Asymptotic study for Stokes-Brinkman model with jump embedded transmission conditions

(1)

HAL Id: hal-01184429

https://hal.inria.fr/hal-01184429

Submitted on 14 Aug 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Asymptotic study for Stokes-Brinkman model with jump embedded transmission conditions

Philippe Angot, Gilles Carbou, Victor Péron

To cite this version:

Philippe Angot, Gilles Carbou, Victor Péron. Asymptotic study for Stokes-Brinkman model with jump embedded transmission conditions. Asymptotic Analysis, IOS Press, 2016, 96 (3-4), pp.223-249.

�10.3233/ASY-151336�. �hal-01184429�

(2)

EMBEDDED TRANSMISSION CONDITIONS

PHILIPPE ANGOT¹, GILLES CARBOU², VICTOR P ´ERON^2,3

ABSTRACT. In this paper, one considers the coupling of a Brinkman model and Stokes equations with jump embedded transmission conditions. In this model, one assumes that the viscosity in the porous region is very small. Then we derive a Wentzel–Kramers–Brillouin (WKB) expansion in power series of the square root of this small parameter for the velocity and the pressure which are solution of the transmission problem. This WKB expansion is justified rigorously by proving uniform errors estimates.

CONTENTS

1. Introduction 1

2. Statement of the Main Results 3

3. Uniform estimates 5

4. Formal Asymptotic Expansion 7

5. Determination, existence and regularity of the asymptotics 11

6. Regularity proof for the asymptotics 18

7. Estimates of remainders 22

References 25

1. INTRODUCTION

We address the problem of fluid flow modeling in complex media which combine porous regions and fluid regions with free flow. This issue holds for instance in the study of aquifer media made up of a porous media containing craks and conduits (see [8]) and also the passive control of the flow around an obstacle covered by a porous thin layer (see [4]).

In this paper the free flow satisfies the linear Stokes equation. In the porous media, we consider two models, the Brinkman and the Darcy models. There are several interface conditions in the literature. In the case of the Stokes-Brinkman coupling, the simpler interface condition is the continuity of the velocity and the normal stress. More accurate models are given by Ochoa- Tapia & Whitaker transmission conditions, by Beavers & Joseph conditions or Beaver, Joseph

Date: June 16, Version 6.

1Laboratoire d’Analyse, Topologie et Probabilit´es (LATP), UMR CNRS 7353 Equipe d’Analyse Appliqu´ee Centre de Mathmatiques et Informatique (CMI), Aix-Marseille Universit, Technople Chteau-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13 FRANCE.

2Laboratoire de Mathématiques et de leurs Applications de Pau, UMR CNRS 5142, Bâtiment IPRA, Université de Pau et des Pays de l’Adour, Avenue de l’Université - BP 1155, 64013 PAU CEDEX.

3Magique 3d, INRIA Bordeaux Sud-Ouest, Universit´e de Pau et des Pays de l’Adour, Avenue de l’Universit´e - BP 1155, 64013 PAU CEDEX.

1

(3)

2 PHILIPPE ANGOT , GILLES CARBOU, VICTOR P ´ERON

& Saffman conditions (see [8]). In this paper, we deal with the more general jump embedded transmission conditions described in [2]. This condition links the jumps and the averages of both the velocity and the shear stress on the interface. A small parameter appears in this system, in particular the equivalent viscosity in the porous part is small, so that when this parameter tends to zero, we expect that the flow in the porous part will be described by the Darcy law. Our interest lies in the asymptotic study justifying the obtention of the limit model. In particular we describe the boundary layer due to the jump conditions appearing in the porous medium.

Let us describe the Stokes-Brinkman model with jump embedded transmission conditions.

The problem is set in the domainΩ⊂R³made of a fluid regionΩ₊and a porous subdomainΩ−. We assume that the domainsΩ− andΩ₊ are Lipschitz and bounded, and thatΩ− ⊂ Ω− ⊂Ω.

Ω−

Ω+

Σ Γ

n

Figure 1 – The domainΩand the subdomainsΩ−,Ω₊

We denoteΣ = ∂Ω− so that se haveΩ = Ω−∪Σ∪Ω+ (see Figure1). We denote bynthe outward unit normal at∂Ω−.

On the fluid regionΩ+, the velocityv⁺_ε and the pressurep⁺_ε satisfy the Stokes equations. In the porous regionΩ−, the velocityv_ε⁻and the pressurep⁻_ε satisfy a Brinkman model. We couple these models by the Beaver-Joseph conditions at the common boundaryΣ. These conditions link the jumps of the velocity and the normal stress vector with the averages of these quantities onΣ.

We denote by σ⁺(v⁺_ε, p⁺_ε) (resp. σ⁻(v⁻_ε, p⁻_ε)) the stress tensor in the fluid (resp. in the porous) medium :

σ⁺(v_ε⁺, p⁺_ε) = 2µd(v⁺_ε)−p⁺_εI, σ⁻(v⁻_ε, p⁻_ε) = 2εd(v_ε⁻)−p⁻_εI, with

• d(v) = 1 2

∇v+∇^⊥v

, that is(d(v))ij = 1 2

∂v^j

∂x_i + ∂vⁱ

∂x_j

,

• µis the viscosity of the fluid andεis the effective viscosity in the porous medium.

The problem writes

(1.1)











−∇ ·σ⁻(v⁻_ε, p⁻_ε) +κv⁻_ε =g⁻ in Ω−

−∇ ·σ⁺(v⁺_ε, p⁺_ε) =g⁺ in Ω+

∇ ·v_ε⁻= 0 in Ω−

∇ ·v_ε⁺= 0 in Ω+

σ⁺(v_ε⁺, p⁺_ε)·n=σ⁻(v_ε⁻, p⁻_ε)·n+M{v_ε}+l on Σ {σ(v_ε, p_ε)·n}=S(v⁺_ε −v_ε⁻) +h on Σ

vε= 0 on Γ

(4)

where:

• κ >0is fixed positive constant.

• In the previous equations set onΣ, we denote by{w}the mean value acrossΣofw.

• The right-hand sides in (1.1) are given data which are defined as follow :g^±: Ω±→R³ are vector fields in L²(Ω±)3

,h∈ H^−1/2(Σ)3

, andl∈ H^−1/2(Σ)3

are vector fields defined onΣ.

• The matrixM is zero onn^⊥and satisfies:

Mξ =β(ξ·n)n withβ >0.

• The matrixSsatisfiesS_|n^⊥ =αI|n^⊥ andS(n) = 1 εn:

S(ξ) = 1

ε(n·ξ)n+α(ξ−(n·ξ)n), withα >0.

We remark that using the divergence free conditions, we can replace the first equation in (1.1) by

(1.2) −ε∆v⁻_ε +∇p⁻_ε +κv⁻_ε =g⁻ in Ω−

Notation 1.1. For any setO ⊂R³we denoteL²(O)the space L²(O)3

andH^s(O)the space (H^s(O))³.

2. STATEMENT OF THEMAINRESULTS

In order to prove existence of weak solutions for (1.1), let us describe the associated variational formulation.

We denote byPH¹(Ω)the space of vector fieldsv ∈L²(Ω)such thatv⁺ :=v_|Ω₊ ∈H¹(Ω₊) andv⁻ := v|Ω− ∈ H¹(Ω−). We introduce a weak formulation of the problem (1.1) for v = (v⁺, v⁻)in the space

V ={u∈PH¹(Ω)|divu^±= 0 in Ω±, u⁺= 0 on Γ},

endowed with the piecewiseH¹norm. Such a variational formulation writes : Findv= (v⁻, v⁺)∈ V such that

(2.1) ∀u∈V, aε(v,u) =b(u),

where

a_ε(v,u) := 2ε Z

Ω−

d(v⁻) :d(u⁻) dx+ Z

Ω−

κv⁻·u⁻dx+ 2µ Z

Ω+

d(v⁺) :d(u⁺) dx +< β{v·n},{u·n}>−1/2,Σ +< α[vτ],[uτ]>−1/2,Σ+ε⁻¹ <[v·n],[u·n]>−1/2,Σ, with < , >_−1/2,Σ being a duality pairing betweenH^−1/2(Σ)³ and H^1/2(Σ)³, [v] = v⁺−v⁻ denoting the jump ofvacrossΣ, and

b(u) = Z

Ω

g·udx+<h,[u]>−1/2,Σ+<l,{u}>−1/2,Σ,

(5)

compare with [1, Th 1.1], [2, Th 2.1]. The duality pairing< , >_−1/2,ΣbetweenH^−1/2(Σ)³ and H^1/2(Σ)³ coincides with the duality pairing inL²(Σ)³.

Forε >0, applying Lax Milgram Theorem, we prove the existence and uniqueness of weak solution for (1.1) and we obtain uniform estimates of the solutions with respect to the parameter ε.

Theorem 2.1. Letg^± ∈ L²(Ω±)³ and h,l ∈ L²(Σ)³. Then, the problem(2.1) has a unique solutionv = v_ε ∈ V for all ε >0. Moreover, forε >0 small enough, the following uniform estimate holds :

(2.2)

εkd(v⁻)k²_0,Ω₋+κ

4kv⁻k²_0,Ω₋+µC²

3 kv⁺k²_1,Ω

++ 1

4εk[v·n]k²_0,Σ+α

4k[v]k²_0,Σ+β

4k{v·n}k²_0,Σ 6c kgk²_0,Ω+khk²_0,Σ+klk²_0,Σ

, with a constantc=c(µ, κ, α, β).

Remark2.2. This result still holds when the datahandlbelong to the spaceH⁻¹²(Σ)³. For the sake of simplicity, we prove this lemma whenhandlbelong to the spaceL²(Σ)³. We eventually compare this stability result with [1,2] where the author prove also energy estimates. In [1, Th 1.1] and [2, Th 2.1], estimates are non-necessary uniform, whereas estimates (2.2) are uniform with respect to the parameterε.

The asymptotic limit of the Stokes-Brinkman model towards the Stokes-Darcy one with Beavers- Joseph interface conditions is studied in [2,8] in the case of a flat interface when the viscosity

˜

µin the porous region is very smallµ˜ =ε 1and when the jump of the normal velocities is penalized. We address in this paper the problem of the convergence of the model (1.1) when the parameterεtends to zero.

When ε tends to zero, we formally converge to the following Stokes Darcy problem with Beavers & Joseph interface conditions

(2.3)











∇p⁻₀ +κv⁻₀ =g⁻ in Ω−

−∇ ·σ(v₀⁺, p⁺₀) =g⁺ in Ω₊

∇ ·v₀⁻= 0 in Ω−

∇ ·v₀⁺= 0 in Ω+

σ(v₀⁺, p⁺₀)·n=−p⁻₀n+ β

2 (v₀⁺+v₀⁻)·n

n+l on Σ

(v₀⁺−v₀⁻)·n= 0 on Σ

v⁺₀ = 0 on Γ.

This problem is incompatible with the limit (as ε tends to zero) of the tangential part of the interface condition{σ(v_ε, pε)·n}=S(v_ε⁺−v_ε⁻) +honΣso that it appears a boundary layer inside the porous medium. We describe this boundary layer by an asymptotic expansion at any order with a WKB method. We derive this expansion in power series of the small parameter√

ε for both the velocityv_εand the pressurep_εwhich are solutions of the transmission problem. This expansion is justified rigorously by proving uniform estimates for remainders of this expansion, Theorem7.1.

An immediate corollary of our asymptotic expansion is the following convergence theorem:

(6)

Theorem 2.3. We assume that the data in(1.1)satisfy:

g⁻∈H⁵(Ω−), g⁺∈H⁴(Ω+), l∈H⁹²(Σ) and h∈H⁹²(Σ).

Then, the solutionv_εfor(1.1)given by Theorem2.1satisfies:

v⁺_ε(x) =v⁺₀(x) +r⁺_ε(x) forx∈Ω+

v⁻_ε(x) =v⁻₀(x) +ev⁻₀(x,^d(x)^√_ε) +r⁻_ε(x) forx∈Ω−

where

• v0 is the solution of (2.3),

• ev⁻₀ is a boundary layer term of the formev⁻₀(x,z) =w0(x) exp(−√ κz),

• d(x)is the euclidean distance toΣ,

• rεis a remainder terms.

This remainder term satisfies the following estimate:

εkd(r⁻_ε)k²_0,Ω₋+κ

4kr⁻_εk²_0,Ω₋+µC²

3 kr⁺_εk²_1,Ω₊ 6Cε¹².

The concept of WKB expansion is rather classical in the modeling of problems arising in fluid mechanics. For instance in [7, 5, 6] the authors derive WKB expansions with boundary layer terms or thin layer asymptotics to describe penalization methods in the context of viscous incompressible flow.

In this work, one difficulty to validate the WKB expansion lies in the proof of both existence and regularity results for one part of the asymptotics which appear in this expansion at any order and which solve Darcy-Stokes problems with non-standard transmission conditions. It is possible to tackle these problems by carefully introducing a Dirichlet–to–Neumann operator which lead us to prove well-posedness results and elliptic regularity results simply for the Stokes operator with mixed boundary conditions.

The outline of the paper proceeds as follows. We prove the well-posedness result for Problem (2.1) together with uniform estimates with respect to the small parameter is Section3. In Sections 4and5, one exhibits a formal WKB expansion for the solution of the transmission problem. The equations satisfied by the asymptotics at any order are explicited in section5and existence and regularity results concerning the asymptotics which satisfy Darcy-Stokes problems with non- standard transmission conditions are claimed in Prop. 5.1. The proof of this proposition is postponed to section6. In section7, one proves uniform errors estimates to validate this WKB expansion.

3. UNIFORM ESTIMATES

In this section, we prove the well-posedness result for Problem (2.1) together with uniform estimates, Theorem2.1.

Notation 3.1. We denote byk · k_s,Othe norm in the Sobolev spaceH^s(O).

Proof of Theorem 2.1. Since Ω+ is a lipschitz bounded domain, since v⁺ = 0 on ∂Ω for v∈V, we have the following Poincar´e inequality inΩ₊:

(3.1) ∃C >0 s.t. ∀v∈V kd(v⁺)k_0,Ω₊ >Ckv⁺k_1,Ω₊ ,

(7)

then, there hold∀v∈V,

a_ε(v, v)>2εkd(v⁻)k²_0,Ω₋+κkv⁻k²_0,Ω₋+ 2µC²kv⁺k²_1,Ω

+ +ε⁻¹k[v·n]k²_0,Σ

+βk{v·n}k²_0,Σ+αkv_τk²_0,Σ. Hence, aε is V-coercive, and according to the Lax-Milgram Lemma, the problem (2.1) has a unique solutionv_ε∈V for allε >0. We also infer : ifvsatisfies (2.1), then∀v∈V,

2εkd(v⁻)k²_0,Ω₋+κkv⁻k²_0,Ω₋+ 2µC²kv⁺k²_1,Ω

++ε⁻¹k[v·n]k²_0,Σ+βk{v·n}k²_0,Σ+αk[v_τ]k²_0,Σ 6kv⁺k_0,Ω₊kg⁺k_0,Ω₊+kv⁻k_0,Ω₋kg⁻k_0,Ω₋+k[v]k_0,Σkhk_0,Σ+k{v}k_0,Σklk_0,Σ. Letε >0such thatα6 _2ε¹. Then, for allη1, η2>0, there holds

(3.2)

++ 1

2εk[v·n]k²_0,Σ+βk{v·n}k²_0,Σ+αk[v]k²_0,Σ 6 η₁

2kvk²_0,Ω+ 1 2η1

kgk²_0,Ω+η₂

2k[v]k²_0,Σ+ 1 2η2

khk²_0,Σ+k{v·n}k_0,Σklk_0,Σ+k{v_τ}k_0,Σklk_0,Σ. We treat hereafter the last two terms : for allη3 >0, there holds

k{v·n}k_0,Σklk_0,Σ 6 η₃

2 k{v·n}k²_0,Σ+ 1 2η3

klk²_0,Σ. Since{v_τ}=v_τ⁺−¹₂[vτ], there holds

k{v_τ}k_0,Σklk_0,Σ 6(kv_τ⁺k_0,Σ+1

2k[v_τ]k_0,Σ)klk_0,Σ, hence, using a trace inequality, there existsc >0such that

k{v_τ}k_0,Σklk_0,Σ6(ckv⁺k_1,Ω₊+1

2k[v_τ]k_0,Σ)klk_0,Σ. We infer : for allη4, η5 >0,

k{v_τ}k_0,Σklk_0,Σ 6 η₄c

2 kv⁺k²_1,Ω

++ c

2η4

klk²_0,Σ+η₅

4k[v_τ]k²_0,Σ+ 1 4η5

klk²_0,Σ. According to (3.2), we obtain

(3.3)

++ 1

2εk[v·n]k²_0,Σ+αk[v]k²_0,Σ+βk{v·n}k²_0,Σ 6 η1

2kvk²_0,Ω+ η2

2 k[v]k²_0,Σ+η3

2k{v·n}k²_0,Σ+η4c

2 kv⁺k²_1,Ω

++ η5

4 k[v_τ]k²_0,Σ + 1

2η₁kgk²_0,Ω+ 1

2η₂khk²_0,Σ+ ( 1 2η₃ + c

2η₄ + 1

4η₅)klk²_0,Σ. We fix constantsη_isuch that

η₄c

2 6 µC² 3 , η₁

2 6min(κ 2,µC²

3 ), η₂ 2 +η₅

4 6 α 2 , η₃

2 6 β 2 .

(8)

According to (3.3), we infer (3.4)

εkd(v⁻)k²_0,Ω₋+κ

4kv⁻k²_0,Ω₋+µC²

3 kv⁺k²_1,Ω

++ 1

4εk[v·n]k²_0,Σ+α

4k[v]k²_0,Σ+β

4k{v·n}k²_0,Σ 6C(µ, κ, α, β) kgk²_0,Ω+khk²_0,Σ+klk²_0,Σ

. 4. FORMALASYMPTOTICEXPANSION

Hypothesis 4.1. From now on, we assume that the surfacesΣ(interface) andΓ(external boundary) are smooth.

Notation 4.2. IfX is a vector field defined onΣ, we denote byX_τ the tangent components of X: Xτ(x) =X(x)−(X(x)·n(x))n(x), so thatX = (X·n)n+Xτ. For example, we denote by(d(v_ε).n)_τ the tangent components of the normal constraintd(v_ε).ndefined on the interface Σ.

Rewriting the transmission conditions set onΣin (1.1), we use Formulation (1.2) to obtain the following equivalent problem,

−ε∆v⁻_ε +∇p⁻_ε +κv⁻_ε =g⁻ in Ω−

(4.1)

− ∇ ·σ⁺(v⁺_ε, p⁺_ε) =g⁺ in Ω₊

(4.2)

∇ ·v⁻_ε = 0 in Ω−

(4.3)

∇ ·v⁺_ε = 0 in Ω₊

(4.4)

2µd(v_ε⁺)·n−p⁺_εn= 2εd(v⁻_ε)·n−p⁻_εn+β

2 (v_ε⁺+v_ε⁻)·n

n+l on Σ

(4.5)

α(v_ε⁺−v_ε⁻)τ = 2µ d(v⁺_ε)·n

τ−(h+1

2l)τ on Σ

(4.6)

ε⁻¹(v_ε⁺−v_ε⁻)·n= 2µ(d(v_ε⁺)·n)·n−p⁺_ε −β

4(v⁺_ε +v⁻_ε)·n−(h+1

2l)·n on Σ (4.7)

v_ε⁺= 0 on Γ

(4.8)

(we recall that∇ ·σ⁺(v⁺_ε, p⁺_ε) =µ∆v⁺_ε − ∇p⁺_ε)

As already said, when ε tends to zero, we formally converge to Problem (2.3). The well posedness of this limit system is establish in Section6. This problem is incompatible with the

(9)

limit of the jump condition (4.6) on the tangential velocity:

α(v⁺₀ −v⁻₀)_τ = 2µ d(v₀⁺)·n

τ −(h+ 1

2l)_τ on Σ.

Hence, it appears a boundary layer that we describe with a multiscale method, namely a WKB expansion: we exhibit series expansions in powers of√

εfor the flowv_ε, and the pressurep_ε. In the fluid part, we do not expect the formation of a boundary layer, therefore we look for an Ansatzon the form:

v⁺_ε(x)≈X

j>0

ε²^jv⁺_j (x), (4.9)

p⁺_ε(x)≈X

j>0

ε^j²p⁺_j (x). (4.10)

In the porous part, we denote byd: Ω−→R⁺the euclidean distance toΣ,d(x) =dist(x,Σ).

We describe the velocity and the pressure on the following way:

v⁻_ε(x)≈X

j>0

ε^j²v_j⁻(x,d(x)

√ε ), with v_j⁻(x,z) = v⁻_j (x) +ev⁻_j (x,z), (4.11)

p⁻_ε(x)≈X

j>0

ε^j²p⁻_j (x,d(x)

√ε ), with p⁻_j(x,z) = p⁻_j(x) +ep⁻_j (x,z). (4.12)

The termsev⁻_j andep⁻_j are boundary layer terms and defined onΣ×R⁺. They are required to tend to0(such as their derivatives) whenz→ ∞.

We use the following notations (see also [7]) : x = (x1, x2, x3)denotes the cartesian coordi- nates inR³,

∂i = ∂

∂xi

, ∂z= ∂

∂z , ∂zz= ∂²

∂z², ∇ep⁻= (∂1ep⁻, ∂2ep⁻, ∂3ep⁻)^T. Forev⁻= (ev^−,1,ev^−,2,ev^−,3)^T,

∇ ·ev⁻=∂₁ev^−,1+∂₂ev^−,2+∂₃ev^−,3, (d(ev⁻))_ij = 1 2

∂ev^−,j

∂x_i + ∂ev^−,i

∂x_j

,

(∇d· ∇)∂_zev⁻=

3

X

j=1

∂d

∂x_j

∂²ev⁻

∂x_j∂z, ∆ev⁻=∂₁²ev⁻+∂₂²ev⁻+∂₃²ev⁻.

We recall thatnis the outward unit normal on∂Ω−= Σ. Since we assume thatΣis a regular manifold, thendis smooth in a neighborhood ofΣ. In this neighborhood,|∇d|= 1. In addition, forx∈Σ,∇d(x) =−n(x)so that∂nd=−1onΣ. We extendnby settingn(x) =−∇d(x). If Xis a vector field defined inΩ−, in a neighborhood ofΣ, we define the tangential part ofXby Xτ(x) =X(x)−(X(x)·n(x))n(x).

By simple calculations there holds

∆

x7→ev⁻(x,d(x)

√ε )

= ε⁻¹∂_zzev⁻(x,d(x)

√ε ) +ε⁻¹² 2(∇d· ∇)∂_zev⁻+ ∆d ∂_zev⁻

(x,d(x)

√ε ) +∆ev⁻(x,d(x)

√ε )

(10)

∇

x7→ep⁻(x;d(x)

√ε )

=ε⁻¹²∇d ∂_zep⁻(x,d(x)

√ε ) +∇ep⁻(x,d(x)

√ε )

∇ ·

x7→ev⁻(x;d(x)

√ε )

=ε⁻¹²∇d·∂_zev⁻(x,d(x)

√ε ) +∇ ·ev⁻(x,d(x)

√ε ) d

x7→ev⁻(x,d(x)

√ε )

·n=−ε⁻¹²1

2 ∂_zev⁻+ (∂_zev⁻·n)n

(x;d(x)

√ε )+d(ev⁻)·n(x;d(x)

√ε ) on Σ. We insert theAnsatz(4.9)-(4.10)-(4.11)-(4.12) in equations (4.1)-(4.8) and we perform the iden- tification of terms with the same power in√

ε.

4.1. Asymptotic expansion of(4.1).

Orderε⁻¹². There holds∇d ∂_zep⁻₀ = 0, and sinceep⁻₀ →0whenz→ ∞, we infer

(4.13) ep⁻₀ = 0 in Ω−×R⁺.

Orderε⁰. According to (4.13),(v₀⁻, p⁻₀)solves

(4.14) −∂zzev⁻₀ +κev⁻₀ +κv₀⁻+∇p⁻₀ +∇d ∂_zep⁻₁ =g⁻ in Ω−×R⁺.

Hence, taking the limit of this equation whenz → ∞, sinceev₀⁻and all its derivatives tends to zero whenztends to+∞, we infer

(4.15) κv⁻₀ +∇p⁻₀ =g⁻ in Ω−,

and by difference with the previous equation, we obtain

(4.16) −∂zzev⁻₀ +κev⁻₀ +∇d ∂_zep⁻₁ = 0 in Ω−×R⁺. Orderε¹².(v₁⁻, p⁻₁)solves

(4.17) −∂zzev⁻₁ +κev⁻₁ +κv⁻₁ +∇p⁻₁(x) +∇ep⁻₁ +∇d ∂_zep⁻₂ = (2(∇d· ∇)∂_zev⁻₀ + ∆d ∂zev₀⁻), and taking the limit of this equation whenz→ ∞, we infer

(4.18) κv⁻₁(x) +∇p⁻₁(x) = 0 in Ω−. By difference with the previous equation, we obtain

(4.19) −∂_zzev⁻₁ +κev⁻₁ +∇ep⁻₁ +∇d ∂_zep⁻₂ = 2(∇d· ∇)ev⁻_0,Z+ ∆d ∂_zev⁻₀ , in Ω−×R⁺. Orderε^j²,j>2.(v⁻_j , p⁻_j)solves

(4.20) −∂_zzev_j⁻+κev_j⁻+κv⁻_j +∇p⁻_j +∇ep⁻_j +∇d ∂_zep⁻_j+1

= (2(∇d· ∇)∂_zev⁻_j−1+ ∆d ∂zev⁻_j−1) + ∆(v⁻_j−2+ev_j−2⁻ ), hence, we infer

(4.21) κv⁻_j +∇p⁻_j = ∆v⁻_j−2 in Ω−, (4.22) −∂zzev_j⁻+κev_j⁻+∇ep⁻_j +∇d ∂_zep⁻_j+1

= (2(∇d· ∇)∂_zev_j−1⁻ + ∆d ∂_zev⁻_j−1) + ∆ev_j−2⁻ in Ω−×R⁺. 4.2. Asymptotic expansion of(4.2).

Orderε⁰.v⁺₀ satisfies

(4.23) − ∇ ·σ⁺(v⁺₀,p⁺₀) =g⁺ in Ω+.

(11)

Orderε^j²,j>1. For allj∈N,v⁺_j satisfies

(4.24) − ∇ ·σ⁺(v⁺_j,p⁺_j) = 0 in Ω₊. 4.3. Asymptotic expansion of(4.3).

Orderε⁻¹². There holds

(4.25) ∇d·∂_zev₀⁻= 0 in Ω−×R⁺. Since∇d=−n, sinceev⁻₀ tends to zero whenztends to+∞, we obtain

(4.26) ev⁻₀ ·n= 0 in Ω−×R⁺.

Orderε^j²,j>0. There holds

(4.27) ∇d·∂zev_j+1⁻ +∇ ·v_j⁻+∇ ·ev⁻_j = 0 in Ω−×R⁺. We infer

∇ ·v⁻_j = 0 in Ω−, (4.28)

∇d·∂_zev⁻_j+1+∇ ·ev_j⁻= 0 in Ω−×R⁺. (4.29)

Orderε^j². For allj∈N,v⁺_j satisfies

(4.30) ∇ ·v⁺_j = 0 in Ω₊.

Orderε⁰.

(4.31) 2µd(v⁺₀)·n−p⁺₀n=−p⁻₀n−ep⁻₀(·,0)n+β

2 (v₀⁺+ v⁻₀ +ev₀⁻(·,0))·n

n+l on Σ Orderε¹².

(4.32) 2µd(v⁺₁)·n−p⁺₁n= −1

2 ∂zev⁻₀(·,0) + (∂zev⁻₀(·,0)·n)n

−p⁻₁n−ep⁻₁(·,0)n +β

2 (v⁺₁ + v⁻₁ +ev⁻₁(·,0))·n

n on Σ Orderε^j²,j>2.

(4.33)

2µd(v_j⁺)·n−p⁺_jn= −1 2

∂zev⁻_j−1(·,0) + (∂zev⁻_j−1(·,0)·n)n

+ 2d(v_j−2⁻ )·n +2d(ev⁻_j−2)(·,0)·n−p⁻_jn−ep⁻_j (·,0)n

+β 2

(v⁺_j + v⁻_j +ev⁻_j (·,0))·n

n on Σ. 4.6. Asymptotic expansion of(4.6).

Orderε⁰.

(4.34) α(v⁺₀ −v₀⁻−ev⁻₀(·,0))_τ = 2µ d(v⁺₀)·n

τ −

h+1 2l

τ

on Σ

(12)

Orderε^j²,j>1.

(4.35) α(v⁺_j −v⁻_j −ev⁻_j (·,0))_τ = 2µ

d(v⁺_j )·n

τ on Σ

Orderε⁰.

(4.36) (v⁺₀ −v₀⁻−ev⁻₀(·,0))·n= 0 on Σ Orderε¹².

(4.37) (v⁺₁ −v₁⁻−ev⁻₁(·,0))·n= 0 on Σ Orderε.

(4.38)

(v⁺₂−v⁻₂−ev₂⁻(·,0))·n= 2µ d(v⁺₀)·n

·n−p⁺₀−β

4 v₀⁺+ v⁻₀ +ev₀⁻(·,0)

·n−(h+1

2l)·n on Σ Orderε^j²,j >2.

(4.39)

(v⁺_j −v⁻_j −ev_j⁻(·,0))·n= 2µ

d(v⁺_j−2)·n

·n−p⁺_j −β 4

v⁺_j−2+ v⁻_j−2+ev⁻_j−2(·,0)

·n on Σ. 4.8. Asymptotic expansion of(4.8).

Orderδ^j. For allj∈N,v⁺_j satisfies the external boundary condition

(4.40) v⁺_j = 0 in Γ.

5. DETERMINATION,EXISTENCE AND REGULARITY OF THE ASYMPTOTICS

Letk>1. We assume that the data in (4.1)-(4.8) satisfy:

(5.1) g⁻ ∈H^k(Ω−), g⁺ ∈H^k−1(Ω₊), l∈H^k−¹²(Σ) and h∈H^k−¹²(Σ).

5.1. Equations satisfied by asymptotics of order 0.

5.1.1. Determination ofep⁻₀ andev₀⁻·n. We have already obtained in (4.13) and (4.26) that (5.2) ep⁻₀ = 0andev⁻₀ ·n= 0 in Ω−×R⁺.

5.1.2. Determination ofv⁻₀,p⁻₀ v₀⁺andp⁺₀. According to (4.36) and (4.26), we infer (v⁺₀ −v⁻₀)·n= 0 on Σ.

According to and (4.13) and (4.31)

2µd(v⁺₀)·n−p⁺₀n=−p⁻₀n+β

2 (v₀⁺+ v⁻₀)·n

n+l on Σ.

(13)

Finally, according to (4.15), (4.28), (4.31), and according to (4.23) and (4.30) whenj = 0, (v⁻₀,v⁺₀), and(p⁻₀,p⁺₀)satisfy the transmission problem :

(5.3)











κv⁻₀ +∇p⁻₀ =g⁻ in Ω−

∇ ·v⁻₀ = 0 in Ω−.

−∇ ·σ⁺(v⁺₀,p⁺₀) =g⁺ in Ω+

∇ ·v⁺₀ = 0 in Ω₊

(v₀⁺−v⁻₀)·n= 0 on Σ

2µd(v⁺₀)·n−p⁺₀n=−p⁻₀n+^β₂ (v⁺₀ + v₀⁻)·n

n+l on Σ

v⁺₀ = 0 on Γ.

Let us claim the following existence and regularity result concerning such a problem

Proposition 5.1. Letk ≥ 1. Let g⁻ ∈ H^k(Ω−), g⁺ ∈ H^k−1(Ω+), l ∈ H^k−¹²(Σ) andh ∈ H^k−¹²(Σ). We assume thathsatisfies the compatibility condition

Z

Σ

hdσ= 0.

We consider the following problem:

(5.4)











κv⁻+∇p⁻=g⁻ in Ω−

∇ ·v⁻= 0 in Ω−

−∇ ·σ⁺(v⁺,p⁺) =g⁺ in Ω+

∇ ·v⁺= 0 in Ω+

(v⁺−v⁻)·n=h on Σ

2µd(v⁺)·n−p⁺n=−p⁻n+^β₂((v⁺+ v⁻)·n)n+l on Σ

v⁺= 0 on Γ.

Then(5.4)admits a weak solution unique up to additive constants forp⁺andp⁻. It satisfies the regularity properties:

v⁺∈H^k+1(Ω₊), v⁻∈H^k(Ω−), p⁺∈H^k(Ω₊) and p⁻∈H^k+1(Ω−).

This proposition will be proved in Section6.

Using Proposition 5.1 with g^± = g^±, l = l and h = 0, using the regularity of the data (5.1), we obtain the existence and uniqueness (up to an additive constant for the pressures) of the asymptoticsv⁺₀ ∈H^k+1(Ω₊), v₀⁻∈H^k(Ω−), p⁺₀ ∈H^k(Ω₊) and p⁻₀ ∈H^k+1(Ω−).

5.1.3. Determination of(ev⁻₀)τ. The tangential component of (4.16) reduce to

−∂_zz(ev⁻₀)_τ +κ(ev₀⁻)_τ = 0 in Ω−×R⁺. Hence, sinceev⁻₀ →0whenz→ ∞, we obtain

ev₀⁻(x,z)τ =ev⁻₀(x,0)τexp(−√

κz) in Ω−×R⁺, According to (4.34), we infer

ev⁻₀(x,0)_τ = (v⁺₀ −v₀⁻)_τ−2µ

α (d(v⁺₀)·n)_τ+ 1 α

h+ 1

2l

τ

on Σ.

(14)

Hereafter, we introducew⁰₀ which is a tangential extension of(v⁺₀ −v⁻₀)τ −^2µ_α(d(v⁺₀)·n)τ+

1

α h+¹₂l

τin the domainΩ−. We choose this extensionw⁰₀such that it has a support in a tubular neighborhood of the interfaceΣ. Sincev⁺₀ ∈ H^k+1(Ω₊), since v⁻₀ ∈ H^k(Ω−), since landh belong to the spaceH^k−¹²(Σ), we can take this extension satisfyingw⁰₀ ∈ H^k(Ω−), and using (5.2) we obtain

(5.5) ev₀⁻(x,z) =w⁰₀(x) exp(−√

κz) in Ω−×R⁺, and thenev₀⁻is completely defined by (5.5).

5.2. Equations satisfied by asymptotics of order 1.

5.2.1. Determination ofep⁻₁. According to (4.26), by taking the normal components in (4.16), we obtainep⁻_1,Z = 0. Hence, sinceep⁻₁ →0whenz→ ∞, we obtain

(5.6) ep⁻₁ = 0 in Ω−×R⁺.

5.2.2. Determination ofev⁻₁ ·n. According to (4.29)

∂zev⁻₁ · ∇d=−∇ ·ev₀⁻=−(∇ ·w⁰₀) exp(−√

κz) in Ω−×R⁺. Hence, sinceev⁻₁ tends to zero whenztends to+∞,

(5.7) ev⁻₁ ·n=ν₁⁰exp(−√

κz) with ν₁⁰= −1

√κ(∇ ·w₀⁰)∈H^k−1(Ω−).

5.2.3. Determination of(v⁻₁,v₁⁺), and(p⁻₁,p⁺₁). According to (4.26) and (5.6), the transmission condition (4.32) writes :

2µd(v₁⁺)·n−p⁺₁n=−1

2∂_zev⁻₀ −p⁻₁n+β

2 (v⁺₁ + v₁⁻+ev⁻₁)·n

n on Σ

Hence, according to (4.18), (4.28), and to (4.24) and (4.30) when j = 1, and according to (4.32), (4.37), (5.5) and (5.7),(v⁻₁,v⁺₁), and(p⁻₁,p⁺₁)satisfy the transmission problem:

(5.8)











κv⁻₁ +∇p⁻₁ = 0 in Ω−

∇ ·v⁻₁ = 0 in Ω−

−∇ ·σ⁺(v⁺₁,p⁺₁) = 0 in Ω+

∇ ·v⁺₁ = 0 in Ω₊

(v⁺₁ −v⁻₁)·n=ν₁⁰ on Σ

2µd(v⁺₁)·n−p⁺₁n=−p⁻₁n+^β₂ (v⁺₁ + v⁻₁)·n

n+¹₂√

κw⁰₀+^β₂ν₁⁰n on Σ

v⁺₁ = 0 on Γ.

We apply Proposition5.1with g^±= 0, l= 1

2

√κw⁰₀+β

2ν₁⁰n∈H^k−³²(Σ), and h=ν₁⁰ ∈H^k−³²(Σ), andk > 2. We remark that ν₁⁰ satisfies the compatibility condition R

Σν₁⁰dσ = 0sinceν₁⁰ is the divergence of a tangential vector field. Indeed, the divergence ofw⁰₀ on the surfaceΣis the divergence operator onΣapplied to a tangent vector field onΣ; hence by the Stokes formula, the integral of∇ ·w⁰₀vanishes sinceΣdoes not have boundary.

(15)

Hence we obtain the existence and uniqueness (up to an additive constant for the pressures) for the asymptoticsv⁻₁,v⁺₁,p⁻₁ andp⁺₁ which satisfy:

(5.9) v⁺₁ ∈H^k(Ω+), v₁⁻∈H^k−1(Ω−), p⁺₁ ∈H^k−1(Ω+) and p⁻₁ ∈H^k(Ω−). 5.2.4. Determination of(ev⁻₁)_τ. Usingep⁻₁ = 0, then (4.19) writes now

(5.10)

−∂_zzev⁻₁ +κev₁⁻+∇d ∂_zep⁻₂ =−√

κ(2(∇d· ∇)w⁰₀+ ∆dw₀⁰) exp(−√

κz), in Ω−×R⁺. Taking the tangential components of the previous equation, we obtain thatev⁻₁(x,z)_τ solves (5.11) −∂_zz(ev₁⁻)_τ+κ(ev⁻₁)_τ =−√

κ(2(∇d· ∇)w⁰₀+ ∆dw₀⁰)_τexp(−√

κz) in Ω−×R⁺, with (see (4.35)) :

(5.12) (ev⁻₁)τ(x,0) = (v₁⁺−v⁻₁)τ− 2µ

α d(v⁺₁)·n

τ on Σ.

On the one hand, we introducew⁰₁which is a tangential extension of(v⁺₁−v₁⁻)τ−2µ

α d(v₁⁺)·n

τ

in the domainΩ−. We choose this extensionw⁰₁ such that it has a support in a tubular neighborhood of the interfaceΣ. Sincev⁺₁ ∈H^k(Ω₊)andv₁⁻ ∈ H^k−1(Ω−), we can take an extension satisfying

w₁⁰∈H^k−1(Ω−). On the other hand, we denote

w¹₁ =−1

2 2∇d· ∇w⁰₀+ ∆dw⁰₀

τexp(−√ κz).

We remark thatw¹₁∈H^k−1(Ω−)sincew₀⁰∈H^k(Ω−).

By solving (5.11) with the boundary condition (5.12) and we obtain that (5.13) (ev₁⁻)_τ(x,z) = w₁⁰(x) +z w¹₁(x)

exp(−√

κz) with w⁰₁ ,w¹₁ in H^k−1(Ω−).

5.3. Characterization of the asymptotics of order j ≥ 2. First let us write the equations satisfied by the asymptotics at orderj.

At order2, from the normal components of (4.19), using (5.5), we have (−∂_zzev⁻₁ +κev⁻₁)·n−∂zep⁻₂ =−√

κ(2(∇d· ∇)w₀⁰+ ∆dw⁰₀)·nexp(−√ κz).

Hence, according to (5.7),ep⁻₂ satisfies

∂zep⁻₂ =√

κ(2(∇d· ∇)w₀⁰)·nexp(−√

κz), in Ω−×R⁺. We infer

(5.14) ep⁻₂ =q⁰₂exp(−√

κz) with q⁰₂ =−(2(∇d· ∇)w⁰₀)·n∈H^k−1(Ω−).

At orderj≥3, from the normal part of (4.22), replacingjbyj−1, we obtain that

∂zep⁻_j =

−∂_zzev⁻_j−1+κev⁻_j−1+∇ep⁻_j−1−2∇d· ∇∂_zev_j−2⁻ −∆d∂zev⁻_j−2−∆ev⁻_j−3

·n,