Feedback boundary stabilization of 2d fluid-structure interaction systems

We study the feedback stabilization of a system composed by an incompressible viscous ﬂuid and a deformable structure located at the boundary of the ﬂuid domain. We stabilize the position and the velocity of the structure and the velocity of the ﬂuid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the ﬂuid domain and with values in a ﬁnite dimensional space. Our result concerns weak solutions for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the ﬂuid domain of the stationary state and of the stabilized solution are diﬀerent. We prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the deformable structure and the velocity of the ﬂuid.


Introduction
We consider the problem of stabilization for a fluid-structure system composed by a viscous incompressible fluid and a deformable structure located at the boundary of the fluid domain. The fluid motion is modeled by the Navier-Stokes system and the structure deformation follows the equation of a "viscous" beam. Such a model is already considered by several authors ( [33], [10], etc.). Our aim consists in showing the boundary stabilization of such a system in the 2d case and for weak solutions. The method used here could be adapted for other fluid-structure systems in the case of a fluid modeled by the Navier-Stokes system. In the 3d case or for strong solutions, the stabilization of such fluid-structure systems could be obtained by using the methodology developed in [6] or in [8].
Such results of stabilization are classical for the classical Navier-Stokes system (without any structure), see for instance, [23], [38], [9], [37], [2], [5], etc. Note that for this problem there is a difference between the dimension 2 and the dimension 3: due to the nonlinearity in the Navier-Stokes system, and to the method developed (stabilization of the linearized system, fixed point), in dimension 2 one can take initial data in L 2 (or H s , s < 1/2), whereas in dimension 3, one needs to take the initial data in H 1 (or at least H s , s 1/2). As a consequence, in dimension 3, we have to impose compatibility condition at t = 0 between the initial condition and the feedback control u (see [4,3,2] for details). Several techniques have been considered to overcome this difficulty: [34], [2], [5], etc. For instance in [5], the solution consists in assuming that the control u satisfies an evolution equation with another feedback control. We are thus reduced to stabilize a system coupling the fluid velocity and the control u. In dimension 2, the method allows to consider classical feedback operators for weak solutions. This is done for the Navier-Stokes system in [38]. Note that in dimension 2, the stabilization of strong solutions leads to the same problem of compatibility conditions.
For the fluid-structure interaction systems, there are few results of stabilization. A first result was obtained in [36] for the system considered in this article. The target velocity v S is zero, the control is acting in the whole structure and the author works with strong solutions (initial data in H 1 for the fluid velocity). The case of a deformable structure immersed in a fluid is considered in [19], [18]. For the case of a rigid body, a 1d simplified model is treated in [7] whereas the 2d and 3d case are considered in [6]. In this last paper, we work with a notion of weak solutions in order to deal with the 2d case without the problem of the compatibility conditions. However, in [6] we need that the initial and the final position of the structure are equal.
The main novelty of this work is to prove stabilizability result for weak solutions of a fluid-structure system. Moreover, we consider a nontrivial target velocity v S to be stabilized. We work in the 2d case only and the method for the stabilization follows the same idea as the papers quoted above. One important difficulty that we need to deal with is that there is no proof in the literature for the existence of weak solutions of a fluid-structure system with a Banach fixed point. In order to do this here, a first step consists in performing a change of variables to work on a cylindrical domain (see Section 2). Such an approach is already considered for strong solutions and there exists changes of variables that allow to keep the divergence free conditions and the form of the boundary conditions. We don't employ such a change of variables on the unknowns but on the test functions. This leads to transform our system in a cylindrical domain with non homogeneous divergence conditions and non homogeneous boundary conditions. We can overcome the corresponding difficulty by using a framework developed in [39] for the Navier-Stokes system. All this work could be adapted to other fluid-structure systems such as the case of rigid bodies moving into a viscous incompressible fluid as in [6]. Indeed, the presence here of the deformable structure that follows a beam equation leads to lots of technical difficulties (see the three sections in the appendix).
Let us also mention that several works have been devoted to the study of the fluid-plate system. The model was proposed in [33]. The existence of weak solutions was proved in [16] and the existence of strong solutions was obtained in [10] and in [29]. For these two results, as in our case, the damping term, that is −δ∂tssη, is essential. In [24], the author manages to prove the existence of weak solutions (in 2d or in 3d) for this problem in the case without damping (δ = 0).
The change of variables also allows us to describe the feedback law satisfied by u in (1.3). We take u under the form Fj(v(t, .), η(t, .), ∂tη(t, .))vj(x) t > 0, x ∈ Γ0, (1.29) with and Now, let us give the definition of a weak solution for our problem.

we have
We refer to Section 2.1 for the precise definition of the functional spaces used above.
Remark 1.3. Note that assumption F(η S ) of class C 1,1 allows us to freely use H 2 -regularity results for the Laplace equation and for the Stokes equations. It is also a natural assumption because, even if the reference domain F ref and η S are regular, the boundary conditions η S = ∂sη S = 0 on {0, 1} do not guarantee a class of regularity for F(η S ) better than C 1,1 .
Remark 1.5. The uniqueness of the controlled weak solution (in the sense of Definition 1.1) is not proved in Theorem 1.2. Since the proof relies on a Banach fixed point argument it is indeed true that the solution is unique within a class of stable solutions sufficiently close to the stationary state. But uniqueness is not obtained in the classical energy space defined by (1.32). The uniqueness of weak solution is not an easy issue, even under the hypothesis of small initial data. It must be the subject of further investigations. Remark 1.6. Using the method developed here, we can obtain the same result for other fluid-structure systems. For instance, we could obtain the stabilization of weak solutions for the case where the structure is a rigid body (see [6]). One could also consider the case of a deformable structure in the case where the equation of deformation is approximated by a finite dimensional method: see [20], [28], [14]. For these cases, the fixed point and the estimates are simpler than here. The case of a deformable structure modeled by the Lame equation or by the wave equation with an adequate damping can also obtained directly from our work. For other damping laws or without damping, even the well-posedness is not always done and the corresponding stabilization problems have to be studied differently. Let us quote some references on the well-posedness of such systems: [15], [12], [13], [30], [40], etc.
The outline of the paper is as follows. In Section 2, we construct the change of variables and we rewrite the system in a fixed domain. We then obtain the system satisfied by the difference between the controlled solution and the stationary state. By linearizing this system, we obtain in Section 3 the coupled system (3.1)-(3.4) that couples an Oseen's type system with a beam type system with dissipation. With this dissipation, we prove that the semigroup associated with system (3.1)-(3.4) is analytic. That allows us to use the general theory developed in [5,8] to deduce in Section 4 the feedback stabilization of our linear system, first in the homogeneous case and then in the non homogeneous case (and in particular with terms corresponding to the non null divergence condition and non null boundary condition). In Section 5, we use a fixed point procedure to obtain the stabilization of the nonlinear system and thus to prove the main result. In the appendix, we postpone technical proofs to the three sections: Section A is devoted to the change of variables, Section B to the linearization, and Section C to some estimates for the fixed point.
2 Notation and change of variables

Notation
The classical Lebesgue and Sobolev spaces are written L α , H k and we denote by C b the continuous and bounded maps. We use the bold notation for the spaces of vector fields: L α = (L α ) 2 , H k = (H k ) 2 etc. For a Hilbert space X and 0 < T +∞, L p (0, T ; X ) and H s (0, T ; X ), p ∈ [1, ∞], s 0, are usual vector-valued Lebesgue and Sobolev spaces and in the case T = +∞, we use the shorter expressions L p (X ) def = L p (0, +∞; X ) and H s (X ) def = H s (0, +∞; X ). We denote by L 2 loc (0, T ; X ) (resp. H s loc (0, T ; X )) the set of functions belonging to L 2 (0, T ; X ) (resp. H s (0, T ; X )) for all T > 0. For two Hilbert spaces X , Y we If Z is a vector-valued function space of the time variable t 0, for σ > 0 we use the subscript σ in Zσ to denote the space Zσ We use the notation (X ) , or simply X , for the dual space of X , we use the notation L(X , Y) for the bounded linear maps from X into Y and the notation X → Y for the continuous embedding of X into Y. Moreover, [X , Y] θ denotes the complex interpolation space of index θ ∈ (0, 1). If X → Y the following continuous embeddings hold for all θ ∈ (0, 1) and s ∈ (1/2, 1]: (2. 2) The first above embedding is an easy consequence of the fact that [X , Y] 1/(2s) is the trace space of L 2 (X ) ∩ H s (Y), see e.g. [25]. The second one comes from the equality [L 2 (X ), H s (Y)] θ = H θs ([X , Y] θ ) (see Theorem 5.1 and (6.8) in [27]) combined with the embedding In order to simplify the notation, we write in what follows and we introduce spaces of free divergence functions in F as well as the corresponding trace spaces on ∂F: We also use functional spaces of type L 2 (0, ∞; H 1 (F(η(t)))). Such a space is defined through a family {X(t, ·)} t 0 of C 1 -diffeomorphisms that transforms F onto F(η(t, ·)) and such that both X and X −1 belong to C b (C 1 (F)). We say that v ∈ L 2 (0, ∞; H 1 (F(η(t)))) if v • X ∈ L 2 (H 1 (F)). It can be seen that the above definition of L 2 (0, ∞; H 1 (F(η(t)))) is independent of the choice of X. For instance, if η ∈ C b (C 1 ([0, 1])) one can choose the family of change of variables {X η(t) } t 0 introduced in Section 2.2 below. Other spaces of functions defined on a non cylindrical domain of R 3 are defined similarly: C b ([0, ∞); L 2 (F(η(t)))), In what follows, C > 0 denotes a generic constant that may change from line to line and which is independent on the other terms of the relation where it is used. We recall that, here and in what follows, we use the simplified notation F def = F(η S ) and Γstr def = Γstr(η S ). Since η S ∈ C 3 ([0, 1]), see (1.33), we can extend η S to a function in C 3 (R).

Rewriting the system (1.3) in a fixed domain
In this section we assume (2.8). Our change of variables is defined by v(t, y) def = ∇X(t, y) * v(t, X(t, y)). (2.11) Remark 2.1. We could have used the change of variables v(t, y) def = v(t, X(t, y)) or, as in [41] or [14], v(t, y) def = Cof(∇X(t, y)) * v(t, X(t, y)). The advantage of the latter choice is that it preserves the divergence free condition. Here we use this formula to transform the test function ϕ, see (2.15) below.
We have the following results (the technical proof is postponed in Section A.1).
We can then transform the weak formulation (1.23): combining Lemma A.1, Lemma A.2, and Lemma A.3 in the appendix (Section A), we obtain that v satisfies the weak formulation Since Using the above relation, (1.27) and (2.18), we deduce that Now, we can decompose the above operators in a linear part and surlinear part. First, we define the sets of type Qi(α1, . . . , α k ) where i, k ∈ N. They are the sets of polynomials in the variables α1, . . . , α k and with coefficients that are Lipschitz continuous functions of y ∈ R 2 and of ξ and that vanish in F\Vα (see (2.4)), and such that the degree of its nonzero monomial of lowest degree is greater or equal to i. For instance, we can write 1 1 + (∂y 2 θ)ξ = 1 − (∂y 2 θ)ξ + (∂y 2 θ) 2 (ξ) 2 1 + (∂y 2 θ)ξ .
The last expression means the total degree with respect to the first and the third variables. For the linear part, we also introduce a notation: we write the linear mappings that depend on y in a Lipschitz continuous way and that vanish in F\Vα (see (2.4)). From Lemma 2.2 and Lemma B.1, we obtain where To avoid a linear operator in the divergence condition, we consider another change of variable: and we write F div (Z) = div(r (2) (ξ, ∂sξ, w)). (2.25) Then the divergence condition for w can be rewritten as: In order to rewrite the boundary condition for w we write where M and T η S are defined by (1.12) and (1.13). Note that the fact that the range of T belongs to V 0 (∂F) follows from the following calculation: Moreover, since the localization operator Ξ is defined by (1.9) from a smooth cut off function ρ supported in Γ0 we can abusively consider Ξ as an element of L(L 2 (F)) and (1.9) becomes: Then the boundary condition for w can be rewritten as In the above expression, we have written γ (1) instead of γ (1) (ξ, ∂sξ) to shorten the formula.
From Lemmas B.2, B.3, B.4, B.5, we can write the above relation as and In what follows we write and we write γ (3) and γ (4) as operators acting on ξ1 = ξ and ξ2 = ∂tξ, namely Hence, using standard arguments, (2.45) can be rewritten as the following dynamical system: for any
The three other relations can be obtained in a similar way.
3 Operators for the linear system

General functional settings
This section is devoted to the study of the linear system where here F , G, F div and F b are given. Let us remark that the results given in this section can be obtained for general operators A1, A2, T , Λ (1) and Λ (2) . More precisely, we only need to assume that A1 : D(A1) ⊂ HS → HS and A2 : D(A2) ⊂ HS → HS are positive, densely defined, self-adjoint and with compact resolvents and that D(A Assumption (3.5) is crucial in our analysis since it allows to invoke [17] and to obtain the analyticity of the semigroup generated by the underlying linear operator of system (3.1)-(3.4) (see Proposition 3.11 below). We suppose that T ∈ L(HS , V 0 (∂F)) satisfies (2.55), (2.57), (2.56), (2.58), that (T ξ)Γ 0 ≡ 0 for any ξ ∈ HS . Finally, the operators Λ (1) , Λ (2) are assumed to satisfy (2.59), (2.60), (2.61), (2.62) and (2.63). Note that the operators A1, A2, T , Λ (1) and Λ (2) defined by (1.16), (1.17), (2.28) and (2.44) satisfy the above conditions. We still assume that Ξ ∈ L(L 2 (∂F)) is the self-adjoint operator defined by (2.29). We need its precise definition to obtain the adjoint of the control operator (see (3.67) below). We recall that Ξ satisfies (2.53). We first consider system (3.1)-(3.4) in the case F = 0, G = 0, F div = 0 and F b = 0: (3.10) The above system is completed with the initial conditions We show that the system (3.6)-(3.11) can be rewritten in the form where A is the infinitesimal generator of an analytic semigroup. This abstract form is quite standard in the study of the stabilizability for the Navier-Stokes system, see [35]. We consider the space L 2 (F) × D(A 1/2 1 ) × HS equipped with the scalar product: 1 , ξ and we introduce the following spaces: 1 , ξ 2 ] ∈ H: Then we have in particular that F w (1) · w (2) dy = 0 for all w (2) ∈ V 1 0 (F) and the De Rham Lemma guarantees that w (1) = ∇p for some p ∈ H 1 (F) such that F p dy = 0, see [42, Chap. I, Prop. 1.1 and Rem 1.4]. Thus, by plugging w (1) = ∇p in (3.15) and integrating by parts, we obtain that 2 ] ∈ D(A 1/2 1 ) × HS , which gives the result.
Proposition 3.2. The orthogonal projection operator P : Proof. First, by using (3.14) we verify that for any [w, where the pressure function p ∈ H 1 (F) obeys F p dy = 0 and is solution to the Neumann problem: . Then, we deduce from (2.56) and from the C 1,1 regularity of ∂F that (T ξ2) · n ∈ H 1/2 (∂F). Similarly, from (2.56) and (2.57) we get T (T * (pn)) · n − w · n ∈ H 1/2 (∂F) and from the regularity of ∂F and standard elliptic properties of the Neumann problem we deduce that Then the conclusion follows by an interpolation argument.
As a consequence, since P is self-adjoint on H, a duality argument yields the following result. (3.18)

The operator A 0
First, we define the linear operator A0 : D(A0) ⊂ H → H as follows: we set and for w, ξ1, ξ2 ∈ D(A0), we set Proof. Standard calculation gives, for all Z ∈ D(A0), A0Z, Z 0 which implies that A0 is dissipative. Then, we show that (λ − A0) is onto for some λ > 0: assume F = [f, g, h] ∈ H, we have to prove the existence and the uniqueness of Z = [w, ξ1, ξ2] ∈ D(A0) such that Let us consider a variational formulation associated to (3.22): find The Riesz theorem gives the existence and uniqueness of [w, ξ2] ∈ V satisfying (3.24). Taking ζ2 = 0 in (3.24) and using the De Rham theorem, we obtain the existence of q such that (w, q) is the weak solution of the Stokes system (the three first equations of (3.22)). From (2.56), we deduce T ξ2 ∈ V 3/2 (∂F) and thus, since f ∈ L 2 (F), standard elliptic results on the Stokes system give w ∈ H 2 (F) and q ∈ H 1 (F). In particular, T(w, q)n ∈ H 1/2 (∂F) and thus T * (T(w, q)n) ∈ HS .
We write ξ1 = λ −1 (ξ2 + g) and we use that (w, q) satisfies the Stokes system to transform (3.24) into for all ζ2 ∈ D(A 1/2 1 ). Note that we have used the continuous embedding D(A 1/2 1 ) → D(A2). The above system implies that A1ξ1 ∈ HS and thus that ξ1 ∈ D(A1). Finally, the fact that A0 is densely defined with compact resolvent is straightforward.
Using that A generates a semigroup of contractions, we have (see, for instance, [32, Corollary 3.6, p.11]) In order to prove the exponential stability of (e tA )t>0, we show the existence of C > 0 such that: The above estimate yields the result: indeed, for τ ∈ R and δ ∈ (0, 1), one can write the formula Taking δ < 1/C where C is the constant in (3.28), it yields (3.25) and (3.26). Now let us prove (3.28): assume λ ∈ C with λ ∈ (0, 1) and assume (λ − A0)[w, ξ1, ξ2] = [f, g, h] ∈ H. This relation can be written as (3.22). Multiplying by [w, ξ1, ξ2], we first obtain Moreover, since Γ0 is a nonempty open subset such that w = 0 on Γ0, we have the Poincaré inequality Combining this relation with the trace inequality, the Korn inequality and (2.55), we deduce that: Then using the above inequality with (3.29) we obtain Combining the above inequality with λξ1 = ξ2 + g and (2.56) yields Next, from the two last equalities in (3.22) we obtain Then by multiplying the above equation by ξ1 and using D(A The above inequality, (2.56) and (3.31) yield Moreover, from the following Green formula and with (3.30) and the first equation in (3.22) we obtain Then combining this last estimate with (3.33) yields Moreover, this last estimates with (3.30) yields and it proves (3.28).
Remark 3.6. Assumption (3.5) is not used in the proof of Proposition 3.5. It remains true even if A2 = 0.
We have the following characterization of the adjoint of A0.
Proposition 3.7. The adjoint of the operator A0 is given by and where [·, ·]· denotes the complex interpolation method. Moreover, we have Proof. Relations Using (2.56) and standard result on the Stokes system, we deduce that for any α ∈ [0, 1], It is clear that We deduce by interpolation that for all α ∈ [0, 1]: Then the conclusion follows from (3.37), from (see [22]) and from the characterization of this last interpolation space (see [25]).
Corollary 3.9. The following continuous embedding holds: Proof. First, from (3.38) we deduce that for α ∈ [0, 1/4), Then for X ∈ H and Y ∈ L 2 (F) × D(A Then it follows, and we conclude with a density argument.
We recall a classical result for analytic semigroups (see [ then (e tA ) is an analytic semigroup on H.
We recall the proof of this lemma for sake of completeness.

20
Thus there exists δ ∈ (0, π/2) such that Applying [32, Thm 5.2, p.61], we deduce that (e tA ) is an analytic semigroup. Proof. We apply Lemma 3.10. We already know from Proposition 3.5 that iR ⊂ ρ(A0). For τ ∈ R * , we consider the equation Multiplying the first equation by w and performing an integration by parts we obtain: Then multiplying by τ and taking the imaginary part of the above equation first gives: and with the Cauchy-Schwarz inequality, we obtain: We now consider the equation of the structure (the last two equations of (3.43)): since the dissipation (the term A2ξ2) is sufficient, the corresponding system is parabolic. More precisely, since A2 is a positive, densely defined, self-adjoint operator on HS with (3.5), Theorem 1.1 in [17] guarantees that Then by combining (3.46), (3.47) and the boundedness of T * : L 2 (∂F) → HS we deduce that: In order to remove the term T(w, p)n L 2 (∂F ) in the above estimate, we first use the trace theorem and regularity results for the Stokes system, for ∈ (0, 1/4): and then with the first equality in (3.43) and the boundedness of T : Combining the above relation with ξ2 = iτ ξ1 − g we deduce Now let us prove that for ∈ (0, 1/4), ) with ∈ (0, 1/4). Using Proposition 3.2 and Proposition 3.8, we have P[ϕ, ζ1, 0] ∈ D((−A0) ). Then we can write Consequently, we deduce (3.51) and combining it with (3.50) yields The above relation and (3.48) imply Recalling (iτ − A0)Z = F, this can be written Then by iterating the argument we finally prove that for all n ∈ N * there exists Cn > 0 such that and for n 1/ , the above relation with iτ (iτ − A0) −1 F = A0(iτ − A0) −1 F + F and (3.26) finally yields iτ (iτ − A0) −1 F H C F H which gives the result.

The operator A
Now we define the operator A of our system: and for w, ξ1, ξ2 ∈ D(A), we set   To characterize the adjoint of A, assume w, ξ1, ξ2 , ϕ, ζ1, ζ2 ∈ D(A) and we observe that: Here we have used that v S = 0 on Γstr and ϕ = 0 on Γ0. Using (2.61) we deduce the result.
Let us fix λ0 > 0 large enough so that λ0 − A is positive and (λ0 − A) α is well defined for α ∈ (0, 1). We deduce from Proposition 3.12 and similarly as for Proposition 3.8 the following result. . Note that to obtain (λ − A)Z, Z 0 we have to control the terms coming from Λ (1) , Λ (2) and this can be done by using (2.59), (2.60) and (2.63). In particular we use the fact that Λ (2,2) ∈ L(HS, (H 1 (F)) ) which follows from (2.59) and (2.63) with an interpolation argument.

The operator B
Next, we introduce the Dirichlet operator DF : V 0 (∂F) → L 2 (F) × D(A 1/2 1 ) × HS defined as follows: for u ∈ V 0 (∂F) we denote by DF u def = [wu ξ1,u ξ2,u] the unique solution of . Then setting w = w + z, we see that (3.60) writes 1 ) solution of (3.62). To prove the case s = −1/2, we recall that in that case DF u is defined by duality as follows: for any where [ϕ, ξ, ζ] ∈ D(A * ) and π ∈ H 1 (F) such that F πdy = 0 satisfy Moreover, the adjoint of B is defined by

Feedback Stabilizability of the linear system 4.1 Stabilizability of the homogeneous linear system
The goal of this subsection is to prove, for a fixed rate of decrease σ > 0, the existence of a feedback control such that solutions of (3.6)-(3.10) tends to zero as t → +∞ with an exponential rate of decrease σ > 0.
For that, we are going to show the existence of families (ϕj, ζ1,j, ζ2,j) and vj, j = 1, . . . , Nσ such that the underlying closed-loop linear operator of (3.6)-(3.10) with (4.1) generates and analytic and exponentially stable semigroup of type lower than −σ (see [  Proof. The proof of the above proposition relies on the Hautus-Fattorini stabilizability criterion, see [5,Theorem 1] or [8]. Since A has compact resolvent and generates an analytic semigroup on H, and since B is relatively bounded with respect to A, then the homogeneous linear system is stabilizable by finite dimensional feedback control for any rate of decrease if and only if the following criterion is satisfied for all λ ∈ C: λΦ − A * Φ = 0 and B * Φ = 0 =⇒ Φ = 0. and ρT(ϕ, π)n = ∂F ρT(ϕ, π)n · ndγ ρn on ∂F.
Then the general framework of [5,8] can be applied and for a given σ > 0, there exist families

Stabilizability of the non homogeneous linear system
The goal of this section is to obtain regularity results for the following nonhomogeneous linear system: Suppose for the moment that (F div , F b ) = (0, 0). By taking into account (4.1) in formulation (3.12)-(3.13) complemented with the nonhomogeneous right-hand terms F , G, we deduce that the above system with F div = 0 and F b = 0 can be rewritten as Here we have used that Fσ vanishes on H ⊥ . The notation F − div G means here the operator In what follows, we recall that we use the notation (2.1). We have the following result. (H 1 (F) ), G ∈ L 2 σ (L 2 (F)) and (F div , F b ) = (0, 0). Then system (4.6)-(4.11) admits a unique solution and we have Proof. We write system (4.6)-(4.11) as (4.12)-(4.13). By using (4.14) we have Since Aσ generates an analytic semigroup on H, from maximal regularity results applied to equation (4.12), we deduce from (4.15) and PZ 0 ∈ H that PZ ∈ Wσ(D((−Aσ) 1/2 ), D((−A * σ ) 1/2 ) ). Finally, from the definition (4.2) of the operator Fσ, from Proposition 3.14 and from Proposition 3.2, equality (4.13) yields ( Let us now consider the case of non zero nonhomegeneous terms F div and F b . For that we need to introduce a lifting operator for the divergence condition which is compatible with the feedback condition, To state regularity properties for L div we need the functional framework introduced in [39]. For s ∈ [−1/2, 2] we define In what follows, we need another assumption than the ones introduced in Section 3.1: for some ε ∈ (0, 1/8): Proposition 4.4. Let ε ∈ (0, 1/8) be given in (4.20). The mapping L div defined above satisfies: Proof. This existence and properties of L div are obtained by a duality argument. First we consider [f, Moreover we have the estimate From (3.56) and (4.2), it means that we have the existence and uniqueness of the solution of (4.23) Using that [ϕj, ζ1,j, ζ2,j] ∈ D(A * ) and Proposition 3.1, we deduce that the pressure χ satisfies (4.24) We can assume χ ∈ L 2 0 (F) and in that case, using the Poincaré-Wirtinger inequality, Lemma 2.4 and (4.22), we obtain Note that χ can be decomposed as Now, let us assume that From (2.56) and (4.22), combined with the above assumption and standard elliptic regularity for the Stokes system, we deduce Using the forth equation of (4.23) and (2.62), we deduce that ζ2 ∈ D(A 1/2+ε/2 1 ). Then combining (2.63), (4.20) and (2.58) and the above regularity for (ϕ, χ, ζ2), we deduce that ζ1 ∈ D(A 1+ε/2 1 ) with the estimate We can now prove the well-posedness of (4.16)-(4.19) by a duality argument. First we rewrite this system as Assume now that [w, ξ1, ξ2] is a regular solution of the above system and [ϕ, ζ1, ζ2] ∈ D(A * ) is the solution (4.23). We multiply the first equation of (4.23) by w and (4.29) by ϕ. After some calculation, we obtain Combining the two above relations with the fact that F hdy = ∂F g · ndγ leads to for any [f, ζ f 1 , ζ f 2 ] satisfying (4.27) and where where (ϕ, χ, ζ1, ζ2) is the solution of (4.23) associated with [f, ζ f 1 , ζ f 2 ]. The existence and uniqueness for this problem is a consequence of (4.28).
Next, for ε ∈ (0, 1/8) given in (4.20) let us define the following functional spaces: Notice that (2.2) yields the following continuous embedding: We are now in position to state the main result of this section.
Then system (4.6)-(4.11) admits a unique solution [w, ξ1, ξ2] ∈ G and we have Proof. We write system (4.6)-(4.11) as (4.39)-(4.42) with Z = [w, ξ1, ξ2]. Since Using Corollary 3.3 and (4.48) we deduce from the above relation Using Proposition 4.1, Proposition 3.13 and (3.38) we deduce from the above relation, From the hypotheses on the initial conditions, and from the above relation, we obtain where Z 0 def = [w 0 , ξ 0 1 , ξ 0 2 ]. Gathering (4.50), (4.51) and applying Proposition 4.5 and Proposition 4.3 with the fact that we deduce that We underline that the last above embedding is well justified by Proposition 4.1. In particular the embedding 1 ) is true since ε < 1/4. We deduce from the above relation, from the definition (4.2) of the operator Fσ, from Proposition 3.14 and from Proposition 3.2 that Combining the above relations, we deduce that with the estimate (4.49). In order to prove Theorem 1.2, we consider the Banach spaces E and G defined by (4.46) and (4.47) and the following mapping defined on a closed ball of E of radius R > 0, where Z = w, ξ1, ξ2 ∈ G is the solution of (4.6)-(4.11) given by Corollary 4.6 and where F (Z), G(Z), F b (Z), F div (Z) are defined by (2.43), (2.25) and (2.27). We remark that if F, G, F b , F div is a fixed point of the mapping Ψ, then the corresponding solution w, ξ1, ξ2 of (4.6)-(4.11) is a solution of (2.48)-(2.52). Consequently, we are reduced to show that Ψ admits a fixed point. We prove that for R small enough, Ψ is well-defined from BE (0, R) onto itself and that the restriction of Ψ on this closed ball is a contraction mapping.
First, we notice that (4.49) implies (2.8) provided that F, G, F b , F div ∈ BE (0, R) with R small enough and that [w 0 , ξ 0 1 , ξ 0 2 ] has a norm small enough in L 2 (F) × D(A 1/2 1 ) × HS. In particular, the changes of variables X and Y are well-defined as well as F (Z), G(Z), F b (Z), F div (Z).
Second, we use several technical results whose proofs are given in the next subsections. To simplify the notation, in what follows, we assume Proposition 5.1. There exists C # > 0 such that for all R > 0 and [w 0 , ξ 0 1 , ξ 0 2 ] satisfying (5.1), and all F, G, F div , F b ∈ BE (0, R), From the above proposition, we remark that if and R is small enough so that 4C # R 1, then Ψ is well-defined from BE (0, R) into itself.
The second important technical result we need is the following: div and F (2) , G (2) , F With the same conditions (5.2) and (5.3), we deduce that the restriction of Ψ on BE (0, R) is a contraction mapping. The classical Banach fixed point theorem allows us to deduce the existence of a solution.
1 , ξ div , then we have from Corollary 4.6, 25) and as in the previous section we thus deduce In the same way as for Lemma 5.3 and Lemma 5.4 we can prove the following Lemma.

A Calculation for the change of variables
In this section, we gather several lemmas and several proofs for the change of variables (Section 2).
Thus on Γstr, and ( and M (4) On the other hand, We can compute the first part of the above right hand side by using The third term in (A.16) is Proof. We first write Therefore, Then, using (2.15), we have and thus The above relation yields We also have Finally, we conclude and B (2) Proof. We first write We have and from (A.13) and (A.21) This yields the result.

B Calculation for the linearization
Here we suppose that (2.7) is satisfied. We recall that where θ ∈ C 3 (R 2 ) is defined in Section 2.2 and where ξ = η − η S , which originally belongs to H 2 0 (0, 1), has been extended by zero outside (0, 1) to a function of H 2 (R) while keeping the same notation.
In what follows, we recall that γ (i) (y, ·) are linear mappings that depend on y in a Lipschitz continuous way and that vanish in F\Vα (see (2.4)). We also recall that Q2(α1, . . . , α k ) where k ∈ N denote the set of polynomials in the variables α1, . . . , α k and with coefficients that are Lipschitz continuous functions of y ∈ R 2 and of ξ and that vanish in F\Vα, and such that the degree of its nonzero monomial of lowest degree is greater or equal to 2.

(B.9)
We use now the above decomposition in order to linearize the operators appearing with the change of variables. We recall that v is defined by (2.11) and w by (2.19). We also recall that (2.7) is satisfied and that we assume v S ∈ W 2,∞ (F), div v S = 0 and f S ∈ W 2,∞ (F).
First we deal with the linearization of the condition on the divergence.
be a rectangle such that F ⊂ T . There exists an extension operator E which is continuous from L 2 (F) into L 2 (T ) as well as from H 1 (F) into H 1 0 (T ). Note that an interpolation argument guarantees that we also have E ∈ L(H s (F), H s 0 (T )) for s ∈ (0, 1). Then using this extension operator, any function in H s (F), s ∈ [0, 1] can be considered as a function in H s 0 (T ): in what follows we will extend wi, i = 1, 2 or some test functions ϕ, but for simplicity we will keep the same name wi and ϕ instead of E(wi) and E(ϕ). We will also freely use the boundedness properties of E without recalling it. Moreover, any function defined on (0, 1) is considered as a function defined on T but only depending on y1 and equal to zero outside (0, 1): by this way we can consider ξ1 and ξ2 as functions defined on T .
Here, 1I j denote the characteristic functions of Ij. Above we have use the fact that g is zero outside (0, 1).
We conclude by using (C.2). Finally, the case j = 2 follows more easily from an integration by parts with (5.13)-(5.15), by taking into account the estimate of ∂y 2 τ in (C.9) and the fact that ξ1 is independent on y2. The details are left to the reader.