PULLBACK ATTRACTOR AND INVARIANT MEASURES FOR THE THREE-DIMENSIONAL REGULARIZED MHD EQUATIONS

. This article studies the three-dimensional regularized Magnetohydrodynamics (MHD) equations. Using the approach of energy equations, the authors prove that the associated process possesses a pullback attractor. Then they establish the unique existence of the family of invariant Borel probability measures which is supported by the pullback attractor.


1.
Introduction. In this article, we consider the following non-autonomous threedimensional (3D) regularized Magnetohydrodynamics (MHD) equations (1.1) u(x, τ ) = u τ (x), b(x, τ ) = b τ (x), x ∈ Ω, (1.5) where τ ∈ R and Ω ⊂ R 3 is a bounded domain with smooth enough boundary ∂Ω, the velocity field u = (u 1 , u 2 , u 3 ), the magnetic field b = (b 1 , b 2 , b 3 ) and the total pressure p are the unknown terms, ν is the kinematic viscosity and µ is the constant magnetic resistivity, f represents volume force applied to the fluid, g is usually zero when Maxwell's displacement currents are ignored. We will assume the constants ν, µ, α and β are all positive. Equations (1.1)-(1.2) are regularization in both the velocity and the magnetic field of the following large eddy simulation model for the turbulent flow of a magnetofluid (see [7]): (1.6) show that a continuous process {U(t, τ )} t τ on a complete metric space X possesses a unique family of Borel invariant probability measures in X if {U(t, τ )} t τ satisfies (i) the process {U(t, τ )} t τ possesses a pullback attractor in X; and (ii) for every u 0 ∈ X and every t ∈ R, the X-valued function τ −→ U(t, τ )u 0 is continuous and bounded on (−∞, t].
Zhao and Yang used this theory to construct the invariant Borel probability measures for the non-autonomous globally modified Navier-Stokes equations in [34].
Here we will also borrow this result to obtain the unique existence of the family of invariant Borel probability measures which are supported by the obtained pullback attractor. The key step is to estimate the difference between two solutions (u (1) , b (1) ) and (u (2) , b (2) ) of equations (1.1)- (1.5). When estimating the following nonlinear terms (u (1) · ∇)u (1) − (u (2) · ∇)u (2) , u (1) − u (2) , (2) , u (1) − u (2) , we also find that the coupled structure of the addressed equations plays a key role. The rest of this article is arranged as follows. The next section is preliminaries concerning some notations and operators, as well as the existence and uniqueness of solutions to the 3D regularized MHD equations. Section 3 is devoted to establishing the existence of pullback attractor for the associated process in space W via the approach of energy equations. In the last section, we prove that there exists a unique family of invariant Borel probability measures on the pullback attractor.
2. Existence and uniqueness of the solutions. In this section, we first introduce some notations and operators. Then we present the existence and uniqueness of solutions to equations (1.1)- (1.5).
In this article, R denotes the set of real numbers and c(·, ·) stands for the generic constant (depending essentially on the quantities in the brackets) that can take different values in different places. L p (Ω) = (L p (Ω)) 3 is the 3D Lebesgue space with norm · L p (Ω) , and · L 2 (Ω) = · for brevity. At the same time, H m (Ω) is the 3D Sobolev space {φ ∈ L 2 (Ω)| ∇ k φ ∈ L 2 (Ω), k m} with norm · H m (Ω) , and We will use (·, ·) for the inner product in H and ·, · for the dual pairing between V and V . We also use the norm in W as where α and β come from (1.1) and (1.2), respectively. By the Poincaré inequality, we see that above norm is equivalent to the usual norm We consider the usual operators in the theory of Navier-Stokes equations. Let P be the Leray-Helmholtz projection from L 2 (Ω) onto H and A : V → V be the linear operator defined as We denote D(A) = {u ∈ V |Au ∈ H}. Since the smoothness of ∂Ω, we have D(A) = H 2 (Ω) ∩ V and Au = −P ∆u, ∀u ∈ D(A). We also define a continuous trilinear form and set B(u), w = B(u, u), w for short. Note that V ⊆ H 1 0 (Ω) is a closed subspace. Thus B(u, v): V ×V → V is continuous. For above introduced operators A and B, we have the following classical results.
Lemma 2.1. ( [4,13]) There exist two positive constants c 1 and c 2 depending only on Ω such that As a consequence of (2.2), we have With the above notations, excluding the pressure p, we can express the weak version of equations (1.1)-(1.5) in the solenoidal vector field as follows (see e.g. [32]): Assume that (u τ , b τ ) ∈ W and both f and g belong to L 2 loc (R; V ). We next specify the definition of weak solutions to (2.7)-(2.9). Definition 2.2. It is said (u, b) be a weak solution to (2.7)-(2.9) if (u, b) ∈ L 2 (τ, T ; W ) for all T > τ and satisfies (2.7)-(2.9).

17)
and λ 1 is the first eigenvalue of the operator A.
Proof. For the existence and regularity of the solutions, we can proceed using the Galerkin scheme as that as in [11]. Here we omit the details. We next prove the energy inequality (2.14).
Multiplying u with (2.7) and b with (2.8), respectively, and then adding the resulting equalities give Let σ be some constant satisfying (2.16). By (2.18), we have, using the Poincaré inequality and the Hölder inequality, where the constant ε is given by (2.17).

Then bothṽ andξ belong to
Similarly, Therefore, for any given T > τ , we set and obtain for any s ∈ [τ, T ] that Hence, by (2.11) and (2.12), and thus for all t ∈ [τ, T ]. From this inequality and the Gronwall inequality we obtain The uniqueness of the weak solutions is proved. Lemma 2.3 shows the globally existence and uniqueness of the weak solutions to (2.7)-(2.9). In Section 4, we will prove that the solutions are continuous with respect to the initial data (see Lemma 4.4). Therefore, we can define the solution operators U (t, τ ) as which generates a continuous process {U (t, τ )} t τ in W , where u(t; τ, u τ ), b(t; τ, b τ ) denotes the solution of (2.7)-(2.8) corresponding to the initial datum (u τ , b τ ) at initial time τ . Moreover, we see from (2.13) (2.29) 3. Existence of the pullback attractors. In this section, we first introduce some definitions concerning the pullback attractors. Then we establish the existence of pullback attractors for {U (t, τ )} t τ in W .
In the sequel, we use P(W ) to denote the family of all nonempty subsets of W , and consider a family of nonempty sets D 0 = {D 0 (t)| t ∈ R} ⊆ P(W ). Let D be a given nonempty class of families parameterized in time D = {D(t)| t ∈ R} ⊆ P(W ). The class D will be called a universe in P(W ).
Definition 3.2. The process {U (t, τ )} t τ is said to be pullback D 0 -asymptotically compact if for any t ∈ R and any sequences {τ n } ⊆ (−∞, t] and {(x n , y n )} ⊆ W satisfying τ n → −∞ and (x n , y n ) ∈ D 0 (τ n ) for all n, the sequence {U (t, τ n )(x n , y n )} is relatively compact in W . {U (t, τ )} t τ is said to be pullback D-asymptotically compact if it is D-asymptotically compact for any D ∈ D.
References [3,11] proved the general existence and minimality results of a pullback attractor and its property for general process. For example, García-Luengo, Marín-Rubio and Real in [11] pointed out that the family A D is minimal in the sense that if C = {C(t)| t ∈ R} ⊆ P(W ) is a family of closed sets such that for any To prove the existence of the pullback attractor for {U (t, τ )} t τ in W , we need the following assumption on the external force functions f (t) and g(t).
From now on, we denote by D σ the class of all families of nonempty subsets Lemma 3.4. Let assumption (H 1 ) hold. Then the process {U (t, τ )} t τ possesses a pullback D σ -absorbing set in space W .
Proof. Set and where B W (0, R 1/2 σ (t)) denotes the family of closed balls in space W centered at zero and with radius R 1/2 σ (t). Then from (2.14) we see that the family D σ (t) defined by (3.4) is the desired pullback D σ -absorbing set for {U (t, τ )} t τ in W .
We next use the approach of energy equation to investigate the pullback asymptotic compactness of the process {U (t, τ )} t τ in W for the universe D σ . To this end, we first introduce two projection operators and then give some convergent relations for the solutions. The projection operators Π 1 and Π 2 are defined as Lemma 3.5. Let assumption (H 1 ) hold and τ < t be given. Consider a sequence {(u τ,n , b τ,n )} ⊂ W weakly converging to (u τ , b τ ) in W as n → ∞. Then the following convergent relations hold: Π 2 U (·, s)(u τ,n , b τ,n ) Π 2 U (·, s)(u τ , b τ ) weakly in L 2 (τ, t; V ), (3.8) Proof. The proof of (3.7)-(3.8) can be done similarly to that of [11, Theorem 4, (2.26)], and the proof of (3.9) can be done analogously to that of [11,Theorem 6], since the a priori estimates follow the similar. The fact that the whole sequence meets the above convergent relations is a consequence of the uniqueness of the weak solution for (2.7)-(2.9).
At this stage, we combine Lemma 3.4 and Lemma 3.6 with García-Luengo et al. [11,Theorem 3.11,Corollary 3.13] to get the main result of this section as follows.
As pointed out in Section 2, Lemma 4.4 shows that the solutions of (2.7)-(2.9) depend continuously on the initial data in the strong topology of space W .
At this stage, we take Theorem 3.7, Lemma 4.3 and Lemma 4.5 into account and obtain the following result.
Theorem 4.6. Suppose assumption (H 1 ) hold. Let {U (t, τ )} t τ be the process associated to the solution operators of (2.7)-(2.9) andÂ Dσ be the pullback D σattractor obtained in Theorem 3.7. Fix a generalized Banach limit LIM T →∞ and let ψ : R → W be a continuous map such that ψ(·) ∈ D σ . Then there exists a unique family of Borel probability measures {m t } t∈R in space W such that the support of the measure m t is contained in A Dσ (t) and for any real-valued continuous functional φ on W . In addition, m t is invariant in the sense that A Dσ (t) φ(w)dm t (w) = A Dσ (τ ) φ(U (t, τ )w)dm τ (w), t τ.