Global weak solutions to magnetic fluid flows with nonlinear Maxwell-Cattaneo heat transfer law

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nonlinear Maxwell-Cattaneo heat transfer law

Fatah Aggoune, Kamel Hamdache, Djamila Hamroun

To cite this version:

Fatah Aggoune, Kamel Hamdache, Djamila Hamroun. Global weak solutions to magnetic fluid flows with nonlinear Maxwell-Cattaneo heat transfer law. 2014. �hal-01023053�

nonlinear Maxwell-Cattaneo heat transfer law

F. Aggoune

, K. Hamdache

and D. Hamroun

Abstract

We discuss the equations describing the dynamic of the heat transfer in a magnetic fluid flow under the action of an applied magnetic field. Instead of the usual heat transfer equation we use a generalization given by the Maxwell-Cattaneo law which is a system satisfied by the temperature and the heat flux. We prove a global existence of weak solutions to the system having a finite energy.

Keywords: Navier-Stokes equations, Bloch-Torrey equation, magnetostatic equation, Maxwell-Cattaneo law, heat transfer, magnetic field, magnetization

AMS subject classifications: 76N10, 35Q35.

1 Introduction

1.1 Statement of the model

In this work, we study the heat transfer in a magnetic incompressible fluid flow under the action of an applied magnetic field. The temperature θ of the fluid is usually described by the linear heat transfer equation

∂tθ+U · ∇θ=−divQ (1)

related to the linear Fourier law

Q=−κ∇θ (2)

Qbeing the heat flux andUthe fluid velocity. To ovoid the paradox of the instantaneous heat propagation inherent to the parabolic type equation, another model was offered in the pioneering work of Vernotte [23] and Cattaneo [6]. In this model, the Fourier law (2)is replaced by the heat-flux equation

τ ∂tQ+Q=−κ∇θ (3)

∗Laboratoire AMNEDP, Faculté de Mathématiques. Université USTHB, Algérie (Email: ag- [email protected])

†Centre de Mathématiques Appliquées, CNRS UMR 7641 Ecole Polytechnique, 91128 Palaiseau cedex, France (Email: [email protected])

‡Laboratoire AMNEDP, Faculté de Mathématiques, Université USTHB, Algérie (Email: djam- [email protected])

1

where τ > 0 is the time relaxation parameter. For τ = 0, we recover equation (2).

Combining the temperature equation and the heat-flux equation we see thatθsatisfies an hyperbolic type equation. System(1)-(3)was generalized by Guyer and Krumhansl see [13] for example, by introducing a diffusion process in (3) so that the heat-flux equation becomes

τ(∂_tQ−γ∆Q) =−Q−κ∇θ (4)

where γ > 0 is a diffusion coefficient. When the heat conductivity is enhanced by radiation effects see [12, 10, 11], the linear Fourier law is replaced by a nonlinear one which writes aqs

Q=−∇K(θ). (5)

In [11], the model of heat transfer by the nonlinear Fourier law in an incompressible fluid flow has been discussed.

In this work we are dealing with the nonlinear Maxwell-Cattaneo law for heat trans- fer which is a generalization of the nonlinear Fourier law, more precisely we consider that the dynamic of the couple(θ, Q) is governed by the system

∂tθ+U· ∇θ=−divQ τ(∂tQ−γ∆Q) =−τ

2curlU ×Q−Q− ∇K(θ). (6) The monotone functionK(θ) discussed in this work is given by

K(θ) =κ θ+α θ³ (7)

whereκ >0andα >0are the heat conductivity coefficients and we refer the reader to [19, 20] for the introduction of the rotation term ¹₂curlU×Q. Notice that the power 3 used in the definition of the functionK is less than the values indicated in [12].

The Maxwell-Cattaneo system (6) is coupled to the incompressible Navier-Stokes equations satisfied by the fluid velocityU and the pressure p as well as to the Bloch- Torrey equation satisfied by the magnetization fieldM and the magnetostatic equation for the magnetic fieldH. Namely, we have

divU = 0

∂_tU + (U.∇)U −η∆U +∇p=−ρ(θ)g+µ₀(M.∇)H+^µ₂⁰curl (M×H)

∂_tM+ (U.∇)M−σ∆M+1

δ(M−χ₀H) = 1

2curlU×H−β₀M×(M×H) div (H+M) =F, curlH= 0

(8)

where the densityρ(θ) is given by the state of law

ρ(θ) =ρ0(1−β(θ−θ⁰)) (9) whereρ₀ is the fluid density at the the temperature θ⁰ and β is a physical coefficient.

The function g represents the force of gravity, F is a function linked to the applied

magnetic field andη, µ₀, σ, δ, χ₀, β₀>0are physical parameters.

When the magnetization M is assumed to be in equilibrium state meaning it is parallel to the magnetic fieldH, the model in consideration is quite different from the one studied in this work. The magnetization law writes in general as

M =χ(θ,|H|)H (10)

In that case, the Maxwell-Cattaneo system becomes

∂tθ+U· ∇θ+µ0θ∂M

∂θ ·(U · ∇)H=−divQ+ηΦ(U) τ

∂tQ−∆Q+ ¹₂curlU×Q

=−Q− ∇K(θ)

(11)

whereµ0θ∂M

∂θ ·(U· ∇)H is the thermal power andΦ(U)is the energy dissipation. The heat transfer problem in an incompressible fluid flow under the above Maxwell-Cattaneo law in a magnetic fluid is an open problem.

LetD⊂R³be an open, bounded, regular and simply connected domain, with boundary Γ. For T >0 fixed, we set DT = (0, T)×D and ΓT = (0, T)×Γ. The equations (6) and (8) are set onDT with the following initial and boundary conditions

U(0) =U0, divU0= 0, M(0) =M0, inD

U = 0, M ·n= 0, curlM×n= 0, H·n= 0,on Γ_T (12) θ(0) =θ₀, Q(0) =Q₀ inD

Q×n= 0, τ γdivQ− K(θ) = 0 on ΓT

(13) wherenrepresents the unit outward normal to the boundary Γ. Problem(6)-(8)-(12)- (13)will be labeled problem (P).

System(8) with the temperature equation (1) has been discussed in [1, 2]. The linear Maxwell-Cattaneo system (1)-(3) has been studied in [14, 15] in the case where the velocityU is fixed.

1.2 Notations and spaces

For1≤q ≤ ∞ands∈R, let L^q(D) and W^s,q(D) be the usual Lebesgue and Sobolev spaces of scalar functions. Ifq = 2, W^s,2(D) is denoted by H^s(D) and k · k and (·;·) denote the norm and the scalar product of the Hilbert spaceL²(D). For vector valued functions we use the notationsL^q(D),W^s,q(D),H^s(D) and the notations of norm and the scalar product of L²(D) are unchanged. If V is a Banach space we denote by h·;·i_V⁰_×V (or simplyh·;·iif no confusion arises) the duality product whereV⁰is the dual space ofV. IfV is an Hilbert space with scalar product(·;·), we set

C([0, T];V weak) ={u: [0, T]→V; (u(·), v)∈ C([0, T]), ∀v∈V}.

Let D(D,R³) the set of functions f : D→ R³ which are infinitely differentiable with compact support in D and H¹₀(D) its closure in H¹(D). Now, we introduce the func- tional spaces used in the theory of Navier-Stokes equations, see [22, 7] for example

D_s(D) ={v∈ D(D,R³); divv= 0 inD}

U =closure of D_s(D) inH¹(D), U₀ =closure of D_s(D) inL²(D).

Then it is well known that

U ={v∈H¹₀(D); divv= 0 inD}

U₀={v∈L²(D); divv= 0 inD, v·n= 0 on Γ}

(14) and identifyingU₀ with its dual, we get as usual the inclusionsU ⊂ U₀ ⊂ U⁰.

For the Bloch-Torrey equation satisfied by M and the heat-flux equation satisfied by Qwe introduce the Hilbert spaces

H¹t(D) ={M ∈H¹(D); M ·n= 0 onΓ}

H¹n(D) ={Q∈H¹(D); Q×n= 0 onΓ}

equipped with the norm ofH¹(D). Then (see [7] for example) there exists C >0 such that for allV in eitherH¹t(D) or H¹n(D) the following estimate holds

k∇Vk ≤C(kVk²+kcurlVk²+kdivVk²)^1/2 (15) hence the norm of H¹(D) is equivalent to the norm (kVk²+kcurlVk²+kdivVk²)^1/2 on the spaces H¹t(D) and H¹n(D). We recall the relation −∆ = curl²− ∇div so that for regular vector fieldsΨand Φthe following Green formula holds

− Z

D

∆Ψ·Φdx= Z

D

curl Ψ·curl Φdx+ Z

D

div Ψ div Φdx +

Z

Γ

curl Ψ·(Φ×n)dΓ− Z

Γ

div Ψ(Φ·n)dΓ.

To deal with the magnetostatic equation, we set L²_] ={ψ∈L²(D);

Z

D

ψ(x)dx= 0} andH_]¹ =H¹(D)∩L²_].

The Hilbert spaceH_]¹ is equipped with the normk∇ψkwhich is equivalent to the usual norm ofH¹(D)thanks to Poincaré-Wirtinger inequality : there existsC >0such that for allψ∈H¹_] we have

kψk ≤Ck∇ψk. (16)

To end these notations, we point out that throughout this paper, C > 0 indicates a generic constant depending only on some bounds of the physical data, which takes different values in different occurrences. The dependency of the constants C >0 with respect to a parameter mis written as Cm.

Now, let us focus our attention on the magnetostatic equation to give some useful continuity results on the solution

1.3 The magnetostatic equation

LetM ∈L²(D) andF ∈L²_], we consider the following problem Findϕ∈H_]¹; ∀ψ∈H_]¹,

Z

D

(∇ϕ+M)· ∇ψ dx=− Z

D

F ψ dx. (17) This problem admits a unique solution ϕinH_]¹ and we have

Z

D

∇ϕ.M dx=−k∇ϕk²− Z

D

F ϕ dx. (18)

then

k∇ϕk ≤(kMk+CkFk). (19)

In particular the application

H: (M, F)7→ϕ (20)

is continuous from L²(D)×L²_] to H_]¹. Furthermore testing equation (17) with ψ− Z

D

ψ dx, ψ ∈H¹(D), we see that Z

D

(∇ϕ+M)· ∇ψ dx=− Z

D

F ψ dx, ∀ψ∈H¹(D) (21) andH =∇ϕsolves the problem







div (H+M) =F, curlH = 0 inD (H+M)·n= 0 onΓ.

Moreover using classical regularity results for elliptic problems, we conclude that if F ∈L²_] and M ∈H¹t(D), thenϕ∈H²(D)∩H_]¹ and we have

kϕk_H2(D)≤C(kdivMk+kFk). (22) ThereforeH =∇ϕ∈H¹t(D)and we have

kHk_H1(D)≤C(kdivMk+kFk). (23) We can see thatH is also continuous from L²(DT)×L²(0, T;L²_]) to L²(0, T;H_]¹) and from H¹(0, T;L²(D))×H¹(0, T;L²_]) to H¹(0, T;H_]¹). Moreover for F ∈H¹(0, T;L²_]) andM ∈H¹(0, T;L²(D)), we have

Z

D

(∇(∂_tϕ) +∂tM)· ∇ψ dx=− Z

D

∂tF ψ dx,∀ψ∈H¹(D), t∈(0, T). (24)

2 Main results

Before stating our main result, let us give the formal energy estimates for problem (P).

2.1 Energy estimates

Let (U, M, H, θ, Q) be a regular solution to system (P). We proceed as in [1, 2] to obtain, forθfixed, the energy estimate satisfied by(U, M, H). Fort∈[0, T], we set

E_nsbt(t) = 1

2kU(t)k²+ µ0

2 (kM(t)k²+kH(t)k²) E_nsbt,0 = 1

2kU₀k²+ µ₀

2 (kM₀k²+kH₀k²)

(25)

whereH₀ =∇ϕ₀andϕ₀ is the unique solution of the following problem (see subsection 1.3)

Findϕ₀ ∈H_]¹ such that Z

D

(∇ϕ₀+M₀)· ∇ψ dx=− Z

D

F(0)ψ dx, ∀ψ∈H_]¹ (26) and

F_nsbt(t) =ηk∇U(t)k²+µ0σ(kcurlM(t)k²+ 2kdivM(t)k²) +µ0

δ kM(t)k²+ µ0

δ (1 + 2χ0)kH(t)k²+β0µ0kM(t)×H(t)k².

(27) Then we get the energy estimate

E_nsbt(t) + Z t

0

F_nsbt(s)ds≤ E_nsbt,0+C Z t

0

kρ(θ(s))k²ds+C Z t

0

kG(s)k²ds (28) for allt≥0where

G(t) =kF(t)k²+k∂_tF(t)k². (29) Now we consider the Maxwell-Cattaneo system(6)satisfied by (θ, Q) for U fixed. Let

$the primitive function of K defined by

$(θ) = κ 2θ²+α

4θ⁴. (30)

Multiplying the temperature equation by K(θ) and the heat-flux equation by Q then integrating by parts and adding both results, we get the energy estimate associated with the Maxwell-Cattaneo system

E_mc(t) + Z t

0

F_mc(s)ds≤ E_mc,0 (31) for allt≥0with

E_mc(t) = Z

D

$(θ(t))dx+ τ

2kQ(t)k², E_mc,0 = Z

D

$(θ₀)dx+ τ

2kQ₀k² (32) F_mc(t) =τ γ(kcurlQ(t)k²+kdivQ(t)k²) +kQ(t)k². (33) The total energy E and the total dissipation energy F of the full problem (P) are defined by

E(t) =E_nsbt(t) +E_mc(t), F(t) =F_nsbt(t) +F_mc(t) (34) and it holds

E(t) + Z t

0

F(s)ds≤ E₀+C Z t

0

kρ(θ(s))k²ds+C Z t

0

G(s)ds. (35)

2.2 Statement of the result

We will use the following hypotheses

U₀∈ U₀, M₀, Q₀ ∈L²(D), divQ₀ ∈L^12/11(D), θ₀ ∈L⁴(D) g∈L^∞(D_T), F ∈H¹(0, T;L²(D)),

Z

D

F(t, x)dx= 0 for allt∈[0, T].

(36)

Let us give now the definition of a global weak solution to problem(P)

Definition 1 We say that (U, M, H, θ, Q) is a global weak solution with finite energy of problem (P) if the following conditions are fulfilled

U ∈L^∞(0, T;U₀)∩L²(0, T;U)

M ∈L^∞(0, T;L²(D))∩L²(0, T;H¹_t(D)) H ∈L^∞(0, T;L²(D))∩ L²(0, T;H¹t(D)) Q∈L^∞(0, T;L²(D))∩L²(0, T;H¹n(D)) θ∈L^∞(0, T;L⁴(D))

(37)

and

(i) the linear momentum equation holds weakly in the sense that for all v∈ U d

dt Z

D

U ·v dx+ Z

D

(U· ∇)U·v dx+η Z

D

∇U· ∇v dx=

− Z

D

ρ(θ)g·v dx+µ₀ Z

D

(M· ∇)H·v dx+µ0

2 Z

D

M ×H·curlv dx U(0) =U₀

(38)

(ii) the magnetization equation satisfies for allw∈H¹_t(D) the weak formulation d

dt Z

D

M·w dx+ Z

D

(U · ∇)M·w dx+σ Z

D

curlU ·curlw dx +σ

Z

D

divMdivw dx+1 δ

Z

D

(M−χ0H)·w dx= 1

2 Z

D

curlU×H·w dx−β₀ Z

D

M×H·M×w dx M(0) =M₀

(39)

(iii) the magnetic field is given by H =∇ϕ where ϕ∈L^∞(0, T;L²_]) and satisfies for all ψ∈H_]¹

Z

D

(∇ϕ(t) +M(t))· ∇ψ dx=− Z

D

F(t)ψ dx (40)

(iv) the couple (θ, Q) satisfies the Maxwell-Cattaneo system in the following sense Z

DT

θ(∂_ta+U · ∇a)dxdt= Z

DT

divQ a dxdt− Z

D

θ₀a(0)dx (41)

for all a∈ D([0, T[×D) and for all b∈H¹n(D) with divb∈L⁴(D) τ d

dt Z

DT

Q·b dx+τ γ Z

D

(curlQ·curlb+ divQdivb)dx+

Z

D

Q·b dx+τ 2

Z

D

curlU ×Q·b dx= Z

D

K(θ) divb dx Q(0) =Q₀.

(42)

Moreover the energy estimates (28) and (31) hold for all t∈(0, T).

Remark 1

1. As usual, we get the pressure p ∈ W^−1,∞(0, T;L²(D)) by using the De Rham theorem.

2. From the weak formulations, we deduce that (∂_tU, ∂_tM, ∂_tQ) ∈ L¹(0, T;U⁰ × (H_t¹(D))⁰ ×(H_n¹(D))⁰) so that (U, M, Q) ∈ C([0, T];U⁰×(H_t¹(D))⁰ ×(H_n¹(D))⁰) and the corresponding initial conditions are meaningful and moreover U, M, Q∈ C([0, T];L²(D) weak).

3. The theory of transport equation leads to the result θ ∈ C([0, T];L⁴(D) weak)∩ C([0, T];L^p(D)), for all1≤p <4 (see [5] for example) which gives a sense to the initial condition.

Theorem 1 Under hypotheses (36), there exists a global weak solution with finite en- ergy of problem(P). Moreoverθ has the regularity

θ∈L^36/11(0, T;L^36/7(D)). (43)

Remark 2 One can relax the condition divb ∈ L⁴(D) on test functions b in (42) to the condition divb∈L^12/5(D).

We will prove existence of solutions to problem (P) in several steps, using a regu- larization method and some compactness results. The paper is organized as follows.

In section 3, we introduce the regularized problem(P_ν)obtained by adding an elliptic term −ν∇ ·(|∇θ|²∇θ) in the temperature equation, ν > 0 being a small parameter together to a regularization of the initial condition θ0. By using the Faedo-Galerkine method, we obtain a sequence of approximated solutions (Un, Mn, Hn, θn, Qn) which converge towards(U_ν, M_ν, H_ν, θ_ν, Q_ν) a global weak solution with finite energy of sys- tem(P_ν).

In section 4, we prove Theorem 1. We first introduce an auxiliary problem satisfied by ζ_ν =τ γdivQ_ν − K(θ_ν) and establish a compacity result verified by ζ_ν which allows to get the limit of the nonlinear termK(θ_ν). Then we get Theorem 1 by passing to the limit as ν→0.

3 The regularized problem (P

)

Letν >0be a small parameter and (θ₀^ν) such that

(θ₀^ν)⊂W^1,4(D), θ^ν₀ →θ₀ strongly in L⁴(D). (44) We define the regularized problem(P_ν) as the system (8)−(12) coupled to the regu- larized Maxwell-Cattaneo system

∂_tθ+ (U · ∇)θ−ν∇ ·(|∇θ|²∇θ) =−divQinD_T τ(∂tQ−γ∆Q) =−τ

2curlU ×Q−Q− ∇K(θ) inD_T ν|∇θ|²∇θ·n= 0, Q×n= 0, τ γdivQ− K(θ) = 0 on Γ_T θ(0) =θ₀^ν, Q(0) =Q0 inD

(45)

Note that we use the nonlinear elliptic operator−ν∇·(|∇θ|²∇θ)instead of−ν∆θwhich is commonly used to regularize a transport equation, owing to obtain approximate solutionsθν belonging toW^1,4(D) and therefore to L^∞(D).

Proceeding as previously the energy associated with(45)takes the form E_mc(t) +

Z t 0

F_mc(s)ds+ν Z t

0

R(s)ds≤ E_mc,0^ν (46) for allt≥0where

R(t) =κk∇θk⁴_L4(D)+ 3α Z

D

θ²|∇θ|⁴ dx (47) which is well defined thanks to the Sobolev embeddingW^1,4(D)⊂ C(D)and

E_mc,0^ν = Z

D

(κ

2|θ^ν₀|²+ Z

D

α

4|θ₀^ν|⁴)dx+kQ₀k²

It is easy to verify that the energy estimate associated with the problem(P_ν) writes as E(t) +

Z t 0

F(s)ds+ν Z t

0

R(s)ds≤C+C Z t

0

kρ(θ(s))k²ds+C Z t

0

kG(s)k²ds (48) whereC >0 does not depend on ν. We will prove the following existence result

Theorem 2 Under hypotheses (36), there exists a global weak solution(Uν, Mν, Hν, θν, Qν) of problem (P_ν) such that

Uν ∈L^∞(0, T;U₀)∩L²(0, T;U)

M_ν, H_ν ∈L^∞(0, T;L²(D))∩L²(0, T;H¹t(D)) Q_ν ∈L^∞(0, T;L²(D))∩L²(0, T;H¹_n(D)) θν ∈L^∞(0, T;L⁴(D))∩L⁴(0, T;W^1,4(D))

(49)

and satisfying the energy estimates (46) and (48) and the problem in the following sense

(i) U_ν(0) =U₀ and for all v∈ U d

dt Z

D

U_ν·v dx+ Z

D

(U_ν · ∇)U_ν ·v dx+η Z

D

∇U_ν · ∇v dx=

− Z

D

ρ(θ_ν)g·v dx+µ₀ Z

D

(M_ν · ∇)H_ν ·v dx+µ₀ 2

Z

D

M_ν ×H_ν·curlv dx (50) (ii) Mν(0) =M0 and for allw∈H¹t(D)

d dt

Z

D

Mν·w dx+ Z

D

(Uν · ∇)M_ν·w dx+σ Z

D

curlMν ·curlw dx +σ

Z

D

divMνdivw dx+1 δ

Z

D

(Mν−χ0Hν)·w dx= 1

2 Z

D

curlUν×Hν ·w dx−β0

Z

D

Mν ×Hν ·Mν×w dx

(51)

(iii) θ_ν(0) =θ₀^ν and for all a∈W^1,4(D) d

dt Z

D

θ_νa dx− Z

D

θ_νU · ∇a dx+ν Z

D

|∇θ_ν|²∇θ_ν· ∇a dx=− Z

D

divQ_νa dx (52) (iv) Qν(0) =Q0 and for all b∈H¹n(D)

τ d dt

Z

D

Qν·b dx+τ γ Z

D

(curlQν ·curlb+ divQνdivb)dx+

τ 2

Z

D

curlUν ×Qν ·b dx=− Z

D

Qν·b dx+ Z

D

K(θ_ν) divb dx

(53)

with Hν =∇ϕ_ν where ϕν =H(M_ν, F) is defined in (20).

3.1 Faedo-Galerkine approximation for (P

)

Letν >0be fixed, consider the weak formulation of problem(P_ν) given in Theorem 2.

In order to solve this problem by the Faedo-Galerkine method, we introduce the Hilbert basis(V_j)j≥1,(W_j)j≥1,(Φ_j)j≥1 of the spacesU,H¹t(D),H¹n(D)respectively and a basis (vj)j≥1 ofW^1,4(D). For simplicity, we assume these basis to be orthonormal inL²(D).

We seek for approximated solutions of the system(P_ν)of the form U_n(t) =

n

X

j=1

α_j(t)V_j, M_n(t) =

n

X

j=1

β_j(t)W_j, θ_n(t) =

n

X

j=1

a_j(t)v_j, Q_n(t) =

n

X

j=1

b_j(t)Φ_j

(54)

satisfying for alln∈N^∗ and1≤j ≤n (i) d

dt Z

D

U_n·V_jdx+ Z

D

(U_n· ∇)U_n·V_jdx+η Z

D

∇U_n· ∇V_jdx=

− Z

D

ρ(θ_n)g·V_jdx+µ₀ Z

D

(M_n· ∇)H_n·V_jdx+ µ₀ 2

Z

D

M_n×H_n·curlV_jdx Un(0) =U0n

(55)

(ii) d dt

Z

D

Mn·Wjdx+ Z

D

(Un· ∇)M_n·Wjdx+σ Z

D

curlMn·curlWjdx +σ

Z

D

divMndivWjdx+1 δ

Z

D

(Mn−χ0Hn)·Wjdx= 1

2 Z

D

curlUn×Hn·Wjdx−β0

Z

D

Mn×Hn·Mn×Wjdx M_n(0) =M_0n

(56)

(iii) d dt

Z

D

θnvjdx− Z

D

θnUn· ∇v_jdx+ν Z

D

|∇θ_n|²∇θ_n· ∇v_jdx=

− Z

D

divQnvjdx θ_n(0) =θ_0n^ν

(57)

(iv) τ d dt

Z

D

Qn·Φjdx+τ γ Z

D

(curlQn·curl Φj+ divQndiv Φj)dx=

−τ 2

Z

D

curlUn×Qn·Φjdx− Z

D

Qn·Φjdx+ Z

D

K(θ_n) div Φjdx Q_n(0) =Q_0n

(58)

where

Hn=∇ϕ_n, ϕn=H(M_n, F)) U0n=

n

X

j=1

α^j_0nVj, M0n=

n

X

j=1

β_0n^j Wj, θ_0n^ν =

n

X

j=1

a^ν,j_0n vj, Q0n=

n

X

j=1

b^j_0nΦj. We assume that

(U_0n, M_0n, Q_0n)→(U₀, M₀, Q₀) strongly in (L²(D))³ θ^ν_0n→θ^ν₀ strongly in W^1,4(D).

(59)

This problem will be labeled (P_νⁿ).

3.2 Solving the system (P

)

Let the vector valued functionsαⁿ= (α₁,· · ·, α_n),βⁿ= (β₁,· · ·, β_n),aⁿ= (a₁,· · ·, a_n) andbⁿ= (b₁,· · ·, b_n), we consider the function

t∈[0, T]→γ_n(t) = (αⁿ(t), βⁿ(t), aⁿ(t), bⁿ(t))∈(Rⁿ)⁴ thenγ_n satisfies the ordinary differential system

γ_n⁰ +Anγn=Zn(t, γn), γn(0) =γ0n (60) whereγ_0n= (α_0n, β_0n, a^ν_0n, b_0n)∈(Rⁿ)⁴,A_nis a n⁴×n⁴ constant matrix involving the terms

η Z

D

∇V_i· ∇V_j dx, σ Z

D

(curlW_i· curlW_j+ divW_idivW_j)dx τ γ

Z

D

(curl Φi·curl Φj+ div Φidiv Φj)dx+ Z

D

Φi·Φjdx

and the vector fieldZ_n= (Z_n¹, Z_n², Z_n³, Z_n⁴)∈(Rⁿ)⁴ is defined as follows Z_nj¹ (t, γn) =−

Z

D

(Un· ∇)U_n·Vjdx− Z

D

ρ(θn)g·Vjdx +µ₀

Z

D

(U_n· ∇)H_n·V_jdx+µ0

2 Z

D

M_n×H_n·curlV_jdx Z_nj² (t, γ_n) =−

Z

D

(U_n· ∇)M_n·W_jdx−1 δ

Z

D

(M_n−χ₀H_n)·W_jdx +1

2 Z

D

curlU_n×H_n·W_jdx−β₀ Z

D

M_n×H_n·M_n×W_jdx Z_nj³ (t, γ_n) =

Z

D

θ_nU_n· ∇v_jdx−ν Z

D

|∇θ_n|²∇θ_n· ∇v_i dx− Z

D

divQ_nv_jdx Z_nj⁴ (t, γ_n) =−τ

2 Z

D

curlU_n×Q_n·Φ_jdx+ Z

D

K(θ_n) div Φ_jdx for 1≤j≤n.

Notice thatZ_nhas the same regularity in the time variabletas the functionF appearing in the magnetostatic equation and it is continuous and locally lipschitz continuous with respect to the variable γn. Hence there exists a unique maximal solution γn of (60) defined on a time interval[0;T_n]satisfying γ_n ∈H¹(0, T_n; (Rⁿ)⁴). We shall prove that T_n=T with the following estimate.

Let(Un, Mn, θn, Qn)be the solution of(P_νⁿ)defined on(0, Tn). We want to verify that sup

t∈[0;Tn]

(kU_nk²+kM_nk²+kθ_nk⁴_L4(D)+kQ_nk²)(t)<∞. (61) We multiply equation (58) byb_j and add these equations for 1≤j≤n, we obtain

τ 2

d

dtkQ_nk²+τ γ(kcurlQ_nk²+kdivQ_nk²) +kQ_nk²= Z

D

K(θ_n) divQ_ndx. (62) We use the equation (57) that we multiply by Θj(t) =

Z

D

K(θ_n)·vj dx and add the equalities for1≤j ≤nto obtain

d dt(κ

2kθ_nk²+α

4kθ_nk⁴_L4(D)) +ν(κk∇θ_nk⁴_L4(D)+ 3α Z

D

θ²_n|∇θ_n|⁴ dx) =

− Z

D

divQ_nK(θ_n)dx.

(63)

Adding (62) and (63) lead to 1

2 d

dt(κkθ_nk²+α

2kθ_nk⁴_L4(D)+τkQ_nk²) +νκk∇θ_nk⁴_L4(D)

+3ναkθ_n|∇θ_n|²k²+τ γ(kcurlQ_nk²+kdivQ_nk²) +kQ_nk² = 0

(64)

Therefore, integrating between 0 andtand using (59), we easily deduce that (κkθ_nk²+α

2kθ_nk⁴_L4(D)+τkQ_nk²)(t) + 2ν Z t

0

κk∇θ_nk⁴_L4(D)ds +2

Z t 0

(3ναkθ_n|∇θ_n|²k²+τ γ(kcurlQ_nk²+kdivQ_nk²) +kQ_nk²)ds= κkθ^ν_0nk²+α

2kθ^ν_0nk⁴_L4(D)+τkQ_0nk² ≤C

(65)

withC independent of n. Similarly, we obtain from equations (55) and (56) 1

2 d

dtkU_nk²+ηk∇U_nk²=− Z

D

ρ(θn)g·Undx

−µ₀ Z

D

(Un· ∇)M_n·Hndx+µ0

2 Z

D

(Mn×Hn)·curlUndx

(66)

1 2

d

dtkM_nk²+σkcurlM_nk²+σkdivM_nk²+1

δkM_nk² = +χ₀

δ Z

D

H_n·M_ndx+ 1 2

Z

D

curlU_n×H_n·M_ndx

(67)

so (66) and (67) lead to 1

2 d

dt(kU_nk²+µ0kM_nk²) +ηk∇U_nk²+µ0σ(kcurlMnk²+kdivMnk²) + µ0

δ kM_nk²

=− Z

D

ρ(θn)g·Undx−µ0

Z

D

(Un· ∇)M_n·Hndx+µ0χ0

δ Z

D

Hn·Mndx.

Using equation (24) for unknownϕ_n and data M_n, and testing withψ=ϕ_n, we get Z

D

H_n·∂_tM_ndx=−1 2

d

dtkH_nk²− Z

D

∂_tF ϕ_ndx.

Now we multiply equation (56) by hj(t) = Z

D

Hn·Wj dx and add the equalities for 1≤j≤nto obtain

Z

D

∂tMn·Hndx+ Z

D

(Un· ∇)M_n·Hndx+σ Z

D

divMndivHndx +1

δ Z

D

(Mn−χ0Hn)·Hndx=−β₀kM_n×Hnk²

(68)

so Z

D

(Un· ∇)M_n·Hndx= 1 2

d

dtkH_nk²+ Z

D

∂tF ϕndx−1 δ

Z

D

Mn·Hndx

−σ Z

D

divMn(F−divMn)dx+χ0

δ kH_nk²−β0kM_n×Hnk².

(69)

>From (21), we see that Z

D

H_n·M_ndx=−kH_nk²− Z

D

F ϕ_ndx therefore

Z

D

(Un· ∇)M_n·Hndx= 1 2

d

dtkH_nk²+ Z

D

(∂tF+1

δF)ϕndx+σkdivMnk²

−σ Z

D

divMnF dx+ 1 +χ0

δ kH_nk²−β0kM_n×Hnk².

(70)

so integrating between 0 andt, we get 1

2(kU_nk²+µ₀(kM_nk²+kH_nk²)(t) +η Z t

0

k∇U_nk²ds +

Z _t

0

(µ₀σ(kcurlM_nk²+ 2kdivM_nk²) +µ0

δ kM_nk²)ds +

Z t 0

µ0(1 + 2χ0)

δ kH_nk² ds+β0µ0

Z t 0

kM_n×Hnk² ds= 1

2(kU_0nk²+µ₀(kM_0nk²+kH_0nk²)+

Z t 0

Z

D

ρ(θn)g·Undxds−µ0(1 +χ0) δ

Z t 0

Z

D

F ϕn dxds

−µ₀ Z t

0

Z

D

∂tF ϕn dxds+µ0σ Z t

0

Z

D

FdivMn dxds

(71)

where

H0n=∇ϕ_0n, ϕ0n=H(M_0n, F(0)). (72) Using the inequalities

| Z _t

0

Z

D

ρ(θ_n)g·U_ndxds| ≤C_T +C Z _t

0

kθ_nk² ds+ 1 2

Z _t

0

kU_nk² ds,

µ0(1 +χ0)

δ |

Z t 0

Z

D

F ϕndxds|+µ0| Z t

0

Z

D

∂tF ϕndxds| ≤ C(kFk²_L2(DT)+k∂_tFk²_L2(DT)) +µ0(1 + 2χ0)

2δ

Z t 0

kH_nk² ds,

µ₀σ|

Z _t

0

Z

D

FdivM_ndxds| ≤CkFk²_L2(DT)+µ₀σ Z _t

0

kdivM_nk² ds.

We get

1

2(kU_nk²+µ₀(kM_nk²+kH_nk²)(t)+

Z t 0

[ηk∇U_nk²+µ0σ(kcurlMnk²+ 2kdivMnk²) +µ0

δ kM_nk²]ds +

Z t 0

[µ₀(1 + 2χ₀)

2δ kH_nk²+β₀µ₀kM_n×H_nk²]ds≤ A_n+C_T +C

Z t 0

kθ_nk²ds+1 2

Z t 0

kU_nk² ds

(73)

where

An= 1

2(kU_0nk²+µ0(kM_0nk²+kH_0nk²)≤C

with C independent of n in view of (59), (72) and (19). Thus thanks to (65) and Gronwall inequality, we deduce that

kU_n(t)k²+kM_n(t)k²+kH_n(t)k² ≤C+ exp(Ct) (74)

then 1

2(kU_nk²+µ0(kM_nk²+kH_nk²)(t)+

Z t 0

[ηk∇U_nk²+µ₀σ(kcurlM_nk²+ 2kdivM_nk²) +µ₀

δ kM_nk²]ds +

Z _t

0

[µ0(1 + 2χ0)

2δ kH_nk²+β₀µ₀kM_n×H_nk²]ds≤C+ exp(Ct).

(75)

This ends the proof of (61) so we conclude thatT_n=T for alln≥1.

3.3 Convergence of the Faedo-Galerkine scheme

Letν be fixed, the estimates (65) and (75) show that Lemma 1

• (U_n)_n is uniformly bounded in L^∞(0, T;U₀)∩L²(0, T;U)

• (Mn)n andHn are uniformly bounded in L^∞(0, T;L²(D))∩L²(0, T;H¹t(D))

• (M_n×H_n)_n is uniformly bounded in L²(0, T;L²(D))

• (Qn)n is uniformly bounded in L^∞(0, T;L²(D))∩L²(0, T;H¹n(D))

• (θ_n)_n is uniformly bounded in L^∞(0, T;L⁴(D))∩L⁴(0, T;W^1,4(D)).

Notice that we get the uniform bound of(H_n)_ninL²(0, T;H¹t(D))using the bound of (M_n)_n and (23). Hence we get the convergence

Lemma 2 Let ν > 0 be fixed. There exists subsequences still denoted (Un), (Mn), (H_n), (Q_n) and (θ_n) such that when n→ ∞

U_n* U_ν weakly−? inL^∞(0, T;L²(D))and weakly in L²(0, T;U)

Mn* Mν, Hn* Hν weakly−?in L^∞(0, T;L²(D))and weakly inL²(0, T;H¹_t(D)) Q_n* Q_ν weakly−? inL^∞(0, T;L²(D)) and weakly inL²(0, T;H¹n(D))

θ_n* θ_ν weakly−?in L^∞(0, T;L⁴(D)) and weakly inL⁴(0, T;W^1,4(D)) Moreover, we have

|∇θ_n|²∇θ_n*Λν weakly inL⁴³(DT). (76) In order to pass to the limit in the nonlinear terms, we need strong convergence for the sequences in some spaces. To apply compactness results, we need to estimate the time derivatives of the solutions.

Let us begin with(∂tθn)n. We multiply equation (57) by a⁰_j(t) and add the resulting equalities for1≤j ≤nto get

k∂_tθnk²+ Z

D

∂tθn∇θ_n·Undx+ν 4

d

dtk∇θ_nk⁴_L4(D) =− Z

D

divQn∂tθn (77)

Since we have| Z

D

∂tθn∇θ_n·Undx| ≤ 1

4k∂_tθnk²+k∇θ_nk²_L4(D)kU_nk²_L4(D) we obtain k∂_tθnk²+ν

2 d

dtk∇θ_nk⁴_L4(D)≤2kdivQnk²+ 2k∇θ_nk²_L4(D)kU_nk²_L4(D) (78) Therefore, integrating between 0 andt, using (59) and (65), we easily deduce that

Z t 0

k∂_tθ_nk²ds+ν

2k∇θ_nk⁴_L4(D)≤ ν

2k∇θ^ν_0nk⁴_L4(D)+ 2

Z T 0

kdivQnk²ds+ 2 Z t

0

k∇θ_nk²_L4(D)kU_nk²_L4(D)ds

≤C+ 2 Z t

0

k∇θ_nk²_L4(D)kU_nk²_L4(D)ds

(79)

withC independent of n.

Settingy(t) = k∇θ_n(t)k⁴_L4(D) and F(t) =kU_nk²_L4(D) then from (79), y(t) satisfies the integral inequality

y(t)≤Cν + 2Mν

Z t 0

py(s)F(s)ds.

Using the Gronwall-Bellman-Bihari inequality (see [3]) we deduce y(t)≤p

Cν +Mν

Z t 0

F(s)ds 2

. Hence we get for all t∈[0, T]the estimate

k∇θ_n(t)k²_L4(D)≤p

Cν +Mν

Z t 0

kU_n(s)k²_L4(D)ds which leads to

Z T 0

k∂_tθnk²ds≤C+ 2p Cν

Z T 0

kU_n(s)k²_L4(D)ds+ 2Mν

Z T 0

kU_n(s)k²_L4(D)ds 2

(80) and we conclude that(∂_tθ_n)_n is uniformly bounded in L²(D_T) with respect to n.

To estimate ∂_tU_n, ∂_tM_n and ∂_tQ_n we need some notations. For a function f defined on [0, T]with values in a space V, let febe the function equal to f on [0, T]and to 0 elsewhere and letfbbe its Fourier transform defined by

fb(τ) = Z

R

exp(−2iπtτ)f(t)e dt= Z T

0

exp(−2iπtτ)f(t)dt, τ ∈R. We will prove that for0< γ <1/4,

Z

R

|τ|^2γkUb_n(τ)k² dτ ≤C. (81) Proceeding as in [22] (see also [16]) and since(Un)nis uniformly bounded inL²(0, T;U), it is enough to verify that

|τ| kUbn(τ)k²≤CkUbn(τ)k_U +CkUbn(τ)k, ∀τ ∈R. (82)

We write the equation (55) ofU_n in the form d

dt Z

D

Un·Vjdx=hL_n, Vji, U_n(0) =U0n (83) for all1≤j≤nwhere the linear form L_n is defined onU by

hL_n, ϕi= Z

D

(U_n· ∇)U_n·ϕ dx+η Z

D

∇U_n· ∇ϕ dx+ Z

D

ρ(θ_n)g·ϕ dx

−µ₀ Z

D

(M_n· ∇)H_n·ϕ dx−µ₀ 2

Z

D

M_n×H_n·curlϕ dx

(84)

We haveL_n∈ U⁰ p.p. t∈(0, T) and

kL_nk_U⁰ ≤C(kU_nk²_H1(D)+kU_nk_H1(D)+kρ(θ_n)k+

kM_nk_H1(D)kH_nk_H1(D)+kM_n×Hnk)

and we conclude thanks to Lemma 1 that (L_n) is uniformly bounded in L¹(0, T;U⁰).

Now we rewrite (99) as follows d

dt Z

D

Uen·Vjdx=hLf_n, Vji+ Z

D

U0n·Vj dx

δ0−Z

D

Un(T)·Vj dx

δT (85) for 1≤j≤nwhereδadenotes the Dirac distribution at a∈R. Therefore, we obtain

2iπτ Z

D

Ub_n·V_jdx=hLc_n, v_ji+ Z

D

U_0n·V_j dx−exp(−2iπT τ) Z

D

U_n(T)·V_j dx (86) Next we multiply equality (86) byαcj(τ) the conjugate ofcαj(τ) and add the equalities for 1≤j≤nto get

2iπτkUb_nk²=hLc_n,Ub_ni+ Z

D

U_0n·Ub_n dx−exp(−2iπT τ) Z

D

U_n(T)·Ub_n dx (87) therefore since for allτ ∈R, we have

kLc_n(τ)k_U⁰ ≤ Z T

0

kL_n(t)k_U⁰dt≤C then using Plancherel identity we get (82).

Similar proofs work for∂tMn and ∂tQn. The above results are summarized in Lemma 3 There exists C_ν >0 such that for all n

Z

R

|τ|^2γ(kUbn(τ)k²+kMcn(τ)k²+kQbn(τ)k²) dτ ≤Cν (88) Moreover we have

k∂_tθ_nk_L2(DT)≤C_ν (89) Combining the bounds of Lemma 1 and Lemma 3 and applying Lions compactness lemma for(U_n, M_n, Q_n) and and Aubin compactness lemma for θ_n we get the strong convergence results we get the strong convergence results