$\displaystyle\left\lVert{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})-{\mathcal{F}}(\mathbf{b}_{0},\mathbf{g})\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
K_{p}^{{}^{1}\\!/_{p}}R\Big{\\{}\left\lVert\mathbf{f}_{1}-\mathbf{g}_{1}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}-g_{2}\right\rVert_{X^{\infty}_{p,q}}\Big{\\}}$
$\displaystyle=\rho_{0}\left\lVert\mathbf{f}-\mathbf{g}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (5.55)

and $\displaystyle{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})$ is a contraction
on the space $\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$ as soon
as

$\rho_{0}=K_{p}^{{}^{1}\\!/_{p}}R<1\quad\text{or}\quad
R<\frac{1}{K_{p}^{{}^{1}\\!/_{p}}}.$ (5.56)

In this case, the map $\displaystyle{\mathcal{F}}(\mathbf{b}_{0},\mathbf{f})$
defined in (5.10) has a fixed point $\displaystyle\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}$ in $\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$:

${\mathcal{F}}\begin{pmatrix}\mathbf{b}_{0},\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}\end{pmatrix}=\begin{bmatrix}\mathbf{z}\\\ h\end{bmatrix},\
\text{or}\ \begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{z}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\mathbf{z}]h(\tau)\end{bmatrix}d\tau,$ (5.57)

and such point $\displaystyle\begin{bmatrix}\mathbf{z}\\\ h\end{bmatrix}$ is
the unique solution of the translated non-linear system (5.2), or (5.3), with
finite dimensional control

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

in feedback form, as described by Eq (5.2). Theorem 5.3 and hence Theorem 5.1
are proved. ∎

######  Remark 5.1.

As
$[\mathbf{u}^{1}_{k},u^{2}_{k}]\in(\widehat{\mathbf{W}}^{q}_{\sigma})^{u}_{N}\subset\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$, we have that the feedback control acting on the fluid variable
$\mathbf{u}$ is of reduced dimension $(d-1)$, see (4.7c).

##  6 Proof of Theorem 2.3: local exponential decay of the non-linear
$[\mathbf{z},h]$ translated dynamics (5.2) = (5.3) with finite dimensional
localized feedback controls and $(d-1)$ dimensional $\mathbf{u}^{1}_{k}$

In this section we return to the feedback problem (5.2) = (5.3), rewritten
equivalently as in (5.4)

$\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{z}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\mathbf{z}]h(\tau)\end{bmatrix}d\tau,$ (6.1)

For $\mathbf{b}_{0}=[\mathbf{z}_{0},h_{0}]$ in a small ball of
$\displaystyle\mathbf{V}^{q,p}(\Omega)=\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$, Theorem 5.1(= Theorem 2.2) provides a unique solution
$\\{\mathbf{z},h\\}$ in a small ball of
$\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$. We
recall from (4.13)

$\left\lVert e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)},t\geq 0$ (6.2)

$\displaystyle\mathbf{V}^{q,p}(\Omega)=\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$. Our goal is to show that for
$\displaystyle[\mathbf{z}_{0},h_{0}]$ in a small ball of
$\displaystyle\mathbf{V}^{q,p}(\Omega)$, problem (6.1) satisfies the
exponential decay

$\left\lVert\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}(t)\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq
C_{a}e^{-at}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)},t\geq 0,\text{for
some constants }a>0,C=C_{a}\geq 1.$ (6.3)

Step 1: Starting from (6.1) and using the decay (6.2), we estimate

$\displaystyle\left\lVert\begin{bmatrix}\mathbf{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$ $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+$
$\displaystyle\hskip 56.9055pt\sup_{0\leq
t\leq\infty}\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{z}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\mathbf{z}]h(\tau)\end{bmatrix}d\tau\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$
(6.4) $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+$
$\displaystyle\hskip
56.9055ptC_{1}\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{z}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\mathbf{z}]h(\tau)\end{bmatrix}d\tau\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (6.5) $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+C_{2}\bigg{[}\left\lVert{\mathcal{N}}_{q}\mathbf{z}\right\rVert_{L^{p}\big{(}0,\infty;\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}}+$
$\displaystyle\hskip
142.26378pt\left\lVert{\mathcal{M}}_{q}[\mathbf{z}]h\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\bigg{]}$
(6.6) $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+C_{3}\bigg{[}\left\lVert\mathbf{z}\right\rVert^{2}_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert
h\right\rVert_{X^{\infty}_{p,q}}\bigg{]}.$ (6.7)

In going from (6.4) to (6.5) we have recalled the embedding
$\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}\hookrightarrow
L^{\infty}\big{(}0,\infty;\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)\big{)}$ from (2.13) or (4.22). Next, in going from (6.5) to
(6.6) we have used the maximal regularity property (i)–(4.20). Finally, to go
from (6.6) to (6.7), we have invoked (5.15) for
$\displaystyle{\mathcal{N}}_{q}\mathbf{z}$ and (5.16) for
$\displaystyle{\mathcal{M}}_{q}[\mathbf{z}]h$. Thus, the conclusion of Step 1
is

$\displaystyle\left\lVert\begin{bmatrix}\mathbf{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$ $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+C_{3}\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\bigg{[}\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}+\left\lVert
h\right\rVert_{X^{\infty}_{p,q}}\bigg{]}$ (6.8)
$\displaystyle=M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+C_{3}\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert[\mathbf{z},h]\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}.$ (6.9)

Step 2: We now return to (6.1) and take the
$\displaystyle\mathbf{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$ norm
across:

$\displaystyle\left\lVert\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}(t)\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq\left\lVert
e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}+$ $\displaystyle\hskip
42.67912pt\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\mathbf{z}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\mathbf{z}]h(\tau)\end{bmatrix}d\tau\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (6.10) $\displaystyle(\text{by (\ref{2.16})})\ $
$\displaystyle\leq M_{1}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+C_{3}\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert[\mathbf{z},h]\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (6.11)

by invoking (4.20) on the first semigroup term in (6.10) and the estimate from
(6.5) to (6.9) on the second integral term in (6.10). Thus (6.11) is
established and implies

$\left\lVert\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\Big{[}1-C_{3}\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\Big{]}\leq
M_{1}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}.$ (6.12)

[This is the counterpart of [L-P-T.1, Eq (9.7)]].

Step 3: The well-posedness Theorem 5.1 says that

$\begin{Bmatrix}\text{ If }\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq
r_{1}\\\\[11.38109pt] \text{for }r_{1}\text{ sufficiently
small}\end{Bmatrix}\implies\begin{Bmatrix}\text{ The solution
}[\mathbf{z},h]\text{ satisfies}\\\\[8.53581pt]
\left\lVert\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\leq r\end{Bmatrix}$ (6.13)

where the constant $r$ satisfies the constraint (5.29) in terms of the
constant $r_{1}$ and some constant $C$ in (5.27) or (5.28) or (5.15) or
(5.16). We seek to guarantee that we have

$\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\leq\frac{1}{2C_{3}}<\frac{1}{2C},\mbox{
hence
}\frac{1}{2}\leq\Big{[}1-C_{3}\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\Big{]},$
(6.14)

where we can always take the constant $C_{3}$ in (6.11) greater than the
constant $C$ in (5.29), (5.30), (5.31).Then (6.14) can be achieved by choosing
$r_{1}>0$ sufficiently small. In fact, as $r_{1}\searrow 0$. Eq. (5.30) shows
that the interval $r_{\min}\leq r\leq r_{\max}$ of corresponding values of $r$
tends to the interval $[0,\frac{1}{C}]$. Thus, (6.14) can be achieved as
$r_{\min}\searrow 0$: $0<r_{\min}<r<\frac{1}{2C}$. Next (6.14) implies that
(6.12) becomes

$\left\lVert\begin{bmatrix}\mathbf{z}\\\
h\end{bmatrix}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\leq 2M_{1}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq 2M_{1}r_{1}.$
(6.15)

by (6.13). Substituting (6.15) in estimate (6.9) then yields

$\displaystyle\left\lVert\begin{bmatrix}\mathbf{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$ $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}+2M_{1}C_{3}\left\lVert\mathbf{z}\right\rVert_{\mathbf{X}^{\infty}_{p,q,\sigma}}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$
$\displaystyle\leq\widehat{M}\Big{[}e^{-\gamma_{0}t}+4\widehat{M}_{1}C_{3}r_{1}\Big{]}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$ (6.16)

again by (6.15) with
$\displaystyle\widehat{M}=\max\\{M_{\gamma_{0}},M_{1}\\}$. This is the
counterpart of [L-P-T.1, Eq (9.16)].

Step 4: We now take $T$ sufficiently large and $r_{1}>0$ sufficiently small so
that

$\beta=\widehat{M}e^{-\gamma_{0}T}+4\widehat{M}C_{3}r_{1}<1.$ (6.17)

Then (6.16) implies

$\left\lVert\begin{bmatrix}\mathbf{z}(T)\\\
h(T)\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq\beta\left\lVert\begin{bmatrix}{\mathbf{z}}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$ (6.18)

$\displaystyle\mathbf{V}^{q,p}(\Omega)=\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$, and hence

$\left\lVert\begin{bmatrix}\mathbf{z}(nT)\\\
h(nT)\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq\beta\left\lVert\begin{bmatrix}\mathbf{z}((n-1)T)\\\
h((n-1)T)\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq\beta^{n}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}.$ (6.19)

Since $\beta<1$, the semigroup property of the evolution implies [Bal] that
there are constants $M_{\widetilde{\gamma}}\geq 1,\widetilde{\gamma}>0$ such
that

$\left\lVert\begin{bmatrix}\mathbf{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq
M_{\widetilde{\gamma}}e^{-\widetilde{\gamma}t}\left\lVert\begin{bmatrix}\mathbf{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\mathbf{V}^{q,p}(\Omega)}$ (6.20)

with
$\displaystyle\left\lVert[\mathbf{z}_{0},h_{0}]\right\rVert_{\mathbf{V}^{q,p}(\Omega)}\leq
r_{1}=$ small. This proves (2.14), i.e Theorem 2.3 ∎

######  Remark 6.1.

The above computations - (6.17) through (6.19) - can be used to support
qualitatively the intuitive expectation that “the larger the decay rate
$\gamma_{0}$ in (2.3) =(4.13)= (6.2) of the linearized feedback
$\mathbf{w}$-dynamics (2.2), the larger the decay rate $\widetilde{\gamma}$ in
(6.20) of the nonlinear feedback $\\{\mathbf{z},h\\}$-dynamics (2.5) = (2.6)
or (5.2),(5.3); hence the larger the rate $\widetilde{\gamma}$ in (2.19) of
the original $\\{\mathbf{y},\theta\\}$-dynamics in (2.17) = (2.18)”.

The following considerations are somewhat qualitative. Let $S(t)$ denote the
non-linear semigroup in the space
$\mathbf{V}^{q,p}(\Omega)\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
L^{q}(\Omega)$, with infinitesimal generator
$\displaystyle\big{[}\mathbb{A}_{{}_{F,q}}-{\mathcal{N}}_{q}\big{]}$
describing the feedback $\\{\mathbf{z},h\\}$-dynamics (2.5) = (2.6), or
(5.2),(5.3) as guaranteed by the well posedness Theorem 2.2 = Theorem 5.1.
Thus, $\begin{bmatrix}\mathbf{z}(t)\\\
h(t)\end{bmatrix}=S(t)\begin{bmatrix}\mathbf{z}_{0}\\\ h_{0}\end{bmatrix}$ on
$\mathbf{V}^{q,p}(\Omega)$. By (6.17), we can rewrite (6.18) as:

$\left\lVert
S(T)\right\rVert_{{\mathcal{L}}\big{(}\mathbf{V}^{q,p}(\Omega)\big{)}}\leq\beta<1.$
(6.21)

It follows from [Bal, p 178] via the semigroup property that

$-\widetilde{\gamma}\ \ \text{is just below}\ \ \frac{\ln\beta}{T}<0.$ (6.22)

Pick $r_{1}>0$ in (6.17) so small that $4\widehat{M}^{2}C_{3}r_{1}$ is
negligible, so that $\beta$ is just above
$\displaystyle\widehat{M}e^{-\gamma_{0}T}$, so $\ln\beta$ is just above
$\displaystyle\big{[}\ln\widehat{M}-\gamma_{0}T\big{]}$, hence

$\frac{\ln\beta}{T}\text{ is just above
}\Big{[}(-\gamma_{0})+\frac{\ln\widehat{M}}{T}\Big{]}.$ (6.23)

Hence, by (6.22), (6.23),

$\widetilde{\gamma}\sim\gamma_{0}-\frac{\ln\widehat{M}}{T}$ (6.24)

and the larger $\gamma_{0}$, the larger is $\widetilde{\gamma}$, as desired.

##  Appendix A Some auxiliary results for the Stokes and Oseen operators:
analytic semigroup generation, maximal regularity, domains of fractional
powers

In this section we collect some known results used in the paper. As a
prerequisite of the present Appendix A, we make reference to the paragraph
Definition of Besov spaces $\mathbf{B}^{s}_{q,p}$ on domains of class $C^{1}$
as real interpolation of Sobolev spaces, Eqts (1.8)-(1.11) and Remark 1.1.

  1. (a)

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

######  Theorem A.1.

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 (1.14), repeated here as

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

generates a s.c. analytic semigroup $e^{-A_{q}t}$ on
$\mathbf{L}^{q}_{\sigma}(\Omega)$. See [Gi.1] and the review paper [H-S,
Theorem 2.8.5 p 17].

    2. (ii)

The Oseen operator ${\mathcal{A}}_{q}$ in (1.18)

${\mathcal{A}}_{q}=-(\nu
A_{q}+A_{o,q}),\quad{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\subset\mathbf{L}^{q}_{\sigma}(\Omega)$
(A.2)

generates a s.c. analytic semigroup $e^{{\mathcal{A}}_{q}t}$ on
$\mathbf{L}^{q}_{\sigma}(\Omega)$. This follows as $A_{o,q}$ is relatively
bounded with respect to $A^{{}^{1}\\!/_{2}}_{q}$, see (1.16): thus a standard
theorem on perturbation of an analytic semigroup generator applies [Pazy,
Corollary 2.4, p 81].

    3. (iii)

$\displaystyle 0\in\rho(A_{q})$ $\displaystyle=\text{ the resolvent set of the
Stokes operator }A_{q}$ (A.3a)
$\displaystyle\begin{picture}(0.0,0.0)\put(-40.0,10.0){
$\left\\{\rule{0.0pt}{18.0pt}\right.$}\end{picture}A_{q}^{-1}$
$\displaystyle:\mathbf{L}^{q}_{\sigma}(\Omega)\longrightarrow\mathbf{L}^{q}_{\sigma}(\Omega)\text{
is compact}$ (A.3b)

    4. (iv)

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

$\left\lVert
e^{-A_{q}t}\right\rVert_{{\mathcal{L}}(\mathbf{L}^{q}_{\sigma}(\Omega))}\leq
Me^{-\delta t},\ t>0.$ (A.4)

  2. (b)

Domains of fractional powers, ${\mathcal{D}}(A_{q}^{\alpha}),0<\alpha<1$ of
the Stokes operator $A_{q}$ on $\mathbf{L}^{q}_{\sigma}(\Omega),1<q<\infty$.
We elaborate on (1.17a-b)

######  Theorem A.2.

For the domains of fractional powers
${\mathcal{D}}(A_{q}^{\alpha}),0<\alpha<1$, of the Stokes operator $A_{q}$ in
(A.1) = (1.14), the following complex interpolation relation holds true [Gi.2]
and [H-S, Theorem 2.8.5, p 18]

$[{\mathcal{D}}(A_{q}),\mathbf{L}^{q}_{\sigma}(\Omega)]_{1-\alpha}={\mathcal{D}}(A_{q}^{\alpha}),\
0<\alpha<1,\ 1<q<\infty;$ (A.5)

in particular

$[{\mathcal{D}}(A_{q}),\mathbf{L}^{q}_{\sigma}(\Omega)]_{\frac{1}{2}}={\mathcal{D}}(A_{q}^{{}^{1}\\!/_{2}})\equiv\mathbf{W}_{0}^{1,q}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega).$
(A.6)

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

$\left\lVert\nabla\ \cdot\ \right\rVert_{L^{q}(\Omega)}\text{ and
}\left\lVert\ \right\rVert_{L^{q}(\Omega)}$ (A.7)

are related via Poincaré inequality.

  3. (c)

The Stokes operator $-A_{q}$ and the Oseen operator
${\mathcal{A}}_{q},1<q<\infty$ generate s.c. analytic semigroups on the Besov
space, from (1.11)

$\displaystyle\Big{(}\mathbf{L}^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}\mathbf{g}\in\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{
div }\mathbf{g}=0,\ \mathbf{g}|_{\Gamma}=0\Big{\\}}\quad\text{if
}\frac{1}{q}<2-\frac{2}{p}<2;$ (A.8a)
$\displaystyle\Big{(}\mathbf{L}^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}\mathbf{g}\in\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{
div }\mathbf{g}=0,\
\mathbf{g}\cdot\nu|_{\Gamma}=0\Big{\\}}\equiv\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$
(A.8b) $\displaystyle\hskip 170.71652pt\text{ if
}0<2-\frac{2}{p}<\frac{1}{q}.$

Theorem A.1 states that the Stokes operator $-A_{q}$ generates a s.c. analytic
semigroup on the space $\mathbf{L}^{q}_{\sigma}(\Omega),\ 1<q<\infty$, hence
on the space ${\mathcal{D}}(A_{q})$ in (1.14) = (A.1), with norm
$\displaystyle\left\lVert\ \cdot\
\right\rVert_{{\mathcal{D}}(A_{q})}=\left\lVert A_{q}\ \cdot\
\right\rVert_{\mathbf{L}^{q}_{\sigma}(\Omega)}$ 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 (A.8). Next, the Oseen operator
${\mathcal{A}}_{q}=-(\nu A_{q}+A_{o,q})$ in (A.2) = (1.18) likewise generates
a s.c. analytic semigroup $\displaystyle e^{{\mathcal{A}}_{q}t}$ on
$\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)$ 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 $\mathbf{L}^{q}_{\sigma}(\Omega)$. 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 (A.8). 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
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, as
this does not contain B.C., Remark 1.1. The objective of the present paper is
precisely to obtain stabilization results on spaces that do not recognize B.C.

######  Theorem A.3.

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

    1. (i)

The Stokes operator $-A_{q}$ in (A.1) = (1.14) generates a s.c. analytic
semigroup $e^{-A_{q}t}$ on the space
$\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ defined in (1.11b) =
(A.8b) which moreover is uniformly stable, as in (A.4),

$\left\lVert
e^{-A_{q}t}\right\rVert_{{\mathcal{L}}\big{(}\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}\leq
Me^{-\delta t},\quad t>0.$ (A.9)

    2. (ii)

The Oseen operator ${\mathcal{A}}_{q}$ in (A.2)=(1.18) generates a s.c.
analytic semigroup $e^{{\mathcal{A}}_{q}t}$ on the space
$\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ in (1.11b) = (A.8).

  4. (d)

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

$\displaystyle\boldsymbol{\varphi}_{t}-\Delta\boldsymbol{\varphi}+\nabla\pi$
$\displaystyle=\mathbf{F}$ $\displaystyle\text{ in }(0,T]\times\Omega\equiv Q$
(A.10a) $\displaystyle div\ \boldsymbol{\varphi}$ $\displaystyle\equiv 0$
$\displaystyle\text{ in }Q$ (A.10b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-80.0,5.0){
$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}\left.\boldsymbol{\varphi}\right\rvert_{\Sigma}$
$\displaystyle\equiv 0$ $\displaystyle\text{ in
}(0,T]\times\Gamma\equiv\Sigma$ (A.10c)
$\displaystyle\left.\boldsymbol{\varphi}\right\rvert_{t=0}$
$\displaystyle=\boldsymbol{\varphi}_{0}$ $\displaystyle\text{ in }\Omega,$
(A.10d)

rewritten in abstract form, after applying the Helmholtz projection $P_{q}$ to
(A.10a) and recalling $A_{q}$ in (A.1) = (1.14) as

$\boldsymbol{\varphi}^{\prime}+A_{q}\boldsymbol{\varphi}=\mathbf{F}_{\sigma}\equiv
P_{q}\mathbf{F},\quad\boldsymbol{\varphi}_{0}\in\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}.$
(A.11)

Next, we introduce the space of maximal regularity for
$\\{\boldsymbol{\varphi},\boldsymbol{\varphi}^{\prime}\\}$ as [H-S, p 2;
Theorem 2.8.5.iii, p 17], [G-G-H.1, p 1404-5], with $T$ up to $\infty$:

$\mathbf{X}^{T}_{p,q,\sigma}=L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap{W}^{1,p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))$
(A.12)

(recall (A.1)= (1.14) for ${\mathcal{D}}(A_{q})$) and the corresponding space
for the pressure as

$Y^{T}_{p,q}=L^{p}(0,T;\widehat{W}^{1,q}(\Omega)),\quad\widehat{W}^{1,q}(\Omega)=W^{1,q}(\Omega)/\mathbb{R}.$
(A.13)

The following embedding, also called trace theorem, holds true [Amann.2,
Theorem 4.10.2, p 180, BUC for $T=\infty$], [P-S].

$\mathbf{X}^{T}_{p,q,\sigma}\subset\mathbf{X}^{T}_{p,q}\equiv
L^{p}(0,T;\mathbf{W}^{2,q}(\Omega))\cap
W^{1,p}(0,T;\mathbf{L}^{q}(\Omega))\hookrightarrow
C\Big{(}[0,T];\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\Big{)}.$ (A.14)

For a function $\mathbf{g}$ such that $div\ \mathbf{g}\equiv 0,\
\mathbf{g}|_{\Gamma}=0$ we have
$\mathbf{g}\in\mathbf{X}^{T}_{p,q}\iff\mathbf{g}\in\mathbf{X}^{T}_{p,q,\sigma}$.
The solution of Eq (A.11) is

$\boldsymbol{\varphi}(t)=e^{-A_{q}t}\boldsymbol{\varphi}_{0}+\int_{0}^{t}e^{-A_{q}(t-s)}\mathbf{F}_{\sigma}(\tau)d\tau.$
(A.15)

The following is the celebrated result on maximal regularity on
$\mathbf{L}^{q}_{\sigma}(\Omega)$ of the Stokes problem due originally to
Solonnikov [Sol.2] reported in [H-S, Theorem 2.8.5.(iii) and Theorem 2.10.1
p24 for $\boldsymbol{\varphi}_{0}=0$], [Saa], [G-G-H.1, Proposition 4.1 , p
1405].

######  Theorem A.4.

Let $1<p,q<\infty,T\leq\infty$. With reference to problem (A.10) = (A.11),
assume

$\mathbf{F}_{\sigma}\in L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega)),\
\boldsymbol{\varphi}_{0}\in\Big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}.$
(A.16)

Then there exists a unique solution
$\boldsymbol{\varphi}\in\mathbf{X}^{T}_{p,q,\sigma},\pi\in Y^{T}_{p,q}$ to the
dynamic Stokes problem (A.10) or (A.11), continuously on the data: there exist
constants $C_{0},C_{1}$ independent of
$T,\mathbf{F}_{\sigma},\boldsymbol{\varphi}_{0}$ such that via (A.14)

$\displaystyle
C_{0}\left\lVert\boldsymbol{\varphi}\right\rVert_{C\big{(}[0,T];\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}$
$\displaystyle\leq\left\lVert\boldsymbol{\varphi}\right\rVert_{\mathbf{X}^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
(A.17)
$\displaystyle\equiv\left\lVert\boldsymbol{\varphi}^{\prime}\right\rVert_{L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))}+\left\lVert
A_{q}\boldsymbol{\varphi}\right\rVert_{L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
$\displaystyle\leq
C_{1}\bigg{\\{}\left\lVert\mathbf{F}_{\sigma}\right\rVert_{L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))}+\left\lVert\boldsymbol{\varphi}_{0}\right\rVert_{\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}}\bigg{\\}}.$

In particular,

    1. (i)

With reference to the variation of parameters formula (A.15) of problem (A.11)
arising from the Stokes problem (A.10), we have recalling (A.12): the map

$\displaystyle\mathbf{F}_{\sigma}$
$\displaystyle\longrightarrow\int_{0}^{t}e^{-A_{q}(t-\tau)}\mathbf{F}_{\sigma}(\tau)d\tau\
:\text{continuous}$ (A.18) $\displaystyle
L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))$
$\displaystyle\longrightarrow\mathbf{X}^{T}_{p,q,\sigma}\equiv
L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap
W^{1,p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega)).$ (A.19)

    2. (ii)

The s.c. analytic semigroup $e^{-A_{q}t}$ generated by the Stokes operator
$-A_{q}$ (see (A.1)= (1.14)) on the space
$\displaystyle\Big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
(see statement below (A.8)) satisfies

$e^{-A_{q}t}:\
\text{continuous}\quad\Big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}\longrightarrow\mathbf{X}^{T}_{p,q,\sigma}\equiv\\\
L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap
W^{1,p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega)).$ (A.20a) In particular via
(A.8b), 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
$\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$,
satisfies $e^{-A_{q}t}:\
\text{continuous}\quad\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow\mathbf{X}^{T}_{p,q,\sigma}.$
(A.20b)

    3. (iii)

Moreover, for future use, for $1<q<\infty,1<p<\frac{2q}{2q-1}$, then (A.17)
specializes to

$\left\lVert\boldsymbol{\varphi}\right\rVert_{\mathbf{X}^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}\leq
C\bigg{\\{}\left\lVert\mathbf{F}_{\sigma}\right\rVert_{L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))}+\left\lVert\boldsymbol{\varphi}_{0}\right\rVert_{\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\bigg{\\}}.$
(A.21)

  5. (e)

Maximal $L^{p}$ regularity on $\mathbf{L}^{q}_{\sigma}(\Omega)$ 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
$\mathbf{L}^{q}_{\sigma}(\Omega)$-asserted in Theorem A.4 into the maximal
regularity of the Oseen operator ${\mathcal{A}}_{q}=-\nu A_{q}-A_{o,q}$ in
(A.2) exactly on the same space $\mathbf{X}^{T}_{p,q,\sigma}$ defined in
(A.12), however only up to $T<\infty$.

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

$\displaystyle\boldsymbol{\varphi}_{t}-\Delta\boldsymbol{\varphi}+L_{e}(\boldsymbol{\varphi})+\nabla\pi$
$\displaystyle=F$ $\displaystyle\text{ in }(0,T]\times\Omega\equiv Q$ (A.22a)
$\displaystyle div\ \boldsymbol{\varphi}$ $\displaystyle\equiv 0$
$\displaystyle\text{ in }Q$ (A.22b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-110.0,7.0){$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}\left.\boldsymbol{\varphi}\right\rvert_{\Sigma}$
$\displaystyle\equiv 0$ $\displaystyle\text{ in
}(0,T]\times\Gamma\equiv\Sigma$ (A.22c)
$\displaystyle\left.\boldsymbol{\varphi}\right\rvert_{t=0}$
$\displaystyle=\boldsymbol{\varphi}_{0}$ $\displaystyle\text{ in }\Omega,$
(A.22d)
$L_{e}(\boldsymbol{\varphi})=(y_{e}.\nabla)\boldsymbol{\varphi}+(\boldsymbol{\varphi}.\nabla)y_{e}\hskip
85.35826pt$ (A.23)

rewritten in abstract form, after applying the Helmholtz projector $P_{q}$ to
(A.22a) and recalling ${\mathcal{A}}_{q}$ in (A.2), as

$\boldsymbol{\varphi}_{t}={\mathcal{A}}_{q}\boldsymbol{\varphi}+P_{q}\mathbf{F}=-\nu
A_{q}\boldsymbol{\varphi}-A_{o,q}\boldsymbol{\varphi}+\mathbf{F}_{\sigma},\quad\boldsymbol{\varphi}_{0}\in\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
(A.24)

whose solution is

$\boldsymbol{\varphi}(t)=e^{{\mathcal{A}}_{q}t}\boldsymbol{\varphi}_{0}+\int_{0}^{t}e^{{\mathcal{A}}_{q}(t-\tau)}\mathbf{F}_{\sigma}(\tau)d\tau,$
(A.25) $\boldsymbol{\varphi}(t)=e^{-\nu
A_{q}t}\boldsymbol{\varphi}_{0}+\int_{0}^{t}e^{-\nu
A_{q}(t-\tau)}\mathbf{F}_{\sigma}(\tau)d\tau-\int_{0}^{t}e^{-\nu
A_{q}(t-\tau)}A_{o,q}\boldsymbol{\varphi}(\tau)d\tau.$ (A.26)

######  Theorem A.5.

Let $1<p,q<\infty,\ 0<T<\infty$. Assume (as in (A.16))

$\mathbf{F}_{\sigma}\in
L^{p}\big{(}0,T;\mathbf{L}^{q}_{\sigma}(\Omega)\big{)},\quad\boldsymbol{\varphi}_{0}\in\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
(A.27)

where ${\mathcal{D}}(A_{q})={\mathcal{D}}({\mathcal{A}}_{q})$, see
(A.2)=(1.18). Then there exists a unique solution
$\boldsymbol{\varphi}\in\mathbf{X}^{T}_{p,q,\sigma},\ \pi\in Y^{T}_{p,q}$ of
the dynamic Oseen problem (A.22), continuously on the data: that is, there
exist constants $C_{0},C_{1}$ independent of
$\mathbf{F}_{\sigma},\boldsymbol{\varphi}_{0}$ such that

$\displaystyle
C_{0}\left\lVert\boldsymbol{\varphi}\right\rVert_{C\big{(}[0,T];\mathbf{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}$
$\displaystyle\leq\left\lVert\boldsymbol{\varphi}\right\rVert_{\mathbf{X}^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
$\displaystyle\equiv\left\lVert\boldsymbol{\varphi}^{\prime}\right\rVert_{L^{p}(0,T;\mathbf{L}^{q}(\Omega))}+\left\lVert
A_{q}\boldsymbol{\varphi}\right\rVert_{L^{p}(0,T;\mathbf{L}^{q}(\Omega))}+\left\lVert\pi\right\rVert_{Y^{T}_{p,q}}$
(A.28) $\displaystyle\leq
C_{T}\left\\{\left\lVert\mathbf{F}_{\sigma}\right\rVert_{L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))}+\left\lVert\boldsymbol{\varphi}_{0}\right\rVert_{\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}}\right\\}$
(A.29)

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

    1. i.

The map

$\displaystyle\mathbf{F}_{\sigma}\longrightarrow\int_{0}^{t}e^{{\mathcal{A}}_{q}(t-\tau)}\mathbf{F}_{\sigma}(\tau)d\tau\
:\text{continuous}$ (A.30) $\displaystyle
L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))$ $\displaystyle\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)}$

where then automatically, see (A.24)

$L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))\longrightarrow
W^{1,p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))$ (A.31)

and ultimately via (A.12)

$L^{p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega))\longrightarrow\mathbf{X}^{T}_{p,q,\sigma}\equiv
L^{p}\big{(}0,T;{\mathcal{D}}(A_{q})\big{)}\cap
W^{1,p}(0,T;\mathbf{L}^{q}_{\sigma}(\Omega)).$ (A.32)

    2. ii.

The s.c. analytic semigroup $e^{{\mathcal{A}}_{q}t}$ generated by the Oseen
operator ${\mathcal{A}}_{q}$ (see (A.2)=(1.18)) on the space
$\displaystyle\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
satisfies for $1<p,q<\infty$

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)}$
(A.33)

and hence automatically by (A.12)

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\big{(}\mathbf{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}\longrightarrow\mathbf{X}^{T}_{p,q,\sigma}.$
(A.34)

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 $\displaystyle\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$,
satisfies

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)},\
T<\infty.$ (A.35)

and hence automatically

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\widetilde{\mathbf{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow\mathbf{X}^{T}_{p,q,\sigma},\
T<\infty.$ (A.36)

A proof is given in [L-P-T.1, Appendix B].

##  Appendix B The UCP of Theorem 1.4 implies the controllability rank
condition (4.10)

We return to equations (4.2) and (4.3) giving eigenvalues/vectors of the
operator $\mathbb{A}_{q}$ in (1.27) and its adjoint $\mathbb{A}_{q}^{*}$:

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

The operator $\mathbb{A}_{q}$ : We recall from (1.27), (1.23), (1.24) that

$\mathbb{A}_{q}=\begin{bmatrix}{\mathcal{A}}_{q}&-{\mathcal{C}}_{\gamma}\\\
-{\mathcal{C}}_{\theta_{e}}&{\mathcal{B}}_{q}\end{bmatrix}:\mathbf{W}^{q}_{\sigma}(\Omega)=\mathbf{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\supset{\mathcal{D}}(\mathbb{A}_{q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q})\\\
=(\mathbf{W}^{2,q}(\Omega)\cap\mathbf{W}^{1,q}_{0}(\Omega)\cap\mathbf{L}^{q}_{\sigma}(\Omega))\times(W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega))\longrightarrow\mathbf{W}^{q}_{\sigma}(\Omega).$ (B.3)
$\displaystyle\quad{\mathcal{C}}_{\gamma}h$ $\displaystyle=-\gamma
P_{q}(h\mathbf{e}_{d}),\quad{\mathcal{C}}_{\gamma}\in{\mathcal{L}}(L^{q}(\Omega),\mathbf{L}^{q}_{\sigma}(\Omega)),$
(B.4) $\displaystyle\quad{\mathcal{C}}_{\theta_{e}}\mathbf{z}$
$\displaystyle=\mathbf{z}\cdot\nabla\theta_{e},\quad{\mathcal{C}}_{\theta_{e}}\in{\mathcal{L}}(\mathbf{L}^{q}_{\sigma}(\Omega),L^{q}(\Omega)).$
(B.5)

With $\boldsymbol{\Phi}=[\boldsymbol{\varphi},\psi],$ the PDE-version of
$\mathbb{A}_{q}\boldsymbol{\Phi}=\lambda\boldsymbol{\Phi}$ is

$\displaystyle-\nu\Delta\boldsymbol{\varphi}+L_{e}(\boldsymbol{\varphi})+\nabla\pi-\gamma\psi\mathbf{e}_{d}$
$\displaystyle=\lambda\boldsymbol{\varphi}$ $\displaystyle\text{in }\Omega$
(B.6a)
$\displaystyle-\kappa\Delta\psi+\mathbf{y}_{e}\cdot\nabla\psi+\boldsymbol{\varphi}\cdot\nabla\theta_{e}$
$\displaystyle=\lambda\psi$ $\displaystyle\text{in }\Omega$ (B.6b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-140.0,3.0){
$\left\\{\rule{0.0pt}{36.0pt}\right.$}\end{picture}\text{div
}\boldsymbol{\varphi}$ $\displaystyle=0$ $\displaystyle\text{in }\Omega$
(B.6c) $\displaystyle\boldsymbol{\varphi}=0,\ \psi$ $\displaystyle=0$
$\displaystyle\text{on }\Gamma.$ (B.6d)

The operator $\mathbb{A}_{q}^{*}$: We return to the Helmholtz decomposition in
(1.5), (1.6) and provide additional information. For
$\mathbf{M}\subset\mathbf{L}^{q}(\Omega),\ 1<q<\infty$, we denote the
annihilator of $\mathbf{M}$ by

$\mathbf{M}^{\perp}=\bigg{\\{}\mathbf{f}\in\mathbf{L}^{q^{\prime}}(\Omega):\int_{\Omega}\mathbf{f}\cdot\mathbf{g}\
d\Omega=0,\text{ for all }\mathbf{g}\in\mathbf{M}\bigg{\\}}$ (B.7)

where $q^{\prime}$ is the dual exponent of $\displaystyle q:\
{}^{1}\\!/_{q}+{}^{1}\\!/_{q^{\prime}}=1$.

######  Proposition B.1.

[H-S, Prop 2.2.2 p6], [Ga.3, Ex. 16 p115], [F-M-M]
Let $\Omega\subset\mathbb{R}^{d}$ be an open set and let $1<q<\infty$.

  1. a)

The Helmholtz decomposition exists for $\mathbf{L}^{q}(\Omega)$ if and only if
it exists for $\mathbf{L}^{q^{\prime}}(\Omega)$, and we have: (adjoint of
$P_{q}$) = $P^{*}_{q}=P_{q^{\prime}}$ (in particular $P_{2}$ is orthogonal),
where $P_{q}$ is viewed as a bounded operator
$\displaystyle\mathbf{L}^{q}(\Omega)\longrightarrow\mathbf{L}^{q}(\Omega)$,
and $\displaystyle P^{*}_{q}=P_{q^{\prime}}$ as a bounded operator
$\displaystyle\mathbf{L}^{q^{\prime}}(\Omega)\longrightarrow\mathbf{L}^{q^{\prime}}(\Omega),\
{}^{1}\\!/_{q}+{}^{1}\\!/_{q^{\prime}}=1$.

  2. b)

Then, with reference to (1.6)

$\Big{[}\mathbf{L}^{q}_{\sigma}(\Omega)\Big{]}^{\perp}=\mathbf{G}^{q^{\prime}}(\Omega)\text{
and
}\Big{[}\mathbf{G}^{q}(\Omega)\Big{]}^{\perp}=\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega).$
(B.8a)

######  Remark B.1.

Throughout the paper we shall use freely that

$\big{(}\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}^{\prime}=\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega),\quad\frac{1}{q}+\frac{1}{q^{\prime}}=1$
(B.8b)

This can be established as follows. From (1.6) write
$\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)$ as a factor space
$\displaystyle\mathbf{L}^{q}_{\sigma}(\Omega)=\mathbf{L}^{q}(\Omega)/\mathbf{G}^{q}(\Omega)\equiv\mathbf{X}/\mathbf{M}$
so that [T-L.1, p 135].

$\big{(}\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}^{\prime}=\big{(}\mathbf{L}^{q}(\Omega)/\mathbf{G}^{q}(\Omega)\big{)}^{\prime}=\big{(}\mathbf{X}/\mathbf{M}\big{)}^{\prime}=\mathbf{M}^{\perp}=\Big{[}\mathbf{G}^{q}(\Omega)\Big{]}^{\perp}=\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega).$
(B.8c)

In the last step, we have invoked (B.8a), which is also established in [Ga.3,
Lemma 2.1, p 116]. Similarly

$\big{(}\mathbf{G}^{q}(\Omega)\big{)}^{\prime}=\big{(}\mathbf{L}^{q}(\Omega)/\mathbf{L}^{q}_{\sigma}(\Omega)\big{)}^{\prime}=\Big{[}\mathbf{L}^{q}_{\sigma}(\Omega)\Big{]}^{\perp}=\mathbf{G}^{q^{\prime}}(\Omega).$
(B.8d)

Next, let
$\boldsymbol{\varphi}^{*}\in\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)$ and
$h\in\L^{q}(\Omega)$. From (B.4), we then compute in the noted duality
pairings, via the above results, omitting at times the symbol $\Omega$:

$\displaystyle\big{(}{\mathcal{C}}_{\gamma}h,\boldsymbol{\varphi}^{*}\big{)}_{\mathbf{L}^{q}_{\sigma}(\Omega),\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)}$
$\displaystyle=\big{(}-\gamma
P_{q}(h\mathbf{e}_{d}),\boldsymbol{\varphi}^{*}\big{)}_{\mathbf{L}^{q}_{\sigma}(\Omega),\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)}=\big{(}(h\mathbf{e}_{d}),-\gamma(P_{q^{\prime}}\boldsymbol{\varphi}^{*})\big{)}_{\mathbf{L}^{q},\mathbf{L}^{q^{\prime}}}$
(B.9)
$\displaystyle=\big{(}h,-\gamma(P_{q^{\prime}}\boldsymbol{\varphi}^{*})\cdot\mathbf{e}_{d})\big{)}_{L^{q},L^{q^{\prime}}}=\big{(}h,{\mathcal{C}}_{\gamma}^{*}\boldsymbol{\varphi}^{*}\big{)}_{L^{q},L^{q^{\prime}}}$
(B.10)

Thus

${\mathcal{C}}_{\gamma}^{*}\boldsymbol{\varphi}^{*}=-\gamma(P_{q^{\prime}}\boldsymbol{\varphi}^{*})\cdot\mathbf{e}_{d},\,\,{\mathcal{C}}_{\gamma}^{*}\in\mathcal{L}(\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega),L^{q^{\prime}}(\Omega)).$
(B.11)

Next, let $\mathbf{z}\in\mathbf{L}^{q}_{\sigma}(\Omega)$, so that
$\text{div}\,\mathbf{z}\equiv 0,\mathbf{z}\cdot\nu=0$ on $\Gamma$ by (1.5).
Let $\psi^{*}\in L^{q^{\prime}}(\Omega)$. For $q>d$, recall
$\nabla\theta_{e}\in\mathbf{W}^{1,q}(\Omega)\hookrightarrow\mathbf{L}^{\infty}(\Omega)$
from Theorem 1.1. Then, from (B.5) we compute

$\displaystyle\big{(}{\mathcal{C}}_{\theta_{e}}\mathbf{z},\psi^{*}\big{)}_{L^{q},L^{q^{\prime}}}$
$\displaystyle=\big{(}\mathbf{z}\cdot\nabla\theta_{e},\psi^{*}\big{)}_{L^{q},L^{q^{\prime}}}=\big{(}(P_{q}\mathbf{z})\cdot\nabla\theta_{e},\psi^{*}\big{)}_{L^{q},L^{q^{\prime}}}$
(B.12)
$\displaystyle=\int_{\Omega}(P_{q}\mathbf{z})\cdot(\psi^{*}\nabla\theta_{e})\,d\Omega=\big{(}P_{q}\mathbf{z},\psi^{*}\nabla\theta_{e}\big{)}_{L^{q},L^{q^{\prime}}}$
(B.13)
$\displaystyle=\big{(}\mathbf{z},P_{q^{\prime}}(\psi^{*}\nabla\theta_{e})\big{)}_{\mathbf{L}^{q}_{\sigma}(\Omega),\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)}=\big{(}\mathbf{z},{\mathcal{C}}_{\theta_{e}}^{*}\psi^{*}\big{)}_{\mathbf{L}^{q}_{\sigma}(\Omega),\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)}.$
(B.14)

Thus,

${\mathcal{C}}_{\theta_{e}}^{*}\psi^{*}=P_{q^{\prime}}(\psi^{*}\nabla\theta_{e})\in\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega).$
(B.15)

The adjoint $\mathbb{A}_{q}^{*}$ of $\mathbb{A}_{q}$ in (B.3) is, by virtue of
(B.11) and (B.15)

$\mathbb{A}_{q}^{*}=\begin{bmatrix}{\mathcal{A}}_{q}^{*}&-{\mathcal{C}}_{\theta_{e}}^{*}\\\
-{\mathcal{C}}_{\gamma}^{*}&{\mathcal{B}}_{q}^{*}\end{bmatrix}:\mathbf{W}^{q^{\prime}}_{\sigma}(\Omega)=\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega)\times
L^{q^{\prime}}(\Omega)\supset{\mathcal{D}}(\mathbb{A}_{q}^{*})={\mathcal{D}}({\mathcal{A}}_{q}^{*})\times{\mathcal{D}}({\mathcal{B}}_{q}^{*})\\\
=(\mathbf{W}^{2,q^{\prime}}(\Omega)\cap\mathbf{W}^{1,q^{\prime}}_{0}(\Omega)\cap\mathbf{L}^{q^{\prime}}_{\sigma}(\Omega))\times(W^{2,q^{\prime}}(\Omega)\cap\mathbf{W}^{1,q^{\prime}}_{0}(\Omega))\longrightarrow\mathbf{W}^{q^{\prime}}_{\sigma}(\Omega).$
(B.16)

With $\boldsymbol{\Phi^{*}}=[\varphi^{*},\psi^{*}],$ the PDE-version of
$\mathbb{A}_{q}^{*}\boldsymbol{\Phi^{*}}=\lambda\boldsymbol{\Phi^{*}}$ is

$\displaystyle-\nu\Delta\boldsymbol{\varphi^{*}}+L^{*}_{e}(\boldsymbol{\varphi^{*}})+\psi^{*}\nabla\theta_{e}+\nabla\pi$
$\displaystyle=\lambda\boldsymbol{\varphi^{*}}$ $\displaystyle\text{in
}\Omega$ (B.17a)
$\displaystyle-\kappa\Delta\psi^{*}+\mathbf{y}_{e}\cdot\nabla\psi^{*}-\gamma\boldsymbol{\varphi^{*}}\cdot\mathbf{e}_{d}$
$\displaystyle=\lambda\psi^{*}$ $\displaystyle\text{in }\Omega$ (B.17b)
$\displaystyle\begin{picture}(0.0,0.0)\put(-138.0,4.0){
$\left\\{\rule{0.0pt}{35.0pt}\right.$}\end{picture}\text{div
}\boldsymbol{\varphi^{*}}$ $\displaystyle=0$ $\displaystyle\text{in }\Omega$
(B.17c) $\displaystyle\boldsymbol{\varphi^{*}}=0,\ \psi^{*}$ $\displaystyle=0$
$\displaystyle\text{on }\Gamma.$ (B.17d)

Proof that the UCP of Theorem 1.4 implies the controllability rank condition
(4.10)). We return to (B.1) and (B.2) and express the eigenvectors in terms of
their coordinates, as $(d+1)$ vectors.

$\boldsymbol{\Phi_{ij}}=\\{\boldsymbol{\varphi}_{ij},\psi_{ij}\\}=\\{\varphi_{ij}^{(1)},\varphi_{ij}^{(2)},...,\varphi_{ij}^{(d-1)},\varphi_{ij}^{(d)},\psi_{ij}\\}\,,\text{
a }(d+1)\text{-vector}$ (B.18)
$\boldsymbol{\Phi^{*}_{ij}}=\\{\boldsymbol{\varphi^{*}_{ij}},\psi^{*}_{ij}\\}=\\{\varphi_{ij}^{*(1)},\varphi_{ij}^{*(2)},...,\varphi_{ij}^{*(d-1)},\varphi_{ij}^{*(d)},\psi^{*}_{ij}\\}\,,\text{
a }(d+1)\text{-vector}.$ (B.19)

With reference to (B.19), we introduce the following corresponding $d$-vector

$\boldsymbol{\widehat{\Phi}^{*}_{ij}=\\{\boldsymbol{\widehat{\varphi}^{*}_{ij}}},\psi^{*}_{ij}\\}=\\{\varphi_{ij}^{*(1)},\varphi_{ij}^{*(2)},...,\varphi_{ij}^{*(d-1)},\psi^{*}_{ij}\\}\,,\text{
a }d\text{-vector}$ (B.20)

obtained from $\boldsymbol{\Phi^{*}_{ij}}$ by omitting the d-component
$\varphi_{ij}^{*(d)}$ of the vector $\boldsymbol{\Phi^{*}_{ij}}$. Next, with
reference to (4.6), construct the following matrix $U_{i}$ of size
$\ell_{i}\times K,\ K=\sup\\{\ell_{i}:i=1,\dots,M\\}$

$U_{i}=\begin{bmatrix}(\mathbf{u}_{1},\widehat{\Phi}_{i1}^{*})_{\omega}&\dots&(\mathbf{u}_{K},\widehat{\Phi}_{i1}^{*})_{\omega}\\\\[2.84526pt]
(\mathbf{u}_{1},\widehat{\Phi}_{i2}^{*})_{\omega}&\dots&(\mathbf{u}_{K},\widehat{\Phi}_{i2}^{*})_{\omega}\\\
\vdots&\ddots&\vdots\\\
(\mathbf{u}_{1},\widehat{\Phi}_{i\ell_{i}}^{*})_{\omega}&\dots&(\mathbf{u}_{K},\widehat{\Phi}_{i\ell_{i}}^{*})_{\omega}\\\
\end{bmatrix}:\ell_{i}\times K.$ (B.21)

Here with

$\mathbf{u}_{k}=[\mathbf{u}_{k}^{1},u_{k}^{2}]=[(u_{k}^{1})^{(1)},(u_{k}^{1})^{(2)}...(u_{k}^{1})^{(d-1)},u_{k}^{2}]\in\mathbf{\widehat{L}}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ (B.22a) $\widehat{\mathbf{L}}^{q}_{\sigma}(\Omega)\equiv\text{
the space obtained from }\mathbf{L}^{q}_{\sigma}(\Omega)\text{ after omitting
the }d\text{-coordinate},$ (B.22b)

we have defined the duality pairing over $\omega$ as

$\displaystyle(\mathbf{u}_{k},\widehat{\Phi}^{*}_{ij})_{\omega}$
$\displaystyle=\begin{pmatrix}\begin{bmatrix}\mathbf{u}^{1}_{k}\\\
u^{2}_{k}\end{bmatrix},\begin{bmatrix}\widehat{\boldsymbol{\varphi}}^{*}_{ij}\\\
\psi^{*}_{ij}\end{bmatrix}\end{pmatrix}_{\omega}=\int_{\omega}[\mathbf{u}^{1}_{k}\cdot\widehat{\boldsymbol{\varphi}}^{*}_{ij}+u^{2}_{k}\psi^{*}_{ij}]d\omega$
$\displaystyle=(\mathbf{u}^{1}_{k},\widehat{\boldsymbol{\varphi}}^{*}_{i1})_{\widehat{\mathbf{L}}^{q}(\omega),\widehat{\mathbf{L}}^{q^{\prime}}(\omega)}+(u^{2}_{k},\psi^{*}_{ij})_{L^{q}(\omega),L^{q^{\prime}}(\omega)}$
(B.23) $\displaystyle=\int_{\omega}\begin{bmatrix}(u_{k}^{1})^{(1)}\\\
(u_{k}^{1})^{(2)}\\\ \vdots\\\ (u_{k}^{1})^{(d-1)}\\\
u_{k}^{2}\end{bmatrix}\cdot\begin{bmatrix}\varphi_{ij}^{*(1)}\\\
\varphi_{ij}^{*(2)}\\\ \vdots\\\ \varphi_{ij}^{*(d-1)}\\\
\psi^{*}_{ij}\end{bmatrix}\,d\omega.$ (B.24)

The controllability Kalman/Hautus algebraic condition of the finite-
dimensional projected $\mathbf{w}_{N}$-equation in (3.6a) is given by (4.10)

$\text{rank }U_{i}=\text{full}=\ell_{i},\quad i=1,\dots,M.$ (B.25)

$M$= number of distinct unstable eigenvalues in (1.33). Thus, given the
(unstable) eigenvalues $\bar{\lambda}_{i}$ of
$\mathbb{A}_{q}^{*}\,,i=1,2,...,M$ we need to show that the corresponding
vectors
$\boldsymbol{\widehat{\Phi}}^{*}_{i1},..\boldsymbol{\widehat{\Phi}}^{*}_{il_{i}}$
(defined in (B.20))

$\bar{\lambda}_{i}$$\widehat{\Phi}^{*}_{i1}$,$\widehat{\Phi}^{*}_{i2},$$\ldots$$\widehat{\Phi}^{*}_{i\ell_{i}}$
(B.26)

are linearly independent in $\widehat{\mathbf{L}}^{q^{\prime}}(\omega)$, where
$\ell_{i}=$ geometric multiplicity of $\lambda_{i}$. Thus, we seek to
establish that

$\sum_{j=1}^{\ell_{i}}\alpha_{j}\widehat{\Phi^{*}}_{ij}\equiv 0\mbox{ in
}\widehat{\mathbf{L}}^{q^{\prime}}(\omega)\times
L^{q^{\prime}}(\omega)\quad\Longrightarrow\quad\alpha_{j}=0,j=1,\ldots,\ell_{i}.$
(B.27)

To this end, define the vector $\Phi=[\boldsymbol{\varphi^{*}},\psi^{*}]$ (we
suppress dependence on $i$) by

$\Phi^{*}=\sum_{j=1}^{\ell_{i}}\alpha_{j}\Phi^{*}_{ij}\quad\mbox{ in
}\mathbf{L}^{q^{\prime}}(\Omega)\times L^{q^{\prime}}(\Omega)$ (B.28)

that is, $\Phi^{*}$ is a linear combination, with the same constants
$\alpha_{j}$ as in (B.27), of the eigenvectors
$\Phi^{*}_{i1},...,\Phi^{*}_{il_{i}}$ in (B.2). Then
$\boldsymbol{\Phi^{*}}=[\boldsymbol{\varphi}^{*},\psi^{*}]$ is itself an
eigenvector of $\mathbb{A}_{q}^{*}$ corresponding to the eigenvalue
$\bar{\lambda}_{i}$. Thus we have

$\mathbb{A}_{q}^{*}\boldsymbol{\Phi}^{*}=\bar{\lambda}_{i}\boldsymbol{\Phi}^{*}\mbox{
in }\widehat{\mathbf{L}}^{q^{\prime}}(\Omega)\times
L^{q^{\prime}}(\Omega);\quad\boldsymbol{\varphi^{*}}\in\mathbf{L}^{q^{\prime}}(\Omega)\,,\psi^{*}\in
L^{q^{\prime}}(\Omega).$ (B.29)

along with

$\widehat{\boldsymbol{\Phi}}^{*}\equiv 0\text{ in }\omega\text{ by
}(\ref{E3-8}).$ (B.30)

Therefore
$\boldsymbol{\boldsymbol{\Phi}^{*}}=[\boldsymbol{\varphi}^{*},\psi^{*}],$
satisfies the eigenvalue problem (B.17a-b-c-d), along with the over-determined
condition in $\omega$, due to (B.30)

$\boldsymbol{\varphi}^{*}=\\{\varphi^{*(1)},\varphi^{*(2)},...,\varphi^{*(d-1)}\\}\equiv
0\text{ in }\omega\,,\qquad\psi^{*}\equiv 0\text{ in }\omega.$ (B.31)

According to Theorem 1.4, then

$\boldsymbol{\Phi^{*}}=[\boldsymbol{\varphi^{*}},\psi]\equiv 0\quad\mbox{ in
}\Omega.$ (B.32)

Indeed, in order to reach conclusion (B.32), one does not need the B.C.
(B.17d) on $\Gamma$, as this is not needed in Theorem 1.4. Going back to
(B.28), we have

$\Phi^{*}=\sum_{j=1}^{\ell_{i}}\alpha_{j}\Phi^{*}_{ij}\equiv 0\mbox{ in
}\Omega;\mbox{ hence, }\alpha_{1}=0,\cdots,\alpha_{\ell_{i}}=0,\quad
i=1,\ldots,M$ (B.33)

since the eigenvectors $\left\\{\Phi^{*}_{ij}\right\\}_{j=1}^{\ell_{i}}$ are
linearly independent in $\mathbf{L}^{q^{\prime}}(\Omega)\times
L^{q^{\prime}}(\Omega)$. Conclusion (B.27) is established. (The fact that
$\bar{\lambda}_{i}$ is unstable, is irrelevant in the above argument.) Due to
the just established property (B.27), we then see that the algebraic
conditions (B.25) can be satisfied by infinitely many choices of the vectors
$\mathbf{u}_{1},\ldots,\mathbf{u}_{K}\in\widehat{\mathbf{L}}^{q^{\prime}}(\Omega)\times
L^{q^{\prime}}(\Omega)$.

Acknowledgment. The authors wish to thank two referees for their thorough
reading of the original version and for their comments that have led to an
improved exposition, with additional references. In particular they have
raised stimulating questions regarding possible generalizations of the
mathematical treatment. Some of their questions are addressed and answered in
Section 2.4 (points 3, 5, 6, 7). The proof in Appendix B that the UCP of
Theorem 1.4 implies the controllability rank condition (4.10) was inserted
upon request of one referee. In addition, the authors would like to thank the
following people for providing references for Theorem 1.1 for $q\neq 2$:
Professors Andrea Giorgini, Giusy Mazzone, Hyunseok Kim.

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