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

and Applied Analysis 3


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: where the vector function (, ) is the velocity of the fluid,  is the external force, the scalar function  represents the pressure, and (()) = (  (())) is the extra stress tensor of the fluid.Ladyzhenskaya [2] formulated a two-dimensional non-Newtonian fluid model with the extra stress tensor: where 2 , and  0 ,  1 , , and  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: where   = 2 1 (  /  ) (, ,  = 1, 2) and n = ( 1 ,  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][2][3][4][5][6][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]. 2

Abstract and Applied Analysis
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 motivation for the present paper is the desire to understand the stability of attractors for the above twodimensional 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 .Consider the problem (4)-( 7) and the following 2D incompressible non-Newtonian fluid with an additive noise: where   = (  )  = (1/2)(((  )  /  ) + ((  )  /  )), () is an independent two-sided Wiener processes, and  is a function satisfying some conditions to be specified below.Caraballo et al. [25][26][27][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 |∇| of the velocity is relatively large [2].Clearly, (4) reduces to Navier-Stokes equations when  =  1 = 0.
The main result in the present paper is the stability of the attractor in the sense that lim  → 0 + dist  (A  , A 0 ) = 0 with probability one, (9) where dist  (⋅, ⋅) is the Hausdorff semidistance on the metric space  (see notation in Section 2) and A  and A 0 are the attractors associated with (8) and ( 4)- (7), respectively.Given a  > 0, we prove that there exists an  0 (depending on , a parameter event in a probability space (Ω, F, P)) sufficiently small, such that the random attractors A  are inside the  neighborhood of the global attractor A 0 for all  ∈ (0,  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  → 0 + dist  (A  (), A 0 ) = 0 with probability one.

Global Existence and Uniqueness of Solutions
In this section, we introduce some notations and recall some results about non-Newtonian fluid dynamics.Define where (⋅, ⋅) denotes the inner product in  and ⟨⋅, ⋅⟩ stands for the dual pairing between  and   .If we identify  with   , then  →  =   →   with continuous and compact embeddings.
We also define the bilinear form Lemma 1 (see [4]).There exist two positive constants  1 and  2 , which depend only on D, such that From the definition of (⋅, ⋅) and Lemma 1 we see that (⋅, ⋅) defines a positive definite symmetric bilinear form on .As a consequence of the Lax-Milgram Lemma, we obtain an operator  ∈ L(,   ), via Moreover, let () = { ∈  :  ∈ }, and then () is a Hilbert space.We have (see [8]) For brevity, we use  1 0 (D) to denote ( 1 0 (D)) 2 in the sequel.
Moreover, A 0 is compact in space  and for any B  ⊂  bounded, 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 = { ∈ C(R; ), (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 -algebra on Ω.Let To obtain the uniqueness and existence of solutions for the problem (8), we need the following assumption.
Using the notations and operators introduced above, we can put (8) into the following abstract form: Let  > 0 be a constant and denote  by the solution of the stationary solution of the Itô equation: The solution is often called an Ornstein-Uhlenbeck process.
In fact, We now make the change Then V  (, ) satisfies the following random abstract evolutionary equation: Now ( 28) can be studied for each  ∈ Ω.

Stability of Attractors
We first prove the following lemma, which plays a key role later on.
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