A dynamic electroviscoelastic problem with thermal effects

DOI: 10.24193/subbmath.2021.4.13

A dynamic electroviscoelastic problem with thermal effects

Sihem Smata and Nemira Lebri

Abstract. We consider a mathematical model which describes the dynamic pro- cess of contact between a piezoelectric body and an electrically conductive foun- dation. We model the material’s behavior with a nonlinear electro-viscoelastic constitutive law with thermal effects. Contact is described with the Signorini condition, a version of Coulomb’s law of dry friction. A variational formulation of the model is derived, and the existence of a unique weak solution is proved. The proofs are based on the classical result of nonlinear first order evolution inequali- ties, the equations with monotone operators, and the fixed point arguments.

Mathematics Subject Classification (2010):74M15, 74M10, 74F05, 49J40.

Keywords: Piezoelectric, frictional contact, thermo-elasto-viscoplastic, fixed point, dynamic process, Coulomb’s friction law, evolution inequality.

1. Introduction

Piezoelectricity is the ability of certain crystals, like the quartz, to produce a voltage when they are subjected to mechanical stress. On a nanoscopic scale, the piezoelectric phenomenon arises from a nonuniform charge distribution within a crys- tal unit cells, and the piezoelectricity is then perceived as the electrical polarization due to mechanical input. Different models have been developed to describe the inter- action between the electric and mechanical fields (see, e.g.[12, 13, 15]). Therefore there is a need to extend the results on models for contact with deformable bodies which include coupling between mechanical and electrical properties. General models for elastic materials with piezoelectric effects can be found in [13, 17] and more recently in [2], viscoelastic piezoelectric materials in [2, 17] or elasto-viscoplastic piezoelectric materials have been studied in [9].

In this paper, we consider a general model for the dynamic process of fric- tional contact between a deformable body and a rigid obstacle. The material obeys an electro-viscoelastic constitutive law with piezoelectric and thermal effects. More- over, the contact and friction are modelled by Signorini’s conditions and a non local

Coulomb’s friction law. We derive a variational formulation of the model, which is set as a system coupling a variational second order evolution inequality. We establish the existence of a unique weak solution of the model. The idea is to reduce the sec- ond order evolution inequality of the system to first order evolution inequality. Then adopting fixed point methods frequently we prove an existence and uniqueness of dis- placement and temperature fields, using monotonicity and convexity properties. The importance of this paper is to make the coupling of an electro-viscoelastic problem with thermal effects. The paper is structured as follows. In Section 2 we describe the mechanical problem and provide comments on the contact boundary conditions. In Section 3 we list the assumptions on the data and derive the variational formulation.

In Sections 4, we present our main existence and uniqueness results, which state the unique weak solvability of the Signorini’s contact electro-visco -elastic problem with non local Coulomb’s friction lawn conditions.

2. Problem statement

We consider a body made of a piezoelectric material which occupies the do- main Ω ⊂ R^d (d≤3) with a Lipschitz boundary Γ. The body is modelled with an electro-visco-elastic constitutive law, allowing piezoelectric effects. Let [0.T] be the time interval where T >0, let Γ be split into three measurable parts Γ1, Γ2 and Γ3

such thatmeasΓ1>0. We assume that the body is fixed on Γ1and surface tractions of densityhact on Γ2.On Γ3, the body may come into contact with a rigid obstacle.

In other hand, Γ be split into two measurable sets Γa and Γb such thatmeasΓb >0 and Γ3 ⊂ Γb . We assume that the electrical potential q0 act on Γa and a surface electric charge of densityq2act on Γb, we assume that the problem is quasistatic.The piezoelectric effect is the apportion of electric charges on surfaces of particular crys- tals after deformation. We denote byS^d the space of second order symmetric tensors on the spaceR^d and use· and |.| for the inner product and the Euclidean norm on the spaceR^d (respectively; S^d). Alsoν represents the unit outward normal on Γ, the classical formulation of the electro-visco-elastic contact friction problem is described by:

Problem P. Find a displacement field u : Ω×[0.T] → R^d, a stress field σ : Ω× [0.T]→S^d, an electric potential fieldϕ: Ω×[0.T]→R, an electric displacement field D: Ω×[0.T]→R^d and a temperature fieldθ: Ω×[0, T]→R+ such that:

σ=Aε u^.

+Gε(u)−ξ^∗E(ϕ)−θM_e in Ω×[0.T], (2.1) D=βE(ϕ) +ξε(u) in Ω×[0.T], (2.2) ρu^..=Div σ+f0in Ω×[0.T], (2.3)

div D=q₀ in Ω×[0.T], (2.4)

. .

θ−div (k∇θ) =−M∇u^.+qein Ω×[0.T], (2.5)

−kij

∂θ

∂υυj =ke(θ−θR) on Γ3×[0.T], (2.6)

θ= 0 on Γ1∪Γ2×(0, T), (2.7)

u= 0 on Γ1×[0.T], (2.8)

σν=hon Γ2 ×[0.T], (2.9) u_ν≤0, σ_ν≤0, u_ν σ_ν= 0 on Γ₃×[0.T], (2.10)







|σ_τ| ≤µp|R σ_ν|

|σ_τ|< µp|R σ_ν|=⇒u^._τ = 0

|στ|=µp|R σν|=⇒ ∃ λ≥0 such thatστ =−λu^.τ

on Γ₃×[0.T], (2.11)

ϕ= 0 on Γ_a×[0.T], (2.12)

Dν=q2on Γb×[0.T], (2.13)

u(0) =u0 ,u^.(0) =v0 andθ(0) =θ0 in Ω×[0.T], (2.14) where (2.1), (2.2) are the thermo-electro -visco-elastic constitutive law of the mate- rial, we denoteε(u) (respectively;E(ϕ) =−∇ϕ,A,G,ξ, ξ^∗, β) the linearized strain tensor (respectively; electric field, the viscosity nonlinear tensor, the elasticity ten- sor, the third order piezoelectric tensor and its transpose, the electric permittivity tensor),θ represent the temperature, Me:= (mij) represents the thermal expansion tansor, (2.3) represents the equation of motion whereρrepresents the mass density, (2.4) represents the equilibrium equation, Equation (2.5) describes the evolution of the temperature field, wherek:= (kij) represents the thermal conductivity tensor,qe

the density of volume heat sources. The associated temperature boundary condition is

given by (2.6),whereθ_r is the temperature of the foundation, andk_eis the exchange coefficient between the body and obstacle. Equation (2.7) means that the temperature vanishes on Γ₁∪Γ₂×(0, T). We mention thatDivσ,divD are the divergence oper- ators, (2.8) and (2.9) are the displacement and traction boundary conditions, (2.10), (2.11) the Signorini’s contact with a non local Coulomb’s friction law conditions. u

ν and uτ(respectively; σν and στ) denote the normal displacement and the tangen- tial displacement (respectively; the normal stress and the tangential stress). R will represent a normal regularization operator that is a linear and continuous operator R :H⁻¹²(Γ) → L²(Γ). We shall need it to regularize the normal trace of the stress which is too rough on Γ. p is a non-negative function, the so-called friction bound, µ ≥ 0 is the coefficient of friction. The friction law was used in some studies with p(r) =r₊ wherer₊ =max{0, r}. Recently, from thermodynamic considerations, a new version of Coulomb’s law is proposed, it consists to take:

p(r) =r(1−αr)+, (2.15)

where α is a small positive coefficient related to the hardness and the wear of the contact surface. (2.12), (2.13) represent the electric boundary conditions. Finally, in (2.14)u0is the given initial displacement,v0is the given initial velocity andθ0is the initial temperature.

3. Variational formulation and preliminaries

For a weak formulation of the problem, first we introduce some notation. The indicesi, j, k, l range from 1 tod and summation over repeated indices is implied.

An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e. g: u_i.j = ^∂u_∂xⁱ

j. We also use the following notations:

H =L²(Ω)^d={u= (u_i)/u_i∈L²(Ω)}, H={σ= (σ_ij)/σ_ij =σ_ji∈L²(Ω)}, H1={u= (ui)/ε(u)∈ H}, H1={σ∈ H/Divσ∈H},

The operators of deformationεand divergence Divare defined by ε(u) = (ε_ij(u)), ε_ij(u) = 1

2(u_i,j+u_j,i), Divσ= (σ_ij,j).

The spacesH,H, H1, andH1are real Hilbert spaces endowed with the canonical inner products.

We denote by |. |H (respectively; | .|H, |. |H₁, and|. |H1) the associated norm on the spaceH ( respectively; H,H1, andH1).

We use standard notation for theL^p and the Sobolev spaces associated with Ω and Γ and, for a functionψ∈ H¹(Ω) we still writeψ to denote it trace on Γ.We recall that the summation convention applies to a repeated index.

For the electric displacement field we use two Hilbert spaces:

W=L²(Ω)^d, W₁=

D∈ W,divD∈L²(Ω) endowed with the inner products:

(D, E)_W =R

ΩD_iE_idx, (D, E)_W

1 = (D, E)_W+ (divD,divE)

L²(Ω). And the associated norm|.|_W(respectively;|.|_W

1). The electric potential field is to be found in:

W =

ψ∈H¹(Ω), ψ = 0 on Γa .

Sincemeas(Γ_a)>0,the following Friedrichs-Poincar´e’s inequality holds, thus:

|∇ψ|_W ≥cF|ψ|_H1(Ω) ∀ψ∈W, (3.1) wherecF >0 is a constant which depends only on Ω and Γa. OnW, we use the inner product given by:

(ϕ, ψ)_W = (∇ϕ,∇ψ)_W,

and let |.|_W be the associated norm. It follows from (3.1) that |.|_H1(Ω) and |.|_W are equivalent norms onW and therefore (W,|.|_W) is a real Hilbert space.

Moreover, by the Sobolev trace Theorem, there exists a constantec₀, depending only on Ω, Γ_a and Γ₃such that:

|ψ|_L2(Γ3)≤ec0|ψ|_W ∀ψ∈W. (3.2) We recall that when D ∈ W1 is a sufficiently regular function, the Green’s type formula holds:

(D,∇ψ)_W+ (divD, ψ)

L²(Ω)= Z

Γ

Dν.ψda. (3.3)

Whenσis a regular function, the following Green’s type formula holds:

(σ, ε(v))_H+ (Divσ, v)_H= Z

Γ

σν.vda ∀v∈H1. (3.4) Next, we define the space:

V ={u∈H1/ u= 0 on Γ1}.

Sincemeas(Γ₁)>0, the following Korn’s inequality holds:

|ε(u)|_H ≥cK|v|_H

1 ∀v∈V, (3.5)

wherecK>0 is a constant which depends only on Ω and Γ1. On the spaceV we use the inner product:

(u, v)_V = (ε(u), ε(v))_H,

let|.|_V be the associated norm. It follows by (3.5) that the norms|.|_H

1 and|.|_V are equivalent norms onV and therefore, (V,|.|_V) is a real Hilbert space. Moreover, by the Sobolev trace Theorem, there exists a constantc0depending only on the domain Ω,Γ1and Γ3 such that:

|v|_L2(Γ3)^d≤c₀|v|_V ∀v∈V. (3.6) In what follows, we assume the following assumptions on the problemP.























(a) :A: Ω×S^d→S^d,

(b) :∃M_A>0 such that :|A(x, ε1)− A(x, ε2)| ≤M_A|ε1−ε2|

∀ε1, ε2∈S^d,a. e.x∈Ω,

(c) :∃mA>0 such that :|A(x, ε1)− A(x, ε2), ε1−ε2| ≥mA|ε1−ε2|²

∀ε1, ε2∈S^d,a. e.x∈Ω,

(d) : the mappingx→ A(x, ε) is lebesgue measurable in Ω for allε∈S^d, (e) : the mappingx→ A(x,0)∈ H,

(3.7)















(a) :G: Ω×S^d→S^d,

(b) :∃M_G >0 such that :|G(x, ξ₁)− G(x, ξ₂)| ≤M_G|ξ1−ξ₂|

∀ξ1, ξ2∈S^d,a. e.x∈Ω,

(d) : the mappingx→ G(x, ξ) is lebesgue measurable in Ω for allξ∈S^d, (e) : the mappingx→ G(x,0)∈ H,

(3.8)







(a) :ξ= (e_ijk) : Ω×S^d→R^d,

(b) :ξ(x, τ) = (e_ijk(x)τ_jk) ∀τ= (τ_ij)∈S^d, a. e. x∈Ω, (c) :e_ijk=e_ikj∈L^∞(Ω),

(3.9)















(a) :β= (β_ij) : Ω×R^d→R^d,

(b) :β(x, E) = (bij(x)Ej) ∀E= (Ei)∈R^d, a.e.x∈Ω, (c) :b_ij =b_ji∈L^∞(Ω),

(d) :∃mβ >0 such that :bij(x)EiEj ≥mβ|E|²

∀E= (Ei)∈R^d, x∈Ω.

(3.10)

From the assumptions (3.9) and (3.10),we deduce that the piezoelectric operator ξ(respectively; the electric permittivity operatorβ) is linear, has measurable bounded

component denotedeijk( respectively;bij) and moreover,βis symmetric and positive definite.

Recall also that the transposed operatorξ^∗ is given byξ^∗ = (e^∗_ijk) wheree^∗_ijk =e_kij and the following equality holds:

ξσ.v=σ.ξ^∗v ∀σ∈S^d, v∈R^d. The friction function satisfies:















p: Γ3×R→R⁺ verifies:

(a) :∃M >0 such that :|p(x, r1)−p(x, r2)| ≤M|r1−r2| For everyr1, r2∈R, a. e. x∈Γ3,

(b) : the mapping :x→p(x, r) is measurable on Γ3, for everyr∈R, (c) :p(x,0) = 0, a. e. x∈Γ3.

(3.11)

We note that (3.11) is satisfied in the case in whichpgiven by (2.12).

We also assume that the body forces and surface tractions have the regularity:

f₀∈L²(0.T;H), h∈L²

0.T;L²(Γ₂)^d

, (3.12)

The thermal tensors and the heat source density satisfy

M = (m_ij), m_ij=m_ji∈L^∞(Ω), q_e∈L²(0, T;L²(Ω)), (3.13) and for somec_k >0, for all (ζ_i)∈R_d :

k = (ki,j), kij =kji∈L^∞(Ω), kijζiζj≥ckζiζj (3.14) as well as the densities of electric charges satisfy:

q0∈L² 0.T;L²(Ω)

, q2∈L² 0.T;L²(Γb)

. (3.15)

We define the functionf : [0.T]→V andq: [0.T]→W by:

(f(t), v)_V =R

Ωf0(t)vdx+R

Γ₂h(t)vda ∀v∈V, t∈[0.T], (3.16) (q(t), ψ)_W =−R

Ωq0(t)ψdx+R

Γ_bq2(t)ψda ∀ψ∈W, t∈[0.T]. (3.17) for all u, v ∈ V, ψ ∈ W and t ∈ [0.T], and note that conditions (3.14) and (3.15) imply that

f ∈L²(0.T;V⁰), q∈L²(0.T;W), (3.18) while the friction coefficientµ, the mass densityρsatisfies

µ∈L^∞(Γ₃), µ(x)≥0, a. e.on Γ₃,

ρ∈L^∞(Ω) there existsρ^∗>0 such thatρ(x)≥ρ^∗, a.e.x∈Ω. (3.19) u₀∈V, v₀∈H, θ₀∈E, θ_R∈W^1,2(0, T;L²(Γ₃)), k_e∈L^∞(Ω,R+) , (3.20) The functionr:V →L2(Ω) satisfies that there exists a constantLr>0 such that

|r(v1)−r(v2)|_L2(Ω)≤Lr|v1−v2|V,∀v1, v2∈V (3.21) We denote by the frictionfunctional j:H ×V →R

j(σ, v) = Z

Γ3

µp|R σν| |vτ|da. (3.22)

We denote byU the convex subset of admissible displacements fields given by U ={v∈H1/v= 0 on Γ1, vν≤0 on Γ3}, (3.23) By a standard procedure based on Green’s formula, we obtain the following formula- tion of the mechanical problem (2.1)−(2.14).

ProblemPV.Find a displacement fieldu: Ω×[0.T]→R^d,an electric potentiel field ϕ: Ω×[0.T]→R, an electric displacement field D : Ω×[0.T]→R^d such that and a temprature fieldθ: Ω×[0, T]→R+ such that:

(u, w^.. −u)^. V⁰×V + (σ(t), ε(w−u(t)))^. _H+j(σ, w)−j(σ,u(t))^.

≥(f(t), w−u(t))^. ∀u, w∈V (3.24)

(D(t),Oψ)_L2(Ω)^d+ (q(t), ψ)_W = 0 ∀ψ∈W (3.25) (

.

θ(t) +Kθ(t) =Re

u.(t) +Q(t) t∈(0, T), ∀ψ∈W (3.26) u(0) =u0 ,u^.(0) =v0andθ(0) =θ0 (3.27) whereQ: [0, T]→E⁰, K:E→E⁰, R:V →E⁰ are given by

(Q(t), µ)_W = Z

Γ₃

k_e(u_ν)θ_Rµda+ Z

Ω

qµdx, (3.28)

(Kτ, µ)E⁰×E=

d

X

i,j=1

Z

Ω

kij

∂τ

∂x_j

∂µ

∂x_idx+ Z

Γ3

keτ µda, (Reµ, v)_{E0 ×E}

=− Z

Ω

(M∇v)dx, (3.29)

for allv∈V, µ, τ ∈E.

4. Existence and uniqueness result

Our main result which states the unique solvability of Problem are the following.

Theorem 4.1. Let the assumptions(3.7)−(3.20)hold. Then, ProblemP V has a unique solution {u, ϕ, D, θ}which satisfies

u∈C¹(0.T;H)∩W^1.2(0.T;V)∩W^2.2(0.T;V⁰) (4.1)

ϕ∈W^1.2(0.T;W) (4.2)

σ∈L²(0.T;H), Divσ∈L²(0.T;V⁰) (4.3)

D∈W^1.2(0.T;W1) (4.4)

θ∈W^1,2(0, T;E⁰)∩L²(0, T;E)∩C(0, T;L²(Ω)) (4.5) We conclude that under the assumptions (3.7)−(3.21), the mechanical problem (2.1)−

(2.14) has a unique weak solution with the regularity (4.1)−(4.5).The proof of this theorem will be carried out in several steps. It is based on arguments of first order evolution nonlinear inequalities (see Refs. [5,7-9]), evolution equations (see Ref. [2]), and fixed point arguments.

Let G ∈ L²(0.T;H) and η ∈ L²(0.T;V⁰) are given, we deduce a variational formulation of ProblemP V.

ProblemP VGη:Find a displacement fielduGη : [0.T]→V such that







u_Gη(t)∈U (u^.._Gη, w−u^._Gη)_V⁰_×V + (Aε(u^._Gη(t)), ε(w−u^._Gη(t))_H+ η, w−u^._Gη(t)

V⁰×V +j(G, w)−j(G,u^._Gη(t))≥(f(t), w−u^._Gη(t))

∀w∈V

(4.6)

u._Gη(0) =v(0) =v₀ (4.7)

We definef_η(t)∈V fora.e.t∈[0.T] by

(f_η(t), w)_V⁰_×V = (f(t)−η(t), w)_V⁰_×V,∀w∈V. (4.8) From (3.18), we deduce that:

f_η∈L²(0.T;V⁰) (4.9)

Let nowuGη: [0.T]→V be the function defined by

u_Gη(t) =

t

Z

0

v_Gη(s)ds+u₀, ∀t∈[0.T] (4.10)

We define the operatorA:V⁰ →V by

(Av, w)V⁰×V = (Aε(v), ε(w))H,∀v, w∈V. (4.11) Lemma 4.2. For allG∈L²(0.T;H)andη ∈L²(0.T;V⁰),P VGη has a unique solution with the regularity:

vGη∈C(0.T;H)∩L²(0.T;V)andv^.Gη∈L²(0.T;V⁰). (4.12) Proof. The proof from nonlinear first order evolution inequalities, given in Refs ([8]).

In the second step, we use the displacement fielduGη to consider the following variational problem.

ProblemP V1Gη : Find an electric potential field ϕ_Gη : Ω×[0.T]→W such that:

(β∇ϕGη(t),Oψ)_L2(Ω)^d−(ξε(uGη(t)),Oψ)_L2(Ω)^d = (q(t), ψ)_W

∀ψ∈W, t∈[0.T] (4.13) We have the following result forP V1_Gη:

Lemma 4.3. There exists a unique solution ϕ_Gη ∈ W^1.2(0.T;W) satisfies (4.13), moreover ifϕ₁andϕ₂ are two solutions to(4.13). Then, there exists a constantsc >0 sach that:

|ϕ1(t)−ϕ2(t)|_W ≤c|u1(t)−u2(t)|_V ∀t∈[0.T]. (4.14)

Proof. See [16].

In the third step, we use the displacement fielduη obtained in Lemma 4.2 to consider the following variational problem.

ProblemP V1θη: Findθη : [0, T]→E⁰ satisfying a.e.t∈(0, T)

.

θη(t) +Kθη(t) =Re

u.η(t) +Q(t) t∈(0, T), inE⁰, (4.15)

θ_η(0) =θ₀. (4.16)

Lemma 4.4. ProblemP V1θη has a unique solution, for allη∈ W,

θη∈W^1,2(0;T;E⁰)∩L²(0;T;E)∩C(0;T;L²(Ω)), C >0, ∀η∈L²(I;V⁰) satisfying

|θη₁−θη₂|²_L2(Ω)≤C Z T

0

|υ1(s)−υ2(s)|²_Vds ∀t∈(0, T). (4.17) Proof. The existence and uniqueness result verifying (4.15) follows from classical re- sult on first order evolution equation, applied to the Gelfand evolution triple

E⊂F ≡F⁰ ⊂E⁰

We verify that the operator K is linear, continuous, strongly monotone, and from the expression of the operator R, vη ∈ W^1,2(0, T;V) ⇒ Rvη ∈ W^1,2(0, T;F), as Q∈W^1,2(0, T;E) thenRvη+Q∈W^1,2(0, T;E). We deduce (4.17) see [1].

We consider the operator

Λ :L²(0.T;H ×V⁰)→L²(0.T;H ×V⁰) be defined as

Λ (G, η) = (Λ1(G),Λ2(η)),∀G∈L²(0.T;H),∀η ∈L²(0.T;V⁰),

|Λ (G₂, η₂)−Λ (G₁, η₂)|²=|(Λ₁(G₂),Λ₂(η₂))−(Λ₁(G₁),Λ₂(η₁))|²,

|Λ1(G2)−Λ1(G1),Λ2(η2)−Λ2(η1)|²=|Λ1(G2)−Λ1(G1)|² +|Λ2(η2)−Λ2(η1)|².

(4.18)

We show that Λ has a unique fixed point.

Lemma 4.5.

Λ (G^∗, η^∗) = (G^∗, η^∗). (4.19) Proof. Let (G_i,η_i) are functions inL²(0.T;H ×V⁰) and denote by (u_i, ϕ_i, θ_i ) the functions obtained in Lemma 4.2, Lemma 4.3 and Lemma 4.4,

for(G, η) = (G_i, η_i)i= 1.2. Lett∈[0.T]. From (2.1) it results

|G₂−G₁|²_H≤c

|v₂(t)−v₁(t)|²_V +|ϕ₂(t)−ϕ₁(t)|²_W +|u2(t)−u1(t)|²_V +|θη₁−θη₂|²_L2(Ω)

(4.20) Therefore (4.14) and (4.17) yields

|G2−G1|²_H≤c |v2(t)−v1(t)|²_V +|u2(t)−u1(t)|²_V + Z T

0

|υ1(s)−υ2(s)|²_Vds

! . (4.21)

Using (4.6),we find

(v^.₂(t)−v^.₁(t), v₂(t)−v₁(t)) + (Aε(v₂(t))− Aε(v₁(t)), v₂(t)−v₁(t))+

(η₂(t)−η₁(t), v₂(t)−v₁(t)) +j(G₂, v₂(t))−j(G₂, v₁(t))

−j(G₁, v₂(t)) +j(G₁, v₁(t))≤0

(4.22) And, we have

j(G2, v2(t))−j(G2, v1(t))−j(G1, v2(t)) +j(G1, v1(t))

≤ Z

Γ3

µp|R G2ν| |v2τ|da− Z

Γ3

µp|R G2ν| |v1τ|da

− Z

Γ₃

µp|RG1ν| |v2τ|da+ Z

Γ₃

µp|R G1ν| |v1τ|da. (4.23) Moreover, from (3.11),(3.19) and using the properties ofR , we find

j(G2, v2(t))−j(G2, v1(t))−j(G1, v2(t)) +j(G1, v1(t))≤c|G2−G1|_H|v2−v1|_V (4.24) So, (4.22) will be

(v^.₂(t)−v^.₁(t), v₂(t)−v₁(t))_V⁰_×V + (Aε(v₂(t))− Aε(v₁(t)), v₂(t)−v₁(t)) + (η2(t)−η1(t), v2(t)−v1(t))≤c|G2−G1|_H|v2−v1|_V (4.25) We integrate this equality with respect to time.

We use the initial conditions v1(0) = v2(0) = v0, the relation (3.7) and Cauchy- Schwarz’s inequality. for allt∈[0, T]. Then, using the inequality

2ab≤ a²

m_A +m_Ab², we obtain

1

2|v2(t)−v1(t)|²_V +mA

2 Z t

0

|v2(s)−v1(s)|²_V ds

≤ 1 2mA

Z t 0

|η2(s)−η₁(s)|²_V0+m_A 2

Z t 0

|v2(s)−v₁(s)|²_V ds +c

Z t 0

|G2(s)−G1(s)|²_H+ Z t

0

|v2(s)−v1(s)|²_V ds.

(4.26) We apply Gronwall’s inequality to obtain

|v2(t)−v1(t)|²_V ≤c Z t

0

|G2(s)−G1(s)|²_Hds+ Z t

0

|η2(s)−η1(s)|²_V0

. (4.27) In other hand

|η₂(t)−η₁(t)|²_V0 ≤c

|ϕ₂(t)−ϕ₁(t)|²_W +|u₂(t)−u₁(t)|²_V +|θ_η1−θ_η2|²_L2(Ω)

(4.28) Therefore (4.14) and (4.17) yields

|η2(t)−η₁(t)|²_V0 ≤c Z T

0

|υ1(s)−υ₂(s)|²_Vds

!

(4.29)

Using (4.6),we find

(v^.2(t)−v^.1(t), v2(t)−v1(t)) + (Aε(v2(t))− Aε(v1(t)), v2(t)−v1(t)) + (η2(t)−η1(t), v2(t)−v1(t)) +j(G2, v2(t))−j(G2, v1(t))

−j(G₁, v₂(t)) +j(G₁, v₁(t))≤0 (4.30)

We integrate this equality with respect to time.

We use the initial conditions v1(0) = v2(0) = v0, the relations (3.7),(4.24) and Cauchy-Schwarz’s inequality, for allt∈[0, T]. Then, using the inequality

ab≤c a²+b² , we obtain

Z t 0

|v2(s)−v1(s)|²_V ds≤c Z t

0

|η2−η1|²_V0ds+ Z t

0

|G2−G1|²_Hds

(4.31) Applying the inequality (4.10) in (4.31).So (4.21) will be

|G2(t)−G₁(t)|²_H≤c Z t

0

|η2(s)−η₁(s)|²_V0ds+ Z t

0

|G2(s)−G₁(s)|²_Hds

. (4.32) From (4.10), (4.29) and (4.31) we find

|η2−η₁|²_V0≤c Z t

0

|η2−η₁|²_V0ds+ Z t

0

|G2−G₁|²_Hds

. (4.33)

Using (4.16),to see that

|Λ (G2, η2)−Λ (G1, η1)|²≤c Z t

0

|(G2, η2)−(G1, η1)|²_H×V0ds. (4.34) And denoting bypthe powers of operator Λ,(4.32) imply by recurrence that

|Λ^p(G₂, η₂)−Λ^p(G₁, η₁)|²_L2(0.T;H×V⁰)

≤ (ct)^p

p! |(G2, η₂)−(G₁, η₁)|²_L2(0.T;H×V⁰). (4.35) This inequality shows that for a sufficiently large pthe operator Λ^p is a contraction on the Banach space L²(0.T;H ×V⁰) and therefor, there exists a unique element:

(G^∗, η^∗)∈L²(0.T;H ×V⁰)such that

Λ (G^∗, η^∗) = (G^∗, η^∗). (4.36) From (4.18), we find

(G^∗, η^∗) = (σG^∗η^∗, ξ^∗∇ϕG^∗η^∗+Gε(uG^∗η^∗)−θG^∗η^∗Me). (4.37) Now, we have all the ingredients to provide the proof of Theorem 4.1.

Proof of Theorem 4.1. Existence. Let (G^∗, η^∗)∈L²(0.T;H ×V⁰) be the fixed point ofP VGη and let (u^∗, ϕ^∗, θ^∗) be the solution to Problems P VGη, P V1Gη andP V1θη

for (G, η) = (G^∗, η^∗), that is, u^∗ =uG^∗η^∗,ϕ^∗ =ϕG^∗η^∗ andθ^∗ =θG^∗η^∗ . It results from(3.24),(3.25) and (3.26) that (u^∗, ϕ^∗, θ^∗) is a solution of ProblemP V. Property (4.1) (4.2) and (4.5) follows from Lemmas 4.2, 4.3 and 4.4.

Uniqueness.The uniqueness of the solution is a consequence of the uniqueness of the

fixed point of operator defined by (4.18).

Acknowledgment. This work has been realized thanks to the: Direction g´en´erale de la Recherche Scientifique et du D´eveloppement Technologique ”DGRSDT”. MESRS Algeria. And Research Project under code: PRFUCOOL03UN1901200180004.

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Sihem Smata

Applied Mathematics Laboratory, Department of Mathematics,

Faculty of Sciences, University of Setif 1, 19000, Algeria

e-mail:[email protected] Nemira Lebri

Applied Mathematics Laboratory,

Department of Mathematics, Faculty of Sciences, University of Setif 1, 19000, Algeria

e-mail:nem [email protected]