Generalized Navier–Stokes equations with non-standard conditions for blood flow in atherosclerotic artery

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Generalized Navier–Stokes equations with non-standard conditions for blood flow in atherosclerotic artery

S. Boujena

, N. El Khatib

& O. Kafi

a

Faculté des Sciences -Ain Chock, Université Hassan II, B.P 5366, Maarif, Casablanca, Morocco.

b

Department of Computer Science and Mathematics, Lebanese American University, P.O. Box: 36, Byblos, Lebanon.

c

CEMAT/IST and Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal.

Published online: 22 Jul 2015.

To cite this article: S. Boujena, N. El Khatib & O. Kafi (2015): Generalized Navier–Stokes equations

with non-standard conditions for blood flow in atherosclerotic artery, Applicable Analysis: An International Journal, DOI: 10.1080/00036811.2015.1068297

To link to this article: http://dx.doi.org/10.1080/00036811.2015.1068297

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http://dx.doi.org/10.1080/00036811.2015.1068297

Generalized Navier–Stokes equations with non-standard conditions for blood flow in atherosclerotic artery

S. Boujena^a, N. El Khatib^b∗and O. Kafi^c

aFaculté des Sciences -Ain Chock, Université Hassan II, B.P 5366, Maarif, Casablanca, Morocco;

bDepartment of Computer Science and Mathematics, Lebanese American University, P.O. Box: 36, Byblos, Lebanon;^cCEMAT/IST and Department of Mathematics, Instituto Superior Técnico, Av.

Rovisco Pais 1, 1049-001 Lisbon, Portugal Communicated by G. Panasenko

(Received 19 December 2014; accepted 29 June 2015)

In this paper, we consider the blood flow in a stenosed artery. We give an analytical study of the equations for a non-Newtonian fluid modeling the blood for which the behavior is obeying to Carreau’s law. The case we treat is different than classic cases where the total pressure is in the natural boundary conditions. For this, we use the Faedo–Galerkin method to prove the existence of a weak solution for the fluid problem. Then we use a coupled approach between the fluid equations and the solid model of the arterial wall and the atheromatous plaque. A special attention is paid to the effects of the wall motion on the local fluid displacement, on the stresses, and on strains in the diseased arterial wall. These relevant quantities are analyzed extensively through numerical results.

Keywords: non-Newtonian fluids; Faedo–Galerkin method; blood flow;

atheromatous plaque

AMS Subject Classifications:35Q30; 92C50; 74F10

1. Introduction

Diseased arteries can create high levels of turbulence, head loss, and a choked-flow condition in which tubes can collapse. Most of these diseases are caused by local factors acting at a specific site. The stress and mass transfer at the vessel wall are important factors that trigger biological responses. One of these responses is the formation of an atherosclerotic stenosis.

The localization of the atherosclerotic plaque in the artery is due to the observation of local variations in the wall shear stress.[1] However, in the particular case of atheromatous plaques, local hamodynamics are not only governed by the geometry of the stenosis and the properties of the arterial wall, but also by the rheological properties of blood. The latter being a complex fluid and primarily composed of red blood cells (RBCs), which occupy (in a healthy human body) about 45% of the blood volume. The rest consists of plasma and other blood particles (white blood cells, platelets, etc.). The descriptions of blood flow properties escape the traditional laws that govern simple fluids. The complex character of

∗Corresponding author. Email: [email protected]

© 2015 Taylor & Francis

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the blood results from an intimate coupling between the shape of RBCs and the ambient plasma, which leads to a rich set of RBC morphologies in the blood circulatory system.[2]

In the literature, there exists a considerable number of studies concerning theoretical and numerical modeling of blood flow through a pipe representing a portion of a blood vessel with or without the presence of a stenosis. Some of these studies consider the blood as a Newtonian fluid and the geometry of the stenosis to be a function of a smooth shape (e.g.

a cosine function). On the one side, to model the blood as a Newtonian fluid that flows inside a largerigid artery, a wide range of results can be used. In chronological order, we can mention first the works of Leray [3–5] and Ladyzhenskaya [6]. Then come to the books of Lions [7] and Temam [8]. Other studies considering the Stokes and Navier–Stokes equations for blood have been done later by Verfürth [9,10] and Conca [11] with different boundary conditions (no stress and no slip at the boundary). We also find results of existence and uniqueness of the solution of the Navier–Stokes problem in the case of a class of non- Newtonian fluids with Dirichlet boundary conditions in the work of Boujena [12]. On the other side, to model the blood flow inside a largecompliantartery, the equations modeling the fluid are coupled with the ones modeling the vessel wall.

This gives us the so-calledfluid–structure interaction(FSI) coupled problem. Modeling the FSI between the blood and the arterial wall is a challenging task. First, because of the highly non-linear nature of the governing equations and because of the problems related to the non-regularity of the boundaries and the fluid–structure interface. (Existence results have been obtained only to understand if the equations describing the model provide a mathematically well-posed problem. For recent overviews on this problem, the reader is refereed to [13,14] and references therein). Second, because of the complexity of the numerical study of the FSI problem. Indeed, it does not only inherit the difficulties associated to the fluid and solid simulation, but the coupling of these two systems is also cumbersome in many situations. Here, we notice that the difficulties arising from this coupled system depend strongly on the physical properties of the case to be simulated. We can select from the literature dealing with the FSI some relevant studies. In [15], the authors study the coupling of a generalized Newtonian fluid describing the blood flow with an elastic structure describing the vessel wall, taking into account the shear-thinning behavior of the blood, to capture the pulse wave due to the interaction between blood and the vessel wall in a three-dimensional (3D) geometry of a healthy artery. In [16], a new algorithm to efficiently predict the hamodynamics in large arteries is addressed and validated. Otherwise, a two- dimensional (2D) model of FSI between blood flow and atherosclerotic plaque is studied in [17] and the conditions of plaque rupture are investigated; then in [18], the author proposes an incorporation of FSI considering the non-Newtonian character of the blood to investigate the recirculations downstream of the plaque and the stress on the latter.

All of these studies investigate either the blood-wall or the blood-plaque interaction, but none of them considers these simultaneously. This is why we decided to take them both into account. In [19], we consider a 2D analytical study of the generalized Navier–

Stokes equations modeling the blood flow and interacting with the vessel wall and the atherosclerotic plaque. We provide an existence result of solutions for the evolutionary case, under suitable hypothesis on the boundary conditions. Then, this coupled problem is solved numerically using the arbitrary Lagrangian–Eulerian (ALE) formulation.

But, the geometry of the 2D model studied in [19] still does not correspond to the geometry of the real blood vessel. It represents a simplified geometry to start with, since the blood vessel is a 3D domain. Therefore, we extend the results of the previous study by

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considering a model with a 3D geometry of anidealizedblood vessel in order to be more realistic, the aim being to analyze the evolutionary non-linear 3D-flow characteristics of blood in a stenosed artery. For that, we consider the arterial segment as a distensible cylin- drical tube and the blood flowing through as a generalized Newtonian fluid with variable viscosity and non-standard boundary conditions. Our choice of this unusual formulation is motivated by the fact that the total normal stress becomes a natural boundary condition once the problem is written in a weak form, as it will be shown below in the text of the proof. The solution of this problem is known as Womersley flow and can be seen as the unsteady part of the Poiseuille flow. It is also a “reference solution” for blood flow in large arteries. Then, when we couple the fluid equations with a model for the vessel wall in the FSI model, the condition of the wall rigidity will be relaxed to investigate the effects of wall compliance on local fluid mechanics, on the stresses, and also on strains in the diseased arterial wall. These relevant quantities are shown and analyzed extensively through the numerical simulations that are presented at the end of the paper along with a comprehensive discussion. We shall mention here that when we consider the 3D case, the computation of the numerical solution is an additional challenging issue. Indeed, the number of degrees of freedom in the 3D case is much more important than the one in the 2D one, then the computational time and the memory needed to perform the simulations are much more important.

The results that we present in this paper could be extended to the realistic 3D geometry, where the computational domain will be a stenotic carotid bifurcation derived from medical imaging.

2. Mathematical model

Blood is a non-Newtonian fluid, mainly because its viscosity depends on its velocity. This behavior, qualified as shear-thinning, is related to the fact that red blood cells can deform and aggregate.[20] These two processes are closely related to the properties of the plasma and red blood cells.[21] Recently, the interest in problems of non-Newtonian fluid has grown considerably. This is particularly due to their medical and clinical applications. Thus, many mathematical models describing the rheology of blood have been extensively developed.

However, there is no single governing constitutive equation describing all the properties of the non-Newtonian fluids (the blood at present). Therefore, we try to develop rheological models, as close as possible to reality, in order to define the viscosity as a function of the tensorDu= ¹₂(∇u+ ∇^Tu), or more precisely as a function ofs(u), the second invariant of the strain rate tensor defined by

(s(u))²=2Du:Du=2

i,j

(Du)i j(Du)ji.

Letf be an open bounded domain ofR³representing a portion of a cylindrical diseased artery. We denote by_wthe portion of the boundary corresponding to the physical arterial wall, whilei n andout represent the so-called artificial boundaries (since they do not correspond to any physical interface) such that∂f = ¯i n∪ ¯_w∪ ¯out withi n∩_w∩ out = ∅: Figure1.

We ignore body forces, i.e. we takef = 0, for the fluid and the structure in the next section. That is, in practice, to ignore the effects of gravity. The evolution problem involves velocityu=(u1,u2,u3)and pressurepboth defined overf ×(0,T). Then, using the vectorial identity

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Figure 1. Geometric configuration.

u· ∇u=1

2∇|u|²+curlu×u, the Navier–Stokes equations read

ρf∂u

∂t +ρf

1

2∇|u|²+curlu×u

− ∇ ·(2μ(s(u))Du)+ ∇p=0, inf ×(0,T), (1)

∇ ·u=0, inf ×(0,T). (2)

These equations are combined with the boundary conditions

σ^{t ot}(u,p)·n=h, oni n×(0,T), (3) σ^{t ot}(u,p)·n=0, onout ×(0,T), (4)

u=0, onw×(0,T), (5)

wherehis a given function (an explicit form ofhis provided in the Section5.2) andσ^{t ot} is the total stress tensor given by

σ^{t ot}(u,p)= − p+ρf

2 |u|²

I₃+2μ(s(u))Du, wherenis the outward normal vector.

We consider also the initial condition

u=u₀, fort =0 inf. (6) Considering the system of Equations (1)–(6), our goal is to give a weak formulation of the problem and then to prove the existence of solution in a new functional framework that we introduce. We also prove some properties of the operators obtained through the analysis of the model.

To carry out the mathematical analysis of the model, we introduce the following func- tional spaces:

V = ϕ∈

C^∞ f

3

| ∇ ·ϕ =0 inf, ϕ=0 on_w

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and

V =V^(H1(f))3 along with the inner product

((u,v))=

f

Du:Dvdx= 1 4

3 i,j=1

f

∂u_i

∂x_j +∂u_j

∂x_i

∂v_i

∂x_j +∂v_j

∂x_i

dx. The associated norm is denoted by|| · ||.

It is clear thatV is a closed subspace of(H¹(f))³, hence it is a Hilbert space.

Moreover

V ⊂

v∈(H¹(f))³| ∇ ·v=0 inf,v=0 onw

.

We recall that the inner product of(L²(f))³is denoted by( , ), with the associated norm| · |, such that for(u,v)∈(L²(f))³, we have

(u,v)= 3 i=1

f

u_iv_idx. Letϕ∈V, for allt∈(0×T), we put

(Av, ϕ)=

f

2μ(s(v))Dv:Dϕdx, (7) b(u,v, ϕ)=(curlu×v, ϕ). (8)

3. Preliminary results

To prove the existence of a weak solution for the problem (1)–(6), we use the following preliminary results.

Le m m a 3.1 Assume that the functionμsatisfies the following assumptions (1) μ:R₊−→R₊,

(2) μ is continuous,

(3) lims→+∞[μ(s)] =μ0 with μ0>0, (4) μ is continuously differentiable, (5) s|μ(s)| ≤μ(s),∀s∈R+,

(6) a =sup_s∈R+μ(s)and b=infs∈R+μ(s).

Then A, as defined in(7), is a hemicontinuous monotone operator, satisfying for all (v,w)∈V³

(Av−Aw,v−w)≥b||v−w||².

Proof See [22] and [12].

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Le m m a 3.2 Letf be an open bounded domain of R³with Lipschitz boundary∂f. Then C^∞(f)is dense in H1(f)and the application

γ0:C^∞(f)−→L²(∂f) v−→γ0(v)=v|_∂f

extends to a continuous linear operatorγ0from H₁(f)to L²(∂f)called trace operator.

Proof See [23].

Le m m a 3.3 For allu,w∈V , the linear form b as defined in(8)verifies:

(1) b(v,v,v)=b(v,w,w)=0, (2) b(v,v,w)= −b(v,w,v).

Proof From the definition ofb, it is straightforward to see that this form is well defined in(H¹(f))³, furthermore it is continuous:

|b(u,v,w)| ≤

f

((v· ∇)u)·wdx +

f

((∇u)v)·wdx

≤2C²||u||_H¹_(f)||v||_H¹_(f)||w||_H¹_(f)

whereCis a positive constant, from the continuous embedding ofH¹(f)inL⁴(f).

Then if we consider two vectors v = (v1,v2,v3),w = (w1,w2,w3)and using the properties of the vector operator Curl, we can easily find that

• curlv×v·v=curlv×w·w=0 hence

b(v,v,v)=b(v,w,w)=0, and

• curlv×v·w= −curlv×w·vhence

b(v,v,w)= −b(v,w,v).

Le m m a 3.4 Let{u_n}n∈N be a sequence in L²(0,T;(L²(f))³)such thatu_n → u in L²(0,T;(L²(f))³)andu_n uin L²(0,T;V), then for allw∈C¹(Q¯)

_T

0

b(u_n(t),u_n(t),w)dt→ _T

0

b(u(t),u(t),w)dt with Q=f × [0,T].

Proof The proof is based on the second part of the previous lemma, then we apply the

same technique as in [8].

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Le m m a 3.5 Assume thatγ ∈ L²(0,T;(L²())³)such that is a part of∂f and consider the following application:

T_γ :L²(0,T;V)−→R w−→

_T

0

γ (x,t)w(x,t)dxdt thenT_γ is well defined, linear and continuous.

Proof The proof can be easily extended from the 2D case (see our previous work [19]) to

3D one.

Le m m a 3.6 For all(u,v)∈V², we have

−∇ ·(2μ(s(u))Du),v_V×V =(Au,v)−2μ(s(u))Du·n,v

H⁻¹²(^∂f)^3×H12(^∂f)³ wherenis the outward normal vector over∂f.

Proof Clearly, the equality holds for(u,v)∈V²after applying Green’s formula.

Now let(u,v)∈V×V. Then there exists a sequencep−→upinVsuch thatup−→u inV and we have

−∇ ·(2μ(s(u))Du),v_V×V =

−∇ ·(2μ(s(u))Du)+ ∇ ·(2μ(s(u_p))Du_p),v

V×V

−

∇ ·(2μ(s(u_p))Du_p),v

V×V

We know that ∇ ·(2μ(s(u_p))Du_p),v

V×V =(Au_p,v)−

2μ(s(u_p))Du_p·n,v

H⁻¹²

∂f 3

×

H¹²

∂f 3

and

−∇ ·(2μ(s(u))Du)+ ∇ ·(2μ(s(u_p))Du_p),v

V×V

=

−∇ ·(2μ(s(u))Du)+ ∇ ·(2μ(s(up))Du)

−∇ ·(2μ(s(u_p))Du)+ ∇ ·(2μ(s(u_p))Du_p),v

V×V.

Then, after passing to the limit forp, we deduce that the equality holds for all(u,v)∈ V ×V.

Finally, let(u,v) ∈ V ×V. Then there exists a sequence p −→ vp inV such that v_p−→ vinV and for all pwe can write

∇ ·(2μ(s(u))Du),v_p

V×V =(Au,v_p)−

2μ(s(u))Du·n,v_p

H⁻¹²

∂f 3

×

H¹²

∂f 3. Therefore

∇ ·(2μ(s(u))Du),v_V×V =

∇ ·(2μ(s(u))Du),v−v_p

V×V

+

∇ ·(2μ(s(u))Du),v_p

V×V.

∇ ·(2μ(s(u))Du),v_V×V =

∇ ·(2μ(s(u))Du),v−v_p

V×V

+

∇ ·(2μ(s(u))Du),v_p

V×V.

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Hence

∇ ·(2μ(s(u))Du),v_V×V =

∇ ·(2μ(s(u))Du),v−vp

V×V +(Au,vp)

−

2μ(s(u))Du·n,vp

H⁻¹²(∂f)³×

H¹²(∂f)³. Then, after passing to the limit onp, the result is deduced.

Le m m a 3.7 For all(u,v)∈V ×V , we have ∇|u|²,v

V×V = |u|²,v·n

H⁻¹²(∂f) 3

×

H¹²(^∂f)³ withnthe outward normal vector over∂f.

Proof It is clear that the equality holds for(u,v)∈ V²after applying Green’s formula.

Let now(u,v)∈V×V. Then there exists a sequencep −→upinV such thatup−→u inV and we have

∇|u|²,v

V×V =

∇|u|²− ∇|up|²,v

V×V +

∇|up|²,v

V×V, but

∇|u|²,v

V×V =

∇|u|²− ∇|u_p|²,v

V×V + |u_p|²,v·n

H⁻¹2(∂f)³×

H¹2(∂f) 3. Passing to the limit onp, we deduce that the equality holds for all(u,v)∈V ×V. Now, let(u,v)∈V×V. Then there exists a sequencep −→v_pinVsuch thatv_p−→v inV, and for allpwe can write

∇|u|²,v_p

V×V = |u|²,v_p·n

(H⁻¹²(∂f))³×

H¹²(^∂f)³, therefore

∇|u|²,v

V×V =

∇|u|²,v−vp

V×V +

∇|u|²,vp

V×V

and

∇|u|²,v

V×V = ∇|u|²,v−vp_V×V + |u|²,vp·n

H⁻¹²(∂f) 3

×

H¹²(∂f)³.

The limit onpallows us to deduce the result.

Le m m a 3.8 For all(p,v)∈L²(f)×V , we have

−∇p,v_V×V = − pn,v

H⁻¹²(∂f) 3

×

H¹²(^∂f)³ withnthe outward normal vector over∂f.

Proof The equality, obviously, holds for all(p,v)∈V×Vthanks to the Green formula.

And if (p,v) ∈ L²(f)×V, then there exists a sequence m −→ pm inV such that p_m −→pinL²(f)and for all p, we can write

−∇pm,v_V×V = − pmn,v

H⁻¹²(∂f) 3

×

H¹²(∂f)³ .

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But

−∇p,v_V×V = −∇(p−pm),v_V×V + −∇pm,v_V×V. i.e.

−∇p,v_V×V = −∇(p−p_m),v_V×V + p_mn,v

H⁻¹2(^∂f)^3×H12(^∂f)³. Then, after passing to the limit onm, the result is deduced.

And if(p,v)∈ L²(f)×V then there exists a sequencem −→ vm inV such that vm −→vinV. Thus, for allp, we can write

−∇p,vm_V×V = − pn,vm

(H⁻¹²(∂f))³×

H¹²(∂f)³. But

−∇p,v_V×V = −∇p,v−v_mV×V+ −∇p,v_mV×V. i.e.

−∇p,v_V×V = −∇p,v−vm_V×V+ pn,v

H⁻¹²(∂f) 3

×

H¹²(∂f)³.

Then, after passing to the limit onm, the result is deduced.

Th e o r e m 3.9 (Compactness theorem, (see [8])) Let B₀and B be two Banach spaces and B₁be a Hilbert space such that B₀ → B → B₁with continuous embeddings and

the embedding B₀→B being a compact one.Let then Z =

v|v∈ L^p0(0,T;B0), ∂v

∂t ∈L¹(0,T;B1)

.

Then, under the previous hypotheses, the embedding of Z in L^p0(0,T;B)is compact for any finite number p₀>1.

4. Existence result

Th e o r e m 4.1 (Existence Theorem) Letu₀∈(L²(f))³,h∈L²(0,T;(L²(i n))³), and μsatisfying the hypothesis of Lemma3.1. Then there exists at least one solutionuof problem (1)–(6)such that

u∈ L²(0,T;V)∩L^∞(0,T;(L²(f))³),

u ∈L¹(0,T;V)

u = du

dt and Vis the dual space of V

Proof In the functional framework mentioned above and using Lemmas3.6–3.8, a weak formulation of the problem (1)–(6) can be obtained as follows

⎧⎨

⎩ ρf

∂u

∂t,v

V×V

+(Au,v)+ρfb(u,u,v)= h,vi n ∀v∈V, u(0)=u0, inf.

(9)

where ·,·γ stands for the duality pairing between

H⁻¹²(γ )3

and

H¹²(γ )3

, for a regular open subsetγ of∂f.

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The classical tool that we introduce here is well known as the Faedo–Galerkin method.

In this method, we define a set of differential equations system for which the existence of a discrete solution follows from Cauchy’s theorem. Different a priori estimates are then derived; they provide uniform bounds on the sequence of solutions. This is the corner stone of the proof, as it allows us to state that there exists a subsequence of discrete solutions that converges to a solution of (9) when the dimension of the finite dimensional subspaces goes to infinity.

4.1. Approximated solutions We consider the spectral problem

((w,v))=λ(w,v) ∀v∈V. (10) SinceV is a closed subspace of(H¹(f))³and the injection of(H¹(f))³in(L²(f))³ is compact, then (10) admits a sequence of eigenvaluesλj corresponding to eigenvectors w_j such that

((w_j,v))=λj(w_j,v) ∀v∈V and{w_j}j∈Nis orthonormal in(L²(f))³and orthogonal inV.

LetN ∈N^∗, if we denote by[w₁, . . .w_N]the subspace spanned byw₁, . . .w_N, then V = [w₁· · ·w_N]^V.

We denote byu_N(t)the approximate solution of (9) defined by

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u_N(x,t)=u_N(t)(x)∈[w1, . . .w_N], u_N(x,t)= ^N

j=1

C^N_j (t)w_j(x), u_N(·,0)=u_0N(·)∈[w1, . . .w_N], whereu_0N=_N

i=1u₀,w_iV×Vw_i andu_0N ^N→^→∞u₀in(L²(f))³. We have then

⎧⎨

⎩

ρfu_N(t),w_jV×V +(Au_N(t),w_j)+ρfb(u_N(t),u_N(t),w_j)=

i n

hw_jdσ, u_N(·,t)=u_0N(·)∈[w1, . . .w_N] 1≤ j ≤N.

(11) This set of equations yields a system of non-linear differential equations in the components C^N_j (t), which definesuN(t)a solution of (11) on[0,TN] ⊂ [0,T].

We will see in the next point that we can takeTN =T for allN ∈N.

4.2. A priori estimates

Multiplying (11) by C^N_j (t)and summing the obtained equations we deduce, thanks to Lemma3.3, that

ρf

1 2

d

dt|u_N(t)|²+(Au_N(t),u_N(t))=

i n

hu_N(t)dσ, (12)

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from the hypothesis of lemma 3.1, we have

f

μ(s(u_N(t)))s²(u_N(t))dx≥b

f

s²(u_N(t))dx and (12) implies

ρf

1 2

d

dt|uN(t)|²+2b

f

s²(uN(t))dx≤

i n

huN(t)dσ. (13) On the other hand, using Hölder inequality, we get

i n

hu_N(t)dσ ≤ |h|_L²_{(i n)}|u_N(t)|_L²_{(i n)} and using the trace theorem, we get

|h|L²(i n)|u_N(t)|L²(i n)≤K|h|L²(i n)||u_N(t)||H¹(f). Applying the Poincaré inequality, we get

K|h|L²(i n)||u_N(t)||H¹(f)≤K C(f)|h|L²(i n)|∇u_N(t)|L²(f), and from Korn inequality, we obtain

K C(f)|h|L²(i n)|∇u_N(t)|L²(f)≤K MC(f)|h|L²(i n)||Du_N(t)||L²(f). Moreover, we can write

K C(f)|h|L²(i n)|∇u_N(t)|L²(f)≤ K²M²C²(f)

6b |h|²_L2(i n)+3b

2 ||Du_N(t)||²_L2(f). From (13), we have

ρf

d

dt|u_N(t)|²+b||Du_N(t)||²_L2(f)≤C|h(t)|²_L2(i n), (14) then if we integrate this inequality over[0,T], we get

ρf|uN(t)|²+b T

0 ||DuN(t)||²_L2(f)dt ≤C T

0 |h(t)|²_L2(i n)dt+ρf|uN0|², thus, there are two constantsC1>0 andC2>0 such that

|u_N(t)| ≤C₁and _T

0

||Du_N(t)||²_L2(f)dt ≤C₂ ∀t ∈ [0,T_N]. (15) We first deduce thatT_N=T and moreover

u_N is a bounded sequence inL²(0,T;V)∩L^∞(0,T;(L²(f))²).

We also prove that

∀N ∈N,u_Nis a bounded sequence inL¹(0,T;V).

To do this, first we show that

AuNis a bounded sequence inL²(0,T;V)

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and

Bu_Nis a bounded sequence inL¹(0,T;V).

Indeed, for allv∈V, sup_s∈R+μ(s)=aand from the hypothesis of lemma 2.1, we have (Au_N(t),v)=

f

(2μ(s(u_N(t)))Du_N(t),Dv)dx≤2au_N(t) v, hence by duality

Au_N(t)_V ≤2au_N(t) and from (15), we deduce that

T 0

Au_N(t)²_Vdt≤4a² T

0

u_N(t)²dt≤C_A. On the other hand, from the definition ofb(u_N(t),u_N(t),v), we can write

b(u_N(t),u_N(t),v)=(Bu_N(t),v)≤K_b||u_N(t)||²_(H1(f))³||v||_(H¹_(f))³, hence by duality

BuN(t)_V ≤Kb||uN(t)||²_(H1(f))³, then

Au_NandBu_N are bounded sequences inL²(0,T;V)andL¹(0,T;V), respectively.

(16) Furthermore, letv ∈ [w1· · ·wN], thenv(x)= ^N

j=1

αjwj(x)for allx ∈f.v(x)is then substituted in (9) so that

ρfu_N(t),vV×V =ρf

N j=1

αju_N(t),wjV×V

= −^N

j=1

αj(Au_N(t),w_j)−ρf

N j=1

αjb(u_N(t),u_N(t),w_j) +^N

j=1

αj

i nh(t)w_jdσ, t ∈ [0,T].

(17)

Thus, according to (16), we deduce from (17)

ρfu_N(t),vV×V ≤ M(u_N(t) + ||u_N(t)||²+ |h|L²(i n))v.

Let nowv ∈ V. As mentioned above, the “special basis” formed by the eigenvectors {w1· · ·wN}spans a subspace which is dense inV, then there exists(vp)p∈ [w1, . . .wn]

such thatvp−→vin(H¹(f))³. By duality, we obtain

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u_N(t)

V ≤M(u_N(t) + ||u_N(t)||²+ |h|L²(∂f)).

Thus, we prove that

u_N is a bounded sequence inL¹(0,T;V). (18)

4.3. Passing to the limit

We now use this compactness theorem in the following situation: p₀=2,p₁=1,B₀=V, B₁ = V, and we choose B = (L²(∂f))³ and Z = {v|v ∈ L²(0,T;V), ^∂_∂^v_t ∈ L¹(0,T;V)}.

From (15), (18) and the compactness theorem, we deduce that we can extract a subse- quence{u_m}m∈Nsuch that

u_m u inL²(0,T;V),

u_m →u weak star inL^∞(0,T;(L²(∂f))³),

u_m →u inL²(0,T;(L²(∂f))³)and i.e. inf × [0,T].

Moreover,umverifies (11) for allm∈N.

Furthermore, using (16), we have

Au_m χ inL²(0,T;V).

Indeed, letψbe a differentiable continuous function in[0,T]such thatψ(T)=0.

By multiplying the Equation (11) by the functionψ(t)and integrating by parts, we get:

−ρf

_T

0 u_m(t), ψ(t)w_jV×Vdt+ _T

0 (Au_m(t),w_jψ(t))dt +ρf

_T

0

b(u_m(t),u_m(t),w_jψ(t))dt =ρfu_0m,w_jV×Vψ(0) +

_T

0

i n

hw_jψ(t)dσdt. (19)

Then using Lemma3.4and passing to the limit, we obtain

−ρf

T 0

u(t), ψ(t)wjV×Vdt+ T

0

(χ(t),wjψ(t))dt +ρf

_T

0

b(u(t),u(t),w_jψ(t))dt=ρfu₀,w_jV×Vψ(0)+ _T

0

i n

hw_jψ(t)dσdt, (20) for allw_j.

By density, (20) holds for allv∈V.

Therefore (20) is true for allv∈V andψ∈D(0,T). After integration by parts in the first integral, we can deduce that

ρf

d

dtu(t),vV×V+ χ(t),vV×V+ρfb(u(t),u(t),v)=

i n

hvdσ, ∀v∈V, (21)

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