Uniform local null control of the Leray-α model

ESAIM: COCV 20 (2014) 1181–1202 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv/2014011 www.esaim-cocv.org UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL ∗,∗∗ F´ agner D. Araruna1, Enrique Fern´ andez-Cara2and Diego A. Souza1,2 Abstract. This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier−Stokes system (αis a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-αequations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α→0+to a null control of the Navier−Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc. Mathematics Subject Classification. 93B05, 35Q35, 35G25, 93B07. Received November 22, 2013. Revised February 8, 2014. Published online August 8, 2014. 1. Introduction. The main results Let Ω⊂RN(N=2,3) be a bounded connected open set whose boundary Γis of class C2.Letω⊂Ωbe a (small) nonempty open set, let γ⊂Γbe a (small) nonempty open subset of Γand assume that T>0. We will use the notation Q=Ω×(0,T)andΣ=Γ×(0,T) and we will denote by n=n(x) the outward unit normal to Ωat the points x∈Γ; spaces of RN-valued functions, as well as their elements, are represented by boldface letters. The Navier−Stokes system for a homogeneous viscous incompressible fluid (with unit density and unit kinematic viscosity) subject to homogeneous Dirichlet boundary conditions is given by ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=fin Q, ∇·y=0 in Q, y=0on Σ, y(0) = y0in Ω, (1.1) where y(the velocity field) and p(the pressure) are the unknowns, f=f(x,t)isaforcingtermandy0=y0(x) is a prescribed initial velocity field. Keywords and phrases. Null controllability, Carleman inequalities, Leray-αmodel, Navier−Stokes equations. ∗Partially supported by INCTMat, CAPES and CNPq (Brazil). ∗∗ Partially supported by CAPES (Brazil) and grants MTM2006-07932, MTM2010-15592 (DGI-MICINN, Spain). 1Departamento de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa PB, Brasil. [email protected] 2Departamento EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. [email protected]; [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2014 1182 F.D. ARARUNA ET AL. In order to prove the existence of a solution to the Navier−Stokes system, Leray in [26]hadtheideaof creating a turbulence closure model without enhancing viscous dissipation. Thus, he introduced a “regularized” variant of (1.1) by modifying the nonlinear term as follows: yt−Δy+(z·∇)y+∇p=fin Q, ∇·y=0 inQ, where yand zare related by z=φα∗y(1.2) and φαis a smoothing kernel. At least formally, the Navier−Stokes equations are recovered in the limit as α→0+,sothatz→y. In this paper, we will consider a special smoothing kernel, associated to the Stokes-like operator Id +α2A, where Ais the Stokes operator (see Sect. 2). This leads to the following modification of the Navier−Stokes equations, called the Leray-αsystem (see [4]): ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(z·∇)y+∇p=fin Q, z−α2Δz+∇π=yin Q, ∇·y=0,∇·z=0 in Q, y=z=0on Σ, y(0) = y0in Ω. (1.3) In almost all previous works found in the literature, Ωis either the N-dimensional torus and the PDE’s in (1.3) are completed with periodic boundary conditions or the whole space RN. Then, zsatisfies an equation of the kind z−α2Δz=y(1.4) and the model is (apparently) slightly different from (1.3). However, since ∇·y= 0, it is easy to see that (1.4), in these cases, is equivalent to the equation satisfied by zand πin (1.3). It has been shown in [4] that, at least for periodic boundary conditions, the numerical solution of the equations in (1.3) matches successfully with empirical data from turbulent channel and pipe flows for a wide range of Reynolds numbers. Accordingly, the Leray-αsystem has become preferable to other turbulence models, since the associated computational cost is lower and no introduction of ad hoc parameters is required. In [19], the authors have compared the numerical solutions of three different α-models useful in turbulence modelling (in terms of the Reynolds number associated to a Navier−Stokes velocity field). The results improve as one passes from the Navier−Stokes equations to these models and clearly show that the Leray-αsystem has the best performance. Therefore, it seems quite natural to carry out a theoretical analysis of the solutions to (1.3). We will be concerned with the following controlled systems ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(z·∇)y+∇p=v1ωin Q, z−α2Δz+∇π=yin Q, ∇·y=0,∇·z=0 in Q, y=z=0on Σ, y(0) = y0in Ω, (1.5) and ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(z·∇)y+∇p=0in Q, z−α2Δz+∇π=yin Q, ∇·y=0,∇·z=0 in Q, y=z=h1γon Σ, y(0) = y0in Ω, (1.6) UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL 1183 where v=v(x,t) (respectively h=h(x,t)) stands for the control, assumed to act only in the (small) set ω(respectively on γ) during the whole time interval (0,T). The symbol 1ω(respectively 1γ) stands for the characteristic function of ω(respectively of γ). In the applications, the internal control v1ωcan be viewed as a gravitational or electromagnetic field. The boundary control h1γis the trace of the velocity field on Σ. Remark 1.1. It is completely natural to suppose that yand zsatisfy the same boundary conditions on Σ since, in the limit, we should have z=y. Consequently, we will assume that the boundary control h1γacts simultaneously on both variables yand z. In what follows, (·,·)and·denote the usual L2scalar products and norms (in L2(Ω), L2(Ω), L2(Q), etc.) and K,C,C1,C2, . . . denote various positive constants (usually depending on ω,Ωand T). Let us recall the definitions of some usual spaces in the context of incompressible fluids: H=u∈L2(Ω):∇·u=0inΩ, u·n=0onΓ, V=u∈H1 0(Ω):∇·u=0inΩ. Note that, for every y0∈Hand every v∈L2(ω×(0,T)), there exists a unique solution (y,p,z,π)for(1.5) that satisfies (among other things) y,z∈C0([0,T]; H); see Proposition 1.2 below. This is in contrast with the lack of uniqueness of the Navier−Stokes system when N=3. The main goals of this paper are to analyze the controllability properties of (1.5)and(1.6) and determine the way they depend on αas α→0+. The null controllability problem for (1.5)attimeT>0 is the following: For any y0∈H, find v∈L2(ω×(0,T)) such that the corresponding state (the corresponding solution to (1.5)) satisfies y(T)=0in Ω. (1.7) The null controllability problem for (1.6)attimeT>0 is the following: For any y0∈H, find h∈L2(0,T;H−1/2(γ)) with γh·ndΓ=0and an associated state (the corresponding solution to (1.6)) satisfying y,z∈C0[0,T]; L2(Ω) and (1.7). Recall that, in the context of the Navier−Stokes equations, Lions conjectured in [27] the global distributed and boundary approximate controllability; since then, the controllability of these equations has been intensively studied, but for the moment only partial results are known. Thus, the global approximate controllability of the two-dimensional Navier−Stokes equations with Navier slip boundary conditions was obtained by Coron in [6]. Also, by combining results concerning global and local controllability, the global null controllability for the Navier−Stokes system on a two-dimensional manifold without boundary was established in Coron and Fursikov [7]; see also Guerrero et al. [24] for another global controllability result. The local exact controllability to bounded trajectories has been obtained by Fursikov and Imanuvilov [17,18], Imanuvilov [25]andFern´andez-Cara et al. [13] under various circumstances; see Guerrero [22] and Gonz´alez- Burgos et al. [21] for similar results related to the Boussinesq system. Let us also mention [3,8,9,14], where analogous results are obtained with a reduced number of scalar controls. For the (simplified) one-dimensional viscous Burgers model, positive and negative results can be found in [12,20,23]; see also [11], where the authors consider the one-dimensional compressible Navier−Stokes system. 1184 F.D. ARARUNA ET AL. Our first main result in this paper is the following: Theorem 1.2. There exists >0(independent of α)such that, for each y0∈Hwith y0≤,thereexist controls vα∈L∞(0,T;L2(ω)) such that the associated solutions to (1.5)fulfill (1.7).Furthermore,these controls can be found satisfying the estimate vαL∞(L2)≤C, (1.8) where Cis also independent of α. Our second main result is the analog of Theorem 1.2 in the framework of boundary controllability. It is the following: Theorem 1.3. There exists δ>0(independent of α)such that, for each y0∈Hwith y0≤δ,thereexist controls hα∈L∞(0,T;H−1/2(γ)) with γhα·ndΓ=0and associated solutions to (1.6)that fulfill (1.7). Furthermore, these controls can be found satisfying the estimate hαL∞(H−1/2)≤C, (1.9) where Cis also independent of α. The proofs rely on suitable fixed-point arguments. The underlying idea has applied to many other nonlinear control problems. However, in the present cases, we find two specific difficulties: •In order to find spaces and fixed-point mappings appropriate for Schauder’s Theorem, the initial state y0must be regular enough. Consequently, we have to establish regularizing properties for (1.5)and(1.6); see Lemmas 2.7 and 4.2 below. •For the proof of the uniform estimates (1.8)and(1.9), careful estimates of the null controls and associated states of some particular linear problems are needed. We will also prove results concerning the controllability in the limit, as α→0+.Itwillbeshownthatthe null-controls for (1.5) can be chosen in such a way that they converge to null-controls for the Navier−Stokes system ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=v1ωin Q, ∇·y=0 in Q, y=0on Σ, y(0) = y0in Ω. (1.10) Also, it will be seen that the null-controls for (1.6) can be chosen such that they converge to boundary null-controls for the Navier−Stokes system ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(y·∇)y+∇p=0in Q, ∇·y=0 in Q, y=h1γon Σ, y(0) = y0in Ω. (1.11) More precisely, our third and fourth main results are the following: Theorem 1.4. Let >0be furnished by Theorem 1.2. Assume that y0∈Hand y0≤,letvαbe a null control for (1.5)satisfying (1.8)and let (yα,p α,zα,π α)be the associated state. Then, at least for a subsequence, one has vα→vweakly-∗in L∞(0,T;L2(ω)), zα→yand yα→ystrongly in L2(Q), as α→0+,where(y,v)is, together with some p, a state-control pair for (1.10)satisfying (1.7). UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL 1185 Theorem 1.5. Let δ>0be furnished by Theorem 1.3. Assume that y0∈Hand y0≤δ,lethαbe a null control for (1.6)satisfying (1.9)and let (yα,p α,zα,π α)be the associated state. Then, at least for a subsequence, one has hα→hweakly-∗in L∞(0,T;H−1/2(γ)), zα→yand yα→ystrongly in L2(Q), as α→0+,where(y,h)is, together with some p, a state-control pair for (1.11)satisfying (1.7). The rest of this paper is organized as follows. In Section 2, we will recall some properties of the Stokes operator and we will prove some results concerning the existence, uniqueness and regularity of the solution to (1.3). Section 3deals with the proofs of Theorems 1.2 and 1.4. Section 4deals with the proofs of Theorems 1.3 and 1.5. Finally, in Section 5, we present some additional comments and open questions. 2. Preliminaries In this section, we will recall some properties of the Stokes operator. Then, we will prove that the Leray-α system is well-posed. Also, we will recall the Carleman inequalities and null controllability properties of the Oseen system. 2.1. The Stokes operator Let P:L2(Ω)→ Hbe the orthogonal projector, usually known as the Leray Projector. Recall that Pmaps Hs(Ω)intoHs(Ω)∩Hfor all s≥0. We will denote by Athe Stokes operator,i.e. the self-adjoint operator in Hformally given by A=−PΔ. For any u∈D(A):=V∩H2(Ω)andanyw∈H,theidentityAu =wholds if and only if (∇u,∇v)=(w,v)∀v∈V. It is well-known that A:D(A)→ Hcan be inverted and its inverse A−1is self-adjoint, compact and positive. Consequently, there exists a nondecreasing sequence of positive numbers λjand an associated orthonormal basis of H, denoted by (wj)∞ j=1, such that Awj=λjwj∀j≥1. Accordingly we can introduce the real powers of the Stokes operator. Thus, for any r∈R,weset D(Ar)=⎧ ⎨ ⎩u∈H:u= ∞  j=1 ujwj,with ∞  j=1 λ2r j|uj|2<+∞⎫ ⎬ ⎭ and Aru= ∞  j=1 λr jujwj,∀u= ∞  j=1 ujwj∈D(Ar). Let us present a result concerning the domains of the powers of the Stokes operator. Theorem 2.1. Let r∈Rbe given, with −1 2<r<1.Then D(Ar/2)=Hr(Ω)∩Hwhenever −1 2<r<1 2, D(Ar/2)=Hr 0(Ω)∩Hwhenever 1 2≤r≤1. Moreover, u→ (u,Aru)1/2is a Hilbertian norm in D(Ar/2), equivalent to the usual Sobolev Hr-norm. In other words, there exist constants c1(r),c 2(r)>0such that c1(r)uHr≤(u,Aru)1/2≤c2(r)uHr∀u∈D(Ar/2). 1186 F.D. ARARUNA ET AL. The proof of Theorem 2.1 can be found in [16]. Notice that, in view of the interpolation K-method of Lions and Peetre, we have D(Ar/2)=D((−Δ)r/2)∩H. Hence, thanks to an explicit description of D((−Δ)r/2), the stated result holds. Now, we are going to recall an important property of the semigroup of contractions e−tAgenerated by A, see [15]: Theorem 2.2. For any r>0,thereexistsC(r)>0such that Are−tAL(H;H)≤C(r)t−r∀t>0.(2.1) In order to prove (2.1), it suffices to observe that, for any u=+∞ j=1 ujwj∈H, one has Are−tAu= +∞  j=1 λr je−tλjujwj. Consequently, Are−tAu2= +∞  j=1 λr je−tλjuj2≤max λ∈Rλre−tλ2 u2 and, since max λ∈Rλre−tλ =(r/e)rt−r, we get easily (2.1). 2.2. Well-posedness for the Leray-αsystem Let us see that, for any α>0, under some reasonable conditions on fand y0,theLeray-αsystem (1.3) possesses a unique global weak solution. Before this, let us introduce σNgiven by σN=2ifN=2, 4/3ifN=3. Then, we have the following result: Proposition 2.3. Assume that α>0is fixed. Then, for any f∈L2(0,T;H−1(Ω)) and any y0∈H,there exists exactly one solution (yα,p α,zα,π α)to (1.3),with yα∈L2(0,T;V)∩C0([0,T]; H),(yα)t∈L2(0,T;V), zα∈L2(0,T;D(A3/2)) ∩C0([0,T]; D(A)).(2.2) Furthermore, the following estimates hold: yαL2(V)+yαC0([0,T];H)≤CB0(y0,f), (yα)tLσN(V)≤CB0(y0,f)(1 + B0(y0,f)), zα2 L∞(H)+2α2zα2 L∞(V)≤CB0(y0,f)2, 2α2zα2 L∞(V)+α4zα2 L∞(D(A)) ≤CB0(y0,f)2. (2.3) Here, Cis independent of αand we have introduced the notation B0(y0,f):=y0+fL2(H−1). UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL 1187 Proof. The proof follows classical and rather well-known arguments; see for instance [10,30]. For completeness, they will be recalled. •Existence: We will reduce the proof to the search of a fixed point of an appropriate mapping Λα3. Thus, for each y∈L2(0,T;H), let (z,π) be the unique solution to ⎧ ⎨ ⎩ z−α2Δz+∇π=yin Q, ∇·z=0 in Q, z=0on Σ. It is clear that z∈L2(0,T;D(A)) and then, thanks to the Sobolev embedding, we have z∈L2(0,T;L∞(Ω)). Moreover, the following estimates are satisfied: z2+2α2z2 L2(V)≤y2, 2α2z2 L2(V)+α4z2 L2(D(A)) ≤y2. From this z, we can obtain the unique solution (y,p) to the linear system of the Oseen kind ⎧ ⎪ ⎨ ⎪ ⎩ yt−Δy+(z·∇)y+∇p=fin Q, ∇·y=0 in Q, y=0on Σ, y(0) = y0in Ω. Since f∈L2(0,T;H−1(Ω)) and y0∈H, it is clear that y∈L2(0,T;V)∩C0([0,T]; H),yt∈L2(0,T;V) and the following estimates hold: yC0([0,T];H)+yL2(V)≤C1B0(y0,f), ytL2(V)≤C2(1 + zL2(D(A)))B0(y0,f)≤C2(1 + α−2y)B0(y0,f).(2.4) Now, we introduce the Banach space W={w∈L2(0,T;V):wt∈L2(0,T;V)}, the closed ball K={y∈L2(0,T;H):y≤C1√TB0(y0,f)} and the mapping ˜ Λα,with ˜ Λα(y)=y, for all y∈L2(0,T;H). Obviously ˜ Λαis well-defined and maps continuously the whole space L2(0,T;H)intoW∩K. Notice that any bounded set of Wis relatively compact in the space L2(0,T;H), in view of the classical results of the Aubin−Lions kind, see for instance [28]. Let us denote by Λαthe restriction to Kof ˜ Λα. Then, thanks to (2.4), Λαmaps Kinto itself. Moreover, it is clear that Λα:K→ Ksatisfies the hypotheses of Schauder’s Theorem. Consequently, Λαpossesses at least one fixed point in K. This immediately achieves the proof of the existence of a solution satisfying (2.2). The estimates (2.3)a,(2.3)cand (2.3)dare obvious. On the other hand, (yα)tLσN(V)≤CfL2(H−1)+yαL2(V)+(zα·∇)yαLσN(H−1) ≤CB0(y0,f)+zαLsN(L4)yαLsN(L4) ≤CB0(y0,f)+zαL∞(H)+zαL2(V)yαL∞(H)+yαL2(V) ≤CB0(y0,f)(1 + B0(y0,f)), 3Alternatively, we can prove the existence of a solution by introducing adequate Galerkin approximations and applying (classical) compactness arguments. 1188 F.D. ARARUNA ET AL. where sN=2σN. Here, the third inequality is a consequence of the continuous embedding L∞(0,T;H)∩L2(0,T;V)→LsN(0,T;L4(Ω)). This estimate completes the proof of (2.3). •Uniqueness: Let (yα,p α,zα,π α)and(y α,p α,z α,πα)betwosolutionsto(1.3) and let us introduce u:= yα−y α,q=pα−p α,m:= zα−z αand h=πα−π α.Then ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ut−Δu+(zα·∇)u+∇q=−(m·∇)y αin Q, m−α2Δm+∇h=uin Q, ∇·u=0,∇·m=0 in Q, u=m=0on Σ, u(0) = 0in Ω. Since u∈L∞(0,T;H), we have m∈L∞(0,T;D(A)) (where the estimate of this norm depends on α). Therefore, we easily deduce from the first equation of the previous system that 1 2 d dtu2+∇u2≤m∞∇y αu for all t.Sincem∞≤CmD(A)≤Cα−2u,weget 1 2 d dtu2+∇u2≤Cα−2∇y αu2. Therefore, in view of Gronwall’s Lemma, we see that u≡0. Accordingly, we also have m≡0and uniqueness holds.  We are now going to present some results concerning the existence and uniqueness of a strong solution. We start with a global result in the two-dimensional case. Proposition 2.4. Assume that N=2and α>0is fixed. Then, for any f∈L2(0,T;L2(Ω)) and any y0∈V, there exists exactly one solution (yα,p α,zα,π α)to (1.3),with yα∈L2(0,T;D(A)) ∩C0([0,T]; V),(yα)t∈L2(0,T;H), zα∈L2(0,T;D(A2)) ∩C0([0,T]; D(A3/2)).(2.5) Furthermore, the following estimates hold: (yα)t+yαC0([0,T ];V)+yαL2(D(A)) ≤B1(y0V,f), zα2 C0([0,T];V)+2α2zα2 C0([0,T];D(A)) ≤yα2 C0([0,T];V), (2.6) where we have introduced the notation B1(r, s):=(r+s)1+(r+s)2eC(r2+s2)2. Proof. First, thanks to Proposition 2.3, we see that there exists a unique weak solution (yα,p α,zα,π α)satisfying (2.2), and (2.3). In particular, zα∈L2(0,T;V) and we have zα(t)≤yα(t)and zα(t)V≤yα(t)V,∀t∈[0,T]. UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL 1189 As usual, we will just check that good estimates can be obtained for yα,(yα)tand zα. Thus, we assume that it is possible to multiply by −Δyαthe motion equation satisfied by yα. Taking into account that N=2,we obtain: 1 2 d dt∇yα2+Δyα2=−(f,Δyα)+((zα·∇)yα,Δyα) ≤f2+1 4Δyα2+zα1/2zα1/2 Vyα1/2 VΔyα3/2 ≤f2+1 2Δyα2+Czα2zα2 Vyα2 V. Therefore, d dt∇yα2+Δyα2≤Cf2+(yα2yα2 V)∇yα2. In view of Gronwall’s Lemma and the estimates in Proposition 2.3, we easily deduce (2.5)and(2.6).  Notice that, in this two-dimensional case, the strong estimates for yαin (2.6) are independent of α; obviously, we cannot expect the same when N=3. In the three-dimensional case, what we obtain is the following: Proposition 2.5. Assume that N=3and α>0is fixed. Then, for any f∈L2(0,T;L2(Ω)) and any y0∈V, there exists exactly one solution (yα,p α,zα,π α)to (1.3),with yα∈L2(0,T;D(A)) ∩C0([0,T]; V),(yα)t∈L2(0,T;H), zα∈L2(0,T;D(A2)) ∩C0([0,T]; D(A3/2)). Furthermore, the following estimates hold: yαL∞(V)+yαL2(D(A)) +(yα)t≤B2(y0V,f,α), zα2 L∞(V)+2α2zα2 L∞(D(A)) ≤yα2 L∞(V), (2.7) where we have introduced B2(r, s, α):=C(r+s)eCα−4(r+s)2. Proof. Thanks to Proposition 2.3, there exists a unique weak solution (yα,p α,zα,π α) satisfying (2.2)and(2.3). In particular, we obtain that zα∈L∞(Q), with zα∞≤C α2y0H+fL2(H−1). On the other hand, y0∈V. Hence, from the usual (parabolic) regularity results for Oseen systems, the solution to (1.3) is more regular, i.e. yα∈L2(0,T;D(A)) ∩C0([0,T]; V)and(yα)t∈L2(0,T;H). Moreover, yαverifies the first estimate in (2.7). This achieves the proof.  Let us now provide a result concerning three-dimensional strong solutions corresponding to small data, with estimates independent of α: Proposition 2.6. Assume that N=3.ThereexistsC0>0such that, for any α>0,anyf∈L∞(0,T;L2(Ω)) and any y0∈Vwith M:= max ∇y02,f2/3 L∞(L2)<1 2(1 + C0)T,(2.8) 1196 F.D. ARARUNA ET AL. Notice that, if Uis bounded set of Wthen it is relatively compact in the space L∞(0,T;D(Aσ)), in view of the classical results of the Aubin−Lions kind, see for instance [28]. Let us denote by Λαthe restriction to Kof ˜ Λα. Then, thanks to Lemma 3.1 and (2.17), if y0D(A)≤ε (independent of α!) Λαmaps Kinto itself. Moreover, it is clear that Λα:K→ Ksatisfies the hypotheses of Schauder’s Theorem. Indeed, this nonlinear mapping is continuous and compact (the latter is a consequence of the fact that, if Bis bounded in L∞(0,T;D(Aσ)), then ˜ Λα(B) is bounded in W). Consequently, Λαpossesses at least one fixed point in K, and this ends the proof of Theorem 1.2. ProofofTheorem1.4.Let vαbe a null control for (1.5) satisfying (1.8)andlet(yα,p α,zα,π α)bethestate associated to vα.From(1.8)andtheestimates(2.3) for the solutions yα,thereexistv∈L∞0,T;L2(ω)and y∈L∞(0,T;H)∩L2(0,T;V)withyt∈LσN(0,T;V) such that, at least for a subsequence vα→vweakly-in L∞0,T;L2(ω), yα→yweakly-in L∞(0,T;H) and weakly in L2(0,T;V), (yα)t→ytweakly in LσN(0,T;V). Since W:= {m∈L2(0,T;V): mt∈LσN(0,T;V)}is continuously and compactly embedded in L2(Q), we have that yα→yin L2(Q)anda.e. This is sufficient to pass to the limit in the equations satisfied by yα,vαand zα. We conclude that yis, together with some pressure p, a solution to the Navier−Stokes equations associated to a control vand satisfies (1.7).  4. The boundary case: Theorems 1.3 and 1.5 This section is devoted to prove the local boundary null controllability of (1.6) and the uniform controllability property in Theorem 1.5. ProofofTheorem1.3.Again, we will use a fixed point argument. Contrarily to the case of distributed controllability, we will have to work in a space ˜ Yof functions defined in an extended domain. Let * Ωbe given, with Ω⊂* Ωand ∂* Ω∩Γ=Γ\γsuch that ∂* Ωis of class C2(see Fig. 1). Let ω⊂* Ω\Ωbe a non-empty open subset and let us introduce * Q:= ˜ Ω×(0,T)and * Σ:= ∂* Ω×(0,T). The spaces and operators associate to the domain * Ωwill be denoted by * H,* V,* A,etc. Remark 4.1. In view of Lemma 2.7, for the Proof of Theorem 1.3 we just need to consider the case in which the initial state y0belongs to Vand possesses a sufficiently small norm in V.Indeed,weonlyhavetotake initially hα≡0and apply Lemma 2.7 to the solution to (1.6).  Let y0∈Vbe given and let us introduce the extension by zero ˜ y0of y0.Then˜ y0∈* V. We will use the following result, similar to Lemma 2.7, whose proof is postponed to the end of the section: Lemma 4.2. There exists a continuous function φ:R+→ R+satisfying φ(s)→0as s→0+with the following property: a) For any y0∈Vand any α>0, there exist times T0∈(0,T), controls hα∈L2(0,T 0;H1/2(Γ)) with γhα·ndΓ≡0, associated solutions (yα,p α,zα,π α)to (1.6)in Ω×(0,T 0)and arbitrarily small times t∗∈ (0,T/2) such that the yαcan be extended to * Ω×(0,T 0)and the extensions satisfy ˜ yα(t∗)2 D(* A)≤φ(y0V). b) The set of these t∗has positive measure. c) The controls hαare uniformly bounded, i.e. hαL∞(0,T0;H1/2(γ)) ≤C. UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL 1197 Figure 1. The domain * Ω. In view of Lemma 4.2, for the proof of Theorem 1.3, we just need to consider the case in which the initial state y0is such that its extension ˜ y0to ˜ Ωbelongs to D(˜ A) and possesses a sufficiently small norm in D(˜ A). We will prove that there exists (˜ yα,˜pα,zα,π α,˜ v), with ˜ v∈L∞(0,T;L2(ω)), satisfying ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˜ yt−Δ˜ y+( ˜ z·∇)˜ y+∇˜p=˜ v1ωin * Q, z−α2Δz+∇π=˜ yin Q, ∇·˜ y=0 in * Q, ∇·z=0 in Q, ˜ y=0on * Σ, z=˜ yon Σ, ˜ y(0) = ˜ y0in * Ω (4.1) and ˜ y(T)=0in * Ω,where˜ zis the extension by zero of z. Obviously, if this were the case, the restriction (y,p) of (˜ y,˜p)toQ, the couple (z,π) and the lateral trace h:= ˜ y|γ×(0,T)would satisfy (1.6)and(1.7). Let us fix σwith N/4<σ<β<1. Then, for each y∈L∞(0,T;D(˜ Aσ)), let w=w(x,t)andπ=π(x,t)be the unique solution to ⎧ ⎨ ⎩ w−α2Δw+∇π=α2Δyin Q, ∇·w=0 in Q, w=0on Σ. Since y∈L∞(0,T;D(˜ Aσ)), its restriction to Qbelongs to L∞(0,T;H2σ(Ω)). Then, Lemma 2.8 implies w∈L∞(0,T;D(Aσ)) and, thanks to Theorem 2.1,wealsohavew∈L∞(Q)and w2 L∞(0,T;D(Aσ)) ≤Cy2 L∞(0,T;D(˜ Aσ)), where Cis independent of α. Let ˜ wbe the extension by zero of wand let us set ˜ z:= y+˜ w. Let us consider the system (2.16)withh replaced by ˜ zand Ωreplaced by * Ω.InviewofTheorem2.9, we can associate to ˜ zthe null control ˜ vof minimal 1198 F.D. ARARUNA ET AL. norm in L∞(0,T;L2(˜ω)) and the corresponding solution (˜ y,˜p)to(2.16). Since ˜ y0∈D(˜ A), ˜ z∈L∞(* Q)and ˜ v∈L∞(0,T;L2(˜ω)), we have ˜ y∈L2(0,T;D(˜ A)) ∩C0([0,T]; ˜ V),˜ yt∈L2(0,T;˜ H) and the following estimate holds: ˜ yt2 L2(˜ H)+˜ y2 L2(D(˜ A)) +˜ y2 L∞(˜ V)≤C!˜ y02 ˜ V+˜ v2 L∞(L2(˜ω))"eC˜ z2 ∞.(4.2) Also, in account of Lemma 3.1, one has ˜ y∈L∞(0,T;D(˜ Aβ)) and ˜ yL∞(D(˜ Aβ)) ≤C(˜ y0D(˜ A)+˜ vL∞(L2(˜ω)))eCyL∞(0,T ;D(˜ Aσ))). Now, let us set W={m∈L∞(0,T;D(˜ Aβ)) : mt∈L2(0,T;˜ H)}, and let us consider the closed ball K={y∈L∞(0,T;D(˜ Aσ)) : yL∞(D(˜ Aσ)) ≤1} and the mapping ˜ Λα,with ˜ Λα(y)=˜ yfor all y∈L∞(0,T;D(˜ Aσ)). Obviously, ˜ Λαis well-defined and maps the whole space L∞(0,T;D(˜ Aσ)) into W. Furthermore, any bounded set U⊂Wthen it is relatively compact in L∞(0,T;D(˜ Aσ)). Let us denote by Λαthe restriction to Kof ˜ Λα. Thanks to Lemma 3.1 and (2.17), there exists ε>0 (independent of α) such that if ˜ y0D(˜ A)≤ε,Λαmaps Kinto itself and it is clear that Λα:K→ Ksatisfies the hypotheses of Schauder’s Theorem. Consequently, Λαpossesses at least one fixed point in Kand (4.1) possesses a solution.This ends the Proof of Theorem 1.3. ProofofTheorem1.5.The proof is easy, in view of the previous uniform estimates. It suffices to adapt the argument in the Proof of Theorem 1.4 and deduce the existence of subsequences that converge (in an appropriate sense)toasolutionto(1.11) satisfying (1.7). For brevity, we omit the details.  ProofofLemma4.2.For instance, let us only consider the case N= 3. We will reduce the proof to the search of a fixed point of another mapping Φα. For any y0∈V,anyT0∈(0,T)andanyy∈L4(0,T 0;* V)), let (w,π) be the unique solution to ⎧ ⎨ ⎩ w−α2Δw+∇π=α2Δyin Ω×(0,T 0), ∇·w=0 in Ω×(0,T 0), w=0on Γ×(0,T 0), let ˜ wbe the extension by zero of w,letusset˜ z:= y+˜ wand let us introduce the Oseen system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˜ yt−Δ˜ y+( ˜ z·∇)˜ y+∇˜p=0in * Ω×(0,T 0), ∇·˜ y=0 in * Ω×(0,T 0), ˜ y=0on ∂* Ω×(0,T 0), ˜ y(0) = ˜ y0in * Ω. It is clear that the restriction of yto Ω×(0,T 0) belongs to L4(0,T 0;H1(Ω)), whence we have from Lemma 2.8 that w∈L4(0,T 0;V)and wL4(0,T0;V)≤CyL4(0,T0;* V). UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL 1199 It is also clear that we can get estimates like those in the proof of Proposition 2.6 for * y.Inotherwords,for any y0∈V, we can find a sufficiently small T0>0 such that * y∈L2(0,T 0;D(* A)) ∩C0([0,T 0]; * V),* yt∈L2(0,T 0;* H) and * yL2(0,T0;D(* A)) +* yC0([0,T0]; * V)+* ytL2(0,T0;* H)≤C!T0,y0V,yL4(0,T0;* V)", where Cis nondecreasing with respect to all arguments and goes to zero as y0V→0. Now, let us introduce the mapping Φα:L4(0,T 0;* V)→ L4(0,T 0;* V), with Φα(y)=* yfor all y∈L4(0,T;* V). This is a continuous and compact mapping. Indeed, from well-known interpolation results, we have that the embedding L2(0,T 0;D(* A)) ∩L∞(0,T 0;* V)→L4(0,T 0;D(* A3/4)) is continuous and this shows that, if * yis bounded in L2(0,T 0;D(* A)) ∩C0([0,T 0]; * V)and* ytis bounded in L2(0,T 0;* H), then * ybelongs to a compact set of L4(0,T 0;* V). Then, as in the Proofs of Theorems 1.2 and 1.3, we immediately deduce that, whenever y0V≤δ(for some δindependent of α), Φαpossesses at least one fixed point. This shows that the nonlinear system (4.1)issolvable for ˜ v≡0andy0V≤δ. Now, the argument in the proof of Lemma 2.7 can be applied in this framework and, as a consequence, we easily deduce Lemma 4.2. 5. Additional comments and questions 5.1. Controllability problems for semi-Galerkin approximations Let {w1,w2,...}be a basis of the Hilbert space V. For instance, we can consider the orthogonal base formed by the eigenvectors of the Stokes operator A. Together with (1.5), we can consider the following semi-Galerkin approximated problems: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(zm·∇)y+∇p=v1ωin Q, (zm(t),w)+α2(∇zm(t),w)=(y(t),w)∀w∈Vm,zm(t)∈Vm,t∈(0,T), ∇·y=0,in Q, y=0on Σ, y(0) = y0in Ω, (5.1) where Vmdenotes the space spanned by w1,...,wm. Arguing as in the proof of Theorem 1.2, it is possible to prove a local null controllability result for (5.1). More precisely, for each m≥1, there exists εm>0 such that, if y0≤εm, we can find controls vmand associated states (ym,p m,zm) satisfying (1.7). Notice that, in view of the equivalence of norms in Vm, the fixed point argument can be applied in this case without any extra regularity assumption on y0; in other words, Lemma 2.7 is not needed here. On the other hand, it can also be checked that the maximal εmare bounded from below by some positive quantity independent of mand αand the controls vmcan be found uniformly bounded in L∞(0,T;L2(ω)). As a consequence, at least for a subsequence, the controls converge weakly-∗in that space to a null control for (1.5). However, it is unknown whether the problems (5.1)areglobally null-controllable; see below for other considerations concerning global controllability. 1200 F.D. ARARUNA ET AL. 5.2. Another strategy: Applying an inverse function theorem There is another way to prove the local null controllability of (1.5)thatreliesonLiusternik’s Inverse Function Theorem, see for instance [1]. This strategy has been introduced in [18] and has been applied successfully to the controllability of many semilinear and nonlinear PDE’s. In the framework of (1.5), the argument is as follows: 1. Introduce an appropriate Hilbert space Yof state-control pairs (yα,p α,zα,π α,vα) satisfying (1.7). 2. Introduce a second Hilbert space Zof right hand sides and initial data and a well-defined mapping F:Y→ Z such that the null controllability of (1.5) with state-controls in Yis equivalent to the solution of the nonlinear equation F(yα,p α,zα,π α,vα)=(0,y0),(yα,p α,zα,π α,vα)∈Y.(5.2) 3. Prove that Fis C1in a neighborhood of (0,0,0,0,0)andF(0,0,0,0,0)isonto. Arguing as in [13], all this can be accomplished satisfactorily. As a result, (5.2) can be solved for small initial data y0and the local null controllability of (1.5)holds. 5.3. On global controllability properties It is unknown whether a general global null controllability result holds for (1.5). This is not surprising, since the same question is also open for the Navier−Stokes system. What can be proved (as well as for the Navier−Stokes system) is the null controllability for large time: for any given y0∈H,thereexistsT∗=T∗(y0) such that (1.5) can be driven exactly to zero with controls vα uniformly bounded in L∞(0,T ∗;L2(ω)). Indeed, let εbe the constant furnished by Theorem 1.2 corresponding to the time T= 1 (for instance). Let us first take vα≡0. Then, since the solution to (1.3)withf=0satisfies yα(t)0, there exists T0(depending on y0but not on α) such that yα(T0)≤ε. Therefore, there exist controls v α∈L∞(T0,T 0+1;L2(ω)) such that the solution to (1.5) that starts from yα(T0)attimeT0satisfies yα(T0+1)=0. Hence, the assertion is fulfilled with T∗=T0+1and vα=0for 0 ≤t<T 0, v αfor T0≤t≤T∗. A similar argument leads to the null controllability of (1.5)forlargeα. In other words, it is also true that, for any given y0∈Hand T>0, there exists α∗=α∗(y0,T) such that, if α≥α∗,then(1.5)canbedriven exactly to zero at time T. 5.4. The Burgers-αsystem There exist similar results for a regularized version of the Burgers equation, more precisely the Burgers-α system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−yxx +zyx=v1(a,b)in (0,L)×(0,T), z−α2zxx =yin (0,L)×(0,T), y(0,t)=y(L, t)=z(0,t)=z(L, t)=0 on (0,T), y(x, 0) = y0(x)in(0,L). (5.3) These have been proved in [2]. This system can be viewed as a toy or preliminary model of (1.5). There are, however, several important differences between (1.5)and(5.3): •The solution to (5.3) satisfies a maximum principle that provides a useful L∞-estimate. •There is no apparent energy decay for the uncontrolled solutions. As a consequence, the large time null controllability of (5.3) is unknown. UNIFORM LOCAL NULL CONTROL OF THE LERAY-αMODEL 1201 •It is known that, in the limit α=0,i.e. for the Burgers equation, global null controllability does not hold; consequently, in general, the null controllability of (5.3) with controls bounded independently of αis impossible. We refer to [2] for further details. 5.5. Local exact controllability to the trajectories It makes sense to consider not only null controllability but also exact to the trajectories controllability problems for (1.5). More precisely, let ˆ y0∈Hbe given and let (ˆ y,ˆp, ˆ z,ˆπ) a sufficiently regular solution to (1.3)forf≡0 and y0=ˆ y0. Then the question is whether, for any given y0∈H, there exist controls vsuch that the associated states, i.e. the associated solutions to (1.5), satisfy y(T)=ˆ y(T)inΩ. The change of variables y=ˆ y+u,z=ˆ z+w, allows to rewrite this problem as the null controllability of a system similar, but not identical, to (1.5). It is thus reasonable to expect that a local result holds. 5.6. Controlling with few scalar controls The local null controllability with N−1 or even less scalar controls is also an interesting question. In view of the achievements in [3]and[9]fortheNavier−Stokes equations, it is reasonable to expect that results similar to Theorems 1.2 and 1.4 hold with controls vsuch that vi≡0forsomei; under some geometrical restrictions, it is also expectable that local exact controllability to the trajectories holds with controls of the same kind, see [14]. 5.7. Other related controllability problems There are many other interesting questions concerning the controllability of (1.5) and related systems. For instance, we can consider questions like those above for the Leray-αequations completed with other boundary conditions: Navier, Fourier or periodic conditions for yand z, conditions of different kinds on different parts of the boundary, etc. We can also consider Boussinesq-αsystems, i.e. systems of the form ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yt−Δy+(z·∇)y+∇p=θk+v1ωin Q, θt−Δθ +z·∇θ=w1ωin Q, z−α2Δz+∇π=yin Q, ∇·y=0,∇·z=0 in Q, y=z=0,θ=0 on Σ, y(0) = y0,θ(0) = θ0in Ω. Some of these results will be analyzed in a forthcoming paper. Acknowledgements. The authors thank J.L. Boldrini for the constructive conversations on the mathematical model. References [1] V.M. Alekseev, V.M. Tikhomirov and S.V. Fomin, Optimal control. Contemporary Soviet Mathematics. Translated from the Russian by V.M. Volosov. Consultants Bureau, New York (1987). [2] F.D. Araruna, E. Fern´andez-Cara and D.A. Souza, On the control of the Burgers-alpha model. Adv. Differ. Eq. 18 (2013) 935–954. 1202 F.D. 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