Local exact controllability for the two- and three-dimensional compressible Navier–Stokes equations

ABSTRACT The goal of this article is to present a local exact controllability result for the two- and three-dimensional compressible Navier–Stokes equations on a constant target trajectory when the controls act on the whole boundary. Our study is then based on the observability of the adjoint system of some linearized version of the system, which is analyzed using a subsystem for which the coupling terms are somewhat weaker. In this step, we strongly use Carleman estimates in negative Sobolev spaces.

(1.1) Here ρ S is the density (assumed to be strictly positive), u S the velocity, and p is the pressure, which follows the standard polytropic law: for some γ ≥ 1 and κ > 0. (Actually, our proof will only require p to be C 3 locally around the target density.) The parameters µ and λ correspond to constant viscosity parameters and are assumed to satisfy µ > 0 and dλ + 2µ ≥ 0 (the only condition required for our result is µ > 0 and λ + 2µ > 0).
In this work, we intend to consider the local exact controllability around constant trajectories (ρ, u) ∈ R * + × R d \ {0}. Here, the controls do not appear explicitly in (1.1) as we are controlling the whole external boundary (0, T) × ∂ for the equation of the velocity and the incoming part u S · n < 0 of the boundary for the equation of the density, n being the unit outward normal on ∂ [26, Chapter 5].
Theorem 1.1 extends the results in [9] to the multidimensional case. As in the onedimensional case, our result proves local controllability to constant states having nonzero velocity. This restriction appears explicitly in the condition (1.3). As expected, this condition is remanent from the transport equation satis ed by the density which allows the information to travel at a velocity (close to) u.
This transport phenomenon and its consequences on the controllability of compressible Navier-Stokes equations have been also developed and explained in the articles [5,6,22,27] focusing on the linearized equations in the case of zero velocity. Using moving controls, [3,23] managed to show that controllability for a system of linear viscoelasticity can be reestablished if the control set travels in the whole domain (among some other geometric conditions, see [3] for further details). Let us also mention the work [4] where the one-dimensional compressible Navier-Stokes equations linearized around a constant state with nonzero velocity are studied thoroughly using a spectral approach and suitable Ingham-type inequalities.
To prove Theorem 1.1, we will deal with system (1.1) and (1.2) thinking to it as a coupling of parabolic and transport equations, and we shall therefore borrow some ideas from previous works studying controllability of systems coupling parabolic and hyperbolic e ects, in particular, the works [1] focusing on a system of linear thermoelasticity, [9] for the one-dimensional compressible Navier-Stokes equation around a constant state with nonzero velocity, [3] for a system of viscoelasticity with moving controls or [2] for nonhomogeneous incompressible Navier-Stokes equations. All these works are all based on suitable Carleman estimates designed simultaneously for the control of a parabolic equation following the ideas in [13] and for the control of the hyperbolic equation.
Our approach will follow this path and use Carleman estimates with weight functions traveling at velocity u similarly as in [2,3,9]. But we will also need to construct smooth trajectories to guarantee that the velocity eld belongs to L 2 (0, T; H 3 ( )). This space is natural as it is included in the space L 1 (0, T; Lip( )), ensuring the existence and continuity of the ow. Therefore, to obtain velocity elds in L 2 (0, T; H 3 ( )), we will use duality and develop observability estimates in negative Sobolev spaces in the spirit of the work [19].
We will not deal with the Cauchy problem for system (1.1) and (1.2), as our strategy directly constructs a solution of (1.1) and (1.2). We refer the interested reader to the pioneering works by Lions [21] and Feireisl et al. [11]. Nevertheless, we emphasize that our approach will use on the adjoint equations a new variable which is similar to the so-called viscous e ective ux introduced by Lions [21] to gain compactness properties.
Let us brie y mention other related works in the literature. In particular, we shall quote the works on the controllability of compressible Euler equations, namely, the ones obtained in [20] in the one-dimensional setting in the context of classical C 1 solutions and the ones developed by the second author in the context of weak entropy solutions obtained in [15] for isentropic one-dimensional Euler equations and [16] for nonisentropic one-dimensional Euler equations. We also refer to the work [25] for a global approximate controllability result for the three-dimensional Euler equations with controls spanned by a nite number of modes. When considering incompressible ows, the literature is large. We refer for instance to the works [12,17,18] for several results on the local exact controllability to trajectories for the (homogeneous) incompressible Navier-Stokes equations, and to the works [7,14] for global exact controllability results for incompressible perfect uids.
Outline. The article is organized as follows. Section 2 presents the general strategy of the proof of Theorem 1.1. Section 3 shows the controllability of a suitable system of one parabolic and one transport equation. Section 4 deduces from Section 3 a controllability result for the linearized Navier-Stokes equations. Section 5 then explains how to perform a xed point argument using the controllability results developed beforehand, thus proving Theorem 1.1. Section 6 provides some open problems.

Main steps of the proof
Since we are controlling the whole external boundary, can be embedded into some torus T L , where T L is identi ed with [0, L] d with periodic conditions. The length L is large enough (for instance L = diam( ) + 5|u|T) and we may consider the control problem in the cube [0, L] d completed with periodic boundary conditions with controls appearing as source terms supported in T L \ . Our control system then reads as follows: We also extend the initial data (ρ 0 , u 0 ) to T L such that With these notations, one needs to solve the following control problem: with initial dataρ and source termsf ρ (ρ,ǔ) = −ρdiv(ǔ), (2.5) satis esρ To take the support of the control functionsv ρ andv u into account, we introduce a smooth cuto function χ ∈ C ∞ (T L ; [0, 1]) satisfying and we will look forv ρ andv u approximatively under the form see (2.15) for the precise form ofv ρ ,v u . Now to solve the controllability problem (2.3)-(2.7), we will use a xed point argument. A di culty arising when building this argument is that the termǔ · ∇ρ in (2.3) (1) cannot be considered as source term, though it is quadratic, as there is no gain of regularity in transport equations. Hence we start by removing this term through a di eomorphism close to the identity. To be more precise, we de ne the ow Xǔ = Xǔ(t, τ , x) corresponding toǔ and de ned for To give a sense to (2.9), we requireǔ ∈ L 1 (0, T; W 1,1 (T L )) and div(ǔ) ∈ L 1 (0, T; W 1,∞ (T L )) [8]. But as we will work in Hilbert spaces with integer indexes, we will rather assume the stronger assumptionǔ ∈ L 2 (0, T; H 3 (T L )), in which case, the ow Xǔ is well de ned classically by Cauchy-Lipschitz's theorem. We then set, which are inverse one from another, i.e., Yǔ(t, Zǔ(t, x)) = Zǔ(t, Yǔ(t, x)) = x for all (t, x) ∈ [0, T] × T L . Forǔ suitably small, both transformations Yǔ(t, ·) and Zǔ(t, ·), t ∈ [0, T] are di eomorphism of T L which are close to the identity map on the torus. This change of variable is reminiscent of the Lagrangian coordinates and allow to straighten the characteristics. We thus set, for (t, x) ∈ [0, T] × T L , ρ(t, x) =ρ(t, Yǔ(t, x)), u(t, x) =ǔ(t, Yǔ(t, x)), (2.11) In the following, to avoid heavy notations, we will write x for the space variable before and a er the change of variable. However, to avoid confusions, we will maintain di erent notations for (ρ,ǔ), which are de ned in the original domain, and (ρ, u) for the functions de ned in the domain a er the change of variable. A er tedious computations developed in Appendix A, our problem can now be reduced to nd controlled solutions (ρ, u) of (2.12) for some ε > 0 small enough, with initial data given by and source terms f ρ (ρ, u) given by f ρ (ρ, u) = −ρDZ tǔ (t, Yǔ(t, x)) : Du − ρ(DZ tǔ (t, Yǔ(t, x)) − I) : Du, and f u (ρ, u) by where δ j,k is the Kronecker symbol (δ j,k = 1 if j = k, δ j,k = 0 if j = k) and satisfying the controllability requirement 14) The corresponding control functions in (2.3) will then be given for (t, Let us then remark that the map Yǔ can be computed starting from u. Indeed, we have the identity Yǔ(t, X 0 (t, T, x)) = Xǔ(t, T, x), so that by di erentiation with respect to the time variable, we get, for all (t, x) ∈ [0, T] × T L , ∂ t Yǔ(t, X 0 (t, T, x)) + u · ∇Yǔ(t, X 0 (t, T, x)) = u +ǔ(t, Xǔ(t, T, x)).
Next we shall introduce a map F : ( ρ, u) → (ρ, u) de ned on a convex subset of some weighted Sobolev spaces, corresponding to some Carleman estimate described later. This xed point map is constructed as follows. Given ( ρ, u) small in a suitable norm, we rst de ne Y = Y(t, x) as the solution of Then we de ne Z = Z(t, x) as follows: for all t ∈ [0, T], Z(t, ·) is the inverse of Y(t, ·) on T L . In other words, for all (t, x) ∈ [0, T] × T L , We will see that for suitably small u, Y(t, ·) is invertible for all t ∈ [0, T], see Proposition 5.1.
Corresponding to the initial data, we introduce 19) and, corresponding to the source terms, and We then look for (ρ, u) solving the controllability problem with source terms f ρ ( ρ, u), f u ( ρ, u) as in (2.20) and (2.21) and satisfying the controllability objective (2.14).
We are therefore reduced to study the controllability of the linear system Since this is a linear system, the controllability of (2.24) (stated in Theorem 4.1 within the framework we will use) is equivalent to the observability property stated in Theorem 4.2 for the adjoint equation (2.25) The main idea to get an observability inequality for (2.25) is to remark that, taking the divergence of the equation of z, the equations of σ and div(z) form a closed coupled system: where ν := λ + 2µ > 0.
Now we are led to introduce a new variable q as follows: (2.28) The advantage of considering system (2.28) rather than (2.26) directly is that now the coupling between the two equations is of lower order. In particular, the observability can now be obtained directly by considering independently the observability inequality for the equation of σ , which is of transport type, and for the equation of z, which is of parabolic type, considering in both cases the coupling term as a source term. Let us emphasize that the quantity q in (2.27) can be seen as a version of the so-called e ective viscous ux νdiv(u) − p(ρ), which has been used for the analysis of the Cauchy problem for compressible uids [11,21], but for the dual operator. Now, let us again remark that as system (2.28) is linear, 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 variables (σ , q). This leads to the following controllability problem, precisely addressed in Theorem 3.1: where in order to add a margin on the control zone, we introduce χ 0 is a smooth cuto function satisfying Now to solve the controllability problem (2.29), we use again another xed point argument and begin by considering the following decoupled controllability problem: Getting suitable estimates on the controllability problem (2.31) will allow us to solve the controllability problem (2.29) by a xed point argument. Note that the control problem for (2.31) simply consists of the control of two decoupled equations, the one in r of transport type, the other one in y of parabolic type. We are then reduced to these two classical problems. It turns out that our main di culty then will be to show the existence of smooth controlled trajectories for smooth source terms. Indeed, this is needed as we would like to consider velocity elds u ∈ L 1 (0, T; Lip(T L )). As Carleman estimates are the basic tools to establish the controllability of parabolic equations and to estimate the regularity of controlled trajectories and since they are based on the Hilbert structures of the underlying functional spaces, it is therefore natural to try getting velocity elds This regularity corresponds to the following regularity properties on the other functions: • f r , f y ,f r ,f y ∈ L 2 (0, T; H 2 (T L )), r ∈ L 2 (0, T; H 2 (T L )), y ∈ L 2 (0, T; H 4 (T L )).

Construction of the weight function
The controllability and observability properties of the systems described in Section 2.1 will be studied using Carleman estimates. These require the introduction of several notations, in particular, to de ne the weight function. We rst construct a functionψ =ψ(t, x) ∈ C 2 ([0, T] × T L ) satisfying the following properties.

First, it is assumed that
(2.32)

2.
We assume thatψ is constant along the characteristics of the target ow, i.e.,ψ solves (2.34)

3.
We nally assume the existence of a subset ω ⋐ {χ 0 = 1} such that The existence of a functionψ satisfying those assumptions is easily obtained for L large enough, e.g., L = diam( ) + 5|u|T: it su ces to choose taking values in [0, 1] and having its critical points in some ω 0 such that dist(ω 0 , T L \ Supp χ 0 ) ≥ 2|u|T and then to propagatẽ ψ with (2.34). This leaves room to de ne ω. Now onceψ is set, we de ne Next we pick T 0 > 0, T 1 > 0, and ε > 0 small enough, so that Now for any α ≥ 2, we introduce the weight function in time θ(t) de ned by (2.38) Then we consider the following weight function ϕ = ϕ(t, x): where s, λ 0 are positive parameters with s ≥ 1, λ 0 ≥ 1 and α is chosen as which is always larger than 2, thus being compatible with the condition θ ∈ C 2 ([0, T)). Actually, in the sequel, we will use that s can be chosen large enough, but for what concerns λ 0 , it can be xed from the beginning as equal to the constant λ 0 obtained in Theorem 3.2 below. Also note that, due to the de nition of ψ in (2.36), to the condition (2.32) and to λ 0 ≥ 1, We emphasize that the weight functions θ and ϕ depend on the parameters s and λ 0 but we will omit these dependences in the following for simplicity of notations.
Remark 2.1. Here, we have chosen to use the Carleman estimates stated in Theorem 3.7 and proved in [2] rather than the parabolic Carleman estimates derived in [13]. This choice allows us to avoid the use of results on the Cauchy theory on (1.1) as our controlled trajectory will be directly obtained through a xed point method.
To motivate this choice, we recall in particular that the usual parabolic Carleman estimates of [13] were used in [9] to derive similar results as Theorem 1.1 in the one-dimensional case. This led us to require that the initial conditions lie in a H 3 × H 3 neighborhood of the target trajectory (ρ, u) in order to be able to li the initial conditions, as we used the result of Matsumura and Nishida [24]. In fact, if instead of using the Carleman estimates in [13], we use the Carleman estimates in [2], one can derive the same result as in [9] for initial data in a H 1 × H 1 neighborhood of the initial data, see for instance [10] where this idea was used in the context of nonconstant target trajectories.
Notations. In the following, we will consider functional spaces depending on the time and space variables. This introduces heavy notations, that we will keep in the statements of the theorems, but that we shall simplify in the proof by omitting the time interval (0, T) and the spatial domain T L as soon as no confusion can occur. Thus, we will use the notations: and so on for the other functional spaces. Similarly, we will o en denote by · H 2 and · H 3 the norms · H 2 (T L ) , · H 3 (T L ) .

The controllability problem (2.29)
The goal of this section is to solve the controllability problem (2.29): . There exists C > 0 and s 0 ≥ 1 large enough such that for all s ≥ s 0 , if f r and f y satisfy the integrability conditions there exists a controlled trajectory (r, y) solving (2.29) and satisfying the following estimate: , and f r and f y satisfy we furthermore have the following estimate: ≤ C f r e s L 2 (0,T;H 2 (T L )) + f y e s L 2 (0,T;H 2 (T L )) + r 0 e s (0)

4)
for some constant C independent of s ≥ s 0 .
As explained in Section 2, Theorem 3.1 will be proved by a xed point theorem based on the understanding of the controllability problem (2.31), which amounts to understand two independent controllability problems, one for the heat equation satis ed by y, the other one for the transport equation satis ed by r.
The section is then organized as follows. First, we recall the controllability properties of the heat equation. Second, we explain how to exhibit a null-controlled trajectory for the transport equation. Third, we explain how these constructions can be combined in order to get Theorem 3.1.

Controllability of the heat equation
In this paragraph, we deal with the following controllability problem: given y 0 andf y , nd a control function v y such that the solution y of

Results
As it is done classically, the study of the controllability properties of (3.5) is based on the observability of the adjoint system, which is obtained with the following Carleman estimate: Theorem 3.2 (Theorem 2.5 in [2]). There exist constants C 0 > 0 and s 0 ≥ 1 and λ 0 ≥ 1 large enough such that for all smooth functions w on [0, T] × T L and for all s ≥ s 0 , where we have set Using Theorem 3.2 and the remark that for some constant C ≥ 1 independent of s, we obtain the following controllability result: and y 0 ∈ L 2 (T L ), there exists a solution (y, v y ) of the control problem (3.5) and (3.6) which furthermore satis es the following estimate: s 3/2 ye sϕ L 2 (0,T;L 2 (T L )) + θ −3/2 χ 0 v y e sϕ L 2 (0,T;L 2 (T L )) + s 1/2 θ −1 ∇ye sϕ L 2 (0,T;L 2 (T L )) ≤ C θ −3/2f y e sϕ L 2 (0,T;L 2 (T L )) + Cs 1/2 y 0 e sϕ(0) . (3.10) Besides, this solution (y, v y ) can be obtained through a linear operator in (y 0 ,f y ). If y 0 ∈ H 1 (T L ), we also have . (3.11) The proof of Theorem 3.3 is done in [2] for an initial data y 0 = 0. We shall therefore not provide extensive details for its proof, but only explain how it should be adapted to the case y 0 = 0, see the proof in Section 3.1.2.
The proof is done below in Section 3.1.2 and is mainly based on regularity results.

Proofs
Sketch of the proof of Theorem 3.3. For later purpose, let us brie y present how the proof of Theorem 3.3 works. It mainly consists in introducing the functional considered on the set Here the overline refers to the completion with respect to the Hilbert norm · obs de ned by (3.14) Thanks to the Carleman estimate (3.7), · obs is a norm. The assumptions y 0 ∈ L 2 and (3.9) imply that J is well de ned, convex and coercive on H obs . Therefore it has a unique minimizer W in H obs and the couple (y, v y ) given by solves the controllability problem (3.5) and (3.6). Using the coercivity of J immediately yields L 2 (L 2 ) estimates on y and v y and on W obs using J(W) ≤ J(0) = 0: The other estimates on y are derived by weighted energy estimates similar to the ones developed in [2, Theorem 2.6], the only di erence coming from the integrations by parts in time leading to new terms involving y 0 . Details of the proof are le to the reader.
We then simply use classical parabolic regularity estimates for y * , item 1 and (3.10) and (3.11).

Controllability of the transport equation
We study the following control problem: Givenf r and r 0 nd a control function v r such that the solution r of    ∂ t r + u · ∇r + p ′ (ρ)ρ ν r =f r + v r χ 0 , in (0, T) × T L , and r 0 ∈ L 2 (T L ), there exists a function v r ∈ L 2 (0, T; L 2 (T L )) such that the solution r of (3.17) satis es the control requirement (3.18). Besides, the controlled trajectory r and the control function v r satisfy . (3.20) If r 0 ∈ H 1 (T L ) andf r satis es (3.17) andf r e 6s /7 ∈ L 2 (0, T; H 1 (T L )), then r furthermore belongs to L 2 (0, T; H 1 (T L )) and satis es re 6s /7 L 2 (0,T;H 1 (T L )) + v r e 6s /7 .
Proof. The proof of Theorem 3.5 consists in an explicit construction solving the control problem (3.17) and (3.18) and then on suitable estimates on it. An explicit construction. Let η 0 be a smooth cuto function taking value 1 on {x ∈ T L , with d(x, ) ≤ 5ε + |u|T 0 } and vanishing on {x ∈ T L , d(x, ) ≥ 6ε + |u|T 0 }. We then introduce η the solution of and the solutions r f and r b (here the subscripts 'f ' and 'b' stand for forward and backward, respectively) of and where a denotes the constant a = p ′ (ρ)ρ ν .
We then set where η 1 (t) is a smooth cuto function taking value 1 on [0, T 0 /2] and vanishing for t ≥ T 0 and η 2 = η 2 (x) is a smooth cuto function taking value 1 for x with d(x, ) ≤ 3ε and vanishing for x with d(x, ) ≥ 4ε. One easily checks that r solves thus corresponding to a control function localized in the support of χ 0 due to the condition on the support of η 2 . Besides, r given by (3.29) Estimates on r. Let us start with estimates on r f . To get estimates on r f , we perform weighted energy estimates on (3.24) on the time interval (0, T − 2T 1 ). Multiplying (3.24) by θ −3 r f e 2sϕ , we obtain d d But, for all t ∈ (0, T − 2T 1 ) and x ∈ T L , We thus conclude .
The estimate on v r in (3.20) is also a simple consequence of its explicit value in (3.28) and the fact that η 0 ∈ W 1,∞ , Regularity results. To obtain regularity results on r and v r , it is then su cient to get regularity estimates on r f solution of (3.24) on the time interval (0, T −2T 1 ) and on r b solution of (3.25) on the time interval (T 0 , T). Of course, these estimates will be of the same nature, so we only focus on r f , the other case being completely similar.
To get weighted estimates in higher norms, we do higher order energy estimates on (3.24). For instance, ∇r f satis es the equation An energy estimate for ∇ 2 r f on (0, T − 2T 1 ) directly yields ∇ 2 r f e 6s /7 L ∞ (0,T−2T 1 ;L 2 ) ≤ C ∇ 2f r e 6s /7 thus concluding the proof of Theorem 3.5.

Proof of Theorem 3.1
Existence of a solution to the control problem. We construct the controlled trajectory using a xed point argument. We introduce the sets C r s = {r ∈ L 2 (0, T; L 2 (T L )) such that θ −3/2 re sϕ ∈ L 2 (0, T; L 2 (T L ))}, (3.34) C y s = {y ∈ L 2 (0, T; H 1 (T L )) such that ye sϕ , θ −1 ∇ye sϕ ∈ L 2 (0, T; L 2 (T L ))}. Forr ∈ C r s andỹ ∈ C y s , we introducẽ As f r and f y satisfy (3.1), for (r,ỹ) ∈ C r s × C y s ,f r satis es (3.19) andf y satis es (3.9). Therefore, one can de ne a map s on C r s ×C y s which to a data (r,ỹ) ∈ C r s ×C y s associates (r, y), where r is the solution of the controlled problem    ∂ t r + u · ∇r + p ′ (ρ)ρ ν r =f r + v r χ 0 , in (0, T) × T L , r(0, ·) = r 0 (·), r(T, ·) = 0, in T L , (3.35) given by Theorem 3.5, and y is the solution of the controlled problem y(0, ·) = y 0 (·), y(T, ·) = 0, in T L , (3.36) given by Theorem 3.3. Then we remark that Theorems 3.3 and 3.5 both yield a linear construction, respectively, for (y 0 ,f y ) → (y, v y ) and for (r 0 ,f r ) → (r, v r ). To apply Banach's xed point theorem, let us show that the map s is a contractive mapping for s large enough.
Let (r a ,ỹ a ) and (r b ,ỹ b ) be elements of C r s × C y s , and call their respective images (r a , y a ) = s (r a ,ỹ a ) and (r b , , while Theorem 3.3 implies .
In particular, we have .
Thus the quantity de nes a norm on C r s × C y s for which the map s satis es Consequently, if s is chosen large enough, the map s is a contractive mapping and by Banach's xed point theorem, s has a unique xed point (r, y) in C r s × C one gets with Theorems 3.3 and 3.5 that (r, y) solution of (2.28) satis es ≤ C θ −3/2 f r e sϕ L 2 (L 2 ) + s −1/2 θ −3/2 f y e sϕ L 2 (L 2 ) + C r 0 e sϕ(0) L 2 + y 0 e sϕ(0) L 2 , that is, the estimate (3.2).
The above regularity results come with estimates. Tracking them yields estimate (3.4).
Theorem 4.1 is deduced by duality from observability estimates for Eq. (2.25), which is the adjoint system of (2.24). Namely, we will show the following observability result for (2.25): where χ = χ(x) is the cuto function de ned in (2.8).
The proofs of the above theorem are provided in next sections. Section 4.2 gives the proof of Theorem 4.2. Section 4.3 then explains how to deduce Theorem 4.1 from Theorem 4.2.

Proof of Theorem 4.1
Proof of Theorem 4.1. Using the observability estimates (4.3) satis ed by solutions (σ , z) of Eq. (2.25), we can again argue by duality to deduce that System (2.24) is controllable and the estimate (4.1) follows immediately.
To conclude (4.2), we look at the equations satis ed by ρe 5s /6 and ue 5s /6 and perform regularity estimates on each equation. To estimate the regularity of ρe 5s /6 , as does not satisfy the transport equation anymore ( is independent of the space variable x), this induces a small loss in the parameter s, which is re ected by the fact that we estimate ρe 5s /6 instead of ρe 6s /7 . This is similar for the estimate on the velocity eld u.

Proof of Theorem 1.1
In this section, we x the parameter s = s 0 so that Theorem 4.1 applies. We introduce the set on which the xed point argument will take place: The precise de nition of our xed point map is then given as follows: where G is de ned in Theorem 4.1, ( ρ 0 , u 0 ) is de ned in (2.19) and f ρ ( ρ, u), f u ( ρ, u) are de ned in (2.20) and (2.21). Therefore, our rst goal is to check that F is well de ned on C R , and for that purpose, we shall in particular show that, for ( ρ, u) ∈ C R one has ρ 0 e s (0) ∈ H 2 (T L ), u 0 e 7s (0)/6 ∈ H 2 (T L ), f ρ ( ρ, u)e s ∈ L 2 (0, T; H 2 (T L )) and f u ( ρ, u)e 7s /6 ∈ L 2 (0, T; H 1 (T L )).

The map F in (5.1) is well de ned on C R
To show that F is well de ned, we rst study the maps Y and Z de ned in (2.17) and (2.18) and prove some of their properties, in particular that they are close to the identity map. We can then de ne the source term f ρ ( ρ, u), f u ( ρ, u) and the initial data ( ρ 0 , u 0 ). Accordingly, we will deduce that the map F in (5.1) is well de ned on C R for R > 0 small enough.

Estimates on Y and Z in (2.17) and (2.18)
We start with the following result: Then the map Y de ned in (2.17) satis es, for some constant C independent of R > 0: Therefore, there exists R 0 ∈ (0, 1) such that for all R ∈ (0, R 0 ) the map Z de ned in where χ is de ned by (2.8).
Proof. Let us consider the equation satis ed by the map Using (2.17), direct computations show that δ satis es Therefore, we immediately get that As for all t ∈ [0, T], we deduce from the above formula and (5.2) that δe 5s /12 C 0 (H 3 ) ≤ CR. Using Eq. (5.8) and the bound (5.2), we also derive ∂ t δe 5s /12 C 0 (H 2 ) ≤ CR, from which, together with |s∂ t | ≤ Ce s /12 independently of s, we immediately deduce (5.3).
In particular, Consequently, Z de ned by (2.18) is well de ned by the inverse function theorem (note that C 0 (H 2 ) ⊂ C 0 (C 0 ) in dimension d ≤ 3), and Z ∈ C 0 (C 1 ) with To get estimates on Z in weighted norms, we start from the formula Z(t, Y(t, x)) = x and di erentiate it with respect to x: We obtain, for all t ∈ [0, T] and x ∈ T L , i.e., D Z(t, Y(t, x))(I + D δ(t, x)) = I.

Estimates on
De ne ( ρ 0 , u 0 ) as in (2.19). Then there exists a constant C > 0 independent of R such that Proof. It is a straightforward consequence of the estimate (5.3) derived in Proposition 5.1.

Conclusion
Putting together the estimates obtained in Proposition 5.1, Lemmas 5.2 and 5.3 and using Theorem 4.1, we get the following result: Proposition 5.4. Let (ρ 0 ,ǔ 0 ) in H 2 (T L )×H 2 (T L ) satisfying (5.12) for some δ > 0, ( ρ, u) ∈ C R for some R ∈ (0, R 0 ) with R 0 given by Proposition 5.1. Then the map F in (5.1) is well de ned, and there exist a constant C such that (ρ, u) = F ( ρ, u) satis es Besides, the condition (5.7) is satis ed for χ de ned in (2.8).
Proposition 5.4 is the core of the xed point argument developed below.
To each u n , we associate the corresponding ow Y n de ned by 16) and the respective inverse Z n as in (2.18) (which is well-de ned according to Proposition 5.1). The sequence Y n is bounded in L ∞ (H 3 ) ∩ W 1,∞ (H 2 ) according to (5.3) and then converge weakly- * in L ∞ (H 3 ) ∩ W 1,∞ (H 2 ). Passing to the limit in Eq. (5.16), we easily get that the weak limit of the sequence Y n is Y de ned by (2.17). Besides, by Aubin-Lions' lemma and the weak convergence of Y n toward Y, we also have the strong convergence of Y n to Y in W 1/4,5 (0, T; H 11/4 (T L )) and therefore in C 0 ([0, T]; C 1 (T L )). Consequently, the sequence Z n strongly converges to Z in C 0 ([0, T]; C 1 (T L )). These strong convergences allow to show that where ( ρ 0,n , u 0,n ) = (ρ 0 ( Y n (0, x)),ǔ 0 ( Y n (0, x))). From the uniform bounds (5.4) and (5.5) on the quantity D Z n (t, Y n (t, x)) − I and Aubin-Lions' Lemma, we also deduce that Using then the uniform bound (5.6), the identity D 2 Z n (t, Y n (t, x)) = D(D Z n (t, Y n (t, x)))D Z n (t, Y n (t, x)), and the convergence (5.18), we also conclude that x)) weakly in L 2 (0, T; H 1 (T L )).

The case of nonconstant trajectories
Our result only considers the local exact controllability around a constant state. The next question concerns the case of local exact controllability around nonconstant target trajectories, similarly as what has been done in the context of nonhomogeneous incompressible Navier-Stokes equations in [2] and in the context of compressible Navier-Stokes equation in one space dimension in the recent preprint [10]. We expect such results to be true provided the target trajectory is su ciently smooth and assuming some geometric condition on the ow of the target velocity eld u. Namely, if we denote by X the ow corresponding to u, i.e., given by d dt X(t, τ , x) = u(t, X(t, τ , x)), X(τ , τ , x) = x, (6.1) it is natural to expect a geometric condition of the form ∀x ∈ , ∃t ∈ (0, T), s.t. X(t, 0, x) / ∈ , (6.2) corresponding to the time condition (1.3) in the case of a constant velocity eld. But considering the case of a nonconstant velocity eld would introduce many new terms in the proof and make it considerably more intricate, including for instance the di culty to control the density when recirculation appears close to the boundary of . This issue needs to be carefully analyzed and discussed.

Controllability from a subset of the boundary
From a practical point of view, it seems more reasonable to control the velocity and the density from some part of the boundary in which the target velocity eld u enters in the domain. But in this case, one needs to make precise what are the boundary conditions on the velocity eld u S . One could think for instance to Dirichlet boundary conditions of the form u S = u on the out ow boundary Ŵ out . But this would mean that one should nd a solution (ρ, u) of the control problem (2.12)-(2.14) with boundary conditions u = 0 on the out ow boundary Ŵ out . This would introduce a lot of additional technicalities as the dual variable z would also satisfy Dirichlet homogeneous boundary conditions on the out ow boundary Ŵ out . The variable q in (2.27) would therefore have nonhomogeneous Dirichlet boundary conditions on (0, T)×Ŵ out and careful estimates should be done to recover estimates on z, for instance based on the delicate Carleman estimates proved in [18].