Jean-Michel Coron
email: coron@ann.jussieu.fr

Frédéric Marbach
email: marbach@ann.jussieu.fr

Franck Sueur
email: franck.sueur@math.u-bordeaux.fr

Ping Zhang

Controllability of the Navier

1

Introduction and statement of the main result

Historical context

In the late 1980's, Jacques-Louis Lions introduced in [START_REF] Lions | Exact controllability for distributed systems. Some trends and some problems[END_REF] (see also [START_REF] Lions | On the controllability of distributed systems[END_REF][START_REF] Lions | Remarks on the control of everything[END_REF][START_REF] Lions | Sur le contrôle des équations de Navier-Stokes[END_REF]) the question of the controllability of uid ows in the sense of how the Navier-Stokes system can be driven by a control of the ow on a part of the boundary to a wished plausible state, say a vanishing velocity. Jacques-Louis Lions' problem has been solved in [START_REF] Coron | Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions[END_REF] by the rst three authors in the particular case of the Navier slip-with-friction boundary condition (see also [START_REF] Coron | On the controllability of the Navier-Stokes equation in spite of boundary layers[END_REF] for a gentle introduction to this result). In its original statement with the no-slip Dirichlet boundary condition, it is still an important open problem in uid controllability.

Statement of the main result

In this paper we consider the case where the ow occupies a rectangle, where controls are applied to the lateral boundaries and the no-slip condition is prescribed on the upper and lower boundaries. We thus consider a rectangular domain Ω := (0, L) × (-1, 1), where L > 0 is the length of the domain. We will use (x, y) as coordinates. Inside this domain, a uid evolves under the Navier-Stokes equation. We will name u = (u 1 , u 2 ) the two components of its velocity. Hence, u satises:

∂ t u + (u • ∇) u + ∇p -∆u = f g , div u = 0, (1.1)
in Ω, where p denotes the uid pressure and f g a force term (to be detailed below). We think of this domain as a river or a tube and we assume that we are able to act on the uid ow at both end boundaries: Γ 0 := {0} × (-1, 1) and Γ L := {L} × (-1, 1).

On the remaining parts of the boundary, Γ ± := (0, L) × {±1}, we assume that we cannot control the uid ow and that it satises null Dirichlet boundary conditions: u = 0 on Γ ± .

(1.2)

We will consider initial data in the space L 2 div (Ω) of divergence free vector elds, tangent to the boundaries Γ ± . The main result of this paper is the following. Theorem 1. Let T > 0 and u * in L 2 div (Ω). For any k ∈ N and for any η > 0, there exists a force f g ∈ L 1 ((0, T ); H k (Ω)) satisfying f g L 1 ((0,T );H k (Ω)) ≤ η (1.3) and an associated weak Leray solution u ∈ C 0 ([0, T ]; L 2 div (Ω)) ∩ L 2 ((0, T ); H 1 (Ω)) to (1.1) and (1.2) satisfying u(0) = u * and u(T ) = 0.

Since the notion of weak Leray solution is classically dened in the case where the null Dirichlet boundary condition is prescribed on the whole boundary, let us detail that we say that u ∈ C 0 ([0, T ]; L 2 div (Ω)) ∩ L 2 ((0, T ); H 1 (Ω)) is a weak Leray solution to (1.1) and (1.2) satisfying u(0, •) = u * and u(T, •) = 0 when it satises the weak formulation

- T 0 Ω u • ∂ t ϕ + T 0 Ω (u • ∇)u • ϕ + 2 T 0 Ω D(u) : D(ϕ) = Ω u * • ϕ(0, •) + T 0 Ω u • f g , (1.4)
for any test function ϕ ∈ C ∞ ([0, T ]× Ω) which is divergence free, tangent to Γ ± , vanishes at t = T and vanishes on the controlled parts of the boundary Γ 0 and Γ L . Thus, (1.4) encodes the no-slip condition on the upper and lower boundaries only. This under-determination encodes that one has control over the remaining part of the boundary, that is on the lateral boundaries. The controls on the lateral boundaries are therefore not explicit in the statement of Theorem 1. Still the proof below will provide some more insights on the nature of possible controls to the interested reader. Once a trajectory is known, one can indeed deduce that the associated controls are the traces on Γ 0 and Γ L of the solution.

Hence, elementary trace theorems and the regularity of weak Leray solutions imply that such controls are at least in L 2 ((0, T ); H 1 2 (Γ 0 ∪ Γ L )).

Comments and references

Remark 1.1 (Relation to the open problem). We view Theorem 1 as an intermediate step towards an answer to Jacques-Louis Lions' problem, which requires to prove that the theorem is still true with a vanishing distributed force f g = 0. Here, we need a non-vanishing force but we can choose it very small even in strong topologies. Our result therefore sug-

gests that the answer to Jacques-Louis Lions' question is very likely positive, at least for this geometry. Nevertheless, new ideas are probably necessary to eliminate the unwanted distributed force we use.

Remark 1.2 (Local vs. global null controllability and Reynolds numbers). The fact that, for any T > 0, one can drive to the null equilibrium state u = 0 in time T without any distributed force (f g = 0) was already known when the initial data u * is small enough in L 2 (Ω) (with a maximal size depending on T ). In this case, one may think of the bilinear term in Navier-Stokes system as a small perturbation term of the Stokes equation so that the controllability can be obtained by means of Carleman estimates and xed point theorems.

Loosely speaking, such an approach corresponds to low Reynolds controllability.

More generally, local null controllability is a particular case of local controllability to trajectories. For Dirichlet boundary conditions, the rst results have been obtained by Imanuvilov who proved local controllability in 2D and 3D provided that the initial state are close in H 1 norm, with interior controls, rst towards steady-states in [START_REF] Imanuvilov | On exact controllability for the Navier-Stokes equations[END_REF] then towards strong trajectories in [START_REF] Imanuvilov | Remarks on exact controllability for the Navier-Stokes equations[END_REF]. Fursikov and Imanuvilov proved large time global null controllability in 2D for a control supported on the full boundary of the domain in [START_REF] Fursikov | On exact boundary zero-controllability of twodimensional Navier-Stokes equations[END_REF]. Still in 2D, they also proved local controllability to strong trajectories for a control acting on a part of the boundary and initial states close in H 1 norm in [START_REF] Fursikov | Exact local controllability of two-dimensional Navier-Stokes equations[END_REF]. Eventually, in [START_REF] Fursikov | Local exact boundary controllability of the Navier-Stokes system[END_REF] they proved in 2D and 3D local controllability to strong trajectories with controls acting on the full boundary, still for initial states close in H 1 norm. More recently, these works have been improved in [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF], where the authors proved local controllability towards less regular trajectories with interior controls and for initial states close in L 2 norm in 2D and L 4 norm in 3D.

In contrast, in Theorem 1, the initial data u * can be arbitrarily large (and T arbitrarily small). This corresponds to controllability of Navier-Stokes system at large Reynolds numbers. In this regime, the rst author and Fursikov proved global null controllability for the Navier-Stokes system in a 2D manifold without boundary in [START_REF] Coron | Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary[END_REF].

Remark 1.3 (Comparison with earlier results). For the large Reynolds regime, let us mention the earlier references [START_REF] Guerrero | Remarks on global approximate controllability for the 2-D Navier-Stokes system with Dirichlet boundary conditions[END_REF][START_REF] Guerrero | A result concerning the global approximate controllability of the NavierStokes system in dimension 3[END_REF] where a related result is obtained in a similar setting.

In this earlier result, the distributed force can be chosen small in L p ((0, T ); H -1 (Ω)), where 1 < p < 4/3. The fact that, in Theorem 1, our phantom force can be chosen arbitrarily small in the space L 1 ((0, T ), H k (Ω)) for any k ≥ 0, is the major improvement of this work.

Remark 1.4 (Geometric setting). Theorem 1 remains true for any rectangular domain (0, L 1 )×(0, L 2 ) and any positive viscosity ν, thanks to a straightforward change of variables.

On the other hand, we consider the case of a rectangle because it provides many crucial simplications. This is not for the sake of clarity of the exposition; we suspect that the case of a general domain requires dierent arguments. A key point is that this geometric setting and the use of well-chosen controls enable us to guarantee that the boundary layer equations we consider will remain linear and well-posed (see (3.14) and Remark 3.5). As we use a ushing strategy in the horizontal direction, we make use of controls on the two lateral sides, and on the two components of the velocity elds, on the contrary to [START_REF] Coron | Local null controllability of the three-dimensional navierstokes system with a distributed control having two vanishing components[END_REF][START_REF] Fernández-Cara | Some controllability results forthe n-dimensional navierstokes and boussinesq systems with n-1 scalar controls[END_REF] where the case of a small initial data is considered.

Moreover there is a very good chance that the result of Theorem 1 also holds in the case where the domain Ω is the parallelepiped (0, L 1 ) × (0, L 2 ) × (-1, 1), where L 1 , L 2 > 0, with only two opposite uncontrolled faces on which the no-slip condition is imposed. Yet another highly likely extension is the case where the domain Ω is the circular duct D × (0, L), where D is the unit disk and L > 0, with controls on the lateral sides D × {0} and D × {L} and with the no-slip condition on T × (0, L).

Remark 1.5 (Additional properties of the phantom force). During the proof, we will check that the phantom force f g we use has C ∞ regularity by parts with respect to time and C ∞ regularity with respect to space for each time. Moreover, we will check that, during the most important step of our strategy (the global approximate control phase, which involves passing through intermediate states of very large size), there exists δ > 0 such that

supp f g (t) ⊂ [0, L] × [-1 + δ, 1 -δ].
(1.5)

In the case where the initial data u * is smooth, the smoothness of the phantom force guarantees the existence of a smooth controlled solution satisfying the conclusion of Theorem 1.

Moreover, we think that this also holds true for the two particular 3D settings mentioned above, ruling out the possibility of a blow up in these cases.

Remark 1.6 (Comparison with results on wild solutions). Let us highlight that the solution mentioned in Theorem 1 is not a wild solution. Indeed, let us recall that, in 2D, without control, weak Leray solutions are unique and regular for positive times. The controlled solution that is considered in Theorem 1 benets from the same regularity than the one in the classical Leray theory. This solution vanishes at some positive time thanks to the action of a well chosen regular source term. This has to be distinguished from the recent results [START_REF] Buckmaster | Wild solutions of the navierstokes equations whose singular sets in time have hausdor dimension strictly less than 1[END_REF][START_REF] Buckmaster | Convex integration and phenomenologies in turbulence[END_REF][START_REF] Buckmaster | Nonuniqueness of weak solutions to the navierstokes equation[END_REF] where wild weak solutions (with a regularity less that the one considered by Leray) of the 3D Navier-Stokes equations are constructed, and these solutions may vanish at some positive nite time too, without any control.

Finally let us foreshadow that the proof of Theorem 1 will make great use of analyticity technics. Let us therefore refer to [START_REF] Herbst | Analyticity estimates for the navierstokes equations[END_REF] for a glimpse of classical and recent analyticity results on the Navier-Stokes equations without control.

Strategy of the proof and plan of the paper

We explain the strategy of the proof of Theorem 1, which is divided into three steps. First, we prove that the initial data can be regularized into an analytic function with arbitrary analyticity radius. Then, we prove that a large analytic initial data with a sucient analyticity radius can be driven approximately to the null equilibrium. Last, we know that small enough states can be driven exactly to the rest state. These three steps are implemented in the three propositions below, where we set the Navier-Stokes equations in the horizontal band

B := R × [-1, 1], (1.6)
with a control supported in the extended region, outside of Ω. Therefore, we look for solutions to

     ∂ t u + (u • ∇)u + ∇p -∆u = f c + f g in (0, T ) × B, div u = 0 in (0, T ) × B, u = 0 on (0, T ) × ∂B, (1.7)
where the force f c is a control supported in B \ Ω and the force f g is the phantom (ghost) force supported in Ω. Restricting such solutions of Navier-Stokes in the band to the physical domain Ω will prove Theorem 1. We also introduce a domain [START_REF] Bahouri | Fourier analysis and nonlinear partial dierential equations[END_REF] (see Fig. 1).

G := [-2L, -L] × [-1,
(1.8)

We will denote by e x and e y the unit vectors of the canonical basis of R 2 .

Proposition 1.7 (Analytic regularization of the initial data). Let T > 0 and ρ b > 0. Let

u * ∈ L 2 div ( Ω) with u * • e x = 0 on Γ 0 and Γ L . For any k ∈ N and η b > 0, there exists an extension u a ∈ L 2 div (B) of u * to the band B, a control force f c ∈ C ∞ ([0, T ] × (B \ Ω)), a phantom force f g ∈ C ∞ ([0, T ] × Ω) satisfying f g L 1 ((0,T );H k (Ω)) ≤ η b , (1.9) a weak Leray solution u ∈ C 0 ([0, T ]; L 2 div (B)) ∩ L 2 ((0, T ); H 1 (B)) to (1.7) associated with the initial data u a , C b > 0 and T b ≤ T such that u b := u(T b ) ∈ L 2 div (B) satises u b|G H k (G) + u b|B\Ω L 2 (B\Ω) + u b|Ω -u * L 2 (Ω) ≤ η b , (1.10) ∀m ≥ 0, ∂ m x u b H 3 (B) ≤ m! ρ m b C b , (1.11) 0≤α+β≤3 ∂ α x ∂ β y u b L 1 x (L 2 y ) ≤ C b .
(1.12) Proposition 1.8 (Global approximate null controllability from any analytic initial data).

Let T > 0. There exists ρ b > 0 such that, for every σ > 0 and each u b ∈ L 2 div (B) for which there exists C b > 0 such that (1.11) and (1.12) hold, for every k ∈ N and δ ∈ (0, 1 2 ), there exist two forces

f c ∈ C ∞ ([0, T ] × (B \ Ω)) and f g ∈ C ∞ ([0, T ] × Ω) satisfying f g L 1 ((0,T );H k (Ω)) ≤ C k,δ u b|G H k (G) , (1.13) supp f g ⊂ (0, T ) × [0, L] × [-1 + δ, 1 -δ], (1.14) supp f c ⊂ (0, T ) × B \ Ω (1.15) and a weak solution u ∈ C 0 ([0, T ]; L 2 loc,div (B)) ∩ L 2 ((0, T ); H 1 loc (B)) to (1.7) associated with the initial data u b , such that there exists T c ≤ T such that u c := u(T c ) ∈ L 2 loc,div (B) satises u c|Ω L 2 (Ω) ≤ σ + u b|{|y|≥1-2δ} L 2 (B) . (1.16)
Moreover, the constant C k,δ only depends on k and δ.

Proposition 1.9 (Local null controllability). Let T > 0. There exists σ > 0 such that, for any u c ∈ L 2 div (Ω) which satises

u c|Ω L 2 (B) ≤ 3σ, (1.17)
there exists a weak solution u ∈ C 0 ([0, T ]; L 2 div (Ω)) ∩ L 2 ((0, T ); H 1 (Ω)) to (1.1) with the initial data u c and f g = 0, which satises u(T ) = 0.

Let us prove that the combination of these three propositions implies Theorem 1. First, thanks to standard arguments, it is sucient to prove Theorem 1 for initial data satisfying u * • e x = 0 on Γ 0 and Γ L . Indeed, applying vanishing boundary controls on the original system for any positive time guarantees that the state gains this property. Therefore, we assume that the initial data already satises this property. We x the quantities step by step in the following manner.

Let T > 0 and u * ∈ L 2 div (Ω) satisfying u * • e x = 0 on Γ 0 and Γ L . Let ρ b > 0 be given by Proposition 1.8 for a time interval of length T /3.

Let σ > 0 be given by Proposition 1.9 for a time interval of length T /3.

Let δ 1 ∈ (0, 1 2 ) small enough such that (u * ) |{|y|≥1-2δ 1 } L 2 (Ω) ≤ σ. Let k ∈ N and η > 0.
Let η b := min{η/2, σ, η/(2C k,δ 1 )}.

We apply Proposition 1.7 with a time interval of length T /3, ρ b , k and η b . Hence, there exists T b ≤ T /3 and a solution u dened on [0, T b ] with u(0) |Ω = u * and such that u b := u(T b ) satises (1.10), (1.11) and (1.12).

We apply Proposition 1.8 with a time interval of length T /3, ρ b , δ 1 , k and σ. This yields a solution u dened on [T b , T c ] with T c ≤ T b +T /3 ≤ 2T /3 such that u c := u(T c ) satises (1.16).

By triangular inequality

, u b|{|y|≥1-2δ 1 } L 2 (B) ≤ u b|B\Ω L 2 (B) + u b|Ω -u * L 2 (Ω) + (u * ) |{|y|≥1-2δ 1 } L 2 (Ω)
. Since η b ≤ σ, (1.10) and (1.16) imply that (1.17) holds.

Finally, we apply Proposition 1.9. This yields a solution u dened on [T c , T 3 ] with T 3 := T c + T /3 ≤ T such that u(T 3 ) = 0.

The concatenated forces f c and f g are C ∞ by parts in time with C ∞ regularity in space.

This concludes the proof of Theorem 1 up to extending the solution and the forces by 0 on [T 3 , T ].

x y 0 L 2L 3L -L -2L -3L Flushing of 2L δ B = R × [-1, 1] Ω G supp f g Figure 1: The horizontal band B = R × [-1, 1], the physical domain Ω = [0, L] × [-1, 1] and the domain G = [-2L, -L] × [-1, 1].
Remark 1.10. The fact that, starting from a nite energy initial data, the solution to the Navier-Stokes equation instantly becomes analytic is well-known. However, in the uncontrolled setting, the analytic radius only grows like √ t. In Proposition 1.7, we use the phantom force to enhance the regularization in short time.

It would be desirable to know if Proposition 1.7 holds with a phantom force f g satisfying a support condition such as (1.14) of Proposition 1.8, since it would imply that Theorem 1 holds with a phantom force whose support never touches the boundary Γ ± .

Remark 1.11. The small-time global approximate null controllability result of Proposition 1.8 will be proved thanks to a return-method argument (see Section 3). A base ow will shift the whole band B of a distance 2L towards the right. Roughly speaking, the main part of the initial data u b|Ω will then be outside of the physical domain and killed by a control force. However, since we need to work in an analytic setting (to establish estimates for a PDE with a derivative loss, see Section 5), this action cannot be exactly localized outside of the physical domain. Its leakage inside the physical domain will be related to the values of u b in G and lead to the unwanted phantom force. This explains estimate (1.13).

Of course, since u b is analytic in the tangential direction x, it cannot satisfy u b|G = 0.

Remark 1.12. Proposition 1.9 is a direct consequence of known results concerning the small-time local null controllability of the Navier-Stokes equation (see Remark 1.2 references). In our context involving phantom forces, one can avoid the use of these technical local results relying on Carleman estimates thanks to the following alternative statement, leveraging the phantom force to drive a small data u c exactly to zero: Let T, η > 0 and k ∈ N. There exists σ > 0 such that, for any u c ∈ L 2 div (Ω) satisfying (1.17), there exists a phantom force

f g ∈ C ∞ ([0, T ] × Ω) satisfying (1.3) and a weak solution u ∈ C 0 ([0, T ]; L 2 div (Ω)) ∩ L 2 ((0, T ); H 1 (Ω)) to (1.1)
with the initial data u c , which satises u(T ) = 0. The sketch of proof of this result would be as follows. Let ū be the solution to the free Navier-Stokes equation on [0, T ] starting from ū(0) = u c . One constructs explicitly u as u(t) := ū(t) for t ≤ T 2 and u(t

) := β(t)ū( T 2 ), where β ∈ C ∞ ([ T 2 , T ]; [0, 1]) with β( T
2 ) = 1 and β(T ) = 0. Plugging this explicit formula in (1.1) gives a denition of the phantom force f g that one needs to use. Hence, there exists

C k,β > 0 such that, if u c is small enough, then f g L 1 ((0,T );H k (Ω)) ≤ C k,β ū( T 2 ) H k+2 (Ω)
. Moreover, thanks to classical regularization result for the 2D Navier-Stokes equation, there exists

C k (T ) > 0 such that ū( T 2 ) H k+2 (Ω) ≤ C k (T ) u c L 2 (Ω) . Hence, choosing 3σC k (T )C k,β ≤ η concludes the proof.
Proposition 1.7 is proved in Section 2. Proposition 1.8 is the main contribution of this paper. We explain our strategy to prove this small-time global approximate null controllability result in Section 3. It is based on the construction of approximate trajectories and the well-prepared dissipation method. We give estimates concerning the boundary layer in Section 4, estimates concerning the remainder in Section 5 and nally analytic estimates on the approximate trajectories themselves as an appendix in Section 6.

Regularization enhancement

This section is devoted to the proof of Proposition 1.7.

Fourier analysis in the tangential direction

We introduce a few notations that will be used throughout this paper.

Tangential Fourier transform. To perform analytic estimates in the tangential direction, we use Fourier analysis in the tangential direction. For a ∈ L 2 (B), we will denote its Fourier transform with respect to the tangential variable by Fa and dene it as Fa(ξ, y) := Rx a(x, y)e -ixξ dx.

(2.1)

We dene similarly the reciprocal Fourier transform F -1 , which obviously also acts only on the tangential variable.

Band-limited functions. Let N > 0. We will sometimes need to consider functions in L 2 (B) whose Fourier transform is supported within the set of tangential frequencies ξ satisfying |ξ| ≤ N . Therefore, we introduce the Fourier multiplier

P N (ξ) := 1 [-N,N ] (ξ) (2.2)
and the associated functional space

L 2 N (B) := a ∈ L 2 (B); a = P N a . (2.3)
For any k ∈ N and a ∈ H k (B), it is clear that

P N a -a H k (B) → 0 as N → +∞.
(2.4)

Regularization enhancement using the phantom and the control

We start with the following lemma concerning the possibility to remove high tangential frequencies from a smooth initial data. We denote by P the usual Leray projector on divergence free vector elds, tangent to ∂B.

Lemma 2.1. There exists a geometric constant C θ > 0 such that the following result holds. Let u a ∈ L 2 div (B) and T b > 0. We denote by u a the solution of the free Navier-Stokes equation at time T b starting from u a . There exists a family indexed by N >

0 of vector elds u N ∈ C 0 ([0, T b ]; L 2 div (B)) ∩ L 2 ((0, T b ); H 1 (B)) associated with forces f N ∈ C 0 ([0, T b ]; H k (B)), which are weak Leray solutions to ∂ t u N -∆u N + P[(u N • ∇)u N ] = Pf N , u N (0) = u a (2.5)
and satisfy, for any k ∈ N and ρ b > 0,

(f N ) |Ω L 1 ((0,T b );H k (Ω)) -→ N →+∞ 0, (2.6) f N ∈ C ∞ ([0, T b ] × B), (2.7) (u N (T b ) -u a ) |Ω L 2 (Ω) -→ N →+∞ 0, (2.8) (u N (T b )) |G H k (G) -→ N →+∞ 0, (2.9) (u N (T b )) |B\ Ω L 2 (B\ Ω) ≤ C θ ( u a ) |B\ Ω L 2 (B\ Ω) + o N →+∞ (1), (2.10) ∃C N > 0, sup m≥0 ∂ m x u N (T b ) H 3 (B) ≤ m! ρ m b C N , (2.11
)

∃C N > 0, 0≤α+β≤3 ∂ α x ∂ β y u N (T b ) L 1 x (L 2 y ) ≤ C N .
(2.12)

Proof. Let u a ∈ L 2 div (B). Let T b > 0. Let v be the weak Leray solution to ∂ t v -∆v + P[(v • ∇)v] = 0, v(0) = u a . (2.13) Hence, by denition, u a = v(T b ). It is classical to prove that v ∈ C ∞ ((0, T b ] × B) (see e.g. [40]). Let β ∈ C ∞ ([0, T b ]; [0, 1]) with β = 1 on [0, T b /3] and β = 0 on [2T b /3, T b ]. Let θ ∈ C ∞ (B; [0, 1]) with θ = 1 for x ∈ [0, L] and θ = 0 for x < -L or x > 2L. Let ψ v (t, x, y) := - y -1 v(t, x, y) • e x dy.
(2.14)

We introduce

u := ∇ ⊥ (βψ v + (1 -β)θψ v ) .
(2.15)

Hence div u = 0. Moreover, since β does not depend on the space variables, one has

u = βv + (1 -β)θv + (1 -β)ψ v ∇ ⊥ θ. (2.16)
Then, u is the weak Leray solution to

∂ t u -∆u + P[(u • ∇)u] = Pg, u(0) = u a , (2.17)
where we set

g := β(1 -θ)v -βψ v ∇ ⊥ θ + (1 -β)∂ t ψ v ∇ ⊥ θ -2(1 -β)(∇θ • ∇)v -(1 -β)∆θv -(1 -β)∆ ψ v ∇ ⊥ θ -β(1 -β)(v • ∇)((1 -θ)v) -(1 -β) 2 (θv • ∇)((1 -θ)v) + (1 -β)(β + (1 -β)θ)(v • ∇)(ψ v ∇ ⊥ θ) + (1 -β)ψ v (∇ ⊥ θ • ∇)u.
(2.18)

Since each term involves, on the one hand β or 1β and, on the other hand, 1θ or a derivative of θ,

supp g ⊂ [T b /3, T b ] × (B \ Ω) and g ∈ C ∞ ([0, T b ] × B). We dene u N (t) := β(t)u(t) + (1 -β(t))P N u(t), (2.19)
where P N is dened in (2.2). In particular, (2.16) implies (2.5), provided that one sets

f N := β(u -P N u) + (1 -β) (P N u • ∇)P N u -P N ((u • ∇)u) + β(1 -β) (u • ∇)(P N u -u) -(P N u • ∇)(P N u -u) + βg + (1 -β)P N g.
(2.20)

Let k ∈ N. Thanks to denition (2.17), there holds (2.6) and (2.7). Indeed, u belongs to C 0 ([T b /3, T b ]; H k+1 (Ω)) and the family P N u converges towards u in this space. From (2.16), at the nal time, one has

u N (T b ) = P N u(T b ) = P N θv(T b ) + ψ v (T b )∇ ⊥ θ = P N θ u a + ψ ua ∇ ⊥ θ , (2.21)
where ψ ua is dened from u a similarly as ψ v is from v in (??). In particular, we deduce from (2.18) that u N (T b ) is entire in x so that, for any ρ b > 0, there exists C b > 0 such that (2.11) holds. We also deduce from (2.18

) that u N (T b ) → θ u a +ψ ua ∇ ⊥ θ in H k (B), which implies (2.8), (2.9) and (2.10) with C θ = 2 θ W 1,∞ (B) .
To obtain (2.12), we change slightly the denition (2.2) of P N . Instead of a rectangular window lter, we dene P N as the Fourier multiplier W N , where

W N ∈ C ∞ (R; [0, 1]) is such that W N (ξ) = 1 for ξ ∈ [-N + 1, N -1]
and W N (ξ) = 0 when |ξ| ≥ N . This preserves the property (2.4) but has a better behavior with respect to L 1 norms in space. Indeed, if φ ∈ S(R, R), one checks that W N φ ∈ L 1 (R). This property implies (2.12) because u(T b ) has a compact support.

Proof of the regularization proposition

We turn to the proof of Proposition 1.7.

Let T, ρ b , η b > 0 and k ∈ N. Let u * ∈ L 2 div (Ω) satisfying u * • e x = 0 on Γ 0 and Γ L . Let u a be the extension by 0 of u * to B. Since u * • e x = 0 on Γ 0 and Γ L , u a ∈ L 2 div (B). For T b ∈ (0, T ) small enough, the free solution starting from u a at time T b , say u a satises

( u a ) |B\ Ω L 2 (B\ Ω) ≤ η b /2C θ and ( u a ) |Ω -u * L 2 (Ω) ≤ η b /5.
We choose N large enough such that (2.8), (2.9) and (2.10) imply that u b := u N (T b ), where the family (u N , f N ) is given by Lemma 2.1, satises (1.10) and such that (2.6) ensures (1.9). Estimates (2.11) and (2.12) prove (1.11) and (1.12). This concludes the proof of Proposition 1.7, provided that we dene f g := (f N ) |Ω and f c := (f N ) |B\Ω , each being smooth within its support.

Strategy for global approximate controllability

We explain our strategy to prove Proposition 1.8. Let T > 0,

u b ∈ L 2 div (B), δ ∈ (0, 1
2 ) and k ∈ N. We intend to construct a family of approximate trajectories depending on a small parameter 0 < ε 1 and driving u b approximately to zero. We detail the construction of this family in the following paragraphs. Then, we prove estimates on boundary layer terms for these approximate trajectories in Section 4. We prove estimates on the remainder in Section 5 and postpone analytic-type estimates for these approximate trajectories to Section 6.

Small-time to small-viscosity scaling

Let κ ∈ (0, 1). Although it might seem like a further complication, our strategy is based on trying to control the system (1.7) at an even shorter time scale, ε 1-κ T , passing through intermediate states (velocities) of order 1/ε. For ε ∈ (0, 1), we introduce the trajectories

U ε (t, x, y) := u ε (t/ε, x, y)/ε, P ε (t, x, y) := p ε (t/ε, x, y)/ε 2 , (3.1) F ε g (t, x, y) := f ε g (t/ε, x, y)/ε 2 and F ε c (t, x, y) := f ε c (t/ε, x, y)/ε 2 . (3.2)
The tuples

(U ε , P ε , F ε c , F ε g ) dene solutions to (1.7) with initial data u b if and only if the new unknowns (u ε , p ε , f ε c , f ε g ) are solutions to the rescaled system            ∂ t u ε + (u ε • ∇) u ε + ∇p ε -ε∆u ε = f ε g + f ε c in (0, T /ε κ ) × B, div u ε = 0 in (0, T /ε κ ) × B, u ε = 0 on (0, T /ε κ ) × ∂B, u ε | t=0 = εu b in B. (3.3)
Observe the three dierences between (3.3) and the original system (1.7):

the Laplace term has a small factor ε in front of it rather than 1,

the system is set on the long time interval (0, T /ε κ ) rather than (0, T ), the initial data is εu b rather than u b .

We construct approximate solutions to (3.3) in the following paragraph.

Return method ansatz

We introduce the following explicit approximate solution to (3.3):

u ε app (t, x, y) := u 0 (t) + χ(y)v 0 t, ϕ(y)/ √ ε + εu 1 (t, x, y)+ε 2 w ε (t, y), (3.4)
p ε app (t, x, y) := p 0 (t, x),

(3.5) f ε c (t, x, y) := εf 1 |B\Ω (t, x, y), (3.6)
f ε g (t, x, y) := εf 1 |Ω (t, x, y).

(3.7)

In the following lines, we dene each of the terms involved in this approximate solution.

We refer to Section 3.4 for comments on the choice of these proles.

Base Euler ow prole. Let n ∈ N satisfying n ≥ 3 and

n > 3 4 1 κ -1 . (3.8) Let h in C ∞ (R + , R) be such that supp h ⊂ (0, T /3] ∪ [2T /3, T ), (3.9)
T /3 0 h(t)dt = 2L, (3.10)
T 0 t k h(t)dt = 0 for 0 ≤ k < n. (3.11)
We dene u 0 (t) := h(t)e x and p 0 (t, x) := -ḣ(t)x.

(3.12)

For a function a ∈ L 2 (B), we will denote its translation along the base ow h by (τ h a)(t, x, y) := a x -t 0 h(s) ds, y .

(3.13) Boundary layer prole. Let ϕ ∈ C ∞ ([-1, 1], [0, 1]) such that ϕ(±1) = 0 and |ϕ (y)| = 1 for |y| ≥ 1/4. Let χ ∈ C ∞ ([-1, 1], [0, 1]) such that χ(y) = 1 for |y| ≥ 2/3, and χ(y) = 0 for |y| ≤ 1/3. Let V (t, z) be the solution to      ∂ t V -∂ zz V = 0 in R + × R + , V (t, 0) = h(t) on R + , V (0, z) = 0 in R + . (3.14)
We dene v 0 (t, z) := -V (t, z)e x .

(3.15)

In the sequel, for any function V(t, z), depending on the fast variable, we will denote its evaluation at z = ϕ(y)/ √ ε by

{V}(t, y) := V t, ϕ(y) √ ε . (3.16) Linearized Euler ow prole. Let β ∈ C ∞ (R + , [0, 1]) non-increasing such that β(t) = 1 for t ≤ T /3 and β(t) = 0 for t ≥ 2T /3. Let χ δ ∈ C ∞ ([-1, 1], [0, 1]
) such that χ δ (y) = 1 for |y| ≤ 1 -2δ and χ δ (y) = 0 for |y| ≥ 1δ. We dene the stream function associated with u b , then u 1 and eventually the force f 1 :

ψ b (x, y) := - y -1 u b (x, y ) • e x dy , (3.17)
u 1 (t, x, y) := β(t)τ h ∇ ⊥ [χ δ ψ b ] + τ h ∇ ⊥ [(1 -χ δ )ψ b ], (3.18)
f 1 (t, x, y) := β(t) χ δ (y)u b (x -2L, y) -χ δ (y)ψ b (x -2L, y)e x . (3.19)
Technical prole. For t ∈ R + and y ∈ [-1, 1], we dene the source

f ε W := - χ ϕ 2 {z 2 V } -2 χ ϕ 3 ϕ {z 3 ∂ z V }. (3.20) Let W ε (t, y) : R + × [-1, 1] → R be the solution to      ∂ t W ε -ε∂ yy W ε = f ε W in R + × [-1, 1], W ε (t, ±1) = 0 on R + , W ε (0, y) = 0 in [-1, 1]. (3.21)
Finally, we let

w ε (t, y) := W ε (t, y)e x . (3.22)
Equation satised by the approximate trajectories. Then (u ε

app , p ε app ) are solutions to            ∂ t u ε app + u ε app • ∇ u ε app -ε∆u ε app + ∇p ε app = f ε c + f ε g + εf ε app in (0, T /ε κ ) × B, div u ε app = 0 in (0, T /ε κ ) × B, u ε app = 0 on (0, T /ε κ ) × ∂B, u ε app | t=0 = εu b in B, (3.23)
where we dene

f ε app := -ε∆u 1 + ε(u 1 • ∇)u 1 +ε 2 W ε ∂ x u 1 + ε 2 (u 1 • e y )∂ y w ε -χ{V }∂ x u 1 - u 1 • e y ϕ √ εχ {zV } + χϕ {z∂ z V } e x .
(3.24)

Estimates and proof of approximate controllability

By construction, the approximate trajectory will be small at the nal time.

Proposition 3.1. There exists a constant C app > 0 such that, for ε > 0 small enough,

1 ε u ε app (T /ε κ ) |Ω L 2 (B) ≤ C app ε 1 4 + ε κ(n-3 4 ( 1 κ -1)) | ln ε| n+ 3 4 + u b|{|y|≥1-2δ} L 2 (B) . (3.25)
Moreover, the approximate trajectory can be arbitrarily close to a true trajectory.

Indeed, we can construct a remainder which is small, provided that the initial data u b is suciently regular (its tangential analytic radius is large enough).

Proposition 3.2. There exists ρ b > 0, depending only on T such that, if u b satises (1.11) and (1.12) for some C b > 0, there exists C r > 0 such that, for ε > 0 small enough, there exists a weak Leray solution

r ε ∈ C 0 ([0, T /ε κ ], L 2 div (B)) ∩ L 2 ((0, T /ε κ ), H 1 (B)) to                ∂ t r ε + u ε app • ∇ r ε + ε (r ε • ∇) r ε + (r ε • ∇) u ε app -ε∆r ε + ∇π ε = -f ε app in (0, T /ε κ ) × B, div r ε = 0 in (0, T /ε κ ) × B, r ε = 0 on (0, T /ε κ ) × ∂B, r ε | t=0 = 0 in B, (3.26)
which moreover satises

r ε L ∞ ((0,T /ε κ );L 2 (B)) ≤ C r (ε 1 4 + ε 1-κ ). (3.27)
It is straightforward to check that Proposition 3.1 and Proposition 3.2 imply Proposition 1.8. Indeed, let ρ b be given by Proposition 3.2 and assume that u b satises (1.11) and (1.12) for some C b > 0. We choose ε > 0 small enough such that the conclusions of both propositions hold. We construct an exact trajectory by setting u ε := u ε app + εr ε and p ε := p ε app + επ ε .

), F g ∈ C ∞ ([0, T c ] × Ω), F c ∈ C ∞ ([0, T c ] × B \ Ω) and moreover supp F g ⊂ (0, T c ) × [0, L] × [-1 + δ, 1 -δ], (3.29) supp F c ⊂ (0, T c ) × B \ Ω,
F ε g (t) L 1 ([0,Tc];H k (Ω)) = 1 ε 2 f ε g (t/ε) L 1 ([0,ε 1-κ T ];H k (Ω)) = 1 ε f ε g (t) L 1 ([0,T /ε κ ];H k (Ω)) = f 1 |Ω L 1 ([0,T /ε κ ];H k (Ω)) ≤ χ δ u b + χ δ ψ b H k (G) , (3.31)
where we recall that the set G is dened in (1.8). This proves the estimate (1.13) concerning the size of the phantom force, for a constant C k,δ which only depends on the norm of χ δ in H k+1 (-1, 1), and thus concludes the proof of the approximate controllability result Proposition 1.8.

We prove Proposition 3.1 in Section 4 (thanks to the well-prepared dissipation method) and Proposition 3.2 in Section 5 (using a long-time nonlinear Cauchy-Kovalevskaya estimate).

Comments and insights on the proposed expansion

Remark 3.3 (Return method and base Euler ow). Since system (3.3) can be seen as a perturbation of the Euler equations a natural idea is to follow the return method introduced by Coron in [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF] (see also [START_REF] Coron | Control and nonlinearity[END_REF]Chapter 6]) to prove the controllability of the Euler equations in the 2D case (see also [START_REF] Coron | On the controllability of 2-D incompressible perfect uids[END_REF], and [START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF] for the 3D case). Loosely speaking the idea is to overcome that the linearized problem around zero is not controllable by introducing, thanks to the boundary control, a velocity u 0 of order O(1) (whereas the initial velocity is only of order O(ε)) solution to the Euler equation satisfying u 0 | t=0 = u 0 | t=T = 0 and such that the corresponding ow ushes all the domain out during the time interval (0, T ). In the present case of a rectangle this step is pretty easy and explicit: it corresponds to the introduction of a ow which ushes out the initial data. From (3.12), we get that (u 0 , p 0 ) indeed solves the incompressible Euler equation:

     ∂ t u 0 + u 0 • ∇ u 0 = -∇p 0 , in R + × B, div u 0 = 0 in R + × B, u 0 • e y = 0 on R + × ∂B, (3.32)
with initial data u 0 (0) = 0 and u 0 (t) = 0 for t ≥ T .

Remark 3.4 (Transport of the initial data). The term u 1 takes into account the initial data u b , which is transported by the ow u 0 . Using (3.12), (3.18) and (3.19), we obtain that u

1 solves            ∂ t u 1 + h(t)∂ x u 1 = f 1 in R + × B, div u 1 = 0 in R + × B, u 1 = 0 on R + × ∂B, u 1 (0) = u b in B. (3.33)
Thanks to assumption (3.10), it is clear that the initial data will be ushed outside of the domain at time T /3. During the time interval [T /3, 2T /3], the initial data u b has been shifted towards the right of a distance 2L. This is the time interval during which the force f 1 kills most of the initial data (for |y| ≤ 1δ).

The key point is that, outside of the physical domain, this force is merely a control.

However, since we need this force to be analytic, it also acts a little bit within the physical domain. This gives rise to an unwanted phantom force. The purpose of the second term v 0 is to recover the Dirichlet boundary condition by introducing the boundary layer generated by u 0 . Thanks to our previous choice of u 0 we will avoid the diculty usually associated with the Prandtl equation. Indeed the boundary layer will also be fully horizontal (tangential) and will not depend on x so that the equation for v 0 will deplete into a linear heat equation with non-homogeneous Dirichlet data depending on u 0 . The quantity ϕ(y)/ √ ε reects quick variations within the boundary layer, where ϕ(y) is the distance to the boundary.

4

Well-prepared dissipation method for the boundary layer

The key argument of the well-prepared dissipation method is that the normal dissipation involved in uid mechanics boundary layer equations can dissipate most of their energy, provided that the created boundary layers are well-prepared in some sense. Roughly speaking, this preparation amounts to ensure that they do not contain energy at low frequencies.

Large time decay of the boundary layer prole

In the work [START_REF] Coron | Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions[END_REF] concerning the case of the Navier slip-with-friction boundary condition, we used boundary controls to import enough vanishing moments thanks to the transport by the Euler ow within the boundary layer. In this work, we cannot use this strategy because we do not want the boundary layer prole to depend on the slow tangential variable, see Remark 3.5. Instead we rely on the assumptions (3.11) on the base Euler ow. We prove below that these conditions entail a good decay for the boundary layer prole. This decay will be used both to prove that the source terms generated by v in equation (3.26) for the remainder are integrable with respect to time and that the boundary prole at the nal time is small enough to apply a local controllability result. For s, m ∈ N and I an interval of R, we introduce the following weighted Sobolev spaces:

H s,m (I) := f ∈ H s (I), s α=0 I 1 + z 2 m f (α) (z) 2 dz < +∞ , (4.1)
which we endow with their natural norm. We will use this denition with I = R or I = R + .

Lemma 4.1. Let T > 0, s, n ∈ N and h ∈ C ∞ (R, R) satisfying (3.9) and (3.11). We consider V the solution to (3.14). For any 0 ≤ m ≤ 2n + 1, there exists a constant C such that the following estimate holds:

|V (t, •)| H s,m (R + ) ≤ C ln(2 + t) 2 + t 1 4 + 2n+1 2 -m 2 . (4.2)
Proof. Estimate (4.2) is straightforward up to time T because its right-hand side is bounded from below for t ∈ [0, T ]. Thus, we focus on large time estimates. We start by explicit computations in the frequency domain using Fourier transform. We consider the auxiliary system

∂ t f -∂ zz f = (h(t) -ḣ(t)) • sgn(z)e -|z| , t ≥ 0, z ∈ R, f (0, z) = 0, t = 0, z ∈ R. (4.3)
Since the source term in (4.3) is odd, its unique solution f satises f (t, 0) = 0 for all t ∈ R + . Hence, thanks to the uniqueness property for the heat equation on the half-line, there holds V (t, z) = f (t, z) + h(t)e -z for t, z ≥ 0 because both sides of this equality solve the same heat equation. Therefore, proving estimates on f will provide estimates on V .

After Fourier transform and solving the ODE, we obtain the formula:

f (t, ζ) := R f (t, z)e -iζz dz = - 2iζ 1 + ζ 2 t 0 e -(t-s)ζ 2 h(s) -ḣ(s) ds. (4.4)
Since h vanishes after T (see (3.9)), the behavior of f (and thus V ) after time T is entirely determined by the initial data f T (z) := f (T, z). Thanks to [START_REF] Coron | Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions[END_REF]Lemma 6], to establish (4.2), it suces to check that, for 0 ≤ j ≤ 2n,

∂ j ζ fT (0) = 0. (4.5)
Thanks to (4.4) and to the Leibniz rule, for j ∈ N, one has:

∂ j ζ fT (ζ) = -i j k=0 j k ∂ j-k ζ 2ζ 1 + ζ 2 T 0 h(t) -ḣ(t) ∂ k ζ e -(T -t)ζ 2 dt. (4.6) First, since ζ → 2ζ/(1 + ζ 2
) is an odd function, only its odd derivatives don't vanish at zero. Second, thanks to the Arbogast rule for the iterated dierentiation of composite functions (also known as Faà di Bruno's formula), one has:

∂ k ζ e -(T -t)ζ 2 = m 1 +2m 2 =k k! m 1 !m 2 ! -2ζ(T -t) 1! m 1 -2(T -t) 2! m 2 e -(T -t)ζ 2 . (4.7)
Hence, this derivative is non null at zero only if k is even, say k = 2k and the only nonvanishing term in the right-hand side of (4.7) is the one corresponding to (m 1 , m 2 ) = (0, k ) and is proportional to (Tt) k . From (4.6) and (4.7) we deduce that ∂ j ζ fT (0) is a linear combination of the moments

T 0 h(t) -ḣ(t) (T -t) k dt, (4.8)
where 0 ≤ 2k ≤ j -1. Thanks to (3.9) and (3.11), the integrals (4.8) vanish for 0 ≤ k < n. So (4.5) holds for j ≤ 2n -1. Last, (4.5) also holds for j = 2n because, when j is even, all the terms vanish. Indeed, in (4.6), either k is odd or jk = 2nk is even. This concludes the proof of the lemma.

Fast variable scaling and Lebesgue norms

Let us prove the following lemma, which is a simpler version of [25, Lemma 3, page 150].

Lemma 4.2. Let γ ∈ C 0 ([-1, 1]) with γ ≡ 0 on -1 3 , 1 3 . For V ∈ L 2 (R + ) and ε > 0:

γ{V} L 2 (-1,1) ≤ 2ε 1 4 γ ∞ V L 2 (R + ) .
(4.9)

Proof. For -1 ≤ y ≤ -1 4 , we assumed ϕ = 1. Thus, ϕ(y) = 1 + y. Recalling the fast variable notation (3.16) and performing an ane change of variables gives

-1 3 -1 γ 2 (y)V 2 ϕ(y) √ ε dy = √ ε 2 3 √ ε 0 γ 2 ( √ εz -1)V 2 (z) dz ≤ √ ε γ 2 ∞ V 2 L 2 (R + ) . ( 4
W ε (t) L ∞ (-1,1) + ∂ y W ε (t) L ∞ (-1,1) ≤ ε -3 4 C W . ( 4
∂ t W ε L ∞ (R + ;L 2 (-1,1)) ≤ 2 ∂ t f ε W L 1 (R + ;L 2 (-1,1)) .
(4.12)

Plugging this estimate in the equation (3.21) yields

∂ yy W ε L ∞ (R + ;L 2 (-1,1)) ≤ 1 ε f ε W L ∞ (R + ;L 2 (-1,1)) + 2 ∂ t f ε W L 1 (R + ;L 2 (-1,1)) . (4.13)
Thanks to estimate (4.9) from Lemma 4.2 applied to the denition (3.20) of f ε W , we obtain, for t ≥ 0,

f ε W (t) L 2 (-1,1) ≤ 2ε 1 4 χ ϕ -2 ∞ z 2 V (t, z) L 2 (R + ) + 2ε 1 4 2χ ϕ ϕ -3 ∞ z 3 ∂ z V (t, z) L 2 (R + ) ≤ Cε 1 4 V (t) H 1,3 (R + ) , (4.14)
where C is a nite constant because, by construction, χ and χ vanish for |y| ≥ 2 3 , so that the division by ϕ which vanishes at y = ±1 is not singular. Proceeding similarly and using the equation (3.14) on V , we obtain

∂ t f ε W (t) L 2 (-1,1) ≤ Cε 1 4 V (t) H 3,3 (R + ) .
f ε W (t) L 2 (-1,1) ≤ Cε 1 4 ln(2 + t) 2 + t 9 4
≤ Cε

∂ t f ε W L 1 (R + ;L 2 (-1,1)) ≤ Cε 1 4 +∞ 0 ln(2 + t) 2 + t 9 4
dt ≤ 2Cε (4.17)

Eventually, plugging (4.16) and (4.17) into (4.13) proves (4.11) thanks to the boundary conditions W ε (t, ±1) = 0 and the Poincaré-Wirtinger inequality for ∂ y W ε .

Proof of the decay of approximate trajectories

We prove Proposition 3.1. Recalling the denition (3.4) of u ε app , we estimate the size of each term at the time T /ε κ .

Thanks to (3.9) and (3.12), u 0 (T /ε κ ) = 0.

Thanks to (4.9) from Lemma 4.2 and (4.2) from Lemma 4.1, there holds

χ{v 0 (T /ε κ )} L 2 y ≤ 2ε 1 4 χ ∞ V (T /ε κ ) L 2 (R + ) ≤ 2ε 1 4 C ln(2 + T /ε κ ) 2 + T /ε κ 3 4 +n ≤ Cε 1+κ(n-3 4 ( 1 κ -1)) | ln ε| n+ 3 4 , (4.18)
for some constant C > 0.

Thanks to (3.18),

u 1 (T /ε κ ) = ∇ ⊥ [(1 -χ δ )ψ b ]. (4.19)
Moreover, since u b satises (1.12), u b ∈ L 1 (B). In particular, since u b is divergencefree, this implies that, for all x ∈ R,

+1 -1 u b (x, y) • e x dy = 0, (4.20)
so that ψ b , which was dened as (3.17) can equivalently be written as

ψ b (x, y) = 1 y u b • e x dy. (4.21)
Thanks to (4.19), this implies that there exists a constant C δ > 0 which only depends on the norm of χ δ in H 1 (-1, 1) such that ε u 1 (T /ε κ ) L 2 (B) ≤ εC δ u b|{|y|≥1-2δ} L 2 (B) .

ε 2 w ε (T /ε κ ) L ∞ y ≤ ε 1+ 1 4 C W . (4.23)
Gathering these estimates concludes the proof of estimate (3.25) of Proposition 3.1.

Estimates on the remainder

This section is devoted to the proof of Proposition 3.2. An important diculty to obtain some uniform energy estimates of r ε from system (3.26) is that the term (r ε • ∇)u ε app contains a term with a factor 1/ √ ε due to the fast variation of the boundary layer term in the normal variable (see the expansion (3.4) of u ε app ). To deal with this diculty we use a reformulation of this term where the singular factor is traded against a loss of derivative on r ε in the tangential direction x (see Section 5.1). Then, we establish a long-time nonlinear Cauchy-Kovalevskaya estimate (see Section 5.3) thanks to some tools from Littlewood-Paley theory which are recalled in Section 5.2.

Dyadic partition of unity. We recall that, for a ∈ L 2 (B), we dened its Fourier transform Fa in the tangential direction as (2.1). We

x χ lp , ϕ lp ∈ C ∞ (R, [0, 1]) such that supp ϕ lp ⊂ τ ∈ R; 3 4 ≤ |τ | ≤ 8 3 , (5.7) supp χ lp ⊂ τ ∈ R; |τ | ≤ 4 3 , (5.8) ∀τ ∈ R * , j∈Z ϕ lp (2 -j τ ) = 1, (5.9) ∀τ ∈ R, χ lp (τ ) + j∈N ϕ lp (2 -j τ ) = 1, (5.10) ∀τ ∈ R * , 1 2 ≤ j∈Z ϕ 2 lp (2 -j τ ) ≤ 1, (5.11)
The existence of such a dyadic partition of unity is proved in [1, Proposition 2.10]. For k ∈ Z, we introduce the Fourier multipliers ∆k and Ṡk by dening, for any a ∈ L 2 (B),

∆k a := F -1 ϕ lp (2 -k ξ)Fa(ξ, y) , (5.12)
Ṡk a := F -1 χ lp (2 -k ξ)Fa(ξ, y) .

(

The operators ∆k and Ṡk are with respect to the horizontal variable only. For a ∈ L 2 (B), one has, thanks to (5.9) and (5.10),

Ṡk a =

j≤k-1 ∆j a. (5.15)

Since we will use such norms for functions whose Fourier transforms in x are compactly supported, we do not provide more details on the denition of the corresponding functional spaces, referring for more to [START_REF] Bahouri | Fourier analysis and nonlinear partial dierential equations[END_REF].

Classical estimates. We recall the following classical estimates, for which we track the constants. First, we will use the following Bernstein type lemma from [1, Lemma 2.1].

Lemma 5.2. There exists a universal constant C B ≥ 2 such that the following properties

hold. Let 1 ≤ p ≤ q ≤ +∞, α ∈ {0, 1}, k ∈ Z and a ∈ L 2 (B). If the support of Fa is included in {(ξ, y); 2 -k |ξ| ≤ 100}, then ∂ α x a L q x (L 2 y ) ≤ C B 2 k α+ 1 p -1 q a L p x (L 2
y ) .

(5.16)

If the support of Fa is included in {(ξ, y);

1 100 ≤ 2 -k |ξ| ≤ 100}, then a L p x (L 2 y ) ≤ C B 2 -kα ∂ α x a L p x (L 2 y ) .
(5.17

) Lemma 5.3. Let a ∈ H 1 0 ([-1, 1] y ). Then a L ∞ y ≤ a 1 2 L 2 y ∂ y a 1 2
L 2 y .

(5.18)

Proof. This is a classical Gagliardo-Nirenberg interpolation inequality (see [START_REF] Nirenberg | On elliptic partial dierential equations[END_REF]). The fact that (5.18) holds with a unit constant for this particular choice of exponents is proved for example in [START_REF] Morosi | On the constants for some fractional GagliardoNirenberg and Sobolev inequalities[END_REF]Corollary 5.12] (which in fact yields a constant 2 -1

2 ).

As a consequence of Lemma 5.2, we have the following embedding. Indeed this is the main motivation for considering the 1 norm rather than the 2 norm in the denition of the homogeneous Besov norms Ḃs .

Lemma 5.4. Let a ∈ H 1 0 (B). There holds,

k∈Z 2 k 2 ∆k a L 2 x (L ∞ y ) ≤ C B ∇a Ḃ0 , (5.19) k∈Z ∆k a L ∞ (B) ≤ C 2 B ∇a Ḃ0 .
(5.20)

Proof. Let a ∈ H 1 0 (B). Hence, for almost every x ∈ R x , a(x, •) ∈ H 1 0 ([-1, 1] y ) and we can apply Lemma 5.3. Using (5.18), Cauchy-Schwarz then (5.17) yields

2 k 2 ∆k a L 2 x (L ∞ y ) ≤ 2 k 2 ∆k a 1 2 L 2 ∆k ∂ y a 1 2 L 2 ≤ C 1 2 B ∆k ∂ x a 1 2 L 2 ∆k ∂ y a 1 2 L 2 ≤ C 1 2 B ∆k ∇a L 2 , ( 5
∆k a L ∞ x (L ∞ y ) ≤ C B 2 k 2 ∆k a L 2
x (L ∞ y ) .

(

(B) such that div a = 0. For each k ∈ Z, ∆k a 2 L 2 x (L ∞ y ) ≤ C B 2 k 2 ∆k a L 2 (B) , (5.23) Proof. Let a ∈ H 1 0 (B). Hence, for almost every x ∈ R x , a(x, •) ∈ H 1 0 ([-
∆k a 2 L 2 x (L ∞ y ) ≤ ∆k a 2 1 2 L 2 ∆k ∂ y a 2 1 2 L 2 .
(5.24)

Then, using that div a = 0 and Lemma 5.2, we observe that ∆k ∂ y a Paraproduct decomposition. We shall use the Bony's decomposition (see [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]) for the horizontal variable:

2 L 2 = ∆k ∂ x a 1 L 2 ≤ C B 2 k ∆k a 1 L 2 .
f g = T f g + T g f + R(f, g), (5.26)
where (5.29)

T f g := k Ṡk-1 f ∆k g,
Thanks to the support properties (5.7) of ϕ lp and (5.8) of χ lp , the following lemma holds.

Lemma 5.6. For any f , g and h in L 2 (B),

T f g, ∆k h = k ∈Z/ |k -k|≤4 ( Ṡk -1 f )( ∆k g), ∆k h , (5.30) R(f, g), ∆k h = k ∈Z/ k ≥k-3
( ∆k f )( ∆k g), ∆k h .

(5.31)

Analyticity by Fourier multipliers. Let |∂ x | denote the Fourier multiplier with symbol |ξ|. We associate with any positive C 1 function of time ρ, the operator e ρ|∂x| mapping any reasonable function f (t, x, y) (say such that f ∈ L 1 loc (L 2 N (B)), for some N ∈ N), to (e ρ|∂x| f )(t, x, y) := F -1 e ρ(t)|ξ| Ff (t, ξ, y) (x).

(5.32)

Recall that F denotes the Fourier transform with respect to the tangential variable x, see (2.1). The function ρ describes the evolution of the radius of analyticity of the considered function. Below we establish a long-time Cauchy-Kovalevskaya estimate, for which the function ρ decays in time but not linearly.

Product estimates for analytic functions. For a ∈ L 2 (B), we introduce the notation a + := F -1 |Fa|. (5.36)

This scalar product is positive and this concludes the proof of (5.35).

Long-time weakly nonlinear Cauchy-Kovalevskaya estimate

In this paragraph, we explain how we will prove a long-time weakly nonlinear Cauchy-Kovalevskaya estimate on the remainder. We start by dening quantities that will enable us to dene the expected prole of analyticity ρ(t). Then, we close the estimate relying on a Grönwall-type argument. In the following paragraphs, we will prove the required estimates.

Remark 5.8. The idea of closing an estimate on a nonlinear function of the solution to control the loss of analyticity dates back to Chemin in [START_REF] Chemin | Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray[END_REF]. It was later used in the context of anisotropic Navier-Stokes equations in [START_REF] Chemin | Global regularity for some classes of large solutions to the Navier-Stokes equations[END_REF] and, more recently, for Prandtl equations in [START_REF] Zhang | Long time well-posedness of Prandtl system with small and analytic initial data[END_REF],

using only analyticity in the tangential direction.

Friedrichs' regularization scheme

In order for our manipulations to make sense, we will restrict (5.5) to a bounded range of frequencies. Then, we establish estimates which are independent on the considered range and we pass to the limit. This process was introduced by Friedrichs in [START_REF] Friedrichs | The identity of weak and strong extensions of dierential operators[END_REF] (see also [START_REF] Paicu | Équation anisotrope de Navier-Stokes dans des espaces critiques[END_REF] for a recent example of the passage to the limit). Let N ∈ N. Instead of (5.5), we consider the modied equation

         ∂ t r ε N + (h -χ{V }+ε 2 W ε )∂ x r ε N -ε∆r ε N + ∇π ε N = f ε N in (0, T /ε κ ) × B, div r ε N = 0 in (0, T /ε κ ) × B, r ε N = 0 on (0, T /ε κ ) × ∂B, r ε N | t=0 = 0 on B, (5.37)
where we introduce

-f ε N := P N f ε app + (M [∂ x r ε N,1 ]){z∂ z v 0 } + r ε N,2 (χ {v 0 }+ε 2 ∂ y w ε ) + εP N (r ε N • ∇) u 1 + εP N u 1 • ∇ r ε N + εP N (r ε N • ∇) r ε N .
(5.38)

In the sequel, to lighten the notations, we will write r instead of r ε N and we will omit the projections P N . It will be clear from our proof that we perform a priori estimates which are independent of N . Therefore, using usual compactness arguments, our proof will also yield the same energy estimate for the initial equation (5.5). Since this argument is quite classical, we will only detail the a priori estimates. Even though this regularization process is transparent in the proof, it is necessary to ensure that all the quantities are well dened.

Denition of the analyticity prole

We start by dening the analyticity radius that we will require on the coecients and the source terms of the equation for the remainder

ρ 0 := 2 + 10 2 C B +∞ 0 z∂ z v 0 (t, z) L ∞ (R + ) dt.
(5.39)

We apply the dyadic operator ∆k to (5.48) and take the L 2 (B) inner product of the resulting equation with ∆k r. We observe, by integration by parts, that the contributions due to the fourth and sixth terms vanish, so that

1 2 d dt ∆k r(t) 2 L 2 -ρ |∂ x | ∆k r, ∆k r + β ∆k r(t) 2 L 2 + ε ∇ ∆k r 2 L 2
= ∆k e ρ|∂x|-β f ε N , ∆k r .

(5.49)

Above and below we simply denote by L 2 the space L 2 (B). Using the denition (5.12) of ∆k and the support property (5.7) of ϕ lp , we know that

|∂ x | ∆k r, ∆k r ≥ 1 2 2 k ∆k r 2 L 2 .
(5.50)

Then, integrating over [0, t], we obtain

1 2 ∆k r(t) 2 L 2 + 1 2 2 k t 0 | ρ| ∆k r 2 L 2 + t 0 β ∆k r 2 L 2 + ε t 0 ∇ ∆k r 2 L 2 ≤ t 0 ∆k e ρ|∂x|-β f ε N , ∆k r .
(5.51)

We take the square roots and sum the resulting inequalities for k ∈ Z to deduce that

k∈Z ∆k r(t) L 2 + k∈Z 2 k 2 t 0 | ρ| ∆k r 2 L 2 1 2 + √ 2 k∈Z t 0 β ∆k r 2 L 2 1 2 + √ 2ε k∈Z t 0 ∆k ∇r 2 L 2 1 2 ≤ 2 √ 2 k∈Z t 0 | ∆k e ρ|∂x|-β f ε N , ∆k r | 1 2 .
(5.52) Proposition 5.10 (Proof in Section 5.4). For t ∈ [0, T * ], there holds

2 √ 2 k∈Z t 0 ∆k e ρ|∂x|-β f ε N , ∆k r 1 2 ≤ k∈Z 2 k 2 t 0 | ρ| ∆k r 2 L 2 1 2 + √ 2 k∈Z t 0 β ∆k r 2 L 2 1 2 + 1 4 √ ε k∈Z t 0 ∆k ∇r 2 L 2 1 2 + √ 2 k∈Z t 0 1 α ε e ρ 0 |∂x| ∆k f ε app 2 L 2 1 2 .
(5.53)

The proof of Proposition 5.10 is given in Section 5.4. Let us admit Proposition 5.10

for the time being and see how to conclude the proof of Proposition 3.2. Combining (5.52) and (5.53) we deduce that

k∈Z ∆k r(t) L 2 + √ ε k∈Z t 0 ∆k ∇r 2 L 2 1 2 ≤ √ 2 k∈Z t 0 1 α ε e ρ 0 |∂x| ∆k f ε app 2 L 2 1 2
.

(5.54) Proposition 5.11 (Proof in Section 6.3). If u b satises (1.11) and (1.12) for a constant C b > 0 and ρ b > ρ 0 , there exists C f > 0 such that, for ε, κ ∈ (0, 1), sup

t∈[0,T /ε κ ] k∈Z t 0 1 α ε e ρ 0 |∂x| ∆k f ε app 2 L 2 1 2 ≤ C f (ε 1 4 + ε 1-κ ).
(5.55)

As long as t ≤ T * , (5.54) holds and, thanks to Proposition 5.11, we obtain

k∈Z ∆k r(t) L 2 + √ ε k∈Z t 0 ∆k ∇r 2 L 2 1 2 ≤ √ 2C f (ε 1 4 + ε 1-κ ).
(5.56)

Moreover, t 0 | ρ| ≤ 10 2 C B +∞ 0 z∂ z v 0 (t, z) L ∞ (R + ) + 10 7 C 4 B εe β t 0 ∇r Ḃ0 (5.57)
and, for t ≤ T /ε κ ,

ε t 0 ∇r Ḃ0 = ε k∈Z t 0 ∆k ∇r L 2 ≤ √ εt • √ ε k∈Z t 0 ∆k ∇r 2 L 2 1 2 ≤ √ 2T C f (ε 1 4 + ε 1-κ ). (5.58)
Combining these estimates yields

ρ(T /ε κ ) ≥ 2 -10 7 C 4 B e β √ 2T C f (ε 1 4 + ε 1-κ ).
(5.59) Thus, for ε small enough, ρ(T /ε κ ) ≥ 1 and thus T * N = T /ε κ and one has

r ε N L ∞ (L 2 (B)) + √ ε ∇r ε N L 2 (L 2 (B)) ≤ √ 2e β C f (ε 1 4 + ε 1-κ ).
(5.60)

This estimate being uniform with respect to N , one can pass to the limit (for xed ε) towards r ε , and then take ε small enough to conclude the proof of Proposition 3.2.

Proof of Proposition 5.10

To prove Proposition 5.10 we estimate separately the terms corresponding to the dierent terms of the decomposition of the source term f ε N in (5.38). Let us start with the term corresponding to a loss of derivative.

Lemma 5.12. For t ∈ [0, T * ], there holds

k∈Z t 0 ∆k e ρ|∂x|-β (M [∂ x r 1 ]){z∂ z v 0 }, ∆k r 1 2 ≤ k∈Z 2 k 2 t 0 2C B z∂ z v 0 L ∞ z ∆k r 2 L 2 1 2
.

Proof. Since {z∂ z v 0 } and the operator M do not depend on the x variable,

∆k e ρ|∂x|-β (M [∂ x r 1 ]){z∂ z v 0 } = (M [ ∆k ∂ x r 1 ]){z∂ z v 0 }.
(5.61) Moreover, using the denition of M in (5.2), Hardy's inequality, and the fact that |χ| ≤ 1, we get that, for any a ∈ L 2 y (-1, 1),

M [a] L 2 y ≤ 2 a L 2 y .
(5.62) Hence, using (5.16) from Lemma 5.2, we obtain

∆k e ρ|∂x|-β (M [∂ x r 1 ]){z∂ z v 0 }, ∆k r ≤ 2 k+1 C B z∂ z v 0 L ∞ z ∆k r 2 L 2 .
(5.63)

The result follows by integration in time and summation over k ∈ Z of the square roots.

Lemma 5.13. For t ∈ [0, T * ], there holds

k∈Z t 0 ∆k e ρ|∂x|-β r 2 (χ {v 0 }+ε 2 ∂ y w ε ), ∆k r 1 2 ≤ k∈Z t 0 χ {v 0 }+ε 2 ∂ y w ε L ∞ (B) ∆k r 2 L 2 1 2
.

(5.64)

Proof. Since χ {v 0 }+ε 2 ∂ y w ε does not depend on x,

∆k e ρ|∂x|-β (r 2 (χ {v 0 }+ε 2 ∂ y w ε )) = (χ {v 0 }+ε 2 ∂ y w ε ) ∆k r 2 , (5.65)
and therefore the result readily follows by the Cauchy-Schwarz inequality.

Lemma 5.14 (Proof in Section 5.5). For t ∈ [0, T * ], there holds

k∈Z t 0 ∆k e ρ|∂x|-β (r • ∇) u 1 + u 1 • ∇ r , ∆k r 1 2 ≤ 10 8 C 2 B k∈Z t 0 2 1 ∆k r 2 L 2 1 2 + 1 20 k∈Z t 0 ∆k ∇r 2 L 2 1 2 .
(5.66) Lemma 5.15 (Proof in Section 5.6). For t ∈ [0, T * ], there holds

k∈Z t 0 ∆k e ρ|∂x|-β ((r • ∇) r), ∆k r 1 2 ≤ 600C 2 B k ∈Z 2 k 2 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2
.

Proposition 5.10 follows from the denition (5.38) of f ε N , Lemma 5.12, Lemma 5.13, Lemma 5.14 and Lemma 5.15. Observe in particular that the sum of the right hand sides of Lemma 5.12 and of Lemma 5.15 can be bounded by the rst term in the right hand sides of the estimate in Proposition 5.10 thanks to (5.45). On the other hand the sum of the right hand sides of Lemma 5.13 and Lemma 5.14 can be bounded by the other terms in the right hand sides of the estimate in Proposition 5.10 thanks to (5.43).

Estimate of

(r • ∇)u 1 + (u 1 • ∇)r.
Proof of Lemma 5.14

Due to div r = 0, we get, integrating by parts, that

∆k e ρ|∂x|-β (r • ∇) u 1 , ∆k r = - 2 i,j=1
∆k e ρ|∂x|-β r i u 1 j , ∆k ∂ i r j .

(5.67)

Due to div u 1 = 0, we get, integrating by parts, that

∆k e ρ|∂x|-β u 1 • ∇ r, ∆k r = - 2 i,j=1
∆k e ρ|∂x|-β u 1 i r j , ∆k ∂ i r j .

(5.68)

This yields a total of 8 scalar terms, which we estimate separately using the same method for each set of indexes. We explain the proof only for the terms in (5.67) (the terms of (5.68) are handled similarly). By Bony's decomposition (5.26), we expand the products as

r i u 1 j = T r i u 1 j + T u 1 j r i + R(r i , u 1 j ).
(5.69)

We explain the three estimates for each term in the right-hand side of (5.69).

First estimate

Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality

(5.30), we write ∆k e ρ|∂x| (T r i u 1 j ), ∆k

∂ i r j = |k -k|≤4
∆k e ρ|∂x| ( Ṡk -1 r i )( ∆k u 1 j ), ∆k ∂ i r j . | ∆k e ρ|∂x|-β ( Ṡk -1 r i )( ∆k

u 1 j ), ∆k ∂ i r j | ≤ Ṡk -1 r + i L ∞ x (L 2 y ) ∆k e ρ|∂x| (u 1 j ) + L 2 x (L ∞ y ) ∆k ∂ i r j L 2 ≤ C 3 2 B 1 2 -k 2 ∆k ∇r L 2 k ≤k -2 2 k 2 ∆k r L 2 .
(5.71) Thanks to the Peter-Paul inequality, we deduce that

| ∆k e ρ|∂x|-β ( Ṡk -1 r i )( ∆k u 1 j ), ∆k ∂ i r j | ≤ 1 (9 • 8 • 60) 2 ∆k ∇r 2 L 2 + 1 4 (9 • 8 • 60) 2 C 3 B 2 k -k   k ≤k -2 1 ∆k r L 2   2 .
(5.72) Summing these estimates and applying Minkowsky's inequality leads to

k∈Z t 0 | ∆k e ρ|∂x|-β (T r i u 1 j ), ∆k ∂ i r j | 1 2 ≤ 2160C 2 B k ∈Z   k ≥k -2 |k-k |≤4 2 k -k 2   t 0 2 1 ∆k r 2 L 2 1 2 + 1 9 • 8 • 60 k∈Z |k -k|≤4 t 0 ∆k ∇r 2 L 2 1 2 ≤ 2 • 10 6 C 2 B k∈Z t 0 2 1 ∆k r 2 L 2 1 2 + 1 8 • 60 k∈Z t 0 ∆k ∇r 2 L 2 1 2
.

(5.73)

Second estimate

Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality (5.30), we write ∆k e ρ|∂x| (T u 1

j r i ), ∆k ∂ i r j = |k -k|≤4 ∆k e ρ|∂x| ( Ṡk -1 u 1 j )( ∆k r i ), ∆k ∂ i r j .
It follows from the decomposition (5.14) of Ṡk -1 , the estimate (5.20) from Lemma 5.4 and the denition (5.42) of 1 that

e ρ|∂x| Ṡk -1 (u 1 ) + L ∞ ≤ C 2 B 1 .
(5.74)

Hence, with the product estimate (5.35) from Lemma 5.7, we infer

| ∆k e ρ|∂x|-β ( Ṡk -1 u 1 j )( ∆k r i ), ∆k

∂ i r j | ≤ e ρ|∂x| Ṡk -1 (u 1 ) + L ∞ ∆k r + i L 2 ∆k ∇r L 2 ≤ C 2 B 1 ∆k r L 2 ∆k ∇r L 2 .
(5.75) Summing these estimates and using the Peter-Paul inequality yields

k∈Z t 0 | ∆k e ρ|∂x|-β (T u 1 j r i ), ∆k ∂ i r j | 1 2 ≤ 2 • 10 5 C 2 B k∈Z t 0 2 1 ∆k r 2 L 2 1 2 + 1 8 • 60 k∈Z t 0 ∆k ∇r 2 L 2 1 2
.

(5.76)

Third estimate

By using the support properties of the paraproduct decomposition as in Lemma 5.6, equality (5.31), and the denition (5.29), we obtain ∆k e ρ|∂x| R(r i , u 1 j ), ∆k

∂ i r j = k ≥k-3 ∆k e ρ|∂x| ( ∆k r i )( ∆k u 1 j ), ∆k ∂ i r j = k ≥k-3 |k -k |≤1
∆k e ρ|∂x| ( ∆k r i )( ∆k u 1 j ), ∆k ∂ i r j .

Thanks to the product estimate (5.35) | ∆k e ρ|∂x|-β ( ∆k r i )( ∆k

u 1 j ), ∆k ∂ i r j | ≤ ∆k r + i L ∞ x (L 2 y ) e ρ|∂x| ∆k (u 1 j ) + L 2 x (L ∞ y ) ∆k ∇r L 2 ≤ 2 k-k 2 C 3 2
B 1 ∆k r L 2 ∆k ∇r L 2 .

(5.77)

Using the Peter-Paul inequality and summing the resulting estimates, we nd (5.88) Thanks to estimate (5.16) from Lemma 5.2,

k∈Z t 0 | ∆k e ρ|∂x|-β R(r i , u 1 j ), ∆k ∂ i r j | 1 2 ≤10 6 C 2 B k∈Z t 0 2 1 ∆k r 2 L 2 1 2 + 1 8 • 60
∆k ∂ x r L 2 ≤ C B 2 k ∆k r L 2 .
(5.89)

Using estimate (5.20) from Lemma 5.4, we obtain that

Ṡk -1 r + 1 L ∞ ≤ C 2 B ∇r + 1 Ḃ0 ≤ C 2 B ∇r Ḃ0 .
(5.90)

Gathering our estimates, we obtain

k∈Z t 0 | ∆k e ρ|∂x|-β (T r 1 ∂ x r), ∆k r | 1 2 ≤ C 3 2 B k∈Z |k -k|≤4 2 k 2 t 0 e β ∇r Ḃ0 ∆k r L 2 ∆k r L 2 1 2 ≤ 10 2 C 2 B k ∈Z 2 k 2 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2
.

(5.91)

This concludes the proof of (5.81).

5.6.2 Proof of (5.82)

Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality

(5.30), we expand ∆k e ρ|∂x| (T ∂xr r 1 ), ∆k r = |k -k|≤4

e ρ|∂x| ( Ṡk -1 ∂ x r)( ∆k r 1 ), ∆k r .

(5.92)

Thanks to the product estimate (5.35) from Lemma 5.7, we obtain

| e ρ|∂x|-β ( Ṡk -1 ∂ x r)( ∆k r 1 ), ∆k r | ≤ e β Ṡk -1 ∂ x r + L ∞ x (L 2 y ) ∆k r + 1 L 2 x (L ∞ y ) ∆k r + L 2 .
(5.93)

Thanks to estimate (5.19) from Lemma 5.4, we get

∆k r + 1 L 2 x (L ∞ y ) ≤ C B 2 -k 2 ∇r Ḃ0 .
(5.94)

Gathering our estimates and using the Peter-Paul inequality,

t 0 Ṡk -1 ∂ x r + L ∞ x (L 2 y ) ∇r Ḃ0 ∆k r + L 2 1 2 ≤ 2 3k 4 t 0 ∇r Ḃ0 ∆k r 2 L 2 1 2 + 2 -3k 4 t 0 ∇r Ḃ0 Ṡk -1 ∂ x r + 2 L ∞ x (L 2 y ) 1 2 , (5.95)
we obtain k∈Z t 0

| ∆k e ρ|∂x|-β (T ∂xr r 1 ), ∆k r |

1 2 ≤ C 1 2 B k∈Z 2 k 2 |k -k|≤4 2 k-k 4 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2 + C 1 2 B k∈Z 2 -k |k -k|≤4 2 k-k 4 t 0 e β ∇r Ḃ0 Ṡk -1 ∂ x r + 2 L ∞ x (L 2 y ) 1 2
.

(5.96)

The rst term is bounded by 10C B E. Using Minkowski's inequality and (5.14), we estimate the second term as

k∈Z 2 -k |k -k|≤4 2 k-k 4 t 0 e β ∇r Ḃ0 Ṡk -1 ∂ x r + 2 L ∞ x (L 2 y ) 1 2 ≤ k∈Z 2 -k |k -k|≤4 2 k-k 4 j≤k -2 t 0 e β ∇r Ḃ0 ∆j ∂ x r + 2 L ∞ x (L 2 y ) 1 2
.

(5.97) Now we deduce from Bernstein's Lemma 5.2, estimate (5.16), that

∆j ∂ x r + 2 L ∞ x (L 2 y ) ≤ C 2 B 2 3j ∆j r 2 L 2 .
(5.98)

We deduce from these estimates that the second term of (5.96) is bounded by 80C 2 B E.

This concludes the proof of (5.82).

Proof of (5.83)

First, using integration by parts in the horizontal direction, we get ∆k e ρ|∂x| ∂ x R(r 1 , r), ∆k r = -∆k e ρ|∂x| R(r 1 , r), ∆k ∂ x r . e ρ|∂x| ( ∆k r 1 )( ∆k r), ∆k ∂ x r .

(5.100)

Thanks to the product estimate (5.35) from Lemma 5.7, we obtain

| e ρ|∂x|-β ( ∆k r 1 )( ∆k r), ∆k ∂ x r | ≤ e β ∆k r + 1 L 2 x (L ∞ y ) ∆k r + L 2 x (L 2 y ) ∆k ∂ x r + L ∞ x (L 2
y ) .

(

∆k r + 1 L 2 x (L ∞ y ) ≤ C B 2 -k 2 ∇r Ḃ0 .
(5.102) Thanks to estimate (5.16) from Lemma 5.2, we have

∆k ∂ x r + L ∞ x (L 2 y ) ≤ C B 2 3k 2
∆k r L 2 .

(5.103)

Gathering our estimates, we obtain

k∈Z t 0 | ∆k e ρ|∂x|-β ∂ x R(r 1 , r), ∆k r | 1 2 ≤ C B k∈Z k ≥k-3 |k -k|≤1 2 3k 4 -k 4 t 0 e β ∇r Ḃ0 ∆k r L 2 ∆k r L 2 1 2 ≤ C B k∈Z k ≥k-3 |k -k|≤1 2 3k 4 -k 4 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2 + C B k∈Z k ≥k-3 |k -k|≤1 2 3k 4 -k 4 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2
(5.104)

Up to reordering the sums, one gets that the rst term is bounded by 32C B E and that the second term is bounded by 51C B E. This concludes the proof of (5.83).

| e ρ|∂x|-β ( Ṡk -1 r 2 )( ∆k ∂ y r), ∆k r | ≤ e β Ṡk -1 r + 2 L ∞ ∆k ∂ y r + L 2 ∆k r + L 2 .
(5.106)

From the denition (5.15) of Ḃ0 , ∆k ∂ y r L 2 ≤ ∇r Ḃ0 .

(5.107)

Summing over k ∈ Z and using Young's inequality, we obtain

k∈Z t 0 | ∆k e ρ|∂x|-β (T r 2 ∂ y r), ∆k r | 1 2 ≤ k∈Z |k -k|≤4 2 -k 2 t 0 e β ∇r Ḃ0 Ṡk -1 r + 2 2 L ∞ 1 2 + k∈Z |k -k|≤4 2 k 2 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2
(5.108) Using Minkowski's inequality and (5.14), the rst sum is bounded as

k∈Z |k -k|≤4 2 -k 2 t 0 e β ∇r Ḃ0 Ṡk -1 r + 2 2 L ∞ 1 2 ≤ k∈Z |k -k|≤4 2 -k 2 j≤k -2 t 0 e β ∇r Ḃ0 ∆j r + 2 2 L ∞ 1 2
(5.109) Thanks to estimates (5.23) from Lemma 5.5 and (5.16) from Lemma 5.2, we get ∆j r 2 L ∞ ≤ C 2 B 2 j ∆j r L 2 .

(5.110)

Gathering these two estimates and reordering the sums we conclude that the rst term in the right hand side of (5.109) is bounded by 36C 2 B E. Since the second term is bounded by 9E this concludes the proof of (5.84).

| e ρ|∂x|-β ( Ṡk -1 ∂ y r)( ∆k r 2 ), ∆k r | ≤ e β Ṡk -1 ∂ y r + L 2 ∆k r + 2 L ∞ ∆k r + L 2 .
(5.112)

On the one hand, thanks to estimate (5.23) from Lemma 5.5 and estimate (5.16) from Lemma 5.2, we have

∆k r + 2 L ∞ ≤ C 2 B 2 k ∆k r L 2 .
(5.113)

On the other hand, from the denition (5.15) of Ḃ0 , Ṡk -1 ∂ y r L 2 ≤ ∇r Ḃ0 .

(5.114)

Gathering our estimates, we obtain

k∈Z t 0 | ∆k e ρ|∂x|-β (T ∂yr r 2 ), ∆k r | 1 2 ≤ C B k∈Z |k -k|≤4 2 k 2 t 0 e β ∇r Ḃ0 ∆k r L 2 ∆k r L 2 1 2 ≤ 10 2 C B k ∈Z 2 k 2 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2
.

(5.115)

This concludes the proof of (5.85).

5.6.6 Proof of (5.86)

| e ρ|∂x|-β ( ∆k r 2 )( ∆k r), ∆k ∂ y r | ≤ e β ∆k r + 2 L 2 x (L ∞ y ) ∆k r + L 2 x (L 2 y ) ∆k ∂ y r + L ∞ x (L 2
y ) .

(

| e ρ|∂x|-β ( ∆k r 2 )( ∆k r), ∆k ∂ y r | ≤ e β C 2 B 2 k 2 + k 2 ∆k r L 2 ∆k r L 2 ∆k ∂ y r L 2 . (5.119)
We use the following crude estimate, which follows from denition (5.15).

∆k ∂ y r L 2 ≤ ∇r Ḃ0 .

(5.120)

Gathering our estimates, we obtain

k∈Z t 0 | ∆k e ρ|∂x|-β ∂ y R(r 2 , r), ∆k r | 1 2 ≤ C B k∈Z k ≥k-3 |k -k |≤1 2 k 4 + k 4 t 0 e β ∇r Ḃ0 ∆k r L 2 ∆k r L 2 1 2 ≤ 6C B k∈Z k ≥k-3 2 k 4 + k 4 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2 ≤ 6C B k ∈Z 2 k 2   k≤k +3 2 k-k 4   t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2 ≤ 10 2 C B k ∈Z 2 k 2 t 0 e β ∇r Ḃ0 ∆k r 2 L 2 1 2
.

(5.121)

This concludes the proof of (5.86).

6

Analytic estimates for the approximate trajectories

Preliminary estimates

We introduce notations and prove preliminary estimates that will be used in the sequel.

In this paragraph, a denotes a function in L 2 (B) for which all the norms and sums that we manipulate are nite.

For s ∈ N, ρ > 0 and p ∈ {2, +∞}, we introduce the tangential analytic-type norm

T /ε κ 0 ∂ y w ε (t) L ∞ (B) dt ≤ T ε 2-κ ∂ y W ε L ∞ (R + ;L ∞ (B)) ≤ C W T ε 1-κ+ 1 4 . (6.10) 2
Recalling the denition (5.42) of 1 and applying estimate (6.8) to a = ∇u 1 (t) and (6.4), there holds for any t ≥ 0, (6.11) thanks to estimate (1.12).

1 (t) ≤ C ρ b ∇u 1 (t) L 1 x (L 2 y ) + G ρ b 1,2 (∇u 1 (t)) ≤ C ρ b ∇u 1 i L 1 x (L 2 y ) + ∇u 1 o L 1 x (L 2 y ) + C 1 ≤ C ρ b (2C b + C 1 ) ,
Recalling the denition (5.41) of α ε , T /ε κ 0 α ε (t) dt ≤ T + π 2 .

(6.12)

Gathering these four estimates concludes the proof of Proposition 5.9.

Estimates for the source terms

We prove Proposition 5.11. We will use the two following inequalities. First, for g ∈ L L 2 (B) Recalling the denition (3.24) of f ε app , we proceed slightly dierently for the rst four terms (using (6.13)), and for the last three terms (using (6.14)).

1 2 ≤ 2ε 1 4 R + (1 + t 2 ) V (t) 2 L 2 (R + ) dt
f ε app := -ε∆u 1 + ε(u 1 • ∇)u 1 +ε 2 W ε ∂ x u 1 + ε 2 (u 1 • e y )∂ y w ε -χ{V }∂ x u 1 -u 1 • e y ϕ √ εχ {zV } + χϕ {z∂ z V } e x .

(6.15)

First term. We apply (6.13) to g := -∆u Second term. We apply (6.13) to g := (u 1 • ∇)u 1 . Similarly, we decompose u 1 thanks to (6.2) to get an estimate which is uniform with respect to time. We get four terms.

As an example, let us bound using (6.8), for some ρ ∈ (ρ 0 , ρ b ), k∈Z e ρ 0 |∂x| ∆k (u

1 i •∇)u 1 i L 2 (B) ≤ Cρ (u 1 i • ∇)u 1 i L 1 x (L 2 y ) + G ρ 1,2 ((u 1 i • ∇)u 1 i ) . (6.19)
The rst term is nite thanks to (1.12). The second term requires an analytic estimate for this quadratic term. For m ≥ 0, thanks to estimate (4.11) from Lemma 4.3. We proceed as above using (6.2) to get uniform bounds on the sums.

Fifth term. We apply (6.14) to g := χ∂ x u 1 and V := V . The integral in time of the boundary layer is nite thanks to (4.2) from Lemma 4.1 because n ≥ 3. The sum in k is also nite using the same techniques as above.

Sixth term. We apply (6.14) to g := χ (u 1 • e y )/ϕ and V := zV . The integral in time of the boundary layer is nite thanks to (4.2) from Lemma 4.1 because n ≥ 3. The sum in k is also nite using the same techniques as above.

Seventh term. We apply (6.14) to g := χϕ (u 1 • e y )/ϕ and V := z∂ z V . The integral in time of the boundary layer is nite thanks to (4.2) from Lemma 4.1 because n ≥ 3.

The sum in k is also nite using the same techniques as above.

(3. 28 )

 28 Combining (3.28) with the equation (3.23) satised by u ε app and the equation (3.26) satised by r ε proves that u ε is a weak solution to (3.3). Let σ > 0. Choosing ε > 0 small enough, summing estimates (3.25) and (3.27) and recalling the denition (3.28) of u ε , the assumption (3.8) on n and the scaling (3.1) proves that (1.16) holds at the time T c := ε 1-κ T < T . Since u b satises (1.11), u b ∈ C ∞ (B). Thus, thanks to (3.2), (3.6), (3.7) and (3.19

( 3 . 30 )

 330 Moreover, using (3.2), (3.7) and (3.19), one has

Remark 3 . 5 (

 35 Boundary layer correction). A major diculty is linked to the discrepancy between the Euler and the Navier-Stokes equations in the vanishing viscosity limit. Indeed, although inertial forces prevail inside the domain, viscous forces play a crucial role near the uncontrolled boundary, and give rise to a boundary layer of order O(1) associated with the velocity u 0 which does not satisfy the tangential part of the Dirichlet condition on the top and bottom boundaries.

(4. 15 )

 15 Combining (4.14) with Lemma 4.1 applied to m = 3, n = 3, s = 1, we obtain

  [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF] with Lemma 4.1 applied to m = 3, n = 3, s = 3, we obtain

(4. 22 )

 22 Thanks to estimate (4.11) from Lemma 4.3,

(5. 14 )

 14 Homogeneous Besov spaces. For a ∈ L 2 (B), we will use for s = 0 and s = 1 2 the following quantity corresponding to a homogeneous Besov norm a Ḃs := k∈Z 2 ks ∆k a L 2 (B) .

(5. 25 )

 25 Gathering (5.24) and (5.25) proves (5.23) since C B ≥ 1.

( 5 . 33 )

 533 Lemma 5.7. Let N ∈ N * and a, b, c ∈ L 2 N (B). There holds a + L 2 = a L 2 , (5.34) e ρ|∂x| P N (ab), c ≤ (e ρ|∂x| a + )(e ρ|∂x| b + ), c + . (5.35) Proof. Equality (5.34) is an immediate consequence of the denition (5.33) and Plancherel's theorem. Moreover, by Plancherel's theorem, the normalization (2.1), the triangle inequality and Plancherel's theorem once more, we have that e ρ|∂x| P N (ab), c = 1 2π y |ξ|≤N Fc(ξ)e ρ|ξ| η Fa(ξη)Fb(η) dη dξ dy ≤ 1 2π y ξ∈R |Fc(ξ)| η e ρ|ξ-η| |Fa(ξη)|e ρ|η| |Fb(η)| dη dξ dy = (e ρ|∂x| a + )(e ρ|∂x| b + ), c + .

( 5 .

 5 70) Thanks to the product estimate (5.35) from Lemma 5.7, the embedding estimate (5.21) from Lemma 5.4, the decomposition (5.14) of Ṡk -1 and estimate (5.16) from Lemma 5.2, and the denition (5.42) of 1 , we obtain

  estimates (5.73), (5.76) and (5.78) for the 8 pairs of indexes concludes the proof of Lemma 5.14 with the claimed constants. 5.6 Estimate of (r • ∇)r. Proof of Lemma 5.15 We prove Lemma 5.15, concerning the estimate of the trilinear term. First, using Bony's paraproduct decomposition (5.26) and the divergence free condition in (3.26), we write the quadratic term as (r • ∇)r = T r 1 ∂ x r + T ∂xr r 1 + ∂ x R(r 1 , r) + T r 2 ∂ y r + T ∂yr r 2 + ∂ y R(r 2 , r).

2 ≤ 10 2 ≤ 10 2 ≤ 10 2 ≤ 10 2 ≤ 10 2 ≤

 2102102102102102 the right hand side of the estimate given in Lemma 5.15. The proof of Lemma 5.15 is obtained by summation of the six following estimates:k∈Z t 0 ∆k e ρ|∂x|-β (T r 1 ∂ x r), ∆k r 1 ρ|∂x|-β (T ∂xr r 1 ), ∆k r 1 ρ|∂x|-β ∂ x R(r 1 , r), ∆k r 1 ρ|∂x|-β (T r 2 ∂ y r), ∆k r 1 e ρ|∂x|-β (T ∂yr r 2 ), ∆k r 1 e ρ|∂x|-β ∂ y R(r 2 , r), ∆k r 1 10 2 C 2 B E.

1

 1 Proof of(5.81) Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality (5.30), we expand ∆k e ρ|∂x| (T r 1 ∂ x r), ∆k r = |k -k|≤4 e ρ|∂x| ( Ṡk -1 r 1 )( ∆k ∂ x r), ∆k r .

( 5 .

 5 87)Thanks to the product estimate (5.35) from Lemma 5.7, we obtain| e ρ|∂x|-β ( Ṡk -1 r 1 )( ∆k ∂ x r), ∆k r | ≤ e β Ṡk -1 r + 1 L ∞ ∆k ∂ x r + L 2 ∆k r + L 2 .

( 5 .

 5 99) Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality (5.31), and the denition (5.29) we expand the term as ∆k e ρ|∂x| R(r 1 , r), ∂ x ∆k r = k ≥k-3 e ρ|∂x| ( ∆k r 1 )( ∆k r), ∆k ∂ x r = k ≥k-3 |k -k |≤1

5. 6 . 4

 64 Proof of(5.84) Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality (5.30), we expand ∆k e ρ|∂x| (T r 2 ∂ y r), ∆k r = |k -k|≤4 e ρ|∂x| ( Ṡk -1 r 2 )( ∆k ∂ y r), ∆k r .

( 5 .

 5 105)Thanks to the product estimate (5.35) from Lemma 5.7, we obtain

5. 6 . 5

 65 Proof of(5.85) Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality (5.30), we expand ∆k e ρ|∂x| (T ∂yr r 2 ), ∆k r = |k -k|≤4 e ρ|∂x| ( Ṡk -1 ∂ y r)( ∆k r 2 ), ∆k r .

( 5 .

 5 111)Thanks to the product estimate (5.35) from Lemma 5.7, we obtain

First, using integration

  by parts in the vertical direction, and the null boundary condition in (5.48), we get ∆k e ρ|∂x| ∂ y R(r 2 , r), ∆k r = -∆k e ρ|∂x| R(r 2 , r), ∆k ∂ y r (5.116) Using the support properties of the paraproduct decomposition as in Lemma 5.6, equality (5.31), and the denition (5.29) we expand the term as ∆k e ρ|∂x| R(r 2 , r), ∆k ∂ y r = k ≥k-3 e ρ|∂x| ( ∆k r 2 )( ∆k r), ∆k ∂ y r = k ≥k-3 |k -k |≤1 e ρ|∂x| ( ∆k r 2 )( ∆k r), ∆k ∂ y r (5.117) Thanks to the product estimate (5.35) from Lemma 5.7, we obtain

  .118) Thanks to estimates (5.23) from Lemma 5.5 and (5.16) from Lemma 5.2, we get

(6. 1 )( 6 . 2 )

 162 In particular, thanks to condition(1.11), one has G ρ b 3,2 (u b ) ≤ C b .We introduce the notationsu 1 i = ∇ ⊥ [χ δ ψ b ] and u 1 o := ∇ ⊥ [(1χ δ )ψ b ].Hence, recalling the translation notation (3.13), the denition (3.18) of u 1 can be recast asu 1 = βτ h u 1 i + τ h u 1 o .From the denition (3.17) of ψ b , the smoothness of χ δ and the conditions (1.11) we deduce that there exists C 1 such that,G ρ b 3,2 (u 1 i ) + G ρ b 3,2 (u 1 o ) ≤ C 1 .

(6. 3 )ρ 0 |∂x| ∆k a L 2 x 2 + 2 L p y dξ 1 2 ( 6 . 5 )|ξ| 1 2

 32222651 Thus, using the relation (6.2), for any t ≥ 0,G ρ b 3,2 (u 1 (t)) ≤ C 1 .(6.4)Let p ∈ {2, +∞}. Thanks to the Peter-Paul inequality, then using the property (5.11) and the support property (5.7) for the operator ∆k , k∈Z e |ξ| -1 2 )e 2ρ 0 |ξ| Fa(ξ)For low frequencies, using a uniform bound for the Fourier transform yields|ξ|≤1 (+ |ξ| -12 )e ρ 0 |ξ| Fa(ξ)

(6. 6 )

 6 For high frequencies, using the elementary inequality e x ≤ 2 cosh x, any ρ > ρ 0 , there exists C ρ > 0 such that,k∈Z e ρ 0 |∂x| ∆k a L 2 x (L p y ) ≤ C ρ a L 1 x (L p y ) + G ρ 1,p (a) .

(6. 8 ) 6 . 2 χVV≤( 6 . 9 )

 86269 Estimates for the amplication termsWe prove Proposition 5.9. Recalling the denition (5.43) of β, we proceed term by term.Recalling the denition (4.1) of the space H 1,1 (R + ) and using the decay estimate (4.2) for the boundary layer from Lemma 4.1 with n = 1, we get+∞ 0 {v 0 (s)} L ∞ (B) ds ≤ χ L ∞ (-1,1) +∞ 0 (s) L ∞ (R + ) ds ≤ 2 χ L ∞ (-1,1) +∞ 0 (s) H 1,1 (R + ) ds ≤ 2C χ L ∞ (-152C χ L ∞ (-1,1).Recalling the denition (3.22) of w ε and using estimate (4.11) from Lemma 4.3, ε

T /ε κ 0 1 α ε e ρ 0 |∂x| ∆k (εg) 2 L 2 (B) 1 2 ≤ ε 1 -κ k∈Z sup t≥0 e ρ 0

 22210 2 ((0, T /ε κ ) × B), thanks to the rst part of the denition (5.41) of α ε k∈Z |∂x| ∆k g(t) L 2 (B) .

(6. 13 )T /ε κ 0 1 α ε e ρ 0

 130 For g ∈ L 2 ((0, T /ε κ )×B) and V ∈ L 2 (R + ×R + ), thanks to the second part of the denition (5.41) of α and estimate (4.9) from Lemma 4.2, k∈Z |∂x| ∆k ({V}g) 2

e ρ 0 |∂x| ∆k ∆ 1 u

 1 (t) L 2 (B) < +∞.

ρ b 1 , 2

 12 and a, b ∈ L 2 (B), thanks the Leibniz dierentiation rule, (a)G ρ b 2,2 (b)≤ CG ρ b 1,2 (a)G ρ b 2,2 (b),(6.20)for some C > 0 independent of m because ρ < ρ b . Hence,k∈Z sup t≥0 e ρ 0 |∂x| ∆k (u 1 (t) • ∇)u 1 (t) L 2 (B) < +∞.

(6. 21 ) 1 4

 211 Third and fourth terms. We apply (6.13) to g := εW ε ∂ x u 1 + ε(u 1 • e y )∂ y w ε . Since W ε does not depend on x, we use the estimatee ρ 0 |∂x| ∆k g(t) L 2 (B) ≤ ε( W ε ∞ + ∂ y w ε ∞ )× ( e ρ 0 |∂x| ∆k ∂ x u 1 (t) L 2 (B) + e ρ 0 |∂x| ∆k u 1 (t) L 2 (B) ) ≤ ε C W ( e ρ 0 |∂x| ∆k ∂ x u 1 (t) L 2 (B) + e ρ 0 |∂x|∆k u 1 (t) L 2 (B) )(6.22)

  Assume that(3.11) holds for some n ≥ 3. There exists C W such that, for every ε ∈ (0, 1), the solution W ε to (3.21) satises, for every t ≥ 0,

				.10)
	Proceeding likewise for	1 3 ≤ y ≤ 1 and bounding	√	2 by 2 yields (4.9),
	4.3 Estimates for the technical prole
	Lemma 4.3.

  from Lemma 5.7, the embedding estimate (5.21) from Lemma 5.4, estimate (5.16) from Lemma 5.2, and the denition (5.42) of 1 , we infer

  1 . For t ≥ 0, thanks to (6.2),e ρ 0 |∂x| ∆k ∆u 1 (t) L 2 (B) = e ρ 0 |∂x| ∆k ∆(β(t)(τ h u 1 i )(t) + (τ h u 1 o )(t)) L 2 (B) = e ρ 0 |∂x| ∆k ∆(β(t)u 1 i + u 1 o ) L 2 (B) ≤ e ρ 0 |∂x|∆k ∆u 1 i L 2 (B) + e ρ 0 |∂x| ∆k ∆u 1

			(6.16)
			o L 2 (B)
	Thanks to (6.8), (6.3) and (1.12), we deduce that
	e ρ 0 |∂x|	∆k ∆u 1 i L 2 (B) + e ρ 0 |∂x|	∆k ∆u 1 o L 2 (B) < +∞.
	k∈N

Acknowledgements

The authors warmly thank Jean-Pierre Puel for his advice concerning the local controllability of the Navier-Stokes equations, and Marius Paicu for his insight on analytic regularization properties for Navier-Stokes systems. The authors also thank an anonymous referee for suggesting Remark 1.12. J.-M. Coron and F. Marbach were supported by ERC Advanced Grant 266907 (CP-DENL) of the 7th Research Framework Programme (FP7). J.-M. Coron also benets from the supports of ANR project Finite4SoS (ANR 15 CE23 0007) and Laboratoire sinofrançais en mathématiques appliquées (LIASFMA). F. Sueur was supported by the Agence Nationale de la Recherche, Project DYFICOLTI, grant ANR-13-BS01-0003-01, Project IFSMACS, grant ANR-15-CE40-0010 and Project BORDS, grant ANR-16-CE40-0027-01, the Conseil Régional d'Aquitaine, grant 2015.1047.CP, and by the H2020-MSCA-ITN-2017 program, Project ConFlex, Grant ETN-765579. P. Zhang is partially supported by NSF of China under Grant 11371347. F. Sueur warmly thanks Beijing's Morningside center for Mathematics for its kind hospitality a stay in September 2017. F. Marbach and F. Sueur are grateful to ETH-ITS and ETH Zürich for their kind hospitality during two stays respectively in October and November 2017.

Remark 5.1. The well-posedness of the Prandtl equations as well as the convergence of the Navier-Stokes equations to the Prandtl equations in the analytic setting dates back to [START_REF] Sammartino | Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations[END_REF][START_REF] Sammartino | Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution[END_REF][START_REF] Maria | Well-posedness of the boundary layer equations[END_REF]. The seminal results of Caisch and Sammartino require analyticity in both spatial directions, and only imply well-posedness of the Prandtl equations on a small time interval. Analytic techniques have been later used in [START_REF] Ignatova | Almost global existence for the Prandtl boundary layer equations[END_REF][START_REF] Zhang | Long time well-posedness of Prandtl system with small and analytic initial data[END_REF] to obtain large-time wellposedness for Prandtl equations by requiring analyticity only in the tangential direction.

Singular amplication to loss of derivative

On the one hand, we use the expansion (3.4)

(5.1)

Let M be the operator which associates with any function a ∈ L 2 (-1, 1), the function M [a] dened for y in (-1, 1) by

where the signs are chosen depending on whether ±y ≥ 0. Using the null boundary condition and the divergence-free condition in (3.26) and the fact that |ϕ | = 1 where χ = 0, we obtain that the rst term in the right-hand side of (5.1) can be recast as

On the other hand we decompose the term (u ε app • ∇)r ε of (3.26), thanks to (3.4), into

(5.4) Thus, using (5.3) and (5.4), the system (3.26) now reads

where we introduce

(5.6)

A few tools from Littlewood-Paley theory

To perform analytic estimates, we use Fourier analysis and Littlewood-Paley decomposition. We refer to [START_REF] Bahouri | Fourier analysis and nonlinear partial dierential equations[END_REF]Chapter 2] for a detailed course on Littlewood-Paley theory. Although all the functions we consider in this section are dened on the band B = R x × [-1, 1] y , we only perform Fourier analysis and Littlewood-Paley decomposition in the tangential direction x ∈ R x . When a confusion is possible, we will use the subscripts x or y to stress the variable involved in the functional spaces.

Recalling the denition (4.1) of the space H 2,2 (R + ), one has, for t ≥ 0,

(5.40) Hence, since n ≥ 2, thanks to the decay estimate (4.2) from Lemma 4.1, ρ 0 < +∞. Up to a normalization constant due to Bernstein-type estimates, this radius corresponds to the total amount of the loss of derivative that we expect. Then, we set, for t ≥ 0,

(5.41)

∆k ∇u 1 (t) L 2 (B) .

(5.42)

These quantities will help us to control the (non singular but long-time) amplication terms in the evolution of the remainder. We set

(5.43) Proposition 5.9 (Proof in Section 6.2). If u b satises (1.11) for a constant C b > 0 and ρ b > ρ 0 , there exists β > 0 such that, for ε, κ ∈ (0, 1),

(5.44)

We consider the local solution ρ N (t) to the following nonlinear ODE:

(5.45) Since, for almost every t, r ε N (t) ∈ L 2 N (B), the right-hand side is Lipschitz continuous with respect to ρ N (with constants that may depend on N ). Hence, we can apply the Cauchy-Lipschitz theorem and consider the maximal solution of (5.45). We set (5.47)

In the sequel, we simply write ρ instead of ρ N and T * instead of T * N and we prove estimates which are uniform with respect to N .

Grönwall-type energy estimate

We start with deducing from (5.37) that: