# Upper Semicontinuity of Attractors for a Non-Newtonian Fluid under Small Random Perturbations

Authors: Jianxin Luo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper investigates the limiting behavior of attractors for a two-dimensional incompressible non-Newtonian fluid under small random perturbations. Under certain conditions, the upper semicontinuity of the attractors for diminishing perturbations is shown.


---

## Body

## 1. Introduction

Fluid flows arise in numerous scientific and industrial endeavors such as aeronautical sciences, meteorology, thermohydraulics, petroleum industry, and plasma physics. The equations describing the motion of the fluid flows are determined by its extra stress tensor. If the extra stress tensor of the fluid depends linearly on its symmetric part of the velocity gradient, the fluid is called Newtonian. Otherwise, the fluid is called non-Newtonian [1]. For instance, gases, water, motor oil, alcohols, and simple hydrocarbon compounds tend to be Newtonian fluids and their motions can be described by the Navier-Stokes equations. Molten plastics, polymer solutions, and paints tend to be non-Newtonian fluids, which may be described by the following system:
(1)
∂
u
∂
t
+
(
u
·
∇
)
u
+
∇
p
=
∇
·
τ
(
e
(
u
)
)
+
g
,
∇
·
u
=
0
,
where the vector function u
(
x
,
t
) is the velocity of the fluid, g is the external force, the scalar function p represents the pressure, and τ
(
e
(
u
)
)
=
(
τ
i
j
(
e
(
u
)
)
) is the extra stress tensor of the fluid.Ladyzhenskaya [2] formulated a two-dimensional non-Newtonian fluid model with the extra stress tensor:
(2)
τ
i
j
(
e
(
u
)
)
=
2
μ
0
(
ε
+
|
e
|
2
)
-
α
/
2
e
i
j
-
2
μ
1
Δ
e
i
j
,
i
,
j
=
1,2
,
where
(3)
e
i
j
=
e
i
j
(
u
)
=
1
2
(
∂
u
i
∂
x
j
+
∂
u
j
∂
x
i
)
,
|
e
|
2
=
∑
i
,
j
=
1
2
|
e
i
j
|
2
,
and μ
0, μ
1, α, and ε are parameters associated with the fluid and generally depend on temperature and pressure. The initial-boundary value problem of (1)-(2) on a 2D bounded domain D (with regular boundary) can be formulated as follows:
(4)
∂
u
∂
t
+
(
u
·
∇
)
u
+
∇
p
=
∇
·
(
2
μ
0
(
ε
+
|
e
|
2
)
-
α
/
2
e
-
2
μ
1
Δ
e
)
+
g
,
(5)
∇
·
u
=
0
,
x
∈
D
,
(6)
u
=
0
,
τ
i
j
l
n
j
n
l
=
0
,
x
∈
∂
D
,
(7)
u
|
t
=
0
=
u
0
,
where τ
i
j
l
=
2
μ
1
(
∂
e
i
j
/
∂
x
l
)
(
i
,
j
,
l
=
1,2
) and n
=
(
n
1
,
n
2
) denotes the exterior unit normal vector to the boundary ∂
D. The first condition in (6) is the usual no-slip condition, while the second one expresses the fact that the first moments of the traction vanish on the boundary. This is a direct consequence of the principle of virtual work. We refer to [1–7] for more physical background. Recent results on well posedness, regularity, and long-term behavior of solutions for (4)–(7) are in, for example, [1–5, 7–10].Attractors are an important concept in the study of infinite dimensional dynamical systems. There are numerous works on the autonomous and nonautonomous equations concerning this subject; see, for example, Chepyzhov and Vishik [11], Hale [12], Robinson [13], and Temam [14]. However, external forces or time-dependent influences in some fluid and materials phenomenon lead to the presence of stochastic terms in the above model equations (see, e.g., [15, 16]).The fundamental theory of random dynamical systems (RDS) was developed in 1990s by many people, including Arnold [15], Crauel and Flandoli [17], Crauel et al. [18], Flandoli and Schmalfuss [19], Caraballo, Langa, Robinson, and their coauthors [20–24].The motivation for the present paper is the desire to understand the stability of attractors for the above two-dimensional non-Newtonian fluid under vanishing small random perturbations.We investigate the relations between the random attractor and its deterministic counterpart when the incompressible non-Newtonian fluid is subject to a small random perturbation, whose strength is measured by a small positive parameterϵ. Consider the problem (4)–(7) and the following 2D incompressible non-Newtonian fluid with an additive noise:
(8)
d
u
ϵ
=
{
(
ε
+
|
e
ϵ
|
2
)
-
α
/
2
-
(
u
ϵ
·
∇
)
u
ϵ
-
∇
p
+
∇
·
[
2
μ
0
(
ε
+
|
e
ϵ
|
2
)
-
α
/
2
e
ϵ
-
2
μ
1
Δ
e
ϵ
]
+
g
}
d
t
+
ϵ
ϕ
d
ω
(
t
)
,
∇
·
u
ϵ
=
0
,
x
∈
D
⊂
R
2
,
where e
ϵ
=
(
e
ϵ
)
i
j
=
(
1
/
2
)
(
(
∂
(
u
ϵ
)
i
/
∂
x
j
)
+
(
∂
(
u
ϵ
)
j
/
∂
x
i
)
), ω
(
t
) is an independent two-sided Wiener processes, and ϕ is a function satisfying some conditions to be specified below. Caraballo et al. [25–28] proved the stability of the attractors for a class of evolution equations under small random perturbations and the results were applied to various physical equations. We note that (4) is regarded as the modified Navier-Stokes equations as the gradient |
∇
u
| of the velocity is relatively large [2]. Clearly, (4) reduces to Navier-Stokes equations when α
=
μ
1
=
0.The main result in the present paper is the stability of the attractor in the sense that(9)
lim
⁡
ϵ
→
0
+
dis
t
H
(
A
ϵ
,
A
0
)
=
0
with
probability
one
,
where dis
t
H
(
·
,
·
) is the Hausdorff semidistance on the metric space H (see notation in Section 2) and A
ϵ and A
0 are the attractors associated with (8) and (4)–(7), respectively. Given a δ
>
0, we prove that there exists an ϵ
0 (depending on ω, a parameter event in a probability space (
Ω
,
F
,
P
)) sufficiently small, such that the random attractors A
ϵ are inside the δ neighborhood of the global attractor A
0 for all ϵ
∈
(
0
,
ϵ
0
) with probability one.The paper is organized as follows. In the next section, we introduce some notations and recall some results from [8, 10]. Section 3 is devoted to prove the stability of solutions of the perturbed random system to the unperturbed deterministic system and then show the stability of the random attractor by showing lim
⁡
ϵ
→
0
+
⁡
dis
t
H
(
A
ϵ
(
ω
)
,
A
0
)
=
0 with probability one.


## 2. Global Existence and Uniqueness of Solutions

In this section, we introduce some notations and recall some results about non-Newtonian fluid dynamics. DefineV
=
{
φ
=
(
φ
1
,
φ
2
)
∈
(
C
0
∞
(
D
¯
)
)
2
,
∇
·
φ
=
0
in
D
,
φ
=
0
on
⁡
∂
D
},
H
=
closure
of
V
in
(
L
2
(
D
)
)
2
with
norm
⁡
∥
·
∥
≐
∥
·
∥
(
L
2
(
D
)
)
2,
V
=
closure
of
V
in
(
H
2
(
D
)
)
2
with
norm
⁡
∥
·
∥
V
≐
∥
·
∥
(
H
2
(
D
)
)
2,
H
′
=
dual
space
of
H, V
′
=
dual
space
of
V,where (
·
,
·
) denotes the inner product in H and 〈
·
,
·
〉 stands for the dual pairing between V and V
′. If we identify H with H
′, then V
↪
H
=
H
′
↪
V
′ with continuous and compact embeddings.We also define the bilinear form(10)
a
(
u
,
v
)
=
∑
i
,
j
,
k
=
1
2
‍
∫
D
‍
∂
e
i
j
(
u
)
∂
x
k
∂
e
i
j
(
v
)
∂
x
k
d
x
,
u
,
v
∈
V
.Lemma 1 (see [4]).
There exist two positive constantsc
1 and c
2, which depend only on D, such that
(11)
c
1
∥
u
∥
V
2
≤
a
(
u
,
u
)
≤
c
2
∥
u
∥
V
2
,
∀
u
∈
V
.From the definition ofa
(
·
,
·
) and Lemma 1 we see that a
(
·
,
·
) defines a positive definite symmetric bilinear form on V. As a consequence of the Lax-Milgram Lemma, we obtain an operator A
∈
L
(
V
,
V
′
), via
(12)
〈
A
u
,
v
〉
=
a
(
u
,
v
)
,
∀
u
,
v
∈
V
.
Moreover, let D
(
A
)
=
{
u
∈
V
:
A
u
∈
H
}, and then D
(
A
) is a Hilbert space. We have (see [8])
(13)
c
1
∥
u
∥
V
≤
∥
A
u
∥
.
For brevity, we use H
0
1
(
D
) to denote (
H
0
1
(
D
)
)
2 in the sequel. We also define a continuous trilinear form on H
0
1
(
D
)
×
H
0
1
(
D
)
×
H
0
1
(
D
) as follows:
(14)
b
(
u
,
v
,
w
)
=
∑
i
,
j
=
1
2
‍
∫
D
‍
u
i
∂
v
j
∂
x
i
w
j
d
x
,
u
,
v
,
w
∈
H
0
1
(
D
)
.
Since V
⊂
H
0
1
(
D
), b
(
·
,
·
,
·
) is continuous on V
×
V
×
V and one can check
(15)
b
(
u
,
v
,
w
)
=
-
b
(
u
,
w
,
v
)
,
b
(
u
,
v
,
v
)
=
0
,
∀
u
,
v
,
w
∈
V
.
Now for any u
∈
V,
(16)
〈
B
(
u
,
u
)
,
w
〉
=
b
(
u
,
u
,
w
)
,
∀
w
∈
V
,
defines a continuous functional B
(
u
) from V
×
V to V
′. Finally, for u
∈
V, we set μ
(
u
)
=
2
μ
0
(
ε
+
|
e
(
u
)
|
2
)
-
α
/
2 and define N
(
u
) as
(17)
〈
N
(
u
)
,
v
〉
=
∑
i
,
j
=
1
2
‍
∫
D
‍
μ
(
u
)
e
i
j
(
u
)
e
i
j
(
v
)
d
x
,
∀
v
∈
V
.
Then the functional N
(
u
) is continuous from V to V
′. When u
∈
D
(
A
), N
(
u
) can be extended to H via
(18)
〈
N
(
u
)
,
v
〉
=
-
∫
D
‍
{
∇
·
[
μ
(
u
)
e
(
u
)
]
}
·
v
d
x
,
∀
v
∈
H
.Eliminating the pressurep by a proper projection, we have the weak version of problem (4)–(7) in the solenoidal vector fields as follows (see [4, 9]):
(19)
∂
u
∂
t
+
2
μ
1
A
u
+
B
(
u
,
u
)
+
N
(
u
)
=
g
,
u
|
t
=
0
=
u
0
.Lemma 2 (see [8, 9]).
Letg
∈
H (independent of time t). Then the semigroup {
S
(
t
)
}
t
≥
0 associated with (19) possesses a global attractor A
0 in H satisfying(1)
(compactness) A
0 is compact in H;
(2)
(invariance) S
(
t
)
A
0
=
A
0, t
≥
0;
(3)
(attractivity) for all B
⊂
H bounded, lim
⁡
t
→
+
∞
⁡
dist
⁡

H
(
S
(
t
)
B
,
A
0
)
=
0.
Moreover,A
0 is compact in space V and for any B
V
⊂
V bounded,
(20)
lim
⁡
t
→
+
∞
dist
⁡

V
(
S
(
t
)
B
V
,
A
0
)
=
0
.In this paper, we use the concepts concerning the metric dynamical system (MDS), random dynamical system (RDS), random closed set, and global random attractor from [15].We now takeΩ
=
{
ω
∈
C
(
R
;
H
)
,
ω
(
0
)
=
0
} and endow it with the compact open topology (see Appendices A
.
2 and A
.
3 in [15]). Take P as the corresponding product measure of two Wiener measures on the negative and the positive time parts of Ω, and denote F
0 by the Borel σ-algebra on Ω. Let
(21)
θ
t
ω
(
s
)
=
ω
(
s
+
t
)
-
ω
(
t
)
,
t
,
s
∈
R
.To obtain the uniqueness and existence of solutions for the problem (8), we need the following assumption.Assumption A.
Ifμ
0
,
μ
1
,
ε
>
0, α
∈
(
0,1
), g
∈
H, and ϕ
∈
D
(
A
), then there exists a positive number δ such that
(22)
|
〈
B
(
u
,
ϕ
)
,
u
〉
|
≤
δ
∥
u
∥
2
,
∀
u
∈
V
.Remark 3.
Since〈
B
(
u
,
ϕ
)
,
u
〉
=
∑
i
,
j
=
1
2
∫
D
u
i
(
∂
ϕ
j
/
∂
x
i
)
u
j
d
x, we see that the assumption (22) is satisfied provided that the function ϕ is Lipschitz continuous on D.Using the notations and operators introduced above, we can put (8) into the following abstract form:
(23)
d
u
ϵ
=
[
-
2
μ
1
A
u
ϵ
-
B
(
u
ϵ
,
u
ϵ
)
-
N
(
u
ϵ
)
+
g
]
d
t
+
ϵ
ϕ
d
ω
(
t
)
,
(24)
u
ϵ
|
t
=
0
=
u
0
.Letη
>
0 be a constant and denote z by the solution of the stationary solution of the Itô equation:
(25)
d
z
=
-
η
z
d
t
+
d
ω
(
t
)
.
The solution is often called an Ornstein-Uhlenbeck process. In fact,
(26)
z
(
t
)
=
∫
-
∞
t
‍
e
-
η
(
t
-
s
)
d
ω
(
s
)
.
We now make the change
(27)
v
ϵ
(
t
,
ω
)
=
u
ϵ
(
t
,
ω
)
-
ϵ
ϕ
z
(
t
,
ω
)
.
Then v
ϵ
(
t
,
ω
) satisfies the following random abstract evolutionary equation:
(28)
d
v
ϵ
d
t
+
2
μ
1
A
v
ϵ
+
B
(
u
ϵ
,
u
ϵ
)
+
N
(
u
ϵ
)
+
ϵ
η
ϕ
z
+
2
μ
1
ϵ
A
(
ϕ
z
)
=
g
.
Now (28) can be studied for each ω
∈
Ω.Lemma 4 (see [10]).
Let Assumption A hold. Then forP-a.s. ω
∈
Ω, the following results hold.(1)
For allt
0
∈
R and v
0
=
v
ϵ
(
0
)
∈
H, there exists a unique solution v
ϵ
∈
C
(
[
t
0
,
+
∞
)
;
H
)
∩
L
loc
⁡
2
(
t
0
,
+
∞
;
V
) of (28) with initial value v
0.
(2)
Ifv
0
∈
V, then the solution belongs to C
(
[
t
0
,
+
∞
)
;
V
)
∩
L
loc
⁡
2
(
t
0
,
+
∞
;
D
(
A
)
).
(3)
For everyτ
>
0, v
ϵ
(
t
)
∈
C
(
[
t
0
+
τ
,
+
∞
)
;
V
)
∩
L
loc
⁡
2
(
t
0
+
τ
,
+
∞
;
D
(
A
)
).
(4)
Denote the solution byv
ϵ
(
t
,
ω
;
t
0
,
v
0
). Then the map v
0
↦
v
ϵ
(
t
,
ω
;
t
0
,
v
0
) is continuous for all t
≥
t
0.By Lemma4 we see that for each ϵ
∈
(
0,1
] there is a continuous mapping from H into itself: v
0
↦
v
ϵ
(
t
,
ω
;
t
0
,
v
0
), where v
ϵ
(
t
,
ω
;
t
0
,
v
0
) is the solution of (28) with initial value v
ϵ
(
t
0
)
=
v
0. Thus for each ϵ
∈
(
0,1
], we can define an RDS ψ
ϵ
(
t
,
ω
) associated with (23) and (24) in H by setting
(29)
ψ
ϵ
(
t
,
ω
)
u
0
=
u
ϵ
(
t
,
ω
)
=
v
ϵ
(
t
,
ω
)
+
ϵ
ϕ
z
(
t
,
ω
)
=
v
ϵ
(
t
,
ω
;
0
,
u
0
-
ϵ
ϕ
z
(
0
,
ω
)
)
+
ϵ
ϕ
z
(
t
,
ω
)
.We end this section with the concept of random attractor forψ
ϵ
(
t
,
ω
) in space H.Definition 5.
Let the RDS{
ψ
(
t
,
ω
)
}
t
≥
0
,
ω
∈
Ω on (
Ω
,
F
,
P
,
(
θ
t
)
t
∈
R
) be defined by (29) with state space H, if ω
↦
A
(
ω
) satisfies the following:(i)
(random compactness)A
(
ω
) is a random compact set of H for a.e. ω
∈
Ω,
(ii)
(invariance) for a.e.ω
∈
Ω and all t
≥
0, S
(
t
,
ω
)
A
(
ω
)
=
A
(
θ
t
ω
),
(iii)
(attracting property) for any bounded subsetB of H and a.e. ω
∈
Ω,
(30)
lim
⁡
t
→
+
∞
Dis
t
H
(
S
(
t
,
θ
-
t
ω
)
B
,
A
(
ω
)
)
=
0
.Then A
(
ω
) is called a global random attractor for {
ψ
(
t
,
ω
)
}
t
≥
0
,
ω
∈
Ω in space H.


## 3. Stability of Attractors

We first prove the following lemma, which plays a key role later on.Lemma 6.
The solutionu
ϵ
(
0
,
ω
;
-
t
0
,
u
0
) of (23) and (24) converges in H
P-a.s. as ϵ
→
0
+ to the solution u
(
t
0
;
u
0
) of the unperturbed problem (19), uniformly on bounded sets of initial conditions. That is, for P almost every ω
∈
Ω, t
0
∈
R
+ and B
⊂
H bounded, the following convergence holds:
(31)
lim
⁡
ϵ
→
0
+
∥
u
ϵ
(
0
,
ω
;
-
t
0
,
u
0
)
-
u
(
t
0
,
u
0
)
∥
=
0
.Proof.
Letw
ϵ
(
t
,
ω
)
=
u
ϵ
(
t
,
ω
)
-
u
(
t
) be the difference between the solutions of the perturbed and unperturbed equations with the same initial condition u
0 at -
t
0. It is clear that w
ϵ satisfies
(32)
d
w
ϵ
+
[
2
μ
1
A
w
ϵ
+
B
(
u
ϵ
,
u
ϵ
)
-
B
(
u
,
u
)
d
w
ε
+
+
N
(
u
ϵ
)
-
N
(
u
)
]
d
t
=
ϵ
ϕ
d
ω
(
t
)
,
w
ϵ
(
-
t
0
)
=
0
.
We now make the change of variables
(33)
z
ϵ
=
w
ϵ
-
ϵ
ϕ
ω
(
t
)
and thus formally obtain
(34)
d
z
ϵ
d
t
+
2
μ
1
A
z
ϵ
+
2
μ
1
ϵ
A
(
ϕ
ω
(
t
)
)
+
B
(
u
ϵ
,
u
ϵ
)
-
B
(
u
,
u
)
+
N
(
u
ϵ
)
-
N
(
u
)
=
0
.
Taking the inner product of the above equation with z
ϵ, we get
(35)
1
2
d
d
t
∥
z
ϵ
∥
2
+
2
μ
1
(
A
z
ϵ
,
z
ϵ
)
+
2
μ
1
ϵ
(
ω
(
t
)
A
ϕ
,
z
ϵ
)
+
〈
B
(
u
ϵ
,
u
ϵ
)
,
z
ϵ
〉
-
〈
B
(
u
,
u
)
,
z
ϵ
〉
+
〈
N
(
u
ϵ
)
,
z
ϵ
〉
-
〈
N
(
u
)
,
z
ϵ
〉
=
0
.
Therefore
(36)
2
μ
1
ϵ
(
ω
(
t
)
A
ϕ
,
z
ϵ
)
≤
2
μ
1
ϵ
|
ω
(
t
)
|
∥
A
ϕ
∥
∥
z
ϵ
∥
≤
2
μ
1
2
ϵ
2
ω
2
(
t
)
∥
A
ϕ
∥
2
+
∥
z
ϵ
∥
2
2
.
By the property of the operator B
(
·
,
·
), we imply that
(37)
B
(
u
ϵ
,
u
ϵ
)
-
B
(
u
,
u
)
=
B
(
w
ϵ
+
u
,
w
ϵ
+
u
)
-
B
(
u
,
u
)
=
B
(
w
ϵ
,
w
ϵ
)
+
B
(
w
ϵ
,
u
)
+
B
(
u
,
w
ϵ
)
=
B
(
z
ϵ
+
ϵ
ϕ
ω
(
t
)
,
z
ϵ
+
ϵ
ϕ
ω
(
t
)
)
+
B
(
z
ϵ
+
ϵ
ϕ
ω
(
t
)
,
u
)
+
B
(
u
,
z
ϵ
+
ϵ
ϕ
ω
(
t
)
)
.
Thus,
(38)
〈
B
(
u
ϵ
,
u
ϵ
)
-
B
(
u
,
u
)
,
z
ϵ
〉
=
〈
B
(
z
ϵ
+
ϵ
ϕ
ω
(
t
)
,
z
ϵ
+
ϵ
ϕ
ω
(
t
)
)
,
z
ϵ
〉
+
〈
B
(
z
ϵ
+
ϵ
ϕ
ω
(
t
)
,
u
)
,
z
ϵ
〉
+
〈
B
(
u
,
z
ϵ
+
ϵ
ϕ
ω
(
t
)
)
,
z
ϵ
〉
=
〈
B
(
z
ϵ
,
z
ϵ
+
ϵ
ϕ
ω
(
t
)
)
,
z
ϵ
〉
+
〈
B
(
ϵ
ϕ
ω
(
t
)
,
ϵ
ϕ
ω
(
t
)
)
,
z
ϵ
〉
+
〈
B
(
z
ϵ
,
u
)
,
z
ϵ
〉
+
〈
B
(
ϵ
ϕ
ω
(
t
)
,
u
)
,
z
ϵ
〉
+
〈
B
(
u
,
ϵ
ϕ
ω
(
t
)
,
u
)
,
z
ϵ
〉
.
We now estimate the right hand side of (38) terms by terms. First note that
(39)
〈
B
(
z
ϵ
,
ϵ
ϕ
ω
(
t
)
,
z
ϵ
)
〉
≤
ϵ
|
ω
(
t
)
|
|
〈
B
(
z
ϵ
,
ϕ
)
,
z
ϵ
〉
|
〈
B
(
z
ε
,
ε
ϕ
ω
(
t
)
,
z
ε
)
〉
≤
δ
ϵ
|
ω
(
t
)
|
∥
z
ϵ
∥
2
,
〈
B
(
ϵ
ϕ
ω
(
t
)
,
ϵ
ϕ
ω
(
t
)
)
,
z
ϵ
〉
≤
ϵ
2
ω
2
(
t
)
〈
B
(
ϕ
,
ϕ
)
,
z
ϵ
〉
≤
ϵ
2
ω
4
(
2
)
∥
ϕ
∥
L
4
(
D
)
∥
∇
ϕ
∥
L
4
(
D
)
∥
z
ϵ
∥
≤
c
ϵ
2
ω
2
(
t
)
∥
ϕ
∥
1
/
2
∥
∇
ϕ
∥
1
/
2
∥
ϕ
∥
1
/
4
∥
Δ
ϕ
∥
3
/
4
∥
z
ϵ
∥
≤
c
ϵ
2
ω
4
(
t
)
∥
∇
ϕ
∥
2
∥
Δ
ϕ
∥
2
+
∥
z
ϵ
∥
2
2
,
〈
B
(
z
ϵ
,
u
)
,
z
ϵ
〉
≤
c
∥
z
ϵ
∥
∥
∇
u
∥
∥
∇
z
ϵ
∥
〈
B
(
z
ε
,
u
)
,
z
ε
〉
≤
3
μ
1
4
∥
z
ϵ
∥
V
2
+
c
∥
∇
u
∥
2
∥
z
ϵ
∥
2
,
〈
B
(
ϵ
ϕ
ω
(
t
)
,
u
)
,
z
ϵ
〉
≤
ϵ
|
ω
(
t
)
|
|
〈
B
(
ϕ
,
u
)
,
z
ϵ
〉
|
≤
ϵ
|
ω
(
t
)
|
∥
ϕ
∥
L
4
(
D
)
∥
∇
u
∥
L
4
(
D
)
∥
z
ϵ
∥
≤
ϵ
|
ω
(
t
)
|
∥
ϕ
∥
1
/
2
∥
∇
ϕ
∥
1
/
2
∥
∇
u
∥
1
/
2
∥
Δ
u
∥
1
/
2
∥
z
ϵ
∥
≤
c
ϵ
2
|
ω
(
t
)
|
2
∥
ϕ
∥
∥
∇
ϕ
∥
∥
u
∥
V
2
+
∥
z
ϵ
∥
2
2
,
〈
B
(
u
,
ϵ
ϕ
ω
(
t
)
)
,
z
ϵ
〉
≤
ϵ
|
ω
(
t
)
|
|
〈
B
(
u
,
ϕ
)
,
z
ϵ
〉
|
≤
ϵ
|
ω
(
t
)
|
∥
u
∥
1
/
2
|
∥
∇
u
∥
1
/
2
|
∥
∇
ϕ
∥
1
/
2
∥
Δ
ϕ
∥
1
/
2
∥
z
ϵ
∥
≤
ϵ
2
|
ω
(
t
)
|
2
∥
u
∥
∥
u
∥
H
1
(
D
)
2
∥
ϕ
∥
H
1
(
D
)
∥
A
ϕ
∥
+
∥
z
ϵ
∥
2
2
.
Then also note that
(40)
〈
N
(
u
ϵ
)
-
N
(
u
)
,
z
ϵ
〉
=
〈
N
(
u
ϵ
)
-
N
(
u
)
,
u
ϵ
-
u
-
ϵ
ϕ
ω
(
t
)
〉
.
By the nonnegativity of N
(
·
) (see [4]), we conclude that
(41)
〈
N
(
u
ϵ
)
-
N
(
u
)
,
u
ϵ
-
u
〉
≥
0
.
Set F
(
s
)
=
2
μ
0
(
ε
+
|
s
|
2
)
-
α
/
2
s, where
(42)
s
=
(
s
1
s
2
s
3
s
4
)
∈
M
2
×
2
,
|
s
|
2
=
∑
i
=
1
4
s
i
2
‍
,
s
i
∈
R
,
i
=
1,2
,
3,4
.
Here M
2
×
2 denotes the matrix of order 2
×
2. Then the first order Fréchet derivative of F
(
s
) is
(43)
DF
(
s
)
=
2
μ
0
(
ε
+
|
s
|
2
)
-
α
/
2
×
(
1
-
α
s
1
2
ε
+
|
s
|
2
-
α
s
1
s
2
ε
+
|
s
|
2
-
α
s
1
s
3
ε
+
|
s
|
2
-
α
s
1
s
4
ε
+
|
s
|
2
-
α
s
1
s
2
ε
+
|
s
|
2
1
-
α
s
2
2
ε
+
|
s
|
2
-
α
s
2
s
3
ε
+
|
s
|
2
-
α
s
2
s
4
ε
+
|
s
|
2
-
α
s
1
s
3
ε
+
|
s
|
2
-
α
s
2
s
3
ε
+
|
s
|
2
1
-
α
s
3
2
ε
+
|
s
|
2
-
α
s
3
s
4
ε
+
|
s
|
2
-
α
s
1
s
4
ε
+
|
s
|
2
-
α
s
2
s
4
ε
+
|
s
|
2
-
α
s
3
s
4
ε
+
|
s
|
2
1
-
α
s
4
2
ε
+
|
s
|
2
)
.
Since 0
<
α
<
1, we have
(44)
|
-
α
s
i
s
j
ε
+
|
s
|
2
|
<
|
s
i
s
j
ε
+
|
s
|
2
|
<
1
,
i
,
j
=
1
,
2
,
3
,
4
,
0
<
1
-
α
s
i
2
ε
+
|
s
|
2
<
1
,
i
=
1
,
2
,
3
,
4
.
Consequently,
(45)
∥
DF
(
s
)
∥
≤
8
μ
0
(
ε
+
|
s
|
2
)
-
α
/
2
≤
8
μ
0
ε
-
α
/
2
,
∀
s
∈
M
2
×
2
.
Now for anya
,
b
∈
R
4,
(46)
F
(
b
)
-
F
(
a
)
=
∫
0
1
‍
DF
(
a
+
τ
(
b
-
a
)
)
(
b
-
a
)
d
τ
.
Taking a
=
e
(
u
ϵ
)
=
(
e
i
j
(
u
ϵ
)
), b
=
e
(
u
)
=
(
e
i
j
(
u
)
), applying the integration by parts first, and then utilizing the above inequality about F
(
s
), we have
(47)
〈
N
(
u
ϵ
)
-
N
(
u
)
,
ϵ
ϕ
ω
(
t
)
〉
=
∑
i
,
j
=
1
2
∫
D
‍
[
F
(
e
i
j
(
u
ϵ
)
)
-
F
(
e
i
j
(
u
)
)
]
·
e
i
j
(
ϵ
ϕ
ω
(
t
)
)
d
x
‍
≤
c
ϵ
|
ω
(
t
)
|
(
∥
∇
(
u
ϵ
-
u
)
∥
)
∥
∇
ϕ
∥
≤
c
ϵ
|
ω
(
t
)
|
∥
∇
(
z
ϵ
+
ϵ
ϕ
ω
(
t
)
)
∥
∥
∇
ϕ
∥
≤
3
μ
1
4
∥
∇
z
ϵ
∥
2
+
c
ϵ
2
ω
2
(
t
)
∥
∇
ϕ
∥
2
.
Combining (35)–(40) and (47), we obtain
(48)
d
d
t
∥
z
ϵ
∥
2
≤
I
(
t
)
+
G
(
t
)
∥
z
ϵ
∥
2
,
where
(49)
I
(
t
)
=
c
ϵ
[
μ
1
2
ϵ
ω
2
(
t
)
∥
A
ϕ
∥
2
+
ϵ
ω
4
(
t
)
∥
∇
ϕ
∥
2
∥
A
ϕ
∥
2
+
ϵ
|
ω
(
t
)
|
∥
ϕ
∥
∥
∇
ϕ
∥
∥
u
∥
V
2
+
ϵ
|
ω
2
(
t
)
|
∥
u
∥
×
∥
∇
u
∥
∥
∇
ϕ
∥
∥
A
ϕ
∥
+
ϵ
|
ω
2
(
t
)
|
∥
A
ϕ
∥
2
]
,
G
(
t
)
=
2
+
ϵ
δ
|
ω
(
t
)
|
+
c
∥
∇
u
∥
2
.
Then, by Gronwall inequality,
(50)
∥
z
ϵ
(
t
)
∥
2
≤
I
(
t
)
+
∫
-
t
0
t
‍
I
(
s
)
G
(
s
)
exp
⁡
(
∫
s
t
‍
G
(
τ
)
d
τ
)
d
s
.
By Lemma 2 we find that ∥
u
(
t
)
∥
V
2 is uniformly bounded with the initial values belonging to a bounded set (see [8] for details). Obviously, ∥
u
(
t
)
∥
≤
∥
u
(
t
)
∥
V
,
∥
∇
u
(
t
)
∥
≤
∥
u
(
t
)
∥
V. Thus, I
(
t
)
→
0 as ϵ
→
0
+ and by (50) we get ∥
z
ϵ
(
t
)
∥
2
→
0 as ϵ
→
0
+ for all t
≥
-
t
0. Therefore,
(51)
lim
⁡
ϵ
→
0
+
∥
w
ϵ
(
t
)
∥
2
≤
lim
⁡
ϵ
→
0
+
2
(
∥
z
ϵ
∥
2
+
ϵ
2
∥
ϕ
∥
ω
2
(
t
)
)
=
0
.
We complete the proof by taking t
=
0.Remark 7.
The RDS (see (29)) is defined using the transformation (27) into a random equation with the Ornstein-Uhlenbeck process. The estimates in Lemma 6 are obtained using a change of variable (33). It is just for the convenience of computation. In fact, we can check that the functions given by the two changes of variable satisfy the same equation (23). Thus the solutions u
ϵ of the stochastic system (23) given by the two changes of variables are the same.Lemma 8.
Let Assumption A hold. Then for eachϵ
∈
(
0,1
], there exists a random attractor A
ϵ
(
ω
) for the RDS ψ
ϵ
(
t
,
ω
) associated with (23) and (24), and in addition there exists a compact set K such that P-a.s.
(52)
lim
⁡
ϵ
→
0
+
dist
⁡

H
(
A
ϵ
(
ω
)
,
K
)
=
0
.Proof.
For eachϵ
∈
(
0,1
], the existence of a random attractor A
ϵ
(
ω
) for the RDS ψ
ϵ
(
t
,
ω
) associated with (23) and (24) can be found in [10]. Also, the existence of a compact set K and (52) can be established, with a similar way to Lemma 4.3 of [10] by showing that ∥
u
(
0
)
∥
V
≤
r
(
ω
) (the radius of random absorbing set in V) and lim
⁡
ϵ
→
0
+
⁡
r
(
ω
)
≤
r
d with r
d independent of ω
∈
Ω. The detailed proof is omitted here.By Lemmas 3.1 and 3.3 and Theorem 2 of [25] and Theorem 1.1 of [8], we obtain the main result of this paper as follows.Theorem 9.
Let Assumption A hold. Then for eachϵ
∈
(
0,1
], the random dynamical system generated by (23) and (24) possesses a compact random attractor A
ϵ and the deterministic dynamical system generated by (19) has a global attractor A
0 in H. Moreover,
(53)
lim
⁡
ϵ
→
0
+
dist
⁡

H
(
A
ϵ
(
ω
)
,
A
0
)
=
0
with probability one.



---

*Source: 546898-2014-06-19.xml*