1 Introduction

In this paper,we consider the following stochastic incompressible non-Newtonian fluids with multiplicative noise,and assume that D ⊂ R² is a two-dimensional bounded smooth open domain,

with the boundary conditions where ◦ denotes the Stratonovich sense in the stochastic term,and ω_j(t),1≤j≤m,are mutually independent two-sided Wiener processes,b_j ∈ R,1≤j≤m,are given. In Eq. (1), vector function u is the unknown denoting the velocity of the fluid,g is the external force function,and the scalar function π represents the pressure,τ_ij (e(u)) is usually called the extra stress tensor of the fluid, where and τ_ijl = (i,j,l = 1,2),κ = (κ₁,κ₂) denotes the exterior unit normal to the boundary ∂D. The first condition represents the usual no-slip condition associated with a viscous fluid, while the second one expresses the fact that the first moments of the traction vanish on ∂D,it is a direct consequence of the principle of virtual work.

In the case of p = 2,μ₁ = b_j = 0,Eq. (1) is a deterministic Navier-Stokes equation. In the case of μ₀ = μ₁ = b_j = 0,Eq. (1) is an Euler equation. They both are called Newtonian fluids. The fluid is called shear thinning while 1<p<2 and is shear thickening while p > 2.

There are many works concerning the deterministic non-Newtonian fluids,involving the questions of existence,uniqueness,and regularity of the solution and the existence of attractor and manifold^{[1, 2, 3, 4, 5, 6, 7]}.

In the past decades,for stochastic Navier-Stokes equation,Burgers equation,etc.,there have been much work and interesting results,one can refer to Refs. ^{[8, 9, 10, 11, 12, 13, 14]},and the references therein. After that,there are a series of papers to investigate stochastic non-Newtonian fluids. Some important results have been obtained,such as Refs. ^{[15, 16, 17, 18]} and so on. Recently,Zhao et al.^[19] proved the existence of global random attractor for two-dimensional stochastic non-Newtonian fluids with multiplicative noise in the case of 1<p<2,and Guo^[20] extended the result to the case of 2<p<3. The operator ∇· (Δe) in Eq. (1) makes the non-Newtonian fluids have a higher four order regularity. For deterministic non-Newtonian fluid system,the H²-regularity attractors have been obtained^{[3, 5, 21]}. We naturally want to know whether the similar result is also true for non-Newtonian fluid with multiplicative noise. That is the reason why the present paper is addressed. In this paper,we prove that there exist H²-regularity random attractors for stochastic non-Newtonian fluids with multiplicative noise in the case of 1<p<2.

The fundamental theory of random dynamical system was developed by Anrold,Crauel, Debussche and Flandoli,and others. They presented a general theory to study random attractor^{[8, 9]}. In this paper,we use the theory to prove the existence of H²-regularity random attractors for stochastic non-Newtonian fluids with multiplicative noise. Firstly,we make use of the Stratonovich transform to change the stochastic equation to a deterministic equation with random parameter. Secondly,we obtain the existence of bounded absorbing sets by some estimates of solution in space V . Thirdly,we use the compact embedding of Sobolev space to obtain the existence of H²-compact random set.

The outline of this paper is as follows. In Section 2,we recall some basic concepts related to random attractors for random dynamical system. In Section 3,we develop all the results needed to prove the existence of random attractors in space V with 1<p<2,g ∈H¹. In Section 4,we establish the existence of a compact random attractor in V by compactness of Sobolev embedding.

Remark 1.1 After making the Stratonovich transform,the larger p,the more difficult to obtain some norm estimates of v. Especially,in the case of p > 2,we need to restrict the upper bound of p in order to obtain the existence of H²-compact absorbing set. We should discuss this problem in a forthcoming paper by elaborate estimates.

2 Preliminaries

We introduce some functional spaces and notations for the convenience of the following contents.

L^q(D)—the Lebesgue space with norm ||·||L^q ,and ||·||L² = ||·||. Particularly,||u||L∞ = |u(x)|,for q = ∞.

H^σ(D)—the Sobolev space {u ∈ L²(D),D^ku ∈ L²(D),k≤σ},and ||·||_H^σ = ||·||_σ.

Define a space of smooth functions

H means the closure of V in L²(D) with norm ||·||. The inner product in H denoted by (·,·).

H₀¹ (D) means the closure of V inH¹(D) with norm ||·||1.

V means the closure of V in H²(D) with norm ||·||₂,and V ′ is the dual space of V .

By simple computation,we can conclude the results ∇·e(u) = ,and ∇·(Δe(u)) = . Thus,2μ₁∇ · (Δe(u)) = μ₁Δ²u.

For notational simplicity,C is a generic constant,and may assume various values from line to line throughout this paper.

Consider the positive definite symmetric bilinear form a(·,·) : V × V → R given by

Applying the Lax-Milgram lemma,we obtain an isometric operator A ∈ L(V,V ′) via where V ′ is the dual space of V ,and the domain of A is In fact,A = PΔ²,where P is the Leray projector from L²(D) to H.

Define the trilinear form b on H¹₀ (D) ×H¹₀ (D) ×H¹₀ (D) given by Next,define a bilinear map B onH¹₀ (D) ×H¹₀ (D) by Define the map N(u) onH¹₀ (D) as follows: where γ(u) = .

From these preparation,Eqs. (1)-(4) can be translated into the following abstract problems in H:

We recall some basic concepts related to randomattractors for random dynamical system^{[8, 9]}. Let (X,d) be a complete separable metric space with Borel σ-algebra B(x) endowed with distance d,and let (Ω,F,P) be a complete probability space. We also denote mappings S(t,s; ω) : X → X,−∞<s≤t<∞ with explicit dependence on ω.

Definition 2.1 A random set K(t,ω) is called attracting in X,if for any bounded non- random sets B ⊂ X and a.e. ω,

where d(A,B) is the semidistance defined by

Definition 2.2 A family A(ω),ω ∈ of closed subsets of X is said to be measurable,if the mapping ω → d(x,A(ω)) is measurable for each x ∈ X.

Definition 2.3 Let {θ_t :Ω → Ω,t ∈ R} be a family of measure preserving transformation of (Ω ,F,P) such that θ₀ = id and θ_t+s = θ_t ◦ θs for all t,s ∈ R. Here,we assume θ_t is ergodic under P. Especially,for all s<t ∈ R,and x ∈ X,

Definition 2.4 Let S(t,s; ω)t>s,ω∈ be a stochastic dynamical system,if A(t,ω) satisfies:

(1) for any bounded subset B of X,t ∈ R,and a.e. ω ∈ (attracting property),

(2) for a.e. ω ∈ and all s≤t (invariance),

Then,A(t,ω) is called to be the random attractor.

Theorem 2.1^[8] Let S(t,s; ω)t>s,ω∈ be a stochastic dynamical system satisfying the fol- lowing conditions:

(1) S(t,r; ω)S(r,s; ω)x = S(t,s; ω)x,for all s≤r≤t and x ∈ X;

(2) S(t,s; ω) is continuous in X,for all s≤t;

(3) for all s<t and x ∈ X,the mapping

is measurable from (Ω ,F) to (X,B(X));

(4) for all t,x ∈ X and P-a.e. ω,the mapping

is right continuous at any point.

Assume that there exists a group θ_t (t ∈ R) of measure preserving mappings such that

holds and for P-a.e. ω,there exists a compact attracting set K(ω) at time 0,for P-a.e. ω ∈ Ω. We set Λ(ω) = ,where the union is taken over all the bounded subsets of X and A(B,ω) is given by Then,Λ(ω) is a random attractor.

Remark 2.1 In application to stochastic Eqs. (6)-(7),let = {ω ∈ C(R,l²)| ω(0) = 0}, with P being the product measure of two Wiener measures on the negative and the positive time parts of Ω . In this case,time shift θ_t is defined as

Therefore,the condition (8) is satisfied.

3 A priori estimate

In this section,we introduce an auxiliary Stratonovich process which enables us to change the stochastic equation to an evolution equation depending on a random parameter. Considering the process η(t) = ,which satisfies the Stratonovich equation

Set v(t) = η(t)u(t),satisfying the following equation:

Similarly (see ^{[4, 14]}),we can use the Galerkin method to prove that for P-a.e. ω ∈ Ω ,there holds:

i) For g ∈ H,v_s ∈ H,s<T ∈ R,there exists a unique weak solution to Eqs. (10)-(11) v ∈ C(s,T ;H)TL²(s,T ; V ) with initial value v(s) = v_s.

ii) For g ∈H¹,vs ∈ V,s<T ∈ R,there exists a unique weak solution to Eqs. (10)-(11) v ∈ C(s,T ; V )TL²(s,T ;D(A)) with initial value v(s) = v_s.

We define the stochastic dynamical system (S(t,s; ω))t>s,ω∈Ω by

The key is to obtain the existence of a compact attracting set at time 0 in V . First,we try to obtain some estimates in H,H¹ ₀ (D),and V . Second,we use the compactness of the embedding to prove the existence of compact random attractors.

Lemma 3.1 Let 1<p<2,g ∈ H,there exist random radii r₁(ω) and r₂(ω),such that ∀ ρ > 0,there exists s(ω)≤−1,such that for all s≤s(ω) and for all u_s ∈ H with ||u_s||≤ρ, the solution of Eqs. (10)-(11) with v_s = η(s)u_s satisfies the following inequalities:

where r₁²(ω) = .

Proof Taking the inner product of Eq. (10) with v in H,and noticing that b(u,u,v) = 0, we obtain

Let I = . Applying the condition v = ηu,namely,e(v) = ηe(u),yields I > 0.

We drop the term involving μ₀ in (12),then deduce

where the ǫ-Young inequality is used,and λ is a constant satisfying the inequality ||∇v||² > λ||v||². Furthermore,we can get Obviously, For s≤−1,and −1≤t≤0,by the Gronwall lemma on the interval [s,t],we can deduce On the other hand,by standard argument, It follows that σ → e^μ₁λ2ση²(σ) is pathwise integrable over (−∞,0], Let ),which is finite P−a.s. by the above conditions. Given ρ > 0,there exists s(ω) such that e^μ₁λ2 η²(s)ρ²≤1,for all s≤s(ω),it follows that Integrating Eq. (15) with respect to t from [−1,0],we get we can easily deduce that

Lemma 3.2 Let 1<p<2,g ∈ H,there exist random radii r₃(ω) and r₄(ω),such that ∀ ρ > 0,there exists s(ω)≤−1,such that for all s≤s(ω),and for all u_s ∈ H,with ||u||≤ρ, the solution of Eqs. (10)-(11) with vs = η(s)u_s satisfies the inequalities

Proof Taking the inner product of Eq. (10) with Δ²v in H,we can get

Obviously,applying the ǫ-Young inequality,we obtain By the H¨older inequality,Gagliardo-Nirenberg inequality,and ǫ-Young inequality,we get where C = . Similar to the derivation in Ref. ^[5],we can obtain Combining the above estimates,we can deduce It can be also obtained that Let L(t) = ,M(t) = . Then, For −1≤ξ≤t≤0,using the Gronwall inequality on the interval [ξ,t],we obtain Integrating the above inequality with respect to ξ over [−1,0],we get To obtain the boundedness of ||v(t)||²,next,we estimate the right-hand side of inequality (25) one by one,i.e., where we have used the conclusion of Lemma 3.1, By the property of Stratonovich process η(t) in Lemma 3.1,we deduce thatL(σ)dσ and M(σ)dσ are bounded. Combining these estimates,we can conclude that Integrating (23) with respect to t over [−1,0] ,then In other words,

Actually,from this Lemma,we get

and from the above results, is bounded. This Lemma implies the existence of bounded absorbing set.

4 Existence of random attractors in V

In this section,we establish the existence of a compact random attractor in V by compactness of Sobolev embedding. First,we need the following Lemma.

Lemma 4.1 Let 1<p<2,g ∈H¹,there exists random radius r₇(ω) such that ∀ρ > 0, there exists s(ω)≤−1,satisfying for all s≤s(ω),and for all u_s ∈ H,with ||u_s||≤ρ,the solution of Eqs. (10)-(11) with v_ss = η(s)u_s satisfies the inequality

Proof Taking the inner product of Eq. (10) with in H,

for the first two terms in the left hand side of Eq. (29),we obtain Applying the ε-Young inequality,we can get Now,we estimate the nonlinear terms in Eq. (29). where we have used the H¨older inequality. Furthermore,we can obtain where we have used the Sobolev embedding H²(D) → L^∞(D),ε-Young inequality,and the condition v = ηu.

For other nonlinear term,using the H¨older inequality yields

Let The estimate of J is similar to Ref. ^[15],by the integration by parts,the assumed condition 1<p<2,the Sobolev embedding,and some technological computation. Then,we can get Thus, where we have used the ε-Young inequality,and the condition v = ηu.

Putting the above inequalities together,we obtain

Then,changing the inequality suitably to use the Gronwall inequality yields Let Thus, For −1≤ζ≤ξ≤0,by the Gronwall inequality on the interval [ζ,ξ],we obtain Integrating the above inequality with respect to ζ over [−1,0] yields Next,we prove the right-hand side of (33) is bounded. where we have used the conclusion of Lemma 3.2.

Note that

where we have used the property of Stratonovich process η(t) and the conclusion of Lemma 3.2.

Similarly,

From Eq. (33) and the above inequalities,we obtain Now,we put ,especially,for ξ = 0,

Theorem 4.1 Let 1<p<2,g ∈H¹,and us ∈ H,there exist random attractors for the stochastic non-Newtonian fluid with multiplicative noise (6)-(7) in V .

Proof Let K(ω) be the ball in H³(D) with radius r₇(ω),we have proved that for any B bounded in V ,there exists s(ω) such that for s≤s(ω),

This clearly implies that K(ω) is an attracting set at time t = 0. Since it is compact in V , Theorem 2.1 applies.

Theorem 4.1 implies that the asymptotic smoothing effect of the fluids in the sense that the solution becomes eventually more regular than the initial data.