# Asymptotic Stability of a Compressible Oseen-Structure Interaction via a
Pointwise Resolvent Criterion

Pelin G. Geredeli
Department of Mathematics
Iowa State University, Ames-IA, USA email address<EMAIL_ADDRESS>
###### Abstract

In this study, we consider a linearized compressible flow structure
interaction PDE model for which the interaction interface is under the effect
of material derivative term. While the linearization takes place around a
constant pressure and density components in structure equation, the flow
linearization is taken with respect to a non-zero, fixed, variable ambient
vector field. This process produces extra “convective derivative” and
“material derivative” terms which causes the coupled system to be
nondissipative.

We analyze the long time dynamics in the sense of asymptotic (strong)
stability in an invariant subspace (one dimensional less) of the entire state
space where the continuous semigroup is “uniformly bounded”. For this, we
appeal to the pointwise resolvent condition introduced in [18] which avoids
many technical complexity and provides a very clean, short and easy-to-follow
proof.

Key terms: Flow-structure interaction, compressible flows, stability,
resolvent, uniformly bounded semigroup, material derivative

##  1  Introduction

The mathematical analysis of fluid structure interaction (FSI) problems
constitutes a broad area of research with applications in aeroelasticity,
biomechanics and fluid dynamics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 19,
20, 21, 22, 29, 30, 38]. Such interactive dynamics between flow/fluid and a
plate (or shell) are mathematically realized by coupled PDE systems with
compressible flow and elastic plate components. The analysis of these PDE
systems is considered from many points of view [7, 8, 10, 21, 29, 30].

In this work, we consider a linearized flow-structure PDE model with respect
to some reference state which results in the appearance of an arbitrary
spatial flow field. In contrast to the incompressible case, having a
compressible flow component presents a great many difficulties due to the
increase in the number of unknown variables. By the nature and physics of
compressible flows, density can change by pressure forces and a new set of
governing equations are necessarily derived along with the equations for the
conservation of mass and momentum. These equations should be valid for the
flows (compressible) whose range of Mach number is

$MachNumber=M=\frac{\text{velocity}}{\text{local speed of sound}}>0.3.$

The cases $M<0.3$ and $0.3<M<0.8$ are subsonic/incompressible and
subsonic/compressible regimes, respectively. Compressible flows can be either
transonic $(0.8<M<1.2)$ or supersonic $(1.2<M<3.0)$. In supersonic flows,
pressure effects are only transported downstream; the upstream flow is not
affected by conditions downstream.

Our principle aim is to consider the long-time behavior of the corresponding
coupled FSI system with a focus of (asymptotic) strong stability properties of
the $C_{0}-$semigroup generated by the solution. This asymptotic decay for
solutions of the compressible flow-structure PDE model will be stated within
the context of the associated semigroup formulation and “frequency domain”
approach.

The FSI Geometry

Let the flow domain $\mathcal{O}\subset\mathbb{R}^{3}$ with Lipschitz boundary
$\partial\mathcal{O}$. We assume that
$\partial\mathcal{O}=\overline{S}\cup\overline{\Omega}$, with
$S\cap\Omega=\emptyset$, and the (structure) domain
$\Omega\subset\mathbb{R}^{3}$ is a _flat_ portion of $\partial\mathcal{O}$
with $C^{2}-$ boundary. In particular, $\partial\mathcal{O}$ has the following
specific configuration:

$\Omega\subset\left\\{x=(x_{1,}x_{2},0)\right\\}\,\text{\ and \ surface
}S\subset\left\\{x=(x_{1,}x_{2},x_{3}):x_{3}\leq 0\right\\}\,.$ (1)

Additionally, the flow domain $\mathcal{O}$ should be curvilinear polyhedral
domain which satisfies the following conditions:

  * •

Each corner of the boundary $\partial\mathcal{O}$ -if any- is diffeomorphic to
a convex cone,

  * •

Each point on an edge of the boundary $\partial\mathcal{O}$ is diffeomorphic
to a wedge with opening $<\pi.$

We note that these additional conditions on the flow domain $\mathcal{O}$ are
necessary for the application of some elliptic regularity results for
solutions of second order boundary value problems on corner domains [24, 27].
We denote the unit outward normal vector to $\partial\mathcal{O}$ by $\
\mathbf{n}(\mathbf{x})$ where $\mathbf{n|}_{\Omega}=[0,0,1]$, and the unit
outward normal vector to $\partial\Omega$ by $\mathbf{\nu}\mathbf{(x)}$. Some
examples of geometries can be seen in Figure 1\.

$\Omega$$S$$\mathcal{O}$

Figure 1: Polyhedral Flow-Structure Geometries

Linearization and the PDE Model.

In what follows we provide some information about the linearization process
and the PDE description of the compressible flow-structure interaction system
under consideration. Firstly, we note that since the linear flow problem here
is already of great technical complexity and mathematical challenge we assume
that the pressure is a linear function of the density; $p(x,t)=C\rho(x,t),$ as
is typically seen in the compressible flow literature, and it is chosen as a
primary variable to solve. For further and detailed explanations of the
physical background concerning the relationship between pressure and density,
the reader is referred to [30, 10].

The linearization takes place around an equilibrium point of the form
$\left\\{p_{\ast},\mathbf{U},\varrho_{\ast}\right\\}$ where the pressure and
density components ${p_{\ast},\varrho_{\ast}}$ are assumed to be scalars (for
simplicity, assume $p_{\ast}=\varrho_{\ast}=1$), and a generally non-zero,
fixed, ambient vector field $\mathbf{U}:\mathcal{O}\rightarrow\mathbb{R}^{3}$

$\mathbf{U}(x_{1},x_{2},x_{3})=[U_{1}(x_{1},x_{2},x_{3}),U_{2}(x_{1},x_{2},x_{3}),U_{3}(x_{1},x_{2},x_{3})].$

At this point, we emphasize that flow linearization is taken with respect to
some inhomogeneous compressible Navier-Stokes system; thus, $\mathbf{U}$ does
not need to be divergence free, generally.

Now, with respect to the above linearization, the small perturbations give the
following physical equations by generalizing the forcing functions:

$(\partial_{t}+\mathbf{U}\cdot\nabla)p+\text{div}(u)+(\text{div}~{}\mathbf{U})p=f(\mathbf{x})\quad{\
\ in\ \ }\mathcal{O}\times\mathbb{R}_{+},$
$(\partial_{t}+\mathbf{U}\cdot\nabla)u-\nu\Delta
u-(\nu+\lambda)\nabla\text{div}u+\nabla p+\nabla\mathbf{U}\cdot
u+(\mathbf{U}\cdot\nabla\mathbf{U})p=\mathbf{F(\mathbf{x})}\quad{\ \ in\
}~{}\mathcal{O}\times\mathbb{R}_{+}.$

(For further discussion, see also [21, 10].) When we delete some of the non-
critical lower order and the benign inhomogeneous terms in the above
equations, this linearization gives rise to the following system of equations,
in solution variables $u(x_{1},x_{2},x_{3},t)$ (flow velocity),
$p(x_{1},x_{2},x_{3},t)$ (pressure), $w_{1}(x_{1},x_{2},t)$ (elastic plate
displacement) and $w_{2}(x_{1},x_{2},t)$ (elastic plate velocity):

$\displaystyle\left\\{\begin{array}[]{l}p_{t}+\mathbf{U}\cdot\nabla
p+\text{div}~{}u\mathbf{+}\text{div}(\mathbf{U)}p=0~{}\text{ in
}~{}\mathcal{O}\times(0,\infty)\\\ u_{t}+\mathbf{U}\cdot\nabla
u+u\cdot\nabla\mathbf{U}-\text{div}\sigma(u)+\eta u+\nabla p=0~{}\text{ in
}~{}\mathcal{O}\times(0,\infty)\\\
(\sigma(u)\mathbf{n}-p\mathbf{n})\cdot\boldsymbol{\tau}=0~{}\text{ on
}~{}\partial\mathcal{O}\times(0,\infty)\\\ u\cdot\mathbf{n}=0~{}\text{ on
}~{}S\times(0,\infty)\\\ u\cdot\mathbf{n}=w_{2}+\mathbf{U}\cdot\nabla
w_{1}\text{ \ \ on }~{}\Omega\times(0,\infty)\text{ }\end{array}\right.$ (7)
$\displaystyle\left\\{\begin{array}[]{l}w_{1_{t}}-w_{2}-\mathbf{U}\cdot\nabla
w_{1}=0\text{ \ \ on }~{}\Omega\times(0,\infty)\\\
w_{2_{t}}+\Delta^{2}w_{1}+\left[2\nu\partial_{x_{3}}(u)_{3}+\lambda\text{div}(u)-p\right]_{\Omega}=0~{}\text{
on }~{}\Omega\times(0,\infty)\\\ w_{1}=\frac{\partial
w_{1}}{\partial\nu}=0~{}\text{ on
}~{}\partial\Omega\times(0,\infty)\end{array}\right.$ (11)
$\displaystyle\begin{array}[]{c}\left[p(0),u(0),w_{1}(0),w_{2}(0)\right]=\left[\overline{p},\overline{u},\overline{w_{1}},\overline{w_{2}}\right]\in\mathcal{H}_{0}.\end{array}$
(13)

Here, $\mathcal{H}_{0}$ is given as follows:

$\mathcal{H}_{0}=\mathcal{\\{}[p_{0},u_{0},w_{1},w_{2}]\in\mathcal{H}:\int\limits_{\mathcal{O}}p_{0}d\mathcal{O}+\int\limits_{\Omega}w_{1}d\Omega=0\mathcal{\\}},\text{\
\ }$ (14)

where

$\mathcal{H}\equiv L^{2}(\mathcal{O})\times\mathbf{L}^{2}(\mathcal{O})\times
H_{0}^{2}(\Omega)\times L^{2}(\Omega)$ (15)

is the associated finite energy (Hilbert) space, topologized by the standard
inner product:

$(\mathbf{y}_{1},\mathbf{y}_{2})_{\mathcal{H}}=(p_{1},p_{2})_{L^{2}(\mathcal{O})}+(u_{1},u_{2})_{\mathbf{L}^{2}(\mathcal{O})}+(\Delta
w_{1},\Delta w_{2})_{L^{2}(\Omega)}+(v_{1},v_{2})_{L^{2}(\Omega)}$ (16)

for any $\mathbf{y}_{i}=(p_{i},u_{i},w_{i},v_{i})\in\mathcal{H},~{}i=1,2.$

It was shown in [7] and [29] that $\mathcal{H}_{0}^{\bot}$ is the null space
of the flow structure semigroup generator closely associated with (2)-(4). It
will be shown below that solutions of (2)-(4), with initial data drawn from
$\mathcal{H}_{0}$, decay asymptotically to the zero state. Also, the terms
$\mathbf{U}\cdot\nabla u+u\cdot\nabla\mathbf{U}$ constitute the so-called
Oseen (linear) approximation of the Navier-Stokes equations [42].

The quantity $\eta>0$ represents a drag force of the domain on the viscous
flow. In addition, the quantity $\mathbf{\tau}$ in (7) is in the space
$TH^{1/2}(\partial\mathcal{O)}$ of tangential vector fields of Sobolev index
1/2; that is,

$\mathbf{\tau}\in
TH^{1/2}(\partial\mathcal{O)=}\\{\mathbf{v}\in\mathbf{H}^{\frac{1}{2}}(\partial\mathcal{O}):\mathbf{v}|_{\partial\mathcal{O}}\cdot\mathbf{n}=0~{}\text{
on }~{}\partial\mathcal{O}\\}.$ (17)

(See e.g., p.846 of [17].) In addition, we take ambient field
$\mathbf{U}\in\mathbf{V}_{0}\cap W$ where

$\mathbf{V}_{0}=\\{\mathbf{v}\in\mathbf{H}^{1}(\mathcal{O})~{}:~{}\left.\mathbf{v}\right|_{\partial\mathcal{O}}\cdot\mathbf{n}=0~{}\text{
on }~{}\partial\mathcal{O}\\},$ (18)
$W=\\{v\in\mathbf{H}^{1}(\mathcal{O}):v\in L^{\infty}(\mathcal{O}),\text{ \ \
}div(v)\in L^{\infty}(\mathcal{O}),\text{ \ \ and \ \ }v|_{\Omega}\in
C^{2}(\overline{\Omega})\\}$ (19)

(The vanishing of the boundary for ambient fields is a standard assumption in
compressible flow literature; see [23],[41],[32],[1].) Moreover, the stress
and strain tensors in the flow PDE component of (7)-(13) are defined
respectively as

$\sigma(\mathbf{\mu})=2\nu\epsilon(\mathbf{\mu})+\lambda[I_{3}\cdot\epsilon(\mathbf{\mu})]I_{3};\text{
\
}\epsilon_{ij}(\mathbf{\mu})=\dfrac{1}{2}\left(\frac{\partial\mathbf{\mu}_{j}}{\partial
x_{i}}+\frac{\partial\mathbf{\mu}_{i}}{\partial x_{j}}\right)\text{, \ }1\leq
i,j\leq 3,$

where Lamé Coefficients $\lambda\geq 0$ and $\nu>0$.

Here, we impose the so called _impermeability condition_ on $\Omega$; namely,
we assume that no fluid passes through the elastic portion of the boundary
during deflection [15, 28]. Also, note that the FSI problem under
consideration has a _material derivative_ term on the deflected interaction
surface which computes the time rate of change of any quantity such as
temperature or velocity (and hence also acceleration) for a portion of a
material in motion. For further details and the physical explanation of the
material derivative boundary conditions, see [10, 30].

##  2 Previous Considerations

The long-time behavior, in particular the stability properties of
incompressible/compressible fluid/flow structure interaction (FSI) systems
have been a popular topic treated by many authors at different levels [3, 4,
5, 6, 7, 9, 13, 19, 20, 22, 29]. The PDE systems under consideration are
generally quite complicated, due to their unbounded hyperbolic-parabolic
coupling mechanisms, both being intrinsic to the underlying physics.

In contrast to the large body of literature on incompressible FSI [3, 4, 5, 6,
9, 11, 12, 13, 14, 19, 20, 22, 38], existing work on compressible flows which
interact with elastic solids is relatively limited due to the inherent
mathematical challenges presented by the extra density (pressure) variable. In
analyzing these compressible FSI PDE models there are the following
challenges: (i) Given that the flow-plate variables are coupled via boundary
interfaces, the FSI geometry is inherently nonsmooth, [21, 23]. Although such
geometries are physically relevant, these domains also give rise to regularity
issues for solutions; (ii) The linearization of the compressible FSI system
under consideration takes place around a rest state which includes a generally
nonzero (not constant) ambient flow field $\mathbf{U}$, and the said pressure
PDE will contain a “convective derivative” term $\mathbf{U}\nabla p$ which is
strictly above the level of finite energy; (iii) The boundary conditions which
couple flow and structure contain nondissipative and unbounded terms which
complicate the PDE analysis.

A preliminary linearized compressible FSI model with $\mathbf{U}\equiv 0$ (and
so without material derivative) was derived by I. Chueshov [21]. In this work,
the author showed the wellposedness and the existence of global attractors in
the case that the structure equation has a von Karman nonlinearity. However,
in this pioneering work, the author noted that his methods (Galerkin approach
and Lyapunov functionals) to show the wellposedness and long-term behavior of
the corresponding system would not accommodate the case of interest; namely
linearization about $\mathbf{U}\neq 0$.

The main reason for the difficulty is the need to control the “convective
derivative” term $\mathbf{U}\nabla p$ which requires a decomposition of the
fluid and pressure solution pair $\\{u,p\\}$ with an eventual appeal to some
elliptic regularity results on nonsmooth domains. Subsequently, the suggested
model was analyzed in [8] via a semigroup formulation and a wellposedness
result with additional interior terms associated to the $\mathbf{U}\neq 0$
under a pure velocity matching condition at the interface was obtained. Then
in [10], the authors re-visited the same problem after a careful derivation of
fluid-structure interface conditions written in terms of “material derivative”
$(\partial t+\mathbf{U}\cdot\nabla)w.$ This material derivative computes the
time rate of change of any quantity such as temperature or velocity (and hence
also acceleration) for a portion of a material in motion. However, since the
material derivative term $\mathbf{U}\nabla w$ is unbounded and nondissipative,
it adds an additional challenge to the analysis. In [10], this lack of
boundedness and dissipativity were ultimately overcome by re-topologizing the
finite energy space, and so the desired wellposedness result was obtained by
an appropriate semigroup formulation. At this point, we should note that in
both papers [8, 10], the obtained semigroup is unfortunately NOT uniformly
bounded (in time) which prevents us to seek long term behavior of the solution
to these corresponding problems.

With a view of looking into stability properties of the flow structure PDE
models considered in [8, 10], it appeared natural to ask: Is it possible to
obtain a semigroup wellposedness and subsequently a stability result, with the
semigroup being bounded uniformly in time, at least in some (inherently
invariant) subspace of the finite energy space? This was indeed a very
important departure point in order to analyze the stability of these FSI
problems since there was not any long time behavior result in the literature
for such classes of FSI systems. Motivated by this question, an initial result
of uniform stability result for the solution to linearized compressible FSI
model (without material derivative) in an appropriate subspace was shown in
[7, 29] by using linear perturbation theory and also some novel multipliers.

##  3 Novelty

Having established the uniform decay result for the solution to the “material
free” compressible FSI system in [7, 29], our next goal was to analyze the
long time behavior of the system, when under the effect of “material term” on
the interaction interface. In order to embark on this stability work, we
firstly needed to have a uniformly bounded (in time) semigroup, again in some
(inherently invariant) subspace of the finite energy space. This result was
obtained in [30]. Now, in the present work, we provide a novel result on the
asymptotic (strong) stability of compressible FSI model under said “material
derivative” boundary conditions. This result ascertains that solution to the
FSI model decays to zero state for all initial data taken from a subspace
$\mathcal{H}_{0}$ which is “almost” the entire phase space $\mathcal{H}$. This
is, to the knowledge of the author, the first such result obtained with
“material derivative” BC.

By way of obtaining the aforesaid asymptotic (strong) stability result, we
operate in the frequency domain which requires us to deal with some static
equations, analogous to (2)-(3), and the resolvent of the generator of the
dynamical system. While we have already been aware of the need to control the
“convective derivative” term $\mathbf{U}\nabla p$ from our previous works, in
this current manuscript the primary challenge is to have tight control of the
material derivative term $\mathbf{U}\nabla w$ in the coupling condition at the
interface since it destroys the dissipative nature of the dynamics. In this
regard, the main challenges associated with the analysis and the novelties are
as follows:

(i) Inner product adjustment for dissipativity: In order to establish a long
time behavior result for the given FSI system, we expect that the energy of
this system is decreasing. However, the presence of the problematic convective
and material derivative terms $\mathbf{U}\nabla p$ and $\mathbf{U}\nabla w$
adds a great challenge to the long term analysis since they can not be treated
as a perturbation and moreover cause a lack of dissipativity property in the
system. To deal with these issues and get the necessary dissipativity estimate
we change the inner-product structure of the state space and we re-topologize
the inherently invariant subspace with an inner product which is equivalent to
the standard inner product defined for the entire state space. In this
construction, we make use of a multiplier which exploits the characterization
of this invariant subspace and the Dirichlet map that extends boundary data
defined on the interaction interface to a harmonic function in the flow
domain. See Theorem 1 below and [30].

(ii) Use of pointwise resolvent criterion for stability: Since the domain of
the associated flow-structure generator is not compactly embedded into finite
energy space $\mathcal{H}$, (see (20)-(21) below) it would seem natural to
appeal to well known strong stability result introduced by Arend-Batty [2].
However, this does not seem practicable for our model since the spectral
analysis of the semigroup generator will require a great amount of
technicalities. For example, to analyze the residual spectrum of the
generator, we need to check that none of the points on the imaginary axis are
the eigenvalues of the adjoint operator of this generator. Unfortunately, it
is not going to be straightforward to get this result since the adjoint,
itself, is given as the sum of three complicated matrix (see Lemma 3).
Instead, we will use the very interesting pointwise resolvent criterion
developed by Chill and Tomilov [18, 40] for the stability of bounded
semigroups which perfectly works for our model (See also [16] which gives a
preliminary version of the resolvent criterion for strong stability.) Hence we
provide a very clean, short and easy-to-follow proof which does not touch the
–very complicated– adjoint operator and the challenges it might cause.

##  4 Preliminaries

Throughout, for a given domain $D$, the norm of corresponding space $L^{2}(D)$
will be denoted as $||\cdot||_{D}$ (or simply $||\cdot||$ when the context is
clear). Inner products in $L^{2}(\mathcal{O})$ or
$\mathbf{L}^{2}(\mathcal{O})$ will be denoted by
$(\cdot,\cdot)_{\mathcal{O}}$, whereas inner products
$L^{2}(\partial\mathcal{O})$ will be written as
$\langle\cdot,\cdot\rangle_{\partial\mathcal{O}}$. We will also denote
pertinent duality pairings as $\left\langle\cdot,\cdot\right\rangle_{X\times
X^{\prime}}$ for a given Hilbert space $X$. The space $H^{s}(D)$ will denote
the Sobolev space of order $s$, defined on a domain $D$; $H_{0}^{s}(D)$ will
denote the closure of $C_{0}^{\infty}(D)$ in the $H^{s}(D)$-norm
$\|\cdot\|_{H^{s}(D)}$. We make use of the standard notation for the boundary
trace of functions defined on $\mathcal{O}$, which are sufficently smooth:
i.e., for a scalar function $\phi\in H^{s}(\mathcal{O})$,
$\frac{1}{2}<s<\frac{3}{2}$, $\gamma(\phi)=\phi\big{|}_{\partial\mathcal{O}},$
which is a well-defined and surjective mapping on this range of $s$, owing to
the Sobolev Trace Theorem on Lipschitz domains (see e.g., [39], or Theorem
3.38 of [37]). Also, $C>0$ will denote a generic constant.

In order to reduce the PDE system (7)-(13) to an ODE setting in the Hilbert
space $\mathcal{H},$ define the operators

$\mathcal{A}=\left[\begin{array}[]{cccc}-\mathbf{U}\mathbb{\cdot}\nabla(\cdot)&-\text{div}(\cdot)&0&0\\\
-\mathbb{\nabla(\cdot)}&\text{div}\sigma(\cdot)-\eta
I-\mathbf{U}\mathbb{\cdot\nabla(\cdot)-}\nabla\mathbf{U}\cdot(\cdot)&0&0\\\
0&0&0&I\\\
\left.\left[\cdot\right]\right|_{\Omega}&-\left[2\nu\partial_{x_{3}}(\cdot)_{3}+\lambda\text{div}(\cdot)\right]_{\Omega}&-\Delta^{2}&0\end{array}\right];$
(20)

and

$B=\left[\begin{array}[]{cccc}-\text{div}(\mathbf{U)(\cdot)}&0&0&0\\\
0&0&0&0\\\ 0&0&\mathbf{U}\mathbb{\cdot}\nabla(\cdot)&0\\\
0&0&0&0\end{array}\right]$ (21)

where $D(\mathcal{A}+B)\subset\mathcal{H}$ is given by

$D(\mathcal{A}+B)=\\{(p_{0},u_{0},w_{1},w_{2})\in
L^{2}(\mathcal{O})\times\mathbf{H}^{1}(\mathcal{O})\times
H_{0}^{2}(\Omega)\times L^{2}(\Omega)~{}:~{}\text{properties
}(A.i)\text{--}(A.vi)~{}~{}\text{hold}\\}:$

  1. (A.i)

$\mathbf{U}\cdot\nabla p_{0}\in L^{2}(\mathcal{O})$

  2. (A.ii)

$\text{div}~{}\sigma(u_{0})-\nabla p_{0}\in\mathbf{L}^{2}(\mathcal{O})$ (So,
$\left[\sigma(u_{0})\mathbf{n}-p_{0}\mathbf{n}\right]_{\partial\mathcal{O}}\in\mathbf{H}^{-\frac{1}{2}}(\partial\mathcal{O})$)

  3. (A.iii)

$-\Delta^{2}w_{1}-\left[2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})\right]_{\Omega}+p_{0}|_{\Omega}\in
L^{2}(\Omega)$ (by elliptic regularity theory $w_{1}\in H^{3}(\Omega))$

  4. (A.iv)

$\left(\sigma(u_{0})\mathbf{n}-p_{0}\mathbf{n}\right)\bot~{}TH^{1/2}(\partial\mathcal{O})$.
That is,

$\left\langle\sigma(u_{0})\mathbf{n}-p_{0}\mathbf{n},\mathbf{\tau}\right\rangle_{\mathbf{H}^{-\frac{1}{2}}(\partial\mathcal{O})\times\mathbf{H}^{\frac{1}{2}}(\partial\mathcal{O})}=0\text{
\ in }\mathcal{D}^{\prime}(\mathcal{O})\text{\ for every }\mathbf{\tau}\in
TH^{1/2}(\partial\mathcal{O})$

  5. (A.v)

$w_{2}+\mathbf{U}\cdot\nabla w_{1}\in H_{0}^{2}(\Omega)$ (and so $w_{2}\in
H_{0}^{1}(\Omega))$

  6. (A.vi)

The flow velocity component $u_{0}=\mathbf{f}_{0}+\widetilde{\mathbf{f}}_{0}$,
where $\mathbf{f}_{0}\in\mathbf{V}_{0}$ and
$\widetilde{\mathbf{f}}_{0}\in\mathbf{H}^{1}(\mathcal{O})$ satisfies111The
existence of an $\mathbf{H}^{1}(\mathcal{O})$-function
$\widetilde{\mathbf{f}}_{0}$ with such a boundary trace on Lipschitz domain
$\mathcal{O}$ is assured; see e.g., Theorem 3.33 of [37].

$\widetilde{\mathbf{f}}_{0}=\begin{cases}0&~{}\text{ on }~{}S\\\
(w_{2}+\mathbf{U}\cdot\nabla w_{1})\mathbf{n}&~{}\text{
on}~{}\Omega\end{cases}$

(and so $\mathbf{f}_{0}|_{\partial\mathcal{O}}\in
TH^{1/2}(\partial\mathcal{O})$).

Then the function $\Phi(t)=\left[p,u,w_{1},w_{2}\right]\in
C([0,T];\mathcal{H})$ that solves the problem (7)-(13) satisfies

$\displaystyle\dfrac{d}{dt}\Phi(t)$ $\displaystyle=$
$\displaystyle(\mathcal{A}+B)\Phi(t);$ $\displaystyle\Phi(0)$ $\displaystyle=$
$\displaystyle\Phi_{0}$ (22)

We should note that the semigroup generator $(\mathcal{A}+B)$ is
$\mathcal{H}_{0}$ \- invariant (see Lemma 5 of [30]). Moreover, in [30] we
showed that the flow-structure PDE system (2)-(4), in the absence of the term
$u\cdot\nabla\mathbf{U}$ in the flow component, is associated with the
generator of a $C_{0}$\- contraction semigroup in $\mathcal{H}_{0}$.
Therefore, by a bounded perturbation argument, $(\mathcal{A}+B)$ generates a
$C_{0}$\- semigroup in $\mathcal{H}_{0},$ which, however, might not be a
contraction. Since the main goal of this manuscript is to show the asymptotic
(strong) stability of the PDE system (7)-(13) for every initial data in the
subspace $\mathcal{H}_{0}$, having a uniformly bounded semigroup will be our
key point to embark on proving our main result, Theorem 4. For the sake of
simplicity, we use the notation

$(\mathcal{A}+B)|_{\mathcal{H}_{0}}=(\mathcal{A}+B).$

In order to analyze the long time behavior of the solution to (7)-(13) in the
reduced space $\mathcal{H}_{0}$ (defined in (14)), we have to guarantee that
the dynamical system (7)-(13) is dissipative. But, unfortunately, the presence
of the generally nonzero ambient vector field $\mathbf{U}$ and the material
derivative term boundary conditions in the model cause the lack of
dissipativity of the generator $(\mathcal{A}+B)$ (with respect to the standard
inner product given in (16).) Hence, to obtain a necessary dissipativity
estimate for each solution variable, we re-topologize the phase space
$\mathcal{H}$ with a new inner product to be used in $\mathcal{H}_{0}$ and
equivalent to the natural inner product given in (16). (See [30]) For the
readers convenience, we provide the details here as well:

With the above notation let us take
$\varphi=\left[p_{0},u_{0},w_{1},w_{2}\right]\in\mathcal{H}_{0},$
$\widetilde{\varphi}=\left[\widetilde{p}_{0},\widetilde{u}_{0},\widetilde{w}_{1},\widetilde{w}_{2}\right]\in\mathcal{H}_{0}.$
Then the new inner product is given as

$((\varphi,\widetilde{\varphi}))_{\mathcal{H}_{0}}=(p_{0},\widetilde{p}_{0})_{\mathcal{O}}+(u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}),\widetilde{u}_{0}-\alpha
D(g\cdot\nabla\widetilde{w}_{1})e_{3}+\xi\nabla\psi(\widetilde{p}_{0},\widetilde{w}_{1}))_{\mathcal{O}}$
$+(\Delta w_{1},\Delta\widetilde{w}_{1})_{\Omega}+(w_{2}+h_{\alpha}\cdot\nabla
w_{1}+\xi
w_{1},\widetilde{w}_{2}+h_{\alpha}\cdot\nabla\widetilde{w}_{1}+\xi\widetilde{w}_{1})_{\Omega},$
(23)

and in turn the norm

$\displaystyle\left\|\left|\varphi\right|\right\|_{\mathcal{H}_{0}}^{2}=\left(\left(\varphi,\varphi\right)\right)_{\mathcal{H}_{0}}$
$\displaystyle=$
$\displaystyle\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|u_{0}-\alpha
D(g\cdot\nabla
w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1})\right\|_{\mathcal{O}}^{2}$ (24)
$\displaystyle+$ $\displaystyle\left\|\Delta
w_{1}\right\|_{\Omega}^{2}+\left\|w_{2}+h_{\alpha}\cdot\nabla w_{1}+\xi
w_{1}\right\|_{\Omega}^{2}$

for every $\varphi=\left[p_{0},u_{0},w_{1},w_{2}\right]\in\mathcal{H}_{0}.$
Here,

$\textbf{a)}\text{ \
}\xi=\frac{(\frac{1}{2}-C_{2}r_{\mathbf{U}})-\sqrt{(\frac{1}{2}-C_{2}r_{\mathbf{U}})^{2}-4C_{2}(C_{1}+C_{2}r_{\mathbf{U}})r_{\mathbf{U}}}}{2(C_{1}+C_{2}r_{\mathbf{U}})},\text{
\ \ \ \ \ {b)} \
}r_{\mathbf{U}}=\left\|\mathbf{U}\right\|_{\ast}+\left\|\mathbf{U}\right\|_{\ast}^{2}+\left\|\mathbf{U}\right\|_{\ast}^{3}$
(25)

where $C_{1},C_{2}>0$ are independent of measurement
$\left\|\mathbf{U}\right\|_{\ast}$ which is given as

$\left\|\mathbf{U}\right\|_{\ast}=\left\|\mathbf{U}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\text{div}(\mathbf{U)}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\mathbf{U}|_{\Omega}\right\|_{C^{2}(\overline{\Omega})}.$
(26)

Also,

(i) the function $\psi=\psi(f,\chi)\in H^{1}(\mathcal{O)}$ is considered to
solve the following BVP for data $f\in L^{2}(\mathcal{O})$ and $\chi\in
L^{2}(\Omega)$

$\left\\{\begin{array}[]{c}-\Delta\psi=f\text{ \ \ \ in \ }\mathcal{O}\\\
\frac{\partial\psi}{\partial n}=0\text{ \ \ on \ }S\\\
\frac{\partial\psi}{\partial n}=\chi\text{ \ \ on \ }\Omega\end{array}\right.$
(27)

with the compatibility condition

$\int\limits_{\mathcal{O}}fd\mathcal{O+}\int\limits_{\Omega}\chi d\Omega=0.$
(28)

We should note that by known elliptic regularity results for the Neumann
problem on Lipschitz domains–see e.g; [31]– we have

$\left\|\psi(f,\chi)\right\|_{H^{\frac{3}{2}}(\mathcal{O)}}\leq\left[\left\|f\right\|_{\mathcal{O}}+\left\|\chi\right\|_{\partial\mathcal{O}}\right].$
(29)

(ii) the map $D(\cdot)$ is the Dirichlet map that extends boundary data
$\varphi$ defined on $\Omega$ to a harmonic function in $\mathcal{O}$
satisfying:

$D\varphi=f\Leftrightarrow\left\\{\begin{array}[]{c}\Delta f=0\text{ \ \ in \
}\mathcal{O}\\\ f|_{\mathcal{\partial O}}=\varphi|_{ext}\text{ \ \ on \ \
}\mathcal{\partial O}\end{array}\right.$ (30)

where

$\varphi|_{ext}=\begin{cases}0&\text{ on }~{}S\\\ \phi&\text{ on
}~{}\Omega\end{cases}$

Then by, e.g., [37, Theorem 3.3.8], and Lax-Milgram Theorem, we deduce that

$D\in\mathcal{L}\big{(}H_{0}^{1/2+\epsilon}(\Omega);H^{1}(\mathcal{O})\big{)}.$
(31)

(iii) the vector field $h_{\alpha}(\cdot)$ is defined as
$h_{\alpha}(\cdot)=\mathbf{U}|_{\Omega}-\alpha g,$ where $g(\cdot)$ is a
$C^{2}$ extension of the normal vector $\mathbf{\nu}(x)$ (recall, with respect
to $\Omega)$ and we specify the parameter $\alpha$ to be

$\alpha=2\left\|\mathbf{U}\right\|_{\ast},$ (32)

where $\left\|\mathbf{U}\right\|_{\ast}$ is as defined in (26).

##  5 Result I: Bounded Semigroup Wellposedness in $\mathcal{H}_{0}$

The bounded semigroup wellposedness is given as follows:

###### Theorem 1

With reference to problem (7)-(13), assume that
$\mathbf{U}\in\mathbf{V_{0}}\cap W$ with

$\left\|\mathbf{U}\right\|_{\ast}=\left\|\mathbf{U}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\text{div}(\mathbf{U)}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\mathbf{U}|_{\Omega}\right\|_{C^{2}(\overline{\Omega})}$

sufficiently small. Then we have the following:
(i) The operator
$(\mathcal{A}+B):D(\mathcal{A}+B)\cap\mathcal{H}_{0}\rightarrow\mathcal{H}_{0}$
is maximal dissipative. In particular, it obeys the following inequality for
all $\phi=\left[p_{0},{u}_{0},w_{1},w_{2}\right]\in
D(\mathcal{A}+B)\cap\mathcal{H}_{0}:$

$\text{Re}(([\mathcal{A}+B]\varphi,\varphi))_{\mathcal{H}_{0}}\leq\left(-\frac{1}{4}+C_{\delta}r_{\mathbf{U}}\right)\left[(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}+\left\|u_{0}\right\|_{\mathcal{O}}^{2}\right]+\left(-\frac{1}{2}+\delta
C^{\ast}\right)\xi\left[\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right],$ (33)

where $0<\delta<\frac{1}{2C^{\ast}}$, and $r_{\mathbf{U}}$ is as given in
(25).
(ii) Consequently, the operator
$(\mathcal{A}+B):D(\mathcal{A}+B)\cap\mathcal{H}_{0}\rightarrow\mathcal{H}_{0}$
generates a strongly continuous semigroup $\\{e^{(\mathcal{A}+B)t}\\}_{t\geq
0}$ on $\mathcal{H}_{0}.$ Hence, for every initial data
$\left[\overline{p},\overline{u},\overline{w_{1}},\overline{w_{2}}\right]\in\mathcal{H}_{0},$
the solution $\left[p(t),{u}(t),w_{1}(t),w_{2}(t)\right]$ of problem (7)-(13)
is given continuously by

$\left[\begin{array}[]{c}p(t)\\\ u(t)\\\ w_{1}(t)\\\
w_{2}(t)\end{array}\right]=e^{(\mathcal{A}+B)t}\left[\begin{array}[]{c}\overline{p}\\\
\overline{u}\\\ \overline{w_{1}}\\\ \overline{w_{2}}\end{array}\right]\in
C([0,T];\mathcal{H}_{0}).$ (34)

Moreover, this semigroup is uniformly bounded in time with respect to the
standard $\mathcal{H}$-inner product. With respect to the special norm in
(24), the semigroup $\\{e^{(\mathcal{A}+B)t}\\}_{t\geq 0}$ is in fact a
contraction.

Proof. Let

$\mathcal{A}_{0}\equiv\mathcal{A}+L_{\mathbf{U}},$ (35)

where

$L_{\mathbf{U}}=\left[\begin{array}[]{cccc}0&0&0&0\\\
0&\nabla\mathbf{U}\cdot(\cdot)&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right]$
(36)

with $D(\mathcal{A}_{0}+B)=D(\mathcal{A}+B).$ Therewith, it is established in
[30] that $\mathcal{A}_{0}+B$ is maximal dissipative in $\mathcal{H}_{0},$ for
all $\left\|\mathbf{U}\right\|_{\ast}$ sufficiently small. In particular, we
have for $\phi=\left[p_{0},{u}_{0},w_{1},w_{2}\right]\in
D(\mathcal{A}_{0}+B)\cap\mathcal{H}_{0}:$

$\text{Re}(([\mathcal{A}_{0}+B]\varphi,\varphi))_{\mathcal{H}_{0}}\leq-\frac{1}{4}(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}-\frac{\eta}{4}\left\|u_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|\Delta
w_{1}\right\|_{\Omega}^{2},$ (37)

where parameter $\xi$ is as given in (25). (For the sake of completion, the
proof of this estimate is given in the Appendix.) Given
$\phi=\left[p_{0},{u}_{0},w_{1},w_{2}\right]\in
D(\mathcal{A}+B)\cap\mathcal{H}_{0},$ invoking (37) we have

$\displaystyle(([\mathcal{A}+B]\varphi,\varphi))_{\mathcal{H}_{0}}$
$\displaystyle=$
$\displaystyle(([\mathcal{A}_{0}+B]\varphi,\varphi))_{\mathcal{H}_{0}}-((L_{\mathbf{U}}\varphi,\varphi))_{\mathcal{H}_{0}}$
(38) $\displaystyle\leq$
$\displaystyle-\frac{1}{4}(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}-\frac{\eta}{4}\left\|u_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}$
$\displaystyle-(\nabla\mathbf{U}\cdot(u_{0}),u_{0}-\alpha D(g\cdot\nabla
w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}))_{\mathcal{O}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}-\frac{\eta}{4}\left\|u_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}$
$\displaystyle-\int\limits_{\partial\mathcal{O}}(u_{0}\cdot\mathbf{n)U}\cdot[u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}))]d\partial\mathcal{O}$
$\displaystyle+\int\limits_{\mathcal{O}}div(u_{0})\mathbf{U}\cdot[u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}))]d\mathcal{O}$
$\displaystyle+\int\limits_{\mathcal{O}}\mathbf{U}\cdot\left(u_{0}\cdot\nabla[u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}))]\right)d\mathcal{O}$

We estimate RHS implicitly using the regularity results of [25, 26, 27], which
are valid under the assumptions made on the geometry.
(I) The mapping in (30) satisfies

$\left\|D(g\cdot\nabla w_{1})\right\|_{H^{1}(\mathcal{O})}\leq
C\left\|w_{1}\right\|_{H_{0}^{2}(\Omega)}$ (39)

Thus, we have
(I-a):

$\displaystyle\left|\alpha\int\limits_{\partial\mathcal{O}}(u_{0}\cdot\mathbf{n)U}\cdot\overline{D(g\cdot\nabla
w_{1})e_{3}}d\partial\mathcal{O}\right|$ $\displaystyle\leq$
$\displaystyle\alpha\left\|\mathbf{U}\right\|_{\ast}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}\left\|w_{1}\right\|_{H_{0}^{2}(\Omega)}$
(40) $\displaystyle=$
$\displaystyle\sqrt{\left\|\mathbf{U}\right\|_{\ast}}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}\frac{\alpha\sqrt{\left\|\mathbf{U}\right\|_{\ast}}}{\sqrt{\xi}}\sqrt{\xi}\left\|\Delta
w_{1}\right\|_{\Omega}$ $\displaystyle\leq$ $\displaystyle
C_{\delta}\left\|\mathbf{U}\right\|_{\ast}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\frac{\delta\alpha^{2}\left\|\mathbf{U}\right\|_{\ast}\xi}{\xi}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}$ $\displaystyle\leq$ $\displaystyle
C_{\delta}\left\|\mathbf{U}\right\|_{\ast}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\delta
C\frac{r_{\mathbf{U}}}{\xi}\xi\left\|\Delta w_{1}\right\|_{\Omega}^{2}$
$\displaystyle\leq$ $\displaystyle
C_{\delta}\left\|\mathbf{U}\right\|_{\ast}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\delta
C\xi\left\|\Delta w_{1}\right\|_{\Omega}^{2}.$

At this point, we should emphasize that $\frac{r_{\mathbf{U}}}{\xi}$ will be
bounded if $\left\|\mathbf{U}\right\|_{\ast}$ is small enough and so $C>0$ is
independent of $\left\|\mathbf{U}\right\|_{\ast}$. Also, we implicitly used
the Sobolev Trace Theorem for the first inequality.
(I-b): Again by (39), similarly

$\left|\alpha\int\limits_{\mathcal{O}}\mathbf{U}\cdot\left(u_{0}\cdot\nabla\overline{D(g\cdot\nabla
w_{1})e_{3}}\right)d\mathcal{O}\right|\leq
C_{\delta}\left\|\mathbf{U}\right\|_{\ast}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\delta
C\xi\left\|\Delta w_{1}\right\|_{\Omega}^{2}$ (41)

(II)  Using (29) and (39), we have as in (I-a),

$\displaystyle\left|\int\limits_{\mathcal{O}}div(u_{0})\mathbf{U}\cdot[u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}))]d\mathcal{O}\right|$
$\displaystyle\leq$ $\displaystyle
C_{\delta}(\left\|\mathbf{U}\right\|_{\ast}^{2}+\left\|\mathbf{U}\right\|_{\ast})\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}$
(42) $\displaystyle+$ $\displaystyle\delta
C\xi\left[\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right]$

(III) The mapping (27) satisfies

$\left|\psi(p_{0},w_{1})\right\|_{H^{2}(\mathcal{O)}}\leq\left[\left\|p_{0}\right|_{\mathcal{O}}+\left\|w_{1}\right|_{H_{0}^{\frac{1}{2}+\epsilon}(\Omega\mathcal{)}}\right]$
(43)

Consequently, we have

$\displaystyle\left|\xi\int\limits_{\partial\mathcal{O}}(u_{0}\cdot\mathbf{n)U}\cdot\overline{\nabla\psi(p_{0},w_{1})}d\partial\mathcal{O}\right|+\left|\xi\int\limits_{\mathcal{O}}\mathbf{U}\cdot\left(u_{0}\cdot\nabla[\nabla\psi(p_{0},w_{1}))]\right)d\mathcal{O}\right|$
(44) $\displaystyle\leq$ $\displaystyle
C\xi\left\|\mathbf{U}\right\|_{\ast}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}\left[\left\|p_{0}\right\|_{\mathcal{O}}+\left\|\Delta
w_{1}\right\|_{\Omega}\right]$ $\displaystyle\leq$ $\displaystyle
C_{\delta}\left\|\mathbf{U}\right\|_{\ast}^{2}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\delta
C\xi\left[\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right]$

Applying the estimates (40),(41), (42) and (44) to RHS of (38), we now have

$\displaystyle\text{Re}(([\mathcal{A}+B]\varphi,\varphi))_{\mathcal{H}_{0}}$
$\displaystyle\leq$
$\displaystyle-\frac{1}{4}(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}-\frac{\eta}{4}\left\|u_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}$
$\displaystyle+C_{\delta}\left\|\mathbf{U}\right\|_{\ast}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+C_{\delta}\left\|\mathbf{U}\right\|_{\ast}^{2}\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}$
$\displaystyle+\delta
C\xi\left[\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right]$

which yields the estimate (33), upon a use of Korn’s inequality and this gives
the dissipativity of $\mathcal{A}+B,$ for $\mathbf{U}$ sufficiently small and
$0<\delta<\frac{1}{2C^{\ast}}.$ Moreover, it is shown in [30] that
$\mathcal{A}_{0}+B$ is maximal dissipative, then by a perturbation
argument–see e.g., p. 211 of [33]– $\mathcal{A}+B$ is likewise maximal
dissipative. This concludes the proof of Theorem 1.

##  6 Result II: Asymptotic (Strong) Stability

This section is devoted to address the issue of asymptotic behavior of the
solution whose existence-uniqueness is guaranteed by Theorem 1. In this
regard, we show that the system given in (7)-(13) is strongly stable in
$\mathcal{H}_{0}$. That is, given any
$\left[\overline{p},\overline{u},\overline{w_{1}},\overline{w_{2}}\right]\in\mathcal{H}_{0},$
the corresponding solution $\left[p(t),{u}(t),w_{1}(t),w_{2}(t)\right]\in
C([0,T];\mathcal{H}_{0})$ of (7)-(13) satisfies

$\lim_{t\rightarrow\infty}\left\||\left[p(t),{u}(t),w_{1}(t),w_{2}(t)\right]|\right\|_{\mathcal{H}_{0}}=0.$

Our proof will be based on an ultimate appeal to the pointwise criterion [18,
pp. 26, Theorem 8.4 (i)]:

###### Theorem 2

Let $A$ be the generator of a bounded $C_{0}$-semigroup on a Hilbert space
$X$. If there exists a dense set $M\subset X$ such that

$\underset{\alpha\rightarrow 0^{+}}{\lim}\sqrt{\alpha}R(\alpha+i\beta;A)x=0$

for every $x\in M$ and every $\beta\in\mathbb{R},$ then the semigroup is
stable.

There are reasons why we use here the resolvent criterion in [18], instead of
the more wellknown approaches in [36] and [2]. For one, the domain of the
flow-structure generator $\mathcal{A}+B$ is not compactly embedded into
$\mathcal{H}_{0},$ and so the classical stabilizability approach in [36] is
not possible. In addition, the spectrum criterion for stability in [2] is not
applicable here since it would entail the elimination of all three parts of
$\sigma(\mathcal{A}+B);$ particularly, the analysis of residual spectrum will
be of great technical complexity and mathematical challenge since it requires
one to prove that $i\beta$ $(\beta\neq 0)$ is not an eigenvalue of the adjoint
of the generator $\mathcal{A}+B$. In order to justify our concern related to a
residual spectrum analysis and to see how technical calculations the adjoint
of the generator will require, we provide here the adjoint operator
$(\mathcal{A}+B)^{\ast}$ (For a detailed proof, the reader is referred to [30,
Lemma 13]):

###### Lemma 3

With reference to problem (7)-(13), the adjoint operator

$(\mathcal{A}+B)^{\ast}:D((\mathcal{A}+B)^{\ast})\cap\mathcal{H}_{0}\subset\mathcal{H}_{0}\rightarrow\mathcal{H}_{0}$

of the semigroup generator $\mathcal{A}+B$ is defined as

$(\mathcal{A}+B)^{\ast}=\mathcal{A}^{\ast}+B^{\ast}$
$=\left[\begin{array}[]{cccc}\mathbf{U}\mathbb{\cdot}\nabla(\cdot)&\text{div}(\cdot)&0&0\\\
\mathbb{\nabla(\cdot)}&\text{div}\sigma(\cdot)-\eta
I+\mathbf{U}\mathbb{\cdot\nabla(\cdot)-\nabla}\mathbf{U\cdot(\cdot)}&0&0\\\
0&0&0&-I\\\
-\left[\cdot\right]_{\Omega}&-\left[2\nu\partial_{x_{3}}(\cdot)_{3}+\lambda\text{div}(\cdot)\right]_{\Omega}&\Delta^{2}&0\end{array}\right]$
$+\left[\begin{array}[]{cccc}\text{div}(\mathbf{U)}(\cdot)&0&0&0\\\
0&\text{div}(\mathbf{U)}(\cdot)&0&0\\\
{{\mathring{A}}^{-1}}\left\\{\text{div}{{([U}_{1},U_{2}]{)+\mathbf{U\cdot}\nabla)}}\right\\}{(\cdot)|}_{\Omega}&{{\mathring{A}}^{-1}\left\\{\text{div}{{[U}_{1},U_{2}]{+\mathbf{U\cdot}\nabla}}\right\\}}\left[2\nu\partial_{x_{3}}(\cdot)_{3}+\lambda\text{div}(\cdot)\right]_{\Omega}&0&0\\\
0&0&0&0\end{array}\right]$
$+\left[\begin{array}[]{cccc}-\text{div}(\mathbf{U)(\cdot)}&0&0&0\\\
0&0&0&0\\\
0&0&-{{\mathring{A}}^{-1}\left\\{(\text{div}{{[U}_{1},U_{2}]{+\mathbf{U\cdot}\nabla)}\Delta}^{2}(\cdot)\right\\}+}\mathbf{U}\mathbb{\cdot}\nabla(\cdot)+\Delta{{\mathring{A}}^{-1}\nabla}^{\ast}(\mathbb{\nabla\cdot}(\mathbf{U}\mathbb{\cdot}\nabla(\cdot))\mathbb{)}&0\\\
0&0&0&0\end{array}\right]$ $=L_{1}+L_{2}+B^{\ast}$ (45)

Here,
$\nabla^{\ast}\in\mathcal{L}(L^{2}(\Omega),[H^{1}(\Omega)]^{{}^{\prime}})$ is
the adjoint of the gradient operator
$\nabla\in\mathcal{L}(H^{1}(\Omega),L^{2}(\Omega))$ and the domain of
$(\mathcal{A}+B)^{\ast}|_{\mathcal{H}_{0}}$ is given as

$D((\mathcal{A}+B)^{\ast})\cap\mathcal{H}_{0}=\\{(p_{0},u_{0},w_{1},w_{2})\in
L^{2}(\mathcal{O})\times\mathbf{H}^{1}(\mathcal{O})\times
H_{0}^{2}(\Omega)\times L^{2}(\Omega)~{}:~{}\text{properties
}\mathbf{(A^{\ast}.i)}\text{--}\mathbf{(A^{\ast}.vii)}~{}~{}\text{hold}\\},$

where

  1. 1.

$\mathbf{(A^{\ast}.i)}$ $\mathbf{U}\cdot\nabla p_{0}\in L^{2}(\mathcal{O})$

  2. 2.

$\mathbf{(A^{\ast}.ii)}$ $\text{div}~{}\sigma(u_{0})+\nabla
p_{0}\in\mathbf{L}^{2}(\mathcal{O})$ (So,
$\left[\sigma(u_{0})\mathbf{n}+p_{0}\mathbf{n}\right]_{\partial\mathcal{O}}\in\mathbf{H}^{-\frac{1}{2}}(\partial\mathcal{O})$)

  3. 3.

$\mathbf{(A^{\ast}.iii)}$
$\Delta^{2}w_{1}-\left[2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})\right]_{\Omega}-p_{0}|_{\Omega}\in
L^{2}(\Omega)$

  4. 4.

$\mathbf{(A^{\ast}.iv)}$
$\left(\sigma(u_{0})\mathbf{n}+p_{0}\mathbf{n}\right)\bot~{}TH^{1/2}(\partial\mathcal{O})$.
That is,

$\left\langle\sigma(u_{0})\mathbf{n}+p_{0}\mathbf{n},\mathbf{\tau}\right\rangle_{\mathbf{H}^{-\frac{1}{2}}(\partial\mathcal{O})\times\mathbf{H}^{\frac{1}{2}}(\partial\mathcal{O})}=0\text{
\ in }\mathcal{D}^{\prime}(\mathcal{O})\text{\ for every }\mathbf{\tau}\in
TH^{1/2}(\partial\mathcal{O})$

  5. 5.

$\mathbf{(A^{\ast}.v)}$ The flow velocity component
$u_{0}=\mathbf{f}_{0}+\widetilde{\mathbf{f}}_{0}$, where
$\mathbf{f}_{0}\in\mathbf{V}_{0}$ and
$\widetilde{\mathbf{f}}_{0}\in\mathbf{H}^{1}(\mathcal{O})$ satisfies

$\widetilde{\mathbf{f}}_{0}=\begin{cases}0&~{}\text{ on }~{}S\\\
w_{2}\mathbf{n}&~{}\text{ on}~{}\Omega\end{cases}$

(and so $\left.\mathbf{f}_{0}\right|_{\partial\mathcal{O}}\in
TH^{1/2}(\partial\mathcal{O})$)

  6. 6.

$\mathbf{(A^{\ast}.vi)}$ $[-w_{2}+\mathbf{U}\mathbb{\cdot}\nabla
w_{1}+\Delta{{\mathring{A}}^{-1}\nabla}^{\ast}(\mathbb{\nabla\cdot}(\mathbf{U}\mathbb{\cdot}\nabla
w_{1})\mathbb{)}]\in H_{0}^{2}(\Omega),$ (and so $w_{2}\in H_{0}^{1}(\Omega)$)

  7. 7.

$\mathbf{(A^{\ast}.vii)}$ $\int\limits_{\mathcal{O}}[\mathbf{U}\cdot\nabla
p_{0}+$div$~{}(u_{0})]d\mathcal{O}$
$+\int\limits_{\Omega}{{\mathring{A}}^{-1}}\left\\{(\text{div}{[U}_{1},U_{2}{]+\mathbf{U\cdot}\nabla)(\left[p_{0}+2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})\right]_{\Omega})}\right\\}d\Omega$
$-\int\limits_{\Omega}{{\mathring{A}}^{-1}\left\\{(\text{div}{{[U}_{1},U_{2}]{+\mathbf{U\cdot}\nabla)}\Delta}^{2}{w}_{1}\right\\}}d\Omega$
$+\int\limits_{\Omega}[\mathbf{U}\mathbb{\cdot}\nabla
w_{1}+\Delta{{\mathring{A}}^{-1}\nabla}^{\ast}(\mathbb{\nabla\cdot}(\mathbf{U}\mathbb{\cdot}\nabla
w_{1})\mathbb{)]}d\Omega$
$=0.$

Looking at the expression for $\mathcal{A}^{\ast}+B^{\ast}$ in (45), it seems
that showing this adjoint operator has empty null space–necessary for ruling
out residual spectrum of $\mathcal{A}+B$–would be a difficult exercise. Now,
our stability result is as follows:

###### Theorem 4

The bounded $C_{0}-$semigroup $\left\\{e^{(\mathcal{A}+B)t}\right\\}_{t\geq
0}$ given in Theorem 1 is strongly stable under the condition that

$\left\|\mathbf{U}\right\|_{\ast}=\left\|\mathbf{U}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\text{div}(\mathbf{U)}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\mathbf{U}|_{\Omega}\right\|_{C^{2}(\overline{\Omega})}$

is small enough . That is, the solution
$\phi(t)=\left[p(t),{u}(t),w_{1}(t),w_{2}(t)\right]$ of the PDE system
(7)-(13) tends asymptotically to the zero state for all initial data
$\phi_{0}\in\mathcal{H}_{0}.$

Proof. The proof relies on the pointwise resolvent criterion given in Theorem
2, and hence to obtain the strong stability result it will be enough to show
that the resolvent operator

$R(a+ib;[\mathcal{A}+B])=((a+ib)I-[\mathcal{A}+B])^{-1}$

obeys the limit estimate:

$\underset{a\rightarrow
0^{+}}{\lim}\left\|\left|\sqrt{a}R(a+ib;[\mathcal{A}+B])\phi^{\ast}\right|\right\|_{\mathcal{H}_{0}}=0$
(46)

where $a>0$, $b\in\mathbb{R}$ and given $\phi^{\ast}\in\mathcal{H}_{0}.$ Here,
we invoke the special inner product $((\cdot,\cdot))_{\mathcal{H}_{0}}$
(defined in (23)) to get the necessary estimates but since the norms
$||\cdot||_{\mathcal{H}}$ and $||\cdot||_{\mathcal{H}_{0}}$ are equivalent, we
obtain the strong stability with respect to the standard inner product as
well. Let

$\phi=\left[\begin{array}[]{c}p_{0}\\\ u_{0}\\\ w_{1}\\\
w_{2}\end{array}\right]=R(a+ib;[\mathcal{A}+B])\phi^{\ast}\in
D(\mathcal{A}+B)\cap\mathcal{H}_{0},\text{ \ \ \ \ \ \
}\phi^{\ast}=\left[\begin{array}[]{c}p_{0}^{\ast}\\\ u_{0}^{\ast}\\\
w_{1}^{\ast}\\\ w_{2}^{\ast}\end{array}\right]\in\mathcal{H}_{0}$

Then $\phi$ solves the following static PDE system:

$\displaystyle\left\\{\begin{array}[]{l}(a+ib)p_{0}+\mathbf{U}\cdot\nabla
p_{0}+\text{div}~{}u_{0}\mathbf{+}\text{div}(\mathbf{U)}p_{0}=p_{0}^{\ast}~{}\text{
in }~{}\mathcal{O}\\\ (a+ib)u_{0}+\mathbf{U}\cdot\nabla
u_{0}+\nabla\mathbf{U\cdot u}_{0}-\text{div}\sigma(u_{0})+\eta u_{0}+\nabla
p_{0}=u_{0}^{\ast}~{}\text{ in }~{}\mathcal{O}\\\
(\sigma(u_{0})\mathbf{n}-p_{0}\mathbf{n})\cdot\boldsymbol{\tau}=0~{}\text{ on
}~{}\partial\mathcal{O}\\\ u_{0}\cdot\mathbf{n}=0~{}\text{ on }~{}S\\\
u_{0}\cdot\mathbf{n}=w_{2}+\mathbf{U}\cdot\nabla w_{1}\text{ \ \ on
}~{}\Omega\text{ }\end{array}\right.$ (52)
$\displaystyle\left\\{\begin{array}[]{l}(a+ib)w_{1}-w_{2}-\mathbf{U}\cdot\nabla
u_{0}=w_{1}^{\ast}\\\
(a+ib)w_{1}+\Delta^{2}w_{1}+\left[2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})-p_{0}\right]_{\Omega}=w_{2}^{\ast}~{}\text{
on }~{}\Omega\\\ w_{1}=\frac{\partial w_{1}}{\partial\nu}=0~{}\text{ on
}~{}\partial\Omega\end{array}\right.$ (56)

We invoke the dissipativity estimate (33) in Theorem 1 (i):

$(a+ib)\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}^{2}-\left(\left([\mathcal{A}+B])\phi,\phi\right)\right)_{\mathcal{H}_{0}}=\left(\left(\phi^{\ast},\phi\right)\right)_{\mathcal{H}_{0}},$

or

$a\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}^{2}-\text{Re}\left(\left([\mathcal{A}+B])\phi,\phi\right)\right)_{\mathcal{H}_{0}}=\text{Re}\left(\left(\phi^{\ast},\phi\right)\right)_{\mathcal{H}_{0}}$

which gives

$\left(\frac{1}{4}-C_{\delta}r_{\mathbf{U}}\right)\left[(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}+\left\|u_{0}\right\|_{\mathcal{O}}^{2}\right]+\left(\frac{1}{2}-\delta
C^{\ast}\right)\xi\left[\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right]\leq\left|\text{Re}\left(\left(\phi^{\ast},\phi\right)\right)_{\mathcal{H}_{0}}\right|.$
(57)

In turn, from the boundary condition
$w_{2}=u_{0}\cdot\mathbf{n}-\mathbf{U}\cdot\nabla w_{1}$ and (57), we get

$\left\|w_{2}\right\|_{\Omega}\leq
C_{\xi,\mathbf{U}}\sqrt{\left|\text{Re}\left(\left(\phi^{\ast},\phi\right)\right)_{\mathcal{H}_{0}}\right|}$
(58)

where we have also used implicitly the Sobolev Trace Theorem. Now, combining
(57) and (58), and using the equivalence of norms
$\left\|\left|\cdot\right|\right\|_{\mathcal{H}_{0}}$ and
$\left\|\cdot\right\|_{\mathcal{H}}$ on $\mathcal{H}_{0},$ we have then

$\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}\leq
C_{\xi,\mathbf{U}}\sqrt{\left|\text{Re}\left(\left(\phi^{\ast},\phi\right)\right)_{\mathcal{H}_{0}}\right|}.$

Scaling the above inequality by $\sqrt{a},$ followed by Young’s Inequality
gives now

$\displaystyle\sqrt{a}\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}$
$\displaystyle\leq$
$\displaystyle\sqrt{a}C_{\xi,\mathbf{U}}\left\|\left|\phi^{\ast}\right|\right\|_{\mathcal{H}_{0}}^{\frac{1}{2}}\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{a}}{2}C_{\xi,\mathbf{U}}^{2}\left\|\left|\phi^{\ast}\right|\right\|_{\mathcal{H}_{0}}+\frac{\sqrt{a}}{2}\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}.$

From the last inequality, we have

$\frac{\sqrt{a}}{2}\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}\leq\frac{\sqrt{a}}{2}C_{\xi,\mathbf{U}}^{2}\left\|\left|\phi^{\ast}\right|\right\|_{\mathcal{H}_{0}},$

and so

$\underset{a\rightarrow
0^{+}}{\lim}\sqrt{a}\left\|\left|\phi\right|\right\|_{\mathcal{H}_{0}}=0.$

This gives the desired limit estimate (46) and concludes the proof of Theorem
4.

##  7 Appendix

In this paper, one of the key points used to analyze the stability properties
of the PDE system (7)-(13) is the dissipativity estimate (33) in the
inherently invariant subspace of the finite energy space. To show this, we
appeal to the dissipativity result given in [30] for an operator which is
closely associated our generator $\mathcal{A}+B$. For the readers convenience,
we also provide its proof.

###### Lemma 5

Let $\mathcal{A}_{0}$ be the operator defined in (35). Then, with reference to
problem (7)-(13), the semigroup generator
$(\mathcal{A}_{0}+B):D(\mathcal{A}_{0}+B)\cap\mathcal{H}_{0}\subset\mathcal{H}_{0}\rightarrow\mathcal{H}_{0}$
is dissipative with respect to inner product
$((\cdot,\cdot))_{\mathcal{H}_{0}}$ for
$\left\|\mathbf{U}\right\|_{\ast}=\left\|\mathbf{U}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\text{div}(\mathbf{U)}\right\|_{L^{\infty}(\mathcal{O})}+\left\|\mathbf{U}|_{\Omega}\right\|_{C^{2}(\overline{\Omega})}$
small enough. In particular, for
$\varphi=\left[p_{0},u_{0},w_{1},w_{2}\right]\in
D(\mathcal{A}_{0}+B)\cap\mathcal{H}_{0},$

$\text{Re}(([\mathcal{A}_{0}+B]\varphi,\varphi))_{\mathcal{H}_{0}}\leq-\frac{(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}}{4}-\frac{\eta\left\|u_{0}\right\|_{\mathcal{O}}^{2}}{4}-\frac{\xi\left\|p_{0}\right\|_{\mathcal{O}}^{2}}{2}-\frac{\xi\left\|\Delta
w_{1}\right\|_{\Omega}^{2}}{2},$ (59)

where $\xi$ is specified in (92).

Proof. Given $\varphi=\left[p_{0},u_{0},w_{1},w_{2}\right]\in
D(\mathcal{A}_{0}+B)\cap\mathcal{H}_{0},$ we have

$(([\mathcal{A}_{0}+B]\varphi,\varphi))_{\mathcal{H}_{0}}=(-\mathbf{U}\nabla
p_{0}-\text{div}(u_{0})-\text{div}(\mathbf{U})p_{0},p_{0})_{\mathcal{O}}$
$+(-\nabla p_{0}+\text{div}\sigma(u_{0})-\eta u_{0}-\mathbf{U}\nabla
u_{0},u_{0}-\alpha D(g\cdot\nabla w_{1})e_{3})_{\mathcal{O}}$ $+(-\nabla
p_{0}+\text{div}\sigma(u_{0})-\eta u_{0}-\mathbf{U}\nabla
u_{0},\xi\nabla\psi(p_{0},w_{1}))_{\mathcal{O}}$
$-\alpha(D(g\cdot\nabla[w_{2}+\mathbf{U}\nabla w_{1}])e_{3},u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}))_{\mathcal{O}}$
$+\xi(\nabla\psi(-\mathbf{U}\nabla
p_{0}-\text{div}(u_{0})-\text{div}(\mathbf{U})p_{0},w_{2}+\mathbf{U}\nabla
w_{1}),u_{0}-\alpha D(g\cdot\nabla w_{1})e_{3})_{\mathcal{O}}$
$+\xi^{2}(\nabla\psi(-\mathbf{U}\nabla
p_{0}-\text{div}(u_{0})-\text{div}(\mathbf{U})p_{0},w_{2}+\mathbf{U}\nabla
w_{1}),\nabla\psi(p_{0},w_{1}))_{\mathcal{O}}$ $+(\Delta w_{2},\Delta
w_{1})_{\Omega}+(\Delta(\mathbf{U}\nabla w_{1}),\Delta w_{1})_{\Omega}$
$+(p_{0}|_{\Omega}-\left[2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})\right]|_{\Omega},w_{2}+h_{\alpha}\cdot\nabla
w_{1}+\xi w_{1})_{\Omega}$ $+(h_{\alpha}\cdot\nabla[w_{2}+\mathbf{U}\nabla
w_{1}],w_{2}+h_{\alpha}\cdot\nabla w_{1}+\xi w_{1})_{\Omega}$
$-(\Delta^{2}w_{1},w_{2}+h_{\alpha}\cdot\nabla w_{1}+\xi w_{1})_{\Omega}$
$+\xi(w_{2}+\mathbf{U}\nabla w_{1},w_{2}+h_{\alpha}\cdot\nabla w_{1}+\xi
w_{1})_{\Omega}.$

After integration by parts we then arrive at

$(([\mathcal{A}_{0}+B]\varphi,\varphi))_{\mathcal{H}_{0}}=-(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}-\eta\left\|u_{0}\right\|_{\mathcal{O}}^{2}+\frac{1}{2}\int\limits_{\mathcal{O}}\text{div}(\mathbf{U})[|u_{0}|^{2}-|p_{0}|^{2}]d\mathcal{O}$
$+2i\text{Im}[(p_{0},\text{div}(u_{0}))_{\mathcal{O}}+(\Delta w_{2},\Delta
w_{1})_{\Omega}]-i\text{Im}[(\mathbf{U}\nabla
p_{0},p_{0})_{\mathcal{O}}+(\mathbf{U}\nabla u_{0},u_{0})_{\mathcal{O}}]$
$+\sum\limits_{j=1}^{8}I_{j},$ (60)

where above the $I_{j}$ are given by:

$I_{1}=(\nabla p_{0}-\text{div}\sigma(u_{0})+\eta u_{0}+\mathbf{U}\nabla
u_{0},\alpha D(g\cdot\nabla w_{1})e_{3})_{\mathcal{O}}$
$-\alpha(p_{0}|_{\Omega}-\left[2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})\right]|_{\Omega},g\cdot\nabla
w_{1})_{\Omega},$ (61) $I_{2}=(-\nabla p_{0}+\text{div}\sigma(u_{0})-\eta
u_{0}-\mathbf{U}\nabla
u_{0},\xi\nabla\psi(p_{0},w_{1}))_{\mathcal{O}}-\xi(\Delta^{2}w_{1},w_{1})_{\Omega}$
$+(p_{0}|_{\Omega}-\left[2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})\right]|_{\Omega},\xi
w_{1})_{\Omega},$ (62) $I_{3}=-\alpha(D(g\cdot\nabla[w_{2}+\mathbf{U}\nabla
w_{1}])e_{3},u_{0}-\alpha D(g\cdot\nabla
w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1}))_{\mathcal{O}},$ (63)
$I_{4}=\xi(\nabla\psi(-\mathbf{U}\nabla
p_{0}-\text{div}(u_{0})-\text{div}(\mathbf{U})p_{0},w_{2}+\mathbf{U}\nabla
w_{1}),u_{0}-\alpha D(g\cdot\nabla w_{1})e_{3})_{\mathcal{O}},$ (64)
$I_{5}=\xi^{2}(\nabla\psi(-\mathbf{U}\nabla
p_{0}-\text{div}(u_{0})-\text{div}(\mathbf{U})p_{0},w_{2}+\mathbf{U}\nabla
w_{1}),\nabla\psi(p_{0},w_{1}))_{\mathcal{O}},$ (65)
$I_{6}=(\Delta(\mathbf{U}\nabla w_{1}),\Delta
w_{1})_{\Omega}-(\Delta^{2}w_{1},h_{\alpha}\cdot\nabla w_{1})_{\Omega},$ (66)
$I_{7}=(h_{\alpha}\cdot\nabla[w_{2}+\mathbf{U}\nabla w_{1}],w_{2})_{\Omega},$
(67) $I_{8}=(h_{\alpha}\cdot\nabla[w_{2}+\mathbf{U}\nabla
w_{1}],h_{\alpha}\cdot\nabla w_{1}+\xi w_{1})_{\Omega}$
$+\xi(w_{2}+\mathbf{U}\nabla w_{1},w_{2}+h_{\alpha}\cdot\nabla w_{1}+\xi
w_{1})_{\Omega}.$ (68)

where we also recall the definition $h_{\alpha}=\mathbf{U}|_{\Omega}-\alpha
g.$ In the course of estimating the terms (61)-(68) above, we will invoke the
polynomial

$r(a)=a+a^{2}+a^{3},$

and for the simplicity, we set

$r_{\mathbf{U}}=r(\left\|\mathbf{U}\right\|_{\ast}).$

We start with $I_{1};$ integrating by parts, we have

$I_{1}=-\alpha(p_{0},\text{div}[D(g\cdot\nabla
w_{1})e_{3}])_{\mathcal{O}}+\alpha(\sigma(u_{0}),\epsilon(D(g\cdot\nabla
w_{1})e_{3})_{\mathcal{O}}$ $+\alpha\eta(u_{0},D(g\cdot\nabla
w_{1})e_{3})_{\mathcal{O}}+\alpha(\mathbf{U}\nabla u_{0},D(g\cdot\nabla
w_{1})e_{3})_{\mathcal{O}}$ (69)

Using the fact that Dirichlet map $D\in
L(H_{0}^{\frac{1}{2}+\epsilon}(\Omega),H^{1}(\mathcal{O}))$, we have

$I_{1}\leq
r_{\mathbf{U}}C\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}$ (70)

We continue with $I_{2};$ using the definition of the map $\psi(\cdot,\cdot)$
in (27) and integrating by parts we get

$I_{2}=-\xi\int\limits_{\mathcal{O}}\left|p_{0}\right|^{2}d\mathcal{O}-\xi(\sigma(u_{0}),\epsilon(\nabla\psi(p_{0},w_{1})))_{\mathcal{O}}$
$+\xi\left\langle\sigma(u_{0})\mathbf{n}-p_{0}\mathbf{n},(\nabla\psi(p_{0},w_{1}),\mathbf{n})\mathbf{n}\right\rangle_{\partial\mathcal{O}}-\eta(u_{0},\xi\nabla\psi(p_{0},w_{1}))_{\mathcal{O}}$
$(-\mathbf{U}\nabla
u_{0},\xi\nabla\psi(p_{0},w_{1}))_{\mathcal{O}}-(\Delta^{2}w_{1},\xi
w_{1})_{\Omega}$
$+(p_{0}|_{\Omega}-\left[2\nu\partial_{x_{3}}(u_{0})_{3}+\lambda\text{div}(u_{0})\right]|_{\Omega},\xi
w_{1})_{\Omega},$

whence we obtain

$I_{2}\leq-\xi\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\xi\left\|\Delta
w_{1}\right\|_{\Omega}^{2}+\xi
r_{\mathbf{U}}C\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}$ $+\xi
C\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}\left[\left\|p_{0}\right\|_{\mathcal{O}}+\left\|\Delta
w_{1}\right\|_{\Omega}\right]\right\\}.$ (71)

For $I_{3}:$ recalling the boundary condition

$(u_{0})_{3}|_{\Omega}=w_{2}+\mathbf{U}\nabla w_{1},$

making use of Lemma 6.1 of [10] and considering the assumptions made on the
geometry, we have

$I_{3}\leq\alpha
C\left\|g\cdot\nabla(u_{0})_{3}\right\|_{H^{-\frac{1}{2}}(\Omega)}\left\|u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}+\xi\nabla\psi(p_{0},w_{1})\right\|_{\mathcal{O}}$
$\leq
C\left[r_{\mathbf{U}}\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}+\xi^{2}\left\\{\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}\right]$ (72)

where we have also implicitly used the Sobolev Embedding Theorem. To continue
with $I_{4}:$

$I_{4}=\xi(\nabla\psi(-\mathbf{U}\nabla
p_{0}-\text{div}(\mathbf{U})p_{0},0),u_{0}-\alpha D(g\cdot\nabla
w_{1})e_{3})_{\mathcal{O}}$
$+\xi(\nabla\psi(-\text{div}(u_{0}),u_{0}\cdot\mathbf{n}),u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3})_{\mathcal{O}}$ $=I_{4a}+I_{4b}$ (73)

Since $\mathbf{U}\cdot\mathbf{n|}_{\partial\mathcal{O}}\mathbf{=0,}$ we have
that $(\mathbf{U}\nabla
p_{0}+$div$(\mathbf{U})p_{0})\in[H^{1}(\mathcal{O})]^{{}^{\prime}}$ with

$\left\|\mathbf{U}\nabla
p_{0}+\text{div}(\mathbf{U})p_{0}\right\|_{[H^{1}(\mathcal{O})]^{{}^{\prime}}}\leq
C\left\|\mathbf{U}\right\|_{\ast}\left\|p_{0}\right\|_{\mathcal{O}}.$ (74)

By Lax-Milgram Theorem, we then have

$I_{4a}\leq C\xi\left\|\nabla\psi(-\mathbf{U}\nabla
p_{0}-\text{div}(\mathbf{U})p_{0},0)\right\|_{\mathcal{O}}\left\|u_{0}-\alpha
D(g\cdot\nabla w_{1})e_{3}\right\|_{\mathcal{O}}$ $\leq C\xi
r_{\mathbf{U}}\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}$ (75)

and similarly

$I_{4b}\leq C\xi
r_{\mathbf{U}}\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}.$ (76)

Now, applying (75)-(76) to (73) gives

$I_{4}\leq C\xi
r_{\mathbf{U}}\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}.$ (77)

Estimating $I_{5}:$ we proceed as before done for $I_{4}$ and invoke (74), Lax
Milgram Theorem and the estimate (29) to have

$I_{5}\leq
C\xi^{2}\left[\left\|\mathbf{U}\right\|_{\ast}\left\\{\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}+\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}\right]$
(78)

For $I_{6},$ in order to estimate the second term in (66), we follow the
standard calculations used for the flux multipliers and the commutator symbol
given by

$[P,Q]f=P(Qf)-Q(Pf)$ (79)

for the differential operators $P$ and $Q$. Hence,

$\displaystyle-(\Delta^{2}w_{1},h_{\alpha}\cdot\nabla w_{1})_{\Omega}=$
$\displaystyle~{}(\nabla\Delta w_{1},\nabla(h_{\alpha}\cdot\nabla
w_{1}))_{\Omega}$ (80) $\displaystyle=$ $\displaystyle~{}-(\Delta
w_{1},\Delta(h_{\alpha}\cdot\nabla
w_{1}))_{\Omega}+\int_{\partial\Omega}(h_{\alpha}\cdot\mathbf{\nu})|\Delta
w_{1}|^{2}d\partial\Omega,$ (81)

where, in the first identity we have directly invoked the clamped plate
boundary conditions, and in the second we have used the fact that
$w_{1}=\partial_{\mathbf{\nu}}w_{1}=0$ on $\partial\Omega$ which yields that

$\frac{\partial}{\partial\mathbf{\nu}}(h_{\alpha}\cdot\nabla
w_{1})=(h_{\alpha}\cdot\mathbf{\nu})\frac{\partial^{2}w_{1}}{\partial\mathbf{\nu}}=(h_{\alpha}\cdot\mathbf{\nu})(\Delta
w_{1}\big{|}_{\partial\Omega}).$

(See [34] or [35, p.305]). Using the commutator bracket $[\cdot,\cdot]$, we
can rewrite the latter relation as

$-(\Delta^{2}w_{1},h_{\alpha}\cdot\nabla w_{1})_{\Omega}=~{}-(\Delta
w_{1},[\Delta,h_{\alpha}\cdot\nabla]w_{1})_{\Omega}-(\Delta
w_{1},h_{\alpha}\cdot\nabla(\Delta
w_{1}))_{\Omega}+\int_{\partial\Omega}(h_{\alpha}\cdot\mathbf{\nu})|\Delta
w_{1}|^{2}d\partial\Omega.$

With Green’s relations once more:

$\displaystyle-(\Delta^{2}w_{1},h_{\alpha}\cdot\nabla w_{1})_{\Omega}=$
$\displaystyle~{}-(\Delta
w_{1},[\Delta,h_{\alpha}\cdot\nabla]w_{1})_{\Omega}-\frac{1}{2}\int_{\partial\Omega}(h_{\alpha}\cdot\mathbf{\nu})|\Delta
w_{1}|^{2}d\partial\Omega$
$\displaystyle+\frac{1}{2}\int_{\Omega}\big{[}\text{div}(h_{\alpha})\big{]}|\Delta
w_{1}|^{2}d\Omega-i\text{Im}(\Delta w_{1},h_{\alpha}\cdot\nabla(\Delta
w_{1}))_{\Omega}$
$\displaystyle+\int_{\partial\Omega}(h_{\alpha}\cdot\mathbf{\nu})|\Delta
w_{1}|^{2}d\partial\Omega.$ (82)

Thus,

$\displaystyle-(\Delta^{2}w_{1},h_{\alpha}\cdot\nabla w_{1})_{\Omega}=$
$\displaystyle~{}-(\Delta
w_{1},[\Delta,h_{\alpha}\cdot\nabla]w_{1})_{\Omega}+\frac{1}{2}\int_{\partial\Omega}(h_{\alpha}\cdot\mathbf{\nu})|\Delta
w_{1}|^{2}d\partial\Omega$
$\displaystyle+\frac{1}{2}\int_{\Omega}\big{[}\text{div}(h_{\alpha})\big{]}|\Delta
w_{1}|^{2}d\Omega-i\text{Im}(\Delta w_{1},h_{\alpha}\cdot\nabla(\Delta
w_{1})).$ (83)

Since $h_{\alpha}=\mathbf{U}\big{|}_{\Omega}-\alpha g$, where $g$ is an
extension of $\mathbf{\nu}(\mathbf{x})$, we will have then

$-\text{Re}(\Delta^{2}w_{1},h_{\alpha}\cdot\nabla
w_{1})_{\Omega}=~{}\dfrac{1}{2}\int_{\partial\Omega}(\mathbf{U}\cdot\mathbf{\nu}-\alpha)|\Delta
w_{1}|^{2}d\partial\Omega+\dfrac{1}{2}\int_{\Omega}\text{div}(h_{\alpha})|\Delta
w_{1}|^{2}d\Omega-\text{Re}(\Delta
w_{1},[\Delta,h_{\alpha}\cdot\nabla]w_{1})_{\Omega}$ (84)

Since we can explicitly compute the commutator

$\displaystyle[\Delta,{h_{\alpha}}\cdot\nabla]w_{1}=$ $\displaystyle(\Delta
h_{1})(\partial_{x_{1}}w_{1})+2(\partial_{x_{1}}h_{1})(\partial_{x_{1}}^{2}w_{1})+2(\partial_{x_{2}}h_{2})(\partial_{x_{2}}^{2}w_{1})+(\Delta
h_{2})(\partial_{x_{2}}w_{1})$
$\displaystyle+2\text{div}(h_{\alpha})(\partial_{x_{1}}\partial_{x_{2}}w_{1}),$

and

$\big{|}\big{|}[\Delta,{h_{\alpha}}\cdot\nabla]w_{1}\big{|}\big{|}_{L^{2}(\Omega)}\leq
r_{\mathbf{U}}||\Delta w_{1}||_{L^{2}(\Omega)}.$ (85)

combining (84)-(85) we eventually get

$-\text{Re}(\Delta^{2}w_{1},h_{\alpha}\cdot\nabla
w_{1})_{\Omega}\leq\frac{1}{2}\int\limits_{\partial\Omega}[\mathbf{U\cdot\nu-}\alpha]\left|\Delta
w_{1}\right|^{2}d\partial\Omega+Cr_{\mathbf{U}}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}.$ (86)

Moreover, for the first term of (66), we have

$(\Delta(\mathbf{U}\nabla w_{1}),\Delta w_{1})_{\Omega}=(\mathbf{U}\nabla
w_{1}),\Delta w_{1})_{\Omega}-([\mathbf{U\cdot}\nabla,\Delta]w_{1},\Delta
w_{1})_{\Omega}$
$=\int\limits_{\partial\Omega}(\mathbf{U\cdot\nu})\left|\Delta
w_{1}\right|^{2}d\partial\Omega-\int\limits_{\partial\Omega}\text{div}(\mathbf{U)}\left|\Delta
w_{1}\right|^{2}d\partial\Omega$ $-([\mathbf{U\cdot}\nabla,\Delta]w_{1},\Delta
w_{1})_{\Omega}-\int\limits_{\Omega}\Delta w_{1}\mathbf{U\cdot}\nabla(\Delta
w_{1})d\Omega$

where we also use the commutator expression in (79). This gives us

$\text{Re}(\Delta(\mathbf{U}\nabla w_{1}),\Delta
w_{1})_{\Omega}\leq\frac{1}{2}\int\limits_{\partial\Omega}(\mathbf{U\cdot\nu})\left|\Delta
w_{1}\right|^{2}d\partial\Omega+Cr_{\mathbf{U}}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}.$ (87)

Now applying (86)-(87) to (66), we obtain

$\text{Re}I_{6}\leq\int\limits_{\partial\Omega}[\mathbf{U\cdot\nu-}\frac{\alpha}{2}]\left|\Delta
w_{1}\right|^{2}d\partial\Omega+Cr_{\mathbf{U}}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}.$ (F)

To estimate $I_{7}:$ since $w_{2}\in H_{0}^{1}(\Omega),$ we have

$\text{Re}(h_{\alpha}\cdot\nabla
w_{2},w_{2})_{\Omega}=-\frac{1}{2}\int\limits_{\Omega}\text{div}(h_{\alpha}\mathbf{)}\left|w_{2}\right|^{2}d\Omega$
$=-\frac{1}{2}\int\limits_{\Omega}\text{div}(h_{\alpha}\mathbf{)}\left|(u_{0})_{3}-\mathbf{U}\nabla
w_{1}\right|^{2}d\Omega$

after using the boundary condition in $\mathbf{(A.v)}.$ Applying the last
relation to RHS of (67) and recalling that
$h_{\alpha}=\mathbf{U|}_{\Omega}-\alpha g,$ we get

$\text{Re}I_{7}=\text{Re}(h_{\alpha}\cdot\nabla
w_{2},w_{2})_{\Omega}+\text{Re}(h_{\alpha}\cdot\nabla(\mathbf{U}\nabla
w_{1}),(u_{0})_{3}-\mathbf{U}\nabla w_{1})_{\mathcal{O}}$ $\leq
Cr_{\mathbf{U}}\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}$ (88)

where we also implicitly use Sobolev Trace Theorem. Lastly, for the term
$I_{8}$, we proceed in a manner similar to that adopted for $I_{7}$ and we
have

$I_{8}=(h_{\alpha}\cdot\nabla(u_{0})_{3},h_{\alpha}\cdot\nabla w_{1}+\xi
w_{1})_{\Omega}$ $+\xi((u_{0})_{3},(u_{0})_{3}-\mathbf{U}\cdot\nabla
w_{1}+h_{\alpha}\cdot\nabla w_{1}+\xi w_{1})_{\Omega}$ $\leq
C\left[r_{\mathbf{U}}+\xi^{2}\right]\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}$
$+C\xi\left[\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+r_{\mathbf{U}}\left\\{\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}\right]$ (89)

Now, if we apply (70)-(89) to RHS of (60), we obtain

$\text{Re}(([\mathcal{A}_{0}+B]\varphi,\varphi))_{\mathcal{H}_{0}}\leq-(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}-\eta\left\|u_{0}\right\|_{\mathcal{O}}^{2}-\xi\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\xi\left\|\Delta
w_{1}\right\|_{\Omega}^{2}$
$+\int\limits_{\partial\Omega}[\mathbf{U\cdot\nu-}\frac{\alpha}{2}]\left|\Delta
w_{1}\right|^{2}d\partial\Omega$ $+C\left[r_{\mathbf{U}}+\xi
r_{\mathbf{U}}+\xi^{2}+\xi\right]\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}$
$+C\left[r_{\mathbf{U}}+\xi
r_{\mathbf{U}}+\xi^{2}+\xi^{2}r_{\mathbf{U}}\right]\left\\{\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}$
$+C\xi\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}\left\\{\left\|p_{0}\right\|_{\mathcal{O}}+\left\|\Delta
w_{1}\right\|_{\Omega}\right\\}.$ (90)

We recall now the value of $\alpha=2\left\|\mathbf{U}\right\|_{\ast}$ to get

$\text{Re}(([\mathcal{A}_{0}+B]\varphi,\varphi))_{H_{N}^{\bot}}\leq-(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}-\eta\left\|u_{0}\right\|_{\mathcal{O}}^{2}-\xi\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\xi\left\|\Delta
w_{1}\right\|_{\Omega}^{2}$
$+\left[(C_{1}+C_{2}r_{\mathbf{U}})\xi^{2}+C_{2}r_{\mathbf{U}}\xi+C_{2}r_{\mathbf{U}}\right]\left\\{\left\|p_{0}\right\|_{\mathcal{O}}^{2}+\left\|\Delta
w_{1}\right\|_{\Omega}^{2}\right\\}$
$+\frac{1}{2}\left\\{(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}+\eta\left\|u_{0}\right\|_{\mathcal{O}}^{2}\right\\}+C_{3}\left[r_{\mathbf{U}}+\xi
r_{\mathbf{U}}+\xi^{2}+\xi\right]\left\|u_{0}\right\|_{H^{1}(\mathcal{O})}^{2}$
(91)

where the positive constants $C_{1},C_{2}$ and $C_{3}$ are obtained with the
application of Holder-Young and Korn’s inequalities and $C_{2}$ depends on the
constant in Korn’s inequality. We now specify $\xi$ be a zero of the equation

$(C_{1}+C_{2}r_{\mathbf{U}})\xi^{2}+(C_{2}r_{\mathbf{U}}-\frac{1}{2})\xi+C_{2}r_{\mathbf{U}}=0.$

Namely,

$\xi=\frac{\frac{1}{2}-C_{2}r_{\mathbf{U}}}{2(C_{1}+C_{2}r_{\mathbf{U}})}-\frac{\sqrt{(\frac{1}{2}-C_{2}r_{\mathbf{U}})^{2}-4C_{2}(C_{1}+C_{2}r_{\mathbf{U}})r_{\mathbf{U}}}}{2(C_{1}+C_{2}r_{\mathbf{U}})}$
(92)

where the radicand is nonnegative for $\left\|\mathbf{U}\right\|_{\ast}$
sufficiently small. Then (91) becomes

$\text{Re}(([\mathcal{A}_{0}+B]\varphi,\varphi))_{\mathcal{H}_{0}}\leq-\frac{(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}}{4}-\eta\frac{\left\|u_{0}\right\|_{\mathcal{O}}^{2}}{4}-\frac{\xi}{2}\left\|p_{0}\right\|_{\mathcal{O}}^{2}-\frac{\xi}{2}\left\|\Delta
w_{1}\right\|_{\Omega}^{2}$
$-\frac{(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}}{4}-\eta\frac{\left\|u_{0}\right\|_{\mathcal{O}}^{2}}{4}$
$+C_{K}\left[r_{\mathbf{U}}+\xi
r_{\mathbf{U}}+\xi^{2}+\xi\right]\left\\{(\sigma(u_{0}),\epsilon(u_{0}))_{\mathcal{O}}+\eta\left\|u_{0}\right\|_{\mathcal{O}}^{2}\right\\}.$

With $\xi$ as prescribed in (92), we now have the dissipativity estimate (59),
for $\left\|\mathbf{U}\right\|_{\ast}$ small enough. (Here we also implicitly
re-use Korn’s inequality and $C_{K}$ is the constant there). This concludes
the proof of Lemma 5.

##  8 Acknowledgement

The author would like to thank the National Science Foundation, and
acknowledge her partial funding from NSF Grant DMS-1907823.

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