reflexivity of the space, that $\displaystyle\mathbb{A}_{{}_{F,q}}$ generates
a s.c. analytic semigroup on the space
$\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)\equiv\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$. ∎

The next result proves the sought-after uniform stabilization of the localized
feedback $\boldsymbol{w}$-system (5.3) or (5.4); that is, that the s.c.
analytic semigroup $\displaystyle e^{\mathbb{A}_{{}_{F,q}}t}$ on
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega),\ t\geq 0$ or on
$\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)$, as guaranteed by Theorem 6.1,
is uniformly stable.

######  Theorem 6.2.

Under the same setting of Theorem 4.2 concerning in particular the choice of
the vectors $\boldsymbol{q}_{k},\boldsymbol{p}_{k},\boldsymbol{u}_{k},f_{k},$
the $\boldsymbol{w}$-problem (5.3), (5.4) in feedback from is uniformly
stable, with a decay rate $\gamma>0,\ Re\ \lambda_{N+1}<-\gamma<0$ either in
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega)\equiv\boldsymbol{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ or else in
$\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)\equiv\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$.

$\left\lVert\begin{bmatrix}\boldsymbol{w}_{f}\\\
w_{h}\end{bmatrix}\right\rVert_{(\cdot)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{w}_{f}(0)\\\
w_{h}(0)\end{bmatrix}\right\rVert_{(\cdot)},\ t\geq 0.$ (6.9a) In short,
recalling the operator $\displaystyle\mathbb{A}_{{}_{F,q}}$ in (5.3b)
$\left\lVert e^{\mathbb{A}_{{}_{F,q}}t}\right\rVert_{{\mathcal{L}}(\cdot)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t},\ t\geq 0.$ (6.9b)

Here, $(\cdot)$ denotes either the space $\boldsymbol{W}^{q}_{\sigma}(\Omega)$
or the space $\boldsymbol{V}_{b}^{q,p}(\Omega)$, see Remark 1.3.

###### Proof.

The proof is similar to that of [62], making critical use of Theorem 4.2 on
the uniform stabilization (4.55) of the $\boldsymbol{w}_{N}$-system (4.46) in
feedback form with arbitrary decay rate $\gamma_{1}>0$, in particular
$-\gamma_{1}<Re\ \lambda_{N+1}<0$. Next, one examines the impact of the
constructive feedback controls in (4.50), (4.51) and (4.56) on the
$\zeta_{N}$-dynamics (3.6b), whose explicit solution is given by the variation
of parameter formula

$\zeta_{N}(t)=e^{\mathbb{A}_{q,N}^{s}}\zeta_{N}(0)+\int_{0}^{t}e^{\mathbb{A}_{q,N}^{s}(t-\tau)}(I-P_{N})\begin{bmatrix}P_{q}(m\boldsymbol{u}_{N})(\tau)\\\\[2.84526pt]
{\mathcal{B}}_{q}Dv_{N}(\tau)\end{bmatrix}d\tau.$ (6.10)

where

$\left\lVert e^{\mathbb{A}_{q,N}^{s}}\right\rVert_{{\mathcal{L}}(\cdot)}\leq
C_{\gamma_{0}}e^{-\gamma_{0}t},\ 0\geq t,\ 0<\gamma_{0}<\left\lvert Re\
\lambda_{N+1}\right\rvert;$ (6.11)

while $\boldsymbol{u}_{N}(t),\ v_{N}(t)$ decay with an arbitrary large
exponential rate $\gamma_{1}>0$ as in (4.55). The validity of (6.11) on
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega)$ follows by (3.3) and (3.4),
$\displaystyle\mathbb{A}_{q,N}^{s}$ being the generator of a s.c. analytic
semigroup $\displaystyle e^{\mathbb{A}_{q}t}$ (Theorem 1.2), restricted to the
stable subspace $\displaystyle(\boldsymbol{W}^{q}_{\sigma})^{s}_{N}$ in (3.2).
The validity of (6.11) on $\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)$
follows by interpolation as
$\displaystyle{\mathcal{D}}(\mathbb{A}_{q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q})={\mathcal{D}}(A_{q})\times{\mathcal{D}}(B_{q})$,
see (1.34). ∎

Details are in [62, Section 10].

##  7 Maximal $L^{p}$-regularity on
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega)$ of the linearized feedback
operator $\displaystyle\mathbb{A}_{{}_{F,q}}$ in (5.3b) up to $T=\infty$.

Consider the following abstract dynamics

$\zeta_{t}=\mathbb{A}_{{}_{F,q}}\zeta+\chi,\quad\zeta(0)=\zeta_{0},\quad\text{in
}\boldsymbol{W}^{q}_{\sigma}(\Omega)$ (7.1)
$\zeta(t)=e^{\mathbb{A}_{{}_{F,q}}}\zeta_{0}+\int_{0}^{t}e^{\mathbb{A}_{{}_{F,q}}(t-s)}\chi(s)ds$
(7.2)

The main result of the present section is

######  Theorem 7.1.

With reference to the $\boldsymbol{w}$-problem (5.3) in feedback form defining
the feedback operator $\displaystyle\mathbb{A}_{{}_{F,q}}$, assume the setting
of Theorem 4.2 regarding the choice of the vectors
$\boldsymbol{q}_{k},\boldsymbol{p}_{k},\boldsymbol{u}_{k},f_{k},$ in (5.1),
(5.2), so that Theorem 6.2 holds true. Then the operator
$\displaystyle\mathbb{A}_{{}_{F,q}}$ has maximal $L^{p}$-regularity on
$\boldsymbol{W}^{q}_{\sigma}(\Omega)$ up to $T=\infty$;

$\mathbb{A}_{{}_{F,q}}\in
MReg\left(L^{p}\left(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega)\right)\right).$
(7.3)

That is,

$(L\chi)(t)=\int_{0}^{t}e^{\mathbb{A}_{{}_{F,q}}(t-s)}\chi(s)ds$ (7.4)

continuous:

$L^{p}(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega))\longrightarrow
L^{p}\left(0,\infty;{\mathcal{D}}\left(\mathbb{A}_{{}_{F,q}}\right)\right)$
(7.5)

so that continuously from (7.1) with $\zeta_{0}=0$:

$\displaystyle\chi\in
L^{p}\left(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega)\right)$
$\displaystyle\longrightarrow\zeta\in\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$ $\displaystyle\hskip 17.07182pt\in
L^{p}\left(0,\infty;{\mathcal{D}}\left(\mathbb{A}_{{}_{F,q}}\right)\right)\cap
W^{1,p}(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega))$ (7.6a)
$\displaystyle\longrightarrow\boldsymbol{X}^{\infty}_{p,q}\times
X^{\infty}_{p,q}$ $\displaystyle\equiv
L^{p}\left(0,\infty;\boldsymbol{W}^{2,q}(\Omega)\right)\times
L^{p}\left(0,\infty;W^{2,q}(\Omega)\right)$ (7.6b)

###### Proof.

Again, as in the proof of Theorem 6.1 about analyticity of $\displaystyle
e^{\mathbb{A}_{{}_{F,q}}t}$, it suffices to show that the streamlined operator
$\displaystyle\hat{\mathbb{A}}_{{}_{F,q}}$ in (5.4), (5.5) has maximal
$L^{p}$-regularity on $\boldsymbol{W}^{q}_{\sigma}(\Omega)$ up to $T<\infty$.
Equivalently, since $\boldsymbol{W}^{q}_{\sigma}(\Omega)$ is reflexive, that
the adjoint operator $\displaystyle\hat{\mathbb{A}}^{*}_{{}_{F,q}}$ has
maximal $L^{p}$-regularity on
$\displaystyle\left(\boldsymbol{W}^{q}_{\sigma}(\Omega)\right)^{*}=\boldsymbol{W}^{q^{\prime}}_{\sigma}(\Omega)$
up to $T<\infty$. To establish this, we return to estimate (6.6) showing
statement (6.3), with
$\displaystyle\theta_{0}=1-\gamma=1-{}^{1}\\!/_{2q}+\varepsilon<1$. Now we
invoke that $\displaystyle\mathbb{A}_{q}^{*}$, hence
$\hat{\mathbb{A}}^{*}_{q}$ in (6.2) has maximal $L^{p}$-regularity on
$\boldsymbol{W}^{q^{\prime}}_{\sigma}(\Omega)$, (by duality on Theorem
1.2(ii), since $\boldsymbol{W}^{q}_{\sigma}(\Omega)$ is a reflexive space.) We
then apply known perturbation theory [23, Theorem 6.2, p311], or [53, Remark
1i, p 426 for $\beta=1$] to conclude that
$\displaystyle\hat{\mathbb{A}}_{{}_{F,q}}^{*}\in
MReg(L^{p}(0,T;\boldsymbol{W}^{q^{\prime}}_{\sigma}(\Omega))),\ T<\infty$, and
hence $\displaystyle\hat{\mathbb{A}}_{{}_{F,q}}\in
MReg(L^{p}(0,T;\boldsymbol{W}^{q}_{\sigma}(\Omega))),\ T<\infty$. Then we
obtain $\displaystyle\mathbb{A}_{{}_{F,q}}\in
MReg(L^{p}(0,T;\boldsymbol{W}^{q}_{\sigma}(\Omega))),\ T<\infty$. But
$\displaystyle e^{\mathbb{A}_{{}_{F,q}}t}$ is uniformly stable on
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega)$ by Theorem 6.2. Hence we
obtain $\displaystyle\mathbb{A}_{{}_{F,q}}\in
MReg(L^{p}(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega)))$, and Theorem 7.1 is
proved. ∎

We next examine the regularity of the term $\displaystyle
e^{\mathbb{A}_{{}_{F,q}}t}\zeta_{0}$ due to the initial condition $\zeta_{0}$
as in (7.2).

######  Theorem 7.2.

  1. (i)

Let $\displaystyle 1<p<\frac{2q^{\prime}}{2q^{\prime}-1},\ 1<q^{\prime}\leq
2,\ 2\leq q,\ \frac{1}{q}+\frac{1}{q^{\prime}}=1$. Consider the adjoint
operator
$\displaystyle\mathbb{A}_{{}_{F,q}}^{*}=\hat{\mathbb{A}}_{{}_{F,q}}^{*}+\Pi^{*}$
of $\displaystyle\mathbb{A}_{{}_{F,q}}$ in (5.4), where
$\displaystyle{\mathcal{D}}\left(\mathbb{A}_{{}_{F,q}}^{*}\right)={\mathcal{D}}\left(\hat{\mathbb{A}}_{{}_{F,q}}^{*}\right)={\mathcal{D}}\left(A^{*}_{q}\right)\times{\mathcal{D}}\left(B^{*}_{q}\right)$
by (6.1b). The adjoint s.c. analytic semigroup $\displaystyle
e^{\mathbb{A}^{*}_{{}_{F,q}}t}$ on
$\displaystyle\boldsymbol{W}^{q^{\prime}}_{\sigma}(\Omega)$ is uniformly
stable, by duality on (6.9b). Then

$e^{\mathbb{A}^{*}_{{}_{F,q}}t}:\text{ continuous
}\equiv\boldsymbol{V}^{q^{\prime},p}_{b}(\Omega)\equiv\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q^{\prime},p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q^{\prime},p}(\Omega)\\\
=(\boldsymbol{L}^{q^{\prime}}_{\sigma}(\Omega),{\mathcal{D}}(A^{*}_{q}))_{1-\frac{1}{p},p}\times(L^{q^{\prime}}(\Omega),{\mathcal{D}}(B^{*}_{q}))_{1-\frac{1}{p},p}\\\
\longrightarrow\boldsymbol{X}^{\infty}_{p,q^{\prime},\sigma}\times
X^{\infty}_{p,q^{\prime}}\equiv
L^{p}(0,\infty;{\mathcal{D}}(\mathbb{A}^{*}_{{}_{F,q}}))\cap
W^{1,p}(0,\infty;\boldsymbol{W}^{q^{\prime}}_{\sigma}(\Omega))$ (7.10)

  2. (ii)

Consider the original s.c. analytic feedback semigroup $\displaystyle
e^{\mathbb{A}_{{}_{F,q}}t}$ on
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega)$, which is uniformly stable
here by (6.9b). Let $\displaystyle 1<p<\frac{2q}{2q-1},\ 2\leq q$.

$e^{\mathbb{A}_{{}_{F,q}}t}:\text{ continuous
}\equiv\boldsymbol{V}^{q,p}_{b}(\Omega)\equiv\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\\\
=\left(\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\right)_{1-\frac{1}{p},p}\times\left(L^{q}(\Omega),{\mathcal{D}}(B_{q})\right)_{1-\frac{1}{p},p}\\\
\longrightarrow\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}\equiv
L^{p}\left(0,\infty;{\mathcal{D}}\left(\mathbb{A}_{{}_{F,q}}\right)\right)\cap
W^{1,p}\left(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega)\right)$ (7.14)

###### Proof.

The proof is similar to that in [62, Section 11]. ∎

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

In this section we return to the translated non-linear
$\displaystyle\begin{bmatrix}\boldsymbol{z}\\\ h\end{bmatrix}$-dynamics (1.28)
or (1.29) and apply to it the feedback control pair $\\{\boldsymbol{u},v\\}$

$\begin{bmatrix}P_{q}m(\boldsymbol{u})\\\
v\end{bmatrix}=\begin{bmatrix}P_{q}\Bigg{(}m\bigg{(}\displaystyle\sum_{k=1}^{K}\left\langle
P_{N}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix},\boldsymbol{q}_{k}\right\rangle\boldsymbol{u}_{k}\bigg{)}\Bigg{)}\\\\[5.69054pt]
\displaystyle\sum_{k=1}^{K}\left\langle P_{N}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix},\boldsymbol{p}_{k}\right\rangle
f_{k}\end{bmatrix}=\begin{bmatrix}J\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\\\\[8.53581pt] F\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\end{bmatrix}$ (8.1)

as in (2.10) that is, of the same structure as the feedback operators $J$ and
$F$ in (5.1), (5.2) identified on the RHS of the linearized
$\displaystyle\boldsymbol{w}=\begin{bmatrix}\boldsymbol{w}_{f}\\\
w_{h}\end{bmatrix}$-dynamics (5.3). These are the feedback operators which
produced the s.c. analytic, uniformly stable semigroup $\displaystyle
e^{\mathbb{A}_{F,q}t}$ on
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega)=\boldsymbol{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$ or on
$\boldsymbol{V}_{b}^{q,p}(\Omega)\equiv\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, (Theorem 6.1 and Theorem 6.2) possessing
$L^{p}$-maximal regularity on $\boldsymbol{W}^{q}_{\sigma}(\Omega)$ up to
$T=\infty$; (Theorem 7.1). Thus, returning to (1.28) or (1.29), in this
section we consider the following feedback nonlinear problem, see (2.12)

$\frac{d}{dt}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}=\mathbb{A}_{F,q}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}-\begin{bmatrix}{\mathcal{N}}_{q}&0\\\
0&{\mathcal{M}}_{q}[\boldsymbol{z}]\end{bmatrix}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix};\quad\mathbb{A}_{F,q}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}=\begin{bmatrix}{\mathcal{A}}_{q}&-{\mathcal{C}}_{\gamma}\\\
-{\mathcal{C}}_{\theta_{e}}&0\end{bmatrix}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\\\ +\begin{bmatrix}J\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\\\ -{\mathcal{B}}_{q}\left(h-DF\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\right)\end{bmatrix},$ (8.4)

specifically as in (2.11)

$\displaystyle\frac{d\boldsymbol{z}}{dt}-{\mathcal{A}}_{q}\boldsymbol{z}+{\mathcal{C}}_{\gamma}h+{\mathcal{N}}_{q}\boldsymbol{z}$
$\displaystyle=P_{q}\left(m\left(\sum_{k=1}^{K}\left\langle
P_{N}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix},\boldsymbol{q}_{k}\right\rangle\boldsymbol{u}_{k}\right)\right)$
(8.5a)
$\displaystyle\frac{dh}{dt}+{\mathcal{C}}_{\theta_{e}}\boldsymbol{z}+{\mathcal{M}}_{q}[\boldsymbol{z}]h$
$\displaystyle=-{\mathcal{B}}_{q}\left(h-D\left(\displaystyle\sum_{k=1}^{K}\left\langle
P_{N}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix},\boldsymbol{p}_{k}\right\rangle f_{k}\right)\right).$ (8.5b)

The variation of parameter formula for Eq (8.4) is

$\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{z}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{z}]h(\tau)\end{bmatrix}d\tau.$ (8.6)

We already know from (2.7) or (6.9b) that for
$\displaystyle\\{\boldsymbol{z}_{0},h_{0}\\}\in\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\equiv\boldsymbol{V}_{b}^{q,p}(\Omega),\
1<p<\frac{2q}{2q-1}$, we have: there is $M_{\gamma_{0}}$ such that

$\left\lVert e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)},\ t\geq 0$
(8.7)

with $M_{\gamma_{0}}$ possibly depending on $p,q$. Maximal $L^{p}$-regularity
properties corresponding to the solution operator formula (8.6) were
established in Theorem 7.1, Theorem 7.2. Accordingly, for

$\displaystyle\boldsymbol{b}_{0}$
$\displaystyle\equiv\\{\boldsymbol{z}_{0},h_{0}\\}\in\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\equiv\boldsymbol{V}_{b}^{q,p}(\Omega)$
(8.8) $\displaystyle\boldsymbol{f}$
$\displaystyle\equiv\\{\boldsymbol{f}_{1},f_{2}\\}\in\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$ $\displaystyle\hskip 28.45274pt\equiv
L^{p}(0,\infty,{\mathcal{D}}(\mathbb{A}_{F,q}))\cap
W^{1,p}(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega))\text{ (see
\eqref{2.15})}$ (8.9) $\displaystyle{\mathcal{D}}(\mathbb{A}_{F,q})$
$\displaystyle\subset{\mathcal{D}}(A_{q})\times W^{2,q}(\Omega)$
$\displaystyle\hskip
28.45274pt=[\boldsymbol{W}^{2,q}(\Omega)\cap\boldsymbol{W}^{1,q}_{0}(\Omega)\cap\boldsymbol{L}^{q}_{\sigma}(\Omega)]\times
W^{2,q}(\Omega)\text{ (see \eqref{5.5b})}$ (8.10)
$\displaystyle\boldsymbol{W}^{q}_{\sigma}(\Omega)$
$\displaystyle=\boldsymbol{L}^{q}_{\sigma}(\Omega)\times L^{q}(\Omega)\text{
(see \eqref{1.30})}$ $\displaystyle\boldsymbol{X}^{\infty}_{p,q}$
$\displaystyle\equiv
L^{p}\left(0,\infty;\left(\boldsymbol{W}^{2,q}(\Omega)\cap\boldsymbol{W}^{1,q}_{0}(\Omega)\cap\boldsymbol{L}^{q}_{\sigma}(\Omega)\right)\times
W^{2,q}(\Omega)\right)$ $\displaystyle\hskip 170.71652pt\cap
W^{1,p}(0,\infty;\boldsymbol{W}^{q}_{\sigma}(\Omega));$ (8.11a) $\displaystyle
X^{\infty}_{p,q}$ $\displaystyle\equiv L^{p}(0,\infty;W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega))\cap W^{1,q}(0,\infty;L^{q}(\Omega)),$ (8.11b)

as in (8.6), we define the operator

${\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})\equiv
e^{\mathbb{A}_{F,q}t}\boldsymbol{b}_{0}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{f}_{1}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}(\tau)\end{bmatrix}d\tau.$ (8.12)

The main result of this section is Theorem 2.2 restated as

######  Theorem 8.1.

Let $\displaystyle d=2,3,\ q>d,\ 1<p<\frac{2q}{2q-1}$. There exists a positive
constant $r_{1}>0$ (identified in the proof below in (8.33) such that if

$\left\lVert\boldsymbol{b}_{0}\right\rVert_{\boldsymbol{V}^{q,p}(\Omega)}=\left\lVert\\{\boldsymbol{z}_{0},h_{0}\\}\right\rVert_{\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}<r_{1},$ (8.13)

then the operator ${\mathcal{F}}$ in (8.12) has a unique fixed point non-
linear semigroup solution in $\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$, see (2.15)-(2.18), or (8.9)

${\mathcal{F}}\begin{pmatrix}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix},\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\end{pmatrix}=\begin{bmatrix}\boldsymbol{z}\\\ h\end{bmatrix},\
\text{or}\ \begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{z}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{z}]h(\tau)\end{bmatrix}d\tau,$ (8.14)

which therefore is the unique solution of problem (8.4) = (8.5) in
$\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$.

The proof of Theorem 2.2 = Theorem 8.1 is accomplished in two steps.

Step 1:

######  Theorem 8.2.

Let $d=2,3,\ q>d$ and $\displaystyle 1<p<\frac{2q}{2q-1}$. There exists a
positive constant $r_{1}>0$ (identified in the proof below in (8.33)) and a
subsequent constant $r>0$ (identified in the proof below in (8.31)) depending
on $r_{1}>0$ and a constant $C$ in (8.30), such that with
$\displaystyle\left\lVert\boldsymbol{b}_{0}\right\rVert_{\boldsymbol{V}^{q,p}(\Omega)}<r_{1}$
as in (8.13), the operator
$\displaystyle{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})$ in (8.12) maps
a ball $B(0,r)$ in $\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$ into itself. ∎

Theorem 8.1 will follow then from Theorem 8.2 after establishing that

Step 2:

######  Theorem 8.3.

Let $d=2,3,\ q>d$ and $\displaystyle 1<p<\frac{2q}{2q-1}$. There exists a
positive constant $r_{1}>0$ such that if
$\displaystyle\left\lVert\boldsymbol{b}_{0}\right\rVert_{\boldsymbol{V}^{q,p}(\Omega)}<r_{1}$
as in (8.13), there exists a constant $0<\rho_{0}<1$ (identified in (8.64)),
such that the operator ${\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})$ in
(8.12) defines a contraction in the ball $B(0,\rho_{0})$ of
$\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$. ∎

The Banach contraction principle then establishes Theorem 8.1, once we prove
Theorems 8.2 and 8.3. These are proved below.

Proof of Theorem 8.2

Step 1: We start from the definition (8.12) of
$\displaystyle{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})$ and invoke the
maximal regularity properties (7.14) of Theorem 7.2 for $\displaystyle
e^{\mathbb{A}_{F,q}t}$ and (7.6) = (2.15) of Theorem 7.1 for the integral term
in (8.12). We then obtain from (8.12)

$\displaystyle\left\lVert{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq\left\lVert
e^{\mathbb{A}_{F,q}t}\boldsymbol{b}_{0}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$
$\displaystyle+\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{f}_{1}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}(\tau)\end{bmatrix}d\tau\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (8.15) $\displaystyle\leq
C\bigg{[}\left\lVert\boldsymbol{b}_{0}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+$
$\displaystyle\left\lVert{\mathcal{N}}_{q}\boldsymbol{f}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega)\big{)}}+\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\bigg{]}.$
(8.16)

Step 2: Regarding the term $\displaystyle{\mathcal{N}}_{q}\boldsymbol{f}_{1}$
we can invoke [60, Eq (8.19)] to obtain

$\left\lVert{\mathcal{N}}_{q}\boldsymbol{f}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega)\big{)}}\leq
C\left\lVert\boldsymbol{f}_{1}\right\rVert^{2}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}},\
\boldsymbol{f}_{1}\in\boldsymbol{X}^{\infty}_{p,q,\sigma}.$ (8.17)

Regarding the term $\displaystyle{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}$,
we can trace the proof in [60, from (8.10) $\longrightarrow$ (8.18)] (which
yielded estimate (8.16)). For the sake of clarity, we shall reproduce the
computations in the present case with
$\displaystyle{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}=\boldsymbol{f}_{1}\cdot\nabla
f_{2}$, see (1.20), mutatis mutandis. We shall obtain

$\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\leq
C\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}},\
\boldsymbol{f}_{1}\in\boldsymbol{X}^{\infty}_{p,q,\sigma},f_{2}\in
X^{\infty}_{p,q}.$ (8.18)

In fact, let us compute from (1.20)

$\displaystyle\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}^{p}$
$\displaystyle\leq\int_{0}^{\infty}\left\lVert\boldsymbol{f}_{1}\cdot\nabla
f_{2}\right\rVert^{p}_{L^{q}(\Omega)}dt$ (8.19)
$\displaystyle\leq\int_{0}^{\infty}\bigg{\\{}\int_{\Omega}\left\lvert\boldsymbol{f}_{1}(t,x)\right\rvert^{q}\left\lvert\nabla
f_{2}(t,x)\right\rvert^{q}d\Omega\bigg{\\}}^{{}^{p}\\!/_{q}}dt$ (8.20)
$\displaystyle\leq\int_{0}^{\infty}\bigg{\\{}\bigg{[}\sup_{\Omega}$
$\displaystyle\left\lvert\nabla
f_{2}(t,x)\right\rvert^{q}\bigg{]}^{{}^{1}\\!/_{q}}\bigg{[}\int_{\Omega}\left\lvert\boldsymbol{f}_{1}(t,x)\right\rvert^{q}d\Omega\bigg{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$
(8.21) $\displaystyle\leq\int_{0}^{\infty}\left\lVert\nabla
f_{2}(t,\cdot)\right\rVert^{p}_{\boldsymbol{L}^{\infty}(\Omega)}\left\lVert\boldsymbol{f}_{1}(t,\cdot)\right\rVert^{p}_{\boldsymbol{L}^{q}_{\sigma}(\Omega)}dt$
(8.22) $\displaystyle\leq\sup_{0\leq t\leq\infty}$
$\displaystyle\left\lVert\boldsymbol{f}_{1}(t,\cdot)\right\rVert^{p}_{\boldsymbol{L}_{\sigma}^{q}(\Omega)}\int_{0}^{\infty}\left\lVert\nabla
f_{2}(t,\cdot)\right\rVert^{p}_{\boldsymbol{L}^{\infty}(\Omega)}dt$ (8.23)
$\displaystyle=\left\lVert\boldsymbol{f}_{1}\right\rVert^{p}_{L^{\infty}(0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega))}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;\boldsymbol{L}^{\infty}(\Omega))}.$
(8.24)

See also [60, Eq (8.14)]. The following embeddings hold true (see the stronger
Eq (2.16)):

  1. (i)

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

$\displaystyle\boldsymbol{f}_{1}\in\boldsymbol{X}^{\infty}_{p,q,\sigma}\hookrightarrow\boldsymbol{f}_{1}$
$\displaystyle\in L^{\infty}(0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega))$
(8.25a) $\displaystyle\text{ so that,
}\left\lVert\boldsymbol{f}_{1}\right\rVert_{L^{\infty}(0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega))}$
$\displaystyle\leq
C\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}};$
(8.25b)

  2. (ii)

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

$W^{1,q}(\Omega)\subset L^{\infty}(\Omega)\text{ for q}>\text{dim }\Omega=d,\
d=2,3,$ (8.26)

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

$\displaystyle\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;\boldsymbol{L}^{\infty}(\Omega))}$
$\displaystyle\leq C\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;\boldsymbol{W}^{1,q}(\Omega))}\leq
C\left\lVert f_{2}\right\rVert^{p}_{L^{p}(0,\infty;W^{2,q}(\Omega))}$ (8.27)
$\displaystyle\leq C\left\lVert f_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}.$
(8.28)

In going from (8.27) to (8.28) we have recalled the definition of $f_{2}\in
X^{\infty}_{p,q}$ in (8.11b) or (2.15), as $f_{2}$ was taken at the outset in
$\displaystyle{\mathcal{D}}({\mathcal{B}}_{q})\subset L^{q}(\Omega)$. Then the
sought-after final estimate (8.18) of the nonlinear term
$\displaystyle{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}$ is obtained from
substituting (8.25b) and (8.28) into the RHS of (8.24).

Step 3: Substituting estimates (8.17) and (8.18) on the RHS of (8.16), we
finally obtain

$\left\lVert{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\leq
C\Big{\\{}\left\lVert\boldsymbol{b}_{0}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\big{(}\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}\big{)}\Big{\\}}.$ (8.29)

See [60, Eqt (8.20)].

Step 4: We now impose restrictions on the data on the RHS of (8.29):
$\boldsymbol{b}_{0}$ is in a ball of radius $r_{1}>0$ in
$\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)=\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ and
$\boldsymbol{f}=\\{\boldsymbol{f}_{1},f_{2}\\}$ lies in a ball of radius $r>0$
in $\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$.
We further demand that the final result
$\displaystyle{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})$ shall lie in a
ball of radius $r>0$ in
$\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$.
Thus, we obtain from (8.29)

$\displaystyle\left\lVert{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
C\Big{\\{}\left\lVert\boldsymbol{b}_{0}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\big{(}\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}\big{)}\Big{\\}}$ $\displaystyle\leq
C(r_{1}+r\cdot r)\leq r.$ (8.30)

This implies

$Cr^{2}-r+Cr_{1}\leq 0\quad\text{or}\quad\frac{1-\sqrt{1-4C^{2}r_{1}}}{2C}\leq
r\leq\frac{1+\sqrt{1-4C^{2}r_{1}}}{2C}$ (8.31)

whereby

$\begin{Bmatrix}\text{ range of values of r
}\end{Bmatrix}\longrightarrow\text{ interval
}\Big{[}0,\frac{1}{C}\Big{]},\text{ as }r_{1}\searrow 0,$ (8.32)

a constraint which is guaranteed by taking

$r_{1}\leq\frac{1}{4C^{2}},\ C\text{ being the constant in }(\ref{8.28})\
\left(\text{w.l.o.g. }C>\frac{1}{4}\right).$ (8.33)

We have thus established that by taking $r_{1}$ as in (8.33) and subsequently
$r$ as in (8.31), then the map

${\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})\text{ takes:
}\begin{Bmatrix}\text{ ball in }\boldsymbol{V}_{b}^{q,p}(\Omega)\\\ \text{of
radius }r_{1}\end{Bmatrix}\times\begin{Bmatrix}\text{ ball in
}\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}\\\ \text{of
radius }r\end{Bmatrix}\text{ into }\\\ \begin{Bmatrix}\text{ ball in
}\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}\\\ \text{of
radius }r\end{Bmatrix},\ d<q,\ 1<p<\frac{2q}{2q-1}.$ (8.36)

This establishes Theorem 8.2. ∎

Proof of Theorem 8.3

Step 1: For
$\boldsymbol{f}=\\{\boldsymbol{f}_{1},f_{2}\\},\boldsymbol{g}=\\{\boldsymbol{g}_{1},g_{2}\\}$
both in the ball of $\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}$ of radius $r$ obtained in the proof of Theorem 8.2, we
estimate from (8.12):

$\displaystyle\left\lVert{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})-{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{g})\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}=$ $\displaystyle\hskip
71.13188pt\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{f}_{1}(\tau)-{\mathcal{N}}_{q}\boldsymbol{g}_{1}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}(\tau)-{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}(\tau)\end{bmatrix}d\tau\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$
$\displaystyle\leq\widetilde{m}\Big{[}\left\lVert{\mathcal{N}}_{q}\boldsymbol{f}_{1}-{\mathcal{N}}_{q}\boldsymbol{g}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega)\big{)}}+\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}\right\rVert_{L^{p}(0,\infty;L^{q}(\Omega))}\Big{]}$
(8.37)

after invoking the maximal regularity property (7.6) of Theorem 7.1, as in
(8.16).

Step 2: As to the first term of the RHS of (8.37), we can invoke [60, Eq
(8.41)] and obtain

$\left\lVert{\mathcal{N}}_{q}\boldsymbol{f}_{1}-{\mathcal{N}}_{q}\boldsymbol{g}_{1}\right\rVert_{L^{p}\big{(}0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega)\big{)}}\leq
2^{{}^{1}\\!/_{p}}C^{{}^{1}\\!/_{p}}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left(\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\right.\\\
\left.+\left\lVert\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\right).$
(8.40)

Regarding the second term on the RHS of (8.37) involving ${\mathcal{M}}_{q}$,
we can track the proof of [60, from (8.28) to (8.41)] (which yielded estimate
(8.40)) mutatis mutandis. For the sake of clarity, we shall reproduce the
computations in the present case, recalling from (1.22) that
$\displaystyle{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}=\boldsymbol{f}_{1}\cdot\nabla
f_{2},{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}=\boldsymbol{g}_{1}\cdot\nabla
g_{2}$. We shall obtain

$\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}\right\rVert^{p}_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\leq
C\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\\\
+\left\lVert\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\Big{\\}}.$ (8.43)

In fact, adding and subtracting

$\displaystyle{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}$
$\displaystyle=\boldsymbol{f}_{1}\cdot\nabla
f_{2}-\boldsymbol{g}_{1}\cdot\nabla g_{2}$
$\displaystyle=\boldsymbol{f}_{1}\cdot\nabla
f_{2}-\boldsymbol{g}_{1}\cdot\nabla f_{2}+\boldsymbol{g}_{1}\cdot\nabla
f_{2}-\boldsymbol{g}_{1}\cdot\nabla g_{2}$
$\displaystyle=(\boldsymbol{f}_{1}-\boldsymbol{g}_{1})\cdot\nabla
f_{2}+\boldsymbol{g}_{1}\cdot\nabla(f_{2}-g_{2})=A+B.$ (8.44)

Thus, using $(*):\ \left\lvert A+B\right\rvert^{p}\leq 2^{p}\big{[}\left\lvert
A\right\rvert^{p}+\left\lvert B\right\rvert^{p}\big{]}$ [82, p. 12], we
estimate

$\displaystyle\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}^{p}$
$\displaystyle=\int_{0}^{\infty}\bigg{\\{}\bigg{[}\int_{\Omega}\left\lvert\boldsymbol{f}_{1}\cdot\nabla
f_{2}-\boldsymbol{g}_{1}\cdot\nabla
g_{2}\right\rvert^{q}d\Omega\bigg{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$ (8.45)
(by (8.44)) $\displaystyle=\int_{0}^{\infty}\bigg{[}\int_{\Omega}\left\lvert
A+B\right\rvert^{q}d\Omega\bigg{]}^{{}^{p}\\!/_{q}}dt$ (8.46)
$\displaystyle\leq
2^{p}\int_{0}^{\infty}\bigg{\\{}\int_{\Omega}\big{[}\left\lvert
A\right\rvert^{q}+\left\lvert
B\right\rvert^{q}\big{]}d\Omega\bigg{\\}}^{{}^{p}\\!/_{q}}dt$ (8.47)
$\displaystyle=2^{p}\int_{0}^{\infty}\bigg{\\{}\Big{[}\int_{\Omega}\left\lvert
A\right\rvert^{q}d\Omega+\int_{\Omega}\left\lvert
B\right\rvert^{q}d\Omega\Big{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$ (8.48)
$\displaystyle=2^{p}\int_{0}^{\infty}\bigg{\\{}\Big{[}\left\lVert
A\right\rVert^{q}_{L^{q}(\Omega)}+\left\lVert
B\right\rVert^{q}_{L^{q}(\Omega)}\Big{]}^{{}^{1}\\!/_{q}}\bigg{\\}}^{p}dt$
(8.49) $\displaystyle\left(\text{by }(*)\mbox{ with
}p\to\frac{1}{q}\right)\qquad$ $\displaystyle\leq 2^{p}\cdot
2^{{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{\\{}\left\lVert
A\right\rVert_{L^{q}(\Omega)}+\left\lVert
B\right\rVert_{L^{q}(\Omega)}\Big{\\}}^{p}dt$ (8.50) $\displaystyle(\text{by
}(*))\qquad$ $\displaystyle\leq
2^{p+p+{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{[}\left\lVert
A\right\rVert^{p}_{L^{q}(\Omega)}+\left\lVert
B\right\rVert^{p}_{L^{q}(\Omega)}\Big{]}dt$ (8.51)
$\displaystyle=2^{p+p+{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{[}\left\lVert(\boldsymbol{f}_{1}-\boldsymbol{g}_{1})\cdot\nabla
f_{2}\right\rVert^{p}_{L^{q}(\Omega)}$ $\displaystyle\hskip
28.45274pt+\left\lVert\boldsymbol{g}_{1}\cdot\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{q}(\Omega)}\Big{]}dt$
(8.52) $\displaystyle\leq
2^{p+p+{}^{p}\\!/_{q}}\int_{0}^{\infty}\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{L}^{q}_{\sigma}(\Omega)}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{q}(\Omega)}$ $\displaystyle\hskip
28.45274pt+\left\lVert\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{L}^{q}_{\sigma}(\Omega)}\left\lVert\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{q}(\Omega)}\Big{\\}}dt,$
(8.53)

recalling the definitions of $A$ and $B$ in (8.44) in passing from (8.51) to
(8.52). [60, This is the counterpart Eq (8.36)]. Next we proceed by majorizing
the $L^{q}(\Omega)$-norm for $\nabla f_{2}$ and $\nabla(f_{2}-g_{2})$ by the
$L^{\infty}(\Omega)$-norm. We obtain

$\displaystyle\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}^{p}$
$\displaystyle\leq\text{RHS of (\ref{8.45})}$ $\displaystyle\leq
c2^{p+p+{}^{p}\\!/_{q}}\bigg{\\{}\sup_{0\leq
t\leq\infty}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{L}^{q}_{\sigma}(\Omega)}\int_{0}^{\infty}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{\infty}(\Omega)}$ $\displaystyle+\sup_{0\leq
t\leq\infty}\left\lVert\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{L}^{q}_{\sigma}(\Omega)}\int_{0}^{\infty}\left\lVert\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{\infty}(\Omega)}dt\bigg{\\}}$
(8.54) $\displaystyle\leq
c2^{p+p+{}^{p}\\!/_{q}}\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{L^{\infty}(0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega))}\left\lVert\nabla
f_{2}\right\rVert^{p}_{L^{p}(0,\infty;L^{\infty}(\Omega))}$
$\displaystyle+\left\lVert\boldsymbol{g}_{1}\right\rVert^{p}_{L^{\infty}(0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega))}\left\lVert\nabla(f_{2}-g_{2})\right\rVert^{p}_{L^{p}(0,\infty;L^{\infty}(\Omega))}\Big{\\}}$
(8.55) by (8.25b) and (8.28) $\displaystyle\leq
C\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}^{p}\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}^{p}+\left\lVert\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}^{p}\left\lVert
f_{2}-g_{2}\right\rVert_{X^{\infty}_{p,q}}^{p}\Big{\\}},$ (8.56)

counterpart of [60, Eq (8.39)].

Step 3: We substitute estimate (8.40) and estimate (8.56) on the RHS of (8.37)
and obtain via $(*)$

$\displaystyle\left\lVert{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})-{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{g})\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
2^{p}\widetilde{m}^{p}\Big{\\{}\left\lVert{\mathcal{N}}_{q}\boldsymbol{f}_{1}-{\mathcal{N}}_{q}\boldsymbol{g}_{1}\right\rVert^{p}_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}$
$\displaystyle+\left\lVert{\mathcal{M}}_{q}[\boldsymbol{f}_{1}]f_{2}-{\mathcal{M}}_{q}[\boldsymbol{g}_{1}]g_{2}\right\rVert^{p}_{L^{p}(0,\infty;L^{q}(\Omega))}\Big{\\}}$
(8.57) $\displaystyle\leq
C_{p}\widetilde{m}^{p}\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\Big{(}\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\Big{)}^{p}$
$\displaystyle+\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}+\left\lVert\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\Big{\\}}.$ (8.58)

This is the counterpart of [60, Eq (8.42)].

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

$\displaystyle\left\lVert\boldsymbol{f}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$
$\displaystyle=\left\lVert\boldsymbol{f}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}\right\rVert_{X^{\infty}_{p,q}}<R$ (8.59)
$\displaystyle\left\lVert\boldsymbol{g}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$
$\displaystyle=\left\lVert\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
g_{2}\right\rVert_{X^{\infty}_{p,q}}<R.$ (8.60)

Then (8.59), (8.60) used (8.58) implies

$\displaystyle\left\lVert{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})-{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{g})\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
C_{p}\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\big{[}(2R)^{p}+R^{p}\big{]}$
$\displaystyle\hskip 56.9055pt+\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}R^{p}\Big{\\}}$ (8.61)
$\displaystyle\leq
C_{p}\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\big{[}(2^{p}+1)R^{p}\big{]}$
$\displaystyle\hskip 56.9055pt+\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}R^{p}\Big{\\}}$
$\displaystyle\big{(}K_{p}=C_{p}(2^{p}+1)\big{)}\quad$ $\displaystyle\leq
K_{p}R^{p}\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert^{p}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}-g_{2}\right\rVert^{p}_{X^{\infty}_{p,q}}\Big{\\}}$
$\displaystyle[a>0,b>0,\ a^{p}+b^{p}\leq(a+b)^{p}]\quad$ $\displaystyle\leq
K_{p}R^{p}\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}-g_{2}\right\rVert_{X^{\infty}_{p,q}}\Big{\\}}^{p}$ (8.62)

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

$\displaystyle\left\lVert{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{f})-{\mathcal{F}}(\boldsymbol{b}_{0},\boldsymbol{g})\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq
K_{p}^{{}^{1}\\!/_{p}}R\Big{\\{}\left\lVert\boldsymbol{f}_{1}-\boldsymbol{g}_{1}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
f_{2}-g_{2}\right\rVert_{X^{\infty}_{p,q}}\Big{\\}}$
$\displaystyle=\rho_{0}\left\lVert\boldsymbol{f}-\boldsymbol{g}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (8.63)

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

$\rho_{0}=K_{p}^{{}^{1}\\!/_{p}}R<1\quad\text{or}\quad
R<\frac{1}{K_{p}^{{}^{1}\\!/_{p}}}.$ (8.64)

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

${\mathcal{F}}\begin{pmatrix}\boldsymbol{b}_{0},\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\end{pmatrix}=\begin{bmatrix}\boldsymbol{z}\\\ h\end{bmatrix},\
\text{or}\ \begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{z}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{z}]h(\tau)\end{bmatrix}d\tau,$ (8.65)

and such point $\displaystyle\begin{bmatrix}\boldsymbol{z}\\\ h\end{bmatrix}$
is the unique solution of the translated non-linear system (8.4), or (8.5),
with finite dimensional control

$\begin{bmatrix}P_{q}m(\boldsymbol{u})\\\\[5.69054pt]
v\end{bmatrix}=\begin{bmatrix}\displaystyle
P_{q}\Bigg{(}m\bigg{(}\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix},\boldsymbol{p}_{k}\bigg{)}\boldsymbol{u}_{k}\bigg{)}\Bigg{)}\\\\[8.53581pt]
\displaystyle\sum_{k=1}^{K}\bigg{(}P_{N}\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix},\boldsymbol{p}_{k}\bigg{)}f_{k}\end{bmatrix}$

in feedback form, as described by Eq (8.1). Theorem 8.3 and hence Theorem 8.1
are proved. ∎

######  Remark 8.1.

Recall from (2.1) and the statement of Theorem 2.1, that
$\displaystyle\boldsymbol{u}_{k}\in(\widehat{\boldsymbol{W}}^{q}_{\sigma})^{u}_{N}\subset\widehat{\boldsymbol{L}}^{q}_{\sigma}(\Omega)$
as established in Appendix C. This means that the feedback control acting on
the fluid variable $\boldsymbol{u}$ is of reduced dimension $(d-1)$ to include
the $d$th component. ∎

##  9 Proof of Theorem 2.3: local exponential decay of the non-linear
$[\boldsymbol{z},h]$ translated dynamics (2.11) = (2.12) with finite
dimensional localized feedback controls and $(d-1)$ dimensional
$\boldsymbol{u}_{K}$.

In this section we return to the feedback problem (2.12), rewritten
equivalently as in (2.14)

$\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}(t)=e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}-\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{z}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{z}]h(\tau)\end{bmatrix}d\tau,$ (9.1)

For $\boldsymbol{b}_{0}=[\boldsymbol{z}_{0},h_{0}]$ in a small ball of
$\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)=\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, Theorem 8.1(= Theorem 2.2) provides a
unique solution $\\{\boldsymbol{z},h\\}$ in a small ball of
$\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$. We
recall from (6.9) of Theorem 6.2

$\left\lVert e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)},t\geq 0$
(9.2)

$\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)=\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$. Our goal is to show that for
$\displaystyle[\boldsymbol{z}_{0},h_{0}]$ in a small ball of
$\displaystyle\boldsymbol{V}_{b}^{q,p}(\Omega)$, problem (9.1) satisfies the
exponential decay

$\left\lVert\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}(t)\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq
C_{a}e^{-at}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)},t\geq 0,\\\
\text{for some constants }a>0,\ C=C_{a}\geq 1.$ (9.5)

Step 1: Starting from (9.1) and using the decay (9.2), we estimate

$\displaystyle\left\lVert\begin{bmatrix}\boldsymbol{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
$\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+$
$\displaystyle\sup_{0\leq t\leq\infty}$
$\displaystyle\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{z}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\boldsymbol{z}]h(\tau)\end{bmatrix}d\tau\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
(9.6) $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
$\displaystyle+C_{1}\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{z}(\tau)\\\\[5.69054pt]
{\mathcal{M}}_{q}[\boldsymbol{z}]h(\tau)\end{bmatrix}d\tau\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (9.7) $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
$\displaystyle+C_{2}\bigg{[}\left\lVert{\mathcal{N}}_{q}\boldsymbol{z}\right\rVert_{L^{p}\big{(}0,\infty;\boldsymbol{L}^{q}_{\sigma}(\Omega)\big{)}}+$
$\displaystyle\hskip
56.9055pt\left\lVert{\mathcal{M}}_{q}[\boldsymbol{z}]h\right\rVert_{L^{p}\big{(}0,\infty;L^{q}(\Omega)\big{)}}\bigg{]}$
(9.8) $\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
$\displaystyle+C_{3}\bigg{[}\left\lVert\boldsymbol{z}\right\rVert^{2}_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert
h\right\rVert_{X^{\infty}_{p,q}}\bigg{]}.$ (9.9)

In going from (9.6) to (9.7) we have recalled the embedding
$\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}\hookrightarrow
L^{\infty}\left(0,\infty;\boldsymbol{V}_{b}^{q,p}(\Omega)\right)$ from
(2.16)-(2.18). Next, in going from (9.7) to (9.8) we have used the maximal
regularity property (7.6) of Theorem 7.1. Finally, to go from (9.8) to (9.9),
we have invoked (8.17) for $\displaystyle{\mathcal{N}}_{q}\boldsymbol{z}$ and
(8.18) for $\displaystyle{\mathcal{M}}_{q}[\boldsymbol{z}]h$. Thus, the
conclusion of Step 1 is

$\displaystyle\left\lVert\begin{bmatrix}\boldsymbol{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
$\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+C_{3}\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\bigg{[}\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}+\left\lVert
h\right\rVert_{X^{\infty}_{p,q}}\bigg{]}$ (9.10)
$\displaystyle=M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+C_{3}\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert[\boldsymbol{z},h]\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}.$ (9.11)

Step 2: We now return to (9.1) and take the
$\displaystyle\boldsymbol{X}^{\infty}_{p,q,\sigma}\times X^{\infty}_{p,q}$
norm across:

$\displaystyle\left\lVert\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}(t)\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ $\displaystyle\leq\left\lVert
e^{\mathbb{A}_{F,q}t}\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}+$ $\displaystyle\hskip
42.67912pt\left\lVert\int_{0}^{t}e^{\mathbb{A}_{F,q}(t-\tau)}\begin{bmatrix}{\mathcal{N}}_{q}\boldsymbol{z}(\tau)\\\\[2.84526pt]
{\mathcal{M}}_{q}[\boldsymbol{z}]h(\tau)\end{bmatrix}d\tau\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (9.12) $\displaystyle(\text{by \eqref{7.8}}\ $
$\displaystyle\leq M_{1}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+C_{3}\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert[\boldsymbol{z},h]\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}$ (9.13)

by invoking (7.14) of Theorem 7.2 on the first semigroup term in (9.12) and
the estimate from (9.7) to (9.11) on the second integral term in (9.12). Thus
(9.13) is established and implies

$\left\lVert\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\Big{[}1-C_{3}\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\Big{]}\leq
M_{1}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}.$ (9.14)

[This is the counterpart of [60, Eq (9.7)]].

Step 3: The well-posedness Theorem 8.1 says that

$\begin{Bmatrix}\text{ If }\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq
r_{1}\\\\[11.38109pt] \text{for }r_{1}\text{ sufficiently
small}\end{Bmatrix}\implies\begin{Bmatrix}\text{ The solution
}[\boldsymbol{z},h]\text{ satisfies}\\\\[8.53581pt]
\left\lVert\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\leq r\end{Bmatrix}$ (9.15)

where the constant $r$ satisfies the constraint (8.31) in terms of the
constant $r_{1}$ and some constant $C$ in (8.29) or (8.30) or (8.17) or
(8.18). We seek to guarantee that we have

$\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\leq\frac{1}{2C_{3}}<\frac{1}{2C},\mbox{
hence
}\frac{1}{2}\leq\Big{[}1-C_{3}\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\Big{]},$
(9.16)

where we can always take the constant $C_{3}$ in (9.16) greater than the
constant $C$ in (8.31), (8.32), (8.33).Then (9.16) can be achieved by choosing
$r_{1}>0$ sufficiently small. In fact, as $r_{1}\searrow 0$. Eq. (8.32) 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, (9.16) can be achieved as
$r_{\min}\searrow 0$: $0<r_{\min}<r<\frac{1}{2C}$. Next (9.16) implies that
(9.14) becomes

$\left\lVert\begin{bmatrix}\boldsymbol{z}\\\
h\end{bmatrix}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}\times
X^{\infty}_{p,q}}\leq 2M_{1}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq
2M_{1}r_{1}.$ (9.17)

by (9.15). Substituting (9.17) in estimate (9.11) then yields

$\displaystyle\left\lVert\begin{bmatrix}\boldsymbol{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
$\displaystyle\leq
M_{\gamma_{0}}e^{-\gamma_{0}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}+2M_{1}C_{3}\left\lVert\boldsymbol{z}\right\rVert_{\boldsymbol{X}^{\infty}_{p,q,\sigma}}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$
$\displaystyle\leq\widehat{M}\Big{[}e^{-\gamma_{0}t}+4\widehat{M}C_{3}r_{1}\Big{]}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$ (9.18)

again by (9.17) with
$\displaystyle\widehat{M}=\max\\{M_{\gamma_{0}},M_{1}\\}$. This is the
counterpart of [60, Eq (9.16)].

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

$\beta=\widehat{M}\left[e^{-\gamma_{0}T}+4\widehat{M}C_{3}r_{1}\right]<1.$
(9.19)

Then (9.18) implies

$\left\lVert\begin{bmatrix}\boldsymbol{z}(T)\\\
h(T)\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq\beta\left\lVert\begin{bmatrix}{\boldsymbol{z}}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$ (9.20)

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

$\left\lVert\begin{bmatrix}\boldsymbol{z}(nT)\\\
h(nT)\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq\beta\left\lVert\begin{bmatrix}\boldsymbol{z}((n-1)T)\\\
h((n-1)T)\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq\beta^{n}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}.$ (9.21)

Since $\beta<1$, the semigroup property of the evolution implies [6] that
there are constants $M_{\widetilde{\gamma}}\geq 1,\widetilde{\gamma}>0$ such
that

$\left\lVert\begin{bmatrix}\boldsymbol{z}(t)\\\
h(t)\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq
M_{\widetilde{\gamma}}e^{-\widetilde{\gamma}t}\left\lVert\begin{bmatrix}\boldsymbol{z}_{0}\\\
h_{0}\end{bmatrix}\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}$ (9.22)

with
$\displaystyle\left\lVert[\boldsymbol{z}_{0},h_{0}]\right\rVert_{\boldsymbol{V}_{b}^{q,p}(\Omega)}\leq
r_{1}=$ small. This proves (2.18), i.e Theorem 2.3. ∎

######  Remark 9.1.

The above computations - (9.19) through (9.21) - can be used to support
qualitatively the intuitive expectation that “the larger the decay rate
$\gamma_{0}$ in (6.9) = (9.2) of the linearized feedback
$\boldsymbol{w}$-dynamics (5.4), the larger the decay rate
$\widetilde{\gamma}$ in (9.22) of the nonlinear feedback
$\\{\boldsymbol{z},h\\}$-dynamics (9.1) or (2.13); hence the larger the rate
$\widetilde{\gamma}$ in (9.22) of the original
$\\{\boldsymbol{y},\theta\\}$-dynamics in (1.1)”

The following considerations are somewhat qualitative. Let $S(t)$ denote the
non-linear semigroup in the space
$\boldsymbol{V}_{b}^{q,p}(\Omega)\equiv\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\times
B^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, with infinitesimal generator
$\displaystyle\big{[}\mathbb{A}_{{}_{F,q}}-{\mathcal{N}}_{q}\big{]}$
describing the feedback $\\{\boldsymbol{z},h\\}$-dynamics, (9.1) or (2.13) as
guaranteed by the well posedness Theorem 2.2 = Theorem 8.1. Thus,
$\begin{bmatrix}\boldsymbol{z}(t)\\\
h(t)\end{bmatrix}=S(t)\begin{bmatrix}\boldsymbol{z}_{0}\\\ h_{0}\end{bmatrix}$
on $\boldsymbol{V}_{b}^{q,p}(\Omega)$. By (9.19), we can rewrite (9.20) as:

$\left\lVert
S(T)\right\rVert_{{\mathcal{L}}\big{(}\boldsymbol{V}_{b}^{q,p}(\Omega)\big{)}}\leq\beta<1.$
(9.23)

It follows from [6, p 178] via the semigroup property that

$-\widetilde{\gamma}\ \ \text{is just below}\ \ \frac{\ln\beta}{T}<0.$ (9.24)

Pick $r_{1}>0$ in (9.19) 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

$\frac{\ln\beta}{T}\text{ is just above
}\Big{[}(-\gamma_{0})+\frac{\ln\widehat{M}}{T}\Big{]}.$ (9.25)

Hence, by (9.24), (9.25),

$\widetilde{\gamma}\sim\gamma_{0}-\frac{\ln\widehat{M}}{T}$ (9.26)

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

##  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 $\boldsymbol{B}^{s}_{q,p}$ on domains of class
$C^{1}$ as real interpolation of Sobolev spaces, Eqts (1.7)-(1.11) and Remark
1.2.

  1. (a)

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

######  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

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

generates a s.c. analytic semigroup $e^{-A_{q}t}$ on
$\boldsymbol{L}^{q}_{\sigma}(\Omega)$. See [33] and the review paper [37,
Theorem 2.8.5 p 17].

  2. (ii)

The Oseen operator ${\mathcal{A}}_{q}$ in (1.17)

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

generates a s.c. analytic semigroup $e^{{\mathcal{A}}_{q}t}$ on
$\boldsymbol{L}^{q}_{\sigma}(\Omega)$. This follows as $A_{o,q}$ is relatively
bounded with respect to $A^{{}^{1}\\!/_{2}}_{q}$, see (1.15): thus a standard
theorem on perturbation of an analytic semigroup generator applies [68,
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}$
$\displaystyle:\boldsymbol{L}^{q}_{\sigma}(\Omega)\longrightarrow\boldsymbol{L}^{q}_{\sigma}(\Omega)\text{
is compact}.$ (A.3b)

  4. (iv)

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

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

* (b)

Domains of fractional powers, ${\mathcal{D}}(A_{q}^{\alpha}),0<\alpha<1$ of
the Stokes operator $A_{q}$ on
$\boldsymbol{L}^{q}_{\sigma}(\Omega),1<q<\infty$. We elaborate on (1.16)

######  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 [34]
and [37, Theorem 2.8.5, p 18]

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

in particular

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

* (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)

$\displaystyle\Big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}\boldsymbol{g}\in\boldsymbol{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{
div }\boldsymbol{g}=0,\ \boldsymbol{g}|_{\Gamma}=0\Big{\\}}$
$\displaystyle\hskip 56.9055pt\text{if }\frac{1}{q}<2-\frac{2}{p}<2;$ (A.8a)
$\displaystyle\Big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),\mathcal{D}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
$\displaystyle=\Big{\\{}\boldsymbol{g}\in\boldsymbol{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega):\text{
div }\boldsymbol{g}=0,\ \boldsymbol{g}\cdot\nu|_{\Gamma}=0\Big{\\}}$
$\displaystyle\equiv\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega),\
\text{ if }0<2-\frac{2}{p}<\frac{1}{q}.$ (A.8b)

Theorem A.1 states that the Stokes operator $-A_{q}$ generates a s.c. analytic
semigroup on the space $\boldsymbol{L}^{q}_{\sigma}(\Omega),\ 1<q<\infty$,
hence on the space ${\mathcal{D}}(A_{q})$ in (1.14) = (A.1), with norm
$\displaystyle\left\lVert\ \cdot\
\right\rVert_{{\mathcal{D}}(A_{q})}=\left\lVert A_{q}\ \cdot\
\right\rVert_{\boldsymbol{L}^{q}_{\sigma}(\Omega)}$ 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) = (1.17) likewise generates
a s.c. analytic semigroup $\displaystyle e^{{\mathcal{A}}_{q}t}$ on
$\displaystyle\boldsymbol{L}^{q}_{\sigma}(\Omega)$ 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
$\boldsymbol{L}^{q}_{\sigma}(\Omega)$. 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
$\displaystyle\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$, as
this does not contain B.C., Remark 1.2. 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
$\widetilde{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ defined in (1.11) = (A.8b)
which moreover is uniformly stable, as in (A.4),

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

  2. (ii)

The Oseen operator ${\mathcal{A}}_{q}$ in (A.2) = (1.17) generates a s.c.
analytic semigroup $e^{{\mathcal{A}}_{q}t}$ on the space
$\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$ in (1.11) =
(A.8).

* (d)

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

$\displaystyle\boldsymbol{\varphi}_{t}-\Delta\boldsymbol{\varphi}+\nabla\pi$
$\displaystyle=\boldsymbol{F}$ $\displaystyle\text{ in
}(0,T]\times\Omega\equiv Q$ (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

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

Next, we introduce the space of maximal regularity for
$\\{\boldsymbol{\varphi},\boldsymbol{\varphi}^{\prime}\\}$ as [37, p 2;
Theorem 2.8.5.iii, p 17], [32, p 1404-5], with $T$ up to $\infty$:

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

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

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

The following embedding, also called trace theorem, holds true [2, Theorem
4.10.2, p 180, BUC for $T=\infty$], [69].

$\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}\subset\widetilde{\boldsymbol{X}}^{T}_{p,q}\equiv
L^{p}(0,T;\boldsymbol{W}^{2,q}(\Omega))\cap
W^{1,p}(0,T;\boldsymbol{L}^{q}(\Omega))\\\ \hookrightarrow
C\Big{(}[0,T];\boldsymbol{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\Big{)}.$ (A.16)

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

$\boldsymbol{\varphi}(t)=e^{-A_{q}t}\boldsymbol{\varphi}_{0}+\int_{0}^{t}e^{-A_{q}(t-s)}F_{\sigma}(\tau)d\tau.$
(A.17)

The following is the celebrated result on maximal regularity on
$\boldsymbol{L}^{q}_{\sigma}(\Omega)$ of the Stokes problem due originally to
Solonnikov [81] reported in [37, Theorem 2.8.5.(iii) and Theorem 2.10.1 p24
for $\boldsymbol{\varphi}_{0}=0$], [73], [32, 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

$F_{\sigma}\in L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega)),\
\boldsymbol{\varphi}_{0}\in\Big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}.$
(A.18)

Then there exists a unique solution
$\boldsymbol{\varphi}\in\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma},\pi\in\widetilde{Y}^{T}_{p,q}$
to the dynamic Stokes problem (A.10) or (A.11), continuously on the data:
there exist constants $C_{0},C_{1}$ independent of
$T,F_{\sigma},\boldsymbol{\varphi}_{0}$ such that via (A.16)

$\displaystyle
C_{0}\left\lVert\boldsymbol{\varphi}\right\rVert_{C\big{(}[0,T];\boldsymbol{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}$
$\displaystyle\leq\left\lVert\boldsymbol{\varphi}\right\rVert_{\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{\widetilde{Y}^{T}_{p,q}}$
(A.19)
$\displaystyle\equiv\left\lVert\boldsymbol{\varphi}^{\prime}\right\rVert_{L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))}$
$\displaystyle+\left\lVert
A_{q}\boldsymbol{\varphi}\right\rVert_{L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))}+\left\lVert\pi\right\rVert_{\widetilde{Y}^{T}_{p,q}}$
$\displaystyle\leq C_{1}\bigg{\\{}\left\lVert
F_{\sigma}\right\rVert_{L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))}$
$\displaystyle+\left\lVert\boldsymbol{\varphi}_{0}\right\rVert_{\big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}}\bigg{\\}}.$

In particular,

  1. (i)

With reference to the variation of parameters formula (A.17) of problem (A.11)
arising from the Stokes problem (A.10), we have recalling (A.12): the map

$\displaystyle F_{\sigma}$
$\displaystyle\longrightarrow\int_{0}^{t}e^{-A_{q}(t-\tau)}F_{\sigma}(\tau)d\tau\
:\text{continuous}$ (A.20) $\displaystyle
L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))$
$\displaystyle\longrightarrow\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}\equiv
L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap
W^{1,p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega)).$ (A.21)

  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
$\displaystyle\Big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}$
(see statement below (A.8)) satisfies

$e^{-A_{q}t}:\
\text{continuous}\quad\Big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\Big{)}_{1-\frac{1}{p},p}\longrightarrow\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}\equiv\\\
L^{p}(0,T;{\mathcal{D}}(A_{q}))\cap
W^{1,p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega)).$ (A.22c) 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
$\displaystyle\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)$,
satisfies $e^{-A_{q}t}:\
\text{continuous}\quad\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}.$
(A.22d)

  3. (iii)

Moreover, for future use, for $1<q<\infty,1<p<\frac{2q}{2q-1}$, then (A.19)
specializes to

$\left\lVert\boldsymbol{\varphi}\right\rVert_{\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{\widetilde{Y}^{T}_{p,q}}\leq
C\bigg{\\{}\left\lVert
F_{\sigma}\right\rVert_{L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))}+\left\lVert\boldsymbol{\varphi}_{0}\right\rVert_{\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)}\bigg{\\}}.$
(A.23)

* (e)

Maximal $L^{p}$ regularity on $\boldsymbol{L}^{q}_{\sigma}(\Omega)$ 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 $\boldsymbol{L}^{q}_{\sigma}(\Omega)$-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
$\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}$ 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
$\boldsymbol{y}_{e}$, see (1.2):

$\displaystyle\boldsymbol{\varphi}_{t}-\Delta\boldsymbol{\varphi}+L_{e}(\boldsymbol{\varphi})+\nabla\pi$
$\displaystyle=\boldsymbol{F}$ $\displaystyle\text{ in
}(0,T]\times\Omega\equiv Q$ (A.24a) $\displaystyle div\ \boldsymbol{\varphi}$
$\displaystyle\equiv 0$ $\displaystyle\text{ in }Q$ (A.24b)
$\displaystyle\left.\boldsymbol{\varphi}\right\rvert_{\Sigma}$
$\displaystyle\equiv 0$ $\displaystyle\text{ in
}(0,T]\times\Gamma\equiv\Sigma$ (A.24c)
$\displaystyle\left.\boldsymbol{\varphi}\right\rvert_{t=0}$
$\displaystyle=\boldsymbol{\varphi}_{0}$ $\displaystyle\text{ in }\Omega,$
(A.24d)
$L_{e}(\boldsymbol{\varphi})=(\boldsymbol{y}_{e}.\nabla)\boldsymbol{\varphi}+(\boldsymbol{\varphi}.\nabla)\boldsymbol{y}_{e}\hskip
85.35826pt$ (A.25)

rewritten in abstract form, after applying the Helmholtz projector $P_{q}$ to
(A.24a) and recalling ${\mathcal{A}}_{q}$ in (A.2), as

$\boldsymbol{\varphi}_{t}={\mathcal{A}}_{q}\boldsymbol{\varphi}+P_{q}\boldsymbol{F}=-\nu
A_{q}\boldsymbol{\varphi}-A_{o,q}\boldsymbol{\varphi}+\boldsymbol{F}_{\sigma},\quad\boldsymbol{\varphi}_{0}\in\big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
(A.26)

whose solution is via (1.18) = (A.2)

$\boldsymbol{\varphi}(t)=e^{{\mathcal{A}}_{q}t}\boldsymbol{\varphi}_{0}+\int_{0}^{t}e^{{\mathcal{A}}_{q}(t-\tau)}\boldsymbol{F}_{\sigma}(\tau)d\tau,$
(A.27) $\boldsymbol{\varphi}(t)=e^{-\nu
A_{q}t}\boldsymbol{\varphi}_{0}+\int_{0}^{t}e^{-\nu
A_{q}(t-\tau)}\boldsymbol{F}_{\sigma}(\tau)d\tau-\int_{0}^{t}e^{-\nu
A_{q}(t-\tau)}A_{o,q}\boldsymbol{\varphi}(\tau)d\tau.$ (A.28)

######  Theorem A.5.

Let $1<p,q<\infty,\ 0<T<\infty$. Assume (as in (A.18))

$\boldsymbol{F}_{\sigma}\in
L^{p}\big{(}0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega)\big{)},\quad\boldsymbol{\varphi}_{0}\in\big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
(A.29)

where ${\mathcal{D}}(A_{q})={\mathcal{D}}({\mathcal{A}}_{q})$, see (A.2) =
(1.18). Then there exists a unique solution
$\boldsymbol{\varphi}\in\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma},\
\pi\in\widetilde{Y}^{T}_{p,q}$ of the dynamic Oseen problem (A.24),
continuously on the data: that is, there exist constants $C_{0},C_{1}$
independent of $\boldsymbol{F}_{\sigma},\boldsymbol{\varphi}_{0}$ such that

$\displaystyle
C_{0}\left\lVert\boldsymbol{\varphi}\right\rVert_{C\big{(}[0,T];\boldsymbol{B}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\big{)}}$
$\displaystyle\leq\left\lVert\boldsymbol{\varphi}\right\rVert_{\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}}+\left\lVert\pi\right\rVert_{\widetilde{Y}^{T}_{p,q}}$
$\displaystyle\equiv\left\lVert\boldsymbol{\varphi}^{\prime}\right\rVert_{L^{p}(0,T;\boldsymbol{L}^{q}(\Omega))}+\left\lVert
A_{q}\boldsymbol{\varphi}\right\rVert_{L^{p}(0,T;\boldsymbol{L}^{q}(\Omega))}+\left\lVert\pi\right\rVert_{\widetilde{Y}^{T}_{p,q}}$
(A.30) $\displaystyle\leq
C_{T}\left\\{\left\lVert\boldsymbol{F}_{\sigma}\right\rVert_{L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))}+\left\lVert\boldsymbol{\varphi}_{0}\right\rVert_{\big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}}\right\\}$
(A.31)

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

  1. i.

The map

$\displaystyle\boldsymbol{F}_{\sigma}\longrightarrow\int_{0}^{t}e^{{\mathcal{A}}_{q}(t-\tau)}\boldsymbol{F}_{\sigma}(\tau)d\tau\
:\text{continuous}$ (A.32) $\displaystyle
L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))$ $\displaystyle\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)}$

where then automatically, see (A.26)

$L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))\longrightarrow
W^{1,p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))$ (A.33)

and ultimately via (A.12)

$L^{p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega))\longrightarrow\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}\equiv
L^{p}\big{(}0,T;{\mathcal{D}}(A_{q})\big{)}\cap
W^{1,p}(0,T;\boldsymbol{L}^{q}_{\sigma}(\Omega)).$ (A.34)

  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
$\displaystyle\big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}$
satisfies for $1<p,q<\infty$

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)}$
(A.35)

and hence automatically by (A.12)

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\big{(}\boldsymbol{L}^{q}_{\sigma}(\Omega),{\mathcal{D}}(A_{q})\big{)}_{1-\frac{1}{p},p}\longrightarrow\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma}.$
(A.36)

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

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow
L^{p}\big{(}0,T;{\mathcal{D}}({\mathcal{A}}_{q})={\mathcal{D}}(A_{q})\big{)},\
T<\infty.$ (A.37)

and hence automatically

$e^{{\mathcal{A}}_{q}t}:\
\text{continuous}\quad\widetilde{\boldsymbol{B}}^{2-{}^{2}\\!/_{p}}_{q,p}(\Omega)\longrightarrow\boldsymbol{\widetilde{X}}^{T}_{p,q,\sigma},\
T<\infty.$ (A.38)

A proof is given in [60, Appendix B].

##  B Material in support of the proof of Theorem 4.1: the required UCP and
$\displaystyle D^{*}{\mathcal{B}}_{q}^{*}f$ in (4.17).

###  B.1 The required UCP.

We return to the operator $\displaystyle\mathbb{A}_{q}$ in (1.34):

$\mathbb{A}_{q}=\begin{bmatrix}{\mathcal{A}}_{q}&-{\mathcal{C}}_{\gamma}\\\
-{\mathcal{C}}_{\theta_{e}}&-{\mathcal{B}}_{q}\end{bmatrix}:\boldsymbol{W}^{q}_{\sigma}(\Omega)=\boldsymbol{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)\supset{\mathcal{D}}(\mathbb{A}_{q})={\mathcal{D}}({\mathcal{A}}_{q})\times{\mathcal{D}}({\mathcal{B}}_{q})\\\
=(\boldsymbol{W}^{2,q}(\Omega)\cap\boldsymbol{W}^{1,q}_{0}(\Omega)\cap\boldsymbol{L}^{q}_{\sigma}(\Omega))\times(W^{2,q}(\Omega)\cap
W^{1,q}_{0}(\Omega))\longrightarrow\boldsymbol{W}^{q}_{\sigma}(\Omega).$ (B.3)

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

$\displaystyle-\nu\Delta\boldsymbol{\varphi}+L_{e}(\boldsymbol{\varphi})+\nabla\pi-\gamma\psi\boldsymbol{e}_{d}$
$\displaystyle=\lambda\boldsymbol{\varphi}$ $\displaystyle\text{in }\Omega$
(B.4a)
$\displaystyle-\kappa\Delta\psi-\boldsymbol{y}_{e}\cdot\nabla\psi+\boldsymbol{\varphi}\cdot\nabla\theta_{e}$
$\displaystyle=\lambda\psi$ $\displaystyle\text{in }\Omega$ (B.4b)
$\displaystyle\text{div }\boldsymbol{\varphi}$ $\displaystyle=0$
$\displaystyle\text{in }\Omega$ (B.4c) $\displaystyle\boldsymbol{\varphi}=0,\
\psi$ $\displaystyle=0$ $\displaystyle\text{on }\Gamma.$ (B.4d)

Several UCP for over-determined versions of the eigenproblem (B.4) are given
in [89]. However, establishing in Theorem 4.1 controllability of the finite
dimensional projected problem (4.26) or (4.34) via verification of the Kalman
rank condition (4.39) involves the following UCP for the adjoint problem.
First, recall from (4.4) that the adjoint $\displaystyle\mathbb{A}_{q}^{*}$ or
$\mathbb{A}_{q}$ in (B.3) is

$\mathbb{A}_{q}^{*}=\begin{bmatrix}{\mathcal{A}}_{q}^{*}&-{\mathcal{C}}_{\theta_{e}}^{*}\\\
-{\mathcal{C}}_{\gamma}^{*}&-{\mathcal{B}}_{q}^{*}\end{bmatrix}:\boldsymbol{W}^{q^{\prime}}_{\sigma}(\Omega)=\boldsymbol{L}^{q^{\prime}}_{\sigma}(\Omega)\times
L^{q^{\prime}}(\Omega)\supset{\mathcal{D}}(\mathbb{A}_{q}^{*})={\mathcal{D}}({\mathcal{A}}_{q}^{*})\times{\mathcal{D}}({\mathcal{B}}_{q}^{*})\\\
=(\boldsymbol{W}^{2,q^{\prime}}(\Omega)\cap\boldsymbol{W}^{1,q^{\prime}}_{0}(\Omega)\cap\boldsymbol{L}^{q^{\prime}}_{\sigma}(\Omega))\times(W^{2,q^{\prime}}(\Omega)\cap\boldsymbol{W}^{1,q^{\prime}}_{0}(\Omega))\longrightarrow\boldsymbol{W}^{q^{\prime}}_{\sigma}(\Omega).$
(B.7)

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

$\displaystyle-\nu\Delta\boldsymbol{\varphi^{*}}+L^{*}_{e}(\boldsymbol{\varphi^{*}})+\psi^{*}\nabla\theta_{e}+\nabla\pi$
$\displaystyle=\lambda\boldsymbol{\varphi^{*}}$ $\displaystyle\text{in
}\Omega$ (B.8a)
$\displaystyle-\kappa\Delta\psi^{*}-\boldsymbol{y}_{e}\cdot\nabla\psi^{*}-\gamma\boldsymbol{\varphi^{*}}\cdot\boldsymbol{e}_{d}$
$\displaystyle=\lambda\psi^{*}$ $\displaystyle\text{in }\Omega$ (B.8b)
$\displaystyle\text{div }\boldsymbol{\varphi^{*}}$ $\displaystyle=0$
$\displaystyle\text{in }\Omega$ (B.8c)
$\displaystyle\boldsymbol{\varphi^{*}}=0,\ \psi^{*}$ $\displaystyle=0$
$\displaystyle\text{on }\Gamma.$ (B.8d)

The UCP invoked in the proof of Theorem 4.1 is

######  Theorem B.1.

[89, Theorem 5] Let
$\displaystyle\\{\boldsymbol{\varphi},\psi,\pi\\}\in\left[\boldsymbol{W}^{2,q}(\Omega)\cap\boldsymbol{L}^{q}_{\sigma}(\Omega)\right]\times
W^{2,q}(\Omega),\ \pi\in W^{1,q}(\Omega),$ be a solution of the following dual
problem

$\displaystyle-\nu\Delta\boldsymbol{\varphi^{*}}+L^{*}_{e}(\boldsymbol{\varphi^{*}})+\psi^{*}\nabla\theta_{e}+\nabla\pi$
$\displaystyle=\lambda\boldsymbol{\varphi^{*}}$ $\displaystyle\text{in
}\Omega$ (B.9a)
$\displaystyle-\kappa\Delta\psi^{*}-\boldsymbol{y}_{e}\cdot\nabla\psi^{*}-\gamma\boldsymbol{\varphi^{*}}\cdot\boldsymbol{e}_{d}$
$\displaystyle=\lambda\psi^{*}$ $\displaystyle\text{in }\Omega$ (B.9b)
$\displaystyle\text{div }\boldsymbol{\varphi^{*}}$ $\displaystyle=0$
$\displaystyle\text{in }\Omega$ (B.9c)
$\displaystyle\left\\{\varphi^{*(1)},\dots,\varphi^{*(d-1)}\right\\}=0,\
\psi^{*}$ $\displaystyle=0$ $\displaystyle\text{on }\omega$ (B.9d)

with over-determination in (B.9d) Then

$\boldsymbol{\varphi}^{*}=0,\ \psi^{*}=0,\ \pi=const\ \text{in }\Omega.\ \qed$
(B.10)

###### Proof of $\displaystyle D^{*}{\mathcal{B}}^{*}f=-\kappa\frac{\partial
f}{\partial\nu}\bigg{|}_{\Gamma},\ f\in{\mathcal{D}}({\mathcal{B}}_{q}^{*})$,
in (4.17) essentially by Green’s formula.

  1. 1.

Let $1<q<\infty$ and define

$B_{o,q}h=\boldsymbol{y}_{e}\cdot h,\
L^{q}(\Omega)\supset{\mathcal{D}}(B_{o,q})=W^{1,q}_{0}(\Omega)\longrightarrow
L^{q}(\Omega)$ (B.11)

Then

$B_{o,q}^{*}h=-\boldsymbol{y}_{e}\cdot f,\
L^{q^{\prime}}(\Omega)\supset{\mathcal{D}}(B_{o,q}^{*})=W^{1,q^{\prime}}_{0}(\Omega)\longrightarrow
L^{q^{\prime}}(\Omega)$ (B.12)

In fact recalling $\boldsymbol{y}_{e}|_{\Gamma}\equiv 0$ and div
$\boldsymbol{y}_{e}\equiv 0$ in $\Omega$ from (1.2c-d), we compute in the
duality pairing $L^{q}(\Omega),\ L^{q^{\prime}}(\Omega)$, with
$h\in{\mathcal{D}}(B_{o,q}),\ f\in{\mathcal{D}}(B^{*}_{o,q})$ by (B.11):

$\displaystyle\left\langle B_{o,q}h,f\right\rangle$
$\displaystyle=\int_{\Omega}f\boldsymbol{y}_{e}\cdot\nabla
hd\Omega=\int_{\Gamma}\cancel{hf\boldsymbol{y}_{e}\cdot\nu
d\Gamma}-\int_{\Omega}h\text{ div}(f\boldsymbol{y}_{e})d\Omega$
$\displaystyle=-\int_{\Omega}h\boldsymbol{y}_{e}\cdot\nabla
fd\Omega=\left\langle h,B^{*}_{o,q}f\right\rangle_{\Omega}$ (B.13)

  2. 2.

We return to the operator ${\mathcal{B}}_{q}$ in (1.18), with
$h\in{\mathcal{D}}({\mathcal{B}}_{q})$

${\mathcal{B}}_{q}h=-\kappa\Delta h+B_{o,q}h,\ \text{so that
}{\mathcal{B}}_{q}^{*}h=-\kappa\Delta f+B_{o,q}^{*}f,\
f\in{\mathcal{D}}({\mathcal{B}}_{q}^{*})$ (B.14)

We shall show that

$\displaystyle D^{*}{\mathcal{B}}^{*}f=-\kappa\frac{\partial
f}{\partial\nu}\bigg{|}_{\Gamma},\
f\in{\mathcal{D}}({\mathcal{B}}_{q}^{*})=W^{2,q^{\prime}}(\Omega)\cap
W^{1,q^{\prime}}_{0}(\Omega)$ (B.15)

In fact, by (B.12) and (B.14) we compute with $v\in L^{2}(\Gamma)$ recalling
the definition of $D$ in (1.23a)

$\displaystyle\left\langle
D^{*}{\mathcal{B}}^{*}_{q}f,v\right\rangle_{\Gamma}$
$\displaystyle=\left\langle{\mathcal{B}}^{*}_{q}f,Dv\right\rangle_{\Omega}=\left\langle\Delta
f,-\kappa Dv\right\rangle_{\Omega}+\left\langle
f,B^{*}_{o,q}Dv\right\rangle_{\Omega}$ (B.16) $\displaystyle=\left\langle
f,\cancel{(-\kappa\Delta-\boldsymbol{y}_{e}\cdot\nabla)}Dv\right\rangle_{\Omega}+\int_{\Gamma}\frac{\partial
f}{\partial\nu}(-\kappa
Dv)d\Gamma+\int_{\Gamma}\cancel{f\left(\kappa\frac{\partial
Dv}{\partial\nu}\right)}d\Gamma$ (B.17)
$\displaystyle=\left\langle-\kappa\frac{\partial
f}{\partial\nu},v\right\rangle_{\Gamma},\quad\text{for all }v\in
L^{q}(\Gamma).$ (B.18)

and (B.15) is established. ∎

##  C Validation of the Kalman controllability conditions (4.39) with fluid
vectors
$\displaystyle\\{\boldsymbol{u}_{1},\dots,\boldsymbol{u}_{\ell_{i}}\\},\
\ell_{i}\leq K,\ i=1,\dots,M$ having only $(d-1)$ components.

For the sake of the clarity, we shall consider separately the cases $d=2$ and
$d=3$.

Case $d=2$. Express the $2$-dimensional vectors $\boldsymbol{u}_{i}$ and
$\boldsymbol{\varphi}_{ij}^{*}$ of Section 4 in terms of their two components.

$\boldsymbol{u}_{i}=\left[u^{(1)}_{i},u^{(2)}_{i}\right],\
\boldsymbol{\varphi}^{*}_{ij}=\left[\varphi_{ij}^{*(1)},\varphi_{ij}^{*(2)}\right],\
i=1,\dots,M,\ j=1,\dots,\ell_{i}$ (C.1)

Here we shall recall the matrix $\boldsymbol{W}_{i}$ in (4.27) but we shall
replace the matrix $U_{i}$ in (4.28) with the following matrix $U_{i}^{(2)}$:

$U^{(2)}_{i}=\begin{bmatrix}\left\langle
u^{(2)}_{1},\varphi_{i1}^{*(2)}\right\rangle_{\omega}&\dots&\left\langle
u^{(2)}_{\ell_{i}},\varphi_{i1}^{*(2)}\right\rangle_{\omega}\\\\[5.69054pt]
\left\langle
u^{(2)}_{2},\varphi_{i2}^{*(2)}\right\rangle_{\omega}&\dots&\left\langle
u^{(2)}_{\ell_{i}},\varphi_{i2}^{*(2)}\right\rangle_{\omega}\\\
\vdots&&\vdots\\\ \left\langle
u^{(2)}_{1},\varphi_{i\ell_{i}}^{*(2)}\right\rangle_{\omega}&\dots&\left\langle
u^{(2)}_{\ell_{i}},\varphi_{i\ell_{i}}^{*(2)}\right\rangle_{\omega}\end{bmatrix}:\ell_{i}\times
K$ (C.2)

where now the duality pairing $\left\langle\ ,\ \right\rangle_{\omega}$
involves two scalar functions. Next, for $d=2$, we shall establish the Kalman
algebraic rank condition for the finite dimensional, unstable, feedback
problem (4.26), by employing not the full strength of the $2$-dimensional
vectors $\boldsymbol{u}_{i}$ and $\boldsymbol{\varphi}_{ij}^{*}$ as in (4.39)
involving the matrix $U_{i}$ in (4.28), but instead replacing $U_{i}$ with
$U_{i}^{(2)}$. This way, only the second (scalar) components $u^{(2)}_{i}$ and
$\varphi_{ij}^{*(2)}$ of the $2$-dimensional vectors $\boldsymbol{u}_{i}$ and
$\boldsymbol{\varphi}_{ij}^{*}$ in (C.1) are needed. Recall that we are
dealing with a pair $\\{\omega,\widetilde{\Gamma}\\}$ as in Fig 1. The
counterpart of Theorem 4.1 is now

######  Theorem C.1.

Let $d=2$. It is possible to select boundary vectors $f_{1},\dots,f_{K}$ in
${\mathcal{F}}\subset W^{2-{}^{1}\\!/_{q},q}(\widetilde{\Gamma})$ with support
on $\widetilde{\Gamma}$, and scalar second components
$\\{u^{(2)}_{1},\dots,u^{(2)}_{\ell_{i}}\\}$ as in (C.1), such that

$\text{rank }\left[W_{i},\ U_{i}^{(2)}\right]=\ell_{i},\ i,\dots,M.$ (C.3)

###### Proof.

We shall appropriately modify the proof of Theorem 4.1. As in this proof, the
crux is to establish that we cannot have simultaneously

$\partial_{\nu}\psi_{i\ell_{i}}^{*}=\sum_{j=1}^{\ell_{i}-1}\alpha_{j}\partial_{\nu}\psi_{ij}^{*}\text{
in }L^{q}(\widetilde{\Gamma})\text{ and
}\varphi^{*(2)}_{i\ell_{i}}=\sum_{j=1}^{\ell_{i}-1}\alpha_{j}\varphi^{*(2)}_{ij}\text{
in }L^{q}_{\sigma}(\omega)$ (C.4)

with the same constant $\alpha_{1},\dots,\alpha_{\ell_{i}-1}$ in both
expression.

Claim: Statement (C.4) is false. By contradiction, suppose that both linear
combinations in (C.4) hold true. Next, as in (4.42) define the $(d+1)$-vector
$\displaystyle\boldsymbol{\Phi}^{*}\equiv\\{\boldsymbol{\varphi}^{*},\psi^{*}\\}$
by

$\boldsymbol{\Phi}^{*}\equiv\begin{bmatrix}\boldsymbol{\varphi}^{*}\\\
\psi^{*}\end{bmatrix}=\sum_{j=1}^{\ell_{i}-1}\begin{bmatrix}\alpha_{j}\boldsymbol{\varphi}_{ij}^{*}\\\
\alpha_{j}\psi_{ij}^{*}\end{bmatrix}-\begin{bmatrix}\boldsymbol{\varphi}_{i\ell_{i}}^{*}\\\
\psi_{i\ell_{i}}^{*}\end{bmatrix}=\sum_{j=1}^{\ell_{i}-1}\alpha_{j}\boldsymbol{\Phi}^{*}_{ij}-\boldsymbol{\Phi}^{*}_{i\ell_{i}},\
i=1,\dots,M;\ q\geq 2,$ (C.5)

in
$\boldsymbol{W}^{q}_{\sigma}(\Omega)\equiv\boldsymbol{L}^{q}_{\sigma}(\Omega)\times
L^{q}(\Omega)$, with
$\displaystyle\boldsymbol{\Phi}^{*}_{ij}\equiv\\{\boldsymbol{\varphi}^{*}_{ij},\psi^{*}_{ij}\\}$
eigenvector of $\displaystyle\mathbb{A}_{q,N}^{*}$ or
$\displaystyle\mathbb{A}_{q}^{*}$, as in (4.3). Then, in view of (C.4), we now
obtain

$\varphi^{*(2)}\equiv 0\text{ in
}\omega,\quad\partial_{\nu}\psi^{*}|_{\widetilde{\Gamma}}\equiv 0.$ (C.6)

(in place of (4.43)). As in the proof of Theorem 4.1, since the
$\displaystyle\Phi^{*}_{ij}$ are eigenvectors of
$\displaystyle\mathbb{A}_{q,N}^{*}$, or $\displaystyle\mathbb{A}_{q}^{*}$, so
is the vector
$\displaystyle\boldsymbol{\Phi}^{*}\equiv\\{\boldsymbol{\varphi}^{*},\psi^{*}\\}$
defined in (C.5). Thus
$\displaystyle\boldsymbol{\Phi}^{*}\equiv\\{\boldsymbol{\varphi}^{*},\psi^{*}\\}$
satisfies the PDE version (4.36a-d) of the eigenproblem for the dual
$\displaystyle\mathbb{A}_{q}^{*}$, which is now augmented with the over-
determined conditions in (C.6). The fact that now
$\displaystyle\boldsymbol{\varphi}^{*}|_{\Gamma}=0$ by (4.44d) on the entire
boundary $\Gamma$, permits a-fortiori the argument of [89, Theorems 6, 7] to
hold true. Namely, invoke $\displaystyle\varphi^{*(2)}\equiv 0$ in $\omega$ in
(C.6) to obtain, as in the proof of Theorem 4.1, the over-determined problem
(4.45) for $\psi^{*}$ on $\omega$, with the conclusion that

$\psi^{*}\equiv 0\text{ in }\omega,\text{ as in }\eqref{4.38}$ (C.7)

Next, the divergence condition (4.44c) yields in view of (C.6): $div\
\boldsymbol{\varphi}^{*}=\varphi^{*(1)}_{x_{1}}+\varphi^{*(2)}_{x_{2}}=\varphi^{*(1)}_{x_{1}}=0$
in $\omega$. Hence $\varphi^{*(1)}(x_{1},x_{2})\equiv c(x_{2})$ in $\omega$,
where $c(x_{2})$ is a function constant w.r.t. $x_{1}$ and depending only on
$x_{2}$ in $\omega$. Next, let $\displaystyle P=\\{x_{1}(P),x_{2}(P)\\}$ be an
arbitrary point of $\omega$. Consider the line $\ell$ passing through the
point $P$ and parallel to the $x_{1}$-axis. On such a line $\ell$, the value
$\displaystyle\varphi^{*(1)}(x_{1},x_{2}(P))\equiv c_{2}(x_{2}(P))$ is
constant w.r.t. $x_{1}$, as long as $\ell$ intersect $\omega$. By definition
of the small set $\omega$ supported by $\widetilde{\Gamma}$, there is a non-
empty open subset $\widetilde{\omega}\subset\omega$, where this happens: for
all points $P$ in $\widetilde{\omega}$, the line $\ell$ remains in
$\widetilde{\omega}$ and hits the boundary $\widetilde{\Gamma}$, where
condition (4.44d) applies for $\displaystyle\varphi^{*(1)}|_{\Gamma}=0$. Thus
$\varphi^{*(1)}\equiv 0$ in $\widetilde{\omega}\subset\omega$. Recalling
(C.6), we finally have

$\boldsymbol{\varphi}^{*}=\\{\varphi^{*(1)},\varphi^{*(2)}\\}\equiv 0\text{ in
}\widetilde{\omega}\subset\omega,\text{ along with }\psi^{*}\equiv 0\text{ in
}\widetilde{\omega}\text{ by }\eqref{C.7}.$ (C.8)

We can then apply [89, Theorem 5 for $\widetilde{\omega}$] and conclude that

$\boldsymbol{\varphi}^{*}=\\{\varphi^{*(1)},\varphi^{*(2)}\\}\equiv 0\text{ in
}\Omega,\ \psi^{*}\equiv 0\text{ in }\Omega,\ p\equiv\text{ const in }\Omega.$
(C.9)

The rest of the proof proceeds as in Theorem 4.1 following (4.48). Theorem C.1
is established. ∎

Case $d=3$. Now we express the $3$-dimensional vectors $\boldsymbol{u}_{i}$
and $\boldsymbol{\varphi}^{*}_{ij}$ of Section 4 in terms of their components
as

$\boldsymbol{u}_{i}=\left[u^{(1)}_{i},u^{(2)}_{i},u^{(3)}_{i}\right],\
\boldsymbol{\varphi}^{*}_{ij}=\left[\varphi_{ij}^{*(1)},\varphi_{ij}^{*(2)},\varphi_{ij}^{*(3)}\right],\
i=1,\dots,M,\ j=1,\dots,\ell_{i}.$ (C.10)

We now distinguish two sub-cases.

Sub-case $\\{1,3\\}$. Here, we extract only the first and third components,
while we omit the second component. Accordingly, we introduce the following
$2$-dimension vectors $\boldsymbol{u}^{(1,3)}_{i}$ and
$\boldsymbol{\varphi}^{*(1,3)}$ and the corresponding matrix $\displaystyle
U^{(1,3)}_{i}$

$\displaystyle\boldsymbol{u}_{i}^{(1,3)}=\left[u^{(1)}_{i},u^{(3)}_{i}\right],\
\boldsymbol{\varphi}^{*(1,3)}_{ij}=\left[\varphi_{ij}^{*(1)},\varphi_{ij}^{*(3)}\right]\hskip
42.67912pt$ $\displaystyle
U^{(1,3)}_{i}=\begin{bmatrix}\left\langle\begin{bmatrix}u^{(1)}_{1}\\\\[2.84526pt]
u^{(3)}_{1}\end{bmatrix},\begin{bmatrix}\varphi_{i1}^{*(1)}\\\\[2.84526pt]
\varphi_{i1}^{*(3)}\end{bmatrix}\right\rangle_{\omega}&\dots&\left\langle\begin{bmatrix}u^{(1)}_{\ell_{i}}\\\\[2.84526pt]
u^{(3)}_{\ell_{i}}\end{bmatrix},\begin{bmatrix}\varphi_{i1}^{*(1)}\\\\[2.84526pt]
\varphi_{i1}^{*(3)}\end{bmatrix}\right\rangle_{\omega}\\\ \dots&\dots&\dots\\\
\left\langle\begin{bmatrix}u^{(1)}_{1}\\\\[2.84526pt]
u^{(3)}_{1}\end{bmatrix},\begin{bmatrix}\varphi_{i\ell_{i}}^{*(1)}\\\\[2.84526pt]
\varphi_{i\ell_{i}}^{*(3)}\end{bmatrix}\right\rangle_{\omega}&\dots&\left\langle\begin{bmatrix}u^{(1)}_{\ell_{i}}\\\\[2.84526pt]
u^{(3)}_{\ell_{i}}\end{bmatrix},\begin{bmatrix}\varphi_{i\ell_{i}}^{*(1)}\\\\[2.84526pt]
\varphi_{i\ell_{i}}^{*(3)}\end{bmatrix}\right\rangle_{\omega}\end{bmatrix}$
(C.11)

Sub-case $\\{2,3\\}$. Here, we extract only the second and third components,
while we omit the first component. Accordingly, we introduce the following
$2$-dimension vectors $\boldsymbol{u}^{(2,3)}_{i}$ and
$\boldsymbol{\varphi}^{*(2,3)}$ and the corresponding matrix $\displaystyle
U^{(2,3)}_{i}$

$\displaystyle\boldsymbol{u}_{i}^{(2,3)}=\left[u^{(2)}_{i},u^{(3)}_{i}\right],\
\boldsymbol{\varphi}^{*(2,3)}_{ij}=\left[\varphi_{ij}^{*(2)},\varphi_{ij}^{*(3)}\right]\hskip
42.67912pt$ $\displaystyle
U^{(2,3)}_{i}=\begin{bmatrix}\left\langle\begin{bmatrix}u^{(2)}_{1}\\\\[2.84526pt]
u^{(3)}_{1}\end{bmatrix},\begin{bmatrix}\varphi_{i1}^{*(2)}\\\\[2.84526pt]
\varphi_{i1}^{*(3)}\end{bmatrix}\right\rangle_{\omega}&\dots&\left\langle\begin{bmatrix}u^{(2)}_{\ell_{i}}\\\\[2.84526pt]
u^{(3)}_{\ell_{i}}\end{bmatrix},\begin{bmatrix}\varphi_{i1}^{*(2)}\\\\[2.84526pt]
\varphi_{i1}^{*(3)}\end{bmatrix}\right\rangle_{\omega}\\\ \dots&\dots&\dots\\\
\left\langle\begin{bmatrix}u^{(2)}_{1}\\\\[2.84526pt]
u^{(3)}_{1}\end{bmatrix},\begin{bmatrix}\varphi_{i\ell_{i}}^{*(2)}\\\\[2.84526pt]
\varphi_{i\ell_{i}}^{*(3)}\end{bmatrix}\right\rangle_{\omega}&\dots&\left\langle\begin{bmatrix}u^{(2)}_{\ell_{i}}\\\\[2.84526pt]
u^{(3)}_{\ell_{i}}\end{bmatrix},\begin{bmatrix}\varphi_{i\ell_{i}}^{*(2)}\\\\[2.84526pt]
\varphi_{i\ell_{i}}^{*(3)}\end{bmatrix}\right\rangle_{\omega}\end{bmatrix}$
(C.12)

Next, for $d=3$, we shall establish the Kalman algebraic rank condition for
the finite dimensional, unstable, feedback problem (4.26), by replacing the
matrix $U_{i}$ in (4.28), with either the matrix $U^{(1,3)}_{i}$ in (C) (Sub-
case $\\{1,3\\}$); or else with the matrix $U^{2,3}_{i}$ in (C) (Sub-case
$\\{2,3\\}$). This way, the third (scalar) components $u^{(3)}_{i}$ and
$\varphi^{*(3)}_{ij}$ of the $3$-dimensional vectors $\boldsymbol{u}_{i}$ and
$\boldsymbol{\varphi}^{*}_{ij}$ in (C.10) are needed in both sub-cases. The
counterpart of Theorem 4.1 (or Theorem C.1) is now

######  Theorem C.2.

Let $d=3$. It is possible to select boundary vectors $f_{1},\dots,f_{K}$ in
${\mathcal{F}}\subset W^{2-{}^{1}\\!/_{q},q}(\widetilde{\Gamma})$ with support
on $\widetilde{\Gamma}$, and

  1. (i)

either $2$-dimensional vectors
$\boldsymbol{u}^{(1,3)}_{i}=\left\\{u^{(1)}_{i},u^{(3)}_{i}\right\\}$ as in
(C)

  2. (ii)

or else $2$-dimensional vectors
$\boldsymbol{u}^{(2,3)}_{i}=\left\\{u^{(2)}_{i},u^{(3)}_{i}\right\\}$ as in
(C),

such that

$\text{rank }\left[W_{i},\ U_{i}^{(1,3)}\right]=\ell_{i},\ i,\dots,M,\text{ in
case }(i)$ (C.13)

or

$\text{rank }\left[W_{i},\ U_{i}^{(2,3)}\right]=\ell_{i},\ i,\dots,M,\text{ in
case }(ii)$ (C.14)

respectively.

###### Proof.

We shall only give a proof in the sub-case $\\{1,3\\}$, as the proof of the
sub-case $\\{2,3\\}$ is the same mutatis mutandis. As in the proof of Theorem
4.1 (or Theorem C.1), the crux is to establish that we cannot have
simultaneously

$\partial_{\nu}\psi_{i\ell_{i}}^{*}=\sum_{j=1}^{\ell_{i}-1}\alpha_{j}\partial_{\nu}\psi_{ij}^{*}\text{
in
}L^{q}(\widetilde{\Gamma})\quad\text{and}\quad\varphi^{*(1,3)}_{i\ell_{i}}=\sum_{j=1}^{\ell_{i}-1}\alpha_{j}\varphi^{*(1,3)}_{ij}\text{
in }\boldsymbol{L}^{q}_{\sigma}(\omega)$ (C.15)

with the same constants $\alpha,\dots,\alpha_{\ell_{i}-1}$ in both
expressions. ∎

Claim: Statement (C.15) is false. By contradiction, suppose that both linear
combinations in (C.15) hold true. Next, define the $(d+1)$-vector
$\displaystyle\boldsymbol{\Phi}^{*}\equiv\\{\boldsymbol{\varphi}^{*},\psi^{*}\\}$
as in (4.42) (or (C.5)). Then, in view of (C.15), we now obtain

$\boldsymbol{\varphi}^{*(1,3)}\equiv\left\\{\varphi^{*(1)},\varphi^{*(3)}\right\\}\equiv
0\text{ in }\omega,\quad\partial_{\nu}\psi^{*}|_{\widetilde{\Gamma}}\equiv 0.$
(C.16)

(in place of (4.43) or (C.6)). As in Theorem 4.1 (or Theorem C.1) the
$\displaystyle\boldsymbol{\Phi}^{*}\equiv\\{\boldsymbol{\varphi}^{*},\psi^{*}\\}$
is an eigenvector of $\mathbb{A}_{q}^{*}$ and thus satisfies the corresponding
PDE-version - that is problem (4.36a-d), augmented this time with the over-
determined conditions in (C.16). Next, invoke $\varphi^{*(3)}\equiv 0$ in
$\omega$ from (C.16) to obtain, as this the proof of Theorem 4.1, the over-
determined problem (4.45) for $\psi^{*}$ in $\omega$, with the conclusion that

$\psi^{*}\equiv 0\text{ in }\omega,$ (C.17)

as in (4.44) (or (C.7)). Next, the divergence condition (4.44c) yields in the
view of (C.16): $div\
\boldsymbol{\varphi}^{*}=\varphi^{*(1)}_{x_{1}}+\varphi^{*(2)}_{x_{2}}+\varphi^{*(3)}_{x_{3}}=\varphi^{*(2)}_{x_{2}}=0$
in $\omega$. Hence $\varphi^{*(2)}(x_{1},x_{2},x_{3})\equiv c(x_{1},x_{3})$ in
$\omega$, where $c(x_{1},x_{3})$ is a function constant w.r.t. $x_{2}$ and
depending only on $x_{1}$ and $x_{3}$ in $\omega$. Next, let $\displaystyle
P=\\{x_{1}(P),x_{2}(P),x_{3}(P)\\}$ be an arbitrary point of $\omega$.
Consider the plane $\pi_{P}$ passing through the point $P$ and parallel to the
$\\{x_{1},x_{3}\\}$-coordinate plane. As the point
$\\{x_{1},x_{2}(P),x_{3}\\}$ of $\omega$ runs over the plane $\pi_{P}$, the
value $\displaystyle\varphi^{*(2)}(x_{1},x_{2}(P),x_{3})=c(x_{1},x_{3})$ is
independent of $x_{2}(P)$, as long as such plane $\pi_{P}$ intersects
$\omega$. By definition of the small $\omega$ supported by
$\widetilde{\Gamma}$, there is a non-empty open subset
$\widetilde{\omega}\subset\omega$, where this happens: for all points $P$ in
$\widetilde{\omega}$, the plane $\pi_{P}$ remains in $\widetilde{\omega}$ and
hits the boundary $\widetilde{\Gamma}$, where condition (4.44d) applies for
$\displaystyle\varphi^{*(2)}|_{\Gamma}=0$. Thus
$\displaystyle\varphi^{*(2)}\equiv 0$ in $\widetilde{\omega}\subset\omega$.
Recalling (C.16) we conclude that

$\boldsymbol{\varphi}^{*}=\left\\{\varphi^{*(1)},\varphi^{*(2)},\varphi^{*(3)}\right\\}\equiv
0\text{ in }\widetilde{\omega}\subset\omega,\text{ along with }\psi^{*}\equiv
0\text{ in }\widetilde{\omega}\text{ by }\eqref{C.17}.$ (C.18)

We can then apply [89, Theorem 5 for $\widetilde{\omega}$] and conclude that

$\boldsymbol{\varphi}^{*}=\left\\{\varphi^{*(1)},\varphi^{*(2)},\varphi^{*(3)}\right\\}\equiv
0\text{ in }\Omega,\ \psi^{*}\equiv 0\text{ in }\Omega,\ p\equiv\text{ const
in }\Omega.$ (C.19)

The rest of the proof proceeds as in Theorem 4.1 following (4.48). Theorem C.2
is established.

The authors wish to thank the referee for much appreciated comments and
suggestions.

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