Local exact controllability for the 2 and 3-d compressible Navier-Stokes equations

The goal of this article is to present a local exact controllability result for the 2 and 3-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 thanks to a subsystem for which the coupling terms are somewhat weaker. In this step, we strongly use Carleman estimates in negative Sobolev spaces.


Introduction
We consider the isentropic compressible Navier-Stokes equation in dimension two or three in space, in a smooth bounded domain Ω ⊂ R d , d = 2 or d = 3: Here ρ S is the density, 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 ) × ∂Ω 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 ∂Ω, see e.g.[24,Chapter 5].
Given e a direction in the unit sphere S d−1 of R d , we define the thickness of some nonempty open set A ⊂ R d in the direction e as the following nonnegative number: Our main result is the following.Theorem 1.1.Let d ∈ {2, 3}, ρ > 0 and u ∈ R d \ {0}.Let L 0 > 0 be larger than the thickness of Ω in the direction u/|u|, and assume T > L 0 /|u|. (
This transport phenomenon and its consequences on the controllability of compressible Navier-Stokes equations have been also developed and explained in the articles [25,5,21] focusing on the linearized equations in the case of zero-velocity.Using then moving controls, [22,3] 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 compressible Navier-Stokes equations in 1d linearized around a constant state with non-zero velocity are studied thoroughly using a spectral approach and suitable Ingham-type inequalities.
In order to prove Theorem 1.1, we will deal with system (1.1)-(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 effects, in particular the works [1] focusing on a system of linear thermoelasticity, [8] for the 1d compressible Navier-Stokes equation around a constant state with non-zero velocity, [3] for a system of viscoelasticity with moving controls, or [2] for non-homogeneous 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 [12] 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 [8,3,2].But we will also need to construct smooth trajectories in order to guarantee that the velocity field belongs to L 2 (0, T ; H 3 (Ω)).This space is natural as it is included in the space L 1 (0, T ; Lip(Ω)) ensuring the existence and continuity of the flow.Therefore, in order to obtain velocity fields in L 2 (0, T ; H 3 (Ω)), we will use duality and develop observability estimates in negative Sobolev spaces in the spirit of the work [18].
We will not deal with the Cauchy problem for system (1.1)-(1.2),as our strategy directly constructs a solution of (1.1)-(1.2).We refer the interested reader to the pioneering works by P.-L.Lions [20] and E. Feireisl et al. in [10].Nevertheless, we emphasize that our approach will use on the adjoint equations a new variable which is similar to the so-called viscous effective flux introduced by P.-L.Lions in [20] in order to gain compactness properties.
Let us briefly 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 [19] in the 1-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 [14] for isentropic 1-d Euler equations and [15] for non-isentropic 1d Euler equations.We also refer to the work [23] for a global approximate controllability result for the 3-d Euler equations with controls spanned by a finite number of modes.When considering incompressible flows, the literature is large.We refer for instance to the works [16,11,17] for several results on the local exact controllability to trajectories for the (homogeneous) incompressible Navier-Stokes equations, and to the works [6,13] for global exact controllability results for incompressible perfect fluids.
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 it a controllability result for the linearized Navier-Stokes equations.Section 5 then explains how to perform a fixed 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, Ω can be embedded into some torus T L , where T L is identified with [0, L] d with periodic conditions.The length L is large enough (for instance L = diam(Ω) + 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 \ Ω.Our control system then reads as follows: where vρ and vu are control functions supported in [0, T ] × (T L \ Ω).Then we set 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: Given (ρ 0 , ǔ0 ) small in H 2 (T L ) × H 2 (T L ), find control functions vρ and vu supported in [0, T ] × (T L \ Ω) such that the solution of with initial data ρ(0, x) = ρ0 (x), ǔ(0, and source terms To take the support of the control functions vρ and vu into account, we introduce a smooth cut-off and we will look for vρ and vu in the form vρ = v ρ χ and vu = v u χ. Now in order to solve the controllability problem (2.3)-(2.7),we will use a fixed point argument.A difficulty arising when building this argument is that the term ǔ • ∇ρ in (2.3) (1) is very singular.Hence we start by removing this term via a diffeomorphism close to the identity.To be more precise, we define the flow X ǔ = X ǔ(t, τ, x) corresponding to ǔ and defined for (t, τ, x) (2.9) In order to give a sense to (2.9), we require ǔ ∈ L 1 (0, T ; W 1,1 (T L )) and div(ǔ) ∈ L 1 (0, T ; W 1,∞ (T L )) (see [7]).But as we will work in Hilbert spaces with integer indexes, we will rather assume the stronger assumption ǔ ∈ L 2 (0, T ; H 3 (T L )), in which case, the flow X ǔ is well-defined classically by Cauchy-Lipschitz's theorem.We then set, for which are inverse one from another, i.e.Y ǔ(t, Z ǔ(t, x)) = Z ǔ(t, Y ǔ(t, x)) = x for all (t, x) ∈ [0, T ]×T L .For ǔ suitably small, both transformations Y ǔ(t, •) and Z ǔ(t, •), t ∈ [0, T ], are diffeomorphism 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 After tedious computations developed in Appendix A, our problem can now be reduced to find controlled solutions (ρ, u) of for some ε > 0 small enough, with initial data given by and source terms f ρ (ρ, u) given by 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 ρ(T ) = 0, u(T ) = 0 in T L . (2.14) The corresponding control functions in (2.12) will then be given for Let us then remark that the map Y ǔ can be computed starting from u. Indeed, we have that Y ǔ(t, X 0 (t, T, x)) = X ǔ(t, T, x) so that by differentiation with respect to the time variable, we get, for all (t, x) In particular, using this equation at the point X 0 (T, t, x), we obtain Next we shall introduce a map F : ( ρ, u) → (ρ, u) defined on a convex subset of some weighted Sobolev spaces, corresponding to some Carleman estimate described later.This fixed point map is constructed as follows.Given ( ρ, u) small in a suitable norm, we first define Y = Y (t, x) as the solution of Then we define 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) (2.18) 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 and, corresponding to the source terms, and We then look for (ρ, u) solving the controllability problem with initial data with source terms f ρ ( ρ, u), f u ( ρ, u) as in (2.20)-(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) is equivalent to the observability property 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: q := ν div(z) + ρσ.
(2.27) System (2.26) can then be rewritten with the unknown (σ, q) as (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 effective viscous flux ν div(u) − p(ρ), which has been used for the analysis of the Cauchy problem for compressible fluids [10,20], 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 controllability problem: where in order to add a margin on the control zone we introduce χ 0 is a smooth cut-off function satisfying Supp χ 0 {χ = 1} and χ 0 (x) = 1 for all x ∈ T L such that d(x, Ω) ≥ 3ε. (2.30) Now in order to solve the controllability problem (2.29), we use again another fixed point argument, and begin by considering the following decoupled controllability problem: (2.31) Getting suitable estimates on the controllability problem (2.31) will allow us to solve the controllability problem (2.29) by a fixed 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 difficulty then will be to show the existence of smooth controlled trajectory for smooth source terms.Indeed, this is needed as we would like to consider velocity fields 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 fields u ∈ L 2 (0, T ; This regularity corresponds to the following regularity properties on the other functions:

Construction of the weight function
The controllability and observability properties of the systems described in Section 2.1 will be studied by using Carleman estimates.These require the introduction of several notations, in particular to define the weight function.We first construct a function ψ = ψ(t, x) ∈ C 2 ([0, T ] × T L ) satisfying the following properties.
1. First, it is assumed that We assume that ψ is constant along the characteristics of the target flow, i.e. ψ solves or equivalently (2.34) 3. We finally assume the existence of a subset ω {χ 0 = 1} such that The existence of a function ψ satisfying those assumptions is easily obtained for L large enough, e.g.L = diam(Ω) + 5|u|T : it suffices to choose Ψ 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 propagate ψ with (2.34).This leaves room to define ω.Now once ψ is set, we define ψ(t, x) := ψ(t, x) + 6. (2.36) Next we pick T 0 > 0, T 1 > 0 and ε > 0 small enough so that (2.37) Now for any α ≥ 2, we introduce the weight function in time θ(t) defined 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 fixed from the beginning as equal to the constant λ 0 obtained in Theorem 3.2 below.Also note that, due to the definition of ψ in (2.36), to the condition (2.32) and to λ 0 ≥ 1, we have for all where Φ(t) := θ(t)λ 0 e 12λ0 . (2.42) 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.
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 often denote by 3 The controllability problem (2.29) The goal of this section is to solve the controllability problem (2.29): There exist 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: + y 0 e sΦ(0) for some constant C independent of s ≥ s 0 .
As explained in Section 2, Theorem 3.1 will be proved by a fixed 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 satisfied by y, the other one for the transport equation satisfied by r.
The section is then organized as follows.Firstly, we recall the controllability properties of the heat equation.Secondly, we explain how to exhibit a null-controlled trajectory for the transport equation.Thirdly, 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 and fy , find a control function v y such that the solution y of satisfies y(T, •) = 0, in T L . (3.6)

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:

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 ξ(t, x) = θ(t) exp(λ 0 ψ(t, x)). (3.8) Using Theorem 3.2 and the remark that for some constant C ≥ 1 independent of s, we obtain the following controllability result: Theorem 3.3 (Inspired by Theorem 2.6 in [2]).There exist positive constants C > 0 and s 0 ≥ 1 such that for all s ≥ s 0 , for all fy satisfying and y 0 ∈ L 2 (T L ), there exists a solution (y, v y ) of the control problem (3.5)-(3.6)which furthermore satisfies the following estimate: + Cs 1/2 y 0 e sϕ(0) L 2 (T L ) . (3.10) Besides, this solution (y, v y ) can be obtained through a linear operator in (y 0 , fy ).
If y 0 ∈ H 1 (T L ), we also have + Cs 1/2 y 0 e sϕ(0) 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 3.1.2and is mainly based on regularity results.

Proofs
Sketch of the proof of Theorem 3.3.For later purpose, let us briefly 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 defined by 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-defined, 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)- (3.6).Using the coercivity of J immediately yields L 2 (L 2 ) estimates on y and v y and on W obs by 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 difference coming from the integrations by parts in time leading to new terms involving y 0 .Details of the proof are left to the reader.
Proof.The proof of Theorem 3.5 consists in an explicit construction solving the control problem (3.17)-(3.18)and then on suitable estimates on it.
An explicit construction.Let η 0 be a smooth cut-off function taking value 1 on {x ∈ T L , with d(x, Ω) ≤ 5ε + |u|T 0 } and vanishing on {x ∈ T L , d(x, Ω) ≥ 6ε + |u|T 0 }.We then introduce η the solution of and the solutions r f and r b (here 'f ' stands for forward, 'b' for backward) of and where a denotes the constant We then set where η 1 (t) is a smooth cut-off function taking value 1 on [0, T 0 /2] and vanishing for t ≥ T 0 and η 2 = η 2 (x) is a smooth cut-off function taking value 1 for x with d(x, Ω) ≤ 3ε and vanishing for x with d(x, Ω) ≥ 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 (3.26) satisfies r(0, due to the conditions on the support of η 0 , η 1 , η 2 and the condition (2.37) on the flow corresponding to u.
Actually, thanks to the choice of ε > 0, T 0 > 0 and T 1 > 0 in (2.37) we have 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 But, for all t ∈ (0, T − 2T 1 ) and x ∈ T L , We thus conclude Similarly, one can show that r b satisfies .
To conclude that , we use the explicit definition of r in (3.26) and identity (3.29), and notice that η 0 , η 1 and η 2 belong to L ∞ , and (η, u) to L ∞ (L ∞ ).
The estimate on v r in (3.20) is also a simple consequence of its explicit value in (3.28) and the fact that Regularity results.To obtain regularity results on r and v r , it is then sufficient 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.

Proof of Theorem 3.1
Existence of a solution to the control problem.We construct the controlled trajectory using a fixed point argument.
We introduce the sets (3.34) For r ∈ C r s and ỹ ∈ C y s , we introduce As f r and f y satisfy (3.1), for (r, ỹ) ∈ C r s × C y s , fr satisfies (3.19) and fy satisfies (3.9).Therefore, one can define a map Λ 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 given by Theorem 3.5, and y is the solution of the controlled problem given by Theorem 3.3.
Then we remark that Theorems 3.3 and 3.5 both yield a linear construction, respectively for (y 0 , fy ) → (y, v y ) and for (r 0 , fr ) → (r, v r ).In order to apply Banach's fixed point theorem, let us show that the map Λ s is a contractive mapping for s large enough.
Let (r a , ỹa ) and (r b , ỹb ) be elements of C r s × C y s , and call their respective images (r a , y a ) = Λ s (r a , ỹa ), and (r b , y and Fy = fy (r a , ỹa ) − fy (r b , ỹb ), by Theorem 3.5 we have , while Theorem 3.3 implies Thus the quantity defines a norm on C r s × C y s for which the map Λ s satisfies Consequently, if s is chosen large enough, the map Λ s is a contractive mapping and by Banach's fixed point theorem, Λ s has a unique fixed point (r, y) in C r s × C y s .By construction, this fixed point (r, y) solves the controllability problem (2.29).Besides, estimating fr (y) and fy (r) by 4 Controllability estimates for (2.24) In this section, we study the controllability of (2.24).This is given in the following statement.
Proof.We first recover observability estimates for (2.28).We write down But by construction, if we associate to (r 0 , y 0 ) ∈ H 2 (T L ) × H 3 (T L ) and f r , f y satisfying f r e sΦ , f y e sΦ ∈ L 2 (0, T ; H 2 (T L )) the controlled trajectory of (2.29) in Theorem 3.1, then we get: Using (3.4), we obtain: .
Then we use the equation of z in (2.25) to recover estimates on z, that we rewrite as follows: Then we use the duality with the following controllability problem for the heat equation: Replacing s by 7s/6 in Proposition 3.4 items 1 & 2, we get that if y 0 ∈ H 2 (T L ) and fy e 7sΦ/6 ∈ L 2 (0, T ; H 1 (T L )), then there exists a controlled trajectory y satisfying (4.4) with control v y with + C y 0 e 7sΦ(0)/6 Arguing by duality, we thus obtain + z(0)e −7sΦ(0)/6 .
Using that (σ, z) satisfies Equation (2.25), we 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 satisfied by ρe 5sΦ/6 and ue 5sΦ/6 and perform regularity estimates on each equation.To estimate the regularity of ρe 5sΦ/6 , as Φ does not satisfy the transport equation anymore (Φ is independent of the space variable x), this induces a small loss in the parameter s, which is reflected by the fact that we estimate ρe 5sΦ/6 instead of ρe 6sΦ/7 .This is similar for the estimate on the velocity field u.
5 Proof of Theorem 1.1 In this section, we fix the parameter s = s 0 so that Theorem 4.1 applies.We introduce the set on which the fixed point argument will take place: The precise definition of our fixed point map is then given as follows: where G is defined in Theorem 4.1, ( ρ 0 , u 0 ) is defined in (2.19) and f ρ ( ρ, u), f u ( ρ, u) are defined in (2.20)-(2.21).Therefore, our first goal is to check that F is well-defined on C R , and for that purpose, we shall in particular show that, for ( ρ, u) ∈ C R one has ρ 0 e sΦ(0) ∈ H 2 (T L ), u 0 e 7sΦ(0)/6 ∈ H 2 (T L ), f ρ ( ρ, u)e sΦ ∈ L 2 (0, T ; H 2 (T L )) and f u ( ρ, u)e 7sΦ/6 ∈ L 2 (0, T ; H 1 (T L )).

5.1
The map F in (5.1) is well-defined on C R In order to show that F is well-defined, we first study the maps Y and Z defined in (2.17)-(2.18)and prove some of their properties, in particular that they are close to the identity map.We can then define 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-defined on C R for R > 0 small enough.
Proof.Let us consider the equation satisfied by the map Using (2.17), direct computations show that δ satisfies Therefore, we immediately get that As for all t ∈ [0, T ], we deduce from the above formula and (5.2) that δe 5sΦ/12 C 0 (H 3 ) ≤ CR.
In particular, D δ ≤ CR, so that for R small enough, for all (t, x) ∈ [0, T ]×T L , D Y (t, x) = I +D δ(t, x) is invertible.Consequently, Z defined by (2.18) is well-defined by the inverse function theorem (note that ≤ CR. In order to get estimates on Z in weighted norms, we start from the formula Z(t, Y (t, x)) = x and differentiate 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.
Proof.In the following, we will repeatedly use the estimates derived in Proposition 5.1 and the crucial remark that 5/6 + 5/12 > 7/6.
• Concerning f ρ ( ρ, u).We perform the following estimates: as and • Concerning f u ( ρ, u).Using that the product is continuous from Using again that H 2 (T L ) is an algebra, In order to estimate the terms coming from the pressure, we write p(ρ where h is a C 1 function depending on the pressure law (here we use that the pressure law p belongs to C 3 locally around ρ), so we have Combining all the above estimates yields Lemma 5.2.

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: 5.11) 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-defined, and there exist a constant C such that (ρ, u) = F ( ρ, u) satisfies Besides, the condition (5.7) is satisfied for χ defined in (2.8).
Proposition 5.4 is the core of the fixed point argument developed below.
We then focus on the continuity of the map F on C R endowed with the (L 2 (0, T ; L 2 (T L ))) 2 topology.Let us consider a sequence ( ρ n , u n ) in C R converging strongly in (L 2 (0, T ; L 2 (T L ))) 2 to some element ( ρ, u).The set C R is closed under the topology of (L 2 (0, T ; L 2 (T L ))) 2 and therefore ( ρ, u) ∈ C R and we have the weak- * convergence of ( Using the bounds defining C R , by Aubin-Lions' lemma we also have the following convergences to ( ρ, u): (5.14) To each u n , we associate the corresponding flow Y n defined by and the respective inverse Z n as in (2.18) (which is well-defined thanks 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 Equation (5.15), we easily get that the weak limit of the sequence Y n is

Further comments
The case of non-constant trajectories.Our result only considers the local exact controllability around a constant state.The next question concerns the case of local exact controllability around non-constant target trajectories, similarly as what has been done in the context of non-homogeneous incompressible Navier-Stokes equations in [2] and in the context of compressible Navier-Stokes equation in one space dimension in the recent preprint [9].We expect such results to be true provided the target trajectory is sufficiently smooth and assuming some geometric condition on the flow of the target velocity field u.Namely, if we denote by X the flow corresponding to u, i.e. given by d dt X(t, τ, x) = u(t, X(t, τ, x)), X(τ, τ, x) = x, ( it is natural to expect a geometric condition of the form ∀x ∈ Ω, ∃t ∈ (0, T ), s.t.X(t, 0, x) / ∈ Ω, (6.2) corresponding to the time condition (1.3) in the case of a constant velocity field.But considering the case of a non-constant velocity field would introduce many new terms in the proof and make it considerably more intricate, including for instance the difficulty to control the density when recirculation appears close to the boundary of Ω.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 field u enters in the domain.But in this case, one needs to make precise what are the boundary conditions on the velocity field u S .
One could think for instance to Dirichlet boundary conditions of the form u S = u on the outflow boundary Γ out .But this would mean that one should find a solution (ρ, u) of the control problem (2.12)-(2.23)-(2.14)with boundary conditions u = 0 on the outflow boundary Γ out .This would introduce a lot of additional technicalities as the dual variable z would also satisfy Dirichlet homogeneous boundary conditions on the outflow boundary Γ out .The variable q in (2.27) would therefore have non-homogeneous Dirichlet boundary conditions on (0, T ) × Γ out and careful estimates should be done to recover estimates on z, for instance based on the delicate Carleman estimates proved in [17].