Uniform stabilization of Boussinesq systems in critical $\mathbf{L}^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will be minimal in number, and of reduced dimension: more precisely, they will be of dimension $(d-1)$ for the fluid component and of dimension $1$ for the heat component. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to $\mathbf{L}^3(\Omega$) for $ d = 3 $) and the space $L^q(\Omega$) for the thermal component, $ q>d $. Thus, this paper may be viewed as an extension of \cite{LPT.1}, where the same interior localized uniform stabilization outcome was achieved by use of finite dimensional feedback controls for the Navier-Stokes equations, in the same Besov setting.


Controlled dynamic Boussinesq equations
In this paper, we consider the following Boussinesq approximation equations in a bounded connected region Ω in R d with sufficiently smooth boundary Γ = ∂Ω. More specific requirements will be given below. Let Q ≡ (0, T ) × Ω and Σ ≡ (0, T ) × ∂Ω where T > 0. Further, let ω be an arbitrary small open smooth subdomain of the region Ω, ω ⊂ Ω, thus of positive measure. Let m denote the characteristic function of ω: m(ω) ≡ 1, m(Ω/ω) ≡ 0.
Notation: Vector-valued functions and corresponding function spaces will be boldfaced. Thus, for instance, for the vector valued (d-valued) velocity field or external force, we shall write say y, f ∈ L q (Ω) rather than y, f ∈ (L q (Ω)) d .
The Boussinesq system under the action of two localized interior controls m(x)u(t, x) and m(x)v(t, x) supported on Q ω ≡ (0, ∞) × ω is then y t − ν∆y + (y · ∇)y − γ(θ −θ)e d + ∇π = m(x)u(t, x) + f (x) in Q (1.1a) y(0, x) = y 0 , θ(0, x) = θ 0 on Ω. (1.1e) In the Boussinesq approximation system, y = {y 1 , . . . , y d } represents the fluid velocity, θ the scalar temperature of the fluid, ν the kinematic viscosity coefficient, κ the thermal conductivity. The scalar function π is the unknown pressure. The term e d denotes the vector (0, . . . , 0, 1). Moreover γ =ḡ/θ whereḡ is the acceleration due to gravity andθ is the reference temperature. The d-vector valued function f (x) and scalar function g(x) correspond to an external force acting on the Navier-Stokes equations and a heat source density acting on the heat equation, respectively. They are given along with the I.C.s y 0 and θ 0 . Note that y · ∇θ = div(θy).
The Boussinesq system models heat transfer in a viscous incompressible heat conducting fluid. It consists of the Navier-Stokes equations (in the vector velocity y) coupled with the convectiondiffusion equation (for the scalar temperature θ). The external body force f (x) and the heat source density g(x) may render the overall system unstable in the technical sense described below by (1.33). The goal of the paper is to exploit the localized controls u and v, sought to be finite dimensional and in feedback form, in order to stabilize the overall system. As an additional benefit of our investigation, the feedback fluid component of u will be of reduced dimension (d − 1) rather than d, while the feedback heat component of u will be of 1-dimensional. This is a consequence of the Unique Continuation Property expressed by Theorem 1.4 for the adjoint static problem. As far as practical applications are concerned, one may consider the situation of controlling the temperature and humidity in a bounded restricted environment -see [B-H], [B-H-H] for an eloquent description of the physical phenomenon. Due to the physical significance of the Boussinesq system, the problem of its stabilization has been considered in the literature -with both localized and boundary controls -following of course prior developments concerning the Navier-Stokes model alone. See subsection 2.4 for a view of the literature.
Motivation: why studying uniform stabilization of the Boussinesq problem (1.1) in the Besov functional setting (in the fluid component) of the present paper?
In short: Stimulated by recent research achievements [L-P-T.1], [L-P-T.2] on the uniform stabilization of the Navier-Stokes equations -to be elaborated below -the present paper sets the stage as a preliminary, needed step toward the authors' final goal of solving the Boussinesq uniform stabilization problem with finite dimensional feedback controls acting tangentially on a part of the boundary in dimension d = 3. We next illustrate the above assertion. While several results have been obtained in 2 and 3 dimensional cases, the main open question after 20 years or so -in both the Navier-Stokes as well as the Boussinesq systems -is whether it is possible at all to uniformly stabilize the 3 − d equations with finite dimensional feedback controls acting on the boundary. Indeed, the main obstacle in this 3 − d case is due to compatibility conditions between boundary terms and initial conditions, resulting from the necessity of using sufficiently regular [differentiable] solutions for d = 3. As it was originally the case with the Navier-Stokes equations, the Boussinesq case in the literature considers until now [B], [Lef], [R-R-R], [Wang] the problem within the Hilbert framework. Due to the Stokes non-linearity in 3 − d, this necessitates in the Navier-Stokes equationswhether alone or as a component of the Boussinesq system -the need of working with sufficiently high order Sobolev spaces which recognize boundary conditions, and thus impose compatibility conditions. This has so far prevented the goal of achieving in the 3 − d case the desired result of finite-dimensionality of the tangential boundary feedback stabilizing control in full generality, beyond the special case of [B-L-T.1], [L-T.4], where the initial conditions are compactly supported. See also [B, below (4.122)]. For Navier-Stokes equations, this 3d-recognized long-standing open problem on the finite dimensionality of the localized tangential boundary feedback stabilizing control in full generality, was recently settled in the affirmative in [L-P-T.2]. Its treatment strongly benefited -technically and conceptually -from the preliminary successful analysis of the test case on uniform stabilization of the (d = 2 and) d = 3 -Navier-Stokes equations, by means of finite dimensional feedback stabilizing controls localized on an arbitrarily subset ω of the interior Ω, precisely in the same Besov setting [L-P-T.1]. To this end, it was necessary to introduce a radically different functional setting. It is based on Besov spaces of tight indices, which do not recognize boundary conditions, see Remark 1.1, "close" to the well-known critical space L 3 in space and L 1 in time for d = 3. As said, it has the virtue of not recognizing boundary conditions, see Remark 1.1, while still being adequate for handling the N-S nonlinearity in 3 − d. Circumventing the obstacle of compatibility conditions on the boundary, which prevents (for d ≥ 3) finite dimensionality of stabilizing boundary controls in all prior Hilbert-based treatments, has been a major predicament in 3d-boundary feedback stabilization of N-S flows. In contrast the Besov setting develops a theory within non-Hilbertian structures, where solutions under considerations have higher integrability but almost no differentiability. Thus, such Besov space setting, which has already been used for wellposedness of systems of incompressible flows, eg [D-M], [M-Z], is here a forced necessity introduced by the authors in order to solve the uniform stabilization problem of the Navier-Stokes equations with a localized tangential boundary feedback control which moreover is finite dimensional also for d = 3. Though the jump from localized interior to localized boundary (tangential) stabilizing finite dimensional (d = 3) controls offers serious additional challenges -conceptual as well as technical -the solution in [L-P-T.1] of the former case proved very useful in achieving the solution of the latter case in [L-P-T.2]. Accordingly, the present paper is the generalization to the Boussinesq system of the local interior stabilization problem of the Navier-Stokes problem in [L-P-T.1]. It is intended to be equally enlightening and useful toward the final goal, which is the uniform stabilization problem of the Boussinesq system by finite dimensional localized tangential boundary controls; that is, the direct counterpart of the Navier-Stokes result in [L-P-T.2].
Accordingly, in seeking a 3d extension of such tangential boundary stabilization problem from the 3d-Navier-Stokes equations to the 3d-Boussinesq system, in the present paper we first consider the new Besov-setting in the context of interior localized controls.

Stationary Boussinesq equations
Our starting point is the following result.

A first quantitative description of the main goal of the present paper
The starting point of the present paper is the following: that under a given external force f (x) for the fluid equations, a given heat source g(x) for the thermal equation, and given viscosity coefficient ν and thermal conductivity κ, the equilibrium solution {y e , θ e } is unstable, in a quantitative sense to be made more precise in sub-section 1.7, specifically in (1.33). This will mean that the free dynamics linear operator A q defined in (1.27) -which has compact resolvent, and is the generator of a s.c. analytic semigroup in the appropriate functional setting -has N unstable eigenvalues.
The main goal of the present paper is then -at first qualitatively -to feedback stabilize the non-linear Boussinesq model (1.1) subject to rough (non-smooth) initial conditions {y 0 , θ 0 }, in the vicinity of an (unstable) equilibrium solution {y e , θ e } in (1.2), by means of a finite dimensional localized feedback control pair {mu, mv}. Thus this paper pertains to the general issue of "turbulence suppression or attenuation" in fluids. The general topic of turbulence suppression (or attenuation) in fluids has been the object of many studies over the years, mostly in the engineering literaturethrough experimental studies and via numerical simulation -and under different geometrical and dynamical settings. The references cited in the present paper by necessity pertain mostly to the mathematical literature, and most specifically on the localized interior control setting of problem (1.1) [Lef], [Wang]. A more precise description thereof is as follows: establish interior localized exponential stabilization of problem (1.1) near an unstable equilibrium solution {y e , θ e } by means of a finite dimensional localized, spectral-based feedback control pair {mu, mv}, in the important case of initial condition y 0 of low regularity, as technically expressed by y 0 being in a suitable Besov space with tight indices, and θ 0 being in a corresponding L q -space q > d. More precisely, the resulting state space for the pair {y, θ}, where uniform stabilization will be achieved is the space as a real interpolation space between L q (Ω) and W 2,q (Ω). This setting will be further elaborated after introducing the Helmhotz decomposition below. In particular, local exponential stability for the velocity field y near an equilibrium solution y e will be achieved in the topology of the Besov subspace B 2− 2 /p q,p (Ω) in (1.11b). Note the tight index: 1 < p < 6 / 5 for q > d = 3, and 1 < p < 4 / 3 for d = 2. It will be documented below in Remark 1.1 that in such setting, the compatibility conditions on the boundary of the initial conditions are not recognized. This feature is precisely our key objective within the stabilization problem. The fundamental reason is that such feature will play a critical role in our aforementioned research project which consists in showing local uniform stabilization of the Boussinesq system, by means of finite dimensional, localized tangential boundary controls also in dimension dimΩ = d = 3; the perfect counterpart of [L-P-T.2]. In the case of the Navier-Stokes equations, uniform stabilization in the vicinity of a steady state solution by means of a localized tangential boundary feedback control which in addition is also finite dimensional for d = 3 was a recognized open problem. It was recently settled in the affirmative in [L-P-T.2] precisely in the Besov subspace B 2− 2 /p q,p (Ω), 1 < p < 6 / 5 for q > d = 3. For d = 3, such space is 'close' to L 3 . Thus this departure from the Hilbert space theory -the latter present in all other publications related to the stabilization of Navier-Stokes or Boussinesq systems -allows one to trade differentiability with integrability. This then avoids the obstacle of compatibility conditions while preserving just enough regularity in order to carry-out the nonlinear analysis in 3 − d.
Criticality of the space L 3 for d = 3: We now expand on the issue regarding the 'criticality' of the space L 3 (Ω). In the case of the uncontrolled N-S equations defined on the full space R 3 , extensive research efforts have recently lead to the discovery that the space L 3 (R 3 ) is a 'critical' space for the issue of well-posedness. Assuming that some divergence free initial data in L 3 (R 3 ) would produce finite time singularity, then there exists a so-called minimal blow-up initial data in Thus, these latter two references manage to solve the uniform stabilization problem for the controlled N-S equations in a correspondingly related low-regularity function space setting. Our present paper is then an extension of [L-P-T.1] to the Boussinesq system (1.1). A further justification of our low-regularity level of the Besov space in (1.4) is provided by the final goal of our line of research. Based critically on said low-regularity level of the Besov space (1.11b), which does not recognize compatibility conditions on the boundary of the initial conditions, Remark 1.1-we shall seek to solve in the affirmative a presently open problem by showing that uniform stabilization of the Boussinesq system is possible by localized tangential boundary feedback controls which moreover are finite dimensional also in dim Ω = d = 3. This result will be an extension of [L-P-T.2] in the case of the 3 − d N-S equations.

Helmholtz decomposition
A first difficulty one faces in extending the local exponential stabilization result for the interior localized problem (1.1) from the Hilbert-space setting in [Lef], [Wang] to the L q -based setting is the question of the existence of a Helmholtz (Leray) projection for the domain Ω in R d . More precisely: Given an open set Ω ⊂ R d , the Helmholtz decomposition answers the question as to whether L q (Ω) can be decomposed into a direct sum of the solenoidal vector space L q σ (Ω) and the space G q (Ω) of gradient fields. Here, (1.5) Both of these are closed subspaces of L q .
Definition 1.1. Let 1 < q < ∞ and Ω ⊂ R n be an open set. We say that the Helmholtz decomposition for L q (Ω) exists whenever L q (Ω) can be decomposed into the direct sum (nonorthogonal) The unique linear, bounded and idempotent (i.e. P 2 q = P q ) projection operator P q : L q (Ω) −→ L q σ (Ω) having L q σ (Ω) as its range and G q (Ω) as its null space is called the Helmholtz projection. Additional information of use in the present paper is given in Appendix B below,say Proposition B.1. This is an important property in order to handle the incompressibility condition div y ≡ 0. For instance, if such decomposition exists, the Stokes equation (say the linear version of (1.1a) with control u ≡ 0 and no coupling, thus θ ≡ 0) can be formulated as an equation in the L q setting. Here below we collect a subset of known results about Helmholtz decomposition. We refer to Section 2.2], in particular to the comprehensive Theorem 2.2.5 in this reference, which collects domains for which the Helmholtz decomposition is known to exist. These include the following cases: (i) any open set Ω ⊂ R d for q = 2, i.e. with respect to the space L 2 (Ω); more precisely, for q = 2, we obtain the well-known orthogonal decomposition (in the standard notation, where ν = unit outward normal vector on Γ) [C-F, Prop 1.9, p 8] On the other hand, on the negative side, it is known that there exist domains Ω ⊂ R d such that the Helmholtz decomposition does not hold for some q = 2 [M -B].
Assumption (H-D). Henceforth in this paper, we assume that the bounded domain Ω ⊂ R d under consideration admits a Helmholtz decomposition for the values of q, 1 < q < ∞, here considered at first, for the linearized problem (1.28) or (1.29) below. The final results Theorem 2.2 through 2.4 for the non-linear problem (1.1) will require q > d, see (5.24), in the case of interest d = 2, 3.
We can now provide further critical information on the Besov space B 2− 2 /p q,p (Ω) which is the fluid component of the state space (1.3) where uniform stabilization takes place.
Definition of Besov spaces B s q,p on domains of class C 1 as real interpolation of Sobolev spaces: Let m be a positive integer, m ∈ N, 0 < s < m, 1 ≤ q < ∞, 1 ≤ p ≤ ∞, then we define [G-G-H.1, p 1398] the Besov space (1.8) as a real interpolation space between L q (Ω) and W m,q (Ω).This definition does not depend on m ∈ N [Wahl, p xx]. This clearly gives (1.9) We shall be particularly interested in the following special real interpolation space of the L q and W 2,q spaces m = 2, s = 2 − 2 p : (1.10) Our interest in (1.10) is due to the following characterization [Amann.2,Thm 3.4], [G-G-H.1, p 1399]: if A q denotes the Stokes operator to be introduced in (1.14) below, then Remark 1.1. Notice that, in (1.11b), the condition g · ν| Γ = 0 is an intrinsic condition of the space L q σ (Ω) in (1.5), not an extra boundary condition as g| Γ = 0 in (1.11a). Orientation: As already noted,ultimately, we shall seek to obtain uniform feedback stabilization of the fluid component y in the Besov subspace B 2− 2 /p q,p (Ω), dim Ω = d < q < ∞, 1 < p < 2q / 2q−1 , defined by real interpolation in (1.4), (1.11b); The reason being that such a space does not recognize boundary conditions, as noted above in Remark 1.1. Analyticity and maximal regularity of the Stokes problem will require p > 1.
By way of orientation, we state at the outset two main points. For the linearized w-problem (1.28) below in the feedback form (2.2) or (4.12), the corresponding well-posedness and global feedback uniform stabilization result, Theorem 2.1 or Theorem 4.1, holds in general for 1 < q < ∞. Instead, the final, main well-posedness and feedback uniform, local stabilization results, Theorems 2.2 through 2.4, the latter for the nonlinear feedback problem (2.17) or (2.18) corresponding to the original problem (1.1), will require q > 3 to obtain the embedding W 1,q (Ω) ֒→ L ∞ (Ω) in our case of interest d = 3, see (5.24), hence 1 < p < 6 / 5 ; and q > 2, in the d = 2-case, hence 1 < p < 4 / 3 .

Translated Nonlinear Boussinesq Problem and its Abstract Model
PDE Model: We return to Theorem 1.1 which provides an equilibrium triplet {y e , θ e , π e }. Then, we translate by {y e , θ e , π e } the original Boussinesq problem (1.1). Thus we introduce new variables z = y − y e (a d-vector), h = θ − θ e (a scalar), χ = π − π e (a scalar) (1.12) and obtain the translated problem L e (z) = (y e · ∇)z + (z · ∇)y e (Oseen perturbation). (1.13f) Abstract Nonlinear Translated Model. First, for 1 < q < ∞ fixed, the Stokes operator A q in L q σ (Ω) with Dirichlet boundary conditions is defined by (1.14) The operator A q has a compact inverse A −1 q on L q σ (Ω), hence A q has a compact resolvent on L q σ (Ω). Next, we define the heat operator B q in L q (Ω) with homogeneous Dirichlet boundary conditions (1.15) Next, we introduce the first order operator A o,q , This leads to the definition of the Oseen operator for the fluid This leads to the definition of the following operator for the heat (1.20) Then, we define the projection of the nonlinear portion of the fluid operator in (1.13a) Kes,p. 74]).
Then, we define the nonlinear coupled term of the heat equation as Finally, we define the coupling linear terms as bounded operators on L q (Ω), L q σ (Ω) respectively, q > d: [from the heat equation] C θe z = z · ∇θ e , C θe ∈ L(L q σ (Ω), L q (Ω)). (1.24) Next we apply the Helmholtz projector P q on the coupled N-S equation (1.13a), invoke the operators introduced above and obtain the following abstract version of the controlled Boussinesq system Properties of the operator A q will be given in the next Section 1.7.

The linearized w-problem of the translated z-model
Next, still for 1 < q < ∞, we introduce the linearized controlled system of the translated model (1.13), or (1.25), (1.26) in the variable w = {w f , w h } ∈ L q σ (Ω) × L q (Ω) ≡ W q σ (Ω): d dt (1.29e) 1.7 Properties of the operator A q in (1.27) We shall use throughout the following notation (recall (1.5), (1.27) and (1.3)) Remark 1.2. By using the maximal regularity of the heat equation, instead of the state space V q,p (Ω) in (1.3), (1.30) we could take the state space V q,p b to be the product of two Besov spaces, (Ω). Here, the second Besov component is the real interpolation between L q (Ω) and D(B q ), see [P-Sch]. This remark applies to all results involving V q,p , but it will not necessarily be noted explicitly case by case, in order not to overload the notation. Accordingly, we shall look at the operator A q in (1.27) as defined on either space (1.31) In Section 3, we shall omit specification of Ω and simply write W q σ . The following result collects basic properties of the operator A q . It is essentially a corollary of Theorems A.3 and A.4 in Appendix A for the Oseen operator A q , as similar results hold for the operator B q , while the operator C γ and C θe in the definition (1.27) of A q are bounded operators, see (1.23), (1.24).
Theorem 1.2. With reference to the Operator A q in (1.27), (1.31), the following properties hold true: (i) A q is the generator of strongly continuous analytic semigroup on either W q σ (Ω) or V q,p (Ω) for t > 0; (ii) A q possesses the L p -maximal regularity property on either W q σ (Ω) or V q,p (Ω) over a finite interval: Analyticity of e Aq t (resp. e Bqt ) in L q σ (Ω) (resp. L q (Ω)) implies analyticity of e Aqt (resp. e Bqt ) on D(A q ) = D(A q ) (resp. D(B q ) = D(B q )), hence analyticity of e Aqt (resp. e Bqt ) on the interpolation (1.4) in the scalar case).
For the notation of, and the results on, maximal regularity, see [Amann.1], [Dore] [Weis], [Wahl], etc. In particular, we recall that on a Banach space, maximal regularity implies analyticity of the semigroup, [DeS] but not conversely.We refer to Appendix A.
Basic assumption: By Theorem 1.2, the operator A q in (1.27) has the eigenvalues (spectrum) located in a triangular sector of well-known type. Then our basic assumption -which justifies the present paper -is that such operator A q is unstable: that is A q has a finite number, say N , of eigenvalues λ 1 , λ 2 , λ 3 , . . . , λ N on the complex half plane {λ ∈ C : Re λ ≥ 0} which we then order according to their real parts, so that . . , N , being an unstable eigenvalue repeated according to its geometric multiplic- Remark 1.3. This remark is inserted upon request of a referee. Condition (1.33) is intrinsic to the notion of 'stabilization', whereby then one seeks to construct a feedback control that transforms an original unstable problem (with no control) into a stable one. However, as is well-known, the same entire procedure can be employed to enhance at will the stability of an originally stable system (Re λ 1 < 0) by feedback control. Regarding the issue of multiplicity of the eigenvalues, we recall the well-known classical result [Mi], are not needed. [If included, the resulting problem would be an eigenproblem for the Boussinesq operator with over/determination as in (1.34d)]. However, the UCP required in Section 4 to establish the controllability of the finite dimensional projected problem (3.6a) via verification of the Kalman rank condition (4.11) involves the following adjoint problem.
Theorem 1.4. (The required UCP). In the same notation {ϕ, ψ, π} of Theorem 1.3 with the same assumed regularity,let We also note that in taking the adjoint of the operator A q in (1.27) the coupling operators C γ and C θe switch places into their adjoints. See A * q in (B.16) of Appendix B. This fact is critical in obtaining in the adjoint UCP of Theorem 1.4 that only the first d − 1 components {ϕ 1 , ..., ϕ d−1 } of ϕ need to be assumed as vanishing in ω in (1.37d). See Appendix B for more details. Notice, in fact, that if condition ψ = 0 in ω is used in (1.37b), it then follows that the last component ϕ d = 0 in ω because of the form of the vector e = (0, 0..., 1). This along with (1.37d) yield the same overdetermined conditions on ω as in (1.34d). In this sense, Theorem 1.4 is essentially a corollary of the proof of Theorem 1.3 and this justifies the insertion of Theorem 1.3 here. See [T-W.1] for proofs via Carleman type estimates. Here, moreover it is shown that for the case where the subset ω is in particular subtended by an arbitrary portion of the boundary Γ and satisfies some additional geometrical conditions, then only (d − 2) components of the vector ϕ need to be assumed to vanish on ω, thus further relaxing condition (1.37d). In these cases, however we also need a boundary condition weaker than ϕ| Γ = 0. Details are given in Remark 1.5. References [Fa], [F-L] provide a UCP for the Stokes problem with implications on approximate controllability. The UCP property of Theorem 1.3 in the case ψ ≡ 0 [i.e. Oseen's operator in ϕ] has been shown via Carleman's estimates in [B-T.1] by first transforming Ω in a "bent" half-space with a parabolic boundary, next selecting the Melrose-Sjostrand form for the Laplacian, and finally applying the Carleman estimates in integral form from [H]. A different proof, directly on Ω, and this time with no use of the condition ϕ ≡ 0 on Γ, was later given in [T.5], also via use of (different) Carleman-type estimates for the Laplacian. The direct (on Ω) Carlemann-type estimate proof in [T.5], with additional technical modification, can be made to work and establish both Theorem 1.3 and Theorem 1.4 ([T-W.1]). In effect, such reference [T-W.1] provides two additional UCPs, say for the original problem (1.34 a-b-c) with over determination this time on a small subdomain ω subtended by an arbitrary portion of the boundary. A proof yielding, say Theorem 1.3 with d = 2 and with limited regularity of the solution was given in [R-R-R]. Theorem 1.4 is in line with an "observability inequality" for the time dependent problem (1.1) needed in the study of local controllability to the origin or to a trajectory given in [FGIP]. It improves on the prior observability inequality in [Gue]. We shall invoke Theorem 1.4 in Section 4, to verify the Kalman rank condition (4.11). The proof of the implication: Theorem 1.4 =⇒ (4.11) is given Appendix B.
We can now state the main results of the present paper.
Recall that the vector {ϕ, ψ} = {ϕ 1 , ϕ 2 , ..., ϕ d , ψ} in L q σ (Ω) has (d + 1) coordinates, the first d co-ordinates correspond to the fluid space, while the last coordinate corresponds to the heat space. Motivated by the UCP of Theorem 1.4, we shall introduce the space L q σ (Ω) , and similarly for L q (Ω), whereby then we consider the extension of the linearized problem (1.28) to such complexified space L q σ (Ω) × L q (Ω). Thus, henceforth, w will mean w+iw, u will mean u+iũ, w 0 will mean w 0 +iw 0 . Our results would be given in this complexified setting. How to return to real-valued formulation of the results was done in these past reference (see e.g. . Because of space constraints, such real-valued statements will not be explicitly listed on the present paper. We refer to the above references. A main additional feature of the results below is that the feedback control u 1 k corresponding to the fluid equation is of reduced dimension: that is, of dimension (d − 1) rather than of dimension d. This is due to the UCP of Theorem 1.4, as noted in Remark 1.4.

Global well-posedness and uniform exponential stabilization of the lin
See also Remark 1.2.
Recall the space L q σ (Ω) from (1.39) and let likewise ( W q σ ) u N be the space obtained from (W q σ ) u N by omitting the d coordinate from the vectors of (W q σ ) u N . Then, one may construct a finite dimensional feedback operator F , such that, with w = {w f , w h }, w N = P N w and , then the w-problem (1.28) can be rewritten in feedback form on W q σ (Ω) as follows 2) has the following properties: (ii) It is uniformly (exponentially) stable in either of these spaces (iii) Finally, A F,q has maximal L p -regularity up to T = ∞ on either of these spaces: Remark 2.1. We note explicitly that the vector u 1 k acting on the fluid d−dimensional component u is of reduced dimension (d − 1), rather than d.
The proof of Theorem 2.1 begins in Section 3 and proceed through Section 4.
2.2 Local well-posedness and uniform (exponential) null stabilization of the translated nonlinear {z, h}-problem (1.13) or (1.25) by means of a finite dimensional explicit, spectral based feedback control pair localized on ω Starting with the present subsection, the nonlinearity of problem (1.1) will impose for d = 3 the requirement q > 3, see (5.24) below. As our deliberate goal is to obtain the stabilization result for the fluid component y in the space B 2− 2 /p q,p (Ω) which does not recognize boundary conditions, Remark 1.1, then the limitation p < 2q / 2q−1 of this space applies, see (1.11b). In conclusion, our well-posedness and stabilization results will hold under the restriction q > 3, 1 < p < 6 / 5 for d = 3, and q > 2, 1 < p < 4 / 3 for d = 2.
in the following feedback form i.e. subject to a feedback control of the same structure as in the linear are the same vectors as those constructed in Theorem 2.1 and appearing in (2.2).In particular u 1 k has (d − 1) components. There exists a positive constant r 1 > 0 (identified in the proof below in (5.31)) such that if then problem (2.6) defines a unique (fixed point) non-linear semigroup solution in the space where we have set, via (1.18), (1.14) for A q and (1.20), (1.15) for B q : Moreover, we have The space X ∞ p,q,σ × X ∞ p,q defined above is the space of L p -maximal regularity for the generator A F,q . X ∞ p,q,σ is the space of L p -maximal regularity of the Stokes operator, A q in (1.14), see (A.12) and also (A.14) in Appendix A. X ∞ p,q is the space of L p -maximal regularity of the generator B q in (1.15). The proof of Theorem 2.2 is given in Section 5. Theorem 2.3. In the situation of Theorem 2.2, we have that such solution is uniformly stable in (2.14) The proof of Theorem 2.3 is given in Section 6.
2.3 Local well-posedness and uniform (exponential) stabilization of the original nonlinear {y, θ}-problem (1.1) in a neighborhood of an unstable equilibrium solution {y e , θ e }, by means of a finite dimensional explicit, spectral based feedback control pair localized on ω The result of this subsection is an immediate corollary of sub-section 2.2.
Theorem 2.4. Let 1 < p < 6 / 5 , q > 3, d = 3; and 1 < p < 4 / 3 , q > 2, d = 2. Consider the original Boussinesq problem (1.1). Let {y e , θ e } be a given unstable equilibrium solution pair as guaranteed by Theorem 1.1 for the steady state problem (1.2). For a constant ρ > 0, let the initial condition If ρ > 0 is sufficiently small, then (i) for each {y 0 , θ 0 } ∈ V ρ , there exists an interior finite dimensional feedback control pair that is, of the same structure as in the translated {z, h}-problem (2.6), with the same vectors p k , u k in (2.2), such that the closed loop problem corresponding to (1.1) rewritten abstractly after application of the Helmholtz projection P q as . See also Remark 1.2. Thus, here as in the proceeding Theorem 2.1 through Theorem 2.3, we have (ii) Moreover, such solution exponentially stabilizes the equilibrium solution {y e , θ e } in the space B 2− 2 /p q,p (Ω) × L q (Ω) ≡ V q,p (Ω): there exist constants γ > 0 and M γ ≥ 1 such that said solution satisfies 2.4 Comparison with the literature 1. With reference to both the "Motivation" of Subsection 1.1 as well as Subsection 1.3, it was already emphasized that all prior literature on the problem of feedback stabilization of either the Navier-Stokes equations or, subsequently, the Boussinesq system is carried out in a Hilbert-Sobolev setting. As already noted, this treatment is inadequate to obtain finite dimensionality in full generality of the localized tangential boundary feedback control for the 3d-Navier-Stokes equations. 2. In the present contribution, following [L-P-T.1, L-P-T.2], the analysis of the original non-linear problem is carried out in the context of the property of maximal regularity of the linearized feedback w-problem (2.2) or (4.12); that is, of the operator A F,q in (4.12). This applies to both well-posedness (in Section 5) as well as stabilization (in Section 6) of the nonlinear feedback problem. This is in contrast with prior treatments, such as [B-T.1], [Ba], which often rely on chopping the nonlinearity and carrying out a limit process. The maximal regularity approach introduced for stabilization problems in [L-P-T.1] is cleaner and more desirable both technically and conceptually. On the other hand, applicability of Maximal Regularity requires well balanced spaces. Recent developments in this particular area [Weis, S-S, Saa, P-S, K-W.1, K-W.2], etc. were critical for carrying out our analysis.
3. We also point out that, as in [L-T.3, L-T.4] and [L-P-T.1, L-P-T.2] for the Navier-Stokes equations, in the case of the Boussinesq system, the number of needed controls will be related to the more desirable geometric multiplicity, not the larger algebraic multiplicity as in prior treatments, [B-T.1], [Ba,p 276] of the unstable eigenvalues, by using the classical test for controllability of a system in Jordan form [Chen.1], [B-M, p 464]. This allows one also to obtain constructively an explicit form of the finite dimensional feedback control; and, moreover, to show that the feedback control acting on the fluid may be taken of reduced dimension: one unit less, i.e.

4.
We now review the literature on the feedback stabilization of the Boussinesq system. The first contribution is due to [Wang] via internal feedback controls in the Hilbert setting H × L 2 (Ω), where H is defined in (1.17b). The feedback controllers are both infinite-dimensional of the type: u = −k(y − y e ) and v = −k(θ − θ e ), for large k, under technical assumptions on the localization of the controls. Paper [Lef] also studies the feedback stabilization problem with, eventually, localized controls, which again are infinite dimensional; eg defined in terms of a sub-differential. These results are in stark contrast with the present work, where -as noted in point 3 above -first, we establish that the stabilizing control is finite dimensional; and, second, we show that its fluid component is of reduced dimension, as expressed only by means of (d − 1)-components. Finally paper [R-R-R] studies the feedback stabilization problem with, this time, mixed boundary and claims to provide a rigorous treatment of problems studied in [B-H].

5.
We now comment on the additional technical and conceptual difficulties in extending the uniform stabilization problem of [L-P-T.1] for the Navier-Stokes equations to the present Boussinesq system. First, the general strategy to attack the problem of feedback stabilization for parabolic-like dynamics with feedback controllers of any type was introduced in [T.1]. It has since become a standard approach in the literature, which was followed in a variety of parabolic problems [L-T.1], [L-T.2]. In its implementation, however, this general strategy encounters an array of technical challenges that are highly feedback-dependent: the most demanding cases are when the feedback controllers are based on the boundary and even with boundary actuator, subject perhaps to additional requirements (arbitrarily small portion of the boundary, etc). This general strategy is used in [L-P-T.1] as well as in the present paper, with the space B 2− 2 /p q,p (Ω) in (1.11b) being the space of the fluid component where the stabilization is established. Such Besov space with tight indices has the critical property of being 'under the radar of compatibility conditions'. The presence of fluid-thermal coupling produces two level of problems: at the finite dimensional level, based an a UCP with special features; at the infinite dimensional level, based on establishing maximal regularity on a non-Hilbert setting, therefore stronger than analyticity of the underlying semigroup. As described below in point 6, the mathematical machinery used in the present paper could handle an array of other settings: first, a variety of different Boundary Conditions rather than just the non-slip Dirichlet boundary conditions (1.1d); second, much more demanding 'seriously unbounded' coupling operators between the fluid and the thermal equations than the bounded coupling operators -C γ and C θe in (1.23), (1.24) -which are offered by the physically based Boussinesq model (1.1). A key feedback -and problem -dependent source of difficulty encountered in the present Boussinesq model over the Navier-Stokes case of [L-P-T.1] arises, as usual, at the level of testing and establishing controllability of the finite dimensional projected w N -equation in (3.6a); that is, in verifying the full rank condition (4.10). It took some time in the literature to realize that in the cases of "challenging" feedbacks such as the localized feedback of [B-T.1] for the Navier-Stokes equation or else boundary feedback (with sensors and/actuators imposed as being boundary traces), a natural way of checking the resulting Kalman rank condition is to fall into an appropriate Unique Continuation Property for an adjoint problem. In doing so, as noted in Remark 1.4, we extract the benefit of obtaining the closed-loop feedback control acting on the fluid equation of reduced dimension: (d − 1) rather than d. As noted in point 3 above, this conclusion is in line with the open-loop controllability results in [C-G.2], [Gue], [C-G-I-P], [C-L]. It was a referee who suggested that we carry out such a study.
6. In response to a referee's suggestion, we address the following Question: Can our mathematical treatment cover other boundary conditions and more general coupled terms between the fluid and the thermal equations?
The answer is in the affirmative. We may e.g. cover all the B.C.s listed in the 2016 Birkhäuser treatise [P-S], page 338. They refer to a very general Stokes problem (Laplacian replaced by normally strongly elliptic differential operator on a bounded domain, in fact even with a domain with compact boundary, say of class C 3 ). They are referred to as (mathematical definitions are given in [P-S] page 338): (i) no-slip (the ones in our present treatment); (ii) pure slip; (iii) outflow; (iv) free. We now justify our assertion. Whatever the boundary conditions and the coupling terms, one ends up with the 2 × 2 operator matrix of the type of A F,q in (2.5): the two operators A q and B q on the main diagonal are the free dynamic operators of each equation, while the two operators C γ and C θe on the secondary diagonal are the coupling terms. What does our mathematical treatment need? What it needs is that the resulting 2 × 2 operator matrix possess the L p maximal regularity on the desired functional setting [(fluid space) × (heat space)]; eventually, V q,p (Ω) ≡ B 2− 2 /p q,p (Ω) × L q (Ω) in (1.3). Hence, a-fortiori, be the generator of a s.c. analytic semigroup on such setting. Thus, as a first point, we need the statement of Theorem 4.2. In the present setting of problem (1.1), the conclusion of Theorem 4.2 is obtained in the basis of the following reasons: 1a) The operators A q and B q on the main diagonal have L p maximal regularity on each appropriate functional setting, on the basis of, ultimately, Solonnikov's old result for the Stokes problem for the fluid and also the corresponding known property for the heat operator. Thus the corresponding diagonal 2×2 operator matrix has such L p maximal regularity on the desired cross functional setting.
1b) Moreover, in the present physical model (1.1), the two coupling operators C γ and C θe on the secondary diagonal are bounded operators, see (1.23), (1.24), But boundedness of these two coupling operators is much more than it is needed mathematically.

Conclusion:
We can take other homogeneous boundary conditions for the fluid and the heat equations as long as they produce the required L p maximal regularity properties, on the space (1.3). Regarding the fluid equation, each of the above B.C.s (i), (ii), (iii), (iv) does yield L p maximal regularity of the Stokes problem and even for much more general elliptic operators than the Laplacian. See [P-S, Theorem 7.3.1, p339].The Stokes operator is enough as the L p maximal regularity can then be transferred to the Oseen operator. As to the thermal equation, other B.C.s such as Neumann or Robin, beside the Dirichlet B.C., will likewise yield L p maximal regularity for the thermal Laplacian. Moreover, the coupling operators C γ and C θe may be much worse than bounded perturbations as in (1.23), (1.24).It is well known [Dore], [K-W.1] that if an operator enjoys L p maximal regularity, then adding a perturbation up to the same level minus epsilon preserve L p maximal regularity (same as for analyticity). This will allow to greatly extend the model, purely in mathematical grounds, not necessarily based on physical grounds. Our model 1.1 is the one which we found in many PDE-papers, beyond the control/stabilization areas.  [FGIP], [F-S-S], [F], [Gue] for an incomplete sample of recent works, using radically different techniques from those of the present paper (save for the underlying UCP of Theorem 1.4, such as Carleman type inequalities for both Navier-Stokes or Boussinesq equations. On the other hand, the issue of deducing stabilization from controllability in the case of finite dimensional non linear ODE was much investigated in the 70-80's. As is well known, there is a revealing, simple counter example of Brockett: a system of dimension n = 3 with m = 2 controls that is locally controllable, (though its linearization is not controllable), symmetric, yet it violates the condition of asymptotic stability [Br], [Za,p 88]. We also point out that there is extensive work at the PDE-level (or infinite-dimensional level) which shows that controllability implies stabilization under an additional condition (which is often true in many PDE-problems but it is challenging to check) [L-T.6], [L-T.7], [L-T.8], [L-T.9]. However these works refer to hyperbolic or hyperbolic-like dynamics (such as wave equations, dynamic system of elasticity, Schrödinger equations or plate equations).
3 Beginning with the proof of Theorem 2.1, Spectral decomposition of the linearized w-problem (1.28) or (1.29) We return to the assumed starting point of the present paper, which is that the free dynamics operator A q in the open-loop controlled linear system (1.28) is unstable. Its properties are collected in Theorem 1.2. Accordingly, its eigenvalues satisfy the statement which includes their location in (1.33). Denote by P N and P * N (which actually depend on q) the projections given explicitly by where, here Γ (respectively, its conjugate counterpartΓ) is a smooth closed curve that separates the unstable spectrum from the stable spectrum of A q (respectively, A * q ). As in [B-L-T.1, Sect 3.4, p 37], following [T.1], [T.2], we decompose the space W q σ = W q σ (Ω) ≡ L q σ (Ω) × L q (Ω) into the sum of two complementary subspaces (not necessarily orthogonal): where each of the spaces (W q σ ) u N and (W q σ ) s N (which depend on q, but we suppress such dependence) is invariant under A q , and let be the restrictions of A q to (W q σ ) u N and (W q σ ) s N respectively. The original point spectrum (eigenvalues) {λ j } ∞ j=1 of A q is then split into two sets and (W q σ ) u N is the generalized eigenspace of A u q,N in (3.3). The system (1.28) on W q σ ≡ L q σ (Ω) × L q (Ω) can accordingly be decomposed as (3.5) After applying P N and (I − P N ) (which commute with A q ) on (1.28), we obtain via (3.3) respectively.
4 Proof of Theorem 2.1. Global well-posedness and uniform exponential stabilization of the linearized w-problem (1.28) on the Orientation: We shall appeal to several technical developments in [L-P-T.1], where the case of N-S equations has been studied. We will have to adopt and transfer some of its procedures to the case of the Boussinesq system. Thus, properties such as maximal regularity and the entire development for uniform stabilization of the linearized Boussinesq dynamics need to be established. It is worth noticing that while analyticity and maximal regularity are equivalent properties in the Hilbert setting [DeS] this is not so in the Banach setting where maximal regularity is a more general and more delicate property. Moreover, it is only after we establish uniform stabilization that we can claim maximal regularity up to infinity. This needs to be asserted by direct analysis of singular integrals. To proceed, we recall the state space W q σ (Ω) = L q σ (Ω) × L q (Ω) ≡ (W q σ ) u N ⊕ (W q σ ) s N . The unstable uncontrolled operator is A q in (1.27).
The same strategy employed in [L-P-T.1] leading to the linearized w-dynamics of the translated non-linear z-problem in terms of finite dimensional feedback controls can be followed now, where w and z are augmented vectors consisting of a fluid vector component and a scalar thermal component. Thus we first seek to establish that the finite dimensional projection -the w N -equation presently in (3.6a) is controllable on (W q σ ) u N , hence exponentially stabilizable with an arbitrarily large decay rate [Za,p. 44]. Next, one then examines the corresponding ζ N -equation, presently (3.6b), where the arbitrarily large decay rate of the feedback w N -equation combined with the exponential stability on (W q σ ) s N of the s.c. analytic semigroup e A s q,N t yields the desired result. All this works thanks to the present corresponding Unique Continuation Property, this time of the Boussinesq system, that is Theorem 1.4. This implies the Kalman algebraic condition (4.10). A proof is given in Appendix B. Such condition (4.10) is equivalent to linear independence of certain vectors occurring in (4.10) below, so that the finite dimensional projected w N -dynamics satisfies the controllability condition of Kalman or Hautus. See corresponding cases in [T.3], [T.4]. More precisely, the counterpart of the analysis of [L-P-T.1, Section 3 and 4] has now some novel aspects which account for the present additional property that the UCP of Theorem 1.4 involves only the first (d − 1) components of the fluid vector. This conceptual advantage will be responsible for heavier notation as described below.
(where in the last equality we have invoked Remark B.1 in Appendix B) corresponding to the M distinct unstable eigenvalues λ 1 , . . . , λ M of A q and λ 1 , . . . , λ M of A * q respectively, either on W q σ or on V q,p : We now express the eigenvectors Φ * ij in terms of their coordinates, as (d + 1) vectors: With reference to (4.4), we introduce the following corresponding d-vector of the vector Φ * ij . Next,we construct the following matrix U i of size ℓ i × K, K = sup{ℓ i : i = 1, . . . , M } (4.6) Here we have set L q σ (Ω) ≡ the space obtained from L q σ (Ω) after omitting the d-coordinate from the vectors of L q σ (Ω) (4.7b) ( W q σ ) u N ≡ the space obtained from (W q σ ) u N after omitting the d-coordinate from the vectors of (W q σ ) u N . (4.7c) In (4.6), we have defined the duality pairing over ω as (4.9) The controllability Kalman/Hautus algebraic condition of the finite-dimensional projected w Nequation in (3.6a) is given by rank U i = full = ℓ i , i = 1, . . . , M. (4.10) It is proved in Appendix B that the UCP of Theorem 1.4 =⇒ rank condition (4.10). As a consequence, we obtain the following most critical result counterpart of [L-P-T.1, Theorem 6.1].
Theorem 4.1. Let the operator A q have N possibly repeated unstable eigenvalues {λ j } N j=1 as in (1.33) of which M are distinct. Let ℓ i denote the geometric multiplicity of λ i . Set K = sup{ℓ i ; i = 1, · · · , M }. Then, one may construct a feedback operator F such that the w-problem (1.28) can be rewritten in feedback form on W q σ (Ω) or V q,p (Ω) as follows (w N = P N w): w(0) = w 0 (4.12) with D(A F,q ) = D(A q ) where the operator A F,q in (2.2) or (4.12) has the following properties: (i) It is the generator of a s.c. analytic semigroup e A F,q t in the space W q σ (Ω) = L q σ (Ω) × L q (Ω) as well as in the space V q,p (Ω) = B 2− 2 /p q,p (Ω) × L q (Ω).
(iii) Finally, A F,q has maximal L p -regularity up to T = ∞ on either of these spaces: (4.14) The PDE version of the closed-loop abstract model (2.2) or (4.12), is (refer to the open-loop (1.29)) What property (iii) in Theorem 4.1 means explicitly is singled out in the next result. This result will be critically used in the subsequent non linear analysis of Section 5 and 6.
Theorem 4.2. With respect to the operator A F,q in (2.2) or (4.12) of Theorem 4.1, recall Then, the following properties hold true.
The proof of Theorem 2.2 = Theorem 5.1 is accomplished in two steps.
The Banach contraction principle then establishes Theorem 5.1, once we prove Theorems 5.2 and 5.3. These are proved below.
Proof of Theorem 5.2 Step 1: We start from the definition (5.10) of F(b 0 , f ) and invoke the maximal regularity properties (4.20) for e A F,q t and (4.19) for the integral term in (5.10). We then obtain from (5.10) (5.14) Step 2: Regarding the term N q f 1 we can invoke [L-P-T.1, Eq (8.19)] to obtain Regarding the term M q [f 1 ]f 2 , we can trace the proof in [L-P-T.1, from (8.10) −→ (8.18)] (which yielded estimate (5.14)). For the sake of clarity, we shall reproduce the computations in the present case with M q [f 1 ]f 2 = f 1 · ∇f 2 , see (1.22), mutatis mutandis. We shall obtain In fact, let us compute See also [L-P-T.1, Eq (8.14)].
Step 5: We now impose restrictions on the data on the RHS of (5.27): b 0 is in a ball of radius r 1 > 0 in V q,p (Ω) = B 2− 2 /p q,p (Ω)×L q (Ω) and f = {f 1 , f 2 } lies in a ball of radius r > 0 in X ∞ p,q,σ ×X ∞ p,q . We further demand that the final result F(b 0 , f ) shall lie in a ball of radius r > 0 in X ∞ p,q,σ × X ∞ p,q . Thus, we obtain from (5.27) This implies whereby range of values of r −→ interval 0, 1 C , as r 1 ց 0, (5.30) a constraint which is guaranteed by taking r 1 ≤ 1 4C 2 , C being the constant in (5.28)(w.l.o.g C > 1/4). (5.31) We have thus established that by taking r 1 as in (5.31) and subsequently r as in (5.29), then the map This establishes Theorem 5.2.
Step 2: As to the first term or the RHS of (5.33), we can invoke [L-P-T.1, Eq (8.41)] and obtain Regarding the second term on the RHS of (5.33) involving M q , we can track the proof of [L-P-T.1, from (8.28) to (8.41)] (which yielded estimate (5.34)) mutatis mutandis. For the sake of clarity, we shall reproduce the computations in the present case, recalling from (1.22) that M q [f 1 ]f 2 = f 1 · ∇f 2 , M q [g 1 ]g 2 = g 1 · ∇g 2 . We shall obtain In fact, adding and subtracting [L-P-T.1, This is the counterpart Eq (8.36)]. Next we proceed by majorizing the L q (Ω)-norm for ∇f 2 and ∇(f 2 − g 2 ) by the L ∞ (Ω)-norm. We obtain p,q p (5.54) using K p = C p (2 p + 1) and a p + b p ≤ (a + b) p for a > 0, b > 0. Finally, and F(b 0 , f ) is a contraction on the space X ∞ p,q,σ × X ∞ p,q as soon as (5.56) In this case, the map F(b 0 , f ) defined in (5.10) has a fixed point z h in X ∞ p,q,σ × X ∞ p,q : in feedback form, as described by Eq (5.2). Theorem 5.3 and hence Theorem 5.1 are proved.  2) provides a unique solution {z, h} in a small ball of X ∞ p,q,σ × X ∞ p,q . We recall from (4.13) , t ≥ 0 (6.2) by invoking (4.20) on the first semigroup term in (6.10) and the estimate from (6.5) to (6.9) on the Since β < 1, the semigroup property of the evolution implies [Bal] that there are constants M γ ≥ 1, γ > 0 such that with [z 0 , h 0 ] V q,p (Ω) ≤ r 1 = small. This proves (2.14), i.e Theorem 2.3 Remark 6.1. The above computations -(6.17) through (6.19) -can be used to support qualitatively the intuitive expectation that "the larger the decay rate γ 0 in (2.3) =(4.13)= (6.2) of the linearized feedback w-dynamics (2.2), the larger the decay rate γ in ( . By (6.17), we can rewrite (6.18) as: It follows from [Bal,p 178] via the semigroup property that − γ is just below ln β T < 0. (6.22) Pick r 1 > 0 in (6.17) so small that 4 M 2 C 3 r 1 is negligible, so that β is just above M e −γ 0 T , so ln β is just above ln M − γ 0 T , hence ln β T is just above (−γ 0 ) + ln M T . (6.23) Hence, by (6.22), (6.23), (6.24) and the larger γ 0 , the larger is γ, as desired.
Appendix A Some auxiliary results for the Stokes and Oseen operators: analytic semigroup generation, maximal regularity, domains of fractional powers In this section we collect some known results used in the paper. As a prerequisite of the present Appendix A, we make reference to the paragraph Definition of Besov spaces B s q,p on domains of class C 1 as real interpolation of Sobolev spaces, Eqts (1.8)-(1.11) and Remark 1.1.
Theorem A.1. Let d ≥ 2, 1 < q < ∞ and let Ω be a bounded domain in R d of class C 3 . Then (i) the Stokes operator −A q = P q ∆ in (1.14), repeated here as generates a s.c. analytic semigroup e −Aqt on L q σ (Ω). See [Gi.1] and the review paper Theorem 2.8.5 p 17].
(ii) The Oseen operator A q in (1.18) generates a s.c. analytic semigroup e Aqt on L q σ (Ω). This follows as A o,q is relatively bounded with respect to A 1 / 2 q , see (1.16): thus a standard theorem on perturbation of an analytic semigroup generator applies [Pazy,Corollary 2.4,p 81].
(c) The Stokes operator −A q and the Oseen operator A q , 1 < q < ∞ generate s.c. analytic semigroups on the Besov space, from (1.11) Theorem A.1 states that the Stokes operator −A q generates a s.c. analytic semigroup on the space L q σ (Ω), 1 < q < ∞, hence on the space D(A q ) in (1.14) = (A.1), with norm · D(Aq) = A q · L q σ (Ω) as 0 ∈ ρ(A q ). Then, one obtains that the Stokes operator −A q generates a s.c. analytic semigroup on the real interpolation spaces in (A.8). Next, the Oseen operator A q = −(νA q +A o,q ) in (A.2) = (1.18) likewise generates a s.c. analytic semigroup e Aqt on L q σ (Ω) since A o,q is relatively bounded w.r.t. A . Moreover A q generates a s.c. analytic semigroup on D(A q ) = D(A q ) (equivalent norms). Hence A q generates a s.c. analytic semigroup on the real interpolation space of (A.8). Here below, however, we shall formally state the result only in the case 2 − 2 / p < 1 / q . i.e. 1 < p < 2q / 2q−1 , in the space B 2− 2 /p q,p (Ω), as this does not contain B.C., Remark 1.1. The objective of the present paper is precisely to obtain stabilization results on spaces that do not recognize B.C. (d) Space of maximal L p regularity on L q σ (Ω) of the Stokes operator −A q , 1 < p < ∞, 1 < q < ∞ up to T = ∞. We return to the dynamic Stokes problem in {ϕ(t, x), π(t, x)} rewritten in abstract form, after applying the Helmholtz projection P q to (A.10a) and recalling A q in (A.1) = (1.14) as X T p,q,σ ⊂ X T p,q ≡ L p (0, T ; W 2,q (Ω)) ∩ W 1,p (0, T ; L q (Ω)) ֒→ C [0, T ]; B 2− 2 /p q,p (Ω) . (A.14) For a function g such that div g ≡ 0, g| Γ = 0 we have g ∈ X T p,q ⇐⇒ g ∈ X T p,q,σ . The solution of Eq (A.11) is The following is the celebrated result on maximal regularity on L q σ (Ω) of the Stokes problem due originally to Solonnikov [Sol.2] reported in Theorem 2.8.5.(iii) and Theorem 2.10.1 p24 for ϕ 0 = 0], [Saa], [G-G-H.1, Proposition 4.1 , p 1405].