Mehdi Badra
email: mehdi.badra@univ-pau.fr

Sylvain Ervedoza
email: sylvain.ervedoza@math.univ-toulouse.fr.

Sergio Guerrero
email: guerrero@ann.jussieu.fr

Local controllability to trajectories for non-homogeneous 2-d incompressible Navier-Stokes equations *

Keywords:

The goal of this article is to show a local exact controllability to smooth (C 2 ) trajectories for the 2-d density dependent incompressible Navier-Stokes equations. Our controllability result requires some geometric condition on the flow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the flow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations.

Introduction

The goal of this article is to discuss the local exact controllability property for the 2-d non-homogeneous Navier Stokes equations.

Setting and main results. Let Ω be a smooth bounded domain of R 2 , T > 0 and denote (0, T ) × Ω by ΩT . Let us consider a trajectory (σ, y) of the non-homogeneous Navier-Stokes equations:        ∂tσ + div(σ y) = f σ in ΩT , σ∂ty + σ(y • ∇)y -ν∆y + ∇q = f y in ΩT , div y = 0 in ΩT , (σ(0), y(0)) = (σ0, y 0 ) in Ω.

(1.1) Here, ν > 0 is the viscosity parameter and the source terms (f σ , f y) are assumed to be known. We will focus on the local exact controllability problem around the trajectory (σ, y) with a control exerted on the boundary (0, T ) × ∂Ω: Given (σ0, y0) close to the initial data (σ0, y 0 ), find control functions (hσ, hy) on (0, T ) × ∂Ω such that the solution (σ, y) of        ∂tσ + div(σy) = f σ in ΩT , σ∂ty + σ(y • ∇)y -ν∆y + ∇q = f y in ΩT , div y = 0 in ΩT , (σ(0), y(0)) = (σ0 + ρ0, y 0 + u0), in Ω, (1.2) with the boundary conditions: σ = σ + hσ for (t, x) ∈ (0, T ) × ∂Ω, with y(t, x) • n(x) < 0, (1.3)

y = y + hy on (0, T ) × ∂Ω, (1.4)
satisfies (σ(T ), y(T )) = (σ(T ), y(T )).

(1.5)

Our goal is to present a positive answer to this control problem under suitable assumptions on the target trajectory (σ, y), and in particular one of hyperbolic nature on the flow corresponding to y. Besides, our strategy will yield a control acting on some suitable subsets of the boundary which correspond, roughly speaking, to the complement of the part of the boundary in which the scalar product of the target velocity y with the normal vector n is positive for all time t ∈ [0, T ]. Going further requires some notations. We denote by L 2 (Ω), L ∞ (Ω), H r (Ω), H r 0 (Ω) etc for r ≥ 0, the usual Lebesgue and Sobolev spaces of scalar functions, and we write in bold the spaces of vector-valued functions: L 2 (Ω) = (L 2 (Ω)) 2 , H r (Ω) = (H r (Ω)) 2 , etc. We also define V 1 0 (Ω) :

def = {v ∈ H 1 0 (Ω) | div v = 0 in Ω}.
In the following, we will always assume that the target velocity y belongs to C 2 (ΩT ). It can thus be extended into a C 2 ([0, T ] × R 2 ) function, still denoted the same for simplicity but not necessarily divergence free outside ΩT . This allows to define the flow X = X(t, τ, x) associated to that velocity y: ∀(t, τ, x) ∈ [0, T ] 2 × R 2 , ∂tX(t, τ, x) = y(t, X(t, τ, x)), X(τ, τ, x) = x.

(1.6)

Thus we define the outgoing subset of Ω for the flow X as follows:

Ω T out : def = x ∈ Ω | ∃t ∈ (0, T ) s.t. X(t, 0, x) ∈ R 2 \Ω .

(1.7)

One of our main assumptions is the following one:

Ω = Ω T out . (1.8)
Note that this assumption does not depend on the extension y on [0, T ]×R 2 and is intrinsic.

This assumption is of hyperbolic nature as it requires the time T to be large enough to guarantee that all the particles that were in Ω at time t = 0 have been transported by the flow outside Ω in a time strictly smaller than T . Of course, this is remanent from the density equation (1.2) [START_REF] Albano | Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system[END_REF] in which the density is transported along the flow corresponding to the velocity of the fluid.

As we said, we will not require the control to be supported on the whole boundary (0, T )× ∂Ω, but only on some part of it (0, T ) × Γc where Γc = ∂Ω\Γ0 and Γ0 (the part without control) is an open subset of ∂Ω satisfying the following conditions:

(i). Γ0 has a finite number of connected components, (ii). sup

[0,T ]×Γ 0 y • n > 0.

(1.9) Note that the above condition garantees the existence of γ > 0 such that y(t, x) • n(x) ≥ γ for all (t, x) ∈ (0, T ) × Γ0.

Our main result states as follows:

Theorem 1.1. Let Ω be a smooth bounded domain of R 2 . Assume that the target trajectory (σ, y) solution of (1.1) satisfies

(σ, y) ∈ C 2 ([0, T ] × Ω) × C 2 ([0, T ] × Ω) and inf [0,T ]×Ω σ > 0.
(1.10)

Assume that the condition (1.8) is satisfied for the time T . Then there exists ε > 0 such that for all (ρ0, u0)

∈ L ∞ (Ω) × V 1 0 (Ω) satisfying ρ0 L ∞ (Ω) + u0 H 1 0 (Ω) ≤ ε, (1.11)
there exists a controlled trajectory

(σ, y) ∈ L ∞ (ΩT ) × H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω))
solution of (1.2)-(1.4) satisfying the control requirement (1.5).

Besides, if Γ0 denotes an open subset of the boundary satisfying (1.9), we may further impose y = y on (0, T ) × Γ0. In particular, in that case, no boundary condition is imposed on the density on Γ0.

Actually, we will only prove Theorem 1.1 when Γ0 = ∅. When Γ0 = ∅, Theorem 1.1 can be proved more easily following the same lines, as the extensions arguments we will perform can be handled much more easily.

Strategy of the proof. The proof of Theorem 1.1 is based on a technical fixed-point procedure, and we briefly explain below its general strategy. Setting ρ :

def = σ -σ, u : def = y -y, (1.12)
and f (ρ, u) : in ΩT , div u = 0 in ΩT , (ρ(0), u(0)) = (ρ0, u0)

in Ω, (1.14) with the boundary conditions u = 0 on (0, T ) × Γ0, (1.15) and with the requirement (ρ(T ), u(T )) = (0, 0) in Ω. (1.16) To construct a solution of (1.14)- (1.16), the strategy consists in finding a fixed-point to some mapping F (ρ 0 ,u 0 ) : u → u defined in such a way that u = F (ρ 0 ,u 0 ) ( u) is a suitable solution of:

               ∂tρ + (y + u) • ∇ρ = -u • ∇σ in ΩT , σ∂tu + σ(y • ∇)u + σ(u • ∇)y -ν∆u + ∇p = f (ρ, u)
in ΩT , div u = 0 in ΩT , u = 0 on (0, T ) × Γ0, (ρ(0), u(0)) = (ρ0, u0) in Ω, (ρ(T ), u(T )) = (0, 0)

in Ω.

(1.17)

The mapping F (ρ 0 ,u 0 ) is defined in two steps. First, for a given u, we define F1( u, ρ0) : def = ρ, where ρ will be constructed as a suitable solution of the following control problem for the equation of the density:

   ∂tρ + (y + u) • ∇ρ = -u • ∇σ in ΩT , ρ(0) = ρ0
in Ω, ρ(T ) = 0 in Ω.

(1.18)

Then, we define F2(f , u0) : def = u, where u is a suitable solution of the following control problem for the equation of the velocity:

           σ∂tu + σ(y • ∇)u + σ(u • ∇)y -ν∆u + ∇p = f in ΩT , div u = 0 in ΩT , u = 0 on (0, T ) × Γ0, u(0) = u0
in Ω, u(T ) = 0 in Ω.

(1. [START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF] The mapping F (ρ 0 ,u 0 ) is then defined as follows:

F (ρ 0 ,u 0 ) ( u) : def = u,
where ρ = F1( u, ρ0), and u = F2(f (ρ, u), u0).

(1.20)

Hence our strategy decouples the control problem (1.2)-(1.5) into two control problems, (1.18) for the equation of the density, and (1. [START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF] for the equation of the velocity, each of which having different behaviors. Indeed, on one hand, the control problem (1. [START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF]) is of parabolic type, and it will be handled by using global Carleman estimates following the general approach of Fursikov and Imanuvilov [START_REF] Fursikov | Controllability of evolution equations[END_REF] for the heat equations: in the case of Navier-Stokes equations, this approach has already been successfully implemented in the works [START_REF] Imanuvilov | Remarks on exact controllability for the Navier-Stokes equations[END_REF][START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF].

On the other hand, the control problem (1.18) involves a transport equation. This can be easily controlled provided the time T > 0 is large enough to allow all the particles in Ω to go outside the domain, i.e. when condition (1.8) is satisfied.

But the problem is that we want the above mapping F (ρ 0 ,u 0 ) to map some convex set into itself. In order to do this, we should be able to get estimates on the above control problems in spaces that behave suitably with respect to both of them. In particular, this will lead us to introduce Carleman weights that follow the dynamics of the transport equation, that is weight functions which are transported by the flow. This strategy then follows the one recently developed in [START_REF] Ervedoza | Local exact controllability for the one-dimensional compressible Navier-Stokes equation[END_REF] for deriving local exact controllability results for the 1d compressible Navier-Stokes equations around constant non-vanishing velocities.

Actually, the Carleman estimates we develop in this article also present the feature of not vanishing at time t = 0. This allows us to construct a solution (ρ, u) of (1.14) without using any property of the Cauchy problem for the non-homogeneous Navier-Stokes equations.

Related references and comments. To our knowledge, control properties for nonhomogeneous Navier-Stokes equations have only been studied in [START_REF] Fernández-Cara | Motivation, analysis and control of the variable density Navier-Stokes equations[END_REF], which proves several optimal control results in that context for various cost functions.

For the homogeneous Navier-Stokes equations, the density is assumed to be constant and thus the equations reduce to the equations on the velocity. In that case, several local exact controllability results have been established in [START_REF] Imanuvilov | Remarks on exact controllability for the Navier-Stokes equations[END_REF][START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF] based on parabolic Carleman estimates, see e.g. [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]. Later on, several different strategies have been proposed, see for instance [START_REF] Fernández-Cara | Some controllability results for the N -dimensional Navier-Stokes and Boussinesq systems with N -1 scalar controls[END_REF][START_REF] González-Burgos | Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation[END_REF][START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF]. We also point out that these results also use the Carleman estimate derived in [START_REF] Imanuvilov | Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems[END_REF] for non-homogeneous elliptic problems in order to handle the pressure term.

But our problem also involves some transport phenomenon, and therefore also shares some features of the thermoelasticity equations [START_REF] Albano | Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system[END_REF], the viscoelasticity models [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF][START_REF] Chaves-Silva | Null controllability of a system of viscoelasticity with a moving control[END_REF], and the compressible Navier-Stokes equations [START_REF] Ervedoza | Local exact controllability for the one-dimensional compressible Navier-Stokes equation[END_REF]. Our approach is actually close to the one developed in [START_REF] Ervedoza | Local exact controllability for the one-dimensional compressible Navier-Stokes equation[END_REF]. Though, the divergence free condition in the model we consider here requires a specific treatment.

In this article, we will not use any result on the Cauchy problem for (1.2), as our strategy will automatically construct a trajectory (σ, y) solving the equations (1.2). However, several results are available in the literature. We refer to the work [START_REF] Fernández-Cara | Motivation, analysis and control of the variable density Navier-Stokes equations[END_REF] for several results and comments on the Cauchy problem for the non-homogeneous incompressible Navier-Stokes equations and to the references therein.

Let us also note that we will need a precise understanding of the transport equation when transported by a flow entering the domain. More precisely, we will use in an essential way the compactness result in [START_REF] Boyer | Outflow boundary conditions for the incompressible nonhomogeneous Navier-Stokes equations[END_REF]Theorem 4], obtained as a consequence of [START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF].

We also underline that Theorem 1.1 does not state the uniqueness of the controlled trajectory (σ, y). This is due to the lack of regularity for the density σ which only belongs to L ∞ (ΩT ), see [START_REF] Desjardins | Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations[END_REF] for the uncontrolled case. Another limitation of this control result is that it is only valid for 2-d geometry. This restriction comes from the treatment of the velocity equation. For that, we prove a new Carleman inequality for the Stokes equations which hardly relies on the use of the stream function of the velocity, see Section 2.

Finally note that our result also allows the use of non-trivial trajectories. For instance, if Γ0 = ∅ and (σ, y) = (1, 0), one may consider the trajectory (σ * (t), y * (t)) = (1, η(t/T )U) for constant vector fields U and η = η(t) ∈ [0, 1] a bump function taking value 0 at t = 0 and t = 1 and with η = 1 on [1/3, 2/3]. Note that (σ * (t), y * (t)) = (1, 0) at time t = 0 and at time t = T . But for T > 0 and large U, (σ * (t), y * (t)) satisfies (1.8) and all the assumptions of Theorem 1.1, while whatever the time T > 0 is, the trajectory (σ(t), y(t)) = (1, 0) clearly does not satisfy (1.8). This suggests that the geometric condition (1.8) may be avoided in some cases using "return method" type ideas, see e.g. [START_REF] Coron | On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions[END_REF][START_REF] Coron | Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary[END_REF].

Outline. This article is organized as follows. Section 2 explains how to solve the control problem (1.19) by the use of Carleman estimates for the Stokes operator. Section 3 shows how to construct a controlled density satisfying (1.18) and to derive weighted estimates on it. Section 4 then focuses on the proof of Theorem 1.1 by putting together the arguments developed in Sections 2 and 3. Finally, the Appendix gives the detailed proofs of some technical results.

Controlling the velocity

This section is dedicated to the construction of a solution of (1.19).

Statement of the result

In order to solve the control problem (1.19), we will consider (1.19) in an extended domain O as follows:

O is a smooth bounded domain of R 2 satisfying Ω ⊂ O, ∂O is of class C 2 , ∂O ∩ ∂Ω ⊃ Γ0.
(2.1)

We then extend (σ, y) on [0, T ] × O, still denoted the same for simplicity, such that

(σ, y) ∈ C 2 ([0, T ] × O) × C 2 ([0, T ] × O) and inf [0,T ]×O σ(t, x) > 0. (2.2)
Remark that this is possible due to the assumption (1.10). As u0 ∈ V 1 0 (Ω), extending it by zero outside Ω, we get an extension, still denoted the same, such that

u0 ∈ H 1 0 (O) and div u0 = 0 in O. (2.3)
By also extending f by zero outside Ω and setting OT = (0, T ) × O, ΓT = (0, T ) × ∂O we then consider the following system

       σ(∂tu + (y • ∇)u + (u • ∇)y) -ν∆u + ∇p = f + h1 O\Ω in OT , div u = 0 in OT , u = 0 on ΓT , u(0) = u0 in O.
(2.4)

Here, 1 O\Ω is the characteristic function of O \ Ω and h ∈ L 2 (OT ) is a control function.

Note that the presence of 1 O\Ω in (2.4) implies that the action of the control is supported in O \ Ω. We thus intend to solve the following control problem: Given u0 ∈ H 1 0 (O) satisfying (2.3) and a source term f in some suitable space, find a control function h ∈ L 2 (OT ) such that the solution u of (2.4) satisfies u(T ) = 0 in O.

(2.5)

Indeed, if we are able to solve this control problem, the restriction of the solution u to Ω would yield a solution of the control problem (1.19). In order to solve the control problem (2.4)-(2.5), as it is classical by now, we are going to establish a suitable observability estimate for the adjoint problem

   -∂t(σv) -D(σv)y -σv div y -ν∆v + ∇p = g in OT , div v = 0 in OT , v = 0 on ΓT , (2.6)
where Dv := ∇v + t ∇v is the symmetrized gradient.

To state our result precisely, let us introduce the weight functions we will use in the Carleman estimate. We assume that we have a function ψ = ψ(t, x) ∈ C 2 (OT ) such that ψ :

def = ψ(t, x) such that        ∀(t, x) ∈ OT , ψ(t, x) ∈ [0, 1], ∀(t, x) ∈ ΓT , ∂n ψ(t, x) ≤ 0, ∀t ∈ [0, T ], ψ(t) |∂O is constant, ∀t ∈ [0, T ], infO ψ(t, •) = ψ(t) |∂O .
(2.7)

We also assume the existence of two open subsets ωT ⋐ ωT of [0, T ] × (O \ Ω) (here and in the following, the symbol ⋐ means that there exists a compact set

KT of [0, T ] × (O \ Ω) such that ωT ⊂ KT ⊂ ωT ) and a constant α > 0 such that inf O T \ ωT {|∇ ψ|} ≥ α > 0. (2.8)
For m ≥ 1, we set ψ(t, x) :

def = ψ(t, x) + 6m. (2.9)
We then set T0 > 0 and T1 > 0 such that T1 ≤ 1/4 and T0 + 2T1 < T and choose a weight function in time θm,µ(t) depending on the parameters m ≥ 1 and µ ≥ 2 defined by θm,µ :

def = θm,µ(t) such that                        ∀t ∈ [0, T0], θm,µ(t) = 1 + 1 - t T0 µ , ∀t ∈ [T0, T -2T1], θm,µ(t) = 1, ∀t ∈ [T -T1, T ), θm,µ(t) = 1 (T -t) m , θm,µ is increasing on [T -2T1, T -T1], θm,µ ∈ C 2 ([0, T )).
(2.10) For simplicity of notations in the following we omit the dependence on m and µ and we simply write θ : def = θm,µ. We will then take the following weight functions ϕ = ϕ(t, x) and ξ = ξ(t, x):

ϕ(t, x) : def = θ(t) λe 6λ(m+1) -exp(λψ(t, x)) , ξ(t, x) : def = θ(t) exp(λψ(t, x)), (2.11)
where s, λ are positive parameters with s ≥ 1, λ ≥ 1 and µ is chosen as

µ = sλ 2 e λ(6m-4) , (2.12)
which is always bigger than 2, thus being compatible with the condition θ ∈ C 2 ([0, T ]). Note that the weight functions ϕ and ξ, depend on s, λ, m, and should rather be denoted by ϕ s,λ,m , resp. ξ s,λ,m , but we drop these indexes for simplicity of notations. Remark that, due to the definition of ψ in (2.9) and the conditions (2.7), we have, for all λ ≥ 1 and (t, x) ∈ OT , Then, for m ≥ 5, there exist some constants s0 ≥ 1, λ0 ≥ 1 and C > 0 such that for all smooth solution v of (2.6) with source term g ∈ L 2 (OT ), for all s ≥ s0 and λ ≥ λ0,

3 4 θ(t)λe 6λ(m+1) ≤ ϕ(t, x) ≤ θ(t)λe 6λ(m+1
s 1/2 λ -1/2 O (ξ * ) 4-2/m |v(0, •)| 2 e -2sϕ * (0) + sλ 2 O T ξ 4 |v| 2 e -2sϕ +s -1 O T ξ 2 |∇v| 2 e -2sϕ + s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * v 2 H 1 (O) ≤ C s 5/2 λ 2 ω T ξ 6 |v| 2 e 2sϕ * -4s ϕ + s 1/2 λ -1/2 O T (ξ) 4-2/m |g| 2 e -2sϕ .
(2.16)

The proof of Theorem 2.1 is done in Sections 2.2 and 2.3. We are first going to prove a slightly improved version of the Carleman estimates (2.16) for solutions v of the simplified version of the adjoint problem (2.6):

   -σ∂tv -ν∆v + ∇p = g in OT , div v = 0 in OT , v = 0 on ΓT .
(2.17)

Our approach then consists first in taking the curl of the equation (2.17) and consider the equation of w = rot v: Among the new features of the Carleman estimate of Theorem 2.1 with respect to those in the literature, let us point out the following facts:

-σ∂tw -ν∆w = rot g + ∂tv • ∇ ⊥ σ in OT . ( 2
• The weight function in time θm,µ in (2.10) does not blow up as the time t goes to 0. However, our proof requires a strong convexity property close to t = 0, tuned by the choice of the parameter µ in (2.10) as a suitable function of the parameters s and λ, see (2.12).

• The weight function ψ depends on both the time and space variables. As we shall explain, this is not a big issue as long as we guarantee that for all t ∈ [0, T ], ψ(t) is constant on the boundary ∂O, thus allowing to apply the Carleman inequality of [START_REF] Imanuvilov | Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems[END_REF] for elliptic equations.

Based on Theorem 2.1, following standard duality arguments, we prove the following control result: Theorem 2.2. Within the setting and assumptions of Theorem 2.1, there exists a constant C > 0 such that for all s ≥ s0 and λ ≥ λ0, if u0 verifies (2.3) and f ∈ L 2 (OT ) satisfies

O T ξ -4 |f | 2 e 2sϕ < ∞, (2.21)
there exists a control function h ∈ L 2 (OT ) supported in ωT and a controlled trajectory u ∈ L 2 (OT ) such that u solves the control problem (2.4)-(2.5) and (u, h) satisfies the estimate

e 3 4 sϕ * u 2 L 2 (H 2 )∩H 1 (L 2 ) + s 1/2 λ 5/2 O T ξ 2/m-4 |u| 2 e 2sϕ + s -3/2 ω T ξ -6 |h| 2 e 4s ϕ-2sϕ * ≤ C O T ξ -4 |f | 2 e 2sϕ + e 5 2 sϕ * (0,•) u0 2 H 1 0 (O) . (2.22)
The proof of Theorem 2.2 is given in Section 2.4.

Carleman estimates for the heat equation

The goal of this section is to show the following estimate: For all M > 0, there exist constants C > 0, s0 and λ0 such that for all s ≥ s0 and λ ≥ λ0, for all smooth functions w in OT , such that

-σ∂tw -ν∆w = a0w + A1 • ∇w + g0 + n i=1 bi∂igi + bn+1∂tgn+1 in OT , with a0 ∈ L ∞ (OT ), A1 ∈ L ∞ (0, T ; W 1,∞ (O)), g0, gi ∈ L 2 (OT ), and coefficients bi ∈ L ∞ (0, T ; W 1,∞ (O)), bn+1 ∈ W 1,∞ (0, T ; L ∞ (O)) satisfying a0 L ∞ (O T ) + A1 L ∞ (0,T ;W 1,∞ (O)) + n i=1 bi L ∞ (0,T ;W 1,∞ (O)) + bn+1 W 1,∞ (0,T ;L ∞ (O)) ≤ M, (2.23)
we have This proof is inspired by the ones in [START_REF] Imanuvilov | Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations[END_REF], see also [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and applications to controllability[END_REF]. Below, we only state Theorems 2.4-2.5, whose proofs are postponed to the appendix.

s 3 λ 4 O T ξ 3 |w| 2 e -2sϕ ≤ C O T |g0| 2 e -2sϕ
Proof of Theorem 2.3. As said above, the proof is done in three steps. An L 2 -Carleman estimate. The first result is the following L 2 -Carleman estimate for the heat equation: Theorem 2.4. Assume the setting of Theorem 2.3. For all m ≥ 1, there exist constants C0 > 0, s0 ≥ 1 and λ0 ≥ 1 such that for all smooth functions z on OT satisfying z = 0 on ΓT , for all s ≥ s0, λ ≥ λ0, we have

O |∇z(0)| 2 e -2sϕ(0) + s 2 λ 3 e 2λ(6m+1) O |z(0)| 2 e -2sϕ(0) + sλ 2 O T ξ|∇z| 2 e -2sϕ + s 3 λ 4 O T ξ 3 |z| 2 e -2sϕ ≤ C0 O T |(-σ∂t -ν∆)z| 2 e -2sϕ + C0s 3 λ 4 ω T ξ 3 |z| 2 e -2sϕ . (2.25)
The proof of Theorem 2.4 is given in Section A.1. It is rather classical except for the weight function ϕ, which does not blow up as t → 0 and for the weight function ψ which depends on both time and space variables. This introduces in the proof of Theorem 2.4 several new technical issues, though our proof follows the lines of [START_REF] Fursikov | Controllability of evolution equations[END_REF].

Estimates on a control problem. We then analyze the following control problem:

for f ∈ L 2 (OT ), find a control function h ∈ L 2 ( ωT ) such that the solution y of    ∂t(σy) -ν∆y = f + h1 ω T , in OT , y = 0, on ΓT , y(0, •) = 0, in O, (2.26)
solves the control problem:

y(T, •) = 0, in O. (2.27)
We claim the following result:

Theorem 2.5. Assume the setting of Theorem 2.3. For all m ≥ 1, there exist positive constants C > 0, s0 ≥ 1 and λ0 ≥ 1 such that for all s ≥ s0 and λ ≥ λ0, for all f satisfying

O T ξ -3 |f | 2 e 2sϕ < ∞, (2.28)
there exists a solution (Y, H) of the control problem (2.26)-(2.27) which furthermore satisfies the following estimate:

s 3 λ 4 O T |Y | 2 e 2sϕ + ω T ξ -3 |H| 2 e 2sϕ + sλ 2 O T ξ -2 |∇Y | 2 e 2sϕ + 1 s O T ξ -4 (|∂tY | 2 + |∆Y | 2 )e 2sϕ + λ Γ T ξ -3 |∂nY | 2 e 2sϕ ≤ C O T ξ -3 |f | 2 e 2sϕ . (2.29)
The proof of Theorem 2.5 is given in Section A.2. Again, the proof is rather classical and is based on the duality between the Carleman estimates, which are weighted observability estimates, and controllability, and then on energy estimates. Note however that these energy estimates have to be derived using the weight functions defined in (2.7)- (2.11), and this introduces some novelties in the computations.

A duality argument.

O T w(f + H1 ω T ) + Γ T wν∂nY = O T (a0wY -w div (A1Y ) + g0Y - n i=1 gi∂i(biY ) -gn+1∂t(bn+1Y )). (2.30)
In particular, as f = ξ 3 we -2sϕ satisfies

O T ξ -3 |f | 2 e 2sϕ = O T ξ 3 |w| 2 e -2sϕ , according to (2.29) we can construct (Y, H) solution of        ∂t(σY ) -ν∆Y = ξ 3 we -2sϕ + H1 ω T , in OT , Y = 0, on ΓT , Y (0, •) = 0, in O, Y (T, •) = 0, in O, (2.31)
for which we have the estimate:

s 3 λ 4 O T |Y | 2 e 2sϕ + ω T ξ -3 |H| 2 e 2sϕ + sλ 2 O T ξ -2 |∇Y | 2 e 2sϕ + 1 s O T ξ -4 (|∂tY | 2 + |∇Y | 2 )e 2sϕ + λ Γ T ξ -3 |∂nY | 2 e 2sϕ ≤ C O T ξ 3 |w| 2 e -2sϕ . (2.32)
Using then the identity (2.30), we infer

O T ξ 3 |w| 2 e -2sϕ ≤ C 1 sλ 2 O T ξ 2 |w| 2 e -2sϕ 1/2 sλ 2 O T ξ -2 (|Y | 2 + |∇Y | 2 )e 2sϕ 1/2 + C 1 s 3 λ 4 O T |g0| 2 e -2sϕ 1/2 s 3 λ 4 O T |Y | 2 e 2sϕ 1/2 + C 1 sλ 2 O T ξ 2 n i=1 |gi| 2 e -2sϕ 1/2 sλ 2 O T ξ -2 (|Y | 2 + |∇Y | 2 )e 2sϕ 1/2 + C s O T ξ 4 |gt| 2 e -2sϕ 1/2 1 s O T ξ -4 (|Y | 2 + |∂tY | 2 )e 2sϕ 1/2 + C 1 λ Γ T ξ 3 |w| 2 e -2sϕ 1/2 λ Γ T ξ -3 |∂nY | 2 e 2sϕ 1/2 + C ω T ξ 3 |w| 2 e -2sϕ 1/2 ω T ξ -3 |H| 2 e 2sϕ 1/2
, which immediately yields the claimed result by (2.32).

Proof of Theorem 2.1

This section aims at proving Theorem 2.1. This will be done in two steps. We first prove the following Carleman estimate for v solution of (2.17):

Theorem 2.6. Within the setting and assumptions of Theorem 2.1, for any m ≥ 5, there exist some constants s0 ≥ 1, λ0 ≥ 1 and C > 0 such that for all solution v of (2.17) with source term g ∈ L 2 (OT ), for all s ≥ s0 and λ ≥ λ0,

s 1/2 λ -1/2 O (ξ * ) 4-2/m |v(0, •)| 2 e -2sϕ * (0) + sλ 2 O T ξ 4 |v| 2 e -2sϕ +s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * v 2 H 1 (O) + O T ξ 3 | rot v| 2 e -2sϕ +s -1 O T ξ 2 |∇v| 2 e -2sϕ ≤ C s 5/2 λ 2 ω T ξ 6 |v| 2 e 2sϕ * -4s ϕ + s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ + s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * g 2 H -1 (O) +s -1/2 λ -3/2 O T (ξ * ) 3-3/m |g| 2 e -2sϕ * .
(2.33)

The proof of Theorem 2.6 is done below in Section 2.3.1. In Section 2.3.2 we then explain how Theorem 2.6 implies Theorem 2.1.

Proof of Theorem 2.6

Let v be a solution of (2.17) with source term g. As w = rot v satisfies (2.18), the Carleman estimate (2.24) applies to w: for all s ≥ s0 and λ ≥ λ0,

O T ξ 3 |w| 2 e -2sϕ ≤ C ω T ξ 3 |w| 2 e -2sϕ + s O T ξ 4 |v| 2 e -2sϕ +λ -1 Γ T ξ 3 |w| 2 e -2sϕ + s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ . (2.34)
Here and in the following ωT = [0, T ] × ω where ω is a Lipschitz subdomain O\Ω such that ω ⋐ ω ⋐ ω. Note in particular that ωT ⋐ ωT ⋐ ωT .

Next, because v is divergence free we also have, for all t ∈ (0, T ),

-∆v(t) = rot w(t) in O, v(t) = 0 on ∂O. (2.35)
Thus, using elliptic Carleman estimates with source term in H -1 (O) with weight e -sϕ(t,•) and integrating in time, see [START_REF] Imanuvilov | Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems[END_REF], we immediately get

s -1 O T ξ 2 |∇v| 2 e -2sϕ + sλ 2 O T ξ 4 |v| 2 e -2sϕ ≤ C O T ξ 3 |w| 2 e -2sϕ + sλ 2 ω T ξ 4 |v| 2 e -2sϕ . (2.36)
Combined with (2.34), and using the fact that w = rot v is bounded by ∂nv on ΓT (recall that v = 0 on ΓT ) and that ξ * = ξ and ϕ * = ϕ on (0, T ) × ∂O, we immediately have that for some s0 > 1 and λ0 > 1, for all s ≥ s0 and λ ≥ λ0,

s -1 O T ξ 2 |∇v| 2 e -2sϕ + O T ξ 3 |w| 2 e -2sϕ + sλ 2 O T ξ 4 |v| 2 e -2sϕ ≤ C ω T ξ 3 |w| 2 e -2sϕ + sλ 2 ω T ξ 4 |v| 2 e -2sϕ +λ -1 Γ T (ξ * ) 3 |∂nv| 2 e -2sϕ * + s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ . (2.37)
We then introduce the stream function ζ associated to v, i.e. v = ∇ ⊥ ζ, which can be computed explicitly as the solution of (2.19) for some constants ci(t) due to the dimension N = 2, see e.g. [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]Corollary 3.1]. Note that, by adding a constant to ζ if necessary, without loss of generality we can assume that (2.20) is also satisfied. Applying the elliptic Carleman estimate to the equation (2.19) (see e.g. [START_REF] Fursikov | Controllability of evolution equations[END_REF]), we obtain that

s 3 λ 4 O T ξ 6 |ζ| 2 e -2sϕ + sλ 2 O T ξ 4 |∇ζ| 2 e -2sϕ ≤ C O T ξ 3 |w| 2 e -2sϕ + s 3 λ 4 ω T ξ 6 |ζ| 2 e -2sϕ . (2.38)
Note that the Carleman estimate of [START_REF] Fursikov | Controllability of evolution equations[END_REF] is obtained for homogeneous Dirichlet boundary conditions. But it is easily seen that it remains true for a boundary data whose tangential derivative at the boundary vanishes, which is the case for ζ.

Of course, estimate (2.34) requires an observation term in ζ in ωT . But Poincaré Wirtinger inequality and condition (2.20) implies, for all t ∈ [0, T ],

ω |ζ(t, •)| 2 ≤ C ω |∇ζ(t, •)| 2 = ω | rot ζ(t, •)| 2 = ω |v(t, •)| 2 ,
and in particular:

ω T ξ 6 |ζ| 2 e -2sϕ ≤ C ω T ξ 6 |v| 2 e -2s ϕ .
(2.39)

Let us stress the fact that the 2-d assumption is also used at this stage since (2.39) relies on the identity

|∇ζ(t, •)| 2 = | rot ζ(t, •)| 2 .
Next, we use (2.38) to derive suitable weighted energy estimates for v, hence for ∂nv on the boundary ∂O. But since we do not have any estimate on the pressure in the Stokes equation (2.17), we are reduced to derive energy estimates for v with weight functions independent of x.

Estimates in L 2 (0, T ; H 1 (O)). We set (va, pa) :

def = θ1(t)(v, p) with θ1(t) : def = s 1/4 λ -1/4 (ξ * ) 2-1/m e -sϕ * (t) . Using ∂tϕ * ≤ Cλ(ξ * ) 1+1/m in OT . (2.40)
and explicit computations, we get

θ ′ 1 ≥ -Cs 5/4 λ 3/4 (ξ * ) 3 e -sϕ * (t) .
(2.41)

The pair (va, pa) satisfies

       -σ∂tva -ν∆va + ∇pa = θ1g -σθ ′ 1 v in OT , div va = 0 in OT , va = 0 on ΓT , va(T ) = 0 in O.
(2.42)

We want to obtain an estimate of the L 2 (H 1 0 )-norm of va, so we multiply the partial differential equation in (2.42) by va, we integrate in OT and we integrate by parts. This yields:

1 2 σ(0, •)va(0, •) 2 L 2 (O) + ν va 2 L 2 (0,T ;H 1 0 (O)) = O T θ1g • va - O T σθ ′ 1 v • va - 1 2 O T ∂tσ |va| 2 . (2.43)
First, we remark that

O T θ1g • va ≤ ν 4 O T |∇va| 2 + C T 0 |θ1| 2 g 2 H -1 (O) .
(2.44)

We then focus on the second term of (2.43) and use (2.41)

- O T σθ ′ 1 v • va ≤ Cs 3/2 λ 1/2 O T (ξ * ) 5-1/m v • ∇ ⊥ ζe -2sϕ * (t) = -Cs 3/2 λ 1/2 O T (ξ * ) 5-1/m rot v ζe -2sϕ * (t) ≤ Cs 5/2 λ 3/2 O T (ξ * ) 6 |ζ| 2 e -2sϕ * (t) + νs 1/2 λ -1/2 4 O T (ξ * ) 4-2/m |∇v| 2 e -2sϕ * (t) ≤ Cs 5/2 λ 3/2 O T (ξ * ) 6 |ζ| 2 e -2sϕ * (t) + ν 4 O T |∇va| 2 .
The last term can be handled similarly:

1 2 O T ∂tσ |va| 2 ≤ Cs 1/2 λ -1/2 O T (ξ * ) 4-2/m |v| 2 e -2sϕ * ≤ Cs 5/2 λ 3/2 O T (ξ * ) 6 |ζ| 2 e -2sϕ * (t) + ν 4 O T |∇va| 2 .
Plugging these three last estimates in (2.43), we obtain

va(0, •) 2 L 2 (O) + va 2 L 2 (0,T ;H 1 0 (O)) ≤ C s 5/2 λ 3/2 O T (ξ * ) 6 |ζ| 2 e -2sϕ * + θ1g 2 L 2 (0,T ;H -1 (O)) . (2.45) Estimate in L 2 (0, T ; H 2 (O)). Let us now set (v b , p b ) : def = θ2(t)(v, p) with θ2(t) : def = s -1/4 λ -3/4 (ξ * ) 3/2-3/(2m) e -sϕ * (t) ,
for which explicit computations yield:

θ ′ 2 ≥ -Cs 3/4 λ 1/4 (ξ * ) 5 2 - 1 2m e -sϕ * (2.46) This pair (v b , p b ) satisfies        -σ∂tv b -∆v b + ∇p b = θ2g -σθ ′ 2 v in OT , div v b = 0 in OT , v b = 0 on ΓT , v b (T ) = 0 in O.
(2.47)

We then multiply the partial differential equation in (2.47) by (-∆v b + ∇p b )/σ, we integrate in OT and we integrate by parts:

1 2 O |∇v b (0, •)| 2 + O T 1 σ |-∆v b + ∇p b | 2 = O T θ2 σ g (-∆v b + ∇p b ) - O T θ2θ ′ 2 |∇v| 2 . (2.48) Using (2.
2) we can estimate the first term as follows:

O T θ2 σ g (-∆v b + ∇p b ) ≤ 1 4 O T 1 σ | -∆v b + ∇p b | 2 + C θ2g 2 L 2 (O T ) .
(2.49)

For the second term, remark that by (2.46), we have

θ2θ ′ 2 ≥ -Cs 1/2 λ -1/2 (ξ * ) 4-2/m e -2sϕ * = -Cθ 2 1 , thus yielding - O T θ2θ ′ 2 |∇v| 2 ≤ C θ1v 2 L 2 (0,T ;H 1 (O)) = C va 2 L 2 (0,T ;H 1 (O)) .
Therefore, using the above estimate and (2.49) into (2.48), we obtain

v b 2 L 2 (0,T ;H 2 (O)) ≤ C O T | -∆v b + ∇p b | 2 ≤ C θ2g 2 L 2 (O T ) + va 2 L 2 (0,T ;H 1 (O)) , (2.50)
where we have used the classical H 2 -estimate for the stationary Stokes system, see e.g.

[4, Theorem IV.5.8].

Global Estimate on v and its normal derivative. Since v = 0 on ΓT , classical estimates yield

∂nv(t, •) 2 L 2 (∂O) ≤ C v(t, •) H 1 0 (O) v(t, •) H 2 (O) + v(t, •) 2 H 1 0 (O)
, and in particular, using the fact that θ2(t) ≤ θ1(t) for all t ∈ (0, T ),

λ -1/2 (ξ * ) 7 4 - 5 4m ∂nve -sϕ * (t, •) 2 L 2 (∂O) ≤ C θ1v(t, •) H 1 0 (O) θ2v(t, •) H 2 (O) + θ1v(t, •) 2 H 1 0 (O) .
Putting together (2.45) and (2.50) with this last estimate, using (2.38) and (2.39) to estimate the term in ζ and taking into account that m ≥ 5, we deduce that

va(0, •) 2 L 2 (O) + θ1v 2 L 2 (0,T ;H 1 0 (O)) + θ2v 2 L 2 (0,T ;H 2 (O)) +λ -1 (ξ * ) 3/2 ∂nve -sϕ * 2 L 2 (Γ T ) ≤ C s -1/2 λ -5/2 O T ξ 3 |w| 2 e -2sϕ + s 5/2 λ 3/2 ω T ξ 6 |v| 2 e -2s ϕ + θ1g 2 L 2 (0,T ;H -1 (O)) + θ2g 2 L 2 (O T ) . (2.51)
Elimination of the boundary term. We come back to the Carleman inequality (2.37) and we combine it with (2.51): for s large enough,

va(0, •) 2 L 2 (O) + θ1v 2 L 2 (0,T ;H 1 0 (O)) + θ2v 2 L 2 (0,T ;H 2 (O)) s -1 O T ξ 2 |∇v| 2 e -2sϕ + O T ξ 3 |w| 2 e -2sϕ + sλ 2 O T ξ 4 |v| 2 e -2sϕ ≤ C ω T ξ 3 |w| 2 e -2sϕ + s 5/2 λ 2 ω T ξ 6 |v| 2 e -2s ϕ + θ1g 2 L 2 (0,T ;H -1 (O)) + θ2g 2 L 2 (O T ) + s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ . (2.52)
Removing the observation on w. We now estimate the local term in |w| 2 . For this purpose, we recall that ωT

= [0, T ] × ω ⋐ ωT = [0, T ] × ω and we consider a positive function χ ∈ C 2 (O) such that χ = 1 in ω, χ = 0 in O \ ω.
Using

ω T ξ 3 |w| 2 e -2sϕ ≤ ω T ξ 3 |w| 2 e -2s ϕ , (2.53)
we are reduced to estimate the right hand side of (2.53):

ω T ξ 3 |w| 2 e -2s ϕ ≤ ω T χ ξ 3 |w| 2 e -2s ϕ ≤ ω T χ ξ 3 |∇v| 2 e -2s ϕ = - ω T χ ξ 3 ∆v ve -2s ϕ + 1 2 ω T ∆χ ξ 3 |v| 2 e -2s ϕ . ≤ εs -1/2 λ -3/2 O T (ξ * ) 3-3/m |∆v| 2 e -2sϕ * + Cεs 1/2 λ 3/2 ω T (ξ * ) -3+3/m ξ 6 |v| 2 e 2sϕ * -4s ϕ ,
where the last estimate follows from Young's identity and where ε > 0. Using the last above inequality in (2.52) with ε small enough and recalling the definition of θ2, we get in particular

s 1/2 λ -1/2 O (ξ * ) 4-2/m |v(0, •)| 2 e -2sϕ * + s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * v 2 H 1 (O) s -1 O T ξ 2 |∇v| 2 e -2sϕ + sλ 2 O T ξ 4 |v| 2 e -2sϕ + O T ξ 3 | rot v| 2 e -2sϕ ≤ C s 5/2 λ 2 ω T ξ 6 |v| 2 e 2sϕ * -4s ϕ +s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ + s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * g 2 H -1 (O) +s -1/2 λ -3/2 O T (ξ * ) 3-3/m |g| 2 e -2sϕ * . (2.54)
This concludes the proof of Theorem 2.6.

Proof of Theorem 2.1

Let v be a smooth solution of (2.6) with source term g. Then v is a solution of (2.17) with source term g = g + ∂tσ v + D(σv)y + σv div (y).

Applying Theorem 2.6 to v with source term g, for all s ≥ s0 and λ ≥ λ0 we get

s 1/2 λ -1/2 O (ξ * ) 4-2/m |v(0, •)| 2 e -2sϕ * (0) + sλ 2 O T ξ 4 |v| 2 e -2sϕ +s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * v 2 H 1 (O) + O T ξ 3 | rot v| 2 e -2sϕ +s -1 O T ξ 2 |∇v| 2 e -2sϕ ≤ C s 5/2 λ 2 ω T ξ 6 |v| 2 e 2sϕ * -4s ϕ + s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ + s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * g 2 H -1 (O) +s -1/2 λ -3/2 O T (ξ * ) 3-3/m |g| 2 e -2sϕ * (2.55)
and we are thus reduced to estimate the last terms of the inequality. But we have

s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ ≤ C s -1 λ -2 O T ξ 2 |g| 2 e -2sϕ + s -1 λ -2 O T ξ 2 |v| 2 e -2sϕ + s -1 λ -2 O T ξ 2 |∇v| 2 e -2sϕ , s -1/2 λ -3/2 O T (ξ * ) 3-3/m |g| 2 e -2sϕ * ≤ C s -1/2 λ -3/2 O T (ξ * ) 3-3/m |g| 2 e -2sϕ * + s -1/2 λ -3/2 O T (ξ * ) 3-3/m |v| 2 e -2sϕ * + s -1/2 λ -3/2 O T (ξ * ) 3-3/m |∇v| 2 e -2sϕ * ,
in which all the terms in v, ∇v can be absorbed by the left-hand side of (2.55) for s and λ large enough. We also have, for all t ∈ (0, T ),

g(t) 2 H -1 (O) ≤ C g(t, •) 2 L 2 (O) + C v(t, •) 2 L 2 (O) .
Hence

s 1/2 λ -1/2 T 0 (ξ * ) 4-2/m e -2sϕ * g 2 H -1 (O) ≤ Cs 1/2 λ -1/2 O T (ξ * ) 4-2/m e -2sϕ * |g| 2 + Cs 1/2 λ -1/2 O T (ξ * ) 4-2/m e -2sϕ * |v| 2 . (2.56)
Plugging these last estimates in (2.55), we obtain (2.16) for s and λ large enough.

Proof of Theorem 2.2

We use the following simplified form of (2.16): for all s ≥ s0 and λ ≥ λ0 and all smooth solutions v of (2.6) with source term g:

O (ξ * ) 4-2/m |v(0, •)| 2 e -2sϕ * (0) + s 1/2 λ 5/2 O T ξ 4 |v| 2 e -2sϕ ≤ C s 2 λ 5/2 ω T ξ 6 |v| 2 e 2sϕ * -4s ϕ + O T ξ 4-2/m |g| 2 e -2sϕ .
(2.57)

Easy density arguments then show that this result extends to all solutions v of (2.6) with source term g ∈ L 2 (OT ) and final data v(T

) = vT ∈ V 1 0 (Ω).
We then follow the proof of Theorem 2.5 and introduce the functional JSt defined by JSt(vT , g) :

def = 1 2 O T ξ 4-2/m |g| 2 e -2sϕ + s 2 λ 5/2 2 ω T ξ 6 |v| 2 e 2sϕ * -4s ϕ - O T f • v - O u0(•) • v(0, •), (2.58) defined for data (vT , g) ∈ V 1 0 (Ω) × L 2 (OT )
, where v solves (2.6) with v(T ) = vT . We then need to define the functional JSt on the set X St,obs :

def = X 0 St,obs • St,obs , where X 0 St,obs : def = (vT , g) ∈ V 1 0 (Ω) × L 2 (OT )} (2.59)
and the norm (vT , g) St,obs is defined by

(vT , g) 2 St,obs : def = O T ξ 4-2/m |g| 2 e -2sϕ + s 2 λ 5/2 ω T ξ 6 |v| 2 e 2sϕ * -4s ϕ ,
where v is the corresponding solution to (2.6).

According to (2.57), the functional JSt can be extended by continuity on X St,obs if f satisfies (2.21). The functional JSt then has a unique minimizer on X St,obs , that we denote (VT , G) and corresponds to a solution V of (2.6). We get, for all smooth solution v of (2.6) corresponding to a source term g,

0 = O T ξ 4-2/m G • ge -2sϕ + s 2 λ 5/2 ω T ξ 6 V • ve 2sϕ * -4s ϕ - O T f • v - O u0(•) • v(0, •). (2.60)
In particular, setting

u = ξ 4-2/m Ge -2sϕ , h = -s 2 λ 5/2 ξ 6 Ve 2sϕ * -4s ϕ 1ω T , (2.61)
we obtain a solution in the sense of transposition of the control problem (2.4)-(2.5) with a control term acting only on ωT . Besides, using again the Carleman estimate (2.57) and the fact that JSt(VT , G) ≤ JSt(0, 0) = 0, one immediately derives that

(VT , G) 2 obs ≤ C s 1/2 λ 5/2 O T ξ -4 |f | 2 e 2sϕ + C O (ξ * ) 2/m-4 |u0| 2 e 2sϕ * (0) .
(2.62) Hence, using (2.61), the controlled trajectory (u, h) satisfies

O T ξ 2/m-4 |u| 2 e 2sϕ + 1 s 2 λ 5/2 ω T ξ -6 |h| 2 e 4s ϕ-2sϕ * ≤ C s 1/2 λ 5/2 O T ξ -4 |f | 2 e 2sϕ + O (ξ * ) 2/m-4 |u0| 2 e 2sϕ * (0) . (2.63)
Finally, we can then derive H 1 (L 2 ) ∩ L 2 (H 2 ) estimates on u by applying regularity results for Stokes equations to the system satisfied by e

Basic properties of the flow

Let y be the extension of y on [0, T ] × R 2 and X the corresponding flow, defined in (1.6). As y ∈ C 2 ([0, T ] × R 2 ), the flow X is continuous with respect to the variables (t, τ, x) ∈ [0, T ] 2 × R 2 . We first discuss the stability of property (1.8):

Lemma 3.1. Assume that y ∈ C 2 ([0, T ] × R 2 )
, and that the flow X defined by (1.6) satisfies (1.8). There exist ε > 0, T * 0 > 0 and T * 1 > 0 such that for all T0 ∈ (0, T * 0 ), for all T1 ∈ (0, T * 1 ) and for all x ∈ Ω, there exists t ∈ [T0, T -2T1] such that d(X(t, T0, x), Ω) ≥ 2ε.

Proof. The proof is done by contradiction. Assume it is false. Then for all ε > 0, there exist T ε 0 > 0 and T ε 1 such that T ε 0 , T ε 1 converge to 0 as ε → 0, and an xε in Ω such that

∀t ∈ [T ε 0 , T -2T ε 1 ], d(X(t, T ε 0 , xε), Ω) < 2ε. (3.1)
But xε is bounded in Ω. Hence, up to a subsequence, it converges to some x in Ω. As the flow X is continuous in [0, T ] 2 × R 2 and the distance function is continuous, for each t ∈ (0, T ), one could then pass to the limit in (3.1):

∀t ∈ (0, T ), d(X(t, 0, x), Ω) = 0.
This is of course in contradiction with (1.8).

For u ∈ L 2 (0, T ; H 2 (R 2 )) we denote by X the flow defined by

∂t X(t, τ, x) = (y + u)(t, X(t, τ, x)), X(τ, τ, x) = x. (3.2)
We then show that, provided u is small enough, the property (1.8) also holds for X:

Lemma 3.2.
Under the setting of Lemma 3.1, there exists ς > 0 such that for all u ∈ L 2 (0, T ;

H 2 (R 2 )), satisfying u L 2 (0,T ;L ∞ (R 2 )) ≤ 2ς, (3.3)
the flow X defined by (3.2) satisfies the following property: for all T0 ∈ (0, T * 0 ), for all T1 ∈ (0, T * 1 ) and for all x ∈ Ω, there exists t ∈ [T0, T -2T1] such that d( X(t, T0, x), Ω) ≥ ε.

Proof. Set L = ∇y L ∞ (0,T ;L ∞ (Ω)) . For τ, t ∈ [0, T ] 2 with t ≥ τ and x ∈ R 2 , we have:

| X(t, τ, x) -X(t, τ, x)| = | X(t, τ, x) -X(τ, τ, x) + X(τ, τ, x) -X(t, τ, x)| = t τ ∂t X(t ′ , τ, x) -∂tX(t ′ , τ, x) dt ′ = t τ u(t ′ , X(t ′ , τ, x)) + y(t ′ , X(t ′ , τ, x)) -y(t ′ , X(t ′ , τ, x))dt ′ ≤ |t -τ | 1/2 u L 2 (τ,t;L ∞ (R 2 )) + L t τ | X(t ′ , τ, x) -X(t ′ , τ, x)|dt ′ .
Then Gronwall's Lemma yields for all t ∈ [0, T ] and x ∈ R 2 :

| X(t, τ, x) -X(t, τ, x)| ≤ T 1/2 e LT u L 2 (τ,t;L ∞ (R 2 )) . (3.4)
According to Lemma 3.1, Lemma 3.2 thus holds by setting ς = T -1/2 e -LT ε/2 in (3.3).

Construction of the controlled density

In this section, we assume that

u ∈ L 2 (0, T ; H 2 (R 2 )) and u L 2 (0,T ;L ∞ (R 2 )) ≤ 2ς, (3.5)
where ς is given by Lemma 3.2. We then choose T0 ∈ (0, T * 0 ) and T1 ∈ (0, T * 1 ), where T * 0 , T * 1 are given by Lemma 3.2.

The construction of the controlled density ρ solution of (1.18) is then done as in [START_REF] Ervedoza | Local exact controllability for the one-dimensional compressible Navier-Stokes equation[END_REF]: we construct a forward solution ρ f and a backward solution ρ b of the transport equation in (1.18) and we glue these two solutions according to the characteristics of the flow. Indeed, we define ρ f as the solution of

       ∂tρ f + (y + u) • ∇ρ f = -u • ∇σ in ΩT , ρ f (t, x) = 0 for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + u(t, x)) • n(x) < 0, ρ f (0) = ρ0 in Ω, (3.6)
and ρ b as the solution of

       ∂tρ b + (y + u) • ∇ρ b = -u • ∇σ in ΩT , ρ b = 0 for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + u(t, x)) • n(x) > 0, ρ b (T ) = 0 in Ω. (3.7)
We also introduce χ the solution of

       ∂tχ + (y + u) • ∇χ = 0 in ΩT , χ = 1 t∈(0,T 0 ) (t) for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + u(t, x)) • n(x) < 0, χ(0) = 1 in Ω. (3.8)
We finally define ρ(t, x) as follows,

ρ(t, x) : def = (1 -χ(t, x))ρ b (t, x) + χ(t, x)ρ f (t, x). (3.9)
It is easy to check that this function ρ satisfies the transport equation (1.18) (1) and the required initial condition (1.18) [START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF] . The final condition ρ(T ) = 0 in (1.18) (3) is satisfied due to the properties of the flow proved in Lemma 3.2, which guarantees that χ(T ) = 0.

In the next subsections, we describe how to get estimates on the function ρ constructed in (3.9) in the weighted spaces adapted to the Carleman estimates derived in Section 2.

Explicit description of the density

To begin with, let us remark that the function χ is explicitly given by:

χ(t, x) =    1 if t < T0 1 if t ≥ T0 and X(τ, t, x) ∈ Ω for all τ ∈ [T0, t], 0 else, (3.10)
so that from Lemma 3.2 we have in particular

χ(t, x) = 0 and ρ(t, x) = ρ b (t, x) for (t, x) ∈ [T -2T1, T ] × Ω. (3.11)
We also give explicit expressions for ρ f and ρ b . In order to do that, for t ∈ [0, T ], we introduce

Ω [0] (t) : def = {x ∈ Ω | X(τ, t, x) ∈ Ω for all τ ∈ [0, t]} Ω [T ] (t) : def = {x ∈ Ω | X(τ, t, x) ∈ Ω for all τ ∈ [t, T ]} (3.12)
and for all (t, x) ∈ [0, T ] × Ω:

tin(t, x) : def = sup{τ ∈ [0, t) | X(τ, t, x) ∈ ∂Ω}, tout(t, x) : def = inf{τ ∈ (t, T ] | X(τ, t, x) ∈ ∂Ω}.
(3.13)

In the above definitions, we use the convention sup ∅ = 0 and inf ∅ = T . This way,

tin(t, x) = 0 iff x ∈ Ω [0] (t) and tout(t, x) = T iff x ∈ Ω [T ] (t).
Using these notations, ρ f and ρ b are explicitly given by

ρ f (t, x) =        ρ0( X(0, t, x)) - t 0 ( u • ∇σ)(τ, X(τ, t, x))dτ if x ∈ Ω [0] (t), - t t in (t,x) ( u • ∇σ)(τ, X(τ, t, x))dτ else, (3.14)
ρ b (t, x) = t tout(t,x) ( u • ∇σ)(τ, X(τ, t, x))dτ for x ∈ Ω. ( 3.15)
We are now in position to derive weighted estimates on ρ.

Weighted estimates on the density

In order to derive weighted estimates on ρ based on the Carleman weights ψ, θ, ϕ, ξ described in (2.7)-(2.9)-(2.10)-(2.11), we will need some further assumptions.

Assumptions on the weights. We assume that T0 and T1 in the definition of θ in (2.10) satisfy T0 ∈ (0,

T * 0 ), T1 ∈ (0, T * 1 ), (3.16)
where T * 0 and T * 1 are given by Lemma 3.1. We also assume that the function ψ in (2.7) satisfies the transport equation ∂tψ + y • ∇ψ = 0 in ΩT .

(3.17)

Assumptions on u. In order to derive estimates on ρ, we shall assume that u is in a weighted Sobolev space. According to Theorem 2.2, it is natural to assume ξ -2 ue sϕ ∈ L 2 (0, T ; L 2 (Ω)), (3.18)

ue 3sϕ * /4 ∈ L 2 (0, T ; H 2 (Ω)) with ue 3sϕ * /4 L 2 (0,T ;H 2 (Ω)) ≤ ς. (3.19)
Extension of u. To fit into the setting of Section 3.2, we extend u on [0, T ] × R 2 that we still denote the same: u = E( u), where E denotes an extension from

H 2 (Ω) to H 2 (R 2 ) such that E(v) H 2 (R 2 ) ≤ 2 v H 2 (Ω) for all v ∈ H 2 (Ω)
. This allows us to define the flow

X by (3.2) for (t, τ, x) ∈ [0, T ] 2 × R 2 .
Note that, for s large enough, this last assumption is stronger than (3.5) and is thus perfectly compatible with the construction of Section 3.2, as it implies in particular that

θ u L 2 (0,T ;L ∞ (R 2 )) ≤ cςe -c 0 sλ , (3.20)
where c0 > 0 is independent of s and λ. For the following we suppose that s ≥ s0 and λ ≥ 1 with s0 large enough such that (3.5) and (3.20) are satisfied.

On the flows X and X. We first establish a lemma on the closeness of X to X.

Lemma 3.3. There exists c > 0 independent of s and λ such that for all (τ, t)

∈ [0, T ] 2 and x ∈ R 2 : | X(τ, t, x) -X(τ, t, x)| ≤ cςe -c 0 sλ . (3.21)
Moreover, if T0 ≤ t ≤ τ ≤ T , we also have

θ(t)| X(τ, t, x) -X(τ, t, x)| ≤ cςe -c 0 sλ . (3.22)
Proof. Estimate (3.21) is an immediate consequence of (3.4) and (3.20). From (3.4), we also have θ

(t)| X(τ, t, x) -X(τ, t, x)| ≤ T 1/2 e LT θ(t) u L 2 (t,τ ;L ∞ (R 2 )) ,
where L = ∇y L ∞ (0,T ;L ∞ (R 2 )) . Using the fact that θ is increasing on [T0, T ],

θ(t)| X(τ, t, x) -X(τ, t, x)| ≤ T 1/2 e LT θ u L 2 (t,τ ;L ∞ (R 2 )) ,
for all T0 ≤ t ≤ τ ≤ T , which concludes the proof of Lemma 3.3 by (3.20).

On the weight functions. Here, we shall deeply use the fact that ψ is assumed to solve the transport equation (3.17), thus implying in particular that

∀(t, τ, x) ∈ [0, T ] 2 × R 2 , ψ(t, X(t, τ, x)) = ψ(τ, x).
(3.23)

We then show the following lemma:

Lemma 3.4. There exist c1 > 0, c2 > 0 and c3 > 0 independent of s and λ, and s0 > 1 such that for all s ≥ s0, λ ≥ 1, the following inequalities hold:

1. For all t ∈ [0, T -2T1], τ ∈ [0, t] and x ∈ R 2 , ϕ(t, x) -ϕ(τ, X(τ, t, x)) ≤ c1ςe -c 2 sλ , (3.24) ξ(τ, X(τ, t, x)) ξ(t, x) ≤ 2e c 1 ςe -c 2 sλ . (3.25) 2. For all t ∈ [T0, T ], τ ∈ [t, T ] and x ∈ R 2 , ϕ(t, x) -ϕ(τ, X(τ, t, x)) ≤ c1ςe -c 2 sλ -c3(θ(τ ) -θ(t)), (3.26)
ξ(τ, X(τ, t, x)) ξ(t, x) ≤ θ(τ ) θ(t) e c 1 ςe -c 2 sλ . (3.27)
Proof. We focus on the proof of item 2, the first one being similar and easier because θ takes value in [START_REF] Albano | Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system[END_REF][START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF] close to t = 0. Estimate (3.26) follows from the following computations: for T0 ≤ t ≤ τ ≤ T , ϕ(t, x)ϕ(τ, X(τ, t, x))

= θ(t) λe 6λ(m+1)e λψ(t,x)θ(τ ) λe 6λ(m+1)e λψ(τ, X(τ,t,x))

= θ(t) e λψ(τ, X(τ,t,x))e λψ(t,x) + (θ(t)θ(τ )) λe 6λ(m+1)e λψ(τ, X(τ,t,x))

≤ θ(t) e λψ(τ, X(τ,t,x))e λψ(t,x) -c3(θ(τ )θ(t)),

for some c3 > 0, where we used in the last estimate that θ is increasing on [T0, T ]. We then use (3.23) and (3. |θ(t) e λψ(τ, X(τ,t,x))e λψ(t,x) | = θ(t) e λψ(τ, X(τ,t,x))e λψ(τ,X(τ,t,x))

≤ cθ(t)λ ∇ψ ∞e λ(6m+1) | X(τ, t, x) -X(τ, t, x)| ≤ c1ςe -c 2 sλ ,
for s large enough, as announced in (3.26). Next, by construction we have ξ(τ, X(τ, t, x)) ξ(t, x) = θ(τ ) θ(t) e λ(ψ(τ, X(τ,t,x))-ψ(τ,X(τ,t,x)))

≤ θ(τ ) θ(t) e λ ∇ψ ∞| X(τ,t,x)-X(τ,t,x)| , (3.28) which immediately yields (3.27) by (3.22).

We immediately deduce from Lemma 3.4 the following: there exist s0 ≥ 1 and c > 0 independent of s and λ such that for all λ ≥ 1, s ≥ s0, for all (τ, t, x)

∈ [0, T ] × [0, T ] × Ω satisfying τ ≤ t ≤ T -2T1 or T0 ≤ t ≤ τ , ℵ(t, x) ≤ cℵ(τ, X(τ, t, x)). ( 3
ℵ(t, x) ≤ θ 2 (τ )e -c 3 sθ(τ ) θ 2 (t)e -c 3 sθ(t) e c 1 ς(s+2)e -sλc 2 ℵ(τ, X(τ, t, x)).
But, for s ≥ 2/c3, the function x → x 2 e -c 3 sx is decreasing on [1, +∞) and then, since θ is increasing on [T0, T ], θ 2 (τ )e -c 3 sθ(τ ) ≤ θ 2 (t)e -c 3 sθ(t) .

On the controlled trajectory ρ. We now derive estimates on the controlled trajectory ρ given by Section 3.2: Theorem 3.6. Let ψ, θ, ϕ, ξ are defined in (2.7)-(2.9)-(2.10)-(2.11) and assume (3.16), (3.17). Further assume that u satisfies (3.18) and (3.19) with s ≥ s0, λ ≥ 1 and s0 large enough such that (3.5) and (3.20) are satisfied.

There exists c > 0 independent of s, λ and u such that the solution ρ given by Section 3.2 satisfies

ℵρ L 2 (Ω T ) ≤ C ℵ u L 2 (0,T ;L 2 (Ω)) + e sϕ * (0) ρ0 L 2 (Ω) , (3.31)
where ℵ is given by (3.29), and

e sλe 6λ(m+1) θ(t)/2 ρ L ∞ (Ω T )
≤ C e sλe 6λ(m+1) θ(t)/2 u L 2 (0,T ;L ∞ (Ω))

+e sλe 6λ(m+1) ρ0 L ∞ (Ω) . Let us begin with the proof of estimate (3.32). On one hand, as t → sλe 6λ(m+1) θ(t) is non-increasing on (0, T -2T1), from (3.14) we get, for all (t, x)

∈ (0, T -2T1) × Ω e sλe 6λ(m+1) θ(t) |ρ f (t, x)| 2 ≤ 2e sλe 6λ(m+1) θ(t) ρ0 2 L ∞ (Ω) + 2 ∇σ 2 L ∞ (Ω T ) t 0 e sλe 6λ(m+1) θ(τ ) u(τ, •) 2 L ∞ (Ω) dτ .
On the other hand, using that t → sλe 6λ(m+1) θ(t) is non-decreasing on (T0, T ), from (3.15), similarly, we have, for all (t, x) ∈ (T0, T ) × Ω,

e sλe 6λ(m+1) θ(t) |ρ b (t, x)| 2 ≤ ∇σ 2 L ∞ (Ω T ) T t e sλe 6λ(m+1) θ(τ ) u(τ, •) 2 L ∞ (Ω) dτ .
Together with the fact that the solution χ of (3.8) takes value in [0, 1] on ΩT and the properties (3.10), these two estimates easily yield (3.32).

We then focus on the proof of (3.31), that mainly relies on the two following estimates: for all time t ∈ (0, T -2T1), we get

Ω |ρ f (t)| 2 ℵ 2 (t)dx ≤ C e 2sϕ * (0) Ω |ρ0| 2 dx + Ω T | u| 2 ℵ 2 dxdτ , (3.33)
and for all time t ∈ (T0, T ),

Ω |ρ b (t)| 2 ℵ 2 (t)dx ≤ c Ω T | u| 2 ℵ 2 dxdτ. ( 3
|ρ f (t, x)| 2 ℵ 2 (t, x) ≤ C |ρ0( X(0, t, x))| 2 ℵ 2 (0, X(0, t, x)) + t 0 | u(τ, X(τ, t, x))| 2 ℵ 2 (τ, X(τ, t, x))dτ , whereas for x ∈ Ω\Ω [0] (t), |ρ f (t, x)| 2 ℵ 2 (t, x) ≤ C t t in (t,x) | u(τ, X(τ, t, x))| 2 ℵ 2 (τ, X(τ, t, x))dτ.
Combining these two estimates, for all t ∈ (0, T -2T1) we get:

Ω |ρ f (t, x)| 2 ℵ 2 (t, x)dx ≤ C Ω [0] (t) |ρ0( X(0, t, x))| 2 ℵ 2 (0, X(0, t, x))dx + C t 0 Ω 1 [t in (t,x),t] (τ )| u(τ, X(τ, t, x))| 2 ℵ 2 (τ, X(τ, t, x))dxdτ. (3.35)
Since y + u is divergence free in ΩT , the Jacobian of x → X(t, τ, x) equals 1 identically. Therefore,

Ω [0] (t) |ρ0( X(0, t, x))| 2 ℵ 2 (0, X(0, t, x))dx = X(0,t,Ω [0] (t)) |ρ0(x)| 2 ℵ 2 (0, x)dx ≤ Ω |ρ0(x)| 2 ℵ 2 (0, x)dx.
Similarly, we get

t 0 Ω 1 [t in (t,x),t] (τ )| u(τ, X(τ, t, x))| 2 ℵ 2 (τ, X(τ, t, x))dxdτ ≤ t 0 Ω | u(τ, x)| 2 ℵ 2 (τ,

Proof of Theorem 1.1

We are now in position to prove Theorem 1.1. The idea is to construct suitable convex sets which are invariant by the mapping F = F (ρ 0 ,u 0 ) in (1.20) and relatively compact for a topology making F continuous. In all this section, we assume the assumptions of Theorem 1.1.

Main steps of the proof of Theorem 1.1

In the introduction, we introduced formally a mapping F . We are now in position to define it precisely.

In order to do this, the first step in the proof of Theorem 1.1 is to construct a weight function ψ which is suitable for both Section 2 and Section 3, i.e. suitable in the same time for controlling the velocity equation and the density equation. We claim the following result, proved in Section 4.2: Lemma 4.1. Let Ω be a smooth bounded domain. Further assume the regularity condition (1.10) on (σ, y), the geometric condition (1.8) and condition (1.9).

Then one can find a smooth (C 2 ) bounded domain O satisfying (2.1) such that there exists a C 2 (OT )-function ψ satisfying the transport equation (3.17) and satisfying assumptions (2.7) to (2.8) for ωT = [0, T ]×ω and ωT = [0, T ]× ω where ω, ω are two subdomains of O\Ω such that ω ⋐ ω.

Next, we take T * 0 , T * 1 and ς > 0 given by Lemma 3.2 and fix T0 ∈ (0, T * 0 ) and T1 ∈ (0, T * 1 ). We then use the function ψ, θ, ϕ and ξ given by (2.9), (2.10), (2.11) for m ≥ 5, s ≥ s0, λ ≥ λ0, and the notations given in (2.14)-(2.15). Moreover, we suppose that s0, λ0 are large enough given by Theorem 2.2 and Theorem 3.6. Now, we define the spaces X s,λ and Y s,λ depending on positive parameters s ≥ s0 and λ ≥ λ0 as follows:

X s,λ : def = {u ∈L 2 (ΩT ),
with div (u) = 0 in ΩT , (4.1)

s 1/4 ξ 1/m-2 e sϕ u ∈ L 2 (ΩT ), e 3sϕ * /4 u ∈ L 2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; L 2 (Ω))},
endowed with the norm

u 2 X s,λ : def = e 3sϕ * /4 u 2 L 2 (H 2 )∩H 1 (L 2 ) + s 1/2 ξ 1/m-2 e sϕ u 2 L 2 (Ω T ) , and
Y s,λ : def = {ρ ∈ L ∞ (ΩT )
, with ξ -2 e sϕ ρ ∈ L 2 (ΩT ) and e sλe 6λ(m+1) θ/2 ρ ∈ L ∞ (ΩT )}, endowed with the norm

ρ Y s,λ : def = ξ -2 e sϕ ρ L 2 (Ω T ) + e sλe 6λ(m+1) θ/2 ρ L ∞ (Ω T )
.

We also introduce the space F s,λ defined by

F s,λ : def = {f ∈ L 2 (0, T ; L 2 (Ω)), with ξ -2 f e sϕ ∈ L 2 (0, T ; L 2 (Ω))}
endowed with the norm f F s,λ :

def = ξ -2 f e sϕ L 2 (L 2 )
. Note that, in the above definitions as well as in the following results, we keep the dependence in both parameters λ and s to be consistent with notations of Section 2. However, only the dependence in s will be needed in this section.

We then derive the following results.

Theorem 4.2 (On the mapping F1). Fix ρ0 ∈ L ∞ (Ω). For all u ∈ X s,λ with u X s,λ ≤ ς, the construction in Section 3.2 yields ρ = F1( u, ρ0) solution of the control problem (1.18). Besides, ρ ∈ Y s,λ and for some constant C independent of s ≥ s0 and λ ≥ λ0,

ρ Y s,λ ≤ C 1 s 1/4 u X s,λ + e sϕ * (0) ρ0 L ∞ (Ω) . (4.2)
Furthermore, the application F1 satisfies the following compactness property: If un is a sequence of functions in X s,λ with un X s,λ ≤ ς which weakly converges to some u in X s,λ , the corresponding sequence ρn = F1( un, ρ0) strongly converges to F1( u, ρ0) in all L q (ΩT ) for q ∈ [1, ∞).

The proof of Theorem 4.2 is done in Section 4.3. Let us point out that the compactness property stated in Theorem 4.2 is of primary importance for our result and follows from [START_REF] Boyer | Outflow boundary conditions for the incompressible nonhomogeneous Navier-Stokes equations[END_REF]Theorem 4].

We then focus on the study of the mapping F2:

Theorem 4.3 (On the mapping F2). We can define a bounded linear mapping F2 : [START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF]) and satisfies, for some constant C > 0 independent of ≥ s0 and λ ≥ λ0,

F s,λ × V 1 0 (Ω) → X s,λ such that for all u0 ∈ V 1 0 (Ω) and f ∈ F s,λ , u = F2(f , u0) solves the control problem (1.
u X s,λ ≤ C f F s,λ + e Theorem 4.
3 is a direct consequence of Theorem 2.2: the mapping F2 is obtained by restricting the controlled trajectory given by Theorem 2.2 to (0, T ) × Ω. Of course, this depends on the extension O of Ω, but this choice is done once for all. Estimate (4.3) is then a rewriting of Theorem 2.2 by taking into account that f and u0 are extended by zero outside Ω.

We are then able to derive the following properties on the mapping F in (1.20), whose proof is postponed to Section 4.4:

Theorem 4.4. Let ρ0 ∈ L ∞ (Ω) and u0 ∈ V 1 0 (Ω)
. Then for all s ≥ s0 and λ ≥ λ0 the mapping F in (1.20) is well-defined for all u ∈ X s,λ with u X s,λ ≤ ς. Besides, for all u ∈ X s,λ with u X s,λ ≤ ς, u = F ( u) belongs to X s,λ , and satisfies, for some constant C0 independent of s and λ,

u X s,λ ≤ C0 1 s 1/4 u X s,λ + u 2 X s,λ +e sϕ * (0) ρ0 L ∞ (Ω) + e 2sϕ * (0) ρ0 2 L ∞ (Ω) + e 5 4
sϕ * (0) u0 H 1 0 (Ω) . (4.4) Moreover, if un is a sequence of functions in X s,λ with un X s,λ ≤ ς which weakly converges to some u in X s,λ , the corresponding sequence un = F ( un) strongly converges to

u = F ( u) in L 2 (0, T ; L 2 (Ω)).
We may then conclude the proof of Theorem 1.1. For R ∈ (0, ς), we introduce the closed convex set X R s,λ = {u ∈ X s,λ with u X s,λ ≤ R}. We then choose R small enough such that C0R ≤ 1/4, where C0 is the constant in (4.4), λ = λ0 and s ≥ s0 large enough to guarantee C0 ≤ s 1/4 /4. We then get from (4.4) that for all u ∈ X R s,λ 0 , u = F ( u) satisfies

u X s,λ 0 ≤ R 2 + C0 e sϕ * (0) ρ0 L ∞ (Ω) + e 2sϕ * (0) ρ0 2 L ∞ (Ω) + e 5 4 sϕ * (0) u0 H 1 0 (Ω)
. Thus, choosing ε > 0 sufficiently small in (1.11), we can guarantee that the mapping F maps X R s,λ 0 to itself. We then check that the set X R s,λ 0 is compact in L 2 (0, T ; L 2 (Ω)) as H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) is compactly embedded in L 2 (0, T ; L 2 (Ω)) due to Rellich's compactness theorem and Aubin-Lions' theorem.

Besides, the mapping F is continuous on X R s,λ 0 endowed with the L 2 (0, T ; L 2 (Ω))topology from Theorem 4.4. Indeed, if un is a sequence of functions in X R s,λ 0 which strongly converges to u in L 2 (0, T ; L 2 (Ω)), it necessarily weakly converges in X R s,λ 0 . Thus, from the last item of Theorem 4.4, un = F ( un) strongly converges to u in L 2 (0, T ; L 2 (Ω)).

Schauder's fixed point theorem then implies the existence of a fixed point to the mapping F , and concludes the proof of Theorem 1.1.

Proof of Lemma 4.1

We do it in several steps.

Construction of O. In a neighborhood of Γc, according to Assumption (1.9), there exists a C 2 extension O of Ω such that

• Ω ⊂ O;
• Γ0 ⊂ ∂Ω ∩ ∂O and for all t ∈ (0, T ) and x ∈ ∂Ω ∩ ∂O, y(t, x) • n ≥ γ/2;

• ∂O ∩ ∂Ω and O \ Ω have a finite number of connected components.

Let ω, ω be two subdomains of O\Ω such that ω ⋐ ω and fix d0 = dist(ω, Ω). Construction of an extension y e of y in O. We then construct an extension y e ∈ C 2 ([0, T ] × R 2 ) of y outside ΩT (i.e y e ≡ y in ΩT ) satisfying

y e C 2 ([0,T ]×O) < ∞, inf [0,T ]×∂O y e • n > 0, (4.5)
and y e ≡ 0 in (0, T ) × ω. (4.6)

Before going into the detailed construction of y e , let us remark that y e cannot be divergence free as it would not be compatible with the condition inf [0,T ]×∂O y e • n > 0.

In order to construct such extension y e , we proceed as follows. First, we consider any extension of y in C 2 ([0, T ] × R 2 ). By continuity, there exists d1 > 0 such that for all (t, x) ∈ (0, T ) × ∂O with d(x, Ω) < d1, y(t, x) • n ≥ γ/3. We also introduce a function m in C 2 ([0, T ] × R 2 ) such that m • n = 1 on the whole boundary ∂O and m ≡ 0 in ω, and a smooth non-negative cut-off function η = η(x) taking value 1 in Ω and 0 for all x ∈ O with d(x, Ω) > min{d0, d1}, and we then consider

y e (t, x) = η(x)y(t, x) + (1 -η(x))m(x).
This function indeed belongs to

C 2 ([0, T ] × R 2 ). Besides, inf [0,T ]×∂O y e • n ≥ min γ 3 , 1 ,
and (4.6) is trivially satisfied as m ≡ 0 and η ≡ 0 in ω. Construction of ψ. We then construct a function ψT = ψT (x) such that

• ψT is a non-negative C 2 (O) function;

• The critical points of ψT all belong to ω;

• ψT satisfies the following conditions on the boundary ∂O:

     ψT (x) = 0 on ∂O, y e (T, x) • ∇ ψT (x) = -1 on ∂O, ∂ty e (T, x) • ∇ ψT (x) -(y e (T, x) • ∇) 2 ψT (x) = 0 on ∂O. (4.7)
• infO ψT = ( ψT ) |∂O = 0.

Note that such function exists according to the construction of Fursikov and Imanuvilov in [START_REF] Fursikov | Controllability of evolution equations[END_REF] suitably modified to handle the conditions on the first and second order derivatives on the boundary of O. This can be done easily following the lines of [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]Appendix III].

We then consider the solution

ψ of      ∂t ψ + y e • ∇ ψ = 0 in OT , ψ(t, x) = t -T on ΓT , ψ(T ) = ψT in O. (4.8)
Note that this problem is well-posed as, by construction, y e (t, x) • n > 0 for all (t, x) ∈ (0, T ) × ∂O. We then want to check that

• ∂n ψ(t, x) ≤ 0 for (t, x) ∈ (0, T ) × ∂O;

• ψ belongs to C 2 ([0, T ] × O);
• For all t ∈ [0, T ], the critical points of ψ(t, •) belong to ω;

• For all t ∈ [0, T ], infO ψ(t, •) = ψ(t) |∂O ;
Indeed, providing these properties are true, one can choose a > 0 and b ∈ R such that the function ψ = a ψ + b is suitable for Lemma 4.1.

Using the equation (4.8) and the fact that tangential derivatives of ψ vanish due to the boundary conditions, we get, for all (t, x) ∈ (0, T ) × ∂O,

y e (t, x) • n ∂n ψ(t, x) = -∂t ψ(t, x) = -1.
Using (4.5), we thus deduce that

∀(t, x) ∈ (0, T ) × ∂O, ∂n ψ(t, x) ≤ -1 inf [0,T ]×∂O y e (t, x) • n < 0. (4.9)
To describe more precisely the function ψ, we will introduce the flow Xe corresponding to y e , i.e. the solution of ∀(t, τ, x) ∈ [0, T ] 2 × R 2 , ∂tXe(t, τ, x) = y e (t, Xe(t, τ, x)), Xe(τ, τ, x) = x.

(4.10)

The fact that ψ ∈ C 2 ([0, T ] × O) follows from the following lemma, whose proof is postponed to Appendix B: We then have to check that the critical points of ψ(t, •) all belong to ω. We first remark that (4.9) implies that there is no critical point on the boundary ∂O. We then remark that ∇ ψ solves the equation Since there is no critical point on the boundary ∂O and thanks to conditions (4.5), for all time tc ∈ [0, T ], the critical points xc of ψ(tc, •) are linked by a trajectory τ → Xe(τ, tc, xc) to a critical point xc,T of ψT , that is xc = Xe(tc, T, xc,T ). By construction of ψT , xc,T necessarily belongs to ω. But, according to condition (4.6), as long as Xe(t, T, xc,T ) ∈ ω, ∂tXe(t, T, xc,T ) = 0, so that Xe(t, T, xc,T ) = xc,T for all t ∈ [0, T ]. This implies that the set of critical points of ψ(t, •) is invariant through the flow Xe and is then included in ω.

∂t∇ ψ + (y e • ∇)∇ ψ + Dy e ∇ ψ = 0 in OT . ( 4
We finally check the condition infO ψ(t, •) = ψ(t) |∂O for all t ∈ [0, T ] by contradiction. If this were wrong, there would exist t ∈ [0, T ] and xt ∈ O such that xt ∈ Argmin ψ(t, •). Thus, xt would be a critical point, and as above, Xe(T, t, xt) would belong to O and be a critical point of ψT . Following, ψ(t, xt) = ψT (Xe(T, t, xt)) would be larger than 0 due to the assumption on ψT . But from the boundary conditions, it follows that infO ψ(t) cannot be strictly smaller than ψ(t) |∂O , which is negative for all time t ∈ [0, T ).

Proof of Theorem 4.2

According to Section 3, the construction in Section 3.2 yields ρ = F1( u, ρ0) solution of the control problem (1.18) for u satisfying (3.5). This condition is indeed satisfied for u ∈ X s,λ with u X s,λ ≤ ς, see (3.18)-(3.19)-(3.20). Theorem 3.6 immediately provides estimate (4.2), as λe 6λ(m+1) θ/2 ≤ 3ϕ * /4, see (2.13). We then focus on the proof of the compactness property. According to the construction in Section 3.2, we introduce ρ f,n the solution of

       ∂tρ f,n + (y + un) • ∇ρ f,n = -un • ∇σ in ΩT , ρ f,n (t, x) = 0 for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + un(t, x)) • n(x) < 0, ρ f,n (0) = ρ0 in Ω, (4.12
)

ρ b,n the solution of        ∂tρ b,n + (y + un) • ∇ρ b,n = -un • ∇σ in ΩT , ρ b,n = 0 for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + un(t, x)) • n(x) > 0, ρ b,n (T ) = 0 in Ω, (4.13)
and χn the solution of

       ∂tχn + (y + un) • ∇χn = 0 in ΩT , χn = 1 t∈(0,T 0 ) (t) for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + un(t, x)) • n(x) < 0, χn(0) = 1 in Ω. (4.14)
Since un is a bounded sequence of H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)), which is compact in L 2 (0, T ; L 2 (Ω)), up to a subsequence still denoted the same for simplicity, un converge to u weakly in H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) and strongly in L 2 (0, T ; L 2 (Ω)). Then [START_REF] Boyer | Outflow boundary conditions for the incompressible nonhomogeneous Navier-Stokes equations[END_REF]Theorem 4] applies and for all q ∈ [1, +∞) the sequence χn strongly converges towards χ in L q (ΩT ) solution of (3.8).

Next, to pass to the limit in (4.12), we notice that σ f,n :

def = σ + ρ f,n solves        ∂tσ f,n + (y + un) • ∇σ f,n = 0 in ΩT , σ f,n (t, x) = 0 for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + un(t, x)) • n(x) < 0, σ f,n (0) = ρ0 in Ω. (4.15)
Thus, by applying again [3, Theorem 4] we deduce that, for all q ∈ [1, +∞), the sequence σ f,n is strongly convergent in L q (ΩT ) to the solution

σ f of        ∂tσ f + (y + u) • ∇σ f = 0 in ΩT , σ f (t, x) = 0 for t ∈ (0, T ), x ∈ ∂Ω, with (y(t, x) + u(t, x)) • n(x) < 0, σ f (0) = ρ0 in Ω. (4.16)
It follows that ρ f,n strongly converges in all L q (ΩT ) for q ∈ [1, ∞) to ρ f = σ fσ, which solves (3.6) by construction.

Of course, the same can be done to show that ρ b,n strongly converges in all L q (ΩT ) for q ∈ [1, ∞) to the solution ρ b of (3.7). Consequently, the sequence ρn = F1( un, ρ0)

converges to ρ = F1( u, ρ0) in L q (ΩT ) for all q ∈ [1, ∞).
un and u are uniformly bounded in X s,λ , so the convergence of un to u actually is weak in X s,λ .

Proof of Theorem 4.4

Let ρ0 ∈ L ∞ (Ω), u0 ∈ V 1 0 (Ω) and u ∈ X s,λ with u X s,λ ≤ ς. According to Theorem 4.2, ρ = F1( u, ρ0) belongs to Y s,λ and is bounded in that space by (4.2). Thus, according to Theorem 4.3, for F to be well-defined, we have to check that f (ρ, u) given in (1.13) belongs to F s,λ , and we will get estimates on u = F ( u) from an estimate of f (ρ, u) in F s,λ according to (4.3). We thus estimate f (ρ, u) in F s,λ term by term from estimates on ρ ∈ Y s,λ and u ∈ X s,λ . We easily check

ξ -2 e sϕ ρ(∂t u + (y + u) • ∇ u + u • ∇y) L 2 (L 2 ) ≤ e sλe 6λ(m+1) θ/2 ρ L ∞ ξ -2 e sϕ-sλe 6λ(m+1) θ/2 (∂t u + ((y + u) • ∇) u + u • ∇y) L 2 (L 2 ) ≤C ρ Y s,λ e 3sϕ * /4 u L 2 (H 2 )∩H 1 (L 2 ) ξ -2 e sϕ-sλe 6λ(m+1) θ/2-3sϕ * /4 L ∞
, where we used that y + u is bounded in L ∞ (0, T ; L 4 (Ω)) due to Sobolev's embedding as u belongs to L 2 (0, T ; H 2 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) and is of norm bounded by ς, and that

e 3sϕ * /4 ∇ u L 2 (L 4 ) ≤ C e 3sϕ * /4 u L 2 (H 2 )∩H 1 (L 2 )
.

According to (2.13), sϕsλe 6λ(m+1) θ/2 -3sϕ * /4 ≤ -sϕ/4, and thus there exists some constant C independent of s and λ such that

ξ -2 e sϕ-sλe 6λ(m+1) θ/2-3sϕ * /4 L ∞ ≤ C. Following, ξ -2 e sϕ ρ(∂t u + (y + u) • ∇ u) L 2 (L 2 ) ≤ C ρ Y s,λ u X s,λ . (4.17)
Next, we estimate σ( u • ∇) u. Similarly as above, we write

ξ -2 e sϕ σ u • ∇ u L 2 (L 2 ) ≤C e 3sϕ * /4 u L ∞ (L 4 ) e 3sϕ * /4 ∇ u L 2 (L 4 ) ξ -2 e sϕ-3sϕ * /2 L ∞ ≤C u 2 X s,λ . (4.18)
Last, we estimate ρ(∂ty + (y • ∇)y):

ξ -2 e sϕ ρ(∂ty + (y • ∇)y) L 2 (L 2 ) ≤ C ξ -2 e sϕ ρ L 2 ≤ C ρ Y s,λ . (4.19)
Putting estimates (4.17)-(4.19) together, we obtain:

f (ρ, u) F s,λ = ξ -2 e sϕ f (ρ, u) L 2 (L 2 ) ≤ C( ρ Y s,λ + ρ 2 Y s,λ + u 2 X s,λ ). (4.20)
Combined with estimates (4.2) and (4.3), this yields the well-posedness of the mapping F for u ∈ X s,λ with u X s,λ ≤ ς and the estimate (4.4). We now focus on the last part of Theorem 4.4. Let un is a sequence of X s,λ with un X s,λ ≤ ς which weakly converges to u. Note that this weak convergence implies that u X s,λ ≤ ς, so that F ( u) is well-defined.

Besides that, according to Theorem 4.2, the sequence ρn = F1( un, ρ0) strongly converges in all L q (ΩT ) with q < ∞ to ρ = F1( u, ρ0) and the sequence ρn is uniformly bounded in Y s,λ .

We then have to check that f (ρn, un) weakly converges in F s,λ to f (ρ, u). But (4.20) shows that the sequence f (ρn, un) is bounded in F s,λ , and thus we only need to prove that the sequence f (ρn, un) weakly converges in D ′ (ΩT ) to f (ρ, u). To obtain this convergence result in D ′ (ΩT ), as ρn strongly converges to ρ in all L q (ΩT ) with q < ∞ and un weakly converges to u in H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)), we only have to focus on the convergence of the term (σ + ρn) un • ∇ un. But, using the compactness of

H 1 (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 2 (Ω)) in L 4 (0, T ; L 4 (Ω)), we have the convergences σ + ρn -→ n→∞ σ + ρ strongly in L q (ΩT ), q ∈ [1, ∞), un -→ n→∞ u strongly in L 4 (0, T ; L 4 (Ω)), ∇ un -→ n→∞ ∇ u weakly in L 2 (0, T ; L 2 (Ω)),
so that, choosing q = 4 for instance, we obtain the weak convergence of (σ + ρn) un • ∇ un to (σ + ρ) u • ∇ u. Following, f (ρn, un) weakly converges in F s,λ to f (ρ, u) and, since F2 : F s,λ ×V 1 0 (Ω) → X s,λ is a linear bounded operator, we obtain that un = F ( un) = F2(f (ρn, un), u0) weakly converges to F2(f (ρ, u), u0) = F ( u) = u in X s,λ . Finally, as X s,λ is compact in L 2 (0, T ; L 2 (Ω)), un strongly converges to u in L 2 (0, T ; L 2 (Ω)).

A Proofs of Theorems 2.4 and 2.5

For simplicity, we make the proof of Theorems 2.4 and 2.5 for ν of equal to 1.This can be done without loss of generality by replacing σ and f by σ/ν and f /ν if needed.

A.1 Proof of Theorem 2.4

Let z be a smooth function on [0, T ] × O satisfying z = 0 on (0, T ) × ∂O and set

f : def = -σ∂tz -∆z, (t, x) ∈ (0, T ) × O, (A.1) Set then w = e -sϕ z. (A.2)
According to the definition of θ in (2.10), w satisfies

w(T, x) = 0, ∇w(T, x) = 0, x ∈ O, (A.3)
in addition to the conditions w(t, x) = 0 on (0, T ) × ∂O.

Besides, with f as in (A.1), w satisfies e -sϕ f = e -sϕ (-σ∂tz -∆z) = e -sϕ (-σ∂t(e sϕ w) -∆(e sϕ w)) = Pϕw, where the operator Pϕ is given by

Pϕw = -σ∂tw -sσ∂tϕw -∆w -2s∇ϕ • ∇w -s 2 |∇ϕ| 2 w -s∆ϕw. (A.4)
We now set P1, P2 and R the operators:

P1w = -σ∂tw -2s∇ϕ • ∇w + 2sλ 2 |∇ψ| 2 ξw, (A.5) P2w = -∆w -sσ∂tϕw -s 2 |∇ϕ| 2 w, (A.6) Rw = sλ∆ψξw -sλ 2 |∇ψ| 2 ξw, (A.7) so that Pϕ = P1 + P2 + R.
We then use that P1w + P2w = f e -sϕ -Rw and then

O T |P1w| 2 + O T |P2w| 2 + 2 O T P1wP2w = O T |f e -sϕ -Rw| 2 ≤ 2 O T |f | 2 e -2sϕ + 2 O T |Rw| 2 . (A.8)
The main part of the proof then consists in computing the scalar product of P1w with P2w and estimate it from below.

Computations. We write

O T P1wP2w = 3 i,j=1 Iij,
where Ii,j is the scalar product of the i-th term of P1w with the j-th term of P2w. Computation of I11.

I11 = O T σ∂tw∆w = - O T σ∂t |∇w| 2 2 - O T ∂tw∇σ • ∇w = 1 2 O σ(0)|∇w(0)| 2 + 1 2 O T ∂tσ|∇w| 2 - O T ∂tw∇σ • ∇w. (A.9) Computation of I12. I12 = s O T σ 2 ∂tw∂tϕw = - s 2 O σ 2 (0)∂tϕ(0)|w(0)| 2 - s 2 O T σ 2 ∂ttϕ|w| 2 -s O T σ∂tσ∂tϕ|w| 2 . (A.10) Computation of I13. I13 = s 2 O T σ∂tw|∇ϕ| 2 w = - s 2 2 O σ(0)|∇ϕ(0)| 2 |w(0)| 2 - s 2 2 O T σ∂t |∇ϕ| 2 |w| 2 (A.11) - s 2 2 O T ∂tσ|∇ϕ| 2 |w| 2 .
Computation of I21.

I21 = 2s O T ∇ϕ • ∇w∆w = 2s Γ T ∂nϕ|∂nw| 2 -2s O T ∇ (∇ϕ • ∇w) • ∇w = 2s Γ T ∂nϕ|∂nw| 2 -2s O T D 2 ϕ(∇w, ∇w) -s O T ∇ϕ • ∇ |∇w| 2 = s Γ T ∂nϕ|∂nw| 2 -2s O T D 2 ϕ(∇w, ∇w) + s O ∆ϕ|∇w| 2 .
(A.12)

Computation of I22. I22 = 2s 2 O T σ∇ϕ • ∇w∂tϕw = -s 2 O T div (σ∂tϕ∇ϕ)|w| 2 = -s 2 O T σ div (∂tϕ∇ϕ)|w| 2 -s 2 O T ∇σ • ∇ϕ∂tϕ|w| 2 . (A.13)
Computation of I23.

I23 = 2s 3 O T ∇ϕ • ∇w|∇ϕ| 2 w = -s 3 O T div |∇ϕ| 2 ∇ϕ |w| 2 . (A.14) Computation of I31. I31 = -2sλ 2 O T |∇ψ| 2 ξw∆w = 2sλ 2 O T |∇ψ| 2 ξ|∇w| 2 + 2sλ 2 O T ∇(|∇ψ| 2 ξ)w • ∇w. (A.15) Computation of I32. I32 = -2s 2 λ 2 O T σ|∇ψ| 2 ξ∂tϕ|w| 2 . (A.16) Computation of I33. I33 = -2s 3 λ 2 O T |∇ψ| 2 ξ|∇ϕ| 2 |w| 2 . (A.17)
Combining the above computations (A.9)-(A.17), we obtain the following:

O T P1wP2w = 1 2 O σ(0)|∇w(0)| 2 + 1 2 O |w(0)| 2 σ(0) -s 2 |∇ϕ(0)| 2 -sσ(0)∂tϕ(0) (A.18) -2s O T D 2 ϕ(∇w, ∇w) + s O T (∆ϕ + 2λ
O T ∂tσ|∇w| 2 + 2sλ 2 O T ∇(|∇ψ| 2 ξ)w • ∇w -s O T σ∂tσ∂tϕ|w| 2 - s 2 2 O T ∂tσ|∇ϕ| 2 |w| 2 -s 2 O T ∇σ∇ϕ∂tϕ|w| 2 - O T ∂tw∇σ • ∇w. (A.24)
Positivity. Our main goal now is to check that the coefficients in the above integrals are positive, except perhaps on the observation set ωT . At this step, we will strongly rely upon the choice of the weight function ϕ in (2.11), and on the formula

∂tϕ = ∂tθ θ ϕ -λ∂tψξ, ∂tξ = ∂tθ θ ξ + λ∂tψξ. (A.25)
In the following, to simplify notations, we will denote by C generic positive large constants that do not depend on s or λ and by c generic positive small constants independent of s and λ. The constants may change from line to line.

Positivity of the terms (A.18) at t = 0. Explicit computations yield -∂tϕ(0) = µ T0 (λe 6λ(m+1)e λψ(0) ) + 2λ∂tψ(0)e λψ(0) ≥ csλ 3 e λ(12m+2)

whereas |∇ϕ(0)| 2 ≤ Cλ 2 |ξ(0)| 2 ≤ Cλ 2 e 2λ(6m+1) .
Thus, with (2.2), for some λ1 > 0, taking

λ ≥ λ1 ≥ 1, inf O -s 2 |∇ϕ(0)| 2 -sσ(0)∂tϕ(0) ≥ cs 2 λ 3 e 2λ(6m+1) , (A.26)
and, following,

1 2 O σ(0)|w(0)| 2 -s 2 |∇ϕ(0)| 2 -sσ(0)∂tϕ(0) ≥ cs 2 λ 3 e 2λ(6m+1
= C(α, D 2 ψ ∞) such that ∀η ∈ R N , ∀(t, x) ∈ ωT , -2sD 2 ϕ(η, η) + s(∆ϕ + 2λ 2 |∇ψ| 2 ξ)|η| 2 ≥ csλ 2 ξ|η| 2 -Csλ 2 ξ|η| 2 .
Hence we obtain, for all λ ≥ λ1, Explicit computations yield:

-2s O T D 2 ϕ(∇w, ∇w) + s O T (∆ϕ + 2λ 2 |∇ψ| 2 ξ)|∇w| 2 ≥ csλ 2 O T ξ|∇w| 2 -
-∂t |∇ϕ| 2 -(∆ϕ + 2λ 2 |∇ψ| 2 ξ)∂tϕ = -λ 3 ξ 2 ∂tψ|∇ψ| 2 -2λ 2 ξ 2 ∇ψ • ∇∂tψ -λ 2 ξ 2 ∂tψ∆ψ (A.35) + ∂tθ θ -λ 2 ξϕ|∇ψ| 2 + λξ∆ψϕ -2λ 2 ξ 2 |∇ψ| 2 . (A.36)
Before going further, let us remark that, using ξ ≥ 1, there exists a positive constant C, only depending on the C 2 -norm of ψ such that for all λ ≥ 1, for all (t, x) ∈ (0,

T ) × O, -λ 3 ξ 2 ∂tψ|∇ψ| 2 -2λ 2 ξ 2 ∇ψ • ∇∂tψ -λ 2 ξ 2 ∂tψ∆ψ -2λ 2 ξ 2 |∇ψ| 2 ≤ Cλ 3 ξ 3 .
This estimate is sufficient to handle the terms in (A.35).

We will then focus on the terms in (A.36). First remark that on (T0, T -2T1), ∂tθ ≡ 0, so the term in (A.36) simply vanishes.

On (T -2T1, T ), we use the fact that there exists a constant C > 0 such that

∀t ∈ (T -2T1, T ), |∂tθ| ≤ Cθ 2 .
Hence there exists C = C( ∇ψ ∞ , ∆ψ ∞ ) such that for all (t, x)

∈ (T -2T1, T ) × O, ∂tθ θ -λ 2 ξϕ|∇ψ| 2 + λξ∆ψϕ -2λ 2 ξ 2 |∇ψ| 2 ≤ Cλ 2 θξϕ ≤ Cλ 3 ξ 3 , (A.37)
where for the last inequality we have used |θϕ| ≤ λξ 2 , which is a consequence of (2.13). On (0, T0), we are going to use that ∂tθ ≤ 0 and θ ∈ [1, 2] and thus the term in (A.36) has the good sign outside ωT . Indeed, using (2.8), we can find λ4 = λ4(α, ∆ψ ∞ ) ≥ λ3 such that for all λ ≥ λ4, for all (t, x) ∈ (0, T0) × O such that (t, x) / ∈ ωT ,

--λ 2 ξϕ|∇ψ| 2 + λξ∆ψϕ -2λ 2 ξ 2 |∇ψ| 2 ≥ cλ 2 ξϕ,
whereas it is bounded by Cλ 2 ξϕ everywhere in OT . We thus derive, for all λ ≥ λ4, Let us first remark that we immediately have

s 2 O T |w| 2 σ -∂t |∇ϕ| 2 -(∆ϕ + 2λ 2 |∇ψ| 2 ξ)∂tϕ ≥ cs 2 λ 2 T 0 0 O |∂tθ|ξϕ|w| 2 -Cs 2 λ 3 O T ξ 3 |w| 2 -
-λ∂ttψξ -λ 2 (∂tψ) 2 ξ ≤ Cλ 2 ξ 3 .
For t ∈ (0, T0), we further have ∀t ∈ (0, T0), |∂ttθ| ≤ Cs 2 λ 4 e λ(12m-8) , |∂tθ| ≤ Csλ 2 e λ(6m-4) , so that, on (0, T0)

|∂ttϕ| ≤ Cs 2 λ 5 e λ(12m-8) e 6λ(m+1) + Csλ 3 e λ(6m-4) ξ + Cλ 2 ξ 3 ≤ Cs 2 λ 2 ξ 3 .

For t ∈ (T -2T1, T ), we have

∀t ∈ (T -2T1, T ), |∂ttθ| ≤ Cθ 3 and |∂tθ| ≤ Cθ 2 .
Hence, using (2.13) and θϕ ≤ λξ 2 , for some positive constant

C = C( ∂tψ ∞), ∀(t, x) ∈ (T -2T1, T ) × O, |∂ttϕ| ≤ Cθ 2 ϕ + Cλθξ + Cλ 2 ξ 3 ≤ Cλ 2 ξ 3 .
Combining all these estimates, we get Here, we only have to remark that ∂nϕ ≥ 0 since ∂nψ ≤ 0 by construction, see (2.7).

s O T σ 2 |w| 2 - 1 2 ∂ttϕ ≥ -Cs 3 λ 2 O T ξ 3 |w| 2 . (A.
A bound on IR in (A.24) We also provide an upper bound on IR. First, we shall of course use the immediate estimate

1 2 O T ∂tσ|∇w| 2 ≤ C O T |∇w| 2 .
Using ∇(|∇ψ| 2 ξ) ≤ Cλξ, one easily checks that Moreover, using |∇ϕ| ≤ Cλξ, (A.25) and θϕ ≤ λξ 2 we also obtain

s 2 2 O T ∂tσ|∇ϕ| 2 |w| 2 ≤ Cs 2 λ 2 O T ξ 2 |w| 2 , s 2
O T ∇σ∇ϕ∂tϕ|w| 2 ≤ Cs 2 λ T 0 0 O ξϕ|∂tθ||w| 2 + Cs 2 λ 2 O T ξ 3 |w| 2 .
Finally, we also have

O T ∂tw∇σ • ∇w ≤ C 1 sλ O T 1 ξ |∂tw| 2 + Csλ O T ξ|∇w| 2 , (A.46)
and combining all the above estimates,

|IR| ≤ C sλ O T 1 ξ |∂tw| 2 + Csλ O T ξ|∇w| 2 + Cs 2 λ T 0 0 O ξ|∂tθ|ϕ|w| 2 + Cs 2 λ 4 O T ξ 3 |w| 2 . (A.47)
A lower bound for the cross-product P1wP2w. This step simply consists in putting together all the above estimates: for all s ≥ s1 and λ ≥ λ5,

2 O T P1wP2w ≥ O |∇w(0)| 2 + cs 2 λ 3 e 12λm+2 O |w(0)| 2 +csλ 2 O T ξ|∇w| 2 -Csλ 2 ωT ξ|∇w| 2 +cs 3 λ 4 O T ξ 3 |w| 2 + cs 2 λ 2 T 0 0 O |∂tθ|ξϕ|w| 2 -Cs 3 λ 4 ωT ξ 3 |w| 2 -Cs 2 λ 2 ωT ∩{t∈(0,T 0 )} |∂tθ|ξϕ|w| 2 -|IR|.
Thus, using (A.47), for some s2 ≥ s1 and λ6 ≥ λ5, for all s ≥ s2 and λ ≥ λ6

2 O T P1wP2w ≥ O |∇w(0)| 2 + cs 2 λ 3 e 12λm+2 O |w(0)| 2 +csλ 2 O T ξ|∇w| 2 -Csλ 2 ωT ξ|∇w| 2 + cs 3 λ 4 O T ξ 3 |w| 2 +cs 2 λ 2 T 0 0 O |∂tθ|ξϕ|w| 2 -Cs 3 λ 4 ωT ξ 3 |w| 2 -Cs 2 λ 2 ωT ∩{t∈(0,T 0 )} |∂tθ|ξϕ|w| 2 - C sλ T 0 O 1 ξ |∂tw| 2 .
(A.48)

Conclusion.

We first derive a Carleman estimate on w with gradient observations, and then explains how to remove this term using a suitable multiplier.

A Carleman estimate on w with gradient observations. According to estimates (A.8) and (A.48), for all s ≥ s2 and λ ≥ λ6,

O T |P1w| 2 + |P2w| 2 + c O |∇w(0)| 2 + cs 2 λ 3 e 12λm+2 O |w(0)| 2 +csλ 2 O T ξ|∇w| 2 + cs 3 λ 4 O T ξ 3 |w| 2 + cs 2 λ 2 T 0 0 O |∂tθ|ϕξ|w| 2 ≤ C O T |f | 2 e -2sϕ + C O T |Rw| 2 + Csλ 2 ωT ξ|∇w| 2 +Cs 3 λ 4 ωT ξ 3 |w| 2 + Cs 2 λ 2 ωT ∩{t∈(0,T 0 )} |∂tθ|ξϕ|w| 2 + C sλ O T 1 ξ |∂tw| 2 .
To handle the term Rw 2 L 2 , we recall that Rw is given by (A.7), hence

O T |Rw| 2 ≤ Cs 2 λ 4 O T ξ 3 |w| 2 .
where

C = C( ∇ψ ∞, ∆ψ ∞ ) is a positive constant. Also note that 1 sλ O T 1 ξ |∂tw| 2 ≤ C sλ O T |P1w| 2 + Csλ O T ξ|∇w| 2 + Csλ 3 O T ξ 3 |w| 2 .
In particular, for some s3 ≥ s2, for all s ≥ s3 and λ ≥ λ6,

O T |P1w| 2 + |P2w| 2 + c O |∇w(0)| 2 + cs 2 λ 3 e 12λm+2 O |w(0)| 2 +csλ 2 O T ξ|∇w| 2 + cs 3 λ 4 O T ξ 3 |w| 2 + cs 2 λ 2 T 0 0 O |∂tθ|ξϕ|w| 2 ≤ C O T |f | 2 e -2sϕ + Csλ 2 ωT ξ|∇w| 2 +Cs 3 λ 4 ωT ξ 3 |w| 2 + Cs 2 λ 2 ωT ∩{t∈(0,T 0 )} |∂tθ|ξϕ|w| 2 .
(A.49)

In (A.49), the observation is done on ωT and concerns both w and ∇w. Below, we shall explain that this observation can be done only on w provided we take an observation set slightly larger.

A Carleman estimate on w without gradient observations. Recall that ωT ⋐ ωT , then there exists a nonnegative smooth function η = η(t, x) taking value in [0, 1] such that η = 1 on ωT , and η = 0 in (0, T ) × O \ ωT . We then compute the scalar product of P2w and ηsλ 2 ξw:

O T P2w(ηsλ 2 ξw) = sλ 2 O T ηξ|∇w| 2 - sλ 2 2 O T ∆(ηξ)|w| 2 -s 2 λ 2 O T ησ∂tϕξ|w| 2 -s 3 λ 2 O T η|∇ϕ| 2 ξ|w| 2 .
In particular, using (A.25) and (2.13),

sλ 2 O T ηξ|∇w| 2 + cs 2 λ 2 T 0 0 O ση|∂tθ|ξϕ|w| 2 ≤ O T P2w(ηsλ 2 ξw) + sλ 2 O T |∆(ηξ)||w| 2 + s 2 λ 3 e 6λ(m+1) T T -2T 1 O ησ|∂tθ|ξ|w| 2 + s 2 λ 3 O T ησ|∂tψ|ξ 2 |w| 2 + s 3 λ 4 O T η|∇ψ| 2 ξ 3 |w| 2 .
Of course, this implies that

sλ 2 ωT ξ|∇w| 2 + s 2 λ 2 ωT ∩{t∈(0,T 0 )} ση|∂tθ|ξϕ|w| 2 ≤ 1 √ s O T |P2w| 2 + Cs 5/2 λ 4 O T η 2 ξ 2 |w| 2 + 2Csλ 2 O T |∆(ηξ)||w| 2 + 2Cs 2 λ 3 e 6λ(m+1) T T -2T 1 O η|∂tθ|ξ|w| 2 + 2Cs 2 λ 3 O T η|∂tψ|ξ 2 |w| 2 + 2Cs 3 λ 4 ω T |∇ψ| 2 ξ 3 |w| 2 .
But there exists a constant

C = C( η L ∞ (C 2 ) , ∇ψ ∞ , ∆ψ ∞ , ∂tψ ∞) such that |∆(ηξ)| ≤ Cλ 2 ξ 2 , sup [T -2T 1 ,T ) |∂tθ| θ 2 ≤ C,
hence, using the fact that η is supported on ωT ,

s 5/2 λ 4 O T η 2 ξ 2 |w| 2 + sλ 2 O T |∆(ηξ)||w| 2 + s 2 λ 3 O T η|∂tψ|ξ 2 |w| 2 ≤ Cs 3 λ 4 ω T ξ 3 |w| 2 , whereas s 2 λ 3 e 6λ(m+1) T T -2T 1 O η|∂tθ|ξ|w| 2 ≤ Cs 2 λ 3 e 6λ(m+1) T T -2T 1 O ηθ 2 ξ|w| 2 ≤ Cs 3 λ 4 ω T ξ 3 |w| 2 .
Hence, by combining above estimates with (A.49), for some s4 ≥ s3 and λ7 ≥ λ6, there exists a constant C such that for all s ≥ s4 and λ ≥ λ7,

O |∇w(0)| 2 + s 2 λ 3 e λ(12m+2) O |w(0)| 2 + sλ 2 O T ξ|∇w| 2 + s 3 λ 4 O T ξ 3 |w| 2 + s 2 λ 2 T 0 0 O |∂tθ|ξϕ|w| 2 ≤ C O T |f | 2 e -2sϕ + Cs 3 λ 4 ω T ξ 3 |w| 2 . (A.50)
Back to the function z. We now go back to the function z = we sϕ . For that, let us first remark that there exists a constant C = C( ∇ψ ∞ ) such that for all (t, x) ∈ (0, T ) × O,

|z| 2 e -2sϕ = |w| 2 , |∇z| 2 e -2sϕ ≤ 2|∇w| 2 + 2s 2 |∇ϕ| 2 |w| 2 ≤ 2|∇w| 2 + 2Cs 2 λ 2 ξ 2 |w| 2 .
We immediately deduce from (A.50) that for all s ≥ s4 and λ ≥ λ7, for some positive constant C,

O |∇z(0)| 2 e -2sϕ(0) + s 2 λ 3 e λ(12m+2) O |z(0)| 2 e -2sϕ(0) + sλ 2 O T ξ|∇z| 2 e -2sϕ + s 3 λ 4 O T ξ 3 |z| 2 e -2sϕ + s 2 λ 2 T 0 0 O |∂tθ|ξϕ|z| 2 e -2sϕ ≤ C O T |f | 2 e -2sϕ + Cs 3 λ 4 ω T ξ 3 |z| 2 e -2sϕ . (A.51)
We conclude the proof of Theorem 2.4 by setting s0 = s4 and λ0 = λ7.

A.2 Proof of Theorem 2.5

We divide the proof in several steps. A duality approach. To solve the control problem (2.26)-(2.27), we first rewrite the control problem under a weak form. Multiplying y solution of (2.26) by smooth functions z on [0, T ] × O such that z = 0 on [0, T ] × ∂O, we get:

O σ(T )y(T )z(T ) + O T y(-σ∂tz -∆z) = O T f z + ω T hz.
(A.52)

In particular, since σ(T ) > 0, the null-controllability requirement (2.27) is satisfied if and only if for all smooth functions z on

[0, T ] × O such that z = 0 on [0, T ] × ∂O O T y(-σ∂tz -∆z) = O T f z + ω T hz.
(A.53)

The trick now is to introduce a functional J whose Euler Lagrange equation coincide with (A.53): For smooth functions z on [0, T ] × O such that z = 0 on [0, T ] × ∂O, we define

J(z) = 1 2 O T |(-σ∂t -∆)z| 2 e -2sϕ + s 3 λ 4 2 ω T ξ 3 |z| 2 e -2sϕ - O T f z. (A.54)
But the set of smooth functions z on [0, T ] × O such that z = 0 on [0, T ] × ∂O is not a Banach space. We thus introduce

X obs = {z ∈ C ∞ ([0, T ] × O) such that z = 0 on [0, T ] × ∂O} • obs (A.55)
where • obs is the Hilbert norm defined by

z 2 obs = O T |(-σ∂t -∆)z| 2 e -2sϕ + s 3 λ 4 ω T ξ 3 |z| 2 e -2sϕ . (A.56)
The set X obs is then endowed with the Hilbert structure given by • obs . Note that here we use the fact that • obs is a norm, which is a consequence of the Carleman estimate (2.25). Also note that X obs and • obs strongly depends on s and λ and we shall follow these dependences carefully in the sequel. The functional J can be extended as a continuous functional on X obs provided (2.28). Indeed, due to (2.25), we easily have, for some constant C > 0 independent of s and λ,

O T f z ≤ C z obs 1 s 3 λ 4 O T ξ -3 |f | 2 e 2sϕ 1/2 . (A.57)
It follows that, if condition (2.28) is satisfied, the functional J can be uniquely extended as a continuous functional (still denoted the same) on X obs . Besides, (A.57) also implies the coercivity of J on X obs . Since it is also strictly convex on X obs since • obs is an Hilbert norm, J admits a unique minimizer Z on X obs . Setting In particular, (A.59) holds for all smooth functions z on [0, T ] × O such that z = 0 on [0, T ] × ∂O with z(T ) ≡ 0, which implies that Y solves the equation (2.26) with h = H in the sense of transposition. By uniqueness of solutions in the sense of transposition, this is the solution of (2.26) in the classical sense. In particular, since H ∈ L 2 (OT ), Y is C([0, T ]; L 2 (O)). Then, using again (A.59), we remark that it coincides with (A.53), hence Y solves the control requirement (2.27).

Y = (-σ∂t -∆)Ze -2sϕ and H = -s 3 λ 4 ξ 3 Ze -2sϕ 1 ω T , (
Besides, using (A.57) and the fact that J(Z) ≤ J(0) = 0, Our goal now is to obtain an estimate on ∇Y . In order to do this, for ε > 0, we introduce ϕε(t, x) : def = θε(t) λe 6λ(m+1)e ψ(t,x) , ξε(t) : def = θε(t)e ψ(t,x) and θε is given by: θε : Since the constant C is independent of ε > 0, we can pass to the limit ε → 0, and using (A.60) and the fact that σ is bounded from below away from 0, we get: Since the constant C does not depend on ε > 0, we can pass to the limit ε → 0: In order to study the regularity of ψ, we will introduce the function tout = tout(t, x) defined for (t, x) ∈ (0, T ) × O as the supremum of the time τ ∈ (t, T ] for which ∀t ′ ∈ (t, τ ), Xe(t ′ , t, x) ∈ O. It is not difficult to check that this time tout can also be characterized as the solution of ), for any ε > 0, there exists a neighborhood Vε of (tout(t, x), xout(t, x)) in [0, T ] × O such that |tout(t ′ , x ′ )tout(t, x)| < ε for all (t ′ , x ′ ) ∈ Vε. In particular, for some tε ∈ (0, T ) close to tout(t, x), Vε is a neighborhood of (tε, Xe(tε, tout(t, x), xout(t, x))) = (tε, Xe(tε, t, x)). Following, {Xe(ttε + t ′ , t ′ , x ′ ), (t ′ , x ′ ) ∈ Vε} is a neighborhood of (t, Xe(t, tε, Xe(tε, t, x))) = (t, x) on which tout is at distance at most ε of tout(t, x).

def=

  -ρ(∂tu + ((y + u) • ∇)u + (u • ∇)y)σ(u • ∇)uρ(∂ty + (y • ∇)y), ∂tρ + (y + u) • ∇ρ = -u • ∇σ in ΩT , σ∂tu + σ(y • ∇)u + σ(u • ∇)y -ν∆u + ∇p = f (ρ, u)

  22):

Proposition 3 . 5 .

 35 Introducing the weight function ℵ(t, x) : def = (ξ(t, x)) -2 e sϕ(t,x) , (3.29)

  (3.32) Proof. The proof of Theorem 3.6 follows from the precise description of ρ f and ρ b given in (3.14)-(3.15).

  .34) Indeed, once estimates (3.33)-(3.34) are proved, we can bound the L 2 (ΩT )-norm of ℵρ by the sum of the L ∞ ((0, T -2T1); L 2 (Ω))-norm of ρ f and of the L ∞ ((T0, T ); L 2 (Ω))-norm of ρ b , and estimate (3.31) immediately follows.Let us first present the proof of (3.33). We fix t ∈ [0, T -2T1]. From (3.14) and (3.30) we deduce that, for x ∈ Ω [0] (t),

Lemma 4 . 5 .

 45 Under the above assumptions, ψ ∈ C 2 ([0, T ] × O).

|w| 2 s 3 -Iw ≥cs 3 λ 4 O T ξ 3 |w| 2 + cs 2 λ 2 -Cs 3 λ 4 ωT ξ 3 |w| 2 -

 342242 40) Positivity of the terms (A.20)-(A.21)-(A.22) involving w. Here we combine the estimates in (A.34), (A.38), (A.40) in order to derive suitable estimates for the sum of the terms in (A.20)-(A.21)-(A.22). To simplify notations, let us set Iw the sum of the terms in (A.20)-(A.21)-(A.22): div (|∇ϕ| 2 ∇ϕ) -2λ 2 |∇ψ| 2 ξ|∇ϕ| 2 + s 2 σ -∂t |∇ϕ| 2 -(∆ϕ + 2λ 2 |∇ψ| 2 ξ)∂tϕ + sσ 2 -1 2 ∂ttϕ . (A.41) Putting together (A.34), (A.38), (A.40), we deduce that there exist s1 ≥ 1 and λ5 ≥ λ4 such that for s ≥ s1 and λ ≥ λ5, Cs 2 λ 2 ωT ∩{t∈(0,T 0 )} |∂tθ|ξϕ|w| 2 . (A.43) Positivity of the boundary terms (A.23).

2sλ 2 O 4 O T ξ 3 |w| 2 + Cλ 2 O T ξ|∇w| 2 .sλ 2 T σ∂tσ∂tϕ|w| 2 ≤

 242222 T ∇(|∇ψ| 2 ξ)w • ∇w ≤ Cs 2 λ e λ(6m-4) λe 6λ(m+1) + Cλξ on (0, T0), Cλξ on (T0, T -2T1), θλe 6λ(m+1) + Cλξ on (T -2T1, T ), so that |∂tϕ| ≤ Csλξ 3 everywhere. Hence s O Cs 2 λ O T ξ 3 |w| 2 . (A.45)

  A.58) writing the Euler Lagrange equation of J at Z, for all smooth functions z on [0, T ] × O such that z = 0 on [0, T ] × ∂O,

s 3 λ 4 O|Y | 2 e 2sϕ + ω T ξ - 3

 43 T |H| 2 e 2sϕ ≤ C O T ξ -3 |f | 2 e 2sϕ .(A.60)Estimates on ∇Y . In the previous step, we found (Y, H) satisfying the equations       ∂t(σY ) -∆Y = f + H1 ω T , in OT , Y = 0, in ΓT , Y (0, •) = 0, in O, Y (T, •) = 0, in O. (A.61)and the estimates (A.60).

def=|Y | 2 σ|Y | 2 ∂t ξ -2 ε e 2sϕε ≥ cs 2 λ 2 T 0 0 Oσ|∂tθ|ξ - 2 ϕ|Y | 2 e 2sϕ -Cs 2 λ 3 T 0 0 O 3 T T 0 O|Y | 2 ∂tσξ -2 ε e 2sϕε ≤ Cs 3 λ 4 O 2 O T ξ - 2 ε

 22022sϕ3030422 θε(t) such that              ∀t ∈ [0, T0], θε(t) = 1 + 1 -t T0 µ , ∀t ∈ [T0, T -2T1 + ε], θε(t) = 1, ∀t ∈ [T -2T1 + ε, T ), θε(t) = θ(tε),µ as in (2.12).We then multiply the equation (A.61) by ξ -2 ε Y e 2sϕε : ∂tσξ -2 ε e 2sϕε . (A.62)We then compute explicitly:e -2sϕε ∂t ξ -2 ε e 2sϕε = 2sλξ -1 ε ∂tψ -2sξ -2 T0), we remove the dependence in ε > 0 as θε = θ on (0, T0). Using (2.13), ∂tθ ≤ 0 and θ ∈[START_REF] Albano | Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system[END_REF][START_REF] Boyer | Trace theorems and spatial continuity properties for the solutions of the transport equation[END_REF] in [0, T0] we have, for all s ≥ s0 and t ∈ (0, T0),-2sξ -2 ∂tθ θ ϕ + 2 ∂tθ θ ξ -2 ≥ cs|∂tθ|ξ -2 ϕ,whereas 2sλξ -1 ∂tψ + 2λ∂tψξ -2 ≤ Csλξ -1 . |Y | 2 e 2sϕ . (A.64) On (T0, T ), from the identity (A.63), using |∂tθε| ≤ Cθ 2 ε σ|Y | 2 ∂t ξ -2 ε e 2sϕε ≤ Cs 2 λ |Y | 2 e 2sϕε . (A.65) Straightforward computations yield ∆ ξ -2 ε e 2sϕε ≤ Cs 2 λ 2 e 2sϕε , from which we get sλ 2 2 O T |Y | 2 ∆ ξ -2 ε e 2sϕε ≤ Cs 3 λ 4 O T |Y | 2 e 2sϕε . T |Y | 2 e 2sϕε , (A.68) combining estimates (A.64)-(A.65)-(A.66)-(A.67)-(A.68) and plugging (A.62), we obtain sλ |∇Y | 2 e 2sϕε + s 2 λ 2 T 0 0 O σ|∂tθ|ξ -2 |Y | 2 e 2sϕ ≤ Cs 3 λ 4 O T |Y | 2 e 2sϕε + C ω T ξ -3 ε |H| 2 e 2sϕε + C O T ξ -3 ε |f | 2 e 2sϕε .

sλ 2 Oξ - 2 |∇Y | 2 e 2sϕ + s 2 λ 2 T 0 0 Oξ - 2 ε e 2sϕε )|∇Y | 2 ≤ 2 O T ξ - 2 ε| 2 e 2sϕε ≤ Csλ 2 O T ξ - 2 ε |∇Y | 2 e 2sϕε + Cs 3 λ 4 Oε |f | 2 e 2sϕε + C ω T ξ - 3 ε

 222022222243 T |∂tθ|ξ -2 ϕ|Y | 2 e 2sϕ ≤ C O T ξ -3 |f | 2 e 2sϕ . (A.69) Estimates on ∆Y , ∂tY . Multiplying the equation (A.61) by -ξ -4 ε ∆Y e 2sϕε /s, V 1 ω T -∂tσY )ξ -4 ε ∆Y e 2sϕε . (A.70) As in (A.63), we compute explicitly -∂t(ξ -4 ε e 2sϕε ). Arguing as in (A.64), we get -|∇Y | 2 e 2sϕ . (A.71) Besides, arguing as in (A.65), we get -Csλ |∇Y | 2 e 2sϕε . (A.73)We then estimate the cross-term of (A.70):1 s O T ∂tY ∇Y • ∇(σξ -4 ε e 2sϕε ) ε |∂tY | 2 e 2sϕε + Csλ 2 O T ξ -2 ε |∇Y | 2 e 2sϕε , (A.74)where σmin : def = min O T σ. From the equation (A.61), ∂tY = 1 σ (∆Y + f + H1ω T -∂tσY ) , H1ω T -∂tσY )ξ -4 ε ∆Y e 2sϕε ≤ |H| 2 e 2sϕε .

ξ - 4 |∆Y | 2 e 2sϕ ≤ Csλ 2 O T ξ - 2 | 2 e 2sϕε + Csλ 2 O T ξ - 2 ε

 42222 |∇Y | 2 e 2sϕ + Cs 3 λ 4 O T |Y | 2 e 2sϕ + C O T ξ -3 |f | 2 e 2sϕ + C ω T ξ -3 |H| 2 e 2sϕ .Using now estimates (A.60), (A.69) and (A.76), we get1 s O T ξ -4 (|∂tY | 2 + |∆Y | 2 )e 2sϕ ≤ C O T ξ -3 |f | 2 e 2sϕ .(A.78)Estimates on ∂nY in L 2 (ΓT ). Let η : O → R N such that η ∈ C 2 (O; R N) and η = n on ∂O. Since Y vanishes on ΓT , we have the following identity: for all ε > 0|∇Y | 2 e 2sϕε .Passing to the limit in ε → 0 and using (A.69) and (A.78) we thus obtainλ T 0 ∂O ξ -3 |∂nY | 2 e 2sϕ ≤ C O T ξ -3 |f |2 e 2sϕ . (A.79) Conclusion. Estimates (A.60), (A.69), (A.78) and (A.79) yield (2.29).B Regularity of the weight functionProof of Lemma 4.5. The first remark is that the flow Xe is C 2 ([0, T ] × [0, T ] × R 2 ) since y e ∈ C 2 ([0, T ] × R 2 ).

OT, 1 =

 1 e • ∇tout = 0 in OT , tout(t) = t on ΓT , tout(T ) = T in O. (B.1)For convenience, we also setxout(t, x) = Xe(tout(t, x), t, x). (B.2)We first prove that tout is continuous in OT . In order to do that, let us remark that Xeis C 2 ([0, T ] × [0, T ] × R 2 ) and for all (t, τ ) ∈ [0, T ] 2 , Xe(t, τ, •) is a C 2 diffeomorphism of R 2 .In particular, OT can be decomposed intoOT = OT,1 ∪ OT,2 ∪ ΣT , {(t, x) ∈ (0, T ) × O, x ∈ Xe(t, T, O)}, OT,2 = {(t, x) ∈ (0, T ) × O, x ∈ Xe(t, T, R 2 \ O)}, ΣT = {(t, x) ∈ (0, T ) × O, x ∈ Xe(t,T, ∂O)}. (B.3) In (B.3), OT,1 and OT,2 are open sets whereas ΣT = OT,1 ∩OT,2 is closed and of dimension 2. For (t, x) ∈ OT,1 ∪ΣT , tout(t, x) = T and tout is thus continuous on OT,1. The continuity on OT,2 is more involved. If (t, x) ∈ OT,2, then xout(t, x) belongs to ∂O. Due to the condition (4.5

  Theorem 2.1. Assume that O is a smooth bounded domain extending Ω as in (2.1), let ω, ω be two subdomains of O\Ω such that ω ⋐ ω and set ωT = [0, T ] × ω and ωT = [0, T ] × ω. Let ψ as in (2.7)-(2.8) and ψ, θ, ϕ, ξ as in (2.9)-(2.10)-(2.11).

					) .	(2.13)
	Finally, we introduce
	ϕ(t) : def = min x∈O	ϕ(t, x),	ϕ * (t) : def = max x∈O	ϕ(t, x) = ϕ |∂O (t),	(2.14)
	ξ(t) : def = max x∈O	ξ(t, x),	ξ * (t) : def = min x∈O	ξ(t, x) = ξ |∂O (t).	(2.15)
	Using these weight functions, we prove the following Carleman estimate for the Stokes
	system (2.6):

  .11) From the equation(4.11), if the point xc is a critical point for ψ(tc, •), then for all t in a neighborhood around tc, Xe(t, tc, xc) is a critical point for ψ(t, •). Note that this neighborhood actually correspond to the set Ic of time t ∈ [0, T ] such that the trajectory τ → Xe(τ, tc, xc) stays in O for τ between t and tc.

  Positivity of the terms (A.[START_REF] Imanuvilov | Carleman estimates for parabolic equations with nonhomogeneous boundary conditions[END_REF]) involving the gradient. For η ∈ R N , we have-2sD 2 ϕ(η, η) + s(∆ϕ + 2λ 2 |∇ψ| 2 ξ)|η| 2 = 2sλ 2 ξ|∇ψ • η| 2 + sλ 2 ξ|∇ψ| 2 |η| 2 + 2sλξD 2 ψ(η, η) -sλξ∆ψ|η| 2 . (A.28)Using (2.8), we get the existence of λ2 = λ2(α, D 2 ψ ∞) ≥ λ1 such that for all λ ≥ λ2 and η ∈ R N ,∀(t, x) ∈ OT \ ωT , -2sD 2 ϕ(η, η) + s(∆ϕ + 2λ 2 |∇ψ| 2 ξ)|η| 2 ≥ csλ 2 |η| 2 ξ,

	)	O	|w(0)| 2 .	(A.27)
				(A.29)
	whereas there exists a positive constant C

  Positivity of the terms (A.20) involving w with scale s 3 . Using ∇ϕ = -λ∇ψξ, we havediv (|∇ϕ| 2 ∇ϕ) = 3λ 4 |∇ψ| 4 ξ 3 + λ 3 ξ 3 div (|∇ψ| 2 ∇ψ), λ 2 |∇ψ| 2 ξ|∇ϕ| 2 = λ 4 |∇ψ| 4 ξ 3 . Hence div (|∇ϕ| 2 ∇ϕ) -2λ 2 |∇ψ| 2 ξ|∇ϕ| 2 = λ 4 |∇ψ| 4 ξ 3 + λ 3 ξ 3 div (|∇ψ|2 ∇ψ). (A.31) Using (2.8), we thus get the existence of λ3 = λ3(α, D 2 ψ ∞) ≥ λ2 such that for λ ≥ λ3, ∀(t, x) ∈ OT \ ωT ,div (|∇ϕ| 2 ∇ϕ) -2λ 2 |∇ψ| 2 ξ|∇ϕ| 2 ≥ cλ 4 ξ 3 . (A.32) whereas there exists a positive constant C = C(α, D 2 ψ ∞) such that ∀(t, x) ∈ ωT ,div (|∇ϕ| 2 ∇ϕ) -2λ 2 |∇ψ| 2 ξ|∇ϕ| 2 ≥ cλ 4 ξ 3 -Cλ 4 ξ 3 . Terms (A.21) involving w in the scale s 2 . We have to estimate -∂t |∇ϕ| 2 -(∆ϕ + 2λ 2 |∇ψ| 2 ξ)∂tϕ.

						(A.33)
	We thus obtain, for all λ ≥ λ3,
	s 3	O T	|w| 2 -div (|∇ϕ| 2 ∇ϕ) -2λ 2 |∇ψ| 2 ξ|∇ϕ| 2
			≥ cs 3 λ 4	O T	ξ 3 |w| 2 -Cs 3 λ 4	ωT	ξ 3 |w| 2 . (A.34)
					Csλ 2	ξ|∇w| 2 . (A.30)
						ωT

  Term (A.22) involving w in the scale s. We have to estimate -∂ttϕ.

					Cs 2 λ 2	ωT ∩{t∈(0,T 0 )}	|∂tθ|ξϕ|w| 2 . (A.38)
	∂ttϕ =	∂ttθ θ	ϕ -2λ	∂tθ θ	∂tψξ -λ∂ttψξ -λ 2 (∂tψ) 2 ξ	(A.39)

sϕ * u. The computations are left to the reader.

This section is devoted to explain how to solve the control problem(1.18). As we said in the introduction, the main difficulty is that we need to provide a controlled trajectory that can be estimated with the use of the weight functions introduced in Section 2.

sϕ * (0) u0 H 1 0 (Ω) .(4.3)

* This work is partially supported by the Agence Nationale de la Recherche (ANR, France), Project CISIFS number NT09-437023.

Thus, tout is continuous in OT . As ψ solution of (4.8) can be written as ψ(t, x) = ψT (xout(t, x)) if tout(t, x) = T, tout(t, x) -T if tout(t, x) < T, (B.4) the continuity of ψ in OT follows from the first compatibility condition in (4.7). Also note that ψ is obviously C 2 in OT,1.

We then focus on the C 1 regularity of ψ. In order to do this, we remark that ∇tout solves

In particular, ∇tout can be computed for any (t, x) ∈ OT,2 by solving for τ between t and tout(t, x) the ODE

.

One then easily obtains that ∇tout is C 0 on OT,2 and from the equation (B.1) we deduce that tout is C 1 in OT,2. From there, we derived immediately from (B.4) that ψ is C 1 on OT,2 and that it can be extended as a C 1 funtion on OT,2 as follows: ∇ ψ can be computed for any (t, x) ∈ ΣT by solving for τ between t and T the ODE:

with ∇ ψ(T, Xe(T, t, x)) = -n(Xe(T, t, x)) y e (T, Xe(T, t, x)) • n(Xe(T, t, x))

.

On the other hand, ψ solves the equation (4.11), and can be extended as a C 1 function on OT,1. For (t, x) ∈ ΣT , this yields ∇ ψ(t, x) as the solution of the ODE (B.6) with ∇ ψ(T, Xe(T, t, x)) given. But, as ψ(T ) is constant on the boundary and satisfies the second compatibility condition in (4.5), we get again (B.7) for (t, x) ∈ ΣT . Following, ∇ ψ is continuous across ΣT , hence on OT . Using the equation (4.8), ψ belongs to C 1 (OT ). The proof of the C 2 regularity follows the same path and is left to the reader.