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

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.

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: 𝜕𝑢+()(𝑒(𝑢𝜕𝑡𝑢⋅∇𝑢+∇𝑝=∇⋅𝜏))+𝑔,∇⋅𝑢=0, (1) 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_𝑖𝑗(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: 𝜏_𝑖𝑗(𝑒(𝑢))=2𝜇₀𝜀+|𝑒|^2−𝛼/2𝑒_𝑖𝑗−2𝜇₁Δ𝑒_𝑖𝑗,𝑖,𝑗=1,2, (2) where 𝑒_𝑖𝑗=𝑒_𝑖𝑗(𝑢)=12𝜕𝑢_𝑖𝜕𝑥_𝑗+𝜕𝑢_𝑗𝜕𝑥_𝑖,|𝑒|²=²_𝑖,𝑗=1𝑒_𝑖𝑗², (3) and 𝜇₀, 𝜇₁, 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: 𝜕𝑢+()𝜕𝑡𝑢⋅∇𝑢+∇𝑝=∇⋅2𝜇₀𝜀+|𝑒|^2−𝛼/2𝑒−2𝜇₁Δ𝑒+𝑔,∇⋅𝑢=0,𝑥∈𝔻,𝑢=0,𝜏_𝑖𝑗𝑙𝑛_𝑗𝑛_𝑙=0,𝑥∈𝜕𝔻,𝑢|_𝑡=0=𝑢₀, (4)(5)(6)(7) where 𝜏_𝑖𝑗𝑙=2𝜇₁(𝜕𝑒_𝑖𝑗/𝜕𝑥_𝑙) (𝑖,𝑗,𝑙=1,2) and 𝐧=(𝑛₁,𝑛₂) 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 parameter 𝜖. Consider the problem (4)–(7) and the following 2D incompressible non-Newtonian fluid with an additive noise: d𝑢_𝜖=−𝑢_𝜖⋅∇𝑢_𝜖−∇𝑝+∇⋅2𝜇₀𝜀+𝑒_𝜖^2−𝛼/2𝑒_𝜖−2𝜇₁Δ𝑒_𝜖(𝑡),+𝑔d𝑡+𝜖𝜙d𝜔∇⋅𝑢_𝜖=0,𝑥∈𝔻⊂ℝ², (8) where 𝑒_𝜖=(𝑒_𝜖)_𝑖𝑗=(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=𝜇₁=0.

The main result in the present paper is the stability of the attractor in the sense that lim_𝜖→0+dist_𝐻𝒜_𝜖,𝒜₀=0withprobabilityone, (9) where dist_𝐻(⋅,⋅) is the Hausdorff semidistance on the metric space 𝐻 (see notation in Section 2) and 𝒜_𝜖 and 𝒜₀ are the attractors associated with (8) and (4)–(7), respectively. Given a 𝛿>0, we prove that there exists an 𝜖₀ (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 𝒜₀ for all 𝜖∈(0,𝜖₀) 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⁺dist_𝐻(𝒜_𝜖(𝜔),𝒜₀)=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. Define 𝒱={𝜑=(𝜑₁,𝜑₂)∈(𝒞^∞₀(𝔻))²,∇⋅𝜑=0in𝔻,𝜑=0on𝜕𝔻},  𝐻=closureof𝒱in(𝐿²(𝔻))²withnorm‖⋅‖≐‖⋅‖_{(𝐿2(𝔻))}²,  𝑉=closureof𝒱in(𝐻²(𝔻))²withnorm‖⋅‖_𝑉≐‖⋅‖_{(𝐻2(𝔻))}²,  𝐻^′=dualspaceof𝐻, 𝑉^′=dualspaceof𝑉,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 𝑎(𝑢,𝑣)=²_{𝑖,𝑗,𝑘=1𝔻}𝜕𝑒_𝑖𝑗(𝑢)𝜕𝑥_𝑘𝜕𝑒_𝑖𝑗(𝑣)𝜕𝑥_𝑘d𝑥,𝑢,𝑣∈𝑉. (10)

Lemma 1 (see [4]). There exist two positive constants 𝑐₁ and 𝑐₂, which depend only on 𝔻, such that 𝑐₁‖𝑢‖²_𝑉≤𝑎(𝑢,𝑢)≤𝑐₂‖𝑢‖²_𝑉,∀𝑢∈𝑉. (11)

From the definition of 𝑎(⋅,⋅) 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 ⟨𝐴𝑢,𝑣⟩=𝑎(𝑢,𝑣),∀𝑢,𝑣∈𝑉. (12) Moreover, let 𝐷(𝐴)={𝑢∈𝑉∶𝐴𝑢∈𝐻}, and then 𝐷(𝐴) is a Hilbert space. We have (see [8]) 𝑐₁‖𝑢‖_𝑉≤‖𝐴𝑢‖. (13) For brevity, we use 𝐻¹₀(𝔻) to denote (𝐻¹₀(𝔻))² in the sequel. We also define a continuous trilinear form on 𝐻¹₀(𝔻)×𝐻¹₀(𝔻)×𝐻¹₀(𝔻) as follows: 𝑏(𝑢,𝑣,𝑤)=²_{𝑖,𝑗=1𝔻}𝑢_𝑖𝜕𝑣_𝑗𝜕𝑥_𝑖𝑤_𝑗d𝑥,𝑢,𝑣,𝑤∈𝐻¹₀(𝔻). (14) Since 𝑉⊂𝐻¹₀(𝔻), 𝑏(⋅,⋅,⋅) is continuous on 𝑉×𝑉×𝑉 and one can check 𝑏(𝑢,𝑣,𝑤)=−𝑏(𝑢,𝑤,𝑣),𝑏(𝑢,𝑣,𝑣)=0,∀𝑢,𝑣,𝑤∈𝑉. (15) Now for any 𝑢∈𝑉, ⟨𝐵(𝑢,𝑢),𝑤⟩=𝑏(𝑢,𝑢,𝑤),∀𝑤∈𝑉, (16) defines a continuous functional 𝐵(𝑢) from 𝑉×𝑉 to 𝑉^′. Finally, for 𝑢∈𝑉, we set 𝜇(𝑢)=2𝜇₀(𝜀+|𝑒(𝑢)|²)^−𝛼/2 and define 𝑁(𝑢) as ⟨𝑁(𝑢),𝑣⟩=²_{𝑖,𝑗=1𝔻}𝜇(𝑢)𝑒_𝑖𝑗(𝑢)𝑒_𝑖𝑗(𝑣)d𝑥,∀𝑣∈𝑉. (17) Then the functional 𝑁(𝑢) is continuous from 𝑉 to 𝑉^′. When 𝑢∈𝐷(𝐴), 𝑁(𝑢) can be extended to 𝐻 via ⟨𝑁(𝑢),𝑣⟩=−_𝔻∇⋅𝜇(𝑢)𝑒(𝑢)⋅𝑣d𝑥,∀𝑣∈𝐻. (18)

Eliminating the pressure 𝑝 by a proper projection, we have the weak version of problem (4)–(7) in the solenoidal vector fields as follows (see [4, 9]): 𝜕𝑢𝜕𝑡+2𝜇₁()(𝑢)𝑢|𝐴𝑢+𝐵𝑢,𝑢+𝑁=𝑔,_𝑡=0=𝑢₀. (19)

Lemma 2 (see [8, 9]). Let 𝑔∈𝐻 (independent of time 𝑡). Then the semigroup {𝑆(𝑡)}_𝑡≥0 associated with (19) possesses a global attractor 𝒜₀ in 𝐻 satisfying(1)(compactness) 𝒜₀ is compact in 𝐻;(2)(invariance) 𝑆(𝑡)𝒜₀=𝒜₀, 𝑡≥0;(3)(attractivity) for all ℬ⊂𝐻 bounded, lim_𝑡→+∞dist_𝐻(𝑆(𝑡)ℬ,𝒜₀)=0.
Moreover, 𝒜₀ is compact in space 𝑉 and for any ℬ^𝑉⊂𝑉 bounded, lim_𝑡→+∞dist_𝑉𝑆(𝑡)ℬ^𝑉,𝒜₀=0. (20)

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 take Ω={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 ℱ₀ by the Borel 𝜎-algebra on Ω. Let 𝜃_𝑡𝜔(𝑠)=𝜔(𝑠+𝑡)−𝜔(𝑡),𝑡,𝑠∈ℝ. (21)

To obtain the uniqueness and existence of solutions for the problem (8), we need the following assumption.

Assumption A. If 𝜇₀,𝜇₁,𝜀>0, 𝛼∈(0,1), 𝑔∈𝐻, and 𝜙∈𝐷(𝐴), then there exists a positive number 𝛿 such that 𝐵𝑢,𝜙,𝑢≤𝛿‖𝑢‖²,∀𝑢∈𝑉. (22)

Remark 3. Since ⟨𝐵(𝑢,𝜙),𝑢⟩=∑²_𝑖,𝑗=1∫_𝔻𝑢_𝑖(𝜕𝜙_𝑗/𝜕𝑥_𝑖)𝑢_𝑗d𝑥, 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: d𝑢_𝜖=−2𝜇₁𝐴𝑢_𝜖−𝐵𝑢_𝜖,𝑢_𝜖−𝑁𝑢_𝜖+𝑔d𝑡+𝜖𝜙d𝜔(𝑡),𝑢_𝜖|_𝑡=0=𝑢₀. (23)(24)

Let 𝜂>0 be a constant and denote 𝑧 by the solution of the stationary solution of the Itô equation: d𝑧=−𝜂𝑧d𝑡+d𝜔(𝑡). (25) The solution is often called an Ornstein-Uhlenbeck process. In fact, 𝑧(𝑡)=^𝑡_−∞𝑒^{−𝜂(𝑡−𝑠)}d𝜔(𝑠). (26) We now make the change 𝑣_𝜖(𝑡,𝜔)=𝑢_𝜖(𝑡,𝜔)−𝜖𝜙𝑧(𝑡,𝜔). (27) Then 𝑣_𝜖(𝑡,𝜔) satisfies the following random abstract evolutionary equation: d𝑣_𝜖d𝑡+2𝜇₁𝐴𝑣_𝜖+𝐵𝑢_𝜖,𝑢_𝜖+𝑁𝑢_𝜖+𝜖𝜂𝜙𝑧+2𝜇₁𝜖𝐴𝜙𝑧=𝑔. (28) Now (28) can be studied for each 𝜔∈Ω.

Lemma 4 (see [10]). Let Assumption A hold. Then for ℙ-a.s. 𝜔∈Ω, the following results hold.(1)For all 𝑡₀∈ℝ and 𝑣₀=𝑣_𝜖(0)∈𝐻, there exists a unique solution 𝑣_𝜖∈𝒞([𝑡₀,+∞);𝐻)∩𝐿²_loc(𝑡₀,+∞;𝑉) of (28) with initial value 𝑣₀.(2)If 𝑣₀∈𝑉, then the solution belongs to 𝒞([𝑡₀,+∞);𝑉)∩𝐿²_loc(𝑡₀,+∞;𝐷(𝐴)).(3)For every 𝜏>0, 𝑣_𝜖(𝑡)∈𝒞([𝑡₀+𝜏,+∞);𝑉)∩𝐿²_loc(𝑡₀+𝜏,+∞;𝐷(𝐴)).(4)Denote the solution by 𝑣_𝜖(𝑡,𝜔;𝑡₀,𝑣₀). Then the map 𝑣₀↦𝑣_𝜖(𝑡,𝜔;𝑡₀,𝑣₀) is continuous for all 𝑡≥𝑡₀.

By Lemma 4 we see that for each 𝜖∈(0,1] there is a continuous mapping from 𝐻 into itself: 𝑣₀↦𝑣_𝜖(𝑡,𝜔;𝑡₀,𝑣₀), where 𝑣_𝜖(𝑡,𝜔;𝑡₀,𝑣₀) is the solution of (28) with initial value 𝑣_𝜖(𝑡₀)=𝑣₀. Thus for each 𝜖∈(0,1], we can define an RDS 𝜓_𝜖(𝑡,𝜔) associated with (23) and (24) in 𝐻 by setting 𝜓_𝜖(𝑡,𝜔)𝑢₀=𝑢_𝜖(𝑡,𝜔)=𝑣_𝜖(𝑡,𝜔)+𝜖𝜙𝑧(𝑡,𝜔)=𝑣_𝜖𝑡,𝜔;0,𝑢₀−𝜖𝜙𝑧(0,𝜔)+𝜖𝜙𝑧(𝑡,𝜔). (29)

We end this section with the concept of random attractor for 𝜓_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_𝑡𝜔),(iii)(attracting property) for any bounded subset ℬ of 𝐻 and a.e. 𝜔∈Ω, lim_𝑡→+∞Dist_𝐻𝑆𝑡,𝜃_−𝑡𝜔ℬ,𝒜(𝜔)=0. (30)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 solution 𝑢_𝜖(0,𝜔;−𝑡₀,𝑢₀) of (23) and (24) converges in 𝐻ℙ-a.s. as 𝜖→0⁺ to the solution 𝑢(𝑡₀;𝑢₀) of the unperturbed problem (19), uniformly on bounded sets of initial conditions. That is, for ℙ almost every 𝜔∈Ω, 𝑡₀∈ℝ₊ and ℬ⊂𝐻 bounded, the following convergence holds: lim_𝜖→0+𝑢_𝜖0,𝜔;−𝑡₀,𝑢₀−𝑢𝑡₀,𝑢₀=0. (31)

Proof. Let 𝑤_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 𝑢₀ at −𝑡₀. It is clear that 𝑤_𝜖 satisfies d𝑤_𝜖+2𝜇₁𝐴𝑤_𝜖+𝐵𝑢_𝜖,𝑢_𝜖()𝑢−𝐵𝑢,𝑢+𝑁_𝜖(𝑢(𝑡),𝑤−𝑁)d𝑡=𝜖𝜙d𝜔_𝜖−𝑡₀=0. (32) We now make the change of variables 𝑧_𝜖=𝑤_𝜖−𝜖𝜙𝜔(𝑡) (33) and thus formally obtain d𝑧_𝜖d𝑡+2𝜇₁𝐴𝑧_𝜖+2𝜇₁𝜖𝐴𝜙𝜔(𝑡)+𝐵𝑢_𝜖,𝑢_𝜖−𝐵(𝑢,𝑢)+𝑁𝑢_𝜖−𝑁(𝑢)=0. (34) Taking the inner product of the above equation with 𝑧_𝜖, we get 12d𝑧d𝑡^𝜖2+2𝜇₁𝐴𝑧_𝜖,𝑧_𝜖+2𝜇₁𝜖𝜔(𝑡)𝐴𝜙,𝑧_𝜖+𝐵𝑢_𝜖,𝑢_𝜖,𝑧_𝜖−𝐵(𝑢,𝑢),𝑧_𝜖+𝑁𝑢_𝜖,𝑧_𝜖−𝑁(𝑢),𝑧_𝜖=0. (35) Therefore 2𝜇₁𝜖𝜔(𝑡)𝐴𝜙,𝑧_𝜖≤2𝜇₁𝜖|𝜔(𝑡)|𝐴𝜙𝑧_𝜖≤2𝜇²₁𝜖²𝜔²(𝑡)𝐴𝜙²+𝑧_𝜖²2. (36) By the property of the operator 𝐵(⋅,⋅), we imply that 𝐵𝑢_𝜖,𝑢_𝜖()𝑤−𝐵𝑢,𝑢=𝐵_𝜖+𝑢,𝑤_𝜖()𝑤+𝑢−𝐵𝑢,𝑢=𝐵_𝜖,𝑤_𝜖+𝐵𝑤_𝜖,𝑢+𝐵𝑢,𝑤_𝜖=𝐵𝑧_𝜖+𝜖𝜙𝜔(𝑡),𝑧_𝜖(𝑡)𝑧+𝜖𝜙𝜔+𝐵_𝜖+𝜖𝜙𝜔(𝑡),𝑢+𝐵𝑢,𝑧_𝜖+𝜖𝜙𝜔(𝑡). (37) Thus, 𝐵𝑢_𝜖,𝑢_𝜖−𝐵(𝑢,𝑢),𝑧_𝜖=𝐵𝑧_𝜖+𝜖𝜙𝜔(𝑡),𝑧_𝜖+𝜖𝜙𝜔(𝑡),𝑧_𝜖+𝐵𝑧_𝜖+𝜖𝜙𝜔(𝑡),𝑢,𝑧_𝜖+𝐵𝑢,𝑧_𝜖+𝜖𝜙𝜔(𝑡),𝑧_𝜖=𝐵𝑧_𝜖,𝑧_𝜖+𝜖𝜙𝜔(𝑡),𝑧_𝜖+𝐵𝜖𝜙𝜔(𝑡),𝜖𝜙𝜔(𝑡),𝑧_𝜖+𝐵𝑧_𝜖,𝑢,𝑧_𝜖+𝐵𝜖𝜙𝜔(𝑡),𝑢,𝑧_𝜖+𝐵𝑢,𝜖𝜙𝜔(𝑡),𝑢,𝑧_𝜖. (38) We now estimate the right hand side of (38) terms by terms. First note that 𝐵𝑧_𝜖,𝜖𝜙𝜔(𝑡),𝑧_𝜖≤𝜖|𝜔(𝑡)|𝐵𝑧_𝜖,𝜙,𝑧_𝜖≤𝛿𝜖|𝜔(𝑡)|𝑧_𝜖²,𝐵𝜖𝜙𝜔(𝑡),𝜖𝜙𝜔(𝑡),𝑧_𝜖≤𝜖²𝜔²(𝑡)𝐵𝜙,𝜙,𝑧_𝜖≤𝜖²𝜔⁴(2)𝜙_𝐿4(𝔻)∇𝜙_𝐿4(𝔻)𝑧_𝜖≤𝑐𝜖²𝜔²(𝑡)𝜙^1/2∇𝜙^1/2𝜙^1/4Δ𝜙^3/4𝑧_𝜖≤𝑐𝜖²𝜔⁴(𝑡)∇𝜙²Δ𝜙²+𝑧_𝜖²2,𝐵𝑧_𝜖,𝑢,𝑧_𝜖≤𝑐𝑧_𝜖‖∇𝑢‖∇𝑧_𝜖≤3𝜇₁4𝑧_𝜖²_𝑉+𝑐‖∇𝑢‖²𝑧_𝜖²,𝐵𝜖𝜙𝜔(𝑡),𝑢,𝑧_𝜖≤𝜖|𝜔(𝑡)|⟨𝐵𝜙,𝑢,𝑧_𝜖⟩≤𝜖|𝜔(𝑡)|𝜙_𝐿4(𝔻)‖∇𝑢‖_𝐿⁴_(𝔻)𝑧_𝜖≤𝜖|𝜔(𝑡)|𝜙^1/2∇𝜙^1/2‖∇𝑢‖^1/2‖Δ𝑢‖^1/2𝑧_𝜖≤𝑐𝜖²|𝜔(𝑡)|²𝜙∇𝜙‖𝑢‖²_𝑉+𝑧_𝜖²2,𝐵𝑢,𝜖𝜙𝜔(𝑡),𝑧_𝜖≤𝜖|𝜔(𝑡)|⟨𝐵𝑢,𝜙,𝑧_𝜖⟩≤𝜖|𝜔(𝑡)|‖𝑢‖^1/2‖∇𝑢‖^1/2∇𝜙^1/2Δ𝜙^1/2𝑧_𝜖≤𝜖²|𝜔(𝑡)|²‖𝑢‖‖𝑢‖²_𝐻1(𝔻)𝜙_𝐻1(𝔻)+𝑧𝐴𝜙^𝜖22. (39) Then also note that 𝑁𝑢_𝜖−𝑁(𝑢),𝑧_𝜖=𝑁𝑢_𝜖−𝑁(𝑢),𝑢_𝜖−𝑢−𝜖𝜙𝜔(𝑡). (40) By the nonnegativity of 𝑁(⋅) (see [4]), we conclude that 𝑁𝑢_𝜖−𝑁(𝑢),𝑢_𝜖−𝑢≥0. (41) Set 𝐹(𝑠)=2𝜇₀(𝜀+|𝑠|²)^−𝛼/2𝑠, where 𝑠𝑠=¹𝑠₂𝑠₃𝑠₄∈ℳ_2×2,|𝑠|²=⁴_𝑖=1𝑠²_𝑖,𝑠_𝑖∈ℝ,𝑖=1,2,3,4. (42) Here ℳ_2×2 denotes the matrix of order 2×2. Then the first order Fréchet derivative of 𝐹(𝑠) is DF(𝑠)=2𝜇₀𝜀+|𝑠|^2−𝛼/2×1−𝛼𝑠²₁𝜀+|𝑠|²−𝛼𝑠₁𝑠₂𝜀+|𝑠|²−𝛼𝑠₁𝑠₃𝜀+|𝑠|²−𝛼𝑠₁𝑠₄𝜀+|𝑠|²−𝛼𝑠₁𝑠₂𝜀+|𝑠|²1−𝛼𝑠²₂𝜀+|𝑠|²−𝛼𝑠₂𝑠₃𝜀+|𝑠|²−𝛼𝑠₂𝑠₄𝜀+|𝑠|²−𝛼𝑠₁𝑠₃𝜀+|𝑠|²−𝛼𝑠₂𝑠₃𝜀+|𝑠|²1−𝛼𝑠²₃𝜀+|𝑠|²−𝛼𝑠₃𝑠₄𝜀+|𝑠|²−𝛼𝑠₁𝑠₄𝜀+|𝑠|²−𝛼𝑠₂𝑠₄𝜀+|𝑠|²−𝛼𝑠₃𝑠₄𝜀+|𝑠|²1−𝛼𝑠²₄𝜀+|𝑠|². (43) Since 0<𝛼<1, we have −𝛼𝑠_𝑖𝑠_𝑗𝜀+|𝑠|²<𝑠_𝑖𝑠_𝑗𝜀+|𝑠|²<1,𝑖,𝑗=1,2,3,4,0<1−𝛼𝑠²_𝑖𝜀+|𝑠|²<1,𝑖=1,2,3,4. (44) Consequently, ‖DF(𝑠)‖≤8𝜇₀𝜀+|𝑠|^2−𝛼/2≤8𝜇₀𝜀^−𝛼/2,∀𝑠∈ℳ_2×2. (45)
Now for any 𝑎,𝑏∈ℝ⁴, 𝐹(𝑏)−𝐹(𝑎)=¹₀DF(𝑎+𝜏(𝑏−𝑎))(𝑏−𝑎)d𝜏. (46) Taking 𝑎=𝑒(𝑢_𝜖)=(𝑒_𝑖𝑗(𝑢_𝜖)), 𝑏=𝑒(𝑢)=(𝑒_𝑖𝑗(𝑢)), applying the integration by parts first, and then utilizing the above inequality about 𝐹(𝑠), we have 𝑁𝑢_𝜖(𝑢)(𝑡=−𝑁,𝜖𝜙𝜔)_{2𝑖,𝑗=1𝔻}𝐹𝑒_𝑖𝑗𝑢_𝜖−𝐹𝑒_𝑖𝑗(𝑢)⋅𝑒_𝑖𝑗(𝑡)|𝜔(𝑡∇𝑢𝜖𝜙𝜔d𝑥≤𝑐𝜖)|_𝜖|𝜔(𝑡∇𝑧−𝑢∇𝜙≤𝑐𝜖)|_𝜖(𝑡)≤+𝜖𝜙𝜔∇𝜙3𝜇₁4∇𝑧_𝜖²+𝑐𝜖²𝜔²(𝑡)∇𝜙². (47) Combining (35)–(40) and (47), we obtain d𝑧d𝑡^𝜖2≤𝐼(𝑡)+𝐺(𝑡)𝑧_𝜖², (48) where 𝐼(𝑡)=𝑐𝜖𝜇²₁𝜖𝜔²(𝑡)𝐴𝜙²+𝜖𝜔⁴(𝑡)∇𝜙²𝐴𝜙²+𝜖|𝜔(𝑡)|𝜙∇𝜙‖𝑢‖²_𝑉+𝜖𝜔²(𝑡𝑢‖×‖𝜔)‖∇𝑢‖∇𝜙𝐴𝜙+𝜖²(𝑡)𝐴𝜙²,𝐺(𝑡)=2+𝜖𝛿|𝜔(𝑡)|+𝑐‖∇𝑢‖². (49) Then, by Gronwall inequality, 𝑧_𝜖(𝑡)²≤𝐼(𝑡)+^𝑡_−𝑡0𝐼(𝑠)𝐺(𝑠)exp^𝑡_𝑠𝐺(𝜏)d𝜏d𝑠. (50) By Lemma 2 we find that ‖𝑢(𝑡)‖²_𝑉 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≥−𝑡₀. Therefore, lim_𝜖→0+𝑤_𝜖(𝑡)²≤lim_𝜖→0+2𝑧_𝜖²+𝜖²𝜙𝜔²(𝑡)=0. (51) 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 each 𝜖∈(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.lim_𝜖→0+dist_𝐻𝒜_𝜖(𝜔),𝒦=0. (52)

Proof. For each 𝜖∈(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 each 𝜖∈(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 𝒜₀ in 𝐻. Moreover, lim_𝜖→0+dist_𝐻𝒜_𝜖(𝜔),𝒜₀=0 (53) with probability one.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper is supported by the special fund of China’s central government for the development of local colleges and universities: the project of national first-level discipline in Oil and Gas Engineering.