Adrien Tendani Soler
email: adrien.tendani-soler@math.u-bordeaux.fr

Local exact controllability for Navier-Stokes-Korteweg model in dimension d ∈ {2, 3}

In this paper, we investigate the local null controllability for a compressible Navier-Stokes-Korteweg with quantum pressure in dimension d ∈ {2, 3}, when the control acts on the whole boundary of a domain or on a subdomain of a torus. We obtain the local null controllability for the velocity and the fluctuation of the density, for all positive times, according to the hidden parabolic structure of the equation. This contrasts with standard control properties for the compressible Navier-Stokes equation. To the best of our knowledge, this is the first work concerning the controlability of Korteweg type system for compressible fluids.

Introduction

We are interested by the local null controllability properties of the Navier-Stokes-Korteweg system which describes a two-phase compressible and viscous fluids, of density ρ and velocity field u. It is generally assumed that phases are separated by a hypersurface and that the jump in the pressure across the interface is proportional to the curvature. Here, we deal with a diffusive interface (DI) model that describes fluids when the change of phase corresponds to a fast but regular transition zone for the density and velocity. This type of models differs from the so-called sharp interface (SI) model when the interface between phases corresponds to a discontinuity in the state space. The basic ideas of the DI model considered here, is to add to the classical compressible fluids equation a capillary term, that penalizes high variations of the density. We refer to [START_REF] Coquel | Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows[END_REF] for the modelling of phase transition. The full derivation of the corresponding equation, that we shall name the compressible Navier-Stokes-Korteweg system is due to J. E. Dunn and J. Serrin (see [START_REF] Dunn | On the thermodynamics of interstitial working[END_REF]).

    

∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) -Au + ∇P (ρ) = div(K), (ρ, u)| t=0 = (ρ 0 , u 0 ), [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] where P (ρ) is the pressure function, Au := div (2µ(ρ)D S (u)) + ∇(ν(ρ) div u)) is the diffusion operator, D S (u) := 1 2 (∇u + t ∇u) is the symmetric gradient and the capillarity tensor is given by ( see [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF])

K := ρ div(κ(ρ)∇ρ)I R d + 1 2 κ(ρ) -ρκ ′ (ρ) |∇ρ| 2 I R d -κ(ρ)∇ρ ⊗ ∇ρ.
This system is due to J. E. Dunn and J. Serrin in [START_REF] Dunn | On the thermodynamics of interstitial working[END_REF]. The density-dependent capillarity function κ is assumed to be positive. If the the capillarity coefficient κ ≡ 0, then System (1) reduces to the classical Navier-Stokes system of compressible fluids. Note that for smooth enough density ρ and capillarity function κ, we have

div K = ρ∇ κ(ρ)∆ρ + 1 2 κ ′ (ρ)|∇ρ| 2 .
The coefficients ν = ν(ρ) and µ = µ(ρ) designate the bulk and shear viscosity, respectively, and are assumed to satisfy in the neighborhood of some reference constant density ρ > 0 the conditions µ > 0 and ν + µ > 0.

We shall assume that the functions λ, µ, κ and P are real analytic in a neighborhood of ρ. To simplify, we set ρ = 1. Introducing a = ρ -1 and denoting by μ = µ(1), ν = ν(1), κ = κ(1), ᾱ = P ′ (1), the system (1) reads

∂ t a + div(u) = f , ∂ t u -Āu + ᾱ∇a -κ∇∆a = g, (2)
where Āu = 2μ div(D S (u)) + ν∇ div u, f = -div(au), g = 4 i=1 gi with The system (2) is a hyperbolic/parabolic coupled system, which is common for compressible Navier-Stokes type systems. Existence of strong solutions was known since the works by Hattori and Li [START_REF] Hattori | Global solutions of a high dimensional system for Korteweg materials[END_REF] in 1996. Then, in 2001, Danchin and Desjardins, studied the well-posedness of the Cauchy problem in critical Besov spaces for global and local solutions and in dimension d ≥ 2 in [START_REF] Danchin | Existence of solutions for compressible fluid models of Korteweg type[END_REF].

             g1 := -u •
In the following subsection, we describe a special case of the compressible Navier-Stokes-Korteweg system, so-called the incompressible Navier-Stokes system with quantum pressure, that will be discussed in this paper.

Compressible Navier-Stokes-Korteweg system with quantum pressure

In this paper, we consider a special case, which is the so-called compressible Navier-Stokes-Korteweg system with quantum pressure. We recall how to derive this equation system from the general system [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF]. Let us set (µ(ρ), ν(ρ), κ(ρ)) = (µρ, νρ, κ/ρ), P (ρ) = αρ, where µ > 0, µ + ν > 0, κ > 0, α > 0 are constants.

Introducing ρ = ρe a ,
where ρ is a constant, the system (1) reads

∂ t a + div(u) = f a (u, a), ∂ t u -µ △u -(µ + ν)∇ div(u) + α∇a -κ∇ △a = f u (u, a), (3)
where f a (a, u) := -u • ∇a,

f u (a, u) := -u • ∇u + µ∇a • ∇u + (µ + ν)∇a • D S u + κ 2 ∇(∇a • ∇a). (4)
The system (3) is called the compressible Navier-Stokes-Korteweg system with quantum pressure. The Quantum Navier-Stokes equation (that is closed to the last system) has been first derived in [START_REF] Brull | Derivation of viscous correction terms for the isothermal quantum Euler model[END_REF] from the Wigner equation by Brull and Méhats (see also [START_REF] Jüngel | Quantum Navier-Stokes equations[END_REF]), and the link with Navier-Stokes-Kortewg system is done in [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension n ≥ 2[END_REF] (see also [START_REF] Jüngel | Global weak solutions to compressible Navier-Stokes equations for quantum fluids[END_REF]). Recently, the analytic smoothing effect in space variable for this equation for both of the velocity field and the density has been discovered by Charve, Danchin and Xu in [START_REF] Charve | Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity[END_REF] (see also [START_REF] Soler | Analytic regularity for Navier-Stokes-Korteweg model on pseudo-measure spaces[END_REF] and [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF]). In all these work, it was shown, that solutions of system [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF] are space analytic at every positive time. This property can be established as consequence of a dissipative estimate on the Fourier modes of the solutions (that we recall in the Lemma 2.0.1) to the linearized system linked with parabolic structure of the system (see Section 2). The space analyticity of the solution pointing out the infinite speed of propagation of the solution, similarly of semi-linear heat equations.

Based on these considerations, we will show that the controllability property of the the Navier-Stokes-Korteweg system are similar to those of parabolic type equation as suggested the instantaneous smoothing effect and the infinite speed of propagation: that is, the local nul controllability at any positive time. This is in contrast with the controllability results for the classical compressible Navier-Stokes system as in [START_REF] Ervedoza | Local exact controllability for the two-and threedimensional compressible Navier-Stokes equations[END_REF], [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] and [START_REF] Ervedoza | Local exact controllability for the onedimensional compressible navier-stokes equation[END_REF] (see also [START_REF] Molina | Local exact boundary controllability for the compressible Navier-Stokes equations[END_REF]) and for the compressible two-phase model as in [START_REF] Tao | Local exact controllability for a viscous compressible two-phase model[END_REF].

When we consider the controllability of the linearized compressible Navier-Stokes system, the transport property of the compressible Navier-Stokes system leads to a restriction of the time of controllability which cannot be arbitrarly small positive time. The consequence of the transport phenomena on the controllability of the compressible Navier-Stokes equation have been developed and explained in the articles [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF], [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF] and [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] focusing on the linearized equation in the case of zero-velocity and is now well-known. Following the strategy of [START_REF] Ervedoza | Local exact controllability for the two-and threedimensional compressible Navier-Stokes equations[END_REF], in order to perform a perturbative argument, we will investigate the linearied Navier-Stokes-Korteweg system. Using a new unknown, we can reduce this last one to a coupled parabolic system which is a closed parabolic subsystem of dimension 2. Under the assumption (ν + 2µ) 2 ≥ 4κ, we can reduce the study to the controllability of two coupled heat equations. For this purpose, we will use the Carleman inequality developed in [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF]. Moreover, in order to obtain the L 2 (H 4 ) × L 2 (H 3 ) regularity for (a, u), we will use the duality between controllability and observability for linear systems and perform observability inequality in negative Sobolev spaces in the spirit of [START_REF] Imanuvilov | Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations[END_REF] or [START_REF] Molina | Local exact boundary controllability for the compressible Navier-Stokes equations[END_REF].

Notation 1.0.1. Throughout this paper, f ≲ a 1 ,...,a k g means that there exists a positive constant C, which depends on the parameters a 1 , . . . , a k such that f ≤ Cg. To simplifies when the context requires it, we will just write f ≲ g, where we omit the dependence on parameter a 1 , . . . , a k .

2 Regularity estimate for linear Navier-Stokes-Korteweg system

For this section we assume that d ≥ 2. We will give some regularity results to the linearized Navier-Stokes-Korteweg system. Namely

∂ t a + div(u) = f, in [0, T ] × T L , ∂ t u -µ △u -(µ + ν)∇ div(u) + α∇a -κ∇ △a = g, in [0, T ] × T L . (5)
We consider the initial value problem

(a, u) | t=0 = (a 0 , u 0 ). (6)
In order to recall some results for the linear system, we will highlight the parabolic property of the linear system. To simplify the argument we assume that α = 0 (in this case the term α∇a, that does not influences overs the dissipation, vanishes) and we consider only the free part (with f = g = 0). Let us introduce the Leray projector, P := Id -∇ div △ and the projector overs potential vector field Q := Id -P (see [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]). Then, the linear system can be rewritten in term of divergence-free part Pu and potential part (also called compressible part) one Qu according to the following subsystem

   ∂ t a + div(Qu) = 0, in (0, T ) × T L , ∂ t Qu -(2µ + ν) △Qu -κ∇ △a = 0, in (0, T ) × T L , ∂ t Pu -µ △Pu = 0, in (0, T ) × T L . (7)
The two last equations has been obtain to apply Q and P respectively to the second line of (5). To reveal the "transfers of parabolicity" phenomenon from the velocity field to the density we begin to remark the devergence-free part that the divergence-free part Pu satisfies a classical heat equation.

On the other hand, we introduce v := |D| -1 div u, so that | Qu| ≤ | v| adn applying |D| -1 div to the second line of (7), we get the following closed subsystem of (7) of dimension 2 × 2

∂ t a v + 0 -|D| κ|D| 3 -(ν + 2µ)|D| 2 a v = 0 0 . (8)
We can interpret the parabolic transfer to making the spectral analysis of the matrix

A(ξ) := 0 -|ξ| κ|ξ| 3 -(ν + 2µ)|ξ| 2 . 1. If (ν + 2µ) 2 ≥ 4k, then A(ξ) has two real eigenvalues -(ν + 2µ) 2 ± (ν + 2µ) 2 -4κ 2 |ξ| 2 , 2. If (ν + 2µ) 2 < 4k, then A(ξ) has two complex eigenvalues -(ν + 2µ) 2 ± i 4κ -(ν + 2µ) 2 2 |ξ| 2 .
According to the dissipation definition for general hyperbolic-parabolic systems with dispersion formulated in [START_REF] Kawashima | Dissipative structure for symmetric hyperbolic-parabolic systems with korteweg-type dispersion[END_REF], we say that the system is purely parabolic in the case 1. In the two cases 1 and 2, the structure of eigenvalue of A(ξ) reveals the dissipative property on the Fourier modes of (a, u).

We recall the Charve-Danchin-Xu estimate (see [START_REF] Charve | Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity[END_REF]) for the solution of linearized system (5) which embodies this dissipative structure.

Lemma 2.0.1. Let d ≥ 2 and T > 0. There exists a positive constant c 0 , depending only on (κ, µ, L, α, ν), such that, for all (a, u) solution of (5) smooth enough on [0, T ] × T L , we have the following inequality

|(⟨ξ⟩ a, u)|(t, ξ) ≲ κ,µ,L,α,ν e -c 0 t|ξ| 2 |(⟨ξ⟩ a, u)|(0, ξ) + t 0 e -c 0 |ξ| 2 (t-τ ) |(⟨ξ⟩ f , g)|(τ, ξ)|dτ, (9)
holds for all ξ ∈ Z d and t ∈ [0, T ].

From this estimate, we deduce here the following regularity result for solutions of the linear Navier-Stokes-Korteweg system.

Lemma 2.0.2. Let d ≥ 0, T > 0 and σ ≥ 2. Let (a 0 , u 0 ) ∈ H σ (T L ) × H σ-1 (T L ) and (f, g) ∈ L 2 (0, T ; H σ-1 (T L )) × L 2 (0, T ; H σ-2 (T L )). If (a, u) is a solution of the Cauchy problem (5)(6), then (a, u) ∈ C([0, T ]; H σ (T L )) ∩ L 2 (0, T ; H σ+1 (T L )) × C([0, T ]; H σ-1 (T L )) ∩ L 2 (0, T ; H σ (T L ))).
Moreover, for all p ∈ [2, +∞[∪{∞}, there exists a positive constant C p such that

∥(a, u)∥ L p (H σ+ 2 p )×L p (H σ-1+ 2 p ) ≤ C p ∥(a 0 , u 0 )∥ H σ ×H σ-1 + ∥(f, g)∥ L 2 (H σ-1 )×L 2 (H σ-2 ) ,
and the constant C p depend only of space and time parameters L and T , of physical parameters κ, µ, ν and of regularity parameters σ and p. Moreover, we have the following H 1 -time regularity

(a, u) ∈ H 1 (0, T ; H σ-1 (T L )) × H 1 (0, T ; H σ-2 (T L )))
and there exist a constante K > 0 that depend only of d, L, T, K, µ, ν and α such that

∥(a, u)∥ H 1 (H σ-1 )×H 1 (H σ-2 ) ≤ K ∥(a 0 , u 0 )∥ H σ ×H σ-1 + ∥(f, g)∥ L 2 (H σ-1 )×L 2 (H σ-2 ) .
The proof of this lemma, from the Charve-Danchin-Xu estimate, is given in the appendix A.

Main results

Let Ω be a smooth bounded domain of R d . First we are controlling the whole external boundary (0, T ) × ∂Ω. For this reason the controls do not appear explicitly in [START_REF] Benzoni-Gavage | Structure of Korteweg models and stability of diffuse interfaces[END_REF]. In this direction, our main result related to the local exact cotrollability of (3) around the stat (0, 0) is the following theorem.

Theorem 3.0.1. Supposed that (2µ + ν) 2 ≥ 4κ. Let d ∈ {2, 3} and T > 0. There exists δ > 0 such that for all (a 0 , u 0 ) ∈ H 3 (Ω) × H 2 (Ω) satisfying ∥(a 0 , u 0 )∥ H 3 (Ω)×H 2 (Ω) ≤ δ, (10)
there exists a solution (a, u) of (3) with initial data

(a, u)| t=0 = (a 0 , u 0 ), in Ω,
and satisfying the control requirement, (a, u)| t=T = (0, 0), in Ω.

Beside, the controlled trajectory (a, u) has the following regularity

(a, u) ∈ C([0, T ]; H 3 (Ω)) ∩ L 2 (0, T ; H 4 (Ω)) × C([0, T ]; H 2 (Ω)) ∩ L 2 (0, T ; H 3 (Ω)).
Since we are controlling the whole external boundary, Ω can be embedded into some torus, where T L is indentified with [0, L] d , with periodic condition. The length L is large enough for Ω ⊂ T L and we may consider the control problem in the cube [0, L] d completeed with periodic boundary condition with controls appearing as some terms supported in T L \ Ω. This leads to study the internal controllability problem associate to (3) on T L . Then we consider the following control system:

∂ t a + div(u) = f a (a, u) + v a 1 T L \Ω , in (0, T ) × T L , ∂ t u -µ △u -(µ + ν)∇ div(u) + α∇a -κ∇ △a = f u (a, u) + v u 1 T L \Ω , in (0, T ) × T L , (11)
where we denote by 1 T L \Ω the indicators function of T L \ Ω. In this system the control (v a , v u ) appear.

Our main results for the local exact controllability of ( 11) is the following theorem.

Theorem 3.0.2. Supposed that (2µ + ν) 2 ≥ 4κ. Let d ∈ {2, 3} and T > 0. There exist δ > 0 such that, for all (a 0 , u 0

) ∈ H 3 (T L ) × H 2 (T L ) satisfying ∥(a 0 , u 0 )∥ H 3 ×H 2 ≤ δ, there exists a control (v a , v u ) ∈ L 2 (H 2 )×L 2 (H 1
) and a corresponding controlled trajectory (a, u) solving [START_REF] Ervedoza | Local exact controllability for the onedimensional compressible navier-stokes equation[END_REF] with initial data (a 0 , u 0 ) satisfying the control requirement

(a, u) | t=T = (0, 0).
Besides, the controlled trajectory (a, u) has the following regularity

(a, u) ∈ C(0, T ; H 3 (T L )) ∩ L 2 (0, T ; H 4 (T L )) × C(0, T ; H 2 (T L )) ∩ L 2 (0, T ; H 3 (T L )).
This theorem is direct consequence of Theorem 9.0.1. The necessity of smoothness assumption on the domain Ω, comes from (at least partially) to the fact that we do expanse the initial data over Ω to the torus T L to deduce the Theorem 3.0.1 from Theorem 3.0.2 and is not intrinsic to the local controllability of [START_REF] Ervedoza | Local exact controllability for the onedimensional compressible navier-stokes equation[END_REF].

General strategy for controllability

We begin to specify some notations using in this paper. Notation 4.0.1. To simplify the text we set, for any (s, σ, p)

∈ Z × R × ([1, +∞[∪{∞}) ∥ • ∥ H s (H σ ) := ∥ • ∥ H s (0,T ;H σ (T L )) , ∥ • ∥ L p (H σ ) := ∥ • ∥ L p (0,T ;H σ (T L )) and ∥ • ∥ H σ := ∥ • ∥ H σ (T L ) .
To take the support of the control functions (v a , v u )1 Ω with more flexibility, we replace

1 Ω by the smooth cut-off function χ ∈ C ∞ c (T L , [0, 1]) satisfying χ(x) = 0, for all x such that d(x, Ω) ≤ ε, χ(x) = 1, for all x such that d(x, Ω) ≥ 3ε (12)
for ε > 0 small enough. We consider the following control problem : Given (a 0 , u 0 ) small in H 3 (T L ) × H 2 (T L ), find control functions v a and v u such that the solution of

∂ t a + div(u) = f a (a, u) + v a χ, in [0, T ] × T L , ∂ t u -µ △u -(µ + ν)∇ div(u) + α∇a -κ∇ △a = f u (a, u) + v u χ, in [0, T ] × T L , (13)
with initial data

(a, u) | t=0 = (a 0 , u 0 ), in T L , (14) satisfies
(a, u) | t=T = (0, 0), in T L . (15)
Now, we are reduced to study the controllability of the linearized system

∂ t a + div(u) = f a + v a χ, in [0, T ] × T L , ∂ t u -µ △u -(µ + ν)∇ div(u) + α∇a -κ∇ △a = f u + v u χ, in [0, T ] × T L . ( 16
)
Since this is a linear system, the controllability of ( 16) is equivalent to the observability property for the adjoint equation

-∂ t σ -α div(z) + κ △ div(z) = g σ , in [0, T ] × T L , -∂ t z -∇σ -µ △z -(µ + ν)∇ div(z) = g z , in [0, T ] × T L . ( 17
)
The main idea to get an observability inequality for ( 17) is to remark that, taking the divergence of the equation of z, the equation of σ and div(z) form a closed coupled system:

-∂ t σ -αq + κ △q = g σ , in [0, T ] × T L , -∂ t q -△σ -(2µ + ν) △q = div(g z ), in [0, T ] × T L , (18)
with q := div(z).

The choice to take only the divergence of the velocity field instead of |D| -1 div(z), which appear in the standard interpretation of the regularity-gain phenomenon for this type of system (see [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF] and [START_REF] Charve | Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity[END_REF]), is related to the difficulty to perform global Carleman inequality with nonlocal operator |D| that appear in [START_REF] Danchin | Existence of solutions for compressible fluid models of Korteweg type[END_REF]. In order to add a margin on the control zone we introduce a smooth cut-off function χ 0 , satisfying: supp{χ 0 } ⊂ {χ = 1} and χ 0 (x) = 1 for all x ∈ T L such that d(x, Ω) ≤ 3ε.

In this paper, we assume that

(2µ + ν) 2 ≥ 4κ. (19)
The hypothesis [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF] is a sufficient condition to ensures that of the matrix

0 κ -1 -(2µ + ν) , (20)
have reals eigenvalues. If [START_REF] Molina | Local exact boundary controllability for the compressible Navier-Stokes equations[END_REF] is diagonalisable, this lead to decoupling the highest order derivative terms. The lower orders terms can be "absorbed" by using Carleman estimate. In the case where case the matrix is non diagonalisable we can also conclude with modified argument from the diagonalisable case.

Diagonalisable case

We use a new unknown σ q := Q σ q ,

where δ := (2µ + ν) 2 -4κ is real number since [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF],

λ + := -(2µ+ν)+δ 2 λ -:= -(2µ+ν)-δ 2 and Q := -λ - δ -κ δ λ + δ -κ δ .
Let us remark that the system [START_REF] Kawashima | Dissipative structure for symmetric hyperbolic-parabolic systems with korteweg-type dispersion[END_REF] can then be rewritten with the new unknown (σ, q) as

-ζ + ∂ t σ -△σ = g σ + αλ - δλ + σ + αλ - δλ + q, in [0, T ] × T L , -ζ -∂ t q -△q = g q -α δ σ -α δ q, in [0, T ] × T L , (21)
where

ζ ± := -1 λ ± > 0 and g σ g q := Q g σ g q .
Now, let us again remark that as system ( 21) is linear, its observability is equivalent to a controlability statement for the adjoint equation written in the dual variables (r, y), where the adjoint is taken with respect to the variable (σ, q). This leads to the controllability problem:

ζ + ∂ t r -△r = f r + αλ - δλ + r -α δ y + χ 0 v r , in [0, T ] × T L , ζ -∂ t y -△y = f y -α δ y + αλ - δλ + r + χ 0 v y , in [0, T ] × T L . (22)

Non-diagonalisable case

Assume that ( 20) is non diagonalisable. Then, since [START_REF] Maity | Some controllability results for linearized compressible Navier-Stokes system[END_REF], there exist R ∈ GL 2 (R) and γ 1 , γ 2 , γ 3 and γ 4 in R such that

σ q := R σ q . satisfy -ζ∂ t σ -△σ = g σ + γ 1 σ + γ 2 q + ζ △q, in [0, T ] × T L , -ζ∂ t q -△q = g q + γ 3 σ + γ 4 q, in [0, T ] × T L , (23)
with ζ := 2 2µ + ν > 0 and g σ g q := R g σ g q .
Let us remark that ( 23) is linear. Thus, by duality, its observability is equivalent to a controllability statement for the adjoint equation written in the dual variables (r, y), where the adjoint is taken with respect to the variable (m, n). This leads to the controllability problem:

ζ∂ t r -△r = f r + γ 1 r + γ 3 y + χ 0 v r + ζ △y, in [0, T ] × T L , ζ∂ t y -△y = f y + γ 2 r + γ 4 y + χ 0 v y , in [0, T ] × T L . ( 24
)
5 Controllability of the heat equation Now, to solve the controllability problem [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF], we use a fixed point argument to reduce to considering the following decoupled controllability problem:

ζ + ∂ t r -△r = fr + v r χ 0 , in [0, T ] × T L , ζ -∂ t y -△y = fy + v y χ 0 , in [0, T ] × T L . (25)

Construction of the weight function

Let ψ ∈ C 2 (T L , R). To fixe the ideas, we assume that, for every

x ∈ T L ψ(x) ∈ [6, 7]. (26)
Additionally, we suppose there exists of a subset ω ⊂⊂ {χ 0 = 1} such that:

inf T L \ω {|∇ψ|} > 0. ( 27
)
We choose T 0 > 0 and T 1 > 0small enough, with T 1 ≤ 1 4 , so that

T 0 + 2T 1 < T.
For any m ≥ 2, we introduce the weight function

θ ∈ C 2 ([0, T [) such that θ = θ(t) =          1 + 1 -t T 0 m , for all t ∈ [0, T 0 ], 1, for all t ∈ [T 0 , T -2T 1 ], θ is increasing, on [T -2T 1 , T -T 1 ], 1
T -t , for all t ∈ [T -T 1 , T [. (28)
Then we consider the following weight function

φ := θ(λe 12λ -e λψ ),
where s, λ are positive parameters with s ≥ 1, λ ≥ 1 and α is chosen as

m = sλ 2 e 12λ ,
which always larger than 2, thus being compatible with the condition θ ∈ C 2 ([0, T [). Note that θ is bounded by below by a positive constant, more precisely

θ ≥ min 1, 1 T 1 . ( 29
)
We point out that, due to the definition of ψ and to the condition [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF], and using that λ 0 ≥ 1, we have the following bounds on [0, T [×T L :

14 15 Φ ≤ φ ≤ Φ, (30)
where Φ := θλe 12λ .

(31)

Carleman inequalities and controllability results for the heat equation

In this subsection we give some tools related the controllability of the heat equation. Let ζ be a real number, such that ζ > 0.

(32)

We consider the following controllability problem: Given w 0 and f , find a control function v w such that the solution w of

ζ∂ t w -△w = f + v w χ 0 , in [0, T ] × T L , w | t=0 = w 0 , in T L , (33)
satisfies w | t=T = 0, in T L . (34)
We use classical method to study the controllability property of (33), which is based on te observability of the adjoint system, obtained here with the following L 2 -Carleman estimate for the heat equation (c.f. [10] Theorem 3.2 or [1] Theorem 2.5 )

Lemma 5.2.1. Let d ≥ 2 and T > 0. There exists constants C > 0, s 0 ≥ 1 and λ 0 ≥ 1, large enough, such that for all smooth complex value function w on [0, T ] × T L and for all s ≥ s 0 and λ ≥ λ 0 , we have

s 3 2 λ 2 ∥ξ 3 2 we -sφ ∥ L 2 (L 2 ) + s 1 2 λ∥ξ 1 2 ∇we -sφ ∥ L 2 (L 2 ) + sλ 3 2 e 7λ ∥w(0)e -sφ(0) ∥ L 2 ≤ C ∥(-ζ∂ t -△)we -sφ ∥ L 2 (L 2 ) + s 3 2 λ 2 ∥ξ 3 2 χ 0 we -sφ ∥ L 2 (L 2 ) ,
where we have set ξ(t, x) := θ(t)e λψ(x) .

The proof is given in [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] for general domain. For the sake of completeness, we give the proof of L 2 -Carleman estimate on T L with the weight φ in the Appendix B, which we will use in this paper. As in [START_REF] Ervedoza | Local exact controllability for the two-and threedimensional compressible Navier-Stokes equations[END_REF], this estimate leads to the following controllability result (c.f. [10] Theorem 3.3). Theorem 5.2.2. Let d ∈ {2, 3} and T > 0. There exist constants C > 0 and s 0 ≥ 1 such that for all s ≥ s 0 , for all f ∈ L 2 (0, T ; L 2 (T L )) satisfying

∥θ -3 2 f e sφ ∥ L 2 (L 2 ) < +∞ (35)
and w 0 ∈ L 2 (T L ), there exists a solution (w, v w ) of the control problem (33)(34) which furthermore satisfies the following estimate:

s 3 2 ∥we sφ ∥ L 2 (L 2 ) + ∥θ -3 2 χ 0 v w e sφ ∥ L 2 (L 2 ) + s 1 2 ∥θ -1 ∇we sφ ∥ L 2 (L 2 ) ≤ C ∥θ -3 2 f e sφ ∥ L 2 (L 2 ) + s 1 2 ∥w 0 e sφ(0) ∥ L 2 . ( 36
)
Moreover, the solution (w, v w ) can be obtained through a linear operator in (w 0 , f ). if w 0 ∈ H 1 (T L ), we also have

s -1 2 ∥θ -2 ∇ 2 we sφ ∥ L 2 (L 2 ) ≤ C ∥θ -3 2 f e sφ ∥ L 2 (L 2 ) + s 1 2 ∥w 0 e sφ(0) ∥ L 2 + s -1 2 ∥∇w 0 e sφ(0) ∥ L 2 . ( 37
)
We need also to know what can be done when the source term f is more regular and lies in

L 2 (0, T ; H 1 (T L )) or in L 2 (0, T ; H 2 (T L )) (c.f. [1] Proposition 3.4).
Lemma 5.2.3. Consider the controlled trajectory (w, v w ) construct in Theorem 5.2.2. Then, with the above notation, for some constant C > 0 independent of s, we have the following properties:

1. (H 2 regularity of the control) we have v w ∈ L 2 (0, T ; H 2 (T L )) and

∥χ 0 v w e 6sΦ/7 ∥ L 2 (H 2 ) ≤ C ∥θ -3 2 f e sφ ∥ L 2 (L 2 ) + ∥w 0 e sΦ(0) ∥ L 2 , 2. (H 3 regularity estimate for the state) if w 0 ∈ H 2 (T L ), f e 6sφ/7 ∈ L 2 (0, T ; H 1 (T L )) and θ -3 2 f e sφ ∈ L 2 (0, T ; L 2 (T L )), then w ∈ L 2 (0, T ; H 3 (T L )) and ∥we 6sΦ/7 ∥ L 2 (H 3 ) ≤ C ∥f e 6sΦ/7 ∥ L 2 (H 1 ) + ∥θ -3 2 f e sφ ∥ L 2 (L 2 ) + ∥w 0 e sΦ(0) ∥ H 2 , 3. (H 4 regularity estimate for the state) if w 0 ∈ H 3 (T L ), f e 6sΦ/7 ∈ L 2 (0, T ; H 2 (T L )) and θ -3 2 f e sφ ∈ L 2 (0, T ; L 2 (T L )), then w ∈ L 2 (0, T ; H 4 (T L )) and ∥we 6sΦ/7 ∥ L 2 (H 4 ) ≤ C ∥f e 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -3 2 f e sφ ∥ L 2 (L 2 ) + ∥w 0 e sΦ(0) ∥ H 3 .
6 Controlability of (22): The diagonalisable case.

This section is devoted to the control of system [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF]. More precisely, we aim to establish the following theorem. We recall that

(2µ + ν) 2 ≥ 4κ Lemma 6.0.1. Let d ∈ {2, 3} and T > 0. Let (r 0 , y 0 ) ∈ L 2 (T L ) × L 2 (T L ).
There exist C > 0 and s 0 ≥ 1 such that for all s ≥ s 0 , if f r and f y satisfy the integrability conditions

∥θ -3 2 (f r , f y )e sφ ∥ L 2 (L 2 ) < +∞, (38)
there exist a controlled trajectory (r, y) solving [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF] and satisfying the following estimate:

∥θ -3 2 (r, y)e sφ ∥ L 2 (L 2 ) + s -1 2 ∥θ -3 2 (χ 0 v r , χ 0 v y )e sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇(r, y)e sφ ∥ L 2 (L 2 ) ≲ s 0 s -1 2 ∥θ -3 2 (f r , f y )e sφ ∥ L 2 (L 2 ) + ∥(r 0 , y 0 )e sφ(0) ∥ L 2 . ( 39
)
Furthermore, if (r 0 , y 0 ) ∈ H 3 (T L ) × H 3 (T L ), and f r and f y satisfy

(f r , f y )e sφ ∈ L 2 (0, T ; H 2 (T L )), (40)
we have the following estimate:

∥(r, y)e 6sΦ/7 ∥ L 2 (H 4 ) + ∥χ 0 (v r , v y )e 6sΦ/7 ∥ L 2 (H 2 ) ≤ C ∥(f r , f y )e sΦ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 , (41)
for some constant C independent of s ≥ s 0 .

Proof. From previous property of controllability of heat equation, we proof the controllability of ( 22) by fixed-point argument.

Existence of the solution to the control problem. We construct the controlled trajectory using a Banach fixed-point argument. We introduce the set

C s := w ∈ L 2 (0, T ; H 1 (T L )) we sφ , θ -1 ∇we sφ ∈ L 2 (0, T ; L 2 (T L )) .
For r and ỹ in C s , we introduce

fr := fr (r, ỹ) = f r + αλ - δλ + r -α δ ỹ, fy := fy (r, ỹ) = f y -α δ ỹ + αλ - δλ + r.
As f r and f y satisfy (38), for every (r, ỹ) in C s × C s , fr and fy the assumption (35) of Theorem 5.2.2. In fact, the Theorem 5.2.2 provided two linear maps (r 0 , fr ) → (r, v r ) and (y 0 , fy ) → (y, v y ). Therefore, one can define a map Λ s on C s × C s which to a data (r, ỹ) in C s × C s associates (r, y) where r and y are respectively solutions of the controlled problem

ζ + ∂ t r -△r = fr + v r χ 0 , in (0, T ) × T L , r | t=0 = r 0 , r | t=T = 0, in T L and ζ -∂ t y -△y = fy + v y χ 0 , in (0, T ) × T L , y | t=0 = y 0 , y | t=T = 0, in T L ,
given by the Theorem 5.2.2. In order to apply Banach's fixed point theorem over Λ s , let us show, using estimate of the Theorem 5.2.2, that the map Λ s is a contractive mapping for s large enough. Let (r a , ỹa ) and (r b , ỹb ) two elements of

C s × C s .We set (R, Y ) := Λ s (r a , ỹa ) -Λ s (r b , ỹb ) is solution of the control problems: ζ + ∂ t R -△R = fR + v R χ 0 , in (0, T ) × T L , R | t=0 = 0, R | t=T = 0, in T L and ζ -∂ t Y -△Y = fY + v Y χ 0 , in (0, T ) × T L , Y | t=0 = 0, Y | t=T = 0, in T L ,
where f R := fr (r a , ỹa ) -fy (r b , ỹb ) and f R := fr (r a , ỹa ) -fy (r b , ỹb ). Let us remark there exist constant C > 0, which not depend of m, such that θ -

s 3 2 ∥(R, Y )e sφ ∥ L 2 (L 2 ) +s 1 2 ∥θ -1 ∇(R, Y )e sφ ∥ L 2 (L 2 ) ≲ ∥(f R , f Y )e sφ ∥ L 2 (L 2 ) ≲ ∥θ -3 2 ((r a , ỹa ) -(r b , ỹb )) e sφ ∥ L 2 (L 2 ) ≲ ∥ ((r a , ỹa ) -(r b , ỹb )) e sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇ ((r a , ỹa ) -(r b , ỹb )) e sφ ∥ L 2 (L 2 ) .
Hence, using s ≥ 1, we get

∥(R, Y )e sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇(R, Y )e sφ ∥ L 2 (L 2 ) ≲ s -1 2 ∥ ((r a , ỹa ) -(r b , ỹb )) e sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇ ((r a , ỹa ) -(r b , ỹb )) e sφ ∥ L 2 (L 2 )
Thus, the quantity

∥(r, y)∥ Cs := ∥(r, y)e sφ ∥ L 2 (0,T ;L 2 (T L )) + ∥θ -1 ∇(r, y)e sφ ∥ L 2 (0,T ;L 2 (T L ))
defines a norm on C s × C s , for which the map Λ s satisfies

∥ (Λ s (r a , ỹa ) -Λ s (r b , ỹb )) ∥ Cs ≤ Cs -1 2 ∥ ((r a , ỹa ) -(r b , ỹb )) ∥ Cs ,
for a constant C > 0 that is independent of s ≥ s 0 . Then, if s is chosen large enough, the map Λ s is a contractive mapping and by the Banach's fixed-point theorem, Λ s has a unique fixed-point (r, y) in

C s × C s .
By construction, this fixed point (r, y) solves the controllability problem [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF]. Furthermore, we get the following estimate for ( fr (r, y), fy (r, y))

∥θ -3 2 fr (r, y), fy (r, y) ∥ L 2 (L 2 ) ≲ ∥θ -3 2 (f r , f y )e sφ ∥ L 2 (L 2 ) + ∥θ -3 2 (r, y)e sφ ∥ L 2 (L 2 ) + ∥θ -3 2 ∇(r, y)e sφ ∥ L 2 (L 2 )
one gets with the Theorem 5.2.2 that (r, y) solution of ( 22) satisfies

∥θ -3 2 (r, y)e sφ ∥ L 2 (L 2 ) + s -1 2 ∥θ -3 2 χ 0 (v r , v y )e sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇(r, y)e sφ ∥ L 2 (L 2 ) ≲ s -1 2 ∥θ -3 2 (f r , f y )e sφ ∥ L 2 (L 2 ) + ∥(r 0 , y 0 )e sφ(0) ∥ L 2 (L 2 ) ,
that is the estimate (39). Regularity estimate. To simplify notations, we set f w := (f r , f y ), f w (w) := (f r (r, y), f y (r, y)), w := (r, y), w 0 := (r 0 , y 0 ) and v w := (v r , v y ). Since (29) and φ ≤ Φ, applying the Lemma 5.2.3 to f r (r, y) and f y (r, y) such that f r (r, y)e 6sΦ/7 , f y (r, y)e 6sΦ/7 ∈ L 2 (H 2 ) and (r 0 , y 0 ) ∈ H 3 × H 3 , we deduce from Lemma 5.2.3 item 1 and 3 that

∥we 6sΦ/7 ∥ L 2 (H 4 ) + ∥χ 0 v w e 6sΦ/7 ∥ L 2 (H 2 ) ≲ ∥f w (w)e 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -3 2 f w (w)e sφ ∥ L 2 (L 2 ) + ∥w 0 e sΦ(0) ∥ H 3 ≲ ∥f w e 6sΦ/7 ∥ L 2 (H 2 ) + ∥w 0 e sΦ(0) ∥ H 3 + ∥θ -3 2 f w e sφ ∥ L 2 (L 2 ) + ∥θ -3 2 we sφ ∥ L 2 (L 2 ) + ∥we 6sΦ/7 ∥ L 2 (H 2 ) ≲ ∥f w e 6sΦ/7 ∥ L 2 (H 2 ) + ∥w 0 e sΦ(0) ∥ H 3 + ∥θ -3 2 we sφ ∥ L 2 (L 2 ) + ∥we 6sΦ/7 ∥ L 2 (H 2 ) .
Then to obtain (41), it remains to estimate the term ∥we 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -3 2 we sφ ∥ L 2 (L 2 ) . For this end purpose, we begin to remark that for all β ≥ 0, since θ is lower bounded, there exist a constant K > 0, independent of s, λ and m such that e 6sΦ/7 ≤ Ke sφ θ -β s -β .

(42)

Indeed, for λ 0 large enough, we have e 6sΦ/7 = e 14 15 sΦ e -sΦ( 1415 -6 7 ) ≤ e sφ e -sΦ( 1415 -6 7 ) = e sφ e -sθλ 0 e 12λ 0 ( 14 15 - 6 7 )

≤ e sφ e -sθλ 0 ≲ e sφ θ -β λ -β 0 s -β ≲ e sφ θ -β s -β .

Furthermore, as Φ depend only of the time variable, we have

∥we 6sΦ/7 ∥ L 2 (H 2 ) ≲ ∥we 6sΦ/7 ∥ L 2 (L 2 ) + ∥∇we 6sΦ/7 ∥ L 2 (L 2 ) + ∥∇ 2 we 6sΦ/7 ∥ L 2 (L 2 ) .
From (42) and using that s ≥ s 0 ≥ 1, we deduce that

∥we 6sΦ/7 ∥ L 2 (L 2 ) ≲ ∥θ -3 2 we sφ ∥ L 2 (L 2 ) (43)
and

∥∇we 6sΦ/7 ∥ L 2 (L 2 ) ≲ ∥θ -1 ∇we sφ ∥ L 2 (L 2 ) . (44)
Applying the estimate (42) with β = 2, and using (37) and that s ≥ 1, we obtain

∥∇ 2 we 6sΦ/7 ∥ L 2 (L 2 ) ≲ s -3 2 ∥θ -3 2 f w e sφ ∥ L 2 (L 2 ) + s -3 2 ∥θ -3 2 we sφ ∥ L 2 (L 2 ) + s -1 ∥w 0 e sφ(0) ∥ L 2 + s -2 ∥∇w 0 e sφ(0) ∥ L 2 ≲ ∥f w e sΦ ∥ L 2 (H 2 ) + ∥w 0 e sΦ(0) ∥ H 3 + ∥θ -3 2 we sφ ∥ L 2 (L 2 ) ,
which combined with (43) and (44), yield that

∥we 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -3 2 we sφ ∥ L 2 (L 2 ) ≲ ∥f w e sΦ ∥ L 2 (H 2 ) + ∥w 0 e sΦ(0) ∥ H 3 + ∥θ -3 2 we sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇we sφ ∥ L 2 (L 2 ) .
Finally, using (39) and the previous inequality, we obtain

∥we 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -3 2 we sφ ∥ L 2 (L 2 ) ≲ ∥f w e sΦ ∥ L 2 (H 2 ) + ∥w 0 e sΦ(0) ∥ H 3 .
That completes the proof of Lemma 6.0.1.

Controlability of (22):

The non-diagonalisable case (2µ + ν) 2 = 4κ

7.1 Estimate for the non-diagonalisable case Theorem 7.1.1. There exist positive constants C > 0 and s 0 ≥ 1 such that for all s ≥ s 0 , for all fw

satisfying ∥θ -1 fw e sφ ∥ L 2 (0,T ;L 2 (T L )) < +∞ (45)
and w 0 ∈ H 1 (T L ), the solution (w, v w ) of the control problem satisfies

s 3 2 ∥θ -1 2 we sφ ∥ L 2 (L 2 ) + ∥θ -1 χ 0 v w e sφ ∥ L 2 (L 2 ) + s 1 2 ∥θ -1 2 ∇we sφ ∥ L 2 (L 2 ) + s -1 2 ∥θ -3 2 ∇ 2 we sφ ∥ L 2 (L 2 ) ≤ C ∥θ -1 fw e sφ ∥ L 2 (L 2 ) + s 1 2 ∥w 0 e sφ(0) ∥ L 2 + s -1 2 ∥∇w 0 e sφ(0) ∥ L 2 . ( 46
)
Moreover, the solution (w, v w ) can be obtain through a linear operator in (w 0 , f ).

Lemma 7.1.2. Consider the controlled trajectory (w, v w ) construct in Theorem 5.2.2. Then, with the above notation, for some constant C > 0 independent of s, we have the following properties:

1. (H 2 regularity of the control) we have v w ∈ L 2 (0, T ; H 2 (T L )) and

∥χ 0 v w e 6sΦ/7 ∥ L 2 (H 2 ) ≤ C ∥θ -1 f e sφ ∥ L 2 (L 2 ) + ∥w 0 e sΦ(0) ∥ L 2 , 2. (H 3 regularity estimate for the state) if w 0 ∈ H 2 (T L ), f e 6sφ/7 ∈ L 2 (0, T ; H 1 (T L )) and θ -1 f e sφ ∈ L 2 (0, T ; L 2 (T L )), then w ∈ L 2 (0, T ; H 3 (T L )) and ∥we 6sΦ/7 ∥ L 2 (H 3 ) ≤ C ∥f e 6sΦ/7 ∥ L 2 (H 1 ) + ∥θ -1 f e sφ ∥ L 2 (L 2 ) + ∥w 0 e sΦ(0) ∥ H 2 , 3. (H 4 regularity estimate for the state) if w 0 ∈ H 3 (T L ), f e 6sΦ/7 ∈ L 2 (0, T ; H 2 (T L )) and θ -1 f e sφ ∈ L 2 (0, T ; L 2 (T L )), then w ∈ L 2 (0, T ; H 4 (T L )) and ∥we 6sΦ/7 ∥ L 2 (H 4 ) ≤ C ∥f e 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -1 f e sφ ∥ L 2 (L 2 ) + ∥w 0 e sΦ(0) ∥ H 3 .
7.2 Controllability for the cascade system Lemma 7.2.1. Let d ∈ {2, 3} and T > 0. Let (r 0 , y 0 ) ∈ L 2 (T L ) × L 2 (T L ). There exist s 0 ≥ 1 such that for all s ≥ s 0 , if (f r , f y ) satisfy the integrability condition

∥θ -1 (f r , f y )e sφ ∥ L 2 (L 2 ) < +∞
there exist a controlled trajectory (r, y) solving (21) and satifying the following estimate:

s -5 6 ∥θ -3 2 χ 0 v r e sφ ∥ L 2 (L 2 ) + s -1 3 ∥θ -1 χ 0 v y e sφ ∥ L 2 (L 2 ) + s∥re sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇re sφ ∥ L 2 (L 2 ) + s 11 6 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s 5 6 ∥θ -1 2 ∇ye sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) ≲ s -5 6 ∥θ -3 2 f r e sφ ∥ L 2 (L 2 ) + s -5 6 ∥r 0 e sφ(0) ∥ L 2 + s 1 3 ∥θ -1 f y e sφ ∥ L 2 (L 2 ) + s 5 6 ∥y 0 e sφ(0) ∥ L 2 + s -1 6 ∥∇y 0 e sφ(0) ∥ L 2 (47) Furthermore, if (r 0 , y 0 ) ∈ H 3 (T L ) × H 2 (T L )
, and f r an f y satisfy

(f r , f y )e sφ ∈ L 2 (0, T ; H 2 (T L )),
we have the following estimate

∥(r, y)e 6sΦ/7 ∥ L 2 (H 4 ) + ∥χ 0 (v r , v y )e 6sΦ/7 ∥ L 2 (H 2 ) ≲ s ∥(f r , f y )e sΦ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 . (48)
Remark that in contrast with (41) we have a dependence on s in (48).

Proof. Let introduce the following two functional spaces

C r s := w ∈ L 2 (0, T ; H 1 (T L )) we sφ , θ -1 ∇we sφ ∈ L 2 (0, T ; L 2 (T L )) and C y s := w ∈ L 2 (0, T ; H 2 (T L )) θ -1 2 we sφ , θ -1 2 ∇we sφ , θ -3 2 △we sφ ∈ L 2 (0, T ; L 2 (T L ))
that equip respectively with norms

∥w∥ C r s := s∥we sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇we sφ ∥ L 2 (L 2 )
and ∥w∥ C y s := s

11 6 ∥θ -1 2 we sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △we sφ ∥ L 2 (L 2 )
for which C r s and C y s are Banach spaces. For r in C r s and ỹ in C y s , we introduce

fr := fr (r, ỹ) = f r + γ 1 r + γ 3 ỹ + ζ △ỹ, fy := fy (r, ỹ) = f y + γ 2 r + γ 4 ỹ.
Using Theorem 7.1.1 and Theorem 5.2.2, one can define a map Λ s on C r s × C y s which to a data (r, ỹ) in C r s × C y s associates (r, y) where r and y are respectively solutions of the controlled problem

   ζ∂ t r -△r = fr + v r χ 0 , in (0, T ) × T L , ζ∂ t y -△y = fy + v r χ 0 , in (0, T ) × T L , (r, y) | t=0 = (r 0 , y 0 ), (r, y) | t=T = (0, 0), in T L
given by Theorem 7.1.1 and Theorem 5.2.2. Let (r a , ỹa ) and (r b , ỹb ) in C r s × C y s . We set (R, Y ) := Λ s (r a , ỹa ) -Λ s (r b , ỹb ), fR := fr (r a , ỹb ) -fr (r b , ỹb ) and fY := fy (r a , ỹb ) -fy (r b , ỹb ) so that (R, Y ) is a solution of the following control problem

   ζ∂ t R -△R = fR R + v R χ 0 , in (0, T ) × T L , ζ∂ t Y -△Y = fY + v Y χ 0 , in (0, T ) × T L , (R, Y ) | t=0 = (0, 0), (R, Y ) | t=T = (0, 0), in T L .
From Theorem 7.1.1 and Theorem 5.2.2, we deduce that

s 3 2 ∥θ -1 2 Y e sφ ∥ L 2 (L 2 ) + s 1 2 ∥θ -1 2 ∇Y e sφ ∥ L 2 (L 2 ) +s -1 2 ∥θ -3 2 △Y e sφ ∥ L 2 (L 2 ) ≲ ∥θ -1 Ỹ e sφ ∥ L 2 (L 2 ) + ∥θ -1 Re sφ ∥ L 2 (L 2 ) ( 49
)
and

s 3 2 ∥Re sφ ∥ L 2 (L 2 ) + s 1 2 ∥θ -1 ∇Re sφ ∥ L 2 (L 2 ) ≲ ∥θ -3 2 Re sφ ∥ L 2 (L 2 ) + ∥θ -3 2 Ỹ e sφ ∥ L 2 (L 2 ) + ∥θ -3 2 △ Ỹ e sφ ∥ L 2 (L 2 ) . (50)
On the one hand, multiplying (49) by s 1 3 , using that θ -1 ≤ θ -1 2 ≲ 1 and s ≥ 1, we get

s 11 6 ∥θ -1 2 Y e sφ ∥ L 2 (L 2 ) + s 5 6 ∥θ -1 2 Y e sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △Y e sφ ∥ L 2 (L 2 )
≲ s

1 3 ∥θ -1 2 Ỹ e sφ ∥ L 2 (L 2 ) + ∥ Re sφ ∥ L 2 (L 2 ) ≲ s -1 3 s 2 3 ∥θ -1 2 Ỹ e sφ ∥ L 2 (L 2 ) + s 2 3 ∥ Re sφ ∥ L 2 (L 2 ) ≲ s -1 3 s 11 6 ∥θ -1 2 Ỹ e sφ ∥ L 2 (L 2 ) + s∥ Re sφ ∥ L 2 (L 2 ) . (51)
On the other hand, multiplying (50) by s -1 2 , using that θ -3 2 ≤ θ -1 2 ≲ 1 and s ≥ 1, we obtain

s∥Re sφ ∥ L 2 (L 2 ) +∥θ -1 ∇Re sφ ∥ L 2 (L 2 ) ≲ s -1 2 ∥ Re sφ ∥ L 2 (L 2 ) + ∥θ -1 2 Ỹ e sφ ∥ L 2 (L 2 ) + ∥θ -3 2 △ Ỹ e sφ ∥ L 2 (L 2 ) ≲ s -1 3 s -1 6 ∥ Re sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -1 2 Ỹ e sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ Ỹ e sφ ∥ L 2 (L 2 ) ≲ s -1 3 s∥ Re sφ ∥ L 2 (L 2 ) + s 11 6 ∥θ -1 2 Ỹ e sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ Ỹ e sφ ∥ L 2 (L 2 ) (52)
Combining (51) and (52), we deduce that

s 11 6 ∥θ -1 2 Y e sφ ∥ L 2 (L 2 ) + s 5 6 ∥θ -1 2 ∇Y e sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △Y e sφ ∥ L 2 (L 2 ) + s∥Re sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇Re sφ ∥ L 2 (L 2 ) ≲ s -1 3 s 11 6 ∥θ -1 2 Ỹ e sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ Ỹ e sφ ∥ L 2 (L 2 ) + s -1 3 s∥ Re sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇ Re sφ ∥ L 2 (L 2 ) (53)
Then (53) can be rewrite follow

∥Λ s (r a , y a ) -Λ s (r b , y b )∥ C r s ×C y s ≤ Cs -1 3 ∥(r a , ỹa ) -(r b , ỹb )∥ C r s ×C y s
where C is a positive constant that not depend of s ≥ s 0 . From the Banach fixed-point theorem, we deduce that for s large enough, Λ s admit an unique fixed-point in C r s × C y s . Let (r, y) ∈ C r s × C y s be the fixed-point of Λ s and let (v r , v y ) be the associated control. From Theorem 7.1.1, we deduce, as for (51), that

s - ∥θ -1 χ 0 v y e sϕ ∥ L 2 (L 2 ) + s 11 6 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s 5 6 ∥θ -1 2 ∇ye sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ye sϕ ∥ L 2 (L 2 ) ≲ s -1 3 s 2 3 ∥θ -1 f r e sφ ∥ L 2 (L 2 ) + s 7 6 ∥y 0 e sφ(0) ∥ L 2 + s 1 6 ∥∇y 0 e sφ(0) ∥ L 2 + s -1 3 s 2 3 ∥θ -1 re sφ ∥ L 2 (L 2 ) + s 2 3 ∥θ -1 ye sφ ∥ L 2 (L 2 ) .
Then using that θ -1 ≤ θ -1 2 ≲ 1, we obtain

s - ∥θ -1 χ 0 v y e sϕ ∥ L 2 (L 2 ) + s 11 6 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s 5 6 ∥θ -1 2 ∇ye sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ye sϕ ∥ L 2 (L 2 )
≲ s

1 3 ∥θ -1 f y e sφ ∥ L 2 (L 2 ) + s 5 6 ∥y 0 e sφ(0) ∥ L 2 + s -1 6 ∥∇y 0 e sφ(0) ∥ L 2 + s 1 3 ∥re sφ ∥ L 2 (L 2 ) + s -1 2 ∥ye sφ ∥ L 2 (L 2 ) . (54)
Otherwise, applying Theorem 5.2.2, we deduce as for (52) that

s -5 6 ∥θ -3 2 v r e sφ ∥ L 2 (L 2 ) + s∥re sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇re sφ ∥ L 2 (L 2 ) ≲ s -1 3 s -1 2 ∥r 0 e sφ(0) ∥ L 2 + s -1 2 ∥θ -3 2 f r e sφ ∥ L 2 (L 2 ) + s -1 3 s -1 6 ∥re sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s -1 3 s -1 2 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) ,
hence, we obtain

s -5 6 ∥θ -3 2 χ 0 v r e sφ ∥ L 2 (L 2 ) + s∥re sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇re sφ ∥ L 2 (L 2 ) ≲ s -5 6 ∥r 0 e sφ(0) ∥ L 2 + s -5 6 ∥θ -3 2 f r e sφ ∥ L 2 (L 2 ) + s -3 6 ∥re sφ ∥ L 2 (L 2 ) + s -5 6 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . (55)
Combining ( 54) and (55), using that θ -3 2 ≤ θ -1 ≲ 1 and s ≥ 1, we obtain

s -5 6 ∥θ -3 2 χ 0 v r ∥ L 2 (L 2 ) + s -1 3 ∥θ -1 χ 0 v y e sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇re sϕ ∥ L 2 (L 2 ) + s 5 6 ∥θ -1 2 ∇ye sφ ∥ L 2 (L 2 ) + s∥re sφ ∥ L 2 (L 2 ) + s 11 6 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) (56) ≲ s -5 6 ∥θ -3 2 f r e sφ ∥ L 2 (L 2 ) + s -5 6 ∥r 0 e sφ(0) ∥ L 2 + s 1 3 ∥θ -1 f y e sφ ∥ L 2 (L 2 )
+ s 5 6 ∥y 0 e sφ(0) ∥ L 2 + s -1 6 ∥∇y 0 e sφ(0) ∥ L 2

+ s -3 6 ∥re sφ ∥ L 2 (L 2 ) + s -3 6 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s -5 6 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) + s 1 3 ∥re sφ ∥ L 2 (L 2 ) + s -1 2 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) ≲ s -5 6 ∥θ -3 2 f r e sφ ∥ L 2 (L 2 ) + s -5 6 ∥r 0 e sφ(0) ∥ L 2 + s 1 3 ∥θ -1 f y e sφ ∥ L 2 (L 2 )
+ s 5 6 ∥y 0 e sφ(0) ∥ L 2 + s -1 6 ∥∇y 0 e sφ(0) ∥ L 2

+ s 1 3 ∥re sφ ∥ L 2 (L 2 ) + s -1 2 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s -5 6 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . (57)
Since the decay on the parameter s is more faster in (57), than in the (56), for s large enough we obtain (47), namely

s -5 6 ∥θ -3 2 χ 0 v r e sφ ∥ L 2 (L 2 ) + s -1 3 ∥θ -1 χ 0 v y e sφ ∥ L 2 (L 2 ) + s∥re sφ ∥ L 2 (L 2 ) + ∥θ -1 ∇re sφ ∥ L 2 (L 2 ) + s 11 6 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s 5 6 ∥θ -1 2 ∇ye sφ ∥ L 2 (L 2 ) + s -1 6 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) ≲ s -5 6 ∥θ -3 2 f r e sφ ∥ L 2 (L 2 ) + s -5 6 ∥r 0 e sφ(0) ∥ L 2 + s 1 3 ∥θ -1 f y e sφ ∥ L 2 (L 2 ) + s 5 6
∥y 0 e sφ(0) ∥ L 2 + s -1 6 ∥∇y 0 e sφ(0) ∥ L 2 .

Regularity estimate. Recall that φ ≤ Φ and θ ≳ 1. Assume that (f r , f y )e sφ ∈ L 2 (H 2 ). We have re 6sΦ/7 ∈ L 2 (H 1 ) and ye 6sΦ/7 ∈ L 2 (H 2 ), hence f y (r, y)e 6sΦ/7 ∈ L 2 (H 1 ). Since Lemma 5.2.3, this in turn show that ye 6sΦ/7 ∈ L 2 (H 3 ). It follows that f r (r, y)e 6sφ/7 ∈ L 2 (H 1 ). Then, from Lemma 7.1.2, it follows that re 6sΦ/7 ∈ L 2 (H 3 ), that give f y (r, y)e 6sφ/7 ∈ L 2 (H 2 ) and using again Lemma 5.2.3, we get ye 6sΦ/7 ∈ L 2 (H 4 ). We deduce that f r e 6sΦ/7 ∈ L 2 (H 2 ), finally (r, y)e 6sΦ/7 ∈ L 2 (H 4 ).

Now, we aim to obtain the estimate (48).

Estimate on ∥ye 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -1 ye sφ ∥ L 2 (H 2 ) . We have

∥ye 6sΦ/7 ∥ L 2 (H 2 ) ≲ ∥ye 6sΦ/7 ∥ L 2 (L 2 ) + ∥∇re 6sΦ/7 ∥ L 2 (L 2 ) + ∥∇ 2 ye 6sΦ/7 ∥ L 2 (L 2 ) . (58)
Furthermore, from (42), it follows that

∥ye 6sΦ/7 ∥ L 2 (L 2 ) ≲ ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) ( 59
)
∥∇ye 6sΦ/7 ∥ L 2 (L 2 ) ≲ s -1 ∥θ -1 2 ∇ye sφ ∥ L 2 (L 2 ) (60)
∥∇ 2 ye 6sΦ/7 ∥ L 2 (L 2 ) ≲ s -2 ∥θ -3 2 ∇ 2 ye sφ ∥ L ( L 2 ) . (61)
Multiplying (46) by s -3 2 , we get

∥θ -1 2 ye sφ ∥ L 2 (L 2 ) + s -1 ∥θ -1 2 ∇ye sφ ∥ L 2 (L 2 ) + s -2 ∥θ -3 2 ∇ 2 ye sφ ∥ L 2 (L 2 ) ≲ ∥f y e sΦ ∥ L 2 (H 2 ) + ∥y 0 e sΦ ∥ H 3 + s -3 2 ∥θ -1 (r, y)e sφ ∥ L 2 (L 2 ) . (62)
Thus, using that θ -1 ≤ θ -1 2 , we deduce from (59), ( 60) and (61) combining with (62) that

∥ye 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -1 ye sφ ∥ L 2 (L 2 ) ≲ ∥f y e sΦ ∥ L 2 (H 2 ) + ∥y 0 e sΦ(0) ∥ H 3 + s -3 2 ∥θ -1 (r, y)e sφ ∥ L 2 (L 2 ) . ( 63
)
Estimate on ∥re 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -1 ye sφ ∥ L 2 (L 2 ) . We have

∥re 6sϕ/7 ∥ L 2 (H 2 ) ≲ ∥re 6sΦ/7 ∥ L 2 (L 2 ) + ∥∇re 6sϕ/7 ∥ L 2 (L 2 ) + ∥∇ 2 re 6sΦ/7 ∥ L 2 (L 2 ) . (64)
Moreover, from (42), we deduce that

∥re 6sΦ/7 ∥ L 2 (L 2 ) ≲ ∥re sφ ∥ L 2 (L 2 ) ( 65
)
∥∇re 6sΦ/7 ∥ L 2 (L 2 ) ≲ s -1 ∥θ -1 ∇re sφ ∥ L 2 (L 2 ) ( 66
)
∥∇ 2 re 6sΦ/7 ∥ L 2 (L 2 ) ≲ s -1 ∥θ -2 ∇ 2 re sφ ∥ L ( L 2 ) . (67)
Multiplying (36) by s -3 2 , we get

∥re sφ ∥ L 2 (L 2 ) + s -1 ∥θ -1 ∇re sφ ∥ L 2 (L 2 ) ≲ ∥f r e sΦ ∥ L 2 (H 2 ) + ∥re sΦ(0) ∥ H 3 + s -3 2 ∥θ -3 2 (r, y)e sφ ∥ L 2 (L 2 ) + s -3 2 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . (68)
Using ( 65) and (66), it follows from (68) that

∥θ -1 re sφ ∥ L 2 (L 2 ) + ∥re 6sΦ/7 ∥ L 2 (L 2 ) + ∥∇re 6sΦ/7 ∥ L 2 (L 2 ) ≲ ∥f r e sΦ ∥ L 2 (H 2 ) + ∥r 0 e sΦ(0) ∥ H 3 + s -3 2 ∥θ -3 2 (r, y)e sϕ ∥ L 2 (L 2 ) + s -3 2 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . (69)
Moreover, by muliplying (37) by s -1 2 , we deduce from (67) that

∥∇ 2 re 6sΦ/7 ∥ L 2 (L 2 ) ≲ ∥f r e sΦ ∥ L 2 (H 2 ) + ∥re sΦ(0) ∥ H 3 + s -1 2 ∥θ -3 2 (r, y)e sφ ∥ L 2 (L 2 ) + s -1 2 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . ( 70
)
Since s ≥ 1, θ -3 2 ≤ θ -1 ≲ 1, φ ≤ Φ and Φ does not depend of the space variable, it follows from (64), ( 69) and (70) that

∥re 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -1 re sφ ∥ L 2 (L 2 ) ≲ ∥f r e sϕ ∥ L 2 (H 2 ) + ∥r 0 e sΦ(0) ∥ H 3 + s -1 2 ∥θ -3 2 (r, y)e sφ ∥ L 2 (L 2 ) + s -1 2 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . ( 71
)
Estimate on ∥(r, y)e 6sΦ ∥ L 2 (L 2 ) + ∥θ -1 (r, y)e sφ ∥ L 2 (L 2 ) . Combining ( 63) and (71), we deduce that for s large enough, we have

∥(r, y)e 6sΦ/7 ∥ L 2 (H 2 ) + ∥θ -1 (r, y)e sφ ∥ L 2 (L 2 ) ≲ ∥(f r , f y )e sΦ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 + s -1 2 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . ( 72
)
Estimate on ∥ye 6sΦ/7 ∥ L 2 (H 4 ) + ∥χ 0 v y e 6sφ/7 ∥ L 2 (H 2 ) . From Lemma 7.1.2, item 1 and 3, and (72), it follows that

∥ye 6dΦ/7 ∥ L 2 (H 4 ) + ∥χ 0 v y e 6sΦ/7 ∥ L 2 (H 2 ) ≲ ∥(f r , f y )e sΦ ∥ L 2 (H 4 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ L 2 (H 2 ) + s -1 2 ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) (73)
Estimate on ∥re 6sΦ/7 ∥ L 2 (H 4 ) + ∥χ 0 v r e 6sΦ/7 ∥ L 2 (H 2 ) . From Lemma 5.2.3 item 1 and 3, and (73), we deduce that

∥re 6sΦ/7 ∥ L 2 (H 4 ) +∥χ 0 v r e 6sΦ/7 ∥ L 2 (H 2 ) (74) ≲ ∥(f r , f y )e sΦ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 + ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . (75)
Estimate on ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . In the one hand, multiplying (46) by s 1 2 , we obtain

∥θ -3 2 △ye 6sΦ/7 ∥ L 2 (L 2 ) + s 2 ∥θ -1 2 ye sφ ∥ L 2 (L 2 ) ≲ s 1 2 ∥(f r , f y )e sφ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 + ∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) . (76)
In other hand, by multiplying (46) by s -1 , we get

s 1 2 ∥θ -1 re sφ ∥ L 2 (L 2 ) ≲ s -1 ∥θ -3 2 f r e sφ ∥ L 2 (L 2 ) + s -1 ∥θ -3 2 (r, y)e sφ ∥ L 2 (L 2 ) + s -1 2 ∥y 0 e sΦ(0) ∥ L 2 . (77)
Using ( 76), (77) and that θ -1 ≤ θ -1 2 , we deduce that for s large enough

∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) + s 2 ∥θ -1 ye sφ ∥ L 2 (L 2 ) ≲ s ∥(f r , f r )e sΦ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 + s 1 2 ∥θ -1 ye sφ ∥ L 2 (L 2 ) + s -1 ∥θ -3 2 ye sφ ∥ L 2 (L 2 ) .
Thus, for s large enough, we have

∥θ -3 2 △ye sφ ∥ L 2 (L 2 ) ≲ s ∥(f r , f y )e sΦ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 (78)
Conclusion. Combining ( 73), ( 75) and ( 78), we conclude that

∥(r, y)e 6sΦ/7 ∥ L 2 (H 4 ) + ∥χ 0 (v r , v y )e 6sφ/7 ∥ L 2 (H 2 ) ≲ s ∥(f r , f y )e sΦ ∥ L 2 (H 2 ) + ∥(r 0 , y 0 )e sΦ(0) ∥ H 3 .
8 Controllability of the linearized system Theorem 8.0.1. Let d ∈ {2, 3}and T > 0.There exists s 0 ≥ 1, such that for all s ≥ s 0 there exist C s > 0, for all (a 0 , u 0 ) ∈ H 3 (T L ) × H 2 (T L ) and f a , f u such that f a e sΦ ∈ L 2 (0, T ; H 2 (T L )) and f u e 7sΦ/6 ∈ L 2 (0, T ; H 2 (T L )), there exist control function v a and v u and a corresponding controlled trajectory (a, u) solving [START_REF] Jüngel | Quantum Navier-Stokes equations[END_REF] with initial data (a 0 , u 0 ), satisfying the controllability requirements [START_REF] Imanuvilov | Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations[END_REF] and depending linearly on the data (a 0 , u 0 , f a , f u ). Besides, we have the estimate

∥(a, u)e 6sΦ/7 ∥ L 2 (H 4 )×L 2 (H 3 ) + ∥χ(v a , v u )e 6sΦ/7 ∥ L 2 (H 2 )×L 2 (H 1 )
≤ C s ∥(f a e sΦ , f u e 6sΦ/7 )∥ L 2 (H 2 )×L 2 (H 1 ) + ∥(a 0 e sΦ(0) , u 0 e 6sΦ(0)/7 ∥ H 3 ×H 2 .

(79)

In particular, this implies

∥(ae 5sΦ/6 , ue 5sΦ/6 )∥ L 2 (H 4 )∩L ∞ (H 3 )∩H 1 (H 2 )×L 2 (H 3 )∩L ∞ (H 2 )∩H 1 (H 1 ) ≤ C s ∥(f a e sΦ , f u e 7sΦ/6 )∥ L 2 (H 2 )×L 2 (H 1 ) + ∥(a 0 e sΦ(0) , u 0 e 7sΦ(0)/6 ∥ H 3 ×H 2 . ( 80
)
This allows to define a linear operator G defined on the set

(a 0 , u 0 , f a , f u ) ∈ H 3 × H 2 × L 2 (H 2 ) × L 2 (H 1 ) f a e sΦ ∈ L 2 (H 2 ) and f u e 7sΦ/6 ∈ L 2 (H 1 ) , by G(a 0 , u 0 , f a , f u ) = (a, u),
where (a, u) is the control trajectory, with initial condition (a 0 , u 0 ) and forces (f a , f u ), satisfying the control requirement [START_REF] Imanuvilov | Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations[END_REF] and estimates (79)-(80).

Proof. Using the duality between ( 22) and ( 21), we will first recover the observability estimate for [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] from the null controllability of the system [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF]. By definition of the dual norm, we have

∥(σ, q)e -sΦ ∥ L 2 (H -2 )×L 2 (H -2 ) + ∥(σ(0), q(0))e -sΦ(0) ∥ H -3 ×H -3 = sup ∥(fr,fy)e sΦ ∥ L 2 H 2 ≤1 ∥(r 0 ,y 0 )e sΦ(0) ∥ H 3 ≤1 {⟨(f r , f y ), (σ, q)⟩ L 2 (H 2 ),L 2 (H -2 ) + ⟨(r 0 , y 0 ), σ(0), q(0)⟩ H 3 ×H 3 ,H -3 ×H -3 }. ( 81
)
Furthermore, by construction, if we associate to (r 0 , y 0

) ∈ H 3 (T L )×H 3 (T L ) and (f r , f y ) ∈ L 2 (H 2 ),
such that (f r , f y )e sΦ ∈ L 2 (H 2 ), the trajectory of ( 22), then we get

⟨(f r , f y ), (σ, q)⟩ L 2 (H 2 ),L 2 (H -2 ) + ⟨(r 0 , y 0 ), σ(0), q(0)⟩ H 3 ×H 3 ,H -3 ×H -3 = ⟨(g σ, g q), (r, y)⟩ L 2 (H -4 )×L 2 (H -4 ),L 2 (H 4 )×L 2 (H 4 ) + ⟨(σ, q), χ 0 (v r , v y )⟩ L 2 (H -2 ),L 2 (H 2 ) .
Consequentlty, using (41), ( 48) and (81), we get

∥(σ, q)e -sΦ ∥ L 2 (H -2 )×L 2 (H -2 ) + ∥(σ(0), q(0))e -sΦ(0) ∥ H -3 ×H -3 ≲ s ∥(g σ, g q)e -6sΦ/7 ∥ L 2 (H -4 )×L 2 (H -4 ) + ∥χ 0 (σ, q)e -6sΦ/7 ∥ L 2 (H -2 )×L 2 (H -2 ) . ( 82
)
We set

Q -1 σ q =: σ q . Since Q -1 g σ g q := g σ div(g z ) ,
we deduce that (σ, q) satisfies ( 18). Then, we obtain the following inequality

∥(σ, q)e -sΦ ∥ L 2 (H -2 )×L 2 (H -2 ) + ∥(σ(0), q(0))e -sΦ(0) ∥ H -3 ×H -3 ≲ s ∥(g σ , g z )e -6sΦ/7 ∥ L 2 (H -4 )×L 2 (H -3 ) + ∥χ 0 (σ, q)e -6sΦ/7 ∥ L 2 (H -2 )×L 2 (H -2 ) , (83)
where (σ, q) satisfies ( 22). Furthermore, (83) is the suitable observability inequality for [START_REF] Song | Global existence and analyticity of L p solutions to the compressible fluid model of korteweg type[END_REF]. Now, we rewrite the equation on z in [START_REF] Jüngel | Quantum Navier-Stokes equations[END_REF] as

-∂ t z -µ △z = g z + ∇σ + (µ + ν)∇q, in (0, T, ) × T L .
To recover the estimate on z, we use the duality with following controllability problem for heat equation

∂ t y -µ △y = fy + v y χ 0 , in [0, T ] × T L , y | t=0 = y 0 and y | t=T = 0, in T L . ( 84
)
Replacing s by 7s 6 in the Lemma 5.2.3 items 1) and 2), with y 0 ∈ H 2 (T L ) and fy e 7sΦ/6 ∈ L 2 (H 1 ), then there exists a controlled trajectory y satisfying (84) with control v y ∈ L 2 (H 1 ) such that ∥y sΦ ∥ L 2 (H 3 ) + ∥v y e sΦ ∥ L 2 (H 1 ) ≲ ∥ fy e 7sΦ/6 ∥ L 2 (H 1 ) + ∥y 0 e 7sΦ(0)/6 ∥ H 2 .

By duality we get

∥ze -7sΦ/6 ∥ L 2 (H -1 ) + ∥z(0)e -7sΦ(0)/6 ∥ H -2 ≲ s ∥(g σ , g z )e -6sΦ/7 ∥ L 2 (H -4 )×L 2 (H -3 ) + ∥χ 0 (σ, q)e -6sΦ/7 ∥ L 2 (H -2) ×L 2 (H -2 ) + ∥χ 0 ze -sΦ ∥ L 2 (H -1 ) .
As χ = 1 on supp(χ 0 ), we have χ 0 χ = χ 0 and χ 0 div(z) = χ 0 div(χz). Therefore, using that the multiplication by χ 0 maps H -2 to itself, we get

∥χ 0 qe -6sΦ/7 ∥ L 2 (H -2 ) ≲ s ∥χze -6sΦ/7 ∥ L 2 (H -2 ) + ∥χσe -6sΦ/7 ∥ L 2 (H -2 )
and combining the above estimate, we obtain

∥σe -sΦ ∥ L 2 (H -2 ) + ∥σ(0)e -sΦ(0) ∥ H -3 + ∥ze -7sΦ/6 ∥ L 2 (H -1 ) + ∥z(0)e -7sΦ(0)/6 ∥ H -2 ≲ s ∥(g σ , g z )e -6sΦ/7 ∥ L 2 (H -4 )×L 2 (H -3 ) + ∥χ(σ, z)e -6sΦ/7 ∥ L 2 (H -2 )×L 2 (H -1 ) .
Since (σ, z) satisfies equation ( 17) and using the above observability inequality, we again argue by duality to deduce that the system ( 16) is controllable and that the following estimate holds

∥(a, u)e 6sΦ/7 ∥ L 2 (H 4 )×L 2 (H 3 ) + ∥χ(v a , v u )e 6sΦ/7 ∥ L 2 (H 2 )×L 2 (H 1 )
≲ s ∥(f a e sΦ , f u e 7sΦ/6 )∥ L 2 (H 2 )×L 2 (H 1 ) + ∥(a 0 e sΦ(0) , u 0 e 7sΦ(0)/6 )∥ H 3 ×H 2 . Now, we are interested to get the regularity estimate (80). We perform the regularity estimate give by the Lemma 2.0.2 to the equation satisfy by (a, u)e 5sΦ/7 , that induce a small loss in the parameter s, which is reflected the fact that we estimate (a, u)e 5sΦ/6 instead of (a, u)e 6sΦ/7 to applying the observability estimate above, and (80) follows.

9 Proof of Theorem 3.0.1

In this section we solve the controllability problem associated to [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension n ≥ 2[END_REF] and deduce the Theorem 3.0.1. Thus we have to establish the following theorem Theorem 9.0.1. Let d ∈ {2, 3} and T > 0. There exist δ > 0 such that, for all (a 0 , u 0

) ∈ H 3 (T L ) × H 2 (T L ) satisfying ∥(a 0 , u 0 )∥ H 3 ×H 2 ≤ δ, there exists a control (v a , v u ) ∈ L 2 (H 2 )×L 2 (H 1
) and a corresponding controlled trajectory (a, u) solving (13) on [0, T ] × T L with initial data (a 0 , u 0 ) satisfying the control requirement

(a, u) | t=T = (0, 0), in T L .
Besides, the controlled trajectory (a, u) has the following regularity

(a, u) ∈ C(0, T ; H 3 (T L )) ∩ L 2 (0, T ; H 4 (T L )) × C(0, T ; H 2 (T L )) ∩ L 2 (0, T ; H 3 (T L )).
Theorem 3.0.1 follows as corollary. Indeed, if (a 0 , u 0 ) ∈ H 3 (Ω) × H 2 (Ω), using that Ω is smooth bounded domain, we can construct an extension (ã 0 , ũ0 ) ∈ H 3 (T L ) × H 2 (T L ), of (a 0 , u 0 ) and this extension satisfies

∥(a 0 , u 0 )∥ H 3 (Ω)×H 2 (Ω) ≤ C(L, Ω)∥(ã 0 , ũ0 )∥ H 3 (T L )×H 2 (T L ) ,
where C(L, Ω) is a positive constant that depend only of Ω and L. Now, applying the Theorem 9.0.1, providing a smallness assumption

∥(a 0 , u 0 )∥ H 3 (Ω)×H 2 (Ω) ≤ C(L, Ω) δ,
for some δ > 0, under which there exists, in particular, a solution

(a, u) ∈ C(0, T ; H 3 (T L )) ∩ L 2 (0, T ; H 4 (T L )) × C(0, T ; H 2 (T L )) ∩ L 2 (0, T ; H 3 (T L ))
of ( 3) with initial data (a 0 , u 0 ) such that (a, u) | t=T = (0, 0).

It remain to choose δ := C(L, Ω) δ (to match with the statement of Theorem 3.0.1) and the desired theorem is establish. Now, we prove the Theorem 9.0.1.

Proof. In order to prove the controllability of the system (13), we will perform a fixed point argument to prove the controllability of the nonlinear system, namely [START_REF] Haspot | Global strong solution for the Korteweg system with quantum pressure in dimension n ≥ 2[END_REF]. We fix the parameter s 0 so that Theorem 8.0.1 holds. On the space X × Y, where

X := L ∞ (H 3 ) ∩ L 2 (H 4 ) ∩ H 1 (H 2 ) and Y := L ∞ (H 2 ) ∩ L 2 (H 3 ) ∩ H 1 (H 2 )
, we define the following closed (convex) subset

B R := (ã, ũ) ∈ X × Y ∥(ã, ũ)e 5s 0 Φ 6 ∥ X×Y ≤ R ,
where R is a positive real number which will be chosen later. Let (a 0 , u 0 ) ∈ H 3 (T L ) × H 2 (T L ). Our goal is to found a fixed-point of the map given as follow

F(ã, ũ) := G (a 0 , u 0 , f a (ã, ũ), f u (ã, ũ)) , (85)
for (ã, ũ) in B R and where according to (4) f a (ã, ũ) and f u (ã, ũ) are given by

f a (ã, ũ) := -ũ • ∇ã, f u (ã, ũ) := f 1 u (ũ, ũ) + f 2 u (ã, ũ) + f 3 u (ã, ã), ũ • ∇ũ + µ∇ã • ∇ũ + (µ + ν)∇ã • Dũ + κ 2 ∇(∇ã • ∇ã). with    f 1 u (ũ, ũ) := ũ • ∇ũ, f 2 u (ã, ũ) := µ∇ã • ∇ũ + (µ + ν)∇ã • Dũ, f 3 u (ã, ã) := κ 2 ∇(∇ã • ∇ã).
For this purpose, we have to prove that :

1. For all R > 0, F is well-defined on B R ; 2. For R > 0 and δ > 0 small enough, F map continuously B R on B R ; 3. For R > 0 small enough, F is a strict contraction maps on B R for some distance.

First, we shall need the estimate the nonlinear terms Lemma 9.0.2. Let s ≥ 0. There exist a constant C > 0, such that for all (ã, ũ) ∈ X × Y such that (ã, ũ)e

5sΦ 6 ∈ X × Y, then ∥(f a (ã, ũ)e sΦ , f u (ã, ũ)e 7sΦ 6 ∥ L 2 (H 2 )×L 2 (H 1 ) ≤ C∥(ã, ũ)e 5sΦ 6 ∥ 2 X×Y . (86)
Moreover, the constant C is independent of s ≥ s 0 .

Proof. We will repeatedly use that Φ is only time variable dependent and, for d ∈ {2, 3} that H 2 (T L ) is Banach algebra and that the product is continuous from

H 1 (T L ) × H 2 (T L ) into H 1 (T L ). Let (ã, ũ) be in X × Y such that (ã, ũ)e 5sΦ 6
∈ X × Y. Using 5 6 + 5 6 > 1, we deduce from ũe

5sΦ 6 ∈ L ∞ (H 2 ) and ∇ãe 5sΦ 6 ∈ L ∞ (H 2 ), that ∥f a (ã, ũ)e sΦ ∥ L 2 (H 2 ) ≲ ∥ũe 5sΦ 6 ∥ L ∞ (H 2 ) ∥∇ãe 5sΦ 6 ∥ L 2 (H 2 ) ≲ ∥ũe 5sΦ 6 ∥ L ∞ (H 2 ) ∥ãe 5sΦ 6 ∥ L 2 (H 3 ) .
Using that 7 6 < 5 6 + 5 6 , we have

• Since ũe 5sΦ 6 ∈ L ∞ (H 2 ) and ∇ũe 5sΦ 6 ∈ L 2 (H 1 ), we have ∥ũ • ∇ũe 7sΦ 6 ∥ L 2 (H 1 ) ≲ ∥ũe 5sΦ 6 ∥ L ∞ (H 2 ) ∥∇ũe 5sΦ 6 ∥ L 2 (H 1 ) ≲ ∥ũe 5sΦ 6 ∥ L ∞ (H 3 ) ∥ũe 5sΦ 6 ∥ L 2 (H 2 ) ,
• Since ∇ãe

5sΦ 6 ∈ L 2 (H 3 ) and ∇ũe 5sΦ 6 ∈ L ∞ (H 1 ), we have ∥∇ã • ∇ũe 6sΦ 7 ∥ L 2 (H 1 ) +∥ã • Dũe 6sΦ 7 ∥ L 2 (H 1 ) ≲ ∥∇ũe 5sΦ 6 ∥ L ∞ (H 1 ) + ∥Dũe 5sΦ 6 ∥ L ∞ (H 1 ) ∥∇ãe 5sΦ 6 ∥ L 2 (H 1 ) ≲ ∥ũe 5sΦ 6 ∥ L ∞ (H 3 ) ∥ãe 5sΦ 6 ∥ L 2 (H 2 ) , • Since ∇ãe 5sΦ ∈ L 2 (H 2 ) ∩ L ∞ (H 2 ) , we have ∥∇|∇ã| 2 e 6sΦ 7 ∥ L 2 (H 1 ) ≲ ∥∇ãe 5sΦ 6 ∥ L ∞ (H 2 ) ∥∇ãe 5sΦ 6 ∥ L 2 (H 2 ) ≲ ∥ãe 5sΦ 6 ∥ L ∞ (H 3 ) ∥ãe 5sΦ 6 ∥ L 2 (H 3 ) .
Thus, we have following estimate for f a (ã, ũ)e sΦ and f u (ã, ũ)e 6sΦ 7

:

∥f u (ã, ũ)e sΦ ∥ L 2 (H 2 ) ≤ C 1 ∥(ã, ũ)e 5sΦ 6 ∥ 2 L 2 (H 4 )∩L ∞ (H 3 )×L 2 (H 3 )∩L ∞ (H 2 )
and ∥f u (ã, ũ)e

6sΦ 7 ∥ L 2 (H 1 ) ≤ C 2 ∥(ã, ũ)e 5sΦ 6 ∥ 2 L 2 (H 4 )∩L ∞ (H 3 )×L 2 (H 3 )∩L ∞ (H 2 )
. where C 1 and C 2 are two positive constants both independent of s ≥ s 0 , which allows us to conclude the proof.

We fix s = s 0 ≥ 1 such that Theorem 8.0.1. Now, we will work the fixed point argument.

Step 1: F is well-define on B R for all R > 0. Let R > 0. From the Lemma 9.0.2, we deduce that

if (ã, ũ)e 5s 0 Φ 6 ∈ X × Y, then f a (ã, ũ)e s 0 Φ ∈ L 2 (H 2 ) and f u (ã, ũ)e 7s 0 Φ 6 ∈ L 2 (H 1 ). Since (a 0 , u 0 ) ∈ H 3 (T L ) × H 2 (T L ),
this show, by using estimate (80) of Theorem 8.0.1, that for all (ã, ũ) ∈ B R the definition of F by 85 is lawful. Furthermore, according to (80) and (86), it follows that for all (ã, ũ) ∈ B R we have ∥F(ã, ũ)e 5s 0 Φ 6 ∥ X×Y ≤ C R 2 + ∥(a 0 e s 0 Φ(0) , u 0 e 6s 0 Φ(0) 7

∥ H 3 ×H 2 . ( 87
)
Step 2 : For R > 0 and δ > 0 small enough, F(B R ) ⊂ B R . First remark that it follows from (9.0.1) and (87) that

∥F(ã, ũ)e 5s 0 Φ 6 ∥ X×Y ≤ C(R 2 + δ),
where C := Ce s 0 Φ(0) . Let R 0 := 1 C . Then for all R 0 > R > 0, we get a positive real number δ R , given by

δ R := R C -R 2 , ( 88
) such that, if ∥(a 0 , u 0 )∥ H 3 ×H 2 ≤ δ R , the bound ∥F(ã, ũ)∥ X×Y ≤ R. This show that for all R 0 > R > 0, if δ R ≥ δ > 0, then F(B R ) ⊂ B R .
From now, at any fixed R ∈ (R 0 , 0), the smallness initial data parameter δ will be automatically δ R . We assume that (a 0 , u 0 ) satisfy

∥(a 0 , u 0 )∥ H 3 ×H 2 ≤ δ R ,
for some R ∈]0, R 0 [ which will be chosen in the next step.

Let

X := L 2 (H 4 ) ∩ L ∞ (H 3 ) and Ỹ := L 2 (H 3 ) ∩ L ∞ (H 2 ). Each B R is contained in X × Ỹ.
We over B R we consider the distance d s given by

d s 0 (V, W ) := ∥(V -W )e 5s 0 Φ 6 ∥ X× Ỹ (V, W ∈ B R ) .
Step 3: F is a strict contraction maps on (B R , d s 0 ). Let (ã 1 , ã2 ) and (ã 2 , ũ2 ) two elements of B R . Taking advantage of the quadratic structure of nonlinearities, we can write

f a (ã 1 , ũ1 ) -f a (ã 2 , ũ2 ) = f a (ã 1 , ũ1 -ũ2 ) + f a (ã 1 -ã2 , ũ2 ), f 1 u (ũ 1 , ũ1 ) -f 1 u (ũ 2 , ũ2 ) = f 1 u (ũ 1 -ũ2 , ũ2 ) + f 1 u (ũ 1 , ũ1 -ũ2 ), f 2 u (ã 1 , ũ1 ) -f 2 u (ã 2 , ũ2 ) = f 1 u (ã 1 , ũ1 -ũ2 ) + f 2 u (ã 1 -ã2 , ũ2 ), f 3 u (ã 1 , ã1 ) -f 3 u (ã 2 , ã2 ) = f 3 u (ã 1 -ã2 , ã2 ) + f 3 u (ã 1 , ã1 -ã2 ).
Now, as the proof of Lemma 9.0.2, keeping mind that 1 < 5 6 + 5 6 and 7 6 < 5 6 + 5 6 , and according to above four identities, we get

∥(f a (ã 1 , ũ1 ) -f a (ã 2 , ũ2 ))e s 0 Φ ∥ L 2 (H 2 ) ≤ C∥ã 1 e 5s 0 Φ 6 ∥ X∥(ũ 1 -ũ2 )e 5s 0 Φ 6 ∥ Ỹ + C∥(ã 1 -ã2 )e 5s 0 Φ 6 ∥ X∥ũ 2 e 5s 0 Φ 6 ∥ Ỹ ≤ 2CRd s 0 ((ã 1 , ũ1 ), (ã 2 , ũ2 )) , ∥(f 1 u (ũ 1 , ũ1 ) -f 1 u (ũ 2 , ũ2 ))e 7s 0 Φ 6 ∥ L 2 (H 1 ) ≤ C∥ũ 1 e 5s 0 Φ 6 ∥ X∥(ũ 1 -ũ2 )e 5s 0 Φ 6 ∥ Ỹ + C∥(ũ 1 -ũ2 )e 5s 0 Φ 6 ∥ X∥ũ 2 e 5s 0 Φ 6 ∥ Ỹ ≤ 2CRd s 0 ((ã 1 , ũ1 ), (ã 2 , ũ2 )) , ∥(f 2 a (ũ 1 , ũ1 ) -f 2 u (ã 2 , ũ2
∥(f 3 u (ã 1 , ã1 ) -f 3 u (ã 2 , ã2 ))e 7s 0 Φ 6 ∥ L 2 (H 1 ) ≤ C∥ã 1 e 5s 0 Φ 6 ∥ X∥(ã 1 -ã2 )e 5s 0 Φ 6 ∥ Ỹ + C∥(ã 1 -ã2 )e 5s 0 Φ 6 ∥ X∥ã 2 e 5s 0 Φ 6 ∥ Ỹ ≤ 2CRd s 0 ((ã 1 , ũ1 ), (ã 2 , ũ2 )) .
Thus, according to the Theorem 79, it follows that

d s 0 (F(ã 1 , ũ1 ), F(ã 2 , ũ2 )) ≤ 6CRd s 0 ((ã 1 , ũ1 ), (ã 2 , ũ2 )) .
Then, we choose

R := 1 12
C and δ := δ R , which satisfy R 0 > R > 0 and such that

d s 0 (F(ã 1 , ũ1 ), F(ã 2 , ũ2 )) ≤ 1 2 d s 0 ((ã 1 , ũ1 ), (ã 2 , ũ2 )) ,
holds for every (ã 1 , ũ1 ) and (ã 2 , ũ2 ) in B R with (a 0 , u 0 ) satisfying

∥(a 0 , u 0 )∥ H 3 ×H 2 ≤ δ.
We conclude that F is a strict contraction of B R endowed the topology induced by the distance d s 0 for which B R is complete.

Step 4: Conclusion. According to the last step, we concluded that F as a fixed-point (a, u) in B R . Moreover, it follow from Theorem 8.0.1 and Lemma 9.0.2 that (a, u) have the wanted regularity.

Remark 9.0.3. After the second step of the proof, if we remark that F maps continuously B R into itself when B R is endowed of the topology of X × Ỹ (or for a more weaker topology as the L 2 (L 2 ) 2 -topology for example). Indeed, since the embedding H 4 (T L ) → H 3 (T L ) and

H 3 (T L ) → H 2 (T L ) are compact and B R is bounded on H 1 (H 2 ) × H 1 (H 1
) and on L 2 (H 4 ) × L 2 (H 3 ), we deduce from the Lion-Aubin theorem (see [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]) that B R is relatively compact on X × Ỹ. Then to apply the Schauder fixed-point theorem1 (over X × Ỹ), its enough to prove that F maps continuously B R into itself. With this remark we aim to point the possibility to tackle more general nonlinearities.

A Proof of the Lemma 2.0.2

Proof. Let (a, u) be a solution of the Cauchy problem (5) [START_REF] Chowdhury | Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension[END_REF]. The strategy consists to obtain firstly L 2 and L ∞ estimate, then use an interpolation inequality to get L p estimates. Finally, we deduce the Sobolev estimates in time from the L 2 -estimates. We set v 0 := (⟨D⟩a 0 , u 0 ), v := (⟨D⟩a, u) and h := (⟨D⟩f, g).

Step 1:

L ∞ estimate. Let t ∈]0, T ]. From Lemma 2.0.1, we deduce that for t ′ ∈ [0, t] and ξ ∈ Z d | v(t ′ , ξ)| ≲ T,c 0 ,L e -t ′ c 0 |ξ| 2 | v 0 (ξ)| + t ′ 0 e -c 0 τ |ξ| | h(τ, ξ)dτ . (89)
Then, if |ξ| ≥ 1, we deduce from Cauchy-Schwarz inequality that

| v(t ′ , ξ)| ≲ T,c 0 ,L e -t ′ c 0 |ξ| 2 | v 0 (ξ)| + 1 |ξ| ∥ h(•, ξ)∥ L 2 (0,T ) .
Taking the supremum over [0, T ] and multiplying on the above inequality by ⟨ξ⟩ 2(s-1) and using that |ξ| ≥ 1, it follow that

sup t ′ ∈[0,t] {⟨ξ⟩ 2(s-1) | v(t, ξ)| 2 } ≲ T,c 0 ,L ⟨ξ⟩ 2(s-1) | v 0 (ξ)| 2 + ⟨ξ⟩ 2(s-1) |ξ| 2 ∥ h(•, ξ)∥ 2 L 2 (0,T ) ≲ T,c 0 ,s,L ⟨ξ⟩ 2(s-1) | v 0 (ξ)| 2 + |ξ| 2(s-2) ∥ h(•, ξ)∥ 2 L 2 (0,T ) ≲ T,c 0 ,s,L ⟨ξ⟩ 2(s-1) | v 0 (ξ)| 2 + ⟨ξ⟩ 2(s-2) ∥ h(•, ξ)∥ 2 L 2 (0,T ) (90)
whereas for ξ = 0, we deduce from (89) that

sup t ′ ∈[0,T ] {| v(t ′ , 0)| 2 } ≲ t,c 0 ,L | v 0 (0)| 2 + ∥ h(•, 0)∥ 2 L 2 (0,T ) . (91)
Combining ( 90) and (91), we deduce that

  ξ∈Z d sup t ′ ∈[0,t] {| v(t ′ , ξ)| 2 }⟨ξ⟩ 2(s-1)   1 2 ≲ T,c 0 ,s ∥v 0 ∥ H s-1 + ∥h∥ L 2 (H s-2 ) . (92)
Step 2: L 2 estimate. Let ξ ∈ Z d . From (89), it follow that

T 0 | v(t, ξ)| 2 dt ′ ≲ t,c 0 ,s T 0 e -2c 0 t ′ |ξ| 2 dt ′ | v 0 (ξ)| 2 + T 0 e -c 0 τ |ξ| 2 dτ ∥ h(•, ξ)∥ L 2 (0,T ) .
Then, if ξ = 0, we have

T 0 | v(t, 0)| 2 dt ′ ≲ t,c 0 ,s | v 0 (0)| 2 + ∥ h(•, 0)∥ L 2 (0,T ) . (93)
Since (93), we deduce that above inequality hold for all ξ ∈ Z d . Thus, summing up the last inequality overs Z d , we deduce that

ξ∈Z d T 0 | v(t, ξ)| 2 ⟨ξ⟩ 2s dt ′ ≲ T,c 0 ,s ξ∈Z d ⟨ξ⟩ 2(s-1) | v 0 (ξ)| 2 + ξ∈Z d ⟨ξ⟩ 2(s-2) ∥ h(•, ξ)∥ L 2 (0,T ) . ( 94
)
Step 3: L p estimate for p / ∈ {2, ∞}. Now, we suppose that p / ∈ {2, ∞}. Let ξ ∈ Z d . Applying the interpolation inequality and Young inequality with 1

( 2 p ) -1 + 1 (1-2 p ) -1 = 1, it follow that T 0 | v(t, ξ)| p dt 2 p ⟨ξ⟩ 4 p ≲ p ∥ v(•, ξ)∥ 4 p L 2 (0,T ) ∥ v(•, ξ)∥ 2(1-2 p ) L ∞ (0,T ) ≲ p ∥ v(•, ξ)∥ 2 L 2 (0,T ) ⟨ξ⟩ 2 + ∥ v(•, ξ)∥ 2 L ∞ (0,T ) .
Then, summing up booth extremal terms of above inequalities over Z d , using Minkowsky inequality, (92) and (94), it follow that

∥v∥ L p (H s-1+ 2 p ) ≲ T,c 0 ,s,p ∥v 0 ∥ H s-1 + ∥h∥ L 2 (H s-2 ) ,
that is the wanted inequality.

Step 4: H 1 -estimates. Using the equation over the fluctuation of the density,that is

∂ t a = -div(u) + f
and the L 2 -estimates established in the first step, we get

∥∂ t a∥ L 2 (H σ-1 ) ≤ ∥ div(u)∥ L 2 (H σ-1 ) + ∥f ∥ L 2 (H σ-1 ) (95)
≲ ∥u∥ L 2 (H σ ) + ∥(f, g)∥ L 2 (H σ-1 )×L 2 (H σ-2 ) (96) ≲ ∥(a 0 , u 0 )∥ H σ ×H σ-1 + ∥(f, g)∥ L 2 (H σ-1 )×L 2 (H σ-2 ) . (97)
Thus ∂ t ∈ L 2 (H σ-1 ) and

∥a∥ H 1 (H σ-1 ) ≲ ∥(a 0 , u 0 )∥ H σ ×H σ-1 + ∥(f, g)∥ L 2 (H σ-1 )×L 2 (H σ-2 ) .
Likewise, using the equation over u, ∂ t u -µ △u -(µ + ν)∇ div(u) + α∇a -κ∇ △a = g, and the first step, we obtain

∥∂ t u∥ L 2 (H σ-1 ) ≲ ∥ △u∥ L 2 (H σ-2 ) + ∥∇ div(u)∥ L 2 (H σ-2 ) + ∥∇a∥ L 2 (H σ-2 ) + ∥∇ △a∥ L 2 (H σ-2 ) + ∥g∥ L 2 (H σ ) ≲ ∥u∥ L 2 (H σ ) + ∥a∥ L 2 (H σ-1 + ∥a∥ L 2 (H σ+1 ) + ∥g∥ L 2 (H σ-2 ) ≲ ∥(a 0 , u 0 )∥ H σ ×H σ-1 + ∥(f, g)∥ L 2 (H σ-1 )×L 2 (H σ-2 ) .
The lemma follow.

B Proof of the Carleman inequality

The proof is adapted to the proof of the Carleman inequality proved in [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] under the assumption ζ > 0.

Let z be a real value smooth function on [0, T ] × T L and set f := -ζ∂ t w -△w.

We shall deal with the function w := e -sφ w.

According to the definition of θ, w satisfies w(T, x) = 0 and ∇w(T, x) = 0, x ∈ T L .

Let define the conjugate of -ζ∂ t -△ by P φ := e -sφ (-ζ∂ t -△)e sφ .

Then P φ w = -ζ∂ t w -sζ∂ t φw -△w -2s∇φ • ∇w -s 2 |∇φ| 2 w -s △φw and e -sφ f = P φ w.

From the standard strategy to prove global Carleman inequality (see [START_REF] Fursikov | Controllability of evolution equations[END_REF] or [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]), we now define quantities P 1 w, P 2 w and Rw by setting

P 1 w := -ζ∂ t w -2s∇φ • ∇w + 2sλ 2 |∇ψ| 2 ξw, P 2 w := -△w -sζ∂ t φw -s 2 |∇φ| 2 w, Rw := sλ △ψξw -sλ 2 |∇ψ| 2 ξw,
so that P φ w = P 1 w + P 2 w + Rw.

Since P 1 w + P 2 w = e -sφ -Rw, we get

[0,T ]×T L |P 1 w| 2 + [0,T ]×T L |P 2 w| 2 + 2 [0,T ]×T L P 1 wP 2 w = [0,T ]×T L |f e -sφ -Rw| 2 ≤ 2 [0,T ]×T L |f | 2 e -2sφ + 2 [0,T ]×T L |Rw| 2 . ( 98
)
The main part of the proof L 2 -Carleman inequality consist to estimate from below the real part of the scalar product of P 1 w with P 2 w. We begin by compute it.

Step 1: computation of the scalar product. We first write

[0,T ]×T L P 1 wP 2 w = 1≤k,l≤3 I k,l ,
where I k,l is the scalar product of the k-th term of P 1 w with the l-th term of P 2 w. We use the L-periodicity to vanish all the boundary terms of integration by parts.

Computing of I 1,1 . Ĩ1,1 = [0,T ]×T L ζ∂ t w △w = - [0,T ]×T L ζ∂ t ∇w • ∇w = - 1 2 [0,T ]×T L ζ∂ t (|∇w| 2 ) = ζ 2 T L |∇w(0)| 2 .
Computing of I 1,2 .

I 1,2 = s [0,T ]×T L |ζ| 2 ∂ t w∂ t φw = |ζ| 2 s 2 [0,T ]×T L ∂ t (|w| 2 )∂ t φ = - s|ζ| 2 2 T L |w(0)| 2 ∂ t φ(0) - s|ζ| 2 2 [0,T ]×T L |w| 2 ∂ tt φ.
Computing I 1,3 .

I 1,3 = s 2 [0,T ]×T L ζ∂ t w|∇φ| 2 w = s 2 ζ [0,T ]×T L ∂ t (|w| 2 )|∇φ| 2 = - s 2 2 T L |∇φ(0)| 2 |w(0)| 2 - s 2 ζ 2 [0,T ]×T L |w| 2 ∂ t (|∇φ| 2 ).
Computing I 2,1 .

I 2,1 = 2s [0,T ]×T L (∇φ • ∇w) △w = -2s [0,T ]×T L ∇(∇φ • ∇w) • ∇w = -2s [0,T ]×T L D 2 φ(∇w, ∇w) -s [0,T ]×T L ∇φ • ∇(|∇w| 2 ) = -2s [0,T ]×T L D 2 φ(∇w, ∇w) + s [0,T ]×T L △φ|∇w| 2 .
Computing I 2,2 .

I 2,2 = 2s 2 ζ [0,T ]×T L ∇φ • ∇w(∂ t φ)w = -s 2 ζ [0,T ]×T L div (∂ t φ∇φ) |w| 2 .
Computing I 2,3 .

I 2,3 = 2s 3 [0,T ]×T L |∇φ| 2 ∇φ • (w∇w) = s 3 [0,T ]×T L |∇φ| 2 ∇φ • ∇|w| 2 = -s 3 [0,T ]×T L div |∇φ| 2 ∇φ |w| 2 .
Computing I 3,1 .

I 3,1 = -2sλ 2 [0,T ]×T L |∇ψ| 2 ξw △w = 2sλ 2 [0,T ]×T L ∇ |∇ψ| 2 ξw • ∇w = -2sλ 2 [0,T ]×T L |∇ψ| 2 ξ|∇w| 2 + 2sλ 2 [0,T ]×T L ∇ |∇ψ| 2 ξ w • ∇w. Computing I 3,2 . I 3,2 = 2s 2 ζλ 2 [0,T ]×T L |∇ψ| 2 ξ∂ t |w| 2 .
Computing I 3,3 .

I 3,3 = -2s 3 λ 2 [0,T ]×T L |∇ψ| 2 ξ|∇φ| 2 |w| 2 = I 3,3 .
From all this formula, we conclude that

[0,T ]×T L P 1 wP 2 w = ζ 2 T L |∇w(0)| 2 - s 2 ζ 2 T L |∇w(0)| 2 |∇φ(0)| 2 - s|ζ| 2 2 T L |w(0)| 2 ∂ t φ(0) (99)
-2s

[0,T ]×T L |w| 2 D 2 φ(∇w, ∇w) + s [0,T ]×T L |∇w| 2 (2λ 2 |∇ψ| 2 ξ + ∇φ) (100) + s 3 [0,T ]×T L |w| 2 div(|∇φ| 2 ∇φ) -2λ 2 |ψ| 2 ξ|∇φ| 2 (101) + s 2 ζ [0,T ]×T L |w| 2 -2λ 2 |ψ| 2 ξ∂ t φ -△ϕ∂ t φ|w| 2 ∂ t (|φ| 2 ) (102) + s|ζ| 2 [0,T ]×T L |w| 2 - 1 2 ∂ 2 t φ (103) + 2sλ 2 [0,T ]×T L ∇(|∇ψ| 2 ζ)w • ∇w. (104)
Step 2: Positivity. In this step, the goal is to check that the coefficients in the above integrals are positives, except perhaps in the observation set [0, T ] × ω. In this perspective we will strongly rely upon the choice of the weight function ϕ, and on the formula

∂ t φ = ∂ t θ θ φ and ∂ t ξ = ∂ t θ θ ξ. (105)
In the following, to simplify notations, we will denote by C ⋆ generic positive large constant the do not depend on s ans λ and by C ⋆ generic positive small constants independent of s and λ. The constant may change from line to line.

Positivity of the term (99). Using (105), for λ large enough, we get

-|ζ| 2 ∂ t φ(0) = m|ζ| 2 T 0 λe 12λ -e λψ(0) = sλ 2 |ζ| 2 e 2λ T 0 (λe 12λ -e λψ(0) ) ≥ C ⋆ |ζ| 2 sλ 3 e 14λ whereas ζ|∇φ(0)| 2 ≤ C ⋆ λ 2 e 14λ |ζ| 2 .
Hence, since (32), there exists λ 1 ≥ 1, such that for all λ ≥ λ 1 we have inf

T L {ζs 2 |∇φ(0)| 2 -s|ζ| 2 ∂ t φ(0)} ≥ C ⋆ s 2 λ 3 e 14λ .
We deduce that

- s 2 ζ 2 T L |w(0)| 2 |∇φ(0)| 2 - s|ζ| 2 2 T L ∂ t φ(0)|w(0)| 2 ≥ C ⋆ s 2 λ 3 e 14λ T L |w(0)| 2 . ( 106
)
Positivity of the term (100). Let η ∈ R d . It follow from the definition of φ that

-2sD 2 φ(η, η) = 2sλD 2 ψ(η, η)ξ + 2sλ 2 |∇ψ • η| 2 , hence -2sD 2 φ(η, η) + s(△φ + 2λ 2 |∇ψ| 2 ξ)|η| 2 = 2sλ 2 ξ|∇ • ψ| 2 + 2sλD 2 ψ(η, η)ξ -sλ| △ψ|ξ + sλ 2 |∇ψ| 2 ξ|η| 2
Since (27), we deduce the existence of λ 2 ≥ λ 1 , such that for all λ ≥ λ 2 and η ∈

C d -2sℜ(D 2 φ(η, η)) + s(△φ + 2sλ|∇ψ| 2 ξ)|η| 2 = -2sD 2 φ(η, η) + s(△φ + 2λ 2 |∇ψ| 2 ξ)|(η)| 2 ≥ C ⋆ sλ 2 ξ|η| 2 = C ⋆ sλ 2 ξ|η| 2 , hold on [0, T ] × (T L \ ω). In addition to, using ψ ∈ C 2 (T L , R), the bound -2sD 2 φ(η, η) + s(△φ + 2sλ|∇ψ| 2 ξ)|η| 2 ≥ C ⋆ sλ 2 ξ|η| 2 -C ⋆ sλ 2 ξ|η| 2 ,
hold on [0, T ] × ω and for all η ∈ C d . Thus, we deduce that, for all λ ≥ λ 2 , -2s

[0,T ]×T L |w| 2 ℜ D 2 φ(∇w, ∇w) + s [0,T ]×T L |∇w| 2 (2λ 2 |∇ψ| 2 ξ + ∇φ) ≥ C ⋆ sλ 2 [0,T ]×T L ξ|∇w| 2 -C ⋆ sλ 2 [0,T ]×ω ξ|∇w| 2 . ( 107
)
Positivity of the terms (101). We have

-div(|∇ϕ| 2 ∇φ) = 3λ 4 |∇ψ| 4 ξ 3 + λ 3 ξ 3 div(|∇ψ| 2 ∇ψ) and λ 2 |∇ψ| 2 ξ|∇φ| 2 = λ 4 |∇ψ| 2 ξ 3 . Therefore -div (|∇φ| 2 ∇φ) -2λ 2 |∇ψ| 2 ξ|∇φ| 2 = λ 4 |∇ψ| 2 ξ 3 + λ 3 ξ 3 div(|∇ψ| 2 ∇ψ).
In consequence, from (27), we deduce there exist of λ 3 ≥ λ 2 , such that for all λ ≥ λ 3 , the following inequality

-div (|∇φ| 2 ∇φ) -2λ 2 |∇ψ| 2 ξ|∇φ| 2 ≥ C ⋆ λ 4 ξ 3 , hold on [0, T ] × (T L \ ω), whereas on [0, T ] × ω, we have -div (|∇φ| 2 ∇φ) -2λ 2 |∇ψ| 2 ξ|∇φ| 2 ≥ C ⋆ λ 4 ξ 3 -C ⋆ λ 4 ξ 3 .
Then, for all λ ≥ λ 3 ,

s 3 [0,T ]×T L |w| 2 (-div(|∇φ| 2 ∇φ)-2λ 2 |∇ψ| 2 ξ|∇φ| 2 ) ≥ C ⋆ s 3 λ 4 [0,T ]×T L ξ 3 |w| 2 -C ⋆ s 3 λ 4 [0,T ]×ω ξ 3 |w| 2 . ( 108
)
Positivity of the term (102). We aim to estimate by below the term

-∂ t |∇φ| 2 -(△φ + 2λ 2 |∇ψ| 2 ξ)∂ t φ = ∂ t θ θ -λ 2 ξφ|∇ψ| 2 + λξ △ψφ -2λ 2 ξ 2 |∇ψ| 2 (109)
with a factor of |∂ t θ|. On [0, T 0 ], we have ∂ t θ ≥ 0 and 1 ≤ θ ≤ 2. Then, using that △ψ is bounded and (27), we deduce that for λ large enough,

∂ t θ -λ 2 ξφ|∇ψ| 2 + λξ △ψφ -2λ 2 ξ 2 |∇ψ| 2 ≥ C ⋆ λ 2 ξφ|∂ t θ|
hold on [0, T 0 ] × (T L \ ω), whereas the following bound

∂ t θ θ -λ 2 ξφ|∇ψ| 2 + λξ △ψφ -2λ 2 ξ 2 |∇ψ| 2 ≥ C ⋆ λ 2 ξφ|∂ t | -C ⋆ λ 2 ξφ|∂ t θ| hold on [0, T 0 ] × ω. On [T 0 , T -2T 1 ],
we have ∂ t θ = 0, hence the term (109) vanish on this interval and is trivially bounded from below by any factor of |∂ t θ|. On [T -2T 1 , T [, we have

|∂ t θ| ≤ C ⋆ θ 2 .
Hence, since |∂ t θ| ≤ λξ 2 , for λ large enough, we have (113)

| ∂ t θ θ (-λ 2 ξφ|∇ψ| 2 + λξ △ψφ -2λ 2 ξ 2 |∇ψ| 2 )| ≤ C ⋆ λ 2 θξφ ≤ C ⋆ λ 3 ξ 3 , on [T -2T
|∂ 2 t φ| ≤ C ⋆ θ 2 φ on [T -2T 1 , T ] × T L . (112)
Step 3: Lower bound for the cross-product P 1 wP 2 w. We aim to estimate by below P 1 wP 2 w with some non negative integrals on [0, T ] × T L and non positive integral over the observability region [0, T ] × ω. Lower bound of (101)-( 102 In the right hands of the above inequality, the negative term which comports an integral over [0, T ]×T L (last three terms) can be "absorbed" by the higher power of s and λ in factor of the corresponding positive integrals, to estimate this member by bellow. Thus, we can found s 2 ≥ s 1 and λ 6 ≥ λ 5 such that for all s ≥ s 2 and λ ≥ λ 6 , the following inequality holds According to estimates (98) and the above inequality we get Beside, the last term of the right hand of the above inequality, satisfies the following estimate

[0,T ]×T L |Rw| 2 ≤ C ⋆ [0,T ]×T L ξ 3 |w| 2 .
Hence, there exist s 3 ≤ s 2 , such that for all s ≤ s 3 and λ ≤ λ 6 , we have Let remark that the solution is done on ω ⊂ {χ 0 = 1} ⊂ supp(χ 0 ). The ideas is to take an observation set slightly larger than [0, T ] × ω, that is supp(χ 0 ). To recover the observability terms of χ 0 |∇w| 2 χ, we begin to compute the scalar product [0,T ]×T L P 2 w(χ 0 λ 2 ξw).

Then we have

  ∇u, g2 := (1 + a) -1 Au -Āu, g3 := -(1 + a) -1 ∇P (1 + a) + ᾱ∇a, g4 := ∇ κ(1 + a) -κ ∆a + 1 2 κ ′ (1 + a)|∇a| 2 .

))e 7s 0 Φ 6 ∥ L 2 (H 1 ) ≤ C∥ã 1 e 5s 0 Φ 6 ∥≤

 6216 X∥(ũ 1 -ũ2 )e CRd s 0 ((ã 1 , ũ1 ), (ã 2 , ũ2 )) , and

Since ∂ 2 2 [⋆ s 3 λ 2 [

 222 t θ = 0 on [T 0 , T -2T 1 ], combining the estimate on [0, T 0 ] × T L (111) and the estimate on [T -2T 1 ] × T L (112), we deduce that s|ζ| 0,T ]×T L ξ 3 |w| 2 .

2 + s 2 ζ|∂ t θ|ξφ|w| 2 -C ⋆ s 3 λ 4 [that 2sλ 2 [ 4 [ 0 ,T ]×T L ξ 3 |w| 2 + C ⋆ λ 2 [ 2 + C ⋆ sλ 2 [ 0 ,T ]×T L ξ|∇w| 2 -C ⋆ sλ 2 [ 0 ,T ]×ω ξ|∇w| 2 + C ⋆ s 3 λ 4 [ 0 ,T ]×T L ξ 3 |w| 2 + C ⋆ s 2 λ 2 [|∂ t θ|ξφ|w| 2 -C ⋆ s 3 λ 4 [ξ 3 |w| 2 -C ⋆ λ 2 [

 222424022220220240222422 )-(103). We denote by I w the sum of the terms (101)-(102)-(103), namelyI w := s 3 [0,T ]×T L |w| 2 div (|∇φ| 2 ∇φ) -2λ 2 |ψ| 2 ξ|∇φ| [0,T ]×T L |w| 2 -2λ|ψ| 2 ξ∂ t φ -△φ∂ t φ|w| 2 ∂ t (|φ| 2 ) ), (110) and (113), we deduce that there exist s 1 ≥ 1 and λ 5 ≥ λ 4 , such that for all s ≥ s 1 and λ ≥ λ 5 , we haveI w ≥ C ⋆ s 3 λ 4 [0,T ]×T L ξ 3 |w| 2 + C ⋆ s 2 λ 2 [0,T 0 ]×T L 0,T ]×ω ξ 3 |w| 2 -C ⋆ s 2 λ 2 [0,T 0 ]×ω |∂ t θ|ξφ|w| 2 . (114)Upper bound of (104). We remark that ∇(|∇ψ| 2 ξ) ≤ C ⋆ λξ, which implies with the elementary Young inequality2 0,T ]×T L ∇(|∇ψ|ξ)w • ∇w ≤ C ⋆ s 2 λLower bound of [0,T ]×T L P 1 wP 2 w. From (106), (107), (114) and (115), we deduce that for all s ≥ s 1 and λ ≥ λ 5 , we have2 [0,T ]×T L P 1 wP 2 w ≥ T L |∇w(0)| 2 + C ⋆ s 2 λ 3 e 14λ T L |w(0)| 0,T 0 ]×T L 0,T ]×ω ξ 3 |w| 2 -C ⋆ s 2 λ 3 [0,T 0 ]×ω |∂ t θ|ξφ|w| 2 -C ⋆ s 2 λ 4 [0,T ]×T L ξ 3 |w| 2 -C ⋆ s 2 λ 3 [0,T ]×T L 0,T ]×T L ξ|∇w| 2 .

2 [ 0 2 + C ⋆ sλ 2 [ 0 ,T ]×T L ξ|∇w| 2 -C ⋆ sλ 2 [ 0 ,T ]×ω ξ|∇w| 2 + C ⋆ s 3 λ 4 [|∂ t θ|ξφ|w| 2 -C ⋆ s 3 λ 4 [ 0 ,T ]×ω ξ 3 2

 20220220242403 ,T ]×T L P 1 wP 2 w ≥ T L |∇w(0)| 2 + C ⋆ s 2 λ 3 e 14λ T L |w(0)| 0,T ]×T L ξ 3 |w| 2 + C ⋆ s 2 λ 2 [0,T 0 ]×T L |w| 2 -C ⋆ s 2 λ 3 [0,T 0 ]×ω |∂ t θ|ξφ|w| 2 .If a and b are two real numbers, then |ab| ≤ a 2 + b 2

|P 2 2 + C ⋆ sλ 2 [ 0 ,T ]×T L ξ|∇w| 2 + C ⋆ s 3 λ 4 [ 0 ,T ]×T L ξ 3 |w| 2 + C ⋆ s 2 λ 2 [|∂ t θ|ξφ|w| 2 ≤ 2 [ 0 ,T ]×ω ξ|∇w| 2 + C ⋆ s 3 λ 4 [ 0 ,T ]×ω ξ 3 |w| 2 + C ⋆ s 2 λ 3 [

 22202402222024023 w| 2 + C ⋆ T L |∇w(0)| 2 + C ⋆ s 2 λ 3 e 14λ T L |w(0)| 0,T 0 ]×T L C ⋆ [0,T ]×T L |f | 2 e -2sφ + C ⋆ sλ

|P 2 2 + C ⋆ sλ 2 [ 0 ,T ]×T L ξ|∇w| 2 + C ⋆ s 3 λ 4 [ 0 ,T ]×T L ξ 3 |w| 2 + C ⋆ s 2 λ 2 [|∂ t θ|ξφ|w| 2 ≤ 2 [+ C ⋆ s 3 λ 4 [ 0 ,T ]×ω ξ 3

 22202402222403 w| 2 + C ⋆ T L |∇w(0)| 2 + C ⋆ s 2 λ 3 e 14λ T L |w(0)| 0,T 0 ]×T L C ⋆ [0,T ]×T L |f | 2 e -2sφ + C ⋆ sλ |w| 2 + C ⋆ s 2 λ 3 [0,T 0 ]×T L |∂ t θ|ξφ|w| 2 .

P 2 w(χ 0 λ 2 ξw) = sλ 2 [χ 0 ξ∂ t φ|w| 2 -s 3 λ 3 [0 ξ |∂ t θ| θ φ|w| 2 +χ 0 ξ∂ t |w| 2 .χ 0 ξ∂ t φ|w| 2 ≥ λ 2 s 2 ζ 2 [ 0 ,T 0 ]×T L χ 0 ξ|∂ t θ|φ|w| 2 -λ 2 s 2 ζλ 2 s 2 ζ 2 [ 2 = s 3 λ 2 [ 0 ,T ]×T L χ 0 ξ|w| 2 |∇φ| 2 . 2 s 2 ζ 2 [ 0 ,T ]×T L χ 0 ξ|∂ t θ|φ|w| 2 + sλ 2 [L ξ 2 |w| 2 + s 2 λ 3 χ 0 ξ 3 |w| 2 |∇ψ| 2 ≤ C ⋆ s 3 λ 4 [ 2 s 2 [ 0 ,T ]×ω ξ|∂ t θ|φ|w| 2 + sλ 2 [|P 2 w| 2 + C ⋆ s 3 λ 4 [

 222322220222220222022324220224 0,T ]×T L χ 0 ξ|w| 2 |∇ϕ| 2 .(117)Furthermore, using that ∂ t θ ≤ 0 on [0, T 0 ] and∂ t θ = 0 on [T 0 , T -2T 1 ], we get [0,T 0 ]×T L χ 0 ξ∂ t φ|w| 2 = -[0,T 0 ]×T L χ [T -2T 1 ,T ]×T L Since 1 ≤ θ ≤ 2 on [0, T 0 ],we deduce that -λ 2 s 2 ζ [0,T ]×T L [T -2T 1 ,T ]×T L χ 0 χ∂ t |w| 2 . 0,T 0 ]×T L χ 0 ξ|∂ t θ|φ|w| 2 + sλ 2 [0,T ]×T L χ 0 ξ|∇w| 2 ≤ sλ 2 [0,T ]×T L |P 2 w||χ 0 ξw| + sλ 2 [0,T ]×T L |w| 2 | △(χ 0 ξ)| + λ 2 s 2 ζ [T -2T 1 ,T ]×T L χ 0 ξ∂ t φ|w| Since | △(χ 0 ξ)| ≤ C ⋆ λ 2 ξ 2 , |∇ϕ| 2 = ξ 2 |∇ψ| 2 and sup [T -2T 1 ,T [ { |∂tθ| θ 2 } ≤ C ⋆and using the elementary Young estimate, we getλ ζe 12λ [T -2T 1 ,T ]×T L χ 0 θ 2 ξ|w| 2 + s 3 λ 3 [0,T ]×T L 0,T ]×T L χ 0 ξ 3 |w| 2 + C ⋆ √ s [0,T ]×T L |P 2 w| 2 .Since ω ⊂ {χ 0 = 1} and ζ > 0, it follow that λ 0,T ]×T L χ 0 ξ 3 |w| 2 .

  [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] , T [×T L . Thus, we can found λ 4 ≥ λ 3 , such that for every λ ≥ λ 4 , we haves 2 ζ [0,T ]×T L |w| 2 (-∂ t |∇φ| 2 -(△φ + 2λ 2 |∇ψ| 2 ξ)∂ t φ) ≥ C ⋆ s 2 λ 2 [0,T 0 ]×T L |w| 2 |∂ t θ|ξφ -C ⋆ s 2 λ 3 On [0, T 0 ], we have |∂ 2 t θ| ≤ C ⋆ s 2 λ 4 e 4λ , hence |∂ 2 t φ| ≤ C ⋆ s 2 λ 5 e 16λ on [0, T 0 ] × T L .(111)as well as, we have|∂ 2 t θ| ≤ C ⋆ θ 3 , on [T -2T 1 , T [.Thus, since (30) and θϕ ≤ λξ 2 , we deduce

				ξ 3 |w| 2
			[0,T ]×T L
			-C ⋆ s 2 λ 2	|∂ t θ|ξφ|w| 2 .	(110)
			[0,T 0 ]×ω
	Positivity of the term of (103). We begin to remark that
	∂ 2 t φ =	∂ 2 t θ θ	φ.

[START_REF] Taylor | Partial Differential Equations III: Nonlinear Equations[END_REF] Theorem B.2, page 302.

Hence, by using (116), we deduce there exist s 4 ≤ s 3 and λ 7 ≤ λ 6 such that

Step 4: Conclusion. It's enough to recover the estimate on z from (119). Since w = we sφ , we get

Combining inequality abuve and (119), the wanted inequality follow.


Alexandre Bouënard

Marcelo M Wanderley

Sylvie Gibet

Analysis of Timpani Preparatory Gesture Parameterization

Keywords: Instrumental Gesture Analysis, Percussion Gesture, Gesture Synthesis

Introduction

The appearance of novel gestural interfaces for instrumental performance, such as virtual character gesture synthesis, yields to a growing interest for the analysis of the expressive components that are at stake during instrumental gestures.

Several quantitative studies on instrumental gestures have been carried out, extracting and evaluating gesture parameters for instance based on classification/recognition methods such as k-Means classifiers [START_REF] Rasamimanana | Gesture Analysis of Violin Bow Strokes[END_REF]. Considering gesture synthesis, data-driven techniques for virtual character animation benefit from the realism of human motion capture data. These are based on large sets of gestures in varying contexts, and rely also on features extraction and data reduction methods for keeping the realism of the produced motion and generating style [START_REF] Hsu | Style Translation for Human Motion[END_REF] with relatively low computational costs.

Our project aims at combining these analysis and synthesis approaches. The present contribution focuses on timpani preparatory gestures, and especially on the evaluation of the relevance of gesture parameters that can be handled by our virtual character animation sytem [START_REF] Bouënard | Enhancing the Visualization of Percussion Gestures by Virtual Character Animation[END_REF].

Gesture Parameterization and Evaluation

We have built a timpani gesture database by using motion camera tracking for three timpani performers [START_REF] Bouënard | Enhancing the Visualization of Percussion Gestures by Virtual Character Animation[END_REF]. The database emphasizes timpani playing characteristics such as stick grips, gesture variations and beat locations. Each captured gesture has been segmented so that the database is made of beat-to-beat units that characterize performer's preparatory gestures.

A descriptive analysis has been conducted [START_REF] Bouënard | Analysis of Percussion Grip for Physically Based Character Animation[END_REF] for discriminating French (F ) and German (G) grips for two timpani players. This has led to the extraction of three parameters E 1 , E 2 and E 3 , the local extrema of the height trajectories of the tip of the stick.

The evaluation of the relevance of these parameters is measured using the kNN classification method (k Nearest Neighbors), with a training set randomly composed of 1/8th of 544 points and a threshold fixed at 10 neighbors. Figure 1 shows the clusters related to the extrema E 1 , E 2 and E 3 . The high recognition rates of these extrema (Table 1) indicate that such a parameterization is wellsuited for characterizing the grips of the two players under study. We are currently working on the evaluation of an extended parameterization that takes into account more characteristics of timpani playing, such as the gesture variations (legato, tenuto, accent, vertical accent and staccato). We plan to present this extended parameterization as part of the final contribution.

Fig. 1 .

 1 Fig. 1. Extrema clusters for French (denoted by • ) and German (denoted by * ) grips.

Table 1 .

 1 kNN recognition results of the parameters E 1 , E 2 and E 3 .

		E 1		E 2		E 3
	T est / T raining	F	G	F	G	F	G
	F	97.1% 2.9% 86.4% 13.6% 83.9% 16.1%
	G	3.4% 96.6% 5.5% 94.5% 24.3% 75.7%

Acknowledgments

The authors would like to thank the timpani performers (McGill University) for their help in this study. This work is partially funded by the Natural Sciences and Engineering Research Council of Canada (Discovery and Special Research Opportunity grants), the Pôle de Compétitivité Bretagne Images & Réseaux and the International Doctoral College of the Université Européenne de Bretagne.