A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations

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Brinkman-Forchheimer equations

Mohammed Louaked, Nour Seloula, Shuyu Sun, Saber Trabelsi

To cite this version:

Mohammed Louaked, Nour Seloula, Shuyu Sun, Saber Trabelsi. A pseudocompressibility method for the incompressible Brinkman-Forchheimer equations. Differential and integral equations, Khayyam Publishing, 2015. �hal-02343592�

incompressible Brinkman- Forchheimer equations

Mohammed Louaked¹, Nour Seloula¹, Shuyu Sun² and Saber Trabelsi³

1Laboratoire de math´ematiques Nicolas Oresme, UMR 6139 CNRS BP 5186, Universit´e de Caen Basse Normandie

2Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia.

3Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia.

E-mail: mohammed.louaked@unicaen.fr,nour-elhouda.seloula@unicaen.fr, shuyu.sun@kaust.edu.sa,saber.trabelsi@kaust.edu.sa

Abstract. In this work, we study the Brinkman-Forchheimer equations for unsteady flows. We prove the continuous dependence of the solution on the Brinkman’s and Forchheimer’s coefficients as well as the initial data and externel forces. Next, we propose and study a perturbed compressible system that approximate the Brinkman- Forchheimer equations. Finally, we propose a time dicretization of the perturbed system by a semi-implicit Euler scheme and next a lowesst-order Raviart-Thomas element is applied for spatial discretization. Some numerical results are given.

AMS classification scheme numbers: 35B30, 35K55, 35Q35

Keywords: Continuous Dependence; Stability; Brinkman-Forchheimer equations, artificial compressibility

1. Introduction

Transport phenomena in porous media are ubiquitous to the extent that it is hardly difficult to find applications without some sort of porous material included. It also spans wide range of scales from small scale applications in confine spaces like bio-organic tissues to large scale applications in subsurface oil and gas reservoirs. With this large number of scales involved, the governing equations that describe conservation laws are, in a sense, simpler than those governing fluid flows. Some remarks, however, worth mentioning. Probably the most important is the fact that the governing laws in porous media have been adapted based on the assumption of the validity of the continuum hypothesis.

This implies that field variables represent continuous functions of space and time and hence enable the conservation laws to be written in the form of partial differential equations [14, 15]. These conservation laws have been postulated based on upscaling the conservation laws at the fluid continuum level using theory of volume averaging, method of homogenization, theory of mixtures, etc. Unfortunaetly, the governing equations based on these theories are usually difficult to solve and they are generally unclosed because they contain terms at the pore scale which are hard to implement. Therefore, to obtain field equations that are workable, researchers suggested terms to the governing equations in addition to some properties of the porous material. In other words, some ad hoc terms are introduced to extend the governing equations to encounter more applications. Then, it remainsto experimentalists and theoreticians to validate these terms. As an example, the simplest Darcy law, [4], suggests that the mass flux and pressure gradient are proportional and the relationship between them is linear. This linear relationship has later been found to be valid for values of Reynolds number less than one. As Reynolds number becomes greater than one, this linear relationship is no longer valid. To account for such nonlinearities, Forchheimer [5, 6] suggested a quadratic term of the velocity to be included when considering momentum balance. Furthermore, Brinkman [1, 2] considers another term to account for the possible no slip condition once a confining wall exists. This produced the widely used Darcy-Brinkman-Forchheimer equations. The main feature of this equation is that it is nonlinear which apparently poses great difficulty to find analytical solution.

The Brinkman model is believed accurate when the flow velocity is too large for Darcy’s law to be valid, and additionally the porosity is not too small. In this article, we are concerned with structural stability for the following Brinkman-Forchheimer (BF in short) equations.

∂_tu −γ∆u+au +b|u|^αu +∇p=f in Ω_T, divu = 0 in Ω_T,

u =0 on Σ_T, u(0) =u₀ in Ω,

(1)

where Ω_T = Ω×]0, T[, Σ_T = Γ×]0, T[. u andprepresent respectively fluid velocity and

pressure. The constantγ >0 is the Brinkman coefficient,a >0 is the Darcy coefficient, bis the Forchheimer coefficient andα∈[1, 2] is a given number. Ω is a bounded domain of R³ with sufficiently smooth boundary Γ. ∆ is the Laplace operator, k · k and hu, vi denote respectively the norm and inner product on L²(Ω) .

2. The existence of solutions for the (BF) equations

The aim of this subsection is to give a variational formulation of problem (1) and to prove the existence of weak solutions. We introduce the spaces

V ={v ∈H¹₀(Ω), divv = 0} and H ={v ∈L²(Ω), divv = 0}.

The variational formulation of (1) can be writen as: Given f ∈ L²(0, T, L²(Ω)) and u0 ∈ V, find u ∈ L²(0, T, V)∩L^∞(0, T, H¹₀(Ω)), ∂tu ∈ L²(0, T, L²(Ω)) satisfying (1) such that for almost all t and v ∈V,

d

dthu(t), vi+γh∇u(t), ∇vi+ahu(t),vi+bh|u(t)|^αu(t),vi=hf,vi. (2) Next, we discuss the solvability of the variational problem (2) by means of Faedo- Galerkin’s method. This will be achieved in several steps that we describe below.

Step 1: Construction of approximating solutions.

Consider an orthonormal basis of V constituted of w₁, w₂, . . . , w_m. Using this basis, we introduce the spaceV_m =hw₁, . . . ,w_mi and define the approximate solutionu_m of (2) as

u_m(t) =

m

X

i=1

g_im(t)w_i (3)

such that d

dthu_m(t), w_ii+γh∇u_m(t), ∇w_ii+ahu_m(t), w_ii+bh|u_m(t)|^αu_m(t), w_ii

=hf, w_ii, t ∈[0, T], i= 1, . . . , m, (4)

u_m(0) =u_0m →u₀ ∈V_m. (5)

Using the theory of ordinary differential equations, it follows that the problem (4) has a solution u_m defined on [0, t_m] with t_m < T.

Step 2: a priori estimates.

First, we multiply (4) byg_im(t) and sum up the obtained set of equations corresponding toi= 1, . . . , m. We obtain

1 2

d

dtku_m(t)k²+γk∇u_m(t)k²+aku_m(t)k²+bku_m(t)k^α+2_α+2 =hf(t), u_m(t)i,

≤ kf(t)k ku_m(t)k,

≤ 1 2

1

af(t)k²+aku_m(t)k² .

Thus, we have d

dtku_m(t)k²+ 2γk∇u_m(t)k²+aku_m(t)k²+ 2bku_m(t)k^α+2_α+2 ≤ 1

akf(t)k². Integrating this inequality from 0 to T with (T ≤t_m) leads to

sup

0≤t≤T

ku_m(t)k²+ 2γ Z T

0

k∇u_m(t)k²dt+a Z T

0

ku_m(t)k²dt+ 2b Z T

0

ku_m(t)k^α+2_α+2dt

≤ 1 a

Z T

0

kf(t)k²dt+ku₀k² <∞. (6) Now, we multiply (4) by g⁰_im(t) and add the obtained equations for i = 1, . . . , m. We obtain

k∂tum(t)k² + 1 2γ d

dtk∇um(t)k²+1 2ad

dtkum(t)k²+ b α+ 2

d

dtkum(t)k^α+2_α+2

=hf(t), ∂tum(t)i,

≤ 1

2kf(t)k²+1

2k∂_tu_m(t)k². (7)

Next, we integrate (7) from 0 to T and get Z T

0

k∂_tu_m(t)k²dt+ γk∇u_m(T)k²+aku_m(T)k²+ 2b

α+ 2ku_m(t)k^α+2_α+2

≤ 1 2

Z T

0

kf(t)k²dt+γk∇u_m0k²+aku_m0k²+ 2b

α+ 2ku_m0k^α+2_α+2. In particular, we obtain

Z T

0

k∂tum(t)k²dt ≤ 1 2

Z T

0

kf(t)k²dt+c <∞, (8)

where

c=γk∇u₀k²+aku₀k²+ 2b

α+ 2ku₀k^α+2_α+2. Eventually, integrating (7) from 0 to t, we get Z t

0

k∂_tu_m(s)k²ds+ γk∇u_m(t)k²+aku_m(t)k²+ 2b

α+ 2ku_m(t)k^α+2_α+2

≤ 1 2

Z T

0

kf(s)k²ds+c. (9)

In particular, we infer sup

0≤t≤T

k∇u_m(t)k² ≤ 1 2γ

Z T

0

kf(s)k²ds+c <∞. (10)

Before going further, let us mention that the abovea priori estimates show that in fact tm =T.

Step 3: Passage to the limit.

We need to pass to the limit whenmapproaches infinity. It follows from (6), (8) and (10) that there is a subsequence of (u_m)_m that we still denote (u_m)_m by abuse of notation and some u such that

u_m −→u weak star in L^∞(0, T, H¹₀(Ω)), (11)

u_m −→u weak in L²(0, T, V), (12)

|u_m|^αu_m −→w weak in L^α+2α+1(0, T, L^α+2α+1(Ω)), (13)

∂_tu_m −→∂_tu weak in L²(0, T, L²(Ω)). (14)

From (12) and (14), we deduce that u is bounded in H¹(Q). Since the imbedding of H¹(Q) inL²(Q) is compact, we can extract a subsequence denoted againu_m such that u_m −→u strongly in L²(0, T, L²(Ω)) and a.e. in Q. (15) Therefore, it remains to prove that w = |u|^αu. For this purpose, we apply the same argument used in [8, Lemma 1.3]. Indeed, using (15), it follows that

|u_m|^αu_m −→ |u|^αu weak in L^α+2α+1(0, T, L^α+2α+1(Ω)).

Thanks to (13), one obtains w = |u|^αu. Now, It is easy to pass to the limit in (4-5) and conclude thatu is solution of (2).

Therefore, we have

Theorem 2.1. For any u₀ ∈ V and f ∈ L²(0, T, L²(Ω)), problem (1) admits at least one solution u satisfying

u∈L^∞(0, T, V)∩L²(0, T, H²(Ω)), ∂_tu∈L²(0, T, L²(Ω)) and p∈L²(0, T, L²(Ω)).

Moreover, for any T > 0, we have sup

0≤t≤T

k∇u(t)k ≤C and

Z T

0

k∂_tu(t)k²dt≤C, (16) where C idenotes a positive constant depending on u₀ and the parameters of problem (1).

3. Continuous dependence on data

In this section, we show that the solution depends continuously on the initial velocity and the external forces. For that purpose, let (u₁, p₁) and (u₂, p₂) such that

∂_tu₁−γ∆u₁ +au₁+b|u₁|^αu₁+∇p₁ =f₁ in Ω_T, divu₁ = 0 in Ω_T,

u₁ =0 on Σ_T, u1(0) =u¹₀ in Ω,

(17)

and

∂tu2−γ∆u2 +au2+b|u2|^αu2+∇p2 =f₂ in ΩT, divu2 = 0 in ΩT,

u₂ =0 on Σ_T, u₂(0) =u²₀ in Ω.

(18)

Now, we are able to state the main result of this section

Theorem 3.1. Letu₁ andu₂ as in (17) and (18). Then, there exists a positive constant K depending on the parameters γ, a and Ω such that

ku1(t)−u2(t)k² ≤e^−Ktku1(0)−u2(0)k²+ Z t

0

e^K(s−t)kf₁(s)−f₂(s)k²ds. (19) Proof. We know that bothu₁andu₂satisfy the variational formulation (2) with respect tou¹₀, f₁ and u²₀,f₂ respectively. Rewriting (2) for u₁ with a test function v−u₁, we obtain

d

dthu₁(t), v−u₁(t)i+γh∇u₁(t), ∇(v −u₁(t))i+ahu₁(t), v −u₁(t)i

+bh|u₁(t)|^αu₁(t), v−u₁(t)i=hf₁, v−u₁(t)i. (20) Equivalently for u₂, we take v −u₂ as a test function and obtain

d

dthu₂(t), v−u₂(t)i+γh∇u₂(t), ∇(v −u₂(t))i+ahu₂(t), v −u₂(t)i

+bh|u₂(t)|^αu₂(t), v−u₂(t)i=hf₂, v−u₂(t)i. (21) Now, letw(t) = u₂(t)−u₁(t) and w(0) =u₂(0)−u₁(0). Choosingv =u₂ in (20) and v =u₁ in (21), summing up both equations, we get

1 2

d

dtkw(t)k²+γk∇w(t)k²+akw(t)k²+ bh|u₂|^αu₂− |u₁|^αu₁, w(t)i

=hf₂(t)−f₁(t), w(t)i

Since the operator T(ξ) = |ξ|^αξ is monotone, we have obviously h|u₁|^αu₁ −

|u₂|^αu₂, u(t)i ≥0. Therefore, d

dtkw(t)k²+ 2γk∇w(t)k²+akw(t)k² ≤ 1

akf₂(t)−f₁(t)k² Thanks to (27), we obtain

d

dtkw(t)k²+ckw(t)k² ≤ 1

akf₂(t)−f₁(t)k²,

where the constant c depends on D(the constant in (27)), γ and a. Eventually, using Gronwall’s lemma, we obtain the estimate (19).

Remark 3.2. Obviously, this last result implies in particular the uniqueness of solutions of problem (1).

4. Continuous dependence on the Brinkman’s and Forchheimer’s coefficients

In this section, we show that the solutions of the BF problem (1) depend continuously on the Brinkman’s and Forchheimer’s coefficients, thereby we extend the result of [3].

Let (u₁, p₁) and (u₂, p₂) such that

∂_tu₁−γ₁∆u₁+au₁+b₁|u₁|^αu₁+∇p₁ = 0 in Ω_T, divu₁ = 0 in Ω_T,

u₁ =0 on Σ_T, u₁(0) =u₀ in Ω,

(22)

and

∂_tu₂−γ₂∆u₂+au₂+b₂|u₂|^αu₂+∇p₂ = 0 in Ω_T, divu₂ = 0 in Ω_T,

u₂ =0 on Σ_T, u₂(0) =u₀ in Ω.

(23)

We set u =u₁−u₂ and p=p₁−p₂. Then, (u, p) satisfies

∂_tu −γ₁∆u−γ∆u₂+au +∇p=−b|u₁|^αu₁−b₂(|u₁|^αu₁− |u₂|^αu₂) in Ω_T, divu = 0 in Ω_T,

u =0 on Σ_T, u(0) =0 in Ω,

(24)

where γ =γ₁−γ₂ and b=b₁−b₂. Now, we claim the following

Theorem 4.1. Let u be the solution of (24). Then, there exist positive constants M and Q, depending on the parametes of (24) such that

k∇u(t)k²+ku(t)k² ≤M γ²+Qb². (25)

Proof. Mulitplying (24) by u and integrating over Ω leads to 1

2 d

dtku(t)k²+γ₁k∇u(t)k²+aku(t)k² = −b₂h|u₁|^αu₁− |u₂|^αu₂, u(t)i

−bh|u₁(t)|^αu₁(t), u(t)i −γh∇u₂(t), ∇u(t)i.

With the monoticity of the operator T(ξ) = |ξ|^αξ, we get 1

2 d

dtku(t)k²+γ₁k∇u(t)k²+aku(t)k²

≤

bh|u₁(t)|^αu₁(t),u(t)i +

γh∇u₂(t), ∇u(t)i

. (26) Next, we define the Sobolev constant D as

∀v ∈H¹₀(Ω), kvk_p ≤Dk∇vk for any 1≤p≤6. (27)

Thus, thanks to H¨older’s inequality and the Sobolev inequality (27), we get

bh|u₁(t)|^αu₁(t), u(t)i

≤ |b|D^α+2k∇u₁(t)k^α+1k∇u(t)k

≤ b²D^2α+4

γ₁ k∇u₁(t)k^2(α+1)+ γ₁

4k∇u(t)k². (28)

Using (28) and the fact that

γh∇u₂(t), ∇u(t)i

≤ |γ|²

γ₁ k∇u₂k²+γ₁

4k∇uk², (29)

we obtain thanks to (26) d

dtku(t)k²+γ₁k∇u(t)k² + 2aku(t)k² ≤2b²D^2α+4

γ₁ k∇u₁(t)k^2α+1+2γ²

γ₁ k∇u₂(t)k²

≤Q₀b²+ 2γ²

γ₁ k∇u₂(t)k², (30)

where Q₀ = 2D^2α+4C^2α+1γ₁⁻¹ and C is the constant in (16). Now, multiplying (24) by

∂_tu and integrating over Ω leads to k∂_tu(t)k²+ 1

2 d dt

γ₁k∇u(t)k²+aku(t)k²

≤

b₂h|u₁(t)|^αu₁(t)− |u₂(t)|^αu₂(t), ∂_tu(t)i

+

γh∆u₂(t), ∂_tu(t)i +

bh|u₁(t)|^αu₁(t), ∂_tu(t)i

. (31) Using the mean value theorem and H¨older’s inequality and (27) and (16), it is easy to show that

b₂h|u₁(t)|^αu₁(t)− |u₂(t)|^αu₂(t), ∂_tu(t)i

≤b₂D^α+1C^αk∇u(t)k k∂_tu(t)k

≤b²₂D^2α+2C^2αk∇u(t)k² +1

3k∂_tu(t)k². (32)

Using the first equation in (23) and the Cauchy-Schwarz inequality, we obtain

γh∆u₂(t), ∂_tu(t)i ≤ γ

γ₂

h∂_tu₂(t) +au₂(t) +b₂|u₂(t)|^αu₂(t), ∂_tu(t)i ,

≤ 1

3k∂_tuk² +3γ² 4γ₂²

∂_tu₂+au₂+b₂|u₂|^αu₂

2

,

≤ 1

3k∂_tuk² +3γ² 4γ₂²

k∂_tu₂k²+a²ku₂k²+b²₂ku₂k^2α+2_2α+2 ,

≤ 1

3k∂_tuk² +3γ² 4γ₂²

k ∂_tu₂k²+a²D²C²+b²₂(DC)^2α+2

. (33)

Similarly, we have

bh|u1(t)|^αu1(t), ∂tu(t)i

≤ |b|D^α+1k∇u1k^α+1k∂tuk,

≤b²3

4D^2(α+1)k∇u₁k^2(α+1)+1

3k∂_tuk²,

≤b²3

4D^2(α+1)C^2(α+1)+1

3k∂_tuk². (34)

Thus, combining (32-34), we obtain from (31) d

dt

γ₁k∇u(t)k²+aku(t)k²

≤M₀k∇u(t)k²+Q₁b²+ 3

2γ₂²γ²k∂_tu₂(t)k²+M₁γ², (35) where

M₀ = 27b²₂D^2α+2C^2α, Q₁ = ³₂D^2α+2C^2α+1, M₁ = ³₂γ₂⁻²(a(DC)²+b²₂(DC)^2α+2).

Multiplying (30) by β = 2^M_γ⁰

1, we obtain d

dt

βku(t)k²

+ 2M0k∇u(t)k²+ 2aβku(t)k² ≤βQ0b²+ 2βγ²

γ₁k∇u2(t)k². (36) Next, we add the bounds (35) and (36) to get

d dt

γ₁k∇u(t)k²+ (a+β)ku(t)k²

+M₀k∇u(t)k²+ 2aβku(t)k²

≤γ²2β

γ₁k∇u₂(t)k² + 9

2γ₂²k∂_tu₂(t)k²+M₁

+b²(Q₁+βQ₀).

Now, we set F(t) = γ₁k∇u(t)k² + (a+β)ku(t)k². It is easy to show that F⁰(t) +Q2F(t)≤γ²

2β

γ₁k∇u2(t)k²+ 9

2γ₂²k∂tu2(t)k²+M1

+b²(Q1 +βQ0), where Q₂ = min (^M_γ⁰

1 , _a+β^2aβ). Therefore, integrating from 0 to t, applying Gronwall’s inequality and using (16), we obtain

F(t)≤γ²

2βγ₁⁻¹C+ 9

2γ₂⁻²C+ M₁ Q₂

+b²(Q₁+βQ₀).

Furtheremore, we can derive (25) with Q= Q₁+βQ₀

min(γ₁, (a+β)) and M =

2βγ₁⁻¹C+_2γ⁹2 2

C+ ^M_Q¹

1

min(γ₁, (a+β)) .

This achieves the proof of the continuous dependence on the Brinkman’s and Forchheimer coefficients.

5. Approximation by the artificial compressibility method

The importance of the treatment of the incompressibility constraint has been recognized since a long time. Towards incompressible flows, density variation is not linked to the pressure. A possible method is to introduce an artificial compressibility, i.e. adding the time-derivation of pressure to the continuity equation, first proposed by Chorin (1967), named pseudo-compressibility. This method introduced waves of finite speed into the incompressible flow, in which the artificial pressure waves would otherwise be infinite. In this section we study a perturbed system that approximates, in the limit, the Brinkman- Forchheimer equations via the artificial compressibility method. We prove the existence of weak solution and show how the solution of the perturbed problem converges to the solution of the Brinkman-Forchheimer problem when ε → 0 where > 0 is arbitrary.

The pseudo-compressibility method that we propose to study is to approximate the

solution (u, p) of the incompressible Brinkman-Forchheimer equations by (u^ε, p^ε) satisfying the following perturbed system:

∂_tu−γ∆u+au+b|u|^αu+∇p =f in Ω_T, divu+∂_tp = 0 in Ω_T,

u =0 on Σ_T, u(0) =u₀ in Ω, p(0) =p0 in Ω,

(37)

where p0 ∈L²(Ω) is arbitrary and independent of . 5.1. Existence of solutions of the perturbed problem

We can easily verify that for a given f ∈ L²(0, T, L²(Ω)) and an u₀ ∈ L²(Ω), the problem (37) is equivalent to the variational formulation: Findu ∈L²(0, T,H¹₀(Ω))∩ L^∞(0, T,L²(Ω)) and p ∈L²(0, T, L²(Ω)) such that

d

dthu^ε(t), vi+γh∇u^ε(t), ∇vi+ahu^ε(t), vi+bh|u^ε(t)|^αu^ε(t),vi

−hp^ε(t), divvi=hf(t), vi, ∀v ∈H₀¹(Ω)∩L^α+2(Ω), (38) hdivu^ε(t), qi=−εh∂tp^ε(t), qi, ∀q∈L²(Ω). (39) Now, we state the existence result as follows

Theorem 5.1. For ε > 0 fixed, given f ∈ L²(0, T, L²(Ω)), u₀ ∈ H and p₀ ∈ L²(Ω), there exists at least one solution (u^ε, p^ε) of the perturbed problems (37). Moreover, (u^ε, p^ε)∈L²(0, T, H₀¹(Ω))∩L^∞(0, T, L²(Ω))×L²(0, T, L²(Ω)).

Proof. To prove the wellposedness of the problem (38-39), we will use the Faedo-Galerkin method. We find the approximate solution of u^ε(t) and p_ε(t) respectively in the forms u^ε_m(t) =

m

X

i=1

gim(t)wi and p^ε_m(t) =

m

X

j=1

ξjm(t)rj, (40)

where the coefficients wi and rj satisfy the system of ODE’s:

d

dthu^ε_m(t), wki+γh∇u^ε_m(t),∇wki+ahu^ε_m(t), wki+bh|u^ε_m(t)|^αu^ε_m(t), wki

−hp^ε_m(t), divw_ki=hf(t), w_ki, k = 1, . . . , m (41) hdivu^ε_m(t), r_li=−εh∂_tp^ε_m(t), r_li, l = 1, . . . , m. (42) Moreover, we have the following initial conditions

u^ε_m(0) =u0m and p^ε_m(0) =p0m, (43)

where u_0m and p_0m are the orthogonal projections of u₀ and p₀ onto the subspaces spanned by w₁, . . . , w_m and r₁, . . . , r_m inL²(Ω) and L²(Ω) respectively.

It is clear that for each m, there exist solutionsu^ε_m(t) andp^ε_m(t) in the form (40) which satisfy (41-43) almost everywhere on 0 ≤ t ≤ T_m for some T_m, 0 < T_m ≤ T. The following estimtes allow us to take T_m =T for all m.

a) First estimtes: We multiply thek^th equation of (41) by g_km(t) and sum up over k. Next we multiply the l^th equation of (42) by ξ_lm(t) and sum up over l. We get

1 2

d

dtku^ε_m(t)k² + γk∇u^ε_m(t)k²+aku^ε_m(t)k²+bku^ε_m(t)k^α+2_α+2+ ε 2

d

dtkp^ε_m(t)k²

=hf(t),u^ε_m(t)i,

≤ a

2ku^ε_m(t)k²+ 1

2akf(t)k², so that

d dt

ku^ε_m(t)k² +εkp^ε_m(t)k²

+ 2γk∇u^ε_m(t)k²+aku^ε_m(t)k²+ 2bku^ε_m(t)k^α+2_α+2

≤ 1

2akf(t)k². (44)

Integrating both sides of (44) over the interval (0, t) we get ku^ε_m(t)k²+εkp^ε_m(t)k²+ 2γ

Z t

0

k∇u^ε_m(s)k²ds+a Z t

0

ku^ε_m(s)k²ds +2b

Z t

0

ku^ε_m(s)k^α+2_α+2ds≤ 1 2a

Z t

0

kf(s)k²ds+ku^ε_m(0)k²+εkp^ε_m(0)k². From this inequality we infer

sup

t∈[0, T]

ku^ε_m(t)k²+εkp^ε_m(t)k²

≤d₁, (45)

Z T

0

k∇u^ε_m(t)k²dt≤ d₁

2γ, (46)

Z T

0

ku^ε_m(t)k²dt≤ d₁

a , (47)

Z T

0

ku^ε_m(t)k^α+2_α+2dt≤ d₁

2b, (48)

where

d1 =ku0k²+εkp0k² + 1 2a

Z T

0

kf(s)k²ds.

b) Second estimtes: In addition to the previous estimates, we want to prove an estimate of the fractional derivative with respect to time ofu^ε_m(t) in order to pass to the limit in the nonlinear term. We let

φ_m(t) =f(t)−γ∆u^ε_m(t)−au^ε_m(t)−b|u^ε_m(t)|^αu^ε_m(t).

Therefore,

kφ_m(t)kH⁻¹(Ω) ≤ kf(t)k+aku^ε_m(t)k+γk∇u^ε_m(t)k+bku^ε_m(t)k^α+1_α+2. (49)

Hence, (41-42) reads now d

dthu^ε_m(t), w_ki − hp^ε_m(t), divw_ki=hφ_m(t), w_ki, k= 1, . . . , m, (50) hdivu^ε_m(t), rli=−εh∂tp^ε_m(t), rli, l = 1, . . . , m. (51) We extend the functions u^ε_m(t), p^ε_m(t), f(t) gkm(t), ξlm(t) and φ^ε_m(t) by 0 outside the interval [0, T]. Then we obtain in the sense of distributions onR:

d

dthue^ε_m(t), w_ki − hpe^ε_m(t), divw_ki

=hφe_m(t),w_ki+hu_0m,w_kiδ(0)− hu^ε_m(T), w_kiδ(T), k = 1, . . . , m, hdivue^ε_m(t), r_li+εh∂_tpe^ε_m(t), r_li

=hp0m, rliδ(0)− hp^ε_m(T), rliδ(T), l = 1, . . . , m,

whereδ(0) andδ(T) are respectively the Dirac distributions at 0 andT. Next, by taking the Fourier transforms, we get

2iπτhub^ε_m(τ), w_ki − hpb^ε_m(τ), divw_ki

=hφb_m(τ), w_ki+hu_0m, w_ki − hu^ε_m(T), w_kiexp(−2iπτ T), k= 1, . . . , m, hdivub^ε_m(τ), rli+ 2iπτ εhpb^ε_m(τ), rli

=εhp_0m, r_li −εhp^ε_m(T), r_liexp(−2iπτ T), l = 1, . . . , m.

Letbg_km(τ) and ξb_lm(τ) be respectively the Fourier transform of g_km(t) and ξ_lm(t). Now, we multiply both last equations by bg_km(τ) (the former) and the ξb_lm(τ) (the latter) and add the results for k = 1, . . . , m and l= 1, . . . , m. The result of this calculation reads 2iπτ

kub^ε_m(τ)k²+εkbp^ε_m(τ)k²

=hφb_m(τ),ub^ε_m(τ)i+hu_0m, ub^ε_m(τ)i+εhp_0m, pb^ε_m(τ)i

−

hu^ε_m(T), ub^ε_m(τ)i+εhp^ε_m(T), pb^ε_m(τ)i

exp(−2iπτ T).

Using estimate (45) we get 2π|τ|

kub^ε_m(τ)k²+εkpb^ε_m(τ)k²

≤ kφbm(τ)kH⁻¹(Ω)kub^ε_m(τ)k+ 2p

d1kub^ε_m(τ)k + 2p

d₁εkpb^ε_m(τ)k. (52)

But kφb_m(τ)k

H⁻¹(Ω)≤ Z T

0

kφ_m(t)k

H⁻¹(Ω)dt. (53)

Using estimates (46-49) and applying the compactness result of [16, Theorem 2.2 page 274], we can show that

Z +∞

−∞

|τ|^2θkub^ε_m(τ)k²dτ ≤Const, for some θ >0. (54) c) Passage to the limit: We need to pass to the limit when m→ ∞and ε→0. At this step, by fixing ε, we are only concerned with the passage to the limit as m → ∞.

Based on (45-48), it is clear that there exist subsequences still denoted u^ε_m and p^ε_m satisfying

u^ε_m −→u^ε in L²(0, T; H¹₀(Ω)) weakly, (55)

u^ε_m −→u^ε in L^∞(0, T; L²(Ω)) weak−star, (56)

p^ε_m −→p^ε in L^∞(0, T; L²(Ω)) weak−star. (57)

u^ε_m −→u^ε in L^α+2(0, T;L^α+2(Ω)) weakly, (58)

Moreover, due to (55), (54) and [16, Theorem 2.3 page 276], we have

u^ε_m −→u^ε in L²(0, T; L²(Ω)) strongly. (59)

In order to prove that (u^ε, p^ε) satisfies (38-39), we shall use the same arguments of [16]. Multiplying both sides of (41-42) by ψ(t) ∈ C^∞(0, T) such that ψ(T) = 0 and integrating over (0, T), we get

− Z T

0

hu^ε_m(t),wkiψ⁰(t)dt+γ Z T

0

h∇u^ε_m(t), ψ(t)∇wkidt+a Z T

0

hu^ε_m(t), wkψ(t)idt +b

Z T

0

h|u^ε_m(t)|^αu^ε_m(t), wkψ(t)idt− Z T

0

h∇p^ε_m(t), wkψ(t)idt

=hu0m, wkiψ(0) + Z T

0

hf(t),wkψ(t)idt, (60)

Z T

0

hdivu^ε_m(t), rlψ(t)idt =εhp0m, rliψ(0) + Z T

0

εhp^ε_m(t), rliψ⁰(t)dt. (61) Using the previous convergence properties, it is easy to pass to the limitm→ ∞ in the linear terms. For the nonlinear term, we apply the following Lemma that we postpone the proof to the end of this paragraph.

Lemma 5.2. We have the following property as m→ ∞ Z T

0

h|u^ε_m(t)|^αu^ε_m(t), w_kψ(t)idt −→

Z T

0

h|u^ε(t)|^αu^ε(t), w_kψ(t)idt. (62) Using lemma 5.2, we can pass to the limit in (60-61) and obtain for k = 1, . . . , m and l = 1, . . . , m

− Z T

0

hu^ε(t), w_kiψ⁰(t)dt+γ Z T

0

h∇u^ε(t), ψ(t)∇w_kidt+a Z T

0

hu^ε(t), w_kψ(t)idt +b

Z T

0

h|u^ε(t)|^αu^ε(t), w_kψ(t)idt− Z T

0

h∇p^ε(t), w_kψ(t)idt

=hu₀, w_kiψ(0) + Z T

0

hf(t), w_kψ(t)idt, (63)

Z T

0

hdivu^ε(t), r_lψ(t)idt=εhp₀, r_liψ(0) + Z T

0

εhp^ε(t), r_liψ⁰(t)dt. (64)

Now, observe that the nonlinear term is continuous with respect tow_k. Indeed, for any w_k∈H¹₀(Ω), we have

Z T

0

h|u^ε(t)|^αu^ε(t),w_kψ(t)idt

≤ sup

t∈[0, T]

|ψ(t)|

Z T

0

Z

Ω

|u^ε(t)|^α+1|w_k|dx dt,

≤Ckw_kkL⁶(Ω)

Z T

0

Z

Ω

|u^ε(t)|^6(α+1)5 dx dt.

Thanks to the embeddingsL^α+2(Ω),→L^6(α+1)5 (Ω) withα ∈[1, 2] andH₀¹(Ω),→L⁶(Ω), we get

Z T

0

h|u^ε(t)|^αu^ε(t),w_kψ(t)idt

≤Cku^εk^α+1

L^α+2(0, T;L^α+2(Ω))kw_kkL⁶(Ω),

≤CkwkkH0¹(Ω). (65)

By linearity, the equation (63) still hlods for any w that is a linear combination of functionswkand by continuity argument this equation is still true for any w ∈H₀¹(Ω).

Similarly, the equation (64) still holds true for any r ∈L²(Ω). Then, we can write

− Z T

0

hu^ε(t), wiψ⁰(t)dt+γ Z T

0

h∇u^ε(t), ψ(t)∇widt+a Z T

0

hu^ε(t), wψ(t)idt +b

Z T

0

h|u^ε(t)|^αu^ε(t), wψ(t)idt− Z T

0

h∇p^ε(t), wψ(t)idt

=hu₀, wiψ(0) + Z T

0

hf(t), wψ(t)idt, (66)

Z T

0

hdivu^ε(t), rψ(t)idt=εhp₀, riψ(0) + Z T

0

εhp^ε(t), riψ⁰(t)dt. (67) In particular, choosing in (66-67)ψ =ϕ∈ D(0, T), we see that (u^ε, p^ε) satisfies (38-39) in the sense of distributions.

It remains to prove thatu^εandp^εsatisfy the initial conditions in (37). For this purpose, we take again, ψ ∈ C^∞(0, T) with ψ(T) = 0, multiply (38-39) by ψ and integrate over (0, T). This yields

− Z T

0

hu^ε(t), wiψ⁰(t)dt+γ Z T

0

h∇u^ε(t), ψ(t)∇widt+a Z T

0

hu^ε(t), wψ(t)idt +b

Z T

0

h|u^ε(t)|^αu^ε(t), wψ(t)idt− Z T

0

h∇p^ε(t), wψ(t)idt

=hu^ε(0), wiψ(0) + Z T

0

hf(t), wψ(t)idt, (68) Z T

0

hdivu^ε(t), rψ(t)idt=εhp^ε(0), riψ(0) + Z T

0

εhp^ε(t), riψ⁰(t)dt. (69) If we compare (66) with (68) and (67) with (69), we observe that

∀w ∈H₀¹(Ω), hu^ε(0)−u₀, wiψ(0) = 0,

∀r∈L²(Ω), hp^ε(0)−p₀, riψ(0) = 0.

Choosingψ(0) = 1 shows that the conditionsu^ε(0) =u₀andp^ε(0) =p₀ are verified.

The proof of the previous Theorem used the property presented in Lemma 5.2 and we prove it here.

Proof of Lemma 5.2. We write the following

Z T

0

h|u^ε_m(t)|^αu^ε_m(t), w_kψ(t)idt − Z T

0

h|u^ε(t)|^αu^ε(t), w_kψ(t)idt

≤ Z T

0

|u^ε_m(t)|^αu^ε_m(t)− |u^ε(t)|^αu^ε(t)

|wk| |ψ(t)|,

≤(α+ 1) sup

x∈Ω

|wk(x)| sup

t∈[0, T]

|ψ(t)|

Z T

0

Z

Ω

u^ε_m(t)−u^ε(t)

|u^ε_m(t)|^α+|u^ε(t)|^α dx dt,

≤(α+ 1) sup

x∈Ω

|wk(x)| sup

t∈[0, T]

|ψ(t)|

Z T

0

ku^ε_m(t)−u^ε(t)k

ku^ε_m(t)k^α_2α+ku^ε(t)k^α_2α dt,

≤Cku^ε_m(t)−u^ε(t)k

L²(0, T ,L²(Ω))

ku^ε_m(t)k^α

L^α+2(0, T ,L^α+2(Ω))+ku^ε(t)k^α

L^α+2(0, T ,L^α+2(Ω))

,

≤Cku^ε_m(t)−u^ε(t)k

L²(0, T ,L²(Ω)).

The last term converges obviously to 0 as m → ∞, this finishes the proof.

5.2. The solutions of perturbed problem converge to solution of BF problem

The aim of this subsection is to consider the limit as → 0. It is useful to establish some a priori estimates for u and p, independent of . Since we are interested in →0, we can always suppose that ≤1. It follows from (45, 46, 55-57) and the lower semi-continuity of the norm that for any ∈(0, 1], it holds

kuk

L^∞(0, T;L²(Ω)) ≤lim inf

m→∞ ku_mk

L^∞(0, T;L²(Ω)) ≤p

d1, (70)

kuk

L²(0, T;H0¹(Ω))≤lim inf

m→∞ ku_mk

L²(0, T;H0¹(Ω)) ≤ s

d₁

2γ, (71)

√εkpk_L^∞_{(0, T;L}²_(Ω))≤√

εlim inf

m→∞ kp_mk_L^∞_{(0, T;L}²_(Ω)) ≤p

d₁. (72)

Eventually, from (54) it holds for 0< θ <1/4 and for any∈(0, 1]

Z +∞

−∞

|τ|^2θkub^ε(τ)k²dτ ≤lim inf

m→∞

Z +∞

−∞

|τ|^2θkub^ε_m(τ)k²dτ ≤Const, (73) where the constant is independent of .

Theorem 5.3. Let (u, p) be the solution of problem (37). Then, there exists a subsequence still denoted by (u, p) so that

u^ε −→u in L²(0, T; H¹₀(Ω)) weakly, (74)

u^ε −→u in L^∞(0, T; L²(Ω)) weak−star, (75)

u^ε −→u in L²(0, T; L²(Ω)) strongly, (76)

p^ε−→p in L^∞(0, T; L²(Ω)) weak−star, (77)

where (u, p) is the solution of the (BF) equations (1).

Proof. By virtue of (70-73), there exists a subsequence still denoted by (u, p) such that

u^ε −→u^∗ in L²(0, T;H¹₀(Ω)) weakly, (78)

u^ε −→u^∗ in L^∞(0, T; L²(Ω)) weak−star, (79)

u^ε −→u^∗ in L²(0, T;L²(Ω)) strongly, (80)

√ p^ε−→χ in L^∞(0, T; L²(Ω)) weak−star, (81)

Passing to the limit → 0 for the subsequence in (39), we obtain in the sense of distributions

√h∂tp(t), qi −→ h∂tχ, qi,

and then h∂tp(t), qi −→0. This limit with (39) gives hdivu^∗(t), qi= 0, ∀q∈L²(Ω),

which in turn implies thatu^∗ is divergence free, i. e. divu^∗ = 0. Therefore u^∗ ∈L²(0, T;V)∩L^∞(0, T; H).

In order to show thatu^∗ verifies the variational formulation (2), we takev ∈V in (38) and we multiply both sides of this last equation by ψ ∈ D(0, T) and we integrate over (0, T), we get

− Z T

0

hu^ε(t), viψ⁰(t)dt+γ Z T

0

h∇u^ε(t), ψ(t)∇vidt+a Z T

0

hu^ε(t), vψ(t)idt +b

Z T

0

h|u^ε(t)|^αu^ε(t), vψ(t)idt= Z T

0

hf(t),vψ(t)idt. (82)

Using the convergence properties (78-79), we can easily pass to the limit ε → 0 in the linear terms in (82). Next, we use the same arguments of the proof of (62) to check that Z T

0

h|u^ε(t)|^αu^ε(t),vψ(t)idt−→

Z T

0

h|u^∗(t)|^αu^∗(t), vψ(t)idt. (83) We obtain in the limit

− Z T

0

hu^∗(t), viψ⁰(t)dt+γ Z T

0

h∇u^∗(t), ψ(t)∇vidt+a Z T

0

hu^∗(t),vψ(t)idt +b

Z T

0

h|u^∗(t)|^αu^∗(t), vψ(t)idt= Z T

0

hf(t),vψ(t)idt, ∀v ∈V. (84) The solution of problem (84) satisfies then (2) in the sense of distributions. It follows that in order to show that u^∗ = u and p^ε = p, it remains only to prove u^∗(0) = u₀ and p^∗(0) =p₀. For this one proceeds exactly as in the proof of the initial conditions of problem (37).