Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations

In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter alpha and consider the difference of behavior between the case alpha in (0,1] and the case alpha=0. The main novel contribution of this paper is to show the continuity of global attractors and exponential attractors with respect to alpha in some sense.


Introduction
In this paper we consider the following models of extensible beams with rotational forces (1 + α(−∆) θ )u tt + ∆ 2 u − M Ω |∇u| 2 dx ∆u = F (x, u, u t , ∆u t ), (1.1) where M is a scalar function, F represents additional damping and forcing terms. When the parameter α > 0 and θ = 1, the rotational inertial momenta of the elements of the beam is taken into account. On the other hand, when α = 0, the kinetic effect of the moment is neglected. The equation (1.1) with α = 0, that is u tt + ∆ 2 u − M Ω |∇u| 2 dx ∆u = F (x, u, u t , ∆u t ), (1.2) has been extensively studied in different contexts since 1950. Woinowsky-Krieger [35] proposed it in the one-dimensional case as a model which describes transverse deflection of an extensible beam, by taking F = 0 and M (s) = as + b, where a, b are positive constants related to the forces applied on the system. See also the papers by Eisley [19], Dickey [15] and Ball [2] for more physical interpretations on extensible beam models. From 1970, other pioneering works related to extensible beams can be found in Ball [3], Dickey [16], Medeiros [28], Brito [6], and Biler [5]. Essentially, these authors studied existence, uniqueness, regularity and stability of solutions. Ever since, several kinds of problems related to these types of equations have been considered by many researchers. Before getting to the main topic, we report a short survey on vibration of extensible beams by pointing out some interesting results on the model (1.2).
Regarding the existence of the decaying solution, Brito [6] established the exponential decaying solutions when F = −δu t (δ > 0). Also, Kouémou Patcheu [22] studied asymptotic behavior of solutions with nonlinear damping F = −g(u t ). Moreover, Vasconcellos and Teixeira [33] and Cavalcanti et al. [7] studied the higher-dimensional cases, by considering nonlinear source and damping terms F = −f (u) − g(u t ) and a nonnegative C 1 -function M . We also refer to the papers [5], [16], [25] and [8], which are related to these works.
With regard to long-time behavior for dynamical systems generated by (1.2), Eden and Milani [17] established the existence of exponential attractors to (1.2) with both hinged and clamped boundary conditions, by taking M to be a given linear function and F = −u t + h(x). Biazutti and Crippa [4] showed the existence of global and exponential attractors to (1.2) with clamped boundary condition, by assuming that M ∈ C 1 ([0, ∞)) is nonnegative and F = −κ(−∆) θ u t + h(x), κ > 0, 0 ≤ θ ≤ 1. Ma and Narciso [26] proved the existence of a global attractor for (1.2) under essentially the same conditions as in [7], with F = −f (u) − g(u t ) + h(x). Also, Yang [36] studied the equation (1.2) with both hinged and clamped boundary conditions, under weaker conditions on the function M and on the nonlinear damping and source terms F = −f (u) − g(u t ) + h(x). He proved the existence of finite-dimensional global and exponential attractors, by assuming that the growth exponent p of the nonlinear source term f (u) is supercritical but is dominated by the growth exponent q of the nonlinear damping g(u t ). Also, Jorge Silva and Narciso [20] showed the existence of a global attractor and an exponential attractor to (1.2) with the supported boundary condition and the initial condition and nonlinear fractional damping term N ( Ω |∇u| 2 dx)(−∆) θ u t (0 ≤ θ ≤ 1). Our study is motivated by their works.
Our main goal in this paper is to study the effect of the fractional rotational inertia on the long-time dynamics. To be more specific and clear, we examine the continuity of attractors when α → 0, that is, we show that the family of global attractors A α,θ and exponential attractors A exp,α,θ are continuous with respect to the parameter α in some sense. Attractors are one of the main objects arising in the study of the asymptotic behavior of infinite-dimensional dissipative dynamical systems and we are able to answer fundamental questions on the properties of limit regimes by studying them. Thus, if attractors A 0.θ and A exp,0,θ are stable with respect to the rotational inertia, we can say that the long-time behavior of each system generated from the problem (1.1) is similar in a sense. We analyze the effect of the fractional rotational term on the long-time dynamics by showing these properties and we present the results in Theorem 2.8.
Before getting into the main subject, we consider the well-posedness and long-time dynamics to the following nonlocal equation related to an extensible beam with nonlinear damping and source terms where Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, 0 ≤ α ≤ 1, 0 ≤ θ ≤ θ ′ ≤ 1, M and N are scalar functions specified later, f (u) is a nonlinear source term and h is an external forcing term. We consider the equation (1.3) with the simply supported boundary condition and initial conditions u(·, 0) = u 0 (·), u t (·, 0) = u 1 (·) in Ω. (1.5) Our concrete aim is further separated into two parts as follows. The first one is to find the bounded set B, which is an absorbing set for all system (H α,θ , S α,θ (t)) generated from the problem (1.3). The second one is to derive an inequality for each system (H α,θ , S α,θ (t)) α∈[0,1] , which is called stability inequality in this paper and leads to a decomposition of a contraction operator part and compact operator part, roughly speaking. These two facts allow us to show the existence of global attractor and construct exponential attractor for each α ≥ 0. The important point here is that we find that the radius of global attractors A α,θ are estimated uniformly with respect α by the first fact, because we show the continuity of the global attractors with respect to α by using the sequentially compactness arguments. To attain these aims, we follow the argument due to [20], where an appropriate perturbation of the energy was introduced. But we need to modify the way of perturbation for handling the rotational inertia. Now we proceed to the consideration for the transition from the case 0 < α ≤ 1 to the case α = 0, but keeping the other conditions on nonlinear terms. In this part we firstly prove two facts to reach our goal. The first one is the continuity of the semiflow with respect to α ∈ [0, 1]. The second one is that the full trajectory which belongs to the global attractors has better regularity than indicated by the topology of the phase space. The second fact is particularly important because compactness arguments in the proof for the upper semi continuity of global attractors is based on the Rellich-Kondrachov compactness theorem. Lastly, we reveal the upper semi continuity for global attractors A α,θ and construct the family of exponential attractors A exp,α,θ , which is continuous at the point α = 0.
The remainder of this paper is organized as follows. In Section 2 we introduce some notations on the function spaces and operators, and state our results. Section 3 is devoted to show that the problem (1.3)-(1.5) is well-posed. In Section 4 we review the basic terminologies and definitions of infinite-dimensional dynamical system and examine the existence of the global attractors and construct the exponential attractors. Finally, Section 5 is dedicated to the proof of the stability with respect to rotational inertia.

Notations and definitions
We first introduce some notation concerning the function spaces and operators that will be used throughout the remainder of this paper.
We denote by L 2 (Ω) the set of square integrable functions with the usual L 2 -inner product (·, ·) and by L p (Ω) the set of p-th power integrable functions with the usual L p (Ω)-norm · p . We set V = H 2 (Ω) ∩ H 1 0 (Ω) with the inner product (∆·, ∆·) and the norm ∆ · 2 . We define the operator A : D(A) → L 2 (Ω); where ∆ is the Laplace operator with the Dirichlet boundary condition. Obviously, A is self-adjoint in D(A) and strictly positive on D(A). Hence as is well-known A has the inverse operator with the domain L 2 (Ω) and it is compact. Thus from the spectral theory there exists an orthonormal basis (ω j ) j∈N in L 2 (Ω) composed by eigenfunctions of A such that Moreover, we can define the fractional powers A s , s ∈ R, of A with domains D(A s ) being Hilbert spaces with the inner products and the norms defined by

and it holds that
In particular, one has D(A 0 ) = L 2 (Ω), D(A 1/4 ) = H 1 0 (Ω), and D(A 1/2 ) = H 2 (Ω) ∩ H 1 0 (Ω) with A 1/2 u = −∆u, u ∈ D(A 1/2 ). Then we can convert the concrete system (1.3) to an abstract evolutional problem given by The long-time dynamics of (2.1) is considered on a Hilbert space H α,θ : and if α = 0, H 0θ is the Hilbert space L 2 (Ω). Now we give the definitions of global attractor, minimal attractor, fractal dimension, exponential attractor and unstable manifold. Definition 2.1. Let X be a complete linear metric space. A bounded set A ⊂ X is said to be a global attractor of the dynamical system (X, S(t)) if and only if the following properties hold: ii. A is uniformly attracting; that is, for all bounded set D ⊂ X Definition 2.2. Let X be a complete linear metric space. A bounded set A min ⊂ X is said to be a minimal attractor of the dynamical system (X, S(t)) if and only if the following properties hold: i. A min is a positively invariant set; that is S(t)A min ⊆ A min , ∀t ≥ 0 ii. A min attracts every point x in X; that is, lim t→∞ dist X (S(t)x, A min ) = 0 for any x ∈ X; iii. A min is minimal; that is, A min has no proper subsets possessing the above properties. Definition 2.3. Let X be a complete linear metric space and K be a compact set in X. The fractal (box-counting) dimension dim f K of K is defined by the formula where, n(K, ǫ) is the minimal number of closed sets of the radius ǫ that cover K.
Definition 2.4. Let X be a complete linear metric space. A compact set A exp ⊂ X is said to be a exponential attractor of the dynamical system (X, S(t)) if and only if A exp is a positively invariant set of finite fractal dimension and for every bounded set D ⊂ X there exists positive constants t D , C D and γ D such that Definition 2.5. Let Y be a subset of the phase space X of the dynamical system (X, S(t)). Then the unstable manifold M u (Y ) emanating from Y is defined as the set of points x ∈ X such that there exists a trajectory γ = {S(t)x = u(t) : t ∈ R} with the properties lim t→−∞ dist(u(t), Y ) = 0.

Stability properties of solutions
The well-posedness of the problem (2.1) provide the family of evolution operators S α,θ (t) : where (u, u t ) is the unique weak solution of (2.1). S α,θ are nonlinear C 0 -semi-groups, and is locally Lipschitz continuous on the phase space H α,θ . Hence, the problem (2.1) generates a dynamical system (H α,θ , S α,θ (t)) and we study the asymptotic behavior of solutions for the problem (2.1) through the dynamical system (H α,θ , S α,θ (t)). Now, we add assumptions for establishing stability properties of solutions: Then we can state the stability properties of solutions as follows: Theorem 2.7. Let us assume that the hypothesis of Theorem 2.6 holds. Besides, we suppose that (H3) and (H4) hold. Then we have i. The dynamical system (H α,θ , S α,θ (t)) generated from the problem (2.1) possesses the global attractor A α,θ ⊂ H α,θ and it is compact and connected.
ii. The global attractor A α,θ is precisely the unstable manifold A α,θ = M u (N ), emanating from the set N consisting of stationary points of S α,θ (t), namely, iii. There exists a global minimal attractor A min to the dynamical system (H α,θ , S α,θ (t)), which is precisely the set of the stationary points, that is, A min = N .

Main Theorem
Theorem 2.8. Let the assumptions of Theorem 2.6 and 2.7 be in force. Then we have i. The family of attractors A α,θ is upper semi-continuous at the point 0, that is, ii. There exist exponential attractors A exp,α,θ for (H α,θ , S α,θ (t)), for which the estimate holds with some exponent 0 < ρ < 1 and constant C > 0.
In the next sections we begin with the proofs of these statements.

Well-posedness for the nonlocal extensible beam equation
Let (ω m ) be the basis in L 2 (Ω), W m the space generated by ω 1 , ..., ω m , and set we consider the following problem: By standard methods in ordinal differential equation, we can prove the existence of C 2 -class solutions to the approximate problem on some interval [0, T m ) and this solution can be extended to the closed interval [0, T ] by using the first energy estimate (3.9) below.

A priori estimates
The First Energy: where the definition of M is given by (2.11). Integrating from 0 to t (≤ T ) we get Now, using Young's inequality with ε = λ1 8 and the condition (2.5) we have Compounding (3.4) with (3.6), we have Going back to (3.4) and combining this uniform boundedness with the easy relation Combining (3.6) with (3.8) we conclude for any t ∈ [0, T ] and m ∈ N, and some constant C 1 > 0 depending on the norm of the initial data in H α,θ .
The Second Energy: Differentiating (3.1) with respect to t and substituting ω = u m tt , it holds that 1 2 , , We shall estimate I 2 , ..., I 5 . First of all, since M and N are C 1 -functions we have from (3.9) that where C 1 is the constant appeared in (3.9). In the following C > 0 denotes various constants which depends on the initial data in H α,θ , but not on T > 0. Using Young's inequality and the self-adjointness of operator A θ (0 ≤ θ ≤ 1), we have Further, utilizing the condition (2.3), generalized Hölder inequality with p 2(p+2) + 1 p+2 + 1 2 = 1, Young's inequality, the estimate (3.9) and the embedding V = H 2 (Ω) ∩ H 1 0 (Ω) ֒→ L p+2 (Ω), we infer It is easy to see that Using these five estimates in (3.10), there exists a constant C > 0 such that (3.11) The estimates (3.9), (3.11) are sufficient to pass the limit in the approximate equation (3.1) to obtain a strong solution satisfying (3.1) and

Weak solutions
For each regular initial data (u k 0 , u k 1 ) there exists a strong solution u k (t) satisfying the estimate (3.9). Furthermore, the difference of strong solutions w(t) := u k (t) − u l (t) satisfies the estimate for all t ∈ [0, T ] and some positive constant C = C( (u 0 , u 1 ) H α,θ , T ). This implies that We omit the details of estimate (3.12) here, because they are identical to that concerning the continuous dependence presented in the below. These estimates are enough to conclude that the limit of approximate solutions satisfy (2.7) and the following weak formulation: (3.14)

Uniqueness of strong and weak solutions
Let z = (u, u t ) andz = (ũ,ũ t ) be two (strong or weak) solutions corresponding to the initial data z 0 = (u 0 , u 1 ) andz 0 = (ũ 0 ,ũ 1 ), respectively. Putting w = u −ũ, we see that the function (w, w t ) = z −z verifies with the initial data (w(0), w t (0)) = z 0 −z 0 , in the strong or weak sense. We first deal with strong solutions. Substituting ω = w t (t) in (3.15), we have . (3.20) Since N (τ ) > 0, the estimate (3.9) implies that N ( Next we estimate J 1 , J 2 , J 3 , and J 4 . Analogously to the estimate of I 2 we have In the next two estimates we shall use the mean value theorem, Young's inequality and the estimate (3.9). Then we get for any ǫ > 0 and From the condition (2.3), we can immediately see that there exists a constant σ 0 > 0 such that and we can estimate J 4 likewise I 5 Using these four estimates in (3.21) and taking ǫ > 0 small enough, there exists a constant C > 0 such that for any t ∈ [0, T ]. From the estimate (3.9) the function 1 + A θ ′ /4 u t (·) 2 2 is integrable on [0, T ]. Then integrating (3.23) on [0, t] and using Gronwall's inequality, we arrive at for some positive constant C T = C T ( (u 0 , u 1 ) H1,1 ). (3.24) shows the continuous dependence of strong solutions on the initial data in H α,θ . The same conclusion holds for weak solutions by using the density argument. In fact, if we consider the initial data z 0 = (u 0 , u 1 ),z 0 = (ũ 0 ,ũ 1 ) ∈ H α,θ , then similarly to (3.13) there exist sequences of strong solutions z k = (u k , u k t ) andz k = (ũ k ,ũ k t ) such that The difference z k −z k := (w k , w k t ) satisfies (3.24) for each k ∈ N, hence the estimate (2.8) holds for the difference of weak solutions z −z after passing the limit as k → ∞.
Particularly, we have uniqueness of both strong and weak solutions. This completes the proof of Theorem 2.6.

Stability and long-time dynamics
In this section we show the existence of attractors and clarifying their properties. First of all, we introduce a couple of notions of infinite-dimensional dynamical systems.

Review on dynamical system
Here, we introduce some important concepts we shall need for proving the statement on stability properties:  iii. Let κ be the Kuratowski measure on X. The semiflow S is said to be uniformly κ-contracting if and only if there is a τ ≥ 0 and a nonnegative function φ(t) with φ(t) → 0, as t → ∞, such that for every bounded set B ⊂ X one has κ(S(t)B) ≤ φ(t)κ(B), for all t > τ . i. The continuous functional Φ defined on B is said to be the Lyapunov function for the dynamical system (X, S(t)) on B if and only if the function t → Φ(S(t)z) is a non-increasing function for any z ∈ B.
ii. The Lyapunov function Φ is said to be strict on B if and only if for z ∈ B, the equation Φ(S(t)z) = Φ(z) for all t > 0 implies that S(t)z = z for all t > 0; that is, z is a stationary point of (X, S(t)).
iii. The dynamical system (X, S(t)) is said to be gradient if and only if there exists a strict Lyapunov function for (X, S(t)) on the whole phase space X.
We note the properties of κ-measure in Proposition 4.4 below, which will be used later on (for the details we refer to [13]). ii vi. If B t is a family of nonempty, closed, bounded sets defined for t > 0 that satisfy B s ⊃ B t whenever (0 ≤)s ≤ t, and κ(B t ) → 0, as t → ∞, then ∩ t>0 B t is a nonempty, compact set in X.
We show the existence of attractors and their structure by using the following criteria: Proposition 4.5. (See [13]) Suppose that the dynamical system (X, S(t)) possesses the following properties: i. The dynamical system (X, S(t)) is dissipative.
ii. The semiflow S is uniformly κ-contracting.
Then the system (X, S(t)) possesses the global attractor. ii. A min = N , where A min is the minimal attractor of the dynamical system (X, S(t)).

Global attractor and minimal attractor
Let us show the existence of the global attractor of (H α,θ , S α,θ (t)) according to the Proposition 4.5.
In the following we use C 0 , C 1 , C 2 to denote several positive constants appearing in the estimates. Firstly, we claim that there exists a constant C > 0 such that In fact, taking derivative of function Ψ α (t), using the weak formulation (3.1), adding and subtracting E α (t) into the resulting expression, we obtain where Now we estimate L 1 , L 2 and L 3 . From the condition (2.11) and embedding D(A 1/2 ) ֒→ D(A 1/4 ), we get Using Young's inequality and the embedding D(A 1/2 ) ֒→ D(A 1/4 ), we obtain From the condition (2.10) we see that Inserting these last three estimates in (4.5) and using the embedding D(A θ ′ /4 ) ֒→ L 2 (Ω), then we get (4.4).
Remark 4.8. From Proposition 4.7, we immediately see that is absorbing set of the system (H α,θ , S α,θ (t)), where δ ′ is an arbitrary positive constant and K 2 = K 2 (h, l 0 , l 1 , l 2 , |Ω|). We can see easily that the system (H α,θ , S α,θ (t)) also has a bounded positively invariant absorbing set B α,θ described as follows where the t B α,θ > 0 is the time such that B α,θ absorbs itself. The definition of the norm · H α,θ implies the inclusion relation such that From this inclusion relation, any ω-limit set of B α,θ is included B 0 : This implies that if the ω-limit set is nonempty, as mentioned above the each system (H α,θ , S α,θ (t)) has compact global attractor A α,θ and attractors are included in the bounded set B 0 . In particular, if (u, u t ) is a solution of the problem (2.1) corresponding to initial data lying in ω(B α,θ ), then it is globally bounded in H 0.θ ; that is The right-hand side of (4.10) does not depend on α, and this fact will play the key roll in the next section.

Existence of the attractors and its structures
There remains to prove the contractivity. For showing this property, we derive the following important inequality, which is said to be a stability inequality.
Proposition 4.9. Let the assumption of Theorem 2.6 be in force. Given a bounded set B ⊂ H α,θ we consider two weak solutions z 1 = (u, u t ), z 2 = (v, v t ) corresponding to initial data z 1 0 = (u 0 , u 1 ), z 2 0 = (v 0 , v 1 ) lying in B. Then the following stability inequality holds: Proof. First of all we fix a bounded set B ⊂ H α,θ and consider two weak solutions where R > 0 depends on the size of B. Putting the difference z 1 − z 2 = (w, w t ) and proceeding exactly as in the proof of the a priori estimates we get the following inequality we can rewrite (4.12) as follows: where we set F α,θ (t) : (4.14) and Next we shall estimate the right-hand side of (4.13). To simplify the notation we use C R to denote various positive constants depending on R > 0, but not on time. Firstly, since M, N ∈ C 1 ([0, ∞)), then the estimate (3.9) implies Likewise the proof of the a priori estimate, applying the mean value theorem, Young's inequality with ǫ > 0, the estimate (3.9) and the embedding D(A 1/2 ) ֒→ D(A 1/4 ) ֒→ L 2 (Ω), we obtain for any ǫ > 0 and Using the same conditions in the proof of the uniqueness, we have p+2 . In addition, from the global estimate (3.9), we get immediately 2 2 . Substituting these four estimates in (4.13) and choosing ǫ > 0 small enough, we see that there exists constants N R , C R > 0 such that Now we define the functional where η > 0 will be fixed later. We first show that there exists s constant C R > 0 such that Indeed, taking derivative of Φ α,θ (t), using the weak formulation for w, adding and subtracting F α,θ (t) in the resulting expression and neglecting unnecessary terms, we arrive at where , (v t (t), A θ ′ /2 w(t)), , w(t)).
First we estimate K 1 . From the estimate (3.9) and the uniform boundedness of N , Young's inequality with ǫ > 0 and the embedding D(A 1/2 ) ֒→ D(A 1/4 ) we get Proceeding almost the same way as J 1 , J 2 , and J 4 , but replacing the function w t by w, we derive for any ǫ > 0 and some positive constant C R . Going back to (4.18) and inserting these four estimates, we result that (4.17) holds, after choosing ǫ > 0 small enough and using the embedding D(A θ ′ /4 ) ֒→ L 2 (Ω).
Combining (4.16) with (4.17), noting that A 1/2 w(t) 2 2 ≤ F α,θ (t), and taking η > 0 small enough such that η < NR CR , there exists a constant C R > 0 such that for all t > 0 and η < NR CR . On the other hand, by taking C 1 := max{1, 1/λ 1 } > 0, it is readily to see that (4.20) Then taking and fixing η > 0 such that η ≤ min{ 1 2C1 , NR CR }, (4.20) implies that Compounding (4.19) with (4.21) we get where we define Applying Gronwall's inequality, we deduce that Moreover, from the estimate (3.9) we also have for some positive constantC R . Thus (4.23) leads to for all t > 0, and some constant C R > 0. Lastly, from (4.14) we note that there exists a constant C R > 0 such that Therefore, combining (4.21) with (4.24)-(4.25), we get the conclusion that the stability inequality (4.11) holds true for some constants C R > 0 and δ R > 0. The proof of Proposition 4.9 is complete. Now we give the proof of the contractivity of the semiflow S α utilizing the stability inequality (4.11): We shall show the following estimate: where φ T is a pseudometric on H α,θ and it satisfies the following property for each T > 0; that is, any bounded sequence in H α,θ has a subsequence which is a Cauchy sequence with respect to φ T .
As a result one has If z 1 , z 2 ∈ B i ∩ D T j , then (4.26) implies that Using the property iii of κ-measure, we have Since this inequality is valid for every ǫ, ǫ ′ > 0, we can send ǫ, ǫ → 0 to obtain κ(S α,θ (T )B) ≤ C B e −δT κ(B), for all T > 0. This inequality draws the conclusion of the contractivity of κ-measure. Our aim, from now on, is to estimate the right-hand side of (4.11) so as to satisfies (4.26). From the interpolation inequality and the estimate (3.9), From Nirenberg-Gagliardo's inequality and the estimate (3.9) we have where ϑ = n 4 1 − 2 p+2 . Taking Θ = min{1/2, 1 − ϑ}, and noting that u i (t) 2 (i = 1, 2) is uniformly bounded, there exists a constant C B > 0 such that (4.29) Using (4.29) we can rewrite (4.11) as follows: Then we can find that φ T satisfies the desire property. Indeed, it is clear that φ T is a pseudometric. Also, given a sequence of initial data z i ∈ B, we write S α,θ (t)z i = (u i (t), u i t (t)). Since B is bounded and the estimate (3.9), the sequence (u i ) i∈N = {u i (τ ) | t ∈ [0, T ] i ∈ N} is uniformly bounded on H α and equicontinuous. Compound these properties with the compactness of the embedding D(A 1/2 ) ֒→ L 2 (Ω), there exists a subsequence still denoted by (u i ) such that (u i ) converges strongly in C([0, T ], L 2 (Ω)), T > 0.
Therefore this completes the proof.
We conclude this section with comment on the structure of the global attractor and minimal attractor. It is easy to check that the energy E α,θ is a Lyapunov function for the dynamical system (H α,θ , S α,θ (t)) and we can immediately see that (H α,θ , S α,θ (t)) is gradient. Thus we are able to get the conclusions on the structures of the global attractor and minimal attractor from the Proposition 4.6.

Exponential attractor
Lastly, we consider the existence of exponential attractors in this subsection in brief. We note that the fractal dimension dim f K of a compact set K can be represented by the formula where, n(K, ǫ) is the minimal number of closed sets of the radius ǫ that cover K.
Lemma 4.10. Let the assumption of Theorem 2.6 be in force. Then the mapping t → S α,θ (t)z is Hölder continuous in H −s α,θ for every z ∈ B α,θ , where 0 < s ≤ 1. Proof. We first prove the assertion for the case s = 1. For z 0 = (u 0 , u 1 ) ∈ B α,θ we can see the from (2.7) and (3.1) that for every t 1 , t 2 ∈ [0, T ], that is, t → S α,θ (t)z 0 is Hölder continuous in H −1 α,θ . Now we consider the case 0 < s < 1. Applying the interpolation theorem in each component of H −s α,θ , using the uniform boundedness of the solutions and the Hölder continuity in H −1 α,θ , we get 5 Stability of attractors with respect to the rotational inertia 5.1 Upper semicontinuity of the global attractors A α when α → 0 In order to show the upper semicontinuity of attractors, we use the following criterion: Proposition 5.1. (See [13]) Assume that a dynamical system (X α , S α (t)) possesses a compact global attractor A α for every α ∈ [0, 1]. Assume that the following conditions hold: • There exists a compact set K ⊂ X such that A α ⊂ K for all α ∈ [0, 1].
Then the family of attractors A α is upper semicontinuous at the point α = 0, that is, In the remainder of this section we check that the system (H α,θ , S α,θ (t)) satisfies the condition of this proposition. for every t ∈ R and for every couple of full trajectories γ andγ taken from the attractor. Now we fix a trajectory γ and we consider the shifted trajectory γ h := {z(t + h) | t ∈ R} for 0 < |h| < 1. Applying (5.5) for this pair of trajectories γ and γ h , we get where z h (t) = (u h (t), u h t (t)), u h (t) = {u(t + h) − u(t)} · h −1 . Let us estimate the right-hand side of (5.6). From the interpolation inequality, Sovolev's embedding, and Young's inequality, From the global estimate (3.9) of the solution we have Taking ǫ > 0 so small that ǫC Aα ≤ sup −∞≤τ ≤t u h (τ ) 2 2 , we get Using the mean value theorem, we have Thus passing with the limit on h → 0 in (5.6), we obtain A 1/2 u t (t) 2 2 + u tt (t) 2 2 + α A θ/4 u tt (t) 2 2 ≤ C A α,θ .
From Remark 4.8, the constant C A α,θ only depend on the radius of B 0 and does not depend on α.
Finally, it is easy to show that A α,θ lies in D(A) × D(A 1/2 ). Actually, in the weak formulation we substitute Au(t) for ω, we have Au(t) 2 ≤ C B0 .
Therefore, we get the conclusion on the regularity of trajectories from the attractor.
From the results of this section, we can conclude the upper semicontinuity of A α,θ . Indeed, from Lemma 5.3 every attractor A α,θ is included in a bounded set K ⊂ D(A) × D(A 1/2 ), that is, Au α 2 2 + A 1/2 v α 2 2 ≤ C, where C does not depend on α and t. Since D(A)×D(A 1/2 ) is compactly embedded into D(A 1/2 )×L 2 (Ω), the set K is compact in D(A 1/2 )×L 2 (Ω). Thus we are able to obtain the conclusion from the Proposition 5.1.

Continuity of exponential attractors when α → 0
At the end of this section, we refer to the continuous of exponential attractors on parameter α. Before getting into the proof, we note the properties of the system we shall use here: • B is a bounded absorbing set for every system (H α,θ , S α,θ (t)).
Proof. Firstly, we recall the proofs for the contractivity of the flow S α and construction of discrete exponential attractor. In what follows, we use the same notations in the subsection 4.3. In a similar way of the proof, we can construct the cover of the set B as follow: where D T j,α ≡ D T j,α (u j ) = {z ∈ B : u − u j C([0,T ];L 2 (Ω)) < ǫ, u is the first component of S α (·)z}.
Then, we can get the cover for B such as and we can construct a discrete exponential attractor for the system (B, V α ): where E r α is the set of an element of covering balls of V r α B: E r α = V r α z r i,j,l : 1 ≤ i ≤ n(B, κ(B)), 1 ≤ j ≤ m r,ν , z r i,j,l ∈ B i ∩ D r α,j ∩ D r l (we note that we can take z r i,j,l from H by the density argument). Then the discrete exponential attractor satisfies the conditions H(A * exp,0 , A * exp .α ) ≤ Cα − log q cT −log q (deriving this estimate is almost the same as the proof in the paper [9], thus we omit the detail here). It is easy to check the same condition holds on exponential attractors {A exp,α } 0≤α≤1 . Therefore we obtain the conclusion H(A exp .α , A exp .0 ) ≤ Cα ρ with ρ = − log q cT −log q .