UNIQUE STRONG SOLUTIONS AND V-ATTRACTOR OF A THREE DIMENSIONAL GLOBALLY MODIFIED MAGNETOHYDRODYNAMIC EQUATIONS

. In this paper, we provide a detailed investigation of the problem of existence and uniqueness of strong solutions of a three-dimensional system of globally modiﬁed magnetohydrodynamic equations which describe the motion of turbulent particles of ﬂuids in a magnetic ﬁeld. We use the ﬂattening prop- erty to establish the existence of the global V -attractor and a limit argument to obtain the existence of bounded entire weak solutions of the three-dimensional magnetohydrodynamic equations with time independent forcing.

1. Introduction. It is well known that the 3D magnetohydrodynamic (MHD) equations describe the motion of electrically conducting fluids and one of these models is given by where u, B and p represent respectively the fluid velocity, the magnetic field and the pressure. f 1 and f 2 are given external forces field. R e and R m are the so-called Reynolds and magnetic Reynolds numbers, respectively, and S = M 2 ReRm is a positive constant, where M is the Hartman number. |B| 2 = B · B and represents the length of the magnetic field. These equations take into account the coupling between Maxwell's equations governing the magnetic field and the Navier-Stokes equations (NSE) governing the fluid motion [3]. They play a fundamental role in astrophysics, geophysics, plasma physics and in many other areas in applied sciences. They have been intensively investigated for many years (see [1,2,3,22,23] just to cite a few), but some very basic issues on their solvability remain unresolved. For example, the problem of uniqueness of weak solution and the global regularity of the solution remain one of the open problems in mathematical physics.
In this paper, we introduce the following system of 3D globally modified magnetohydrodynamic equations (GMMHDE) on a bounded domain M ⊂ R 3 with smooth boundary ∂M endowed with the following initial and boundary conditions where n is the unit outward normal on ∂M, N ∈ (0, ∞) is fixed and F N : [0, ∞) → (0, 1] is defined by F N (r) = min{1, N/r}, r ∈ R + .
The spaces V 1 and V together with u V1 and (u, B) V will be defined later. One notes that when B=0, system (2)-(3) becomes the 3D globally modified Navier-Stokes equations (GMNSE) which have been the object of intense investigations over the last years [4,6,7,8,10,14,15,18,19,21,31]. The 3D GMNSE were introduced and studied in [4]. Contrary to the original 3D NSE, this modified model of the 3D NSE has some good properties such as global existence, uniqueness and regularity. These results are interesting in their own right, but also the GMNSE are useful in obtaining new results about the 3D Navier-Stokes equations. For example, they were used in [4] to establish the existence of bounded entire weak solutions for them. Also in [15], the GMNSE were used to show that the attainability set of weak solutions of the three dimensional Navier-Stokes equations satisfying an energy inequality, are weakly compact and weakly connected. For convergence results of solutions of the GMNSE to solutions of the three dimensional Navier-Stokes equations, see [4,19]. These results were extended in [24,25] to the case of the 3D globally modified Cahn-Hilliard-Navier-Stokes equations and the 3D globally modified Allen-Cahn-Navier-Stokes equations. The GMMHDE (2)-(3) is inspired from the globally modified Navier-Stokes equations (GMNSE) studied in [4]. As noted in [4] concerning the GMNSE, the GMMHDE are indeed globally modified. The factors F N ( u V1 ) and F N ( (u, B) V ) which depend, respectively, on the norms u V1 and (u, B) V are used only to prevent large values of u V1 and (u, B) V dominating the dynamics. Just like the GMNSE, the GMMHDE violate the basic laws of mechanics, but mathematically the model is well defined.
Motivated by the above references, we propose to analyze the globally modified magnetohydrodynamic equations (2)- (3). The following three points are our main contribution in this work: (a) we prove the existence, uniqueness of the weak and strong solutions for the GMMHDE, (b) the flattening property is used to establish the existence of the global Vattractor, (c) using a limiting argument, we obtain the existence of bounded entire weak solutions of the three-dimensional MHD equations with time independent forcing. The paper is structured as follows. In Section 2, we introduce the mathematical setting of our problem. In Section 3, we prove the existence, uniqueness of the weak and strong solutions for the 3D GMMHDE. In addition, we establish the continuous dependence of the solution on N and on the initial value in the space V . We also investigate the relationship between the Galerkin approximations of the 3D GMMHDE and the 3D MHD for a fixed finite dimension. In Section 4, we study the asymptotic behavior of the strong solutions when the forcing terms are time independent and we prove the existence of the global attractor in V . In Section 5, we prove that solutions to the 3D GMMHDE converge to a weak solution of the 3D MHD equations. For a time-independent forcing, we also prove the existence of bounded entire weak solutions of the 3D MHD equations.
2. The mathematical setting. Let us now recall from [22,26] the functional set up of the model (2)-(3) and its abstract formulation.
Bold notation will represent a vector or a tensor. We consider the well-known Thus H 2 = H 1 . We endow H i , i = 1, 2 with the inner product of L 2 (M) and the norm of L 2 (M) denote respectively by (., .) L 2 and |.| L 2 . We equip V 1 with the following inner product We equip V 2 with the scalar product We note that by Poincaré's inequality, the scalar product ((., .)) 1 defines in (7) coincides with the well known inner product in H 1 0 (M). The norm generated by ((., .)) 2 is equivalent to the norm induced by H 1 (M) on V 2 (see [11,Chapter VII]). Hereafter, we set The dual space of V is denoted by V . We endow H with the inner products defined as: They generate equivalent norms (for 0 < S < ∞) We also endow V with the inner products which in turn generate the equivalent norms on V In order to give an abstract formulation of problem (2)-(3), we introduce the operators Aφ, ψ = ((φ, ψ)), for all φ, ψ ∈ V.
A 1 can be defined as an unbounded operator generated by the Stokes problem For more detail concerning the operator A 1 we refer the reader to [9,17,26]. The operator A 2 can also be defined as an unbounded operator generated by the boundary value problem For more information about the operator A 2 , we refer the reader to [11]. A is then an unbounded operator with domain From the continuity of the injection of V i , in H i , i = 1, 2, there exist constant κ i , i = 1, 2 such that The best constant κ i is equal to 1 √ As in [22], we introduce the trilinear form which satisfies the following relations The operator b also satisfies the following estimate Proof. To simplify the notations, we assume the summation over repeated indexes. Let u, v, w ∈ H 1 (M). We have, by using the generalized Hölder inequality with exponents (6, 2, 4, 12) ≤ |u| L 6 |∇v| L 2 |w| 1/2 which proves (19).
From (18), we infer that Now we introduce the continuous bilinear form B : We introduce a diagonal matrix M= (m ij ) 1≤i,j≤6 defined by: Note that It follows from (18) and (27) that We recall that (see [22] ) B 0 , B and b satisfy the following estimates Hereafter we set Arguing similarly as in the proof of (29), we can check that the following inequalities hold We note that in (35) and (32) we have also used (18). Using the operators A, B N previously defined, we rewrite (2)-(3) in the form where y = (u, B) is the solution of (2)-(3) and F = (f 1 , f 2 ). We now define the concept of solution associated with (36).

Remark 2.
By taking ψ = My in (38) 1 and using the fact that B N 0 (y, y, My) = 0 (see (28) 1 ), we derive that y satisfies the energy equality Note that the weak formulation of (2) with F N replaced by 1 is studied in [22], where the existence and uniqueness of solution was proved in the two-dimensional case.
Now we recall from [4,21,24] the following properties of F N , where the proof can be found in [4,21].
In the rest of this paper we will denote by c, a generic positive constant (possibly depending on S, R e , R m , κ 1 , M, κ 2 ), which can vary even within the same line. However, this constant is always independent of time and initial data.
there exists a unique weak solution y = (u, B) of (36), which is in fact a strong solution in the sense that (ii) If the initial condition (u 0 , B 0 ) ∈ V then every weak solution y = (u, B) of (36) is a strong solution, in the sense that then there exists at least one weak solution (u, B) of (36).
Proof. (ii). The proof of (ii) is the consequence of assertion (i).
.., (w n , ψ n )} and denote by P n = (P 1 n , P 2 n ), the orthogonal projector in H onto V n for the scalar product (., .) defines in (10) 1 . Note that P n is also the orthogonal projector in We look for y n = (u n , B n ) ∈ H n solution to the ordinary differential equations dy n dt + Ay n + P n B N (y n , y n ) = P n F , Since P n F = P n (f 1 , f 2 ) is a localy Lipschitz function in (u, B), it follows from the theory of ordinary differential equation that the system (62) has a solution y n , (see also Theorem A 1 of [5]). The function y n exists on an interval [0, T n ]. It will follows from a priori estimates that y n exists on the interval [0, T ]. We now derive needed a priori estimates.
Using the same techniques as in Remark 2, we see that y n satisfies the following energy equality for all t ≥ 0. Note that since P n L(V,V ) ≤ 1. Inserting these estimates (64)-(65) in (63), we infer that for all t ≥ 0. Hence, we have for all t ≥ 0. Also, from (63)-(65) and using also (16), we infer that from which we obtain by the technique of Gronwall's lemma for all t ≥ 0. This proves that the sequence y n = (u n , B n ) remains in a bounded set of L ∞ (0, T ; H) ∩ L 2 (0, T ; V ). Hence, we can use a compactness argument (see [28]) to see that a subsequence y n = (u n , B n ) satisfies with y = (u, B) ∈ L ∞ (0, T ; H) ∩ L 2 (0, T ; V ).
As the weak convergence in L 2 (0, T ; V ) is not enough to ensure that we need to derive stronger a priori estimates. Hence, taking the inner product in H of (62) with Ay n , we obtain It follows from (72)-(74) that Now we distinguish two cases: We mention that by the choice of the orthonormal basis {(w i , ψ i )} of H. Now, dropping the term |Ay n | 2 H in (75), we have the following differential inequality from which we obtain by using Gronwall's lemma Hence, we derive from (67), (75), and (77) that (y n ) = (u n , B n ) satisfies Note that in (78), K and K 1 are positive constants independent of n and depending only on data such as M, R e , R m , S, f 1 , f 2 , T, u 0 and B 0 .
We set |f i | ∞ = |f i | L ∞ (0,∞;Hi) , i = 1, 2. From (75), we obtain From (69), we also have By integration in t from t to t + τ of (66), we obtain, after dropping unnecessary terms, where we have also use (82). Hence, from (83), we deduce that where For t > 0 and τ > 0 given, let us define ρ > 0 by and consider the sets and let us denote |D n | the Lebesgue measure of D n . Using (84), we infer that therefore |D n | ≤ τ /2.
Hence we can ensure that for any t ≥ 0, τ > 0 and n ≥ 1 there exists a t 0 ∈ (t, t + τ ) such that From this property, we have for any given δ > 0 and any t ≥ δ, there exists a t 0 ∈ (t − δ, t) such that From (87) and (81), we have for all t ≥ δ.
Since (u n , B n ) converges strongly to (u, B) in L 2 (δ, T ; V ) for all T > δ > 0, we can assume, eventually extracting a subsequence, that (71) is also satisfied in this case. Hence from (71) and (89) we can take the limit in (62) and prove that (u, B) is a solution to (36) satisfying (61).

Continuous dependence on initial values and N.
Here, we prove that the semi flows generated by the solutions (u N , B N )(t, (u 0 , B 0 )) of the GMMHDE (36) with the parameter N depend continuously on the parameter N as well as on the initial value (u 0 , B 0 ). We begin by given an important result. Theorem 3.3. Assume that f i ∈ L 2 (0, T ; H i ), i = 1, 2 for all T > 0, and let N i > 0, y 0i = (u 0i , B 0i ) ∈ V, i = 1, 2 be given. Let y i = (u i , B i ) be the solution to (36) corresponding to the parameter N i and the initial value y 0i = (u 0i , B 0i ), i = 1, 2.
Then, there exists a positive constant c depending only on R e , R m , M and S such that From [4], we have Making similar reasoning as in (93), we can also check that: Also, we can easily check that Hence, taking the scalar product in H of (92) with Ay, we obtain We can easily check that (see [4]) (99) We will now estimate SK i 2 , K i 3 and K i 4 , i = 1, 2, 3 as follows: Now inserting these estimates (99)-(108) in (98), we obtain It follows from the Gronwall lemma and (109) that which proves (90) 1 and where Z 1 = |A 1 u 2 | 2 L 2 + |A 2 B 2 | 2 L 2 . Now using (109) and (110), we get which end the proof of Theorem 3.3.
As a consequence of Theorem 3.3, we have a continuous dependence on the initial value and N . More precisely, if we denote by (u N , B N )(., (u 0 , B 0 )) the solution to (36) corresponding to the parameter N and the initial value (u 0 , B 0 ), then we have the following result.

3.2.
Comparison of Galerkin solutions of the 3D GMMHDE and 3D MHD. We introduce the following Galerkin ODE for the GMMHDE with parameter N .
In the following proposition, we check that the Galerkin ODE (113)  ) of the sequence y N n which converges uniformly in C(0, T ; R 3 ) to a function y which is the corresponding solution of the n-dimensional Galerkin ODE for the 3D MHD. More precisely, there exists y (∞) n and a constant κ 3 such that y Proof. We first note that where λ 1 j and λ 2 j are the corresponding eigenvalues of the operators A 1 and A 2 . We set |f i | ∞ = |f i | L ∞ (0,T ;Hi) , i = 1, 2. The corresponding energy inequality for the ODE (113) reads where we have also used (114) 3 and the Young inequality. Now from (115), we infer that for all t ∈ [0, T ].
From (113), using (29) 1 and (114), we obtain that a.e. in (0, T ) Hence from (117) n ) is in fact the solution of the n-dimensional Galerkin ODE for the 3D MHD. Indeed, this follows from the uniqueness of solutions of the Galerkin ODE for a given initial value (which is proved in [22,Theorem 3.1]) and the fact that so, n ). We note that in (118) and (119), we have use the following estimates:

4.
Existence of global attractor in V of the 3D GMMHDE. We now assume that the forcing terms f i ∈ H i , i = 1, 2 do not depend on time. We fix N > 0 and we denote by (u, B) ∈ V the unique strong solution to (36). By setting S N (t)(u 0 , B 0 ) = (u, B)(t), it will follows from Theorems 3.1, 3.2 and 3.3 that {S N (t)} t≥0 is a C 0 semigroup in V .

Absorbing set in H.
Here we prove that S N (t) has an absorbing set B H which absorbs bounded sets of V .
Making now similar reasoning as in (66), using also (16), we can check that where Hence from this last inequality, we infer that We conclude that S N (t) has an absorbing set B H in H given by 4.2. Absorbing set in V. Making similar reasoning as in the derivation of (75), taking into account the fact that , i = 1, 2) and using also (32) and the Young inequality with exponents where . From (123), we infer that where C N = max C N , C N S and [(., .)] H defined as in (12).
By integration in t from 0 to t of (125), we deduce that From (126) or (127), we infer that S N (t) has an absorbing set B in V which is given by: 4.3. Asymptotic compactness of the semigroup in V. Here, we prove the asymptotic compactness of the semigroup S N (t). We recall that with the view to proof that the semigroup S N (t) is asymptotically compact, it is enough to very the following flattening property (see [12,20,13] for more details) of the semigroup S N (t).
Proposition 2 (Flattening property). For any bounded set B of V and any σ > 0, there exists T σ (B) > 0 and a finite dimensional subspace V σ of V such that {P σ S N (t)B, t ≥ T σ (B)} is bounded and where P σ : V → V σ is the projection operator.
Proof. Without loss of generality, we can restrict ourselves to B=B (N ) V , the absorbing set of S N (t) in V given by (128). Let σ > 0. We will find an integer N σ > 0 such that the flattening property holds for the N σ -dimensional subspace V σ of V spanned by the first eigenfunctions (w k , ψ k ), k = 1, 2, ..., N σ , where (w k , ψ k ) are the eigenfunctions used in the proof of Theorem 3.2. Let us denote by β 1 k , β 2 k the eigenvalues defined by Nσ . From the fact that B is a bounded absorbing set and P σ (u, We now move to the proof of (129).
For t large enough, we know that (u, B)(t) is uniformly bounded in V .
We now make the following observation: given λ and similar reasoning as in deriving (75) Let y σ (t) = (I − P σ )(u, B)(t) = (I − P σ )S N (t)(u 0 , B 0 ) = (w, ψ). Taking the inner product in H of (36) with We note that (for t large enough) and where we have use (31), the Young inequality and κ N is a monotone nondecreasing function of the parameter N . Also, we note that Inserting these estimates (133)-(134) in (132), we obtain d dt and We note that in (137), we have also use (131). Therefore, for t and N large enough in such a way that the initial condition term becomes smaller than σ/2 and so that the sum of the terms involving λ in the denominator is smaller that σ/2, we derive that y σ (t) 2 V ≤ σ, which proves the flattening property of S N (t).
where dist V is the Hausdorff semi-distance on V .
Proof. The existence of the global attractor follows from the existence of the absorbing set in V as well as the flattening property proved in Proposition 2. Making similar reasoning as in [4], we can check the upper semi-continuity (138). Note that for each N > 0, we have ∧ N ⊂ B

5.
Convergence to weak solution of the 3D MHD equations. We assume that f i ∈ L 2 (0, T ; H i ), i = 1, 2 for all T > 0. Let (u N , B N )(t) be the weak solution to (36) with initial value (u 0
As in [4], we will prove that (u, B) is a weak solution to the 3D MHD (1).
Proof. For the proof of (142) 1 we refer the reader to [4], and arguing similarly as in [4], we can check (142) 2 .
Remark 3. We note that the variational equation (145) differs from the variational equation (38) for the 3D GMMHDE by the absence of the F N ( u V1 ) and F N ( (u, B) V ) factors multiplying the nonlinear term b.

5.1.
Existence of bounded entire weak solutions of the 3D MHD equations. In this part, we suppose that the forcing terms f i ∈ H i , i = 1, 2. Following similar steps as in [4], we prove the existence of a bounded entire weak solution of the 3D MHD equations.
Theorem 5.2. There exists a bounded entire weak solution of the 3D MHD equations (1). More precisely, there exists a bounded entire weak solutions of (1) with the initial value (u 0 , B 0 ) ∈ U 0 , where U 0 is a subset of H consisting of the weak H-cluster points of a sequence in ∧ N .
Proof. For the proof, we will arguing similarly as to the proof of Theorem 14 in [4]. Thus, we omit the details and only give a sketch. We consider a sequence (u 0