Imene Aicha Djebour
email: imene.djebour@univ-lorraine.fr

Local boundary feedback stabilization of a uid-structure interaction problem under Navier slip boundary conditions with time delay

Keywords: Navier-Stokes system, damped beam equation, strong solutions, stabilization, time delay, Navier boundary conditions 2010 Mathematics Subject Classication. 35Q30, 76D05, 76D03, 74F10

We consider a uid-structure interaction system composed by a three-dimensional viscous incompressible uid and an elastic plate located on the upper part of the uid boundary. The uid motion is governed by the Navier-Stokes system whereas the structure displacement satises the damped plate equation. We consider here the Navier slip boundary conditions. The main result of this work is the feedback stabilization of the strong solutions of the corresponding system around a stationary state for any exponential decay rate by means of a time delayed control localized on the xed uid boundary. This work relies on the Fattorini-Hautus criterion. Then, the main tool in this work is to show the unique continuation property of the associate solution to the adjoint system.

Introduction

We suppose that the uid ow occupies a 3D periodic domain and an elastic plate that is disposed on the upper part of the uid boundary. The force exerted by the uid on the plate inuences the deformation of the elastic structure and then the uid domain depends on the plate displacement and hence on time. Consequently, the interaction between the deformable structure and the uid is modeled by a strongly coupled non linear system set in a moving domain. Our aim is to stabilize the position and the velocity of the structure as well as the velocity of the uid around a stationary state using a time delayed control that acts on a local part of the boundary of the uid domain provided that the initial data are close enough to the stationary state in some norm. First, we give some important notations in order to write the system.

Let ω be the rectangular torus

ω = (R/L 1 Z) × (R/L 2 Z) L 1 > 0, L 2 > 0.
For any function η : ω → (-1, ∞), we dene

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}.
(see the gure 1). In particular, we have ∂Ω(η) = Γ(η) ∪ Γ 0 .

x 1

x 2 We consider the following system describing the evolution of the uid that is governed by the Navier-Stokes equations and the displacement of the elastic plate by the damped beam equation

x 3 η Γ(η) Γ 0 L 1 L 2 1
   ∂ t U + (U • ∇)U -∇ • T(U, P ) = f S in (0, ∞) × Ω(η),
∇ • U = 0 in (0, ∞) × Ω(η), ∂ tt η + α∆ 2 η -δ∆∂ t η = H η (U, P ) + h S in (0, ∞) × ω.

(1. [START_REF] Badra | Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system[END_REF] In (1.1), we have denoted by U the uid velocity, P the uid pressure and η the transversal displacement of the elastic structure. The functions f S and h S are time independent data.

The Cauchy stress tensor T(U, P ) is given by T(U, P ) = -P I 3 + 2νD(U ), D(U ) i,j = 1 2

∂U i ∂x j + ∂U j ∂x i .
The coecients α, δ and ν correspond respectively to the rigidity, the damping on the structure and the uid viscosity.

We denote by n the unit exterior normal vector on ∂Ω(η) given as follows:

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)) =   -∂ s1 η(s) -∂ s2 η(s) 1   , s ∈ ω.
Here and in what follows, | • | stands for the Euclidean norm of R k , k 1. Also, we select two tangent vectors τ i , i = 1, 2 linearly independent on ∂Ω(η) such that

τ i = e i , i = 1, 2 on Γ 0 , (1.2)
and

τ 1 (s, 1 + η(s)) =   1 0 ∂ s1 η(s)   , τ 2 (s, 1 + η(s)) =   0 1 ∂ s2 η(s)   on Γ(η).
(1.

3)

The function H η is the contact force exerted by the uid on the interface which is dened by

H η (U, P ) = -1 + |∇η| 2 (T(U, P ) n • e 3 ) .
We complete (1.1) by the Navier slip boundary conditions. Let a ∈ R 3 , we denote by a n and a τ the normal and the tangential components of a :

a n = (a • n) n, a τ = a -a n = -n × ( n × a) .
(1.4)

The Navier slip boundary conditions write as follows:

       U n = (Mv) n on (0, ∞) × Γ 0 , [2νD(U ) n + β 1 U ] τ = β 1 (Mv) τ on (0, ∞) × Γ 0 ,
(U -∂ t ηe 3 ) n = 0 on (0, ∞) × Γ(η), [2νD(U ) n + β 2 (U -∂ t ηe 3 )] τ = 0 on (0, ∞) × Γ(η).

(1.5) Here, v is the control of the system (1.1), (1.5) acting on the xed boundary Γ 0 . In order to preserve the compatibility condition due to the incompressibility of the uid, we use the operator M dened by

Mv = mv - Γ0 mv • n dΓ m n, (1.6)
where m ∈ C 2 (Γ 0 ) is compactly supported in Γ 0 which satises Γ0 m ds = 1, see for example [START_REF] Badra | Stabilization of parabolic nonlinear systems with nite dimensional feedback or dynamical controllers: application to the Navier-Stokes system[END_REF], [START_REF] Raymond | Feedback boundary stabilization of the two-dimensional Navier-Stokes equations[END_REF].

Thus, the operator M localizes the action of the control in a relatively compact subset of Γ 0 . We notice that M ∈ L([L 2 (Γ 0 )] 3 ) and Γ0 Mv • n dΓ = 0.

(1.7)

We assume that the friction coecients β 1 and β 2 are two non negative constants

β 1 0, β 2 0.
Since ω is a rectangular torus, we complement the system (1.1), (1.5) with periodic boundary conditions for the uid and for the elastic plate on the remaining boundaries of Ω(η). Then, we consider data and solutions which are periodic in the both directions e 1 and e 2 , for example :

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 ).
Since U is divergence free and taking into account the equation (1.7), the velocity ∂ t η should satisfy the condition ω ∂ t η ds = 0.

(1.8)

Integrating the plate equation on ω and using (1.8) and the fact that η is periodic, we nd ω H η (U, P ) + h S ds = 0.

(1.9)

The above condition can be satised if we write

P = P 0 + c, (1.10)
where P 0 satises Ω(η)

P 0 dx = 0,
and c is a constant that is chosen conveniently in such a way (1.9) is veried. Consequently, the pressure P will be uniquely dened. In order to impose the condition (1.9) without considering the normalizing constant c, we dene the projection operator M on L 2 0 (ω) where

L 2 0 (ω) = η ∈ L 2 (ω) ; ω η ds = 0 . (1.11)
Since we look for solutions such that (1.8) is veried, it is convenient to consider the restriction of the associated semigroup to the plate equation to L 2 0 (ω). Then, in what follows, we assume that η ∈ L 2 0 (ω). To impose the condition (1.9), we substitute the plate equation by its projection on L 2 0 (ω) using the operator M ∂ tt η + α∆ 2 η -δ∆∂ t η = H η (U, P ) + M h S , where H η (U, P ) = M H η (U, P ).

We complement the system (1.1), (1.5) with the following initial conditions

   U (0, •) = U 0 in Ω(η 0 ), η(0, •) = η 0 in ω, ∂ t η(0, •) = η 1 in ω.
(1.12) Let (w S , p S , η S ) be a stationary state of the system (1.1), (1.5) that is a solution of the system

           (w S • ∇)w S -∇ • T(w S , p S ) = f S in Ω(η S ), ∇ • w S = 0 in Ω(η S ), α∆ 2 η S = H η S (w S , p S ) + M h S in ω, w S n = 0 on ∂Ω(η S ), 2νD(w S )n + βw S τ = 0 on ∂Ω(η S ), (1.13)
where

β(y) = β 1 if y ∈ Γ 0 , β 2 if y ∈ Γ(η S ).
The vector n stands for the unitary exterior normal vector and τ i , i = 1, 2 designate two tangent vectors on ∂Ω(η S ):

n = -e 3 , τ i = e i , i = 1, 2 on Γ 0 , (1.14)
and on Γ(η S ):

n(s, 1 + η S (s)) = N (s, 1 + η S (s)) |N (s, 1 + η S (s))| where N (s, 1 + η S (s)) =   -∂ s1 η S (s) -∂ s2 η S (s) 1   s ∈ ω, (1.15
)

τ 1 (s, 1 + η S (s)) =   1 0 ∂ s1 η S (s)   , τ 2 (s, 1 + η S (s)) =   0 1 ∂ s2 η S (s)   .
(1.16)

Our aim is to stabilize the solution (U, P, η) of (1.1), (1.5), (1.12) around the stationary state (w S , p S , η) that is a solution of (1.13) by means of a feedback boundary control v(t) that appears in (1.5) depending on the state (U, η). In practice, due to a calculus time issues, we include a time delay t 0 > 0 in the control and the main goal is to construct this feedback control in such a way it depends at time t on the values of the state (U (t ), η(t )) for t t -t 0 and the associated strong solution (U, P, η) to the system (1.1), (1.5), (1.12) goes to (w S , p S , η S ) exponentially. This context of stabilization has an important interest in the control theory of PDE: The analysis of the eect of the time delay on the feedback stabilization of some partial dierential equations is given for instance in [START_REF] Datko | An example on the eect of time delays in boundary feedback stabilization of wave equations[END_REF] and [START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF]. Recently, in [START_REF] Bresch-Pietri | New formulation of predictors for nite-dimensional linear control systems with input delay[END_REF] a feedback control is constructed for a nite dimensional system with input delay and in [START_REF] Prieur | Feedback stabilization of a 1-D linear reaction-diusion equation with delay boundary control[END_REF] the authors manage to obtain a stabilizing feedback boundary control to the one-dimensional reaction-diusion system considering a constant delay, their method relies essentially on the Arstein transform and the fact that the generator operator of the system is self-adjoint. We mention also the work [START_REF] Lhachemi | Boundary input-to-state stabilization of a damped Euler-Bernoulli beam in the presence of a state-delay[END_REF] where the author considered the stabilization of a damped Euler-Bernoulli beam equation with a time delayed control, here the system is parabolic but it is not generated by a self-adjoint operator. In [START_REF] Djebour | Feedback stabilization of parabolic systems with input delay[END_REF], the authors extend the theory developed in [START_REF] Prieur | Feedback stabilization of a 1-D linear reaction-diusion equation with delay boundary control[END_REF] and construct a nite dimensional feedback stabilizing control with input delay to a general class of parabolic systems. The challenge in the present work is to apply the theory of [START_REF] Djebour | Feedback stabilization of parabolic systems with input delay[END_REF] to deduce the stabilization of the uid-structure problem (1.1), (1.5) since it describes a parabolic system thanks to the damping term -δ∆∂ t η that appears in the structure equation in (1.1).

In order to state the main result, we give some notations. As it is standard in the studies of uid-structure interaction systems, one of the main diculties lies in the fact that the spatial domain of the uid is variable and unknown. Since the problem consists to compare the asymptotic behavior of the solution (U, P, η) to the stationary state (w S , p S , η S ), we are led to introduce a change of variable to transform the functions (U, P ) into ( u, p) dened in the xed domain Ω(η S ). More precisely, we set

X η 1 ,η 2 : Ω(η 1 ) -→ Ω(η 2 ),   y 1 y 2 y 3   -→     y 1 y 2 1 + η 2 (y 1 , y 2 ) 1 + η 1 (y 1 , y 2 ) y 3     .
(1.17)

We consider the change of variables

X(t, •) = X η S ,η(t,•) , Y (t, •) = X η(t,•),η S , (1.18)
and we set Ω = Ω(η S ).

We dene the new functions

u(t, y) = (Cof ∇X(t, y)) * U (t, X(t, y)), p(t, y) = P (t, X(t, y)) (t 0, y ∈ Ω). (1.19)
We observe that we have

u 0 = (Cof ∇X(0, •)) * U 0 (X(0, •)).
In what follows, we recall that n is the unitary exterior normal vector and τ i , i = 1, 2 are two tangent vectors on ∂Ω already dened in (1.14), (1.16) and (1.15).

The operators (A 1 , D(A 1 )) and (A 2 , D(A 2 )) are dened by

A 1 η = α∆ 2 η, D(A 1 ) = H 4 (ω) ∩ L 2 0 (ω), (1.20)
A 2 η = -δ∆η, D(A 2 ) = H 2 (ω) ∩ L 2 0 (ω). (1.21)
Let X 1 , X 2 be two Banach spaces with the norms . X1 and . X2 respectively. For s 0, we dene

W s (0, ∞; X 1 , X 2 ) = v ∈ L 2 (0, ∞; X 1 ) ; v ∈ H s (0, ∞; X 2 ) , with . W s (0,∞;X1,X2) = . L 2 (0,∞;X1) + . H s (0,∞;X2) . For s = 1, the space W 1 (0, ∞; X 1 , X 2 ) is denoted by W (0, ∞; X 1 , X 2 ).
Let f (t) be a vector valued function. For γ > 0, we dene f γ by

f γ : t → e γt f (t).
Then, for γ > 0, we consider the following spaces

L p γ (0, ∞; X 1 ) = {f ∈ L p (0, ∞; X 1 ) ; f γ ∈ L p (0, ∞; X 1 )}, p ∈ [1, +∞], (1.22)
and

W s γ (0, ∞; X 1 , X 2 ) = {f ∈ W s (0, ∞; X 1 , X 2 ) ; f γ ∈ W s (0, ∞; X 1 , X 2 )}, (1.23)
with the norms

f L p γ (0,∞;X1) = f γ L p (0,∞;X1) , f W s γ (0,∞;X1,X2) = f γ W s (0,∞;X1,X2) . We set X ∞,γ = W γ (0, ∞; [H 2 (Ω)] 3 , [L 2 (Ω)] 3 ) × L 2 γ (0, ∞; H 1 (Ω)/R) × W 2 γ (0, ∞; D(A 1 ), L 2 0 (ω)), (1.24)
such that

(u, p, η) X∞,γ = u Wγ (0,∞;[H 2 (Ω)] 3 ,[L 2 (Ω)] 3 ) + u L ∞ γ (0,∞;[H 1 (Ω)] 3 ) + ∇p L 2 γ (0,∞,[L 2 (Ω)] 3 ) + η W 2 γ (0,∞;D(A1),L 2 0 (ω)) + η L ∞ γ (0,∞;H 3 (ω)) + ∂ t η L ∞ γ (0,∞;H 1 (ω)) . (1.25) We dene the space H = [L 2 (Ω)] 3 × D(A 1/2 1 ) × L 2 0 (ω), furnished with the inner product   φ ζ 1 ζ 2   ,   ψ ξ 1 ξ 2   H = Ω φ • ψ dy + ω A 1/2 1 ζ 1 • A 1/2 1 ξ 1 ds + ω ζ 2 • ξ 2 ds.
We set

W =   u -w S η -η S ∂ t η   , (1.26)
where the function u is dened by (1.19). Finally, we suppose that the initial conditions verify

( u 0 , η 0 , η 1 ) ∈ [H 1 (Ω)] 3 × H 3 (ω) × H 1 (ω) (1.27) ∇ • u 0 = 0 in Ω, ( u 0 -η 1 e 3 ) n = 0 on Γ(η S ), u 0 n = 0 on Γ 0 . (1.28)
Now, we state the main result of this paper.

Theorem 1.1. Let t 0 > 0, γ > 0,

f S ∈ [W 2,∞ (Ω)] 3 , h S ∈ L 2 (ω), (1.29)
(w S , p S , η S ) ∈ [W 2,∞ (Ω)] 3 × W 1,∞ (Ω)/R × C 4 (ω), (1.30)
with

1 + η S > 0, (1.31)
such that the system (1.13) is veried and ( u 0 , η 0 , η 1 ) satisfying (1.27), (1.28). Then, there exist

N γ ∈ N * , K ∈ L ∞ loc (R 2 ; L(H)), (φ k , ζ k 1 , ζ k 2 ) ∈ [H 2 (Ω)] 3 × D(A 1 ) × D(A 1/2 1 ), v k ∈ [H 3/2 (Γ 0 )] 3 , k = 1, . . . , N γ ,
and R > 0, such that if

u 0 -w S [H 1 (Ω)] 3 + η 0 -η S H 3 (ω) + η 1 H 1 (ω) R,
there exists a unique strong solution (U, P, η) of (1.1), (1.5) and (1.12) associated to

v(t) = 1 [t0,+∞) (t) Nγ k=1   W (t -t 0 ) + t-t0 0 K(t -t 0 , s) W (s) ds,   φ k ζ k 1 ζ k 2     H v k , (1.32)
where W is given by (1.26). Moreover, ( u -w S , p -p S , η -η S ) ∈ X ∞,γ where the functions u, p are dened by (1.19) and we have the following estimate

( u -w S , p -p S , η -η S ) X∞,γ C u 0 -w S [H 1 (Ω)] 3 + η 0 -η S H 3 (ω) + η 1 H 1 (ω) . (1.33)
Remark 1.2. We can obtain analogous results if we consider the classical Dirichlet boundary conditions

U = Mv on (0, ∞) × Γ 0 , (U -∂ t ηe 3 ) = 0 on (0, ∞) × Γ(η).
(1.34)

We underline that if β 1 = 0, the theorem 1.1 implies that the system (1.1), (1.5) and (1.12) is exponentially stabilizable by considering only a scalar control in the impermeability boundary condition . Remark 1.3. Here, the regularity of η S ensures to have a sucient regular domain to apply elliptic estimates of the Stokes problem with Navier-slip boundary conditions proved in [START_REF] Beirão Da | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF].

In this paper, we are interested in the feedback stabilization of the system (1.1), (1.5) and (1.12) that is similar to the system studied in [START_REF] Badra | Feedback boundary stabilization of 2D uid-structure interaction systems[END_REF] and [START_REF] Raymond | Feedback stabilization of a uid-structure model[END_REF] except that here, we consider the Navier boundary conditions. These conditions were introduced in [START_REF] Navier | Mémoire sur les lois du mouvement des uides[END_REF] by Navier in 1823. The consideration of this type of boundary conditions is signicant in many physical aspects, see for instance [START_REF] Liakos | Finite-element approximation of viscoelastic uid ow with slip boundary condition[END_REF][START_REF] Verfürth | Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition[END_REF][START_REF] Kistler | Coating ow theory by nite element and asymptotic analysis of the Navier-Stokes system[END_REF]. Note that the existence and uniqueness of the strong solution for the system (1.1), (1.5), (1.12) have been obtained recently in [START_REF] Djebour | On the existence of strong solutions to a uid structure interaction problem with Navier boundary conditions[END_REF]. Concerning the stabilization without delay for uid structure problems, there are few results in the literature. In [START_REF] Raymond | Feedback stabilization of a uid-structure model[END_REF], the author considered a coupling system that models the interaction between an incompressible uid governed by the Navier-Stokes system and an elastic structure that enjoys the damped Euler-Bernoulli equation in a rectangular domain considering the Dirichlet boundary conditions, the control in this case is distributed along the structure. The author shows the exponential stabilization of the strong solution of the considered system around zero. The same system was studied in [START_REF] Badra | Feedback boundary stabilization of 2D uid-structure interaction systems[END_REF], where the authors considered a boundary control acting on a located part of the uid boundary. The authors obtained a stabilization result of the weak solution around a stationary state. In the case where the structure is a rigid body, we have [START_REF] Badra | Feedback stabilization of a simplied 1d uid-particle system[END_REF], [START_REF] Badra | Feedback stabilization of a uid-rigid body interaction system[END_REF].

In this work, we construct a boundary feedback control v with a time delay t 0 > 0 (in the direction of [START_REF] Djebour | Feedback stabilization of parabolic systems with input delay[END_REF]) such that the strong solution of our system is exponentially stabilizable around a xed stationary state. Due to regularity concerns, compatibility conditions at t = 0 between v and the initial conditions should be imposed, see for example [START_REF] Badra | Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system[END_REF][START_REF] Badra | Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations[END_REF][START_REF] Badra | Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control[END_REF]. To overcome this diculty, some strategies was developed in [START_REF] Raymond | Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations[END_REF][START_REF] Badra | Stabilization of parabolic nonlinear systems with nite dimensional feedback or dynamical controllers: application to the Navier-Stokes system[END_REF]. In the case of a time delay, the control acts on the system from t 0 > 0. Consequently, the control vanishes at t = 0. Then, the compatibility conditions are automatically satised and this is a good advantage of the time delay.

The paper is organized as follows: in section 2, we write the system (1.1), (1.5), (1.12) in a xed domain using the change of variables (1.18) and (1.19). Next, we study the associated linear system in section 3 that can be written as

z = Az + Bv + f, z(0) = z 0 . (1.35)
Taking into account the remark 2.1, the linearized problem is written as a system coupling the Oseen equations and the damped plate equation, disturbed by some linear terms appearing in the whole system even in the boundary conditions. We show that the innitesimal operator associated to the linear system is analytic and we establish a rst result of a time delayed stabilization of the linearized problem. In fact, we show that the Fattorini-Hautus criterion is satised

∀ε ∈ D(A * ), ∀λ ∈ C, Re λ -σ A * ε = λε et B * ε = 0 =⇒ ε = 0. (UC σ )
More precisely, the condition (UC σ ) is obtained using the unique continuation property of an auxiliary Oseen system with Neumann boundary condition. Finally, we prove the theorem 1.1 applying the Banach xed point argument.

Change of variables

Let us dene T η

(T η ξ)(y) = 0 if y ∈ Γ 0 , ξ(s)e 3 if y = (s, 1 + η(s)) ∈ Γ(η), (2.1)
for any function η : ω -→ (-1, ∞). We notice that T η ∈ L(L 2 (ω); [L 2 (∂Ω(η))] 3 ) and

(T * η ζ)(s) = 1 + |∇η| 2 ζ(s, 1 + η(s)) • e 3 , ∀ζ ∈ [L 2 (∂Ω(η))] 3 .
We dene T by

T = T η S M,
where we recall that M is the orthogonal projection on the space L 2 0 (ω) dened by (1.11). Using (1.18) and (1.19), the system (1.1), (1.5), (1.12) is equivalent to

       ∂ t u -∇ • T( u, p) = det(∇X)f S (X) + (Id -K η )∂ t u -ν(∆ -L η ) u +(∇ -G η ) p + M η u + N η u in (0, ∞) × Ω, ∇ • u = 0 in (0, ∞) × Ω, ∂ tt η + A 1 η + A 2 ∂ t η = H η S ( u, p) + H( u, η) + M h S in (0, ∞) × ω, (2.2)
with the boundary conditions

[ u -T ∂ t η] n = 1 Γ0 (Mv) n on (0, ∞) × ∂Ω, [2νD( u)n + β( u -T ∂ t η)] τ = 1 Γ0 β(Mv) τ + G( u, η) on (0, ∞) × ∂Ω, (2.3)
and with the initial conditions

   u(0, •) = (Cof ∇X(0, .)) * U 0 (X(0, •)) = u 0 in Ω, η(0, •) = η 0 in ω, ∂ t η(0, •) = η 1 in ω. (2.4)
We precise here that 1 Γ0 is the characteristic function of the set Γ 0 . To simplify the notations, we set

a = (Cof(∇Y )) * , b = (Cof(∇X)), K η u = (∇X) u, G η p = b∇ p.
We have the following formulas

[-ν(∆ -L η ) u] i = ν j,k,l,m det(∇X)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 j,k,l,m det(∇X)a jk (X) ∂Y m ∂x i (X) ∂Y l ∂x j (X) -δ jk δ mi δ jl ∂ 2 u k ∂y l ∂y m + det(∇X) ν j,k,l ∂a ik ∂x j (X) ∂Y l ∂x j (X) + ∂a jk ∂x i (X) ∂Y l ∂x j (X) ∂ u k ∂y l + ν j,k,l ∂a ik ∂x j (X) ∂Y l ∂x j (X) + a ik (X) ∂ 2 Y l ∂x 2 j (X) + ∂a jk ∂x j (X) ∂Y l ∂x i (X) + a jk (X) ∂ 2 Y l ∂x j ∂x i (X) ∂ u k ∂y l + ν k ∂ 2 a ik ∂x 2 j (X) + ∂ 2 a jk ∂x j ∂x i (X) u k , (2.5) [(∇ -G η ) p] i = k (δ ik -b ik ) ∂ p ∂y k = k (δ ik -det(∇X) ∂Y k ∂x i (X)) ∂ p ∂y k , (2.6
)

[N η u] i = - k,l,j det(∇X)a kl (X) ∂a ij (X) ∂x k u l u j - k,l,j,m det(∇X)a kl (X)a ij (X) ∂Y m ∂x k (X) u l ∂ u j ∂y m , (2.7)
[M η u] i = - l,k det(∇X)a ik (X) ∂ u k ∂y l ∂ t Y l (X) - k det(∇X)∂ t a ik (X) u k , i = 1, 2, 3. (2.8)
The non linear term H appearing in the plate equation writes

H( u, η) = νM - j,k ∂a 3k ∂x j (X) + ∂a jk ∂x 3 (X) N j u k + j,k,l δ 3k δ jl (N ) j -a 3k (X) ∂Y l ∂x j (X) N j ∂ u k ∂y l + δ 3l δ jk (N ) j -a jk (X) ∂Y l ∂x 3 (X) N j ∂ u k ∂y l . (2.9)
Now let us deal with the boundary conditions. Let W be the operator dened by

[W( u, η)] k = ν j,m n j (X) ∂a km ∂x j (X) u m + ∂a jm ∂x k (X) u m + β   j a kj (X) u j -T ∂ t η • e k   + ν j,m,q n j (X) 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. (2.10)
The boundary conditions (1.5) become after the change of variables (1.18) and (1.19)

( u -T ∂ t η) n = 1 Γ0 (Mv) n on (0, ∞) × ∂Ω, W( u, η) • ( τ i (X)) = 1 Γ0 β(Mv) • ( τ i (X)), i = 1, 2 on (0, ∞) × ∂Ω, (2.11)
where τ i are dened by (1.2) and (1.3). The second boundary condition in (2.11) can be written as

(2νD( u)n + β( u -T ∂ t η)) • τ i = V i ( u, η) + 1 Γ0 β(Mv) • τ i , i = 1, 2, on ∂Ω,
where τ i are the two tangent vectors on ∂Ω dened by (1.14), (1.16) and V i is given by

V i ( u, η) = (2νD( u)n + β( u -T ∂ t η)) • (τ i -τ i (X)) + (2νD( u)n + β( u -T ∂ t η) -W( u, η)) • τ i (X), i = 1, 2, on ∂Ω. (2.12)
We construct an operator G such that G

• n = 0 and G • τ i = V i on ∂Ω. Consequently, G is dened on Γ(η S ) by G 1 = V 1 ((∂ s2 η S ) 2 + 1) -V 2 (∂ s1 η S ∂ s2 η S ) |N | 2 , G 2 = V 2 ((∂ s1 η S ) 2 + 1) -V 1 (∂ s1 η S ∂ s2 η S ) |N | 2 , G 3 = ∂ s1 η S V 1 + ∂ s2 η S V 2 |N | 2 , (2.13)
and

G is dened on Γ 0 by G = 2 i=1 V i τ i . We set u = u -w S , p = p -p S , ξ = η -η S .
Then, (u, p, ξ) satises the system

   ∂ t u -∇ • T(u, p) + (w S • ∇)u + (u • ∇)w S = F (u, p, ξ) in (0, ∞) × Ω, ∇ • u = 0 in (0, ∞) × Ω, ∂ tt ξ + A 1 ξ + A 2 ∂ t ξ = H η S (u, p) + H(u + w S , ξ + η S ) in (0, ∞) × ω, (2.14)
where

F (u, p, ξ) = (Id -K ξ+η S )∂ t u -ν(∆ -L ξ+η S )(u + w S ) + (∇ -G ξ+η S )(p + p S ) + M ξ+η S (u + w S ) + N ξ+η S (u + w S ) + (u • ∇)w S + (w S • ∇)u + (w S • ∇)w S + det(∇X)f S (X) -f S , (2.15)
with the boundary conditions

[u -T ∂ t ξ] n = 1 Γ0 (Mv) n on (0, ∞) × ∂Ω, [2νD(u)n + β(u -T ∂ t ξ)] τ = 1 Γ0 β(Mv) τ + G(u + w S , ξ + η S ) on (0, ∞) × ∂Ω, (2.16)
and the initial conditions

   u(0, •) = u 0 -w S in Ω, ξ(0, •) = η 0 -η S in ω, ∂ t ξ(0, •) = η 1 in ω.
(2.17)

Remark 2.1. For the linearization of the system (2.14), (2.16), (2.17), we need to nd all the linear terms coming from the change of variables. For example, we observe the term

det(∇X)f S (X) -f S = det(∇X)(f S (X) -f S ) + (det(∇X) -1)f S .
We recall that det(∇X) = 1 + η 1 + η S . We notice that

X(y) = y + y 3 ξ(y 1 , y 2 ) 1 + η S (y 1 , y 2 ) e 3 = y + θξ(y 1 , y 2 )e 3 .
We apply Taylor's formula since

f S ∈ [W 2,∞ (Ω)] 3 , we get f S (X(y)) -f S (y) = θξ(y 1 , y 2 )∇f S (y)e 3 + 1 0 (1 -s)∇ 2 f S (y + sθξ(y 1 , y 2 )e 3 )θξ(y 1 , y 2 )e 3 • θξ(y 1 , y 2 )e 3 ds. (2.18) Then det(∇X(y))f S (X(y)) -f S (y) = ξ(y 1 , y 2 ) 1 + η S (y 1 , y 2 ) f S (y) + θξ(y 1 , y 2 )∇f S (y)e 3 + (ξ(y 1 , y 2 )) 2 1 + η S θ∇f S (y)e 3 + 1 + ξ(y 1 , y 2 ) 1 + η S (y 1 , y 2 ) 1 0 (1 -s)∇ 2 f S (y + sθξ(y 1 , y 2 )e 3 )θξ(y 1 , y 2 )e 3 • θξ(y 1 , y 2 )e 3 ds. (2.19) Now, let assume that ξ L 2 γ (0,∞;L 2 (ω))
CR, then, we have for example

ξ 1 + η S f S L 2 γ (0,∞;[L 2 (Ω)] 3 ) CR.
Thus, the terms of this kind should be considered in the linear part of the system or else it will constitute a problem in the xed point procedure.

To overcome this diculty, we follow the same approach described in [START_REF] Badra | Feedback boundary stabilization of 2D uid-structure interaction systems[END_REF]. More precisely, we write for example det(∇X(y))f S (X(y)) -f S (y) = (12) (ξ) + (12) (ξ), with (12) (ξ)

L 2 γ (0,∞;[L 2 (Ω)] 3 ) C ξ 2 L 2 γ (0,∞;L 2 (ω)) (1 + ξ m L 2 γ (0,∞;L 2 (ω)) ), m 1. (2.20)
where (12) is the linear operator (12) (ξ) = ξ 1 + η S f S , and (12) is the remaining non linear terms appearing in (2.19). Then, the strategy consists to integrate the linear operator (12) in the left hand side of the uid equation in (2.14) and to inject (12) in the right hand side that will be handled in the xed point section.

Due to remark 2.1, we need to decompose the functions F , H and G into a linear part in ξ and a nonlinear part.

The notation ∂ s ξ (resp. ∂ 2 ss ξ, ∂ 3 sss ξ), designates in what follows the rst derivative ∂ sj ξ (resp. the second derivative ∂ 2 sisj ξ, the third derivative ∂ 3 sisj s k ξ ). We underline that the linear part is not given in a explicit way, we just need to write formally the terms that appear precising the derivative order of ξ. The linear parts denoted usually by (i) will have the form that formally writes

(i) (Z 1 , ..., Z m ) = P (i) 1 Z 1 + .... + P (i) m Z m , (2.21)
where

P (i) j
are coecients that depend only on the stationary state (w S , p S , η S ), consequently, they are regular enough. The goal now is to decompose each terms appearing in the formulas of F (u, p, ξ), H(u + w s , ξ + η S ), G(u + w S , ξ + η S ) as it is done in remark 2.1, the formal calculus are done in lemma 4.2. Note that since our system is coupled, we need to obtain a convenient match between the linear operators that appear in the uid equation, the structure equation and even in the boundary conditions. That is given in the following lemma. Lemma 2.2. Suppose that

(w S , p S , η S ) ∈ [W 2,∞ (Ω)] 3 × W 1,∞ (Ω)/R × C 4 (ω).

Then,

• there exist linear operators

L 1 (•) ∈ L(D(A 3/4 1 ), [H 1 (Ω)] 3×3 ), L 2 (•, •) ∈ L(D(A 1/2 1 ) × D(A 1/4 1 ), [L 2 (Ω)] 3 ), L 3 (•) ∈ L(D(A 1/2 1 ), [H 1/2 (∂Ω)] 3 ),
• there exist non linear functions N E , N F , N G and F, such that the system (2.14), (2.16), (2.17) is equivalent to

           ∂ t u -∇ • T(u, p) + (w S • ∇)u + (u • ∇)w S -∇ • L 1 (ξ) + L 2 (ξ, ∂ t ξ) = ∇ • N E (ξ) + N F (ξ) + F(u, p, ξ) in (0, ∞) × Ω, ∇ • u = 0 in (0, ∞) × Ω, ∂ tt ξ + A 1 ξ + A 2 ∂ t ξ + T * (L 1 (ξ)n) = H η S (u, p) + H(u, ξ + η S ) -T * (N E (ξ)n) in (0, ∞) × ω, (2.22)
with the boundary conditions

   [u -T ∂ t ξ] n = 1 Γ0 (Mv) n on (0, ∞) × ∂Ω, 2νD(u)n + L 1 (ξ)n + L 3 (ξ) + β(u -T ∂ t ξ) τ = 1 Γ0 β(Mv) τ -[N E (ξ)n] τ + [N G (ξ)] τ + G(u, ξ + η S ) on (0, ∞) × ∂Ω, (2.23)
and the initial conditions

   u(0, •) = u 0 -w S in Ω, ξ(0, •) = η 0 -η S in ω, ∂ t ξ(0, •) = η 1 in ω. (2.24)
Proof. Decomposition of F (u, p, ξ):

Let decompose the term -ν(∆ -L η ). We notice that

-ν(∆ -L η )( u) = -ν(∆ -L ξ+η S )(u + w S ) = -ν(∆ -L ξ+η S )(u) -ν(∆ -L ξ+η S )(w S ). (2.25)
The term -ν(∆ -L ξ+η S )(u) is nonlinear whereas -ν(∆ -L ξ+η S )(w S ) contains linear terms in ξ, then, from remark 2.1, we should write

-ν(∆ -L ξ+η S )(w S ) = (13) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ) + (13) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ), (2.26)
where (13) is a linear operator of the form (2.21) and (13) is the nonlinear part. To write the linearized system in an appropriate form, we dene for all u

(E η ( u)) im = ν j,k,l det(∇X)a ik (X) ∂Y m ∂x j (X) ∂Y l ∂x j (X) -δ ik δ mj δ jl ∂ u k ∂y l + ν j,k,l det(∇X)a jk (X) ∂Y m ∂x i (X) ∂Y l ∂x j (X) -δ jk δ mi δ jl ∂ u k ∂y l + det(∇X)   ν j,k ∂a ik ∂x j (X) ∂Y m ∂x j (X) + ∂a jk ∂x i (X) ∂Y m ∂x j (X) u k   . (2.27)
Observe that

(∇ • E η ) i = -ν[(∆ -L η )] i + (F 1 η ) i , (2.28)
with

[(F 1 η )( u)] i = ν ∂ det(∇X) ∂y m j,k,l,m a ik (X) ∂Y m ∂x j (X) ∂Y l ∂x j (X) + a jk (X) ∂Y m ∂x i (X) ∂Y l ∂x j (X) ∂ u k ∂y l + j,k ∂a ik ∂x j (X) ∂Y m ∂x j (X) + ∂a jk ∂x i (X) ∂Y m ∂x j (X) u k + ν det(∇X) j,k,l,m,q ∂ 2 Y m ∂x j ∂x q (X) ∂X q ∂y m a ik (X) ∂Y l ∂x j (X) + a jk (X) ∂Y l ∂x j (X) ∂ u k ∂y l + ∂a ik ∂x j (X) + ∂a jk ∂x i (X) u k . (2.29)
From the lemma 4.2, we have

(E ξ+η S (w S )) = L 1 (ξ) + N E (ξ), (2.30)
where L 1 is a linear operator that writes

L 1 (ξ) = Q 1 ξ + Q 2 ∂ s ξ + Q 3 ∂ 2 ss ξ,
where Q i , i = 1, .., 3 are operators that depend only on the stationary state and are of regularity W 1,∞ (Ω). The quantity N E (ξ) is the remaining nonlinear part that contains the terms of the form

c 1 (w S , η S ) ξ m1 (∂ s ξ) m2 (1 + ξ + η S ) α1 , c 2 (w S , η S ) ξ n1 (∂ s ξ) n2 ∂ 2 ss ξ (1 + ξ + η S ) α2 , m 1 + m 2 2, n 1 + n 2 1, α i 1, (2.31)
where c 1 (w S , η S ) and c 2 (w S , η S ) are two quantities of regularity W 1,∞ (Ω). In the other hand, from lemma 4.2, we have also

(F 1 ξ+η S )(w S ) = -(9) (ξ, ∂ s ξ, ∂ 2 ss ξ) -(9) (ξ, ∂ s ξ, ∂ 2 ss ξ), (2.32)
where the expression of the nonlinear part (9) (ξ, ∂ s ξ, ∂ 2 ss ξ) contains terms of the form (2.31). Then, from (2.26), (2.28), (2.30) and (2.32), we deduce (13)

(ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ) = ∇ • (L 1 (ξ)) + (9) (ξ, ∂ s ξ, ∂ 2 ss ξ), (2.33)
(13) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ) = (9) (ξ, ∂ s ξ, ∂ 2 ss ξ) + ∇ • (N E (ξ)). ( 2
L ξ+η S )(u + w S ) writes -ν(∆ -L ξ+η S )(u + w S ) = ∇ • (L 1 (ξ)) + (9) (ξ, ∂ s ξ, ∂ 2 ss ξ) + (9) (ξ, ∂ s ξ, ∂ 2 ss ξ) + ∇ • (N E (ξ)) -ν(∆ -L ξ+η S )(u). (2.35)
Furthermore, using again lemma 4.2, we obtain

[(∇ -G ξ+η S )(p S )] i = k (δ ki -det(∇X) ∂Y k ∂x i (X)) ∂p S ∂y k = (5) i (ξ, ∂ s ξ) + (5) i (ξ, ∂ s ξ), (2.36)
where (5) (ξ, ∂ s ξ) contains the terms

c 3 (p S , η S ) ξ m2 (∂ s ξ) m2 (1 + ξ + η S ) α1 , m 1 + m 2 2, α 1 1, (2.37)
where c 3 is a function that depends on the stationary state and bounded in Ω. Then, we get

[(∇ -G ξ+η S )(p + p S )] i = (5) i (ξ, ∂ s ξ) + (5) i (ξ, ∂ s ξ) + [(∇ -G ξ+η S )(p)] i . (2.38) Moreover, we have l,k det(∇X)a ik (X) ∂w S k ∂y l ∂ t Y l (X) = (∇X∇w S ∂ t Y (X)) i ,
where

(∇X∇w S ∂ t Y (X)) i = ∂ t Y 3 ∂w S i ∂y 3 , i = 1, 2, (∇X∇w S ∂ t Y (X)) 3 = ∂ t Y 3 ∂X 3 ∂y 1 ∂w S 1 ∂y 3 + ∂X 3 ∂y 2 ∂w S 2 ∂y 3 + 1 + η 1 + η S ∂w S 3 ∂y 3 .
We notice that

∂ t Y 3 = -y 3 ∂ t ξ 1 + η , ∂X 3
∂y i = y 3 ∂ si ξ(1 + η S ) -∂ si η S ξ (1 + η S ) 2 , i = 1, 2.
Using (4.14), we nd

-(∇X∇w S ∂ t Y (X)) i = (4) i (∂ t ξ) + (4) i (ξ, ∂ s ξ, ∂ t ξ), where (4) (ξ, ∂ s ξ, ∂ t ξ) has the terms c 4 (w S , η S ) ξ m1 ∂ t ξ∂ s ξ (1 + ξ + η S ) α1 , c 5 (w S , η S ) ξ m2 ∂ t ξ (1 + ξ + η S ) α2 , m 1 0, m 2 1, α i 1, (2.39)
with c 4 and c 5 are W 1,∞ (Ω) functions. Moreover, using lemma 4.2, we get

- k det(∇X)∂ t a ik (X)w S k = (6) i (∂ t ξ, ∂ 2 ts ξ) + (6) i (ξ, ∂ t ξ, ∂ 2 ts ξ), (2.40)
where [START_REF] Badra | Feedback stabilization of a simplied 1d uid-particle system[END_REF] (ξ, ∂ t ξ, ∂ 2 ts ξ) contains terms of the form

c 6 (w S , η S ) ξ m1 ∂ 2 ts ξ (1 + ξ + η S ) α1 , c 7 (w S , η S ) ξ m2 ∂ t ξ (1 + ξ + η S ) α2 , c 8 (w S , η S ) ξ m3 ∂ t ξ∂ s ξ (1 + ξ + η S ) α3 , m 1 1, m 2 1, m 3 0, α i 1, (2.41)
where c i , i = 6, ..., 8 are W 2,∞ (Ω) functions. Then

M ξ+η S (u + w S ) = (4) (∂ t ξ) + (6) (∂ t ξ, ∂ 2 ts ξ) + (6) (ξ, ∂ t ξ, ∂ 2 ts ξ) + (4) (ξ, ∂ s ξ, ∂ t ξ) + M ξ+η S (u). (2.42)
We have,

[N ξ+η S (u + w S ) + u • ∇w S + w S • ∇u + w S • ∇w S ] i = - k,l,j det(∇X)a kl (X) ∂a ij (X) ∂x k w S l w S j + k,l,j,m δ ij δ kl δ km -det(∇X)a kl (X)a ij (X) ∂Y m ∂x k (X) w S l ∂w S j ∂y m + F 2 i (u, ξ + η S ) = F 2 i (u, ξ + η S ) + F 3 i (w S , ξ + η S ), i = 1, 2, 3. (2.43)
Using lemma 4.2, we obtain

F 3 i (w S , ξ + η S ) = (8) i (ξ, ∂ s ξ, ∂ 2 ss ξ) + (8) i (ξ, ∂ s ξ, ∂ 2 ss ξ), (2.44)
where [START_REF] Beirão Da | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF] (ξ, ∂ s ξ, ∂ 2 ss ξ) admits terms of the form (2.31). Then, we get

[N ξ+η S (u + w S ) + u • ∇w S + w S • ∇u + w S • ∇w S ] i = (8) i (ξ, ∂ s ξ, ∂ 2 ss ξ) + (8) i (ξ, ∂ s ξ, ∂ 2 ss ξ) + F 2 i (u, ξ + η S ), i = 1,
F (u, p, ξ) = ∇ • L 1 (ξ) + L 2 (ξ, ∂ t ξ) + ∇ • N E (ξ) + N F (ξ) + F(u, p, ξ). (2.46)
where

L 2 (ξ, ∂ t ξ) = (9) (ξ, ∂ s ξ, ∂ 2 ss ξ) + (5) (ξ, ∂ s ξ) + (4) (∂ t ξ) + (6) (∂ t ξ, ∂ 2 ts ξ) + (8) (ξ, ∂ s ξ, ∂ 2 ss ξ). (2.47) More precisely, L 2 (ξ, ∂ t ξ) = Q 4 ξ + Q 5 ∂ s ξ + Q 6 ∂ 2 ss ξ + Q 7 ∂ t ξ + Q 8 ∂ 2 st ξ, where Q i , i = 4, .
., 8 are operators of regularity W 1,∞ (Ω) and we also have

N F (ξ) = (9) (ξ, ∂ s ξ, ∂ 2 ss ξ) + (5) (ξ, ∂ s ξ) + (4) (ξ, ∂ s ξ, ∂ t ξ) + (6) (ξ, ∂ t ξ, ∂ 2 ts ξ) + (8) (ξ, ∂ s ξ, ∂ 2 ss ξ), (2.48) F(u, p, ξ) = (Id -K ξ+η S )∂ t u -ν(∆ -L ξ+η S )(u) + (∇ -G ξ+η S )(p) + M ξ+η S (u) + F 2 (u, ξ + η S ). (2.49) Decomposition H(u, ξ + η S ):
We have

H(u + w S , ξ + η S ) = H(u, ξ + η S ) + H(w S , ξ + η S ).
We notice that

H(w S , ξ + η S ) = -T * ((E ξ+η S (w S ))n), (2.50)
where T * is dened by

T * (ζ)(s) = M 1 + |∇η S | 2 ζ(s, 1 + η S (s)) • e 3 , ∀ζ ∈ [L 2 (∂Ω)] 3 .
(2.51) Using (2.30), we obtain

H(u + w S , ξ + η S ) = -T * (L 1 (ξ)n) + H(u, ξ + η S ) -T * (N E (ξ)n). (2.52)
Decomposition of G(u, ξ + η S ):

Using the expression of W in (2.10) and of V i in (2.12), we nd

V i (w S , ξ + η S ) = 2νD(w S )n + βw S • (τ i -τ i (X)) + ν j,m,q (n) j δ km δ qj -( n(X)) j a km (X) ∂Y q ∂x j (X) ∂w S m ∂y q -ν j,m,q ( n(X)) j a jm (X) ∂Y q ∂x k (X) -(n) j δ jm δ qk ∂w S m ∂y q -ν j,m ( n(X)) j ∂a km ∂x j (X)w S m + ∂a jm ∂x k (X)w S m + β j (δ kj -a kj (X))w S j τ i k (X), k = 1, 2, 3.
We notice that ν j,m,q

n j δ km δ qj -n j (X)a km (X) ∂Y q ∂x j (X) ∂w S m ∂y q + ν j,m,q n j δ jm δ qk -n j (X)a jm (X) ∂Y q ∂x k (X) ∂w S m ∂y q -ν j,m n j (X) ∂a km ∂x j (X)w S m + ∂a jm ∂x k (X)w S m = - |N | | N | (E ξ+η S (w S ))n k + ν j (n) j - N j | N | ∂w S k ∂y j + ν j (n) j - N j | N | ∂w S j ∂y k . (2.53)
In the other hand, using Taylor's formula

1 | N | = 1 |N | - ∇η S • ∇ξ |N | 3 - |∇ξ| 2 2|N | 3 + 3θ 2 4|N | 1 0 (1 -s) (1 + sθ) 5/2 ds, where θ = 2∇η S • ∇ξ + |∇ξ| 2 |N | 2
. Then, we deduce

1 | N | = 1 |N | - ∇η S • ∇ξ |N | 3 + (10) (∂ s ξ), (2.54)
with (10)

(∂ s ξ) = - |∇ξ| 2 2|N | 3 + 3θ 2 4|N | 1 0 (1 -s) (1 + sθ) 5/2 ds.
Then, using (2.30) and (2.54), we obtain ν j,m,q n j δ km δ qj -n j (X)a km (X) ∂Y q ∂x j (X) ∂w S m ∂y q + ν j,m,q

n j δ jm δ qk -n j (X)a jm (X) ∂Y q ∂x k (X) ∂w S m ∂y q -ν j,m n j (X) ∂a km ∂x j (X)w S m + ∂a jm ∂x k (X)w S m = [-L 1 (ξ)n + (14) (∂ s ξ) -N E (ξ)n + (14) (ξ, ∂ s ξ, ∂ 2 ss ξ)] k , (2.55)
where ( 14)

k (∂ s ξ) = ν ∇η S • ∇ξ |N | 2 j ∂w S k ∂y j + ∂w S j ∂y k ,
and (14) admits the terms (2.31). Moreover, From lemma 4.2, we get

β j (δ kj -a kj (X))w S j = (15) k (ξ, ∂ s ξ) + (15) k (ξ, ∂ s ξ),
where

k had the terms c 9 (w S , η S ) ξ m1 (1 + ξ + η S ) α1 , c (10) (w S , η S )

ξ m2 ∂ s ξ (1 + ξ + η S ) α2 , m 1 2, m 2 1, α i 1, i = 1, 2,
where c i , i = 9, 10 are of regularity W 3/2,∞ (∂Ω).

V i (w S , η) = -[L 1 (ξ)n + L 3 (ξ)] • τ i -[N E (ξ)n] • τ i + (11) i (ξ, ∂ s ξ, ∂ 2 ss ξ), (2.56)
where L 3 is an operator dened by

L 3 (ξ) • τ i = -2νD(w S )n + βw S • (τ i -τ i (X)) -(14) (∂ s ξ) • τ i , L 3 (ξ) • n = 0,
then L 3 can be expressed as the following

L 3 (ξ) = Q 9 ξ + Q 10 ∂ s ξ, (2.57)
where Q 9 and Q 10 are of regularity W 1/2,∞ (∂Ω) and (11)

(ξ, ∂ s ξ, ∂ 2 ss ξ) = -(14) (∂ s ξ) • ( τ i (X) -τ i ) + (14) (ξ, ∂ s ξ, ∂ 2 ss ξ) • τ i (X) -[N E (ξ)n] • ( τ i (X) -τ i ).
Here, [START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF] contains terms of the type (2.31). Finally, we dene N G such that

N G (ξ) • τ i = (11) i (ξ, ∂ s ξ, ∂ 2 ss ξ), N G (ξ) • n = 0.
Then, the system (2.14), (2.16), (2.17) becomes

           ∂ t u -∇ • T(u, p) + (w S • ∇)u + (u • ∇)w S -∇ • L 1 (ξ) + L 2 (ξ, ∂ t ξ) = ∇ • N E (ξ) + N F (ξ) + F(u, p, ξ) in (0, ∞) × Ω, ∇ • u = 0 in (0, ∞) × Ω, ∂ tt ξ + A 1 ξ + A 2 ∂ t ξ + T * (L 1 (ξ)n) = H η S (u, p) + H(u, ξ + η S ) -T * (N E (ξ)n) in (0, ∞) × ω,
and the boundary conditions

   [u -T ∂ t ξ] n = 1 Γ0 (Mv) n on (0, ∞) × ∂Ω, 2νD(u)n + L 1 (ξ)n + L 3 (ξ) + β(u -T ∂ t ξ) τ = 1 Γ0 β(Mv) τ -[N E (ξ)n] τ + [N G (ξ)] τ + G(u, ξ + η S ) on (0, ∞) × ∂Ω,
and the initial conditions

   u(0, •) = u 0 -w S in Ω, ξ(0, •) = η 0 -η S in ω, ∂ t ξ(0, •) = η 1 in ω.
3 Feedback stabilization of the linear system

We consider the linear system associated to (2.14) and (2.16)

   ∂ t w + (w S • ∇) w + ( w • ∇)w S -∇ • T( w, q) -∇ • L 1 (ξ) + L 2 (ξ, ∂ t ξ) = f in (0, ∞) × Ω, ∇ • w = 0 in (0, ∞) × Ω, ∂ tt ξ + A 1 ξ + A 2 ∂ t ξ + T * (L 1 (ξ)n) = -T * (T( w, q)n) + h in (0, ∞) × ω, (3.1)
with the boundary conditions

[ w -T ∂ t ξ] n = 1 Γ0 (Mv) n on (0, ∞) × ∂Ω, 2νD( w)n + β ( w -T ∂ t ξ) + L 1 (ξ)n + L 3 (ξ) τ = 1 Γ0 β(Mv) τ + g on (0, ∞) × ∂Ω, (3.2)
and the initial conditions

   w(0, •) = w 0 = u 0 -w S in Ω, ξ(0, •) = ξ 0 = η 0 -η S in ω, ∂ t ξ(0, •) = ξ 1 = η 1 in ω, (3.3)
where

L 1 (ξ) = Q 1 ξ + Q 2 ∂ s ξ + Q 3 ∂ 2 ss ξ, L 2 (ξ, ∂ t ξ) = Q 4 ξ + Q 5 ∂ s ξ + Q 6 ∂ 2 ss ξ + Q 7 ∂ t ξ + Q 8 ∂ 2 st ξ, L 3 (ξ) = Q 9 ξ + Q 10 ∂ s ξ. (3.4) We set L 2,1 (ξ) = Q 4 ξ + Q 5 ∂ s ξ + Q 6 ∂ 2 ss ξ, L 2,2 (∂ t ξ) = Q 7 ∂ t ξ + Q 8 ∂ 2 ts ξ. (3.5)
Here, we recall that Q i , i = 1, ..., 8 are some operators that depend on the stationary state, of regularity W 1,∞ (Ω) and Q 9 , Q 10 are of regularity W 1/2,∞ (∂Ω). The next step consists to reformulate the system (3.1), (3.2) to get an evolution problem. To do so, we need to lift g in the boundary conditions. Then, we set the following lemma that concerns the instationary Stokes problem with Navier boundary conditions with a non negative friction. Lemma 3.1. Let β 0 and γ 0 > 0. Let ( v, π) that veries the system

           ∂ t v + γ 0 v -∇ • T( v, π) = 0 in (0, ∞) × Ω, ∇ • v = 0 in (0, ∞) × Ω, v n = 0 on (0, ∞) × ∂Ω, [2νD( v)n + β v] τ = g on (0, ∞) × ∂Ω, v(0, •) = 0 in Ω. (3.6) If g ∈ W 1/4 γ (0, ∞; [H 1/2 (∂Ω)] 3 , [L 2 (∂Ω)] 3 ), g n0 = 0, (3.7)
for γ ∈ [0, γ 0 [, then the problem (3.6) admits a unique solution that satises the estimate

v 2 Wγ (0,∞,[H 2 (Ω)] 3 ,[L 2 (Ω)] 3 ) + ∇ π 2 L 2 γ (0,∞;[L 2 (Ω)] 3 ) C g 2 W 1/4 γ (0,∞;[H 1/2 (∂Ω)] 3 ,[L 2 (∂Ω)] 3 ) , (3.8)
where C is a positive constant.

Remark 3.2. Since γ 0 > 0, then the system (3.6) admits a unique solution for any β 0 that is exponentially stable. If γ 0 = β = 0 the system is no more stable hence the importance of taking γ 0 > 0.

Proof. In the proof, we use the fact that the condition (3.7) is equivalent to

g ∈ W 1/2 γ (0, ∞; [H 1 (Ω)] 3 , [L 2 (Ω)] 3 ), g n0 = 0.
This fact is obtained in [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]p.21,Theorem 2.3]. This type of demonstration can be found in [START_REF] Shimada | On the L p -L q maximal regularity for Stokes equations with Robin boundary condition in a bounded domain[END_REF], for the sake of completeness, we recall the main steps of the proof here and see how we can adapt some points of the proof in our case. We consider the following problem

       γ 0 z -∇ • T(z, χ) = 0 in (0, ∞) × Ω, ∇ • z = 0 in (0, ∞) × Ω, z n = 0 on (0, ∞) × ∂Ω, [2νD(z)n + βz] τ = g on (0, ∞) × ∂Ω.
(3.9)

By a density argument, we assume that g ∈ C ∞ 0 (R + , [H 1 (Ω)] 3 ) and g(t) = 0 if t < 0. Then, the system (3.9), admits a unique solution [START_REF] Beirão Da | Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions[END_REF]). We set then, z = v -z, χ = π -χ. Then, ( z, χ) satises the system

(z, χ) ∈ C ∞ 0 (R + , [H 2 (Ω)] 3 ) × C ∞ 0 (R + , H 1 (Ω)/R) (see
           ∂ t z + γ 0 z -∇ • T( z, χ) = -∂ t z in (0, ∞) × Ω, ∇ • z = 0 in (0, ∞) × Ω, z n = 0 on (0, ∞) × ∂Ω, [2νD( z)n + β z] τ = 0 on (0, ∞) × ∂Ω, z(0, •) = 0 in Ω.
(3.10)

The system above admits a solution using the fact that the operator

A = P∆ -γ 0 Id, D(A) = {z ∈ [H 2 (Ω)] 3 ; z • n = 0, [2νD(z)n + βz] τ = 0},
generates an analytic semigroup exponentially stable where P designates the Leray projection. This is a direct consequence of [9, Theorem 2.12, p.115]. We deduce then the existence of a unique solution

( v, π) ∈ C ∞ 0 (R + , [H 2 (Ω)] 3 ) × C ∞ 0 (R + , H 1 (Ω)/R) of the system (3.6). Now, we show (3.8). Let φ ∈ C ∞ 0 (R + , [L 2 (Ω)] 3
) such that ∇ • φ = 0, φ • n = 0, and we suppose that φ = 0 on (T, ∞) × Ω where T > 0. We consider the system

           ∂ t ψ -(γ 0 -γ)ψ + ∇ • T(ψ, θ) = φ in R × Ω, ∇ • ψ = 0 in R × Ω, ψ n = 0 on R × ∂Ω, [2νD(ψ)n + βψ] τ = 0 on R × ∂Ω, ψ(T ) = 0 in Ω. (3.11)
We observe that ψ = 0, θ = 0 on (T, ∞) × Ω. The system (3.11) admits a unique solution

(ψ, θ) ∈ W (R; [H 2 (Ω)] 3 , [L 2 (Ω)] 3 ) × L 2 (R; H 1 (Ω)/R),
exponentially stable satisfying

ψ W (R;[H 2 (Ω)] 3 ,[L 2 (Ω)] 3 ) + θ L 2 (R;H 1 (Ω)/R) C φ L 2 (R;[L 2 (Ω)] 3 ) . (3.12)
We have

R Ω e γt v • φ dydt = R Ω e γt v • (∂ t ψ -(γ 0 -γ)ψ + ∇ • T(ψ, θ)) dydt = - R Ω e γt (∂ t v + γ 0 v -∇ • T( v, π)) • ψ dydt - R ∂Ω e γt g τ • ψ τ dΓdt = - R ∂Ω e γt g τ • ψ τ dΓdt.
Using (3.12), we get

e γt v L 2 (R;[L 2 (Ω)] 3 ) C e γt g L 2 (R;[H 1 (Ω)] 3 ) . (3.13)
In the other hand, we extend n on Ω and we obtain

∂Ω e γt ∂ t g • ψ dΓ = i Ω e γt ∂ i ((n) i ∂ t g) • ψ dy + i Ω e γt ((n) i ∂ t g) • ∂ i ψ dy. Then, R Ω e γt ∂ t v • φ dydt = - i R Ω e γt ∂ i ((n) i ∂ t g) • ψ + e γt ((n) i ∂ t g) • ∂ i ψ dydt. (3.14) To estimate t -→ e γt ∂ t v(t) in L 2 (R, [L 2 (Ω)] 3 ), we notice that R Ω e γt ∂ i ((n) i ∂ t g) • ψ dydt = -γ R Ω e γt ∂ i ((n) i g) • ψ dydt - R Ω e γt ∂ i ((n) i g) • ∂ t ψ dydt. (3.15)
Then,

e γt ∂ i ((n) i ∂ t g), ψ L 2 (R;[L 2 (Ω)] 3 ) C e γt g L 2 (R;[H 1 (Ω)] 3 ) ∂ t ψ L 2 (R,[L 2 (Ω)] 3 ) + γ ψ L 2 (R,[L 2 (Ω)] 3 ) . (3.16)
Moreover,

e γt ((n) i ∂ t g), ∂ i ψ L 2 (R;[L 2 (Ω)] 3 ) = ∂ t (e γt g), (n) i ∂ i ψ L 2 (R;[L 2 (Ω)] 3 ) -γ e γt g, (n) i ∂ i ψ L 2 (R;[L 2 (Ω)] 3 ) .
Then, using [27, Proposition 2.3, Proposition 2.8], we obtain

e γt ((n) i ∂ t g), ∂ i ψ L 2 (R;[L 2 (Ω)] 3 ) C g W 1/2 (R;[H 1 (Ω)] 3 ,[L 2 (Ω)] 3 ) ∇ψ W 1/2 (R;[H 1 (Ω)] 9 ,[L 2 (Ω)] 9 ) . C g W 1/2 (R;[H 1 (Ω)] 3 ,[L 2 (Ω)] 3 ) ψ W (R;[H 2 (Ω)] 3 ,[L 2 (Ω)] 3 ) . (3.17)
Combining (3.17), (3.16), (3.14) and (3.12) we have

e γt ∂ t v L 2 (R;[L 2 (Ω)] 3 ) C e γt g W 1/2 (R;[H 1 (Ω)] 3 ,[L 2 (Ω)] 3 ) . (3.18)
Finally, from the classical elliptic estimates of the Stokes problem, we have

v(t) [H 2 (Ω)] 3 + ∇ π(t) [L 2 (Ω)] 3 C ∂ t v(t) [L 2 (Ω)] 3 + g(t) [H 1 (Ω)] 3 . (3.19)
Using (3.19) and (3.18), we obtain (3.8).

We set w = w -v, q = q -π.

Then, from lemma 3.1, we only need to consider the linear system

   ∂ t w + (w S • ∇)w + (w • ∇)w S -∇ • T(w, q) -∇ • L 1 (ξ) + L 2 (ξ, ∂ t ξ) = f in (0, ∞) × Ω, ∇ • w = 0 in (0, ∞) × Ω, ∂ tt ξ + A 1 ξ + A 2 ∂ t ξ + T * (L 1 (ξ)n) = -T * (T(w, q)n) + h in (0, ∞) × ω, (3.20)
with the boundary conditions

[w -T ∂ t ξ] n = 1 Γ0 (Mv) n on (0, ∞) × ∂Ω, 2νD(w)n + β (w -T ∂ t ξ) + L 1 (ξ)n + L 3 (ξ) τ = 1 Γ0 β(Mv) τ on (0, ∞) × ∂Ω, (3.21)
and the initial conditions

   w(0, •) = w 0 in Ω, ξ(0, •) = ξ 0 in ω, ∂ t ξ(0, •) = ξ 1 in ω, (3.22) where f = f -(w S • ∇) v -( v • ∇)w S + γ 0 v, h = h -T * (T( v, π)n). (3.23)
In this section, we prove that the system (3.20), (3.21) and (3.22) is exponentially stabilisable.

We dene

H = {(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 on ∂Ω},
with the inner product

  w η 1 η 2   ,   v ξ 1 ξ 2   H = w, v [L 2 (Ω)] 3 + A 1/2 1 η 1 , A 1/2 1 ξ 1 L 2 (ω) + η 2 , ξ 2 L 2 (ω) .
We dene also

V = [H 1 (Ω)] 3 × D(A 3/4 1 ) × D(A 1/4 1 ) ∩ H, (3.24)
and the orthogonal projection

P P : [L 2 (Ω)] 3 × D(A 1/2 1 ) × L 2 0 (ω) -→ H.

Finally, we set the operator

A S A S   w η 1 η 2   =   ν∆w -(w S • ∇)w -(w • ∇)w S + ∇ • L 1 (η 1 ) -L 2 (η 1 , η 2 ) η 2 -A 1 η 1 -A 2 η 2 -T * (2νD(w)n + L 1 (η 1 )n)   , (3.25) D(A S ) = {(w, η 1 , η 2 ) ∈ [H 2 (Ω)] 3 × D(A 1 ) × D(A 1/2 1 ) ∩ V, 2νD(w)n + β(w -T η 2 ) + L 1 (η 1 )n + L 3 (η 1 ) τ = 0 on ∂Ω}, (3.26)
and the operator A S dened as follows

A S = PA S , D(A S ) = D(A S ).
(3.27) Proposition 3.3. There exists λ 0 > 0 such that the operator λ 0 Id -A S is the innitesimal generator of a strongly continuous semigroup of contractions on H.

Proof. Let W = (w, ξ 1 , ξ 2 ) ∈ D(A S ). We show that λ 0 Id -A S is dissipative.

(λ 0 Id -A S )W, W = λ 0 Ω |w| 2 dy + Ω (∇w S ) * w • w dy + 2ν Ω |Dw| 2 dy + Ω L 1 (ξ 1 ) : ∇w dy + Ω L 2 (ξ 1 , ξ 2 ) • w dy + ∂Ω β |(w -T ξ 2 ) τ | 2 dΓ + ∂Ω [L 3 (ξ 1 )] τ • (w -T ξ 2 ) τ dΓ + ω A 1/2 2 ξ 2 2 ds + λ 0 ω A 1/2 1 ξ 1 2 ds + λ 0 ω |ξ 2 | 2 ds. (3.28)
From (3.4) and the fact that

Q i , i = 1, ..., 8 are of regularity W 1,∞ (Ω), then L 1 (ξ 1 ) 2 [L 2 (Ω)] 9 C A 1/2 1 ξ 1 2 L 2 0 (ω) , (3.29) L 2 (ξ 1 , ξ 2 ) 2 [L 2 (Ω)] 3 C A 1/2 1 ξ 1 2 L 2 0 (ω) + A 1/2 2 ξ 2 2 L 2 0 (ω) . (3.30)
Since ξ 1 is a periodic function, we get

ξ 1 L 6 (ω) + ∂ s ξ 1 L 6 (ω) C ξ 1 H 2 (ω) C A 1/2 1 ξ 1 L 2 0 (ω)
.

Using that Q 9 and Q 10 are of regularity W 1/2,∞ (∂Ω) → L 3 (∂Ω), then we obtain

L 3 (ξ 1 ) 2 [L 2 (Ω)] 3 C A 1/2 1 ξ 1 2 L 2 0 (ω) . ( 3
+ Ω L 1 (ξ 1 ) : ∇w dy + Ω L 2 (ξ 1 , ξ 2 ) • w dy + ∂Ω [L 3 (ξ 1 )] τ • (w -T ξ 2 ) τ dΓ -ε w 2 [H 1 (Ω)] 3 + A 1/2 2 ξ 2 2 L 2 (Ω) -C w 2 [L 2 (Ω)] 3 + ξ 2 2 L 2 (ω) + A 1/2 1 ξ 1 2 L 2 (ω)
. (3.32)

Thanks to the classical Korn's inequality, we obtain for λ 0 large enough

(λ 0 Id -A S )W, W 0.
It remains to show that the oprator λ 0 Id -A S is m-dissipative: we prove that λ 0 Id -A S is onto.

Let F =   f g h  
∈ H and we consider the equation

(λ 0 Id -A S )   w η 1 η 2   = F, (3.33) λ 0 w -∇ • T(w, q) + (w S • ∇)w + (w • ∇)w S -∇ • L 1 (η 1 ) + L 2 (η 1 , η 2 ) = f in Ω, (3.34a) ∇ • w = 0 in Ω, (3.34b) λ 0 η 1 -η 2 = g in ω, (3.34c)
λ 0 η 2 + A 1 η 1 + A 2 η 2 = -T * (T(w, q)n + L 1 (η 1 )n) + h in ω, (3.34d) [w -T η 2 ] n = 0 on ∂Ω, (3.34e) 2νD(w)n + β(w -T η 2 ) + L 1 (η 1 )n + L 3 (η 1 ) τ = 0 on ∂Ω. (3.34f)
To solve the system above, we introduce the space

V = (φ, ξ) ∈ [H 1 (Ω)] 3 × D(A 1/2 1 ) | ∇ • φ = 0 in Ω, [φ -T ξ] n = 0 on ∂Ω .
Then, solving the equation (3.33) is reduced to solve the following variational problem: one need to nd

(w, η 2 ) ∈ V such that a w η 2 , φ ξ = L φ ξ φ ξ ∈ V , (3.35)
with a : V × V -→ R the bilinear form given by

a w η 2 , φ ξ = λ 0 Ω w • φ dy + Ω ((w S • ∇)w + (w • ∇)w S ) • φ dy + 2ν Ω D(w) : D(φ) dy + 1 λ 0 Ω L 1 (η 2 ) : ∇φ dy + 1 λ 0 Ω L 2,1 (η 2 ) • φ dy + Ω L 2,2 (η 2 ) • φ dy + λ 0 ω η 2 • ξ ds + ω (A 2 η 2 ) • ξ ds + 1 λ 0 ω (A 1/2 1 η 2 ) • (A 1/2 1 ξ) ds + ∂Ω β[w -T (η 2 )] τ • [φ -T (ξ)] τ dΓ + 1 λ 0 ∂Ω [L 3 (η 2 )] τ • [φ -T (ξ)] τ dΓ,
and L : V -→ R the linear form dened by

L φ ξ = Ω f • φ dy + ω h • ξ ds - 1 λ 0 ω (A 1/2 1 g) • (A 1/2 1 ξ) ds - 1 λ 0 Ω L 1 (g) : ∇φ dy - 1 λ 0 Ω L 2,1 (g) • φ dy - 1 λ 0 ∂Ω [L 3 (g)] τ • [φ -T (ξ)] τ dΓ.
The bilinear form a is continuous and coercive on V thanks to the classical Korn's inequality and L is continuous on V. Using the Lax-Milgram theorem, there exists a unique (w, η 2 ) ∈ V that is solution to (3.35). Now, taking ξ = 0 and φ ∈ D σ (Ω), the equation (3.35) becomes

λ 0 Ω w • φ dy + Ω ((w S • ∇)w + (w • ∇)w S ) • φ dy + 2ν Ω D(w) : D(φ) dy + Ω L 1 (η 1 ) : ∇φ dy + Ω L 2 (η 1 , η 2 ) • φ dy = Ω f • φ dy,
that is equivalent to

λ 0 w + (w S • ∇)w + (w • ∇)w S -ν∆w -∇ • L 1 (η 1 ) + L 2 (η 1 , η 2 ) -f, φ = 0, ∀φ ∈ D σ (Ω).
Using the De Rham theorem [29, Proposition 1.2, p.14], we deduce the existence of a unique element q ∈ L 2 (Ω)/R such that (3.34a) is satised. In particular, we have ∇•T(w, q) ∈ [L 2 (Ω)] 3 and T(w, q) ∈ [L 2 (Ω)] 9 . Consequently, we get T(w, q)n ∈ [H -1/2 (∂Ω)] 3 and

Ω T(w, q) : D(φ) dy -T(w, q)n, φ H -1/2 ,H 1/2 = Ω (f -λ 0 w -(w S • ∇)w -(w • ∇)w S -L 2 (η 1 , η 2 )) • φ dy - Ω L 1 (η 1 ) : ∇φ dy + ∂Ω [L 1 (η 1 )n] τ • φ τ dΓ, (3.36) for all φ ∈ [H 1 (Ω)] 3 , φ n = 0. Taking ξ = 0 in (3.35), we obtain λ 0 Ω w • φ dy + 2ν Ω D(w) : D(φ) dy + Ω L 1 (η 1 ) : ∇φ dy + Ω L 2 (η 1 , η 2 ) • φ dy + Ω ((w S • ∇)w + (w • ∇)w S ) • φ dy + [β(w -T (η 2 )) + L 3 (η 1 )] τ , φ H -1/2 ,H 1/2 = Ω f • φ dy, (3.37)
for any φ ∈ [H 1 (Ω)] 3 , ∇ • φ = 0, φ n = 0. Comparing (3.36) and (3.37) and taking into account the fact

Ω T(w, q) : D(φ) dy = 2ν Ω D(w) : D(φ) dy, ∀φ ∈ [H 1 (Ω)] 3 , ∇ • φ = 0, φ n = 0, we obtain for all φ ∈ [H 1 (Ω)] 3 satisfying ∇ • φ = 0 and φ n = 0 -T(w, q)n, φ H -1/2 ,H 1/2 = [β(w -T η 2 ) + L 1 (η 1 )n + L 3 (η 1 )] τ , φ H -1/2 ,H 1/2 . (3.38)
Then, (3.38) is also satised for all φ ∈ [H 1 (Ω)] 3 , φ n = 0 since we can always construct a divergence free function on Ω that coincides with φ at the boundary. Plugging (3.38) in (3.36), we obtain

Ω ((w S • ∇)w + (w • ∇)w S ) • φ dy + 2ν Ω D(w) : D(φ) dy - Ω q∇ • φ dy + [β(w -T η 2 ) + L 3 (η 1 )] τ , φ τ H -1/2 ,H 1/2 = Ω (f -λ 0 w -L 2 (η 1 , η 2 )) • φ dy - Ω L 1 (η 1 ) : ∇φ dy, (3.39) for any φ ∈ [H 1 (Ω)] 3 , φ n = 0.
Then, we deduce that (w, q) is a weak solution of (3.34a), (3.34b), (3.34e) and (3.34f) in the sens of the [8, Denition, p.10]. Since

η 2 ∈ H 2 (ω) then T η 2 ∈ [H 2 (∂Ω)] 3 , we can apply [8, Théorème 1.2] to get that (w, q) ∈ [H 2 (Ω)] 3 × H 1 (Ω)/R.
Then, going back to (3.35), we get

ω (A 1/2 1 η 1 ) • (A 1/2 1 ξ) ds = -λ ω η 2 • ξ ds - ω (A 2 η 2 ) • ξ ds - ω T * (T(u, q)n + L 1 (η 1 )n) • ξ ds + ω h • ξ ds, (3.40)
for all ξ ∈ D(A

1/2 1 ) where η 1 = 1 λ (g + η 2
). We notice that we have η 1 ∈ D(A 1/2 ), then T * (L 1 (η 1 )n) ∈ L 2 0 (ω). In the other hand, we have T(w, q)n ∈ [H 1/2 (∂Ω)] 3 , then it implies that T * (T(w, q)n) ∈ L 2 0 (ω). Moreover, since η 2 ∈ H 2 (ω), we deduce that η 2 ∈ D(A 2 ). Thus, using the fact that D(A

1/2 1 ) is dense in L 2 0 (ω), we obtain A 1 η 1 ∈ L 2 0 (ω).
Applying the Lumer-Phillips theorem, we deduce that λ 0 Id -A S generates a strongly continuous semigroup of contractions on H.

Proposition 3.4. The adjoint operator of A S is given by

A * S   w η 1 η 2   = P   ν∆w + (w S • ∇)w -(∇w S ) * w -η 2 -A -1 1 (L 1 ) * )(∇w) + (L 2,1 ) * w + (L 3 ) * w -((L 3 ) * (T η 2 ) A 1 η 1 -A 2 η 2 -T * (2νD(w)n) -(L 2,2 ) * w   , (3.41)
and

D(A * S ) = {(w, η 1 , η 2 ) ∈ [H 2 (Ω)] 3 × D(A 1 ) × D(A 1/2 1 ) ∩ V, [2νD(w)n + β(w -T η 2 )] τ = 0 on ∂Ω}. (3.42) Proof. Let (w, η 1 , η 2 ) ∈ D(A S ) and (ϕ, ζ 1 , ζ 2 ) ∈ D(A * S ). We have A S (w, η 1 , η 2 ), (ϕ, ζ 1 , ζ 2 ) = -2ν Ω D(w) : D(ϕ) dy + Ω (w S • ∇)ϕ • w - Ω (∇w S ) * ϕ • w dy - Ω L 1 (η 1 ) : ∇w dy - Ω L 2 (η 1 , η 2 ) • w dy - ∂Ω β(w -T η 2 ) τ • (ϕ -T ζ 2 ) τ dΓ - ∂Ω [L 3 (η 1 )] τ • (ϕ -T ζ 2 ) τ dΓ - ω (A 2 ζ 2 ) • η 2 ds. (3.43) Then A S (w, η 1 , η 2 ), (ϕ, ζ 1 , ζ 2 ) = ν Ω ∆ϕ • w dy + Ω (w S • ∇)ϕ • w - Ω (∇w S ) * ϕ • w dy - Ω L 1 (η 1 ) : ∇ϕ dy - Ω L 2 (η 1 , η 2 ) • ϕ dy - ∂Ω [L 3 (η 1 )] τ • (ϕ -T ζ 2 ) τ dΓ - ω (A 2 ζ 2 ) • η 2 ds - ω T * (2νD(ϕ)n)η 2 ds. (3.44)
Using (3.4) and (3.5), we obtain (3.41).

Proposition 3.5. For θ ∈ [0, 1], we have

D((λ 0 Id -A S ) θ ) = [D(A), H] 1-θ , D((λ 0 Id -A * S ) θ ) = [D(A * ), H] 1-θ , (3.45)
for λ 0 > 0 large enough.

Proof. The proof is a direct consequence of proposition 3.3 and of [9, Proposition

Proposition 3.6. The operator A S dened by (3.25) and (3.27) is the innitesimal generator of an analytical semigroup on H.

Proof. We notice that A * S = -A * + O S where A is the operator dened by

D(A) = {(w, η 1 , η 2 ) ∈ [H 2 (Ω)] 3 × D(A 1 ) × D(A 1/2 1 ) ∩ V, [2νD(w)n + β(w -T η 2 )] τ = 0 on ∂Ω}, (3.46)
A   w η 1 η 2   =   -ν∆w -η 2 A 1 η 1 + A 2 η 2 + T * (2νD(w)n)   , (3.47)
D(A) = D(A), A = PA, (3.48)
and

O S   w ξ 1 ξ 2   = P   (w S • ∇)w -(∇w S ) * w -A -1 1 (L 1 ) * )(∇w) + (L 2,1 ) * w + (L 3 ) * w -((L 3 ) * (T ξ 2 ) -(L 2,2 ) * w   .
We precise here D((A * ) 1/2 ) = V.

From [START_REF] Djebour | On the existence of strong solutions to a uid structure interaction problem with Navier boundary conditions[END_REF], A is the innitesimal generator of an analytical semigroup on H. Then, from [21, Corollaire 2.4, p.81], we only need to show that V ⊂ D(O S ). In fact, let (w, ξ 1 , ξ 2 ) ∈ V and (φ, η 1 , η 2 ) ∈ H. First, we use (3.5) and we observe that for

η 2 ∈ H 1 (ω) ∩ L 2 0 (ω) - ω (L 2,2 ) * w • η 2 ds = - Ω w • L 2,2 (η 2 ) dy = - Ω w • (Q 7 η 2 + Q 8 ∇η 2 ) dy = - Ω w • Q 7 η 2 dy - Ω w • Q 8 ∇η 2 dy. (3.49)
By integration by parts, we nd that

Ω w • Q 7 η 2 dy + Ω w • Q 8 ∇η 2 dy C w [H 1 (Ω)] 3 η 2 L 2 (ω) .
Then, by density, we get

O S   w ξ 1 ξ 2   ,   ϕ η 1 η 2   H C w [H 1 (Ω)] 3 ϕ [L 2 (Ω)] 3 + A 1/2 η 1 L 2 (ω) + A 1/2 η 1 L 2 (ω) ξ 2 L 2 (ω) + w [H 1 (Ω)] 3 η 2 L 2 (ω) C (w, ξ 1 , ξ 2 ) V (ϕ, η 1 , η 2 ) H . (3.50)
We deduce that A S is the innitesimal generator of an analytical semigroup on H. Moreover, A S admits a compact resolvent.

Let with

V s n (Γ 0 ) = {v ∈ H s (Γ 0 ) ; Γ0 v • n dΓ = 0}, s 0.
ξ 1 = ξ, ξ 2 = ∂ t ξ, W =   w ξ 1 ξ 2   , F =   f 0 h   , W 0 =   w 0 ξ 0 ξ 1   , where D ∈ L(V 0 n (Γ 0 ), [L 2 (Ω)] 3 × D(A 1/2 1 ) × L 2 0 (ω)) such that Dv = (w, η 1 , η 2 ) veries the system                λ 0 w -∇ • (T(w, π)) + (w S • ∇)w + (w • ∇)w S -∇ • L 1 (η 1 ) + L 2 (η 1 , η 2 ) = 0 in Ω, ∇ • w = 0 in Ω, λ 0 η 1 -η 2 = 0 in ω, λ 0 η 2 + A 1 η 1 + A 2 η 2 + T * (T(w, π)n + L 1 (η 1 )n) = 0 in ω, [w -T η 2 ] n = 1 Γ0 v n on ∂Ω, 2νD(w)n + β(w -T η 2 ) + L 1 (η 1 )n + L 3 (η 1 ) τ = 1 Γ0 (βv τ ) on ∂Ω.
with λ 0 ∈ ρ(A S ), and the operator B is dened by

B : v ∈ U = [H 3/2 (Γ 0 )] 3 -→ (λ 0 Id -A S )PD(Mv) ∈ (D(A * S )) .
We notice that (λ 0 Id -A S ) -1+ε B ∈ L([H 3/2 (Γ 0 )] 3 , H), ∀ε ∈ (0, 1/4).

The adjoint operator

D * ∈ L([L 2 (Ω)] 3 × D(A 1/2 1 ) × L 2 0 (ω), [L 2 (Γ 0 )] 3 ) is dened for all (f , g, h) ∈ [L 2 (Ω)] 3 × D(A 1/2 1 ) × L 2 0 (ω) by D * (f , g, h) = ((T(φ, r)n) n + (T(φ, r)n) τ )| Γ0 , if β 1 > 0, D * (f , g, h) = (T(φ, r)n) n | Γ0 , if β 1 = 0, such that (φ, r, ζ 1 , ζ 2 ) ∈ [H 2 (Ω)] 3 × H 1 (Ω)/R × D(A 1 ) × D(A 1/2 1
) is solution of the system

               λ 0 φ -∇ • (T(φ, r)) -(w S • ∇)φ + (∇w S ) * φ = f in Ω, ∇ • φ = 0 in Ω, λ 0 ζ 1 + ζ 2 + A -1 1 ((L 1 ) * )(∇φ) + (L 2,1 ) * φ + (L 3 ) * φ -(L 3 ) * (T ζ 2 ) = g in ω, λ 0 ζ 2 -A 1 ζ 1 + A 2 ζ 2 + T * (T(φ, r)n) + (L 2,2 ) * φ = h in ω, (φ -T ζ 2 ) n = 0 on ∂Ω, [2νD(φ)n + β(φ -T ζ 2 )] τ = 0 on ∂Ω.
We precise that (φ, ζ 1 , ζ 2 ) is solution of the system

(λ 0 -A * S )(φ, ζ 1 , ζ 2 ) = P(f , g, h). (3.52)
If β 1 > 0, the operator B * is given by

B * (φ, ζ 1 , ζ 2 ) = m (T(φ, r)n) • n - Γ0 mT(φ, r)n • n dΓ n + m(T(φ, r)n) τ Γ0 ,
and if β 1 = 0, we get We deduce that T(φ, r -c(φ, r))n = 0, on Γ 0 . Then, from [START_REF] Fabre | Prolongement unique des solutions de l'equation de Stokes[END_REF], we get φ = 0, in Ω, r = c(φ, r), on Ω.

B * (φ, ζ 1 , ζ 2 ) = m (T(φ, r)n) • n - Γ0 mT(φ,
               λφ -∇ • (T(φ, r)) -(w S • ∇)φ + (∇w S ) * φ = 0 in Ω, ∇ • φ = 0 in Ω, λζ 1 + ζ 2 + A -1 1 ((L 1 ) * )(∇φ) + (L 2,1 ) * φ -(L 3 ) * φ + (L 3 ) * (T ζ 2 ) = 0 in ω, λζ 2 -A 1 ζ 1 + A 2 ζ 2 + T * (T(φ, r)n) + (L 2,2 ) * φ = 0 in ω, (φ -T ζ 2 ) n = 0 on ∂Ω, [2νD(φ)n + β(φ -T ζ 2 )] τ = 0 on ∂Ω.
In particular T(φ, r -c(φ, r))n = 0, on ∂Ω. Theorem 3.7. Let t 0 > 0, σ > 0 and W 0 ∈ V. Then, there exists

N + ∈ N * , K ∈ L ∞ loc (R 2 ; L(H)), (φ k , ζ k 1 , ζ k 2 ) ∈ D(A * S ) and v k ∈ B * (D(A * S )), k = 1, . . . , N + , such that v(t) = 1 [t0,+∞) (t) N+ k=1   W (t -t 0 ) + t-t0 0 K(t -t 0 , s)W (s) ds,   φ k ζ k 1 ζ k 2     H v k , (3.59)
stabilizes exponentially the system (3.51) and we have Corollary 3.8. Let γ 0 > 0. For all γ ∈ [0, γ 0 [, the solution ( w, q, ξ) of the system (3.1), (3.2), (3.3) veries the estimate

W L 2 σ (0,∞;D(A S ))∩C 0 σ ([0,∞);V)∩H 1 σ (0,∞;H) C W 0 V + F L 2 σ (0,∞;H) . ( 3
( w, q, ξ) X∞,γ C (w 0 , ξ 0 , ξ 1 ) V + f L 2 γ (0,+∞;[L 2 (Ω)] 3 ) + h L 2 γ (0,+∞;L 2 (ω)) + g W 1/4 γ (0,+∞;[H 1/2 (∂Ω)] 3 ,[L 2 (∂Ω)] 3 ) . (3.61) 4 Fixed point Let Y ∞ and B ∞,R be given respectively by Y ∞ = L 2 γ (0, ∞; [L 2 (Ω)] 3 ) × W 1/4 γ (0, ∞; [H 1/2 (∂Ω)] 3 , [L 2 (∂Ω)] 3 ) × L 2 γ (0, ∞; L 2 (ω)). (4.1) B ∞,R = {( f , g, h) ∈ Y ∞ | ( f , g, h) Y∞ R}. (4.2)
Let ( f , g, h) ∈ B ∞,R and let (u, p, ξ) be the solution of the system (3.1), (3.2), (3.3) associated to ( f , g, h). From (3.61), we obtain (u, p, ξ) X∞,γ CR,

where we supposed that (w 0 , ξ 0 , ξ 1 ) V R.

From (4.3), we have the following estimates

ξ L ∞ γ (0,∞;L ∞ (ω)) + ∂ sj ξ L ∞ γ (0,∞;L ∞ (ω)) + ∂ 2 sj s k ξ L ∞ γ (0,∞;L 2 (ω)) + ∂ 3 sj s k si ξ L ∞ γ (0,∞;L 2 (ω)) C ξ L ∞ γ (0,∞;H 3 (ω))
CR, (4.4)

∂ t ξ L 4 γ (0,∞;L ∞ (ω)) + ∂ 2 tsj ξ L 6 γ (0,∞;L 2 (ω)) CR, (4.5
)

ξ H 7/8 γ (0,∞;L ∞ (ω)) + ∂ sj ξ H 7/8 γ (0,∞;L ∞ (ω)) + ∂ 2 sj s k ξ H 7/8 γ (0,∞;L 8/3 (ω)) CR, (4.6)
ξ L ∞ γ (0,∞;H 2 (ω)) + ∂ sj ξ L ∞ γ (0,∞;H 3/2 (ω)) + ∂ 2 sj s k ξ L ∞ γ (0,∞;H 1/2 (ω)) CR. (4.7)
In particular, taking into account the condition (1.31), there exists R 0 > 0 such that, if R R 0 , then

1 1 + η L ∞ (0,∞;L ∞ (ω)) C(R 0 ). (4.8)
In fact, if 1 + η S > δ, then R 0 is chosen such that 0 < R 0 < δ.

Here, we recall a well known result about product in Sobolev spaces.

Proposition 4.1. Let s 0, s 1 s and s 2 s such that s 1 + s 2 > s + 1/2. Let X 1 , X 2 and X 3 be three Banach spaces such that for all f ∈ X 2 and g ∈ X 3 , we have

f g X1 C f X2 g X3 . Then ∀u ∈ H s1 (0, ∞; X 2 ), ∀v ∈ H s2 (0, ∞; X 3 ), uv H s (0,∞;X1) C u H s 1 (0,∞;X2) v H s 2 (0,∞;X3) .
Proof. The proof is similar to the one given in [START_REF] Zolesio | Multiplication dans les espaces de Besov[END_REF]Theorem 2] extending data by reexion on R.

Lemma 4.2. Let X and Y given in (1.18). Then, we have

∇Y (X) -I 3 = (1) (ξ, ∂ s ξ) + (1) (ξ, ∂ s ξ), (4.9)
∂ 2 Y 3 ∂x k ∂x j (X) = (2) (ξ, ∂ s ξ, ∂ 2 ss ξ) + (2) (ξ, ∂ s ξ, ∂ 2 ss ξ), (4.10)
∂ 2 a ik ∂x m ∂x l (X) = (3) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ) + (3) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ), (4.11)
∂ t Y = (13) (∂ t ξ) + (13) (ξ, ∂ t ξ), (4.12)
∂ t a ik (X) = (14)

(∂ t ξ, ∂ 2 ts ξ) + (14) (ξ, ∂ t ξ, ∂ 2 ts ξ), (4.13)
where the operators l (i) have the form (2.21) and (i) are given formally by

(1) (ξ, ∂ s ξ) = O(ξ m1 (∂ s ξ) m2 ), m 1 + m 2 2, (2) (ξ, ∂ s ξ, ∂ 2 ss ξ) = O(ξ m 1 (∂ s ξ) m2 (∂ 2 ss ξ) m3 ), m 1 + m 2 + m 3 2, (3) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ) = O(ξ m 1 (∂ s ξ) m2 (∂ 2 ss ξ) m3 (∂ 3 sss ξ) m4 ), m 1 + m 2 + m 3 + m 4 2, (13) (ξ, ∂ t ξ) = O(ξ m1 (∂ t ξ) m2 ), m 1 + m 2 2, (14) (ξ, ∂ t ξ, ∂ 2 ts ξ) = O(ξ m1 (∂ t ξ) m2 (∂ 2 ts ξ) m3 ), m 1 + m 2 + m 3 2.
Proof. We recall that ξ = η -η S , then we have

1 1 + η = 1 1 + η S - ξ (1 + η S ) 2 + ξ 2 (1 + η S ) 2 (1 + η) . (4.14)
After standard calculations, ∇Y (X) -I 3 writes

∂Y 3 ∂x 3 (X) -1 = η S -η 1 + η = - ξ 1 + η S + ξ 2 (1 + η S ) 2 - ξ 3 (1 + η S ) 2 (1 + η) , (4.15)
∂Y 3 ∂x j (X) = -y 3 ∂ sj ξ 1 + η S + y 3 ∂ sj η S ξ (1 + η)(1 + η S ) + y 3 ∂ sj ξ ξ (1 + η)(1 + η S ) , j = 1, 2. (4.16)
Using (4.14), we obtain (4.9) where (1) (ξ, ∂ s ξ) contains the terms of the form

γ 1 (η S ) ξ m1 (1 + ξ + η S ) α1 , γ 2 (η S ) ξ m2 ∂ s ξ (1 + ξ + η S ) α2 , m 1 2, m 2 1, α i 1, i = 1, 2, (4.17)
where γ i , i = 1, 2 are functions of regularity C 3 (ω). We have for all k, j ∈ {1, 2},

∂ 2 Y 3 ∂x k ∂x j (X) = -y 3 ∂ 2 sj s k ξ (1 + η S ) + y 3 ∂ s k η ∂ sj ξ (1 + η)(1 + η S ) + y 3 ∂ sj η ∂ s k ξ (1 + η)(1 + η S ) + y 3 ξ ∂ 2 s k sj η (1 + η S )(1 + η) -2 ∂ s k η∂ sj η (1 + η S )(1 + η) 2 . (4.18)
Then, we deduce (4.10) where (2) (ξ, ∂ s ξ, ∂ 2 ss ξ) had the terms

γ 3 (η S ) ξ m1 (∂ s ξ) m2 (1 + ξ + η S ) α1 , m 1 + m 2 2, γ 4 (η S ) ξ m3 ∂ 2 ss ξ (1 + ξ + η S ) α2 , m 3 1, α i 1, i = 1, 2, (4.19)
where γ i , i = 3, 4 are C 2 (ω) functions. The third derivative ∂ 3 Y 3 ∂x i ∂x k ∂x j (X) admits the terms:

y 3 ∂ 3 sj s k si ξ (1 + η S ) , y 3 ∂ si η∂ 2 sj s k ξ (1 + η)(1 + η S ) , y 3 ∂ 2 sis k η∂ sj ξ (1 + η)(1 + η S ) , y 3 ∂ s k η∂ si η∂ sj ξ (1 + η) 2 (1 + η S ) , y 3 ∂ 2 sj s k η∂ si ξ (1 + η) 2 (1 + η S ) , y 3 ξ∂ 3 sj s k si η (1 + η S )(1 + η) , y 3 ξ∂ 2 s k sj η∂ si η (1 + η) 2 (1 + η S ) , y 3 ξ∂ s k η∂ sj η∂ si η (1 + η) 4 (1 + η S ) . (4.20)
Then, we get (4.11) where (3) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ) contains the terms

γ 5 (η S ) ξ m1 (∂ s ξ) m2 (1 + ξ + η S ) α1 , γ 6 (η S ) ξ n1 (∂ s ξ) n2 ∂ 2 ss ξ (1 + ξ + η S ) α2 , γ 7 (η S ) ξ m3 ∂ 3 sss ξ (1 + ξ + η S ) α3 , m 1 + m 2 2, n 1 + n 2 2, m 3 1, α i 1, i = 1...3, (4.21)
with γ i , i = 5, ..., 7 are bounded functions on ω.

We also have,

∂ t Y 3 = -y 3 ∂ t ξ 1 + η .
We deduce (4.12) where (13) (ξ, ∂ t ξ) had the terms

ξ m1 ∂ t ξ (1 + ξ + η S ) α1 , m 1 1, α 1 1. (4.22)
The terms appearing in ∂ t a ik (X) have the form

y 3 ∂ t ξ∂ sj η (1 + η) 2 , y 3 ∂ t ξ∂ sj η S (1 + η)(1 + η S ) , y 3 ∂ 2 tsj ξ (1 + η) , - (1 + η S )∂ t ξ (1 + η) 2 .
Then, we get (4.13) with (14) (ξ, ∂ t ξ, ∂ 2 ts ξ) contains the terms

γ 8 (η S ) ξ m1 ∂ 2 ts ξ (1 + ξ + η S ) α1 , γ 9 (η S ) ξ m2 ∂ t ξ (1 + ξ + η S ) α2 , γ 10 (η S ) ξ m3 ∂ t ξ∂ s ξ (1 + ξ + η S ) α3 , m 1 1, m 2 1, m 3 0, α i 1, i = 1, ..., 3, (4.23)
where γ i , i = 8, ..., 10 are C 3 (ω) functions.

Let suppose that R < 1.

Lemma 4.3. Let (i) given in lemma 4.2 and let (w S , p S , η S ) ∈ [W 2,∞ (Ω)] 3 × W 1,∞ (Ω) × C 4 (ω).

(4.24)

Then, we have (1) (ξ, ∂ s ξ)

L ∞ γ (0,∞;[L ∞ (Ω)] 9 )
+ (2) (ξ, ∂ s ξ, ∂ 2 ss ξ)

L ∞ γ (0,∞;L 2 (Ω))

+ (3) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ) L ∞ γ (0,∞;L 2 (Ω))
+ (13) (ξ, ∂ t ξ)

L 4 γ (0,∞;[L ∞ (Ω)] 3 )
+ (14) (ξ, ∂ t ξ, ∂ 2 ts ξ)

L 6 γ (0,∞;L 2 (Ω))
CR 2 , (4.25)

(1) (ξ, ∂ s ξ)

L ∞ γ (0,∞;[H 3/2 (∂Ω)] 9 )

+ (2) (ξ, ∂ s ξ, ∂ 2 ss ξ)

L ∞ γ (0,∞;H 1/2 (∂Ω))

+ (1) (ξ, ∂ s ξ) Proof. Using (4.24), (4.4),(4.8) and (4.17), we get (1) (ξ, ∂ s ξ) Moreover, from (2.48), N F (ξ) has the terms a 1 (η S , w S ) ξ m1 (∂ s ξ) m2 (1 + ξ + η S ) α1 , a 2 (η S , w S ) ξ n1 (∂ s ξ) n2 ∂ 2 ss ξ (1 + ξ + η S ) α2 , a 3 (η S , w S ) ξ m3 ∂ 2 ts ξ (1 + ξ + η S ) α3 , a 4 (η S , w S ) ξ m4 ∂ t ξ (1 + ξ + η S ) α4 , a 5 (η S , w S ) ξ m5 ∂ t ξ∂ s ξ (1 + ξ + η S ) α5 , m 1 + m 2 2, n 1 + n 2 2, m 3 1, m 3 1, m 4 1, m 5 0, α i 1, i = 1...5, where a i , i = 1, ..., 5 are bounded functions in Ω.

L ∞ γ (0,∞;[L ∞ (Ω)]
Thus, using (4.8), (4.24), (4.4) and (4.5), we obtain

N F (ξ) L 2 γ (0,∞;[L 2 (Ω)] 9 ) CR 2 .
The functions N E (ξ) and N G (ξ) admit terms of the form (4.47). Then, from lemma 4.3 and (4.24), we have the estimate (4.45). Finally, using lemma 4.3, (4.3) and the same procedure described in [START_REF] Djebour | On the existence of strong solutions to a uid structure interaction problem with Navier boundary conditions[END_REF], we get (4.46). In this case, we show that for R small enough Φ(B ∞,R ) ⊂ B ∞,R and Φ |B ∞,R is a strict contraction. Let ( f , g, h) ∈ B ∞,R , from lemma 4.4, we obtain Φ( f , g, h)

Y∞ CR 2 .
We get similarly Φ( f (1) , g (1) , h (1) ) -Φ( f (2) , g (2) , h (2) )

Y∞ CR ( f (1) , g (1) , h (1) ) -(f (2) , g (2) , h (2) )

Y∞ .

for ( f , g, h), ( f (i) , g (i) , h (i) ) ∈ B ∞,R . Thus, we deduce the principal result of this paper.

Figure 1 :

 1 Figure 1: Conguration of the domain corresponding to the plate displacement η.

. 31 )

 31 Using (3.29), (3.30) and (3.31), we nd Ω (∇w S ) * w • w dy

From [ 26 ]

 26 , the system (3.20), (3.21), (3.22) is equivalent to PW = A S PW + PF + Bv, PW (0) = PW 0 , (Id -P)W = (Id -P)D(Mv), (3.51)

  r)n • n dΓ n Γ0 , where r ∈ H 1 (Ω)/R, is the solution of the problem establish a rst result of stabilization of the linear system (3.20), (3.21) and (3.22). To do so, it suces to verify that (A S , B) satises the Fattorini-hautus criterion (UC σ ) and to apply [14, Theorem 1.1 ]. Let σ > 0 and λ ∈ C such that Re λ -σ. Let (φ, r, ζ 1 , ζ 2 ) solution of the system

  and assume that B * (φ, ζ 1 , ζ 2 ) = 0, it implies that for β 1 0, we havem (T(φ, r)n) • n -Γ0 mT(φ, r)n • n dΓ n + m(T(φ, r)n) τ = 0. r)n • n dΓ.The equation (3.54) becomes m(T(φ, r -c(φ, r))n) n + m(T(φ, r -c(φ, r))n) τ = 0.(3.55)

  obtain T * (T(φ, r)n) = -c(φ, r)T * (n), on ∂Ω. (3.57) From the denition of the operator T * given in (2.51), we have that T * (n) = 0. Then, from (3.57), we obtain T * (T(φ, r)n) = 0, on ∂Ω. (3.58) Moreover, we have T ζ 2 = 0 on ∂Ω, then ζ 2 = 0. The equation (3.58) implies that A 1 ζ 1 = 0 and taking into account the periodicity of ζ 1 , we nd ζ 1 = 0. Then, the condition (UC σ ) is veried. We deduce then the following theorem.

  .60) Finally, using (3.60), (3.8) and (3.23), we obtain the following corollary.

CR 2 .

 2 ∞;[L ∞ (∂Ω)] 9 ) + (2) (ξ, ∂ s ξ, ∂ 2 ss ξ) (4.26)

H

  ∞;L ∞ (ω)) ∂ s ξ H 7/8 γ (0,∞;L ∞ (ω))CR 2 . (4.39) Since 7/8 > 1/2, we use the proposition 4.1, we obtain(∂ s ξ) m2 H 7/8 (0,∞;L ∞ (ω))

∂ 2 ss ξ ( 1 +

 1 ξ + η S ) α2 , d 3 (η S , w S ) ξ m3 ∂ 3 sss ξ (1 + ξ + η S ) α3 , m 1 + m 2 2, n 1 + n 2 2, m 3 1, α i 1, i = 1...3,where d i , i = 1, ..., 3 are bounded functions Ω.From (4.8), (4.24) and (4.4), we obtain∇ • N E (ξ) L 2 γ (0,∞;[L 2 (Ω)] 3 )

4 . 1 Proof of theorem 1. 1 We

 411 dene the application Φ : B ∞,R -→ Y ∞ by Φ( f , g, h) = (∇ • N E (ξ) + N F (ξ) + F(u, p, ξ), -[N E (ξ)n] τ + [N G (ξ)] τ + G(u, ξ + η S ), H(u, ξ + η S ) -T * (N E (ξ)n)).(4.50)

  9 ) Using (4.27), (4.28), (4.29), (4.30) and (4.31), we deduce (4.25). In the other hand, we haveξ∂ s ξ L ∞ γ (0,∞;H 3/2 (ω)) C ξ L ∞ γ (0,∞;H 2 (ω)) ∂ s ξ L ∞ γ (0,∞;H 3/2 (ω))Then, using the fact that∂ s ξ ∈ L ∞ γ (0, ∞; H 2 (ω)) and ∂ s ξ L ∞

						CR 2 .	(4.32)
	Thus, using (4.8) and (4.4), we have
	1 (1 + ξ + η S ) αi	L ∞ γ (0,∞;H 2 (ω))	C, α i 1.	(4.33)
	Then,	ξ m1 ∂ s ξ (1 + ξ + η S ) αi	L ∞ γ (0,∞;H 3/2 (ω))	CR 2 .	(4.34)
						γ (0,∞;H 2 (ω))	CR, we get
		ξ m1 (∂ s ξ) m2	L ∞ γ (0,∞;H 3/2 (ω))	CR 2 .	(4.35)
	We have also
	ξ∂ 2 ss ξ L ∞ γ (0,∞;H 1/2 (ω))	C ξ L ∞ γ (0,∞;H 3/2 (ω)) ∂ 2 ss ξ L ∞ γ (0,∞;H 1/2 (ω))	CR 2 ,	(4.36)
	Then, using (4.33), we obtain
		ξ m1 ∂ 2 ss ξ (1 + ξ + η S ) αi	L ∞ γ (0,∞;H 1/2 (ω))	CR 2 ,	(4.37)
	Furthermore, using (4.8), we get
	ξ
	(1 + ξ + η S ) αi
						CR 2 .	(4.27)
	From (4.8), (4.24), (4.4) and (4.19), we nd
		(2) (ξ, ∂ s ξ, ∂ 2 ss ξ)	L ∞ γ (0,∞;L 2 (Ω))	CR 2 .	(4.28)
	From (4.8), (4.24), (4.4) and (4.21), we obtain
	(3) (ξ, ∂ s ξ, ∂ 2 ss ξ, ∂ 3 sss ξ)	L ∞ γ (0,∞;L 2 (Ω))	CR 2 .	(4.29)
	From (4.8), (4.24), (4.5), (4.4) and (4.22), we get
		(13) (ξ, ∂ t ξ)	L 4 γ (0,∞;[L ∞ (Ω)] 3 )	CR 2 .	(4.30)
	Using (4.8), (4.24), (4.4), (4.5) and (4.23), we get
		(14) (ξ, ∂ t ξ, ∂ 2 ts ξ)	L 6 γ (0,∞;L 2 (Ω))	CR 2 .	(4.31)

  • N E (ξ) L 2 γ (0,∞;[L 2 (Ω)] 3 ) + N F (ξ) L 2 γ (0,∞;[L 2 (Ω)] 3 ) (∂Ω)] 3 ;[L 2 (∂Ω)] 3 ) + N G (ξ) W 1/4 γ (0,∞;[H 1/2 (∂Ω)] 3 ;[L 2 (∂Ω)] 3 ) (F(u, p, ξ), G(u, ξ + η S ), H(u, ξ + η S )) Y∞ CR 2 . (4.46)Proof. From (2.30) and lemma 4.2, N E (ξ) admits the following termsc 1 (η S , w S ) ξ m1 (∂ s ξ) m2 (1 + ξ + η S ) α1 , c 2 (η S , w S ) ξ n1 (∂ s ξ) n2 ∂ 2 ss ξ (1 + ξ + η S ) α2 , m 1 + m 2 2, n 1 + n 2 2, α i 1, i = 1...2, (4.47)where c i , i = 1, 2 are W 1,∞ (Ω) functions. From (4.8), (4.24) and (4.4), we obtainN E (ξ) L 2 γ (0,∞;[L 2 (Ω)] 3 )From (2.28), (2.30), (2.32), (4.24) and lemma 4.2, the terms that appear in ∇ • N E (ξ) are the followingd 1 (η S , w S ) ξ m1 (∂ s ξ) m2 (1 + ξ + η S ) α1 , d 2 (η S , w S ) ξ n1 (∂ s ξ) n2

	Lemma 4.4. We have
										CR 2 ,	(4.43)
					N E (ξ) L 2 γ (0,∞;[L 2 (Ω)] 9 )	CR 2 ,	(4.44)
	N E (ξ)n W 1/4 γ	(0,∞;[H 1/2 CR 2 ,	(4.45)
										CR 2 .	(4.48)
										C (∂ s ξ)	m2 H 7/8 (0,∞;L ∞ (ω)) .
	We deduce				ξ m1 (∂ s ξ) m2 (1 + ξ + η S ) αi	H	7/8 γ	(0,∞;L ∞ (ω))	CR 2 .	(4.40)
	Moreover,
	ξ∂ 2 ss ξ (1 + ξ + η S ) αi	H	7/8 γ	(0,∞;L 8/3 (ω))	C ∂ 2 ss ξ H 7/8 γ	(0,∞;L 8/3 (ω)) ξ H 5/4 γ	(0,∞;L ∞ (ω))	CR 2 .	(4.41)
	From (4.38), we have
					ξ m1 ∂ 2 ss ξ (1 + ξ + η S ) αi	H	7/8 γ	(0,∞;L 8/3 (ω))	CR 2 ,	(4.42)
	Using (4.35), (4.37), (4.40) and (4.42), we obtain (4.26).

∇


Hans Georg Beyer

Gerhard Czeplak°

Uwe Terzenbach°

Lucien Wald

Assessement of the method used to construct clearness index maps for the new European Solar Radiation Atlas ESRA

We present an assessement of the methods used to construct maps for a new Solar Radiation Atlas for Europe. For this atlas station data and satellite derived data are used in an interpolation/merging process to derive maps of the long term monthly global radiation that cover an area ranging from 30° W to 70° E and from 25° to 75° N. Our focus is on the discussion of the accuracy of the method applied -a co-kriging technique. Special emphasis is put on the discussion whether the use of satellite derived radiation maps with a low spatial resolution brings benefits.

1.Introduction

. The area of interest ranges from 30° West to 70° East and from 25° to 75° North. The subtasks of this project comprise the collection of radiation observations for the reference period 1981-1990, the setup of a toolbox to derive the radiation parameters of relevance for solar energy system design from the available data sets,

 Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997.Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy,61,6,[389][390][391][392][393][394][395][396][397] and the construction of maps of monthly means for the whole area. In this paper we give an assesement of the method used to establish these radiation maps. The interpolation procedure applied to derive estimates of the radiation from both the set of station observations and additional radiation maps derived from satellite images is a co-kriging technique. A short introduction to this techique is given in an appendix.

The next section will give an overview of existing approaches for the construction of radiation maps. Section 3 and 4 present the input data sets and their preparation. The quality of the interpolation procedure is discussed in section 5. Special emphasis will be put on the discussion of the benefits of the use of the satellite maps available.

2.Methods applied for the interpolation of solar radiation fields

As the network of meteorological stations with a registration of solar radiation data is in general sparse, there had always been a need for interpolation procedures. The main techniques used are spline function based interpolations and various weighted average procedures comprising the kriging technique (see e.g. [START_REF] Journel | Mining Geostatistics[END_REF].

Spline functions are e.g. applied by [START_REF] Hutchinson | Estimating monthly mean values of daily total solar raditaion for Australia[END_REF] for the generation of maps of monthly mean irradiance over Australia and by [START_REF] Hulme | A 1961-1990 climatology for Europe for climate change modelling and impact applications[END_REF] for the interpolation of long term (30 years) monthly means of sunshine duration data for Europe.

The application of different weighted average methods for daily radiation in Switzerland and various regions in the USA is discussed by [START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF]. Looking at the interpolation error, they reported only marginal differences between interpolations based on methods using weights according to the inverse squared distance for the neighboring stations and kriging methods. However from a general consideration of the methods it was concluded that the kriging technique is recommendable for the task of interpolation of radiation data from network stations. Further application of kriging methods for solar radiation fields are described by [START_REF] Zelenka | Supplementing network global irradiance data, Advances in Solar Energy Technology[END_REF] and [START_REF] Zelenka | Combining METEOSAT and surface network data: a data fusion approach for mapping solar irradiation[END_REF]. Kriging is also used for the generation of maps of the Meteonorm database for Switzerland [START_REF] Kunz | METEONORM -a comprehensive meteorological database and planing tool[END_REF]. Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy,61,6,[389][390][391][392][393][394][395][396][397] As discussed by [START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF] the use of additional information on the solar radiation gained from the analysis of satellite images may improve the quality of radiation maps. This was proven both for daily and long term monthly mean data [START_REF] D'agostino | Estimating solar global irradiance by integration of satellite and network data: the cokriging approach[END_REF][START_REF] Zelenka | Combining METEOSAT and surface network data: a data fusion approach for mapping solar irradiation[END_REF]. A co-kriging technique or simplified versions of it are recommended.

The work of Zelenka and his coauthors was mostly focussed on the area of Switzerland. In this special case they had to deal with a small region of considerable topographic complexity but with a very dense station network as compared to other countries. As satellite data are concerned they disposed of maps with a high spatial resolutin (pixel size of the maps approx. 10*10 km 2 ) .

For the task of the ESRA we have to deal with a largely increased area. The respective station grid that is very sparse in the outer region of the area of interest (see section 3). In addition satellite data that cover the whole region are only available at a low spatial resolution. Nevertheless, an application of the co-kriging procedure has been attempted. A sketch of this method is given in an appendix. For the application of the method of ordinary co-kriging as described in the appendix some pre-treatment of the data was necessary. These procedures, will be given in section 4.

3.Data base

Ground data

Ground data were made available for this project by the WMO radiation data center in St.

Petersbourg, Russia and several national weather services..

Monthly means of the daily radiation sum H for 226 stations with global radiation measurements are computed from daily means covering the years 1981-1990 according to the criterion that at least 7 years of measured data are available for each station.

In addition a set of radiation data derived from daily means of sunshine duration data is taken into account. The derivation of the monthly mean of global irradiance from the sunshine duration data is based on Angstrom correlations and is described by [START_REF] Terzenbach | Estimation of monthly means of global radiation at sunshine measuring sites[END_REF]. This Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy,61,6,[389][390][391][392][393][394][395][396][397] set comprises data for 392 stations. Also data for 6 stations in Tunisia are extracted from a data base made available to us by the Joule Project 'SOLARGIS' [START_REF] Vandenbergh | Integration of renewables in a Tunesian region, Final report project 'SOLARGIS[END_REF] extraterrestrial radiation sum for the corresponding location). Due to the smoothing operation these maps show a continous variation in space (in constrast to the 2.5###x 2.5### resolution of the originla data base). However the spatial frequency of the variations remains low.

To assess the accuracy of the satellite derived data the clearness index values from monthly maps K s are compared to the 10-years averaged ground data (more precise: mean monthly clearness index values K g derived for the stations with at least 7 years of measured data) on a pixel to station basis. The rms-of relative error given by ∆ = -

( ) K K K s g g , )
)

is in the range of 8% (May) to 24% (January).

4.Data preparation and variogram estimation

The application of the ordinary kriging and co-kriging techniques as given in the appendix requires a field that is free of trends and isotropic. Therefore the data base presented above was pretreated to approach these conditions.

To remove the systematic variation of the irradiance data caused by the variation of the astronomic conditions all monthly irradiance data given as mean monthly radiation sum H are transformed into monthly clearness index values.

The clearness index is still affected by a trend with respect to the latitude. For both the ground data and the satellite data at the respective locations the mean clearness index values are extracted on a monthly bases using latitude classes of 3### width. In this calculation stations with an altitude greater then 1000m are not used. In doing this, the specific climatic contitions at mountain sites are not mixed with systematic trends due to pure latitude effects. An example for the month of August is given in fig. 3 for both the ground data and for the satellite derived values. This example shows that for the month presented the satellite data have an offset varying with the latitude.

To remove the latitudinal trend and the offset of the satellite data a simple correction is used:

K x K x K lat K x K x K lat g lat long g lat long g t s lat long s lat long s t ' , , ' , , , ( ) ( ) ( ( ) ( ) ( =
- = - (1)
Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy,61,6,[389][390][391][392][393][394][395][396][397] where the trend functions K g, t (lat) and K s, t (lat) are empirically determined for each month by piecewise linerar interpolations of the data as given in fig. 3.

As described in the appendix for the kriging and co-kriging procedure the spatial structure of the data field is presented by auto-( ) and cross-variograms . For the trend corrected K' values the variograms are defined as:

γ i d ( ) γ 1 2 , ( ) d γ i i i d K x K x d ( ) ( ( ) ( )) ' ' = - + 2 with i : g, s (2) γ g s g g s s d K x K x d K x K x d , ' ' ' ' ( ) ( ( ) ( )) ( ( ) ( ) = - + ⋅ - +
The variogram describes the expectation of the squared difference of the field values separated by a distance d. For an isotropic field the variogram should depend on the absolute value of d only.

As the spatial variation of radiation data is not only affected by the relative horizontal position of locations but also by the difference in altitude [START_REF] Zelenka | Supplementing network global irradiance data, Advances in Solar Energy Technology[END_REF] have proposed the use of an effective distance. The effective separation of locations is described by: co-kriging procedure performed for various values of f (0.1, 0.2, 0.3, 0.4 and 0.5 km/m). Fig. 4 shows the root mean square of relative errors calculated by the cross test as a function of f. In this calculation the full ground data set is involved. It can be seen that the smallest error is attained for the value of f = 0.3 km/m. It should be stressed however that the relative variation of the rms error with f is very weak: less than 5%, and that the results are only weakly dependent upon f. The figure also gives a first indication that the differences in the interpolation quality of kriging and co-kriging procedures are quite small. This will be discussed in more details in section 6.

d d h f with d h f eff = + ⋅ = 2
The variograms used in the co-kriging are calculated thus for the detrended data K' and applying the effective distance d eff according to eqn 2 with f set to 0.3 km/m. The effect of the trend correction is shown in fig. 5. The variogram for uncorrected and corrected ground data has been analysed with respect to the orientation of the station separation. This was done by selecting only station pairs with differences of less than 3° either in latitude (station pairs with distance domminated by their east-west separation) or in longitude (station pairs with distance dominated by their north-south separation). Before the trend correction (i.e the variograms calculated for K g ), the differences of the variograms for north-south and east-west orientated station pairs are significant. For the trend corrected set both variograms are allmost identical.

It may thus be concluded that the trend corrected field approaches isotropy. The effect of the use of the effective distance for the shape of the variograms is shown in fig. 6. When applying the pure horizontal distance the variogram shows almost no decrease for small separation distances (below 200 km). This is caused by station pairs representing some valley/mountain combinations. The increase of the separation of these pairs induced by using the effective distance leads to a variogram with a more pronounced tendency to approach 0 for zero separation.

For the application of the procedure the variograms must be described by analytical functions.

Here an exponential model is used, whose parameters a and b are fitted to the empirical variograms.

Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy, 61, 6, 389-397.

γ ( ) ( exp( / )) d a d b = - - 1 (4)
The fit was performed on the samples calculated for the effective distance range of 0-1500 km with a class width of 50 km. Fig. 7 gives the auto-and cross-variograms for May. The autovariograms indicate that, as expected, the detrended satellite data show less variance than the respective ground data. The ratio of the standard deviations is about 1.5. The cross-variogram reflects that the cross-correlation for detrended satellite and ground data for this month is about 0.5. For other months (January, February) this correlation decreases below 0.1. In the co-kriging technique the low correlation will results in the fact that the satellite data will have a small influence on the results only.

7.Application and results of the co-kriging

The data set as discussed above was used as basis for both kriging and co-kriging interpolations. Cross-tests are performed to estimate the quality of the interpolation and to judge the benefits of the use of the satellite data set.

To keep the expense in calculation time reasonable, the interpolation was performed by using the I = 25 closest stations at each location. An increase of the number of stations to 100 leads only to marginal differences in the results. In the co-kriging procedure, in addition to the I ground station values, J = I+1 satellite derived values are used. The satellite values are taken at the I pixels referring to the positions of the selected ground stations and the pixel referring to the location of interest.

A cross validation was performed in the following way. Each of the stations was for once removed from the data set and the corresponding interpolated value was calculated from the remaining set of stations. This procedure includes the removal of the station of interest from the calculation of the latitude means of the clearness index. However the variograms are not recalculated for every individual estimation. Thus the cross validation is not 'rigorous', but it Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy,61,6,[389][390][391][392][393][394][395][396][397] is thought that the removal of one station would not affect the variogram models to a large extent.

In the following examples only the stations with at least 7 years of measured global radiation data (stations marked black in fig. 1) are taken into account. Thus the results also reflect the effects of the use of irradiance data derived from sunshine duration data in the interpolation procedure.

The differences of the estimated clearness index values K* as compared to the measured values of K g are expressed in terms of the relative error:

∆ = - K K K g g * (5)
For the basic test involving about 200 stations with at least 7 years of measured global radiation data the performances of the kriging and the co-kriging interpolation are almost identical. As shown in fig. 8 both methods result in a root mean square (rms) of the relative errors of about 0.08 in winter and of 0.05 in the summer months. An overall advantage of the co-kriging intepolation of about 2% may be deduced. In addition, the results for a gravity interpolation (see Appendix) are indicated. As already found by [START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF] and [START_REF] Zelenka | Combining METEOSAT and surface network data: a data fusion approach for mapping solar irradiation[END_REF] the loss of accuracy compared to kriging interpolation is only marginal (here:

4% with respect to the co-kriging results). The results for the gravity interpolation are derived using the 7 closest neighbors for all the stations. This value was found to give the lowest overall error for this technique. In general it may be stated that the performance of gravity interpolation is less even with respect to different regions and months than the results of the kriging procedures discussed in the following. These results are close to those found for spline interpolations of 30 year means of monthly sunshine duration data by [START_REF] Hulme | A 1961-1990 climatology for Europe for climate change modelling and impact applications[END_REF] (rmse values of 10% in winter and 5% in summer). Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy,61,6,[389][390][391][392][393][394][395][396][397] Looking at selected groups of stations from areas with a high station density the differences of kriging and co-riging results are even more diminished (see fig. 9). The areas inspected refer to 47.5° < latitude < 55.0° and 6.0° East < longitude <15.0° East ('Germany') and a region including the main part of the Alps (45.5° < latitude < 48.0°; 6.0° East < longitude < 15.0° East). This result is obvious as the satellite data with their slow variation in space can bring no additional information to the field structure as presented by the sets of nearby neighbors. It should be noted, that for the region 'Germany' the precision of the interpolation is very high.

The rms of the relative errors ranges from 0.06 (January) to below 0.03 (July). For the region 'Alps' the rms values are increased due to the complexity of the terrain. For the 20 sites inspected -which include 6 stations at altitudes above 1500m as well a the lowland site Milan (altitude = 20m) the rms values are in the range of 0.11 (January) to 0.05 (July).

The comparison looks different if stations are selected which show are large separation to the next neighbor. As example the cross test results for a set of all stations with a distance of more than 200 km to the next neighbor (8 -13 stations, depending on the month) are shown in fig. 10. Here the differences between the kriging and co-kriging results are more pronounced.

The advantage of the co-kriging interpolation is visible for all months except November. The overall difference is about 9%.

A similar pattern arises for a test performed for all stations using an artificial restriction (fig. 11). As 'neighbors' for kriging and co-kriging only stations with a distance of more than 500 km to the site of interest are used. For the summer months, the rms of the relative errors for the co-kriging is about 6%. This values is close to the rms of the relative error describing the precision of the trend corrected satellite data for the respective month`s. From this we may conclude that for regions sparsely covered by ground stations even the low resolution satellite maps may add valuable information.

Conclusions

Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy, 61, 6, 389-397.

Monthly clearness index maps produced using the co-kriging interpolation described will be presented in the new European Solar radiation Atlas ESRA. Each map will give the monthly clearness index for each of the 1200 x 600 pixels of a 5´x 5´maps. The altitude values for the pixels are taken from a digital elevation model with the same spatial resolution.

Concerning this construction procedure some conclusion may be drawn. We think that the results presented here using the data base discussed, give an idea about the limits for interpolation methods for the whole region studied based on a common characterization of the spatial structure of the stochastic component of the field. Regional improvements may be possible by investigating the best method and structure representation for respective subsets of data.

Besides those regions with almost no ground station data, the most problematic regions are the mountaneous areas. Only for the Swiss alps the data network is dense enough to present some of the complex radiation field structure caused by this type of orographie. In reverse the orograpic structure will be recognizable in the irradiance maps. For this region however the Meteonorm maps [START_REF] Kunz | METEONORM -a comprehensive meteorological database and planing tool[END_REF] are already available. This example uses a different procedure was applied to deal with the altitude dependance of the radiation field by assigning different altitude trend functions for as many as 11 separated subregions. As those functions have to be based on a small number of station data in each subregion and show considerable variation over the subregions this approach was not applied to the European Atlas. For all other high mountain areas the data base is not sufficient to represent detailed orographic effects. Again for the reason of diversity of altitude trend functions no 'ad hoc' altitude dependence is introduced here. Due to the use of the effective interstation distance single stations at higher elevations induce a weak dependance of the irradiances on the altitude for the surrounding regions. The respective data presented on the ESRA maps must be handled with care. Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). Solar Energy,61,6,[389][390][391][392][393][394][395][396][397] A major shortcoming for the preparation of radiation maps seems to be the lack of deterministic models linking orographic features, e.g. mountain ridges, with changes in the radiation field. Only few attemps in this direction are known to us. One is the empirical extraction of dependence on altitude as discussed above. Others include the test of using the distance from the shoreline as a characteristic site parameter in interpolation procedures or the empirical identification of climatic barriers decoupling sites on opposite sides of them (van der [START_REF] Van Der Voet | Spatial interpolation of daily meteorological data, A knowledge-based procedure for the Region of the European Communities[END_REF]. Establishing such empirical relationship requires a large station density for the whole region of interest. Alternatively, satellite derived maps with a high spatial resolution may be used as a data source to pin down systematic small scale variations.

It must be stated that in the present study the benefits due to the use of the available satellite derived maps are very limited. Obviously, due to their low resolution, they do not help to account for small scale orographic effects. As the overall correlation of satellite and ground data is not as good as expected, especially the uncertainty of the final maps for the winter month at the outer bounds of the inspected region is still in the range of 10-15 %.

Satellite derived maps from other sources have been taken into consideration. Maps prepared by the SUNSAT project [START_REF] Raschke | Solar Radiation Atlas of Africa[END_REF] having a resolution of about 50 km*50 km and covering the two years 1985 and 1986 have been tested. When detrended, the respective 2year averaged monthly maps yield correlations with the 10 years ground data that are inferior to those calculated for the SRB maps. Here the variability of individual monthly radiation sums compared to the 10 years mean (see. e.g. [START_REF] Beyer | Time series analyses of the long term monthly horizontal solar radiation[END_REF] plays an important role.

Using merged SRB and SUNSAT maps does not lead to any significant improvement of the co-kriging precision. To overcome these problems satellite derived maps with a reasonable spatial coverage and a suitable resolution must be available for the whole period of interest. )

Appendix

A short introduction to kriging and co-kriging techniques

Kriging was developed in the framework of geostatistics for the analysis of mineral ore resources (see e.g. [START_REF] Journel | Mining Geostatistics[END_REF]. We will give a brief summary of the basic equations used in our application of these methods.

It is assumed that an estimate of the value of a field Y at a location x o may be given by a linear combination of the field values measured at locations x i (x representing the coordinates of a point).

Y x wY x

o i i I i * ( ) ( = ∑ (A1)
In simple methods the weight w i are set directly according to the distance d 0i between x 0 and x i . For example the inverse squared distance methods (in the following also refered to as gravity interpolation) the weights are set as:

w d d i oi oj j = ⋅ ∑ 1 1 1 2 2 (A2)
The kriging procedure is based on the idea to select the weights w i to give a minimum error of the estimator Y*(x o ) to the actual value Y(x o ). This process requires information on the spatial structure of the field which is provided by the (auto-)variogram (or structure function).

γ ( ) ( ( ) ( )) d Y x Y x d = + + 2 (A3)
The variogram describes the expectation of the squared difference of the field values separated by a distance d. For simplicity we assume that the field is isotropic and thus that the variogram depends on the absolute value of d only. In practice this function may be estimated from a given data set by grouping station pairs in classes of distances and calculating the [START_REF] Zelenka | Techniques for supplementing solar radiation network data[END_REF] later used in this study is given by:

:

γ µ γ µ (A5)
The Lagrangian multiplier enters here to cope with the condition that the sum of the weights should be equal to 1, which assures that the estimate is bias free.

A additional source of information on the field may be given by data that are at least positively correlated with the studied field and sampled for a larger number of locations. In our context these are the radiation data estimated from satellite images. A tool for merging both information sources is given by co-kriging .

For the co-kriging procedure the cross-structure of two data fields is characterised by the cross-variogram of the different data sets. The system of equations for the determination of the weights w i and v j using the auto (### 1 ,###

γ
2 )-and cross-variograms (### 1,2 )is given by: { } The last two equations again give the conditions for a bias free estimate.

  on the variogram, minimizing the error of the estimates results in the following equations which determine the weight w i if Y*(x o ) is to be estimated from a sample of I observations. Here the abreviated notation γ γ

b

  To estimate the cross-variogram the field data Y 1 and the additional data Y 2 must be cosampled at a sufficiently large number of locations.Using the information on the auto and cross-structure the interpolated value Y * at location x o may be derived from a weighted sum of I samples for Y 1 and J samples for Y 2 values where J should be larger than I.

  H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). SolarEnergy, 61, 6,[389][390][391][392][393][394][395][396][397]

			(	∆	)	2
			:
	:	horizontal distance	(3)
	∆	:	difference in location altitude
			relative scale factor: equivalent horizontal distance / vertical distance

While Zelenka and Lazic have set the factor f to 0.1 km/m, we used a value of 0.3 km/m. This was done on the basis of the results of cross-validation tests (see below) for both kriging and Beyer

  Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). SolarEnergy, 61, 6,[389][390][391][392][393][394][395][396][397]

  Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). SolarEnergy, 61, 6,[389][390][391][392][393][394][395][396][397] mean squared difference for each class. For the subsequent calculations the empirical variogram is in general fitted by analytical functions. The functions should reflect the expected characteristic of the variogram to increase with increasing distance d. An example for a model out of a family of suitable functions (see e.g. discussion in

  Beyer H.-G., Czeplak G., Terzenbach U., Wald L., 1997. Assessment of the method used to construct clearness index maps for the new european solar radiation atlas (ESRA). SolarEnergy, 61, 6,[389][390][391][392][393][394][395][396][397]

	1 2 , ( ) ( ( ) 1 d Y x Y x d 1 ( = -+	2 )) ( ( ) Y x Y x d 2 ( ⋅ -+	)	(A6)

Acknowledgements

This work was supported by the Commission of the European Communities under Contract

Number JOU 2 -CT -94 6 0305. Discussions with the colleagues participating in this project and with Antoine Zelenka of the Swiss Meteorological service are greatfully acknowledged.