Three dimensional system of globally modified magnetohydrodynamics equations with infinite delays

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Three dimensional system of globally modified magnetohydrodynamics equations with infinite delays

G Deugoue, Jules Djoko, A Fouape

To cite this version:

G Deugoue, Jules Djoko, A Fouape. Three dimensional system of globally modified magnetohydro- dynamics equations with infinite delays. [Research Report] African Peer Review Mechanism. 2021.

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Three dimensional system of globally modified magnetohydrodynamics equations with infinite delays

G. Deugoue

, J. K. Djoko

, and A.C. Fouape

1,3Universite de Dschang, departement de Mathematiques et Informatique, Cameroun

2African peer review mechanism, Johannesburg, South Africa

July 25, 2021

Abstract

Existence and uniqueness of strong solutions for three dimensional system of globally modi- fied magnetohydrodynamics equations containing infinite delays terms are established together with some qualitative properties of the solution in this work. The existence is proved by mak- ing 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.

Keywordsmagnetohydrodynamics equations; globally modified; strong solutions; infinite delays.

Contents

1 Introduction and statement of the problem 1

2 Preliminaries 3

3 Existence and uniqueness result 8

4 Stationary solution 20

4.1 Existence result . . . 21 4.2 Stability of the stationary solution . . . 23

1 Introduction and statement of the problem

Let Ω⊂R³ be an open bounded set with regular boundary Γ =∂Ω,andN >0 be fixed. We defineFN: (0,+∞)→(0,1] by

FN(r) = min

1,N r

, r∈R⁺

and consider the following system of globally modified magnetohydrodynamics equations (GMMHDE)































∂u

∂t +FN(kukV₁)[(u· ∇)u]− 1 Re

∆u−SFN(k(u,B)kV) [(B· ∇)B] +∇

p+S|B|² 2

=f₁(t) in (τ, T)×Ω,

∂B

∂t +FN(k(u,B)kV) [(u· ∇)B−(B· ∇)u] + 1 Rm

curl(curlB) =f₂(t) in (τ, T)×Ω, divu= 0, divB= 0 in(τ, T)×Ω,

u(τ, x) =u0(x), B(τ, x) =B0(x) for allx∈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 pres- sure. f₁ andf₂ are given external forces fields. Re and Rm are the so-called Reynolds and magnetic Reynolds numbers, respectively andS = _R^M2

eR_m 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 andτ the initial time. It is manifest that the system of equa- tions in (1.1) does not represent the MHD model due to un-physical terms introduced such asFN(kukV₁), FN(k(u,B)kV). These terms can find their existence in the original model of globally modified Navier Stokes introduced in [5]. As clearly demonstrated in [5],FN(kukV₁) 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 [13, 14, 15]. 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. [9]), 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 [14] 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 [28], but we also note the contribution in [2, 3, 4, 29, 32, 38]. 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(kukV₁)[(u· ∇)u]− 1 Re

∆u−SFN(k(u,B)kV) [(B· ∇)B]

+∇

p+S^|B₂^|2

=f₁(t) +g₁(t,(ut,Bt)) in (τ, T)×Ω,

∂B

∂t +FN(k(u,B)kV) [(u· ∇)B−(B· ∇)u] + 1 Rm

curl(curlB) =f₂(t)+

g₂(t,(ut,Bt)) in (τ, T)×Ω, divu= 0, divB= 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,

(1.2)

where g₁(t,(ut,Bt)) andg₂(t,(ut,Bt)) are another external forces containing some heredi- tary characteristic (delays terms), whereut and Bt are functions defined on (−∞,0] by the relationsut(s) =u(t+s) andBt(s) =B(t+s) respectively. φ1 andφ2 are given functions defined in the interval (−∞,0].Since the initial time isτ,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 magne- tohydrodynamics (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 [28]. Such models (with delays) have been intensively investigated for many years ( see [2, 3, 4, 29, 32, 38], just to cite some); but globally modified MHD with delays remain to be explored. This work follow our initial works [13, 14, 15] where the focus is on dynamics of globally modified Navier-Stokes coupled with magnetic field or the heat. We should also men- tioned that the inspiration from this work comes from the work of J. Real and the coauthors in [28]. 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 [2, 3, 4, 28, 29, 32, 38]. 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.

2 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 [1, 43]. 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 spacesL²(Ω), H^m(Ω), H0^m(Ω) and we set

L²(Ω) := (L²(Ω))³, H^m(Ω) := (H^m(Ω))³, H^m0 (Ω) := (H0^m(Ω))³,L²0(Ω) := (L²0(Ω))³ (2.1) whereL²0(Ω) =

q∈L²(Ω);

Z

Ω

q(x)dx= 0

.It is noted that for a vectorwwe set kwk^r_Lr(Ω)=

Z

Ω

|w(x)|^rdx ,

where|·|denotes the Euclidean norm|w|²=w·w. We shall frequently use Sobolev imbedding:

for a real numberp∈R, 1≤p≤6, the spaceH¹(Ω) is imbedded intoL^p(Ω). In particular, there exists a constantcp(that depends only onp, Ω andd= 3) such that

for all v∈H¹0, kvk_L^p(Ω)≤cpk∇vk_L^p(Ω) . (2.2) Whenp = 2, this is Poincare’s inequality and c2 is Poincare’s constant. In the case of the maximum norm, the following imbedding holds

for allr > d= 3, W^1,r(Ω)⊂L^∞(Ω) in particular, for eachr > d= 3, there existsc∞,r such that

for all v∈H¹0(Ω)∩W^1,r, kvk_L^∞(Ω)≤c∞,rk∇vk_L^r(Ω). (2.3) Owing to Poincare’s inequality, the semi-norm| · |is a norm onH¹0(Ω), equivalent to the full norm. As it is directly related gradient operator, we take this semi-norm as norm onH¹0(Ω), and we use it to define the dual norm on its dual spaceH⁻¹(Ω):

for all f ∈ ⁻¹(Ω), kfk = sup hf,vi ,

where h·i is the duality pairing between H⁻¹(Ω) and H¹0(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, sayY. More precisely, we denote byk · kY the norm onY and for any numberr with 1≤r≤ ∞, we define

L^r(a, b;Y) ={wmeasurable in (a, b) ; Z b

a

kw(t)k^rYdt <∞}

equipped with the norm

kwk^r_Lr(^a,b;Y)= Z b

a

kw(t)k^rYdt

with the usual modification ifr=∞. It is a Banach space ifY is a Banach space, and when r= 2, it is a Hilbert space ifY is also a Hilbert space.

We also introduce the following spaces V1=

u∈(Cc^∞(Ω))³: divu= 0 , V1= the closure ofV1 inH¹0(Ω), H1 =

u∈L²(Ω) : divu= 0 andu·n= 0 on Γ , V2=

B∈(C^∞(Ω))³: divB= 0, B·n= 0 on Γ , V2=

B∈H¹(Ω) : divB= 0; B·n= 0 on Γ , H2 = the closure ofV2 inL²(Ω).

(2.4)

ThusH2=H1. We endowHi, i= 1,2 with the inner product ofL²(Ω) and the norm ofL²(Ω) denote respectively by (., .)_L2 and|.|L².

We equipV1 with the following inner product ((u,v))1=

3

X

i=1

∂u

∂xi

, ∂v

∂xi

L²

. (2.5)

We equipV2 with the scalar product

((u,v))2= (curlu,curlv)_L2. (2.6) Where curlu=∇ ∧u. We note that by Poincar´e’s inequality, the scalar product ((., .))1

defined in (2.5) coincides with the well known inner product inH¹0(Ω). The norm generated by ((., .))2 is equivalent to the norm induced byH¹(Ω) onV2 (see [16, Chapter VII]). Hereafter, we set

H=H1×H2, V =V1×V2. (2.7)

The dual space ofV is denoted byV⁰. We endowH with the inner products defined as: for allϕ= (u,B), ψ= (v,C)∈H.

(ϕ, ψ) = (u,v)_L2+ (B,C)_L2, [ϕ, ψ] = (u,v)_L2+S(B,C)_L2. They generate equivalent norms (for 0< S <∞)

|ϕ|²H= (ϕ, ϕ) =|u|²_L2+|B|²_L2, [ϕ]²H = [ϕ, ϕ] =|u|²_L2+S|B|²_L2. (2.8) We also endowV 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 onV

kϕk²V = ((ϕ, ϕ)), [[ϕ]]²V = [[ϕ, ϕ]]. (2.10)

In order to give an abstract formulation of problem (1.2), we introduce the operators A1∈ L(V1, V1⁰), A2∈ L(V2, V2⁰), andA ∈ L(V, V⁰) defined by

hA1u,vi= ((u,v))1, for allu,v∈V1, hA2B,Ci= ((B,C))2, for allB,C∈V2, hAϕ, ψi= ((ϕ, ψ)), for allϕ, 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) =H²∩V.From the continuity of the embedding ofVi intoHi, i= 1,2, there exists constantκi, i= 1,2 such that

|u|_L2≤κ1kukV1 for allu∈V1, |B|_L2 ≤κ2kBkV2 for allB∈V2. (2.12) The best constant κi is equal to √¹

λⁱ₁, where λⁱ1 is the first eigenvalue of the compact operatorA⁻¹_i fromHiinto itself. As in [36], we introduce the trilinear formB0 onV ×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 allϕi = (ui,Bi)∈V(i= 1,2,3), whereb(·,·,·) is a continuous trilinear form defined on H¹(Ω)×H¹(Ω)×H¹(Ω) by

b(u,v,w) =

3

X

i,j=1

Z

Ω

ui

∂vj

∂xi

wjdx,

and satisfies the following standard relations, b(u,v,v) = 0, ∀u∈V1, v∈H¹(Ω),

b(u,v,w) =−b(u,w,v), ∀u∈V1, v,w∈H¹(Ω),

|b(u,v,w)| ≤ckuk^1/2_V1 |A1u|^1/2_L2kvkV₁|w|L², ∀u∈D(A1),v∈V1,w∈H1

|b(b1,b2,u)| ≤c|b1|^1/4_L2kb1k^3/4_V

2 kukV1kb2kV2, ∀b1,b2∈V2,u∈V1,

|b(b1,b2,u)| ≤ckb1kV₂|A2b2|L²|u|L², ∀b1∈V2,b2∈D(A2),u∈H1,

|b(b1,u1,b2)| ≤ckb1kV₂|A1u1|_L2|b2|_L2, ∀b1∈V2,u1∈D(A1),b2∈H2.

|b(u,v,w)| ≤ |u|L⁶|∇v|L²|w|^1/2_L2|w|^1/2_L6, ∀u,v,w∈H¹(Ω).

(2.14)

Remark 2.1 Using the inclusion of H¹(Ω)in L^p(Ω)for 1 ≤p≤6, we infer that trilinear formb(·,·,·) also satisfies

|b(u,v,w)| ≤ kukV1kvkV1|w|^1/2_L2kwk^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 formB:V ×V →V⁰ by

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







mii= 1 if 1≤i≤3, mii=Sif 4≤i≤6, mij= 0 ifi6=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)]. (2.19) It follows from (2.14) and (2.19) that

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 [36] )B0 andBsatisfy the following estimates

|B0(ϕ1, ϕ2, ϕ3)| ≤ckϕ1kVkϕ2k^1/2_V |Aϕ2|^1/2_H |ϕ3|H, ∀ϕ1∈V, ϕ2∈D(A), ϕ3∈H, kB(ϕ, ϕ)kV⁰ ≤c|ϕ|^1/2_H kϕk^3/2_V .

(2.21) Hereafter we set

B^N0 (ϕ1, ϕ2, ϕ3) =FN(ku2kV1)b(u1,u2,u3)−SFN(k(u2,B2)kV)b(B1,B2,u3) +FN(k(u2,B2)kV)b(u1,B2,B3)−FN(k(u2,B2)kV)b(B1,u2,B3), DB^N(ϕ1, ϕ2), ϕ3

E

=B^N0 (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

|B₀^N(ϕ1, ϕ2, ϕ3)| ≤cNkϕ1k^1/2_V |Aϕ1|^1/2_H |ϕ3|H

+cSNkϕ1k^1/2_V |Aϕ1|^1/2_H |ϕ3|H, ∀ϕ1∈V, ϕ2∈D(A), ϕ3∈H . (2.23) Secondly

|B0^N(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

kB^N(ϕ1, ϕ2)kV⁰ ≤c|ϕ1|^1/4_H kϕ1k^3/4_V |ϕ2|^1/4_H kϕ2k^3/4_V

+cS|ϕ1|^1/4_H kϕ1k^3/4_V |ϕ2|^1/4_H kϕ2k^3/4_V , ∀ϕi= (ui,Bi)∈V , (2.25) next

kB^N(ϕ1, ϕ2)kV⁰ ≤cNkϕ1kV +cN Skϕ1kV, (2.26) and finally

|B^N0 (ϕ1, ϕ1, ϕ2)| ≤cNkϕ1k^1/2_V |Aϕ1|^1/2_H |ϕ2|H

+ckϕ1k^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 [35]) Lemma 2.1 Let T > 0 and letκ be a non-negative function in L¹(0, T). Let c > 0 be a constant andψ∈ C⁰(0, T)a function that satisfies

for allt∈[0, T], 0≤ψ(t)≤c+ Z t

0

κ(s)ψ(s)ds, thenψsatisfies the bound

ψ(t)≤ce Z t

0

κ(s)ds . Here,C⁰(0, T)denotes the set of continuous functions on[0, T].

LetX a Banach space, we defineBX(a, r) as an open ball of centeraand the radiusrin the spaceX.

One possibility to deal with infinite delays is to follow [28, 29, 30]), which entails to consider, for anyγ >0,the space

Cγ(H) =

ϕ∈ C((−∞,0];H) : such that lim

s→−∞e^γsϕ(s) is well defined, and an element ofH

.

This is a Banach space with the norm kϕk_γ:= sup

s∈(−∞,0]

e^γs|ϕ(s)|_H .

Following [28], more assumptions are required. For that purpose, we assume fori= 1, 2 and for some fixedγ >0 that

gi: (τ, T)× Cγ(H)→L²(Ω) satisfies

(h1) For anyξ= (ξ1, ξ2)∈ Cγ(H),the mapping g_i(., ξ) : (τ, T) → L²(Ω)

t 7→ g_i(t, ξ) is measurable.

(h2)g_i(t,0) = 0 for allt∈(τ, T).

(h3) there exists a constantLg_i>0 such that for anyt∈(τ, T) and for allξ, η∈ Cγ(H),

|g_i(t, ξ)−g_i(t, η)|_L2≤Lg_ikξ−ηk_γ .

Remark 2.2 (h2) and (h3) imply that for all ξ ∈ Cγ(H) |g_i(t, ξ)|_L2 ≤ Lg_ikξk_γ so that

|g_i(., ξ)| ∈L^∞(τ, T).

If we setg= (g₁,g₂), then from(h3), g(t, .) is Lipschitz-continuous onCγ(H).

Using the notations above, we can rewrite (1.2) in the form ( dy

dt +Ay+B^N(y, y) =F+Gt on D⁰(τ, T;V⁰), y(τ+s, x) =φ(s, x), s∈(−∞,0], x∈Ω

(2.28) where y= (u,B), F = (f₁,f₂), φ= (φ1, φ2) andGt = (g₁(t,(yt)),g₂(t,(yt))) withyt = (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∈L²(τ, T;Vi⁰)andg_i: (τ, T)× Cγ(H))→ L²(Ω)satisfies(h1)−(h3)for some fixedγ >0, i= 1, 2.

A weak solution of(2.28)is any pairy= (u,B)∈L²(τ, T;V) such that ( dy

dt +Ay+B^N(y, y) =F+Gt on D⁰(τ, T;V⁰) y(τ+s, x) =φ(s, x), s∈(−∞,0], x∈Ω

(2.29) or equivalently for allϕ= (v,C)∈V





 dy

dt, ϕ

+ ((y, ϕ)) +B^N0(y, y, ϕ) =hf₁,vi+hf₂,Ci+hg₁(t, yt),vi+hg₂(t, yt),Ci, 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) ∈ L²(τ, T;V⁰) satisfies (2.29)1, it follows from (2.26),(2.27) and (h1) that dy

dt ∈L²(τ, T;V⁰), and consequently (see [41]),y∈ C([τ, T);H)so thaty(τ) exists.

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

|u(t)|²_L2 +S|B(t)|²_L2+_R²

e

Z t τ

ku(ξ)k²V₁dξ+ 2S Rm

Zt τ

kB(ξ)k²V₂dξ

=|u0|²_L2+S|B0|²_L2+ 2 Z t

τ

(f₁(ξ),u(ξ))dξ+ 2S Z t

τ

(f₂(ξ),B(ξ))dξ +2

Z t τ

(g₁(ξ,(uξ,Bξ)),u(ξ))dξ+ 2S Z t

τ

(g₂(ξ,(uξ,Bξ)),B(ξ))dξ .

(2.31)

3 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 [5, 34, 37] the following properties ofFN, where the proof can be found in [5, 34]. 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⁺, r6= 0,

|FN(kukV₁)−FN(kvkV₁)| ≤ ^ku−vk_V 1 kvk_V

1

, u,v∈V1, v6= 0,

|FM(p)−FN(r)| ≤ ^|M−N|_r +^|p−r|_r , ∀p, r, M, N∈R⁺, r6= 0

|FN(kukV1)−FN(kvkV1)| ≤ _N¹FN(kukV1)FN(kvkV1)ku−vkV1, u,v∈V1.

(3.1)

In the rest of this paper we will denote byc, a generic positive constant (possibly depending onS, Re, Rm, κ1, κ2,Ω, Lg₁, Lg₂), 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 defini- tion 2.1.

Proof. Letyi= (ui,Bi), i= 1,2 be weak solutions to (2.29) that belong toL²(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 inH of (3.2) withM δy, we obtain

dY dt + 2

Re

kδuk²_V1+ 2S Rm

kδBk²_V2 =−2(B^N(y1, y1)− B^N(y2, y2), M δy)+

2 (G(t,(u1t,B1t))−G(t,(u2t,B2t)),Mδy)

(3.3)

withY=|δu|²_L2+S|δB|²_L2 and 2(−B^N(y1, y1) +B^N(y2, y2), M δy) satisfies the following (see [13] for the details)

2

−B^N(y1, y1) +B^N(y2, y2),My

≤ cN⁴+cN⁸

Y. (3.4)

Using (3.4) and hypothesis (h3) in (3.3), we obtain dY

dt + 2 Re

kδuk²_V1+ 2S Rm

kδBk²_V2 ≤ cN⁴+cN⁸ Y+ 2

Lg₁+SLg₂ kδytk_γ|δy|_H (3.5) Observe thatδy(s) = (0,0) ifs≤τ,

kδytk_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 _R²

ekδuk²V₁+_R^2S

mkδBk²V₂ in (3.5), we have Y(t) ≤ cN⁴+cN⁸

Z t τ

Y(ξ)dξ+ 2η Z t

τ

sup

s∈[τ−ξ,0]

|δy(ξ+s)|_H|δy|_Hdξ

≤ cN⁴+cN⁸ Z t

τ

Y(ξ)dξ+ 2η Z t

τ

sup

s∈[τ,ξ]

|δy(s)|²_Hdξ

≤ (cN⁴+cN⁸) max{1, S}+ 2η Z t

τ

sup

s∈[τ,ξ]

|δy(s)|²_Hdξ,

(3.7)

whereη=Lg₁+SLg₂.

From (3.7), we have for anyt∈[τ, T] min{1, S} sup

s∈[τ,t]

|δy(s)|²_H ≤ (cN⁴+cN⁸) max{1, S}+ 2η Z t

τ

sup

s∈[τ,ξ]

|δy(s)|²_Hdξ . (3.8) The use of Lemma 2.1 leads to sup

s∈[τ,t]

|δy(s)|²_H ≤0 from which we infer thatu1 =u2 and

B1 =B2.

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 [42]. This by the way is the object of our next investigation.

Now, we state the existence result.

Theorem 3.2 We suppose(u(τ),B(τ)) = (φ1(0), φ2(0))∈H, f_i∈L²(τ, T;Hi)and g_i : (τ, T)× Cγ(H)) → L²(Ω) satisfies (h1)−(h3) for some fixed γ > 0, i = 1, 2. Let φ = (φ1, φ2) ∈ Cγ(H) be given, with R :=kφk_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)∩L²(τ+, T;D(A1)×D(A2)) for all0< < T−τ. (3.9) Moreover, if(φ1(0), φ2(0))∈V, then(u,B)satisfies

(u,B)∈ C(τ, T;V)∩L²(τ, T;D(A1)×D(A2)). (3.10) Proof We split it in several steps.

Step1: A Galerkin scheme. Since the injectionV ⊂H is compact,

let{(wi, ψi), i= 1,2, ...} ⊂V be an orthonormal basis ofH, where{wi, i= 1,2, ....}, {ψi, i= 1,2, ....}are eigenfunctions ofA1 andA2, respectively. We set

Vn=Hn= span{(w1, ψ1), ...,(wn, ψn)}and denote byPn= (P_n¹, P_n²), the orthogonal projec- tor from H onto Vn for the scalar product (., .) defined by (2.8)1. Note thatPn is also the orthogonal projector fromD(A), V, V⁰ ontoVn. We look foryn=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_n¹φ1(s), P_n²φ2(s)), s∈(−∞,0].

(3.11) According to (h1)−(h3),the above system of the ordinary differential equations with in- finite delay satisfies the conditions for existence and uniqueness of solutionyn on an interval [τ, Tn], Tn≤T (see Theorem 1.1 of [17]). It will follow from a priori estimates thatynexists on the interval [τ, T].

Step2: A priori estimates. As in remark 2.3,ynsatisfies the following energy inequal- ity:

d

dt|un(t)|²_L2+S d

dt|Bn(t)|²_L2+ 2 Re

kun(t)k²V₁+ 2S Rm

kBn(t)k²V₂

≤2(Pn¹f₁(t),un(t)) + 2S(Pn¹f₂(t),Bn(t)) + 2(Pn¹g₁(t,(un,t,Bn,t)),un(t)) + 2S(Pn²g₂(t,(un,t,Bn,t)),Bn(t)).

(3.12)

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|(Pn¹f₁(t),un(t))| ≤2kf₁(t)kV₁⁰kun(t)kV₁ ≤_2R¹

ekun(t)k²V₁+ckf₁(t)k²_V0

1, (3.13) 2S|(Pn²f₂(t),Bn(t))| ≤2Skf₂(t)kV₂⁰kBn(t)kV₂≤ _2R^S

mkBn(t)k²V₂+ckf₂(t)k²_V0

2, (3.14) 2|(Pn¹g₁(t,(un,t,Bn,t)),un(t))| ≤2kg₁(t,(un,t,Bn,t))kV₁⁰kun(t)kV₁

≤2ckg₁(t,(un,t,Bn,t))kH₁kun(t)kV₁

≤2ck(un,t,Bn,t)kγkun(t)kV₁

≤_2R¹

ekun(t)k²_V1+ck(un,t,Bn,t)k²_γ,

(3.15)

2S|(P_n²g₂(t,(un,t,Bn,t)),Bn(t))| ≤2Skg₂(t,(un,t,Bn,t))k_V⁰

2kBn(t)kV₂

≤_2R^S

mkBnk²_V2+ck(un,t,Bn,t)k²γ. (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 fromτ to someτ≤t≤T,we obtain

|un(t)|²_L2+S|Bn(t)|²_L2+ 1 Re

Z t τ

kun(ξ)k²V₁dξ+ S Rm

Zt τ

kBn(ξ)k²V₂dξ≤ |φ1(0)|²_L2

+S|φ2(0)|²_L2+c Z t

τ

kf₁(ξ)k²_V⁰

1dξ+c Z t

τ

kf₂(ξ)k²_V⁰

2dξ+c Zt

τ

k(un,ξ,Bn,ξ)k²γdξ . (3.17) Furthermore,

k(un,t,Bn,t)k²_γ = sup

θ∈(−∞,0]

e^2γθ|(un(t+θ),Bn(t+θ))|²_H

= sup

θ∈(−∞,0]

e^2γθ

|un(t+θ)|²_L2+|Bn(t+θ))|²_L2

≤ sup

θ∈(−∞,0]

e^2γθ

|φ1(0)|²_L2+S|φ2(0)|²_L2+c Z t+θ

τ

kf₁(ξ)k²_V0 1dξ+

c Zt+θ

τ

kf₂(ξ)k²V₂⁰dξ+c Z t+θ

τ

k(un,ξ,Bn,ξ)k²γdξ

≤ max (

sup

θ∈(−∞,τ−t]

e^2γθ[φ(θ+t−τ)]², sup

θ∈[τ−t,0]

e^2θγ |φ1(0)|²_L2+S|φ2(0)|²_L2+c

t+θ

R

τ

kf₁(ξ)k²_V0 1dξ+

c

t+θ

Z

τ

kf₂(ξ)k²_V⁰

2dξ+c

t+θ

Z

τ

k(un,ξ,Bn,ξ)k²γdξ











≤ max (

sup

θ∈(−∞,τ−t]

e^2γθ[φ(θ+t−τ)]² ,

[φ(0)]²_H+c

t

Z

τ

kf₁(ξ)k²_V0 1dξ+c

t

Z

τ

kf₂(ξ)k²_V0 2dξ+c

t

Z

τ

k(un,ξ,Bn,ξ)k²_γdξ





 . (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)

≤ kφk_γ . and [(u(τ),B(τ))] = [φ(0)]≤ kφk_γ,we deduce from (3.18)

k(un,t,Bn,t)k²_γ≤R²+c Z t

τ

kf₁(ξ)k²_V0 1dξ+c

Zt τ

kf₂(ξ)k²_V0 2dξ+c

Z t τ

k(un,ξ,Bn,ξ)k²γdξ . (3.19) By the Lemma 2.1, we have

k(un,t,Bn,t)k²_γ≤R²e^c(t−τ)+c Z t

τ

kf₁(ξ)k²_V⁰

1dξ+kf₂(ξ)k²_V⁰

2

dξe^c(t−τ). (3.20) Thus, there exists a constantK1=K1(R, τ, Lg₁, Lg₂, T,f₁,f₂)>0 such that

k(un,t,Bn,t)k²_γ≤ K1, (3.21) which together with (3.17) gives

| un(t)|²_L2+S|Bn(t)|²_L2

+ 1

Re

Z T τ

kun(ξ)k²_V1dξ+ S Rm

Z T τ

kBn(ξ)k²_V2dξ

≤R²+c Z T

τ

kf₁(ξ)k²_V0 1dξ+c

ZT τ

kf₂(ξ)k²_V0

2dξ+cK1(T−τ).

(3.22)

(3.22) proves that the sequenceyn= (un,Bn) remains in a bounded set ofL^∞(τ, T;H)∩ L²(τ, T;V)∩ Cγ(H). Hence, we can use a compactness argument (see [42]) to extract a subsequence fromyn= (un,Bn) still denoted byyn= (un,Bn) satisfying

yn→y











weak-star inL^∞(τ, T;H), weakly inL²(τ, T;V), strongly inL²(τ, T;H), a.e., in (τ, T)×Ω,

(3.23)

withy= (u,B)∈L^∞(τ, T;H)∩L²(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 theG(t,(un,t,Bn,t)),this will be done on Step 3; secondly, we need to prove that

FN(kunkV₁)→FN(kukV₁) asn→ ∞,

FN(k(un,Bn)kV)→FN(k(u,B)kV) asn→ ∞, (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 inH between the first equation of (3.11) withAyn, we obtain d

dtkynk²V + 2|Ayn|²H= 2(f₁,A1un) + 2(f₂,A2Bn)−2B0^N(yn, yn,Ayn) + 2(g₁(t,(un,t,Bn,t),A1un)) + 2(g₂(t,(un,t,Bn,t),A2Bn)).

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

In addition, by Young’s inequality, (h2)−(h3), (3.21) one obtains

2|(f₁,A1un)|+ 2|(f₂,A2Bn)| ≤¹₄|A1un|²_L2+¹₄|A2Bn|²_L2+c|f₁|²_L2+c|f₂|²_L2

=¹₄|Ayn|²H+c|f₁|²_L2+c|f₂|²_L2

(3.27) and

2|(g₁(t,(un,t,Bn,t),A1un)) + (g₂(t,(un,t,Bn,t),A2Bn))|

≤1

2|A1un|²_L2+1

2|A2Bn|²_L2+cK₁²= 1

2|Ayn|²_H+cK₁².

(3.28) It follows from (3.26)-(3.28) that

d

dtkynk²V+|Ayn|²H ≤c|f₁|²_L2+c|f₂|²_L2+cK²1+cN⁴kynk²V. (3.29) Now we distinguish two cases:

Case 1: y(τ) = (u(τ),B(τ))∈H.

Integrating (3.29) betweensandtforτ < s≤t≤T,we obtain

kyn(t)k²_V+ Z t

s

|Ayn(ξ)|²_Hdξ

≤ kyn(s)k²V+c Z t

s

|f₁(ξ)|²_L2+|f₂(ξ)|²_L2+K²1

dξ+cN⁴ Z t

s

kyn(ξ)k²Vdξ.

≤ kyn(s)k²V+c Z t

τ

|f₁(ξ)k²_L2+|f₂(ξ)|²_L2+K²1

dξ+cN⁴ Z t

τ

kyn(ξ)k²Vdξ .

(3.30)

Momentarily dropping the term Z t

s

|Ayn(ξ)|²Hdξ in (3.30) and integrating once more be- tweenτ andτ+for some∈(0, T−τ),we have

Zτ+ τ

kyn(t)k²Vds≤ Zτ+

τ

kyn(s)k²Vds+ Zτ+

τ

c

Z t τ

|f₁(ξ)k²_L2+|f₂(ξ)|²_L2+K²₁ dξ

ds+cN⁴ Z τ+

τ

Z t τ

kyn(ξ)k²_Vdξ

ds . (3.31)

Sinceτ+≤T,it follows from (3.31) that

kyn(t)k²V ≤ Z T

τ

kyn(s)k²Vds+c(T−τ) Z T

τ

|f₁(ξ)|²_L2+|f₂(ξ)|²_L2+K²1

dξ+

c(T−τ)N⁴ ZT

τ

kyn(ξ)k²Vdξ . (3.32)

From the estimate (3.22), we infer that the right hand side of (3.32) is bounded independently ofn.Coming back to (3.30) and dropping the termkyn(t)k²_V,we get for some∈[0, T−τ]

ZT τ+

|Ayn(ξ)|²Hdξ≤ kyn(s)k²V +c Z T

τ

|f₁(ξ)|²_L2+|f₂(ξ)|²_L2+K1²

dξ+cN⁴ Z T

τ

kyn(ξ)k²Vdξ . (3.33) We then deduce thatyn∈L^∞(τ+, T;V). Therefore

yn∈L^∞(τ+, T;V)∩L²(τ+, T;D(A1)×D(A2)) for all 0< < T−τ . (3.34) Note from (3.11) that

dyn

dt =−Ayn−PnB^N(yn, yn) +PnF+PnG.

Then using (2.23) we deduce that the sequence n

PnB^N(yn, yn)o

is bounded inL²(τ+, T;H). Therefore, from (3.21) and (3.34), we infer that the sequence

d

dt(un,Bn) is also bounded inL²(τ+, T;H). (3.35) SinceD(A) =D(A1)×D(A2)⊂V ⊂Hwith compact injection, we derive from [25, Theorem 5.1,Chapter 1] that there exists an element (u,B)∈L^∞(τ+, T;V)∩L²(τ+, T;D(A)), and a subsequence of (un,Bn) (still) denoted (un,Bn) such that for allT > τ+, we have

(un,Bn)→(u,B)











weak-star inL^∞(τ+, T;V), weakly inL²(τ+, T;D(A)), strongly inL²(τ+, T;V), a.e., in (τ+, T)×Ω,

(3.36)

and d

dt(un,Bn)→ d

dt(u,B) weakly inL²(τ+, T;H). (3.37) From (3.36), we can assume, eventually extracting a subsequence ofynstill denotedynsuch that

kunkV₁→ kukV₁ a.e. in (τ+, T),

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

FN(kunkV₁)→FN(kukV₁) a.e. in (τ+, T),

FN(k(un,Bn)kV)→FN(k(u,B)kV) a.e. in (τ+, T). (3.39) Case 2: (φ1(0), φ2(0))∈V.

We mention that

k(φ1n(0), φ2n(0))kV =kPn(φ1(0), φ2(0))kV ≤ ky(τ)kV.

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

dtkynk²V ≤c|f₁|²_L2+c|f₂|²_L2+cK²1+cN⁴kynk²V, (3.40) from which we obtain by using Lemma 2.1

kyn(t)k²V ≤ ky(τ)k²Vexp

cN⁴(t−τ)

+cexp{cN⁴(t−τ)}

Z t τ

|f₁(ξ)|²_L2+|f₂(ξ)|²_L2+cK²1

dξ . (3.41) Hence, we derive from (3.29) and (3.32) that (yn) = (un,Bn) satisfies

k(un,Bn)(t)k²V ≤ K2, Z T

τ

|A1un(ξ)|²_L2+|A2Bn(ξ)|²_L2

dξ≤ K3, (3.42) which proves that (un,Bn) is bounded inL^∞(τ, T;V)∩L²(τ, T;D(A1)×D(A2)).

Note that in (3.42), K2 andK3 are positive constants independent ofn and depending only on data Ω, Re, Rm, S,f₁,f₂, T,u0,B0, Lg₁ andLg₂.

Note that from (3.11) that dyn

dt =−Ayn−PnB^N(yn, yn) +PnF+PnG . Then using (2.23) we deduce that the sequence

n

PnB^N(yn, yn)o

n

is bounded inL²(τ, T;H). Therefore, from (3.33) and (3.21), we infer that the sequence

d

dt(un,Bn) is also bounded inL²(τ, T;H). (3.43) 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)∩L²(τ, T;D(A)), and a subsequence of (un,Bn) (still) denoted (un,Bn) such that for allT > τ, we have

(un,Bn)→(u,B)











weak-star inL^∞(τ, T;V), weakly inL²(τ, T;D(A)), strongly inL²(τ, T;V), a.e., in (τ, T)×Ω,

(3.44)

and d

dt(un,Bn)→ d

dt(u,B) weakly inL²(τ, T;H). (3.45) From (3.44), we infer that

kunkV1 → kukV1 a.e. in (τ, T),

k(un,Bn)kV → k(u,B)kV a.e. in (τ, T), (3.46) and therefore

FN(kunkV₁)→FN(kukV₁) a.e. in (τ, T),

FN(k(un,Bn)kV)→FN(k(u,B)kV) a.e. in (τ, T). (3.47) Step3: Passage to the limit.

We want to take the limit in (3.11) whenn goes to +∞.We focus our attention on the term G(t,(un,t,Bn,t)) we refer the reader to [5, 13] 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 [28] where the globally modified Navier-Stokes with infinite delays is investigated. We start by proving that

(un,t,Bn,t)→(ut,Bt) inCγ(H), ∀t∈(−∞, T]. (3.49)