Some controllability results for the N-Dimensional Navier-Stokes and Boussinesq systems with N-1 scalar controls

SIAM J. CONTROL OPTIM.c 2006 Society for Industrial and Applied Mathematics Vol. 45, No. 1, pp. 146–173 SOME CONTROLLABILITY RESULTS FOR THE N-DIMENSIONAL NAVIER–STOKES AND BOUSSINESQ SYSTEMS WITH N−1 SCALAR CONTROLS∗ ENRIQUE FERN´ ANDEZ-CARA†, SERGIO GUERRERO†, OLEG YU. IMANUVILOV‡,AND JEAN-PIERRE PUEL§ Abstract. In this paper we deal with some controllability problems for systems of the Navier– Stokes and Boussinesq kind with distributed controls supported in small sets. Our main aim is to control N-dimensional systems (N+ 1 scalar unknowns in the case of the Navier–Stokes equations) with N−1 scalar control functions. In a first step, we present some global Carleman estimates for suitable adjoint problems of linearized Navier–Stokes and Boussinesq systems. In this way, we obtain null controllability properties for these systems. Then, we deduce results concerning the local exact controllability to the trajectories. We also present (global) null controllability results for some (truncated) approximations of the Navier–Stokes equations. Key words. Navier–Stokes system, exact controllability, Carleman inequalities AMS subject classifications. 34B15, 35Q30, 93B05, 93C10 DOI. 10.1137/04061965X 1. Introduction and examples. Let Ω ⊂RN(N= 2 or 3) be a bounded connected open set whose boundary ∂Ω is regular enough (for instance of class C2). Let O⊂Ω be a (small) nonempty open subset and let T>0. We will use the notation Q=Ω×(0,T) and Σ = ∂Ω×(0,T) and we will denote by n(x) the outward unit normal to Ω at the point x∈∂Ω. On the other hand, we will denote by C,C1,C2, . . . various positive constants (usually depending on Ω and O). We will be concerned with the following controlled Navier–Stokes and Boussinesq systems: (1) ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=v1O,∇·y=0 inQ, y= 0 on Σ, y(0) = y0in Ω and (2) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=v1O+θe N,∇·y=0 inQ, θt−Δθ+y·∇θ=h1Oin Q, y=0,θ= 0 on Σ, y(0) = y0,θ(0) = θ0in Ω (in both dimensions N= 2 and N= 3). ∗Received by the editors November 25, 2004; accepted for publication (in revised form) October 20, 2005; published electronically February 21, 2006. http://www.siam.org/journals/sicon/45-1/61965.html †Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain ([email protected], [email protected]). Partially supported by grant BFM2003–06446 of the D.G.E.S. (Spain). ‡Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064 (vik[email protected]). This work is supported by NSF grant DMS 0205148. §Laboratoire de Math´ematiques Appliqu´ees, Universit´e de Versailles - St. Quentin, 45 Avenue des Etats Unis, 78035 Versailles, France ([email protected]). 146 Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 147 For N= 2, we will also consider the following approximation of the Navier–Stokes system with boundary conditions of the Navier kind: (3) ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)TM(y)+∇p=v1O,∇·y=0 inQ, y·n=0,∇×y= 0 on Σ, y(0) = y0in Ω, where M>0, TM(y)=(TM(y1),T M(y2)) and TMis given by TM(s)=⎧ ⎨ ⎩ −Mif s≤−M, sif −M≤s≤M, Mif s≥M. In systems (1), (2) and (3), v=v(x, t) and h=h(x, t) stand for the control functions. They act during the whole time interval (0,T) over the set O. The symbol 1Ostands for the characteristic function of Oand eNis the Nth vector of the canonical basis of RN. The controllability of Navier–Stokes systems has been the objective of considerable work over the last years. Up to our knowledge, the strongest results have been given in [7], where a strategy based on the methods in [13] and [14] has been followed. Recently, the techniques in [7] have been adapted in [12] to cover Boussinesq systems (see also [3], [4], [8] and [10] for other results). This paper can be viewed as a continuation of [7]. We will present some new results which show that the N-dimensional systems (1) and (2) can be controlled, at least under some geometrical assumptions, with only N−1 scalar controls in L2(O×(0,T)). In particular, the Boussinesq system (2) in dimension N= 2 can be controlled by an action performed only on the temperature equation. We will also prove that the two-dimensional system (3) can be controlled with controls of the form v1Owhere vis the curl of a function in L2(0,T;H1(O)). In this paper, we will have to impose some regularity assumptions on the initial data. To this purpose, we introduce the spaces H,Eand V, with (4) H={w∈L2(Ω)N:∇·w= 0 in Ω,w·n=0on∂Ω}, E=Hif N=2, L4(Ω)3∩Hif N=3 and V={w∈H1 0(Ω)N:∇·w= 0 in Ω}. For system (1), we will assume that the control region Ois adjacent to the boundary ∂Ω (see assumption (11) below) and we will deal with the local exact controllability to the trajectories. More precisely, our task will be to prove that, for any bounded and sufficiently regular solution (y,p) of the uncontrolled Navier–Stokes equations (5) ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=0,∇·y=0 inQ, y= 0 on Σ, y(0) = y0in Ω, Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 148 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL there exists δ>0 such that, whenever y0∈Eand y0−y0E≤δ, we can find L2controls vwith vk≡0 for at least one kand associated states (y,p) satisfying (6) y(T)=y(T)inΩ. Notice that, under these circumstances, after time t=Twe can switch off the control and let the system follow the “ideal” trajectory (y,p). For the Boussinesq system (2), we will assume that Ois adjacent to ∂Ω near a point x0such that nk(x0)= 0 for some k<N. We will also be concerned with the local exact controllability to the trajectories. Now, a trajectory is a bounded and sufficiently regular solution (y,p, θ)of (7) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=θe N,∇·y=0 inQ, θt−Δθ+y·∇θ=0 inQ, y=0, θ = 0 on Σ, y(0) = y0, θ(0) = θ0in Ω. The goal will be to prove that there exists δ>0 such that, whenever (y0,θ0)∈ E×L2(Ω) and (y0,θ0)−(y0, θ0)E×L2≤δ, we can find L2controls vand hwith vk≡vN≡0 and associated states (y, p, θ) satisfying (8) y(T)=y(T) and θ(T)=θ(T)inΩ. In this context, the results established in [12] will be fundamental. Notice that, in particular, when N= 2, we try to control the whole system (2) with just one scalar control h. As far as (3) is concerned, our goal will be to prove the (global) null controllability. That is to say, for each y0∈H, we will try to find controls of the form v1O, where v belongs to the Hilbert space (9) W={∇ × z=(∂2z,−∂1z):z∈L2(0,T;H1(O))}, such that the associated solutions (y,p) satisfy (10) y(T)=0 inΩ. Approximate controllability results have been established for analogous systems in [4]. Observe that in this system the boundary conditions are of the Navier kind as in [3] (for their physical meaning, see, for instance, [11]). This and the fact that N=2 will be essential in the arguments presented below. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 149 Similarly to the previous situation, an extension by zero of the control after time t=Twill keep (y,p) at rest. As mentioned above, some hypotheses will be imposed on the control domain and the trajectories. More precisely, we will frequently assume that (11) ∃x0∈∂Ω,∃ε>0 such that O∩∂Ω⊃B(x0;ε)∩∂Ω (B(x0;ε) is the ball centered at x0of radius ε), (12) y∈L∞(Q)N, yt∈L2(0,T;Lσ(Ω)N)σ>1ifN=2 σ>6/5ifN=3  and (13) θ∈L∞(Q), θt∈L2(0,T;Lσ(Ω)) σ>1ifN=2 σ>6/5ifN=3 . Let us now present our main results in a precise form. The first one concerns the local exact controllability to the trajectories of system (1). Theorem 1. Assume that Osatisfies (11). Then, for any T>0,(1) is locally exactly controllable at time Tto the trajectories (y,p)satisfying (12) with controls v∈L2(O×(0,T))Nhaving one component identically zero. The second main result concerns the controllability of (2). Theorem 2. Assume that Osatisfies (11) with nk(x0)=0for some k<N. Then, for any T>0,(2) is locally exactly controllable at time Tto the trajectories (y,p, θ)satisfying (12)–(13) with L2controls vand hsuch that vk≡vN≡0.In particular, if N=2, we have local exact controllability to the trajectories with controls v≡0and h∈L2(O×(0,T)). The last main result we present in this paper follows in Theorem 3. Theorem 3. Let N=2. Then, for any T>0and any M>0,(3) is null controllable at time Twith controls of the form v1O, where v∈W. For the proofs of these results, following a standard approach, we will first deduce null controllability results for suitable linearized versions of (1), (2) and (3), namely, (14) ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+(y·∇)y+∇p=f+v1O,∇·y=0 inQ, y= 0 on Σ, y(0) = y0in Ω, (15) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(y·∇)y+(y·∇)y+∇p=f+v1O+θe Nin Q, ∇·y=0 inQ, θt−Δθ+y·∇θ+y·∇θ=k+h1Oin Q, y=0,θ= 0 on Σ, y(0) = y0,θ(0) = θ0in Ω and (16) ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=v1O,∇·y=0 inQ, y·n=0,∇×y= 0 on Σ, y(0) = y0in Ω. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 150 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL Then, appropriate arguments will be used to deduce the controllability of the nonlinear systems (1)–(3). Remark 1. When N= 3, it is very natural to ask whether a result similar to Theorem 1 holds with controls having two zero components. In general, the answer is no. In fact, it seems difficult to identify the open sets Ω and Osuch that one has null controllability for all T>0 with controls of this kind. This is unknown even for the classical Stokes equations for which, up to now, the only known results concern approximate controllability; see [16]. Remark 2. Assume that N= 2. The arguments in [7] implicitly show that, under hypotheses (12), we can find controls v1Owith v∈Wsuch that the associated solutions to (1) satisfy y(T)=y(T). Observe that the assumption (11) on the control domain is not necessary here. This paper is organized as follows. We will first establish all the technical results needed in this work in section 2. Section 3 will deal with null controllability results for the linear control systems (14)–(16). Finally, the proofs of Theorems 1, 2 and 3 will be given in section 4. 2. Some previous results. In this section we will establish all the technical results needed in this paper. More precisely, we will present and prove the required Carleman estimates for the backward systems (19), (20) and (21), given below. To do this, let us first introduce some weight functions: (17) α(x, t)=e5/4λmη0∞−eλ(mη0∞+η0(x)) t4(T−t)4, ξ(x, t)=eλ(mη0∞+η0(x)) t4(T−t)4, α(t) = min x∈Ω α(x, t)=e5/4λmη0∞−eλ(m+1) η0∞ t4(T−t)4, α∗(t) = max x∈Ω α(x, t)=e5/4λmη0∞−eλmη0∞ t4(T−t)4,  ξ(t) = max x∈Ω ξ(x, t)=eλ(m+1)η0∞ t4(T−t)4,ξ ∗(t) = min x∈Ω ξ(x, t)= eλmη0∞ t4(T−t)4, where m>4 is a fixed real number. Here, η0is a function verifying (18) η0∈C2(Ω),|∇η0|>0inΩ\O 0,η 0>0 in Ω and η0≡0on∂Ω with O0a nonempty open subset of Othat will be determined below. For any O0, the existence of such a function η0is proved in [9]. Note that these weights have already been used in [7] and [12]. We will be dealing in this section with the adjoint systems to (14) and (15), that is to say, (19) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −ϕt−Δϕ−(Dϕ)y+∇π=g, ∇·ϕ=0 inQ, ϕ= 0 on Σ, ϕ(T)=ϕ0in Ω Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 151 and (20) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −ϕt−Δϕ−(Dϕ)y+∇π=g+θ∇ψ, ∇·ϕ=0 inQ, −ψt−Δψ−y·∇ψ=q+ϕNin Q, ϕ=0,ψ= 0 on Σ, ϕ(T)=ϕ0,ψ(T)=ψ0in Ω (where Dϕ =∇ϕ+∇ϕt) as well as with the adjoint system of ω:= ∇×y(where y is the solution of (16)), which is (21) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −ρt−Δρ−∇×((y·∇×)∇γ)=0,Δγ=ρin Q, γ=0,ρ= 0 on Σ, ρ(T)=ρ0in Ω. Here, g∈L2(Q)N,q∈L2(Q), ϕ0∈H,ψ0∈L2(Ω) and ρ0∈H−1(Ω) (of course, ϕN stands for the last component of the vector field ϕ). 2.1. New Carleman estimates for system (19). We will establish some new Carleman estimates for the solutions of (19). We will assume that Oand ysatisfy (11)–(12). To fix ideas, we will also assume for the moment that N= 3 and n1(x0)=0 (x0appears in assumption (11)). The desired Carleman inequalities will have the form I(ϕ)≤CQ ρ2 1|g|2dx dt +O×(0,T ) ρ2 2|ϕ2|2+|ϕ3|2dx dt, where I(ϕ) contains global weighted integrals of |ϕ|2,|∇ϕ|2, etc. and ρ1and ρ2are appropriate weights that vanish exponentially as t→T. This will suffice to prove in section 3 the null controllability of (14) with controls v1Osatisfying v1≡0. Lemma 1. Assume that N=3,n1(x0)=0and Oand yverify (11)–(12). Then there exists a positive constant Csuch that, for any g∈L2(Q)3and any ϕ0∈H, the associated solution to (19) satisfies: (22) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ I(ϕ):=Q e −2α t4(T−t)4t−12(T−t)−12 |ϕ|2dx dt +Q e −2α t4(T−t)4t−4(T−t)−4|∇ϕ|2)dx dt +Q e −2α t4(T−t)4t4(T−t)4|Δϕ|2+|ϕt|2dx dt ≤CQ e −4eα+2α t4(T−t)4t−30(T−t)−30 |g|2dx dt +O×(0,T ) e −16 eα+14α t4(T−t)4t−132(T−t)−132 (|ϕ2|2+|ϕ3|2)dx dt. Here, αand αare constants only depending on Ω,O,Tand ysatisfying 0<α<α and 8α−7α>0. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 152 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL Proof. Let us first recall a Carleman inequality for the solutions of (19) which has been proved in [7] whenever (12) is fulfilled: (23) s3λ4Q e−2sαξ3|ϕ|2dx dt +sλ2Q e−2sαξ|∇ϕ|2dx dt +s−1Q e−2sαξ−1(|ϕt|2+|Δϕ|2)dx dt ≤C0(1 + T2)s15/2λ20 Q e−4sbα+2sα∗ ξ15/2|g|2dx dt +s16λ40 O0×(0,T ) e−8sbα+6sα∗ ξ16|ϕ|2dx dt. Here, s≥s0and λ≥λ0are arbitrarily large and C0,s0and λ0are suitable constants depending on Ω, O0,Tand y; see Theorem 1 in [7]. Recall that an inequality like (23) had already been proved in [13] using stronger properties on ythan (12). It is immediate from (23) that, for some C1,αand αdepending on Ω, O0,Tand y,wehave: (24) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Q e −2α t4(T−t)4t−12(T−t)−12 |ϕ|2+t−4(T−t)−4|∇ϕ|2dx dt +Q e −2α t4(T−t)4t4(T−t)4|Δϕ|2+|ϕt|2dx dt ≤C1Q e −4eα+2α t4(T−t)4t−30(T−t)−30 |g|2dx dt +O0×(0,T ) e −8eα+6α t4(T−t)4t−64(T−t)−64 |ϕ|2dx dt. Indeed, it suffices to choose (25) ⎧ ⎪ ⎨ ⎪ ⎩ α=s0e5/4λ0mη0∞−eλ0mη0∞, α=s0e5/4λ0mη0∞−eλ0(m+1)η0∞ and C1=C0(1 + T2)s17 0λ40 0e17λ0(m+1)η0∞. Notice that 0 <α<α. Moreover, it can be assumed that 8α−7α>0 (it suffices to notice that λ0is large enough in (25)). We will apply (24) for the open set O0⊂Odefined as follows. We choose κ>0 such that n1(x)=0 ∀x∈B(x0;κ)∩∂O∩∂Ω and we denote this set by Γκ. Then, we define (26) O0={x∈Ω:x=w+τe 1,w∈Γκ,|τ|<τ 0}, with κ, τ0>0 small enough so that we still have (27) O0⊂O and d0:= dist(O0,∂O∩Ω) >0. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 153 Observe that, with this choice, each P∈O 0verifies that one of the two points where the straight line {P+Re1}intersects ∂Ω belongs to ∂O0. Once O0is defined, we apply inequality (24) in this open set and we try to bound the term O0×(0,T ) e −8eα+6α t4(T−t)4t−64(T−t)−64 |ϕ1|2dx dt in terms of local integrals of ϕ2and ϕ3. To this end, for each (x, t)∈O 0×(0,T) we denote by l(x, t) (resp., l(x, t)) the segment that starts from (x, t) with direction e1in the positive (resp. negative) sense and ends at ∂O0. Then, since ϕis divergence-free, it is not difficult to see that ϕ1(x, t)=l(x,t) (∂2ϕ2+∂3ϕ3)(y1,x 2,x 3,t)dy1 for each (x, t)∈O 0×(0,T). For simplicity, let us introduce the notation β(t)=e −8eα+6α t4(T−t)4t−64(T−t)−64 ∀t∈(0,T). Applying at this point H¨older’s inequality and Fubini’s formula, we obtain (28) O0×(0,T ) β(t)|ϕ1|2dx dt ≤C2O0×(0,T ) β(t)l(x,t) (|∂2ϕ2|2+|∂3ϕ3|2)dy1dx dt =C2O0×(0,T ) (|∂2ϕ2|2+|∂3ϕ3|2)e l(y1) β(t)dx1dy1dx2dx3dt ≤C3O0×(0,T ) β(t)(|∂2ϕ2|2+|∂3ϕ3|2)dx dt, where l(y1) stands for the segment l(y1,x 2,x 3,t). Then, we introduce a function ζ∈C2(O) such that ζ≡1inO0,0≤ζ≤1 and ζ(x) = 0 at any point x∈Osatisfying dist(x, ∂O∩Ω) ≤d0/2(d0was defined in (27)). This and the fact that ϕ|Σ≡0 imply O0×(0,T ) β(t)|∂iϕi|2dx dt ≤O×(0,T ) ζβ(t)|∂iϕi|2dx dt =1 2O×(0,T ) ∂2 iiζβ(t)|ϕi|2dx dt −O×(0,T ) ζβ(t)∂2 iiϕiϕidx dt for i=2,3. Finally, in view of Young’s inequality and regularity estimates for ϕiin Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 154 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL Ω(ϕi∈H2(Ω) and ϕiH2≤CΔϕiL2), we also have: O0×(0,T ) β(t)|∂iϕi|2dx dt ≤C4O×(0,T ) e −16 eα+14α t4(T−t)4t−132(T−t)−132 |ϕi|2dx dt +1 2C1C3Q e −2α t4(T−t)4t4(T−t)4|Δϕi|2dx dt, which, combined with (24) and (28), yields (22). Let us now present another Carleman inequality for (19) with weight functions not vanishing at time t=0. Lemma 2. Assume that N=3,n1(x0)=0and Oand yverify (11)–(12). Then there exist positive constants C,αand αwith 0<α<αand 8α−7α>0depending on Ω,O,Tand ysuch that, for any g∈L2(Q)3and any ϕ0∈H, the associated solution to (19) satisfies: (29) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Q e −2α (t)4(t)−12 |ϕ|2+(t)−4|∇ϕ|2dx dt ≤CQ e −4eα+2α (t)4(t)−30 |g|2dx dt +O×(0,T ) e −16 eα+14α (t)4(t)−132 (|ϕ2|2+|ϕ3|2)dx dt, where is the C1function given by (30) (t)=⎧ ⎨ ⎩ T2 4for 0≤t≤T/2, t(T−t)for T/2≤t≤T. To prove (29), it suffices to use (22) and the classical parabolic estimates for the Stokes system satisfied by ϕ. The argument has already been used in [9], [13] and [7] in several similar situations, so we omit it for simplicity. For completeness, let us state the similar result that can be established when N= 2. Here, we assume again that n1(x0)=0. Lemma 3. Assume that N=2,n1(x0)=0and Oand yverify (11)–(12). Then there exist positive constants C,αand αwith 0<α<αand 8α−7α>0depending on Ω,O,Tand ysuch that, for any g∈L2(Q)2and any ϕ0∈H, the associated solution to (19) satisfies: (31) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Q e −2α (t)4(t)−12 |ϕ|2+(t)−4|∇ϕ|2dx dt ≤CQ e −4eα+2α (t)4(t)−30 |g|2dx dt +O×(0,T ) e −16 eα+14α (t)4(t)−132 |ϕ2|2dx dt, where is the function given by (30). Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 161 for τ≥τand λ≥ λ. Here, we have introduced the function η, with η(x)=eλη0(x). In fact, the inequality one can find in [14] contains local integrals of |∂jγ|2and |∇(∂jγ)|2in the right-hand side. But it can be written for a smaller set O⊂⊂ O. Using localizing arguments together with the fact that we actually have a global weighted integral of |Δ(∂jγ)|2in the left-hand side, (47) is easily found. Following the same steps of [7], we set τ=s t4(T−t)4eλmη0∞, we multiply (47) by exp −2se5/4λmη0∞ t4(T−t)4 and we integrate in time over (0,T). This gives s4λ4Q e−2sαξ4|∂jγ|2dx dt +s2λ2Q e−2sαξ2|∇(∂jγ)|2dx dt ≤C17 sQ e−2sαξ|∂jρ|2dx dt +s4λ4O×(0,T ) e−2sαξ4|∂jγ|2dx dt +s5/2λ2T 0 e−2sα∗(ξ∗)5/2∂jγ2 H1/2(∂Ω) for s≥s5and λ≥ λ. Combining this estimate and (46), we have s4λ4Q e−2sαξ4|∂jγ|2dx dt +sλ2Q e−2sα ξ|∇ρ|2dx dt +s3λ4Q e−2sαξ3|ρ|2dx dt ≤C18 s5/2λ2T 0 e−2sα∗(ξ∗)5/2∂jγ2 H1/2(∂Ω) +s5λ6O×(0,T ) e−2sαξ5|∇γ|2dx dt for any s≥s6and λ≥λ5. On the other hand, the boundary term can readily be bounded using the continuity of the trace operator: ∂jγ(t)2 H1/2(∂Ω) ≤C19(∂jγ(t)2 L2+∇(∂jγ)(t)2 L2). Furthermore, since γ|Σ≡0, we know that there exists a positive constant C20 such that ∇(∂jγ)(t)L2≤C20Δγ(t)L2a.e. in (0,T) for j=1,2. Consequently, s4λ4Q e−2sαξ4|∂jγ|2dx dt +sλ2Q e−2sα ξ|∇ρ|2dx dt +s3λ4Q e−2sαξ3|ρ|2dx dt ≤C21 s5λ6O×(0,T ) e−2sαξ5|∇γ|2dx dt for s≥s6. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 162 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL This implies (44) and ends the proof of Lemma 7. Remark 3. An almost immediate consequence of the Carleman estimate (44) is the following observability inequality: (48) (∇γ)(0)2 L2≤CO×(0,T ) |∇γ|2dx dt. Proof. All comes to prove a dissipation result for the L2norm of ∇γ. Indeed, if we can prove that (49) ∇γ(t1)2 L2≤C∇γ(t2)2 L2∀0≤t1<t 2≤T, then using the properties of the weight function e−2sα and estimate (44), we readily deduce (48). Thus, we multiply the equation in (43) by −γand we integrate in Ω. Taking into account that γand ρvanish on ∂Ω, this yields: −1 2 d dt Ω |∇γ|2dx +Ω |Δγ|2dx −Ω ((y·∇×)∇γ)·∇×γdx=0, from which the dissipation estimate (49) follows. In fact, this is what will be used in section 3 to prove the null controllability of system (16). 3. Null controllability of the linearized systems (14), (15) and (16). 3.1. Null controllability of (14). We are dealing here with the following system: (50) ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+(y·∇)y+∇p=f+v1O,∇·y=0 inQ, y= 0 on Σ, y(0) = y0in Ω, where Osatisfies (11) and ysatisfies (12). Our goal will be to find a control vsuch that y(T) = 0 in Ω. Let us introduce some weight functions: β2(t) = exp α (t)4(t)6,β 3(t) = exp 2α−α (t)4(t)15 and β4(t) = exp 8α−7α (t)4(t)66 (recall that was defined in (30)), where αand αare the constants provided by Lemma 2 when N= 3 and Lemma 3 when N= 2. Recall that, in particular, 0<α<αand 8α−7α>0. Of course, we will need some specific conditions on fand y0to get the null controllability of (50). We will use the arguments in [7]. Thus, let us set (51) Ly =yt−Δy+(y,∇)y+(y,∇)y Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 163 and let us introduce the spaces E2={(y, p, v):(y,v)∈E0, −4β2(Ly +∇p−v1O)∈L2(0,T;H−1(Ω)2)} when N= 2 and E3={(y, p, v):(y,v)∈E0, −2β1/2 2y∈L4(0,T;L12(Ω)3), −4β2(Ly +∇p−v1O)∈L2(0,T;W−1,6(Ω)3)} when N= 3, where E0={(y,v):β3y, β4v1O∈L2(Q)N,v 1≡0, −2β1/2 2y∈L2(0,T;V)∩L∞(0,T;H)}. It is clear that ENis a Banach space for the norm · EN, where (y, p, v)E2=β3y2 L2+β4v1O2 L2+−2β1/2 2y2 L2(0,T ;V) +−2β1/2 2y2 L∞(0,T ;H)+−4β2(Ly +∇p−v1O)2 L2(0,T ;H−1)1/2 and (y, p, v)E3=β3y2 L2+β4v1O2 L2+−2β1/2 2y2 L2(0,T ;V) +−2β1/2 2y2 L∞(0,T ;H)+−2β1/2 2y2 L4(0,T ;L12) +−4β2(Ly +∇p−v1O)2 L2(0,T ;W−1,6)1/2. Remark 4. The spaces Ej(j=0,2,3) are natural spaces where solutions of the null controllability of (50) must be found in order to preserve these properties for the nonlinear term (y·∇)y. More details are provided in subsection 4.1. Proposition 1. Assume that n1(x0)=0and Oand yverify (11)–(12).Let y0∈Eand let us assume that −4β2f∈L2(0,T;H−1(Ω)2)if N=2, L2(0,T;W−1,6(Ω)3)if N=3. Then, we can find a control vsuch that the associated solution (y,p)to (50) satisfies (y, p, v)∈EN. In particular, v1≡0and y(T)=0. Sketch of the proof. The proof of this proposition is very similar to the one of Proposition 2 in [7], so we will just give the main ideas. For simplicity, we will only consider the case N= 3. When N= 2, the proof is even easier. Following the arguments in [9] and [13], let us introduce the auxiliary optimal control problem (52) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ inf 1 2Q |β3y|2dxdt +O×(0,T ) |β4v|2dxdt subject to v∈L2(Q)3,supp v⊂O×(0,T),v 1≡0 and ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Ly +∇p=f+v1Oin Q, ∇·y=0 inQ, y= 0 on Σ, y(0) = y0,y(T)=0 inΩ. Notice that a solution (y, p, v) to (52) is a good candidate to satisfy (y, p, v)∈E3. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 164 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL For the moment, let us assume that (52) possesses a solution (y, p, v). Then, by virtue of Lagrange’s principle, there must exist dual variables zand qsuch that (53) ⎧ ⎪ ⎨ ⎪ ⎩y=β−2 3(L∗z+∇q),∇·z=0 inQ, v1≡0,vi=−β−2 4zi(i=2,3) in O×(0,T), z= 0 on Σ, where L∗is the adjoint operator of L, i.e., L∗z=−zt−Δz−(Dz)y. At least formally, the couple (z, q) satisfies (54) a((z, q),(w,h)) = G, (w,h)∀(w,h)∈P0, where P0is the space P0={(w,h)∈C2(Q)4:∇·w=0,w= 0 on Σ,O h(x, t)dx =0} and we have used the notation a((z, q),(w,h)) = Q β−2 3(L∗z+∇q)·(L∗w+∇h)dx dt +O×(0,T ) β−2 4(z2w2+z3w3)dx dt and G, (w,h)=T 0 f(t),w(t)H−1,H1 0dt +Ω y0·w(0) dx. Conversely, if we are able to “solve” (54) and then use (53) to define (y, p, v), we will probably have found a solution to (52). Thus, let us consider the linear space P0. It is clear that a(·,·):P0×P0→ Ris a symmetric, definite positive bilinear form on P0. We will denote by Pthe completion of P0for the norm induced by a(·,·). Then a(·,·) is well-defined, continuous and again definite positive on P. Furthermore, in view of the Carleman estimate (29), the linear form (w,h)→G, (w, h)is well-defined and continuous on P. Hence, from Lax-Milgram’s lemma, we deduce that the variational problem (55) a((z, q),(w,h)) = G, (w,h) ∀(w,h)∈P, (z, q)∈P, possesses exactly one solution (z, q). Let yand vbe given by (53). Then, it is readily seen that they verify Q β2 3|y|2dx dt +O×(0,T ) β2 4|v|2dx dt < +∞ and, also, that yis, together with some pressure p, the weak solution (belonging to L2(0,T;V)∩L∞(0,T;H)) of the Stokes system in (52) for v=v. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 165 In order to prove that (y, p, v)∈E3, it only remains to check that −2β1/2 2yis, together with −2β1/2 2p, a weak solution of a Stokes problem of the kind (50) with a right-hand side in L2(0,T;W−1,6(Ω)3) that belongs to L4(0,T;L12(Ω)3). To this end, we define the functions y∗=−2β1/2 2y, p∗=−2β1/2 2pand f∗=−2β1/2 2(f+v1O). Then (y∗,p ∗) satisfies (56) ⎧ ⎪ ⎨ ⎪ ⎩ Ly∗+∇p∗=f∗+(−2β−1/2 2)ty, ∇·y∗=0 inQ, y∗= 0 on Σ, y∗(0) = −2(0)β1/2 2(0)y0in Ω. From the fact that f∗∈L2(0,T;H−1(Ω)3) and y0∈H, we have indeed y∗∈L2(0,T;V)∩L∞(0,T;H). Finally, we deduce that y∗∈L4(0,T;L12(Ω)3) from Lemma 2 in [7]. This ends the sketch of the proof of Proposition 1. 3.2. Null controllability of system (15). We will establish the null controllability of the linear system (57) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(y·∇)y+(y·∇)y+∇p=f+v1O+θe Nin Q, ∇·y=0 inQ, θt−Δθ+y·∇θ+y·∇θ=k+h1Oin Q, y=0,θ= 0 on Σ, y(0) = y0,θ(0) = θ0in Ω, where Osatisfies (11) and yand θsatisfy (12) and (13), for suitable right-hand sides fand k. The arguments we present here are completely analogous to those in [12] and subsection 3.1 of this paper, so that we will only give a sketch. Thus, we restrict ourselves again to the three-dimensional case with n1(x0)=0. Let us introduce the weight functions β5(t) = exp α (t)4(t)6,β 6(t) = exp 2α−α (t)4(t)15, β7(t) = exp 16α−15α (t)4(t)126,β 8(t) = exp 8α−7α (t)4(t)66 and β9(t) = exp 16α−15α (t)4(t)134, where the constants αand αare furnished by Lemma 5 when N= 3 and Lemma 6 when N= 2 (and, in particular, 0 <α<αand 16α−15α>0). Let us set (58) Pθ =θt−Δθ+y·∇θ Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 166 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL and let us introduce the spaces  E2={(y,p,θ,v,h):(y,θ,v,h)∈ E0, −4β5(Ly +∇p−v1O)∈L2(0,T;H−1(Ω)2), −4β5(Pθ+y·∇θ−h1O)∈L2(0,T;H−1(Ω))} when N= 2 and  E3={(y,p,θ,v,h):(y,θ,v,h)∈ E0, −2β1/2 5y∈L4(0,T;L12(Ω)3), −4β5(Ly +∇p−v1O)∈L2(0,T;W−1,6(Ω)3), −4β5(Pθ+y·∇θ−h1O)∈L2(0,T;H−1(Ω))} when N= 3, where  E0={(y,θ,v,h):(β6y)i,β 7θ, (β8v1O)i,β 9h1O∈L2(Q)(1≤i≤N), v1≡vN≡0, −2β1/2 5y∈L2(0,T;V)∩L∞(0,T;H), −2β1/2 5θ∈L2(0,T;H1 0(Ω)) ∩L∞(0,T;L2(Ω))}. It can be readily seen now that  E0, E2and  E3are Banach spaces for the norms (y,θ,v,h)e E0=β6y2 L2+β7θ2 L2+β8v2 L2 +β9h2 L2+−2β1/2 5y2 L2(0,T ;V)+−2β1/2 5y2 L∞(0,T ;H) +−2β1/2 5θ2 L2(0,T ;H1 0)+−2β1/2 5θ2 L∞(0,T ;L2)1/2, (y,p,θ,v,h)e E2=(y,θ,v,h)2 e E0 +−4β5(Ly +∇p−v1O)2 L2(0,T ;H−1) +−4β5(Pθ+y·∇θ−h1O)2 L2(0,T ;H−1)1/2 and (y,p,θ,v,h)e E3=(y,θ,v,h)2 e E0+−2β1/2 5y2 L4(0,T ;L12) +−4β5(Ly +∇p−v1O)2 L2(0,T ;W−1,6) +−4β5(Pθ+y·∇θ−h1O)2 L2(0,T ;H−1)1/2. Proposition 2. Assume that n1(x0)=0and Oand (y,θ)satisfy (11)–(13).Let y0∈E,θ0∈L2(Ω) and let us assume that −4β1(f,k)∈L2(0,T;H−1(Ω)2)×L2(0,T;H−1(Ω)) if N=2, L2(0,T;W−1,6(Ω)3)×L2(0,T;H−1(Ω)) if N=3. Then, we can find controls vand hsuch that the associated solution to (57) satisfies (y,p,θ,v,h)∈ EN. In particular, v1≡vN≡0and y(T)=θ(T)=0. We omit the proof of this proposition, since it is essentially the same as the one of Proposition 2 in [12] and follows the steps of Proposition 1 above. As we have already indicated, the main ideas come from [13]. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 167 3.3. Null controllability of system (16). We will prove the null controllability of the linear system (59) ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=v1O,∇·y=0 inQ, y·n=0,∇×y= 0 on Σ, y(0) = y0in Ω, where N= 2 and y∈L∞(Q)2. For this purpose, we first rewrite this system using the streamline-vorticity formulation. Thus, setting ω=∇×y,wehave (60) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ωt−Δω+∇×((∇×ψ·∇)y)=∇×(v1O),Δψ=ωin Q, ψ=0,ω= 0 on Σ, ω(0) = ∇×y0in Ω. Proposition 3. Assume that y0∈Hand y∈L∞(Q)2. Then, there exists a constant C(Ω,O,T)>0and controls v1Owith v∈W(Wwas defined in (9)), such that (61) vL2≤Cy0H and the associated solutions of (59) satisfy (62) y∈L2(0,T;H1(Ω)2)∩C0([0,T]; L2(Ω)2),y t∈L2(0,T;H−1(Ω)2), and y(T)=0, with (63) yL2(0,T ;H1)+yC0([0,T ];L2)+ytL2(0,T ;H−1)≤Cy0H. Proof. We first establish the null controllability property for y. This can be done in several ways. One of them is the following. We first define for each ε>0 the functional ⎧ ⎪ ⎨ ⎪ ⎩ Jε(γ0)=1 2O×(0,T ) |∇ × γ|2dx dt +ε∇γ0L2+((∇×γ)(0),y0)L2 ∀γ0∈H1 0(Ω), where γis given by (43) with ρ0=Δγ0∈H−1(Ω). It is not difficult to see from the observability inequality (48) that this functional possesses a unique minimizer γ0 ε∈H1 0(Ω) (see Proposition 2.1 in [5]). Now, from the necessary conditions for Jεto reach a minimum, we have (64) Q ((∇×γε)1O)·(∇×γ)dx dt +ε(∇×γ0 ε ∇ × γ0 εL2 ,∇×γ0)L2 +((∇×γ)(0),y0)L2=0 ∀γ0∈H1 0(Ω). Thus, setting vε=(∇×γε)1Oand putting γ0=γ0 ε, we find from (48) and (64) that (61) holds for vεfor some Cindependent of ε: (65) ∇ × γεL2(O×(0,T ))2=vεL2(O×(0,T ))2≤C. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 168 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL Let us denote by (ωε,ψ ε) the solution to (60) for v=vε. Then, taking into account the systems satisfied by (ρ, γ) and (ωε,ψ ε), we deduce that Q ∇×(vε1O)γdxdt+(∇×γ0,(∇×ψε)(T))L2 −((∇×γ)(0),y0)L2=0 ∀γ0∈H1 0(Ω). Combining this and (64), we obtain (66) (∇×ψε)(T)L2≤ε. From (65) and (66) written for each ε>0, we deduce that, at least for a subsequence, vε→vweakly in L2(O×(0,T))2, where the control v1Ois such that the corresponding solution (ω,ψ) to (60) satisfies (∇×ψ)(T)=y(T)=0 inΩ. Since v∈L2(O×(0,T))2and ∇·v=0inO×(0,T), we necessarily have v∈W (from De Rham’s lemma applied to (v2,−v1)). In order to obtain the desired regularity for y, we will consider again the equations satisfied by ψand ωand we will check that (67) ψ∈L2(0,T;H2(Ω)) ∩C0([0,T]; H1 0(Ω)) and ψt∈L2(Q), with appropriate estimates. For simplicity, we will only present the estimates. The rigorous argument relies on introducing a standard Galerkin approximation of (60) with a “special” basis of H1 0(Ω) (more precisely, the basis formed by the eigenfunctions of the Laplacian- Dirichlet operator in Ω) and deducing for the associated approximate solutions the estimates below. Thus, let us multiply the first equation in (60) by ψand let us integrate by parts. We find that 1 2Ω |∇ψ(t)|2dx +t 0Ω |Δψ|2dx dτ =t 0O v·(∇×ψ)dx dτ −t 0Ω (((∇×ψ),∇)y)·(∇×ψ)dx dτ +1 2(∇ψ)(0)2 L2 for all t∈(0,T). If we integrate by parts in the last integral, we also have −t 0Ω (((∇×ψ)·∇)y)·(∇×ψ)dx dτ =t 0Ω ((∇×ψ)·∇)(∇×ψ)·ydxdτ. Since ψ|Σ≡0, we deduce that (68) ψ∈L2(0,T;H2(Ω)) ∩L∞(0,T;H1 0(Ω)) and (69) ψL2(0,T ;H2)+ψL∞(0,T ;H1 0)≤Cy0L2. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php N-DIMENSIONAL NAVIER–STOKES WITH N−1 CONTROLS 169 Now, let us introduce for each tthe function ψ∗(t)=Δ −1ψt(t), i.e., the solution to −Δψ∗(t)=ψt(t)inΩ ψ∗(t)=0 on∂Ω. Observe that, whenever ψt(t)∈L2(Ω), this function satisfies ψ∗(t)∈H2(Ω) ∩ H1 0(Ω) and (70) ψ∗(t)H2≤Cψt(t)L2. Then, we multiply the first equation of (60) by ψ∗and we integrate by parts. This gives Q |ψt|2dx dt =Q (Δψ)ψtdx dt −Q ((∇×ψ)·∇)(∇×ψ∗)·ydxdt +O×(0,T ) v·(∇×ψ∗)dx dt. Using that v∈L2(Q)2and we already have (69) and (70), we conclude that ψt∈ L2(Q) and (71) ψtL2≤Cy0L2. From (69) and (71), we immediately obtain (67), (62) and (63). This ends the proof of Proposition 3. 4. Proofs of the controllability results for the nonlinear systems. In this last section, we will give the proofs of Theorems 1, 2 and 3. For the proofs of Theorems 1 and 2 we employ an inverse mapping theorem, while a fixed point argument is used for Theorem 3. 4.1. Proof of Theorem 1. We also follow here the steps in [7]. Thus, we set y=y+zand p=p+χand we use these identities in (1). Taking into account that (y,p) solves (5), we find: (72) ⎧ ⎪ ⎨ ⎪ ⎩ Lz +(z·∇)z+∇χ=v1O,∇·z=0 inQ, z= 0 on Σ, z(0) = y0−y0in Ω (recall that Lwas defined in (51)). This way, we have reduced our problem to a local null controllability result for the solution (z,χ)tothenonlinear problem (72). We will use the following inverse mapping theorem (see [1]). Theorem 4. Let B1and B2be two Banach spaces and let A:B1→ B2satisfy A∈C1(B1;B2). Assume that b0∈B1,A(b0)=d0and also that A(b0):B1→ B2is surjective. Then there exists δ>0such that, for every d∈B2satisfying d−d0B2< δ, there exists a solution of the equation A(b)=d, b ∈B1. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 170 FERN´ ANDEZ-CARA, GUERRERO, IMANUVILOV AND PUEL We will apply this result with B1=EN, B2=L2(−4β2;0,T;H−1(Ω)2)×Hif N=2, L2(−4β2;0,T;W−1,6(Ω)3)×(H∩L4(Ω)3)ifN=3 and A(z, χ, v)=(Lz +(z·∇)z+∇χ−v1O,z(0)) ∀(z, χ, v)∈EN. From the facts that −2β1/2 2y∈L4(0,T;L12(Ω)3) and Ais bilinear, it is not difficult to check that A∈C1(B1;B2); more details can be found in [13] or [7]. Let b0be the origin in B1. Notice that A(0,0,0) : B1→ B2is given by A(0,0,0)(z, χ, v)=(Lz +∇χ−v1O,z(0)) ∀(z, χ, v)∈EN and is surjective, in view of the null controllability result for (14) given in Proposition 1. Consequently, we can indeed apply theorem 4 with these data and there exists δ>0 such that, if z(0)E≤δ, then we find a control vsatisfying v1≡0 such that the associated solution to (72) verifies z(T) = 0 in Ω. This concludes the proof of Theorem 1. 4.2. Proof of Theorem 2. Again, we follow here the ideas of [12]. Therefore, we set y=y+z,p=p+χand θ=θ+ρ, so from (2) and (7), we find: (73) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Lz +(z·∇)z+∇χ=v1O+ρe N,∇·z=0 inQ, Pρ+(z·∇)ρ+z·∇ θ=h1Oin Q, z=0,ρ= 0 on Σ, z(0) = y0−y0,ρ(0) = θ0−θ(0) in Ω (Land Pwere respectively defined in (51) and (58)). We are thus led to prove the local null controllability of (73). To this end, we will use again Theorem 4, which was presented in subsection 4.1. Using the same notation as there, we set B1= EN, B2=L2(−4β5;0,T;H−1(Ω)3)×H×L2(Ω) if N= 2 and B2=L2(−4β5;0,T;W−1,6(Ω)3×H−1(Ω)) ×(L4(Ω)3∩H)×L2(Ω) if N=3. Let us introduce A, with A(z, χ, ρ, v, h)=(A1(z, χ, ρ, v),A2(z, ρ, h),z(0),ρ(0)), A1(z, χ, ρ, v)=Lz +(z·∇)z+∇χ−v1O−ρeN and A2(z, ρ, h)=Pρ+(z·∇)ρ+z·∇θ−h1O for every (z, χ, ρ, v, h)∈ EN. Downloaded 05/20/16 to 150.214.230.47. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php