G Deugoue

J K Djoko

A C Fouape

Three dimensional system of globally modified magnetohydrodynamics equations with infinite delays

Keywords: magnetohydrodynamics equations, globally modified, strong solutions, infinite delays

Existence and uniqueness of strong solutions for three dimensional system of globally modified magnetohydrodynamics equations containing infinite delays terms are established together with some qualitative properties of the solution in this work. The existence is proved by making use of; Galerkin's method, Cauchy-Lipshitz's theorem, a priori estimates, the Aubin-Lions compactness theorem. Moreover, we study the asymptotic behavior of the solution.

Introduction and statement of the problem

Let Ω ⊂ R 3 be an open bounded set with regular boundary Γ = ∂Ω, and N > 0 be fixed. We define FN : (0, +∞) → (0, 1] by

FN (r) = min 1, N r , r ∈ R + 1
and consider the following system of globally modified magnetohydrodynamics equations (GMMHDE)

                           ∂u ∂t + FN ( u V 1 )[(u • ∇)u] - 1 Re ∆u -SFN ( (u, B) V ) [(B • ∇)B] +∇ p + S |B| 2 2 = f 1 (t) in (τ, T ) × Ω, ∂B ∂t + FN ( (u, B) V ) [(u • ∇)B -(B • ∇)u] + 1 Rm curl(curlB) = f 2 (t) in (τ, T ) × Ω,
div u = 0, div B = 0 in(τ, T ) × Ω, u(τ, x) = u0(x), B(τ, x) = B0(x) for all x ∈ Ω, u = 0, B • n = 0 and curlB × n = 0 on Γ, (1.1) 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 fields. Re and Rm 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, n is the unit outward normal on Γ and τ the initial time. It is manifest that the system of equations in (1.1) does not represent the MHD model due to un-physical terms introduced such as FN ( u V 1 ), FN ( (u, B) V ). These terms can find their existence in the original model of globally modified Navier Stokes introduced in [START_REF] Caraballo | Unique strong solutions and V -attractors of a three dimensional system of globally modified Navier-Stokes equations[END_REF]. As clearly demonstrated in [START_REF] Caraballo | Unique strong solutions and V -attractors of a three dimensional system of globally modified Navier-Stokes equations[END_REF], FN ( u V 1 ) prevent the rapid grow of velocity gradient and help to obtain uniqueness of weak solution in 3d, property which is lacking for Navier Stokes in 3d. Hence Mathematically, there is a merit of studying this system. Recently globally modified Navier Stokes coupled with the magnetic field or the heat equation have been proposed and analysed in [START_REF] Deugoué | Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamics equations[END_REF][START_REF] Deugoué | On the Convergence of Solutions of Globally Modified Magnetohydrodynamics Equations with Locally Lipschitz Delays Terms[END_REF][START_REF] Deugoué | Globally modified Navier Stokes equations coupled with the heat equation: existence and time discrete approximation[END_REF]. It is clearly observed in those later works that the "perturbation terms" added play a crucial role in describing the unique solvability of the system. Just like the MHD model (cf. [START_REF] Chandrasekhar | Hydrodynamic and Hydromagnetic Stability[END_REF]), the expressions describing the coupling between the velocity and magnetic fields are represented. The question we would like to investigate in this work is simple and summarizes as follows: The system of equations (1.1) has a unique strong solution and stable, what happens if there is a delay ? This question has been answered in [START_REF] Deugoué | On the Convergence of Solutions of Globally Modified Magnetohydrodynamics Equations with Locally Lipschitz Delays Terms[END_REF] where finite delays were considered.

Many problems in applied science, physics, and engineering are modeled mathematically by delay differential equations. The reason of introducing the time delay in (1.1) followed the work of [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF], but we also note the contribution in [START_REF] Caraballo | Navier-Stokes equations with delays[END_REF][START_REF] Caraballo | Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays[END_REF][START_REF] Caraballo | Attractors for 2D-Navier-Stokes model with delays[END_REF][START_REF] Marín-Rubio | Pullback attractors for two-dimensional Navier-Stokes equations in an infinite delay case[END_REF][START_REF] Márquez | Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay[END_REF][START_REF] Medjo | A two-phase flow model with delays[END_REF]. It is observed that delays terms may appear when we want to control the system by applying a force which takes into account not only the present state but the complete history of the system. In this paper, we introduce the following system of 3d globally modified magnetohydrodynamics equations with infinite delays terms (GMMHDED)

                             ∂u ∂t + FN ( u V 1 )[(u • ∇)u] - 1 Re ∆u -SFN ( (u, B) V ) [(B • ∇)B] +∇ p + S |B| 2 2 = f 1 (t) + g 1 (t, (ut, Bt)) in (τ, T ) × Ω, ∂B ∂t + FN ( (u, B) V ) [(u • ∇)B -(B • ∇)u] + 1 Rm curl(curlB) = f 2 (t)+
g 2 (t, (ut, Bt)) in (τ, T ) × Ω, div u = 0, div B = 0 in(τ, T ) × Ω, u = 0, B • n = 0 and curlB × n = 0 on Γ, u(τ + s, x) = φ1(s, x), B(τ + s, x) = φ2(s, x), s ∈ (-∞, 0], x ∈ Ω, (

where g 1 (t, (ut, Bt)) and g 2 (t, (ut, Bt)) are another external forces containing some hereditary characteristic (delays terms), where ut and Bt are functions defined on (-∞, 0] by the relations ut(s) = u(t + s) and Bt(s) = B(t + s) respectively. φ1 and φ2 are given functions defined in the interval (-∞, 0]. Since the initial time is τ, we deduce from the last line of (1.2) that (u(τ ), B(τ )) = (φ1(0), φ2(0)). This system as we see is a modification of the magnetohydrodynamics (MHD) equations with delays, for an incompressible resistive viscous fluid subjected to a Lorentz force due to the presence of a magnetic field. The GMMHDED (1.2) is inspired from the globally modified Navier-Stokes equations (GMNSE) with infinite delays studied in [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF]. Such models (with delays) have been intensively investigated for many years ( see [START_REF] Caraballo | Navier-Stokes equations with delays[END_REF][START_REF] Caraballo | Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays[END_REF][START_REF] Caraballo | Attractors for 2D-Navier-Stokes model with delays[END_REF][START_REF] Marín-Rubio | Pullback attractors for two-dimensional Navier-Stokes equations in an infinite delay case[END_REF][START_REF] Márquez | Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay[END_REF][START_REF] Medjo | A two-phase flow model with delays[END_REF], just to cite some); but globally modified MHD with delays remain to be explored. This work follow our initial works [START_REF] Deugoué | Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamics equations[END_REF][START_REF] Deugoué | On the Convergence of Solutions of Globally Modified Magnetohydrodynamics Equations with Locally Lipschitz Delays Terms[END_REF][START_REF] Deugoué | Globally modified Navier Stokes equations coupled with the heat equation: existence and time discrete approximation[END_REF] where the focus is on dynamics of globally modified Navier-Stokes coupled with magnetic field or the heat. We should also mentioned that the inspiration from this work comes from the work of J. Real and the coauthors in [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF]. It is worth mentioning that our work differ from the one of J. Real and co-authors because we are dealing here with more equations, and there are more nonlinearities in our context, implying that the investigations are more involved even though some of the Proofs presented here are inspired from the works in [START_REF] Caraballo | Navier-Stokes equations with delays[END_REF][START_REF] Caraballo | Asymptotic behaviour of two-dimensional Navier-Stokes equations with delays[END_REF][START_REF] Caraballo | Attractors for 2D-Navier-Stokes model with delays[END_REF][START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF][START_REF] Marín-Rubio | Pullback attractors for two-dimensional Navier-Stokes equations in an infinite delay case[END_REF][START_REF] Márquez | Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay[END_REF][START_REF] Medjo | A two-phase flow model with delays[END_REF]. This work is mainly concerned about the existence and uniqueness of solution of system (1.2) and its long term behavior when the forcing terms are independent of time.

The rest of the paper is structured as follows: in section 2, we recall some spaces useful for the variational formulation of problem (1.2). We also present some mathematical properties and estimates related to the operators involved in the model. In section 3 we establish the existence and the uniqueness of the solutions of the model. Section 4 (the last one) is devoted to the asymptotic behavior of that solution.

Preliminaries

In order to write down in mathematical terms (1.2), some notations and preliminaries need to be introduced. The material is borrowed mainly from [START_REF] Brezis | Functional Analysis, Sobolev Spaces and partial differential equations[END_REF][START_REF] Temam | Navier-Stokes equations, Theory and Numerical Analysis[END_REF]. We recall the abstract spaces for model (1.2) and its abstract formulation. Bold notations will denote a vector or a tensor. We consider the well known Hilbert spaces L 2 (Ω), H m (Ω), H m 0 (Ω) and we set

L 2 (Ω) := (L 2 (Ω)) 3 , H m (Ω) := (H m (Ω)) 3 , H m 0 (Ω) := (H m 0 (Ω)) 3 , L 2 0 (Ω) := (L 2 0 (Ω)) 3 (2.1)
where L 2 0 (Ω) = q ∈ L 2 (Ω); Ω q(x)dx = 0 . It is noted that for a vector w we set

w r L r (Ω) = Ω |w(x)| r dx ,
where |•| denotes the Euclidean norm |w| 2 = w•w. We shall frequently use Sobolev imbedding: for a real number p ∈ R, 1 ≤ p ≤ 6, the space H 1 (Ω) is imbedded into L p (Ω). In particular, there exists a constant cp (that depends only on p, Ω and d = 3) such that

for all v ∈ H 1 0 , v L p (Ω) ≤ cp ∇v L p (Ω) . (2.2)
When p = 2, this is Poincare's inequality and c2 is Poincare's constant. In the case of the maximum norm, the following imbedding holds

for all r > d = 3, W 1,r (Ω) ⊂ L ∞ (Ω)
in particular, for each r > d = 3, there exists c∞,r such that

for all v ∈ H 1 0 (Ω) ∩ W 1,r , v L ∞ (Ω) ≤ c∞,r ∇v L r (Ω) . (2.3)
Owing to Poincare's inequality, the semi-norm | • | is a norm on H 1 0 (Ω), equivalent to the full norm. As it is directly related gradient operator, we take this semi-norm as norm on H 1 0 (Ω), and we use it to define the dual norm on its dual space H -1 (Ω):

for all f ∈ H -1 (Ω), f H -1 (Ω) = sup v∈H 1 0 (Ω) f , v |v| ,
where • is the duality pairing between H -1 (Ω) and H 1 0 (Ω). As usual for handling time dependent problems, it is convenient to consider functions defined on a time interval (a, b) with values in a functional space, say Y . More precisely, we denote by • Y the norm on Y and for any number r with 1 ≤ r ≤ ∞, we define

L r (a, b; Y ) = {w measurable in (a, b) ; b a w(t) r Y dt < ∞} equipped with the norm w r L r (a,b;Y ) = b a w(t) r Y dt
with the usual modification if r = ∞. It is a Banach space if Y is a Banach space, and when r = 2, it is a Hilbert space if Y is also a Hilbert space. We also introduce the following spaces

V1 = u ∈ (C ∞ c (Ω)) 3 : divu = 0 , V1 = the closure of V1 in H 1 0 (Ω), H1 = u ∈ L 2 (Ω) : divu = 0 and u • n = 0 on Γ , V2 = B ∈ (C ∞ (Ω)) 3 : divB = 0, B • n = 0 on Γ , V2 = B ∈ H 1 (Ω) : divB = 0; B • n = 0 on Γ , H2 = the closure of V2 in L 2 (Ω) .
(2.4) Thus H2 = H1. We endow Hi, i = 1, 2 with the inner product of L 2 (Ω) and the norm of L 2 (Ω) denote respectively by (., .) L 2 and |.| L 2 . We equip V1 with the following inner product

((u, v))1 = 3 i=1 ∂u ∂xi , ∂v ∂xi L 2 . (2.5)
We equip V2 with the scalar product

((u, v))2 = (curlu, curlv) L 2 .
(2.6)

Where curlu = ∇ ∧ u. We note that by Poincaré's inequality, the scalar product ((., .))1 defined in (2.5) coincides with the well known inner product in H 1 0 (Ω). The norm generated by ((., .))2 is equivalent to the norm induced by H 1 (Ω) on V2 (see [START_REF] Duvaut | Inequalities in Mechanics and Physics[END_REF]Chapter VII]). Hereafter, we set

H = H1 × H2, V = V1 × V2 . (2.7)
The dual space of V is denoted by V . We endow H with the inner products defined as: for all ϕ = (u, B), ψ = (v, C) ∈ H.

(ϕ, ψ) = (u, v) L 2 + (B, C) L 2 , [ϕ, ψ] = (u, v) L 2 + S(B, C) L 2 .
They generate equivalent norms (for 0 < S < ∞)

|ϕ| 2 H = (ϕ, ϕ) = |u| 2 L 2 + |B| 2 L 2 , [ϕ] 2 H = [ϕ, ϕ] = |u| 2 L 2 + S|B| 2 L 2 . (2.8)
We also endow V with the inner products

((ϕ, ψ)) = 1 Re ((u, v))1 + 1 Rm ((B, C))2, [[ϕ, ψ]] = 1 Re ((u, v))1 + S Rm ((B, C))2 , (2.9)
which in turn generate the equivalent norms on V

ϕ 2 V = ((ϕ, ϕ)), [[ϕ]] 2 V = [[ϕ, ϕ]] .
(2.10)

In order to give an abstract formulation of problem (1.2), we introduce the operators A1 ∈ L(V1, V 1 ), A2 ∈ L(V2, V 2 ), and A ∈ L(V, V ) defined by A1u, v = ((u, v))1, for all u, v ∈ V1 , A2B, C = ((B, C))2, for all B, C ∈ V2 , Aϕ, ψ = ((ϕ, ψ)), for all ϕ, ψ ∈ V .

(2.11) with domains

D(A1) = {u ∈ V1 : A1u ∈ H1} , D(A2) = {u ∈ V2 : A2u ∈ H2} , D(A) = D(A1) × D(A2) .
By the regularity of Γ, D(A) = H 2 ∩ V. From the continuity of the embedding of Vi into Hi, i = 1, 2, there exists constant κi, i = 1, 2 such that

|u| L 2 ≤ κ1 u V 1 for all u ∈ V1, |B| L 2 ≤ κ2 B V 2 for all B ∈ V2 .
(2.12)

The best constant κi is equal to 1 √

λ i 1
, where λ i 1 is the first eigenvalue of the compact operator A -1 i from Hi into itself. As in [START_REF] Sermange | Some mathematical questions related to the MHD equations[END_REF], we introduce the trilinear form

B0 on V × V × V by B0(ϕ1, ϕ2, ϕ3) = b(u1, u2, u3) -Sb(B1, B2, u3) + b(u1, B2, B3) -b(B1, u2, B3) , (2.13)
for all ϕi = (ui, Bi) ∈ V (i = 1, 2, 3), where b(•, •, •) is a continuous trilinear form defined on

H 1 (Ω) × H 1 (Ω) × H 1 (Ω) by b(u, v, w) = 3 i,j=1 Ω ui ∂vj ∂xi wjdx,
and satisfies the following standard relations,

b(u, v, v) = 0, ∀u ∈ V1, v ∈ H 1 (Ω), b(u, v, w) = -b(u, w, v), ∀u ∈ V1, v, w ∈ H 1 (Ω), |b(u, v, w)| ≤ c u 1/2 V 1 |A1u| 1/2 L 2 v V 1 |w| L 2 , ∀u ∈ D(A1), v ∈ V1, w ∈ H1 |b(b1, b2, u)| ≤ c|b1| 1/4 L 2 b1 3/4 V 2 u V 1 b2 V 2 , ∀b1, b2 ∈ V2, u ∈ V1, |b(b1, b2, u)| ≤ c b1 V 2 |A2b2| L 2 |u| L 2 , ∀b1 ∈ V2, b2 ∈ D(A2), u ∈ H1, |b(b1, u1, b2)| ≤ c b1 V 2 |A1u1| L 2 |b2| L 2 , ∀b1 ∈ V2, u1 ∈ D(A1), b2 ∈ H2. |b(u, v, w)| ≤ |u| L 6 |∇v| L 2 |w| 1/2 L 2 |w| 1/2 L 6 , ∀u, v, w ∈ H 1 (Ω) .
(2.14)

Remark 2.1 Using the inclusion of H 1 (Ω) in L p (Ω) for 1 ≤ p ≤ 6, we infer that trilinear form b(•, •, •) also satisfies |b(u, v, w)| ≤ u V 1 v V 1 |w| 1/2 L 2 w 1/2 V 1 , ∀u, v, w ∈ V1 . (2.15)
From (2.14), we infer that

B0(ϕ1, ϕ2, ϕ2) = 0, ∀ϕ1, ϕ2 ∈ V, B0(ϕ1, ϕ2, ϕ3) = -B0(ϕ1, ϕ3, ϕ2), ∀ϕi ∈ V, i = 1, 2, 3. (2.16)
Now we introduce the continuous bilinear form B : V × V → V by B(ϕ1, ϕ2), ϕ3 = B0(ϕ1, ϕ2, ϕ3) .

(2.17)

We also introduce a diagonal matrix M= (mij) 1≤i,j≤6 ∈ M6(R) defined by:

     mii = 1 if 1 ≤ i ≤ 3, mii = S if 4 ≤ i ≤ 6, mij = 0 if i = j.
(2.18)

Note that

B0(ϕ1, ϕ2, Mϕ2) = b(u1, u2, u2) + Sb(u1, B2, B2) -S[b(B1, B2, u2) + b(B1, u2, B2)] . ( 2
B0(ϕ1, ϕ2, Mϕ2) = 0 ∀ϕ1, ϕ2 ∈ V, B0(ϕ1, ϕ2, Mϕ3) = -B0(ϕ1, ϕ3, Mϕ2), ∀ϕi ∈ V, i = 1, 2, 3. (2.20)
We recall that (see [START_REF] Sermange | Some mathematical questions related to the MHD equations[END_REF] ) B0 and B satisfy the following estimates

|B0(ϕ1, ϕ2, ϕ3)| ≤ c ϕ1 V ϕ2 1/2 V |Aϕ2| 1/2 H |ϕ3|H , ∀ϕ1 ∈ V, ϕ2 ∈ D(A), ϕ3 ∈ H, B(ϕ, ϕ) V ≤ c|ϕ| 1/2 H ϕ 3/2 V .
(2.21)

Hereafter we set

B N 0 (ϕ1, ϕ2, ϕ3) = FN ( u2 V 1 )b(u1, u2, u3) -SFN ( (u2, B2) V )b(B1, B2, u3) + FN ( (u2, B2) V )b(u1, B2, B3) -FN ( (u2, B2) V )b(B1, u2, B3) , B N (ϕ1, ϕ2), ϕ3 = B N 0 (ϕ1, ϕ2, ϕ3), ∀ϕi = (ui, Bi) ∈ V, i = 1, 2, 3.
(2.22) Arguing similarly as in the proof of (2.21), we can check that the following inequalities hold

|B N 0 (ϕ1, ϕ2, ϕ3)| ≤ cN ϕ1 1/2 V |Aϕ1| 1/2 H |ϕ3|H +cSN ϕ1 1/2 V |Aϕ1| 1/2 H |ϕ3|H , ∀ϕ1 ∈ V, ϕ2 ∈ D(A), ϕ3 ∈ H . (2.23) Secondly |B N 0 (ϕ1, ϕ1, ϕ2)| ≤ cN |ϕ1| 1/4 H |Aϕ1| 3/4 H |ϕ2|H +cSN |ϕ1| 1/4 H |Aϕ1| 3/4 H |ϕ2|H , ∀ϕ1 ∈ D(A), ϕ2 ∈ H , (2.24) thirdly B N (ϕ1, ϕ2) V ≤ c|ϕ1| 1/4 H ϕ1 3/4 V |ϕ2| 1/4 H ϕ2 3/4 V +cS|ϕ1| 1/4 H ϕ1 3/4 V |ϕ2| 1/4 H ϕ2 3/4 V , ∀ϕi = (ui, Bi) ∈ V , (2.25) next B N (ϕ1, ϕ2) V ≤ cN ϕ1 V + cN S ϕ1 V , (2.26)
and finally

|B N 0 (ϕ1, ϕ1, ϕ2)| ≤ cN ϕ1 1/2 V |Aϕ1| 1/2 H |ϕ2|H +c ϕ1 3/2 V |Aϕ1| 1/2 H |ϕ2|H , ∀ϕ1 ∈ D(A), ϕ2 ∈ H . (2.27)
The analysis of (1.2) will also require the following version of Gronwall's lemma (see [START_REF] Sever | Some Gronwall Type Inequalities and Applications[END_REF]) Lemma 2.1 Let T > 0 and let κ be a non-negative function in L 1 (0, T ). Let c > 0 be a constant and ψ ∈ C 0 (0, T ) a function that satisfies

for all t ∈ [0, T ], 0 ≤ ψ(t) ≤ c + t 0 κ(s)ψ(s)ds, then ψ satisfies the bound ψ(t) ≤ ce t 0 κ(s)ds .
Here, C 0 (0, T ) denotes the set of continuous functions on [0, T ].

Let X a Banach space, we define BX (a, r) as an open ball of center a and the radius r in the space X.

One possibility to deal with infinite delays is to follow [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF][START_REF] Marín-Rubio | Pullback attractors for two-dimensional Navier-Stokes equations in an infinite delay case[END_REF][START_REF] Marín-Rubio | Pullback attractors for globally modified Navier -Stokes equations with infinite delays[END_REF]), which entails to consider, for any γ > 0, the space

Cγ(H) = ϕ ∈ C((-∞, 0]; H) : such that lim s→-∞
e γs ϕ(s) is well defined, and an element of H .

This is a Banach space with the norm

ϕ γ := sup s∈(-∞,0]
e γs |ϕ(s)| H .

Following [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF], more assumptions are required. For that purpose, we assume for i = 1, 2 and for some fixed γ > 0 that gi : (τ, T ) × Cγ(H) → L 2 (Ω) satisfies (h1) For any ξ = (ξ1, ξ2) ∈ Cγ(H), the mapping

g i (., ξ) : (τ, T ) → L 2 (Ω) t → g i (t, ξ) is measurable .
(h2) g i (t, 0) = 0 for all t ∈ (τ, T ). (h3) there exists a constant Lg i > 0 such that for any t ∈ (τ, T ) and for all ξ, η ∈ Cγ(H),

|g i (t, ξ) -g i (t, η)| L 2 ≤ Lg i ξ -η γ . Remark 2.2 (h2) and (h3) imply that for all ξ ∈ Cγ(H) |g i (t, ξ)| L 2 ≤ Lg i ξ γ so that |g i (., ξ)| ∈ L ∞ (τ, T ). If we set g = (g 1 , g 2 ), then from (h3), g(t, .) is Lipschitz-continuous on Cγ(H).
Using the notations above, we can rewrite (1.2) in the form

dy dt + Ay + B N (y, y) = F + Gt on D (τ, T ; V ), y(τ + s, x) = φ(s, x), s ∈ (-∞, 0], x ∈ Ω (2.28)
where y = (u, B), F = (f 1 , f 2 ), φ = (φ1, φ2) and Gt = (g 1 (t, (yt)), g 2 (t, (yt))) with yt = (ut, Bt). We can now define a concept of solution associated to (2.28).

Definition 2.1 We suppose (u(τ ), B(τ )) ∈ H, f i ∈ L 2 (τ, T ; V i ) and g i : (τ, T ) × Cγ(H)) → L 2 (Ω) satisfies (h1) -(h3) for some fixed γ > 0, i = 1, 2.
A weak solution of (2.28) is any pair y = (u, B) ∈ L 2 (τ, T ; V ) such that

dy dt + Ay + B N (y, y) = F + Gt on D (τ, T ; V ) y(τ + s, x) = φ(s, x), s ∈ (-∞, 0], x ∈ Ω (2.29)
or equivalently for all ϕ

= (v, C) ∈ V    dy dt , ϕ + ((y, ϕ)) + B N 0 (y, y, ϕ) = f 1 , v + f 2 , C + g 1 (t, yt), v + g 2 (t, yt), C , y(τ + s, x) = φ(s, x), s ∈ (-∞, 0], x ∈ Ω . (2.30) Remark 2.3 Definition 2.1 provides also the variational formulation of problem (1.2). If y = (u, B) ∈ L 2 (τ, T ; V ) satisfies (2.
29)1, it follows from (2.26),(2.27) and (h1) that dy dt ∈ L 2 (τ, T ; V ), and consequently (see [START_REF] Temam | Navier-Stokes Equations and Nonlinear Functional Analysis[END_REF]), y ∈ C([τ, T ); H) so that y(τ ) exists.

In addition, by taking ϕ = M y in (2.30)1 and using (2.20)1 we infer that y satisfies the following energy equality

|u(t)| 2 L 2 +S|B(t)| 2 L 2 + 2 Re t τ u(ξ) 2 V 1 dξ + 2S Rm t τ B(ξ) 2 V 2 dξ = |u0| 2 L 2 + S|B0| 2 L 2 + 2 t τ (f 1 (ξ), u(ξ))dξ + 2S t τ (f 2 (ξ), B(ξ))dξ +2 t τ (g 1 (ξ, (u ξ , B ξ )), u(ξ)) dξ + 2S t τ (g 2 (ξ, (u ξ , B ξ )), B(ξ)) dξ .
(2.31)

Existence and uniqueness result

In this section, we prove that problem (2.29) has a unique weak solution which is, under some conditions a strong solution. Before doing this, we recall from [START_REF] Caraballo | Unique strong solutions and V -attractors of a three dimensional system of globally modified Navier-Stokes equations[END_REF][START_REF] Romito | The uniqueness of weak solutions of the Globally modified Navier-Stokes equations[END_REF][START_REF] Medjo | Unique strong and V-attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model[END_REF] the following properties of FN , where the proof can be found in [START_REF] Caraballo | Unique strong solutions and V -attractors of a three dimensional system of globally modified Navier-Stokes equations[END_REF][START_REF] Romito | The uniqueness of weak solutions of the Globally modified Navier-Stokes equations[END_REF]. These properties are the main tools in the proof of the uniqueness result. We first recall that;

|FN (p) -FN (r)| ≤ |p-r| r , ∀p, r ∈ R + , r = 0, |FN ( u V 1 ) -FN ( v V 1 )| ≤ u-v V 1 v V 1 , u, v ∈ V1, v = 0, |FM (p) -FN (r)| ≤ |M -N | r + |p-r| r , ∀p, r, M, N ∈ R + , r = 0 |FN ( u V 1 ) -FN ( v V 1 )| ≤ 1 N FN ( u V 1 )FN ( v V 1 ) u -v V 1 , u, v ∈ V1 . (3.1)
In the rest of this paper we will denote by c, a generic positive constant (possibly depending on S, Re, Rm, κ1, κ2, Ω, Lg 1 , Lg 2 ), which can vary even within the same line. However, this constant is always independent of time and initial data. We start by proving the uniqueness result; for this purpose, we have the following.

Theorem 3.1 There exists at most one weak solution (u, B) of (2.29) in the sense of definition 2.1.

Proof. Let yi = (ui, Bi), i = 1, 2 be weak solutions to (2.29) that belong to L 2 (0, T ; V ). We set δy = (δu, δB) = y1 -y2, uit(s) = ui(t + s), Bit(s) = Bi(t + s), s ∈ (-∞, 0]. Then (δu, δB) satisfies

   dδy dt + Aδy = -B N (y1, y1) -B N (y2, y2) + (G(t, (u1t, B1t)) -G(t, (u2t, B2t))) , δy(τ ) = 0. (3.2)
Taking the scalar product in H of (3.2) with M δy, we obtain

dY dt + 2 Re δu 2 V 1 + 2S Rm δB 2 V 2 = -2(B N (y1, y1) -B N (y2, y2), M δy)+ 2 (G(t, (u1t, B1t)) -G(t, (u2t, B2t)), M δy) (3.3) with Y = |δu| 2 L 2 + S|δB| 2 L 2 and 2(-B N (y1, y1) + B N (y2, y2
), M δy) satisfies the following (see [START_REF] Deugoué | Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamics equations[END_REF] for the details)

2 -B N (y1, y1) + B N (y2, y2), M y ≤ cN 4 + cN 8 Y . (3.4)
Using (3.4) and hypothesis (h3) in (3.3), we obtain Remark 3.1 It is worth mentioning that the uniqueness of solution is one of the important property of this model because precisely we do not have that property for the corresponding 3d magnetohydrodynamics version. We can now thing about a complete study of attractor in classical way [START_REF] Temam | Infinite dimensional dynamical systems in mechanics and physics[END_REF]. This by the way is the object of our next investigation. Now, we state the existence result.

dY dt + 2 Re δu 2 V 1 + 2S Rm δB 2 V 2 ≤ cN 4 + cN 8 Y + 2 Lg 1 + SLg 2 δyt γ |δy| H (3.5) Observe that δy(s) = (0, 0) if s ≤ τ, δyt γ = sup s∈(-∞,0] e γs |δy(t + s)| H ≤ sup s∈[τ -t,0] |δy(t + s)| H . (3.6) Dropping momentarily the term 2 Re δu 2 V 1 + 2S Rm δB 2 V 2 in (3.
Theorem 3.2 We suppose (u(τ ), B(τ )) = (φ1(0), φ2(0)) ∈ H, f i ∈ L 2 (τ, T ; Hi) and g i : (τ, T ) × Cγ(H)) → L 2 (Ω) satisfies (h1) -(h3) for some fixed γ > 0, i = 1, 2. Let φ = (φ1, φ2) ∈ Cγ(H) be given, with R := φ γ .
Then there exists a unique weak solution (u, B) of (2.29), which is in fact a strong solution in the sense that it belongs to

C(τ, T ; V ) ∩ L 2 (τ + , T ; D(A1) × D(A2)) for all 0 < < T -τ.
(3.9)

Moreover, if (φ1(0), φ2(0)) ∈ V , then (u, B) satisfies (u, B) ∈ C(τ, T ; V ) ∩ L 2 (τ, T ; D(A1) × D(A2)). (3.10)
Proof We split it in several steps. Step1: A Galerkin scheme. Since the injection V ⊂ H is compact, let {(wi, ψi), i = 1, 2, ...} ⊂ V be an orthonormal basis of H, where {wi, i = 1, 2, ....}, {ψi, i = 1, 2, ....} are eigenfunctions of A1 and A2, respectively. We set Vn = Hn = span {(w1, ψ1), ..., (wn, ψn)} and denote by Pn = (P 1 n , P 2 n ), the orthogonal projector from H onto Vn for the scalar product (., .) defined by (2.8)1. Note that Pn is also the orthogonal projector from D(A), V, V onto Vn. We look for yn = Pn(u, B) = (un, Bn) ∈ Hn solution to the ordinary differential equations with delay

dyn dt + Ayn + PnB N (yn, yn) = PnF + PnGt yn(τ + s) = Pn(φ1(s), φ2(s)) = (P 1 n φ1(s), P 2 n φ2(s)), s ∈ (-∞, 0]. (3.11)
According to (h1) -(h3), the above system of the ordinary differential equations with infinite delay satisfies the conditions for existence and uniqueness of solution yn on an interval [τ, Tn], Tn ≤ T (see Theorem 1.1 of [START_REF] Hino | Functional Differential Equations with Infinite Delay[END_REF]). It will follow from a priori estimates that yn exists on the interval [τ, T ].

Step2: A priori estimates. As in remark 2.3, yn satisfies the following energy inequality: (3.12)

d dt |un(t)| 2 L 2 + S d dt |Bn(t)| 2 L 2 + 2 Re un(t) 2 V 1 + 2S Rm Bn(t) 2 V 2 ≤ 2(P 1 n f 1 (t)
We need to estimate the terms on the right hand side of (3.12). First by Young's and Cauchy-Schwartz's inequalities, we have

2|(P 1 n f 1 (t), un(t))| ≤ 2 f 1 (t) V 1 un(t) V 1 ≤ 1 2Re un(t) 2 V 1 + c f 1 (t) 2 V 1 , (3.13)
2S|(P 2 n f 2 (t), Bn(t))| ≤ 2S f 2 (t) V 2 Bn(t) V 2 ≤ S 2Rm Bn(t) 2 V 2 + c f 2 (t) 2 V 2 , (3.14)
2|(P 1 n g 1 (t, (un,t, Bn,t)), un(t

))| ≤ 2 g 1 (t, (un,t, Bn,t)) V 1 un(t) V 1 ≤ 2c g 1 (t, (un,t, Bn,t)) H 1 un(t) V 1 ≤ 2c (un,t, Bn,t) γ un(t) V 1 ≤ 1 2Re un(t) 2 V 1 + c (un,t, Bn,t) 2 γ , (3.15)
2S|(P

|un(t)| 2 L 2 + S|Bn(t)| 2 L 2 + 1 Re t τ un(ξ) 2 V 1 dξ + S Rm t τ Bn(ξ) 2 V 2 dξ ≤ |φ1(0)| 2 L 2 +S|φ2(0)| 2 L 2 + c t τ f 1 (ξ) 2 V 1 dξ + c t τ f 2 (ξ) 2 V 2 dξ + c t τ (u n,ξ , B n,ξ ) 2 γ dξ . (3.17) Furthermore,
(un,t, Bn,t)

2 γ = sup θ∈(-∞,0] e 2γθ |(un(t + θ), Bn(t + θ))| 2 H = sup θ∈(-∞,0] e 2γθ |un(t + θ)| 2 L 2 + |Bn(t + θ))| 2 L 2 ≤ sup θ∈(-∞,0] e 2γθ |φ1(0)| 2 L 2 + S|φ2(0)| 2 L 2 + c t+θ τ f 1 (ξ) 2 V 1 dξ+ c t+θ τ f 2 (ξ) 2 V 2 dξ + c t+θ τ (u n,ξ , B n,ξ ) 2 γ dξ ≤ max sup θ∈(-∞,τ -t] e 2γθ [φ(θ + t -τ )] 2 , sup θ∈[τ -t,0] e 2θγ |φ1(0)| 2 L 2 + S|φ2(0)| 2 L 2 + c t+θ τ f 1 (ξ) 2 V 1 dξ+ c t+θ τ f 2 (ξ) 2 V 2 dξ + c t+θ τ (u n,ξ , B n,ξ ) 2 γ dξ      ≤ max sup θ∈(-∞,τ -t] e 2γθ [φ(θ + t -τ )] 2 , [φ(0)] 2 H + c t τ f 1 (ξ) 2 V 1 dξ + c t τ f 2 (ξ) 2 V 2 dξ + c t τ (u n,ξ , B n,ξ ) 2 γ dξ    . (3.18) Observing that sup θ∈(-∞,τ -t] e γθ [φ(θ + t -τ )] = sup s∈(-∞,0] e γ(s-(t-τ )) [φ(s)] = sup s∈(-∞,0] e γs [φ(s)] e -(t-τ ) ≤ φ γ .
and [(u(τ ), B(τ ))] = [φ(0)] ≤ φ γ , we deduce from (3.18)

(un,t, Bn,t) 2 γ ≤ R 2 + c t τ f 1 (ξ) 2 V 1 dξ + c t τ f 2 (ξ) 2 V 2 dξ + c t τ (u n,ξ , B n,ξ ) 2 γ dξ . (3.19)
By the Lemma 2.1, we have

(un,t, Bn,t) 2 γ ≤ R 2 e c(t-τ ) + c t τ f 1 (ξ) 2 V 1 dξ + f 2 (ξ) 2 V 2 dξe c(t-τ ) . (3.20)
Thus, there exists a constant

K1 = K1(R, τ, Lg 1 , Lg 2 , T, f 1 , f 2 ) > 0 such that (un,t, Bn,t) 2 γ ≤ K1, (3.21)
which together with (3.17) gives

| un(t)| 2 L 2 + S|Bn(t)| 2 L 2 + 1 Re T τ un(ξ) 2 V 1 dξ + S Rm T τ Bn(ξ) 2 V 2 dξ ≤ R 2 + c T τ f 1 (ξ) 2 V 1 dξ + c T τ f 2 (ξ) 2 V 2 dξ + cK1(T -τ ) . (3.22)
(3.22) proves that the sequence yn = (un, Bn) remains in a bounded set of L ∞ (τ, T ; H) ∩ L 2 (τ, T ; V ) ∩ Cγ(H). Hence, we can use a compactness argument (see [START_REF] Temam | Infinite dimensional dynamical systems in mechanics and physics[END_REF]) to extract a subsequence from yn = (un, Bn) still denoted by yn = (un, Bn) satisfying

yn → y          weak-star in L ∞ (τ, T ; H) , weakly in L 2 (τ, T ; V ) , strongly in L 2 (τ, T ; H) , a.e., in (τ, T ) × Ω , (3.23) with y = (u, B) ∈ L ∞ (τ, T ; H) ∩ L 2 (τ, T ; V ) ∩ Cγ(H).
But the estimates (3.22) are not enough to pass to the limit in (2.29) and deduce the solution of (1.2). More precisely, we have two main difficulties, firstly, we need to pass to the limit on the G(t, (un,t, Bn,t)), this will be done on Step 3; secondly, we need to prove that

FN ( un V 1 ) → FN ( u V 1 ) as n → ∞ , FN ( (un, Bn) V ) → FN ( (u, B) V ) as n → ∞ , (3.24)
To overcome the second difficulty, we need to find a stronger estimate and it is the aim of the lines below. Taking the inner product in H between the first equation of (3.11) with Ayn, we obtain

d dt yn 2 V + 2|Ayn| 2 H = 2(f 1 , A1un) + 2(f 2 , A2Bn) -2B N 0 (yn, yn, Ayn)
+ 2(g 1 (t, (un,t, Bn,t), A1un)) + 2(g 2 (t, (un,t, Bn,t), A2Bn)) .

(3.25)

Now using (2.23) and Young's inequality with the exponents (4, 4/3), we have

2|B N 0 (yn, yn, Ayn)| ≤ cN yn 1/2 V |Ayn| 3/2 H ≤ 1 4 |Ayn| 2 H + cN 4 yn 2 V . (3.26)
In addition, by Young's inequality, (h2) -(h3), (3.21) one obtains

2|(f 1 , A1un)| + 2|(f 2 , A2Bn)| ≤ 1 4 |A1un| 2 L 2 + 1 4 |A2Bn| 2 L 2 + c|f 1 | 2 L 2 + c|f 2 | 2 L 2 = 1 4 |Ayn| 2 H + c|f 1 | 2 L 2 + c|f 2 | 2 L 2 (3.
d dt yn 2 V + |Ayn| 2 H ≤ c|f 1 | 2 L 2 + c|f 2 | 2 L 2 + cK 2 1 + cN 4 yn 2 V . (3.29)
Now we distinguish two cases:

Case 1: y(τ ) = (u(τ ), B(τ )) ∈ H.
Integrating (3.29) between s and t for τ < s ≤ t ≤ T, we obtain

yn(t) 2 V + t s |Ayn(ξ)| 2 H dξ ≤ yn(s) 2 V + c t s |f 1 (ξ)| 2 L 2 + |f 2 (ξ)| 2 L 2 + K 2 1 dξ + cN 4 t s yn(ξ) 2 V dξ. ≤ yn(s) 2 V + c t τ |f 1 (ξ) 2 L 2 + |f 2 (ξ)| 2 L 2 + K 2 1 dξ + cN 4 t τ yn(ξ) 2 V dξ . (3.30)
Momentarily dropping the term

t s |Ayn(ξ)| 2
H dξ in (3.30) and integrating once more between τ and τ + for some ∈ (0, T -τ ), we have

τ + τ yn(t) 2 V ds ≤ τ + τ yn(s) 2 V ds + τ + τ c t τ |f 1 (ξ) 2 L 2 + |f 2 (ξ)| 2 L 2 + K 2 1 dξ ds + cN 4 τ + τ t τ yn(ξ) 2 V dξ ds . (3.31)
Since τ + ≤ T, it follows from (3.31) that

yn(t) 2 V ≤ T τ yn(s) 2 V ds + c(T -τ ) T τ |f 1 (ξ)| 2 L 2 + |f 2 (ξ)| 2 L 2 + K 2 1 dξ+ c(T -τ )N 4 T τ yn(ξ) 2 V dξ . (3.32)
From the estimate (3.22), we infer that the right hand side of (3.32) is bounded independently of n. Coming back to (3.30) and dropping the term yn(t) 2 V , we get for some

∈ [0, T -τ ] T τ + |Ayn(ξ)| 2 H dξ ≤ yn(s) 2 V + c T τ |f 1 (ξ)| 2 L 2 + |f 2 (ξ)| 2 L 2 + K 2 1 dξ + cN 4 T τ yn(ξ) 2 V dξ . (3.33) We then deduce that yn ∈ L ∞ (τ + , T ; V ). Therefore yn ∈ L ∞ (τ + , T ; V ) ∩ L 2 (τ + , T ; D(A1) × D(A2)) for all 0 < < T -τ . (3.
         weak-star in L ∞ (τ + , T ; V ) , weakly in L 2 (τ + , T ; D(A)) , strongly in L 2 (τ + , T ; V ) , a.e., in (τ + , T ) × Ω , (3.36)
and

d dt (un, Bn) → d dt (u, B) weakly in L 2 (τ + , T ; H) . (3.37)
From (3.36), we can assume, eventually extracting a subsequence of yn still denoted yn such that

un V 1 → u V 1 a.e. in (τ + , T ) , (un, Bn) V → (u, B) V a.e. in (τ + , T ) , (3.38)
and therefore

FN ( un V 1 ) → FN ( u V 1 ) a.e. in (τ + , T ), FN ( (un, Bn) V ) → FN ( (u, B) V ) a.

e. in (τ + , T ). (3.39)

Case 2: (φ1(0), φ2(0)) ∈ V .

We mention that (φ1n(0), φ2n(0)) V = Pn(φ1(0), φ2(0)) V ≤ y(τ ) V .

Now, dropping the term |Ayn| 2 H in (3.29), we have the following differential inequality

d dt yn 2 V ≤ c|f 1 | 2 L 2 + c|f 2 | 2 L 2 + cK 2 1 + cN 4 yn 2 V , (3.40)
from which we obtain by using Lemma 2.1 Since D(A) = D(A1) × D(A2) ⊂ V ⊂ H with compact injection, we derive from [25, Theorem 5.1, Chapter 1] that there exists an element (u, B) ∈ L ∞ (τ, T ; V ) ∩ L 2 (τ, T ; D(A)), and a subsequence of (un, Bn) (still) denoted (un, Bn) such that for all T > τ , we have From (3.44), we infer that

yn(t) 2 V ≤ y(τ ) 2 V exp cN 4 (t -τ ) + c exp{cN 4 (t -τ )} t τ |f 1 (ξ)| 2 L 2 + |f 2 (ξ)| 2 L 2 + cK
(un, Bn) → (u, B)          weak-star in L ∞ (τ, T ; V ) , weakly in L 2 (τ, T ; D(A)) , strongly in L 2 (τ, T ; V ) , a.e., in (τ, T ) × Ω ,
un V 1 → u V 1 a.e. in (τ, T ) , (un, Bn) V → (u, B) V a.e. in (τ, T ) , (3.46)
and therefore

FN ( un V 1 ) → FN ( u V 1 ) a.e. in (τ, T ) , FN ( (un, Bn) V ) → FN ( (u, B) V ) a.e. in (τ, T ) . ( 3

.47)

Step3: Passage to the limit.

We want to take the limit in (3.11) when n goes to +∞. We focus our attention on the term G(t, (un,t, Bn,t)) we refer the reader to [START_REF] Caraballo | Unique strong solutions and V -attractors of a three dimensional system of globally modified Navier-Stokes equations[END_REF][START_REF] Deugoué | Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamics equations[END_REF] for the other terms. More precisely, we want to prove that G(t, (un,t, Bn,t)) → G(t, (ut, Bt)) when n → +∞ .

(3.48)

We proceed like in [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF] where the globally modified Navier-Stokes with infinite delays is investigated. We start by proving that We start by proving (3.50). Indeed, if we assume that it is not held, then there exists > 0 and a subsequence (θn)n such that e γθn |Pnφ(θn) -φ(θn)| H > .

(un,t, Bn,t) → (ut, Bt) in Cγ(H), ∀t ∈ (-∞, T ] . ( 3
(3.52)

Hence following [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF], we assume that θn → -∞, otherwise if θn → θ = -∞, then Pnφ(θn) → φ(θ). In fact, Next, we will prove (3.51). From the convergence of yn to y in L 2 (τ, T ; H) given by (3.23), we deduce that yn(t) → y(t) in H a.e. t ∈ (τ, T ]. Since

|Pnφ(θn) -φ(θ)| H ≤ |Pnφ(θn) -Pnφ(θ)| H + |Pnφ(θ) -φ(θ)| H → 0 as n → +∞ . Assume that θn → -∞ as n → +∞, if we set x := lim
yn(t) -yn(s) = t s y (ξ)dξ ∀s, t ∈ [τ, T ] we have yn(t) -yn(s) V ≤ T τ y (ξ) V dξ .
Then from remark 2.3, we deduce that T τ y (ξ) V dξ is bounded. Thus yn is equi-continuous on [τ, T ] with values in V . In addition, yn(t) is bounded in V . In fact, due to the convergence of yn(t) to y(t) in H for a.e.t ∈ [τ, T ], we infer that yn(t) is bounded in H and by the compactness injection of H in to V , we claim that yn(t) is bounded in V . By the Ascoli-Arzela theorem, we have yn → y in C((τ, T ]; V ).

(3.53)

Also by the convergence of yn to y in L ∞ (τ, T ; H), we obtain that for any sequence {tn} ⊂ [τ, T ] with tn → t, one has yn(tn) → y(t) weakly in H. To prove that this is absurd, we will use an energy method. Observe that the sequence yn also satisfies the following energy inequality

|yn(t)| 2 H + 1 Re t s un(ξ) 2 V 1 dξ + S Re t s Bn(ξ) 2 V 2 dξ ≤ |yn(s)| 2 H + c t s f 1 (ξ) 2 V 1 + f 2 (ξ) 2 V 2 dξ + K1(t -s) . (3.56)
On the other hand, since the functions yn are bounded in L ∞ (τ, T ; H), we deduce the existence of ηG = (ηg 1 , ηg 2 ) ∈ L 2 (τ, T ; H) such that (g 1 (t, yn,t), g 2 (t, yn,t)) converges weakly to ηG in L 2 (τ, T ; H). Then passing to the limit in (2.30), we deduce that y is a solution of

d dt (y(t), (v, C)) H + 1 Re ((u(t), v))) V 1 + S Re ((B(t), C))) V 2 + B N (y(t), y(t), (v, C)) = (f 1 (t), v) + (f 2 (t), C) + (ηg 1 (t, (ut, Bt)), v) + (ηg 2 (t, (utBt)), C) . (3.57)
Thus we deduce that, y satisfies the energy inequality

|y(t)| 2 H + 1 Re t s u(ξ) 2 V 1 dξ + S Re t s B(ξ) 2 V 2 dξ ≤ |y(s)| 2 H + c t s f 1 (ξ) 2 V 1 + f 2 (ξ) 2 V 2 dξ + t s |ηG(ξ)| 2 H dξ . (3.58) Since t s |ηG(ξ)| 2 H dξ ≤ lim inf n→∞ t s |(g 1 (ξ, y n,ξ ), g 2 (ξ, y n,ξ ))| 2 H dξ ≤ K1(t -s) .
It is noted that y satisfies also the inequality (3.56) with the same constant K1. Now we consider the functions Jn, J : [τ, T ] → R defined by

Jn(t) = 1 2 |yn(t)| 2 H -c t τ f 1 (ξ) 2 V 1 + f 2 (ξ) 2 V 2 dξ -K1t , and
J(t) = 1 2 |y(t)| 2 H -c t τ f 1 (ξ) 2 V 1 + f 2 (ξ) 2 V 2 dξ -K1t .
Jn and J are continuous and non-increasing functions. Next, we show that for t ≥ s, Jn(t) -Jn(s) ≤ 0. From a direct definition of Jn, we have

Jn(t) -Jn(s) = 1 2 |yn(t)| 2 H - 1 2 |yn(s)| 2 H -c t τ f 1 (ξ) 2 V 1 + f 2 (ξ) 2 V 2 dξ -K1t + K1s = 1 2 |yn(t)| 2 H - 1 2 |yn(s)| 2 H -c t s f 1 (ξ) 2 V 1 + f 2 (ξ) 2 V 2 dξ + K1(s -t) ≤ 0 .
Similarly, we can prove that J is a non-increasing function. this is important since we will approach this value t0 by a sequence tk , this means that lim k→∞ tk → t0, with tk being taken only when (3.59) is valid. Since y(•) is continuous at t0 and tk → t0, for any > 0, there exists k > 0 such that

J( tk ) -J(t0) < 2 .
On the other hand, taking n > n(k ) such that tn > tk , as Jn is non-increasing and for all tk , the convergence (3.59) holds, one has that

Jn(tn) -J(t0) ≤ Jn( tk ) -J( tk ) + J( tk ) -J(t0)
and obviously taking n > n (k ), it is possible due to (3.59) to obtain Jn( tk ) -J( tk ) ≤ 2 .

Moreover, we deduce from (3.23)

tn τ (F (ξ), yn(ξ))dξ → t 0 τ (F (ξ), y(ξ))dξ,
so we conclude that (3.63) holds. Thus (3.60) and finally (3.51) are also true as we wanted to prove.

Now we are ready to pass to the limit in (3.26). Assume initially that y(τ ) = φ(0) ∈ H, the first consequence of the convergence proved above since g i satisfies (h3) is that (g 1 (., (un,., Bn,.)).g 2 (., (un,., Bn,.))) → (g 1 (., (u., B.)).g 2 (., (u., B.))) in L 2 (τ, T ; H).

Hence, we can identify (ηg 1 , ηg 2 ) = (g 1 , g 2 ) in (3.57) so that y is a solution of (2.29).

In the next lines, we prove that the solution of (2.29) given by Theorem 3.2 is continuous in respect to the initial data as well as in the parameter N . More precisely, we prove the following result. Theorem 3.3 Assume that f i ∈ L 2 (τ, T ; Hi) and g i : (τ, T ) × Cγ(H)) → L 2 (Ω) satisfies (h1) -(h3) for some fixed γ > 0, i = 1, 2. Let φi = (φi,1, φi,2) ∈ Cγ(H) be given, with Ri := φi γ and Ni > 0, yi(τ ) = (ui(τ ), Bi(τ )) ∈ V, i = 1; 2 be given. Let yi = (ui, Bi) be the solutions of (2.29) corresponding to the parameter Ni and the initial values yi(τ ) = (ui(τ ), Bi(τ )), i = 1; 2. Then (u1, B1) → (u2, B2) in C(τ, T ; V ) ∩ D(τ, T ; D(A1) × D(A2)) when N1 → N2, (u1(τ ), B1(τ )) → (u2(τ ), B2(τ )) and φ1 → φ2. More precisely, let y = y1 -y2 and φ = φ1 -φ2, the following estimates hold true.

sup θ∈[τ,t] y(θ) 2 V ≤ y(τ ) 2 V + (t -τ ) φ 2 γ + c(N1 -N2) 2 t τ Z1(ξ)dξ × exp c(η + N 4 1 )(t -τ ) + c t τ Z1(ξ)dξ , (3.64)
and

t τ |Ay(ξ)| 2 H dξ ≤ y(τ ) 2 V + (t -τ ) φ 2 γ + c(N1 -N2) 2 t τ Z1(ξ)dξ × 1 + c(N 4 1 + η)(t -τ ) + c t τ Z1(ξ)dξ × exp c(N 4 1 + η)(t -τ ) + c t τ Z1(ξ)dξ , (3.65) for all t ∈ [τ, T ] with Z1 = |A1u2| 2 L 2 + |A2B2| 2 L 2 . Proof. Since y = y1 -y2 = (u1, B1) -(u2, B2) = (δu, δB) and φ = φ1 -φ2 = (φ1,1 - φ2,1, φ1,2 -φ2,2), then y = (δu, δB) satisfies dy dt + Ay + B N 1 (y1, y1) -B N 2 (y2, y2) = G(t, (u1t, B1t)) -G(t, (u2t, B2t)) . (3.66)
From [START_REF] Caraballo | Unique strong solutions and V -attractors of a three dimensional system of globally modified Navier-Stokes equations[END_REF][START_REF] Deugoué | Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamics equations[END_REF], we have

R1 ≡ FN 1 ( u1 V 1 )b(u1, u1, A1δu) -FN 2 ( u2 V 1 )b(u2, u2, A1δu) = FN 1 ( u1 V 1 )b(δu, u1, A1δu) + FN 2 ( u2 V 1 )b(u2, δu, A1δu) + [FN 1 ( u1 V 1 ) -FN 2 ( u2 V 1 )] b(u2, u1, A1δu) .
(3.67)

Making similar reasoning as in (3.67), we can also check that:

R2 ≡ FN 1 ( (u1, B1) V )b(B1, B1, A1δu) -FN 2 ( (u2, B2) V )b(B2, B2, A1δu) = FN 1 ( (u1, B1) V )b(δB, B1, A1δu) + FN 2 ( (u2, B2) V )b(B2, δB, A1δu) + [FN 1 ( (u1, B1) V ) -FN 2 ( (u2, B2) V )] b(B2, B1, A1δu) ≡ R 1 2 + R 2 2 + R 3 2 , (3.68) R3 ≡ FN 1 ( (u1, B1) V )b(u1, B1, A2δB) -FN 2 ( (u2, B2) V )b(u2, B2, A2δB) = FN 1 ( (u1, B1) V )b(δu, B1, A2δB) + FN 2 ( (u2, B2) V )b(u2, δB, A2δB) + [FN 1 ( (u1, B1) V ) -FN 2 ( (u2, B2) V )] b(u2, B1, A2δB) ≡ R 1 3 + R 2 3 + R 3 3 , (3.69)
R4 ≡ FN 1 ( (u1, B1) V )b(B1, u1, A2δB) -FN 2 ( (u2, B2) V )b(B2, u2, A2δB) = FN 1 ( (u1, B1) V )b(δB, u1, A2δB) + FN 2 ( (u2, B2) V )b(B2, δu, A2δB) + [FN 1 ( (u1, B1) V ) -FN 2 ( (B2, u1) V )] b(B2, u1, A2δB) ≡ R 1 4 + R 2 4 + R 3 4 .
(3.70) Also, we can check that

B N 1 (y1, y1) -B N 2 (y2, y2), Ay = R1 -SR2 + R3 -R4 . (3.71) d dt y 2 V + |Ay| 2 H ≤ c(N 4 1 + |A1u2| 2 L 2 + |A2B2| 2 L 2 + η) sup θ∈[τ,t] y(θ) 2 V + c(|A1u2| 2 L 2 + |A2B2| 2 L 2 )(N1 -N2) 2 + φ 2 γ ≡ c(N 4 1 + η + Z1) sup θ∈[τ,t] y(θ) 2 V + cZ1(N1 -N2) 2 + φ 2 γ .
(3.85) Integrating (3.85) from τ to t, we have

y(t) 2 V + t τ |Ay(ξ)| 2 H dξ ≤ y(τ ) 2 V + (t -τ ) φ 2 γ + c(N1 -N2) 2 t τ Z1(ξ)dξ + t τ c(η + N 4 1 + Z1(ξ)) sup θ∈[τ,ξ] y(θ) 2 V dξ , (3.86)
which leads to

sup θ∈[τ,t] y(θ) 2 V + t τ |Ay(ξ)| 2 H dξ ≤ y(τ ) 2 V + (t -τ ) φ 2 γ + c(N1 -N2)
y(θ) 2 V ≤ y(τ ) 2 V + (t -τ ) φ 2 γ + c(N1 -N2) 2 t τ Z1(ξ)dξ × exp c(η + N 4 1 )(t -τ ) + c t τ Z1(ξ)dξ , (3.88)
which proves (3.64). Now using (3.87) and (3.88), we get

t τ |Ay(ξ)| 2 H dξ ≤ y(τ ) 2 V + (t -τ ) φ 2 γ + c(N1 -N2) 2 t τ Z1(ξ)dξ × 1 + c(N 4 1 + η)(t -τ ) + c t τ Z1(ξ)dξ × exp c(N 4 1 + η)(t -τ ) + c t τ Z1 (

Stationary solution

In this section, we are interested in proving that problem (1.2) with some restrictions, admits stationary solutions. Also, we prove under additional assumptions that the stationary solution is unique and is globally asymptotically exponentially stable. The restrictions we impose to give sense to a stationary solution are that f 1 , f 2 , g 1 and g 2 are independent on time. One of the worries is that how (g 1 , g 2 ) acts over a fixed element of H. Inspired by what was done in [START_REF] Marín-Rubio | Three dimensional system of globally modified Navier -Stokes equations with infinite delays[END_REF], we consider (g 1 (w), g 2 (w)) as (g 1 ( w), g 2 ( w)), where w ∈ H is the element that has the only value w for all time t ≤ 0. w is an element of Cγ(H) and w γ = |w| H for some fixed γ > 0; so we will continue denoting w instead of w since no confusion arises. The hypothesis (h3) is reformulated as follows: There exists a constant Lg i > 0 such that for all ξ, η ∈ H,

|g i (ξ) -g i (η)| L 2 (Ω) ≤ Lg i |ξ -η| H . (4.1)
We consider the following system

dy dt + Ay + B N (y, y) = F + G on D (τ, T ; V ) y(τ + s) = (φ1(s), φ2(s)), s ∈ (-∞, 0] (4.2)
where A and B N are defined in section 3. By a stationary solution to (4.2) we mean an element (u * , B * ) ∈ V such that for all ϕ = (v, C) ∈ V,

((y, ϕ)) + B N 0 (y, y, ϕ) = f 1 , v + f 2 , C + g 1 (u * , B * ), v + g 2 (u * , B * ), C . (4.3)
We will prove the existence result by Galerkin's method before studiying its asymptotic behavior.

Existence result

Theorem 4.1 We assume that f 1 , f 2 , g 1 , g 2 are independent on time, and (4.1) holds. If

1 > cLg 1 ( Re λ 1 1 ) 1/2 + cLg 2 ( Rm λ 2 1 ) 1/2 , Problem (4.3) has at least one solution (u * , B * ) which belongs to D(A). In addition, if min {1, S} ≥ max Re(λ 1 1 ) -1 , SRm(λ 2 1 ) -1 cN 4 + cN 8 + cLg 1 ( Re λ 1 1 ) 1/2 + ScLg 2 ( Rm λ 2 1
) 1/2 , (4.4) this solution is unique.

Proof. Like dealing with the evolutionary case, the existence of a solution y * = (u * , B * ) of problem (4.3) is proved by the Galerkin's method as follows. Since the injection V ⊂ H is compact, let {(wi, ψi), i = 1, 2, ...} ⊂ V be an orthonormal basis of H, where {wi, i = 1, 2, ....}, {ψi, i = 1, 2, ....} are eigenfunctions of A1 and A2 respectively. We set Vm = Hm = span {(w1, ψ1), ..., (wm, ψm)} and denote by Pm = (P 1 m , P 2 m ), the orthogonal projector from H onto Vm for the scalar product (., .) defined by (2.8)1. Note that Pm is also the orthogonal projector from D(A), V, V onto Vm. We look for y

* m = (u * m , B * m ), verifying for ϕ = (v, C) ∈ Vm Ay * m , ϕ + PnB N (y * m , y * m ), ϕ = f 1 , v + f 2 , C + g 1 (u * m , B * m ), v + g 2 (u * m , B * m ), C . (4.5)
Since we will apply a consequence of Brouwer's fixed point theorem, see ( [START_REF] Zhang | Renormalized notes for Navier-Stokes equations by Roger Temam[END_REF], Lemma 41, page 23), we define the operators m : Vm → Vm by, for all y

= (u, B), ϕ = (v, C) ∈ Vm . (( my, ϕ)) = Ay, ϕ + PmB N (y, y), ϕ -f 1 , v -f 2 , C -g 1 (u, B), v -g 2 (u, B), C . (4.6)
Since the right hand side of (4.6) is a linear continuous map from Vm to R, by the Riez theorem, each my ∈ Vm is well defined. We now prove that m is continuous. Let (yn) ⊂ Vm be a sequence which converges to y in Vm, since A and B N are continuous from Vm to V m then Ayn → Ay and B N (yn, yn) → B N (y, y). In addition,

g i (yn) -g i (y) V i,m ≤ |g i (yn) -g i (y)| L 2 ≤ Lg i |yn -y| H ≤ cLg i |yn -y| V .
By the compactness of the embedding Vm → Hm and (4.6), we infer that Rmyn → Rmy in Vm as n → ∞. On the other hand, for all y = (u, B) ∈ Vm,

(( my, y)) = Ay, y -f 1 , u -f 2 , B -g 1 (u, B), u -g 2 (u, B), B ≥ y 2 V -(λ 1 1 ) -1/2 |f 1 | L 2 u V 1 -(λ 2 1 ) -1/2 |f 2 | L 2 B V 2 - c(λ 1 1 ) -1/2 Lg 1 y V v V 1 -c(λ 2 1 ) -1/2 Lg 2 y V C V 2 ≥ y V y V 1 -cLg 1 ( Re λ 1 1 ) 1/2 + cLg 2 ( Rm λ 2 1 ) 1/2 - ( Re λ 1 1 ) 1/2 |f 1 | L 2 -( Rm λ 2 1 ) 1/2 |f 2 | L 2 (4.7) where c = max λ 1 1 Re, λ 2 1 Rm 1/2 .
Then, (( my, y)) is non negative on the sphere of Vm with radius

β ≥ ( Re λ 1 1 ) 1/2 |f 1 | L 2 + ( Rm λ 2 1 ) 1/2 |f 2 | L 2 1 -cLg 1 ( Re λ 1 1 ) 1/2 + cLg 2 ( Rm λ 2
| 2 H = (f 1 , A1u * m ) + (f 2 , A2B * m ) -B N 0 (y * m , y * m , Ay * m ) + (g 1 (u * m , B * m ), A1u * m )) + (g 2 (u * m , B * m ), A2B * m )) . (4.8)
Now using (2.23) and Young's inequality with the exponents (4, 4/3), we have From (4.12), we infer that the sequence y * m is bounded in D(A); consequently, using the compact injection of D(A) in V, we can extract a subsequence of y * m still denoted by y * m which converges weakly in D(A) and strongly in V to an element (u * , B * ) ∈ D(A). Finally taking the limit in (4.5), we prove that (u * , B * ) is a solution of a stationary problem (4. Taking the inner product in H of (4.13) with M y and proceed like proving the uniqueness result in the non-stationary case and taking into account (4.1), we obtain Seeing how (3.17)

|B N 0 (y * m , y * m , Ay * m )| ≤ cN y * m 1/2 V |Ay * m | 3/2 H ≤ 1 8 |Ay * m | 2 H + cN 4 ym 2 V . ( 4

Conflict of interest:

The authors declare that there is no conflict of interest

  B) weakly in L 2 (τ, T ; H) .(3.45)

e

  γθ φ(θ), we obtain e γθn |Pnφ(θn) -φ(θn)| H = Pn(e γθn φ(θn)) -e γθn φ(θn) H ≤ Pn(e γθn φ(θn)) -Pnx H + |Pnx -x| H + x -e γθn φ(θn) H → 0 as n → +∞ which is a contradiction with (3.52), hence we have (3.50).

  we prove (3.51) by a contradiction argument. If (3.51) does not hold, then using the fact that yn ∈ C([τ, T ]; H), there would exist > 0, a value t0 ∈ [τ, T ] and subsequences labelled the same yn and tn ⊂ [τ, T ] with t0 = limn→+∞ tn such that |yn(tn) → y(t0)| H ≥ .(3.55)

Remark 3 . 2

 32 ξ)dξ . (3.89) Theorem 3.3 also provides the uniqueness of the strong solution of problem (1.2).

  3). For the uniqueness, let y * = (u * , B * ) and y * = ( u * , B * ) two solutions of (4.3), we set y = y * -y * ; u = u * -u * and B = u * -u * then Ay = -B N (y * , y * ) -B N (ỹ * , ỹ * ) + G(u * , B * ) -G(ũ * , B * )) . (4.13)

e 2 H dξ = t 0 2 H dξ ≤ t 0 ee 2 H 2 H≤ 2 H.e 1 Re (λ 1 1 ) + L 2 g 2 Rme 1 Re (λ 1 1 ) + L 2 g 2 Rm≤ 1 Re (λ 1 1 ) + L 2 g 2 Rm

 2020222112112112 2γθ |(w(ξ + θ)| 2 H ; sup θ∈[-ξ,0] e 2γθ |(w(ξ + θ)| max e -(2γ-β)ξ φ -y * 2 γ ; sup θ∈[-s,0] e (2γ-β)θ e β(ξ+θ) |(w(ξ + θ)| -(2γ-β)ξ φ -y * 2 γ + sup r∈[0,ξ] e rβ |(w(r)| 2 H dξ (4.24)where we have used β ≤ 2γ to deduce the last inequality. Using this in (4.23), we obtaine βt [w(t)] 2 ≤ [w(0)] 2 + L 2We can choose β such that the coefficient of the last integral in (4.25) is negative. Doing that, we deduce (4.18). On the other hand, 2γθ |w(t+ θ)| = max sup θ≤-t e 2γθ |(w(t + θ)| 2 H ; sup θ∈[-t,0] e 2γθ |(w(t + θ)| max e -2γt φ -y * 2 γ , sup θ∈[-t,0] e 2γθ |(w(t + θ)| 2γθ e -β(t+θ) [φ(0) -y * ] 2 + L 2 g (2γ-β)θ e -βt [φ(0) -y * ] 2 + L 2 g e -βt [φ(0) -y * ] 2 + L 2 gsince e (2γ-β)θ ≤ 1 for θ ∈ [-t, 0]. Hence, we deduce(4.19).

  2 n g 2 (t, (un,t, Bn,t)), Bn(t))| ≤ 2S g 2 (t, (un,t, Bn,t)) V 2 Bn(t) V 2 ≤ S

	2Rm Bn 2 V 2 + c (un,t, Bn,t) 2 γ .	(3.16)

where (h2) -(h3) have been used to derive (3.15) and

(3.16)

. Inserting the estimates (3.13)-(3.16) in

(3.11)

and integrating from τ to some τ ≤ t ≤ T, we obtain

  PnB N (yn, yn) + PnF + PnG.

			34)
	Note from (3.11) that
	dyn dt = -Ayn -Then using (2.23) we deduce that
	the sequence PnB Therefore, from (3.21) and (3.34), we infer that the sequence
	d dt	(un, Bn) is also bounded in L 2 (τ + , T ; H) .	(3.35)
	Since D(A) = D(A1) × D(A2) ⊂ V ⊂ H with compact injection, we derive from [25, Theorem
	5.1,Chapter 1] that there exists an element (u,

N (yn, yn) is bounded in L 2 (τ + , T ; H) . B) ∈ L ∞ (τ + , T ; V ) ∩ L 2 (τ + , T ; D(A))

, and a subsequence of (un, Bn) (still) denoted (un, Bn) such that for all T > τ + , we have (un, Bn) → (u, B)

  Bn) is bounded in L ∞ (τ, T ; V ) ∩ L 2 (τ, T ; D(A1) × D(A2)).Note that in (3.42), K2 and K3 are positive constants independent of n and depending only on data Ω, Re, Rm, S, f 1 , f 2 , T, u0, B0, Lg 1 and Lg 2 . PnB N (yn, yn) + PnF + PnG .

		T
	(un, Bn)(t) 2 V ≤ K2,	|A1un(ξ)| 2 L 2 + |A2Bn(ξ)| 2 L 2 dξ ≤ K3 ,	(3.42)
		τ
	which proves that (un, Note that from (3.11) that
	dyn dt = -Ayn -Then using (2.23) we deduce that the sequence
	PnB N (yn, yn)
	Therefore, from (3.33) and (3.21), we infer that the sequence
	d dt	(un,
				2 1 dξ .
				(3.41)
	Hence, we derive from (3.29) and (3.32) that (yn) = (un, Bn) satisfies

n is bounded in L 2 (τ, T ; H) . Bn) is also bounded in L 2 (τ, T ; H) . (

3

.43)

  Moreover by the convergence of yn to y a.e. in time with value on H, it holds that

	Jn(t) → J(t) a.e.in H.	(3.59)
	Now, we want to prove that
	yn(tn) → y(t0) in H,	(3.60)
	which contradicts (3.55).
	Firstly, from (3.54) we recall that
	yn(tn) → y(t0) weakly in H,	(3.61)
	then
	|y(t0)| H ≤ lim inf n→∞	|yn(tn)| H .	(3.62)
	Therefore if we prove that
	lim sup n→∞	|yn(tn)| H ≤ |y(t0)| H	(3.63)
	we obtain lim

n→∞ |yn(tn)| H → |y(t0)| which jointly with (3.61) imply (3.60). If t0 = τ, it follows from (3.50) and (3.58) that lim sup n→∞ |yn(tn)| H ≤ |y(τ )| . So we may assume that t0 > τ ;

  L 2 + (cLg 1 ) 2 + (cLg 2 ) 2 y * H ≤ c|f 1 | 2 L 2 + c|f 2 | 2 L 2 + (cLg 1 ) 2 + (cLg 2 ) 2 + cN 4 y *

					.9)
	In addition, by Young's inequality and (4.1) one obtains
		|(f 1 , A1u * m )| + |(f 2 , A2B * m )| ≤ 1 8 |A1u * m | 2 L 2 + 1 8 |A2B * m | 2 L 2 + c|f 1 | 2 L 2 + c|f 2 | 2 L 2 = 1 8 |Ay * m | 2 H + c|f 1 | 2 L 2 L 2 + c|f 2 | 2	(4.10)
	and	|(g 1 (u * m , B * m ), A1u * m )) + (g 2 (t, (u * m , B * m ), A2B * m ))|
		≤ 1 4 |A1u * m | 2 L 2 + 1 4 |A2B * m | 2	m	2 V	(4.11)
		= 1 4 |Ay * m | 2 H + (cLg 1 ) 2 + (cLg 2 ) 2 y * m	2 V .
	It follows from (4.9)-(4.11) that
		|Ay * m | 2		m	2 V .	(4.12)

  is obtained, we have Introducing in(4.22) an exponential term e βt with a positive value 0 < β < 2γ and integrating from 0 to t, we obtain e βt [w(t)] 2 + min {1, S} -2 cN 4 + cN 8 max

		d dt	|un(t)| 2 L 2 + S|Bn(t)| 2 L 2 +	2 Re		un(t) 2 V 1 +	2S Rm	Bn(t)	2 V 2
		≤ L 2 g 1	Re λ 1 1	wt 2 γ +	1 Re	u 2 V 1 + L 2 g 2	Rm Sλ 2 1	wt 2 γ +	S Rm	B	2 V 2	(4.21)
		+ 2 cN 4 + cN 8 max	1 λ 1 1	,	S 1 λ 2			(u, B) 2 V ,
	which leads to
	d dt	[w(t)] 2 + min {1, S} -2 cN 4 + cN 8 max	1 λ 1 1	,	S 1 λ 2	w(t) 2 V .
					≤ L 2 g 2	Rm Sλ 2 1	+ L 2 g 1	Re 1 λ 1	wt 2 γ .	(4.22)
										1 λ 1 1	,	S λ 2 1	-β max	1 λ 1 1	,	1 1 λ 2	0	t	e βξ w(ξ) 2 V dξ .
			≤ [w(0)] 2 + L 2 g 2	Rm Sλ 2 1	+ L 2 g 1	Re 1 λ 1	0	t	e βξ w ξ	2 γ dξ .

Hence, taking the scalar product in H of (3.66) with Ay, we obtain d dt y 2 V + 2|Ay| 2 H = -2R1 + 2SR2 -2R3 + 2R4 + 2(G(t, (u1t, B1t)) -G(t, (u2t, B2t)), Ay).

(3.72) We can check that (see [START_REF] Caraballo | Unique strong solutions and V -attractors of a three dimensional system of globally modified Navier-Stokes equations[END_REF][START_REF] Deugoué | Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamics equations[END_REF])

3 and R i 4 , i = 1, 2, 3 satisfy the following estimates (see [START_REF] Deugoué | Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamics equations[END_REF] for more details):

where we have used also the embedding of V into H and set η = max Lg 1 , Lg 2 . Then

Now inserting these estimates (3.73)-(3.84) in (3.72), we obtain

On the other hand,

Stability of the stationary solution

As announced before, we prove here that the unique solution of (4.3) given by Theorem 4.1 is globally asymptotically exponentially stable. More precisely, we prove the following result.

Theorem 4.2 Assume that f 1 , f 2 , g 1 , g 2 are independent on time and (4.4) is valid. Assume that the assumptions in Theorem 3.2 are valid. Then for some fixed γ > 0, there exists a value 0 < β < 2γ such that for the solution y(., 0, φ) = (u(., 0, φ), B(., 0, φ)) of problem (1.2) with τ = 0 and φ ∈ Cγ(H),the following estimates hold for all t ≥ 0 :

and

where y * is the unique solution of (4.5) given by Theorem 4.1.

Proof. Just for simplification, we denote y(t) = y(., 0, φ). We also denote w(t) = y(t) -y * . From equations (4.2) and ( 4.