Global controllability of the bousiinesq system with Navier-slip-with-friction and Robin boundary conditions

\mathrm{S}\mathrm{I}\mathrm{A}\mathrm{M} \mathrm{J}. \mathrm{C}\mathrm{O}\mathrm{N}\mathrm{T}\mathrm{R}\mathrm{O}\mathrm{L} \mathrm{O}\mathrm{P}\mathrm{T}\mathrm{I}\mathrm{M}.©2023 \mathrm{S}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{t}\mathrm{y} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{I}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{A}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s} \mathrm{V}\mathrm{o}\mathrm{l}. 61, \mathrm{N}\mathrm{o}. 2, \mathrm{p}\mathrm{p}. 484--510 GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM WITH NAVIER-SLIP-WITH-FRICTION AND ROBIN BOUNDARY CONDITIONS* F. W. CHAVES-SILVA\dagger , E. FERN\' ANDEZ-CARA\ddagger , K. LE BALC'H\S , J. L. F. MACHADO\P ,AND D. A. SOUZA\| Abstract. In this paper, we deal with the global exact controllability to the trajectories of the Boussinesq system posed in 2D or 3D smooth bounded domains. The velocity field of the fluid must satisfy a Navier-slip-with-friction boundary condition, and a Robin boundary condition is imposed to the temperature. We assume that one can act on the velocity and the temperature on a small part of the boundary. For the proof, we first transform the boundary control problem into a distributed control problem. Then, we prove a global approximate controllability result by adapting the strategy of Coron, Marbach, and Sueur [J. Eur. Math. Soc. (JEMS), 22 (2020), pp. 1625--1673]; this relies on the controllability properties of the inviscid Boussinesq system and the analysis of appropriate asymptotic boundary layer expansions. Finally, we conclude with a local controllability result; as in many other cases, this can be established as a consequence of the null controllability of a linearized system through a fixed-point argument. Our contribution can be viewed as an extension of the results in [J. Eur. Math. Soc. (JEMS), 22 (2020), pp. 1625--1673], where thermal effects were not considered. Thus, we prove that the ideas behind the controllability properties of the Euler system and the well-prepared dissipation technique can be adapted to the present situation. Furthermore, we cover all the classical boundary conditions for the temperature, that is, those of the Robin, Neumann, and Dirichlet kinds. Key words. Boussinesq system, Navier-slip-with-friction boundary conditions, global controllability, boundary layers, global Carleman inequalities MSC codes. 35Q35, 76D55, 93B05, 93C10 DOI. 10.1137/21M1425566 1. Introduction. Let \Omega \subset \BbbR n(n= 2 or 3) be a smooth bounded domain with \Gamma := \partial \Omega , and let \Gamma c\subset \Gamma be a nonempty open subset which intersects all connected components of \Gamma . It will be said that \Gamma cis the control boundary. Let us set *Received by the editors June 9, 2021; accepted for publication (in revised form) September 16, 2022; published electronically March 24, 2023. https://doi.org/10.1137/21M1425566 Funding: The second and fifth authors were partially supported by grant PID2020-114976GBI00 funded by MCIN/AEI/10.13039/501100011033. The fifth author was partially supported by grant IJC2018--037863-I funded by MCIN/AEI/10.13039/501100011033. The first author was partially supported by CNPq-Brazil, by grant 2019/0014 of the Para\'{\i}ba State Research Foundation (FAPESQ), and the project CAPES-MathAmSud: ACIPDE. The third author was partially supported by the project TRECOS ANR-20-CE40-0009 funded by the ANR (2021--2024). The fourth author was partially supported by CNPq-Brazil. \dagger Department of Mathematics, Federal University of Para\'{\i}ba, UFPB, CEP 58050-085, Jo\~ao PessoaPB, Brazil (fchav[email protected]). \ddagger University of Sevilla, Dpto EDAN and IMUS, Aptdo 1160, 41080 Sevilla, Spain, and Department of Mathematics, Federal University of Para\'{\i}ba, UFPB, CEP 58050-085, Jo\~ao Pessoa-PB, Brazil ([email protected]). \S Inria, Sorbonne Universit\'e, Universit\'e de Paris, CNRS, Laboratoire Jacques-Louis Lions, Paris, France ([email protected]). \P Federal Institute of Cear\'a, IFCE, CEP 62320-000, Tiangu\'a, CE, Brazil (lucas.machado@ ifce.edu.br). \| University of Sevilla, Dpto. EDAN and IMUS, Aptdo 1160, 41080 Sevilla, Spain (desouza@ us.es). Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. 484 Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 485 Hc:= \{ u\in L2(\Omega )n: div u= 0 in \Omega , u \cdot \nu = 0 on \Gamma \setminus \Gamma c\} , where \nu =\nu (x) is the outward unit normal vector to \Omega at the points x\in \Gamma . Here, the equality u\cdot \nu = 0 on \Gamma \setminus \Gamma cmust be understood in the following sense: \langle u\cdot \nu ,g\rangle H - 1/2(\Gamma ),H1/2(\Gamma ) = 0 \forall g\in H1/2(\Gamma ) with g\equiv 0 on \Gamma c. For a given vector field f, we denote by [f]tan,D(f), and N(f) the tangential part of f, the deformation tensor, and the tangential Navier boundary operator, respectively, given as follows: [f]tan := f - (f\cdot \nu )\nu , D(f) := 1 2\bigl( \nabla f+\nabla ft\bigr) , N(f) := [D(f)\nu +Mf]tan. (1) Here and henceforth, it is assumed that M=M(t,x) is a smooth, symmetric matrixvalued function. It will be called the friction matrix and will be viewed as a measure of the boundary rugosity. We will also set R(\theta ) := \partial \theta \partial \nu +m\theta , where m=m(t,x) is another smooth function, again related to the properties of the boundary, known as the heat transfer coefficient. Let T > 0 be a final time. We will consider the (incomplete) Boussinesq system \left\{ \partial tu - \Delta u+ (u\cdot \nabla )u+\nabla p=\theta en,div u= 0 in (0,T)\times \Omega , \partial t\theta - \Delta \theta +u\cdot \nabla \theta = 0 in (0,T)\times \Omega , u\cdot \nu = 0, N(u) = 0, R(\theta ) = 0 on (0,T)\times (\Gamma \setminus \Gamma c), u(0,\cdot ) = u0, \theta (0,\cdot ) = \theta 0in \Omega , (2) where the functions u,\theta , and pmust be respectively viewed as the velocity field, the temperature, and the pressure of a viscous Newtonian fluid subject to thermal effects, and enis the nth vector of the canonical basis of \BbbR n. Regarded as a control system, we will interpret that the state is (u, \theta ) and the control is the lateral trace of (u,\theta ) on (0,T)\times \Gamma c. 1.1. Main result. Let us introduce the notation XT(\Omega ) := [C0 w([0,T];Hc)\cap L2(0,T;H1(\Omega )n)] \times [C0 w([0,T];L2(\Omega )) \cap L2(0,T;H1(\Omega ))]. Here, for any Banach space B,C0 w([0,T];B) denotes the space of weakly continuous B-valued functions, that is, the functions \phi : [0,T]\mapsto \rightarrow Bsuch that t\in [0,T]\rightarrow \langle \psi ,\phi (t)\rangle B\prime ,B is continuous for every \psi \in B\prime . We have the following result. Theorem 1.1. Let T > 0be a positive time, let (u0,\theta 0)\in Hc\times L2(\Omega ) be a given initial state, and let (u,\theta )\in XT(\Omega ) be a weak trajectory of (2). Then, there exists a controlled weak solution to (2) in XT(\Omega ) satisfying (u,\theta )(T,\cdot ) = \bigl( u,\theta \bigr) (T,\cdot ).(3) Remark 1.1. For the precise notions of weak trajectory and controlled weak solution, see Definition 2.1 below. Essentially, what we require of (\=u, \= \theta ) and (u,\theta ) is to belong to XT(\Omega ) and satisfy (2) in the weak (distributional) sense. Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 486 CHAVES-SILVA ET AL. Remark 1.2. In Theorem 1.1, we do not indicate explicitly which are the controls. As already said, once the controlled solution is found, the associated control is the lateral trace of the solution on (0,T)\times \Gamma c. Remark 1.3. Theorem 1.1 is stated as an existence result. The lack of uniqueness is for two main reasons. First, there exist many controls that drive the solution to (2) to the desired trajectory. Second, even if we select a criterion in order to fix the control without ambiguity, it is obviously unknown whether the associated state is unique in the 3D case (in two dimensions, it is known that the corresponding weak solution is unique; see, for instance, [2, 26] for the Navier--Stokes case). 1.2. Bibliographical comments. We now recall some existing results in the literature related to Theorem 1.1. There are several papers where the controllability properties of the Boussinesq equations are investigated. Most of them are local results covering boundary conditions of various kinds. For instance, in [14] the local exact boundary controllability to the trajectories was obtained with boundary controls acting over the whole boundary; in [15], the exact controllability with distributed controls and periodic boundary conditions was analyzed; in [19], the author proved the local exact controllability to the trajectories with Dirichlet boundary conditions; this situation is also handled with a reduced number of controls in [10, 17]. For nonviscous Boussinesq fluids, this subject has been investigated by Fern\'andez-Cara, Santos, and Souza [11]. On the other hand, the literature on the Navier--Stokes and Boussinesq equations with Navier-slip boundary conditions is scarce. Let us recall some controllability results obtained for the Navier--Stokes system: in [7], a small-time global result for the 2D equations has been proved where the exact controllability can be achieved in the interior of the spatial domain; the residual boundary layers are apparently too strong to be handled satisfactorily during the control design strategy. Guerrero proved in [18] the local exact controllability to the trajectories with general nonlinear Navier boundary conditions. Finally, the small-time global exact controllability with Navierslip-with-friction boundary conditions towards weak trajectories was proved in [8] by Coron, Marbach and Sueur; this article provides a positive answer to the famous open question by J.-L. Lions concerning global null controllability of the Navier--Stokes equations when the boundary conditions are of this kind. Recently, in [25], this result was extended from Leray weak controlled solutions to the case of smooth controlled solutions. For what concerns the Boussinesq system with Navier-slip boundary conditions, see [23, 29] for some local results. 1.3. Strategy of the proof and plan of the paper. Let us briefly indicate the main ideas and results needed for the proof of Theorem 1.1. In section 2, we will reduce the task to the solution of a distributed controllability problem by applying a classical domain extension technique. Then, we will limit our considerations to smooth initial data by using the smoothing effect of the uncontrolled Boussinesq system. In section 3, starting from sufficiently smooth initial data, we prove a global approximate controllability result by adapting the strategy introduced by Coron, Marbach, and Sueur in [8] in the Navier--Stokes case. In section 4, we prove a local controllability result. Here, we use an appropriate Carleman inequality for the adjoint of a linearized system (which leads to the null control of a related linearized system) and a fixed-point strategy. In section 5, we combine all these arguments and achieve the proof. Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 487 2. Domain extension and smoothing effect. 2.1. Domain extension. Let us consider a smooth extended bounded domain \scrO such that \Omega \cup \Gamma c\subset \scrO and \Gamma \setminus \Gamma c\subset \Gamma \scrO := \partial \scrO . In what follows, if there is no ambiguity, we will also denote by \nu (x) the outward unit normal vector to \scrO at the points x\in \partial \scrO . Let us introduce the following notations: \scrO T:= (0,T)\times \scrO and \Kappa T:= (0,T)\times \partial \scrO . In what follows, we will assume that Mand mare extended to [0,T]\times \partial \scrO as smooth functions in such a way that Mis symmetric on (0,T)\times \partial \scrO . This will allow us to speak of N(u) and R(\theta ) on \Kappa T. In general, the notation will be abridged. For instance, if u\in H2(\scrO )nand \theta \in H1(\scrO ), \| (u,\theta )\| H2\times H1will stand for the norm of (u,\theta ) in the space H2(\scrO )n\times H1(\scrO ). The scalar product and norm in L2spaces will be respectively denoted by (\cdot ,\cdot ) and \| \cdot \| . The symbol Cwill stand for a generic positive constant. We will need the space H:= \{ u\in L2(\scrO )n: div u= 0 in \scrO , u \cdot \nu = 0 on \partial \scrO \} . The following proposition enables us to extend the initial conditions to the whole domain \scrO : Proposition 2.1. Let (u0,\theta 0)\in Hc\times L2(\Omega ) be given. Then, there exist (u\ast , \theta \ast )\in L2(\scrO )n+1 and \sigma \ast \in C\infty (\scrO )with Supp \sigma \ast \subset \scrO \setminus \Omega such that u\ast =u0and \theta \ast =\theta 0in \Omega ,div u\ast =\sigma \ast in \scrO , u\ast \cdot \nu = 0 on \partial \scrO , \| u\ast \| +\| \sigma \ast \| \leq C\| u0\| and \| \theta \ast \| \leq C\| \theta 0\| . (4) Proof. Let \theta \ast \in L2(\scrO ) be the extension by zero of \theta 0to the whole domain \scrO . Then, we have \| \theta \ast \| \leq \| \theta 0\| . Next, in order to find an appropriate extension of u0, we first note that the space Hc:= \{ \phi \in C1(\Omega ;\BbbR n) : div \phi = 0 in \Omega , \phi \cdot \nu = 0 on \Gamma \setminus \Gamma c\} is dense in Hc. Let us put \Gamma c=\cup k i=1\Gamma i c,where the \Gamma i cdenote the intersections of \Gamma c with the connected components of \Gamma ,and let (\scrO \setminus \Omega )idenote the subset of \scrO \setminus \Omega for which \partial (\scrO \setminus \Omega )i\cap \partial \Omega = \Gamma i c. Also, let \omega i\subset \subset (\scrO \setminus \Omega )ibe a nonempty open subset, and let \sigma i \ast \in C\infty c(\omega i) be given with \int (\scrO \setminus \Omega )i \sigma i \ast = 1 for i= 1,...,k. Let us assume that u0\in Hc, and let (u0,m)m\geq 1be a sequence in Hcwith u0,m \rightarrow u0 in Hc. For every i\in \{ 1,...,k\} and m\geq 1, the following nonhomogeneous elliptic problem admits a unique solution wi m\in H1((\scrO \setminus \Omega )i): \left\{ - \Delta wi m= - ai m\sigma i \ast in (\scrO \setminus \Omega )i,\int (\scrO \setminus \Omega )i wi m= 0, \partial wi m \partial \nu =u0,m \cdot \nu on \Gamma i c, \partial wi m \partial \nu = 0 on \partial (\scrO \setminus \Omega )i\setminus \Gamma i c, Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 488 CHAVES-SILVA ET AL. where ai m:= \int \Gamma c(u0,m \cdot \nu )d\Gamma . It is clear that, for every i\in \{ 1,...,k\} ,ai mconverges to some ai\in \BbbR and wi mconverges to some wi\in H1((\scrO \setminus \Omega )i) as m\rightarrow +\infty . Let us set u\ast := \biggl\{ u0in \Omega , \nabla wiin (\scrO \setminus \Omega )ifor i= 1,...,k. It is then clear that u\ast \in L2(\scrO )n, div u\ast =\sigma \ast in \scrO for some \sigma \ast \in C\infty c(\scrO \setminus \Omega ), and u\ast \cdot \nu = 0 on \partial \scrO . On the other hand, we see that, by construction, (4) is satisfied. Let us introduce the following notation: WT(\scrO ) :=[C0 w([0,T];L2(\scrO )n)\cap L2(0,T;H1(\scrO )n)] \times [C0 w([0,T];L2(\scrO )) \cap L2(0,T;H1(\scrO ))]. The notion of solution used throughout the paper is the following. Definition 2.1. Let T > 0be a positive time, and let (u0,\theta 0)\in Hc\times L2(\Omega ) be given. It will be said that (u,\theta )\in XT(\Omega ) is a controlled weak trajectory of (2) with initial condition (u0,\theta 0)if (u,\theta )is the restriction to (0,T)\times \Omega of a weak Leray solution, still denoted by (u,\theta ), in the space WT(\scrO ), to the nonlinear system \left\{ \partial tu - \Delta u+ (u\cdot \nabla )u+\nabla p=\theta en+v, div u=\sigma in \scrO T, \partial t\theta - \Delta \theta +u\cdot \nabla \theta =win \scrO T, u\cdot \nu = 0, N(u) = 0, R(\theta ) = 0 on \Kappa T, u(0,\cdot ) = u\ast , \theta (0,\cdot ) = \theta \ast in \scrO , (5) where \bullet v\in C0([0,T];H1(\scrO )n)\cap H1(0,T;L2(\scrO )n),w\in C0([0,T]; H1(\scrO )) \cap H1(0,T;L2(\scrO )), and \sigma \in C\infty (\scrO T)are supported by (0,T)\times (\scrO \setminus \Omega ), and \bullet (u\ast ,\theta \ast )is an extension of (u0,\theta 0)furnished by Proposition 2.1, satisfying div u\ast =\sigma (0,\cdot ). Let us recall an existence result of weak solution to (5); it is taken from [27] (see Proposition 3.7 in that reference) and see also [4, Proposition 2.2]. Proposition 2.2. Let us assume that T > 0and v,\sigma ,w, and (u\ast ,\theta \ast )are as in Definition 2.1. Then there exists at least one weak Leray solution (u, \theta )to (5). 2.2. Smoothing effect of the uncontrolled Boussinesq system. The goal of this section is to show that, starting from L2initial data, at small time the solution is smooth. For convenience, this property will be stated as follows. Lemma 2.1. Let us assume that T > 0and (u,\theta )\in C\infty (\scrO T)n+1 is such that div u= 0 in \scrO Tand u\cdot \nu = 0 on \Kappa T. Then, there exists a smooth function \Psi T: \BbbR +\mapsto \rightarrow \BbbR +with \Psi T(0) = 0 such that, for any (r\ast ,q\ast )\in H\times L2(\scrO )and any weak Leray solution (r,q)\in WT(\scrO )to \left\{ \partial tr - \Delta r+ (r\cdot \nabla )r+ (u\cdot \nabla )r+ (r\cdot \nabla )u+\nabla \pi =qen,div r= 0 in \scrO T, \partial tq - \Delta q+ (r+u)\cdot \nabla q+r\cdot \nabla \theta = 0 in \scrO T, r\cdot \nu = 0, N(r) = 0, R(q) = 0 on \Kappa T, r(0,\cdot ) = r\ast , q(0,\cdot ) = q\ast in \scrO , (6) the following property holds: \exists t0\in [0,T]; \| (r,q)(t0,\cdot )\| H3\times H3\leq \Psi T(\| (r\ast ,q\ast )\| ). Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 489 The proof of this lemma is quite classical but, for completeness, will be given in Appendix A. 3. Approximate controllability problem. In this section, we prove an approximate controllability result starting from sufficiently smooth initial data. Proposition 3.1. Let us assume that T > 0, and let (u, \theta ),v,w, and \sigma be as in Definition 2.1. Suppose that (u, \theta )is, together with p, an associated solution, and assume that the triplet (u,p,\theta )belongs to C\infty (\scrO T;\BbbR n+2). Let (u\ast , \theta \ast )\in [H3(\scrO )n\cap H]\times H3(\scrO )be an initial state. Then, for any \delta > 0, there exist regular controls v,w, and \sigma , again supported in \scrO \setminus \Omega , and an associated weak solution to (5) satisfying \| (u,\theta )(T,\cdot ) - (u,\theta )(T, \cdot )\| \leq \delta . For the proof, we will follow the strategy introduced by Coron, Marbach, and Sueur in [8]. Let us explain how it works. First, a change of scale associated to a small parameter \varepsilon > 0 is introduced and (5) is transformed into a Boussinesq system with small viscosity \varepsilon that must be controlled in the (long) time interval [0,T/\varepsilon ], starting from a small initial state; see (7). The advantage of this scaling is that we can benefit from the nonlinear terms (u\cdot \nabla )uand u\cdot \nabla \theta . Formally, by taking \varepsilon = 0, we obtain the inviscid Boussinesq system; see (11). For this hyperbolic system, we can construct a very particular nontrivial trajectory that connects (0,0) \in \BbbR n+1 to itself and sends any particle outside the physical domain before the final time T. By linearizing the inviscid Boussinesq system around the previous trajectory, we obtain a new hyperbolic linear system that is small-time globally null-controllable (actually, what we do is apply the so-called return method, due to Coron; see [5]; note that the linearization around the trivial state leads to a noncontrollable system). In the particular case of a ``special slip"" boundary condition for the velocity and a Neumann boundary condition for the temperature, that is, with Msuch that [\nabla \times u]tan = 0 on \Kappa Tand m\equiv 0, we immediately conclude by estimating the remainder terms. We do not need to use the long interval time [0,T/\varepsilon ] to control in this case, since the solution is already small at intermediate times T\ast \in (0,T/\varepsilon ). Unfortunately, in the general case, a boundary layer appears. This phenomenon was already taken into account in [12, 22] for the Navier--Stokes PDEs. Thus, we have to introduce some corrector terms in the asymptotic expansion of the solution depending on \varepsilon , in order to estimate the residual layers. It is found that the boundary layer decays but not enough. Hence, the corrector is not sufficiently small at the final time T/\varepsilon and we still cannot conclude. In order to overcome this difficulty, we adapt the well-prepared dissipation method, introduced by Marbach in [28]. Here, the idea is to design a control strategy that reinforces the action of the natural dissipation of the boundary layer after an intermediate time. A desired small state is obtained at final time, and we can finally achieve the proof. In what follows, we will frequently need vector functions (u, p,\theta ,v, w, \sigma ) representing adequate states (u,p,\theta ), controls (v,w), and auxiliary functions \sigma , corresponding to some linear or nonlinear systems. In all cases, it will be implicitly assumed that v, w, and \sigma vanish outside [0,T ]\times (\scrO \setminus \Omega ). 3.1. Time scaling. Let us introduce \biggl\{ u\varepsilon (t,x) := \varepsilon u(\varepsilon t,x), p\varepsilon (t,x) := \varepsilon 2p(\varepsilon t, x), \theta \varepsilon (t, x) := \varepsilon 2\theta (\varepsilon t,x), v\varepsilon (t,x) := \varepsilon 2v(\varepsilon t,x), w\varepsilon (t,x) := \varepsilon 3w(\varepsilon t,x), \sigma \varepsilon (t,x) := \varepsilon \sigma (\varepsilon t,x). Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 490 CHAVES-SILVA ET AL. In these new variables, (5) reads as (7) \left\{ \partial tu\varepsilon - \varepsilon \Delta u\varepsilon + (u\varepsilon \cdot \nabla )u\varepsilon +\nabla p\varepsilon =\theta \varepsilon en+v\varepsilon ,div u\varepsilon =\sigma \varepsilon in (0,T/\varepsilon )\times \scrO , \partial t\theta \varepsilon - \varepsilon \Delta \theta \varepsilon +u\varepsilon \cdot \nabla \theta \varepsilon =w\varepsilon in (0,T/\varepsilon )\times \scrO , u\varepsilon \cdot \nu = 0, N(u\varepsilon ) = 0, R(\theta \varepsilon ) = 0 on (0,T/\varepsilon )\times \partial \scrO , u\varepsilon (0,\cdot ) = \varepsilon u\ast , \theta \varepsilon (0,\cdot ) = \varepsilon 2\theta \ast in \scrO . Thus, we work along a large time interval [0,T/\varepsilon ], starting from the small initial data (\varepsilon u\ast ,\varepsilon 2\theta \ast ). The counterpart is the small viscosity. Accordingly, (7) must be viewed as a singular perturbation of a nonlinear inviscid system. In order to prove Proposition 3.1, it is sufficient to check that \| u\varepsilon (T/\varepsilon ,\cdot ) - \varepsilon u(T, \cdot )\| =o(\varepsilon ) and \bigm\| \bigm\| \theta \varepsilon (T/\varepsilon ,\cdot ) - \varepsilon 2\theta (T, \cdot )\bigm\| \bigm\| =o(\varepsilon 2). 3.2. A special slip boundary condition. In this section, we consider a special situation where the fluid perfectly slips and the proof of Proposition 3.1 is much simpler (there is no boundary layer). For the moment, we will assume that the target trajectory is zero, i.e., (u, p,\theta ,v,w, \sigma )\equiv 0, and we will try to control (7) during the time interval [0,T] instead of [0, T/\varepsilon ]. The goal is to prove \| u\varepsilon (T,\cdot )\| =o(\varepsilon ) and \| \theta \varepsilon (T,\cdot )\| =o(\varepsilon 2).(8) Thus, let us assume that the friction coefficient Mis the Weingarten map (or shape operator) Mw. Thanks to [8, Lemma 1], on the uncontrolled boundary one has zero normal velocity and zero tangential vorticity, that is, u\cdot \nu = 0 and [\nabla \times u]tan = 0 on \Kappa T.(9) 3.2.1. Ansatz with no correction term. Let us introduce an asymptotic expansion of the solution to (7): \biggl\{ u\varepsilon =u0+\varepsilon u1+\varepsilon r\varepsilon , p\varepsilon =p0+\varepsilon p1+\varepsilon \pi \varepsilon , \theta \varepsilon =\theta 0+\varepsilon 2\theta 1+\varepsilon 2q\varepsilon , v\varepsilon =v0+\varepsilon v1, w\varepsilon =w0+\varepsilon 2w1, \sigma \varepsilon =\sigma 0. (10) There is some intuition behind (10). The first term (u0,p0,\theta 0,v0,w0,\sigma 0) is the solution to an inviscid system; take \varepsilon = 0 in (7). It models a reference trajectory around which we linearize the original system, exactly as is done when applying Coron's return method; see [5]. It will be chosen in such a way that the associated flow flushes the particles out of the physical domain before t=T; see (13) below for a more precise explanation. The second term (u1, p1,\theta 1,v1,w1) takes into account the initial data (u\ast ,\theta \ast ) and will be controlled to zero in the physical domain \Omega ; see Lemma 3.2 below. Then, (r\varepsilon , \pi \varepsilon ,q\varepsilon ) contains higher order terms; see (19). At the end, using (10), what we have to prove is that \| (r\varepsilon ,q\varepsilon )(T, \cdot )\| =o(1), in order to be able to conclude (8). 3.2.2. Inviscid flow. By taking \varepsilon = 0 in (7), we obtain the following system: \left\{ \partial tu0+ (u0\cdot \nabla )u0+\nabla p0=\theta 0en+v0,div u0=\sigma 0in \scrO T, \partial t\theta 0+u0\cdot \nabla \theta 0=w0in \scrO T, u0\cdot \nu = 0 on \Kappa T, u0(0,\cdot ) = u0(T,\cdot ) = 0, \theta 0(0,\cdot ) = \theta 0(T,\cdot ) = 0 in \scrO , (11) where v0,w0, and \sigma 0are spatially supported in \scrO \setminus \Omega . Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 491 Let us introduce the flow function \Phi 0:= \Phi 0(s;t,x) associated to u0. That is, for any (t,x), \Phi 0(\cdot ;t,x) solves the ODE problem \biggl\{ \partial s\Phi 0(s;t,x) = u0(s,\Phi 0(s;t,x)), \Phi 0(s;t,x)\bigm| \bigm| s=t=x. (12) Then, we look for trajectories such that \forall x\in \scrO ,\exists tx\in (0,T) such that \Phi 0(tx;0,x)\in \scrO \setminus \Omega .(13) This property is obvious for the points xalready located in \scrO \setminus \Omega . For the points x\in \Omega , we use the following result, whose proof can be found in [6, 8] in the 2D case and [8, 16] in the 3D case: Lemma 3.1. There exists a nonzero solution to (11) (u0,p0,\theta 0,v0,w0,\sigma 0)\in C\infty ([0,T]\times \scrO ;\BbbR 2n+4)such that the associated flow \Phi 0, defined in (12), satisfies (13). Moreover, we can choose u0,\theta 0, and w0such that \theta 0=w0= 0 and \nabla \times u0= 0 in [0, T]\times \scrO (14) and u0,p0,v0,and \sigma 0are compactly supported in time in (0,T). Note that, in the proof of this result, the assumption that \Gamma cintersects all connected components of \Gamma must be used. In what follows, when needed, it will be assumed that (u0,p0,\theta 0,v0,w0, \sigma 0) has been extended by zero after time T. 3.2.3. Flushing. In accordance with Lemma 3.1, we take \theta 0=w0= 0 in (10). Let (u1,\theta 1) be the solution to the linear problem \left\{ \partial tu1+ (u0\cdot \nabla )u1+ (u1\cdot \nabla )u0+\nabla p1= \Delta u0+v1,div u1= 0 in \scrO T, \partial t\theta 1+u0\cdot \nabla \theta 1=w1in \scrO T, u1\cdot \nu = 0 on \Kappa T, u1(0,\cdot ) = u\ast , \theta 1(0,\cdot ) = \theta \ast in \scrO , (15) where v1and w1are forcing terms, spatially supported in \scrO \setminus \Omega . Thanks to (14), we have \Delta u0=\nabla ( div u0) + \nabla \times (\nabla \times u0) = \nabla \sigma 0. Thus, this term can be absorbed by v1. Of course, (15) is a linear uncoupled system. Lemma 3.2. Let us assume that (u\ast ,\theta \ast )\in [H3(\scrO )n\cap H]\times H3(\scrO ). There exist forcing terms v1\in C1([0,T];H1(\scrO )n)\cap C0([0,T];H2(\scrO )n),(16) w1\in C1([0,T];H2(\scrO )) \cap C0([0,T];H3(\scrO )) with Supp (v1,w1)\subset \subset [0,T]\times \scrO \setminus \Omega ,(17) such that the associated solution (u1, \theta 1)to (15) satisfies (u1,\theta 1)(T,\cdot ) = (0,0) in \Omega . Moreover, u1belongs to C0([0,T];H2(\scrO )n)\cap L\infty (0,T;H3(\scrO )n)and a similar property holds for \theta 1. Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 492 CHAVES-SILVA ET AL. Proof. First, note that the result for u1is proved in [8, Lemma 3]. Second, for \theta 1 we have a similar situation and we can apply the same arguments. For completeness, let us sketch the main ideas. We will use the smooth partition of unity \{ \eta \ell : 1 \leq \ell \leq L\} defined in [8, Appendix A], which is related to \Phi 0as follows: thanks to (13), we can find \gamma > 0 and open balls B\ell for 1 \leq \ell \leq Lcovering \scrO with the following property: \biggl\{ \forall \ell , \exists t\ell \in (\gamma ,T - \gamma ),\exists m\ell \in \{ 1,...,\scrM \} such that \Phi 0(s;0,B\ell )\subset Qm\ell \forall s\in (t\ell - \gamma ,t\ell +\gamma ), (18) where the Qm\ell are squares (or cubes) that never intersect \Omega that cover a compact set Kin \scrO such that K\cap \Omega = \emptyset and \forall x\in \scrO ,\exists tx\in (0,T) such that \Phi 0(tx;0,x)\in K and \scrM \in \BbbN is the number of such a cubes; hence, every ball spends a positive amount of time within a given square (cube) where we can use a localized control to act on the \theta 1. Here, it is assumed that the \eta \ell satisfy 0 \leq \eta \ell \leq 1, \sum \eta \ell \equiv 1 and Supp (\eta \ell )\subset B\ell . Let us introduce a smooth function \varrho :\BbbR \mapsto \rightarrow [0,1], with \varrho = 1 on ( - \infty , - \gamma ) and \varrho = 0 on (\gamma ,+\infty ). For each \ell , consider the solution \theta \ell to \biggl\{ \partial t\theta \ell +u0\cdot \nabla \theta \ell = 0 in (0,T)\times \scrO , \theta \ell (0,\cdot ) = \eta \ell \theta \ast in \scrO , and set \theta \ell (t,x) := \varrho (t - t\ell )\theta \ell (t,x). Since \varrho (T - t\ell ) = 0 and \varrho ( - t\ell ) = 1, \theta \ell solves the linear problem \biggl\{ \partial t\theta \ell +u0\cdot \nabla \theta \ell =w\ell in (0,T)\times \scrO , \theta \ell (0,\cdot ) = \eta \ell \theta \ast , \theta \ell (T,\cdot ) = 0 in \scrO , where w\ell (t,x) := \varrho t(t - t\ell )\theta \ell . Since \varrho tvanishes outside ( - \gamma ,\gamma ), one has (18), and \eta \ell is compactly supported in B\ell , it is easy to see that w\ell is supported in [0,T]\times Qm\ell . At this point, we take \theta 1:= \sum \ell \theta \ell and w1:= \sum \ell w\ell and we see that the second PDE and the second initial condition in (15) are satisfied. Thanks to this explicit construction, the spatial regularity of w1and \theta \ell are the same. Therefore, w1\in C1([0,T],H2(\scrO ))\cap C0([0,T], H3(\scrO )). The fact that \theta 1belongs to C0([0,T];H2(\scrO ))\cap L\infty (0,T;H3(\scrO )) readily comes from the fact that the \theta \ell satisfy the same. This ends the proof. Lemma 3.2 is a null controllability result. Thanks to the linearity and reversibility of (15), it leads to an exact controllability result: Lemma 3.3. Let us assume that (u\ast ,\theta \ast ),(uT, \theta T)\in [H3(\scrO )n\cap H]\times H3(\scrO ). Then, there exist v1and w1as in (16) and (17) such that the associated solution to (15) satisfies (u1,\theta 1)(T,\cdot ) = (uT,\theta T). Moreover, u1belongs to C0([0,T ];H2(\scrO )n)\cap L\infty (0,T;H3(\scrO )n)and a similar property holds for \theta 1. 3.2.4. Equations and estimates for the remainder. The equations for r\varepsilon , \pi \varepsilon , and q\varepsilon in the extended domain \scrO Tare (19) \left\{ \partial tr\varepsilon - \varepsilon \Delta r\varepsilon + (u\varepsilon \cdot \nabla )r\varepsilon +\nabla \pi \varepsilon =f\varepsilon - A\varepsilon r\varepsilon +\varepsilon q\varepsilon en+\varepsilon \theta 1en,div r\varepsilon = 0 in \scrO T, \partial tq\varepsilon - \varepsilon \Delta q\varepsilon +u\varepsilon \cdot \nabla q\varepsilon =h\varepsilon - B\varepsilon r\varepsilon in \scrO T, r\varepsilon \cdot \nu = 0,[\nabla \times r\varepsilon ]tan = - [\nabla \times u1]tan, R(q\varepsilon ) = - R(\theta 1) on \Kappa T, r\varepsilon (0,\cdot ) = 0, q\varepsilon (0,\cdot ) = 0 in \scrO , Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 499 obviously u1,\varepsilon and \theta 1,\varepsilon depend on \varepsilon . However, since the reference trajectory is of class C\infty , all the required estimates can be made independent of \varepsilon . In a second step, for large times t\geq T, we modify the expansions and set \biggl\{ u\varepsilon =\surd \varepsilon \{ \rho \} +\varepsilon u(\varepsilon t, \cdot ) + \varepsilon \nabla \zeta \varepsilon +\varepsilon \{ \beta \} +\varepsilon r\varepsilon , p\varepsilon =\varepsilon 2p(\varepsilon t, \cdot ) + \varepsilon \mu \varepsilon +\varepsilon \pi \varepsilon , \theta \varepsilon =\varepsilon 2\theta (\varepsilon t, \cdot ) + \varepsilon 2q\varepsilon , v\varepsilon =\surd \varepsilon v\rho +\varepsilon 2v, w\varepsilon =\varepsilon 3w . (38) Note that, for t\geq T, we have u0= 0 and (u1,\theta 1) is the ``main"" trajectory. Changing (37) by (38) allows us to get rid of some terms in the equations satisfied by the remainder. Indeed, terms such as \varepsilon \Delta u1,\varepsilon (u1\cdot \nabla )u1,\varepsilon u1\cdot \nabla \theta 1, and \varepsilon \Delta \theta 1will no longer appear in (30) and (31) because they are already taken into account by (u,\theta ). Actually, despite the presence of the profile (u1,\theta 1) in both steps, the estimates obtained for the remainder profile are as in section 3.3.4. Let us introduce u(\varepsilon )(t,x) := 1 \varepsilon u\varepsilon \biggl( t \varepsilon ,x\biggr) and \theta (\varepsilon )(t,x) := 1 \varepsilon 2\theta \varepsilon \biggl( t \varepsilon ,x\biggr) . Then, thanks to (26), (27), and (36), we see that \bigm\| \bigm\| \bigm\| u(\varepsilon )(T,\cdot ) - u(T,\cdot )\bigm\| \bigm\| \bigm\| =\bigm\| \bigm\| \bigm\| \varepsilon - 1/2\{ \rho \} (T/\varepsilon ,\cdot ) + \nabla \zeta \varepsilon (T/\varepsilon ,\cdot ) + \{ \beta \} (T/\varepsilon , \cdot ) + r\varepsilon (T/\varepsilon ,\cdot )\bigm\| \bigm\| \bigm\| \leq \varepsilon - 1/2\| \{ \rho \} (T/\varepsilon ,\cdot )\| +\varepsilon 1/4\| \beta (T/\varepsilon ,\cdot ,\cdot )\| H2 x(\scrO ;H0,0 z(\BbbR +)) +\| \rho (T/\varepsilon ,\cdot ,\cdot )\| H1 x(\scrO ;H0,1 z(\BbbR +)) +\| \{ \beta \} (T/\varepsilon ,\cdot )\| +\| r\varepsilon (T/\varepsilon ,\cdot )\| \leq \varepsilon - 1/2\| \rho (T/\varepsilon ,\cdot ,\cdot )\| H0 x(\scrO ;H0,0 z(\BbbR +)) +\varepsilon 1/4\| \rho (T/\varepsilon ,\cdot ,\cdot )\| H3 x(\scrO ;H1,2 z(\BbbR +)) +\| \rho (T/\varepsilon ,\cdot ,\cdot )\| H1 x(\scrO ;H0,1 z(\BbbR +)) +\| \rho (T/\varepsilon ,\cdot ,\cdot )\| H1 x(\scrO ;H1,2 z(\BbbR +)) +O(\varepsilon 1/8). We can use (23) to estimate the terms containing \rho in the estimates above. First, recall that there exists a positive constant C > 0 such that (log s)/s \leq Cs - 1/2for every s\geq 1. Then, by taking \varepsilon sufficiently small, the following is found for k\geq 2: \varepsilon - 1 2\| \rho (T/\varepsilon ,\cdot ,\cdot )\| H0 x(\scrO ;H0,0 z(\BbbR +)) \leq C\varepsilon - 1/2\bigm| \bigm| \bigm| \bigm| log(2 + T/\varepsilon ) 2 + T/\varepsilon \bigm| \bigm| \bigm| \bigm| 1/4+k/2 \leq C\varepsilon - 3/8+k/4, \varepsilon 1 4\| \rho (T/\varepsilon ,\cdot ,\cdot )\| H3 x(\scrO ;H1,2 z(\BbbR +)) \leq C\varepsilon 1/4\bigm| \bigm| \bigm| \bigm| log(2 + T/\varepsilon ) 2 + T/\varepsilon \bigm| \bigm| \bigm| \bigm| - 3/4+k/2 \leq C\varepsilon - 1/8+k/4, \| \rho (T/\varepsilon ,\cdot ,\cdot )\| H1 x(\scrO ;H0,1 z(\BbbR +)) \leq C\bigm| \bigm| \bigm| \bigm| log(2 + T/\varepsilon ) 2 + T/\varepsilon \bigm| \bigm| \bigm| \bigm| - 1/4+k/2 \leq C\varepsilon - 1/8+k/4, | \rho (T/\varepsilon ,\cdot ,\cdot )| H1 x(\scrO ;H1,2 z(\BbbR +)) \leq C\bigm| \bigm| \bigm| \bigm| log(2 + T/\varepsilon ) 2 + T/\varepsilon \bigm| \bigm| \bigm| \bigm| - 3/4+k/2 \leq C\varepsilon - 3/8+k/4. Finally, we choose klarge enough, we conclude that \| u(\varepsilon )(T,\cdot ) - u(T,\cdot )\| = O(\varepsilon 1/8), and, from (36), we have \| \theta (\varepsilon )(T,\cdot ) - \theta (T,\cdot )\| =\| q\varepsilon (T/\varepsilon , \cdot )\| =O(\varepsilon 1/8). This concludes the proof of Proposition 3.1. 4. Local controllability of the Boussinesq system. The results in this section are relatively well known. For clarity, they will be specified and their proof will be sketched to some extent. Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 500 CHAVES-SILVA ET AL. Let \omega cand \omega be two nonempty open sets such that \omega c\subset \subset \omega \subset \subset \scrO \setminus \Omega , and let \chi \omega be a cut-off function such that \chi \omega = 0 outside \omega and \chi \omega = 1 in \omega c. The goal of this section is to prove the local exact controllability to the trajectories of the following Boussinesq system with distributed controls: \left\{ \partial tu - \Delta u+ (u\cdot \nabla )u+\nabla p=\theta en+v\chi \omega ,div u= 0 in \scrO T, \partial t\theta - \Delta \theta +u\cdot \nabla \theta =w\chi \omega in \scrO T, u\cdot \nu = 0, N(u) = 0, R(\theta ) = 0 on \Kappa T, u(0,\cdot ) = u\ast , \theta (0,\cdot ) = \theta \ast in \scrO . (39) Since (39) is nonlinear, we first begin by proving a (global) null controllability result for the following system: (40) \left\{ \partial tz - \Delta z+ ((a+b)\cdot \nabla )z+ (z\cdot \nabla )b+\nabla q=hen+v\chi \omega ,div z= 0 in \scrO T, \partial th - \Delta h+ (a+b)\cdot \nabla h+z\cdot \nabla c=w\chi \omega in \scrO T, z\cdot \nu = 0, N(z) = 0, R(h) = 0 on \Kappa T, z(0,\cdot ) = z\ast , h(0,\cdot ) = h\ast in \scrO , where the vector fields a,b, and Mand the scalar functions cand msatisfy the following assumptions: (41) (a,b,c)\in L\infty (0,T;H\times H\times L2(\scrO )) \cap L\infty (\scrO T)2n+1,(at,bt, ct)\in L2(0,T ;Lr(\scrO )2n+1), M\in E:= H1 - \ell (0,T;W\vargamma 1,\vargamma 1+1(\partial \scrO )n\times n)\cap H(3 - \ell )/2(0,T ;H\vargamma 2(\partial \scrO )n\times n), m\in F:= H1 - \ell (0,T;W\vargamma 1,\vargamma 1+1(\partial \scrO )) \cap H(3 - \ell )/2(0,T ;H\vargamma 2(\partial \scrO )), where \ell \in (0,1/2) is arbitrarily close to 1/2, r= 2n,\vargamma 2= (1/2)(3 - n) + (1 - \ell )(n - 2), and \vargamma 1>1 (arbitrarily small) if n= 3 and \vargamma 1= 1 if n= 2. From well-known Sobolev embeddings, we deduce at once that E \lhook \rightarrow L\infty ((0,T)\times \partial \scrO )n\times nand F \lhook \rightarrow L\infty ((0,T)\times \partial \scrO ). It is well known that the null controllability of (40) is equivalent to the observability of the adjoint system \left\{ - \partial t\phi - \Delta \phi - (a\cdot \nabla )\phi - D(\phi )b+\nabla \pi =c\nabla \psi , div \phi = 0 in \scrO T, - \partial t\psi - \Delta \psi - (a+b)\cdot \nabla \psi =\phi \cdot enin \scrO T, \phi \cdot \nu = 0, N(\phi ) = 0, R(\psi ) = 0 on \Kappa T, \phi (T,\cdot ) = \phi \ast , \psi (T,\cdot ) = \psi \ast in \scrO . (42) The desired observability inequality will be a consequence of a global Carleman inequality for (42); see Proposition 4.1 below. 4.1. Carleman estimates. Before stating the required inequalities, let us introduce several classical weights in the study of Carleman estimates for parabolic equations; see [13]. The basic weight will be a function \eta 0\in C2(\scrO ) verifying \eta 0>0 in \scrO , \eta 0\equiv 0 on \partial \scrO ,| \nabla \eta 0| >0 in \scrO \setminus \omega \prime , where \omega \prime \subset \subset \omega cis a nonempty open set. The existence of \eta 0is proved in [13]. Thus, for any \lambda > 0 we set \alpha (x,t) = e2\lambda \| \eta 0\| \infty - e\lambda \eta 0(x) t4(T - t)4, \alpha \ast (t) = max x\in \scrO \alpha (x,t),\widehat \alpha (t) = min x\in \scrO \alpha (x,t), \xi (x,t) = e\lambda \eta 0(x) t4(T - t)4, \xi \ast (t) = min x\in \scrO \xi (x,t),\widehat \xi (t) = max x\in \scrO \xi (x,t). Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 501 We also introduce the following notation: I(s,\lambda ;\phi ) = \int \int \scrO T e - 2s\alpha \bigl[ s3\lambda 4\xi 3| \phi | 2+s\lambda 2\xi | \nabla \phi | 2+s - 1\xi - 1(| \phi t| 2+| \Delta \phi | 2)\bigr] dxdt, where sand \lambda are positive real numbers and \phi =\phi (t,x). The following Carleman inequality holds: Proposition 4.1. Assume that the assumptions (41) are fulfilled. There exist positive constants \widetilde \lambda ,\widetilde s, and C=C(\scrO , \omega c)such that, for any (\phi \ast ,\psi \ast )\in H\times L2(\scrO ), the corresponding solution to (42) verifies (43) I(s,\lambda ;\phi ) + I(s,\lambda ;\psi )\leq C(1 + T2)s15/2\lambda 8\int \int (0,T )\times \omega c e - 4s\^\alpha +2s\alpha \ast \^ \xi 15/2(| \phi | 2+| \psi | 2)dxdt for all \lambda \geq \widetilde \lambda and s\geq \widetilde s. Furthermore, \widetilde \lambda and \widetilde shave the form \widetilde \lambda =\widetilde \lambda 0e \widetilde \lambda 1Tand \widetilde s= \widetilde s0e\lambda \widetilde s1(T4+T8), where \widetilde \lambda 0,\widetilde \lambda 1, and \widetilde s0only depend on \| a\| \infty ,\| b\| \infty ,\| c\| \infty ,\| at\| L2(Lr), \| bt\| L2(Lr),\| ct\| L2(Lr), and \| M\| E, and \| m\| Fand \widetilde s1only depend on \scrO and \omega c. The proof of Proposition 4.1 consists of three steps: (i) global Carleman estimates for \phi and \psi (see [18, Proposition 2.1] and [27, Appendix D]); (ii) estimates of the pressure by a local term using elliptic Carleman inequalities (see [24]); (iii) estimates of local integrals of \Delta \phi and \phi tby using global energy estimates. For more details, we refer the reader to [27, Appendix E] and [4, Appendix C]. 4.2. Null controllability of the linearized system. In what follows, we take s=\widetilde sand \lambda =\widetilde \lambda . In this section, we prove the null controllability of the linear system (40) as a consequence of the inequality (43). To this end, let us introduce the space where the controls are searched for: \scrH := [H1(0,T;L2(\scrO )n)\cap C0([0,T];H1(\scrO )n)] \times [H1(0, T;L2(\scrO )) \cap C0([0, T]; H1(\scrO ))]. Proposition 4.2. Let (z\ast , h\ast )\in H\times L2(\scrO )be given, and suppose that (41) holds. Then, there exist controls (v,w)\in \scrH such that the corresponding solution to (40) satisfies z(T,\cdot ) = 0 and h(T,\cdot ) = 0. Moreover, the following estimate holds: \| \kappa 1/2v\chi \omega \| +\| \kappa 1/2w\chi \omega \| +\| v\| H1(L2)+\| v\| L\infty (H1)+\| w\| H1(L2)+\| w\| L\infty (H1) \leq C(\| z\ast \| +\| h\ast \| ), where the positive constant C depends only on \scrO ,\omega ,T,\| a\| \infty ,\| b\| \infty ,\| c\| \infty , \| at\| L2(Lr),\| bt\| L2(Lr),\| ct\| L2(Lr), and \| M\| Eand \| m\| F,\kappa (t) = e4s\^\alpha - 2s\alpha \ast \^ \xi - 15/2, \^\alpha ,\alpha \ast , and \^ \xi are defined in section 4.1. The proof of Proposition 4.2 is based on a penalized Hilbert uniqueness method; it follows the ideas of [18, section 3.1]. The details can be found in [27, Proposition 3.17]. 4.3. Local exact controllability to the trajectories of the Boussinesq system. We now prove the local exact controllability to the trajectories of (39). Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 502 CHAVES-SILVA ET AL. Let (u,p,\theta ) be an uncontrolled solution to (39), that is, a triplet satisfying \left\{ \partial tu - \Delta u+ (u\cdot \nabla )u+\nabla p=\theta en,div u= 0 in \scrO T, \partial t\theta - \Delta \theta +u\cdot \nabla \theta = 0 in \scrO T, u\cdot \nu = 0, N(u) = 0, R(\theta ) = 0 on \Kappa T, u(0,\cdot ) = u\ast , \theta (0,\cdot ) = \theta \ast in \scrO . Let us assume that the following holds: u\in X:= H(3 - \ell )/2(0,T;H\vargamma 2+1/2(\scrO )n\cap H)\cap H1 - \ell (0,T;W\vargamma 1+1/2,\vargamma 1+1(\scrO )n), u\ast \in H3(\scrO )n\cap H, N(u\ast ) = 0 on \partial \scrO , \theta \in Y:= H(3 - \ell )/2(0,T;H\vargamma 2+1/2(\scrO )) \cap H1 - \ell (0,T;W\vargamma 1+1/2,\vargamma 1+1(\scrO )), \theta \ast \in H3(\scrO ), R(\theta \ast ) = 0 on \partial \scrO , (44) with \ell ,r,\vargamma 1, and \vargamma 2as in the beginning of section 4. Proposition 4.3. Assume that T > 0and (u,u\ast , \theta , \theta \ast )satisfies (44). Then, there exists \delta T>0such that, for every (u\ast , \theta \ast )\in [H3(\scrO )n\cap H]\times H3(\scrO )satisfying \| u\ast - u\ast \| H3\leq \delta T,\| \theta \ast - \theta \ast \| H3\leq \delta Tand the compatibility conditions N(u\ast ) = 0, R(\theta \ast ) = 0 on \partial \scrO , one can find controls (v,w)\in \scrH and associated solutions (u,p,\theta )to (39) with u(T,\cdot ) = u(T,\cdot )and \theta (T, \cdot ) = \theta (T,\cdot )in \scrO . The proof is based on a Kakutani's fixed-point theorem. It is a straightforward adaptation of the argument in [18, section 3.2]. The details can be found in [27, Proposition 3.18] and [4, Proposition 4.3]. See also [29] for a similar result. 5. Global controllability to the trajectories. Let us explain how the previous arguments can be chained in order to prove the main result, that is, Theorem 1.1. First, we reduce the controllability to weak trajectories to the controllability to smooth trajectories as follows. Despite (u,p,\theta ) only being a weak solution in [0,T], there exists a time interval [\tau 1,\tau 2]\subset (0,T) such that (u,p,\theta ) is smooth in [\tau 1,\tau 2]. This statement follows from classical results; indeed, one can easily adapt [21, Theorems 2, 3, and 9] or [30, Remark 3.2] (written for the Navier--Stokes equations with Dirichlet boundary conditions and source terms) to our context. Then, we can start our control strategy by doing nothing in [0,\tau 1], that is, taking v=w=\sigma = 0 in (5). The weak trajectory will move from (u\ast ,\theta \ast ) to some (u,\theta )(\tau 1,\cdot ), which must be viewed as the new initial data. Hence, without loss of generality, we can work with a smooth reference trajectory. We split the control strategy into four steps. Step 1 - Regularization of the data: We begin by extending \Omega to a new domain \scrO , as explained in section 2.1. We also use Proposition 2.1 to guarantee the existence of (u\ast ,\theta \ast )\in H\times L2(\scrO ) and \sigma \ast \in C\infty c(\omega 0) satisfying (4). We set \sigma (t,x) := \varsigma (t/T)\sigma \ast (x) with \varsigma a smooth nonnegative decreasing function such that \varsigma \equiv 1 near 0 and \varsigma \equiv 0 near 1/8. The function \sigma must satisfy the compatibility condition div u\ast = \sigma (0,\cdot ). Then, we let the system (5) evolve with v=w= 0 in the time interval (0,T/8) in order to reach some data (u,\theta )(T/8,\cdot )\in H\times L2(\scrO ). Next, by using Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 503 the smoothing effect of the uncontrolled Boussinesq system starting from divergencefree data (see Lemma 2.1), we deduce that there exists T1\in (T/8,T/4) such that (u,\theta )(T1,\cdot )\in [H3(\scrO )n\cap H]\times H3(\scrO ). Accordingly, we can apply Lemma 3.3. Step 2 - Global approximate controllability result in L2(\scrO )n+1:Let us set T2:= T/2. Starting from the new initial data (u, \theta )(T1,\cdot ), we use the global approximate controllability result stated in Proposition 3.1 in a time interval of size T2 - T1\geq T/4. Thus, for any \delta > 0, we can build a trajectory starting from (u, \theta )(T1,\cdot ) such that \| (u,\theta )(T2,\cdot ) - (u,\theta )(T2,\cdot )\| \leq \delta . Step 3 - Regularizing argument: Now, we use again Lemma 2.1 to deduce the existence of a time T3\in (T2,3T/4) such that \| (u,\theta )(T3,\cdot ) - (u,\theta )(T3,\cdot )\| H3\times H3\leq \Psi T/4(\delta ). In particular, we can take \delta small enough such that \Psi T/4(\delta )\leq \delta T/4, where \delta T/4is the radius of local controllability result given in Proposition 4.3 and the function \Psi T/4appears in the regularity result for the free Boussinesq system; see Lemma 2.1. Step 4 - Local controllability in H3(\scrO )n+1:Finally, we use the local controllability result in [T3, T3+T/4] and get (u,\theta )(T3+T/4,\cdot ) = (u,\theta )(T3+T/4,\cdot ). Then, extending the control by zero for t\in [T3+T/4,T], we get (3) and the proof is complete. Remark 5.1. A detailed analysis of the proofs of the results in sections 3 to 4 shows that the (intermediate) global approximate controllability result holds as soon as the components of \=uand \= \theta belong to L\infty (0,T;H3(\scrO )) \cap C0([0,T]; H2(\scrO )) and the local exact controllability result holds as soon as (\=u, \= \theta ) satisfies (44). 6. Additional comments and open questions. 6.1. Controlling with fewer controls. A natural extension of the main result would be the global exact controllability with a reduced number of controls acting on a small part of the boundary. Unfortunately, in order to solve this problem, we cannot use the extension domain technique. However, in the spirit of [10, 29] one could try to establish a small-time global null controllability for the internal control system (5) in two dimensions by acting only on the temperature. The intuition behind a result of this kind is the following: the temperature \theta is directly controlled by w; then, \theta acts through the coupling term \theta e2to control the component u2and then u2acts as bilinear control through the term u2\partial x2u1to control the component u1. One can also try get a global control result acting only on the motion, that is, with w= 0 in (5); for some local results in this direction, see [3, 10]. Note that, in the case of Neumann conditions on \theta , i.e., with m\equiv 0, we have an obstruction: Indeed, the total thermal energy associated with \theta is conserved and we have \int \Omega \theta (T,x)dx =\int \Omega \theta 0(x)dx. Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 504 CHAVES-SILVA ET AL. However, one could try to control to zero any initial data for the temperature \theta 0\in L0, where L0is a closed linear subspace of L2(\Omega ) given by L0=\biggl\{ \theta \ast \in L2(\Omega ) : \int \Omega \theta \ast (x)dx = 0\biggr\} , which is invariant for the equation of temperature. Results of these kinds will be analyzed in the near future 6.2. Other boundary conditions for the velocity field. Another natural question is whether Theorem 1.1 holds with usubject to other boundary conditions. By imposing Dirichlet boundary (no-slip) conditions on the velocity, we face a very well known and challenging open problem related to a conjecture by Jacques-Louis Lions. As pointed out in [8], the boundary layer found in the presence of Dirichlet conditions has a behavior which is not as ``good"" as in the case of Navier boundary conditions. This implies many difficulties to estimate the boundary layer profiles and the remainder terms. As an attempt to deal with this problem, we refer the reader to [9], where the authors prove that a kind of global boundary null controllability result holds if we allow a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space; see also [20] for related results. 6.3. Other boundary conditions for the temperature. Let us see that Theorem 1.1 holds with Dirichlet boundary conditions on the temperature. To prove this, we can adapt the strategy of the proof of Theorem 1.1 (see section 5). After the extension and regularization steps, the initial temperature \theta \ast vanishes on the whole boundary \partial \scrO . Then, the temperature \theta 1, which solves (15)2, preserves this property in [0,T/\varepsilon ]. Indeed, since the flow u0is parallel to the boundary, particles on the boundary cannot enter in the domain \scrO , i.e., \Phi 0(s;t,x)\in \partial \scrO for all s,t \in [0,T/\varepsilon ] and x\in \partial \scrO (see [1, Theorem 5.1]). Thus, in the estimates of the reminder q\varepsilon (see section 3.2.4), we get a zero boundary integral \int \partial \scrO q\varepsilon \partial q\varepsilon \partial \nu d\Gamma = \int \partial \scrO \theta 1\partial q\varepsilon \partial \nu d\Gamma = 0. Finally, a local control result for the Boussinesq system with Navier-slip-with-friction boundary conditions for the velocity field and Dirichlet boundary conditions on the temperature can also be deduced, and this completes the argument. 6.4. Some possible extensions. Theorem 1.1 can be easily extended to cover global control properties of a few systems of the Navier--Stokes and Boussinesq kinds. For example, it can be applied to some pollution models, where the motion and temperature PDEs are coupled to one or several additional transport-diffusion-reaction equations. Some results will be given in the near future. Nevertheless, there are other situations where the extension of the result seems more (or much more) delicate. One of them concerns ``complete"" or ``full"" Boussinesq systems. By this we mean the equations \biggl\{ \partial tu - \Delta u+ (u\cdot \nabla )u+\nabla p=\theta en,div u= 0 in (0,T)\times \Omega , \partial t\theta - \Delta \theta +u\cdot \nabla \theta = (\nabla u+\nabla ut)\cdot \nabla uin (0,T)\times \Omega , completed with initial conditions and boundary control requirements as before. Another one is the variable density Navier--Stokes system \biggl\{ \partial t\rho +u\cdot \nabla \rho = 0 in (0,T)\times \Omega , \rho (\partial tu+ (u\cdot \nabla )u) - \Delta u+\nabla p= 0,div u= 0 in (0,T)\times \Omega , Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 505 this time completed with initial conditions for uand \rho and, again, boundary controls acting on u. Appendix A. Regularity of the uncontrolled Boussinesq system. Let us present the proof of Lemma 2.1. In the followinglowing, let us assume that Mand mare regular enough. In what follows, we will use Korn's inequality recurrently. Lemma A.1 (second Korn inequality). There exist two positive constants C1,C2>0such that, for every u\in H1(\scrO )n, one has C1(\| u\| +\| D(u)\| )\leq \| u\| H1\leq C2(\| u\| +\| D(u)\| ). We will also need the following results. Lemma A.2. There exist positive constants Cl,Cr,K > 0such that, for every u\in H1(\scrO )n, we have Cl\| u\| K,M \leq \| u\| H1\leq Cr\| u\| K,M , where \| u\| K,M := (K\| u\| 2+\int \partial \scrO Mu \cdot u+\| D(u)\| 2)1/2. Lemma A.3. There exist positive constants Cl,Cr,\gamma > 0such that, for every \theta \in H1(\scrO ), we have Cl\| \theta \| \gamma ,m \leq \| \theta \| H1\leq Cr\| \theta \| \gamma ,m, where \| \theta \| \gamma ,m := (\gamma \| \theta \| 2+\int \partial \scrO m| \theta | 2+\| \nabla \theta \| 2)1/2. The proofs of these two lemmas rely on the interpolation inequality [2, Theorem III.2.36]. In particular, it is used that there exists a positive constant Csuch that \| u\| L2(\partial \scrO )\leq C\| u\| 1/2\| u\| 1/2 H1\forall u\in H1(\scrO ). Lemma A.4 (Proposition III.2.35 in [2]). Let p\in [1,+\infty ]and q\in [p,p\ast ], where p\ast is the critical exponent associated with p. Then, there exists C > 0such that \| u\| Lq\leq C\| u\| 1+n/q - n/p Lp\| u\| n/p - n/q W1,p \forall u\in W1,p(\scrO ). Lemma A.5 (pages 490--494 in [18]). Let f\in L2(\scrO )nand g\in H1/2(\partial \scrO )n. Then, there exists a unique strong solution (u, p)\in H2(\scrO )n\times H1(\scrO )to the Stokes problem \biggl\{ - \Delta u+\nabla p=f, \nabla \cdot u= 0 in \scrO , u\cdot \nu = 0, N(u) = gon \partial \scrO , and there exists a positive constant C > 0such that \| u\| H2+\| p\| H1\leq C(\| f\| +\| g\| H1/2). Moreover, if f\in Hk(\scrO )nand g\in Hk+1/2(\partial \scrO )nfor some k\geq 0, then (u,p)\in Hk+2(\scrO )n\times Hk+1(\scrO )and we have \| u\| Hk+2 +\| p\| Hk+1 \leq C(\| f\| Hk+\| g\| Hk+1/2). Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 506 CHAVES-SILVA ET AL. Lemma A.6. Let S:D(S)\rightarrow L2 div(\scrO )nbe the Stokes operator, where D(S) = \{ v\in H2(\scrO )n\cap L2 div(\scrO )n:N(v)=0\} and S:= - \BbbP \Delta . There exists a positive constant C > 0such that, for every u\in D(S), we have \| u\| H2\leq C(\| Su\| +\| u\| H1). Moreover, if Su \in Hk(\scrO )nfor some k\geq 0, then u\in Hk+2(\scrO )nand we have \| u\| Hk+2 \leq C(\| Su\| Hk+\| u\| Hk+1 ). Lemma A.7. Let u\in H1(\scrO )satisfying \Delta u\in L2(\scrO )and \partial u \partial \nu +mu = 0 on \partial \scrO . Then, there exists a constant C > 0, only depending on \scrO , such that \| u\| H2\leq C(\| \Delta u\| +\| mu\| H1/2(\partial \scrO )). Moreover, if \Delta u\in Hk(\scrO )for some k\geq 0, then u\in Hk+2(\scrO )and we have \| u\| Hk+2 \leq C(\| \Delta u\| Hk+\| mu\| Hk+1/2(\partial \scrO )). This last result is a consequence of [2, Theorem III.4.3]. Throughout the proof of Lemma 2.1, we will accept that the constants Ccan increase from line to line and depend on Tand the trajectory (u, \theta ). For simplicity, we will only consider the 3D case. The proof is split in several steps. Step 1 - Weak estimates in (0,T/3).Let us first multiply (6)1by rand (6)2 by q, integrate by parts, and sum. We get 1 2 d dt \bigl( \| r\| 2+\| q\| 2\bigr) + 2\| Dr\| 2+\| \nabla q\| 2+ 2\int \partial \scrO Mr \cdot r+\int \partial \scrO m| q| 2 = (qen,r) - \int \scrO (r\cdot \nabla )u\cdot r - \int \scrO r\cdot \nabla \theta q. From the Cauchy--Schwarz and Young inequalities, we obtain 1 2 d dt \bigl( \| r\| 2+\| q\| 2\bigr) + 2\| Dr\| 2+\| \nabla q\| 2+ 2\int \partial \scrO Mr \cdot r+\int \partial \scrO m| q| 2\leq C(\| r\| 2+\| q\| 2). Using Lemmas A.2 and A.3, we deduce that 1 2 d dt \bigl( \| r\| 2+\| q\| 2\bigr) +2 C2 l (\| r\| 2 H1+\| q\| 2 H1)\leq (C+ 2K)\| r\| 2+ (C+ 2\gamma )\| q\| 2.(45) By applying Gronwall's lemma, we have for a.e. t\in [0, T] that \| r(t, \cdot )\| 2+\| q(t, \cdot )\| 2+\int t 0\bigl( \| r(s, \cdot )\| 2 H1+\| q(s, \cdot )\| 2 H1\bigr) ds \leq eCt \bigl( \| r\ast \| 2+\| q\ast \| 2\bigr) . (46) Therefore, from the mean value theorem, we deduce by contradiction that there exists 0\leq t1\leq T/3 such that \| r(t1,\cdot )\| 2 H1+\| q(t1,\cdot )\| 2 H1\leq C1\bigl( \| r\ast \| 2+\| q\ast \| 2\bigr) (47) for a positive constant C1independent of t1. Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy GLOBAL CONTROLLABILITY OF THE BOUSSINESQ SYSTEM 507 Step 2 - Strong estimates in (t1,2T/3).Let \BbbP be the classical Leray projector. We multiply (6)1and (6)2by - Sr and - \Delta q, respectively, and then integrate by parts. Since Mis symmetric, we obtain d dt \biggl( \| Dr\| 2+\int \partial \scrO Mr \cdot r\biggr) +\| Sr\| 2 =\int \partial \scrO (Mt)r\cdot r+\int \scrO \Biggl( (r\cdot \nabla )r\cdot Sr + (u\cdot \nabla )r\cdot Sr + (r\cdot \nabla )u\cdot Sr - (qen,Sr)\Biggr) \leq C\| r\| 2 H1+1 2\| Sr\| 2+C\| q\| 2+\| r\| 2 L6\| \nabla r\| 2 L3. Also, 1 2 d dt \biggl( \| \nabla q\| 2+\int \partial \scrO m| q| 2\biggr) +\| \Delta q\| 2=1 2\int \partial \scrO (mt)q\cdot q+ (r\cdot \nabla q,\Delta q) + (u\cdot \nabla q,\Delta q) + (r\cdot \nabla \theta ,\Delta q) \leq C\| q\| 2 H1+1 2\| \Delta q\| 2+C\| r\| 2+\| r\| 2 L6\| \nabla q\| 2 L3. Multiplying (45) by \varsigma = max\{ K,\gamma \} , adding the above inequalities, and using Lemmas A.2--A.7, we deduce the following: d dt \bigl( \| r\| 2 \varsigma ,M +\| q\| 2 \varsigma ,m\bigr) +\| r\| 2 H2+\| q\| 2 H2 (48) \leq C(\| r\| 2 \varsigma ,M +\| q\| 2 \varsigma ,m +\| r\| 2 L6\| \nabla r\| 2 L3+\| r\| 2 L6\| \nabla q\| 2 L3) \leq C\bigl[ (\| r\| 2 \varsigma ,M +\| q\| 2 \varsigma ,m) + (\| r\| 2 \varsigma ,M +\| q\| 2 \varsigma ,m)3\bigr] . Introducing Y(t) := \| r(t, \cdot )\| 2 \varsigma ,M +\| q(t,\cdot )\| 2 \varsigma ,m, we see that Yis a.e. differentiable and, from (48), we have that Y\prime \leq C(Y3+Y).(49) In view of (49), we obtain Y(t)2\leq eC(t - t1)Y(t1)2 Y(t1)2+ 1 - eC(t - t1)Y(t1)2. Let us take t - t1\leq \tau 1small enough such that eC(t - t1)\leq 1 + 1 2Y(t1)2. Then, Y(t)2\leq 2eC(t - t1)Y(t1)2and, from (47), we deduce that Y(t)\leq CY\ast , where Y\ast := \| r\ast \| 2+\| q\ast \| 2. Therefore, \| r(t,\cdot )\| 2 \varsigma ,M +\| q(t,\cdot )\| 2 \varsigma ,m +\int t t1 (\| r(s,\cdot )\| 2 H2+\| q(s,\cdot )\| 2 H2)ds \leq CY\ast +C(Y\ast +Y3 \ast )\tau 1. Taking \tau 1small enough such that \tau 1\leq (1+Y2 \ast ) - 1, we have that CY\ast +C(Y\ast +Y3 \ast )\tau 1\leq C2Y\ast . Therefore, one has \| r(t,\cdot )\| 2 \varsigma ,M +\| q(t,\cdot )\| 2 \varsigma ,m +\int t t1 (\| r(s,\cdot )\| 2 H2+\| q(s,\cdot )\| 2 H2)ds \leq C2\bigl( \| r\ast \| 2+\| q\ast \| 2\bigr) (50) for t1\leq t\leq t1+\tau 1. This ensures the existence of t1\leq t2<min\{ 2T/3,t1+\tau 1\} such that \| r(t2,\cdot )\| 2 H2+\| q(t2,\cdot )\| 2 H2\leq C2 \tau 1\bigl( \| r\ast \| 2+\| q\ast \| 2\bigr) . Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy 508 CHAVES-SILVA ET AL. Step 3 - Third energy estimate in (t2,T).At this point, we differentiate (6) with respect to time and multiply by \partial trand \partial tq. Then, we integrate by parts to obtain 1 2 d dt\| rt\| 2+ 2\| Drt\| 2+ 2\int \partial \scrO Mrt\cdot rt = - 2\int \partial \scrO Mtr\cdot rt+ (qten,rt) - (rt\cdot \nabla )r\cdot rt - (ut\cdot \nabla )r\cdot rt - (rt\cdot \nabla )u\cdot rt - (r\cdot \nabla )ut\cdot rt \leq C\bigl( \| r\| H1\| rt\| H1+\| qt\| 2+\| rt\| 2+\| rt\| 3\| \nabla r\| \| rt\| 6+\| r\| 2 H1\bigr) and 1 2 d dt\| qt\| 2+\| \nabla qt\| 2+\int \partial \scrO m| qt| 2 = - \int \partial \scrO mtqqt - ((rt+u)\cdot \nabla q,qt) - (rt\cdot \nabla \theta ,qt) - (r\cdot \nabla \theta t,qt) \leq C\bigl( \| q\| H1\| qt\| H1+\| q\| 2 H1+\| qt\| 2+\| rt\| 2+\| rt\| 3\| \nabla q\| \| rt\| 6\bigr) . Consequently, using Lemmas A.2--A.4 and adding the two above inequalities, we have d dt \bigl( \| rt\| 2+\| qt\| 2\bigr) +\| rt\| 2 H1+\| qt\| 2 H1 \leq C\bigl( \bigl( \| r\| 4 H1+\| q\| 4 H1+ 1\bigr) \| rt\| 2+\| qt\| 2+\| r\| 2 H1+\| q\| 2 H1\bigr) . Now, introducing Z(t) := \| rt(t, \cdot )\| 2+\| qt(t, \cdot )\| 2, we find from (50) that Z\prime \leq C[(1 + Y2 \ast )Z+Y\ast ] for t2\leq t\leq t1+\tau 1. By applying Gronwall's lemma, we have for a.e. t\in [t2,t1+\tau 1] Z(t)\leq eC(1+Y2 \ast )(t - t2)(Z(t2) + CY\ast (t - t2)). Since we have Z(t2)\leq \Psi 1(Y\ast ) for some nonnegative regular \Psi 1with \Psi 1(0) = 0, we find that Z(t)\leq \Psi 2(Y\ast ), with \Psi 2(s) := eC(1+s2)(\Psi 1(s) + Cs)\forall s\geq 0. Therefore, \| rt(t, \cdot )\| 2+\| qt(t\cdot )\| 2+\int t t2\bigl( \| rt(s, \cdot )\| 2 H1+\| qt(s, \cdot )\| 2 H1\bigr) ds \leq \Psi 3(Y\ast )\forall t\in [t2,t1+\tau 1], (51) where \Psi 3(s) := C[(1 + s2)\Psi 2(s) + s]. In particular, this yields the existence of t3\in (t2,t1+\tau 1) such that \| rt(t3,\cdot )\| 2 H1+\| qt(t3,\cdot )\| 2 H1\leq \Psi 3(Y\ast ) (t1 - t2+\tau 1).(52) Actually, it is not difficult to check that the set of times t3\in (t2, t1+\tau 1) satisfying (52) has a positive measure. Step 4 - Conclusion. Using (50) and (51), we deduce an estimate of rin L\infty (H2). It suffices to view (6)1as a family of Stokes problems (see Lemma A.5 and the arguments presented in [30, Theorem 3.8]). Then, looking at (6)2as a family of elliptic problems, we also find L\infty (H2) estimates for q; see Lemma A.7. Both estimates depend on Y\ast continuously. Therefore, repeating the procedure, we see that (r(t3),q(t3)) \in H3\times H3with an estimate of the form \Psi (Y\ast ). Copyright ©by SIAM. Unauthorized reproduction of this article is prohibited. Downloaded 01/30/24 to 150.214.182.233 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy