# Uniform stabilization of Navier-Stokes equations in critical $L^{q}$-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 2- or 3-dimensional incompressible Navier-Stokes equations defined
on a bounded domain $\Omega$, with no-slip boundary conditions and subject to
an external force, assumed to cause instability. We then seek to uniformly
stabilize such N-S system, in the vicinity of an unstable equilibrium
solution, in critical $L^{q}$-based Sobolev and Besov spaces, by finite
dimensional feedback controls. These spaces are ‘close’ to $L^{3}(\Omega)$ for
$d=3$. This functional setting is significant. In fact, in the case of the
uncontrolled N-S dynamics, extensive research efforts have recently lead to
the space $L^{3}(\mathbb{R}^{3})$ as being a critical space for the issue of
well-posedness in the full space. Thus, our present work manages to solve the
stated uniform stabilization problem for the controlled N-S dynamics in a
correspondingly related function space setting. In this paper, the feedback
controls are localized on an arbitrarily small open interior subdomain
$\omega$ of $\Omega$. In addition to providing a solution of the uniform
stabilization problem in such critical function space setting, this paper
manages also to much improve and simplify, at both the conceptual and
computational level, the solution given in the more restrictive Hilbert space
setting in the literature. Moreover, such treatment sets the foundation for
the authors’ final goal in a subsequent paper. Based critically on said low
functional level where compatibility conditions are not recognized, the
subsequent paper solves in the affirmative a presently open problem: whether
uniform stabilization by localized tangential boundary feedback controls,
which-in addition-are finite dimensional, is also possible in dim $\Omega=3$.

##  1 Introduction

###  1.1 Controlled Dynamic Navier-Stokes Equations

Let, at first, $\Omega$ be an open connected bounded domain in
$\mathbb{R}^{d},d=2,3$ with sufficiently smooth boundary
$\Gamma=\partial\Omega$. More specific requirements will be given below. Let
$\omega$ be an arbitrarily small open smooth subset of the interior $\Omega$,
$\omega\subset\Omega$, of positive measure. Let $m$ denote the characteristic
function of $\omega$: $m(\omega)\equiv 1,\ m(\Omega\backslash\omega)\equiv 0$.
Consider the following controlled Navier-Stokes Equations with no-slip
Dirichlet boundary conditions, where $Q=(0,\infty)\times\Omega,\
\Sigma=(0,\infty)\times\Gamma$:

$\displaystyle y_{t}(t,x)-\nu\Delta y(t,x)+(y\cdot\nabla)y+\nabla\pi(t,x)$
$\displaystyle=m(x)u(t,x)+f(x)$ $\displaystyle\text{ in }Q$ (1.1a)
$\displaystyle\text{div}\ y$ $\displaystyle=0$ $\displaystyle\text{ in }Q$
(1.1b) $\displaystyle y$ $\displaystyle=0$ $\displaystyle\text{ on }\Sigma$
(1.1c) $\displaystyle y(0,x)$ $\displaystyle=y_{0}(x)$ $\displaystyle\text{ in
}\Omega$ (1.1d)

Notation: As already done in the literature, for the sake of simplicity, we
shall adopt the same notation for function spaces of scalar functions and
function spaces of vector valued functions. Thus, for instance, for the vector
valued ($d$-valued) velocity field $y$ or external force $f$, we shall simply
write say $y,f\in L^{q}(\Omega)$ rather than $y,f\in(L^{q}(\Omega))^{d}$ or
$y,f\in\mathbf{L}^{q}(\Omega)$. This choice is unlikely to generate confusion.
By way of orientation, we state at the outset two main points. For the
linearized $w$-problem (1.13) below in the feedback form (2.16), the
corresponding well-posedness and global feedback uniform stabilization result,
Theorem 2.2, holds in general for $1<q<\infty$. Instead, the final, main well-
posedness and feedback uniform, local stabilization result, Theorem 2.5, for
the nonlinear problem (2.27) or (2.28) corresponding to the original problem
(1.1) will require $q>3$, see (8.16), in the $d=3$-case, hence $\displaystyle
1<p<\frac{6}{5}$; and $q>2$, in the $d=2$-case, hence $1<p<\frac{4}{3}$; see
(1.16). Let $u\in L^{p}(0,T;L^{q}(\Omega))$ be the control input and
$y=(y_{1},\dots,y_{d})$ be the corresponding state (velocity) of the system.
Let $\nu>0$ be the viscosity coefficient. The function $v(t,x)=m(x)u(t,x)$ can
be viewed as an interior controller with support in
$Q_{\omega}=(0,\infty)\times\omega$. The initial condition $y_{0}$ and the
body force $f\in L^{q}(\Omega)$ are given. The scalar function $\pi$ is the
unknown pressure.

###  1.2 Stationary Navier-Stokes equations

The following result represents our basic starting point.

######  Theorem 1.1.

Consider the following steady-state Navier-Stokes equations in $\Omega$

$\displaystyle-\nu\Delta y_{e}+(y_{e}.\nabla)y_{e}+\nabla\pi_{e}$
$\displaystyle=f$ $\displaystyle\text{ in }\Omega$ (1.2a)
$\displaystyle\text{div}\ y_{e}$ $\displaystyle=0$ $\displaystyle\text{ in
}\Omega$ (1.2b) $\displaystyle y_{e}$ $\displaystyle=0$ $\displaystyle\text{
on }\Gamma$ (1.2c)

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

For the Hilbert case $q=2$, see [12, Thm 7.3 p 59] . For the general case
$1<q<\infty$, see [4, Thm 5.iii p 58].

######  Remark 1.1.

It is well-known [32], [37], [55] that the stationary solution is unique when
“the data is small enough, or the viscosity is large enough” [55, p 157; Chapt
2] that is, if the ratio $\displaystyle{}^{\left\lVert
f\right\rVert}\\!/_{\nu^{2}}$ is smaller than some constant that depends only
on $\Omega$ [18, p 121]. When non-uniqueness occurs, the stationary solutions
depend on a finite number of parameters [18, Theorem 2.1, p 121]
asymptotically in the time dependent case.

######  Remark 1.2.

The case where $f(x)$ in (1.1a) is replaced by $\displaystyle\nabla g(x)$ is
noted in the literature as arising in certain physical situations, where $f$
is a conservative vector field. The analysis of this relevant case is
postponed to Remark 1.4, at the end of Section 1.

###  1.3 Main goal of the present paper

For a given external force $f$, if the Reynolds number $\frac{1}{\nu}$ is
sufficiently large, then the steady state solution $y_{e}$ in (1.2) becomes
unstable (in a quantitative sense to be made more precise in Section 2.2
below) and turbulence occurs.

The main goal of the present paper is then - at first qualitatively - to
feedback stabilize the non-linear N-S model (1.1) subject to rough (non-
smooth) initial condition $y_{0}$, in the vicinity of an (unstable)
equilibrium solution $y_{e}$ in (1.2). Thus this paper pertains to the general
context 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 refer mostly to the mathematical literature, and most
specifically on the localized interior control setting of problem (1.1). A
more precise description thereof is as follows: establish interior localized
exponential stabilization of problem (1.1) near an unstable equilibrium
solution by means of a finite dimensional localized, spectral-based feedback
control, in the important case of initial conditions $y_{0}$ of low
regularity, as technically expressed by $y_{0}$ being in suitable
$L^{q}$/Besov space with tight indices. In particular, local exponential
stability for the velocity field $y$ near an equilibrium solution $y_{e}$ will
be achieved in the topology of the Besov space

$\big{(}L^{q}(\Omega),W^{2,q}(\Omega)\big{)}_{1-{}^{1}\\!/_{p},p}=B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega),\quad
1<p<\frac{2q}{2q-1};\ q>d,\ d=2,3.$ (1.3)

See more precisely (1.16a). We note the tight indices: e.g. $\displaystyle
1<p<{}^{6}\\!/_{5}$ for $q>d=3$. 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 the successor paper
[35] in showing that: local tangential boundary feedback stabilization near an
unstable equilibrium solution with finite dimensional controls is possible
also in dimension $d=3$, thus solving in the affirmative a recognized open
problem in the stabilization area. This point will be more appropriately
expanded in Section 1.6 below. For $d=3$, the space is (1.3) in ‘close’ to
$L^{3}(\Omega)$.

Criticality of the space $L^{3}$: We now expand on one message of the abstract
regarding the ‘criticality’ of the space $L^{3}(\Omega)$. 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
$L^{3}(\mathbb{R}^{3})$ is a ‘critical’ space for the issue of well-posedness.
Assuming that some divergence free initial data in $L^{3}(\mathbb{R}^{3})$
would produce finite time singularity, then there exists a so-called minimal
blow-up initial data in $L^{3}(\mathbb{R}^{3})$ [22], [27]. More precisely,
let $y$ now be a solution of the N-S equations (1.1a-b-d) with $m\equiv
0,f\equiv 0$, as defined on the whole space $\mathbb{R}^{3}$. For any
divergence free I.C. $y_{0}\in L^{3}(\mathbb{R}^{3})$, denote by
$T_{max}(y_{0})$ the maximal time of existence of the mild solution starting
from $y_{0}$. Define

$\rho_{max}=\sup\big{\\{}\rho:\ T_{max}(y_{0})=\infty\text{ for every
divergence free }y_{0}\in L^{3}(\mathbb{R}^{3}),\text{ with }\left\lVert
y_{0}\right\rVert_{L^{3}(\mathbb{R}^{3})}<\rho\big{\\}}.$

The following result holds [27, Theorem 4.1, p 14]: Suppose
$\rho_{max}<\infty$. Then there exists some $\displaystyle y_{0}\in
L^{3}(\mathbb{R}^{3})$, $\left\lVert
y_{0}\right\rVert_{L^{3}(\mathbb{R}^{3})}=\rho_{max}$, whose
$T_{max}(y_{0})<\infty$, i.e. the corresponding solution blows up in finite
time. Of the numerous works that followed the pioneering work of [36] along
this line of research, we quote in addition [46], [47], [16], [56].

Thus, our present work manages to solve the uniform stabilization problem for
the controlled N-S equations as (1.1a-b-d) in a correspondingly related low-
regularity function space setting. A further justification of our low-
regularity level of the Besov space in (1.3) is provided by the final goal of
our line of research in the subsequent Part II [35]. Based critically on said
low-regularity level of the Besov space (1.3), which does not recognize
compatibility conditions on the boundary of the initial conditions - such Part
II solves in the affirmative a presently open problem by showing that uniform
stabilization by localized tangential boundary feedback controls which
moreover are finite dimensional is possible also in dim $\Omega=d=3$.

The successor [35] to the present work will extensively review the literature
as it pertains to the boundary stabilization case, particularly with
tangential control action. Accordingly, below we shall review the present
study mostly in comparison with the prior solution of the localized interior
stabilization in the Hilbert-based treatment of the original work [7] (via a
Riccati-based finite dimensional control), followed by [10] (via a spectral
based finite dimensional control) and textbook versions thereof [6, Chapter
4].

###  1.4 Qualitative Orientation

####  1.4.1 On the local, interior, feedback stabilization problem: Past
Literature

We start with an unstable steady state solution $y_{e}$, given an external
force $f$, and a sufficiently large Reynold number $\frac{1}{\nu}$, as
described in Section 1.3. We deliberately aim at rough (non-smooth) initial
conditions $y_{0}$. We then seek a finite-dimensional interior localized
feedback control $u$, such that the corresponding N-S problem is well-posed in
a suitable function space setting (depending on the I.C. $y_{0}$) and its
solution $y$ in (1.1) is locally exponentially stable near the equilibrium
solution $y_{e}$, in a suitably corresponding norm. This problem was
originally posed and solved in the Hilbert space setting in [7, Theorem 2.2 p
1449] by means of a finite dimensional Riccati-based feedback control $u$,
where exponential decay is obtained in the
${\mathcal{D}}(A^{{}^{1}\\!/_{4}})$-topology. Here $A$ is the positive self-
adjoint Stokes operator in (1.17) with $q=2$ in the space $H$ with
$L^{2}(\Omega)$-topology. See (1.6) below. A similar exponential decay result,
in the same ${\mathcal{D}}(A^{{}^{1}\\!/_{4}})$-topology, is given in [10, Thm
5.1, p 42], this time by means of a finite-dimensional, spectral-based
feedback control $u$. These results are reproduced in textbook form in [6,
Chapter 4].

Regarding the solution given in these Hilbert-space based references, we point
out (at present) two defining, linked characteristics of their finite
dimensional treatment:

  1. (i)

The number of stabilizing (localized) controls for the (complex-valued)
nonlinear dynamics (1.1) is $N=\sup\\{N_{i};\ i=1,\dots,M\\}$, that is, the
max of the algebraic multiplicity $N_{i}$ of the $M$ distinct unstable
eigenvalues $\lambda_{i}$, see (2.2), of the projected Oseen operator
${\mathcal{A}}_{N}^{u}$ in (2.5).

  2. (ii)

in the fully general case, the algebraic (Kalman rank) conditions for
controllability under which the finite dimensional feedback control is
explicitly constructed involve the Grahm-Schmidt orthogonalization of the
generalized eigenfunctions of the adjoint $({\mathcal{A}}_{N}^{u})^{*}$,
making the test difficult to verify. Only in the case where the restriction
${\mathcal{A}}_{N}^{u}$ in (2.5) of the Oseen operator ${\mathcal{A}}$ in
(1.10) is semisimple (algebraic and geometric multiplicity of the unstable
eigenvalues coincide), are the controllability tests given in terms of
eigenfunctions of $({\mathcal{A}}_{N}^{u})^{*}$.

####  1.4.2 Additional goals of the present paper as definite improvements
over the literature

We list these main additional goals of the present work aimed at markedly
improving both the results and the approach of the original reference [7],
followed by [10] and their textbook version [6, Chapter 4]. They are:

  1. (i)

With reference to part (i) in Section 1.4.1, our next goal is to obtain (in
the general case of Theorem 4.1) that the number of finite dimensional
stabilizing controls needed for the (complex valued version of the) dynamics
(1.1) is $K=\sup\\{\ell_{i};\ i=1,\dots,M\\}$, where $\ell_{i}$ is the
geometric multiplicity of the $M$ distinct unstable eigenvalues $\lambda_{i}$
in (2.2). This is a notable reduction in the number of needed controls over
the max algebraic multiplicity $N$ in (i) of Section 1.4.1.

  2. (ii)

Intimately linked to goal (i) is the next goal to obtain the controllability
Kalman rank condition of Section 3 be expressed in terms of only the
eigenfunctions - not the generalized eigenfunctions: as in the past literature
[8], [9], [10], [6, Chapter 4] \- of the adjoint
$({\mathcal{A}}_{N}^{u})^{*}$.

  3. (iii)

An important additional goal is to simplify and make more transparent the
well-posedness and local stabilization arguments for the non-linear problem,
in particular through a direct analysis of the nonlinear operator
${\mathcal{N}}_{q}$ in (1.11), called $B$ in [7], not its approximation
sequence $B_{\varepsilon}$ as in [7, Section 4, p1480] or [6, proof of
Therorem 3.4 p 107-110]. More precisely, unlike these references, the present
paper carries out an analysis of the critical issues based on the maximal
regularity property of the linearized feedback operator
$\displaystyle\mathbb{A}_{{}_{F}}(=\mathbb{A}_{{}_{F,q}})$ in (6.1) in the
critical $L^{q}$/Besov setting. This point also is further expanded in Section
1.6 below.

  4. (iv)

A final goal - in line with goal (iii) above - is to obtain corresponding
results for the pressure $\pi$ in (1.1a), as part of the same maximal
regularity property of the linearized feedback operator
$\displaystyle\mathbb{A}_{{}_{F}}(=\mathbb{A}_{{}_{F,q}})$ in (6.1), unlike
[7, Theorem 2.3, p1450].

###  1.5 What is one key motivation for seeking interior localized feedback
exponential stabilization of problem (1.1) in the topology of the Besov space
in (1.3)?

As already discussed in Section 1.3, obtaining the resulting stabilization in
a non-Hilbertian setting is of theoretical interest in itself, and is in line
with recent developments in 3-d N-S equations defined on the entire space
$\mathbb{R}^{3}$, where recent breakthroughs have identified the space
$L^{3}(\mathbb{R}^{3})$ as a critical space for well-posedness, possessing the
minimal blow up initial data property [22], [27]. However, our main original
motivation for the present study is another. The present paper intends to test
$L^{q}$/Besov spaces techniques initially in the interior localized feedback
stabilization problem (1.1) in the critical low regularity setting of (1.3).
The true aim is, however, to export them with serious additional technical
difficulties, to solve the presently recognized open problem of the local
feedback exponential stabilization of the N-S equations with finite-
dimensional feedback tangential boundary controllers in the case of dimension
$d=3$ [35]. In fact, present state-of-the-art has succeeded [33], [34] in
establishing local exponential stabilization (asymptotic turbulence
suppression) by means of finite-dimensional tangential feedback boundary
control in the Hilbert setting and with no assumptions whatsoever on the Oseen
operator in two cases:

  1. (i)

when the dimension $d=2$,

  2. (ii)

when the dimension $d=3$ but the initial condition $y_{0}$ in (1.1.d) is
compactly supported.

In the general $d=3$ case, the non-linearity of the N-S problem forces a
Hilbert space setting with a high-topology
$H^{{}^{1}\\!/_{2}+\epsilon}(\Omega)$ for the initial conditions, whereby the
compatibility conditions on the boundary kick in. These then cannot allow the
stabilizing feedback control to be finite-dimensional in general. More
precisely, even at the level of the linearized boundary problem for $d=3$,
open loop exponential stabilization [10, Proposition 3.7.1 Remark 3.7.1], [33,
Proposition 2.5, eq (2.48)] provide a boundary control consisting of a finite-
dimensional term plus the term $e^{-2\gamma_{1}t}((I.C.)|_{\Gamma})$, with
$\gamma_{1}>0$ preassigned, which spoils the finite-dimensionality, unless the
initial condition is compactly supported. These limitations are in subsequent
literature. In contrast, the Besov space in (1.3) above, which is “close” to
the space $L^{3}(\Omega)$ for $d=3$, has the key, fundamental advantage of not
recognizing the boundary conditions. That is why this paper is interested in a
stabilization result in such a low regularity space for $d=3$, at first in the
case of interior localized control.

###  1.6 Comparison with prior work, once the present treatment is specialized
to the Hilbert setting $(q=2)$

A comparison with the original prior work [7], [10], reproduced in [6, Chapter
4] which was carried out in the Hilbert setting, is in order.

Orientation Even when specialized to the Hilbert space setting $(q=2)$, the
present treatment offers distinct, notable advantages – both conceptual and
computational over prior literature quoted above. These include not only
definitely simpler and more direct arguments but also transparent
simplifications in the actual construction of the finite dimensional
stabilizing controllers as well as their number. Qualitative details are given
below. The main conceptual approach and the final results of the present paper
are (when specialized to the Hilbert setting) qualitatively in line with those
in [7]: local uniform stabilization of the non-linear $y$-problem (1.1) near
an unstable equilibrium solution $y_{e}$ by means of finite dimensional,
arbitrarily localized controllers is based on the corresponding result on
(global) uniform stabilization of the linearized $w$-system (1.13). This in
turn rests on the space decomposition technique introduced in [57] for
parabolic problems (and also for differentiable semigroups): its foundational
starting point is the controllability of the finite dimensional unstable
projected system $w_{N}$ in (2.8a). However, in the implementation of these
two fundamental phases, linear analysis - in particular, its finite
dimensional $w_{N}$ component - and nonlinear analysis, the present paper
provides a much more attractive, more powerful and mature treatment. We
mention the most relevant new features. They are:

  1. 1.

finite dimensional analysis leading, through a much more simplified and more
direct approach, to a lower (optimal) number of feedback controls;

  2. 2.

infinite dimensional analysis on the nonlinear effects (the operator
${\mathcal{N}}_{q}$ in (1.11)) handled by critical and clean use of maximal
regularity of the linearized feedback operator
$\displaystyle\mathbb{A}_{{}_{F}}(\equiv\mathbb{A}_{{}_{F,q}})$ in (6.1),
rather than by the approximation argument as in [7]. This refers to both $y$
and $\pi$.

These two points are explained below.

  1. 1.

Stabilization of the linearized $w$-problem (1.13). The key foundational
algebraic test for controllability of the finite dimensional $w_{N}$-system
(2.8a) on the finite dimensional unstable subspace $W^{u}_{N}$ is much
simplified, sharper and leads, in principle, to checkable conditions and to an
implementable procedure to obtain constructively the finite dimensional
stabilizing vectors $u_{k}\in W^{u}_{N},\ p_{k}\in(W^{u}_{N})^{*}$. In fact,
the present treatment shows that (for the complexified version of the N-S) the
required number of feedback stabilizing finite dimensional controllers is
$K=\sup\\{\ell_{i},\ i=1,\dots,M\\}$, the max of the geometric multiplicity
$\ell_{i}$ of the $M$ distinct unstable eigenvalues $\lambda_{i}$ of the Oseen
operator $\displaystyle{\mathcal{A}}$; not the larger $\sup\\{N_{i},\
i=1,\dots,M\\}$, the max of the algebraic multiplicity $N_{i}$ of its distinct
unstable eigenvalues, as in [7], [8], [10], [6, Chapter 4]. Let alone
$\displaystyle N=\sum_{i=1}^{M}N_{i}=$ dim $W^{u}_{N}$ (dimension of the
generalized eigenspace of the unstable eigenvalues) as in the treatment of [6,
assumption K.2 p 123], where, in addition, the simplifying assumption that
algebraic and geometric multiplicities coincide for the unstable eigenvalues.
Moreover, the entire analysis of the present paper rests only on the (true)
eigenvectors corresponding to the unstable eigenvalues of the adjoint operator
in (3.1); not only under the Finite Dimensional Spectral Condition
(semisimplicity) as in [7] where $W^{u}_{N}$ has a basis of such (true)
eigenvectors, but also in the most general case where the projected Oseen
operator $\displaystyle{\mathcal{A}}_{N}^{u}$ is in Jordan form, and hence the
basis on $W^{u}_{N}$ consists instead of all generalized eigenvectors
corresponding to the unstable eigenvalues. As first noted in [33] in the study
of tangential boundary stabilization of the N-S equations, even in the general
case possessing only a basis of generalized eigenfunctions arising from the
Jordan form, the final test for controllability involves only the true
eigenfunctions of the adjoint operator: the algebraic test (4.13), (4.14) for
controllability in the general case is exactly the same as the algebraic test
(3.18) in the semisimple (diagonalizable)-case; and only the true
eigenfunctions count. This justifies why the number $K$ of (complex valued)
stabilizing controllers as in Theorem 2.1 is equal to the supremum of the
geometric multiplicity of the unstable eigenvalues, not the supremum of their
larger algebraic multiplicity as in past references [7], [8], [10], [6,
Chapter 4] as noted above. Moreover, in [7] repeated in [6, Chapter 4] the
procedure for testing controllability in the general case was much more
cumbersome and far less implementable: the original basis of generalized
eigenfunctions of the adjoint operator in the general case was transformed
into an orthonormal basis of $W^{u}_{N}$ via the Schmidt orthogonalization
process, and the test for the finite dimensional controllability was then
based on such transformed, and thus in principle difficult to check,
orthogonalized system: a much more complicated test than the one using just
the true eigenfunctions as in (4.13).

  2. 2.

Local Stabilization of the nonlinear translated $z$-equation (1.7) near the
origin, hence of the original $y$-equation (1.1) near an equilibrium solution
$y_{e}$. Treatment of the nonlinearity in the present work is much more
transparent and direct than the one performed in [7]. Here the analysis is
directly in terms of the nonlinear operator ${\mathcal{N}}_{q}$ in (1.11) and
makes use of maximal regularity properties of the linearized feedback operator
$\displaystyle\mathbb{A}_{{}_{F}}(\equiv\mathbb{A}_{F,q})$ in (6.1) (maximal
regularity is equivalent to analyticity of the semigroup in the Hilbert
setting. Instead, in the Banach setting, maximal regularity implies, but is
not necessarily implied by, analyticity of the semigroup). In contrast, in [7]
with $q=2$, an approximation argument of the nonlinear operator
${\mathcal{N}}_{q}$, denoted by $B$, was used, by introducing a sequence of
approximating operators $B_{\varepsilon}$ thereof [7, Section 4, p1480]. A
critical step in [7] is that the nonlinearity $B$ (or its approximation) be
controlled by the topology of the $A^{{}^{3}\\!/_{4}}$-power; and this in turn
is achieved by using an optimal control approach with
$A^{{}^{3}\\!/_{4}}$-penalization of the solution via Riccati equations. There
is no need of this in the present treatment (the analysis of the optimization
problem and Riccati equation in a non-Hilbert setting is not the right tool).
We likewise note that our present treatment of the passage from the
$w$-linearized problem (2.16) to the fully non-linear $z$-system (2.20) is
also different from the one employed in [8], [34] which was also direct in
terms of the nonlinear operator ${\mathcal{N}}$. It was however not maximal
regularity - based, as in the present paper.

  3. 3.

Well-posedness of the pressure $\pi$ for the original $y$-problem in the
feedback form as in (2.22) in the vicinity of the equilibrium pressure
$\pi_{e}$ in (1.2a). The well-posedness result Theorem 10.2 on the pressure
$\pi$ of the original $y$-problem on feedback form as given by (2.26), (2.27)
is the $L^{q}$/Besov space counterpart of the Hilbert ($L^{2}$)-version given
by [7, Theorem 2.3 p1450]. The present proof is much more direct as, again, is
based on maximal regularity properties. In contrast, the proof in [7, p 1484]
is based on the approximation of the original problem.

###  1.7 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 [7], [8] to the $L^{q}$ 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 $L^{q}(\Omega)$ can be
decomposed into a direct sum of the solenoidal vector space
$L^{q}_{\sigma}(\Omega)$ and the space $G^{q}(\Omega)$ of gradient fields.
Here,

$\displaystyle L^{q}_{\sigma}(\Omega)$ $\displaystyle=\overline{\\{y\in
C_{c}^{\infty}(\Omega):\text{div}\ y=0\text{ in
}\Omega\\}}^{\left\lVert\cdot\right\rVert_{q}}$ (1.4) $\displaystyle=\\{g\in
L^{q}(\Omega):\text{div}\ g=0;\ g\cdot\nu=0\text{ on }\partial\Omega\\},$
$\displaystyle\hskip 85.35826pt\text{ for any locally Lipschitz domain
}\Omega\subset\mathbb{R}^{d},d\geq 2$ $\displaystyle G^{q}(\Omega)$
$\displaystyle=\\{y\in L^{q}(\Omega):y=\nabla p,\ p\in
W_{loc}^{1,q}(\Omega)\\}\ \text{where }1\leq q<\infty.$

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

######  Definition 1.1.

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

$L^{q}(\Omega)=L^{q}_{\sigma}(\Omega)\oplus G^{q}(\Omega).$ (1.5)

The unique linear, bounded and idempotent (i.e. $P_{q}^{2}=P_{q}$) projection
operator $P_{q}:L^{q}(\Omega)\longrightarrow L^{q}_{\sigma}(\Omega)$ having
$L^{q}_{\sigma}(\Omega)$ as its range and $G^{q}(\Omega)$ as its null space is
called the Helmholtz projection. Additional information is given in Appendix
A.

This is an important property in order to handle the incompressibility
condition $div\ y\equiv 0$. For instance, if such decomposition exists, the
Stokes equation (say the linear version of (1.1) with control $u\equiv 0$) can
be formulated as an equation in the $L^{q}$ setting. Here below we collect a
subset of known results about Helmholtz decomposition. We refer to [26,
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 $L^{2}(\Omega)$; more precisely, for $q=2$, we obtain the well-known
orthogonal decomposition (in the standard notation, where $\nu=$unit outward
normal vector on $\Gamma$) [12, Prop 1.9, p 8]

$\displaystyle L^{2}(\Omega)$ $\displaystyle=H\oplus H^{\perp}$ (1.6a)
$\displaystyle H$ $\displaystyle=\\{\phi\in L^{2}(\Omega):div\ \phi\equiv
0\text{ in }\Omega;\ \phi\cdot\nu\equiv 0\text{ on }\Gamma\\}$ (1.6b)
$\displaystyle H^{\perp}$ $\displaystyle=\\{\psi\in L^{2}(\Omega):\psi=\nabla
h,\ h\in H^{1}(\Omega)\\};$ (1.6c)

  2. (ii)

a bounded $C^{1}$-domain in $\mathbb{R}^{d}$ [17], $1<q<\infty$ [21, 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 [17];

  4. (iv)

a bounded convex domain $\Omega\subset\mathbb{R}^{d},d\geq 2,1<q<\infty$ [17].

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 $q\neq 2$ [38].

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 (1.13) below. The final result Theorem 2.5 for the
non-linear problem (1.1) will require $q>d$, see (8.16), in the case of
interest $d=2,3$.

###  1.8 Translated nonlinear Navier-Stokes $z$-problem: reduction to zero
equilibrium

We return to Theorem 1.1 which provides an equilibrium pair
$\\{y_{e},\pi_{e}\\}$. Then, as in [7], [8], [33] we translate by
$\\{y_{e},p_{e}\\}$ the original N-S problem (1.1). Thus we introduce new
variables

$\displaystyle z=y-y_{e},\quad\chi=\pi-\pi_{e}$ (1.7a) and obtain the
translated problem $\displaystyle z_{t}-\nu\Delta
z+(y_{e}\cdot\nabla)z+(z\cdot\nabla)y_{e}+(z\cdot\nabla)z+\nabla\chi$
$\displaystyle=mu$ $\displaystyle\text{ in }Q$ (1.7b)
$\displaystyle\text{div}\ z$ $\displaystyle=0$ $\displaystyle\text{ in }Q$
(1.7c) $\displaystyle z$ $\displaystyle=0$ $\displaystyle\text{ on }\Sigma$
(1.7d) $\displaystyle z(0,x)$ $\displaystyle=y_{0}(x)-y_{e}(x)$
$\displaystyle\text{ on }\Omega$ (1.7e)

We shall accordingly study the local null feedback stabilization of the
$z$-problem (1.7), that is, feedback stabilization in a neighborhood of the
origin. As usual, we next apply the projection $P_{q}$ below (1.5) to the
translated N-S problem (1.7) to eliminate the pressure $\chi$. We thus proceed
to obtain the corresponding abstract setting for the problem (1.7) as in [7]
except in the $L^{q}$-setting rather than in the $L^{2}$-setting as in this
reference. Note that $P_{q}z_{t}=z_{t}$, since $z\in L^{q}_{\sigma}(\Omega)$
in (1.4).

###  1.9 Abstract nonlinear translated model

First, for $1<q<\infty$ fixed, the Stokes operator $A_{q}$ in
$L^{q}_{\sigma}(\Omega)$ with Dirichlet boundary conditions is defined by [23,
p 1404], [26, p 1]

$A_{q}z=-P_{q}\Delta z,\quad\mathcal{D}(A_{q})=W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega)\cap L^{q}_{\sigma}(\Omega).$ (1.8)

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

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

$A_{o,q}z=P_{q}[(y_{e}\ .\ \nabla)z+(z\ .\
\nabla)y_{e}],\quad\mathcal{D}(A_{o,q})=\mathcal{D}(A_{q}^{{}^{1}\\!/_{2}})\subset
L^{q}_{\sigma}(\Omega),$ (1.9)

where the $\displaystyle{\mathcal{D}}(A_{q}^{{}^{1}\\!/_{2}})$ is defined
explicitly in (1.22) below. Thus, $A_{o,q}A_{q}^{-{}^{1}\\!/_{2}}$ is a
bounded operator on $L^{q}_{\sigma}(\Omega)$, 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

${\mathcal{A}}_{q}=-(\nu
A_{q}+A_{o,q}),\quad{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\subset
L^{q}_{\sigma}(\Omega).$ (1.10)

Finally, we define the projection of the nonlinear portion of the static
operator in (1.7b)

${\mathcal{N}}_{q}(z)=P_{q}[(z\cdot\nabla)z],\quad{\mathcal{D}}({\mathcal{N}}_{q})=W^{1,q}(\Omega)\cap
L^{\infty}(\Omega)\cap L^{q}_{\sigma}(\Omega).$ (1.11)

[As shown in (8.16) in the analysis of the non-linear problem, at the end we
shall use $\displaystyle W^{1,q}(\Omega)\subset L^{\infty}(\Omega)$ for $q>$
dim $\Omega=3$ [30, Theorem 2.4.4, p74, requiring $C^{1}$ boundary.]]

Thus, the Navier-Stokes translated problem (1.7), after application of the
Helmholtz projector $P_{q}$ in Definition 1.1 and use of (1.8)-(1.11), can be
rewritten as the following abstract equation in $L^{q}_{\sigma}(\Omega)$:

$\displaystyle\frac{dz}{dt}+\nu A_{q}z+A_{o,q}z+P_{q}[(z\cdot\nabla)z]$
$\displaystyle=P_{q}(mu)\quad\text{ or
}\quad\frac{dz}{dt}-{\mathcal{A}}_{q}z+{\mathcal{N}}_{q}z=P_{q}(mu)\text{ in
}L^{q}_{\sigma}(\Omega)$ (1.12a)
$\displaystyle\begin{picture}(0.0,0.0)\put(-135.0,12.0){
$\left\\{\rule{0.0pt}{25.0pt}\right.$}\end{picture}z(x,0)$
$\displaystyle=z_{0}(x)=y_{0}(x)-y_{e}\text{ in }L^{q}_{\sigma}(\Omega).$
(1.12b)

###  1.10 The linearized problem of the translated model

Next, still for $1<q<\infty$, we consider the following linearized system of
the translated model (1.7) or (1.12):

$\displaystyle\frac{dw}{dt}+\nu A_{q}w+A_{o,q}w$
$\displaystyle=P_{q}(mu)\quad\text{ or
}\quad\frac{dw}{dt}-{\mathcal{A}}_{q}w=P_{q}(mu)\text{ in
}L^{q}_{\sigma}(\Omega)$ (1.13a)
$\displaystyle\begin{picture}(0.0,0.0)\put(-80.0,12.0){
$\left\\{\rule{0.0pt}{25.0pt}\right.$}\end{picture}w_{0}(x)$
$\displaystyle=y_{0}(x)-y_{e}\text{ in }L^{q}_{\sigma}(\Omega).$ (1.13b)

###  1.11 Some auxiliary results for problem (1.13): analytic semigroup
generation, maximal regularity, domains of fractional powers

In this subsection we collect some known results to be used in the sequel.

  1. (a)

Definition of Besov spaces $B^{s}_{q,p}$ on domains of class $C^{1}$ as real
interpolation of Sobolev spaces: Let $m$ be a positive integer,
$m\in\mathbb{N},0<s<m,1\leq q<\infty,1\leq p\leq\infty,$ then we define [23, p
1398]

$B^{s}_{q,p}(\Omega)=(L^{q}(\Omega),W^{m,q}(\Omega))_{\frac{s}{m},p}$ (1.14a)
This definition does not depend on $\displaystyle m\in\mathbb{N}$ [63, p xx].
This clearly gives $W^{m,q}(\Omega)\subset B_{q,p}^{s}(\Omega)\subset
L^{q}(\Omega)\quad\text{ and }\quad\left\lVert
y\right\rVert_{L^{q}(\Omega)}\leq C\left\lVert
y\right\rVert_{B_{q,p}^{s}(\Omega)}.$ (1.14b)

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

$B^{2-\frac{2}{p}}_{q,p}(\Omega)=\big{(}L^{q}(\Omega),W^{2,q}(\Omega)\big{)}_{1-\frac{1}{p},p}.$
(1.15)

Our interest in (1.15) is due to the following characterization [3, Thm 3.4],
[23, p 1399]: if $A_{q}$ denotes the Stokes operator introduced in (1.8), then

$\displaystyle\Big{(}L^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}g\in B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{ div
}g=0,\ g|_{\Gamma}=0\Big{\\}}\quad\text{if }\frac{1}{q}<2-\frac{2}{p}<2$
(1.16a)
$\displaystyle\Big{(}L^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}g\in B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{ div
}g=0,\
g\cdot\nu|_{\Gamma}=0\Big{\\}}\equiv\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$
(1.16b) $\displaystyle\hskip 113.81102pt\text{ if
}0<2-\frac{2}{p}<\frac{1}{q};\text{ or }1<p<\frac{2q}{2q-1}.$

Notice that, in (1.16b), the condition $\displaystyle g\cdot\nu|_{\Gamma}=0$
is an intrinsic condition of the space $\displaystyle L^{q}_{\sigma}(\Omega)$
in (1.4), not an extra boundary condition as $\displaystyle g|_{\Gamma}=0$ in
(1.16a).

######  Remark 1.3.

In the analysis of well-posedness and stabilization of the nonlinear N-S
problem (1.1), with control $u$ in feedback form - such as the non linear
translated feedback problem (2.20) = (8.1)- we shall need to impose the
constrain $q>3$, see Eq (8.16), to obtain the embedding $\displaystyle
W^{1,q}\hookrightarrow L^{\infty}(\Omega)$ in our case of interest $d=3$, as
already noted below (1.11). What is then the allowable range of the parameter
$p$ in such case $q>3$? The intended goal of the present paper is to obtain
the sought-after stabilization result in a function space, such as a
$\displaystyle B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-space, that does not
recognize boundary conditions of the I.C. Thus, we need to avoid the case in
(1.16a), as this implies a Dirichlet homogeneous B.C. Instead, we need to fit
into the case (1.16b). We shall then impose the condition $\displaystyle
2-\frac{2}{p}<\frac{1}{q}<\frac{1}{3}$ and then obtain that $p$ must satisfy
$\displaystyle p<\frac{6}{5}$. Moreover, analyticity and maximal regularity of
the Stokes problem will require $p>1$. Thus, in conclusion, the allowed range
of the parameters $p,q$ under which we shall solve the well-posedness and
stabilization problem of the nonlinear N-S feedback system (2.20) = (8.1) for
$d=3$, in the space
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ which - as
intended - does not recognize boundary conditions is: $\displaystyle q>3,\
1<p<\frac{6}{5}$. See Theorems 2.3 through 2.5.

  2. (b)

The Stokes and Oseen operators generate a strongly continuous analytic
semigroup on $L^{q}_{\sigma}(\Omega)$, $1<q<\infty$.

######  Theorem 1.2.

Let $d\geq 2,1<q<\infty$ and let $\Omega$ be a bounded domain in
$\mathbb{R}^{d}$ of class $C^{3}$. Then

    1. (i)

the Stokes operator $-A_{q}=P_{q}\Delta$ in (1.8), repeated here as

$-A_{q}\psi=P_{q}\Delta\psi,\quad\psi\in\mathcal{D}(A_{q})=W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega)\cap L^{q}_{\sigma}(\Omega)$ (1.17)

generates a s.c analytic semigroup $e^{-A_{q}t}$ on $L^{q}_{\sigma}(\Omega)$.
See [24] and the review paper [26, Theorem 2.8.5 p 17].

    2. (ii)

The Oseen operator ${\mathcal{A}}_{q}$ in (1.10)

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

generates a s.c analytic semigroup $e^{{\mathcal{A}}_{q}t}$ on
$L^{q}_{\sigma}(\Omega)$. This follows as $A_{o,q}$ is relatively bounded with
respect to $A^{{}^{1}\\!/_{2}}_{q}$, defined in (1.22), see below (1.9): thus
a standard theorem on perturbation of an analytic semigroup generator applies
[39, Corollary 2.4, p 81].

    3. (iii)

$\displaystyle 0\in\rho(A_{q})$ $\displaystyle=\text{ the resolvent set of the
Stokes operator }A_{q}$ (1.19a)
$\displaystyle\begin{picture}(0.0,0.0)\put(-40.0,10.0){
$\left\\{\rule{0.0pt}{18.0pt}\right.$}\end{picture}A_{q}^{-1}$
$\displaystyle:L^{q}_{\sigma}(\Omega)\longrightarrow
L^{q}_{\sigma}(\Omega)\text{ is compact}$ (1.19b)

    4. (iv)

The s.c. analytic Stokes semigroup $e^{-A_{q}t}$ is uniformly stable on
$L^{q}_{\sigma}(\Omega)$: there exist constants $M\geq 1,\delta>0$ (possibly
depending on $q$) such that

$\left\lVert
e^{-A_{q}t}\right\rVert_{{\mathcal{L}}(L^{q}_{\sigma}(\Omega))}\leq
Me^{-\delta t},\ t>0.$ (1.20)

  3. (c)

Domains of fractional powers, ${\mathcal{D}}(A_{q}^{\alpha}),0<\alpha<1$ of
the Stokes operator $A_{q}$ on $L^{q}_{\sigma}(\Omega),1<q<\infty$,

######  Theorem 1.3.

For the domains of fractional powers
${\mathcal{D}}(A_{q}^{\alpha}),0<\alpha<1$, of the Stokes operator $A_{q}$ in
(1.8) = (1.17), the following complex interpolation relation holds true [25]
and [26, Theorem 2.8.5, p 18]

$[{\mathcal{D}}(A_{q}),L^{q}_{\sigma}(\Omega)]_{1-\alpha}={\mathcal{D}}(A_{q}^{\alpha}),\
0<\alpha<1,\ 1<q<\infty;$ (1.21)

in particular

$[{\mathcal{D}}(A_{q}),L^{q}_{\sigma}(\Omega)]_{\frac{1}{2}}={\mathcal{D}}(A_{q}^{{}^{1}\\!/_{2}})\equiv
W_{0}^{1,q}(\Omega)\cap L^{q}_{\sigma}(\Omega).$ (1.22)

Thus, on the space ${\mathcal{D}}(A_{q}^{{}^{1}\\!/_{2}})$, the norms

$\left\lVert\nabla\ \cdot\ \right\rVert_{L^{q}(\Omega)}\text{ and
}\left\lVert\ \right\rVert_{L^{q}(\Omega)}$ (1.23)

are equivalent via Poincaré inequality.

  4. (d)

The Stokes operator $-A_{q}$ and the Oseen operator
${\mathcal{A}}_{q},1<q<\infty$ generate s.c. analytic semigroups on the Besov
space

$\displaystyle\Big{(}L^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}g\in B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{ div
}g=0,\ g|_{\Gamma}=0\Big{\\}}\quad\text{if }\frac{1}{q}<2-\frac{2}{p}<2;$
(1.24a)
$\displaystyle\Big{(}L^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}g\in B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{ div
}g=0,\
g\cdot\nu|_{\Gamma}=0\Big{\\}}\equiv\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$
(1.24b) $\displaystyle\hskip 199.16928pt\text{ if
}0<2-\frac{2}{p}<\frac{1}{q}.$

Theorem 1.2 states that the Stokes operator $-A_{q}$ generates a s.c analytic
semigroup on the space $L^{q}_{\sigma}(\Omega),\ 1<q<\infty$, hence on the
space ${\mathcal{D}}(A_{q})$ in (1.17), with norm $\displaystyle\left\lVert\
\cdot\ \right\rVert_{{\mathcal{D}}(A_{q})}=\left\lVert A_{q}\ \cdot\
\right\rVert_{L^{q}_{\sigma}(\Omega)}$ as $0\in\rho(A_{q})$. Then, one obtains
that the Stokes operator $-A_{q}$ generates a s.c. analytic semigroup on the
real interpolation spaces in (1.24). Next, the Oseen operator
${\mathcal{A}}=-(\nu A_{q}+A_{o,q})$ likewise generates a s.c. analytic
semigroup $\displaystyle e^{{\mathcal{A}}_{q}t}$ on $\displaystyle
L^{q}_{\sigma}(\Omega)$ since $A_{o,q}$ is relatively bounded w.r.t.
$A_{q}^{{}^{1}\\!/_{2}},$ as $A_{o,q}A_{q}^{-{}^{1}\\!/_{2}}$ is bounded on
$L^{q}_{\sigma}(\Omega)$. Moreover ${\mathcal{A}}_{q}$ generates a s.c.
analytic semigroup on
$\displaystyle{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})$
(equivalent norms). Hence ${\mathcal{A}}_{q}$ generates a s.c. analytic
semigroup on the real interpolation space of (1.24). Here below, however, we
shall formally state the result only in the case $\displaystyle
2-{}^{2}\\!/_{p}<{}^{1}\\!/_{q}$. i.e. $\displaystyle 1<p<{}^{2q}\\!/_{2q-1}$,
in the space $\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, as
this does not contain B.C. The objective of the present paper is precisely to
obtain stabilization results on spaces that do not recognize B.C.

######  Theorem 1.4.

Let $1<q<\infty,1<p<{}^{2q}\\!/_{2q-1}$.

    1. (i)

The Stokes operator $-A_{q}$ in (1.17) generates a s.c. analytic semigroup
$e^{-A_{q}t}$ on the space $\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$
defined in (1.16) = (1.24) which moreover is uniformly stable, as in (1.20),

$\left\lVert
e^{-A_{q}t}\right\rVert_{{\mathcal{L}}\big{(}\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}\leq
Me^{-\delta t},\quad t>0.$ (1.25)

    2. (ii)

The Oseen operator ${\mathcal{A}}_{q}$ in (1.18) generates a s.c. analytic
semigroup $e^{{\mathcal{A}}_{q}t}$ on the space
$\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ in (1.16) = (1.24).

  5. (e)

Space of maximal $L^{p}$ regularity on $L^{q}_{\sigma}(\Omega)$ of the Stokes
operator $-A_{q},\ 1<p<\infty,\ 1<q<\infty$ up to $T=\infty$. We return to the
dynamic Stokes problem in $\\{\varphi(t,x),\pi(t,x)\\}$

$\displaystyle\varphi_{t}-\Delta\varphi+\nabla\pi$ $\displaystyle=F$
$\displaystyle\text{ in }(0,T]\times\Omega\equiv Q$ (1.26a) $\displaystyle
div\ \varphi$ $\displaystyle\equiv 0$ $\displaystyle\text{ in }Q$ (1.26b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-70.0,5.0){
$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}\left.\varphi\right\rvert_{\Sigma}$
$\displaystyle\equiv 0$ $\displaystyle\text{ in
}(0,T]\times\Gamma\equiv\Sigma$ (1.26c)
$\displaystyle\left.\varphi\right\rvert_{t=0}$ $\displaystyle=\varphi_{0}$
$\displaystyle\text{ in }\Omega,$ (1.26d)

rewritten in abstract form, after applying the Helmholtz projection $P_{q}$ to
(1.26a) and recalling $A_{q}$ in (1.17) as

$\varphi^{\prime}+A_{q}\varphi=F_{\sigma}\equiv
P_{q}F,\quad\varphi_{0}\in\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
(1.27)

Next, we introduce the space of maximal regularity for
$\\{\varphi,\varphi^{\prime}\\}$ as [26, p 2; Theorem 2.8.5.iii, p 17], [23, p
1404-5], with $T$ up to $\infty$:

$X^{T}_{p,q,\sigma}=L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap
W^{1,p}(0,T;L^{q}_{\sigma}(\Omega))$ (1.28)

(recall (1.8) for ${\mathcal{D}}(A_{q})$) and the corresponding space for the
pressure as

$Y^{T}_{p,q}=L^{p}(0,T;\widehat{W}^{1,q}(\Omega)),\quad\widehat{W}^{1,q}(\Omega)=W^{1,q}(\Omega)/\mathbb{R}.$
(1.29)

The following embedding, also called trace theorem, holds true [3, Theorem
4.10.2, p 180, BUC for $T=\infty$], [40].

$X^{T}_{p,q,\sigma}\subset X^{T}_{p,q}\equiv L^{p}(0,T;W^{2,q}(\Omega))\cap
W^{1,p}(0,T;L^{q}(\Omega))\hookrightarrow
C\Big{(}[0,T];B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\Big{)}.$ (1.30)

For a function $g$ such that $div\ g\equiv 0,\ g|_{\Gamma}=0$ we have $g\in
X^{T}_{p,q}\iff g\in X^{T}_{p,q,\sigma}$, by (1.4).
The solution of Eq(1.27) is

$\varphi(t)=e^{-A_{q}t}\varphi_{0}+\int_{0}^{t}e^{-A_{q}(t-s)}F_{\sigma}(\tau)d\tau.$
(1.31)

The following is the celebrated result on maximal regularity on
$L^{q}_{\sigma}(\Omega)$ of the Stokes problem due originally to Solonnikov
[49] reported in [26, Theorem 2.8.5.(iii) and Theorem 2.10.1 p24 for
$\varphi_{0}=0$], [41], [23, Proposition 4.1 , p 1405].

######  Theorem 1.5.

Let $1<p,q<\infty,T\leq\infty$. With reference to problem (1.26) = (1.27),
assume

$F_{\sigma}\in L^{p}(0,T;L^{q}_{\sigma}(\Omega)),\
\varphi_{0}\in\Big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}.$
(1.32)

Then there exists a unique solution $\varphi\in X^{T}_{p,q,\sigma},\pi\in
Y^{T}_{p,q}$ to the dynamic Stokes problem (1.26) or (1.27), continuously on
the data: there exist constants $C_{0},C_{1}$ independent of
$T,F_{\sigma},\varphi_{0}$ such that via (1.30)

$\displaystyle
C_{0}\left\lVert\varphi\right\rVert_{C\big{(}[0,T];B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}$
$\displaystyle\leq\left\lVert\varphi\right\rVert_{X^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
(1.33)
$\displaystyle\equiv\left\lVert\varphi^{\prime}\right\rVert_{L^{p}(0,T;L^{q}_{\sigma}(\Omega))}+\left\lVert
A_{q}\varphi\right\rVert_{L^{p}(0,T;L^{q}_{\sigma}(\Omega))}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
$\displaystyle\leq C_{1}\bigg{\\{}\left\lVert
F_{\sigma}\right\rVert_{L^{p}(0,T;L^{q}_{\sigma}(\Omega))}+\left\lVert\varphi_{0}\right\rVert_{\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}}\bigg{\\}}.$

In particular,

    1. (i)

With reference to the variation of parameters formula (1.31) of problem (1.27)
arising from the Stokes problem (1.26), we have recalling (1.28): the map

$\displaystyle F_{\sigma}$
$\displaystyle\longrightarrow\int_{0}^{t}e^{-A_{q}(t-\tau)}F_{\sigma}(\tau)d\tau\
:\text{continuous}$ (1.34) $\displaystyle L^{p}(0,T;L^{q}_{\sigma}(\Omega))$
$\displaystyle\longrightarrow X^{T}_{p,q,\sigma}\equiv
L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap W^{1,p}(0,T;L^{q}_{\sigma}(\Omega))$
(1.35)

    2. (ii)

The s.c. analytic semigroup $e^{-A_{q}t}$ generated by the Stokes operator
$-A_{q}$ (see (1.17)) on the space
$\displaystyle\Big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
(see statement below (1.24)) satisfies

$e^{-A_{q}t}:\
\text{continuous}\quad\Big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}\longrightarrow
X^{T}_{p,q,\sigma}\equiv L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap
W^{1,p}(0,T;L^{q}_{\sigma}(\Omega))$ (1.36a) In particular via (1.24b), for
future use, for $1<q<\infty,1<p<\frac{2q}{2q-1}$, the s.c. analytic semigroup
$\displaystyle e^{-A_{q}t}$ on the space
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, satisfies
$e^{-A_{q}t}:\
\text{continuous}\quad\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow
X^{T}_{p,q,\sigma}.$ (1.36b)

    3. (iii)

Moreover, for future use, for $1<q<\infty,1<p<\frac{2q}{2q-1}$, then (1.33)
specializes to

$\left\lVert\varphi\right\rVert_{X^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}\leq
C\bigg{\\{}\left\lVert
F_{\sigma}\right\rVert_{L^{p}(0,T;L^{q}_{\sigma}(\Omega))}+\left\lVert\varphi_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\bigg{\\}}.$
(1.37)

  6. (f)

Maximal $L^{p}$ regularity on $L^{q}_{\sigma}(\Omega)$ of the Oseen operator
${\mathcal{A}}_{q},\ 1<p<\infty,\ 1<q<\infty$, up to $T<\infty$. We next
transfer the maximal regularity of the Stokes operator $(-A_{q})$ on
$L^{q}_{\sigma}(\Omega)$-asserted in Theorem 1.5 into the maximal regularity
of the Oseen operator ${\mathcal{A}}_{q}=-\nu A_{q}-A_{o,q}$ in (1.18) exactly
on the same space $X^{T}_{p,q,\sigma}$ defined in (1.28), however only up to
$T<\infty$.

Thus, consider the dynamic Oseen problem in $\\{\psi(t,x),\pi(t,x)\\}$ with
equilibrium solution $y_{e}$, see (1.2):

$\displaystyle\psi_{t}-\Delta\psi+L_{e}(\psi)+\nabla\pi$ $\displaystyle=F$
$\displaystyle\text{ in }(0,T]\times\Omega\equiv Q$ (1.38a) $\displaystyle
div\ \psi$ $\displaystyle\equiv 0$ $\displaystyle\text{ in }Q$ (1.38b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-100.0,7.0){$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}\left.\psi\right\rvert_{\Sigma}$
$\displaystyle\equiv 0$ $\displaystyle\text{ in
}(0,T]\times\Gamma\equiv\Sigma$ (1.38c)
$\displaystyle\left.\psi\right\rvert_{t=0}$ $\displaystyle=\psi_{0}$
$\displaystyle\text{ in }\Omega,$ (1.38d)
$L_{e}(\psi)=(y_{e}.\nabla)\psi+(\psi.\nabla)y_{e}\hskip 170.71652pt$ (1.39)

rewritten in abstract form, after applying the Helmholtz projector $P_{q}$ to
(1.38a) and recalling ${\mathcal{A}}_{q}$ in (1.18), as

$\psi_{t}={\mathcal{A}}_{q}\psi+P_{q}F=-\nu A_{q}\psi-
A_{o,q}\psi+F_{\sigma},\quad\psi_{0}\in\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
(1.40)

whose solution is

$\psi(t)=e^{{\mathcal{A}}_{q}t}\psi_{0}+\int_{0}^{t}e^{{\mathcal{A}}_{q}(t-\tau)}F_{\sigma}(\tau)d\tau.$
(1.41) $\psi(t)=e^{-\nu A_{q}t}\psi_{0}+\int_{0}^{t}e^{-\nu
A_{q}(t-\tau)}F_{\sigma}(\tau)d\tau-\int_{0}^{t}e^{-\nu
A_{q}(t-\tau)}A_{o,q}\psi(\tau)d\tau.$ (1.42)

######  Theorem 1.6.

Let $1<p,q<\infty,\ 0<T<\infty$. Assume (as in (1.32))

$F_{\sigma}\in
L^{p}\big{(}0,T;L^{q}_{\sigma}(\Omega)\big{)},\quad\psi_{0}\in\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
(1.43)

where ${\mathcal{D}}(A_{q})={\mathcal{D}}({\mathcal{A}}_{q})$, see (1.18).
Then there exists a unique solution $\psi\in X^{T}_{p,q,\sigma},\ \pi\in
Y^{T}_{p,q}$ of the dynamic Oseen problem (1.38), continuously on the data:
that is, there exist constants $C_{0},C_{1}$ independent of
$F_{\sigma},\psi_{0}$ such that

$\displaystyle
C_{0}\left\lVert\varphi\right\rVert_{C\big{(}[0,T];B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}$
$\displaystyle\leq\left\lVert\varphi\right\rVert_{X^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
$\displaystyle\equiv\left\lVert\varphi^{\prime}\right\rVert_{L^{p}(0,T;L^{q}(\Omega))}+\left\lVert
A_{q}\varphi\right\rVert_{L^{p}(0,T;L^{q}(\Omega))}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
(1.44) $\displaystyle\leq C_{T}\bigg{\\{}\left\lVert
F_{\sigma}\right\rVert_{L^{p}(0,T;L^{q}_{\sigma}(\Omega))}+\left\lVert\varphi_{0}\right\rVert_{\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}}\bigg{\\}}$
(1.45)

where $T<\infty$. Equivalently, for $1<p,q<\infty$

    1. i.

The map

$\displaystyle
F_{\sigma}\longrightarrow\int_{0}^{t}e^{{\mathcal{A}}_{q}(t-\tau)}F_{\sigma}(\tau)d\tau\
:\text{continuous}$ (1.46) $\displaystyle L^{p}(0,T;L^{q}_{\sigma}(\Omega))$
$\displaystyle\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)}$

where then automatically, see (1.40)

$L^{p}(0,T;L^{q}_{\sigma}(\Omega))\longrightarrow
W^{1,p}(0,T;L^{q}_{\sigma}(\Omega))$ (1.47)

and ultimately

$L^{p}(0,T;L^{q}_{\sigma}(\Omega))\longrightarrow X^{T}_{p,q,\sigma}\equiv
L^{p}\big{(}0,T;{\mathcal{D}}(A_{q})\big{)}\cap
W^{1,p}(0,T;L^{q}_{\sigma}(\Omega)).$ (1.48)

    2. ii.

The s.c. analytic semigroup $e^{{\mathcal{A}}_{q}t}$ generated by the Oseen
operator ${\mathcal{A}}_{q}$ (see (1.18)) on the space
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
satisfies for $1<p,q<\infty$

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)}$
(1.49)

and hence automatically by (1.28)

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}\longrightarrow
X^{T}_{p,q,\sigma}.$ (1.50)

In particular, for future use, for $1<q<\infty,1<p<\frac{2q}{2q-1}$, we have
that the s.c. analytic semigroup $\displaystyle e^{{\mathcal{A}}_{q}t}$ on the
space $\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, satisfies

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)},\
T<\infty.$ (1.51)

and hence automatically

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow
X^{T}_{p,q,\sigma},\ T<\infty.$ (1.52)

A proof is given in Appendix B.

######  Remark 1.4.

The literature reports physical situations where the volumetric force $f$ is
actually replaced by $\nabla g(x)$; that is, $f$ is a conservative vector
field. Thus, returning to Eq (1.2a) with $f(x)$ replaced now by $\nabla g(x)$
we see that a solution of such stationary problem is $y_{e}=0,\ \pi_{e}=g$,
hence $L_{e}(\cdot)\equiv 0$ by (1.39). Returning to Eq (1.1a) with $f$
replaced by $\nabla g(x)$ and applying to the resulting equation the
projection operator $P_{q}$, one obtains in this case the projected equation

$y_{t}-\nu P_{q}\Delta
y+P_{q}\big{[}(y\cdot\nabla)y\big{]}=P_{q}(mu)\quad\text{in }Q.$ (1.53)

This, along with the solenoidal and boundary conditions (1.1b), (1.1c), yields
the corresponding abstract form recalling also (1.11)

$y_{t}+\nu A_{q}y+{\mathcal{N}}_{q}y=P_{q}(mu)\quad\text{in
}L^{q}_{\sigma}(\Omega).$ (1.54)

Then $y$-problem (1.54) is the same as the $z$-problem (1.12a), except without
the Oseen term $A_{o,q}$. The linearized version of problem (1.54) is then

$\eta_{t}+\nu A_{q}\eta=P_{q}(mu)\quad\text{in }L^{q}_{\sigma}(\Omega),$
(1.55)

which is the same as the $w$-problem (1.13a), except without the Oseen term
$A_{o,q}$. The s.c. analytic semigroup $\displaystyle e^{-\nu A_{q}t}$ driving
the linear equation (1.55) is uniformly stable in $L^{q}_{\sigma}(\Omega)$,
see (1.20), as well as in
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, see (1.25).
Then, in the case of the present Remark, the present paper may be used to
enhance at will the uniform stability of the corresponding problem with $u$
given in feedback form as in the RHS of Eq (2.20) as to obtain a decay rate
much bigger than the original $\delta>0$ in (1.20) or (1.25). Thus there is no
need to perform the translation $y\longrightarrow z$ of Section 1.8, when $f$
in (1.2a) is replaced by $\nabla g(x)$; i.e. $y_{e}=0$ in this case. The
important relevance of the present Remark will be pointed out in the follow-
out paper [35] where only finitely many localized tangential boundary feedback
controls will be employed to the so far open case dim $\Omega=3$. The
corresponding required “unique continuation property” holds true for the
Stokes problem ($y_{e}=0$), see [59], [60].

##  2 Main results

###  2.1 Orientation

All the main results of this paper, Theorems 2.1 through 2.5, are stated (at
first) in the complex state space setting $\displaystyle
L^{q}_{\sigma}(\Omega)+iL^{q}_{\sigma}(\Omega)$. Thus, the finitely many
stabilizing feedback vectors $p_{k},u_{k}$ constructed in the subsequent
proofs belong to the complex finite dimensional unstable subspace
$(W^{u}_{N})^{*}$ and $W^{u}_{N}$ respectively. The question then arises as to
transfer back these results into the original real setting. This issue was
resolved in [7]. Here, such translation, taken from [7], from the results in
the complex setting (Theorems 2.1 through 2.5) into corresponding results in
the original real setting is given in Section 2.7.

Step 1: First, we will show in Theorem 2.2 that the linearized Navier-Stokes
problem $w_{t}={\mathcal{A}}_{q}w+P_{q}(mu)$ in (1.13) can be uniformly
(exponentially) stabilized in the basic space $\displaystyle
L^{q}_{\sigma}(\Omega),1<q<\infty$ in fact, in the space
$\displaystyle{\mathcal{D}}(A_{q}^{\theta}),\ 0\leq\theta\leq 1$, or
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$,
in particular $\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ by
means of an explicitly constructed, finite dimensional spectral-based feedback
controller $mu$, localized on $\omega$, whose structure is given in (2.16).

Step 2: Next, we proceed to the non-linear translated Navier-Stokes
$z$-problem (1.12) with a control $u$ having the same structure as the finite-
dimensional, spectral based stabilizing control used in the linearized problem
(1.13). This strategy leads to the non-linear feedback $z$-problem (2.20). We
then establish two results for problem (2.20):

(i) The first, Theorem 2.3, is that problem (2.20) is locally well-posed, i.e.
for small initial data $z_{0}$, in the desired space
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$. It will require
the constraint $q>3$, see (8.16), to obtain $\displaystyle
W^{1,q}(\Omega)\hookrightarrow L^{\infty}(\Omega)$ for $d=3$. In achieving
this result, we must factor in what is the deliberate, sought-after goal of
the present paper: that is, to obtain (well-posedness and) uniform
stabilization of the original non-linear problem (1.1) near an equilibrium
solution, in a function space that does not recognize boundary conditions.
This is the space
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ having only the
boundary condition $\displaystyle\left.g\cdot\nu\right\rvert_{\Gamma}=0$
inherited from the basic $\displaystyle L^{q}_{\sigma}(\Omega)$-space, see
(1.16b) and statement below it. In contrast, we deliberately exclude then the
space in (1.16a), $\displaystyle p>{}^{2q}\\!/_{2q-1}$, having an explicit
additional B.C. In conclusion, for the nonlinear problem, we need to work with
the space $\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ in
(1.16b), and this requires for $d=3$ the range $\displaystyle
q>3,1<p<{}^{2q}\\!/_{2q-1}$, that is $\displaystyle 1<p<{}^{6}\\!/_{5}$ where
the boundary conditions are not recognized. In this case the space
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)=\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-{}^{1}\\!/_{p},p}$
with index $\displaystyle 1-{}^{1}\\!/_{p}$ close to zero is “close” to the
space $\displaystyle L^{q}(\Omega)$, for $q>3$. Accordingly, with reference to
the feedback $z$-problem (2.20), we take
$z_{0}\in\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega),\
q>3,1<p<{}^{6}\\!/_{5}$ sufficiently small, and show that (2.20) is well-posed
in the function space $X^{\infty}_{p,q,\sigma}$ in (1.28). To this end, we use
critically the maximal regularity result Theorem 7.1. This is Theorem 2.3.

(ii) Second, we address the stabilization problem and show that such Navier-
Stokes feedback problem (2.20) is, in fact, locally exponentially stabilizable
in a neighborhood of the zero equilibrium solution in the state space
$\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$. This is Theorem 2.4.

Such results, Theorem 2.3 and the Theorem 2.4 for the translated Navier-Stokes
$z$-problem (2.20) in feedback form then at once translate into counterpart
results of local well-posedness and local interior stabilization of the
original $y$-problem (1.1) in a neighborhood of the equilibrium solution
$y_{e}$, with an explicit finite dimensional feedback control localized on
$\omega$ whose structure is given in (2.28b). Thus Theorem 2.5 gives the main
result of the present paper.

###  2.2 Introducing the problem of feedback stabilization of the linearized
$w$-problem (1.13) on the complexified $L^{q}_{\sigma}(\Omega)$ space.

Preliminaries: In this subsection we take $q$ fixed, $1<q<\infty$ throughout.
Accordingly, to streamline the notation in the preceding setting of Section 1,
we shall drop the dependence on $q$ of all relevant quantities and thus write
$P,A,A_{o},{\mathcal{A}}$ instead of $P_{q},A_{q},A_{o,q},{\mathcal{A}}_{q}$.
We return to the linearized system (1.13).

Moreover, as in [7], [8], we shall henceforth let $L^{q}_{\sigma}(\Omega)$
denote the complexified space
$L^{q}_{\sigma}(\Omega)+iL^{q}_{\sigma}(\Omega)$, whereby then we consider the
extension of the linearized problem (1.13) to such complexified space. Thus,
henceforth, $w$ will mean $w+i\tilde{w}$, $u$ will mean $u+i\tilde{u}$,
$w_{0}$ will mean $w_{0}+i\tilde{w}_{0}$:

$\frac{dw}{dt}+\nu
Aw+A_{0}w=P(mu),\quad\text{or}\quad\frac{dw}{dt}-{\mathcal{A}}w=P(mu),\
w(0)=w_{0}\text{ on }L^{q}_{\sigma}(\Omega).$ (2.1)

As noted in Theorem 1.2(iii), the Oseen operator ${\mathcal{A}}$ has compact
resolvent on $L^{q}_{\sigma}(\Omega)$. It follows that ${\mathcal{A}}$ has a
discreet point spectrum $\sigma({\mathcal{A}})=\sigma_{p}({\mathcal{A}})$
consisting of isolated eigenvalues $\\{\lambda_{j}\\}_{j=1}^{\infty}$, which
are repeated according to their (finite) algebraic multiplicity $\ell_{j}$.
However, since ${\mathcal{A}}$ generates a $C_{0}$ analytic semigroup on
$L^{q}_{\sigma}(\Omega)$, its eigenvalues $\\{\lambda_{j}\\}_{j=1}^{\infty}$
lie in a triangular sector of a well-known type.

The case of interest in stabilization occurs where ${\mathcal{A}}$ has a
finite number, say $N$, of eigenvalues
$\lambda_{1},\lambda_{2},\lambda_{3},\dots,\lambda_{N}$ on a complex half
plane $\\{\lambda\in\mathbb{C}:Re~{}\lambda\geq 0\\}$ which we then order
according to their real parts, so that

$\ldots\leq Re~{}\lambda_{N+1}<0\leq Re~{}\lambda_{N}\leq\ldots\leq
Re~{}\lambda_{1},$ (2.2)

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 $\lambda_{j}$ of ${\mathcal{A}}$, 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.

There are results in the literature [28] that quantify the number of unstable
eigenvalues in terms of the system parameters. Denote by $P_{N}$ and
$P_{N}^{*}$ the projections given explicitly by [29, p 178], [7], [8]

$\displaystyle P_{N}$ $\displaystyle=-\frac{1}{2\pi
i}\int_{\Gamma}\left(\lambda
I-{\mathcal{A}}\right)^{-1}d\lambda:L^{q}_{\sigma}(\Omega)\text{ onto
}W^{u}_{N}$ (2.3a) $\displaystyle P_{N}^{*}$ $\displaystyle=-\frac{1}{2\pi
i}\int_{\bar{\Gamma}}\left(\lambda
I-{\mathcal{A}}^{*}\right)^{-1}d\lambda:(L^{q}_{\sigma}(\Omega))^{*}\text{
onto }(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega),$ (2.3b)

by (A.2c), where $\Gamma$ (respectively, its conjugate counterpart
$\bar{\Gamma}$) is a smooth closed curve that separates the unstable spectrum
from the stable spectrum of ${\mathcal{A}}$ (respectively,
${\mathcal{A}}^{*}$). As in [8, Sect 3.4, p 37], following [57],we decompose
the space $L^{q}_{\sigma}(\Omega)$ into the sum of two complementary subspaces
(not necessarily orthogonal):

$L^{q}_{\sigma}(\Omega)=W^{u}_{N}\oplus W^{s}_{N};\quad W^{u}_{N}\equiv
P_{N}L^{q}_{\sigma}(\Omega);\quad
W^{s}_{N}\equiv(I-P_{N})L^{q}_{\sigma}(\Omega);\quad\text{ dim }W^{u}_{N}=N$
(2.4)

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

${\mathcal{A}}^{u}_{N}=P_{N}{\mathcal{A}}={\mathcal{A}}|_{W^{u}_{N}};\quad{\mathcal{A}}^{s}_{N}=(I-P_{N}){\mathcal{A}}={\mathcal{A}}|_{W^{s}_{N}}$
(2.5)

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

$\sigma({\mathcal{A}}_{N}^{u})=\\{\lambda_{j}\\}_{j=1}^{N};\quad\sigma({\mathcal{A}}_{N}^{s})=\\{\lambda_{j}\\}_{j=N+1}^{\infty},$
(2.6)

and $W_{N}^{u}$ is the generalized eigenspace of ${\mathcal{A}}^{u}_{N}$ in
(2.1). The system (2.1) on $L^{q}_{\sigma}(\Omega)$ can accordingly be
decomposed as

$w=w_{N}+\zeta_{N},\quad w_{N}=P_{N}w,\quad\zeta_{N}=(I-P_{N})w.$ (2.7)

After applying $P_{N}$ and $(I-P_{N})$ (which commute with ${\mathcal{A}}$) on
(2.1), we obtain via (2.5)

$\text{on }W_{N}^{u}:w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}=P_{N}P(mu);\
w_{N}(0)=P_{N}w_{0}$ (2.8a) $\text{on
}W_{N}^{s}:\zeta^{\prime}_{N}-{\mathcal{A}}^{s}_{N}\zeta_{N}=(I-P_{N})P(mu);\
\zeta_{N}(0)=(I-P_{N})w_{0}$ (2.8b)

respectively.

Main Result: We may now state the main feedback stabilization result of the
linearized problem (1.13) (=(2.1)) on the complexified space
$L^{q}_{\sigma}(\Omega)$). The proof is constructive. How to construct the
finitely many stabilizing vectors will be established in the proof.

We anticipate the fact (noted in (3.2) and (4.0)) below that, for
$1<p,q<\infty$:

$\begin{aligned} W^{u}_{N}=&\text{ space of generalized}\\\ &\text{
eigenfunctions of ${\mathcal{A}}_{q}(={\mathcal{A}}^{u}_{N})$}\\\ &\text{
corresponding to its distinct}\\\ &\text{ unstable
eigenvalues}\end{aligned}\subset\begin{cases}\begin{aligned}
&\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}\\\
&\big{[}{\mathcal{D}}(A_{q}),L^{q}_{\sigma}(\Omega)\big{]}_{1-\alpha}={\mathcal{D}}(A^{\alpha}_{q}),\
0\leq\alpha\leq 1\end{aligned}\end{cases}\subset L^{q}_{\sigma}(\Omega).$
(2.9)

###  2.3 Uniform (exponential) stabilization of the linear finite-dimensional
$w_{N}$-problem (2.8a) in the space $W^{u}_{N}$ by means of a finite-
dimensional, explicit, spectral based feedback control localized on $\omega$.

######  Theorem 2.1.

Let $\lambda_{1},.,\lambda_{i},.,\lambda_{M}$ be the unstable distinct
eigenvalues of the Oseen operator ${\mathcal{A}}(={\mathcal{A}}_{q})$ (see
(1.10)) with geometric multiplicity $\ell_{i}$ and set $K=\sup\
\\{\ell_{j};j=1,\dots,M\\}.$ Let $\omega$ be an arbitrarily small open portion
of the interior with sufficiently smooth boundary $\partial\omega$. Then:
Given $\gamma>0$ arbitrarily large, one can construct suitable interior
vectors $[u_{1},\dots,u_{K}]$ in the smooth subspace $W^{u}_{N}$ of
$L^{q}_{\sigma}(\omega),1<q<\infty$, and accordingly obtain a K-dimensional
interior controller $u=u_{N}$ acting on $\omega$, of the form

$u=\sum_{k=1}^{K}\mu_{k}(t)u_{k},\quad u_{k}\in W^{u}_{N}\subset
L^{q}_{\sigma}(\Omega),\quad\mu_{k}(t)=\text{scalar},$ (2.10)

such that, once inserted in the finite dimensional projected $w_{N}$-system in
(2.8), yields the system

$w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}=P_{N}P\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}\mu_{k}(t)u_{k}\Bigg{)}\Bigg{)},$
(2.11)

whose solution then satisfies the estimate

$\left\lVert w_{N}(t)\right\rVert_{L^{q}_{\sigma}(\Omega)}+\left\lVert
u_{N}(t)\right\rVert_{L^{q}_{\sigma}(\omega)}\leq C_{\gamma}e^{-\gamma
t}\left\lVert P_{N}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)},\ t\geq 0.$
(2.12)

In (2.12) we may replace the $L^{q}_{\sigma}(\Omega)$-norm, $1<q<\infty$,
alternatively either with the
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
norm, $1<q<\infty$; or else with the
$\displaystyle\big{[}{\mathcal{D}}(A_{q}),L^{q}_{\sigma}(\Omega)\big{]}_{1-\alpha}={\mathcal{D}}(A^{\alpha}_{q})$-norm,
$0\leq\alpha\leq 1,\ 1<q<\infty$. In particular, we also have

$\left\lVert
w_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+\left\lVert
u_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\leq
C_{\gamma}e^{-\gamma t}\left\lVert
P_{N}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\
t\geq 0,$ (2.13)

in the $\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-norm, $1<q<\infty,\
p<{}^{2q}\\!/_{2q-1}$.
[Estimate (2.13) will be invoked in the nonlinear stabilization proof of
Section 9.]

Moreover, the above control $\displaystyle
u=u_{N}=\sum_{k=1}^{K}\mu_{k}(t)u_{k}$, the terms $u_{k}\in W^{u}_{N},$ in
(2.10) can be chosen in feedback form: that is, of the form
$\mu_{k}(t)=(w_{N}(t),p_{k})_{\omega}$ for suitable vectors
$p_{k}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega)$ depending on
$\gamma$. Here and henceforth
$\displaystyle(v_{1},v_{2})_{\omega}=\int_{\omega}v_{1}\cdot\bar{v}_{2}\
d\omega,\ v_{1}\in W^{u}_{N}\subset L^{q}_{\sigma}(\Omega),\
v_{2}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega)$. In
conclusion, $w_{N}$ in (2.11) satisfying (2.12),(2.13) is the solution of the
following equation on $W^{u}_{N}$ (see (2.8)):

$w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}=P_{N}P\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}(w_{N}(t),p_{k})_{\omega}u_{k}\Bigg{)}\Bigg{)},\
u_{k}\in W^{u}_{N}\subset L^{q}_{\sigma}(\Omega),\
p_{k}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega),$ (2.14a)

rewritten as

$w^{\prime}_{N}=\bar{A}^{u}w_{N},\quad
w_{N}(t)=e^{\bar{A}^{u}t}P_{N}w_{0},\quad w_{N}(0)=P_{N}w_{0}.$ (2.14b)

A proof of Theorem 2.1 is given in Section 5.

###  2.4 Global well-posedness and uniform exponential stabilization on the
linearized $w$-problem (2.1) in various $L^{q}_{\sigma}(\Omega)$-based spaces,
by means of the same feedback control obtained for the $w_{N}$-problem in
Section 2.3.

Again, $1<q<\infty$ throughout this section.

######  Theorem 2.2.

With reference to the unstable, possibly repeated, eigenvalues
$\\{\lambda\\}_{j=1}^{N}$ in (2.2), $M$ of which are distinct, let
$\varepsilon>0$ and set $\gamma_{0}=|Re~{}\lambda_{N+1}|-\varepsilon$. Then
the same K-dimensional feedback controller

$u=u_{N}=\sum_{k=1}^{K}(w_{N}(t),p_{k})_{\omega}u_{k},\quad u_{k}\in
W^{u}_{N}\subset L^{q}_{\sigma}(\Omega),\ p_{k}\in(W^{u}_{N})^{*}\subset
L^{q^{\prime}}_{\sigma}(\Omega),$ (2.15)

constructed in Theorem 2.1, (2.14a) and yielding estimate (2.12), (2.13) for
the finite-dimensional projected $w_{N}$-system (2.8), once inserted, this
time in the full linearized $w$-problem (2.1), yields the linearized feedback
dynamics $(w_{N}=P_{N}w)$:

$\frac{dw}{dt}={\mathcal{A}}w+P\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}(P_{N}w,p_{k})_{\omega}u_{k}\Bigg{)}\Bigg{)}\equiv\mathbb{A}_{{}_{F}}w$
(2.16)

where $\mathbb{A}_{{}_{F}}$ is the generator of a s.c. analytic semigroup in
the space $L^{q}_{\sigma}(\Omega)$. Here,
$\displaystyle{\mathcal{A}}={\mathcal{A}}_{q},\ P=P_{q},\
\mathbb{A}_{{}_{F}}=\mathbb{A}_{{}_{F,q}}$Moreover, such dynamics $w$ in
(2.16) (equivalently, such generator $\mathbb{A}_{{}_{F}}$ in (2.16)) is
uniformly stable in the space $L^{q}_{\sigma}(\Omega)$:

$\left\lVert
e^{\mathbb{A}_{{}_{F}}t}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)}=\left\lVert
w(t;w_{0})\right\rVert_{L^{q}_{\sigma}(\Omega)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert
w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)},\quad t\geq 0$ (2.17)

or for $0<\theta<1$ and $\delta>0$ arbitrarily small

$\displaystyle C_{\gamma_{0},\theta}e^{-\gamma_{0}t}\left\lVert
A^{\theta}_{q}\ w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)},\quad t\geq 0,\
w_{0}\in{\mathcal{D}}(A^{\theta}_{q})$ (2.18a) $\left\lVert A^{\theta}_{q}\
e^{\mathbb{A}_{{}_{F}}t}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)}=\left\lVert
A^{\theta}_{q}\
w(t;w_{0})\right\rVert_{L^{q}_{\sigma}(\Omega)}\leq\left\\{\rule{0.0pt}{25.0pt}\right.$
$\displaystyle C_{\gamma_{0},\theta,\delta}e^{-\gamma_{0}t}\left\lVert
w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)},\quad t\geq\delta>0.$ (2.18b)

As in the case of Theorem 2.1, we may replace the
$L^{q}_{\sigma}(\Omega)$-norm in (2.17), $1<q<\infty$, with the
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$-norm,
$1<p,q<\infty$; in particular, with the
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-norm

$\displaystyle\left\lVert
e^{\mathbb{A}_{{}_{F}}t}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}=\left\lVert
w(t;w_{0})\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$
$\displaystyle\leq C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert
w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\ t\geq 0$
(2.19) $\displaystyle\hskip 113.81102pt1<q<\infty,\ 1<p<\frac{2q}{2q-1}.$

A proof of Theorem 2.2 is given in Section 6.

###  2.5 Local well-posedness and uniform (exponential) null stabilization of
the translated nonlinear $z$-problem (1.7) or (1.12) by means of a finite
dimensional explicit, spectral based feedback control localized on $\omega$.

Starting with the present section, the nonlinearity of problem (1.1) will
impose for $d=3$ the requirement $q>3$, see (8.16) below. As our deliberate
goal is to obtain the stabilization result in the space
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ which does not
recognize boundary conditions, then the limitation $p<{}^{2q}\\!/_{2q-1}$ of
this space applies. In conclusion, our well-posedness and stabilization
results will hold under the restriction $q>3,1<p<{}^{6}\\!/_{5}$ for $d=3$,
and $q>2,1<p<{}^{4}\\!/_{3}$ for $d=2$.

######  Theorem 2.3.

For $d=3$, let $1<p<{}^{6}\\!/_{5}$ and $q>3$, while for $d=2$, let
$1<p<{}^{4}\\!/_{3}$ and $q>2$. Consider the nonlinear $z$-problem (1.12) in
the following feedback form

$\frac{dz}{dt}-{\mathcal{A}}_{q}z+{\mathcal{N}}_{q}z=P_{q}\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}(P_{N}z,p_{k})_{\omega}u_{k}\Bigg{)}\Bigg{)}$
(2.20)

i.e. subject to a feedback control of the same structure as in the linear
$w$-dynamics (2.16), Here $p_{k},u_{k}$ are the same vectors as constructed in
Theorem 2.1, and appearing in (2.14) or (2.16). There exists a positive
constant $\rho>0$ such that, if the initial condition $z_{0}$ satisfies

$\left\lVert
z_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}<\rho,$
(2.21)

then problem (2.20) defines a unique solution $z$ in the space (see (1.28),
(1.30))

$\displaystyle z\in X^{\infty}_{p,q,\sigma}$ $\displaystyle\equiv
L^{p}(0,\infty;{\mathcal{D}}(A_{q}))\cap
W^{1,p}(0,\infty;L^{q}_{\sigma}(\Omega))$ (2.22) $\displaystyle\hookrightarrow
C([0,\infty);\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)),$ (2.23)

where ${\mathcal{D}}(A_{q})$ is topologically $W^{2,q}(\Omega)\cap
L^{q}_{\sigma}(\Omega)$, see (1.8).

A proof of Theorem 2.3 is given in Section 8.

######  Theorem 2.4.

In the situation of Theorem 2.3, we have that such solution is uniformly
stable on the space $\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$: there
exist constants $\widetilde{\gamma}>0,M_{\widetilde{\gamma}}\geq 1$, such that
said solution satisfies

$\displaystyle\left\lVert
z(t;z_{0})\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$
$\displaystyle\leq M_{\widetilde{\gamma}}e^{-\widetilde{\gamma}t}\left\lVert
z_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\ t\geq 0.$
(2.24)

A proof of Theorem 2.4 is given in Section 9. It will be critically based on
the maximal regularity of the semigroup $\displaystyle
e^{\mathbb{A}_{{}_{F}}t}$ giving the solution of the feedback $w$-problem
(2.16), $\displaystyle\mathbb{A}_{{}_{F}}=\mathbb{A}_{{}_{F,q}}$. Remark 9.1,
at the end of Section 9, will provide insight on the relationship between
$\widetilde{\gamma}$ in the nonlinear case in (2.24) and $\gamma_{0}$ in the
corresponding linear case in (2.17).

###  2.6 Local well-posedness and uniform (exponential) stabilization of the
original nonlinear $y$-problem (1.1) in a neighborhood of an equilibrium
solution $y_{e}$, by means of a finite dimensional explicit, spectral based
feedback control localized on $\omega$.

The result of this subsection is an immediate corollary of section 2.5.

######  Theorem 2.5.

Let $1<p<{}^{6}\\!/_{5},q>3,d=3$; and $1<p<{}^{4}\\!/_{3},q>2,d=2$. Consider
the original N-S problem (1.1). Let $y_{e}$ be a given equilibrium solution as
guaranteed by Theorem 1.1 for the steady state problem (1.2). For a constant
$\rho>0$, let the initial condition $y_{0}$ in (1.1d) be in
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ and satisfy

${\mathcal{V}}_{\rho}\equiv\Big{\\{}y_{0}\in\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\left\lVert
y_{0}-y_{e}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\leq\rho\Big{\\}},\quad\rho>0.$
(2.25)

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

  1. (i)

for each $y_{0}\in{\mathcal{V}}_{\rho}$, there exists an interior finite
dimensional feedback controller

$u=F(y-y_{e})=\sum_{k=1}^{K}(P_{N}(y-y_{e}),p_{k})_{\omega}u_{k}$ (2.26)

that is, of the same structure as in the translated N-S $z$-problem (2.20),
with the same vectors $\displaystyle p_{k},u_{k}$ in (2.14) or (2.16), such
that the closed loop problem corresponding to (1.1)

$\displaystyle y_{t}-\nu\Delta y+(y\cdot\nabla)y+\nabla\pi$
$\displaystyle=m(F(y-y_{e}))+f(x)$ $\displaystyle\text{ in }Q$ (2.27a)
$\displaystyle div\ y$ $\displaystyle=0$ $\displaystyle\text{ in }Q$ (2.27b)
$\displaystyle\begin{picture}(120.0,0.0)\put(-18.0,9.0){$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}y$
$\displaystyle=0$ $\displaystyle\text{ on }\Sigma$ (2.27c) $\displaystyle
y|_{t=0}$ $\displaystyle=y_{0}$ $\displaystyle\text{ in }\Omega$ (2.27d)

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

$\displaystyle y_{t}+\nu A_{q}y+{\mathcal{N}}_{q}y$
$\displaystyle=P_{q}\Big{[}m\big{(}F(y-y_{e})\big{)}+f(x)\Big{]}$ (2.28a)
$\displaystyle=P_{q}\bigg{[}m\bigg{(}\sum_{k=1}^{K}\big{(}P_{N}(y-y_{e}),p_{k}\big{)}_{\omega}u_{k}\bigg{)}+f(x)\bigg{]}$
(2.28b) $\displaystyle y(0)$
$\displaystyle=y_{0}\in\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ (2.28c)

has a unique solution $\displaystyle y\in
C\big{(}[0,\infty);\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}.$

  2. (ii)

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

$\left\lVert
y(t)-y_{e}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\leq
M_{\widetilde{\gamma}}e^{-\widetilde{\gamma}t}\left\lVert
y_{0}-y_{e}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\quad
t\geq 0,\ y_{0}\in{\mathcal{V}}_{\rho}.$ (2.29)

Once the neighborhood ${\mathcal{V}}_{\rho}$ 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 9.1 comparing $\widetilde{\gamma}$ in (2.29) with $\gamma_{0}$ in
(2.17).

###  2.7 Results on the real space setting

Here we shall complement the results of Theorems 2.1 through 2.5 by giving
their version in the real space setting. We shall quote from [7]. In the
complexified setting $\displaystyle
L^{q}_{\sigma}(\Omega)+iL^{q}_{\sigma}(\Omega)$ we have that the complex
unstable subspace $W^{u}_{N}$ is,

$\displaystyle W^{u}_{N}$ $\displaystyle=W^{1}_{N}+iW^{2}_{N}$ (2.30)
$\displaystyle=\text{space of generalized eigenfunctions
}\\{\phi_{j}\\}_{j=1}^{N}\text{ of the operator
}{\mathcal{A}}_{q}(={\mathcal{A}}_{q}^{u})\text{ corresponding to}$
$\displaystyle\quad\text{ its }N\text{ unstable eigenvalues.}$ (2.31)

Set $\displaystyle\phi_{j}=\phi_{j}^{1}+i\phi_{j}^{2}$ with
$\phi_{j}^{1},\phi_{j}^{2}$ real. Then:

$W^{1}_{N}=Re\ W^{u}_{N}=\text{span}\\{\phi_{j}^{1}\\}_{j=1}^{N};\quad
W^{2}_{N}=Im\ W^{u}_{N}=\text{span}\\{\phi_{j}^{2}\\}_{j=1}^{N}.$ (2.32)

The stabilizing vectors $p_{k},u_{k},\ k=1,\dots,K$ are complex valued with
$u_{k}\in W^{u}_{N}\subset L^{q}_{\sigma}(\Omega)$, and
$p_{k}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega)$, as in
(2.15).

The complex-valued uniformly stable linear $w$-system in (2.16) with $K$
complex valued stabilizing vectors admits the following real-valued uniformly
stable counterpart

$\frac{dw}{dt}={\mathcal{A}}_{q}w+P_{q}\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}\text{Re
}(w_{N}(t),p_{k})_{\omega}\text{ Re }u_{k}-\sum_{k=1}^{K}\text{Im }\
(w_{N}(t),p_{k})_{\omega}\text{ Im }u_{k}\Bigg{)}\Bigg{)}$ (2.33)

with $2K\leq N$ real stabilizing vectors, see [7, Eq 3.52a, p 1472]. If
$K=\text{sup }\\{\ell_{i},i=1,\dots,M\\}$ is achieved for a real eigenvalue
$\lambda_{i}$ (respectively, a complex eigenvalue $\lambda_{i}$), then the
effective number of stabilizing controllers is $K\leq N$, as the generalized
functions are then real, since $y_{e}$ is real; respectively, $2K\leq N$, for,
in this case, the complex conjugate eigenvalue $\bar{\lambda}_{j}$ contributes
an equal number of components in terms of generalized eigenfunctions
$\displaystyle\phi_{\bar{\lambda}_{j}}=\bar{\phi}_{\lambda_{j}}$. In all
cases, the actual (effective) upper bound $2K$ is $2K\leq N$. For instance, if
all unstable eigenvalues were real and simple then $K=1$, and only one
stabilizing controller is actually needed.

Similarly, the complex-valued locally (near $y_{e}$) uniformly stable
nonlinear $y$-system (2.28) with $K$ complex-valued stabilizing vectors admits
the following real-valued locally uniformly stable counterpart

$\frac{dy}{dt}-\nu
A_{q}y+{\mathcal{N}}_{q}y=P_{q}\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}\text{Re
}(y-y_{e},p_{k})_{\omega}\text{ Re }u_{k}-\sum_{k=1}^{K}\text{Im
}(y-y_{e},p_{k})_{\omega}\text{ Im }u_{k}\Bigg{)}\Bigg{)}$ (2.34)

with $2K\leq N$ real stabilizing vectors, see [8, p 43].

##  3 Algebraic rank condition for the $w_{N}$-dynamics in (2.8a) under the
(preliminary) Finite-Dimensional Spectral Assumption (FDSA)

Preliminaries: For $i=1,\dots,M$, we now denote by
$\\{\phi_{ij}\\}_{j=1}^{\ell_{i}}$, $\\{\phi_{ij}^{*}\\}_{j=1}^{\ell_{i}}$ the
normalized linearly independent eigenfunctions (on $L^{q}_{\sigma}(\Omega)$
and
$\displaystyle(L^{q}_{\sigma}(\Omega))^{\prime}=L^{q^{\prime}}_{\sigma}(\Omega)$,
respectively $\displaystyle{}^{1}\\!/_{q}+{}^{1}\\!/_{q^{\prime}}=1$ invoking
property (A.2b) of Appendix A ) corresponding to the unstable distinct
eigenvalues $\lambda_{1},\ldots,\lambda_{M}$ of ${\mathcal{A}}$ and
$\bar{\lambda}_{1},\ldots,\bar{\lambda}_{M}$ of ${\mathcal{A}}^{*}$,
respectively:

$\displaystyle{\mathcal{A}}\phi_{ij}$
$\displaystyle=\lambda_{i}\phi_{ij}\in{\mathcal{D}}({\mathcal{A}}_{q})=W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega)\cap L^{q}_{\sigma}(\Omega)\in L^{q}(\Omega)$ (3.1a)
$\displaystyle{\mathcal{A}}^{*}\phi_{ij}^{*}$
$\displaystyle=\bar{\lambda}_{i}\phi_{ij}^{*}\in{\mathcal{D}}({\mathcal{A}}_{q}^{*})=W^{2,q^{\prime}}(\Omega)\cap
W^{1,q^{\prime}}_{0}(\Omega)\cap L^{q^{\prime}}_{\sigma}(\Omega)\in
L^{q^{\prime}}(\Omega).$ (3.1b)

FDSA: We henceforth assume in this section that for each of the distinct
eigenvalues $\lambda_{1},\ldots,\lambda_{M}$ of ${\mathcal{A}}$, algebraic and
geometric multiplicity coincide:

$W_{N,i}^{u}\equiv
P_{N,i}L^{q}_{\sigma}(\Omega)=\text{span}\\{\phi_{ij}\\}_{j=1}^{\ell_{i}};\quad(W_{N,i}^{u})^{*}\equiv
P_{N,i}^{*}(L^{q}_{\sigma}(\Omega))^{*}=\text{span}\\{\phi_{ij}^{*}\\}_{j=1}^{\ell_{i}};$
(3.2)

Here $P_{N,i},P_{N,i}^{*}$ are the projections corresponding to the
eigenvalues $\lambda_{i}$ and $\bar{\lambda}_{i}$, respectively. For instance,
$P_{N,i}$ is given by an integral such as the one on the RHS of (2.3a), where
now $\Gamma$ is a closed smooth curve encircling the eigenvalue $\lambda_{i}$
and no other. Similarly $P_{N,i}^{*}$. The space $W_{N,i}^{u}=$ range of
$P_{N,i}$ is the algebraic eigenspace of the eigenvalues $\lambda_{i}$, and
$\ell_{i}=$ dim $W_{N,i}^{u}$ is the algebraic multiplicity of $\lambda_{i}$,
so that $\ell_{1}+\ell_{2}+\dots+\ell_{M}=N$. As a consequence of the FDSA, we
obtain

$W_{N}^{u}\equiv
P_{N}L^{q}_{\sigma}(\Omega)=\text{span}\\{\phi_{ij}\\}_{i=1,j=1}^{M\ \
\ell_{i}};\quad(W_{N}^{u})^{*}\equiv
P_{N}^{*}(L^{q}_{\sigma}(\Omega))^{*}=\text{span}\\{\phi_{ij}^{*}\\}_{i=1,j=1}^{M\
\ \ell_{i}}.$ (3.3)

Without the FDSA, $W_{N}^{u}$ is the span of the generalized eigenfunctions of
${\mathcal{A}}$, corresponding to its unstable distinct eigenvalues
$\\{\lambda_{j}\\}_{j=1}^{M}$; and similarly for $(W_{N}^{u})^{*}$ (see the
subsequent section). In other words, the FDSA says that the restriction
${\mathcal{A}}_{N}^{u}$ in (2.5) is diagonalizable or that
${\mathcal{A}}_{N}^{u}$ is semisimple on $W_{N}^{u}$ in the terminology of
[29, p 43]. Under the FDSA, any vector $w\in W_{N}^{u}$ admits the following
unique expansion [29, p 12, Eq (2.16)], [7, p 1453], in terms of the basis
$\\{\phi_{ij}\\}_{i=1,j=1}^{M\ \ \ell_{i}}$ in $L^{q}_{\sigma}(\Omega)$ and
its adjoint basis [29, p 12] $\\{\phi_{ij}^{*}\\}_{i=1,j=1}^{M\ \ \ell_{i}}$
in $(L^{q}_{\sigma}(\Omega))^{*}$:

$W_{N}^{u}\ni
w=\sum_{i,j}^{M,\ell_{i}}(w,\phi_{ij}^{*})\phi_{ij};\quad(\phi_{ij},\phi_{hk}^{*})=\begin{cases}1&\text{if
$i=h,j=k$}\\\ 0&\text{otherwise}\\\ \end{cases}$ (3.4)

that is, the system consisting of $\\{\phi_{ij}\\}$ and $\\{\phi_{ij}^{*}\\}$,
$i=1,\ldots,M,\ j=1,\ldots,\ell_{i}$, can be chosen to form bi-orthogonal
sequences. Here $(\ ,\ )$ denotes the scalar product between $W^{u}_{N}$ and
$(W^{u}_{N})^{*}$ [29, p 12]. i.e. ultimately, the duality pairing in $\Omega$
between $L^{q}_{\sigma}(\Omega)$ and $(L^{q}_{\sigma}(\Omega))^{*}$. Next, we
return to the $w_{N}$-dynamics in (2.8a), rewritten here for convenience

$\text{on }W_{N}^{u}:w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}=P_{N}P(mu);\
w_{N}(0)=P_{N}w_{0}.$ (3.5)

The term $P_{N}P(mu)$ expressed in terms of adjoint bases. Next, let $mu\in
L^{q}(\omega)$ where $q>1$. Here below we compute the RHS of the term
$P_{N}P(mu)$ via the adjoint bases expansion in (3.4), where we notice that
$P^{*}P_{N}^{*}\phi_{ij}^{*}=\phi_{ij}^{*}$ because
$\phi_{ij}^{*}\in{\mathcal{D}}({\mathcal{A}}^{*})$, so that $\phi_{ij}^{*}$ is
invariant under the projections $P^{*}$ and $P_{N}^{*}$. With
$\displaystyle(f,g)_{\omega}=\int_{\omega}f\bar{g}\ d\omega$, we obtain

$W_{N}^{u}\ni
P_{N}P(mu)=\sum_{i,j=1}^{M,\ell_{i}}(P_{N}P(mu),\phi_{ij}^{*})\phi_{ij}=\sum_{i,j=1}^{M,\ell_{i}}(mu,\phi_{ij}^{*})\phi_{ij}=\sum_{i,j=1}^{M,\ell_{i}}(u,\phi_{ij}^{*})_{\omega}\phi_{ij},$
(3.6)

so that the dynamics (3.5) on $W^{u}_{N}$ becomes

$\text{on
}W_{N}^{u}:w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}=\sum_{i,j=1}^{M,\ell_{i}}(u,\phi_{ij}^{*})_{\omega}\phi_{ij}.\
$ (3.7)

Selection of the scalar interior control function $u$ in finite dimensional
separated form (with respect to $K$ coordinates). Next, we select the control
$u$ of the form given in (2.10)

$u=\sum_{k=1}^{K}\mu_{k}(t)u_{k},\quad u_{k}\in W^{u}_{N}\subset
L^{q}_{\sigma}(\Omega),\quad\mu_{k}(t)=\text{scalar}$ (3.8)

so that the term in (3.6) in $W^{u}_{N}$ specializes to

$W_{N}^{u}\ni
P_{N}P(mu)=\sum_{i,j=1}^{M,\ell_{i}}\Bigg{\\{}\sum_{k=1}^{K}(u_{k},\phi_{ij}^{*})_{\omega}\mu_{k}(t)\Bigg{\\}}\phi_{ij}.\\\
$ (3.9)

Substituting (3.9) on the RHS of (3.5), we finally obtain

$\text{on
}W_{N}^{u}:w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}=\sum_{i,j=1}^{M,\ell_{i}}\Bigg{\\{}\sum_{k=1}^{K}(u_{k},\phi_{ij}^{*})_{\omega}\mu_{k}(t)\Bigg{\\}}\phi_{ij}.$
(3.10)

The dynamics (3.10) in coordinate form on $W^{u}_{N}$. Our next goal is to
express the finite dimensional dynamics (3.10) on the $N$-dimensional space
$W^{u}_{N}$ in a component-wise form. To this end, we introduce the following
ordered bases $\beta_{i}$ and $\beta$ of length $\ell_{i}$ and $N$
respectively:

$\displaystyle\beta_{i}$ $\displaystyle=[\phi_{i1},\dots,\phi_{i\ell_{i}}]:\
\text{ basis on }W^{u}_{N,i}$ (3.11) $\displaystyle\beta$
$\displaystyle=\beta_{1}\cup\beta_{2}\cup\dots\cup\beta_{M}=[\phi_{11},\dots,\phi_{1\ell_{1}},\phi_{21},\dots,\phi_{2\ell_{2}},\dots,\phi_{M1},\dots,\phi_{M\ell_{M}}]:\
\text{ basis on }W^{u}_{N}.$

Thus, we can represent the $N$-dimensional vector $w_{N}\in W^{u}_{N}$ as
column vector $\hat{w}_{N}=[w_{N}]_{\beta}$ as,

$w_{N}=\sum_{i,j=1}^{M,\ell_{i}}w_{N}^{ij}\phi_{ij};\text{ and set
}\hat{w}_{N}=\text{col}[w_{N}^{1,1},\dots,w_{N}^{1,\ell_{1}},\dots,w_{N}^{i,1},\dots,w_{N}^{i,\ell_{i}},\dots,w_{N}^{M,1},\dots,w_{N}^{M,\ell_{M}}].$
(3.12)

######  Remark 3.1.

The eigenfunction $\phi_{ij}$ belongs to $L^{q}_{\sigma}(\Omega)$ as well as
to $\displaystyle{\mathcal{D}}(A_{q})={\mathcal{D}}({\mathcal{A}}_{q})$. Thus,
by real/complex interpolation, see (1.16)/(1.21) they also belong to
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
as well as to
$\displaystyle\big{[}{\mathcal{D}}(A_{q}),L^{q}_{\sigma}(\Omega)\big{]}_{1-\alpha}={\mathcal{D}}(A^{\alpha}_{q}),\
0\leq\alpha\leq 1$; in particular,
$\phi_{ij}\in\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$. See (B.11) or
(B.12) in Appendix B. Thus, exponential decay in $\mathbb{C}^{N}$ of the
$\mathbb{C}^{N}$-vector $\hat{w}_{N}$ translates at once into exponential
decay with the same rate in any of the spaces $\displaystyle
L^{q}_{\sigma}(\Omega),\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p},{\mathcal{D}}(A^{\alpha}_{q})$,
in particular, $\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$
for the vector $w_{N}$, views as a vector on any one of these spaces. This
remark applies to $w_{N}(t)$ and $u_{N}(t)$ in Theorem 2.1, Eqts (2.12),
(2.13) as well as Theorem 2.2, Eqts (2.17)- (2.19).

######  Lemma 3.1.

In $\mathbb{C}^{N}$, with respect to the ordered basis $\beta:\
\\{\phi_{ij}\\}_{i=1,j=1}^{M\ \ \ell_{i}}$ of normalized eigenfunctions of
${\mathcal{A}}_{N}^{u}$, we may rewrite system (3.10) = (3.12) = (2.8a) as

$(\hat{w}_{N})^{\prime}-\Lambda\hat{w}_{N}=U\hat{\mu}_{K}$ (3.13)

where

$\Lambda=\begin{bmatrix}\lambda_{1}I_{1}&&&\text{\huge 0}\\\
&\lambda_{2}I_{2}&&&\\\ &&\ddots&&\\\ \text{\huge
0}&&&\lambda_{M}I_{M}\end{bmatrix}:N\times N,\quad
I_{i}:\ell_{i}\times\ell_{i}\ \text{identity}$ (3.14)

$U_{i}=\begin{bmatrix}(u_{1},\phi_{i1}^{*})_{\omega}&\dots&(u_{K},\phi_{i1}^{*})_{\omega}\\\
(u_{1},\phi_{i2}^{*})_{\omega}&\dots&(u_{K},\phi_{i2}^{*})_{\omega}\\\
\vdots&\ddots&\vdots\\\
(u_{1},\phi_{i\ell_{i}}^{*})_{\omega}&\dots&(u_{K},\phi_{i\ell_{i}}^{*})_{\omega}\\\
\end{bmatrix}:\ell_{i}\times K;\quad U=\begin{bmatrix}U_{1}\\\ U_{2}\\\
\vdots\\\ U_{M}\end{bmatrix}:N\times
K;\quad\hat{\mu}_{K}=\begin{bmatrix}\mu_{1}\\\ \mu_{2}\\\ \vdots\\\
\mu_{K}\end{bmatrix}:K\times 1;$ (3.15)

where $\displaystyle(f,g)_{\omega}=\int_{\omega}f\bar{g}\ d\omega$ and we take
$K\geq\ell_{i},\ i=1,\dots,M$. Thus (3.13) gives the dynamics on $W^{u}_{N}$
as a linear $N$-dimensional ordinary differential equation in coordinate form
in $\mathbb{C}^{N}$.

###### Proof.

Recalling the basis $\beta_{i}$ and the definitions of $U_{i}$ in (3.15), we
can rewrite the term in (3.9) with respect to this basis as

$[P_{N}P(mu)]_{\beta_{i}}=U_{i}\hat{\mu}_{K}:\ell_{i}\times 1;$ (3.16)

Then with respect to the basis $\beta$ in (3.11) and recalling the definition
$U$ in (3.15), we can rewrite the term (3.9) with respect to this basis as

$[P_{N}P(mu)]_{\beta}=\begin{bmatrix}U_{1}\\\ U_{2}\\\ \vdots\\\
U_{M}\end{bmatrix}\hat{\mu}_{K}=\begin{bmatrix}U_{1}\hat{\mu}_{K}\\\
U_{2}\hat{\mu}_{K}\\\ \vdots\\\
U_{M}\hat{\mu}_{K}\end{bmatrix}=U\hat{\mu}_{K}:N\times 1.$ (3.17)

Finally, clearly ${\mathcal{A}}_{N}^{u}$ becomes the diagonal matrix $\Lambda$
in (3.14) with respect to the basis $\beta$, recalling its eigenvalues in
(3.1). ∎

The following is the main result of the present section.

######  Theorem 3.1.

Assume the FDSA. It is possible to select vectors $u_{1},\dots,u_{K}\in
L^{q}_{\sigma}(\omega),q>1,K=\sup\ \\{\ell_{i}:i=1,\dots,M\\}$, such that the
matrix $U_{i}$ of size $\ell_{i}\times K$ in (3.15) satisfies

$rank[U_{i}]=full=\ell_{i}\text{ or }\text{
rank}\begin{bmatrix}(u_{1},\phi_{i1}^{*})_{\omega}&\dots&(u_{K},\phi_{i1}^{*})_{\omega}\\\
(u_{1},\phi_{i2}^{*})_{\omega}&\dots&(u_{K},\phi_{i2}^{*})_{\omega}\\\
\vdots&&\vdots\\\
(u_{1},\phi_{i\ell_{i}}^{*})_{\omega}&\dots&(u_{K},\phi_{i\ell_{i}}^{*})_{\omega}\\\
\end{bmatrix}=\ell_{i};\ \ell_{i}\times K\ \text{for each }i=1,\dots,M.$
(3.18)

###### Proof.

Step 1. By selection, see (3.1) and statement preceding it, the set of vectors
$\phi^{*}_{i1},\dots,\phi^{*}_{i\ell_{i}}$ is linearly independent in
$L^{q^{\prime}}_{\sigma}(\Omega)$, $q^{\prime}$ is the Hölder conjugate of
$\displaystyle q,{}^{1}\\!/_{q}+{}^{1}\\!/_{q^{\prime}}=1$, for each
$i=1,\dots,M$. Next, if the set of vectors
$\\{\phi_{i1}^{*},\dots,\phi_{i\ell_{i}}^{*}\\}$ were linearly independent in
$L^{q^{\prime}}_{\sigma}(\Omega)$, $i=1,\dots,M$, the desired conclusion
(3.18) for the matrix $U_{i}$ to be full rank, would follow for infinitely
many choices of the vectors $u_{1},\dots,u_{K}\in L^{q}_{\sigma}(\Omega)$.

Claim: The set $\\{\phi_{i1}^{*},\dots,\phi_{i\ell_{i}}^{*}\\}$ is linearly
independent on $L^{q^{\prime}}_{\sigma}(\omega)$, for each $i=1,\dots,M$.

The proof will critically depend on a unique continuation result [59] see also
[7, Lemma 3.7 p1466]. By contradiction, let us assume that the vectors
$\\{\phi_{i1}^{*},\dots,\phi_{i\ell_{i}}^{*}\\}\in
L^{q^{\prime}}_{\sigma}(\Omega)$ are instead linearly dependent, so that

$\phi_{i\ell_{i}}^{*}=\sum_{j=1}^{\ell_{i}-1}\alpha_{j}\phi_{i\ell_{j}}^{*}\text{
in }L^{q^{\prime}}_{\sigma}(\Omega)$ (3.19)

with constants $\alpha_{j}$ not all zero. We shall then conclude by [7, Lemma
3.7] and [59] below, that in fact $\phi_{i\ell_{i}}^{*}\equiv 0$ on all of
$\Omega$ as well, thereby making the system
$\\{\phi_{ij}^{*},j=1,\dots,\ell_{i}\\}$ linearly dependent on $\Omega$, a
contradiction. To this end, define the following function (depending on $i$)
in $L^{q^{\prime}}_{\sigma}(\Omega)$

$\phi^{*}=\Bigg{[}\sum_{j=1}^{\ell_{i}-1}\alpha_{j}\phi_{i\ell_{j}}^{*}-\phi_{i\ell_{i}}^{*}\Biggl{]}\in
L^{q^{\prime}}_{\sigma}(\Omega),\quad i=1,\dots,M.$ (3.20)

As each $\phi_{ij}^{*}$ is an eigenvalue of ${\mathcal{A}}^{*}$ (or
$({\mathcal{A}}_{N}^{u})^{*}$) corresponding to the eigenvalue
$\bar{\lambda}_{i}$, see (3.1), so is the linear combination $\phi^{*}$. This
property, along with (3.19) yields that $\phi^{*}$ satisfies the following
eigenvalue problem for the operator ${\mathcal{A}}^{*}$ (or
$({\mathcal{A}}_{N}^{u})^{*}$):

${\mathcal{A}}^{*}\phi^{*}=\bar{\lambda}\phi^{*},\text{ div }\phi^{*}=0\text{
in }\Omega;\quad\phi^{*}=0\text{ in }\omega,\text{ by (\ref{I-3.19})}.$ (3.21)

But the linear combination $\phi^{*}$ in (3.20) of the eigenfunctions
$\phi_{ij}^{*}\in{\mathcal{D}}({\mathcal{A}}^{*})$ satisfies itself the
Dirichlet B.C $\displaystyle\left.\phi^{*}\right\rvert_{\partial\Omega}=0$.
Thus the explicit PDE version of problem (3.21) is

$\displaystyle-\nu\Delta\phi^{*}-(L_{e})^{*}\phi^{*}+\nabla p^{*}$
$\displaystyle=\bar{\lambda}_{i}\phi^{*}\text{ in }\Omega;$ (3.22a)
$\displaystyle\begin{picture}(0.0,0.0)\put(-100.0,0.0){
$\left\\{\rule{0.0pt}{30.0pt}\right.$}\end{picture}\text{div }\phi^{*}$
$\displaystyle=0\text{ in }\Omega;$ (3.22b)
$\displaystyle\phi^{*}\rvert_{\partial\Omega}=0;\quad\phi^{*}$
$\displaystyle=0\text{ in }\omega;$ (3.22c)

$\phi^{*}\in{\mathcal{D}}({\mathcal{A}}^{*});\quad(L_{e})^{*}\phi^{*}=(y_{e}.\nabla)\phi^{*}+(\phi^{*}.\nabla)^{*}y_{e},$
(3.23)

where $(f.\nabla)^{*}y_{e}$ is a $d$-vector whose $i$th component is
$\displaystyle\sum_{j=1}^{d}(D_{i}y_{e_{j}})f_{j}$ [8, p 55].

Step 2. The critical point is now that the over-determined problem (3.22)
implies the following unique continuation result:

$\phi^{*}=0\text{ in }L^{q^{\prime}}_{\sigma}(\Omega);\ \text{or by
(\ref{I-3.20})}\
\phi_{i\ell_{i}}^{*}=\alpha_{1}\phi_{i1}^{*}+\alpha_{2}\phi_{i2}^{*}+\dots+\alpha_{\ell_{i}-1}\phi_{i\ell_{i}-1}^{*}\text{
in }L^{q^{\prime}}_{\sigma}(\Omega),$ (3.24)

i.e. the set $\\{\phi_{i1}^{*},\dots,\phi_{i\ell_{i}}^{*}\\}$ in linearly
dependent on $L^{q^{\prime}}_{\sigma}(\Omega)$. But this is false, by the very
selection of such eigenvectors, see (3.1) and statement preceding it. Thus,
the condition (3.24) cannot hold.

The required unique continuation result is established in [7, Lemma 3.7] or
[59]. The original proof is done in the Hilbert setting but we may invoke the
same result because $\phi^{*}$ has more regularity and integrability than
required since $\phi^{*}$ is an eigenfunction of ${\mathcal{A}}^{*}$. Thus the
claim is established. In conclusion: it is possible to select, in infinitely
many ways, interior functions $u_{1},\dots,u_{K}\in L^{q}_{\sigma}(\Omega)$
such that the algebraic full rank condition (3.18) holds true for each
$i=1,\dots,M$. ∎

##  4 Algebraic rank conditions for the dynamics $w_{N}$ in (2.8a) in the
general case

In the present section we dispense with the FDSA (3.2). More precisely, we
shall obtain Theorem 2.1 without assuming the FDSA (3.2). Thus now

$\displaystyle W^{u}_{N}=$ space of generalized eigenfunctions of
${\mathcal{A}}_{q}(={\mathcal{A}}^{u}_{N})$ (4.0) corresponding to its
distinct unstable eigenvalues.

Warning: In this section we shall denote by $\ell_{i}$ the geometric
multiplicity of the eigenvalue $\lambda_{i}$ and by $N_{i}$ its algebraic
multiplicity.

Step 1: To treat this computationally more complicated case we shall,
essentially invoke the classical result on controllability of a finite-
dimensional, time-invariant system
$\\{\mathbb{A},\mathbb{B}\\},\mathbb{A}:N\times N,\mathbb{B}:N\times p$ where
$\mathbb{A}$ is given in Jordan form $J$. Let again
$\lambda_{1},\lambda_{2},\dots,\lambda_{M}$ be the distinct eigenvalues of
$\mathbb{A}=J$. Let $\mathbb{A}_{i}$ denote all the Jordan blocks associated
with the eigenvalue $\lambda_{i}$; let $\ell_{i}$ be the number of Jordan
blocks of $\mathbb{A}$ (i.e the number of linearly independent eigenvectors
associated with the eigenvalue $\lambda_{i}$). Let $\mathbb{A}_{ij}$ be
$j\textsuperscript{th}$ Jordan block in $\mathbb{A}_{i}$ corresponding to a
Jordan cycle of length $N_{j}^{i}$. That is:

$\mathbb{A}=\text{diag}\\{\mathbb{A}_{1},\mathbb{A}_{2},\dots,\mathbb{A}_{M}\\};\quad\mathbb{A}_{i}=\text{diag}\\{\mathbb{A}_{i1},\mathbb{A}_{i2},\dots,\mathbb{A}_{i\ell_{i}}\\}.$
(4.1)

Partition the matrix $\mathbb{B}$ accordingly:

$\underset{(N\times
N)}{\mathbb{A}}=\begin{bmatrix}\mathbb{A}_{1}&&&\text{\huge 0}\\\
&\mathbb{A}_{2}&&&\\\ &&\ddots&&\\\ \text{\huge
0}&&&\mathbb{A}_{M}\end{bmatrix};\quad\underset{(N\times
p)}{\mathbb{B}}=\begin{bmatrix}\mathbb{B}_{1}\\\ \mathbb{B}_{2}\\\ \vdots\\\
\mathbb{B}_{M}\end{bmatrix}$ (4.2)
$\underset{(N_{i}\times
N_{i})}{\mathbb{A}_{i}}=\begin{bmatrix}\mathbb{A}_{i1}&&&\text{\huge 0}\\\
&\mathbb{A}_{i2}&&&\\\ &&\ddots&&\\\ \text{\huge
0}&&&\mathbb{A}_{i\ell_{i}}\end{bmatrix};\quad\underset{(N_{i}\times
p)}{\mathbb{B}_{i}}=\begin{bmatrix}\mathbb{B}_{i1}\\\ \mathbb{B}_{i2}\\\
\vdots\\\ \mathbb{B}_{i\ell_{i}}\end{bmatrix}$ (4.3)
$\underset{(N_{j}^{i}\times
N_{j}^{i})}{\mathbb{A}_{ij}}=\begin{bmatrix}\lambda_{i}&1&&\text{\huge 0}\\\
&\lambda_{i}&1&\\\ &&\ddots&1\\\ \text{\huge
0}&&&\lambda_{i}\end{bmatrix};\quad\underset{(N_{j}^{i}\times
p)}{\mathbb{B}_{ij}}=\begin{bmatrix}b_{1ij}\\\ b_{2ij}\\\ \vdots\\\
b_{Lij}\end{bmatrix}.$ (4.4)

If $E_{\lambda_{i}}$ and $K_{\lambda_{i}}$ denote the eigenspace and the
generalized eigenspace associated with the eigenvalue $\lambda_{i},\
i=1,\dots,M$, then dim $E_{\lambda_{i}}=\ell_{i}=\\#$ of Jordan blocks in
$\mathbb{A}_{i}$, dim $K_{\lambda_{i}}=N_{i},N_{j}^{i}=$ length of jth-cycle
associated with $\lambda_{i};\ j=1,\dots,\ell_{i}$. We have dim $\displaystyle
W_{N}^{u}=N=\sum_{i=1}^{M}N_{i}=\sum_{i=1}^{M}\sum_{j=1}^{\ell_{i}}N_{j}^{i}$.
In (4.4), the last row of $\mathbb{B}_{ij}$ is denoted by $b_{Lij}$. The
following result is classical [11, p 165].

######  Theorem 4.1.

[33, Theorem 3.1] The pair $\\{J,\mathbb{B}\\}$, $J:N\times N$, Jordan form,
$\mathbb{B}:N\times p$ is controllable if and only if, for each $i=1,\dots,M$
(that is for each distinct eigenvalue) the rows of the $\ell_{i}\times p$
matrix constructed with all “last” rows $b_{Li1},\dots,b_{Li\ell_{i}}$

$\mathbb{B}_{i}^{L}=\begin{bmatrix}\mathbb{B}_{Li1}\\\ \mathbb{B}_{Li2}\\\
\vdots\\\ \mathbb{B}_{Li\ell_{i}}\end{bmatrix}:\ell_{i}\times p$ (4.5)

are linearly independent (on the field of complex numbers). [A direct proof
uses Hautus criterion for controllability [11].]

We next apply the above Theorem 4.1 to the $w_{N}$-problem (2.8a) and (3.5).
To this end, we select a Jordan basis $\beta_{i}$ for the operator
$({\mathcal{A}}_{N}^{u})_{i}$ on $W^{u}_{N,i}$ given by

Jordan Basis:

$\beta_{i}=\Big{\\{}e^{1}_{1}(\lambda_{i}),e^{1}_{2}(\lambda_{i}),\dots,e^{1}_{N^{i}_{1}}(\lambda_{i})\vdots
e^{2}_{1}(\lambda_{i}),e^{2}_{2}(\lambda_{i}),\dots,e^{2}_{N^{i}_{2}}(\lambda_{i})\vdots\dots\vdots
e^{\ell_{i}}_{1}(\lambda_{i}),e^{\ell_{i}}_{2}(\lambda_{i}),\dots,e^{\ell_{i}}_{N^{i}_{\ell_{i}}}(\lambda_{i})\Big{\\}}.$
(4.6a)

Here the first vectors of each cycle:
$e^{1}_{1}(\lambda_{i}),e^{2}_{1}(\lambda_{i}),\dots,e^{\ell_{i}}_{1}(\lambda_{i})$
are eigenvectors of $({\mathcal{A}}_{N}^{u})_{i}$ corresponding to the
eigenvalue $\lambda_{i}$, while the remaining vectors in $\beta_{i}$ are
corresponding generalized eigenvectors. Thus, in the notation (3.1), we have

$\phi_{i1}=e^{1}_{1}(\lambda_{i});\ \phi_{i2}=e^{2}_{1}(\lambda_{i});\dots;\
\phi_{i\ell_{i}}=e^{\ell_{i}}_{1}(\lambda_{i}).$ (4.6b)

Next, we can choose a bi-orthogonal basis $\beta^{*}_{i}$ of
$(({\mathcal{A}}^{u}_{N})^{*})_{i}$ corresponding to its eigenvalue
$\bar{\lambda_{i}}$ given by

Bi-orthogonal Basis:

$\beta^{*}_{i}=\Big{\\{}{\Phi}^{1}_{1}(\bar{\lambda}_{i}),{\Phi}^{1}_{2}(\bar{\lambda}_{i}),\dots,{\Phi}^{1}_{N^{i}_{1}}(\bar{\lambda}_{i})\vdots{\Phi}^{2}_{1}(\bar{\lambda}_{i}),{\Phi}^{2}_{2}(\bar{\lambda}_{i}),\dots,{\Phi}^{2}_{N^{i}_{2}}(\bar{\lambda}_{i})\vdots\dots\vdots{\Phi}^{\ell_{i}}_{1}(\bar{\lambda}_{i}),{\Phi}^{\ell_{i}}_{2}(\bar{\lambda}_{i}),\dots,{\Phi}^{\ell_{i}}_{N^{i}_{\ell_{i}}}(\bar{\lambda}_{i})\Big{\\}}.$
(4.7a)

Thus, in the notation (3.1), we have

$\phi^{*}_{i1}={\Phi}^{1}_{1}(\bar{\lambda}_{i});\
\phi^{*}_{i2}={\Phi}^{2}_{1}(\bar{\lambda}_{i});\dots;\
\phi^{*}_{i\ell_{i}}={\Phi}^{\ell_{i}}_{1}(\bar{\lambda}_{i}).$ (4.7b)

In the bi-orthogonality relationship between the vectors in (4.6) and those in
(4.7), the first eigenvector $e^{1}_{1}(\lambda_{i})$ of the first cycle in
$\beta_{i}$ is associated with the last generalized eigenvector
$\Phi^{1}_{N^{i}_{1}}(\bar{\lambda}_{i})$ of the first cycle in
$\beta^{*}_{i}$; etc, the last generalized eigenvector
$e^{1}_{N^{i}_{1}}(\lambda_{i})$ of the first cycle in $\beta_{i}$ is
associated with the first eigenvector $\Phi^{1}_{1}(\bar{\lambda}_{i})$ of the
first cycle in $\beta^{*}_{i}$; etc.

$e_{1}^{1}(\lambda_{i})$$e_{2}^{1}(\lambda_{i})$$\cdots$$e_{N_{1}^{i}}^{1}(\lambda_{i})$$\Phi_{1}^{1}(\bar{\lambda}_{i})$$\Phi_{2}^{1}(\bar{\lambda}_{i})$$\cdots$$\Phi_{N_{1}^{i}}^{1}(\bar{\lambda}_{i})$
, $\cdots$ (4.8) Figure 1: Relation between the generalized eigenvectors of
${\mathcal{A}}^{u}_{N}$ and $({\mathcal{A}}^{u}_{N})^{*}$

Thus, if $f\in W_{N,i}^{u}$, the following expression holds true:

$\displaystyle
f=(f,\Phi_{N_{1}^{i}}^{1}(\bar{\lambda}_{i}))e_{1}^{1}(\lambda_{i})+\cdots+(f,\Phi_{1}^{1}(\bar{\lambda}_{i}))e^{1}_{N_{1}^{i}}(\lambda_{i})$
$\displaystyle+\cdots+(f,\Phi_{N^{i}_{\ell_{i}}}^{\ell_{i}}(\bar{\lambda}_{i}))e_{1}^{\ell_{i}}(\lambda_{i})$
$\displaystyle+\cdots+(f,\Phi_{1}^{\ell_{i}}(\bar{\lambda}_{i}))e^{\ell_{i}}_{N_{\ell_{i}}^{i}}(\lambda_{i}).$
(4.9)

This expansion is the counterpart of
$\displaystyle\sum_{j=1}^{\ell_{i}}(w,\phi_{ij}^{*})\phi_{ij}\in W^{u}_{N,i}$
in (3.4) under the FDSA. Next, we apply (4) to $f=P_{N}P(mu)$. More
specifically, we shall write the vector representation of $P_{N}P(mu)$ with
respect to the basis $\beta_{i}$ in (4.6a), and moreover, in line with Theorem
4.1, we shall explicitly note only the coordinates corresponding to the
vectors
$e^{1}_{N^{i}_{1}}(\lambda_{i}),e^{2}_{N^{i}_{2}}(\lambda_{i}),\dots,e^{\ell_{i}}_{N^{i}_{\ell_{i}}}(\lambda_{i})$,
each being the last vector of each cycle in (4.6a).

$[P_{N}P(mu)]_{\beta_{i}}=\begin{bmatrix}\times\times\times\\\
\Big{(}u,\Phi_{1}^{1}(\bar{\lambda}_{i})\Big{)}_{\omega}\\\ {1}\\\
\times\times\times\\\
\Big{(}u,\Phi_{1}^{2}(\bar{\lambda}_{i})\Big{)}_{\omega}\\\ {1}\\\
\times\times\times\\\
\Big{(}u,\Phi_{1}^{\ell_{i}}(\bar{\lambda}_{i})\Big{)}_{\omega}\\\
\end{bmatrix}\begin{aligned} \\\ \xleftarrow{}\text{ last row of the $1^{st}$
cycle }\\\ \\\ \\\ \xleftarrow{}\text{ last row of the $2^{nd}$ cycle }\\\ \\\
\\\ \xleftarrow{}\text{ last row of the $\ell_{i}^{th}$ cycle }\end{aligned}$
(4.10)

The symbol $\times\times\times$ corresponds to terms which we do not care
about. In fact, to exemplify, since
$P^{*}P_{N}^{*}\Phi^{1}_{1}(\bar{\lambda}_{i})=\Phi^{1}_{1}(\bar{\lambda}_{i})$
see above (3.6)

$\Big{(}P_{N}P(mu),\Phi^{1}_{1}(\bar{\lambda}_{i})\Big{)}_{\Omega}=\big{(}mu,\Phi^{1}_{1}(\bar{\lambda}_{i})\big{)}_{\Omega}=\big{(}u,\Phi^{1}_{1}(\bar{\lambda}_{i})\big{)}_{\omega}.$
(4.11)

This is the relevant counterpart of expansion $\displaystyle
P_{N}P(mu)=\sum_{i,j=1}^{M,\ell_{i}}(u,\Phi_{ij}^{*})_{\omega}\Phi_{ij}$ in
(3.6) under the FDSA. Notice that (4.10) involves only the eigenvectors
$\Phi^{1}_{1}(\bar{\lambda}_{i}),\Phi^{2}_{1}(\bar{\lambda}_{i}),\dots,\Phi^{\ell_{i}}_{1}(\bar{\lambda}_{i})$
of $({\mathcal{A}}^{u}_{N})^{*}$ corresponding to the eigenvalue
$\bar{\lambda}_{i}$. Next, recalling (2.10): $\displaystyle
u=\sum_{k=1}^{K}\mu_{k}(t)u_{k}$, we obtain that the corresponding counterpart
of (3.15) is

$U_{i}=\begin{bmatrix}\times&\times&&\times\\\
(u_{1},\Phi_{1}^{1})_{\omega}&(u_{2},\Phi_{1}^{1})_{\omega}&\dots&(u_{K},\Phi_{1}^{1})_{\omega}\\\
{4}\\\ \times&\times&&\times\\\
(u_{1},\Phi_{1}^{2})_{\omega}&(u_{2},\Phi_{1}^{2})_{\omega}&\dots&(u_{K},\Phi_{1}^{2})_{\omega}\\\
{4}\\\ \times&\times&&\times\\\
(u_{1},\Phi_{1}^{\ell_{i}})_{\omega}&(u_{2},\Phi_{1}^{\ell_{i}})_{\omega}&\dots&(u_{K},\Phi_{1}^{\ell_{i}})_{\omega}\\\
\end{bmatrix}\begin{aligned} \\\ \xleftarrow{}\text{ row }b_{Li1}(u)\\\ \\\
\xleftarrow{}\text{ row }b_{Li2}(u)\\\ \\\ \xleftarrow{}\text{ row
}b_{Li\ell_{i}}(u)\end{aligned}$ (4.12)

Again, the relevant rows exhibited in (4.12) correspond to the last rows of
each Jordan sub-block
$\\{\mathbb{A}_{i1},\mathbb{A}_{i2},\dots,\mathbb{A}_{i\ell_{i}}\\}$ in (4.3).
In (4.12) we have displayed only such relevant rows:
$b_{Li1},b_{Li2},\dots,b_{Li\ell_{i}}$ According to Theorem 4.1, the test for
controllability as applied to the system (3.5), i.e to the pair
$\\{{\mathcal{A}}^{u}_{N},B\\},B$ = col $[B_{1},B_{2},\dots,B_{M}]$, is

$\text{ rank }\begin{bmatrix}\vspace{2mm}\text{ row }b_{Li1}\text{ of
}B_{i}\\\ \vspace{2mm}\text{ row }b_{Li2}\text{ of }B_{i}\\\
\vspace{2mm}\vdots\\\ \vspace{2mm}\text{ row }b_{Li\ell_{i}}\text{ of
}B_{i}\end{bmatrix}=\text{ rank
}\begin{bmatrix}\vspace{2mm}\big{(}u_{1},\Phi_{1}^{1}\big{(}\bar{\lambda}_{i}\big{)}\big{)}_{\omega}&\dots&\big{(}u_{K},\Phi_{1}^{1}\big{(}\bar{\lambda}_{i}\big{)}\big{)}_{\omega}\\\
\vspace{2mm}\big{(}u_{1},\Phi_{1}^{2}\big{(}\bar{\lambda}_{i}\big{)}\big{)}_{\omega}&\dots&\big{(}u_{K},\Phi_{1}^{2}\big{(}\bar{\lambda}_{i}\big{)}\big{)}_{\omega}\\\
\vspace{2mm}\vdots&&\vdots\\\
\vspace{2mm}\big{(}u_{1},\Phi_{1}^{\ell_{i}}\big{(}\bar{\lambda}_{i}\big{)}\big{)}_{\omega}&\dots&\big{(}u_{K},\Phi_{1}^{\ell_{i}}\big{(}\bar{\lambda}_{i}\big{)}\big{)}_{\omega}\\\
\end{bmatrix}=\ell_{i}$ (4.13)

$i=1,\dots,M$. But this is exactly the test obtained in (3.18) via the
identification in (4.7b):

$\phi^{*}_{i1}=\Phi^{1}_{1}(\bar{\lambda}_{i}),\phi^{*}_{i2}=\Phi^{2}_{1}(\bar{\lambda}_{i}),\dots,\phi^{*}_{i\ell_{i}}=\Phi^{\ell_{i}}_{1}(\bar{\lambda}_{i})$
(4.14)

involving only eigenvectors, not generalized eigenvectors. Thus the remainder
of the proof in section 3 past (3.18) applies and shows Theorem 3.1 without
the FDSA. We have

######  Theorem 4.2.

With reference to $U_{i}$ in (4.12), it is possible to select interior vectors
$u_{1},\dots,u_{K}\in W^{u}_{N}\subset L^{q}_{\sigma}(\Omega),\ K=\sup\
\\{\ell_{i}:i=1,\dots,M\\}$, such that the algebraic conditions (4.13) hold
true, $i=1,\dots,M$.

We close this section by writing down the counterpart of the expansion (3.10)
for the $w_{N}$-dynamics in terms of the basis
$\displaystyle\beta=\beta_{1}\cup\beta_{2}\cup\dots\cup\beta_{M}$, see (3.11),
(4.6a), (4) of the generalized eigenvectors in the present general case.

on $\displaystyle W_{N}^{u}:w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}$
$\displaystyle=\sum_{i=1}^{M}\Bigg{\\{}\sum_{k=1}^{K}\bigg{[}(u_{k},\Phi_{N^{i}_{1}}^{1}(\bar{\lambda}_{i}))_{\omega}\mu_{k}(t)\bigg{]}e^{1}_{1}(\lambda_{i})+\dots+\dots+\sum_{k=1}^{K}\bigg{[}(u_{k},\Phi_{1}^{1}(\bar{\lambda}_{i}))_{\omega}\mu_{k}(t)\bigg{]}e^{1}_{N^{i}_{1}}(\lambda_{i})$
$\displaystyle+\dots+\dots$
$\displaystyle+\sum_{k=1}^{K}\bigg{[}(u_{k},\Phi_{N^{i}_{\ell_{i}}}^{\ell_{i}}(\bar{\lambda}_{i}))_{\omega}\mu_{k}(t)\bigg{]}e^{\ell_{i}}_{1}(\lambda_{i})+\dots+\dots+\sum_{k=1}^{K}\bigg{[}(u_{k},\Phi^{\ell_{i}}_{1}(\bar{\lambda}_{i}))_{\omega}\mu_{k}(t)\bigg{]}e_{N^{i}_{\ell_{i}}}^{\ell_{i}}(\lambda_{i})\Bigg{\\}}.$

##  5 Proof of Theorem 2.1: arbitrary decay rate of the $w_{N}$-dynamics (3.5)
or (4) (or (3.13) under the FDSA) by a suitable finite-dimensional interior
localized feedback control u

We are now in a position to obtain Theorem 2.1, which we restate for
convenience. Let $1<q<\infty.$

######  Theorem 5.1.

Let $\lambda_{1},\dots,\lambda_{M}$ be the unstable distinct eigenvalues of
${\mathcal{A}}$ and let $\omega$ be an arbitrarily small open portion of the
interior with smooth boundary $\partial\omega$. By virtue of Theorem 4.2, pick
interior vectors $[u_{1},\dots,u_{K}]$ in $W^{u}_{N}\subset
L^{q}_{\sigma}(\Omega)$ such that the rank conditions (4.13) hold true, with
$K=\sup\ \\{\ell_{i}:\ i=1,\dots,M\\}$ (respectively, Theorem 3.1 and the
(same) rank conditions (3.18) under FDSA).

Then: Given $\gamma>0$ arbitrarily large, there exists a $K$-dimensional
interior controller $u=u_{N}$ acting on $\omega$, of the form given by (3.8),
with the vectors $u_{k}$ given by Theorem 4.2 via the rank conditions (4.13),
such that, once inserted in (4) yield the estimate

$\left\lVert w_{N}(t)\right\rVert_{L^{q}_{\sigma}(\Omega)}+\left\lVert
u_{N}(t)\right\rVert_{L^{q}_{\sigma}(\omega)}\leq C_{\gamma}e^{-\gamma
t}\left\lVert P_{N}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)},\ t\geq 0,$
(5.1a) where the $L^{q}_{\sigma}(\Omega)$-norm in (5.1a) may be replaced by
the
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$\-
norm , $1<p,q<\infty$; in particular the
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-norm,
$\displaystyle 1<q<\infty,\ 1<p<\frac{2q}{2q-1}$: $\left\lVert
w_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+\left\lVert
u_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\leq
C_{\gamma}e^{-\gamma t}\left\lVert
P_{N}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\
t\geq 0.$ (5.1b)

Here, $w_{N}$ is the solution of (4) (respectively (3.10) under the FDSA) ,
i.e., (3.5) corresponding to the control $u=u_{N}$ in (3.8). Moreover, such
controller $u=u_{N}$can be chosen in feedback form: that is, with reference to
the explicit expression (3.8) for $u$, of the form
$\mu_{k}(t)=(w_{N}(t),p_{k})_{\omega}$ for suitable vectors
$p_{k}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega)$ depending on
$\gamma$. In conclusion, $w_{N}$ in (5.1) is the solution of the equation on
$W^{u}_{N}$ (see (3.5)) specialized as (4)

$w^{\prime}_{N}-{\mathcal{A}}^{u}_{N}w_{N}=P_{N}P\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}(w_{N}(t),p_{k})_{\omega}u_{k}\Bigg{)}\Bigg{)},\
u_{k}\in W^{u}_{N}\subset L^{q}_{\sigma}(\Omega),\
p_{k}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega)$ (5.2)

rewritten as

$w^{\prime}_{N}=\bar{A}^{u}w_{N},\quad
w_{N}(t)=e^{\bar{A}^{u}t}P_{N}w_{0},\quad w_{N}(0)=P_{N}w_{0}.$ (5.3)

###### Proof.

Step 1: Following [33] the proof consists in testing controllability of the
linear, finite-dimensional system (3.5), in short, the pair

$\\{J,B\\},\ B=U:\ N\times K,K=\text{sup }\\{\ell_{i};i=1.\dots,M\\}$ (5.4)

$U=[U_{1},\dots,U_{M}]^{\text{tr}}$, $U_{i}$ given by (4.12) (or by (3.15)
under FDSA). $J$ is the Jordan form of ${\mathcal{A}}^{u}_{N}$ with respect to
the Jordan basis $\beta=\beta_{1}\cup\dots\cup\beta_{M},\ \beta_{i}$ being
given by (4.6a). But the rank conditions (4.13) precisely asserts such
controllability property of the pair $\\{{\mathcal{A}}^{u}_{N}=J,B\\}$, in
light of Theorem 4.1.

Step 2: Having established the controllability criterion for the pair
$\\{{\mathcal{A}}^{u}_{N}=J,B\\}$ then by the well-known Popov’s criterion in
finite-dimensional theory, there exists a real feedback matrix $Q=K\times N$,
such that the spectrum of the matrix $(J+BQ)=(J+UQ)$ may be arbitrarily
preassigned; in particular, to lie in the left half-plane $\\{\lambda:\text{Re
}\lambda<-\gamma<-\text{Re }\lambda_{N+1}\\}$, as desired. The resulting
closed-loop system

$(\hat{w}^{\prime}_{N})-J\hat{w}_{N}=Uu_{N},$ (5.5)

is obtained with $\mathbb{C}^{N}$-vector $u_{N}=Q\hat{w}_{N}$, Q being the
$K\times N$ matrix with row vectors $[\hat{p}_{1},\dots,\hat{p}_{K}],\
\mu^{k}_{N}=(\hat{w}_{N},\hat{p}_{k})$ in the ${\mathbb{C}}^{N}$-inner product
and hence decays with exponential rate

$\left\lvert\hat{w}_{N}(t)\right\rvert_{\mathbb{C}^{N}}\leq
C_{\gamma}e^{-\gamma
t}\left\lvert\hat{w}_{N}(0)\right\rvert_{\mathbb{C}^{N}},\ t\geq 0.$ (5.6)

But the $N$-dimensional vector $\displaystyle w_{N}\in W^{u}_{N}\subset
L^{q}_{\sigma}(\Omega)$ is represented by the $\mathbb{C}^{N}$-vector
$\displaystyle\hat{w}_{N}=[w_{N}]_{\beta}$, where in the general case of
Section 4, $\beta$ is a Jordan basis of generalized eigenfunctions of
$\displaystyle{\mathcal{A}}_{q}(={\mathcal{A}}_{N}^{u})$ corresponding to its
$M$ distinct unstable eigenvalues. Such basis is given by
$\displaystyle\beta=\beta_{1}\cup\beta_{2}\cup\cdots\cup\beta_{M}$, where a
representative $\beta_{i}$ is given in (4.6a). The whole basis can be read off
from (4). In the special case of Section 3 where the FDSA holds, the basis
$\beta$ in $W^{u}_{N}$ is given by the eigenfunctions of the
${\mathcal{A}}_{N}^{u}$ corresponding to its $M$ distinct eigenvalues, see
(3.11). But such eigenfunctions/generalized eigenfunctions are in
${\mathcal{D}}({\mathcal{A}}_{q})$, hence smooth. Thus, the exponential decay
in (5.6) of the coordinate vector $\hat{w}_{N}$ in $\mathbb{C}^{N}$ translates
in same exponential decay of the vector $w_{N}(t)\in W^{u}_{N}$ not only in
the $L^{q}_{\sigma}(\Omega)$-norm but also in the
${\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})$-norm, hence in the
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$-norm,
in particular in the
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-norm. See also
Remark 3.1. Thus, returning from ${\mathbb{C}}^{N}\times{\mathbb{C}}^{N}$ back
to $W^{u}_{N}\times(W^{u}_{N})^{*}$, there exist suitable
$p_{1},\dots,p_{K}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega)$,
such that $\mu^{k}_{N}=(w_{k},p_{k})$, whereby the closed-loop system (5.2)
corresponds precisely to (4) via $P_{N}P(mu)$ written in terms of the Jordan
basis of eigenvectors $\beta$ in (4.6a).

Thus not only we obtain in view of (5.2), (5.3) and (5.6)

$\left\lVert w_{N}(t)\right\rVert_{L^{q}_{\sigma}(\Omega)}=\left\lVert
e^{\bar{A}^{u}t}P_{N}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)}\leq
C_{\gamma}e^{-\gamma t}\left\lVert
P_{N}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)},\ t\geq 0,$ (5.7)

but also, say $\displaystyle 1<q<\infty,1<p<\frac{2q}{2q-1}$

$\left\lVert
w_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}=\left\lVert
e^{\bar{A}^{u}t}P_{N}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\leq
C_{\gamma}e^{-\gamma t}\left\lVert
P_{N}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\
t\geq 0.$ (5.8)

Hence with $u_{N}=Qw_{N}$, we obtain not only

$\displaystyle\left\lVert
w_{N}(t)\right\rVert_{L^{q}_{\sigma}(\Omega)}+\left\lVert
u_{N}(t)\right\rVert_{L^{q}_{\sigma}(\omega)}$ $\displaystyle=\left\lVert
w_{N}(t)\right\rVert_{L^{q}_{\sigma}(\Omega)}+\left\lVert
Qw_{N}(t)\right\rVert_{L^{q}_{\sigma}(\Omega)}$ (5.9)
$\displaystyle\leq\big{(}\left\lvert Q\right\rvert+1\big{)}\left\lVert
e^{\bar{A}^{u}t}P_{N}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)}\leq
C_{\gamma}e^{-\gamma t}\left\lVert
P_{N}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)}$ (5.10)

but also, say

$\left\lVert
w_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+\left\lVert
u_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\leq
C_{\gamma}e^{-\gamma t}\left\lVert
P_{N}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\
t\geq 0.$ (5.11)

∎

######  Remark 5.1.

Under the FDSA, checking controllability of the system (3.13) is easier. To
this end, we can pursue, as usual, two strategies.

A first strategy invokes the well-known Kalman controllability criterion by
constructing the $N\times KN$ Kalman controllability matrix

$\mathcal{K}=[B,\Lambda
B,\Lambda^{2}B,\dots,\Lambda^{N-1}B]=\begin{bmatrix}B_{1}&J_{1}B_{1}&\dots&J_{1}^{N-1}B_{1}\\\
B_{2}&J_{2}B_{2}&\dots&J_{2}^{N-1}B_{2}\\\ {4}\\\
B_{M}&J_{M}B_{M}&\dots&J_{M}^{N-1}B_{M}\end{bmatrix},$ (5.12)
$B=\text{col }[B_{1},B_{2},\dots,B_{M}],\quad
B_{i}=U_{i}:\ell_{i}\times\ell_{i}$ (5.13)

of size $N\times KN,\ N=\text{dim }W^{u}_{N},\
J_{i}=\lambda_{i}I_{i}:\ell_{i}\times\ell_{i},\
B_{i}=U_{i}:\ell_{i}\times\ell_{i}$, and requiring that it be full rank.

$\text{rank }\mathcal{K}=\text{full}=N.$ (5.14)

In view of generalized Vandermond determinants, we then have

$\text{rank }{\mathcal{K}}=N\quad\text{if and only if rank
}U_{i}=\ell_{i}\text{ (full) }i=1,\dots,M,$ (5.15)

precisely as guaranteed by (3.18). A second strategy invokes the Hautus
controllability criterion:

$rank[\Lambda-\lambda_{i}I,B]=rank[\Lambda-\lambda_{i}I,U]=N\ (\text{full})$
(5.16)

for all unstable eigenvalues $\lambda_{i},1,\dots,M$, yielding again the
condition that rank $[U_{i}]=\ell_{i},1,\dots,M$, as generated by (3.18).

##  6 Proof of Theorem 2.2: Feedback stabilization of the original linearized
$w$-Oseen system (1.13) by a finite dimensional feedback controller

The main result on the feedback stabilization of the linearized $w$-system
(1.13) = (2.1) by a finite dimensional controller is Theorem 2.2, here
reformulated in part for convenience in the context of the development of the
present proof. Throughout this section $1<q<\infty$.

######  Theorem 6.1.

Let the Oseen operator ${\mathcal{A}}$ have N possibly repeated unstable
eigenvalues $\\{\lambda_{j}\\}_{j=1}^{N}$ of which $M$ are distinct. Let
$\varepsilon>0$ and set $\gamma_{0}=|Re\ \lambda_{N+1}|-\varepsilon$. Consider
the setting of Theorem 5.1 so that, in particular, the feedback finite-
dimensional control $u=u_{N}$ is given by $\displaystyle
u=u_{N}=\sum_{k=1}^{K}(w_{N}(t),p_{k})u_{k}$ and satisfies estimates (5.1)
with $\gamma>0$ arbitrarily large, for vectors
$p_{1},\dots,p_{k}\in(W^{u}_{N})^{*}\subset L^{q^{\prime}}_{\sigma}(\Omega)$
and vectors $u_{1},\dots,u_{k}\in W^{u}_{N}\subset L^{q}_{\sigma}(\Omega)$
given by Theorem 5.1. Thus, the linearized problem (2.1) specializes to (2.16)

$\frac{dw}{dt}={\mathcal{A}}w+P\Bigg{(}m\Bigg{(}\sum_{k=1}^{K}(w_{N}(t),p_{k})_{\omega}u_{k}\Bigg{)}\Bigg{)}\equiv\mathbb{A}_{{}_{F}}w.$
(6.1)

Here $\displaystyle\mathbb{A}_{{}_{F}}=\mathbb{A}_{{}_{F,q}}$ is the generator
of a s.c. analytic semigroup on either the space $\displaystyle
L^{q}_{\sigma}(\Omega),\ 1<q<\infty$, or on the space
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p},\
1<p,q<\infty,$ in particular on the space
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega),1<q,1<p<{}^{2q}\\!/_{2q-1}$.
Moreover, such dynamics $w$ (equivalently, generator
$\displaystyle\mathbb{A}_{{}_{F}}$) in (6.1) is uniformly stable in each of
these spaces, say

$\left\lVert
e^{\mathbb{A}_{{}_{F}}t}w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)}=\left\lVert
w(t,w_{0})\right\rVert_{L^{q}_{\sigma}(\Omega)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert
w_{0}\right\rVert_{L^{q}_{\sigma}(\Omega)},\quad t\geq 0.$ (6.2)

or say

$\left\lVert
e^{\mathbb{A}_{{}_{F}}t}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}=\left\lVert
w(t,w_{0})\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert
w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\quad t\geq
0.$ (6.3)

###### Proof.

Step 1: According to Theorem 5.1, the finite-dimensional system $w_{N}$ in
(2.8a) = (3.5) is uniformly stabilized by the finite dimensional feedback
controller $u=u_{N}$ given in the RHS of (5.2) = RHS of (6.1) with an
arbitrary preassigned decay rate $\gamma>0$, as given, either in the
$L^{q}_{\sigma}(\Omega)$-norm, or in the
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$-norm
in (5.1a), or in particular, in the
$\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-norm as in (5.1b).

Step 2: Next, we examine the impact of such constructive feedback control
$u_{N}$ on the $\zeta_{N}$-dynamics (2.8b), whose explicit solution can be
given by a variation of parameter formula,

$\zeta_{N}(t)=e^{{\mathcal{A}}^{s}_{N}t}\zeta(0)+\int_{0}^{t}e^{{\mathcal{A}}^{s}_{N}(t-r)}(I-P_{N})P(mu_{N}(r))dr.$
(6.4)

in the notation ${\mathcal{A}}_{N}^{s}=(I-P_{N}){\mathcal{A}},\
{\mathcal{A}}={\mathcal{A}}_{q}$, of (2.5). We now recall from Section 1.10
(d) that the Oseen operator ${\mathcal{A}}_{q}$ generates a s.c. analytic
semigroup not only on $L^{q}_{\sigma}(\Omega)$ but also on
$\displaystyle\big{(}L^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$,
in particular on $\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$. Hence the
feedback operator $\displaystyle\mathbb{A}_{{}_{F}}=\mathbb{A}_{{}_{F,q}}$
similarly generates a s.c. analytic semigroup on these spaces, being a bounded
perturbation of the Oseen operator
$\displaystyle{\mathcal{A}}={\mathcal{A}}_{q}$. So we can estimate (6.4) in
the norm of either of these spaces. Furthermore, the (point) spectrum of the
generator ${\mathcal{A}}_{N}^{s}$ on $W^{s}_{N}$ satisfies $\sup\\{Re\
\sigma({\mathcal{A}}^{s}_{N})\\}<-\left\lvert\lambda_{N+1}\right\rvert<-\gamma_{0}$
by assumption. We shall carry our the supplemental computations explicitly in
the space $\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ for the case of
greatest interest in the nonlinear analysis of sections 8, 9. In the norm of
$\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, we obtain from (6.4) since
the operators $(I-P_{N}),P$ are bounded

$\displaystyle\left\lVert\zeta_{N}(t)\right\rVert$
$\displaystyle\leq\left\lVert
e^{{\mathcal{A}}_{N}^{s}t}\zeta(0)\right\rVert+C\int_{0}^{t}\left\lVert
e^{{\mathcal{A}}_{N}^{s}(t-\tau)}\right\rVert\left\lVert
u_{N}(\tau)\right\rVert d\tau$ (6.5)
$\displaystyle\left\lVert\zeta_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$
$\displaystyle\leq
Ce^{-\gamma_{0}t}\left\lVert\zeta(0)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+C\int_{0}^{t}e^{-\gamma_{0}(t-r)}e^{-\gamma
r}dr\left\lVert
P_{N}w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}.$
(6.6)

recalling estimate (2.13) or (5.11) for $\displaystyle\left\lVert
u_{N}\right\rVert$ in the
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-norm. Since we
may choose $\gamma>\gamma_{0}$ by Theorem 2.1 (or Theorem 5.1), we then obtain

$\displaystyle\left\lVert\zeta_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$
$\displaystyle\leq
C\left[e^{-\gamma_{0}t}+e^{-\gamma_{0}t}\frac{1-e^{-(\gamma-\gamma_{0})t}}{\gamma-\gamma_{0}}\right]\left\lVert
w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$ (6.7)
$\displaystyle\leq Ce^{-\gamma_{0}t}\left\lVert
w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)},\quad\forall
t>0.$ (6.8)

Then, estimate (6.8) for $\zeta_{N}(t)$ along with estimate (2.13) = (5.11)
for $w_{N}(t)$ with $\gamma>\gamma_{0}$ yields the desired estimate (6.3) for
$w=w_{N}+\zeta_{N}$ in the
$\displaystyle\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$-norm:

$\displaystyle\left\lVert
w(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$
$\displaystyle\leq\left\lVert\zeta_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}+\left\lVert
w_{N}(t)\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$ (6.9)
$\displaystyle\leq\big{[}\widetilde{C}_{\gamma_{0}}e^{-\gamma_{0}t}+C_{\gamma}e^{-\gamma
t}\big{]}\left\lVert
w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$ (6.10)
$\displaystyle\leq C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert
w_{0}\right\rVert_{\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}$ (6.11)

and (6.3) is proved. Similar computations from (6.4) to (6.8) apply in the
$\displaystyle L^{q}_{\sigma}(\Omega)$-norm for $\zeta_{N}(t)$, as the Oseen
operator generates a s.c. analytic semigroup on $L^{q}_{\sigma}(\Omega)$ from
Section 1.10 (d). This, coupled with estimate (2.12) for $w_{N}(t)$, yields
estimate (6.2) for the $\displaystyle w=w_{N}+\zeta_{N}$ with
$L^{q}_{\sigma}(\Omega)$-norm. Theorem 6.1 is established. ∎

######  Remark 6.1.

Computations such as those in [7, p 1473] using the analyticity of the Oseen