ON RELATION BETWEEN ATTRACTORS FOR SINGLE AND MULTIVALUED SEMIFLOWS FOR A CERTAIN CLASS OF PDES

. Sometimes it is not possible to prove the uniqueness of the weak solutions for problems of mathematical physics, but it is possible to bootstrap their regularity to the regularity of strong solutions which are unique. In this paper we formulate an abstract setting for such class of problems and we provide the conditions under which the global attractors for both strong and weak solutions coincide and the fractal dimension of the common attractor is ﬁnite. We present two problems belonging to this class: planar Rayleigh– B´enard ﬂow of thermomicropolar ﬂuid and surface quasigeostrophic equation on torus.

1. Introduction. In recent years a lot of effort has been put to the study of global attractors for problems without uniqueness of solutions. Several theories have been developed: theory of generalized semiflows [4], multivalued semiflows [29,30], trajectory attractors [7], and of evolutionary systems [9]. The comparison of the first three of these theories with each other was done in [6,24]. Within all these frameworks a large number of results has been obtained on the existence of global attractors, and their upper semicontinuous dependence on the problem data. Fairly recent overview of key results and open problems has appeared in the review paper [3].
While the question of the global attractor finite dimensionality has been studied very thoroughly for the problems where the solutions are unique, there are very few results available for the situation without the solution uniqueness. In fact, Balibrea, Caraballo, Kloeden, and Valero in their review article [3,Section 4 In this paper we define a class of problems where the weak solution can possibly be nonunique and hence those weak solutions must be described by means of a multivalued theory. Moreover, we assume that in the considered class of problems we can bootstrap the regularity of weak solutions to the regularity of strong solutions, and we have the weak-strong uniqueness. In such case the attractor for the strong solutions can be described using the single-valued theory and we prove that the global attractors for weak and strong solutions coincide.

.3 B)] state that
We illustrate the abstract scheme by two problems: Rayleigh-Bénard problem in two dimensions for thermomicropolar fluids and surface quasigeostrophic equation on torus. To show that the dimension of the attractor-for both problems-is finite we use the uniqueness of the strong solutions and a standard semigroup approach.
We remark here that for the class of problems that we consider, the dynamics on the global attractor is in fact single-valued, and hence the general question of attractor dimension for multivalued systems remains unanswered (there is a strong evidence that the dimension can be in fact infinite [1]). In this sense our examples are similar to that of [39] and give only a partial answer to the still open and difficult question of the finiteness of the fractal dimension for problems with multivalued dynamics on the attractor.
2.1. Existence and relations between attractors for strong and weak solutions. We assume that V and H are two Banach spaces such that V ⊂ H with a continuous embedding. By P(H) (respectively, B(H)) we denote the family of nonempty (respectively, nonempty and bounded) subsets of H. Similar notation is used for V. The Hausdorff semidistance in the Banach space X, where either X = H of X = V is denoted by We consider two dynamical systems: If, in definition of a multivalued semiflow, we replace the inclusion with the equality, i.e. we require that S H (s + t)x = S H (s)S H (t)x, then we say that the m-semiflow is strict. Note, however, that in the present paper we never make any assumptions on the m-semiflow strictness. In applications the multivalued semiflow S H is defined by weak solutions of a given problem for which we do not know the uniqueness, and the semigroup S V is given by the strong solutions which are known to be unique. To establish the existence and relations between attractors for strong and weak solutions we make the following three assumptions.
(i) The semigroup of weak solutions has a compact absorbing set in V, i.e. there exists a set B 0 ⊂ V compact in V such that for any B ∈ B(H) there exists Note that under assumptions (i), (ii) and (iii), the hypotheses of the above lemma are satisfied, and hence these hypotheses guarantee the existence of the set A V -the global attractor for {S V (t)} t≥0 . The proof of the next result can be found in [23,Theorem 9.4] for the case of thermomicropolar fluids. Here, we provide the abstract version of this theorem.

Proof. As
Hence dist H (A V , A H ) = 0, and, as both sets are closed in H it follows that A V ⊂ A H . We will show that A V is attracting with respect to S H (t). Then A H ⊂ A V , since A V is compact in H and A H is the smallest compact attracting set. We take B ∈ B(H), t ≥ t 0 (B) and compute where we have used (i) and (iii). As B 0 is bounded in V, we can pass with t to ∞ and the right-hand side of the last expression tends to zero. Hence and the assertion is proved.
Finally, for B ∈ B(H) and t ≥ t 0 (B) and the right-hand side converges to zero, which concludes the proof of the theorem.
In applications, instead of verifying the assumption (i), it will be more convenient to verify the following two assumptions. (iv) We can bootstrap the regularity of weak solutions, i.e. for any B ∈ B(H) there is > 0 such that S H ( )B ∈ B(V). (v) There exists a compact absorbing set for strong solutions, i.e. there exists Lemma 2.4. If the semiflows S H and S V satisfy (iii), (iv), and (v), then S H satisfies (i).
Proof. Take B ∈ B(H). Let = (B) be as in (iv). We take t > and get Denote B 1 = S H ( )B and take t 0 = t 0 (B 1 ) from (v). We have The assertion is proved.
The following result is a straightforward consequence of Theorem 2.3 and Lemma 2.4.

2.2.
Comparison with the theory of bi-space attractors. The idea of bi-space attractors originates from the work of Babin and Vishik [2] who define them in the following way.
Definition 2.6. Let H be a Banach space and let {S H (t)} t≥0 be a semigroup of operators. Let V be another Banach space. A set A is called an (H, V) global attractor of the semigroup {S H (t)} t≥0 is the following properties hold: Note that we always assume that V ⊂ H, but in Babin and Vishik's theory this embedding is not required. In fact, it may even hold that H ⊂ V or neither of these spaces needs to be included in another one [11,10]. We relax the above definition by allowing the semigroup to be multivalued. Using the above definition we can reformulate Theorem 2.3 in the language of bi-space attractors.
Theorem 2.8. Let V, H be two Banach spaces such that V ⊂ H with a continuous embedding. Let {S H (t)} t≥0 be a multivalued semigroup of operators on H. Define S V (t) = S H (t)| V . If the following assumptions hold: • for every t ≥ 0 the operator S V (t) : V → V is single valued, • the multivalued semigroup {S H (t)} t≥0 has an absorbing set in B 0 which is compact in V, • the mappings S V (t) are continuous on B 0 for every t ≥ 0, then {S H (t)} t≥0 has the (H, V) global attractor.
2.3. Fractal dimension. We will provide the sufficient condition which guarantees that the fractal dimension in V (and thus also in H) of the obtained global attractor A is finite. To this end we reinforce the previous assumptions with the following one.
There exists T > 0 such that (vi) The mapping S V (T ) is Lipschitz in the space V on the attractor, i.e.
(vii) The mapping S V (T ) satisfies the squeezing condition in the space V on the attractor, i.e. V is a Hilbert space and there exists the orthogonal projection P leading from V into its finite dimensional subspace and a constant δ ∈ (0, 1) such that for every u, v ∈ A either Under the stated assumptions the attractor A is a finite dimensional subset of the space V, see [19,Chapter 2] where a stronger result is obtained, namely it is proved that the squeezing condition guarantees the existence of a so called exponential attractor, or [12,Theorem 2.15], where the attractor finite dimensionality has been obtained with (vii) replaced by a weaker assumption valid not only in Hilbert spaces, but also in Banach spaces. The finite dimensionality result of [12,19] is stated in the following lemma.
with N V (A, ) being the minimal cardinality of covering of the compact set A by the closed balls in V of radius ε.
Observe that, as V ⊂ H, then , and hence we deduce, that under assumptions of the above lemma also the fractal dimension in H of the global attractor for the m-semiflow {S H (t)} t≥0 is finite. We will use Lemma 2.9 in Section 3 to demonstrate that the fractal dimension of the global attractor for the problem considered there is finite.
3. Rayleigh-Bénard problem for thermomicropolar fluids. Micropolar fluids are fluids that consist of particles which undergo the intrinsic spin independent of the motion of fluid itself. The constitutive equations of these fluids were introduced in [20]. For the mathematical investigation of micropolar fluid model, see [26,27,28].
Study of thermal processes in those fluids leads to the so-called thermomicropolar fluid model, introduced in [21]. A simplified version of this model was studied in [35,36], and, recently, in [22], cf. also Straughan et al. [31,34].
In this section we consider the Rayleigh-Bénard problem of a planar flow of thermomicropolar fluids. The existence of strong and weak solutions for the problem, the relation between these two classes of solutions as well as the existence of the global attractor have been recently proved in [23]. In particular, in [23] we have proved, using the single valued theory, that for the problem under consideration there exists the global attractor for the strong solutions, and, using the multivalued theory, that there exists the global attractor for the weak solutions. We have also shown in [23] that both attractors coincide. We recall these results in Subsection 3.2. The new results are contained in Subsection 3.3, where we show that the obtained global attractor has finite fractal dimension. We use the method based on the squeezing condition, cf. [19,38], to show the finite dimensionality of global attractor for the strong solutions, and since both attractors coincide, we deduce the finite dimensionality of the global attractor for the weak solutions.
We will formulate the problem in dimensionless variables. In its classical formulation the problem is governed by the following system of equations, cf. [23] 1 Pr together with boundary conditions and initial conditions The physical properties of the fluid are described by six positive constants: the Rayleigh number Ra, the Prandtl number Pr, the micropolar constants K, L, M , and the thermomicropolar constant D, see [23] for some discussion. We note here that the key difficulty in the study of the above system of equations comes from the term Drot ω · ∇θ in equation (4). Because of this term it appears impossible to get the weak solution uniqueness even in two space dimensions.

Throughout this section we will denote
and κ 2 = LPr M for short. We equip the Hilbert space H with the following scalar product, for The associated norm will be denoted by | · |. We note here that the fact that the weights 1/κ 1 and 1/κ 2 appear in the definition of the norm | · | will be of key importance in the proof of the squeezing condition in step 2 in Section 3.3 below. The Hilbert space V will be equipped with the following scalar product, for The associated norm will be denoted by · .
We define the standard trilinear forms (see [33] or [37]) Let A be the minus Laplace operator associated with boundary conditions (5)-(6), The eigenvectors {w k } k≥1 ⊂ D(A) of A form the orthonormal basis of H and We introduce the fractional power of the Laplacian, see [33,Chapters 3 and 6]. Let To make D(A 3/2 ) a Hilbert space we equip it with the inner product which gives the corresponding norm equivalent to the norm of H 3 (Ω).
Now, we introduce the Stokes operator A S , see [33] or [37] for details. Let It is known that A S = −Π∆, where Π stands for the Helmholtz-Leray projector from L 2 (Ω) 2 onto H S . Similar as in the case of the Laplace operator, the eigenfunctions {v k } ∞ k=1 of the operator A S given by form the orthonormal basis of H S and they are smooth. Using these eigenfunctions it is possible to define, similar as for the Laplace operator, S ) is a Hilbert space equipped with the following inner product and norm, equivalent to that of H 3 (Ω) 2 , S u|. We are in position to define weak and strong solutions for the formulated problem. We also recall the results on their existence obtained in [23].
1 Pr for every η ∈ D(A 3/2 ), in the sense of scalar distributions on (0, τ ). Additionally, we require that |θ(·)| is upper-semicontinuous from the right at t = 0 Existence of the weak solution satisfying the above definition has been established in [23,Theorem 5.1].
By a strong solution of problem (1)- (7) we mean a triple of functions (u, ω, θ), for every η ∈ H, in the sense of scalar distributions on (0, τ ).
Existence and uniqueness of strong solutions satisfying the above definition has been proved in [23, Theorem 7.2].

Attractors for single and multivalued semiflows and their equality.
In [23] we proved that weak and strong solutions of problem (1)-(7) given by Definitions 3.1 and 3.2, respectively, exist and that their corresponding attractors coincide. Our results show, in particular, that abstract assumptions (i), (ii), (iii) from Section 2.1 hold in this case. We recall the results of [23] in Lemmas 3.3, 3.4, 3.5, 3.6, and 3.7.
First we have to define the corresponding semiflows. The family of multivalued maps weak solution of problem (1)- (7) with the initial data (u 0 , ω 0 , θ 0 ) in H, given by Definition 3.1} We also define the family of single valued operators is a unique strong solution of problem (1)- (7) with the initial data (u 0 , ω 0 , θ 0 ) in V, given by Definition 3.2} We remind the following two Lemmas.  We shall use Theorem 2.3 to prove existence and equality of global attractors for weak and strong solutions. Thus, we have to show that {S H (t)} t≥0 and {S V (t)} t≥0 satisfy hypotheses (i), (ii), and (iii) from Section 2.1. Below we recall respective lemmas from [23].
In consequence, the hypothesis (i) is satisfied.
Continuity of S V (t) on the absorbing ball B 0 also follows from the a priori estimates proved in [23]. In fact we also have the Lipschitz continuity on B 0 which we establish in the following Lemma.
In consequence, assumptions (ii) and (vi) of Section 2.1 are satisfied.
Proof. A priori estimates which use the uniform Gronwall lemma proved in [23,Lemma 8.3] imply the existence of the absorbing ball The same argument implies that if the initial data belongs to the ball B 0 , then the following bounds hold The assertion follows from these bounds by using the estimates on the difference of two strong solutions obtained in [23, Theorem 6.4, Theorem 7.2].
Finally, we have: [23, Theorem 7.2] Each strong solution given by Definition 3.2 is also a weak solution given by Definition 3.1. Moreover, if (u 0 , ω 0 , θ 0 ) ∈ V then the strong solution with the initial data (u 0 , ω 0 , θ 0 ) is unique also in the class of weak solutions. In consequence, the hypothesis (iii) is satisfied.
Using Lemmas 3.5, 3.6, and 3.7 it follows that all assumptions of Theorem 2.3 hold, whence we get the following result.

3.3.
Finite dimensionality of the global attractor. The attractor finite dimensionality will follow from Lemma 2.9. Since we have already established hypothesis (vi) in Lemma 3.6, it suffices to prove that the squeezing condition given in hypothesis (vii) holds. The proof of the latter consists of two steps. In the first step we establish an additional result on the attractor regularity. We need to show that the functions ω(t) and θ(t) are uniformly bounded in D(A 3/2 ) on the attractor. This result will be used in the second step of the proof to show the squeezing condition by the method of [19,38]. To use this method we need to appropriately define both the linear operator A : D(A) ⊂ H → H and the scalar product on the space H. We have already defined the scalar product on H by the formula (8), and the operator A : D(A) ⊂ H → H will be defined by the formula (34) below.
First, we take the Laplacian of each term of (3) to get Using the formula and substituting η = ∆ω in (20) we get the equation We multiply (21) by η and integrate over Ω, Now, we estimate each term on the right-hand side by using (18), Hence, after choosing a suitable > 0, we get Integrating the above inequality from t ≥ 0 to t + 1 and obtain where we have used |η(t)| = |Aω(t)| and (18). Now, we take the Laplacian of each term of (1) and get 1 Pr For i = 1, 2, we use the formula, and set ξ = −A S u in (24) to get, after taking the scalar product with ξ and integrating over Ω 1 2Pr We proceed with the above equation in a similar way as with equation (22) and we end up with the estimate valid for every t ≥ 0. Now, we multiply (21) by Aη and integrate over Ω,

PIOTR KALITA, GRZEGORZ LUKASZEWICZ AND JAKUB SIEMIANOWSKI
We estimate the terms on the right-hand side as follows, whence, and by (27), we obtain Inequalities (26) and (23) enable us to use the uniform Gronwall lemma which yields We recall that η = |A 3/2 ω|, hence In particular we have proved that ω 1 D(A 3/2 ) ≤ C holds for every (u 1 , ω 1 , θ 1 ) ∈ A.
Then, we use the formulas and set ζ = ∆θ in the above evolution equation to obtain rot (∂ xi ω) · ∇(∂ xi θ) We multiply the above equation by ζ and integrate over Ω to get We estimate the terms on the right-hand side of the above equation using (18) and the already established uniform estimate of ω(t) in D(A 3/2 ), cf. (29). We get, respectively, The above estimates and (31) yield the bound for a suitable constant C. We integrate this bound from t ≥ 0 to t + 1 to get t+1 t ζ(s) 2 ds ≤ C.
In the final step we multiply (30) by Aζ and integrate over Ω, whence We estimate the terms on the right-hand side of the above equation in the following way, Using the above estimates in (33), we obtain d dt ζ 2 ≤ C ξ 2 + ζ 2 + 1 .
In view of (26) and (32), and by the uniform Gronwall lemma, we get which completes the proof of (19) and thus of the theorem.
Step 2. We are about to show that the semiflow S V satisfies the squeezing condition. We denote by P m S : V S → V S the orthogonal projection onto the subspace spanned by the first m eigenvectors of the Stokes operator, i.e.
and by P k : V → V the orthogonal projection onto the subspace spanned by the first k eigenvectors of the Laplace operator A, i.e.
Let A be the differential operator acting from for (u, ω, θ) ∈ D(A). 2 Clearly, A is positive self-adjoint operator. We can relabel the countable set and (e j ) is the orthonormal basis in H consisting of eigenvectors of A and satisfying Ae j = λ j e j , where (λ j ) is the increasing sequence of eigenvalues of A. Then we have V = D(A 1/2 ) and, for (u i , ω i , θ i ) ∈ V we have Thus, we have e j 2 = A 1/2 e j , A 1/2 e j = (Ae j , e j ) = λ j |e j | 2 = λ j , since each e j is normalized in H. We consider the orthogonal projection P n in V onto the first n eigenvectors of A, i.e. P n (u, ω, θ) = n j=1 ((u, ω, θ), e j )e j , for (u, ω, θ) ∈ V.
Observe that, for every (u, ω, θ) ∈ V and n ≥ 1, there are nonnegative integers m, k and l such that n = m + k + l and P n (u, ω, θ) = P m S u, P k ω, P l θ . In the sequel we will simply write P = P n , P S = P m S , P = P k , and P = P l , avoiding the superindexes for the sake of the ease of notation. Note that it is always clear from the context which projection P is considered.
We take the scalar product of the above equation and A S P m S u = A S P S u, and integrate over Ω 1 2Pr We end with 1 2Pr We end with 1 2 for a suitable constant c > 0. If we take the inner product of (36) and A S (I − P S )u, and integrate over Ω then the similar estimates yield (38) Now, we proceed as above with the equation for ω M Pr We multiply the above equation by AP k ω = AP ω and integrate over Ω M 2Pr We estimate as above We end with 1 2 for a suitable constant c > 0. If we replace AP ω by A(I − P )ω and repeat the calculations we get 1 2 Lastly, we test the equation for θ d dt with the function AP l θ = AP θ to obtain We estimate as follows Note that the third and fourth of the above estimates were the reason why we needed to derive in Step 1 the estimates (19). This yields We take n, m, k and l such that n = m + k + l and P = P n = P m S ⊕ P k ⊕ P l . Let us denote p = P(u, ω, θ) and q = (I − P)(u, ω, θ) . By the definition of scalar product we have Similarly, we add (38). (40) and (42) and, modifying c > 0 if necessary, we obtain 1 2 d dt q 2 ≤ −|A(I − P)(u, ω, θ)| 2 + c|A(I − P)(u, ω, θ)| (u, ω, θ) .
If (1) is satisfied then, from (45) and (46), we get This yields and appropriate choice of n ends the argument.
4. Surface quasigeostrophic equation on torus. The second problem from mathematical physics which will serve as an example for our abstract framework will be the forced and critically damped surface quasigeostrophic (SQG) equation on two-dimensional torus T 2 . The equation models the temperature θ on the boundary of a rapidly rotating half space with small Rosby and Eckmann numbers and constant potential vorticity, see [14]. We consider the vanishing viscosity weak solutions, namely, the class of the weak solutions being the limits of the vanishing viscosity approximations. For this class of weak solutions, although their uniqueness is unknown, Cafarelli and Vasseur [5] have proved that the multivalued semigroup has the regularizing effect (cf. also [25]) and that using the De Giorgi iteration method it is possible to bootstrap the regularity from L 2 (T 2 ) to L ∞ (T 2 ). Existence of the global attractor for the multivalued semiflow governed by the viscosity weak solutions have been proved by Cheskidov and Dai [8] using the formalism of evolutionary systems developed by Cheskidov and Foiaş [9]. On the other hand, the method of nonlinear lower bounds developed by Constantin and Vicol [16] turned out useful to prove the existence of the global attractor for the unique strong solutions [15,13]. In [18] it was proved that the global attractor for weak solutions obtained in [8] coincides with the global attractor of [15,13] for the strong solutions but additional requirement of the strong convergence in L 2 (T 2 ) of the initial data needed for the multivalued semiflow to be strict was added in the definition of the weak solution. Using the developed formalism we show in this chapter that the requirement of the strong convergence of the initial data is unnecessary and actually the global attractor for the class of vanishing viscosity weak solutions considered in [5] (for the unforced case) and [8] coincides with the global attractor for the strong solutions considered in [15,13].

4.1.
Problem formulation and definition of weak and strong solutions. We briefly remind the setup of the problem. The two dimensional torus T 2 is defined as (−π, π) 2 and C ∞ p (T 2 ) is the space of restrictions to T 2 of smooth functions which are 2π-periodic in both variables and mean free. In the sequel we will always use the shorthand notation for various spaces of functions defines on T 2 , for example we will write H σ for the closure of C ∞ p (T 2 ), in H σ (T 2 ) norm. All spaces consist of functions which are mean free and 2π-periodic with respect to both variables. The scalar product on L 2 = H 0 will be denoted by ·, · and the same notation will be used for the duality between various spaces and their duals.
For a periodic and mean free distribution φ on T 2 we can define its Fourier coefficients by where Z 2 * = Z 2 \ {(0, 0)}, and if the distribution φ is defined by a function, then its Fourier coefficients are given by the integrals Using the Fourier coefficients it is very easy to define the scale of Sobolev spaces H σ using the formula For σ ∈ R one can define the fractional Laplacian by means of the Fourier transform by the formula The fractional Laplacian is a linear and bounded operator from H σ to its dual H −σ and the norm on H σ can be equivalently defined by the formula φ H σ = (−∆) σ/2 φ L 2 . The Zygmund operator is defined as Λ = (−∆) 1/2 , and, in general, Λ σ = (−∆) σ/2 . Finally the j-th Riesz transform for j = 1, 2 is defined in terms of the Fourier coefficients as Moreover, there holds The considered initial and boundary value problem has the form Note, that the function f is always assumed to be mean free and time independent. We are in position to define the weak solution for the problem (50) is called a weak solution of (50) if for every ϕ ∈ C ∞ 0 ([0, ∞); C ∞ p ) there holds for every T > 0 The existence of weak solutions (for θ 0 ∈ L 2 and f ∈ L 2 ) has been proved in [32]. We will be interested in the subclass of all weak solutions, namely the vanishing viscosity weak solutions. They are defined as the limits of the following auxiliary problems parameterized by ε > 0 The class of vanishing viscosity weak solutions is defined in the following way.
is called a vanishing viscosity weak solution of (50) if it is a weak solution given by (4.1) and there exists a sequence ε n → 0 and corresponding solutions θ εn to (51) with ε = ε n such that θ εn → θ in C w ([0, T ]; L 2 ) for every T > 0.
Remark 1. In [18] the class of vanishing viscosity weak solutions with strongly converging initial data is considered. This restriction is needed to guarantee the strictness of the multivalued semiflow. Since in the abstract framework of the present article we do not need the strictness, we can work with the class of weak solutions given in Definition 4.2 which is possibly broader than that of [18]. Also note that Definition 4.2 is consistent with the corresponding definitions of [5,8].
The proof of the existence of a vanishing viscosity weak solutions for every θ 0 ∈ L 2 and f ∈ L ∞ follows the lines of [5, Appendix C]. The following important property of vanishing viscosity weak solutions have been proved in [8].
Theorem 4.3. Let θ be a vanishing viscosity weak solution with f ∈ L ∞ and θ 0 ∈ L 2 . Then for every t > 0 there holds θ(t) ∈ L ∞ . Moreover, the following estimate holds for t > 0 and the next energy equation holds for every vanishing viscosity weak solution and every 0 ≤ t 0 ≤ t Remark 2. It is clear that the above result implies that every vanishing viscosity weak solution belongs to C([0, ∞); L 2 ).

Attractors for single and multivalued semiflow and their equality.
Define the (possibly multivalued) operator S L 2 (t) : L 2 → P(L 2 ) by the formula  Proof. The fact that S L 2 (t)θ 0 is nonempty follows from the existence of the vanishing viscosity weak solution for every θ 0 ∈ L 2 . On the other hand, the inclusion S L 2 (s + t)θ 0 ⊂ S L 2 (s)S L 2 (t)θ 0 follows from the so called translation property. Indeed, directly from Definitions 4.1 and 4.2 it follows that if θ is a vanishing viscosity weak solution with θ(0) = θ 0 then θ t = θ(·+t) is a vanishing viscosity weak solution with θ t (0) = θ(t) and the proof is complete.
To use the framework of [13,15] we will assume that f ∈ L ∞ ∩ H 1 , and under this regularity the class of strong solutions is defined on H 1 . In fact we have the following result, cf. [15,Theorem 4.5], [13, Proposition 2.1].
Theorem 4.5. Let f ∈ L ∞ ∩ H 1 and let θ 0 ∈ H 1 . There exists a unique strong solution θ ∈ C([0, ∞); H 1 ) ∩ L 2 loc (0, ∞; H 3/2 ) of the initial value problem (50). Hence, we can define the single-valued operator S H 1 (t) : H 1 → H 1 by means of the formula We will use the abstract Theorem 2.5 proved in Section 2.1 with V = H 1 and H = L 2 to get the equality of global attractors for strong and weak solutions. To use this theorem we need to show that its assumptions (ii), (iii), (iv), and (v) hold for S L 2 and S H 1 . Hence, the next part of this chapter is devoted to the verification of these assumptions. Two of them follow directly from the results of [13]. The assertion (v) is expressed in the following Lemma 4.6. [13, Theorem 6.1] There exists a constant R = R( f L ∞ ∩H 1 ) such that for every B ∈ B(H 1 ) there exists t 0 = t 0 (B) such that the ball On the other hand, the assertion (ii) directly follows from In the next result we prove the assertion (iv). Proof. The result in fact follows from the consecutive a priori estimates obtained in [13,15,18] and uses the same bootstrapping argument as one which was used in [18]. Note that all estimates are obtained for the solutions of the problem with the added vanishing viscosity term, and, in consequence, they are preserved in the weak limit. The result is obtained by the bootstrap argument which consists of four steps.
Step 1. Bound in L ∞ . We have already stated the result which is the first step of the bootstrapping argument, namely the L ∞ bound established in Theorem 4.3. So, after t ≥ /4 the trajectories starting from B are absorbed by a bounded set in L ∞ .
Proof. Fix 0 < t 1 < t. Lemma 4.8 implies that θ ∈ L ∞ (t 1 , t; L ∞ ∩ H 1 ). Choose the test function ϕ ∈ C ∞ 0 ([t 1 , t]; C ∞ p ). Then (55) Hence, in the sense of distributions, Note that u · ∇θ ∈ L 2 (t 1 , t; H −1/2 ). Indeed, using the fact that the Riesz transform is a linear and bounded operator from L 4 to L 4 , t t1 u(s) · ∇θ(s)ϕ(s) dt ≤ t t1 u(s) L 4 ∇θ(s) L 2 ϕ(s) L 4 ds ≤ C θ 2 L ∞ (t1,t;H 1 ) ϕ L 2 (t1,t;H 1/2 ) . It follows that θ t ∈ L 2 (t 1 , t; H −1/2 ). Since at least the same regularity of θ t holds for the strong solutions we subtract (55) written for θ and θ and test by ϕ = θ − θ. This It follows that The Gronwall lemma implies that H 3/2 dt . The assertion follows after passage to the limit as t 1 → 0 + . In Lemmas 4.6, 4.7, 4.9, and 4.8 we have established that the assertions (ii), (iii), (iv), and (v) of Section 2.1 hold. Hence, we are in position to use Theorem 2.5 to deduce the following result. Moreover A ⊂ H 1 and the dynamics on this attractor is single valued and given by S H 1 . This attractor is compact in H 1 , and invariant, i.e. S L 2 (t)A = S H 1 (t)A = A for every t ≥ 0. The attractor is minimal in the class of L 2 -closed sets attracting in L 2 with respect to S L 2 and maximal in the class of L 2 -bounded sets which are invariant with respect to either S L 2 or S H 1 .