# Uniform stabilization of Boussinesq systems in critical
$\mathbf{L}^{q}$-based Sobolev and Besov spaces by finite dimensional interior
localized feedback controls ††thanks: The research of I.L. and R.T. was
partially supported by the National Science Foundation under Grant
DMS-1713506. The research of B.P. was supported by the ERC advanced grant
668998 (OCLOC) under the EU’s H2020 research program.

Irena Lasiecka Department of Mathematical Sciences, University of Memphis,
Memphis, TN 38152 USA and IBS, Polish Academy of Sciences, Warsaw, Poland.
Buddhika Priyasad Institute for Mathematics and Scientific Computing,
University of Graz, Heinrichstrasse 36, A-8010 Graz, Austria. (b.sembukutti-
liyanage@uni-graz.at). Roberto Triggiani Department of Mathematical Sciences,
University of Memphis, Memphis, TN 38152 USA.

###### Abstract

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
[L-P-T.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.

##  1 Introduction

###  1.1 Controlled dynamic Boussinesq equations

In this paper, we consider the following Boussinesq approximation equations in
a bounded connected region $\Omega$ in $\mathbb{R}^{d}$ with sufficiently
smooth boundary $\Gamma=\partial\Omega$. More specific requirements will be
given below. Let $Q\equiv(0,T)\times\Omega$ and
$\Sigma\equiv(0,T)\times\partial\Omega$ where $T>0$. Further, let $\omega$ be
an arbitrary small open smooth subdomain of the region $\Omega$,
$\omega\subset\Omega$, thus of positive measure. Let $m$ denote the
characteristic function of $\omega$: $m(\omega)\equiv 1,\
m(\Omega/\omega)\equiv 0$.

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
$\mathbf{y,f}\in\mathbf{L}^{q}(\Omega)$ rather than
$y,f\in(L^{q}(\Omega))^{d}$.

The Boussinesq system under the action of two localized interior controls
$m(x)\mathbf{u}(t,x)$ and $m(x)v(t,x)$ supported on
$Q_{\omega}\equiv(0,\infty)\times\omega$ is then

$\displaystyle\mathbf{y}_{t}-\nu\Delta\mathbf{y}+(\mathbf{y}\cdot\nabla)\mathbf{y}-\gamma(\theta-\bar{\theta})\mathbf{e}_{d}+\nabla\pi$
$\displaystyle=m(x)\mathbf{u}(t,x)+\mathbf{f}(x)$ $\displaystyle\text{ in }Q$
(1.1a)
$\displaystyle\theta_{t}-\kappa\Delta\theta+\mathbf{y}\cdot\nabla\theta$
$\displaystyle=m(x)v(t,x)+g(x)$ $\displaystyle\text{ in }Q$ (1.1b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-175.0,-1.0){
$\left\\{\rule{0.0pt}{40.0pt}\right.$}\end{picture}\text{div }\mathbf{y}$
$\displaystyle=0$ $\displaystyle\text{ in }Q$ (1.1c)
$\displaystyle\mathbf{y}=0,\ \theta$ $\displaystyle=0$ $\displaystyle\text{ on
}\Sigma$ (1.1d) $\displaystyle\mathbf{y}(0,x)=\mathbf{y}_{0},\quad\theta(0,x)$
$\displaystyle=\theta_{0}$ $\displaystyle\text{ on }\Omega.$ (1.1e)

In the Boussinesq approximation system, $\mathbf{y}=\\{y_{1},\dots,y_{d}\\}$
represents the fluid velocity, $\theta$ the scalar temperature of the fluid,
$\nu$ the kinematic viscosity coefficient, $\kappa$ the thermal conductivity.
The scalar function $\pi$ is the unknown pressure. The term $\mathbf{e}_{d}$
denotes the vector $(0,\dots,0,1)$. Moreover
$\displaystyle\gamma={}^{\bar{g}}\\!/_{\bar{\theta}}$ where $\bar{g}$ is the
acceleration due to gravity and $\bar{\theta}$ is the reference temperature.
The $d$-vector valued function $\mathbf{f}(x)$ 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 $\mathbf{y}_{0}$ and $\theta_{0}$. Note that
$\mathbf{y}\cdot\nabla\theta=\text{div}(\theta\mathbf{y})$.

The Boussinesq system models heat transfer in a viscous incompressible heat
conducting fluid. It consists of the Navier-Stokes equations (in the vector
velocity $\mathbf{y}$) coupled with the convection-diffusion equation (for the
scalar temperature $\theta$). The external body force $\mathbf{f}(x)$ 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 $\mathbf{u}$ 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
$\mathbf{u}$ will be of reduced dimension $(d-1)$ rather than $d$, while the
feedback heat component of $\mathbf{u}$ 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
equations - whether 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 $\omega$ of the interior $\Omega$, 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 $\mathbf{L}^{3}$ 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 $d\geq 3$) 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 well-posedness 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.

As in these references, the analysis of seeking stabilizing finite dimensional
controls is spectral based [B-L-T.3], [L-T.3], [L-T.4], unlike the Riccati-
based Hilbert approaches originally used in the Navier-Stokes equations
[B-T.1], [B-L-T.1], [B-L-T.2] and also in the Boussinesq case [B-H-H],
following [L-T.5].

###  1.2 Stationary Boussinesq equations

Our starting point is the following result.

######  Theorem 1.1.

Consider the following steady-state Boussinesq system in $\Omega$

$\displaystyle-\nu\Delta\mathbf{y}_{e}+(\mathbf{y}_{e}\cdot\nabla)\mathbf{y}_{e}-\gamma(\theta_{e}-\bar{\theta})\mathbf{e}_{d}+\nabla\pi_{e}$
$\displaystyle=\mathbf{f}(x)$ $\displaystyle\text{in }\Omega$ (1.2a)
$\displaystyle-\kappa\Delta\theta_{e}+\mathbf{y}_{e}\cdot\nabla\theta_{e}$
$\displaystyle=g(x)$ $\displaystyle\text{in }\Omega$ (1.2b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-165.0,5.0){
$\left\\{\rule{0.0pt}{34.0pt}\right.$}\end{picture}\text{div }\mathbf{y}_{e}$
$\displaystyle=0$ $\displaystyle\text{in }\Omega$ (1.2c)
$\displaystyle\mathbf{y}_{e}=0,\ \theta_{e}$ $\displaystyle=0$
$\displaystyle\text{on }\partial\Omega.$ (1.2d)

Let $1<q<\infty$. For any
$\mathbf{f},g\in\mathbf{L}^{q}(\Omega),L^{q}(\Omega)$, there exists a solution
(not necessarily unique)
$(\mathbf{y}_{e},\theta_{e},\pi_{e})\in(\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega))\times(W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega))\times(W^{1,q}(\Omega)/\mathbb{R})$.

See [Ac], [A-A-C.1], [A-A-C.2] for $q\neq 2$. In the Hilbert space setting,
see [C-F], [FT], [S-R-R], [V-R-R], [Kim].

###  1.3 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 $\mathbf{f}(x)$ for the fluid equations, a given heat source
$g(x)$ for the thermal equation, and given viscosity coefficient $\nu$ and
thermal conductivity $\kappa$, the equilibrium solution
$\\{\mathbf{y}_{e},\theta_{e}\\}$ 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 $\mathbb{A}_{q}$ 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 $\\{\mathbf{y}_{0},\theta_{0}\\}$, in the
vicinity of an (unstable) equilibrium solution
$\\{\mathbf{y}_{e},\theta_{e}\\}$ in (1.2), by means of a finite dimensional
localized feedback control pair $\\{m\mathbf{u},mv\\}$. 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
literature – through 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 $\\{\mathbf{y}_{e},\theta_{e}\\}$ by means of
a finite dimensional localized, spectral-based feedback control pair
$\\{m\mathbf{u},mv\\}$, in the important case of initial condition
$\mathbf{y}_{0}$ of low regularity, as technically expressed by
$\mathbf{y}_{0}$ being in a suitable Besov space with tight indices, and
$\theta_{0}$ being in a corresponding $L^{q}$-space $q>d$. More precisely, the
resulting state space for the pair $\\{\mathbf{y},\theta\\}$, where uniform
stabilization will be achieved is the space

$\quad\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega),\ 1<p<\frac{2q}{2q-1};\ q>d,\,d=2,3,$ (1.3)

where $\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ is a suitable
subspace, see below in (1.11b), of the Besov space.

$\big{(}\mathbf{L}^{q}(\Omega),\mathbf{W}^{2,q}(\Omega)\big{)}_{1-{}^{1}\\!/_{p},p}=\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega),\quad
1<p<\frac{2q}{2q-1};\ q>d,\ d=2,3,$ (1.4)

as a real interpolation space between $\mathbf{L}^{q}(\Omega)$ and
$\mathbf{W}^{2,q}(\Omega)$. This setting will be further elaborated after
introducing the Helmhotz decomposition below. In particular, local exponential
stability for the velocity field $\mathbf{y}$ near an equilibrium solution
$\mathbf{y}_{e}$ will be achieved in the topology of the Besov subspace
$\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ in (1.11b). Note the
tight index: $\displaystyle 1<p<{}^{6}\\!/_{5}$ 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 dim$\Omega=d=3$; 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
$\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, $\displaystyle
1<p<{}^{6}\\!/_{5}$ for $q>d=3$. For $d=3$, such space is ‘close’ to
$\mathbf{L}^{3}$. 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 $\mathbf{L}^{3}$ for $d=3$: We now expand on the
issue regarding the ‘criticality’ of the space $\mathbf{L}^{3}(\Omega)$. In
the case of the uncontrolled N-S equations defined on the full space
$\mathbb{R}^{3}$, extensive research efforts have recently lead to the
discovery that the space $\mathbf{L}^{3}(\mathbb{R}^{3})$ is a ‘critical’
space for the issue of well-posedness. Assuming that some divergence free
initial data in $\mathbf{L}^{3}(\mathbb{R}^{3})$ would produce finite time
singularity, then there exists a so-called minimal blow-up initial data in
$\mathbf{L}^{3}(\mathbb{R}^{3})$ [G-K-P], [J-S], [E-S-S]. More details in the
context of the controlled 3-d Navier-Stokes equations are given in [L-P-T.1],
[L-P-T.2].

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 $\Omega=d=3$. This
result will be an extension of [L-P-T.2] in the case of the $3-d$ N-S
equations.

###  1.4 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 $\mathbf{L}^{q}$-based setting is the question of the
existence of a Helmholtz (Leray) projection for the domain $\Omega$ in
$\mathbb{R}^{d}$. More precisely: Given an open set
$\Omega\subset\mathbb{R}^{d}$, the Helmholtz decomposition answers the
question as to whether $\mathbf{L}^{q}(\Omega)$ can be decomposed into a
direct sum of the solenoidal vector space $\mathbf{L}^{q}_{\sigma}(\Omega)$
and the space $\mathbf{G}^{q}(\Omega)$ of gradient fields. Here,

$\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)$
$\displaystyle=\overline{\\{\mathbf{y}\in\mathbf{C}_{c}^{\infty}(\Omega):\text{div}\
\mathbf{y}=0\text{ in }\Omega\\}}^{\left\lVert\cdot\right\rVert_{q}}$ (1.5)
$\displaystyle=\\{\mathbf{g}\in\mathbf{L}^{q}(\Omega):\text{div}\
\mathbf{g}=0;\ \mathbf{g}\cdot\nu=0\text{ on }\partial\Omega\\},$
$\displaystyle\hskip 85.35826pt\text{ for any locally Lipschitz domain
}\Omega\subset\mathbb{R}^{d},d\geq 2$ $\displaystyle\mathbf{G}^{q}(\Omega)$
$\displaystyle=\\{\mathbf{y}\in\mathbf{L}^{q}(\Omega):\mathbf{y}=\nabla p,\
p\in W_{loc}^{1,q}(\Omega)\\}\ \text{where }1\leq q<\infty.$

Both of these are closed subspaces of $\mathbf{L}^{q}$.

######  Definition 1.1.

Let $1<q<\infty$ and $\Omega\subset\mathbb{R}^{n}$ be an open set. We say that
the Helmholtz decomposition for $\mathbf{L}^{q}(\Omega)$ exists whenever
$\mathbf{L}^{q}(\Omega)$ can be decomposed into the direct sum (non-
orthogonal)

$\mathbf{L}^{q}(\Omega)=\mathbf{L}^{q}_{\sigma}(\Omega)\oplus\mathbf{G}^{q}(\Omega).$
(1.6)

The unique linear, bounded and idempotent (i.e. $P_{q}^{2}=P_{q}$) projection
operator
$P_{q}:\mathbf{L}^{q}(\Omega)\longrightarrow\mathbf{L}^{q}_{\sigma}(\Omega)$
having $\mathbf{L}^{q}_{\sigma}(\Omega)$ as its range and
$\mathbf{G}^{q}(\Omega)$ 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 $\text{div}\ \mathbf{y}\equiv 0$. For instance, if such
decomposition exists, the Stokes equation (say the linear version of (1.1a)
with control $\mathbf{u}\equiv 0$ and no coupling, thus $\theta\equiv 0$) can
be formulated as an equation in the $\mathbf{L}^{q}$ setting. Here below we
collect a subset of known results about Helmholtz decomposition. We refer to
[H-S, 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:

  1. (i)

any open set $\Omega\subset\mathbb{R}^{d}$ for $q=2$, i.e. with respect to the
space $\mathbf{L}^{2}(\Omega)$; more precisely, for $q=2$, we obtain the well-
known orthogonal decomposition (in the standard notation, where $\nu$ = unit
outward normal vector on $\Gamma$) [C-F, Prop 1.9, p 8]

$\displaystyle\mathbf{L}^{2}(\Omega)$
$\displaystyle=\mathbf{H}\oplus\mathbf{H}^{\perp}$ (1.7a)
$\displaystyle\mathbf{H}$
$\displaystyle=\\{\boldsymbol{\phi}\in\mathbf{L}^{2}(\Omega):\text{div}\
\boldsymbol{\phi}\equiv 0\text{ in }\Omega;\ \boldsymbol{\phi}\cdot\nu\equiv
0\text{ on }\Gamma\\}$ (1.7b) $\displaystyle\mathbf{H}^{\perp}$
$\displaystyle=\\{\boldsymbol{\psi}\in\mathbf{L}^{2}(\Omega):\boldsymbol{\psi}=\nabla
h,\ h\in H^{1}(\Omega)\\};$ (1.7c)

  2. (ii)

a bounded $C^{1}$-domain in $\mathbb{R}^{d}$ [F-M-M], $1<q<\infty$ [Ga.3,
Theorem 1.1 p 107, Theorem 1.2 p 114] for $C^{2}$-boundary

  3. (iii)

a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{d}\ (d=3)$ and for
$\frac{3}{2}-\epsilon<q<3+\epsilon$ sharp range [F-M-M];

  4. (iv)

a bounded convex domain $\Omega\subset\mathbb{R}^{d},d\geq 2,1<q<\infty$
[F-M-M].

On the other hand, on the negative side, it is known that there exist domains
$\Omega\subset\mathbb{R}^{d}$ such that the Helmholtz decomposition does not
hold for some $q\neq 2$ [M-B].

Assumption (H-D). Henceforth in this paper, we assume that the bounded domain
$\Omega\subset\mathbb{R}^{d}$ under consideration admits a Helmholtz
decomposition for the values of $q,\ 1<q<\infty$, 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
$\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ which is the fluid
component of the state space (1.3) where uniform stabilization takes place.

Definition of Besov spaces $\mathbf{B}^{s}_{q,p}$ on domains of class $C^{1}$
as real interpolation of Sobolev spaces: Let $m$ be a positive integer,
$m\in\mathbb{N},0<s<m,1\leq q<\infty,1\leq p\leq\infty,$ then we define
[G-G-H.1, p 1398] the Besov space

$\mathbf{B}^{s}_{q,p}(\Omega)=(\mathbf{L}^{q}(\Omega),\mathbf{W}^{m,q}(\Omega))_{\frac{s}{m},p}$
(1.8)

as a real interpolation space between $\mathbf{L}^{q}(\Omega)$ and
$\mathbf{W}^{m,q}(\Omega)$.This definition does not depend on $\displaystyle
m\in\mathbb{N}$ [Wahl, p xx]. This clearly gives

$\mathbf{W}^{m,q}(\Omega)\subset\mathbf{B}_{q,p}^{s}(\Omega)\subset\mathbf{L}^{q}(\Omega)\quad\text{
and }\quad\left\lVert\mathbf{y}\right\rVert_{\mathbf{L}^{q}(\Omega)}\leq
C\left\lVert\mathbf{y}\right\rVert_{\mathbf{B}_{q,p}^{s}(\Omega)}.$ (1.9)

We shall be particularly interested in the following special real
interpolation space of the $\mathbf{L}^{q}$ and $\mathbf{W}^{2,q}$ spaces
$\Big{(}m=2,s=2-\frac{2}{p}\Big{)}$:

$\mathbf{B}^{2-\frac{2}{p}}_{q,p}(\Omega)=\big{(}\mathbf{L}^{q}(\Omega),\mathbf{W}^{2,q}(\Omega)\big{)}_{1-\frac{1}{p},p}.$
(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

$\displaystyle\Big{(}\mathbf{L}^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}\mathbf{g}\in\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{
div }\mathbf{g}=0,\ \mathbf{g}|_{\Gamma}=0\Big{\\}}\quad\text{if
}\frac{1}{q}<2-\frac{2}{p}<2$ (1.11a)
$\displaystyle\Big{(}\mathbf{L}^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}\mathbf{g}\in\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{
div }\mathbf{g}=0,\
\mathbf{g}\cdot\nu|_{\Gamma}=0\Big{\\}}\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$
(1.11b) $\displaystyle\hskip 99.58464pt\text{ if
}0<2-\frac{2}{p}<\frac{1}{q};\text{ or }1<p<\frac{2q}{2q-1}.$

######  Remark 1.1.

Notice that, in (1.11b), the condition
$\displaystyle\mathbf{g}\cdot\nu|_{\Gamma}=0$ is an intrinsic condition of the
space $\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)$ in (1.5), not an extra
boundary condition as $\displaystyle\mathbf{g}|_{\Gamma}=0$ in (1.11a).

Orientation: As already noted,ultimately, we shall seek to obtain uniform
feedback stabilization of the fluid component $\mathbf{y}$ in the Besov
subspace $\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, dim
$\displaystyle\Omega=d<q<\infty$, $\displaystyle 1<p<{}^{2q}\\!/_{2q-1}$,
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 $\mathbf{w}$-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
$1<q<\infty$. 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
$\displaystyle\mathbf{W}^{1,q}(\Omega)\hookrightarrow\mathbf{L}^{\infty}(\Omega)$
in our case of interest $d=3$, see (5.24), hence $\displaystyle
1<p<{}^{6}\\!/_{5}$; and $q>2$, in the $d=2$-case, hence $\displaystyle
1<p<{}^{4}\\!/_{3}$.

###  1.5 Translated Nonlinear Boussinesq Problem and its Abstract Model

PDE Model: We return to Theorem 1.1 which provides an equilibrium triplet
$\\{\mathbf{y}_{e},\theta_{e},\pi_{e}\\}$. Then, we translate by
$\\{\mathbf{y}_{e},\theta_{e},\pi_{e}\\}$ the original Boussinesq problem
(1.1). Thus we introduce new variables

$\displaystyle\mathbf{z}=\mathbf{y}-\mathbf{y}_{e}\ \ (\mbox{a
$d$-vector}),\quad h=\theta-\theta_{e}\ \ (\mbox{a
scalar}),\quad\chi=\pi-\pi_{e}\ \ (\mbox{a scalar})$ (1.12)

and obtain the translated problem

$\displaystyle\mathbf{z}_{t}-\nu\Delta\mathbf{z}+(\mathbf{y}_{e}\cdot\nabla)\mathbf{z}+(\mathbf{z}\cdot\nabla)\mathbf{y}_{e}+(\mathbf{z}\cdot\nabla)\mathbf{z}-\gamma
h\mathbf{e}_{d}+\nabla\chi$ $\displaystyle=m\mathbf{u}\quad\text{ in }Q$
(1.13a) $\displaystyle h_{t}-\kappa\Delta h+\mathbf{y}_{e}\cdot\nabla
h+\mathbf{z}\cdot\nabla h+\mathbf{z}\cdot\nabla\theta_{e}$
$\displaystyle=mv\quad\text{ in }Q$ (1.13b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-240.0,-1.0){
$\left\\{\rule{0.0pt}{40.0pt}\right.$}\end{picture}\text{div}\ \mathbf{z}$
$\displaystyle=0\quad\text{ in }Q$ (1.13c) $\displaystyle\mathbf{z}=0,\ h$
$\displaystyle=0\quad\text{ on }\Sigma$ (1.13d)
$\displaystyle\mathbf{z}(0,x)=\mathbf{z}_{0}=\mathbf{y}_{0}-\mathbf{y}_{e},\quad
h(0,x)=h_{0}$ $\displaystyle=\theta-\theta_{e}\ \text{ on }\Omega$ (1.13e)
$\displaystyle
L_{e}(\mathbf{z})=(\mathbf{y}_{e}\cdot\nabla)\mathbf{z}+(\mathbf{z}\cdot\nabla)\mathbf{y}_{e}\quad(\mbox{Oseen
perturbation}).$ (1.13f)

Abstract Nonlinear Translated Model. First, for $1<q<\infty$ fixed, the Stokes
operator $A_{q}$ in $\mathbf{L}^{q}_{\sigma}(\Omega)$ with Dirichlet boundary
conditions is defined by

$A_{q}\mathbf{z}=-P_{q}\Delta\mathbf{z},\quad\mathcal{D}(A_{q})=\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega).$
(1.14)

The operator $A_{q}$ has a compact inverse $A_{q}^{-1}$ on
$\mathbf{L}^{q}_{\sigma}(\Omega)$, hence $A_{q}$ has a compact resolvent on
$\mathbf{L}^{q}_{\sigma}(\Omega)$.

Next, we define the heat operator $B_{q}$ in $L^{q}(\Omega)$ with homogeneous
Dirichlet boundary conditions

$B_{q}h=-\Delta h,\quad\mathcal{D}(B_{q})=W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega).$ (1.15)

Next, we introduce the first order operator $A_{o,q}$,

$A_{o,q}\mathbf{z}=P_{q}[(\mathbf{y}_{e}\cdot\nabla)\mathbf{z}+(\mathbf{z}\cdot\nabla)\mathbf{y}_{e}],\quad\mathcal{D}(A_{o,q})=\mathcal{D}(A_{q}^{{}^{1}\\!/_{2}})\subset\mathbf{L}^{q}_{\sigma}(\Omega),$
(1.16)

where the $\displaystyle{\mathcal{D}}(A^{{}^{1}\\!/_{2}}_{q})$ is defined
explicitly by complex interpolation

$[{\mathcal{D}}(A_{q}),\mathbf{L}^{q}_{\sigma}(\Omega)]_{1-\alpha}={\mathcal{D}}(A_{q}^{\alpha}),\
0<\alpha<1,\ 1<q<\infty;$ (1.17a)
$[{\mathcal{D}}(A_{q}),\mathbf{L}^{q}_{\sigma}(\Omega)]_{\frac{1}{2}}={\mathcal{D}}(A_{q}^{{}^{1}\\!/_{2}})\equiv\mathbf{W}_{0}^{1,q}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega).$
(1.17b)

Thus, $A_{o,q}A_{q}^{-{}^{1}\\!/_{2}}$ is a bounded operator on
$\mathbf{L}^{q}_{\sigma}(\Omega)$, and thus $A_{o,q}$ is bounded on
${\mathcal{D}}(A_{q}^{{}^{1}\\!/_{2}})$

$\left\lVert A_{o,q}f\right\rVert=\left\lVert
A_{o,q}A_{q}^{-{}^{1}\\!/_{2}}A_{q}^{-{}^{1}\\!/_{2}}A_{q}f\right\rVert\leq
C_{q}\left\lVert A_{q}^{{}^{1}\\!/_{2}}f\right\rVert,\quad
f\in{\mathcal{D}}(A_{q}^{{}^{1}\\!/_{2}}).$

This leads to the definition of the Oseen operator for the fluid

${\mathcal{A}}_{q}=-(\nu
A_{q}+A_{o,q}),\quad{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\subset\mathbf{L}^{q}_{\sigma}(\Omega).$
(1.18)

Next, we introduce the first order operator $B_{o,q}$, corresponding to
$B_{q}$:

$B_{o,q}h=\mathbf{y}_{e}\cdot\nabla
h,\quad\mathcal{D}(B_{o,q})=\mathcal{D}(B_{q}^{{}^{1}\\!/_{2}})\subset
L^{q}(\Omega).$ (1.19)

Thus, $B_{o,q}B_{q}^{-{}^{1}\\!/_{2}}$ is a bounded operator on
$L^{q}(\Omega)$, and thus $B_{o,q}$ is bounded on
${\mathcal{D}}(B_{q}^{{}^{1}\\!/_{2}})$

$\left\lVert B_{o,q}f\right\rVert=\left\lVert
B_{o,q}B_{q}^{-{}^{1}\\!/_{2}}B_{q}^{-{}^{1}\\!/_{2}}B_{q}f\right\rVert\leq
C_{q}\left\lVert B_{q}^{{}^{1}\\!/_{2}}f\right\rVert,\quad
f\in{\mathcal{D}}(B_{q}^{{}^{1}\\!/_{2}}).$

This leads to the definition of the following operator for the heat

${\mathcal{B}}_{q}=-(\kappa
B_{q}+B_{o,q}),\quad{\mathcal{D}}({\mathcal{B}}_{q})={\mathcal{D}}(B_{q})\subset
L^{q}(\Omega).$ (1.20)

Then, we define the projection of the nonlinear portion of the fluid operator
in (1.13a)

${\mathcal{N}}_{q}(\mathbf{z})=P_{q}[(\mathbf{z}\cdot\nabla)\mathbf{z}],\quad{\mathcal{D}}({\mathcal{N}}_{q})=\mathbf{W}^{1,q}(\Omega)\cap\mathbf{L}^{\infty}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega).$
(1.21)

(Recall that $W^{1,q}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ for
$q>d=\mbox{dim }\Omega$ [Kes, p. 74]).

Then, we define the nonlinear coupled term of the heat equation as

${\mathcal{M}}_{q}[\mathbf{z}](h)=\mathbf{z}\cdot\nabla
h,\quad{\mathcal{D}}({\mathcal{M}}_{q}[\mathbf{z}])=W^{1,q}(\Omega)\cap
L^{\infty}(\Omega).$ (1.22)

Finally, we define the coupling linear terms as bounded operators on
$L^{q}(\Omega),\mathbf{L}^{q}_{\sigma}(\Omega)$ respectively, $q>d$:

$\displaystyle\text{[from the NS equation]}\quad{\mathcal{C}}_{\gamma}h$
$\displaystyle=-\gamma
P_{q}(h\mathbf{e}_{d}),\quad{\mathcal{C}}_{\gamma}\in{\mathcal{L}}(L^{q}(\Omega),\mathbf{L}^{q}_{\sigma}(\Omega)),$
(1.23) $\displaystyle\text{[from the heat
equation]}\quad{\mathcal{C}}_{\theta_{e}}\mathbf{z}$
$\displaystyle=\mathbf{z}\cdot\nabla\theta_{e},\quad{\mathcal{C}}_{\theta_{e}}\in{\mathcal{L}}(\mathbf{L}^{q}_{\sigma}(\Omega),L^{q}(\Omega)).$
(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

$\displaystyle\frac{d\mathbf{z}}{dt}-{\mathcal{A}}_{q}\mathbf{z}+{\mathcal{N}}_{q}\mathbf{z}+{\mathcal{C}}_{\gamma}h$
$\displaystyle=P_{q}(m\mathbf{u})$ $\displaystyle\text{ in
}\mathbf{L}^{q}_{\sigma}(\Omega)$ (1.25a)
$\displaystyle\quad\frac{dh}{dt}-{\mathcal{B}}_{q}h+{\mathcal{M}}_{q}[\mathbf{z}]h+{\mathcal{C}}_{\theta_{e}}\mathbf{z}$
$\displaystyle=mv$ $\displaystyle\text{ in }{L}^{q}(\Omega)$ (1.25b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-120.0,15.0){
$\left\\{\rule{0.0pt}{45.0pt}\right.$}\end{picture}\mathbf{z}(x,0)$
$\displaystyle=\mathbf{z}_{0}(x)$ $\displaystyle\text{ in
}\mathbf{L}^{q}_{\sigma}(\Omega)$ (1.25c) $\displaystyle h(x,0)$
$\displaystyle=h_{0}(x)$ $\displaystyle\text{ in }L^{q}(\Omega).$ (1.25d)

or in matrix form in $\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{W}^{q}_{\sigma}(\Omega)$

$\displaystyle\frac{d}{dt}\begin{bmatrix}\mathbf{z}\\\ h\end{bmatrix}$
$\displaystyle=\begin{bmatrix}{\mathcal{A}}_{q}&-{\mathcal{C}}_{\gamma}\\\
-{\mathcal{C}}_{\theta_{e}}&{\mathcal{B}}_{q}\end{bmatrix}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}-\begin{bmatrix}{\mathcal{N}}_{q}&0\\\
0&{\mathcal{M}}_{q}[\mathbf{z}]\end{bmatrix}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}+\begin{bmatrix}P_{q}(m\mathbf{u})\\\ mv\end{bmatrix}$ (1.26a)
$\displaystyle\begin{picture}(0.0,0.0)\put(-20.0,14.0){
$\left\\{\rule{0.0pt}{40.0pt}\right.$}\end{picture}\begin{bmatrix}\mathbf{z}(0)\\\
h(0)\end{bmatrix}$ $\displaystyle=\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\in\mathbf{W}^{q}_{\sigma}(\Omega)$ (1.26b)

$\mathbb{A}_{q}=\begin{bmatrix}{\mathcal{A}}_{q}&-{\mathcal{C}}_{\gamma}\\\
-{\mathcal{C}}_{\theta_{e}}&{\mathcal{B}}_{q}\end{bmatrix}:\mathbf{W}^{q}_{\sigma}(\Omega)=\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\supset{\mathcal{D}}(\mathbb{A}_{q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q})\\\
=(\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega))\times(W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega))\longrightarrow\mathbf{W}^{q}_{\sigma}(\Omega).$ (1.27)

Properties of the operator $\displaystyle\mathbb{A}_{q}$ will be given in the
next Section 1.7.

###  1.6 The linearized $\mathbf{w}$-problem of the translated
$\mathbf{z}$-model

Next, still for $1<q<\infty$, we introduce the linearized controlled system of
the translated model (1.13), or (1.25), (1.26) in the variable
$\mathbf{w}=\\{\mathbf{w}_{f},w_{h}\\}\in\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{W}^{q}_{\sigma}(\Omega)$:

$\frac{d}{dt}\begin{bmatrix}\mathbf{w}_{f}\\\
w_{h}\end{bmatrix}=\mathbb{A}_{q}\begin{bmatrix}\mathbf{w}_{f}\\\
w_{h}\end{bmatrix}+\begin{bmatrix}P_{q}(m\mathbf{u})\\\ mv\end{bmatrix}\text{
in }\mathbf{W}^{q}_{\sigma}(\Omega),$ (1.28)

with I.C.
$\\{\mathbf{w}_{f}(0),w_{h}(0)\\}\in\mathbf{W}^{q}_{\sigma}(\Omega)=\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$. Its PDE version is, recalling (1.13)

$\displaystyle\frac{d}{dt}\mathbf{w}_{f}-\nu\Delta\mathbf{w}_{f}+L_{e}(\mathbf{w}_{f})-\gamma
w_{h}\mathbf{e}_{d}+\nabla\chi$ $\displaystyle=m\mathbf{u}$
$\displaystyle\text{ in }Q$ (1.29a)
$\displaystyle\frac{d}{dt}w_{h}-\kappa\Delta w_{h}+\mathbf{y}_{e}\cdot\nabla
w_{h}+\mathbf{w}_{f}\cdot\nabla w_{h}+\mathbf{w}_{f}\cdot\nabla\theta_{e}$
$\displaystyle=mv$ $\displaystyle\text{ in }Q$ (1.29b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-200.0,10.0){
$\left\\{\rule{0.0pt}{50.0pt}\right.$}\end{picture}\text{div }\mathbf{w}_{f}$
$\displaystyle=0$ $\displaystyle\text{ in }Q$ (1.29c)
$\displaystyle\mathbf{w}_{f}\equiv 0,\ w_{h}$ $\displaystyle\equiv 0$
$\displaystyle\text{ on }\Sigma$ (1.29d)
$\displaystyle\mathbf{w}_{f}(0,\cdot)=\mathbf{w}_{f,0};\quad w_{h}(0,\cdot)$
$\displaystyle=w_{h,0}$ $\displaystyle\text{ on }\Omega.$ (1.29e)

###  1.7 Properties of the operator $\mathbb{A}_{q}$ in (1.27)

We shall use throughout the following notation (recall (1.5), (1.27) and
(1.3))

$\mathbf{W}^{q}_{\sigma}(\Omega)\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\quad\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega).$ (1.30)

######  Remark 1.2.

By using the maximal regularity of the heat equation, instead of the state
space $\mathbf{V}^{q,p}(\Omega)$ in (1.3), (1.30) we could take the state
space $\mathbf{V}_{b}^{q,p}$ to be the product of two Besov spaces, i.e.
$\mathbf{V}_{b}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}\times
B^{2-2/p}_{q,p}(\Omega)$. Here, the second Besov component is the real
interpolation between $L^{q}(\Omega)$ and ${\mathcal{D}}({\mathcal{B}}_{q})$,
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 $\mathbb{A}_{q}$ in (1.27) as
defined on either space

$\mathbb{A}_{q}:\mathbf{W}^{q}_{\sigma}(\Omega)\supset{\mathcal{D}}(\mathbb{A}_{q})\to\mathbf{W}^{q}_{\sigma}(\Omega)\qquad\mbox{or}\qquad\mathbb{A}_{q}:\mathbf{V}^{q,p}(\Omega)\supset{\mathcal{D}}(\mathbb{A}_{q})\to\mathbf{V}^{q,p}(\Omega).$
(1.31)

In Section 3, we shall omit specification of $\Omega$ and simply write
$\mathbf{W}^{q}_{\sigma}$. The following result collects basic properties of
the operator $\mathbb{A}_{q}$. It is essentially a corollary of Theorems A.3
and A.4 in Appendix A for the Oseen operator ${\mathcal{A}}_{q}$, as similar
results hold for the operator ${\mathcal{B}}_{q}$, while the operator
${\mathcal{C}}_{\gamma}$ and ${\mathcal{C}}_{\theta_{e}}$ in the definition
(1.27) of $\mathbb{A}_{q}$ are bounded operators, see (1.23), (1.24).

######  Theorem 1.2.

With reference to the Operator $\mathbb{A}_{q}$ in (1.27), (1.31), the
following properties hold true:

  1. (i)

$\displaystyle\mathbb{A}_{q}$ is the generator of strongly continuous analytic
semigroup on either $\mathbf{W}^{q}_{\sigma}(\Omega)$ or
$\mathbf{V}^{q,p}(\Omega)$ for $t>0$;

  2. (ii)

$\mathbb{A}_{q}$ possesses the $L^{p}$-maximal regularity property on either
$\mathbf{W}^{q}_{\sigma}(\Omega)$ or $\mathbf{V}^{q,p}(\Omega)$ over a finite
interval:

$\mathbb{A}_{q}\in MReg_{p}(L^{p}(0,T;*)),\
0<T<\infty,\quad(*)=\mathbf{W}^{q}_{\sigma}(\Omega)\mbox{ or
}\mathbf{V}^{q,p}(\Omega).$ (1.32)

  3. (iii)

$\displaystyle\mathbb{A}_{q}$ has compact resolvent on either
$\mathbf{W}^{q}_{\sigma}(\Omega)$ or $\mathbf{V}^{q,p}(\Omega)$.

Analyticity of $\displaystyle e^{{\mathcal{A}}_{q}t}$ (resp.
$e^{{\mathcal{B}}_{q}t}$) in $\mathbf{L}^{q}_{\sigma}(\Omega)$ (resp.
$L^{q}(\Omega)$) implies analyticity of $\displaystyle e^{{\mathcal{A}}_{q}t}$
(resp. $\displaystyle e^{{\mathcal{B}}_{q}t}$) on
$\displaystyle{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})$ (resp.
$\displaystyle{\mathcal{D}}({\mathcal{B}}_{q})={\mathcal{D}}(B_{q})$), hence
analyticity of $\displaystyle e^{{\mathcal{A}}_{q}t}$ (resp. $\displaystyle
e^{{\mathcal{B}}_{q}t}$) on the interpolation space
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ in
(1.11b). (or in $\displaystyle\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$) in
(1.10) (resp. (1.4) in the scalar case).

For the notation of, and the results on, maximal regularity, see [Amann.1],
[Dore], [G-G-H.1], [H-S], [K-W.1], [K-W.2], [P-S], [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 $\mathbb{A}_{q}$ 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 $\displaystyle\mathbb{A}_{q}$ is unstable: that is $\mathbb{A}_{q}$
has a finite number, say $N$, of eigenvalues
$\lambda_{1},\lambda_{2},\lambda_{3},\dots,\lambda_{N}$ on the complex half
plane $\\{\lambda\in\mathbb{C}:Re~{}\lambda\geq 0\\}$ which we then order
according to their real parts, so that

$\ldots\leq Re~{}\lambda_{N+1}<0\leq Re~{}\lambda_{N}\leq\ldots\leq
Re~{}\lambda_{1},$ (1.33)

each $\lambda_{i},\ i=1,\dots,N$, being an unstable eigenvalue repeated
according to its geometric multiplicity $\ell_{i}$. Let $M$ denote the number
of distinct unstable eigenvalues $\lambda_{i}$ of $\mathbb{A}_{q}$, so that
$\ell_{i}$ is equal to the dimension of the eigenspace corresponding to
$\lambda_{i}$. Instead, $\displaystyle N=\sum_{i=1}^{M}N_{i}$ is the sum of
the corresponding algebraic multiplicity $N_{i}$ of $\lambda_{i}$, where
$N_{i}$ is the dimension of the corresponding generalized eigenspace.

######  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 ($Re~{}\lambda_{1}<0$) by feedback control. Regarding
the issue of multiplicity of the eigenvalues, we recall the well-known
classical result [Mi], [U.1] that the eigenvalues of the Laplacian are simple
generically with respect to the domain.

The ability to stabilize the Boussinesq system (1.1) in the vicinity of the
(unstable) equilibrium solution $\\{\mathbf{y}_{e},\theta_{e}\\}$, by means of
a finite dimensional feedback control pair $\\{\mathbf{u},v\\}$ localized on
the arbitrarily small sub-domain $\omega$, depends (non-surprisingly,
according to past experience [B-L-T.1], [L-T.3], [L-T.4], [T.3]–[T.5]) on an
appropriate Unique Continuation Property. To this end, we quote two results.
They are part of several UCP proved in [T-W.1].

######  Theorem 1.3.

Let $\displaystyle\Phi=\begin{bmatrix}\boldsymbol{\varphi}\\\
\psi\end{bmatrix}\in[\mathbf{W}^{2,q}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega)]\times
W^{2,q}(\Omega),\ \pi\in W^{1,q}(\Omega)$,
$\boldsymbol{\varphi}=[\varphi_{1},\ldots,\varphi_{d}],\psi=\mbox{scalar,
satisfy the following overdetermined problem}$

$\displaystyle-\nu\Delta\boldsymbol{\varphi}+L_{e}(\boldsymbol{\varphi})+\nabla\pi-\gamma\psi\mathbf{e}_{d}$
$\displaystyle=\lambda\boldsymbol{\varphi}$ $\displaystyle\text{in }\Omega$
(1.34a)
$\displaystyle-\kappa\Delta\psi+\mathbf{y}_{e}\cdot\nabla\psi+\boldsymbol{\varphi}\cdot\nabla\theta_{e}$
$\displaystyle=\lambda\psi$ $\displaystyle\text{in }\Omega$ (1.34b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-130.0,6.0){
$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}\text{div}\
\boldsymbol{\varphi}$ $\displaystyle=0$ $\displaystyle\text{in }\Omega$
(1.34c) $\displaystyle\boldsymbol{\varphi}=0,\ \psi$ $\displaystyle=0$
$\displaystyle\text{on }\omega.$ (1.34d)

Then

$\boldsymbol{\varphi}=0,\psi=0,\pi=const\text{ in }\Omega.$ (1.35)

We point out that the B.C

$\boldsymbol{\varphi}|_{\Gamma}=0,\ \psi|_{\Gamma}=0\text{ on }\Gamma$ (1.36)

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 $\\{\boldsymbol{\varphi},\psi,\pi\\}$
of Theorem 1.3 with the same assumed regularity,let

$\displaystyle-\nu\Delta\boldsymbol{\varphi}+L^{\ast}_{e}(\boldsymbol{\varphi})+\nabla\pi+\psi\nabla\theta_{e}$
$\displaystyle=\lambda\boldsymbol{\varphi}$ $\displaystyle\text{in }\Omega$
(1.37a)
$\displaystyle-\kappa\Delta\psi+\mathbf{y}_{e}\cdot\nabla\psi-\gamma\boldsymbol{\varphi}\cdot\mathbf{e}_{d}$
$\displaystyle=\lambda\psi$ $\displaystyle\text{in }\Omega$ (1.37b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-125.0,4.0){
$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}\text{div
}\boldsymbol{\varphi}$ $\displaystyle=0$ $\displaystyle\text{in }\Omega$
(1.37c) $\displaystyle\psi=0,\,\\{\varphi_{1},...,\varphi_{d-1}\\}$
$\displaystyle=0,$ $\displaystyle\text{on }\omega.$ (1.37d)

Then

$\boldsymbol{\varphi}=0,\psi=0,\pi=const\text{ in }\Omega.$ (1.38)

######  Remark 1.4.

Again, for problem (1.37a-b-c), the B.Cs such as (1.36) are not needed. We
note that in (1.37a) the definition of the adjoint of
$L_{e}(\mathbf{z})=(\mathbf{y}_{e}\cdot\nabla)\mathbf{z}+(\mathbf{z}\cdot\nabla)\mathbf{y}_{e}$
in (1.13f) is
$L^{\ast}_{e}(\boldsymbol{\varphi})\equiv(\mathbf{y}_{e}\cdot\nabla)\boldsymbol{\varphi}+\nabla^{\perp}\mathbf{y}_{e}\cdot\boldsymbol{\varphi}$.
We also note that in taking the adjoint of the operator $\mathbb{A}_{q}$ in
(1.27) the coupling operators ${\mathcal{C}}_{\gamma}$ and
${\mathcal{C}}_{\theta_{e}}$ switch places into their adjoints. See
$\mathbb{A}_{q}^{*}$ 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 $\\{\varphi_{1},...,\varphi_{d-1}\\}$ of $\boldsymbol{\varphi}$
need to be assumed as vanishing in $\omega$ in (1.37d). See Appendix B for
more details. Notice, in fact, that if condition $\psi=0\text{ in }\omega$ is
used in (1.37b), it then follows that the last component $\varphi_{d}=0$ in
$\omega$ because of the form of the vector $\mathbf{e}=(0,0...,1)$. This along
with (1.37d) yield the same overdetermined conditions on $\omega$ 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 $\omega$ is in particular subtended by an arbitrary
portion of the boundary $\Gamma$ and satisfies some additional geometrical
conditions, then only $(d-2)$ components of the vector $\boldsymbol{\varphi}$
need to be assumed to vanish on $\omega$, thus further relaxing condition
(1.37d). In these cases, however we also need a boundary condition weaker than
$\boldsymbol{\varphi}|_{\Gamma}=0$. Details are given in [T-W.1, Theorems 1.6
and 1.7]. These results for the adjoint Boussinesq static problem are in line
with the open-loop controllability results in [C-G.2], [Gue], [C-G-I-P] [C-L].

######  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
$\psi\equiv 0$ [i.e. Oseen’s operator in $\boldsymbol{\varphi}$] has been
shown via Carleman’s estimates in [B-T.1] by first transforming $\Omega$ 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 $\Omega$, and this
time with no use of the condition $\boldsymbol{\varphi}\equiv 0$ on $\Gamma$,
was later given in [T.5], also via use of (different) Carleman-type estimates
for the Laplacian. The direct (on $\Omega$) 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 $\omega$
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 $\implies$(4.11) is given Appendix B. We can now state the main
results of the present paper.

Recall that the vector
$\\{\boldsymbol{\varphi},\psi\\}=\\{\varphi_{1},\varphi_{2},...,\varphi_{d},\psi\\}$
in $\mathbf{L}^{q}_{\sigma}(\Omega)$ 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 $\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)$

$\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)\equiv\text{ the space obtained from
}\mathbf{L}^{q}_{\sigma}(\Omega)\text{ after omitting the }\\\
d\text{-coordinate from the vectors of }\mathbf{L}^{q}_{\sigma}(\Omega).$
(1.39)

##  2 Main results

As in our past work [B-T.1], [B-L-T.1], [L-T.4], [L-T.5], [L-P-T.1],
[L-P-T.2], we shall henceforth let $\mathbf{L}^{q}_{\sigma}(\Omega)$ denote
the complexified space
$\mathbf{L}^{q}_{\sigma}(\Omega)+i\mathbf{L}^{q}_{\sigma}(\Omega)$, and
similarly for $L^{q}(\Omega)$, whereby then we consider the extension of the
linearized problem (1.28) to such complexified space
$\mathbf{L}^{q}_{\sigma}(\Omega)\times L^{q}(\Omega)$. Thus, henceforth,
$\mathbf{w}$ will mean $\mathbf{w}+i\tilde{\mathbf{w}}$, $\mathbf{u}$ will
mean $\mathbf{u}+i\tilde{\mathbf{u}}$, $\mathbf{w}_{0}$ will mean
$\mathbf{w}_{0}+i\tilde{\mathbf{w}}_{0}$. 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. [B-T.1], [B-L-T.1], [L-T.4],
[L-P-T.1, Section 2.7]). 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 $\mathbf{u}^{1}_{k}$ 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.

###  2.1 Global well-posedness and uniform exponential stabilization of the
linearized $\mathbf{w}$-problem (1.28) on either the space
$\mathbf{W}^{q}_{\sigma}(\Omega)\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ or the space
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}\times
L^{q}(\Omega)$, $1<q<\infty,1<p<\frac{2q}{2q-1}$.

See also Remark 1.2.

######  Theorem 2.1.

Let the operator $\mathbb{A}_{q}$ in (1.27) have $N$ possibly repeated
unstable eigenvalues $\displaystyle\\{\lambda_{j}\\}_{j=1}^{N}$ as in (1.33),
of which $M$ are distinct. Let $\ell_{i}$ denote the geometric multiplicity of
$\lambda_{i}$. Set $K=\sup\\{\ell_{i};i=1,\cdots,M\\}$. Let
$\displaystyle(\mathbf{W}^{q}_{\sigma})^{u}_{N}$ be the $N$-dimensional
subspace of $\mathbf{W}^{q}_{\sigma}(\Omega)$ defined in (3.2) below. Recall
the space $\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)$ from (1.39) and let
likewise $\displaystyle(\widehat{\mathbf{W}}^{q}_{\sigma})^{u}_{N}$ be the
space obtained from $(\mathbf{W}^{q}_{\sigma})^{u}_{N}$ by omitting the $d$
coordinate from the vectors of
$\displaystyle(\mathbf{W}^{q}_{\sigma})^{u}_{N}$. Then, one may construct a
finite dimensional feedback operator $F$, such that, with
$\displaystyle\mathbf{w}=\\{\mathbf{w}_{f},w_{h}\\},\mathbf{w}_{N}=P_{N}\mathbf{w}$
and

$\begin{bmatrix}\mathbf{u}\\\\[5.69054pt]
v\end{bmatrix}=\begin{bmatrix}F^{1}(\mathbf{w})\\\\[5.69054pt]
F^{2}(\mathbf{w})\end{bmatrix},\
F(\mathbf{w})=\begin{bmatrix}P_{q}(mF^{1}(\mathbf{w}))\\\\[5.69054pt]
mF^{2}(\mathbf{w})\end{bmatrix}=\begin{bmatrix}P_{q}\Big{(}m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}\mathbf{u}^{1}_{k}\big{)}\Big{)}\\\\[5.69054pt]
m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}u^{2}_{k}\big{)}\end{bmatrix},$
(2.1)

with vectors
$[\mathbf{u}^{1}_{k},u^{2}_{k}]\in(\widehat{\mathbf{W}}^{q}_{\sigma})^{u}_{N}\subset\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$, and
$\mathbf{p}_{k}\in((\mathbf{W}^{q}_{\sigma})^{u}_{N})^{*}\subset\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)\times
L^{q^{\prime}}(\Omega)$ (Remark B.1), then the $\mathbf{w}$-problem (1.28) can
be rewritten in feedback form on $\mathbf{W}^{q}_{\sigma}(\Omega)$ as follows

$\frac{d\mathbf{w}}{dt}=\frac{d}{dt}\begin{bmatrix}\mathbf{w}_{f}\\\\[5.69054pt]
w_{h}\end{bmatrix}=\mathbb{A}_{q}\mathbf{w}+F(\mathbf{w})=\mathbb{A}_{q}\mathbf{w}+\begin{bmatrix}P_{q}\Big{(}m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}\mathbf{u}^{1}_{k}\big{)}\Big{)}\\\\[5.69054pt]
m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}u^{2}_{k}\big{)}\end{bmatrix}=\mathbb{A}_{F,q}\mathbf{w};\\\
\mathbf{w}(0)=\mathbf{w}_{0},$ (2.2)

with
$\displaystyle{\mathcal{D}}(\mathbb{A}_{F,q})={\mathcal{D}}(\mathbb{A}_{q})$
where the operator $\displaystyle\mathbb{A}_{F,q}$ in (2.2) has the following
properties:

  1. (i)

It is the generator of a s.c. analytic semigroup $\displaystyle
e^{\mathbb{A}_{F,q}t}$ in the space
$\mathbf{W}^{q}_{\sigma}(\Omega)\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ as well as in the space
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}\times
L^{q}(\Omega)$, see also Remark 1.2.

  2. (ii)

It is uniformly (exponentially) stable in either of these spaces

$\left\lVert e^{\mathbb{A}_{F,q}t}\mathbf{w}_{0}\right\rVert_{(\cdot)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\mathbf{w}_{0}\right\rVert_{(\cdot)}$
(2.3)

where $(\cdot)$ denotes either
$\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{W}^{q}_{\sigma}(\Omega)$ or else
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)$. In (2.3), $\gamma_{0}$ is any
positive number such that $Re~{}\lambda_{N+1}<-\gamma_{0}<0$.

  3. (iii)

Finally, $\mathbb{A}_{F,q}$ has maximal $L^{p}$-regularity up to $T=\infty$ on
either of these spaces:

$\mathbb{A}_{F,q}\in MReg_{p}(L^{p}(0,\infty;\ \cdot\ )),\text{ where
}(\cdot)\text{ denotes}\begin{picture}(0.0,0.0)\put(-250.0,-10.0){
$\left\\{\rule{0.0pt}{15.0pt}\right.$}\end{picture}$ (2.4)

either $\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{W}^{q}_{\sigma}(\Omega)$ or else
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)$. ∎

######  Remark 2.1.

We note explicitly that the vector $\mathbf{u}^{1}_{k}$ acting on the fluid
$d-$dimensional component $\mathbf{u}$ 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 $\\{\mathbf{z},h\\}$-problem (1.13) or (1.25) by
means of a finite dimensional explicit, spectral based feedback control pair
localized on $\omega$

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
$\mathbf{y}$ in the space
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ which
does not recognize boundary conditions, Remark 1.1, then the limitation
$\displaystyle p<{}^{2q}\\!/_{2q-1}$ of this space applies, see (1.11b). In
conclusion, our well-posedness and stabilization results will hold under the
restriction $\displaystyle q>3,1<p<{}^{6}\\!/_{5}$ for $d=3$, and
$\displaystyle q>2,1<p<{}^{4}\\!/_{3}$ for $d=2$.

######  Theorem 2.2.

Let $\displaystyle d=2,3,\ q>d,\ 1<p<\frac{2q}{2q-1}$. Consider the nonlinear
$\\{\mathbf{z},h\\}$-problem (1.25) in the following feedback form

$\frac{d}{dt}\begin{bmatrix}\mathbf{z}\\\\[5.69054pt]
h\end{bmatrix}=\mathbb{A}_{F,q}\begin{bmatrix}\mathbf{z}\\\\[5.69054pt]
h\end{bmatrix}-\begin{bmatrix}{\mathcal{N}}_{q}&0\\\\[5.69054pt]
0&{\mathcal{M}}_{q}[\mathbf{z}]\end{bmatrix}\begin{bmatrix}\mathbf{z}\\\\[5.69054pt]
h\end{bmatrix};\quad\mathbb{A}_{F,q}=\begin{bmatrix}{\mathcal{A}}_{q}&-{\mathcal{C}}_{\gamma}\\\\[5.69054pt]
-{\mathcal{C}}_{\theta_{e}}&{\mathcal{B}}_{q}\end{bmatrix}+F=\mathbb{A}_{q}+F$
(2.5)

see (2.2). Specifically,

$\displaystyle\frac{d\mathbf{z}}{dt}-{\mathcal{A}}_{q}\mathbf{z}+{\mathcal{C}}_{\gamma}h+{\mathcal{N}}_{q}\mathbf{z}$
$\displaystyle=P_{q}\Bigg{(}m\bigg{(}\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix},\mathbf{p}_{k}\bigg{)}_{\omega}\mathbf{u}^{1}_{k}\bigg{)}\Bigg{)}$
(2.6a) $\displaystyle\begin{picture}(0.0,0.0)\put(-20.0,20.0){
$\left\\{\rule{0.0pt}{40.0pt}\right.$}\end{picture}\frac{dh}{dt}-{\mathcal{B}}_{q}h+{\mathcal{C}}_{\theta_{e}}\mathbf{z}+{\mathcal{M}}_{q}[\mathbf{z}]h$
$\displaystyle=m\bigg{(}\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix},\mathbf{p}_{k}\bigg{)}_{\omega}u^{2}_{k}\bigg{)}$ (2.6b)

i.e. subject to a feedback control of the same structure as in the linear
$\mathbf{w}=\\{\mathbf{w}_{f},w_{h}\\}$-dynamics (2.2). Here,
$\mathbf{p}_{k},\mathbf{u}_{k}=[\mathbf{u}_{k}^{1},u_{k}^{2}]$ are the same
vectors as those constructed in Theorem 2.1 and appearing in (2.2).In
particular $\mathbf{u}^{1}_{k}$ has $(d-1)$ components. There exists a
positive constant $r_{1}>0$ (identified in the proof below in (5.31)) such
that if

$\left\lVert\\{\mathbf{z}_{0},h_{0}\\}\right\rVert_{\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)}<r_{1},$ (2.7)

then problem (2.6) defines a unique (fixed point) non-linear semigroup
solution in the space

$\\{\mathbf{z},h\\}\in\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}\equiv
L^{p}(0,\infty;{\mathcal{D}}(\mathbb{A}_{F,q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q}))\cap
W^{1,p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega)),$ (2.8)

where we have set, via (1.18), (1.14) for ${\mathcal{A}}_{q}$ and (1.20),
(1.15) for ${\mathcal{B}}_{q}$:

$\displaystyle\mathbf{z}\in\mathbf{X}^{\infty}_{p,q,\sigma}$
$\displaystyle\equiv L^{p}(0,\infty;{\mathcal{D}}({\mathcal{A}}_{q})\cap
W^{1,p}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega))$ (2.9) $\displaystyle\equiv
L^{p}(0,\infty;\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega))\cap
W^{1,p}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega))$ $\displaystyle h\in
X^{\infty}_{p,q}$ $\displaystyle\equiv
L^{p}(0,\infty;{\mathcal{D}}({\mathcal{B}}_{q})\cap
W^{1,p}(0,\infty;L^{q}(\Omega))$ (2.10) $\displaystyle\equiv
L^{p}(0,\infty;W^{2,q}(\Omega)\cap W^{1,q}_{0}(\Omega))\cap
W^{1,p}(0,\infty;L^{q}(\Omega)).$

Moreover, we have

$\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}$ $\displaystyle\hookrightarrow
C([0,\infty];\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega))$ (2.11)
$\displaystyle X^{\infty}_{p,q}$ $\displaystyle\hookrightarrow
C([0,\infty];L^{q}(\Omega))$ (2.12)

so that

$\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}\hookrightarrow
C([0,\infty];\mathbf{V}^{q,p}(\Omega)),$ (2.13)

recalling the embedding, called trace theorem [Amann.2, Theorem 4.10.2, p 180,
BUC for $T=\infty$], [P-S], as in [L-P-T.1, Eq. (1.30)]. See also (A.14) in
Appendix A.

The space $\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$ defined
above is the space of $L^{p}$-maximal regularity for the generator
$\mathbb{A}_{F,q}$. $\mathbf{X}^{\infty}_{p,q,\sigma}$ 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. $X^{\infty}_{p,q}$ is the space of
$L^{p}$-maximal regularity of the generator ${\mathcal{B}}_{q}$ 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 the space
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$, see also Remark 1.2: there exists
$\widetilde{\gamma}>0,M_{\widetilde{\gamma}}>0$ such that said solution
satisfies

$\left\lVert\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}(t)\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq
M_{\widetilde{\gamma}}e^{-\widetilde{\gamma}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)},\quad t\geq 0.$
(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 $\\{\mathbf{y},\theta\\}$-problem (1.1) in a neighborhood
of an unstable equilibrium solution $\\{\mathbf{y}_{e},\theta_{e}\\}$, by
means of a finite dimensional explicit, spectral based feedback control pair
localized on $\omega$

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 $\\{\mathbf{y}_{e},\theta_{e}\\}$
be a given unstable equilibrium solution pair as guaranteed by Theorem 1.1 for
the steady state problem (1.2). For a constant $\rho>0$, let the initial
condition $\\{\mathbf{y}_{0},\theta_{0}\\}$ in (1.1e) be in
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$, see also Remark 1.2, and satisfy

$\boldsymbol{{\mathcal{V}}}_{\rho}\equiv\Big{\\{}\\{\mathbf{y}_{0},\theta_{0}\\}\in\mathbf{V}^{q,p}(\Omega):\left\lVert\mathbf{y}_{0}-\mathbf{y}_{e}\right\rVert_{\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+\left\lVert\theta_{0}-\theta_{e}\right\rVert_{L^{q}(\Omega)}\leq\rho\Big{\\}},\quad\rho>0.$
(2.15)

If $\rho>0$ is sufficiently small, then

  1. (i)

for each
$\\{\mathbf{y}_{0},\theta_{0}\\}\in\boldsymbol{{\mathcal{V}}}_{\rho}$, there
exists an interior finite dimensional feedback control pair

$\begin{bmatrix}\mathbf{u}\\\\[17.07164pt]
v\end{bmatrix}=\begin{bmatrix}F^{1}\left(\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix}\right)\\\\[8.53581pt]
F^{2}\left(\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix}\right)\end{bmatrix}=F\left(\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix}\right)=\sum_{k=1}^{K}\left(P_{N}\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix},\mathbf{p}_{k}\right)_{\omega}\mathbf{u}_{k}$
(2.16)

that is, of the same structure as in the translated
$\\{\mathbf{z},h\\}$-problem (2.6), with the same vectors $\displaystyle
p_{k},\mathbf{u}_{k}$ in (2.2), such that the closed loop problem
corresponding to (1.1)

$\displaystyle\mathbf{y}_{t}-\nu\Delta\mathbf{y}+(\mathbf{y}\cdot\nabla)\mathbf{y}-\gamma(\theta-\bar{\theta})\mathbf{e}_{d}+\nabla\pi$
$\displaystyle=m\left(F^{1}\left(\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix}\right)\right)+\mathbf{f}(x)\text{ in }Q$
(2.17a)
$\displaystyle\theta_{t}-\kappa\Delta\theta+\mathbf{y}\cdot\nabla\theta$
$\displaystyle=m\left(F^{2}\left(\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix}\right)\right)+g(x)\text{ in }Q$ (2.17b)
$\displaystyle\text{div}\ \mathbf{y}$ $\displaystyle=0\text{ in }Q$ (2.17c)
$\displaystyle\begin{picture}(125.0,-30.0)\put(-20.0,40.0){$\left\\{\rule{0.0pt}{60.0pt}\right.$}\end{picture}\mathbf{y}=0,\quad\theta$
$\displaystyle=0\text{ on }\Sigma$ (2.17d)
$\displaystyle\mathbf{y}|_{t=0}=\mathbf{y}_{0},\quad\theta|_{t=0}$
$\displaystyle=\theta_{0}\text{ in }\Omega,$ (2.17e)

rewritten abstractly after application of the Helmholtz projection $P_{q}$ as

$\displaystyle\mathbf{y}_{t}+\nu
A_{q}\mathbf{y}+{\mathcal{N}}_{q}\mathbf{y}+C_{\gamma}(\theta-\bar{\theta})$
$\displaystyle=P_{q}\left[m\left(F^{1}\left(\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix}\right)\right)+\mathbf{f}(x)\right]$ (2.18a)
$\left\\{\rule{0.0pt}{60.0pt}\right.$
$\displaystyle=P_{q}\left[m\left(\sum_{k=1}^{K}\left(P_{N}\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix},\mathbf{p}_{k}\right)_{\omega}\mathbf{u}^{1}_{k}\right)+\mathbf{f}(x)\right]$
(2.18b) $\displaystyle\theta_{t}-\kappa
B_{q}\theta+{\mathcal{M}}_{q}[\mathbf{y}]\theta$
$\displaystyle=m\left(\sum_{k=1}^{K}\left(P_{N}\begin{bmatrix}\mathbf{y}-\mathbf{y}_{e}\\\
\theta-\theta_{e}\end{bmatrix},\mathbf{p}_{k}\right)_{\omega}u^{2}_{k}\right)+g(x)$
(2.18c) $\displaystyle\\{\mathbf{y}(0)=\mathbf{y}_{0},\
\theta(0)=\theta_{0}\\}$
$\displaystyle\in\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)$ (2.18d)

has a unique solution $\\{\mathbf{y},\theta\\}\in
C\big{(}[0,\infty);\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\big{)}.$ See also Remark 1.2. Thus, here as in the proceeding
Theorem 2.1 through Theorem 2.3, we have

$\displaystyle
K=\sup\\{\ell_{i};i=1,\cdots,M\\},\mathbf{u}_{k}=[\mathbf{u}^{1}_{k},u^{2}_{k}]$
(2.19) $\left\\{\rule{0.0pt}{25.0pt}\right.$
$\displaystyle\mathbf{u}^{1}_{k}=[u^{(1)}_{k},u^{(2)}_{k},...,u^{(d-1)}_{k}].$

  2. (ii)

Moreover, such solution exponentially stabilizes the equilibrium solution
$\\{\mathbf{y}_{e},\theta_{e}\\}$ in the space
$\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)$: there exist constants
$\widetilde{\gamma}>0$ and $M_{\widetilde{\gamma}}\geq 1$ such that said
solution satisfies

$\left\lVert\mathbf{y}(t)-\mathbf{y}_{e}\right\rVert_{\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+\left\lVert\theta(t)-\theta_{e}\right\rVert_{L^{q}(\Omega)}\leq
M_{\widetilde{\gamma}}e^{-\widetilde{\gamma}t}\left(\left\lVert\mathbf{y}_{0}-\mathbf{y}_{e}\right\rVert_{\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+\right.\\\
\left.\left\lVert\theta_{0}-\theta_{e}\right\rVert_{L^{q}(\Omega)}\right),$
(2.20)

$t\geq 0,\\{\mathbf{y}_{0},\theta_{0}\\}\in\boldsymbol{{\mathcal{V}}}_{\rho}$.
Once the neighborhood $\boldsymbol{{\mathcal{V}}}_{\rho}$ is obtained to
ensure the well-posedness, then the values of $M_{\widetilde{\gamma}}$ and
$\widetilde{\gamma}$ do not depend on ${\mathcal{V}}_{\rho}$ and
$\widetilde{\gamma}$ can be made arbitrarily large through a suitable
selection of the feedback operator $F$.

See Remark 6.1 comparing $\widetilde{\gamma}$ in (2.20) with $\gamma_{0}$ in
(2.3).

###  2.4 Comparison with the literature

$\mathbf{1}$. 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. This obstacle then
motivated the introduction of the $\mathbf{L}^{q}$-Sobolev/Besov setting in
[L-P-T.2] with tight indices, see (1.4), (1.11b), that does not recognize
boundary conditions, see Remark 1.1 , in order to solve affirmatively such
open problem on the finite dimensionality in 3d-Navier-Stokes tangential
boundary feedback stabilization. Reference [L-P-T.1] on localized interior
controls sets the preparatory stage for the more demanding boundary control
case in [L-P-T.2].

$\mathbf{2}$. 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
$\mathbf{w}$-problem (2.2) or (4.12); that is, of the operator
$\mathbb{A}_{F,q}$ 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.

$\mathbf{3}$. 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. $d-1$, than the fluid component of dimension $d$. This is due to the
Unique Continuation Property of the adjoint problem, as stated in Theorem 1.4,
Remark 1.4. This an additional contribution of the present paper. This
dimension reduction in the closed-loop feedback stabilizing fluid control is
in line with the open-loop controllability results in [C-G.2], [Gue], [C-G-
I-P] [C-L]. The authors thank a referee for suggesting to investigate this
possibility, which turned out to be successful. Such investigation also led to
paper [T-W.1].

$\mathbf{4}$. 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 $\mathbf{H}\times L^{2}(\Omega$),
where $\mathbf{H}$ is defined in (1.17b). The feedback controllers are both
infinite-dimensional of the type: $\mathbf{u}=-k(\mathbf{y}-\mathbf{y}_{e})$
and $v=-k(\theta-\theta_{e}),$ 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].

$\mathbf{5}$. 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
$\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ 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 -
${\mathcal{C}}_{\gamma}$ and ${\mathcal{C}}_{\theta_{e}}$ 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 $\mathbf{w}_{N}$\- 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.

$\mathbf{6}$. 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 $\mathcal{C}^{3}$). 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 $2\times 2$ operator matrix of the type of
$\mathbb{A}_{F,q}$ in (2.5): the two operators ${\mathcal{A}}_{q}$ and
${\mathcal{B}}_{q}$ on the main diagonal are the free dynamic operators of
each equation, while the two operators ${\mathcal{C}}_{\gamma}$ and
${\mathcal{C}}_{\theta_{e}}$ on the secondary diagonal are the coupling terms.
What does our mathematical treatment need? What it needs is that the resulting
$2\times 2$ operator matrix possess the $L^{p}$ maximal regularity on the
desired functional setting [(fluid space) $\times$ (heat space)]; eventually,
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$ 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 ${\mathcal{A}}_{q}$ and ${\mathcal{B}}_{q}$ 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 $2\times 2$ 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
${\mathcal{C}}_{\gamma}$ and ${\mathcal{C}}_{\theta_{e}}$ 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
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$ in (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 ${\mathcal{C}}_{\gamma}$ and
${\mathcal{C}}_{\theta_{e}}$ 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.

$\mathbf{7.}$ Question: With boundary conditions leading to global
controllability, is it possible - a referee asked - to get global
stabilization with controls as in this paper? It is not easy to answer this
question in full generality. First, there are several works on local exact
controllability to the origin or to trajectories, see [C-G-I-P], [C-1], [C-2],
[C-G.1], [C-G.2], [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 $\mathbf{w}$-problem (1.28) or (1.29)

We return to the assumed starting point of the present paper, which is that
the free dynamics operator $\mathbb{A}_{q}$ 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 [K-1, p 178], [B-T.1], [B-L-T.1]

$\displaystyle P_{N}$ $\displaystyle=-\frac{1}{2\pi
i}\int_{\boldsymbol{\Gamma}}\left(\lambda
I-\mathbb{A}_{q}\right)^{-1}d\lambda:\mathbf{W}^{q}_{\sigma}\text{ onto
}(\mathbf{W}^{q}_{\sigma})^{u}_{N}\subset\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ (3.1a) $\displaystyle P_{N}^{*}$ $\displaystyle=-\frac{1}{2\pi
i}\int_{\bar{\boldsymbol{\Gamma}}}\left(\lambda
I-\mathbb{A}_{q}^{*}\right)^{-1}d\lambda:(\mathbf{W}^{q}_{\sigma})^{*}\text{
onto
}[(\mathbf{W}^{q}_{\sigma})^{u}_{N}]^{*}\subset\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)\times
L^{q^{\prime}}(\Omega),$ (3.1b)

where, here ${\boldsymbol{\Gamma}}$ (respectively, its conjugate counterpart
$\bar{{\boldsymbol{\Gamma}}}$) is a smooth closed curve that separates the
unstable spectrum from the stable spectrum of $\mathbb{A}_{q}$ (respectively,
$\mathbb{A}_{q}^{*}$). As in [B-L-T.1, Sect 3.4, p 37], following [T.1],
[T.2], we decompose the space
$\mathbf{W}^{q}_{\sigma}=\mathbf{W}^{q}_{\sigma}(\Omega)\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ into the sum of two complementary subspaces (not necessarily
orthogonal):

$\mathbf{W}^{q}_{\sigma}=(\mathbf{W}^{q}_{\sigma})^{u}_{N}\oplus(\mathbf{W}^{q}_{\sigma})^{s}_{N};\
(\mathbf{W}^{q}_{\sigma})^{u}_{N}\equiv P_{N}\mathbf{W}^{q}_{\sigma};\
(\mathbf{W}^{q}_{\sigma})^{s}_{N}\equiv(I-P_{N})\mathbf{W}^{q}_{\sigma};\\\
\text{ dim }(\mathbf{W}^{q}_{\sigma})^{u}_{N}=N$ (3.2)

where each of the spaces $(\mathbf{W}^{q}_{\sigma})^{u}_{N}$ and
$(\mathbf{W}^{q}_{\sigma})^{s}_{N}$ (which depend on $q$, but we suppress such
dependence) is invariant under $\mathbb{A}_{q}$, and let

$\mathbb{A}^{u}_{q,N}=P_{N}\mathbb{A}_{q}=\mathbb{A}_{q}|_{(\mathbf{W}^{q}_{\sigma})^{u}_{N}};\quad\mathbb{A}^{s}_{q,N}=(I-P_{N})\mathbb{A}_{q}=\mathbb{A}_{q}|_{(\mathbf{W}^{q}_{\sigma})^{s}_{N}}$
(3.3)

be the restrictions of $\mathbb{A}_{q}$ to $(\mathbf{W}^{q}_{\sigma})^{u}_{N}$
and $(\mathbf{W}^{q}_{\sigma})^{s}_{N}$ respectively. The original point
spectrum (eigenvalues) $\\{\lambda_{j}\\}_{j=1}^{\infty}$ of $\mathbb{A}_{q}$
is then split into two sets

$\sigma(\mathbb{A}^{u}_{q,N})=\\{\lambda_{j}\\}_{j=1}^{N};\quad\sigma(\mathbb{A}^{s}_{q,N})=\\{\lambda_{j}\\}_{j=N+1}^{\infty},$
(3.4)

and $(\mathbf{W}^{q}_{\sigma})^{u}_{N}$ is the generalized eigenspace of
$\mathbb{A}^{u}_{q,N}$ in (3.3). The system (1.28) on
$\mathbf{W}^{q}_{\sigma}\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ can accordingly be decomposed as

$\mathbf{w}=\mathbf{w}_{N}+\boldsymbol{\zeta}_{N},\quad\mathbf{w}_{N}=P_{N}\mathbf{w},\quad\boldsymbol{\zeta}_{N}=(I-P_{N})\mathbf{w}.$
(3.5)

After applying $P_{N}$ and $(I-P_{N})$ (which commute with $\mathbb{A}_{q}$)
on (1.28), we obtain via (3.3)

$\text{on
}(\mathbf{W}^{q}_{\sigma})^{u}_{N}:\mathbf{w}^{\prime}_{N}-\mathbb{A}^{u}_{q,N}\mathbf{w}_{N}=P_{N}\begin{bmatrix}P_{q}(m\mathbf{u})\\\
mv\end{bmatrix};\quad\mathbf{w}_{N}(0)=P_{N}\begin{bmatrix}\mathbf{w}_{f}(0)\\\
w_{h}(0)\end{bmatrix}$ (3.6a) $\text{on
}(\mathbf{W}^{q}_{\sigma})^{s}_{N}:\boldsymbol{\zeta}^{\prime}_{N}-\mathbb{A}^{s}_{q,N}\boldsymbol{\zeta}_{N}=(I-P_{N})\begin{bmatrix}P_{q}(m\mathbf{u})\\\
mv\end{bmatrix};\quad\boldsymbol{\zeta}_{N}(0)=(I-P_{N})\begin{bmatrix}\mathbf{w}_{f}(0)\\\
w_{h}(0)\end{bmatrix}$ (3.6b)

respectively.

##  4 Proof of Theorem 2.1. Global well-posedness and uniform exponential
stabilization of the linearized $\mathbf{w}$-problem (1.28) on the space
$\mathbf{W}^{q}_{\sigma}(\Omega)\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ or the space
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$

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
$\displaystyle\mathbf{W}^{q}_{\sigma}(\Omega)=\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\equiv(\mathbf{W}^{q}_{\sigma})^{u}_{N}\oplus(\mathbf{W}^{q}_{\sigma})^{s}_{N}$.
The unstable uncontrolled operator is $\displaystyle\mathbb{A}_{q}$ in (1.27).

The same strategy employed in [L-P-T.1] leading to the linearized
$\mathbf{w}$-dynamics of the translated non-linear $\mathbf{z}$-problem in
terms of finite dimensional feedback controls can be followed now, where
$\mathbf{w}$ and $\mathbf{z}$ 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
$\mathbf{w}_{N}$-equation presently in (3.6a) is controllable on
$(\mathbf{W}^{q}_{\sigma})^{u}_{N}$, hence exponentially stabilizable with an
arbitrarily large decay rate [Za, p. 44]. Next, one then examines the
corresponding $\boldsymbol{\zeta}_{N}$-equation, presently (3.6b), where the
arbitrarily large decay rate of the feedback $\mathbf{w}_{N}$-equation
combined with the exponential stability on $(\mathbf{W}^{q}_{\sigma})^{s}_{N}$
of the s.c. analytic semigroup $\displaystyle e^{\mathbb{A}_{q,N}^{s}t}$
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 $\mathbf{w}_{N}$-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.

For each $i=1,\dots,M$, we denote by
$\displaystyle\\{\boldsymbol{\Phi}_{ij}\\}_{j=1}^{\ell_{i}},\\{\boldsymbol{\Phi}^{*}_{ij}\\}_{j=1}^{\ell_{i}}$
the normalized, linearly independent eigenfunctions of $\mathbb{A}_{q}$,
respectively $\mathbb{A}_{q}^{*}$, say, on

$\mathbf{W}^{q}_{\sigma}(\Omega)\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\mbox{ and }\\\
(\mathbf{W}^{q}_{\sigma}(\Omega))^{*}\equiv(\mathbf{L}^{q}_{\sigma}(\Omega))^{\prime}\times(L^{q}(\Omega))^{\prime}=\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)\times
L^{q^{\prime}}(\Omega),\quad\frac{1}{q}+\frac{1}{q^{\prime}}=1,$ (4.1)

(where in the last equality we have invoked Remark B.1 in Appendix B)
corresponding to the $M$ distinct unstable eigenvalues
$\lambda_{1},\ldots,\lambda_{M}$ of $\mathbb{A}_{q}$ and
$\overline{\lambda}_{1},\ldots,\overline{\lambda}_{M}$ of $\mathbb{A}_{q}^{*}$
respectively, either on $\mathbf{W}^{q}_{\sigma}$ or on $\mathbf{V}^{q,p}$:

$\displaystyle\mathbb{A}_{q}\boldsymbol{\Phi}_{ij}$
$\displaystyle=\lambda_{i}\boldsymbol{\Phi}_{ij}\in{\mathcal{D}}(\mathbb{A}_{q})=[\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega)]\times[W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega)]$ (4.2)
$\displaystyle\mathbb{A}_{q}^{*}\boldsymbol{\Phi}_{ij}^{*}$
$\displaystyle=\bar{\lambda}_{i}\boldsymbol{\Phi}_{ij}^{*}\in{\mathcal{D}}(\mathbb{A}_{q}^{*})=[\mathbf{W}^{2,q^{\prime}}(\Omega)\cap\mathbf{W}^{1,q^{\prime}}_{0}(\Omega)\cap\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)]\times[W^{2,q^{\prime}}(\Omega)\cap
W^{1,q^{\prime}}_{0}(\Omega)].$ (4.3)

We now express the eigenvectors $\boldsymbol{\Phi^{*}_{ij}}$ in terms of their
coordinates, as $(d+1)$ vectors:

$\boldsymbol{\Phi^{*}_{ij}}=\\{\boldsymbol{\varphi^{*}_{ij}},\psi^{*}_{ij}\\}=\\{\varphi_{ij}^{*(1)},\varphi_{ij}^{*(2)},...,\varphi_{ij}^{*(d-1)},\varphi_{ij}^{*(d)},\psi^{*}_{ij}\\}\,,\text{
a }(d+1)\text{-vector}.$ (4.4)

With reference to (4.4), we introduce the following corresponding d-vector

$\boldsymbol{\widehat{\Phi}^{*}_{ij}=\\{\boldsymbol{\widehat{\varphi}^{*}_{ij}}},\psi^{*}_{ij}\\}=\\{\varphi_{ij}^{*(1)},\varphi_{ij}^{*(2)},...\varphi_{ij}^{*(d-1)},\psi^{*}_{ij}\\}\,,\text{
a }d\text{-vector},$ (4.5)

obtained from $\boldsymbol{\Phi^{*}_{ij}}$ by omitting the d-component
$\varphi_{ij}^{*(d)}$ of the vector $\boldsymbol{\Phi^{*}_{ij}}$. Next,we
construct the following matrix $U_{i}$ of size $\ell_{i}\times K,\
K=\sup\\{\ell_{i}:i=1,\dots,M\\}$

$U_{i}=\begin{bmatrix}(\mathbf{u}_{1},\widehat{\Phi}_{i1}^{*})_{\omega}&\dots&(\mathbf{u}_{K},\widehat{\Phi}_{i1}^{*})_{\omega}\\\\[2.84526pt]
(\mathbf{u}_{1},\widehat{\Phi}_{i2}^{*})_{\omega}&\dots&(\mathbf{u}_{K},\widehat{\Phi}_{i2}^{*})_{\omega}\\\
\vdots&\ddots&\vdots\\\
(\mathbf{u}_{1},\widehat{\Phi}_{i\ell_{i}}^{*})_{\omega}&\dots&(\mathbf{u}_{K},\widehat{\Phi}_{i\ell_{i}}^{*})_{\omega}\\\
\end{bmatrix}:\ell_{i}\times K.$ (4.6)

Here we have set

$\mathbf{u}_{k}=[\mathbf{u}_{k}^{1},u_{k}^{2}]=[(u_{k}^{1})^{(1)},(u_{k}^{1})^{(2)}...(u_{k}^{1})^{(d-1)},u_{k}^{2}]\in(\widehat{\mathbf{W}}^{q}_{\sigma})^{u}_{N}\subset\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ (4.7a) $\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)\equiv\text{
the space obtained from }\mathbf{L}^{q}_{\sigma}(\Omega)\text{ after omitting
the }\\\ d\text{-coordinate from the vectors of
}\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)$ (4.7b)
$(\widehat{\mathbf{W}}^{q}_{\sigma})^{u}_{N}\equiv\text{ the space obtained
from }(\mathbf{W}^{q}_{\sigma})^{u}_{N}\text{ after omitting the }\\\
d\text{-coordinate from the vectors of }(\mathbf{W}^{q}_{\sigma})^{u}_{N}.$
(4.7c)

In (4.6), we have defined the duality pairing over $\omega$ as

$\displaystyle(\mathbf{u}_{k},\widehat{\Phi}^{*}_{ij})_{\omega}$
$\displaystyle=\begin{pmatrix}\begin{bmatrix}\mathbf{u}^{1}_{k}\\\
u^{2}_{k}\end{bmatrix},\begin{bmatrix}\widehat{\boldsymbol{\varphi}}^{*}_{ij}\\\
\psi^{*}_{ij}\end{bmatrix}\end{pmatrix}_{\omega}=\int_{\omega}[\mathbf{u}^{1}_{k}\cdot\widehat{\boldsymbol{\varphi}}^{*}_{ij}+u^{2}_{k}\psi^{*}_{ij}]d\omega$
$\displaystyle=(\mathbf{u}^{1}_{k},\widehat{\boldsymbol{\varphi}}^{*}_{i1})_{\widehat{\mathbf{L}}^{q}(\omega),\widehat{\mathbf{L}}^{q^{\prime}}(\omega)}+(u^{2}_{k},\psi^{*}_{i1})_{L^{q}(\omega),L^{q^{\prime}}(\omega)}$
(4.8) $\displaystyle=\int_{\omega}\begin{bmatrix}(u_{k}^{1})^{(1)}\\\
(u_{k}^{1})^{(2)}\\\ \vdots\\\ (u_{k}^{1})^{(d-1)}\\\
u_{k}^{2}\end{bmatrix}\cdot\begin{bmatrix}\varphi_{ij}^{*(1)}\\\
\varphi_{ij}^{*(2)}\\\ \vdots\\\ \varphi_{ij}^{*(d-1)}\\\
\psi^{*}_{ij}\end{bmatrix}\,d\omega.$ (4.9)

The controllability Kalman/Hautus algebraic condition of the finite-
dimensional projected $\mathbf{w}_{N}$-equation in (3.6a) is given by

$\text{rank }U_{i}=\text{full}=\ell_{i},\quad i=1,\dots,M.$ (4.10)

It is proved in Appendix B that the UCP of Theorem 1.4$\implies$ 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 $\mathbb{A}_{q}$ have $N$ possibly repeated unstable
eigenvalues $\\{\lambda_{j}\\}_{j=1}^{N}$ as in (1.33) of which $M$ are
distinct. Let $\ell_{i}$ denote the geometric multiplicity of $\lambda_{i}$.
Set $K=\sup\\{\ell_{i};i=1,\cdots,M\\}$. Then, one may construct a feedback
operator $F$

$F(\mathbf{w})=\begin{bmatrix}P_{q}\Big{(}m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}\mathbf{u}^{1}_{k}\big{)}\Big{)}\\\\[5.69054pt]
m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}u^{2}_{k}\big{)}\end{bmatrix}$
(4.11)

with vectors
$[\mathbf{u}^{1}_{k},u^{2}_{k}]\in(\widehat{\mathbf{W}}^{q}_{\sigma})^{u}_{N}\subset\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$, (so that $\mathbf{u}^{1}_{k}$ is $(d-1)$ dimensional, see
(4.7c)) and
$\mathbf{p}_{k}\in((\mathbf{W}^{q}_{\sigma})^{u}_{N})^{*}\subset\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)\times
L^{q^{\prime}}(\Omega)$ such that the $\mathbf{w}$-problem (1.28) can be
rewritten in feedback form on $\mathbf{W}^{q}_{\sigma}(\Omega)$ or
$\mathbf{V}^{q,p}(\Omega)$ as follows ($\mathbf{w}_{N}=P_{N}\mathbf{w}$):

$\frac{d\mathbf{w}}{dt}=\frac{d}{dt}\begin{bmatrix}\mathbf{w}_{f}\\\\[5.69054pt]
w_{h}\end{bmatrix}=\mathbb{A}_{q}\mathbf{w}+F(\mathbf{w})=\mathbb{A}_{q}\mathbf{w}+\begin{bmatrix}P_{q}\Big{(}m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}\mathbf{u}^{1}_{k}\big{)}\Big{)}\\\\[5.69054pt]
m\big{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}u^{2}_{k}\big{)}\end{bmatrix}=\mathbb{A}_{F,q}\mathbf{w};\\\
\mathbf{w}(0)=\mathbf{w}_{0}$ (4.12)

with
$\displaystyle{\mathcal{D}}(\mathbb{A}_{F,q})={\mathcal{D}}(\mathbb{A}_{q})$
where the operator $\displaystyle\mathbb{A}_{F,q}$ in (2.2) or (4.12) has the
following properties:

  1. (i)

It is the generator of a s.c. analytic semigroup $\displaystyle
e^{\mathbb{A}_{F,q}t}$ in the space
$\mathbf{W}^{q}_{\sigma}(\Omega)=\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ as well as in the space
$\mathbf{V}^{q,p}(\Omega)=\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$.

  2. (ii)

It is uniformly (exponentially) stable in either of these spaces

$\left\lVert e^{\mathbb{A}_{F,q}t}\mathbf{w}_{0}\right\rVert_{(\cdot)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\mathbf{w}_{0}\right\rVert_{(\cdot)}\begin{picture}(0.0,0.0)\put(-230.0,-10.0){
$\left\\{\rule{0.0pt}{22.0pt}\right.$}\end{picture}$ (4.13)

where $(\cdot)$ denotes either
$\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{W}^{q}_{\sigma}(\Omega)$ or else
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)$.
In (4.13), $\gamma_{0}$ is any positive number such that
$Re~{}\lambda_{N+1}<-\gamma_{0}<0$.

  3. (iii)

Finally, $\mathbb{A}_{F,q}$ has maximal $L^{p}$-regularity up to $T=\infty$ on
either of these spaces:

$\mathbb{A}_{F,q}\in MReg(L^{p}(0,\infty;\ \cdot\ ))\text{ where
}(\cdot)\text{ denotes}\begin{picture}(0.0,0.0)\put(-240.0,-10.0){
$\left\\{\rule{0.0pt}{18.0pt}\right.$}\end{picture}$ (4.14)

either $\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{W}^{q}_{\sigma}(\Omega)$ or else
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)$ ∎

The PDE version of the closed-loop abstract model (2.2) or (4.12), is (refer
to the open-loop (1.29))

$\displaystyle\frac{d}{dt}\mathbf{w}_{f}-\nu\Delta\mathbf{w}_{f}+L_{e}(\mathbf{w}_{f})-\gamma
w_{h}\mathbf{e}_{d}+\nabla\chi$
$\displaystyle=m\bigg{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}\mathbf{u}^{1}_{k}\bigg{)}\text{
in }Q$ (4.15a) $\displaystyle\frac{d}{dt}w_{h}-\kappa\Delta
w_{h}+\mathbf{y}_{e}\cdot\nabla w_{h}+\mathbf{w}_{f}\cdot\nabla
w_{h}+\mathbf{w}_{f}\cdot\nabla\theta_{e}$
$\displaystyle=m\bigg{(}\sum_{k=1}^{K}(\mathbf{w}_{N},\mathbf{p}_{k})_{\omega}u^{2}_{k}\bigg{)}\text{
in }Q$ (4.15b) $\displaystyle\begin{picture}(0.0,0.0)\put(-205.0,30.0){
$\left\\{\rule{0.0pt}{70.0pt}\right.$}\end{picture}\text{div }\mathbf{w}_{f}$
$\displaystyle=0\text{ in }Q$ (4.15c) $\displaystyle\mathbf{w}_{f}\equiv 0,\
w_{h}$ $\displaystyle\equiv 0\text{ on }\Sigma$ (4.15d)
$\displaystyle\mathbf{w}_{f}(0,\cdot)=\mathbf{w}_{f,0};\quad w_{h}(0,\cdot)$
$\displaystyle=w_{h,0}\text{ on }\Omega.$ (4.15e)

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 $\displaystyle\mathbb{A}_{F,q}$ in (2.2) or
(4.12) of Theorem 4.1, recall

$\mathbf{W}^{q}_{\sigma}(\Omega)\equiv\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega),\quad\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega),\ 1<p<\frac{2q}{2q-1}.$ (4.16)

Then, the following properties hold true.

  1. (i)

$\displaystyle\mathbf{F}$
$\displaystyle\longrightarrow\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\mathbf{F}(\tau)d\tau:\text{
continuous }$ $\displaystyle L^{p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))$
$\displaystyle\longrightarrow
L^{p}(0,\infty;{\mathcal{D}}(\mathbb{A}_{F,q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q}))$
(4.17)

whereby then automatically from (4.12)

$\hskip
85.35826ptL^{p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))\longrightarrow
W^{1,p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))$ (4.18)

and ultimately, on the space of maximal regularity for
$\mathbf{w}=\\{\mathbf{w}_{f},w_{h}\\}$ w.r.t. the operator
$\mathbb{A}_{F,q}$, see (2.8)

$\hskip
28.45274ptL^{p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))\longrightarrow\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}\equiv L^{p}(0,\infty;{\mathcal{D}}(\mathbb{A}_{F,q}))\cap
W^{1,p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega)).$ (4.19)

  2. (ii)

The s.c. analytic uniformly stable semigroup $\displaystyle
e^{\mathbb{A}_{F,q}t}$ on the space
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$ satisfies

$e^{\mathbb{A}_{F,q}t}:\text{ continuous
}\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)\longrightarrow\\\
\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}\equiv
L^{p}(0,\infty;{\mathcal{D}}(\mathbb{A}_{F,q}))\cap
W^{1,p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))$ (4.20)

or equivalently, recalling
${\mathcal{D}}(\mathbb{A}_{F,q})={\mathcal{D}}(\mathbb{A}_{q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q})$
in (1.27), or (2.9),(2.10):

$\displaystyle\longrightarrow\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$ (4.21) $\displaystyle\equiv
L^{p}(0,\infty,[\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega)]\times$
$\displaystyle\qquad[W^{2,q}(\Omega)\cap W^{1,q}_{0}(\Omega)])\cap
W^{1,p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))\hookrightarrow
C([0,\infty];\mathbf{V}^{q,p}(\Omega)).$ (4.22)

see (2.11)-(2.13). Equivalently, in summary, with reference to the
$\mathbf{w}$-equation (2.2), $\mathbf{w}=\\{\mathbf{w}_{f},w_{h}\\}$ for the
following estimates hold true for $\displaystyle 1<p<\frac{2q}{2q-1}$:

$\displaystyle
C_{0}\left\lVert\mathbf{w}\right\rVert_{C(0,\infty;\mathbf{V}^{q,p}(\Omega))}$
$\displaystyle\leq\left\lVert\mathbf{w}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}+\left\lVert\pi\right\rVert_{Y^{\infty}_{p,q}}$
$\displaystyle\leq\left\lVert\mathbf{w}_{t}\right\rVert_{L^{p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))}+\left\lVert\mathbb{A}_{F,q}w\right\rVert_{L^{p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))}+\left\lVert\pi\right\rVert_{Y^{\infty}_{p,q}}$
$\displaystyle\leq
C_{1}\big{\\{}\left\lVert\mathbf{F}\right\rVert_{L^{p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))}+\left\lVert\mathbf{w}_{0}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\big{\\}}.$
(4.23)

where $Y^{\infty}_{p,q}$ is defined in (A.13) of Appendix A.

For the first inequality in (4.23) we have recalled the embedding, called
trace theorem [Amann.2, Theorem 4.10.2, p. 180, BUC for $T=\infty$], [P-S], as
in [L-P-T.1, Eq. (1.30)]. See also (5.23) below.

##  5 Proof of Theorem 2.2. Well-posedness on
$\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$ of the non-linear
$\begin{bmatrix}\mathbf{z}\\\ h\end{bmatrix}$ -dynamics in feedback form

In this section we return to the translated non-linear
$\displaystyle\begin{bmatrix}\mathbf{z}\\\ h\end{bmatrix}$-dynamics (1.25) or
(1.26) and apply to it the feedback control pair $\\{\mathbf{u},v\\}$

$\begin{bmatrix}P_{q}m(\mathbf{u})\\\\[14.22636pt]
m(v)\end{bmatrix}=\begin{bmatrix}P_{q}\Bigg{(}m\bigg{(}\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix},\mathbf{p}_{k}\bigg{)}_{\omega}\mathbf{u}^{1}_{k}\bigg{)}\Bigg{)}\\\\[5.69054pt]
m\bigg{(}\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix},\mathbf{p}_{k}\bigg{)}_{\omega}u^{2}_{k}\bigg{)}\end{bmatrix}=F\left(\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}\right)$ (5.1)

that is, of the same structure as the feedback $F$ in (4.11) identified on the
RHS of the linearized
$\displaystyle\mathbf{w}=\begin{bmatrix}\mathbf{w}_{f}\\\
w_{h}\end{bmatrix}$-dynamics (4.12). This is the feedback operator which
produced the s.c. analytic, uniformly stable semigroup $\displaystyle
e^{\mathbb{A}_{F,q}t}$ on
$\displaystyle\mathbf{W}^{q}_{\sigma}(\Omega)=\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ or on
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}\times
L^{q}(\Omega)$, (Theorem 4.1) possessing $L^{p}$-maximal regularity on these
spaces up to $T=\infty$; see (2.4) or (4.14). Thus, returning to (1.25) or
(1.26), in this section we consider the following feedback nonlinear problem,
see (2.5)

$\frac{d}{dt}\begin{bmatrix}\mathbf{z}\\\\[5.69054pt]
h\end{bmatrix}=\mathbb{A}_{F,q}\begin{bmatrix}\mathbf{z}\\\\[5.69054pt]
h\end{bmatrix}-\begin{bmatrix}{\mathcal{N}}_{q}&0\\\\[5.69054pt]
0&{\mathcal{M}}_{q}[\mathbf{z}]\end{bmatrix}\begin{bmatrix}\mathbf{z}\\\\[5.69054pt]
h\end{bmatrix};\quad\mathbb{A}_{F,q}=\begin{bmatrix}{\mathcal{A}}_{q}&-{\mathcal{C}}_{\gamma}\\\\[5.69054pt]
-{\mathcal{C}}_{\theta_{e}}&{\mathcal{B}}_{q}\end{bmatrix}+F=\mathbb{A}_{q}+F$
(5.2)

specifically

$\displaystyle\frac{d\mathbf{z}}{dt}-{\mathcal{A}}_{q}\mathbf{z}+{\mathcal{C}}_{\gamma}h+{\mathcal{N}}_{q}\mathbf{z}$
$\displaystyle=P_{q}\Bigg{(}m\bigg{(}\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix},\mathbf{p}_{k}\bigg{)}_{\omega}\mathbf{u}^{1}_{k}\bigg{)}\Bigg{)}$
(5.3a) $\displaystyle\begin{picture}(0.0,0.0)\put(-20.0,20.0){
$\left\\{\rule{0.0pt}{40.0pt}\right.$}\end{picture}\frac{dh}{dt}-{\mathcal{B}}_{q}h+{\mathcal{C}}_{\theta_{e}}\mathbf{z}+{\mathcal{M}}_{q}[\mathbf{z}]h$
$\displaystyle=m\bigg{(}\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix},\mathbf{p}_{k}\bigg{)}_{\omega}u^{2}_{k}\bigg{)}.$ (5.3b)

The variation of parameter formula for Eq (5.2) is

$\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{z}(\tau)\\\
{\mathcal{M}}_{q}[\mathbf{z}]h(\tau)\end{bmatrix}d\tau.$ (5.4)

We already know from (2.3) or (4.13) that for
$\displaystyle\\{\mathbf{z}_{0},h_{0}\\}\in\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega),\ 1<p<\frac{2q}{2q-1}$, we have:
there is $M_{\gamma_{0}}$ such that

$\left\lVert e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)},\ t\geq 0$ (5.5)

with $M_{\gamma_{0}}$ possibly depending on $p,q$. Maximal $L^{p}$-regularity
properties corresponding to the solution operator formula (5.4) were
established in Theorem 4.2. Accordingly, for

$\displaystyle\mathbf{b}_{0}$
$\displaystyle\equiv\\{\mathbf{z}_{0},h_{0}\\}\in\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\equiv\mathbf{V}^{q,p}(\Omega)$ (5.6) $\displaystyle\mathbf{f}$
$\displaystyle\equiv\\{\mathbf{f}_{1},f_{2}\\}\in\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}\equiv L^{p}(0,\infty,{\mathcal{D}}(\mathbb{A}_{F,q}))\cap
W^{1,p}(0,\infty;\mathbf{W}^{q}_{\sigma}(\Omega))$ (5.7)
$\displaystyle{\mathcal{D}}(\mathbb{A}_{F,q})$
$\displaystyle={\mathcal{D}}(\mathbb{A}_{q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q})=$
$\displaystyle\hskip
56.9055pt[\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega)]\times[W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega)]$ (5.8) $\displaystyle\mathbf{W}^{q}_{\sigma}(\Omega)$
$\displaystyle=\mathbf{L}^{q}_{\sigma}(\Omega)\times L^{q}(\Omega)$
$\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}$ $\displaystyle\equiv
L^{p}(0,\infty;\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega))\cap
W^{1,p}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega));$ (5.9a) $\displaystyle
X^{\infty}_{p,q}$ $\displaystyle\equiv L^{p}(0,\infty;W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega))\cap W^{1,q}(0,\infty;L^{q}(\Omega)),$ (5.9b)

as in (2.8)-(2.10) or (4.21), we define the operator

${\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})\equiv
e^{\mathbb{A}_{F,q}t}\mathbf{b}_{0}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{f}_{1}(\tau)\\\
{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}(\tau)\end{bmatrix}d\tau.$ (5.10)

The main result of this section is Theorem 2.2 restated as

######  Theorem 5.1.

Let $\displaystyle d=2,3,\ q>d,\ 1<p<\frac{2q}{2q-1}$. There exists a positive
constant $r_{1}>0$ (identified in the proof below in (5.31)) such that if

$\left\lVert\mathbf{b}_{0}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}=\left\lVert\\{\mathbf{z}_{0},h_{0}\\}\right\rVert_{\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)}<r_{1},$ (5.11)

then the operator ${\mathcal{F}}$ in (5.10) has a unique fixed point non-
linear semigroup solution in $\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$, see (2.8)-(2.10) or (4.19)–(4.21) or (5.9a)–(5.9b)

${\mathcal{F}}\begin{pmatrix}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix},\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}\end{pmatrix}=\begin{bmatrix}\mathbf{z}\\\ h\end{bmatrix},\
\text{or}\ \begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{z}(\tau)\\\
{\mathcal{M}}_{q}[\mathbf{z}]h(\tau)\end{bmatrix}d\tau,$ (5.12)

which therefore is the unique solution of problem (5.2) = (5.3) in
$\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$ defined
in (2.8)-(2.10) or (4.19)–(4.21) or (5.9a)–(5.9b).

The proof of Theorem 2.2 = Theorem 5.1 is accomplished in two steps.

Step 1:

######  Theorem 5.2.

Let $d=2,3,\ q>d$ and $\displaystyle 1<p<\frac{2q}{2q-1}$. There exists a
positive constant $r_{1}>0$ (identified in the proof below (5.31)) and a
subsequent constant $r>0$ (identified in the proof below in (5.29)) depending
on $r_{1}>0$ and a constant $C$ in (5.28), such that with
$\displaystyle\left\lVert\mathbf{b}_{0}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}<r_{1}$
as in (5.11), the operator
$\displaystyle{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})$ in (5.10) maps a ball
$B(0,r)$ in $\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$ into itself. ∎

Theorem 5.1 will follow then from Theorem 5.2 after establishing that

Step 2:

######  Theorem 5.3.

Let $d=2,3,\ q>d$ and $\displaystyle 1<p<\frac{2q}{2q-1}$. There exists a
positive constant $r_{1}>0$ such that if
$\displaystyle\left\lVert\mathbf{b}_{0}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}<r_{1}$
as in (5.11), there exists a constant $0<\rho_{0}<1$ (identified in (5.56)),
such that the operator ${\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})$ in (5.10)
defines a contraction in the ball $B(0,\rho_{0})$ of
$\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$. ∎

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
$\displaystyle{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})$ and invoke the maximal
regularity properties (4.20) for $\displaystyle e^{\mathbb{A}_{F,q}t}$ and
(4.19) for the integral term in (5.10). We then obtain from (5.10)

$\displaystyle\left\lVert{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq\left\lVert
e^{\mathbb{A}_{F,q}t}\mathbf{b}_{0}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}+$ $\displaystyle\hskip
28.45274pt\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{f}_{1}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}(\tau)\end{bmatrix}d\tau\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (5.13) $\displaystyle\leq
C\bigg{[}\left\lVert\mathbf{b}_{0}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+$
$\displaystyle\left\lVert{\mathcal{N}}_{q}\mathbf{f}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}}+\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\bigg{]}.$
(5.14)

Step 2: Regarding the term $\displaystyle{\mathcal{N}}_{q}\mathbf{f}_{1}$ we
can invoke [L-P-T.1, Eq (8.19)] to obtain

$\left\lVert{\mathcal{N}}_{q}\mathbf{f}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}}\leq
C\left\lVert\mathbf{f}_{1}\right\rVert^{2}_{\mathbf{X}^{\infty}_{p,q,\sigma}},\
\mathbf{f}_{1}\in\mathbf{X}^{\infty}_{p,q,\sigma}.$ (5.15)

Regarding the term $\displaystyle{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}$, we
can trace the proof in [L-P-T.1, from (8.10) $\longrightarrow$ (8.18)] (which
yielded estimate (5.14)). For the sake of clarity, we shall reproduce the
computations in the present case with
$\displaystyle{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}=\mathbf{f}_{1}\cdot\nabla
f_{2}$, see (1.22), mutatis mutandis. We shall obtain

$\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\leq
C\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}},\
\mathbf{f}_{1}\in\mathbf{X}^{\infty}_{p,q,\sigma},f_{2}\in X^{\infty}_{p,q}.$
(5.16)

In fact, let us compute

$\displaystyle\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}^{p}$
$\displaystyle\leq\int_{0}^{\infty}\left\lVert\mathbf{f}_{1}\cdot\nabla
f_{2}\right\rVert^{p}_{L^{q}(\Omega)}dt$ (5.17)
$\displaystyle\leq\int_{0}^{\infty}\bigg{\\{}\int_{\Omega}\left\lvert\mathbf{f}_{1}(t,x)\right\rvert^{q}\left\lvert\nabla
f_{2}(t,x)\right\rvert^{q}d\Omega\bigg{\\}}^{{}^{p}\\!/_{q}}dt$ (5.18)
$\displaystyle\leq\int_{0}^{\infty}\bigg{\\{}\bigg{[}\sup_{\Omega}\left\lvert\nabla
f_{2}(t,x)\right\rvert^{q}\bigg{]}^{{}^{1}\\!/_{q}}\bigg{[}\int_{\Omega}\left\lvert\mathbf{f}_{1}(t,x)\right\rvert^{q}d\Omega\bigg{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$
(5.19) $\displaystyle\leq\int_{0}^{\infty}\left\lVert\nabla
f_{2}(t,\cdot)\right\rVert^{p}_{\mathbf{L}^{\infty}(\Omega)}\left\lVert\mathbf{f}_{1}(t,\cdot)\right\rVert^{p}_{\mathbf{L}^{q}_{\sigma}(\Omega)}dt$
(5.20) $\displaystyle\leq\sup_{0\leq
t\leq\infty}\left\lVert\mathbf{f}_{1}(t,\cdot)\right\rVert^{p}_{\mathbf{L}_{\sigma}^{q}(\Omega)}\int_{0}^{\infty}\left\lVert\nabla
f_{2}(t,\cdot)\right\rVert^{p}_{\mathbf{L}^{\infty}(\Omega)}dt$ (5.21)
$\displaystyle=\left\lVert\mathbf{f}_{1}\right\rVert^{p}_{L^{\infty}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega))}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;\mathbf{L}^{\infty}(\Omega))}.$ (5.22)

See also [L-P-T.1, Eq (8.14)].

Step 3: The following embeddings hold true (see the stronger Eq (2.11)):

  1. (i)

[G-G-H.1, Proposition 4.3, p 1406 with $\mu=0,s=\infty,r=q$] so that the
required formula reduces to $1\geq{}^{1}\\!/_{p}$, as desired

$\displaystyle\mathbf{f}_{1}\in\mathbf{X}^{\infty}_{p,q,\sigma}\hookrightarrow\mathbf{f}_{1}$
$\displaystyle\in L^{\infty}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega))$
(5.23a) $\displaystyle\text{ so that,
}\left\lVert\mathbf{f}_{1}\right\rVert_{L^{\infty}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega))}$
$\displaystyle\leq
C\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}};$
(5.23b)

  2. (ii)

[Kes, Theorem 2.4.4, p 74 requiring $C^{1}$-boundary]

$W^{1,q}(\Omega)\subset L^{\infty}(\Omega)\text{ for q}>\text{dim }\Omega=d,\
d=2,3,$ (5.24)

so that, with $p>1,q>d$:

$\displaystyle\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;\mathbf{L}^{\infty}(\Omega))}$
$\displaystyle\leq C\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;\mathbf{W}^{1,q}(\Omega))}\leq
C\left\lVert f_{2}\right\rVert^{p}_{L^{p}(0,\infty;W^{2,q}(\Omega))}$ (5.25)
$\displaystyle\leq C\left\lVert f_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}.$
(5.26)

In going from (5.25) to (5.26) we have recalled the definition of $f_{2}\in
X^{\infty}_{p,q}$ in (5.9b) or (2.10), as $f_{2}$ was taken at the outset in
$\displaystyle{\mathcal{D}}({\mathcal{B}}_{q})\subset L^{q}(\Omega)$. Then the
sought-after final estimate (5.16) of the nonlinear term
$\displaystyle{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}$ is obtained from
substituting (5.23b) and (5.26) into the RHS of (5.22).

Step 4: Substituting estimates (5.15) and (5.16) on the RHS of (5.14), we
finally obtain

$\left\lVert{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\leq
C\Big{\\{}\left\lVert\mathbf{b}_{0}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\big{(}\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}\big{)}\Big{\\}}.$ (5.27)

See [L-P-T.1, Eqt (8.20)].

Step 5: We now impose restrictions on the data on the RHS of (5.27):
$\mathbf{b}_{0}$ is in a ball of radius $r_{1}>0$ in
$\displaystyle\mathbf{V}^{q,p}(\Omega)=\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$ and $\mathbf{f}=\\{\mathbf{f}_{1},f_{2}\\}$ lies in a ball of
radius $r>0$ in $\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$. We further demand that the final result
$\displaystyle{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})$ shall lie in a ball of
radius $r>0$ in $\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$. Thus, we obtain from (5.27)

$\displaystyle\left\lVert{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
C\Big{\\{}\left\lVert\mathbf{b}_{0}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\big{(}\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}\big{)}\Big{\\}}$ $\displaystyle\leq
C(r_{1}+r\cdot r)\leq r.$ (5.28)

This implies

$Cr^{2}-r+Cr_{1}\leq 0\quad\text{or}\quad\frac{1-\sqrt{1-4C^{2}r_{1}}}{2C}\leq
r\leq\frac{1+\sqrt{1-4C^{2}r_{1}}}{2C}$ (5.29)

whereby

$\begin{Bmatrix}\text{ range of values of r
}\end{Bmatrix}\longrightarrow\text{ interval
}\Big{[}0,\frac{1}{C}\Big{]},\text{ as }r_{1}\searrow 0,$ (5.30)

a constraint which is guaranteed by taking

$r_{1}\leq\frac{1}{4C^{2}},\ C\text{ being the constant in
}(\ref{3.28})(\text{w.l.o.g }C>1/4).$ (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

${\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})\text{ takes: }\begin{Bmatrix}\text{
ball in }\mathbf{V}^{q,p}(\Omega)\\\ \text{of radius
}r_{1}\end{Bmatrix}\times\begin{Bmatrix}\text{ ball in
}\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}\\\ \text{of radius
}r\end{Bmatrix}\text{ into }\\\ \begin{Bmatrix}\text{ ball in
}\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}\\\ \text{of radius
}r\end{Bmatrix},\ d<q,\ 1<p<\frac{2q}{2q-1}.$ (5.32)

This establishes Theorem 5.2. ∎

Proof of Theorem 5.3

Step 1: For
$\mathbf{f}=\\{\mathbf{f}_{1},f_{2}\\},\mathbf{g}=\\{\mathbf{g}_{1},g_{2}\\}$
both in the ball of $\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$ of radius $r$ obtained in the proof of Theorem 5.2, we
estimate from (5.10):

$\displaystyle\left\lVert{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})-{\mathcal{F}}(\mathbf{b}_{0},\mathbf{g})\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}=$ $\displaystyle\hskip
71.13188pt\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{f}_{1}(\tau)-{\mathcal{N}}_{q}\mathbf{g}_{1}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}(\tau)-{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}(\tau)\end{bmatrix}d\tau\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$
$\displaystyle\leq\widetilde{m}\Big{[}\left\lVert{\mathcal{N}}_{q}\mathbf{f}_{1}-{\mathcal{N}}_{q}\mathbf{g}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}}+\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}\right\rVert_{L^{p}(0,\infty;L^{q}(\Omega))}\Big{]}$
(5.33)

after invoking the maximal regularity property (4.19)–(4.21).

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

$\left\lVert{\mathcal{N}}_{q}\mathbf{f}_{1}-{\mathcal{N}}_{q}\mathbf{g}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}}\leq
2^{{}^{1}\\!/_{p}}C^{{}^{1}\\!/_{p}}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\big{(}\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\big{)}.$
(5.34)

Regarding the second term on the RHS of (5.33) involving ${\mathcal{M}}_{q}$,
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
$\displaystyle{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}=\mathbf{f}_{1}\cdot\nabla
f_{2},{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}=\mathbf{g}_{1}\cdot\nabla g_{2}$.
We shall obtain

$\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}\right\rVert^{p}_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\leq
C\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\\\
+\left\lVert\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\Big{\\}}.$ (5.35)

In fact, adding and subtracting

$\displaystyle{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}$
$\displaystyle=\mathbf{f}_{1}\cdot\nabla f_{2}-\mathbf{g}_{1}\cdot\nabla
g_{2}$ $\displaystyle=\mathbf{f}_{1}\cdot\nabla
f_{2}-\mathbf{g}_{1}\cdot\nabla f_{2}+\mathbf{g}_{1}\cdot\nabla
f_{2}-\mathbf{g}_{1}\cdot\nabla g_{2}$
$\displaystyle=(\mathbf{f}_{1}-\mathbf{g}_{1})\cdot\nabla
f_{2}+\mathbf{g}_{1}\cdot\nabla(f_{2}-g_{2})=A+B.$ (5.36)

Thus, using $(*):\ \left\lvert A+B\right\rvert^{p}\leq 2^{p}\big{[}\left\lvert
A\right\rvert^{p}+\left\lvert B\right\rvert^{p}\big{]}$ [T-L.1, p. 12], we
estimate

$\displaystyle\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}^{p}$
$\displaystyle=\int_{0}^{\infty}\bigg{\\{}\bigg{[}\int_{\Omega}\left\lvert\mathbf{f}_{1}\cdot\nabla
f_{2}-\mathbf{g}_{1}\cdot\nabla
g_{2}\right\rvert^{q}d\Omega\bigg{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$ (5.37)
(by (5.36)) $\displaystyle=\int_{0}^{\infty}\bigg{[}\int_{\Omega}\left\lvert
A+B\right\rvert^{q}d\Omega\bigg{]}^{{}^{p}\\!/_{q}}dt$ (5.38)
$\displaystyle\leq
2^{p}\int_{0}^{\infty}\bigg{\\{}\int_{\Omega}\big{[}\left\lvert
A\right\rvert^{q}+\left\lvert
B\right\rvert^{q}\big{]}d\Omega\bigg{\\}}^{{}^{p}\\!/_{q}}dt$ (5.39)
$\displaystyle=2^{p}\int_{0}^{\infty}\bigg{\\{}\Big{[}\int_{\Omega}\left\lvert
A\right\rvert^{q}d\Omega+\int_{\Omega}\left\lvert
B\right\rvert^{q}d\Omega\Big{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$ (5.40)
$\displaystyle=2^{p}\int_{0}^{\infty}\bigg{\\{}\Big{[}\left\lVert
A\right\rVert^{q}_{L^{q}(\Omega)}+\left\lVert
B\right\rVert^{q}_{L^{q}(\Omega)}\Big{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$
(5.41) $\displaystyle(\text{by }(*)\mbox{ with }p\to\frac{1}{q})\qquad$
$\displaystyle\leq 2^{p}\cdot
2^{{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{\\{}\left\lVert
A\right\rVert_{L^{q}(\Omega)}+\left\lVert
B\right\rVert_{L^{q}(\Omega)}\Big{\\}}^{p}dt$ (5.42) $\displaystyle(\text{by
}(*))\qquad$ $\displaystyle\leq
2^{p+p+{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{[}\left\lVert
A\right\rVert^{p}_{L^{q}(\Omega)}+\left\lVert
B\right\rVert^{p}_{L^{q}(\Omega)}\Big{]}dt$ (5.43)
$\displaystyle=2^{p+p+{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{[}\left\lVert(\mathbf{f}_{1}-\mathbf{g}_{1})\cdot\nabla
f_{2}\right\rVert^{p}_{L^{q}(\Omega)}$ $\displaystyle\hskip
56.9055pt+\left\lVert\mathbf{g}_{1}\cdot\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{q}(\Omega)}\Big{]}dt$
(5.44) $\displaystyle\leq
2^{p+p+{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{L}^{q}_{\sigma}(\Omega)}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{q}(\Omega)}$ $\displaystyle\hskip
56.9055pt+\left\lVert\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{L}^{q}_{\sigma}(\Omega)}\left\lVert\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{q}(\Omega)}\Big{\\}}dt,$
(5.45)

recalling the definitions of $A$ and $B$ in (5.36) in passing from (5.43) to
(5.44). [L-P-T.1, This is the counterpart Eq (8.36)]. Next we proceed by
majorizing the $L^{q}(\Omega)$-norm for $\nabla f_{2}$ and
$\nabla(f_{2}-g_{2})$ by the $L^{\infty}(\Omega)$-norm. We obtain

$\displaystyle\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}^{p}$
$\displaystyle\leq\text{RHS of (\ref{3.45})}$ $\displaystyle\leq
c2^{p+p+{}^{p}\\!/_{q}}\bigg{\\{}\sup_{0\leq
t\leq\infty}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{L}^{q}_{\sigma}(\Omega)}\int_{0}^{\infty}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{\infty}(\Omega)}$ $\displaystyle+\sup_{0\leq
t\leq\infty}\left\lVert\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{L}^{q}_{\sigma}(\Omega)}\int_{0}^{\infty}\left\lVert\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{\infty}(\Omega)}dt\bigg{\\}}$
(5.46) $\displaystyle\leq
c2^{p+p+{}^{p}\\!/_{q}}\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{L^{\infty}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega))}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;L^{\infty}(\Omega))}$
$\displaystyle+\left\lVert\mathbf{g}_{1}\right\rVert^{p}_{L^{\infty}(0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega))}\left\lVert\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{p}(0,\infty;L^{\infty}(\Omega))}\Big{\\}}$
(5.47) by (5.23b) and (5.26) $\displaystyle\leq
C\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}^{p}\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}^{p}+\left\lVert\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}^{p}\left\lVert
f_{2}-g_{2}\right\rVert_{X^{\infty}_{p,q}}^{p}\Big{\\}},$ (5.48)

counterpart of [L-P-T.1, Eq (8.39)].

Step 3: We substitute estimate (5.34) and estimate (5.48) on the RHS of (5.33)
and obtain via $(*)$

$\displaystyle\left\lVert{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})-{\mathcal{F}}(\mathbf{b}_{0},\mathbf{g})\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
2^{p}\widetilde{m}^{p}\Big{\\{}\left\lVert{\mathcal{N}}_{q}\mathbf{f}_{1}-{\mathcal{N}}_{q}\mathbf{g}_{1}\right\rVert^{p}_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}$
$\displaystyle+\left\lVert{\mathcal{M}}_{q}[\mathbf{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\mathbf{g}_{1}]g_{2}\right\rVert^{p}_{L^{p}(0,\infty;L^{q}(\Omega))}\Big{\\}}$
(5.49) $\displaystyle\leq
C_{p}\widetilde{m}^{p}\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}\Big{(}\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\Big{)}^{p}$
$\displaystyle+\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}+\left\lVert\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\Big{\\}}.$ (5.50)

This is the counterpart of [L-P-T.1, Eq (8.42)].

Step 4: Next pick the points $\mathbf{f}=\\{\mathbf{f}_{1},f_{2}\\}$ and
$\mathbf{g}=\\{\mathbf{g}_{1},g_{2}\\}$ in a ball of
$\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$ of
radius $R$:

$\displaystyle\left\lVert\mathbf{f}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$
$\displaystyle=\left\lVert\mathbf{f}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}<R$ (5.51)
$\displaystyle\left\lVert\mathbf{g}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$
$\displaystyle=\left\lVert\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
g_{2}\right\rVert_{X^{\infty}_{p,q}}<R.$ (5.52)

Then (5.51),(5.52) used (5.50) implies

$\displaystyle\left\lVert{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})-{\mathcal{F}}(\mathbf{b}_{0},\mathbf{g})\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
C_{p}\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}\big{[}(2R)^{p}+R^{p}\big{]}$
$\displaystyle\hskip 56.9055pt+\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}R^{p}\Big{\\}}$ (5.53)
$\displaystyle\leq
C_{p}\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}\big{[}(2^{p}+1)R^{p}\big{]}$
$\displaystyle\hskip 56.9055pt+\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}R^{p}\Big{\\}}$
$\displaystyle\big{(}K_{p}=C_{p}(2^{p}+1)\big{)}\quad$ $\displaystyle\leq
K_{p}R^{p}\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert^{p}_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\Big{\\}}$
$\displaystyle[a>0,b>0,\ a^{p}+b^{p}\leq(a+b)^{p}]\quad$ $\displaystyle\leq
K_{p}R^{p}\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}-g_{2}\right\rVert_{X^{\infty}_{p,q}}\Big{\\}}^{p}$ (5.54)

using $K_{p}=C_{p}(2^{p}+1)$ and $a^{p}+b^{p}\leq(a+b)^{p}$ for $a>0,b>0$.
Finally,