Pullback dynamic behavior for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid on 2D bounded domains. We show existence of the pullback exponential attractor introduced by Langa, Miranville and Real [ 27 ], moreover, give existence of the global pullback attractor with finite fractal dimension and reveal the relationship between the global pullback attractor and the pullback exponential attractor. These results improve our previous associated results in papers [ 29 , 40 ] for the non-Newtonian fluid.

In spirit of the papers [8,27,29,40], we will construct the pullback exponential attractor and the global pullback attractor with finite fractal dimension for the non-Newtonian fluid. Although the method used in this article is similar to the one used in [8,27], we emphasize that the non-Newtonian fluid addressed in this paper contain the additional nonlinear term ∇ · (2µ 0 (η + |e(u)| 2 ) −α/2 e(u)) and a higher order derivative ∇ · (2µ 1 ∆e(u)), which lead to an addition nonlinear term N (u) and a different linear term Au in the abstract equation (see (6)). Due to these differences, in the procedure of the a prior estimates and verifying the hypotheses of solutions for the non-Newtonian fluid, we will meet more trouble which can not be worked out trivially. More delicate estimates and analysis for the solutions are required. We also want to point out that the results within this paper improves the results of [29,40]. In [29,40], they have just focused on the existence of the pullback attractors without the exponential retraction rate. Moreover, it is the first time that the finite dimensional global pullback attractor is obtained for the non-Newtonian fluid in this paper.
The paper is organized as follows. Section 2 is notations and preliminaries. In section 3, we derive some a prior estimates. In section 4, we establish existence of the pullback exponential attractor and the global pullback attractor with finite fractal dimensions for the non-Newtonian fluid (1)-(5).
2. Notations and preliminaries. In this section, we first remark the fundamental notations.
According to the above notations, we further denote L 2 loc (R; H) := space of locally integrable functions from R to H. Then we introduce some operators associated to the non-Newtonian fluid and recall some properties of the operators, previous results for the non-Newtonian fluid.
|b(u, v, w)| ≤ c u 1/2 ∇u 1/2 ∇v w 1/2 ∇w 1/2 , ∀ u, v, w ∈ V, In addition, if u ∈ D(A), then N (u) can be extended to H via (2) A is a linear continuous operator from V to V . Moreover, if D(A) := {ϕ ∈ V : Aϕ ∈ H} = V ∩ H 4 (Ω), then A is an isomorphism both from V to V and from D(A) to H, in fact, A = P ∆ 2 , where P is the Leray projector from L 2 (Ω) into H. Also B(·, ·) and N (·) are continuous from V to V .
holds in the sense of D (τ, T ). Then u is called a solution of equations (6)-(7) in the interval (τ, T ). To end this section, we restate the notation of process and definitions of the pullback exponential attractor and the global pullback attractor for non-autonomous dynamical systems in the following (see [8,10,11,12,27]). Denote by P(X) the family of all nonempty subsets of X.
Definition 2.4. It is said that a biparametric family of maps {U (t, τ )} t≥τ is a process on X, which satisfies the following: Moreover, if for any t ≥ τ , U (t; τ ) is continuous on X, then {U (t, τ )} t≥τ is a continuous process on X.
Definition 2.5. A family of sets M = {M(t) : t ∈ R} ⊂ P(X) is called a pullback exponential attractor for the process {U (t, τ )} t≥τ on X if it has the following properties: • Compactness: for any t ∈ R, M(t) is a nonempty compact subset of X; • Pullback exponentially attracting: for any t ∈ R, bounded subset D ⊂ X, some c > 0 lim s→∞ e cs dist X (U (t, t − s)D, M(t)) = 0.

GUOWEI LIU AND RUI XUE
where N (M(t), X) is the minimal number of cubes with length in X which are necessary to cover M(t).
is called a global pullback attractor for the process {U (t, τ )} t≥τ on X if it has the following properties: • Compactness: for any t ∈ R, A(t) is a nonempty compact subset of X;

A prior estimates.
In this section, we give several estimates for weak solutions of (6)- (7). Let u be the weak solution of system (6)- (7) in Lemma 2.3, then (i) u satisfies the "energy equality", for u 0 ∈ H, (ii) u satisfies the "enstrophy equality", for u 0 ∈ H, (iii) From Lemma 2.3, the maps defined by generate a process {U (t, τ )} t≥τ in H and V , respectively. Moreover, we give a weaker assumption of f (x, t).
Then we have the following estimates of solutions for (6)-(7).
Lemma 3.1. Suppose that Assumption 3.1 holds, D ⊂ H is a bounded subset, then for any t 0 ∈ R, u 0 ∈ D, t ≤ t 0 , the following inequalities hold, where Proof. Inserting (8), (11) and non-negativeness of the term N (u), u into the energy equality (14), it follows Multiplying both sides of (22) with e c1µ1θ and using the Cauchy inequality give Integrating (23) with respect to θ, we obtain It follows by (17) in Assumption 3.1, and (24) that for all t ≥ τ , From (25), we can easily deduce that By (22) and the Cauchy inequality, we get

GUOWEI LIU AND RUI XUE
Integrating (26) with respect to θ, we obtain for t − 1 > τ , Therefore, by (17) in Assumption 3.1 and (18), it follows For the enstrophy equality (15), by the Cauchy inequality, we obtain Using the Hölder inequality, the Gagliardo-Nirenberg inequality and the Young inequality, we have For the term N (u(θ)), Au(θ) , since u ∈ L 2 (τ + , T ; D(A)), we have that To estimate (30), we set and M 2×2 means the set of all the matrices of order 2 × 2. By some computations (see [37,38]) we see that the first order and second order Fréchet derivatives of F (S) satisfy where c is a positive constant depending only on µ 0 , ε and α. For any S 1 , S 2 ∈ M 2×2 , we have Taking S 1 = e(u) = (e ij (u(θ))), S 2 = e(0) = (e ij (0)), applying the integration by parts first and (31) and Young inequality, we have Inserting (28), (29) and (32) into the enstrophy equality (15) and by (8), we obtain Setting Then (34) gives Applying the Gronwall inequality to (35), we have for all Integrating (36) with respect to s between t − 1 and t, we obtain It then follows from (8) and (27) that Similarly, we have By (8) and (37)- (39), we obtain Therefore, by (18) and (40), we get Now, integrating (33) with respect to θ, then by (8), we have Therefore, by (19) and (20), we obtain t t−1 Moreover, we give the estimates for the difference of two solutions of (6)-(7) for different initial data.
Without loss of generality, we let τ = 0. Taking inner product in (54) with θAw(θ), we obtain Similar to the estimate (61), we obtain Similar also to the estimate (62), we obtain It follows from (63)-(65) and (8) where Applying the Gronwall inequality to (66), we get On one hand, By (42) and (27), we deduce On the other hand, by (27), we get Therefore, inserting (71) and (72) into (67) and by (8), we get From the inequalities (43) and (44) in Lemma 3.2, we immediately gain Lemma 3.3. Assume f ∈ L 2 loc (R; H), u 01 , u 02 ∈ V . Then there exists a positive function L = L(r 1 , r 2 , r 3 , r 4 ), depending on Ω, but not on f , such that L ∈ C ∞ (R 4 ), is increasing in each of the four variables r i (i = 1, 2, 3, 4) and satisfies for all τ ≤ t, 4. Pullback dynamic behavior for the non-Newtonian fluid. This section is to construct the pullback exponential attractor and the pullback attractor for the non-Newtonian fluid. Following the theoretical framework in [8,27], it is natural to verify the following hypotheses.
(CP)(Existence of a continuous process) The solution operator U (t, τ ) of non-Newtonian fluid defined by (PAS)(Existence of a fixed pullback absorbing set) For any U (t, τ ) ∈ U t0 , where there exists a bounded and closed set B ⊂ V satisfying for any bounded D ⊂ V , there exists a time s D > 0 such that for all s > s D , (SP)(A smooth property) For any U (t, τ ) ∈ U t0 , there exist τ 0 > 0, σ > 0, K > 0 such that (H1) Past continuity w.r.t. time : For any U (t, τ ) ∈ U t0 and t ≤ t 0 , there exist some positive constants C 0 , ε 0 and γ such that ε 0 ≤ τ 0 for all τ 0 ≤ r ≤ 2τ 0 , 0 ≤ s ≤ ε 0 and v ∈ O 1 (B), (H2) Past continuity w.r.t. initial data : For any U (t, τ ) ∈ U t0 and t ≤ t 0 , there exists some positive constant C B such that for all v, w ∈ B, for any 0 ≤ r ≤ 2τ 0 , (H3) Past continuity w.r.t. finial time: For any U (t, τ ) ∈ U t0 and t ≤ t 0 , there exist positive constants C 0 and γ such that for all τ 0 ≤ r ≤ 2τ 0 , 0 ≤ s ≤ ε 0 and v ∈ B, (H4) Future continuity w.r.t. initial data: For any U (t, τ ) ∈ U t0 and t > t 0 and D 1 , D 2 bounded subsets of V , there exists a positive constant L(t, D 1 , D 2 ) such that Next, we will verify the above hypotheses step by step. Hypothesis (CP) is proved using following Lemma.
Proof. It is the immediate result of (16) and (43).
Hypothesis (PAS) is proved using the following Lemma.
Hypothesis (H2) is proved using the following Lemma.
Lemma 4.4. Suppose that Assumption 3.1 holds, then for any U (t, τ ) ∈ U t0 and t ≤ t 0 , there exists some positive constant C B such that for all v, w ∈ B, for any 0 ≤ r ≤ 2τ 0 , Proof. By (73), for all t ∈ R, r ≥ 0, v, w ∈ V , we have In particular, for all r ∈ [0, 2τ 0 ], v, w ∈ B, we gain by Lemma 4.2 and the fact that the function L is increasing in all its entries, which implies U (t, τ ) satisfies the Hypothesis (H2) with Hypothesis (H4) is proved using the following Lemma.
Proof. It is obtained immediately from (73) in Lemma 3.3.
The remainder is to prove the Hypotheses (H1) and (H3). It is necessary to improve the assumption on f and get a higher regularity estimate of U (t, τ ) by the Giga-Sohr argument (see [19,27]).