On the existence of strong solutions to a fluid structure interaction problem with Navier boundary conditions

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On the existence of strong solutions to a fluid structure interaction problem with Navier boundary conditions

Imene Djebour, Takéo Takahashi

To cite this version:

Imene Djebour, Takéo Takahashi. On the existence of strong solutions to a fluid structure interaction

problem with Navier boundary conditions. Journal of Mathematical Fluid Mechanics, Springer Verlag,

In press, 21, �10.1007/s00021-019-0440-7�. �hal-02061542�

On the existence of strong solutions to a fluid structure interaction problem with Navier boundary conditions

Imene A. Djebour ¹ and Takéo Takahashi ^∗1

1 Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy March 8, 2019

Abstract

We consider a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas we add a damping in the plate equation. We use here Navier-slip boundary conditions instead of the standard no-slip boundary conditions. The main results are the local in time existence and uniqueness of strong solutions of the corresponding system and the global in time existence and uniqueness of strong solutions for small data and if we assume the presence of frictions in the boundary conditions.

Keywords: Navier-Stokes system, damped beam equation, strong solutions.

2010 Mathematics Subject Classification. 35Q30, 76D05, 76D03, 74F10

Contents

1 Introduction 2

2 Notation 5

3 Change of variables 7

4 Regularity properties of the Stokes system 10

5 Linear System 15

6 Fixed point 22

6.1 Estimates on the change of variables . . . . 23 6.2 Estimates of F , G, H . . . . 27 6.3 Proof of Theorem 6.1 . . . . 29

A Technical results 30

∗

Takéo Takahashi acknowledges the support of the Agence Nationale de la Recherche, project IFSMACS, ANR-15-CE40-0010

1 Introduction

The aim of this work is to analyze the interaction between a viscous incompressible fluid and a viscous elastic plate. Let us start by presenting the corresponding model. We denote by ω the rectangular torus

x ₁

x 2

x 3

Γ(η)

Γ 0

L ₁

L 2

Figure 1: Configuration of the domain at time t.

ω = ( R /L ₁ Z ) × ( R /L ₂ Z ) L ₁ > 0, L ₂ > 0. (1.1) For any function η : ω → (−1, ∞), we define (see Figure 1)

Ω(η) = {(x 1 , x 2 , x 3 ) ∈ ω × R | 0 < x 3 < 1 + η(x 1 , x 2 )} , Γ(η) = {(x 1 , x ₂ , x ₃ ) ∈ ω × R | x ₃ = 1 + η(x ₁ , x ₂ )} ,

Γ 0 = ω × {0}.

In particular

∂Ω(η) = Γ(η) ∪ Γ 0 . (1.2)

We consider the following system describing the evolution of the fluid governed by the incompressible Navier- Stokes equations, and the movement of the elastic plate







∂ _t U + (U · ∇)U − ∇ · T (U, P ) = 0 t > 0, x ∈ Ω(η(t, ·)),

∇ · U = 0 t > 0, x ∈ Ω(η(t, ·)),

∂ _tt η + α∆ ² η − κ∆η + ση − δ∆∂ _t η = H e η (U, P ) t > 0, s ∈ ω.

(1.3)

In the above system, we have denoted by U the fluid velocity, P the fluid pressure and η the transversal plate displacement.

The Cauchy stress tensor T (U, P ) is defined by

T (U, P ) = −P I 3 + 2νD(U ), D(U ) i,j = 1 2

∂U i

∂x _j + ∂U j

∂x _i

.

The function H e η is the fluid strain on the structure and is defined by H e ^η (U, P ) = − p

1 + |∇η| ² ( T (U, P )n · e 3 ) . We assume

ν > 0, α > 0, σ > 0, κ > 0 and δ > 0. (1.4)

These constants correspond respectively to the rigidity (α), the stretching (κ), the damping on the structure (δ) and the viscosity (ν).

We have denoted by n the unitary exterior normal of ∂Ω(η):

n = −e 3 on Γ 0 , and on Γ(η):

n(s, 1 + η(s)) = N(s, 1 + η(s))

|N(s, 1 + η(s))| , where N(s, 1 + η(s)) =





−∂ s

1

η(s)

−∂ _s

₂

η(s) 1



 , s ∈ ω. (1.5)

Here and in what follows, | · | denotes the Euclidian norm of R ^k , k > 1.

We complete (1.3) by the Navier slip boundary conditions. In order to write these boundary conditions, we need to introduce some notations. We denote by a _n and a _τ the normal and the tangential parts of a ∈ R ³ :

a _n = (a · n)n, a _τ = a − a _n = −n × (n × a) . (1.6) Then, our boundary conditions write as follows



 



 



U n = 0 t > 0, x ∈ Γ 0 , [2D(U )n] _τ + β 1 U τ = 0 t > 0, x ∈ Γ 0 , (U (t, s, 1 + η(t, s)) − ∂ t η(t, s)e 3 ) n = 0 t > 0, s ∈ ω, [2D(U )n] _τ (t, s, 1 + η(t, s)) + β 2 (U(t, s, 1 + η(t, s) − ∂ t η(t, s)e 3 ) τ = 0 t > 0, s ∈ ω.

(1.7)

In what follows, we write the above equations in the following more compact way



 



 



U n = 0 t > 0, x ∈ Γ 0 , [2νD(U )n + β 1 U ] _τ = 0 t > 0, x ∈ Γ 0 ,

(U − ∂ t ηe 3 ) n = 0 t > 0, x ∈ Γ(η), [2νD(U)n + β ₂ (U − ∂ _t ηe ₃ )] _τ = 0 t > 0, x ∈ Γ(η).

(1.8)

We assume that the friction coefficients β 1 and β 2 are constants satisfying β ₁ > 0, β ₂ > 0.

These boundary conditions can be compared with the standard no-slip boundary conditions usually considered with the Navier-Stokes system. In our case, these conditions would write as

U = 0 t > 0, x ∈ Γ 0 ,

U = ∂ t ηe 3 t > 0, x ∈ Γ(η). (1.9)

The Navier slip boundary condition was proposed by Navier in 1823 [28] and is relevant in several physical contexts, see for instance [24, 35, 22].

To complete the system (1.3),(1.8), we add the following initial conditions







η(0, ·) = η ⁰ in ω,

∂ _t η(0, ·) = η ¹ in ω, U (0, ·) = U ⁰ in Ω(η ⁰ ).

(1.10) Let us remark that we don’t need to consider boundary conditions on the “lateral” boundaries since we work with the torus ω (see (1.1) and (1.2)). This means that we are considering periodic boundary conditions for U , P and η:

U (t, x ₁ + L ₁ , x ₂ , x ₃ ) = U(t, x ₁ , x ₂ , x ₃ ), U(t, x ₁ , x ₂ + L ₂ , x ₃ ) = U (t, x ₁ , x ₂ , x ₃ ),

η(t, s 1 + L 1 , s 2 ) = η(t, s 1 , s 2 ), η(t, s 1 , s 2 + L 2 ) = η(t, s 1 , s 2 ),

and a similar relations for P .

Several works have been devoted to the study of the system (1.3), (1.10) with the Dirichlet boundary conditions (1.9): existence of strong solutions ([3], [23]), feedback stabilization ([30], [2]), global existence of strong solutions ([15]). Let us point out that in this latter work, the authors manage to obtain in particular that there is no contact between the plate and the bottom of the domain in finite time for the system (1.3),(1.9),(1.10).

This result, as previous works on fluid-structure interaction systems, shows that the standard no-slip boundary conditions may lead to some paradoxal results as the distance between two structures is going to 0: in the case of rigid bodies immersed into a viscous incompressible fluid, it is shown that in particular geometries there is no contact in finite time of two structures ([18], [19]) and in general, if there is contact, then it occurs with null relative velocity and null relative acceleration ([31]). In [9] and [10], the author considered boundary conditions involving the pressure. Here, our aim is to analyze the same system (1.3) with the Navier-slip boundary conditions (1.8) instead of the Dirichlet boundary conditions. Such a system was already considered in [17] and [27] where the existence of weak solutions is proved in dimension 2 (global existence as long as the deformable structure does not touch the fixed bottom). The uniqueness of weak solutions for this system has been obtained in [16].

Our objective is to prove the existence and uniqueness of strong solutions for small time or for small data.

This is the first work on strong solutions for such a system in the case of Navier-slip boundary conditions and to our knowledge, it is also the first work on strong solutions for this kind of systems in the 3D case.

In the case where the structures are rigid bodies immersed into a viscous incompressible fluid, several authors have already considered the Navier-slip boundary conditions: existence of weak solutions [29] and [12], existence of contact in finite time [13], existence of strong solutions and study of contacts in finite time [36], uniqueness of weak solutions [7]. Let us also mention the work of [8] where they consider a nonlinear boundary condition of Tresca’s type.

The main result of this article is Theorem 1.1.

1. Assume β i > 0 for i = 1, 2 and (1.4). Suppose η ⁰ ∈ H ³ (ω), η ¹ ∈ H ¹ (ω) and U ⁰ ∈ [H ¹ (Ω(η ⁰ ))] ³ such that 1 + η ⁰ > 0, ∇ · U ⁰ = 0 in Ω(η ⁰ ), (U ⁰ − η ¹ e ₃ ) _n = 0 on Γ(η ⁰ ), U _n ⁰ = 0 on Γ ₀ .

There exists a time T ₀ such that the system (1.3),(1.8), (1.10) admits a unique strong solution (U, P, η) on (0, T 0 ):

η ∈ L ² (0, T 0 ; H ⁴ (ω)) ∩ C ⁰ ([0, T 0 ]; H ³ (ω)) ∩ H ¹ (0, T 0 ; H ² (ω)) ∩ C ¹ ([0, T 0 ]; H ¹ (ω)) ∩ H ² (0, T 0 ; L ² (ω)), U ∈ L ² (0, T ₀ ; [H ² (Ω(η(t))] ³ ) ∩ C ⁰ ([0, T ₀ ]; [H ¹ (Ω(η(t)))] ³ ) ∩ H ¹ (0, T ₀ ; [L ² (Ω(η(t)))] ³ ),

∇P ∈ L ² (0, T 0 ; [L ² (Ω(η(t)))] ³ ).

2. Assume β i > 0 for i = 1, 2 with β 1 + β 2 > 0 and (1.4). There exist γ 0 > 0 and R 0 > 0 such that if η ⁰ ∈ H ³ (ω), η ¹ ∈ H ¹ (ω) and U ⁰ ∈ [H ¹ (Ω(η ⁰ ))] ³ satisfy

1 + η ⁰ > 0, ∇ · U ⁰ = 0 in Ω(η ⁰ ), (U ⁰ − η ¹ e 3 ) n = 0 on Γ(η ⁰ ), U _n ⁰ = 0 on Γ 0 . and

U ⁰

_[H

₁

_(Ω)]

₃

+ η ⁰

_H

₃

_(ω) + η ¹

_H

₁

_(ω) 6 R ₀ ,

then the system (1.3),(1.8), (1.10) admits a unique strong solution (U, P, η) on (0, ∞):

η ∈ L ² _γ (0, ∞; H ⁴ (ω)) ∩ BC _γ ⁰ ([0, ∞]; H ³ (ω)) ∩ H _γ ¹ (0, ∞; H ² (ω)) ∩ BC _γ ¹ ([0, ∞]; H ¹ (ω)) ∩ H _γ ² (0, ∞; L ² (ω)), U ∈ L ² _γ (0, ∞; [H ² (Ω(η(t))] ³ ) ∩ BC _γ ⁰ ([0, ∞]; [H ¹ (Ω(η(t)))] ³ ) ∩ H _γ ¹ (0, ∞; [L ² (Ω(η(t)))] ³ ),

∇P ∈ L ² _γ (0, ∞; [L ² (Ω(η(t)))] ³ ),

for γ ∈ [0, γ 0 ].

In the above statement, the spaces L ^p , H ^s are the classical Lebesgue, Sobolev spaces. We use the notation BC ⁰ = C ⁰ ∩ L

^∞

and BC ¹ = C ¹ ∩ W ^1,∞ . The notation · γ is explained below in (2.2), (2.3) and corresponds to an exponential decay of order γ. Finally, the notation L ² (0, T ; H ¹ (Ω(η(t)))) corresponds to the fact that the fluid velocity and pressure are written in a moving domain depending on η. To obtain our result, we thus need to use a change of variables for U and P and the fluid velocity and pressure after change of variables are obtained in spaces of the form L ² (0, T ; H ¹ (Ω)) with a fixed Ω. The precise definition of strong solutions is given in Section 3 (Definition 3.1) and we reformulate the above result in a more precise way in Theorem 6.1.

Remark 1.2. We can write a bi-dimensional version of the system (1.3),(1.8), (1.10) and for such a system, one can prove a similar result as Theorem 1.1. In fact, in that case, one could obtain a global in time existence of strong solutions up to a possible contact between the beam and the bottom of the domain by following the arguments in [15].

Remark 1.3. For the sake of simplicity in the proof of Theorem 1.1 and in the remaining part of this article, we assume κ = σ = 0 since these constants do not play any role in the analysis.

The plan of this paper is as follows: In Section 2, we give some notation. In Section 3, we remap the problem into a fixed domain using a change of variables like it was introduced in [21], and we restate Theorem 1.1. We obtain some regularity properties of the Stokes system in domains of class H ³ in Section 4. In Section 5, we study the linearized problem by writing it as an evolution equation. We prove in particular that the associated semigroup is analytic and in Section 6, we prove the main result using a fixed-point argument.

2 Notation

During the course of our analysis, we will use some functional spaces that we introduce in this section.

First, let us note that due to the incompressibility of the fluid and to the boundary conditions (1.8) 1 and (1.8) 3 , we have

d dt

Z

ω

η ds = 0.

For simplicity, we assume throughout the paper that Z

ω

η ⁰ ds = 0 so that

Z

ω

η(t, ·)ds = 0 (t > 0).

It yields to consider the following space L ² ₀ (ω) =

ξ ∈ L ² (ω) | Z

ω

ξds = 0

,

and the orthogonal projection M : L ² (ω) → L ² ₀ (ω). Applying M on the plate equation (1.3) 3 , we find

∂ _tt η + A ₁ η + A ₂ ∂ _t η = H η (U, P ), where

A ₁ η = α∆ ² η, D(A 1 ) = H ⁴ (ω) ∩ L ² ₀ (ω), A 2 η = −δ∆η, D(A 2 ) = H ² (ω) ∩ L ² ₀ (ω), and

H ^η (U, P ) = M ( H e ^η (U, P )).

The projection of (1.3) 3 onto L ² ₀ (ω)

^⊥

leads to impose the choice of the constant normalizing the pressure, see for instance [15].

We denote by H ^s (0, T ; X) the usual Sobolev spaces with values in a Banach space X. For s > 0, s / ∈ N , the norm of these spaces can be defined by using

bξc s,2,(0,T),X = Z

(0,T )×(0,T)

kξ(t) − ξ(t

⁰

)k ²

_X

|t − t

⁰

| ^2s+1 dtdt

⁰

! ^1/2 .

More precisely, the norm k.k _H

s

(0,T;X) for s ∈ (0, 1) is given by kξk _H

s

(0,T;X) =

kξk ² _L

2

(0,T ;X) + bξc ² s,2,(0,T),X

^1/2

. (2.1)

We recall (see [6]) that if s ∈ 1

2 , 1

, then the norm b·c s,2,(0,T),X is equivalent to the norm defined in (2.1) in the space {ξ ∈ H ^s (0, T ; X) | ξ(0) = 0}.

Let X ₁ , X ₂ be two Banach spaces endowed with the norm k.k

_X

1

respectively k.k

_X

2

. For s > 0, we define the following space

W ^s (0, T ; X ₁ , X ₂ ) =

v ∈ L ² (0, T ; X ₁ ) | v ∈ H ^s (0, T ; X ₂ ) , endowed with norm

k.k _W

s

(0,T ;X

1

,X

2

) = k.k _L

2

(0,T;X

1

) + k.k _H

s

(0,T;X

2

) . For s = 1, we will denote W ¹ (0, T ; X 1 , X 2 ) by W (0, T ; X 1 , X 2 ).

For γ > 0, we also consider the spaces

L ^p _γ (0, ∞; X ₁ ) = {v ∈ L ^p (0, ∞; X ₁ ) ; t 7→ v _γ (t) = e ^γt v(t) ∈ L ^p (0, ∞; X ₁ )}, p ∈ [1, +∞], (2.2) and

W _γ ^s (0, ∞; X ₁ , X ₂ ) = {v ∈ W ^s (0, ∞; X ₁ , X ₂ ) ; t 7→ v _γ (t) = e ^γt v(t) ∈ W ^s (0, ∞; X ₁ , X ₂ )}. (2.3) For these spaces, we use the norms defined by

kvk _L

p

γ

(0,∞;X

1

) = kv γ k _L

p

(0,∞;X

1

) , kvk _W

s

γ

(0,∞;X

1

,X

2

) = kv _γ k _W

s

(0,∞;X

1

,X

₂

) . In what follows, we set

Ω = Ω(η ⁰ ), (2.4)

for the local existence and

Ω = Ω(0), (2.5)

for the global existence.

In order to differentiate the normal or the normal and tangential component of a vector v in Ω and in Ω(t), we use the notation n ₀ , v _n

₀

and v _τ

₀

for the configuration Ω.

We denote by

D σ (Ω) = {φ ∈ [C ₀

^∞

(Ω)] ³ , div φ = 0}.

the space of infinitely differentiable functions with free divergence in Ω with compact support . Let us also define the following space

X T = W (0, T ; [H ² (Ω)] ³ , [L ² (Ω)] ³ ) × L ² (0, T ; H ¹ (Ω)/ R ) × W ² (0, T ; D(A 1 ), L ² ₀ (ω)), (2.6)

endowed with the norm k(u, p, η)k

_X

T

= kuk _{W (0,T;[H}

2

(Ω)]

³

,[L

²

(Ω)]

³

) + kuk _L

∞

(0,T ;[H

¹

(Ω)]

³

) + k∇pk _L

2

(0,T ,[L

²

(Ω)]

³

)

+ kηk _W

2

(0,T;D(A

1

),L

²₀

(ω)) + kηk _L

∞

(0,T;H

³

(ω)) + k∂ t ηk _L

∞

(0,T;H

¹

(ω)) . (2.7) If T = +∞ and γ > 0, we will write

X

_∞,γ

= W γ (0, ∞; [H ² (Ω)] ³ , [L ² (Ω)] ³ ) × L ² _γ (0, ∞; H ¹ (Ω)/ R ) × W _γ ² (0, ∞; D(A 1 ), L ² ₀ (ω)), (2.8) endowed with the norm

k(u, p, η)k

_X

∞,γ

= kuk _W

γ

(0,∞;[H

²

(Ω)]

³

,[L

²

(Ω)]

³

) + kuk _L

∞

γ

(0,∞;[H

¹

(Ω)]

³

) + k∇pk _L

2

γ

(0,∞,[L

²

(Ω)]

³

)

+ kηk _W

2

γ

(0,∞;D(A

₁

),L

²₀

(ω)) + kηk _L

∞

γ

(0,∞;H

³

(ω)) + k∂ t ηk _L

∞

γ

(0,∞;H

¹

(ω)) . (2.9) To write the boundary conditions, we also introduce the operator T defined as follows (see [2]):

T _η

0

ξ(y) =

0 if y ∈ Γ 0 ,

ξ(s)e 3 if y = (s, 1 + η ⁰ (s)) ∈ Γ(η ⁰ ).

We can verify that T _η

0

∈ L(L ² (ω); [L ² (∂Ω)] ³ ) and that T _η

^∗0

ζ = p

1 + |∇η ⁰ | ² ζ · e ₃ , ∀ζ ∈ [L ² (∂Ω)] ³ . We set

T = T η

⁰

M.

We also define

β =

β 1 if y ∈ Γ 0 , β 2 if y ∈ Γ(η ⁰ ).

3 Change of variables

For η ¹ , η ² ∈ H ³ (ω) with

η ¹ , η ² > −1 in ω, we can consider the change of variables X _η

1

,η

²

defined below

X _η

1

,η

²

: Ω(η ¹ ) −→ Ω(η ² ),



 y 1

y 2

y 3



 7−→









y 1

y 2

1 + η ² (y 1 , y 2 ) 1 + η ¹ (y ₁ , y ₂ ) y ₃









. (3.1)

The mapping X _η

1

,η

²

is invertible of inverse X _η

2

,η

¹

. Moreover, using the Sobolev embedding H ³ (ω) , → C ¹ (ω) and that

det(∇X η

¹

,η

²

) = 1 + η ² 1 + η ¹ , we deduce that X _η

1

,η

²

is a C ¹ -diffeomorphism from Ω(η ¹ ) onto Ω(η ² ).

In the case Ω = Ω(η ⁰ ) (see (2.4)), we set

X (t, ·) = X _η

0

,η(t,·) , Y (t, ·) = X _η(t,·),η

0

(3.2) and in the case Ω = Ω(0) (see (2.5)), we set

X(t, ·) = X _0,η(t,·) , Y (t, ·) = X _η(t,·),0 (3.3)

We have in both cases that Y (t, ·) = [X(t, ·)]

⁻¹

.

We consider the following transformation of u and p:

u(t, y) = (Cof ∇X(t, y))

^∗

U (t, X (t, y)), p(t, y) = P(t, X (t, y)) (t > 0, y ∈ Ω). (3.4) Here, (Cof ∇X (t, y))

^∗

denotes the transpose of (Cof ∇X (t, y)). After some standard calculations (see, for instance, [21]), the system (1.3), (1.8), (1.10) can be written as







∂ t u − ∇ · T (u, p) = F (u, p, η) t > 0, y ∈ Ω,

∇ · u = 0 t > 0, y ∈ Ω,

∂ _tt η + A ₁ η + A ₂ ∂ _t η = H η

⁰

(u, p) + H(u, η) t > 0,

(3.5) with the boundary conditions

[u − T ∂ _t η] _n

0

= 0 t > 0, y ∈ ∂Ω, [2νD(u)n 0 + β(u − T ∂ t η)] _τ

0

= G(u, η) t > 0, y ∈ ∂Ω, (3.6) and with the initial conditions







u(0, ·) = u ⁰ = U ⁰ in Ω, η(0, ·) = η ⁰ in ω,

∂ t η(0, ·) = η ¹ in ω.

(3.7) In order to write the nonlinearities F, H, G, we first set

(Cof ∇Y )

^∗

= (a _ik ) _ik . (3.8)

Then

F _i (u, p, η) = X

k

(δ _ik − a _ik (X))∂ _t u _k − X

l,k

a _ik (X) ∂u _k

∂y l

(X )∂ _t Y _l (X) − X

k

∂ _t a _ik (X)u _k

+ ν X

j,k,l,m

a ik (X) ∂Y m

∂x _j (X ) ∂Y l

∂x _j (X ) − δ ik δ mj δ jl

∂ ² u k

∂y _l ∂y _m

+ ν X

j,k,l

2 ∂a ik

∂x j

(X) ∂Y l

∂x j

(X ) + a ik (X ) ∂ ² Y l

∂x ² _j (X )

! ∂u k

∂y l

+ ν X

k

∂ ² a ik

∂x ² _j (X)u k

+ X

k

(δ _ki − ∂Y _k

∂x i

(X )) ∂p

∂y k

− X

k,l,j

a _kl (X) ∂a _ij (X )

∂x k

u _l u _j

+ X

k,l,j,m

δ ij δ kl δ km − a kl (X )a ij (X ) ∂Y m

∂x k

(X)

u l

∂u j

∂y m

, i = 1, 2, 3, (3.9) and

H (u, η) = νM

"

− X

j,k

∂a 3k

∂x j

(X ) + ∂a jk

∂x 3

(X)

u k N j + X

j,k,l

δ 3k δ jl (N 0 ) j − a 3k (X) ∂Y l

∂x j

(X )N j

∂u k

∂y l

+

δ _3l δ _jk (N ₀ ) _j − a _jk (X) ∂Y _l

∂x 3

(X )N _j ∂u _k

∂y l

#

. (3.10) To define G, we introduce the following notations.

τ ¹ =



 1 0

∂ _s

₁

η



 , τ ² =



 0 1

∂ _s

₂

η



 , (3.11)

W k = ν X

j,m

n j

∂a km

∂x j

(X)u m + ∂a jm

∂x k

(X )u m

+ β



 X

j

a kj (X )u j − T ∂ t η · e k





+ ν X

j,m,q

n j

a km (X ) ∂u m

∂y q

∂Y q

∂x j

(X ) + a jm (X) ∂u m

∂y q

∂Y q

∂x k

(X )

, k = 1, 2, 3, (3.12) and

V ⁱ = (2νD(u)n ₀ + β(u − T ∂ _t η)) · τ ₀ ⁱ − W · τ ⁱ , i = 1, 2. (3.13) Then G(u, η) is given by

G 1 (u, η) = V ¹ ((∂ s

₂

η ⁰ ) ² + 1) − V ² (∂ s

₁

η ⁰ ∂ s

₂

η ⁰ )

|N 0 | ² ,

G 2 (u, η) = V ² ((∂ s

₁

η ⁰ ) ² + 1) − V ¹ (∂ s

₁

η ⁰ ∂ s

₂

η ⁰ )

|N 0 | ² ,

G ₃ (u, η) = ∂ _s

₁

η ⁰ V ¹ + ∂ _s

₂

η ⁰ V ²

|N 0 | ² .

(3.14)

More precisely, let us note that

[2νD(U )n + β(U − T ∂η)] _τ = 0 t > 0, x ∈ ∂Ω(η) (3.15) writes as

(2νD(u)n 0 + β (u − T ∂ t η)) · τ ₀ ⁱ = V ⁱ , i = 1, 2. (3.16) The formula (3.14) for G is such that

G · τ ₀ ⁱ = V ⁱ , i = 1, 2, G · n 0 = 0 so that (3.16) is equivalent to the second condition of (3.6), with G tangential.

Using the above transformation, we can now introduce our definition of strong solutions for system (1.3),(1.8), (1.10)

Definition 3.1. The triplet (U, P, η) is a strong solution of (1.3),(1.8), (1.10) if the following conditions are satisfied

η ∈ W ² (0, T ; D(A 1 ), L ² ₀ (ω)), (D1)

1 + η > 0 in [0, T ], (D2)

X and Y are given by (3.2) and (u, p) are given by (3.4), (D3) (u, p) ∈ W (0, T ; [H ² (Ω)] ³ , [L ² (Ω)] ³ ) × L ² (0, T ; H ¹ (Ω)/ R ), (D4) (u, p, η) satisfies the system (3.5),(3.6), (3.7). (D5) Following this definition, in order to prove Theorem 1.1, we have to prove the existence and uniqueness of

(u, p, η) ∈ W (0, T ; [H ² (Ω)] ³ , [L ² (Ω)] ³ ) × L ² (0, T ; H ¹ (Ω)/ R ) × W ² (0, T ; D(A 1 ), L ² ₀ (ω))

solution of the system (3.5),(3.6), (3.7) and satisfying (D2).

4 Regularity properties of the Stokes system

In this section, we obtain some results on the stationary system in Ω(η) for η = η ⁰ (see (2.4)) or for η = 0 (see (2.5)):



 



 



αu − ν∆u + ∇p = f in Ω(η),

∇ · u = g in Ω(η), u _n = a on ∂Ω(η), [2νD(u)n + βu] _τ = b on ∂Ω(η).

(4.1)

Let define the following space

H _τ ¹ = {φ ∈ [H ¹ (Ω(η))] ³ | φ n = 0 on ∂Ω(η)}.

We give the definition of a weak solution of the system (4.1).

Definition 4.1. We say that (u, p) is a weak solution of (4.1) if (u, p) ∈ [H ¹ (Ω(η))] ³ × L ² (Ω(η))/ R and the following variational equation is satisfied:

α Z

Ω(η)

u · φ dy + 2ν Z

Ω(η)

D(u) : D(φ) dy − Z

Ω(η)

p∇ · φdy + Z

∂Ω(η)

βv · φdΓ = Z

Ω(η)

f · φdy + Z

∂Ω(η)

b · φdΓ,

for all φ ¹ ∈ H _τ ¹ .

We have the following result

Theorem 4.2. Assume β > 0 and α > 0 with β 1 + β 2 > 0 or α > 0. Let η ∈ H ³ (Ω(η)) and δ 0 > 0 such that 1 + η > δ 0 on ω. For any

f ∈ (H _τ ¹ )

⁰

, g ∈ L ² (Ω(η)), a ∈ [H ^1/2 (∂Ω(η))] ³ , b ∈ [H

^−1/2

(∂Ω(η))] ³ , such that

Z

Ω(η)

gdy = Z

∂Ω(η)

a · ndΓ, b · n = 0, (4.2)

there exists a unique weak solution (u, p) ∈ [H ¹ (Ω(η))] ³ × L ² ₀ (Ω(η)) to the Stokes system (4.1). Moreover, we have the following estimates:

kuk _[H

1

(Ω(η))]

³

+ k∇pk _[H

−1

(Ω(η))]

³

6 C f

_(H

₁

τ

)

⁰

+ kgk _L

2

(Ω(η)) + kak _[H

1/2

(∂Ω(η))]

³

+ b

_[H

_−1/2

_{(∂Ω(η))]}

₃

, (4.3) where C is a constant which depends on kηk _H

3

(ω) and δ 0 .

Moreover, if

f ∈ [L ² (Ω(η))] ³ , g ∈ H ¹ (Ω(η)), a ∈ [H ^3/2 (∂Ω(η))] ³ , b ∈ [H ^1/2 (∂Ω(η))] ³ ,

such that (4.2) holds, then (u, p) ∈ [H ² (Ω(η))] ³ × (H ¹ (Ω(η)) ∩ L ² ₀ (Ω(η))) and we have the following estimates:

kuk _[H

2

(Ω(η))]

³

+ k∇pk _[L

2

(Ω(η))]

³

6 C f

_[L

₂

_(Ω(η))]

₃

+ kgk _H

1

(Ω(η)) + kak _[H

3/2

(∂Ω(η))]

³

+ b

_[H

_1/2

_{(∂Ω(η))]}

₃

, (4.4) where C is a constant which depends on kηk _H

3

(ω) and δ 0 .

In the case where η ∈ C ^1,1 (ω) such a result is already known, see [1] (see also [4]). Here, we manage to

obtain the result for η ∈ H ³ (ω) by following an idea of [14] and [15].

Proof of Theorem 4.2. The proof follows closely the proof of Lemma 6 in [15]. We assume here β 1 + β 2 > 0, the proof is similar with α > 0.

First, we write the system (4.1) in the domain

Ω = Ω(0) by using the change of variables X 0,η defined by (3.1). Then we set

B _η = Cof(∇X _0,η ), A _η = 1

det(∇X 0,η ) B _η

^∗

B _η , and we define

u = u ◦ X _0,η , p = p ◦ X _0,η , (4.5)

f = det(∇X _0,η )f ◦ X _0,η , g = det(∇X _0,η )g ◦ X _0,η , a = a ◦ X 0,η , b i = B _η

⁻¹

(b i ◦ X 0,η ) · e i , i = 1, 2.

Then system (4.1) is transformed into the following system



 

 



 

 



−∇ · (∇uA η ) + B η ∇p = f in Ω,

∇ · (B

^∗

_η u) = g in Ω, (B _η

^∗

u) · n 0 = (B _η

^∗

a) · n 0 on ∂Ω, ν

|N |

(B

⁻¹

_η ∇uA η )n ₀ + 1

det(∇X 0,η ) ((∇u)

^∗

B _η )n ₀

+ βB

⁻¹

_η u

· e _i = b _i , i = 1, 2 on ∂Ω,

(4.6)

where N is defined by (1.5) and n 0 is the unit exterior normal to Ω (that is ±e 3 ).

Since η ∈ H ³ (ω), we deduce that

B _η , A _η ∈ H ² (ω; H ^s (0, 1)),

for all s > 0 and the corresponding norms depend on kηk _H

3

(ω) and δ ₀ . Moreover, using the embeddings H ¹ (ω) , → L ^p (ω) for all p > 1 and H ² (ω) , → L

^∞

(ω), we deduce that it is sufficient to prove that the solution of (4.6) satisfies

kuk _[H

2

(Ω)]

³

+ k∇pk _[L

2

(Ω)]

³

6 C

kf k _[L

2

(Ω)]

³

+ kgk _H

1

(Ω) + kak _[H

3/2

(∂Ω)]

³

+ kbk _[H

1/2

(∂Ω)]

³

. (4.7)

Step 1: Weak solutions. Let note that the solution of (4.6) verifies

∇ ·

1

det(∇X 0,η ) B _η (∇u)

^∗

B _η

= B _η ∇

∇ · (B _η

^∗

u) det(∇X 0,η )

= B _η ∇

g det(∇X 0,η )

. Let λ > 0 and consider the following system



 

 

 



 

 

 



−∇ · (∇uA η + 1

det(∇X _0,η ) B η (∇u)

^∗

B η ) + B η ∇p = f e in Ω, λp + ∇ · (B

^∗

_η u) = g in Ω, (B _η

^∗

u) · n ₀ = (B _η

^∗

a) · n ₀ on ∂Ω, ν

|N |

(B

⁻¹

_η ∇uA η )n 0 + 1

det(∇X 0,η ) ((∇u)

^∗

B η )n 0

+ βB

⁻¹

_η u

· e i = b i , i = 1, 2 on ∂Ω,

(4.8)

with

f e = f − B η ∇

g det(∇X 0,η )

.

To simplify the notations, we set

D η (u) = ∇uA η + 1

det(∇X 0,η ) B η (∇u)

^∗

B η . We define

V = {v ∈ [H ¹ (Ω)] ³ | (B

^∗

_η v) · n 0 = 0 on ∂Ω}.

We look for weak solutions to the system (4.8). Let f ∈ V

⁰

, g ∈ L ² (Ω), a ∈ [H ^1/2 (∂Ω)] ³ and b ∈ [H

^−1/2

(∂Ω)] ³ . We have B _η ∇

g det(∇X 0,η )

∈ V

⁰

:

B η ∇

g det(∇X 0,η )

, v

V

⁰

,V

= − Z

Ω

g

det(∇X 0,η ) ∇ · (B _η

^∗

v) dy.

Therefore f e ∈ V

⁰

and we multiply the first equation of (4.8) by v ∈ V and the second equation of (4.8) by ψ ∈ L ² (Ω) to obtain

Z

Ω

D _η (u) : ∇v dy − Z

Ω

p∇ · (B _η

^∗

v)dy + λ Z

Ω

p · ψ + (∇ · (B _η

^∗

u)) · ψdy + Z

∂Ω

|N |

ν βu · v dΓ

= h f , vi e V

⁰

,V + Z

Ω

g · ψdy +

b, |N|(det(∇X 0,η )

ν v

H

^−1/2

,H

^1/2

. (4.9) We consider a lifting w satisfying

∇ · (B

^∗

_η w) = g in Ω,

(B _η

^∗

w) · n 0 = (B _η

^∗

a) · n 0 on ∂Ω. (4.10) In order to this, we use [4, Corollary 8.2] and (4.2) to deduce the existence of w ∈ [H ¹ (Ω)] ³ such that

∇ · w = g in Ω, w · n 0 = (B _η

^∗

a) · n 0 on ∂Ω.

Then w = (B _η

^∗

)

⁻¹

w satisfies (4.10) and the estimate

kwk _[H

1

(Ω)]

³

6 C(kgk _L

2

(Ω) + kak _[H

1/2

(∂Ω)]

³

). (4.11) We set u = u b + w. Then, a couple (u, p) is a weak solution of the system (4.8) if and only if ( u, p) b verifies the following variational formulation

Z

Ω

D η ( u) : b ∇v dy − Z

Ω

p∇ · (B _η

^∗

v) dy + λ Z

Ω

p · ψ + (∇ · (B _η

^∗

u)) b · ψ dy + Z

∂Ω

|N |

ν β u b · v dΓ

= − Z

Ω

D η (w) : ∇v dy + h f , vi e V

⁰

,V +

b, |N |(det(∇X 0,η )

ν v

H

^−1/2

,H

^1/2

− Z

∂Ω

|N |

ν βw · v dΓ (v ∈ V, ψ ∈ L ² (Ω)). (4.12) We have that

Z

Ω

D _η (v) : ∇v dy = Z

Ω

∇vB _η

^∗

+ B η (∇v)

^∗

2

det(∇X 0,η ) dy, (4.13)

and writing

v = v ◦ X η,0 ,

we deduce

Z

Ω

∇vB _η

^∗

+ B _η (∇v)

^∗

2

det(∇X _0,η ) dy = Z

Ω(η)

|D(v)| ² dx, ∀v ∈ V, with v · n = 0 on ∂Ω(η). Applying a Korn inequality (see Proposition 4.5 below):

Z

Ω

D η (v) : ∇v dy + Z

∂Ω

|N|

ν β |v| ² dΓ > Ckvk _H

1

(Ω) (v ∈ V ). (4.14) Hence, we can apply the Lax-Milgram theorem and using (4.11), we deduce the existence of a unique solution of (u, p) = (u λ , p λ ) ∈ [H ¹ (Ω)] ³ × L ² (Ω) for (4.8) which verifies the estimates

kuk _[H

1

(Ω)]

³

+ λ kpk _L

2

(Ω) 6 C

kf k _V

0

+ kbk _[H

−1/2

(∂Ω)]

³

+ kgk _L

2

(Ω) + kak _[H

1/2

(∂Ω)]

³

. (4.15)

Taking ψ = 0 and v ∈ [H ₀ ¹ (Ω)] ³ in (4.9), we obtain Z

Ω

D η (u) : ∇v dy + Z

Ω

∇p · (B _η

^∗

v) dy = h f , vi e V

⁰

,V .

This shows that ∇p ∈ [H

⁻¹

(Ω)] ³ and using standard result (see, for instance [4, Proposition 1.1]), we deduce kpk _L

2

(Ω)/

R

6 C

kf k _V

0

+ kvk _[H

1

(Ω)]

³

+ kwk _[H

1

(Ω)]

³

. (4.16)

Then, combining (4.15), (4.16) and (4.11), we obtain the estimate independent of λ:

kuk _[H

1

(Ω)]

³

+ kpk _L

2

(Ω)/

R

6 C

kf k _V

0

+ kbk _[H

−1/2

(∂Ω)]

³

+ kgk _L

2

(Ω) + kak _[H

1/2

(∂Ω)]

³

. (4.17)

We can thus pass to the limit as λ → 0 in (4.8) to obtain a weak solution (u, p) of (4.6). Using the above coercivity argument we also deduce the uniqueness of the weak solution of (4.6).

Step 2: Strong solutions. We use an argument developed in [14] and [15]: if we approximate η by η _ε ∈ C ^1,1 (ω), and the corresponding u _ε , p _ε are H ² and H ¹ . We show below that their norms depend only on the H ³ norm of η _ε so that we can pass to the limit. To simplify, we do not write any ε below.

We first differentiate system (4.6) with respect to y ₁ and y ₂ to obtain a similar problem as (4.6) with source and boundary terms corresponding to the differentiates of f , g, a and b and to terms coming from the A η and B η . We only need to estimate these terms, that is

k∇ · (∇u∂ y

_i

A η ) − ∂ y

_i

B η ∇pk V

⁰

, k∇ · (∂ y

_i

B _η

^∗

u)k _L

2

(Ω) , kB _η

⁻¹

∂ y

_i

B

^∗

_η uk _[H

1/2

(∂Ω)]

³

, k∂ y

_i

B _η

⁻¹

∇uA η k _[H

−1/2

(∂Ω)]

⁹

, kB _η

⁻¹

∇u∂ y

_i

A η k _[H

−1/2

(∂Ω)]

⁹

,

∂ y

_i

1

det(∇X 0,η ) (∇u)

^∗

B η

_[H

_−1/2

_(∂Ω)]

₉

,

1

det(∇X 0,η ) (∇u)

^∗

∂ y

_i

B η

_H[

_−1/2

_(∂Ω)]

₉

.

Here we use a nice idea proposed in [14] and [15]: we estimate the above terms by using the H ² regularity of u and the H ¹ regularity of p. More precisely, using the embeddings H ^1/2 (ω) ⊂ L ⁴ (ω) and H ^1/4 (ω) ⊂ L ^8/3 (ω), we deduce that the above terms are estimated by

kηk _H

3

(ω)

kuk ^1/4 _[H

1

(Ω)]

³

kuk ^3/4 _[H

2

(Ω)]

³

+ kpk ^1/4 _L

2

(Ω) kpk ^3/4 _H

1

(Ω)

. (4.18)

Using the first part of this proof and in particular (4.17), we obtain for i = 1, 2 k∂ y

i

uk _[H

1

(Ω)]

³

+ k∂ y

i

pk _L

2

0

(Ω) 6 C

kf k _[L

2

(Ω)]

³

+ kbk _[H

1/2

(∂Ω)]

³

+ kgk _H

1

(Ω) + kak _[H

3/2

(∂Ω)]

³

+ Ckηk H

³

(ω)

kuk ^1/4 _[H

1

(Ω)]

³

kuk ^3/4 _[H

2

(Ω)]

³

+ kpk ^1/4 _L

2

(Ω) kpk ^3/4 _H

1

(Ω)

. (4.19)

We differentiate (4.6) 2 with respect to y 3 , we obtain −y ₃ ∂ _y

₁

η∂ _y ²

3

u ₁ − y ₃ ∂ _y

₂

η∂ ² _y

3

u ₂ + ∂ ² _y

3

u ₃

L

²

(Ω) 6 C

kf k _[L

2

(Ω)]

³

+ kbk _[H

1/2

(∂Ω)]

³

+ kgk _H

1

(Ω) + kak _[H

3/2

(∂Ω)]

³

+ Ckηk _H

3

(ω)

kuk ^1/4 _[H

1

(Ω)]

³

kuk ^3/4 _[H

2

(Ω)]

³

+ kpk ^1/4 _L

2

(Ω) kpk ^3/4 _H

1

(Ω)

. (4.20) Then, going back to (4.6) 1 , we also obtain

A 33 ∂ ² _y

₃

u 1 − y 3 ∂ y

₁

η∂ y

₃

p

L

²

(Ω) +

A 33 ∂ _y ²

₃

u 2 − y 3 ∂ y

₂

η∂ y

₃

p

L

²

(Ω) +

A 33 ∂ _y ²

₃

u 3 + ∂ y

₃

p L

²

(Ω)

6 C

kf k _[L

2

(Ω)]

³

+ kbk _[H

1/2

(∂Ω)]

³

+ kgk _H

1

(Ω) + kak _[H

3/2

(∂Ω)]

³

+ Ckηk _H

3

(ω)

kuk ^1/4 _[H

₁

_(Ω)]

₃

kuk ^3/4 _[H

₂

_(Ω)]

₃

+ kpk ^1/4 _L

₂

_(Ω) kpk ^3/4 _H

₁

_(Ω)

. (4.21) Since A ₃₃ = 1

1 + η (1 + (y ₃ ∂ _y

₁

η) ² + (y ₃ ∂ _y

₂

η) ² ) > 0, we deduce ∂ _y ²

₃

u

_L

₂

_(Ω)

₃

+ k∂ y

₃

pk _L

2

(Ω) 6 C

kf k _[L

2

(Ω)]

³

+ kbk _[H

1/2

(∂Ω)]

³

+ kgk _H

1

(Ω) + kak _[H

3/2

(∂Ω)]

³

+ Ckηk H

³

(ω)

kuk ^1/4 _[H

1

(Ω)]

³

kuk ^3/4 _[H

2

(Ω)]

³

+ kpk ^1/4 _L

2

(Ω) kpk ^3/4 _H

1

(Ω)

. Combining this with (4.19), we deduce the result.

We also need the following theorem which is proved in [32].

Theorem 4.3. Assume β > 0 with β 1 + β 2 > 0. Let η ∈ H ³ (ω) and δ 0 > 0 such that 1 + η > δ 0 on ω. Let us consider the following non stationary Stokes system:



 

 



 

 



∂ _t v − ∇ · T (v, π) = 0 t > 0, y ∈ Ω,

∇ · v = 0 t > 0, y ∈ Ω, v n

₀

= 0 t > 0, y ∈ ∂Ω, [2νD(v)n 0 + βv] _τ

0

= e g t > 0, y ∈ ∂Ω, v(0, ·) = 0 y ∈ Ω.

(4.22)

There exists γ 0 > 0 such that if

e g ∈ W _γ ^1/4 (0, ∞; [H ^1/2 (∂Ω)] ³ , [L ² (∂Ω)] ³ ), e g n

₀

= 0. (4.23) for some γ ∈ [0, γ 0 ]. Then the problem (4.22) admits a unique solution which satisfies the estimate

kvk ² _W

γ

(0,∞,[H

²

(Ω)]

³

,[L

²

(Ω)]

³

) + k∇πk ² _L

2

γ

(0,∞;[L

²

(Ω)]

³

) 6 C k e gk ² _W

1/4

γ

(0,∞;[H

^1/2

(∂Ω)]

³

,[L

²

(∂Ω)]

³

) (4.24) where C is a positive constant.

We recall that the spaces W _γ ^s (0, ∞; X 1 , X 2 ) and L ² _γ (0, ∞; L ² (Ω)) are defined by (2.3), (2.2).

Remark 4.4. In [32], the author assumes that η is more regular but such an assumption is only used to obtain a lift of the boundary condition by taking a stationary Stokes system of the form (4.1), see relation (75) in [32].

Note also that in [32], the condition (4.23) is replaced by the equivalent condition e g ∈ W _γ ^1/2 (0, ∞; [H ¹ (Ω)] ³ , [L ² (Ω)] ³ ), e g n

₀

= 0.

Such an equivalence can be obtained by using the surjectivity of the trace operator (see [25, p.21, Theorem 2.3]).

We end this section by proving a Korn’s type inequality (that we used in the above proof).

Proposition 4.5. Assume η ∈ W ^1,∞ (ω). Assume that β 1 + β 2 6= 0. There exists a positive constant C > 0, such that

kuk _[H

1

(Ω(η))]

³

6 C

kD(u)k _[L

2

(Ω(η))]

⁹

+

p βu

_[L

₂

_{(∂Ω(η))]}

₃

, (4.25)

for all u ∈ [H ¹ (Ω(η))] ³ .

Proof. We first show by contradiction that kuk _[L

2

(Ω(η))]

³

6 C

kD(u)k _[L

2

(Ω(η))]

⁹

+

p βu

_[L

₂

_{(∂Ω(η))]}

₃

. (4.26)

Assume u k ∈ [H ¹ (Ω(η))] ³ with

ku k k _[L

2

(Ω(η))]

³

= 1, (4.27)

and

kD(u k )k _[L

2

(Ω(η))]

⁹

+

p βu k

_[L

₂

_{(∂Ω(η))]}

₃

→ 0.

Using the classical Korn inequality (see, for instance, [20]), the above relations imply that (u k ) k converges weakly to some u ∈ [H ¹ (Ω(η))] ³ with D(u) = 0 and p

βu = 0 on ∂Ω(η). In particular, see [34, Lemma 1.1 p.18], there exist a, b ∈ R ³ , such that for any y ∈ Ω(η), u(y) = a + b ∧ y. Using that

u(y + L ₁ e ₁ ) = u(y), u(y + L ₂ e ₂ ) = u(y), (y ∈ Ω(η)), we deduce that b = 0, then u = a in Ω(η). Since p

βu = 0 on ∂Ω(η), we obtain that u = 0 in Ω(η). Up to a subsequence u k → u strongly in [L ² (Ω(η))] ³ and thus from (4.27), we get kuk _[L

2

(Ω(η))]

³

= 1 which leads to a contradiction. In order to prove (4.25), we combine (4.26) and the classical Korn inequality (using that Ω(η) is Lipschitz continuous).

5 Linear System

Let us consider a linearized system of (3.5), (3.6), (3.7):







∂ t u − ∇ · T (u, p) = f t > 0, y ∈ Ω,

∇ · u = 0 t > 0, y ∈ Ω,

∂ tt η + A 1 η + A 2 ∂ t η = −T

^∗

( T (u, p)n 0 ) + h t > 0,

(5.1) with the boundary conditions

[u − T ∂ t η] _n

0

= 0 t > 0, y ∈ ∂Ω, [2νD(u)n ₀ + β(u − T ∂ _t η)] _τ

0

= e g t > 0, y ∈ ∂Ω, (5.2) and with the initial conditions







u(0, ·) = u ⁰ in Ω, η(0, ·) = η ⁰ in ω,

∂ t η(0, ·) = η ¹ in ω.

(5.3) Let us consider (v, π) the solution of (4.22) associated with e g. Then w = u − v and q = p − π satisfy







∂ _t w − ∇ · T (w, q) = f t > 0, y ∈ Ω,

∇ · w = 0 t > 0, y ∈ Ω,

∂ _tt η + A ₁ η + A ₂ ∂ _t η = −T

^∗

( T (w, q)n ₀ ) − T

^∗

( T (v, π)n ₀ ) + h t > 0,

(5.4) with the boundary conditions

[w − T ∂ t η] _n

0

= 0 t > 0, y ∈ ∂Ω, [2νD(w)n 0 + β(w − T ∂ t η)] _τ

0

= 0 t > 0, y ∈ ∂Ω, (5.5)

and with the initial conditions







w(0, ·) = u ⁰ in Ω, η(0, ·) = η ⁰ in ω,

∂ t η(0, ·) = η ¹ in ω.

(5.6) To solve (5.4)-(5.6), we use a semigroup approach. We endow the space [L ² (Ω)] ³ × D(A ^1/2 ₁ ) × L ² ₀ (ω) with the scalar product

* 

 w η 1

η 2



 ,



 v ξ 1

ξ 2



 +

= hw, vi _[L

2

(Ω)]

³

+ D

A ^1/2 ₁ η 1 , A ^1/2 ₁ ξ 1

E

L

²

(ω) + hη 2 , ξ 2 i _L

2

(ω) . We consider the following functional spaces

H = n

(w, η 1 , η 2 ) ∈ [L ² (Ω)] ³ × D(A ^1/2 ₁ ) × L ² ₀ (ω) | ∇ · w = 0 in Ω, [w − T η 2 ] _n

0

= 0 on ∂Ω o ,

V =

[H ¹ (Ω)] ³ × D(A ^3/4 ₁ ) × D(A ^1/4 ₁ )

∩ H . (5.7)

We also denote by P the orthogonal projector

P : [L ² (Ω)] ³ × D(A ^1/2 ₁ ) × L ² ₀ (ω) −→ H . Finally, we define

D(A) = n

(w, η 1 , η 2 ) ∈

[H ² (Ω)] ³ × D(A 1 ) × D(A ^1/2 ₁ )

∩ V | [2νD(w)n 0 + β(w − T η 2 )] _τ

0

= 0 on ∂Ω o , (5.8)

A



 w η 1

η 2



 =





−ν∆w

−η 2

A 1 η 1 + A 2 η 2 + T

^∗

(2νD(w)n 0 )



 , (5.9)

and

D(A) = D(A), A = P A. (5.10)

Using the above definition, we can write (5.4)–(5.6) as

W

⁰

+ AW = P F, W (0) = W ⁰ , (5.11)

with

W =



 w η

∂ t η



 , F =



 f 0 h



 .

Proposition 5.1. Assume that β 1 +β 2 6= 0. The operator A defined by (5.8)–(5.10) is the infinitesimal generator of a strongly continuous semigroup of contraction on H .

Proof. First we show that the operator A is dissipative: assume W =



 w η 1

η 2



 ∈ D(A). Then, by integration by parts, we obtain:

hAW, W i = hAW, W i = 2ν Z

Ω

|D(w)| ² dy − Z

∂Ω

2νD(w)n ₀ · [w − T (η ₂ )] dΓ + Z

ω

A ^1/2 ₂ η ₂

2

ds.

We write

− Z

∂Ω

2νD(w)n 0 · [w − T (η 2 )] dΓ = − Z

∂Ω

2ν [D(w)n 0 ] τ

₀

· [w − T (η 2 )] τ

₀

dΓ = Z

∂Ω

β |[w − T (η 2 )] τ

₀

| ² dΓ,

and we deduce

hAW, W i = 2ν Z

Ω

|D(w)| ² dy + Z

ω

A ^1/2 ₂ η 2

2

ds + Z

∂Ω

β|[w − T (η 2 )] τ

₀

| ² dΓ > 0.

Second, we show that the operator A is m-dissipative: we prove that for some λ > 0 the operator λI + A is onto. Let F =



 f g h



 ∈ H . The problem is to find



 w η 1

η 2



 ∈ D(A) solution of the equation

(λI + A)



 w η 1

η 2



 = F, (5.12)

which is equivalent to the system

λw − ∇ · T (w, q) = f in Ω, (5.13a)

∇ · w = 0 in Ω, (5.13b)

λη ₁ − η ₂ = g on ω, (5.13c)

λη 2 + A 1 η 1 + A 2 η 2 = −T

^∗

( T (w, q)n 0 ) + h on ω, (5.13d)

[w − T η 2 ] n

₀

= 0 on ∂Ω, (5.13e)

[2νD(w)n ₀ + β(w − T η ₂ )] _τ

0

= 0 on ∂Ω. (5.13f)

To solve the above system, we use that η ₁ = 1

λ (g + η ₂ ) to obtain a system in (u, η ₂ ) and we introduce the space V = n

(φ, ξ) ∈ [H ¹ (Ω)] ³ × D(A ^1/2 ₁ ) | ∇ · φ = 0 in Ω, [φ − T ξ] n

₀

= 0 on ∂Ω o . We can thus write the equation (5.12) in a variational form: find (w, η ₂ ) ∈ V such that

a w

η 2

,

φ ξ

= L φ

ξ

φ ξ

∈ V

, (5.14)

with a : V × V −→ R given by a

w η 2

,

φ ξ

= λ Z

Ω

w · φ dy + 2ν Z

Ω

D(w) : D(φ) dy + λ Z

ω

η 2 · ξ ds + Z

ω

(A 2 η 2 ) · ξ ds + 1

λ Z

ω

(A ^1/2 ₁ η ₂ ) · (A ^1/2 ₁ ξ) ds + Z

∂Ω

β[w − T (η ₂ )] _τ

₀

· [φ − T (ξ)] _τ

₀

dΓ, and L : V −→ R given by

L φ

ξ

= Z

Ω

f · φ dy + Z

ω

h · ξ ds − 1 λ

Z

ω

(A ^1/2 ₁ g) · (A ^1/2 ₁ ξ) ds.

The bilinear form a is continuous and coercive on V thanks to the classical Korn inequality. We can also check that L is linear and continuous on V. By the Lax-Milgram theorem, there exists a unique (u, η 2 ) ∈ V solution of (5.14).

Now, taking ξ = 0 and φ ∈ D σ (Ω), the equation (5.14) becomes λ

Z

Ω

w · φ dy + 2ν Z

Ω

D(w) : D(φ) dy = Z

Ω

f · φ dy, which is equivalent to

hλw − ν∆w − f, φi = 0, ∀φ ∈ D σ (Ω).

Using the De Rham theorem [33, Proposition 1.2, p.14] , we deduce the existence of a unique q ∈ L ² (Ω)/ R such that (5.13a) holds. In particular, we have ∇ · T (w, q) ∈ [L ² (Ω)] ³ and T (w, q) ∈ [L ² (Ω)] ⁹ . Therefore, we deduce that T (w, q)n 0 ∈ [H

^−1/2

(∂Ω)] ³ and

Z

Ω

T (w, q) : D(φ)dy − h T (w, q)n ₀ , φi _H

−1/2

,H

^1/2

= Z

Ω

(f − λw) · φdy, (5.15)

for all φ ∈ [H ¹ (Ω)] ³ , ∇ · φ = 0, φ _n

₀

= 0. On the other hand, taking ξ = 0 in (5.14) yields λ

Z

Ω

w · φ dy + 2ν Z

Ω

D(w) : D(φ) dy + hβ[w − T (η 2 )] τ

₀

, φ i _H

−1/2

,H

^1/2

= Z

Ω

f · φ dy, (5.16) for all φ ∈ [H ¹ (Ω)] ³ , ∇ · φ = 0, φ n

0

= 0. Comparing (5.15) and (5.16) and taking into account that

Z

Ω

T (w, q) : D(φ)dy = 2ν Z

Ω

D(w) : D(φ)dy, ∀φ ∈ [H ¹ (Ω)] ³ , ∇ · φ = 0, φ n

₀

= 0, we obtain

− h T (w, q)n 0 , φi _H

−1/2

,H

^1/2

= h[β(w − T η 2 )] τ

₀

, φi _H

−1/2

,H

^1/2

= 0, ∀φ ∈ [H ¹ (Ω)] ³ , ∇ · φ = 0, φ n

₀

= 0. (5.17) Let φ ∈ [H ^1/2 (∂Ω)] ³ such that φ n

₀

= 0, and let consider the system







−∇ · T ( b g, q) = 0 b in Ω,

∇ · b g = 0 in Ω, b g = φ on ∂Ω.

The above system admits a unique solution ( b g, q) b ∈ [H ¹ (Ω)] ³ × L ² ₀ (Ω) such that ∇ · b g = 0 and b g| ∂Ω = φ. This implies that (5.17) holds for all φ ∈ [H ¹ (Ω)] ³ , φ n

₀

= 0. Inserting (5.17) in (5.15) we get

Z

Ω

2νD(w) : D(φ)dy − Z

Ω

q∇ · φdy + hβ (w − T η ₂ ) _τ

₀

, φ _τ

₀

i _H

−1/2

,H

^1/2

= Z

Ω

(f − λw) · φdy, (5.18) for all φ ∈ [H ¹ (Ω)] ³ , φ n

₀

= 0 .

Thus, we deduce that (w, q) is a weak solution of (5.13a), (5.13b), (5.13e) and (5.13f) in the sense of Definition 4.1. Since η 2 ∈ H ² (ω), T η 2 ∈ [H ² (∂Ω)] ³ we can apply Theorem 4.2 and obtain (w, q) ∈ [H ² (Ω)] ³ × H ¹ (Ω)/ R .

Going back to the variational formulation (5.14), we deduce Z

ω

(A ^1/2 ₁ η ₁ ) · (A ^1/2 ₁ ξ) ds = −λ Z

ω

η ₂ · ξ ds − Z

ω

(A ₂ η ₂ ) · ξ ds − Z

ω

T

^∗

( T (u, q)n ₀ ) · ξ ds + Z

ω

h · ξ ds,

for any ξ ∈ D(A ^1/2 ₁ ) and where η 1 = 1

λ (g + η 2 ). We have T (w, q)n 0 ∈ [H ^1/2 (∂Ω)] ³ and thus T

^∗

( T (w, q)n 0 ) ∈ L ² ₀ (ω). Moreover since η 2 ∈ H ² (ω), we deduce that η 2 ∈ D(A 2 ). Thus A 1 η 1 ∈ L ² ₀ (ω).

Applying Lumer-Phillips theorem, we conclude that (e

^−tA

) t

>

0 is a semigroup of contractions on H .

In order to prove that (e

^−tA

) _t

_>

₀ is an analytical semigroup, we use Lemma 3.10 in [2]. We first need to show that (e

^−tA

) _t

_>

₀ is exponentially stable.

Proposition 5.2. Assume that β ₁ + β ₂ 6= 0. The semigroup (e

^−tA

) _t

_>

₀ is exponentially stable.

Proof. Since (e

^−tA

) t

>

0 is a semigroup of contraction, we apply the classical result of Huang-Gearhart (see for instance [26, Theorem 1.3.2, p.4]). We have to show that

i R ⊂ ρ(A) and sup

λ∈

R

(iλI + A)

⁻¹

< ∞.