Rémi Buffe
email: remi.buffe@univ-lorraine.fr.

Ludovick Gagnon
email: ludovick.gagnon@inria.fr.

Spectral inequality for an Oseen operator in a two dimensional channel

Keywords: Spectral inequality, Navier-Stokes equation, Null-controllability, Carleman estimate. Contents

, and prove the spectral inequaltiy through the derivation of a proper Carleman estimate. In the spirit of [4], we handle the vorticity near the boundary by using the characteristics sets of Pϕ or Qϕ 0 in the different microlocal regions of the cotangent space. As a consequence of the spectral inequality, we derive a new estimate of the cost of the control for the small-time null-controllability.

Introduction

In this paper, we consider the spectral inequality for the Navier-Stokes equation, with no-slip condition, in a 2-dimensional channel linearized around a Poiseuille flow. Denote Ω = T × (0, L) the spatial domain, where L > 0 and T is the one dimensional torus. For T > 0, the incompressible Navier-Stokes equation with no-slip boundary conditions is

        
∂ t u -∆u + (u • ∇)u + ∇p = 0, (x, y, t) ∈ Ω × (0, T ), div (u) = 0, (x, y, t) ∈ Ω × (0, T ), u(x, 0, t) = u(x, L, t) = 0, (x, t) ∈ T × (0, T ), u(x, y, 0) = u 0 (x, y), (x, y) ∈ Ω, [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF] where the intial data u 0 belongs to the space H, defined in [START_REF] Fursikov | Controllability of evolution equations[END_REF]. It is well-known that there exists a corresponding weak Leray solution to (1) that belongs to C 0 ([0, T ]; H) ∩ L 2 ((0, T ); V ) (see for instance [START_REF] Boyer | Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF], Thoerem V.I.4), where V is also defined in [START_REF] Fursikov | Controllability of evolution equations[END_REF] below. The Poiseuille flow is a laminar velocity field satisfying the no-slip condition,

U(y) = U 1 (y) 0 , U 1 (0) = U 1 (L) = 0, (2)
with U ∈ (C ∞ (0, L)) 2 . The linearized Navier-Stokes equation around the 2-dimensional Poiseuille flow writes

               ∂ t u -∆u + U 1 (y)∂ x u + u 2 ∂ y U 1 (y) 0
+ ∇p = 0, (x, y, t) ∈ Ω × (0, T ), div (u) = 0, (x, y, t) ∈ Ω × (0, T ), u(x, 0, t) = u(x, L, t) = 0, (x, t) ∈ T × (0, T ), u(x, y, 0) = u 0 (x, y), (x, y) ∈ Ω,

with u 0 ∈ H. The spatial operator of (3) is the Oseen operator [START_REF] Peter | Stability and transition in shear flows[END_REF]. Note that up to changing u by e -κt u, for κ > 0, one may write

               ∂ t u -∆u + U 1 (y)∂ x u + u 2 ∂ y U 1 (y) 0
+ ∇p + κu = 0, (x, y, t) ∈ Ω × (0, T ), div (u) = 0, (x, y, t) ∈ Ω × (0, T ), u(x, 0, t) = u(x, L, t) = 0, (x, t) ∈ T × (0, T ), u(x, y, 0) = u 0 (x, y), (x, y) ∈ Ω.

(

) 4
From now on, we refer to (4) as the Oseen equation. The shift by κ of the Oseen operator is introduced so as to gain the monotonicity of the Oseen operator (see Section 1.2), and has no consequences on the nature of our control results. Indeed, let us first introduce the null-controllability of the Oseen equation.

Let ω be an open subset of Ω, and consider

               ∂ t u -∆u + U 1 (y)∂ x u + u 2 ∂ y U 1 (y) 0 + ∇p + κu = χ ω f, (x, y, t) ∈ Ω × (0, T ),
div (u) = 0, (x, y, t) ∈ Ω × (0, T ), u(x, 0, t) = u(x, L, t) = 0, (x, t) ∈ T × (0, T ), u(x, y, 0) = u 0 (x, y), (x, y) ∈ Ω, [START_REF] Coron | Control and nonlinearity[END_REF] where χ ω denotes the characteristic function of the open set Ω. The small-time null-controllability can be stated as follows: for any control time T > 0 and any initial data u 0 ∈ H, does there exist a control function f ∈ L 2 ((0, T ); L 2 (ω)) such that the solution u of (5) satisfies u(T ) = 0 ? In the affirmative case, one is generally led to study the cost of the control, that is the estimation of the constant C T > 0 (depending on T > 0 and ω) of the continuity map between the control and the initial data, ||f || L 2 ((0,T );L 2 (ω)) ≤ C T ||u 0 || H .

Following the Lebeau-Robbiano strategy [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] (see also [START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF][START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF][START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF]) for the controllability of parabolic operators using the spectral inequality, we shall prove, Theorem 1.1 Let ε > 0. Then, there exists C > 0 such that for all T > 0, for all initial data u 0 ∈ H, there exists f ∈ L 2 ((0, T ); L 2 (ω)) such that the solution u of (5) satisfies u(T ) = 0, and such that the following estimate holds

||f || 2 L 2 ((0,T );L 2 (ω)) ≤ Ce C T 1+ε ||u 0 || 2 L 2 (Ω) . (6)
The null-controllability result stated in Theorem 1.1 is not new as it falls within the framework of Fernández-Cara, Guerrero, Yu, Imanuvilov and Puel [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF]. Indeed, considering the linearization of the Navier-Stokes equation ( 1) around smooth solutions U satisfying U ∈ L ∞ ((0, T ) × Ω)) n and ∂ t U ∈ L 2 ((0, T ); L σ (Ω)) n for some σ > 1, they proved notably, using the strategy of Fursikov and Imanuvilov [START_REF] Fursikov | Controllability of evolution equations[END_REF], the small-time null-controllability of the linearized equation. The optimal control cost was however not reached for small time in [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF].

Recently, Chaves-Silva and Lebeau have obtained in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] the optimal cost of controllability C T = Ce C T for the Stokes system in dimension greater or equal to 2, that is the Navier-Stokes equation linearized around zero, improving the cost of control of [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF] in this specific case. The control cost obtained of [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] is a consequence of the spectral inequality for the Stokes system in combination of the Lebeau-Robbiano strategy [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF].

Concerning small-time global controllability to trajectories of the nonlinear Navier-Stokes equation with no-slip boundary condition (1), we refer to the work of Coron, Marbach, Sueur and Zhang [START_REF] Coron | Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force[END_REF] (see also the references therein). In [START_REF] Coron | Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force[END_REF], the return method [START_REF] Coron | Control and nonlinearity[END_REF] was used, with a little help of a phantom force, to drive the solution to zero in arbitrarly small time, by using a trajectory with good transport properties and by analyzing the solution around this trajectory. One of the principal difficulties of [START_REF] Coron | Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force[END_REF] is to deal with the boundary layers. One way of removing this little phantom force is to understand the stability (or instability) of such as Poiseuille (or even Couette), and having a precise knowledge of the cost of control. The present paper can be seen as a first step in this direction, although the present work does not allow us to track the dependency of C T with respect to the size of the potentials.

Our main result is to derive a spectral inequality for the Oseen equation ( 4) in the spirit of [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]. We refer to Section 1.3 for a precise statement and Section 1.4 for insights on the nature of the result. We begin by considering the equation on the vorticity

v = ∂ x u 2 -∂ y u 1 for (4),          ∂ t v -∆v + U 1 (y)∂ x v + (∂ 2 y U 1 )u 2 + κv = 0, (x, y, t) ∈ Ω × (0, T ), div (u) = 0, (x, y, t) ∈ Ω × (0, T ), v(x, 0, t) = -∂ y u 1 (x, 0, t), v(x, L, t) = -∂ y u 1 (x, L, t), (x, t) ∈ T × (0, T ), v(x, y, 0) = rot u 0 (x, y), (x, y) ∈ Ω. (7)
At first sight, the Oseen equation ( 4) can be thought to be a perturbation of the Stokes equation (or equivalently [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF] a perturbation of the equation on the vorticity for the Stokes system), and classical arguments could allow one to absord these lower terms in the Carleman estimates established in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]. However, the divergence free condition on the velocity field u reveals a strong non-local term linking the vorticity to the velocity for [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF] of the form u 2 = ∆ -1 ∂ x v. Furthermore, this internal coupling does not allow us to prove unique continuation property for the augmented elliptic operator on the vorticity in the s direction as opposed to [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] (see Section 1.4 for the definition of the augmented elliptic operator and Remark 5.6 for the lack of propagation of smallness in the s direction). This lead us to place ourselves in the original framework of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], where the obtained spectral inequality is expressed with a local integral in the s-variable, as it is the case here (see Theorem 1.8). Since the works of [START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF][START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF] for the heat operator, using propagation of smallness in the s direction, spectral inequalities has been reformulated with an observation from {s = 0} × ω, which would yield, in the present setting

u 2 L 2 (Ω) ≤ Ce K √ Λ ω |u| 2 dx, ∀u ∈ Π Λ H.
The lack of unique continuation in the s-direction forces us to prove only Theorem 1.8. Finally, the Oseen operator is non self-adjoint, and therefore we shall use the work of [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF], where the Lebeau-Robbiano strategy is generalized for a class of non self-adjoint operators. Before stating the spectral inequality, let us introduce the functional framework as well as some spectral properties of the Oseen operator.

Functional framework

Let us recast equation ( 4) into a semi-group formalism. First we introduce the following Hilbert spaces,

V := {u ∈ H 1 0 (Ω) 2 , div u = 0}, H := {u ∈ L 2 (Ω) 2 , div u = 0, u • ν | ∂Ω = 0}, (8)
endowed by their usual norms, and where ν denotes the outward normal vector at the boundary. We define the Stokes operator,

A 0 u := -P∆u, A 0 : D(A 0 ) ⊂ H -→ H,
where P : L 2 (Ω) 2 → H stands for the standard Leray projection (the orthogonal projection of L 2 (Ω) on H), with domain

D(A 0 ) := V ∩ H 2 (Ω) 2 .
We moreover define,

Ã1 u = U 1 (y)∂ x u + u 2 ∂ y U 1 (y) 0 + κu,
and the operator

A 1 : D(A 1 ) ⊂ H -→ H defined by A 1 u := P Ã1 , of domain, D(A 1 ) := V ∩ H 1 (Ω) 2 .
Finally, the Oseen operator is defined by A :

D(A) = D(A 0 ) ⊂ H → H, A := A 0 + A 1 , (9)
and one can write (4) with the semi-group formalism,

d dt u + Au = 0, u |t=0 = u 0 .

Spectral properties of the Oseen operator

We recall the Weyl asymptotic formula for the two dimensional Stokes operator, obtained in [START_REF] Métivier | Valeurs propres de systèmes elliptiques définis sur des sous espaces[END_REF]. The operator A 0 being positive, self-adjoint and with compact resolvent, there exists a sequence of positive eigenvalues (µ k ) k∈N going to +∞, and a sequence of eigenfunctions

φ k ∈ H satisfying A 0 φ k = µ k φ k . Theorem 1.2 Let N (µ) = {µ k < µ}.
Then, there exists C > 0 such that,

N (µ) ∼ k→+∞ Cµ.
We recall here [12, Proposition 2.1], in the particular case A 1 is q-subordinate to A 0 , with q = 1/2, that we shall use in what follows.

Proposition 1.3 Assume that • Au, u ≥ 0;
• A 0 is a positive self-adjoint with compact resolvent and densely defined operator;

• There exists C > 0 such that for every u ∈ D(A

1/2 0 ) we have, A 1 u, u ≤ C 2 ||A 1/2 0 u|| 1/2 H ||u|| 1/2 H . (10)
Then

• D(A) = D(A 0 ) ⊂ D(A 1 )
⊂ H and has a compact resolvent;

• Sp(A) ⊂ {z ∈ C, | z| ≤ C|z| 1/2 }.
The Oseen operator falls within the hypothesis of Proposition 1.3, provided κ > 0 large enough to ensure the positivity.

Proposition 1.4 Let U ∈ (C ∞ (0, L)) 2 defined by (2). Then, there exists κ > 0 sufficiently large such that the operator A defined by (9

) satisfies Sp(A * ) ⊂ {z ∈ C, | z| ≤ C|z| 1/2 }.
From Proposition 1.4, we have that there exists

C 0 > 0 such that Sp(A) ⊂ {z ∈ C, | z| < C 0 ( z) 1/2 }.
We fix such C 0 > 0 here and below.

We are now ready to define the spectral projectors on the eigenspaces of A * , following [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF].

Definition 1.5 Let Λ 1 < Λ 2 < • • • < Λ k < .
. . a sequence of real numbers going to infinity and such that Λ k / ∈ Sp(A) for all k ∈ N * . We define the following contours in the complex plane

γ k := {z ∈ C, z = Λ k , | z| < C 0 Λ k } ∪ {z ∈ C, z ≤ Λ 1/2 k , | z| = C 0 ( z) 1/2 }.
We also define the spectral projectors,

Π Λ k : H -→ H u -→ 1 2iπ γ k (A * -z) -1 udz.
We finally recall the following estimate of the resolvent along the contours γ k (see [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF]Theorem 2.5] with p = 1, q = 1/2). Note that we can take p = 1, because of the Weyl formula of Theorem 1.2.

Theorem 1.6 There exists C > 0 such that for all k ∈ N, for all z ∈ γ k ,

||(A * -z) -1 || L(H,H) ≤ Ce C √ Λ k
As a corollary, we shall use the following result, at several places.

Corollary 1.7 For every holomorphic function f , for all u ∈ Π Λ k H, there exists C > 0 such that,

||f (A * )u|| L 2 (Ω) ≤ sup z∈γ k |f (z)|Ce C √ Λ k ||u|| L 2 (Ω) .
Proof. We follow here [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF]. One has, by holomorphic functional calculus,

||f (A * )u|| L 2 (Ω) = ||f (A * )Π Λ k u|| L 2 (Ω) = 1 2iπ γ k f (z) A * -z udz L 2 (Ω) ≤ sup z∈γ k |f (z)|meas(γ k ) sup z∈γ k ||(A * -z) -1 || L(H,H) ||u|| L 2 (Ω) ≤ sup z∈γ k |f (z)|Ce C √ Λ k ||u|| L 2 (Ω) . 2

Main result

Our main result is to establish the following spectral inequality for the Oseen operator.

Theorem 1.8 Let S 0 > 0. Let ω ⊂ Ω be a nonempty open set. Then, there exists C, K > 0, and ϕ ∈ C ∞ 0 (0, S 0 ) such that, for every Λ > 1, we have

u 2 L 2 (Ω) ≤ Ce K √ Λ (0,S0)×ω ϕ(s)(A * ) -1/2 sinh(s(A * ) 1/2 )u 2 dxds,
for all u ∈ Π Λ H.

Remark 1.9 As we believe that Lebeau-Robbiano spectral inequalities are interesting on their own, independently on applications, it is important to note that Theorem 1.8 also hold for the operator A. Indeed this spectral inequality is obtained through a direct application of the Carleman estimates proved below, and a careful inspection of the proofs show that it is also valid for the operator A. Indeed, A * and A only differs by their lower order terms, that is A 1 and A * 1 respectively. Proofs are written with A * 1 having in mind applications to control, but also work with A 1 .

The augmented operator

The proof of Theorem 1.8 is an adaptation of the work of Chaves-Silva and Lebeau [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] to our framework. First it is now classical to prove such spectral inequality by deriving a Carleman estimate for an augmented elliptic operator. More precisely, we introduce,

X Λ := (A * ) -1/2 sinh(s(A * ) 1/2 )ũ, ũ ∈ Π Λ H . (11)
Then, the augmented operator corresponds to, for any u ∈ X Λ ,

   (-∂ 2 s + A * )u = 0, (s, x, y) ∈ Z, u |s=0 = 0, (x, y) ∈ Ω, ∂ s u |s=0 = ũ, (x, y) ∈ Ω, (12)
where, Z = (0, S 0 ) × Ω, for some S 0 > 0. However, as pointed out in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF], unique continuation property does not hold for system [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF], mainly due to the pressure. One of the main ideas of [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF], that we shall follow here, is to remove the pressure term by considering instead the equation on the vorticity. But that approach has two major difficulties. The first one is the lack of information of the trace of the vorticity at the boundary. This difficulty is handled, as in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF], with techniques coming from the microlocal analysis for boundary value problems. More specifically, we exploit crucially the fact that the conjugation of the augmented elliptic operator by two different weights yields two different characteristic sets. Hence, we are able to recover the desired estimates using the appropriate conjugated operator in each microlocal region (see Section 3.3). This requires to make sure that these estimates with different weights can be patched together, which is accomplished in Section 4.2.4. The second one is the coupling between the vorticity and the velocity in the interior of the domain due to the low order terms. To overcome this difficulty, we apply successively two Carleman estimates. The drawback however is that we lose the propagation of smallness in the s-direction (as opposed to [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]). That is precisely the reason for using the original strategy of Lebeau-Robbiano [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], developped for non-self-adjoint operators in [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF], as it is the case here.

We conclude this section with some results on the regularity of the augmented elliptic operator. First, note that by elliptic regularity, one has the following lemma.

Lemma 1.10 Let u ∈ X Λ . Then, ∀k ∈ N, one have u ∈ H k (Z).

Proof. Let u = (A * ) -1/2 sinh(s(A * ) 1/2 )ũ, for some ũ ∈ Π Λ H. By the classical elliptic regularity and the resolvent estimate of Corollary 1.7, there exist C 1 > 0, and C 2 > 0 (that depends on Λ) such that,

||u(s, .)|| H 2 (Ω) ≤ C||A * u|| L 2 (Ω) ≤ C 2 ||ũ|| L 2 (Ω) .
From Definition 1.5 and by holomorphic functionnal calculus, for all k ∈ N, one has (A * ) k u ∈ D(A * ). and consequently, by elliptic regularity, one has

||u|| H 2k (Ω) ||A * u|| H k-2 (Ω) ||(A * ) k u|| L 2 (Ω) ||ũ|| L 2 (Ω) .
from the resolvent estimate of Corollary 1.7. Derivatives in s of u can also be estimated in L 2 ((0, S 0 ); H k (Ω)) by again using the definition of u and the resolvent estimate. 2 We shall also use a different formulation of [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF] in the core of the proof of the Carleman estimate, by introducing the pressure term. Lemma 1.11 For all u ∈ X Λ , there exists q ∈ C ∞ (Z) such that,

-∂ 2 s u -∆u + Ã * 1 u + ∇q = 0, (s, x, y) ∈ Z, div u = 0, (s, x, y) ∈ Z. (13)
Proof. We follow [START_REF] Boyer | Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF]. Let φ ∈ V be a test function. One has, for s ∈ (0, S 0 ),

0 = (-∂ 2 s + A * )u, φ H = -∂ 2 s u, φ L 2 + (A * u, φ) V ,V = -∂ 2 s u, φ L 2 + (-P∆u, φ) V ,V + (A * 1 u, φ) V ,V = -∂ 2 s u, φ L 2 + (-∆u, φ) H -1 ,H 1 0 + (A * 1 u, φ) V ,V = -∂ 2 s u -∆u + A * 1 u, φ H -1 ,H 1 0 .
As a result, using Theorem IV.2.3 in [START_REF] Boyer | Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF], there exists q ∈ L 2 such that,

-∂ 2 s u -∆u + A * 1 u + ∇q = 0.
Moreover, the regularity of q follows from Lemma 1.10. 2

Notations and semi-classical norms

We conclude the introduction by introducing notations that shall be used throughout the article. We also recall the standard definitions for semi-classical operators as well as semi-classical norms. We also introduce the definition of a particular class of tangential semi-classical operators that shall be used in Section 4.2.2.

Notations

We recall that,

Z = (0, S 0 ) × Ω,
for some S 0 > 0. Let us also define for later conveniences,

Y = ( S 0 , S 0 -S 0 ) × Ω, for S0 < S 0 /2.
Since Y denote a localization in the s variable, we will use the abuse of notation ∂Y to denote {{y = 0} ∪ {y = L}} ∩ Y . Now let u ∈ X Λ (hence, u solves ( 12)), and define here and below,

v := rot(u), (14)
which satisfies in Ω,

-∂ 2 s v -∆v + rot (A * 1 u) = 0. (15)
In what follows, w(s, x, y) = e ϕ(s,y)/h v(s, x, y).

where the weight function ϕ is defined below in (24). We also define the localized functions we shall work with throughout the article, with q defined by Lemma 1.11,

W (s, x, y) = χ(s, x, y)w(s, x, y), V (s, x, y) = χ(s, x, y)v(s, x, y), U (s, x, y) = χ(s, x, y)u(s, x, y), Q(s, x, y) = χ(s, x, y)q(s, x, y) U(s, x, y) = χ 1 (s)u(s, x, y), V(s, x, y) = χ 1 (s)v(s, x, y), W(s, x, y) = χ 1 (s)w(s, x, y), Q(s, x, y) = χ 1 (s)q(s, x, y), (16)
where χ(s, x, y) = χ 1 (s)χ 2 (x)χ 3 (y) and where χ 1 (s), χ 2 (x), χ 3 (y) are smooth cut-offs in each variable satisfying near a point (s 0 , x 0 , y = 0) ∈ (0, S 0 ) × T × {y = 0} :

χ 1 (s) = 1 if |s -s 0 | < S 1 , 0 if |s -s 0 | > 2S 1 , χ 3 (y) = 1 if y < Y 1 , 0 if y > Y 1 , (17)
for some 0

< 2S 1 < S 0 (that is, supp χ ⊂ Y ) and some Y 1 < L/4. The localization χ(s, x, y) is defined likewise near a point (s 0 , x 0 , y = L) ∈ (0, S 0 ) × T × {y = L}.
Finally, C > 0 is a generic constant which may change from one line to another.

Pseudo-differential operators and semi-classical norms

We introduce the variable z = (s, x, y) ∈ R × T × R and the tangential variable z to be defined as z = (s, x) ∈ R × T, as well as their Fourier counterparts

ζ = (σ, ξ, η) ∈ R × N × R and ζ = (σ, ξ). Note that (z, ζ) ∈ R × T × R × R × N × R := T * (R × T × R) and (z , ζ ) ∈ R × T × R × N := T * (R × T).
We emphasize that the differential operators ∇, ∆ and rot only act on the physical variables x, y. We shall also use the notation ∇ z := t (∂ s , ∂ x , ∂ y ), as well as ∆ z = ∇ 2 z when needed.

Semi-classical operators

Here we recall some facts on semi-classical pseudo-differential operators with a small parameter h, say 0 < h ≤ h 0 . We shall denote by S m sc the space of smooth functions a(z, ζ, h), with 0 < h ≤ h 0 as a small parameter, that satisfy the following behavior at infinity: for all multi-indices α, β there exists C α,β > 0 such that,

∂ α z ∂ β ζ a(z, ζ, h) ≤ C α,β (1 + |ζ| 2 ) (m-|β|)/2 , for all (z, ζ) ∈ T * (R × T × R). For a ∈ S m
sc , we define pseudo-differential operator of order m, denoted by A = Op(a):

Au(z) := 1 (2hπ) 3 ξ∈ 1 2πh Z R 2 R×T×R e i (z-z)•ζ h a(z, ζ, h)u(z)dzdσdη,
One says that a is the symbol of A. We shall denote by Ψ m sc the set of pseudo-differential operators of order m and denote by σ(A) (resp. σ(a)) the principal symbol of the operator A (resp. the symbol a). We refer to [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] to precise definitions of pseudo-differential operators. Thus, define D = h∂/i, then σ(D) = ξ. We shall also denote by D m sc the space of semi-classical differential operators, i.e the case when the symbol a(x, ξ, h) is a polynomial function of order m in ξ. We recall here the composition formula of pseudo-differential operators. Let a ∈ S m sc and b

∈ S m sc , m, m ∈ R, we have Op(a) • Op(b) = Op(c),
for some c ∈ S m+m sc .

Tangential semi-classical operators

In the section we consider pseudo-differential operators which only acts in the tangential direction z , viewing the variable y as a parameter. We define S m T,sc as the set of smooth functions b(z, ζ , h) defined for h a small parameter, say 0 < h < h 0 , satisfying the following behavior at infinity: for all multi-indices

α ∈ N n , β ∈ N n-1 there exists a constant C α,β > 0 such that, ∂ α x ∂ β ζ b(z, ζ , h) ≤ C α,β (1 + |ζ | 2 ) (m-|β|)/2 , for all (z, ζ , h) ∈ T * (R × T)×]0, 1[. For b ∈ S m
T,sc , we define a tangential pseudo-differential operator B := Op T (b) of order m by,

Bu(x) := 1 (2hπ) 2 ξ∈h -1 Z R R×T e i (z -z )•ζ h b(z, ζ , h)u(z )dz dσ
As in the previous section, we define Ψ m T,τ as the set of tangential pseudo-differential operators of order m, and D m T,τ the set of tangential differential operators of order m. We shall use this class of pseudodifferential operators in Section 4.2.1, that is the region where |ξ| is small.

Semi-classical norms

We first define the following semi-classical Sobolev tangential norms, for traces of functions on

R × T at {y = 0} or {y = L} |u| m,sc := | Op T ((1 + |ζ | 2 ) m 2 )u| L 2 (R×T) . For m ∈ N, this semi-classical norm is equivalent to |α|≤m h |α| |∂ |α| z u| L 2 (R×T)
, uniformly for h ∈ (0, 1). We also define the following semi-classical norms, in the interior of the domain R × Ω,

||u|| 2 m,sc := m k=0 ||∂ k y Op T ((1 + |ζ | 2 ) m-k 2 )u|| 2 L 2 (R×T×[0,L]) . As above, if m ∈ N, this semi-classical norm is equivalent to |α|≤m h |α| ||∂ |α| z u|| L 2 (R×T×[0,L]) , uniformly for h ∈ (0, 1).
Below, we shall the following semi-classical trace lemma, that can be found in [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]Section 5 (eq. 44)].

Lemma 1.12 There exists C > 0 such that for all u ∈ H 1 (R × T × [0, L]), for h ∈ (0, 1),

|u | y=L | 0,sc + |u |y=0 | 0,sc ≤ Ch -1/2 ||u|| 1,sc . (18)
2 From the spectral inequality to the cost of control

In this section, using the works of [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF], we show that the spectral inequality of Theorem 1.8 allows one to prove an observability inequality for the Oseen equation, with optimal cost of control. We consider the following controlled system :

d dt u + Au = Pχ ω f, (x, y, t) ∈ Ω × (0, T ), u |t=0 = u 0 ∈ H, (x, y) ∈ Ω. ( 19
)
where χ is the characteristic function of an open subset ω ⊂ Ω, and f is the control function.

Theorem 2.1 Let ε > 0.There exists C > 0 such that for all T > 0, for all initial data u 0 ∈ H, there exists f ∈ L 2 (0, T ; Ω) such that the solution u of (19) satisfies

u(T ) = 0,
and such that the following estimate holds

||f || 2 L 2 (0,T,L 2 (Ω)) ≤ Ce C T 1+ε ||u 0 || 2 L 2 (Ω) (20)
The constant appearing in (20) is referred as the cost of controllability. Controllability of linearized Navier-Stokes equations is already known (see for instance [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF], where the author used a global Carleman estimate for the parabolic operator) but the cost of control is not optimal. The work of [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] exhibited for the first time the optimal behavior of this constant as T → 0 for the Stokes equation. The above spectral inequality of Theorem 1.8 allows us to obtain almost the same result for the Oseen operator, using an adaptation of the arguments of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF][START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF]. The proof relies on the well-known Lebeau-Robbiano strategy. In [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF], in a abstract setting, the author showed that spectral inequality of the form of Theorem 1.8 for non-self-adjoint operators satisfying the properties of Proposition 1.4 implies null-controllability for the parabolic problem [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]. However, cost of controllability was not derived in this article. Here, we then adopt the strategy of [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] in Lemma 2.7 and Theorem 2.6 to obtain the cost of control (20). Note that there is a little loss in the power of T in (20), as we may expect to have ε = 0 as in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF].

Controllability of the low frequencies

We first recast Theorem 1.8 into the formalism of [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF].

Corollary 2.2 Let S 0 > 0 and ω ⊂ Ω be a nonempty open set. Then, there exists C, K > 0, and ϕ ∈ C ∞ 0 (0, S 0 ) such that, for every Λ > 1, we have

u 2 L 2 (Ω) ≤ Ce K √ Λ (0,S0)×ω ϕ(s) sinh(s(A * ) 1/2 )u 2 dxds, for all u ∈ Π Λ H.
Proof. Note that √ A * is an isomorphism on Π λ H. Using the resolvent estimate of Corollary 1.7, we obtain

|| √ A * u|| L 2 (Ω) = || √ A * Π Λ u|| L 2 (Ω) = 1 2iπ γ k (A * -z) -1 √ A * udz L 2 (Ω) ≤ Ce C √ Λ ||u|| L 2 (Ω) .
Hence, applying Theorem 1.8 to v = √ A * u proves Corollary 2.2. 2 Now we consider the parabolic control problem for the low frenquencies,

d dt u + Au = Pχ ω f, (x, y, t) ∈ Ω × (0, T ), u |t=0 = u 0 ∈ Π Λ H. ( 21
)
The controllability of parabolic equations (21) of the lower frequencies for non-self-adjoint operators has been already developed in [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF] in an abstract setting. From Proposition 1.4 and Corollary 2.2, we obtain Theorem 2.3, as they fall into the hypothesis of [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF]. More precisely, Theorem 2.3 is [12, Theorems 4.9 and 4.10] in our setting, with q = 1/2.

Theorem 2.3 Let γ > 1 and let T * > 0. There exists C > 0 such that for all 0 < T < T * , for all u 0 ∈ Π Λ H, there exists f ∈ L 2 ((0, T ); L 2 (Ω)) such that the solution of (21

) satisfies Π Λ u(T ) = 0, with control cost ||f || 2 L 2 ((0,T );L 2 (Ω)) ≤ Ce C( √ Λ+ 1 T γ ) ||u 0 || 2 L 2 (Ω) .
Let us consider the dual counterpart of (21) (along with the change of variables t → (T -t)).

d dt u + A * u = 0 u |t=0 = u 0 ∈ Π Λ H. ( 22
)
The following observability inequality then holds, coming from the duality between controllability and observability, noting that the adjoint of the control operator Pχ ω satisfies (Pχ ω ) * u = u |ω .

Corollary 2.4 Let γ > 1 and let T * > 0. There exists C > 0 such that for all 0 < T < T * , and for all Λ > 0,

||u(T )|| 2 H ≤ Ce C( √ Λ+ 1 T γ ) T 0 ω |u(t, x)| 2 dtdx,
for all u solution of (22).

Decay of the semigroup

Following the idea of the Lebeau-Robbiano strategy, one shall need the following estimate of the natural dissipation of the higher frequencies (see [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF], Proposition 4.12 with θ = 1/2 for a proof : note that one can take θ = 1/2 since in our case, q = 1/2 from Proposition 1.4, and p = 1 from Theorem 1.2).

Proposition 2.5 There exists C > 0, Λ 0 > 0, such that for all Λ ≥ Λ 0 and for all t > 1 √ Λ ,

||S A (t) (I -Π Λ ) || 2 L(H) ≤ Ce C √ Λ-tΛ .

Derivation of an observability inequality

Let u be solution of the following Oseen equation

d dt u + A * u = 0, u |t=0 = u 0 ∈ H. ( 23
)
Theorem 2.6 Let γ > 1 and T * > 0. There exists C > 0 such that for every 0 < T < T * ,

||u(T )|| 2 L 2 (Ω) ≤ Ce C T γ T 0 ω |u(t, x)| 2 dtdx,
with u solution of (23).

Proof. We only provide a sketch of the proof, as it is classical, and sufficient to apply Lemma 2.7 in combination with Lemmata 4.1 and 4.2 in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]. The main idea is to consider of proper partition of the interval [0, T ] = ∪ ∞ k=0 [a j , a j+1 ], and apply Lemma 2.7 between t = a j and t = a j+1 and sum over j. As the left hand side provides a telescopic sum, one obtains the result. We refer to [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF][START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] for more details. 2 Lemma 2.7 There exist T * > 0 and m ∈ (0, 1) such that,

F (t)||u(t)|| 2 H -F (mt)||u 0 || 2 H ≤ t 0 ω |u(t, x)| 2 dtdx,
holds for all u solution of (23), for all t ∈ [0, T * ], with F of the form

F (t) = K 1 e -K 2 t γ , with γ > 1.
Proof. One shall denote by u = v + w where v = Π Λ and w = (1 -Π Λ )u. Note that Corollary 2.4 applies for v and Proposition 2.5 applies for w. To this end, one sets,

f (t) = C 1 e C1( √ Λ+ 1 T γ ) , g(t) = C 2 e C2 √ Λ-tΛ .
Hence, applying Corollary 2.4 and Proposition 2.5, up to translating time 0 → T 1 , for 0 < T 1 < T < T * , for every Λ > Λ 0 , with Λ 0 > 0 sufficiently large, and

T * < Λ -1 2 0 , ||u(T )|| 2 H ≤ C ||v(T )|| 2 H + ||w(T )|| 2 H ≤ C f (T -T 1 ) T T1 ω |v(t, x)| 2 dtdx + ||w(T )|| 2 H ≤ C f (T -T 1 ) T T1 ω |u(t, x)| 2 dtdx + 2f (T -T 1 ) T T1 Ω |w(t, x)| 2 dtdx + ||w(T )|| 2 H ≤ C f (T -T 1 ) T T1 ω |u(t, x)| 2 dtdx + 2f (T -T 1 )(T -T 1 )g(T -T 1 )||w(T 1 )|| 2 H + ||w(T )|| 2 H ≤ C f (T -T 1 ) T T1 ω |u(t, x)| 2 dtdx + (2f (T -T 1 )(T -T 1 )g(T -T 1 )g(T 1 ) + g(T )) ||w(T )|| 2 H ≤ C f (T -T 1 ) T T1 ω |u(t, x)| 2 dtdx + (2f (T -T 1 )(T -T 1 ) + 1) g(T )||w(T )|| 2 H . Let us set T 1 = (1 -ε)T , which implies, (Cf (εT )) -1 ||u(T )|| 2 H -(Cf (εT )) -1 (2f (εT )εT + 1) g(T )||w(T )|| 2 H ≤ T (1-ε)T ω |u(t, x)| 2 dtdx.
Setting (εT ) 1+γ Λ = 1 (note that it is in accordance with hypothesis of Proposition 2.5) yields,

(Cf (εT )) -1 (2f (εT )εT + 1) g(T ) = 2 C 2 C (εT )e C 2 (εT ) γ+1 2 -1 ε(εT ) γ + 1 CC 1 e C 2 (εT ) γ+1 2 -1 ε(εT ) γ -C1( C 1 (εT ) γ+1 2 + 1 (εT ) γ )
.

Hence, there exists ε 0 > 0 and C 3 , C 4 , C 5 , C 6 > 0 such that for every ε < ε 0 , one has,

(Cf (εT )) -1 (2f (εT )εT + 1) g(T ) ≤ C 3 e -C 4 ε(εT ) γ , and (Cf (εT )) -1 ≥ C 5 e -C 6 (εT ) γ .
Taking again ε 0 sufficiently small, there exist K 1 , K 2 > 0 such that for all 0 < ε < ε 0 ,

F (T )||u(T )|| 2 H -F T 2 ||w(T )|| 2 H ≤ T 0 ω |u(t, x)| 2 dtdx, with F (t) = K 1 e -K 2
T γ . This ends the proof. 2

Conjugated operators and weight function properties

In this section, we begin by introducing the different weight functions that shall be used in the Carleman estimates, as well as the conjugated operators. We then set the ground for the analysis of the boundary terms arising in the Carleman estimates (dealt in Section 4.2) by splitting the cotangent space of ∂Y in different regions and by analyzing the roots of the principal symbols of Q ϕ0 and P ϕ .

Augmented operator conjugated by the weight function

Near the boundaries {y = 0} and {y = L}, we shall work with the following weight function, ϕ(s, y) := e γψ (s,y) ,

where ψ (s, y) is smooth and satisfies,

ψ (s, y) = y -ε(s -s 0 ) 2 in a neighborhood of {y = 0}, (L -y) -ε(s -s 0 ) 2 in a neighborhood of {y = L}. ( 25
)
where > 0 shall be fixed small and γ > 0 shall be fixed large in what follows. In order to prove a proper Carleman estimate in Section 4, one shall work with three slightly different conjugated operators. We shall denote by,

P ϕ := h 2 e ϕ/h (-∂ 2 s -∆)e -ϕ/h = -h 2 ∂ 2 s -h 2 ∆ -((∂ y ϕ) 2 + (∂ s ϕ) 2 ) + 2h(∂ y ϕ∂ y + ∂ s ϕ∂ s ) + h 2 (∂ 2 y ϕ + ∂ 2 s ϕ
), with semiclassical principal symbol given by,

p ϕ (s, y, ξ, η, σ) = σ 2 + η 2 + ξ 2 -((∂ y ϕ) 2 + (∂ s ϕ) 2 ) + 2i(∂ y ϕη + ∂ s ϕσ), (26)
where (σ, ξ, η) denotes the Fourier variables of the space variables (s, x, y). Defining the weight function

ϕ 0 , ϕ 0 (s) := ϕ(y = 0, s) = ϕ(y = L, s), (27)
we shall also use the following conjugated operator,

Q ϕ0 := h 2 e ϕ0/h (-∂ 2 s -∆)e -ϕ0/h = -h 2 ∂ 2 s -h 2 ∆ -(∂ s ϕ 0 ) 2 + 2h(∂ s ϕ 0 )∂ s + h 2 ∂ 2 s ϕ 0 , (28)
with semiclassical principal symbol given by,

q ϕ0 (s, ξ, η, σ) = σ 2 + η 2 + ξ 2 -(∂ s ϕ 0 ) 2 + 2i(∂ s ϕ 0 )σ.
Finally, we introduce a third conjugated operator that shall be needed in Section 5,

-∆ ϕ := h 2 e ϕ/h (-∆)e -ϕ/h = -h 2 ∆ -|∇ϕ| 2 + 2h∇ϕ • ∇ + h 2 ∆ϕ, (29)
with a semiclassical principal symbol,

δ ϕ (s, x, y, ξ, η) = η 2 + ξ 2 -|∇ϕ| 2 + 2i∇ϕ • ξ η .

Sub-ellipticity condition

Under the action of the conjugaison, elliptic operators such as P , Q or -∆ defined in the previous section are not elliptic. To handle characteristics sets, one shall use the sub-ellipticity property, defined in what follows.

Definition 3.1 Let U be an open set of R 3 . We say that (P, ϕ) satisfies the Hörmander's sub-ellipticity condition in

U if |∇ z ϕ| = |(∂ s ϕ, ∇ϕ)| ≥ C > 0 on U and if p ϕ (s, x, y, σ, ξ, η) = 0 =⇒ 1 2i {p ϕ , p ϕ } > 0. ( 30
)
In what follows, one shall also need a slightly different sub-ellipticity definition, with the same weight function but for the (not-augmented) -Laplace operator given by (29). Indeed, when considering [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF], one observes that the vorticity v is coupled with the velocity u due to the lower-order terms. One shall pay attention that, for the same weight function ϕ, the Carleman estimates given below applies both for v (associated with the operator P ϕ ) and for u (associated with the operator -∆).

Definition 3.2 We say that (-∆, ϕ) satisfies the sub-ellipticity condition in

U if |∇ϕ| ≥ C > 0 on U and if, δ ϕ (s, x, y, ξ, η) = 0 =⇒ 1 2i {δ ϕ , δ ϕ } > 0. ( 31
)
It is now well-known (see [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF] for instance) that the convexified weight given by (24) satisfies the subellipticity conditions of definitions 3.

Microlocal regions

Following the ideas of [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF], we introduce three different microlocal regions in the cotangent space of ∂Y . First we define,

E + := {(σ, ξ) ∈ R 2 , |ξ| 2 > 2 sup(∂ s ϕ 0 ) 2 }, E -:= {(σ, ξ) ∈ R 2 , |ξ| 2 < 3 sup(∂ s ϕ 0 ) 2 },
and we shall also split

E -= E - 1 ∪ E - 2
where,

E - 1 := {(σ, ξ) ∈ E -, |σ| < 2δ}, E - 2 := {(σ, ξ) ∈ E -, |σ| > δ},
for a small δ > 0 that shall be fixed in what follows. This microlocal decomposition of the phase space are represented in Figure 1, and are motivated by the position of the roots of Q ϕ0 and P ϕ obtained in Section 3.4 and the strategy of the proof detailed in the beginning of Section 4.2.

Roots properties

It is classical when dealing with boundary value problems to consider the principal symbols of the operators P ϕ and Q ϕ0 as polynomials in the conormal variable η. We first analyse the behavior of the roots of the principal symbol q ϕ0 .

Analysis of Q ϕ0

As the weight function ϕ 0 does not depend on the x, y variables, the symbol q ϕ0 can be decomposed as follows,

q ϕ0 (s, σ, ξ, η) = (η -r + (s, σ, ξ))(η -r -(s, σ, ξ)). ( 32
) Lemma 3.3 We have in E + , • -(r ± (s, σ, ξ)) 2 = σ 2 + ξ 2 -(∂ s ϕ 0 ) 2 + 2i∂ s ϕ 0 σ, r ± ∈ S1 T,sc , • r + = -r -, • | Im r ± | ≥ C > 0,
From the last point, we make the following convention ± Im r ± ≥ C > 0.

Proof. The first two points come from the definition of q ϕ0 . For the last point, notice that, in E + , we have, Re((r

± (s, σ, ξ)) 2 ) = (∂ s ϕ 0 ) 2 -σ 2 -ξ 2 ≤ -c < 0, which is sufficient to conclude that ± Im r ± ≥ C > 0.
2 Note in particular from the third point that in the region E + , the operator Q ϕ0 is elliptic, since there is no real characteristic set (characteristics sets are represented in Figure 1).

Analysis of P ϕ

Second, we analyze the roots of P ϕ . As above, we write,

p ϕ (s, y, σ, ξ, η) = (η -ρ + (s, y, σ, ξ))(η -ρ -(s, y, σ, ξ)), (33)
where ρ ± (s, y, σ, ξ) := -i∂ y ϕ ± m(s, y, σ, ξ) ∈ S1 T,sc , with -m 2 = ξ 2 + σ 2 -(∂ s ϕ) 2 + 2i(∂ s ϕ)σ, and by convention Re(m) ≥ 0. We shall use this polynomial form in the region E - 1 . Hence, we shall need the following lemma, that describes the position of the roots in the complex plane in that microlocal region. Lemma 3.4 In E - 1 , we have, • for y ∈ (0, L) in a neighborhood of {y = 0}, we have Im ρ ± ≤ -c < 0,

• for y ∈ (0, L) in a neighborhood of {y = L}, we have Im ρ ± ≥ c > 0.

Proof. On E - 1 , we have |σ| < 2δ and, using the definition of ϕ, |ξ| 2 < 3c 2 . Hence, |m 2 | ≤ c(δ 2 + 2 ). Therefore, for δ and chosen sufficiently small, we have that Im ρ ± has the same sign than -∂ y ϕ. Yet, ∂ y ϕ > 0 near y = 0, and ∂ y ϕ < 0 (see (25) for an explicit expression of ϕ in a neighborhood of the boundaries.) 2 One note that in fact, ∂ ν ϕ | y∈{0,L} < 0, where ν is the unit outward normal vector of the boundary {y = 0} ∪ {y = L}. From now on, δ is fixed in what follows such that Lemma 3.4 holds true. Note that Lemma 3.4 shows that the operator P ϕ is elliptic in the microlocal region E - 1 (a projection on (ξ, σ) of characteristic set of P ϕ is represented in Figure 1).

A Carleman estimate near the boundary 4.1 Estimates for the volume terms

We shall use the following result in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF] as a black box. It is worth noticing that their result applies with the weight function defined in (27) since the weight function satisfies the sub-ellipticity condition of Definition 3.1 and moreover satisfies ∂ ν ϕ ≤ -c < 0 on the support of W (defined in ( 16)). We recall that we denoted by ∂Y the boundaries {y = 0} and {y = L}. Theorem 4.1 There exist h 0 > 0, C > 0 such that,

h||e ϕ/h V || 2 L 2 (Y ) + h 3 ||e ϕ/h ∇ z V || 2 L 2 (Y ) ≤ C h 4 ||e ϕ/h (∂ 2 s + ∆)V || 2 L 2 (Y ) + h|e ϕ0/h V | ∂Y | 2 L 2 (∂Y ) + h 3 |e ϕ/h ∇ z V | ∂Y | 2 L 2 (∂Y ) ,
for all 0 < h ≤ h 0 , for all V of the form [START_REF] Métivier | Valeurs propres de systèmes elliptiques définis sur des sous espaces[END_REF].

Using the particular form of the function V and the divergence free condition, one may recover an estimate on the velocity U . Theorem 4.2 There exist h 0 > 0, C > 0 such that,

h||e ϕ/h V || 2 L 2 (Y ) + h 3 ||e ϕ/h ∇ z V || 2 L 2 (Y ) + ||e ϕ/h U || 2 L 2 (Y ) + h 2 ||e ϕ/h ∇U || 2 L 2 (Y ) ≤ C h 4 ||e ϕ/h (∂ 2 s + ∆)V || 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, χ]u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∇, χ]v|| 2 L 2 (Y ) + h|e ϕ0/h V | ∂Y | 2 L 2 (∂Y ) + h 3 |e ϕ/h ∇ z V | ∂Y | 2 L 2 (∂Y ) , (34)
for all 0 < h ≤ h 0 , for all U, V of the form [START_REF] Métivier | Valeurs propres de systèmes elliptiques définis sur des sous espaces[END_REF].

Proof. Using commutation with cut-off functions, and the divergence free condition on u, we obtain,

h 3 ||e ϕ/h ∆U || 2 L 2 (Y ) h 3 ||e ϕ/h χ∆u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, χ]u|| 2 L 2 (Y ) h 3 ||e ϕ/h χ∇ z v|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, χ]u|| 2 L 2 (Y ) h 3 ||e ϕ/h ∇ z V || 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, χ]u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∇ z , χ]v|| 2 L 2 (Y ) . (35)
Now observe that classical the Carleman estimate holds for ∆ ϕ U , near the boundary ∂Y , as U satisfies homogeneous Dirichlet boundary condition, and falls into the scope of Definition 3.2. We recall it here, adapted to our setting (see for instance [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF][START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]).

Theorem 4.3 Let z 0 ∈ ∂Y and let ϕ ∈ C ∞ (Y ) such that the pair (-∆, ϕ) satisfies Definition 3.2 in a neighborhood of z 0 in Y . Then, there exists h 0 > 0 such that,

||e ϕ/h U || 2 L 2 (Y ) + h 2 ||e ϕ/h ∇U || 2 L 2 (Y ) h 3 ||e ϕ/h ∆U || 2 L 2 (Y ) , (36)
for all U defined in [START_REF] Métivier | Valeurs propres de systèmes elliptiques définis sur des sous espaces[END_REF].

Using (36) to the right-hand side of (35) (up to shrink the cut-off function χ if necessary), we obtain the sought result. 2

Estimates of the boundary terms

The main difficulty in using estimate (34) lies in the estimation of the trace terms in the right-hand side, since we do not have any straightforward information on the traces of v (or equivalently V ). To overcome this difficulty, we shall perform several microlocalizations and use the splitting of the cotangent space of ∂Y : E + , E - 1 and E - 2 . Each region shall have a different treatment. • E - 1 is the low-frequency region. It is now well-known (see [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]) that one can estimate the two boundary traces of the solution by the source term, without any prescribed boundary conditions, since from Lemma 3.4, near {y = 0}, both roots have negative imaginary part and the domain is on the side {y > 0}, and near {y = L} both roots have positive imaginary parts and the domain is on the side {y < L}. For such discussion, one also refers to [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: boundary value problems[END_REF][START_REF] Buffe | Stabilization of the wave equation with ventcel boundary condition[END_REF].

• E + is a high-frequency region in the ξ variable. There, we shall perform an elliptic estimate on u (recall the relation rot u = v) using the ellipticity of Q ϕ0 in that region (see Lemma 3.3) and the knowledge of a Dirichlet boundary condition on the vector field u.

• E - 2 is the remaining region. Using the arguments of [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF], we shall prove an analytic estimate in the (s, σ) variable by a Paley-Wiener type theorem. This section deals with this program. Before doing so, we first recall that, due to the homogeneous boundary condition U | ∂Y = 0, we have W | ∂Y = ∂ y U 1 , where U = t (U 1 , U 2 ) denote the tangential and perpendicular component of U respectively.

Estimates in E -

1

We start with the low frequency region. Let θ - 1 (ξ, σ) = α 1 (ξ)β 1 (σ), with α, β bounded smooth functions of order zero such that supp θ - 1 ⊂ E - 1 . From classical elliptic boundary value problems methods (see for instance [START_REF] Buffe | Stabilization of the wave equation with ventcel boundary condition[END_REF], Proposition 6.1-8), we have the following result. Proposition 4.4 There exists h 0 > 0 and C > 0 such that,

h| Op(θ - 1 )W | 2 L 2 (∂Y ) + h 3 |∇ z Op(θ - 1 )W | 2 L 2 (∂Y ) ≤ C h 4 ||P ϕ W || 2 L 2 (Y ) + h 4 ||∇ z W || 2 L 2 (Y ) + h 2 ||W || 2 L 2 (Y ) , (37)
for all 0 ≤ h ≤ h 0 and for all W of the form (16).

Estimates in E +

We continue with the high frenquency region in the variable ξ. Let θ + (ξ) be a bounded smooth function such that supp θ + ⊂ E + . In the present section, we shall work globally with respect to the (x, y) variables, but locally with respect to the s variable, in order to avoid truncation coming from ∇χp to appear. To do this, we recall ( 16)

U = χ 1 (s)u, Q = χ 1 q,
and we shall consider the following system,

-∂ 2 s U -∆U + ∇Q = F, (s, x, y) ∈ R × Ω, div U = 0, (s, x, y) ∈ R × Ω, ( 38
)
and obtain estimates on U, since u satisfies [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. Introduce the following functions U := e ϕ0/h U, F := h 2 e ϕ0/h F, Q := he ϕ0/h Q. As a result, we shall work with the operator Q ϕ0 (defined in (28)). Hence, U satisfies the following conjugated system,

Q ϕ0 U + h∇Q = F, (s, x, y) ∈ R × Ω, div U = 0, (s, x, y) ∈ R × Ω.
We emphasize that the weight function ϕ 0 only depends on the variable s. A key point of this section is that Q ϕ0 is elliptic on E + (see Lemma 3.3). Note also that we have homogeneous boundary condition.

Proposition 4.5 There exists h 0 > 0 and C > 0 such that,

|| Op T (θ + )U|| 2 2,sc + h 3 |e ϕ0/h Op T (θ + )(∂ y u 1 ) | ∂Y | 2 1,sc ≤ Ch 4 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc , (39)
for all 0 ≤ h ≤ h 0 and for all U of the form [START_REF] Métivier | Valeurs propres de systèmes elliptiques définis sur des sous espaces[END_REF].

Remark that the estimate of the boundary term e ϕ0/h Op T (θ + )(∂ y u 1 ) | ∂Y is precisely the boundary terms appearing in the right-hand side of (34), microlocalized in the E + region.

Proof. The proof is based on integration by parts, noting that the computations are valid because of the regularity of the function u (see Lemma 1.10). We recall that

Q ϕ0 = -h 2 ∆ z -(∂ s ϕ 0 ) 2 +2h(∂ s ϕ 0 )∂ s + h(∂ 2 s ϕ 0 ). Let us denote by U 1 = Op T (θ + )U, Q 1 = Op T (θ + )Q and F 1 = Op T (θ + )F.
As θ + is a Fourier multiplier in the x direction, it commutes with all the involved operators and we have the following system,

     Q ϕ0 U 1 + h∇Q 1 = F 1 , (s, x, y) ∈ R × Ω, div U 1 = 0, (s, x, y) ∈ R × Ω, U 1|y=0 = U 1| y=L = 0, (s, x) ∈ R × T, (40)
Multiplying the first line of (40) by U 1 and integration by parts yields,

h 2 Z |∇ z U 1 | 2 - Z (∂ s ϕ 0 ) 2 |U 1 | 2 ||F 1 || 2 L 2 (Z) .
Note that on the support of θ + , we have h|D x | > 2 sup |∂ s ϕ 0 | 2 , and consequently, there exists C > 0 such that,

||U 1 || 2 1,sc ≤ C||F 1 || 2 L 2 (Z) . (41)
Denoting

G 1 := F 1 + (∂ s ϕ 0 ) 2 U 1 -2h(∂ s ϕ 0 )∂ s U 1 -h(∂ 2 s ϕ 0 )U 1 , (40) reads      -∆ z U 1 + h∇Q 1 = G 1 , (s, x, y) ∈ R × Ω, div U 1 = 0, (s, x, y) ∈ R × Ω, U 1|y=0 = U 1| y=L = 0, (s, x) ∈ R × T, (42)
and G 1 satisfies, using (41),

||G 1 || 2 L 2 (Z) ≤ C||F 1 || 2 L 2 (Z) . ( 43
)
Differentiating the second line of (42), we have div ∂ 2 s U 1 = 0, and consequently, multiplying the first line by ∂ 2 s U 1 and integrating by parts, we have

h 2 ||∂ 2 s U 1 || 2 L 2 (Z) ≤ C||F 1 || 2 L 2 (Z) . ( 44
)
Setting

H 1 := G 1 + h 2 ∂ 2 s U 1 , we have,      -∆U 1 + h∇Q 1 = H 1 , (s, x, y) ∈ R × Ω, div U 1 = 0, (s, x, y) ∈ R × Ω, U 1|y=0 = U 1| y=L = 0, (s, x) ∈ R × T, (45)
with the estimate

||H 1 || 2 L 2 (Z) ≤ C||F 1 || 2 L 2 .
Note that we reduce the problem to the derivation of an elliptic estimate of a semi-classical Stokes system. Hence, it is well-known by ellipticity, that there exists C > 0 such that

||U 1 || 2 2,sc ≤ C||F 1 || 2 L 2 (Z) . (46)
Hence, by the trace formula [START_REF] Peter | Stability and transition in shear flows[END_REF], we deduce that

h 3 |(∂ y U 1 ) | ∂Y | 2 L 2 ((0,S0)×T) h 2 ||∂ y U 1 || 2 1,sc ||U 1 || 2 2,sc ||F 1 || 2 L 2 (Z) .
It remains to estimate

h 5 |∇ s,x (∂ y U 1 ) | ∂Y | 2 L 2 ((0,S0)×T) .
To do this, we differentiate (40) in the tangential (s, x) direction, and then

     Q ϕ0 U 2 + h∇Q 2 = F 2 , (s, x, y) ∈ R × Ω, div U 2 = 0, (s, x, y) ∈ R × Ω, U 2|y=0 = U 2| y=L = 0, (s, x) ∈ R × T, (47)
where 46), we can estimate,

U 2 := ∇ s,x U 1 , F 2 = ∇ s,x F 1 + [(∂ s ϕ 0 ) 2 , ∇ s,x ]U 2 + 2h[∂ s ϕ 0 , ∇ s,x ]∂ s U 1 + h[∂ 2 s ϕ 0 , ∇ s,x ]U 2 and Q 2 = ∇ s,x Q 1 . From (
||F 2 || 2 L 2 (Z) ||F 1 || 2 L 2 (Z) + ||∇ s,x F 1 || 2 L 2 (Z) .
Yet, arguing as above, one can obtain an elliptic estimate on U 2 given by,

||U 2 || 2 2,sc ||F 2 || 2 L 2 (Z) ||F 1 || 2 L 2 (Z) + ||∇ s,x F 1 || 2 L 2 (Z) ,
which yields, using again [START_REF] Peter | Stability and transition in shear flows[END_REF],

h 5 |∇ s,x (∂ y U 1 ) | ∂Y | 2 L 2 ((0,S0)×T) = h 5 |(∂ y U 2 ) | ∂Y | 2 L 2 ((0,S0)×T) ≤ h 4 ||∂ y U 2 || 2 1,sc h 2 ||U 2 || 2 2,sc h 2 ||F 2 || 2 L 2 (Z) h 2 ||F 1 || 2 L 2 (Z) + h 2 ||∇ s,x F 1 || 2 L 2 (Z) .
Summing up, we have proved that there exists C > 0 such that,

||U 1 || 2 2,sc + h 2 ||∇ s,x U 1 || 2 2,sc + h 3 |(∂ y U 1 ) | ∂Y | 2 1,sc ≤ C ||F 1 || 2 L 2 (Z) + h 2 ||∇ s,x F 1 || 2 L 2 (Z) .
We end the proof by noting that

F 1 := h 2 A * 1 U 1 + h 2 [∂ 2 s , χ 1 ]
U, and the sought result follows by taking h > 0 sufficiently small. 2

Estimates in E -

In this section, we shall use the particular form of the function U , mainly the analyticity in the s variable, to estimate the boundary traces in the remaining region E - 2 . Analyticity allow us to gain exponential decay as h → 0 here. Let θ - 2 (σ, ξ)

:= α 2 (ξ)β 2 (σ) ∈ S0 T,sc such that supp θ - 2 ⊂ E - 2 .
Proposition 4.6 There exists h 0 , c 0 > 0 and C such that,

| Op(θ - 2 )W | ∂Ω | 2 1,sc Ce CΛ 1/2 +2 1-c 0 h ||ũ|| 2 L 2 (Ω) . ( 48
)
for all h ∈ (0, h 0 ] and where ũ and W are defined in [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF] and [START_REF] Métivier | Valeurs propres de systèmes elliptiques définis sur des sous espaces[END_REF].

Proof. The beginning of the proof mimmicks the one in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]. First note that, one has, for all n ∈ N,

Op T (θ - 2 )W | ∂Ω = 1 2πh e isσ h β 2 (σ) e -itσ h χ 1 (t)e ϕ0(t)/h Op T (α 2 )v(t, x, 0)dt dσ = 1 2πh e isσ h β 2 (σ) d n (dt) n h n e -itσ h (-iσ) n χ 1 (t)e ϕ0(t)/h Op T (α 2 )v(t, x, 0)dt dσ = - 1 2πh e isσ h β 2 (σ) h n e -itσ h (-iσ) n d n (dt) n χ 1 (t)e ϕ0(t)/h Op T (α 2 )v(t, x, 0) dt dσ = - 1 2πh e isσ h β 2 (σ) h n e -itσ h (-iσ) n e ϕ 0 (t,σ) h M n (t, x)dt dσ,
where we wrote M n (t, x) = e -ϕ 0 (t)

h d n (dt) n χ 1 (t)e ϕ0(t)/h Op T (α 2 )v(t, x, 0) .
In what follows, we shall focus on,

I := e -itσ h e ϕ0(t)/h M n (t, x)dt = e iφ h M n (t, x)dt,
with φ(t, σ) := -tσ -iϕ 0 (t). By definition of the weight function ϕ 0 , we have,

φ(t, σ) = -tσ -i ϕ 0 (s 0 ) -εγ(t -s 0 ) 2 + O(ε 2 ) . ( 49
)
Note that the integrand of I is analytic on the set where χ 1 = 1. We thus introduce the following change of contour in the complex plane,

t → g(t) := t -i σ < σ > κ(t), κ ∈ C ∞ 0 (R), (50)
with supp κ ⊂ {χ 1 = 1}, κ(t) ∈ [0, κ 0 ] and κ(t) = κ 0 on an interval of the type (-ε 0 + s 0 , ε 0 + s 0 ) ⊂ {χ 1 = 1}. We recall that < σ >= (1 + σ 2 ) 1/2 stands for the Japanese bracket. We have,

Im φ(g(t), σ) = σ 2 < σ > κ(t) -ϕ 0 (s 0 ) + εγ(t -s 0 ) 2 -εγ σ 2 < σ > 2 κ(t) 2 + O(ε 2 ),
and we see that for ε > 0 and κ 0 > 0 chosen sufficiently small (and fixed from now on), there exists c 0 > 0 such that, Im φ(g(t), σ) ≥ c 0 -ϕ 0 (s 0 ), since |σ| bounded from below by a strictly positive constant on the support of β 2 (σ). Hence, we shall perform the following change of contour in the integral of I using the analyticity,

|I| = R e iφ h M n (t, x)dt = γ e iφ h M n (t, x)dt = R e iφ(g(t),σ) h M n (g(t), x)g (t)dt e (ϕ0(s0)-c0)/h R |M n (g(t), x)| dt.
As a result, using the fact that |ξ| is uniformly bounded from above in that microlocal region, using trace formulas, and taking n = 4 for instance, we have, by definition of the semi-classical norms,

| Op T (θ - 2 )W | ∂Ω | 2 1,sc | Op T (θ - 2 )W | ∂Ω | 2 0 + h 2 | Op T (θ - 2 )∂ s W | ∂Ω | 2 0 ∂Ω h 2n-2 R e i sσ h σ -n Idσ 2 dsdx + ∂Ω h 2n-2 R e i sσ h σ -n+2 Idσ 2 dsdx h 2n ∂Ω sup σ |I| 2 dxds e 2 ϕ 0 (s 0 )-c 0 h ∂Ω R M 4 (t, x)dt 2 dxds e 2 ϕ 0 (s 0 )-c 0 h |M 4 | 2 0 e 2 ϕ 0 (s 0 )-c 0 h ||M 4 || 2 H 1 (Z) ,
where we used in the last line a trace formula. Note that there exist m ∈ R and C > 0 such that,

|M 4 | ≤ Ch m 4 k=0 |∂ k s v(g(s), x, y)|, |∇ z M 4 | ≤ Ch m 4 k=0 |∂ k s ∇ z v(g(s), x, y)|, Hence, | Op(θ - 2 )W | ∂Ω | 2 1,sc Ch m e 2 1-c 0 h 4 k=0 ||∂ k s v(g(.), ., .)|| 2 H 1 (Z) .
Using the particular form of v given by ( 14), one obtains,

| Op(θ - 2 )W | ∂Ω | 2 1,sc Ch m e 2 1-c 0 h 4 k=0 ||∂ k s u(g(.), ., .)|| 2 H 2 (Z) .
Note that by elliptic estimates of the Stokes operator A 0 = A * 0 , one has for all k ∈ {0, . . . , 4}, ), ., .), ∂ k s u(g(.), ., .) ||∂ k s A * u(g(.), ., .)|| L 2 (Z) + ∂ k s A * u(g(.), ., .), ∂ k s u(g(.), ., .) -∂ k s A * 1 u(g(.), ., .), ∂ k s u(g(.), ., .)

||∂ k s u(g(.), ., .)|| H 2 (Z) ||∂ k s A * 0 u(g(.), ., .)|| L 2 (Z) ||∂ k s A * u(g(.), ., .)|| L 2 (Z) + ||∂ k s A * 1 u(g(.), ., .)|| L 2 (Z) ||∂ k s A * u(g(.), ., .)|| L 2 (Z) + ||∂ k s u(g(.), ., .)|| H 1 (Z) ||∂ k s A * u(g(.), ., .)|| L 2 (Z) + ∂ k s A * 0 u(g(.
||∂ k s A * u(g(.), ., .)|| L 2 (Z) + ||∂ k s u(g(.), ., .)|| L 2 (Z) + ||∂ k s A * 1 u(g(.), ., .)|| L 2 (Z) ||∂ k s u(g(.), ., .)|| L 2 (Z) ,
which yields, using the Young inequality,

||∂ k s u(g(.), ., .)|| H 2 (Z) ||∂ k s A * u(g(.), ., .)|| L 2 (Z) + ||∂ k s u(g(.), ., .)|| L 2 (Z) .
Now we use the particular form of u given by [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF], and the resolvent estimate of Corollary 1.7 to obtain for j = 0, 1,

||∂ k s (A * ) j u(g(.), ., .)|| L 2 = 1 2 ||(A * ) j-1 2 ∂ k s e g(s)(A * ) 1/2 -e -g(s)((A * ) 1/2 ũ|| L 2 (Z) Λ mC e CΛ 1/2 ||ũ|| L 2 Ce CΛ 1/2 ||ũ|| L 2 ,
for some C > 0 independent on Λ. Summing up, up to taking c 0 slightly smaller and h 0 sufficiently small, one has,

| Op(θ - 2 )W | ∂Ω | 1,sc Ce CΛ 1/2 + 1-c 0 h ||ũ|| L 2 ,
which is the sought result. 2

Patching boundary estimates together

In this section, we patch all the above microlocal estimates and we absorb the remainder by taking the Carleman parameter h > 0 small enough. First, we start by patching the estimates on the boundary trace W | ∂Ω .

We set Y = {z ∈ Y, d(z, ∂Y ) > 1 }. One shall use Y for taken sufficiently large, in order to obtain an observation inequality from the interior up to the boundary. The large parameter depends on the small parameter ε > 0, and shall be fixed below. We recall that χ 1 is defined in Section 1.5.1. Proposition 4.7 There exists c 0 > 0, h 0 > 0, > 0, and C > 0 such that,

||e ϕ/h U|| 2 L 2 (Y ) + h 2 ||e ϕ/h ∇U|| 2 L 2 (Y ) + h||e ϕ/h V|| 2 L 2 (Y ) + h 3 ||e ϕ/h ∇ z V|| 2 L 2 (Y ) ≤C h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h [∂ 2 s , χ 1 ]v|| 2 L 2 (Y ) + Ce CΛ 1/2 e 2 1-c 0 h ||ũ|| 2 L 2 (Z) + h||e ϕ/h V|| 2 L 2 (Y ) + h 3 ||e ϕ/h ∇V|| 2 L 2 (Y ) + ||e ϕ/h U|| 2 L 2 (Y ) + h 2 ||e ϕ/h ∇U|| 2 L 2 (Y ) , (51)
for all h ∈ (0, h 0 ].

Note that the two last terms correspond to an observation from the interior of Z.

Proof. Let (Φ i ) i∈I be a finite partition of unity of a neighborhood N of the boundary ∂Y , satisfying i∈I Φ i = 1 on N such that Theorem 4.2 holds for every Φ i . Note that this partition does exist, by compactness of N . Applying Theorem 4.2 for such χ = (χ 1 Φ i ) yields,

h||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) + h 3 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + ||e ϕ/h (χ 1 Φ i )u|| 2 L 2 (Y ) + h 2 ||e ϕ/h ∇(χ 1 Φ i )u|| 2 L 2 (Y ) ≤ C h 4 ||e ϕ/h (∂ 2 s + ∆)(χ 1 Φ i )v|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, (χ 1 Φ i )]u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∇, (χ 1 Φ i )]v|| 2 L 2 (Y ) + h|e ϕ0/h ((χ 1 Φ i )v) | ∂Y | 2 L 2 (∂Y ) + h 3 |e ϕ/h ∇ z ((χ 1 Φ i )v) | ∂Y | 2 L 2 (∂Y ) , (52)
for all 0 < h ≤ h 0 , for all u ∈ X Λ , and v = rot u. Now we combine propositions 4.4, 4.5 and 4.6 to estimate the boundary terms by,

h|e ϕ0/h ((χ 1 Φ i )v) | ∂Y | 2 L 2 (∂Y ) + h 3 |e ϕ/h ∇ z ((χ 1 Φ i )v) | ∂Y | 2 L 2 (∂Y ) Ce CΛ 1/2 +2 1-c 0 h ||ũ|| 2 L 2 (Ω) + h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h P (χ 1 Φ i )v|| 2 L 2 (Y ) + h 4 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + h 2 ||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) . (53)
Note that the power h 2 in front of ||e ϕ0/h [∂ 2 s , Φ]u|| 2 1,sc is natural since the relation between u and v given by v = rot v is not semi-classical. Combining (52) and (53) implies,

h||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) + h 3 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + ||e ϕ/h (χ 1 Φ i )u|| 2 L 2 (Y ) + h 2 ||e ϕ/h ∇(χ 1 Φ i )u|| 2 L 2 (Y ) ≤C h 4 ||e ϕ/h (∂ 2 s + ∆)(χ 1 Φ i )v|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, (χ 1 Φ i )]u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∇, (χ 1 Φ i )]v|| 2 L 2 (Y ) + Ce CΛ 1/2 +2 1-c 0 h ||ũ|| 2 L 2 (Ω) + h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + h 2 ||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) . (54)
We have,

||e ϕ/h (∂ 2 s + ∆)(χ 1 Φ i )v|| 2 L 2 (Y ) ||e ϕ/h (χ 1 Φ i )(∂ 2 s + ∆)v|| 2 L 2 (Y ) + ||e ϕ/h [∂ 2 s + ∆, χ 1 Φ i ]v|| 2 L 2 (Y ) ||e ϕ/h (χ 1 Φ i ) rot A * 1 u|| 2 L 2 (Y ) + ||e ϕ/h [∂ 2 s + ∆, χ 1 Φ i ]v|| 2 L 2 (Y ) ||e ϕ/h (χ 1 Φ i )A * 1 v|| 2 L 2 (Y ) + ||e ϕ/h (χ 1 Φ i )[rot , A * 1 ]u|| 2 L 2 (Y ) + ||e ϕ/h [∂ 2 s + ∆, χ 1 Φ i ]v|| 2 L 2 (Y ) . (55)
Combining ( 54) and (55), as A * 1 and [rot , A * 1 ] are differential operators of order 1, one obtains, by taking 0 < h < h 0 sufficiently small,

h||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) + h 3 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + ||e ϕ/h (χ 1 Φ i )u|| 2 L 2 (Y ) + h 2 ||e ϕ/h ∇(χ 1 Φ i )u|| 2 L 2 (Y ) ≤C h 4 ||e ϕ/h [∂ 2 s + ∆, (χ 1 Φ i )]v|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, (χ 1 Φ i )]u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∇, (χ 1 Φ i )]v|| 2 L 2 (Y ) + Ce CΛ 1/2 +2 1-c 0 h ||ũ|| 2 L 2 (Ω) + h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + h 2 ||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) .
Hence, by summing over i ∈ I, we obtain,

h||e ϕ/h χ 1 v|| 2 L 2 (N ) + h 3 ||e ϕ/h ∇ z χ 1 v|| 2 L 2 (N ) + ||e ϕ/h χ 1 u|| 2 L 2 (N ) + h 2 ||e ϕ/h ∇χ 1 u|| 2 L 2 (N ) ≤C i∈I h 4 ||e ϕ/h [∂ 2 s + ∆, (χ 1 Φ i )]v|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, (χ 1 Φ i )]u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∇, (χ 1 Φ i )]v|| 2 L 2 (Y ) + Ce CΛ 1/2 +2 1-c 0 h ||ũ|| 2 L 2 (Ω) + h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + h 2 ||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) .
Remark that all commutators of of differential operator with Φ i provides lower order terms, and can be absorbed by the left-hand side when evaluated in the neighborhood N of the boundary ∂Y , by taking 0 < h < h 0 sufficiently small. Thus,

h||e ϕ/h χ 1 v|| 2 L 2 (N ) + h 3 ||e ϕ/h ∇ z χ 1 v|| 2 L 2 (N ) + ||e ϕ/h χ 1 u|| 2 L 2 (N ) + h 2 ||e ϕ/h ∇χ 1 u|| 2 L 2 (N ) ≤C i∈I h 4 ||e ϕ/h [∂ 2 s , χ 1 ]v|| 2 L 2 (Y ) + h 4 ||e ϕ/h [∆, Φ i ]χ 1 v|| 2 L 2 (Y \N ) + h 3 ||e ϕ/h [∆, Φ i ]χ 1 u|| 2 L 2 (Y \N ) + h 3 ||e ϕ/h [∇, (χ 1 Φ i )]v|| 2 L 2 (Y \N ) + Ce CΛ 1/2 +2 1-c 0 h ||ũ|| 2 L 2 (Ω) + h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y \N ) + h 2 ||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y \N ) .
Now we observe that by definition U = χ 1 and V = χ 1 v, and moreover there exists > 0 sufficiently large satisfying supp(χ 1 ) ∩ (Y \ N ) ⊂ Y . As a result we can write,

h||e ϕ/h V|| 2 L 2 (N ) + h 3 ||e ϕ/h ∇ z V|| 2 L 2 (N ) + ||e ϕ/h U|| 2 L 2 (N ) + h 2 ||e ϕ/h ∇U|| 2 L 2 (N ) ≤C h 4 ||e ϕ/h [∂ 2 s , χ 1 ]v|| 2 L 2 (Y ) + h 4 ||e ϕ/h [∆, Φ i ]χ 1 v|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∆, Φ i ]χ 1 u|| 2 L 2 (Y ) + h 3 ||e ϕ/h [∇, (χ 1 Φ i )]v|| 2 L 2 (Y ) + Ce CΛ 1/2 +2 1-c 0 h ||ũ|| 2 L 2 (Ω) + h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h ∇ z (χ 1 Φ i )v|| 2 L 2 (Y ) + h 2 ||e ϕ/h (χ 1 Φ i )v|| 2 L 2 (Y ) .
All the commutators in the right-hand side, are at maximum H 1 norms, and as the estimate outside N is immediate, we finally obtain,

||e ϕ/h U|| 2 L 2 (Y ) + h 2 ||e ϕ/h ∇U|| 2 L 2 (Y ) + h||e ϕ/h V|| 2 L 2 (Y ) + h 3 ||e ϕ/h ∇ z V|| 2 L 2 (Y ) ≤C h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc + h 4 ||e ϕ/h [∂ 2 s , χ 1 ]v|| 2 L 2 (Y ) + Ce CΛ 1/2 e 2 1-c 0 h ||ũ|| 2 L 2 (Z) + h||e ϕ/h V|| 2 L 2 (Y ) + h 3 ||e ϕ/h ∇V|| 2 L 2 (Y ) + ||e ϕ/h U|| 2 L 2 (Y ) + h 2 ||e ϕ/h ∇U|| 2 L 2 (Y ) ,
which is the sought result. 2

Proof of the spectral inequality

The strategy to prove the spectral inequality is divided into two parts. The first one rely on the derivation of an observation of a neighborhood of the boundary from an arbitrary large domain in the interior. The second one is more classical, and rely on the propagation of smallness in the interior, where Carleman estimates are known. However, due to the lack of unique continuation in the s variable, one has to be cautious. The two subsections below are respectively devoted to each part.

An observation from an arbitrary large interior domain up to the boundary

Lemma 5.1 For all > 0 there exists C > 0 such that

||ũ|| L 2 (Ω) ≤ Ce C √ Λ ||u|| L 2 (Y ) ,
for all u ∈ X Λ of the form [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF].

Proof. We recall that the cut-off functions are defined by χ(s, x, y) = χ 1 (s)χ 2 (x)χ 3 (y) and where χ 1 (s), χ 2 (x), χ 3 (y) are smooth cut-offs in each variable satisfying near a point (s 0 , x 0 , y = 0) ∈ (0, S 0 ) × T × {y = 0} :

χ 1 (s) = 1 if |s -s 0 | < S 1 , 0 if |s -s 0 | > 2S 1 , χ 3 (y) = 1 if y < Y 1 , 0 if y > 2Y 1 , (56)
for some 0 < 2S 1 < S 0 (that is, supp χ ⊂ Y ) and some Y 1 < L/4. The localization χ(s, x, y) is defined likewise near a point (s 0 , x 0 , y = L) ∈ (0, S 0 ) × (0, L 1 ) × {y = L}.

We then define the following regions of the open set Z by,

O 1 := supp χ 1 , O 2 := (s, x, y) ∈ O 1 | d(y, ∂Ω) < 1 2 ε(S 1 -s 0 ) 2 , O 3 := supp χ 1 ∩ (s, x, y) ∈ O 1 | d(y, ∂Ω) ≥ 1 2 ε(S 1 -s 0 ) 2 .
and

W 0 := (s, x, y) ∈ Y | ψ ε (s, y) ≥ -(S 1 -s 0 ) 2 εδ ,
with δ ∈ (0, 1) that shall be fixed in the proof of the following lemma. Note that U = u on W 0 .

Lemma 5.2 We have,

sup O2 ϕ -inf W0 ϕ < 0 and 1 -c 0 -inf W0 ϕ < 0.
for δ taken sufficiently small.

Proof. We first estimate,

sup O2 ψ ε = sup O2 (-ε(s -s 0 ) 2 + d(y, ∂Ω)) = -ε(S 1 -s 0 ) 2 + sup O2 d(y, ∂Ω) < - 1 2 ε(S 1 -s 0 ) 2 ,
and inf

W0 ψ ε ≥ -(S 1 -s 0 ) 2 εδ,
which proves the first inequality of Lemma 5.2. We also have

1 -c 0 -inf W0 ϕ < 1 -c 0 -e -λ(S1-s0) 2 εδ < 0,
for δ taken sufficiently small. 2 We analyze the first two terms of the right-hand side of (51). We have on the one hand,

h 2 ||e ϕ0/h [∂ 2 s , χ 1 ]u|| 2 1,sc h 2 e sup O 1 ϕ 0 h ||u|| 2 H 1 (Z) + ||∂ s u|| 2 H 1 (Z) . (57)
On the other hand,

h 4 ||e ϕ/h [∂ 2 s , χ 1 ]v|| 2 L 2 (Y ) = h 4 ||e ϕ/h [-∂ 2 s , χ 1 ]v|| 2 L 2 (O2) + h 4 ||e ϕ/h [-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) h 4 e sup O 2 ϕ h ||v|| 2 L 2 (Z) + ||∂ s v|| 2 L 2 (Z) + h 4 ||e ϕ/h [-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) . (58)
Note that terms located in O 3 are interior terms, which corresponds to observation terms. Plugging these two estimates in (51) yields,

||e ϕ/h U|| 2 L 2 (Y ) +h 2 ||e ϕ/h ∇ z U|| 2 L 2 (Y ) + h 3 ||e ϕ/h V|| 2 L 2 (Y ) + h||e ϕ/h ∇ z V|| 2 L 2 (Y ) ≤C h 2 e sup O 1 ϕ 0 h ||u|| 2 H 1 (Z) + ||∂ s u|| 2 H 1 (Z) + h 4 e sup O 2 ϕ h ||v|| 2 L 2 (Z) + ||∂ s v|| 2 L 2 (Z) + h 4 ||e ϕ/h [-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) + Ce CΛ 1/2 e 2 1-c 0 h ||ũ|| 2 L 2 (Z) + h||e ϕ/h V|| L 2 (Y ) + h 3 ||e ϕ/h ∇ z V|| L 2 (Y ) + ||e ϕ/h U|| L 2 (Y ) + h 2 ||e ϕ/h ∇ z U|| L 2 (Y ) . (59)
Remark that by definition, sup

O1 ϕ 0 ≤ sup O2 ϕ. (60)
Now, we estimate, using the particular form of u given by [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF],

||u|| 2 H 1 (Z) + ||∂ s u|| 2 H 1 (Z) (A * 0 u, u) + (A * 0 ∂ s u, ∂ s u) (A * u, u) + (A * ∂ s u, ∂ s u) -(A * 1 u, u) -(A * 1 ∂ s u, ∂ s u) (A * u, u) + (A * ∂ s u, ∂ s u) + ||u|| H 1 (Z) ||u|| L 2 (Z) + ||∂ s u|| H 1 (Z) ||∂ s u|| L 2 (Z) ,
which finally yields, using the Young inequality,

||u|| 2 H 1 (Z) + ||∂ s u|| 2 H 1 (Z) (A * u, u) + (A * ∂ s u, ∂ s u) + ||u|| L 2 + ||∂ s u|| L 2 . ( 61
)
From the particular form of u given by [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF] and from the resolvent estimate of Corollary 1.7, one has,

||u|| 2 H 1 (Z) + ||∂ s u|| 2 H 1 (Z) Ce C √ Λ ||ũ|| L 2 (Ω) . (62)
Note that we also have,

||v|| 2 L 2 (Z) + ||∂ s v|| 2 L 2 (Z) ≤ ||u|| 2 H 1 (Z) + ||∂ s u|| 2 H 1 (Z) (A * u, u) + (A * ∂ s u, ∂ s u) + ||u|| L 2 + ||∂ s u|| L 2 Ce C √ Λ ||ũ|| L 2 (Ω) . (63)
From ( 59), (60), ( 62) and (63), one deduces,

||e ϕ/h U|| 2 L 2 (Y ) e sup O 2 ϕ h Ce C √ Λ + Ce CΛ 1/2 e 2 1-c 0 h ||ũ|| L 2 (Ω) + h||e ϕ/h V|| L 2 (Y ) + h 3 ||e ϕ/h ∇ z V|| L 2 (Y ) + ||e ϕ/h U|| L 2 (Y ) + h 2 ||e ϕ/h ∇ z U|| L 2 (Y ) + ||e ϕ/h [-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) . (64)
Let us obtain a lower bound for the left hand side of (64). We have,

||u|| 2 L 2 (W0) e - inf W 0 ϕ h e sup O 2 ϕ h Ce C √ Λ + Ce CΛ 1/2 e 2 1-c 0 h ||ũ|| L 2 (Ω) + e - inf W 0 ϕ h h||e ϕ/h V|| L 2 (Y ) + h 3 ||e ϕ/h ∇ z V|| L 2 (Y ) + ||e ϕ/h U|| L 2 (Y ) + h 2 ||e ϕ/h ∇ z U|| L 2 (Y ) + ||e ϕ/h [-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) . (65)
Using again the resolvent estimate of Corollary 1.7, one obtains,

||ũ|| 2 L 2 (Ω) Ce C √ Λ e - inf W 0 ϕ h (e sup O 2 ϕ h Ce C √ Λ + Ce CΛ 1/2 +2 (1-c 0 ) h )||ũ|| L 2 (Ω) + e - inf W 0 ϕ h h||e ϕ/h V|| L 2 (Y ) + h 3 ||e ϕ/h ∇ z V|| L 2 (Y ) + ||e ϕ/h U|| L 2 (Y ) + h 2 ||e ϕ/h ∇ z U|| L 2 (Y ) + ||e ϕ/h [-∂ 2 s , χ 1 ]v|| 2 L 2 (O3)
.

From Lemma 5.2, setting 1 h = k √ Λ, for k ∈ R + sufficiently large, yields that there exists C > 0 such that,

||ũ|| 2 L 2 (Ω) ≤ Ce C √ Λ ||V|| 2 L 2 (Y ) + ||∇ z V|| 2 L 2 (Y ) + ||U|| 2 L 2 (Y ) + ||∇ z U|| 2 L 2 (Y ) + ||[-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) . (66)
Remark that only observation terms remains in the right hand side of (66). Let Y ⊂ Ỹ Z and let Θ ∈ C ∞ 0 ( Ỹ ) such that Θ = 1 on Y . Note that,

||[-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) ||v|| 2 H 1 (Y ) , (67)
for a sufficiently large parameter . Let us finish the proof by writing,

||∇ z V|| 2 L 2 (Y ) ||Θ∇ z v|| 2 L 2 ( Ỹ ) + ||v|| 2 L 2 (Y ) ≤ Ỹ Θ∇ z v • ∇ z vdz + ||v|| 2 L 2 (Y ) = - 1 2 Ỹ ∇ z Θ • ∇ z |v| 2 dz - Ỹ Θ∆ z vvdz + ||v|| 2 L 2 (Y ) = 1 2 Ỹ ∆Θ|v| 2 dz + Ỹ Θ rot A * 1 uvdz + ||v|| 2 L 2 (Y ) = 1 2 Ỹ ∆Θ|v| 2 dz + Ỹ ΘA * 1 vvdz + Ỹ Θ[A * 1 , rot ]uvdz + ||v|| 2 L 2 (Y ) ||v|| L 2 ( Ỹ ) + ||Θ∇ z v|| L 2 ( Ỹ ) ||v|| L 2 ( Ỹ ) + ||u|| H 1 ( Ỹ ) ||v|| L 2 ( Ỹ ) + ||v|| 2 L 2 (Y ) ,
which implies, using the Young inequality,

||∇ z V|| 2 L 2 (Y ) ≤ C ||v|| 2 L 2 ( Ỹ ) + ||u|| 2 H 1 ( Ỹ ) ≤ 2C||u|| 2 H 1 ( Ỹ ) .
Using the same arguments than above, in combinaison with (67), allows us to write, up to taking Ŷ slightly larger than Ỹ ,

||V|| L 2 (Y ) +||∇ z V|| L 2 (Y ) +||U|| L 2 (Y ) +||∇U|| L 2 (Y ) +||[-∂ 2 s , χ 1 ]v|| 2 L 2 (O3) Ce C √ Λ ||u|| L 2 ( Ŷ ) ||ũ|| L 2 (Ω) . (68) Indeed, let Ỹ ⊂ Ŷ Z and let Θ ∈ C ∞ 0 ( Ŷ ) such that Θ = 1 on Ỹ , ||∇ z u|| 2 L 2 ( Ỹ ) = ||∇u|| 2 L 2 ( Ỹ ) + ||∂ s u|| 2 L 2 ( Ỹ ) = Θ∇u, ∇u L 2 (Z) + Θ∂ s u, ∂ s u L 2 (Z)
Θu, ∆u

L 2 (Z) + ∇ Θ • ∇u, u L 2 (Z) + Θ∂ 2 s u, u L 2 (Z) + (∂ s Θ)∂ s u, u L 2 (Z) || Θu|| L 2 (Y ) ||∆u|| L 2 (Z) + (∆ Θ)u, u L 2 (Z) + ΘA * u, u L 2 (Z) + (∂ 2 s Θ)u, u L 2 (Z) ||u|| L 2 ( Ŷ ) ||A * u|| L 2 (Z) ,
where we used in the last inequality the elliptic regularity of the Stokes operator. Thus, by the resolvent estimate of Corollary 1.7, we have

||∇ z u|| 2 L 2 ( Ỹ ) Ce C √ Λ ||u|| L 2 ( Ŷ ) ||ũ|| L 2 (Ω) . (69)
Estimates (66) and now implies, by the Young inequality

||ũ|| L 2 (Ω) ≤ Ce C √ Λ ||u|| L 2 ( Ŷ ) ,
which is the sought result. 2

Observability in the interior

We construct the weight function by using the following theorem (see for instance [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]).

Theorem 5.3 There exists ψ ∈ C ∞ (Ω) such that,

• ψ > 0, in Ω,

• ψ| ∂Ω = 0,

• |∇ ψ| ≥ C > 0 on Ω \ ω.

One then defines ψ β (s, x, y) = ψ(x, y) -β|s -s 0 | 2 and ϕ = e λψ β , where β, λ are large parameters, that shall be fixed in what follows. Let s 1 < s 2 < s 3 < s 4 , such that s 0 ± s 4 ∈ (0, S 0 ), that shall be fixed in what follows. We define (see Figure 2 for a sketch of the different sets),

• A 1 := ((s 0 -s 3 , s 0 -s 2 ) ∪ (s 0 + s 2 , s 0 + s 3 )) × Ω, • A 2 := (s 0 -s 2 , s 0 + s 2 ) × ω, • A 3 := (s 0 -s 2 , s 0 + s 2 ) × Ã3 , where Ã3 := {(x, y) ∈ Ω, d((x, y), ∂Ω) < 1 2 }, • A 4 := (s 0 -s 1 , s 0 + s 1 ) × Ã4 , where Ã4 := {(x, y) ∈ Ω; d((x, y), ∂Ω) > 3 4 } \ ω, • Ã5 := {(x, y) ∈ Ω; d((x, y), ∂Ω) > 3 4 }.
Lemma 5.4 Let 0 < s 1 < s 2 < s 3 < s 4 as above. There exist µ > 0 and C > 0 such that,

||u|| H 1 (A4) ≤ C||u|| 1-µ H 2 (Z) ||u|| µ H 2 (A2) ,
for all u of the form [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF].

Proof. Let γ ∈ (0, 1). Let us set δ = -1 2 sup Ã3 ψ -inf Ã4 ψ , and fix β ≥ 1 such that δ β γ-1 > δ + sup Ω ψ -inf Ã4 ψ which is always possible by taking β > 1 sufficiently large. Remark that δ > 0 for > 0 taken sufficiently large. One now set s 1 and s 2 such that, Hence, one moreover can remark that s 3 > 0 can be chosen as small as we want (and so it is for s 1 and s 2 ) by taking β > 0 sufficiently large. From now on, parameters δ, β and γ are fixed, and all the constants involved in the computations below depend on them. Let us define the smooth cut-off function Θ(s, x, y) = Θ 1 (s)Θ 2 (x, y) ∈ C ∞ 0 (Z) such that Θ 1 = 1 on (s 0 -s 2 , s 0 + s 2 ), 0 on (0, S 0 ) \ (s 0 -s 3 , s 0 + s 3 ), Θ 2 = 1 on {d((x, y), ∂Ω) ≥ 1 2 } \ ω, 0 on ω ∪ {d((x, y), ∂Ω) < 1 4 }.

|s 1 | 2 = δ β , |s 2 | 2 = δ β γ . 0 S 0 - s 0 - s 0 + s 1 - s 0 + s 2 - s 0 + s 3 - s 0 -s 1 A 3 A 3 A 4 A 4 - s 0 -s 2 - s 0 -s 3 A 2 | | ω A 1 A 1
(73)

Let V be the interior of supp Θ. One have that there exists λ such that the weight function ϕ satisfies the two sub-ellipticity conditions of definitions 3.1 and 3.2 on the open set V . The parameter λ is fixed from now on. Thus, the following two Carleman estimates holds on V for such a weight function ϕ.

Theorem 5.5 Let V be an open set in Y . Let ϕ be a weight function satisfying the conditions of Definition 3.1 on V . Then, there exist h 0 > 0 and C > 0 such that,

h||e 1 h ϕ f || 2 L 2 (V ) + h 3 ||e 1 h ϕ ∇ z f || 2 L 2 (V ) ≤ Ch 4 ||e 1 h ϕ (-∂ 2 s -∆)f || 2 L 2 (V ) , (74)
for all f ∈ C ∞ 0 (V ), h ≤ h 0 , and such that h||e

1 h ϕ f || 2 L 2 (V ) + h 3 ||e 1 h ϕ ∇f || 2 L 2 (V ) ≤ Ch 4 ||e 1 h ϕ ∆f || 2 L 2 (V ) , (75)
for all f ∈ C ∞ 0 (V ), h ≤ h 0 .

Remark 5.6

The key point here is that the two above Carleman estimates will allow us to handle the low order terms. However, to have (74) and (75) at the same time, one needs |∇ z ϕ| = 0 and |∇ϕ| = 0 on V , which is satisfied only by removing (0, S 0 ) × ω from the support of Θ (see (73)). This is precisely why we will not be able here to propagate the smallness in the s direction.

Let us apply the estimate (74) to f = Θv := Θ rot (u), where u is of the form [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]. We recall that the function v satisfies, -∂ Optimizing this estimate with respect to h > 0 implies that there exists C > 0 and µ ∈ (0, 1) such that ||u|| H 1 (A4) ≤ C||u|| 1-µ H 2 (Z) ||u|| µ H 2 (A2) . The estimate over A 2 being immediate this ends the proof of Lemma 5.4.

2 As Lemma 5.4 only provides an estimate on domain A 4 which may be arbitrary small, we use the following lemma to recover the full estimate on Y that we need having in mind to use Lemma 5.1.

Lemma 5.7 There exist and ˜ > , µ > 0 and C > 0 such that,

||u|| H 1 (Y ) ≤ C||u|| 1-µ H 2 (Z) ||u|| µ H 2 (( 1 ˜ ,S0-1 ˜ )×ω)
, for all u of the form [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]. Thus, applying Lemma 5.4 for each A k 2 and A k 5 , k = 1, . . . , M , yields the result, by noting that A k 2 ⊂ ( 1 ˜ , S 0 -1 ˜ ) × ω, for all k ∈ {1, . . . , M }. 2

Proof. Let

End of the proof

Now we finish the proof of Theorem 1.8. It is sufficient to combine Lemmata 5.1 and 5.4, to obtain,

||ũ|| L 2 (Ω) ≤ Ce C √ Λ ||u|| L 2 (Y ) ≤ Ce C √ Λ ||u|| 1-µ H 2 (Z) ||u|| µ H 2 (( 1 ˜ ,S0-1 ˜ )×ω) .
Assuming,

||u|| H 1 (Z) Ce C√ Λ ||ũ|| L 2 (Ω) , (81)
yields immediatly the result by taking ϕ ∈ C ∞ 0 (0, S 0 ) such that ϕ = 1 on ( 1 ˜ , S 0 -1 ˜ ). The estimate (81) follows from the resolvent estimate of Corollary 1.7.

2

1 Introduction 1 . 1 4 A 1 .

 11141 Functional framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Spectral properties of the Oseen operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The augmented operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Notations and semi-classical norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Pseudo-differential operators and semi-classical norms . . . . . . . . . . . . . . . .2 From the spectral inequality to the cost of control 2.1 Controllability of the low frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Decay of the semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Derivation of an observability inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conjugated operators and weight function properties 3.1 Augmented operator conjugated by the weight function . . . . . . . . . . . . . . . . . . . 3.2 Sub-ellipticity condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Microlocal regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Roots properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Analysis of Q ϕ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Analysis of P ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carleman estimate near the boundary 4.1 Estimates for the volume terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Estimates of the boundary terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Estimates in E - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Estimates in E + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Estimates in E - 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Patching boundary estimates together . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: Representation of the microlocal regions of the cotangent space of ∂Y and illustration of the characteristic sets Char Q ϕ0 ⊂ U and Char P ϕ ⊂ V .

Figure 2 :

 2 Figure 2: The geometry in the interior of Ω × (0, S 0 ).

  A 2 := (s k 0 -s k 2 , s 0 + s 2 ) × ω and A k 5 := (s k 0 -s k 1 , s 0 + s 1 ) × Ã5, where Ã5 := {(x, y) ∈ Ω; d((x, y), ∂Ω) >3 4 }. Remark that there exists a finite number M , and a sequence of (s k 0 ) k∈{1,...,M ⊂ Y , such that, Y ⊂ k=1 A k 5 .

  Θv|| 2 L 2 (V ) + h 3 ||e 1 h ϕ Θ∇ z v|| 2 L 2 (V ) h 4 ||e 1 h ϕ -Θ rot A * 1 u + [Θ, ∂ 2 s + ∆]v || 2 L 2 (V ) h 4 ||e 1 h ϕ Θ rot A * 1 u|| 2 L 2 (V ) + h 4 ||e ΘA * 1 v|| 2 L 2 (V ) + h 4 ||e 1 h ϕ Θ[rot , A * 1 ]u|| 2 L 2 (V ) + h 4 ||e 1 h ϕ [Θ, ∂ 2 s + ∆]v|| 2 L 2 (V )(77) Now, remark that, using the divergence free condition, Note that, taking h > 0 sufficiently small, one can absorb the low order terms in the right hand side of the above inequality to have, [Θ,∂ 2 s + ∆]v|| 2 L 2 (V ) + h 3 ||e 1 h ϕ [∆, Θ]u|| 2 L 2 (V ) . (80)The two remaining terms in the right hand side of the above inequality are localized in regions where Θ varies. Estimate (80) implies,e 1 h inf A 4 ϕ ||u|| 2 H 1 (A4) + ||v|| 2 -inf A 4 ϕ) ||u|| 2 H 1 (A k ) + ||v|| 2 H 1 (A k ) .

	Estimate (74) yields, for h sufficiently small,
	h||e	1 h ϕ 1 h ϕ [Θ, ∂ 2 s + ∆]v|| 2 L 2 (V )
				h 4 ||e h ϕ ||e 1 1 h ϕ ∆ (Θu) || 2 L 2 (V )	||e	1 h ϕ Θ∇ z v|| 2 L 2 (V ) + ||e	1 h ϕ [∆, Θ]u|| 2 L 2 (V ) .	(78)
	Estimate (75) for f = Θu yields, for h sufficiently small,
				||e	1 h ϕ Θu|| 2 L 2 (V ) + h 2 ||e	1 h ϕ Θ∇u|| 2 L 2 (V )	h 3 ||e	1 h ϕ ∆ (Θu) || 2 L 2 (V ) .	(79)
	Combining (77), (78) and (79), we obtain,
	||e	1 h ϕ Θu|| 2 L 2 (V ) + h 2 ||e	1 h ϕ Θ∇u|| 2 L 2 (V ) + h||e	1 h ϕ Θv|| 2 L 2 (V ) + h 3 ||e	1 h ϕ Θ∇ z v|| 2 L 2 (V )	h 4 ||e	1 h ϕ ΘA * 1 v|| 2 L 2 (V )
							+ h 4 ||e	1 h ϕ Θ[rot , A * 1 ]u|| 2 L 2 (V ) + h 4 ||e	1 h ϕ [Θ, ∂ 2 s + ∆]v|| 2 L 2 (V ) + ||e	1 h ϕ [∆, Θ]u|| 2 L 2 (V ) .
	||e	1 h ϕ Θu|| 2 L 2 (V ) + h 2 ||e	1 h ϕ Θ∇u|| 2 L 2 (V ) + h||e	1 h ϕ Θv|| 2 L 2 (V ) + h 3 ||e	1 h ϕ Θ∇ z v|| 2 L 2 (V )
								h 4 ||e h ϕ H 1 (A4) 1 3	e	1 h sup A k
								k=1
	which is equivalent to,
								3
				||u|| 2 H 1 (A4) + ||v|| 2 H 1 (A4)	e	1 h (sup A k
								k=1
	From (70), (71) and (72), there exist ν 1 , ν 2 > 0 such that,
				||u|| 2 H 1 (A4) + ||v|| 2 H 1 (A4)	e	ν 1 h	||u|| 2 H 1 (A2) + ||v|| 2 H 1 (A2) + e -ν 2 h	||u|| 2 H
								(76)

2 s v -∆v + rot A * 1 u = 0, which implies that f satisfies, -∂ 2 s f -∆f = -Θ rot A * 1 u + [Θ, ∂ 2 s + ∆]v. ϕ ||u|| 2 H 1 (A k ) + ||v|| 2 H 1 (A k ) , ϕ1 (Z) + ||v|| 2 H 1 (Z) .

ODISEE (ANR-19-CE48-0004-01) and by the French Grant ANR TRECOS (ANR-20-CE40-0009-01).


Anna Tzanakaki

CTTC Raul Muñoz

Reza Nejabati

Noboru Yoshikane

Markos Anastasopoulos

Johann M Marquez-Barja

Reza Spain

Nejabati

Hercules Avramopoulos

Dimitris Syvridis

Markos Chair

Anastasopoulos

Lena Wosinska

Piero Castoldi

Pablo Pavón

Mariño Upct

Spain Tibor Cinkler

Marco Ruffini

Rafael Montero

Fernando Agraz

Albert Pagès

Salvatore Spadaro

Sanjay K Bose

Francesco Musumeci

Jiawei Zhang

Didier Khorsandi

Wouter Colle

Carla Tavernier

Raffaelli

Dimitra Simeonidou

Konstantinos Christodoulopoulos

Polizois Ippokratis Sartzetakis

Ricard Vilalta

Ramon Casellas

Ricardo Martínez

TPC Chairs

Preface

It was an honor and a pleasure to welcome everyone to ONDM 2019, the 23rd edition of the Optical Network Design and Modelling conference, which was held at the National and Kapodistrian University of Athens, Greece, May 13-16, 2019.

ONDM 2019 focused on cutting-edge research in established areas of optical networking as well as their adoption in support of a wide variety of new services and applications. This involved the most recent trends in networking, including 5G and beyond, big data and network data analytics, cloud/edge computing, autonomic networking, artificial intelligence assisted networks, secure and resilient networks, etc., that drive the need for increased capacity, efficiency, flexibility, and adaptability in the functions that the network can perform.

In this context new disaggregated optical network architectures were discussed, exploiting and integrating novel multidimensional photonic technology solutions, as well as adopting open hardware and software platforms relying on software defined networking (SDN) and network function virtualization (NFV) to allow support of new business models and opportunities. Some of the thematic areas that were covered include:

• • The importance of network user increases, as armed with increasingly-capable smart devices drive network level specifications and design/operation choices -The concept of service is being expanded beyond the traditional end-user context to a variety of end-user and operational services that are ICT or vertical industry related -This imposes the requirement for newer infrastructures, e.g., 5G, 6G and beyond consisting of (optical) communication networks, (edge and cloud) data centers, distributed network functions, etc., in order to become more agile and facilitate more application-centric services with improved user experience

  Novel and multidimensional optical network architectures • Optical network control, management, and orchestration, including SDN and NFV solutions • Slicing, virtualization, and multitenancy techniques for optical networks • Optical networking supporting low latency and high bandwidth network function virtualization • Optical and wireless network convergence, including radio-over-fiber access networks • Optical networks in support of intra-/inter-data center connectivity and cloud/edge computing • Artificial intelligence and data analytics techniques for optical networks • Advances in optical network modeling and optimization • Routing and spectrum assignment in fixed and flex-grid optical networks • Novel optical node designs including disaggregation and open optical line systems • Optical network availability, resilience, survivability, security, and privacy • Optical networking in support of vertical industries • Field trials and interoperability demonstrations of optical networks • Techno-economic studies of optical networks with emphasis on 5G consideration Some of the key take away messages of the conference include:

•

  Infrastructures equipped with additional/newer resources supporting higher bandwidth and lower latency focusing only on the lower layers of the protocol stack are not sufficient anymore -Instead, novel, flexible, open, and scalable architectural approaches supporting technology convergence, not only across network domains but also across network and compute/storage domains, involving inter-and intra-Data Center connectivity, are becoming essential -Methods developing real-time network-, service-, and application-centric analytics, able to process huge volumes of data, adopting artificial intelligence and machine-learning methods are needed to offer improved network and network service performance as well as user-experience, in support of the vision of future IT infrastructures• Significant experimentation activities in real-life test-beds (5G UK) have already provided proof of principle demonstrations of the use and potential of optical networking in support of the new 5G and beyond network paradigm to facilitate end-user services with emphasis in improved experience and quality of leaving • Significant attention has been put towards resilience and security considerations in the context of optical networking and the role optical technologies can play to facilitate more resilient and secure future network infrastructures ONDM 2019 was a single-track conference. The conference included two workshops, one industrial panel, and one summer school co-organized by the H2020 5G PPP project 5G-PICTURE and the Erasmus Mundus programme SMARTNET:• Workshop I: Optical Networking an enabler for 5G Solutions, Trials and Demonstrators • Workshop II: Optical Intra-and Inter-Data Center Networks • Industry Panel: 5G in 2020 and beyond: reality check, rollout plans and path ahead

	Attendance: 90 Attendees
	May 2019	Anna Tzanakaki
		Manos Varvarigos
		Raul Muñoz
		Reza Nejabati
		Noboru Yoshikane
		Markos Anastasopoulos
		Johann Marquez-Barja

Contents xv

TPC Members Slavisa Aleksic Hochschule für Telekommunikation Leipzig, Germany Markos Anastasopoulos University of Bristol, UK Bigomokero Bagula University of the Western Cape, South Africa Johan Bauwelinck Ghent University and imec, Belgium Andrea Bianco Politecnico di Torino, Italy Luiz Bonani Universidade Federal do ABC, Brazil Aparicio Carranza New York City College of Technology, USA Isabella Cerutti Nokia Bell Labs, Italy Jiajia Chen KTH Royal Institute of Technology, Sweden Kostas Christodoulopoulos Nokia Bell Labs, Germany Didier Colle Ghent University and imec, Belgium David Coudert Université Côte d'Azur, Inria, CNRS, and I3S, France Filippo Cugini CNIT, Italy Sandip Das Trinity College Dublin, Ireland Georgios Ellinas University of Cyprus, Cyprus Marija Furdek KTH Royal Institute of Technology, Sweden Maurice Gagnaire Télécom Paris, France Alessio Giorgetti Scuola Superiore Sant'Anna, Italy Philippe Gravey Télécom Bretagne, France Hiroaki Harai National Institute of Information and Communications Technology, Japan Hiroshi Hasegawa Nagoya University, Japan Yusuke Hirota National Institute of Information and Communications Technology, Japan Weisheng Hu Shanghai Jiao Tong University, China Brigitte Jaumard Concordia University, Canada Wojciech Kabacinski Poznan University of Technology, Poland Ezhan Karasan Bilkent University, Turkey Daniel Kilper University of Arizona, USA Ken-ichi Kitayama The Graduate School for the Creation of New Photonics Industries, Japan Nattapong Kitsuwan The University of Electro-Communications, Japan Panagiotis Kokkinos National Technical University of Athens, Greece Yao Li University of Arizona, USA Andrew Lord British Telecom, UK Guido Maier Politecnico di Milano, Italy Ricardo Martinez Centre Tecnològic de Telecomunicacions de Catalunya, Spain Barbara Martini CNIT, Italy Xavier Masip-Bruin Universitat Politècnica de Catalunya, Spain Francesco Matera Fondazione Ugo Bordoni, Italy Paolo Monti KTH Royal Institute of Technology, Sweden Francesco Musumeci Politecnico di Milano, Italy Antonio Napoli Infinera, Germany