
is
bounded
in
L
2
(
I
;
L
2
(
Ω
)
)
,
(145)
(
w
m
)
is
bounded
in
L
∞
(
I
;
H
0
1
(
Ω
)
)
.
In order to prove uniqueness of solution to the cased
=
3 we observe that by assuming d
=
3 we have H
0
1
↪
L
6
(
Ω
). From Lemma 7 with s
=
4, r
=
6 and Young’s inequality we derive
(146)
b
u
~
,
u
1
,
u
~
+
b
u
~
,
w
2
,
w
~
≤
c
u
~
L
6
Ω
u
1
u
~
1
/
2
u
~
1
/
2
+
c
u
~
L
6
Ω
w
2
w
~
1
/
2
w
~
1
/
2
≤
c
u
1
u
~
1
/
2
u
~
3
/
2
+
c
w
2
w
~
1
/
2
u
~
w
~
1
/
2
≤
ν
4
u
~
2
+
c
ν
u
1
4
u
~
2
+
ν
4
u
~
2
+
c
ν
w
2
2
w
~
w
~
≤
ν
4
u
~
2
+
c
ν
u
1
4
u
~
2
+
ν
4
u
~
2
+
ν
1
w
~
2
+
c
ν
1
w
2
4
w
~
2
.
It follows from (103) that
(147)
1
2
d
d
t
u
~
2
+
w
~
2
≤
c
1
+
u
1
4
+
w
2
4
u
~
2
+
w
~
2
.
Integrating from 0 to t we obtain
(148)
u
~
t
2
+
w
~
t
2
≤
c
∫
0
t
‍
1
+
u
1
(
s
)
4
+
w
2
(
s
)
4
(
u
~
s
2
+
w
~
s
2
)
d
s
.
Applying Gronwall’s inequality in (148), we deduce by using (47) and (145) that
(149)
u
1
t
=
u
2
t
,
w
1
t
=
w
2
t
,
∀
t
∈
0
,
T
.



---

*Source: 481754-2015-02-19.xml*