# Controllability of a simplified fluid-structure interaction system

Rémi Buffe  Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
Takéo Takahashi  Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy,
France

###### Abstract

We are interested by the controllability of a fluid-structure interaction
system where the fluid is viscous and incompressible and where the structure
is elastic and located on a part of the boundary of the fluid’s domain. In
this article, we simplify this system by considering a linearization and by
replacing the wave/plate equation for the structure by a heat equation. We
show that the corresponding system coupling the Stokes equations with a heat
equation at its boundary is null-controllable. The proof is based on Carleman
estimates and interpolation inequalities. One of the Carleman estimates
corresponds to the case of Ventcel boundary conditions. This work can be seen
as a first step to handle the real system where the structure is modeled by
the wave or the plate equation.

###### Contents

  1. 1 Introduction
  2. 2 From a spectral inequality to the null-controllability
    1. 2.1 Controllability of the first modes
    2. 2.2 Proof of 2.1
  3. 3 Functional framework and spectral inequality
    1. 3.1 Functional framework
    2. 3.2 Spectral inequality
  4. 4 A global observability estimate on the pressure
    1. 4.1 Choice of the weight and conjugated operators
    2. 4.2 A first estimate on the pressure
    3. 4.3 Estimates in the high frequency regime
    4. 4.4 Proof of 4.1
  5. 5 Proof of the spectral inequality
    1. 5.1 Estimates on the velocity
    2. 5.2 Patching the estimates together
    3. 5.3 From the interpolation inequality to the spectral inequality
  6. A Proof of 5.2
    1. A.1 A Carleman estimate
    2. A.2 Interpolation estimates for the Ventcel boundary condition

Keywords: Null controllability, Navier-Stokes systems, Carleman estimates

2010 Mathematics Subject Classification. 76D05, 35Q30, 93B05, 93B07, 93C10

##  1 Introduction

Fluid-structure interaction systems are important systems for many
applications such as aerodynamics, medicine (for instance the study of the
motion of the blood in veins or in arteries), biology (animal locomotion in a
fluid), civil engineering (design of bridges), naval architecture (design of
boats and submarines), etc. Moreover, their mathematical studies can be very
challenging due to several difficulties: in particular, the complexity of
fluid equations such as the Navier-Stokes system, the strong coupling between
the fluid system and the structure system and the free-boundary corresponding
to the structure displacement.

In this article we consider a simplified fluid-structure interaction system.
The corresponding system without simplification has been proposed in [41] as a
model for the blood flow in a vessel. It writes as follows: we denote by
$\displaystyle\mathcal{I}$ the torus (in order to consider periodic boundary
conditions):

$\displaystyle\mathcal{I}:=\mathbb{R}/(2\pi\mathbb{Z}),$

and for any deformation $\displaystyle\ell:\mathcal{I}\to(-1,\infty)$, we
consider the corresponding fluid domain

$\Omega_{\ell}=\left\\{(x_{1},x_{2})\in\mathcal{I}\times\mathbb{R}\ ;\
x_{2}\in(0,1+\ell(x_{1}))\right\\}.$ (1.1)

Then the equations of motion are

$\left\\{\begin{array}[]{rl}\partial_{t}w+(w\cdot\nabla)w-\operatorname{div}\mathbb{T}(w,\pi)=0&t>0,\
x\in\Omega_{\ell(t)},\\\ \operatorname{div}w=0&t>0,\ x\in\Omega_{\ell(t)},\\\
w(t,x_{1},1+\ell(t,x_{1}))=(\partial_{t}\ell)(t,x_{1})e_{2}&t>0,\
x_{1}\in\mathcal{I},\\\ w=0&t>0,\ x\in\Gamma_{0},\\\
\partial_{tt}\ell+\alpha_{1}\partial_{x_{1}}^{4}\ell-\alpha_{2}\partial_{x_{1}}^{2}\ell-\delta\partial_{t}\partial_{x_{1}}^{2}\ell=-\widetilde{\mathbb{H}}_{\ell}(w,\pi)&t>0,\
x_{1}\in\mathcal{I},\end{array}\right.$ (1.2)

where

$\displaystyle\Gamma_{0}=\mathcal{I}\times\\{0\\},\quad\Gamma_{1}=\mathcal{I}\times\\{1\\}.$

In the above system, we have used the following notations:
$\displaystyle(e_{1},e_{2})$ is the canonical basis of
$\displaystyle\mathbb{R}^{2}$ and

$\mathbb{T}(w,\pi)=2D(w)-\pi I_{2},\quad D(w)=\frac{1}{2}\left(\nabla
w+(\nabla w)^{*}\right),$ (1.3)
$\widetilde{\mathbb{H}}_{\ell}(w,\pi)(t,x_{1})=\left[(1+|\partial_{x_{1}}\ell|^{2})^{1/2}\left[\mathbb{T}(w,\pi)n\right](t,x_{1},1+\ell(t,x_{1}))\cdot
e_{2}\right].$ (1.4)

The two first lines of (1.2) correspond to the Navier-Stokes system for the
fluid velocity $\displaystyle w$ and the pressure $\displaystyle\pi$. The last
line of (1.2) is a beam equation satisfied by the deformation
$\displaystyle\ell$. We have used the standard no-slip boundary conditions
(third and forth equations). To simplify, we assume that the viscosity of the
fluid is constant and equal to 1. The vector fields $\displaystyle n$
corresponds to the unit exterior normal to $\displaystyle\Omega_{\ell(t)}$.

This system has been studied by many authors: [13] (existence of weak
solutions), [7], [30], [21] and [33] (existence of strong solutions), [43]
(stabilization of strong solutions), [4] (stabilization of weak solutions
around a stationary state). There are also some works in the case
$\displaystyle\delta=0$, that is without damping on the beam equation: the
existence of weak solutions is proved in [20] and in [37] (see also [52]). In
[22], the existence of local strong solutions is obtained for a structure
described by either a wave equation ($\displaystyle\alpha_{1}=\delta=0$ and
$\displaystyle\alpha_{2}>0$) or a beam equation with inertia of rotation
($\displaystyle\alpha_{1}>0$, $\displaystyle\alpha_{2}=\delta=0$ and with an
additional term $\displaystyle-\partial_{ttss}\ell$). In [5] and [6], the
authors show the existence and uniqueness of strong solutions in the case
$\displaystyle\alpha_{1}>0$, $\displaystyle\alpha_{2}\geqslant 0$ and
$\displaystyle\delta=0$. Using similar techniques they also analyze the case
of the wave equation ($\displaystyle\alpha_{1}=\delta=0$ and
$\displaystyle\alpha_{2}>0$) in [1] showing in particular that the semigroup
of the linearized system is analytic. Let us mention also some results for
more complex models: [29, 28] (linear elastic Koiter shell), [38] (dynamic
pressure boundary conditions), [39, 40] (3D cylindrical domain with nonlinear
elastic cylindrical Koiter shell), [49] and [50] (nonlinear elastic and
thermoelastic plate equations), [32], [34] (compressible fluids), etc.

The advantage of the damping in the beam equation is that the term
$\displaystyle-\delta\partial_{t}\partial_{x_{1}}^{2}\ell$ is a structural
damping so that the corresponding beam equation becomes a parabolic equation
(see, for instance, [15]). In this work, we consider a simplified model
associated with (1.2). We neglect the deformation of the fluid domain due to
the elastic déformation and we also linearized the Navier-Stokes system by
only considering the Stokes system. Moreover, we replace the damped beam
equation by a heat equation. By setting

$\displaystyle\Omega=\mathcal{I}\times(0,1)$

(see Figure 1), we are thus considering the following system

$\left\\{\begin{array}[]{rl}\partial_{t}w-\Delta
w+\nabla\pi=1_{\omega}f&\text{in}\ (0,T)\times\Omega,\\\
\operatorname{div}w=0&\text{in}\ (0,T)\times\Omega,\\\ w=0&\text{on}\
(0,T)\times\Gamma_{0},\\\ w=\zeta e_{2}&\text{on}\ (0,T)\times\Gamma_{1},\\\
\partial_{t}\zeta-\partial_{x_{1}x_{1}}\zeta=-\mathbb{T}(w,\pi)n\cdot
e_{2}&\text{in}\ (0,T)\times\mathcal{I},\\\ w(0,\cdot)=w^{0}\quad\text{in}\
\Omega,&\zeta(t,0)=\zeta^{0}\quad\text{in}\ \mathcal{I}.\end{array}\right.$
(1.5)

In the above system, $\displaystyle\zeta$ corresponds to the displacement
velocity $\displaystyle\partial_{t}\ell$ in (1.2) and we do not consider
anymore the displacement position $\displaystyle\ell$. We have added a control
$\displaystyle f$ localized in the fluid domain, in an arbitrary small
nonempty open set $\displaystyle\omega$ of $\displaystyle\Omega$. Our goal is
to show the null-controllability of this simplified system and to do this, as
it is standard (see, for instance, [51, Theorem 11.2.1, p.357]), we prove an
observability inequality on the adjoint system:

$\displaystyle\Omega$$\displaystyle\Gamma_{0}$$\displaystyle-$$\displaystyle-$$\displaystyle-$$\displaystyle-$$\displaystyle\Gamma_{1}$
(“beam”)$\displaystyle 0$$\displaystyle 2\pi$$\displaystyle 1$ Figure 1: Our
geometry

$\left\\{\begin{array}[]{rl}\partial_{t}u-\Delta u+\nabla p=0&\text{in}\
(0,T)\times\Omega,\\\ \operatorname{div}u=0&\text{in}\ (0,T)\times\Omega,\\\
u=0&\text{on}\ (0,T)\times\Gamma_{0},\\\ u=\eta e_{2}&\text{on}\
(0,T)\times\Gamma_{1},\\\
\partial_{t}\eta-\partial_{x_{1}x_{1}}\eta=-\mathbb{T}(u,p)n_{|\Gamma_{1}}\cdot
e_{2}&\text{in}\ (0,T)\times\mathcal{I},\\\ u(0,\cdot)=u^{0}\quad\text{in}\
\Omega,&\eta(0,\cdot)=\eta^{0}\quad\text{in}\ \mathcal{I}.\end{array}\right.$
(1.6)

We set

$\displaystyle L^{2}_{0}(\mathcal{I}):=\left\\{f\in L^{2}(\mathcal{I})\ ;\
\int_{0}^{2\pi}f(x_{1})\ dx_{1}=0\right\\}$

and we define the space

$\mathcal{H}:=\left\\{\left[u,\eta\right]\in{L}^{2}(\Omega)\times
L^{2}_{0}(\mathcal{I})\ ;\ u_{2}=0\ \text{on}\ \Gamma_{0},\quad u_{2}=\eta\
\text{on}\ \Gamma_{1},\ \operatorname{div}u=0\;\text{in}\ \Omega\right\\}.$
(1.7)

Then, our main result states as follows:

######  Theorem 1.1.

Let $\displaystyle\gamma>1$ and let $\displaystyle\omega$ be a nonempty open
set of $\displaystyle\Omega$. Then, there exists $\displaystyle C_{0}>0$ such
that for any $\displaystyle T\in(0,1)$ and for any
$\displaystyle\left[u^{0},\eta^{0}\right]\in\mathcal{H}$, the solution
$\displaystyle\left[u,\eta\right]$ of (1.6) satisfies

$\displaystyle\left\|\left[u(T,\cdot),\eta(T,\cdot)\right]\right\|_{\mathcal{H}}^{2}\leqslant
C_{0}\exp\left(\frac{C_{0}}{T^{\gamma}}\right)\iint_{(0,T)\times\omega}|u|^{2}\
dtdx.$

In particular, for any
$\displaystyle\left[w^{0},\zeta^{0}\right]\in\mathcal{H}$ and for any
$\displaystyle T>0$, there exists a control $\displaystyle f\in
L^{2}((0,T)\times\omega)$ such that the solution
$\displaystyle\left[w,\zeta\right]$ of (1.5) satisfies

$\displaystyle w(T,\cdot)=0,\quad\zeta(T,\cdot)=0.$

######  Remark 1.2.

As explained above, 1.1 can be seen as a first control result on a simplified
model. We expect to extend some of the tools developed here to handle the
control properties of the system (1.2) in future works. The controllability
properties of fluid-structure interaction systems have been tackled mainly in
the case where the structure is a rigid body (see, [42], [18], [31], [23],
[9], [8], [45], [17], [16], etc.). In [36], the author shows an observability
inequality for the adjoint of a linearized and simplified fluid-structure
interaction system in the case of a compressible viscous fluid and of a damped
beam. Note that the corresponding control problem involves two controls, one
for the fluid and one for the structure.

For the stabilization of fluid-structure interaction systems, one can quote
some results: [44], [4] (for the case of a damped beam), [48], [46], [3], [2]
(for the case of a rigid body).

######  Remark 1.3.

The method proposed here to control a system involving the Stokes equations is
quite different from the method used in a large part of the literature for the
controllability of fluid systems. In general, the method is based on “global
Carleman inequalities” (see, for instance [19, 24]). Here, we follow another
strategy as in [27, 14]. Such a method is based on local Carleman inequalities
for an ”augmented” elliptic operator, from which one deduces a spectral
inequality, in the spirit of [27, 25]. However, as it is pointed out in [14],
unique continuation property does not hold for the augmented operator in the
direction of the additional variable, due to the pressure. We then use an
adaptation of the original strategy of [27, 26] in our context. This type of
spectral inequality has already been used in the context of fluids in [12]. We
also recall that one can use 1.1 to handle nonlinear controllability issues by
applying the general method proposed in [31].

######  Remark 1.4.

Using the particular geometry considered here, we can simplify the adjoint
system. First on $\displaystyle\Gamma_{1}$, $\displaystyle n=e_{2}$ and using
(1.3), we deduce

$-\mathbb{T}(u,p)n\cdot
e_{2}=-2\partial_{x_{2}}u_{2}+p=2\partial_{x_{1}}u_{1}+p=p,$ (1.8)

since $\displaystyle u_{1}(x_{1},1)=0$ for $\displaystyle
x_{1}\in\mathcal{I}$.

Moreover, using the incompressibility of the fluid and the boundary
conditions, we deduce that

$\displaystyle 0=\int_{\Omega}\operatorname{div}u\ dx=\int_{\mathcal{I}}\eta\
dx_{1}.$

Using this condition on the heat equation on the boundary and (1.8) yields

$\int_{\mathcal{I}}p(x_{1},1)\ dx_{1}=0.$ (1.9)

In particular, in contrast with the standard Stokes system, the pressure is
not determined up to a constant.

The outline of the article is as follows: in Section 2, we show how to obtain
the observability inequality stated in 1.1 from a spectral inequality. Such a
result is quite standard, but here we show that for a self-adjoint operator,
we do not need the usual assumption that is made on the eigenvalues of the
main operator. Then using this general result, we are reduced to show a
spectral inequality that we state in Section 3 along with the functional
framework. The spectral inequality is itself the consequence of an
interpolation inequality that we obtain in Section 5. One of the main
difficulties to obtain such an inequality comes from the fact that we need to
estimate the pressure. Section 4 is devoted to such an estimate which is one
of the main parts of this article. The proof of the spectral inequality and
thus of 1.1 is obtained at the end of Section 5. In the appendix, we show an
interpolation estimate for the Ventcel boundary condition that is mainly a
consequence of a Carleman estimate obtained in [11].

######  Notation 1.5.

In the whole paper, we use $\displaystyle C$ as a generic positive constant
that does not depend on the other terms of the inequality. The value of the
constant $\displaystyle C$ may change from one appearance to another. We also
use the notation $\displaystyle X\lesssim Y$ if there exists a constant
$\displaystyle C>0$ such that we have the inequality $\displaystyle X\leqslant
CY$.

##  2 From a spectral inequality to the null-controllability

This section is devoted to a “classical” result showing that a spectral
inequality implies the final-state observability and thus the null-
controllability. The proof follows closely the proof in [26, 27] and we only
prove it here for sake of completeness and also to show that we do not need
any assumption on the asymptotic behavior of the spectrum of the operator
(which is used in the above references).

More precisely, we assume here that $\displaystyle
A:\mathcal{D}(A)\to\mathcal{H}$ is a positive self-adjoint operator with
compact resolvents in an Hilbert space $\displaystyle\mathcal{H}$. We denote
by $\displaystyle(\lambda_{j})$ the nondecreasing sequence of eigenvalues and
by $\displaystyle(w_{j})$ an orthonormal basis of $\displaystyle\mathcal{H}$
composed by eigenvectors of $\displaystyle A$: $\displaystyle
Aw_{j}=\lambda_{j}w_{j}$ for $\displaystyle j\geqslant 1$. We also consider a
control operator $\displaystyle B\in\mathcal{L}(\mathcal{U},\mathcal{H}).$

######  Theorem 2.1.

Assume the above hypotheses. Assume moreover the existence of $\displaystyle
S_{0}>0$, $\displaystyle C>0$ and $\displaystyle\kappa\in
C^{\infty}_{0}(0,S_{0})$ such that that for any $\displaystyle\Lambda>0$, and
for any $\displaystyle(a_{j})_{j}\in\mathbb{C}^{\mathbb{N}}$,

$\sum_{\lambda_{j}\leqslant\Lambda}\left|a_{j}\right|^{2}\leqslant
Ce^{C\sqrt{\Lambda}}\int_{0}^{S_{0}}\kappa^{2}(s)\left\|\sum_{\lambda_{j}\leqslant\Lambda}a_{j}\cosh(s\sqrt{\lambda}_{j})B^{*}w^{(j)}\right\|_{\mathcal{U}}^{2}\
ds.$ (2.1)

Then for all $\displaystyle\gamma>1$, there exists $\displaystyle C_{0}>0$
such that for any $\displaystyle T\in(0,1)$ and for any $\displaystyle
z^{0}\in\mathcal{H}$,

$\left\|e^{-TA}z^{0}\right\|_{\mathcal{H}}^{2}\leqslant
C_{0}\exp\left(\frac{C_{0}}{T^{\gamma}}\right)\int_{0}^{T}\left\|B^{*}e^{-tA}z^{0}\right\|^{2}_{\mathcal{U}}\
dt.$ (2.2)

We recall that relation (2.2) implies the null-controllability of the system

$\left\\{\begin{array}[]{l}\frac{d\theta}{dt}+A\theta=Bg\quad\text{in}\
(0,T),\\\\[5.69054pt] \theta(0)=\theta^{0}\in\mathcal{H}.\end{array}\right.$
(2.3)

###  2.1 Controllability of the first modes

We define

$\displaystyle\mathcal{H}_{\Lambda}=\operatorname{span}\left\\{w_{j},\
\lambda_{j}\leqslant\Lambda\right\\},\quad\Pi_{\Lambda}:\mathcal{H}\to\mathcal{H}_{\Lambda}\
\text{the orthogonal projection}.$

We are interested here by the control problem

$\left\\{\begin{array}[]{l}\frac{d\theta}{dt}+A\theta=\Pi_{\Lambda}Bg\quad\text{in}\
(0,\tau),\\\\[5.69054pt]
\theta(0)=\theta^{0}\in\mathcal{H}_{\Lambda},\end{array}\right.$ (2.4)

for some $\displaystyle\tau>0$. We consider the linear operator

$\displaystyle
G_{\Lambda}:=\int_{0}^{S_{0}}\kappa^{2}(s)\cosh(s\sqrt{A})\Pi_{\Lambda}BB^{*}\cosh(s\sqrt{A})\Pi_{\Lambda}\
ds.$

From (2.1), $\displaystyle G_{\Lambda}\in\mathcal{L}(\mathcal{H}_{\Lambda})$
is symmetric, positive and invertible with

$\displaystyle\left\|G_{\Lambda}^{-1}\right\|_{\mathcal{L}(\mathcal{H}_{\Lambda})}\leqslant
Ce^{C\sqrt{\Lambda}}.$

We set

$\displaystyle\sigma:=2-\frac{1}{\gamma}\in(1,2).$

From [26, Lemma A.1], there exists $\displaystyle e\in C^{\infty}(\mathbb{R})$
such that for some constants $\displaystyle c_{j}$

$\operatorname{supp}e=[0,1],$ (2.5) $\left|\widehat{e}(z)\right|\leqslant
c_{1}e^{-c_{2}|z|^{1/\sigma}}\quad\text{if}\ \operatorname{Im}(z)\leqslant 0,$
(2.6) $\left|\widehat{e}(z)\right|\geqslant
c_{3}e^{-c_{4}|z|^{1/\sigma}}\quad\text{if}\ z\in i\mathbb{R}^{-}.$ (2.7)

From (2.7), we have that $\displaystyle\widehat{e}(-i\tau
A)\in\mathcal{L}(\mathcal{H}_{\Lambda})$ is invertible and

$\left\|\widehat{e}(-i\tau
A)^{-1}\right\|_{\mathcal{L}(\mathcal{H}_{\Lambda})}\leqslant\frac{1}{c_{3}}e^{c_{4}(\Lambda\tau)^{1/\sigma}}.$
(2.8)

Then we define

$\displaystyle
h_{\Lambda}(s):=-\frac{1}{2}\kappa^{2}(s)B^{*}\cosh(s\sqrt{A})G_{\Lambda}^{-1}\widehat{e}(-i\tau
A)^{-1}e^{-\tau A}\theta^{0}\quad(s\in\mathbb{R}).$

We have that $\displaystyle h_{\Lambda}\in
C^{\infty}_{0}(\mathbb{R},\mathcal{U})$ with
$\displaystyle\operatorname{supp}h_{\Lambda}\subset(0,S_{0})$ and

$\left\|h_{\Lambda}\right\|_{L^{\infty}(\mathbb{R},\mathcal{U})}\leqslant
Ce^{C\sqrt{\Lambda}+c_{4}(\Lambda\tau)^{1/\sigma}}\left\|\theta^{0}\right\|_{\mathcal{H}}.$
(2.9)

Thus,
$\displaystyle\widehat{h}_{\Lambda}\in\mathrm{Hol}(\mathbb{C};\mathcal{U})$
and

$\left\|\widehat{h}_{\Lambda}(z)\right\|_{\mathcal{U}}\leqslant
Ce^{C\sqrt{\Lambda}+c_{4}(\Lambda\tau)^{1/\sigma}}e^{S_{0}\left|\operatorname{Im}(z)\right|}\left\|\theta^{0}\right\|_{\mathcal{H}}.$
(2.10)

As in [47], we introduce $\displaystyle
Q_{\Lambda}\in\mathrm{Hol}(\mathbb{C};\mathcal{U})$ such that

$\displaystyle
Q_{\Lambda}(-iz^{2})=\widehat{h}_{\Lambda}(iz)+\widehat{h}_{\Lambda}(-iz)\quad(z\in\mathbb{C}).$

We deduce from the above relation and (2.10) that

$\left\|Q_{\Lambda}(z)\right\|_{\mathcal{U}}\leqslant
Ce^{C\sqrt{\Lambda}+c_{4}(\Lambda\tau)^{1/\sigma}}e^{S_{0}\sqrt{\left|z\right|}}\left\|\theta^{0}\right\|_{\mathcal{H}}.$
(2.11)

We define
$\displaystyle\mathfrak{g}_{\Lambda}\in\mathrm{Hol}(\mathbb{C};\mathcal{U})$
by

$\displaystyle\mathfrak{g}_{\Lambda}(z):=\widehat{e}(\tau z)Q_{\Lambda}(z).$

From (2.5), (2.6) and (2.11), we have

$\left\|\mathfrak{g}_{\Lambda}(z)\right\|_{\mathcal{U}}\leqslant
Ce^{C\sqrt{\Lambda}+c_{4}(\Lambda\tau)^{1/\sigma}}e^{S_{0}\sqrt{\left|z\right|}}e^{\tau|\operatorname{Im}z|}\left\|\theta^{0}\right\|_{\mathcal{H}}\quad(z\in\mathbb{C})$
(2.12)

and

$\left\|\mathfrak{g}_{\Lambda}(z)\right\|_{\mathcal{U}}\leqslant
Ce^{C\sqrt{\Lambda}+c_{4}(\Lambda\tau)^{1/\sigma}}e^{S_{0}\sqrt{\left|z\right|}}e^{-c_{2}\tau^{1/\sigma}|z|^{1/\sigma}}\left\|\theta^{0}\right\|_{\mathcal{H}}\quad\text{if}\
\operatorname{Im}z\leqslant 0.$ (2.13)

Since $\displaystyle\sigma<2$, we can use a Paley-Wiener type theorem (see
[26, Proposition A.3]) and deduce the existence of $\displaystyle
g_{\Lambda}\in C^{\infty}_{0}((0,\tau);\mathcal{U})$ such that

$\displaystyle\widehat{g}_{\Lambda}(z)=\mathfrak{g}_{\Lambda}(z).$

In particular, from (2.13) and the Laplace method, for all $\displaystyle
t\in(0,\tau)$,

$\left|g_{\Lambda}(t)\right|\leqslant\left\|\mathfrak{g}_{\Lambda}\right\|_{L^{1}(\mathbb{R};\mathcal{U})}\leqslant
Ce^{C\sqrt{\Lambda}+c_{4}(\Lambda\tau)^{1/\sigma}+\frac{C}{\tau^{1/(2-\sigma)}}}\left\|\theta^{0}\right\|_{\mathcal{H}}.$
(2.14)

Now, for any $\displaystyle j$ such that
$\displaystyle\lambda_{j}\leqslant\Lambda$,

$\left(\int_{0}^{\tau}e^{-(\tau-t)A}Bg_{\Lambda}(\tau-t)\
dt,w_{j}\right)_{\mathcal{H}}=\left(B\widehat{g}_{\Lambda}(-i\lambda_{j}),w_{j}\right)_{\mathcal{H}}\\\
=\left(\widehat{e}(-i\tau
A)B\left(\widehat{h}_{\Lambda}(i\sqrt{\lambda_{j}})+\widehat{h}_{\Lambda}(-i\sqrt{\lambda_{j}})\right),w_{j}\right)_{\mathcal{H}}\\\
=\left(\widehat{e}(-i\tau
A)B\int_{0}^{S_{0}}h_{\Lambda}(s)2\cosh(s\sqrt{\lambda_{j}})\
ds,w_{j}\right)_{\mathcal{H}}=-\left(e^{-\tau
A}\theta^{0},w_{j}\right)_{\mathcal{H}}$

so that the solution $\displaystyle\theta$ of (2.4) with the control
$\displaystyle g(t)=g_{\Lambda}(\tau-t)$ satisfies
$\displaystyle\theta(\tau)=0$.

By a duality argument and (2.14), this implies that

$\left\|\Pi_{\Lambda}e^{-\tau A}z^{0}\right\|^{2}\leqslant
C_{1}\exp\left(C_{1}\left(\frac{1}{\tau^{\gamma}}+\sqrt{\Lambda}+(\Lambda\tau)^{\gamma/(2\gamma-1)}\right)\right)\int_{0}^{\tau}\left\|B^{*}\Pi_{\Lambda}e^{-tA}z^{0}\right\|^{2}_{\mathcal{U}}\
dt.$ (2.15)

###  2.2 Proof of 2.1

We are now in a position to prove 2.1, adapting the method of [35].

###### Proof of 2.1.

We set

$\displaystyle z(t)=e^{-tA}z^{0}.$

Assume

$\displaystyle T^{(1)}>0,\quad\tau>0,\quad\text{and}\quad
T^{(2)}=T^{(1)}+\tau.$

From (2.15)

$\left\|\Pi_{\Lambda}z\left(T^{(2)}\right)\right\|^{2}\leqslant
C_{1}\exp\left(C_{1}\left(\frac{1}{\left(\varepsilon\tau\right)^{\gamma}}+\sqrt{\Lambda}+(\Lambda\varepsilon\tau)^{\gamma/(2\gamma-1)}\right)\right)\int_{T^{(2)}-\varepsilon\tau}^{T^{(2)}}\left\|B^{*}\Pi_{\Lambda}z(t)\right\|^{2}_{\mathcal{U}}\
dt.$ (2.16)

We set

$\Lambda=\frac{1}{\left(\varepsilon\tau\right)^{1+\gamma}}$ (2.17)

so that for

$\displaystyle\tau,\varepsilon\in(0,1),$

(2.16) becomes

$2\rho(\tau)\left\|\Pi_{\Lambda}z\left(T^{(2)}\right)\right\|^{2}\leqslant\int_{T^{(2)}-\varepsilon\tau}^{T^{(2)}}\left\|B^{*}\Pi_{\Lambda}z(t)\right\|^{2}_{\mathcal{U}}\
dt$ (2.18)

with

$\rho(\tau):=\frac{1}{2C_{1}}\exp\left(-\frac{3C_{1}}{\left(\varepsilon\tau\right)^{\gamma}}\right).$
(2.19)

Then from (2.18), we deduce

$\rho(\tau)\left\|z\left(T^{(2)}\right)\right\|^{2}\leqslant\int_{T^{(2)}-\varepsilon\tau}^{T^{(2)}}\left\|B^{*}z(t)\right\|^{2}_{\mathcal{U}}\
dt+C\left\|z\left(T^{(1)}\right)\right\|_{\mathcal{H}}^{2}\varepsilon\tau
e^{-2\Lambda\tau(1-\varepsilon)}+\rho(\tau)\left\|z\left(T^{(1)}\right)\right\|_{\mathcal{H}}^{2}e^{-2\Lambda\tau}$
(2.20)

For $\displaystyle\varepsilon>0$ small enough, the above relation yields

$\rho(\tau)\left\|z\left(T^{(2)}\right)\right\|_{\mathcal{H}}^{2}\leqslant\int_{T^{(1)}}^{T^{(2)}}\left\|B^{*}z(t)\right\|^{2}_{\mathcal{U}}\
dt+\rho(\tau/2)\left\|z\left(T^{(1)}\right)\right\|_{\mathcal{H}}^{2}$ (2.21)

Assume

$\displaystyle T\in(0,1).$

Then for all $\displaystyle k\geqslant 0$, (2.21) implies

$\rho\left(\frac{T}{2^{k+1}}\right)\left\|z\left(\frac{T}{2^{k}}\right)\right\|_{\mathcal{H}}^{2}\leqslant\int_{\frac{T}{2^{k+1}}}^{\frac{T}{2^{k}}}\left\|B^{*}z(t)\right\|^{2}_{\mathcal{U}}\
dt+\rho\left(\frac{T}{2^{k+2}}\right)\left\|z\left(\frac{T}{2^{k+1}}\right)\right\|_{\mathcal{H}}^{2}$
(2.22)

and thus

$\rho\left(\frac{T}{2}\right)\left\|z\left(T\right)\right\|_{\mathcal{H}}^{2}\leqslant\int_{0}^{T}\left\|B^{*}z(t)\right\|^{2}_{\mathcal{U}}\
dt.$ (2.23)

Thus for some constant $\displaystyle C_{2}>0$,

$\left\|z\left(T\right)\right\|_{\mathcal{H}}^{2}\leqslant
C_{2}\exp\left(\frac{C_{2}}{T^{\gamma}}\right)\int_{0}^{T}\left\|B^{*}z(t)\right\|^{2}_{\mathcal{U}}\
dt.$ (2.24)

∎

##  3 Functional framework and spectral inequality

In order to prove 1.1, we are going to apply 2.1. In this section, we first
give the functional framework associated with (1.6). Then we write the
spectral inequality that will be proven in the remaining part of the article.

###  3.1 Functional framework

We recall that $\displaystyle\mathcal{H}$ is defined by (1.7). We also define

$\displaystyle\mathcal{V}:=\left\\{\left[u,\eta\right]\in\left({H}^{1}(\Omega)\times
H^{1}(\mathcal{I})\right)\cap\mathcal{H}\ ;\ u_{1}=0\ \text{on}\
\partial\Omega\right\\}.$

We denote by $\displaystyle P_{0}$ the orthogonal projection from
$\displaystyle{L}^{2}(\Omega)\times L^{2}_{0}(\mathcal{I})$ onto
$\displaystyle\mathcal{H}$. We now define the linear operator $\displaystyle
A_{0}:\mathcal{D}(A_{0})\subset\mathcal{H}\to\mathcal{H}$ by

$\mathcal{D}(A_{0}):=\mathcal{V}\cap\Big{[}{H}^{2}(\Omega)\times
H^{2}(\mathcal{I})\Big{]},$ (3.1)

and for
$\displaystyle\begin{bmatrix}u,\eta\end{bmatrix}\in\mathcal{D}(A_{0})$, we set

${A}_{0}\begin{bmatrix}u\\\ \eta\end{bmatrix}:=P_{0}\begin{bmatrix}\Delta
u\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\displaystyle\partial_{x_{1}}^{2}\eta\end{bmatrix}.$ (3.2)

Then one can check that (1.6) writes

$\displaystyle\left\\{\begin{array}[]{rl}\frac{dz}{dt}=A_{0}z\quad\text{in}\
(0,T),\\\ z(0)=z^{0}.\end{array}\right.$

with $\displaystyle z=[u,\eta]$, $\displaystyle z^{0}=[u^{0},\eta^{0}]$. In
the next proposition, we show in particular that $\displaystyle A_{0}$ is the
infinitesimal generator of a semigroup so that $\displaystyle
z(t)=e^{tA_{0}}z^{0}$ for $\displaystyle t\geqslant 0$.

######  Proposition 3.1.

The operator $\displaystyle A_{0}$ defined by (3.1)–(3.2) has compact
resolvents, and is self-adjoint negative on $\displaystyle\mathcal{H}$.

###### Proof.

By definition of $\displaystyle A_{0}$, we have for any
$\displaystyle[u,\eta],[v,\zeta]\in\mathcal{D}(A_{0})$,

$\displaystyle\left\langle A_{0}\begin{bmatrix}u\\\
\eta\end{bmatrix},\begin{bmatrix}v\\\
\zeta\end{bmatrix}\right\rangle_{\mathcal{H}}=\int_{\Omega}\Delta u\cdot v\
dx+\int_{0}^{2\pi}\left(\partial_{x_{1}}^{2}\eta\right)\zeta\
dx_{1}=-\int_{\Omega}\nabla u:\nabla v\
dx-\int_{0}^{2\pi}\left(\partial_{x_{1}}\eta\right)\left(\partial_{x_{1}}\zeta\right)\
dx_{1}.$

Thus $\displaystyle A_{0}$ is symmetric and negative (by using the Poincaré
inequalities).

In order to show that $\displaystyle A_{0}$ is self-adjoint it is sufficient
to show that it is onto. Assume $\displaystyle[f,g]\in\mathcal{H}$ and let us
solve the equation

$-A_{0}\begin{bmatrix}u\\\ \eta\end{bmatrix}=[f,g].$ (3.3)

Multiplying the above equation by $\displaystyle[v,\zeta]\in\mathcal{V}$ leads
to the weak formulation

$\int_{\Omega}\nabla u:\nabla v\
dx+\int_{0}^{2\pi}\left(\partial_{x_{1}}\eta\right)\left(\partial_{x_{1}}\zeta\right)\
dx_{1}=\int_{\Omega}f\cdot v\ dx+\int_{0}^{2\pi}g\zeta\
dx_{1}\quad([v,\zeta]\in\mathcal{V}).$ (3.4)

Using the Poincaré inequalities, we see that we can apply the Riesz theorem
and deduce the existence and uniqueness of
$\displaystyle[u,\eta]\in\mathcal{V}$ solution of (3.4). Then if
$\displaystyle v\in C^{\infty}_{c}(\Omega)$ with
$\displaystyle\operatorname{div}v=0$ and $\displaystyle\zeta=0$ in (3.4), we
obtain that

$\int_{\Omega}\nabla u:\nabla v\ dx=\int_{\Omega}f\cdot v\ dx\quad(v\in
C^{\infty}_{c}(\Omega),\ \operatorname{div}v=0).$ (3.5)

Using the De Rham theorem, we deduce the existence of $\displaystyle p$ such
that

$\begin{cases}-\Delta u+\nabla p=f\quad\text{in}\ \Omega,\\\
\operatorname{div}u=0\quad\text{in}\ \Omega,\\\ u=0\quad\text{on}\
\Gamma_{0}\\\ u=\eta e_{2}\quad\text{on}\ \Gamma_{1}.\end{cases}$ (3.6)

Using the elliptic regularity of the Stokes system, we deduce that
$\displaystyle(u,p)\in H^{3/2}(\Omega)\times H^{1/2}(\Omega)$. Multiplying the
first above equation by $\displaystyle v$, with
$\displaystyle[v,\zeta]\in\mathcal{V}$, we deduce that for any
$\displaystyle\zeta\in H^{1}(\mathcal{I})\cap L^{2}_{0}(\mathcal{I})$,

$\int_{0}^{2\pi}\left(\partial_{x_{1}}\eta\right)\left(\partial_{x_{1}}\zeta\right)\
dx_{1}=-\langle p_{|\Gamma_{1}},\zeta\rangle+\int_{0}^{2\pi}g\zeta\ dx_{1}.$
(3.7)

Since $\displaystyle p_{|\Gamma_{1}}=\left(\left[pI_{3}-\nabla
u\right]n\right)_{|\Gamma_{1}}\in H^{-1/2}(\mathcal{I})$, we deduce that
$\displaystyle\eta\in\mathcal{H}^{3/2}(\mathcal{I})$ and from (3.6) that
$\displaystyle(u,p)\in H^{2}(\Omega)\times H^{1}(\Omega)$. Thus $\displaystyle
p_{|\Gamma_{1}}\in H^{1/2}(\mathcal{I})$, and from (3.7), we deduce that
$\displaystyle\eta\in H^{2}(\mathcal{I})$. We conclude that
$\displaystyle[u,\eta]\in\mathcal{D}(A_{0})$ and satisfies (3.3).

The fact that $\displaystyle A_{0}$ has compact resolvents is coming from the
compact embedding of $\displaystyle H^{2}$ into $\displaystyle L^{2}$ for
bounded domains. ∎

In particular, the eigenvalues $\displaystyle\lambda_{j}>0$ of $\displaystyle-
A_{0}$ satisfy $\displaystyle\lambda_{j}\to\infty$ and there exists

$\left(\begin{bmatrix}u^{(j)}\\\
\eta^{(j)}\end{bmatrix}\right)_{j}\quad\text{orthonormal basis
of}\quad\mathcal{H}$ (3.8)

composed by eigenvectors of $\displaystyle A_{0}$:

$-A_{0}\begin{bmatrix}u^{(j)}\\\
\eta^{(j)}\end{bmatrix}=\lambda_{j}\begin{bmatrix}u^{(j)}\\\
\eta^{(j)}\end{bmatrix}$ (3.9)

The above system can be written as

$\begin{cases}-\Delta u^{(j)}+\nabla p^{(j)}=\lambda_{j}u^{(j)}\\\
\operatorname{div}u^{(j)}=0\\\ u^{(j)}=0\quad\text{on}\ \Gamma_{0}\\\
u^{(j)}=\eta^{(j)}e_{2}\quad\text{on}\ \Gamma_{1}\\\
-\partial_{x_{1}}^{2}\eta^{(j)}-p^{(j)}=\lambda_{j}\eta^{(j)}\quad\text{in}\
\mathcal{I}\end{cases}$ (3.10)

and more precisely as

$\begin{cases}-\lambda_{j}u_{1}^{(j)}-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})u_{1}^{(j)}+\partial_{x_{1}}p^{(j)}=0\\\
-\lambda_{j}u_{2}^{(j)}-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})u_{2}^{(j)}+\partial_{x_{2}}p^{(j)}=0\\\
\partial_{x_{1}}u_{1}^{(j)}+\partial_{x_{2}}u_{2}^{(j)}=0\\\
u_{1}^{(j)}=0\quad\text{on}\ \partial\Omega\\\ u_{2}^{(j)}=0\quad\text{on}\
\Gamma_{0}\\\
-\lambda_{j}u_{2}^{(j)}-\partial_{x_{1}}^{2}u_{2}^{(j)}=p^{(j)}\quad\text{on}\
\Gamma_{1}.\end{cases}$ (3.11)

###  3.2 Spectral inequality

We are now in a position to state the spectral inequality for the operator
$\displaystyle A_{0}$ defined in the previous section.

######  Theorem 3.2.

Let $\displaystyle\omega_{0}$ be a nonempty open subset of
$\displaystyle\Omega$ and $\displaystyle S_{0}>0$. There exist $\displaystyle
C>0$ and $\displaystyle\kappa\in C^{\infty}_{0}(0,S_{0})$ such that for any
$\displaystyle\Lambda>0$, and for any
$\displaystyle(a_{j})_{j}\in\mathbb{C}^{\mathbb{N}}$,

$\sum_{\lambda_{j}\leqslant\Lambda}\left|a_{j}\right|^{2}\leqslant
Ce^{C\sqrt{\Lambda}}\int_{0}^{S_{0}}\kappa^{2}(s)\left\|\sum_{\lambda_{j}\leqslant\Lambda}a_{j}\cosh(s\sqrt{\lambda}_{j})u^{(j)}\right\|_{L^{2}(\omega)}^{2}\
ds.$ (3.12)

In order to prove 3.2, we define for $\displaystyle s\in(0,S_{0})$ and
$\displaystyle x\in\Omega$,

$U(s,x):=\sum_{\lambda_{j}\leqslant\Lambda}a_{j}\cosh(\sqrt{\lambda}_{j}s)u^{(j)}(x),\quad
P(s,x):=\sum_{\lambda_{j}\leqslant\Lambda}a_{j}\cosh(\sqrt{\lambda}_{j}s)p^{(j)}(x)+c_{P}(s)$
(3.13)

and the domains

$Z:=(0,S_{0})\times\Omega=(0,S_{0})\times\mathcal{I}\times(0,1),\quad
J_{i}:=(0,S_{0})\times\Gamma_{i}=(0,S_{0})\times\mathcal{I}\times\\{i\\}\quad(i=0,1).$
(3.14)

From (3.11), we deduce that

$\left\\{\begin{array}[]{l}-\partial_{s}^{2}U_{1}-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})U_{1}+\partial_{x_{1}}P=0\quad\text{in}\
Z,\\\
-\partial_{s}^{2}U_{2}-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})U_{2}+\partial_{x_{2}}P=0\quad\text{in}\
Z,\\\ \partial_{x_{1}}U_{1}+\partial_{x_{2}}U_{2}=0\quad\text{in}\ Z,\\\
U_{1}=0\quad\text{on}\ J_{0}\cup J_{1}\\\ U_{2}=0\quad\text{on}\ J_{0},\\\
-\partial_{s}^{2}U_{2}-\partial_{x_{1}}^{2}U_{2}=P-m_{\mathcal{I}}(P)\quad\text{on}\
J_{1}.\end{array}\right.$ (3.15)

In the above system, we write

$\displaystyle m_{\mathcal{I}}(P):=\frac{1}{2\pi}\int_{0}^{2\pi}P(x_{1},1)\
dx_{1}$

and by using this notation in the last equation of (3.15), we can replace the
pressure that should satisfies a relation of the form (1.9) by the pressure
$\displaystyle P$ defined up to a function $\displaystyle c_{P}$ of
$\displaystyle s$. In that way, we can, in what follows, impose another
condition on $\displaystyle P$ (typically that its mean on an open set is
zero).

To show 3.2, we first truncate $\displaystyle U$ and $\displaystyle P$ in a
neighborhood of $\displaystyle\\{s=s_{0}\\}$, with

$\displaystyle s_{0}:=\frac{S_{0}}{2}.$

We thus consider $\displaystyle\chi\in C^{\infty}_{0}((0,S_{0}))$, satisfying
$\displaystyle 0\leqslant\chi\leqslant 1$ and

$\chi(s)=\begin{cases}1&\text{if }|s-s_{0}|\leqslant S_{0}/8,\\\ 0&\text{if
}|s-s_{0}|\geqslant S_{0}/6.\end{cases}$ (3.16)

We work with the following localized solutions

$u(s,x_{1},x_{2}):=\chi(s)U(s,x_{1},x_{2}),\quad
p(s,x_{1},x_{2}):=\chi(s)P(s,x_{1},x_{2})$ (3.17)

that satisfy

$\left\\{\begin{array}[]{l}-\partial_{s}^{2}u_{1}-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})u_{1}+\partial_{x_{1}}p=f_{1}\quad\text{in}\
Z\\\
-\partial_{s}^{2}u_{2}-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})u_{2}+\partial_{x_{2}}p=f_{2}\quad\text{in}\
Z\\\ \partial_{x_{1}}u_{1}+\partial_{x_{2}}u_{2}=0\quad\text{in}\ Z\\\
u_{1}=0\quad\text{on}\ J_{0}\cup J_{1}\\\ u_{2}=0\quad\text{on}\ J_{0}\\\
-\partial_{s}^{2}u_{2}-\partial_{x_{1}}^{2}u_{2}=f_{3}+p-m_{\mathcal{I}}(p)\quad\text{on}\
J_{1},\end{array}\right.$ (3.18)

where

$f_{1}:=-\chi^{\prime\prime}U_{1}-2\chi^{\prime}\partial_{s}U_{1},\quad
f_{2}:=-\chi^{\prime\prime}U_{2}-2\chi^{\prime}\partial_{s}U_{2},\quad\quad
f_{3}:=-\chi^{\prime\prime}(U_{2})_{|_{J_{1}}}-2\chi^{\prime}(\partial_{s}U_{2})_{|_{J_{1}}}.$
(3.19)

We also have that

$u=0\quad\text{and}\quad p=0\quad\text{if}\
s\notin\left[\frac{1}{3}S_{0},\frac{2}{3}S_{0}\right].$ (3.20)

As usual, we can use the three first equations to obtain the following
equation for the pressure:

$-\Delta
p=-(\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2})p=\partial_{x_{1}}f_{1}+\partial_{x_{2}}f_{2}=0.$
(3.21)

##  4 A global observability estimate on the pressure

In this section, we prove a global estimate on the pressure. We first
introduce our weight and the corresponding conjugated operators. We then state
our main result, that is 4.1. Then we show a first estimate on the pressure
involving high frequency pressure terms at the boundary. Such terms are then
estimated by showing some a priori estimates and this allows us to prove 4.1.

###  4.1 Choice of the weight and conjugated operators

Let us consider a nonempty open set $\displaystyle\omega_{0}$ such that
$\displaystyle\overline{\omega_{0}}\subset\omega$. Let

$\displaystyle\lambda>0,\tau>0.$

Then we consider $\displaystyle\widetilde{\psi}\in
C^{\infty}(\overline{\Omega};\mathbb{R}^{+})$, such that

$\widetilde{\psi}(x_{1},x_{2})=1-x_{2}\text{ in a neighborhood of
}\\{x_{2}=1\\},\text{ and }\widetilde{\psi}(x_{1},x_{2})=x_{2}\text{ in a
neighborhood of }\\{x_{2}=0\\}$ (4.1)

and such that all its critical points belong to $\displaystyle\omega_{0}$:

$\nabla\widetilde{\psi}(x)=0\implies x\in\omega_{0}.$ (4.2)

We set

$\varphi(s,x):=e^{\lambda\widetilde{\psi}(x)-(s-s_{0})^{2}},\quad\varphi_{0}(s):=e^{-(s-s_{0})^{2}}.$
(4.3)

Note that with our above choices,

$\displaystyle\varphi_{0}(s)=\varphi(s,\cdot,0)=\varphi(s,\cdot,1)=\min_{x\in\overline{\Omega}}\varphi(s,x).$

We recall that we define $\displaystyle(u,p)$ from $\displaystyle(U,P)$ by
(3.17) (truncation in $\displaystyle s$) and that the source $\displaystyle
f_{i}$ are defined by (3.19). We then define

$v:=e^{\tau\varphi}u,\quad q:=e^{\tau\varphi}p,\quad
g_{i}:=e^{\tau\varphi}f_{i}\quad(i\in\\{1,\dots,3\\}).$ (4.4)

In order to take into account the dependence in $\displaystyle s$, we write

$\displaystyle z=(s,x)=(s,x_{1},x_{2})\in
Z,\quad\nabla_{z}=\begin{bmatrix}\partial_{s}\\\ \partial_{x_{1}}\\\
\partial_{x_{2}}\end{bmatrix},\quad\Delta_{z}=\partial_{s}^{2}+\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2},$

and their tangential counterparts

$\displaystyle\nabla_{s,x_{1}}=\begin{bmatrix}\partial_{s}\\\
\partial_{x_{1}}\end{bmatrix},\quad\Delta_{s,x_{1}}=\partial_{s}^{2}+\partial_{x_{1}}^{2}.$

We keep our previous notation

$\displaystyle\nabla=\begin{bmatrix}\partial_{x_{1}}\\\
\partial_{x_{2}}\end{bmatrix},\quad\Delta=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}.$

The equations satisfied by $\displaystyle v$ and $\displaystyle q$ can be
written with the introduction of the following conjugated operators:

$\displaystyle\displaystyle Q_{\varphi}$
$\displaystyle\displaystyle:=-e^{\tau\varphi}\Delta_{z}e^{-\tau\varphi}=-\Delta_{z}+2\tau\nabla_{z}\varphi\cdot\nabla_{z}-\tau^{2}|\nabla_{z}\varphi|^{2}+\tau(\Delta_{z}\varphi),$
(4.5) $\displaystyle\displaystyle D_{\varphi}$
$\displaystyle\displaystyle:=-e^{\tau\varphi}\Delta
e^{-\tau\varphi}=-\Delta+2\tau\left(\nabla\varphi\right)\cdot\nabla-\tau^{2}|\nabla\varphi|^{2}+\tau(\Delta\varphi),$
(4.6) $\displaystyle\displaystyle S_{\varphi}$
$\displaystyle\displaystyle:=-e^{\tau\varphi_{0}}\Delta_{s,x_{1}}e^{-\tau\varphi_{0}}=-\Delta_{s,x_{1}}+2\tau(\partial_{s}\varphi_{0})\partial_{s}-\tau^{2}(\partial_{s}\varphi_{0})^{2}+\tau(\partial_{s}^{2}\varphi_{0}).$
(4.7)

Then we deduce from (3.18) and (3.21) the following conjugated system:

$\left\\{\begin{array}[]{l}Q_{\varphi}v_{1}+e^{\tau\varphi}\partial_{x_{1}}p=g_{1}\quad\text{in}\
Z,\\\ Q_{\varphi}v_{2}+e^{\tau\varphi}\partial_{x_{2}}p=g_{2}\quad\text{in}\
Z,\\\ D_{\varphi}q=0\quad\text{in}\ Z,\\\ v_{1}=0\quad\text{on}\ J_{0}\cup
J_{1},\\\ v_{2}=0\quad\text{on}\ J_{0},\\\
S_{\varphi}v_{2}=g_{3}+q-m_{\mathcal{I}}(q)\quad\text{on}\
J_{1}.\end{array}\right.$ (4.8)

We define $\displaystyle h_{i}$ by

$h_{i}:=e^{\tau\varphi_{0}}f_{i}\quad(i=1,2,3).$ (4.9)

The main result of this section is the following result.

######  Theorem 4.1.

There exist $\displaystyle\lambda_{0}=\lambda_{0}(\widetilde{\psi},S_{0})>0$
and $\displaystyle\tau_{0}=\tau_{0}(\widetilde{\psi},S_{0})>0$ such that for
any $\displaystyle\lambda\geqslant\lambda_{0}$ and
$\displaystyle\tau\geqslant\tau_{0}$, there exists $\displaystyle
C=C(\lambda,\widetilde{\psi},S_{0})>0$ such that for any
$\displaystyle\Lambda>0$ and for any
$\displaystyle(a_{j})_{j}\in\mathbb{C}^{\mathbb{N}}$, the function
$\displaystyle q$ defined by (3.13), (3.17) and (4.4) satisfies

$\tau^{3}\|q\|_{L^{2}(Z)}^{2}+\tau\|\nabla
q\|^{2}_{L^{2}(Z)}+\tau^{3}\left|q-m_{\mathcal{I}}(q)\right|_{L^{2}(J_{1})}^{2}+\tau\left|\nabla
q\right|_{L^{2}((0,S_{0})\times\partial\Omega)}^{2}\\\ \leqslant
C\left(\tau^{3}\|q\|^{2}_{L^{2}((0,S_{0})\times\omega_{0})}+\tau\|\nabla
q\|^{2}_{L^{2}((0,S_{0})\times\omega_{0})}+\tau\left(\left\|\partial_{x_{1}}h_{1}\right\|_{L^{2}(Z)}^{2}+\left\|\partial_{x_{1}}h_{2}\right\|_{L^{2}(Z)}^{2}+\left|\partial_{x_{1}}h_{3}\right|_{L^{2}(J_{1})}^{2}\right)\right).$
(4.10)

We prove this theorem in the remainder of this section.

###  4.2 A first estimate on the pressure

In order to prove 4.1, we exploit that $\displaystyle q$ satisfies the third
equation of (4.8), where $\displaystyle D_{\varphi}$ is defined by (4.6).
Since we do not have any boundary condition, we need to split the boundary
value of $\displaystyle q$ into high and low frequencies. More precisely, for
$\displaystyle\mathcal{Q}\in H^{2}(\Omega)$, we introduce the Fourier
coefficients of the trace of $\displaystyle\mathcal{Q}:$

$a_{k}(\mathcal{Q}):=\begin{bmatrix}\frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{Q}(x_{1},0)e^{-ikx_{1}}\
dx_{1}\\\ \frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{Q}(x_{1},1)e^{-ikx_{1}}\
dx_{1}\end{bmatrix}\quad(k\in\mathbb{Z}).$ (4.11)

We then define the sets of low tangential frequencies and high tangential
frequencies:

$\displaystyle\mathrm{LF}_{\tau}:=\\{k\in\mathbb{Z},\;k^{2}\leqslant\frac{\tau^{2}}{2}\inf|\partial_{x_{2}}\varphi|^{2}\\},\quad\mathrm{HF}_{\tau}:=\\{k\in\mathbb{Z},\;k^{2}>\frac{\tau^{2}}{2}\inf|\partial_{x_{2}}\varphi|^{2}\\}.$

In the above definition, the infimum of $\displaystyle\partial_{x_{2}}\varphi$
is taken for $\displaystyle x\in\partial\Omega$ and $\displaystyle
s\in[0,S_{0}]$.

Due to (4.1) and (4.3), we have

$\displaystyle\inf|\partial_{x_{2}}\varphi|=\lambda e^{-s_{0}^{2}}.$

######  Proposition 4.2.

There exist $\displaystyle\lambda_{0}=\lambda_{0}(\widetilde{\psi},S_{0})>0$
and $\displaystyle\tau_{0}=\tau_{0}(\widetilde{\psi},S_{0})>0$ such that for
any $\displaystyle\lambda\geqslant\lambda_{0}$ and
$\displaystyle\tau\geqslant\tau_{0}$, there exists $\displaystyle
C=C(\lambda,\widetilde{\psi},S_{0})>0$ such that for any $\displaystyle
s\in[0,S_{0}],$ and any $\displaystyle\mathcal{Q}\in H^{2}(\Omega)$,

$\tau^{3}\|\mathcal{Q}\|_{L^{2}(\Omega)}^{2}+\tau\|\nabla\mathcal{Q}\|^{2}_{L^{2}(\Omega)}+\tau|\partial_{x_{2}}\mathcal{Q}|_{L^{2}(\partial\Omega)}^{2}+\sum_{k\in\mathrm{LF}_{\tau}}\tau(\tau^{2}+k^{2})|a_{k}(\mathcal{Q})|^{2}\\\
\leqslant
C\left(\|D_{\varphi}\mathcal{Q}\|_{L^{2}(\Omega)}^{2}+\tau^{3}\|\mathcal{Q}\|^{2}_{L^{2}(\omega_{0})}+\tau\|\nabla\mathcal{Q}\|^{2}_{L^{2}(\omega_{0})}+\sum_{k\in\mathrm{HF}_{\tau}}\tau(\tau^{2}+k^{2})|a_{k}(\mathcal{Q})|^{2}\right).$
(4.12)

###### Proof.

We can decompose the operator $\displaystyle D_{\varphi}$ (see (4.6)) as
follows $\displaystyle D_{\varphi}=\mathcal{S}+\mathcal{A}+\mathcal{R}$, where

$\displaystyle\mathcal{S}=-\Delta-\tau^{2}|\nabla\varphi|^{2},\quad\mathcal{A}=2\tau\nabla\varphi\cdot\nabla+2\tau(\Delta\varphi),\quad\mathcal{R}=-\tau(\Delta\varphi).$

Then, after some standard computation, we can obtain that

$\int_{\Omega}(\mathcal{S}\mathcal{Q})(\mathcal{A}\mathcal{Q})dx=\tau\int_{\Omega}\left(2\nabla^{2}\varphi(\nabla\mathcal{Q},\nabla\mathcal{Q})+\Delta\varphi|\nabla\mathcal{Q}|^{2}\right)dx+\tau^{3}\int_{\Omega}\left(2\nabla^{2}\varphi(\nabla\varphi,\nabla\varphi)-|\nabla\varphi|^{2}\Delta\varphi\right)|\mathcal{Q}|^{2}dx\\\
-\tau\int_{\Omega}(\Delta^{2}\varphi)|\mathcal{Q}|^{2}dx+\mathcal{B},$ (4.13)

where

$\displaystyle\nabla^{2}\varphi=\left(\frac{\partial^{2}\varphi}{\partial
x_{i}\partial x_{j}}\right)_{i,j}$

and where $\displaystyle\mathcal{B}$ corresponds to the boundary terms:

$\mathcal{B}=-2\tau\int_{\partial\Omega}\partial_{n}\mathcal{Q}(\nabla\varphi\cdot\nabla\mathcal{Q})d\Gamma+\tau\int_{\partial\Omega}(\partial_{n}\varphi)|\nabla\mathcal{Q}|^{2}d\Gamma-2\tau\int_{\partial\Omega}(\Delta\varphi)(\partial_{n}\mathcal{Q})\mathcal{Q}d\Gamma\\\
+\tau\int_{\partial\Omega}(\partial_{n}\Delta\varphi)\mathcal{Q}^{2}d\Gamma-\tau^{3}\int_{\partial\Omega}|\nabla\varphi|^{2}(\partial_{n}\varphi)\mathcal{Q}^{2}d\Gamma.$

Using (4.1), we have that $\displaystyle\partial_{n}\varphi<0$ and we can
simplify the above quantity:

$\mathcal{B}=\tau\int_{\partial\Omega}\left|\partial_{x_{2}}\varphi\right|\left(\partial_{x_{2}}\mathcal{Q}\right)^{2}d\Gamma-\tau\int_{\partial\Omega}\left|\partial_{x_{2}}\varphi\right|\left(\partial_{x_{1}}\mathcal{Q}\right)^{2}d\Gamma-2\tau\int_{\partial\Omega}(\partial_{x_{2}}^{2}\varphi)(n\cdot
e_{2})(\partial_{x_{2}}\mathcal{Q})\mathcal{Q}d\Gamma\\\
+\tau\int_{\partial\Omega}(\partial_{x_{2}}^{3}\varphi)(n\cdot
e_{2})\mathcal{Q}^{2}d\Gamma+\tau^{3}\int_{\partial\Omega}\left|\partial_{x_{2}}\varphi\right|^{3}\mathcal{Q}^{2}d\Gamma.$

Combining the above relation with (4.3), there exists
$\displaystyle\tau_{1}=\tau_{1}(S_{0})>0$ such that for any
$\displaystyle\tau\geqslant\tau_{1}$, we have

$\mathcal{B}\geqslant\frac{1}{2}\tau\lambda\varphi_{0}\int_{\partial\Omega}\left(\partial_{x_{2}}\mathcal{Q}\right)^{2}d\Gamma-\tau\lambda\varphi_{0}\int_{\partial\Omega}\left(\partial_{x_{1}}\mathcal{Q}\right)^{2}d\Gamma+\frac{3}{4}\tau^{3}\lambda^{3}\varphi_{0}^{3}\int_{\partial\Omega}\mathcal{Q}^{2}d\Gamma\\\
\geqslant\frac{1}{2}\tau\lambda\varphi_{0}\int_{\partial\Omega}\left(\partial_{x_{2}}\mathcal{Q}\right)^{2}d\Gamma+2\pi\tau\lambda\varphi_{0}\sum_{k\in\mathbb{Z}}\left(\frac{3}{4}\tau^{2}\lambda^{2}\varphi_{0}^{2}-k^{2}\right)|a_{k}(\mathcal{Q})|^{2}.$
(4.14)

Using (4.3) and (4.2), there exist $\displaystyle
C_{1}=C_{1}(\widetilde{\psi})$, $\displaystyle C_{2}=C_{2}(\widetilde{\psi})$,
$\displaystyle\tau_{2}=\tau_{2}(S_{0},\widetilde{\psi})$ and
$\displaystyle\lambda_{1}=\lambda_{1}(\widetilde{\psi})$ such that for
$\displaystyle\lambda\geqslant\lambda_{1}$ and
$\displaystyle\tau\geqslant\tau_{2}$,

$\tau\int_{\Omega}\left(2\nabla^{2}\varphi(\nabla\mathcal{Q},\nabla\mathcal{Q})+\Delta\varphi|\nabla\mathcal{Q}|^{2}\right)dx+\tau^{3}\int_{\Omega}\left(2\nabla^{2}\varphi(\nabla\varphi,\nabla\varphi)-|\nabla\varphi|^{2}\Delta\varphi\right)|\mathcal{Q}|^{2}dx-\tau\int_{\Omega}(\Delta^{2}\varphi)|\mathcal{Q}|^{2}dx\\\
\geqslant
C_{1}\int_{\Omega}\left(\tau\lambda^{2}\varphi\left|\nabla\mathcal{Q}\right|^{2}+\tau^{3}\lambda^{4}\varphi^{3}\left|\mathcal{Q}\right|^{2}\right)dx-
C_{2}\int_{\omega_{0}}\left(\tau\lambda^{2}\varphi\left|\nabla\mathcal{Q}\right|^{2}+\tau^{3}\lambda^{4}\varphi^{3}\left|\mathcal{Q}\right|^{2}\right)dx.$
(4.15)

Finally, combining (4.13), (4.14) and (4.15), we deduce the existence of
$\displaystyle\lambda_{0}>0$ and $\displaystyle\tau_{0}>0$ such that for any
$\displaystyle\lambda\geqslant\lambda_{0}$ and
$\displaystyle\tau\geqslant\tau_{0}$, there exist $\displaystyle
C_{3}=C_{3}(\lambda,\widetilde{\psi},S_{0})>0$ and $\displaystyle
C_{4}=C_{4}(\lambda,\widetilde{\psi},S_{0})>0$ such that

$\|D_{\varphi}\mathcal{Q}\|_{L^{2}(\Omega)}^{2}\geqslant\frac{1}{2}\|(\mathcal{S}+\mathcal{A})\mathcal{Q}\|^{2}_{L^{2}(\Omega)}-\|\mathcal{R}\mathcal{Q}\|^{2}_{L^{2}(\Omega)}\geqslant\operatorname{Re}\left(\mathcal{S}\mathcal{Q},\mathcal{A}\mathcal{Q}\right)_{L^{2}(\Omega)}-\|\mathcal{R}\mathcal{Q}\|^{2}_{L^{2}(0,1)}\\\
\geqslant
C_{3}\left(\int_{\Omega}\left(\tau|\nabla\mathcal{Q}|^{2}+\tau^{3}|\mathcal{Q}|^{2}\right)\
dx+\tau\int_{\partial\Omega}\left(\partial_{x_{2}}\mathcal{Q}\right)^{2}d\Gamma+\tau\sum_{k\in\mathrm{LF}_{\tau}}(\tau^{2}+k^{2})|a_{k}(\mathcal{Q})|^{2}\right)\\\
-C_{4}\left(\int_{\omega_{0}}\left(\tau|\nabla\mathcal{Q}|^{2}+\tau^{3}|\mathcal{Q}|^{2}\right)\
dx+\tau\sum_{k\in\mathrm{HF}_{\tau}}k^{2}|a_{k}(\mathcal{Q})|^{2}\right).$

∎

Applying the above result to $\displaystyle\mathcal{Q}=q$ solution of (4.8)
and integrating into $\displaystyle(0,S_{0})$, we deduce that

$\tau^{3}\|q\|_{L^{2}(Z)}^{2}+\tau\|\nabla
q\|^{2}_{L^{2}(Z)}+\tau|\partial_{x_{2}}q|_{L^{2}((0,S_{0})\times\partial\Omega)}^{2}+\sum_{k\in\mathrm{LF}_{\tau}}\tau(\tau^{2}+k^{2})|a_{k}(q)|_{L^{2}(0,S_{0})}^{2}\\\
\leqslant
C\left(\tau^{3}\|q\|^{2}_{L^{2}((0,S_{0})\times\omega_{0})}+\tau\|\nabla
q\|^{2}_{L^{2}((0,S_{0})\times\omega_{0})}+\sum_{k\in\mathrm{HF}_{\tau}}\tau(\tau^{2}+k^{2})|a_{k}(q)|_{L^{2}(0,S_{0})}^{2}\right).$
(4.16)

We recall that $\displaystyle q(s,\cdot)\equiv 0$ if
$\displaystyle|s-s_{0}|\geqslant S_{0}/6$ due to the support of
$\displaystyle\chi$ (see (3.16)). Next, we will estimate the high tangential
frequencies of the pressure.

###  4.3 Estimates in the high frequency regime

We define

$\displaystyle Y:=(0,S_{0})\times(0,1),\quad I_{i}:=(0,S_{0})\times\\{i\\},\
i=0,1.$

We recall that $\displaystyle(u,p)$ is defined by (3.17) and
$\displaystyle(f_{1},f_{2},f_{3})$ is defined by (3.19). We define
$\displaystyle(u^{k},p^{k},f_{1}^{k},f_{2}^{k},f_{3}^{k})$ the Fourier
coefficients of $\displaystyle(u,p,f_{1},f_{2},f_{3})$ in the $\displaystyle
x_{1}$ direction. For instance

$\displaystyle
u^{k}(s,x_{2}):=\frac{1}{2\pi}\int_{0}^{2\pi}u(s,x_{1},x_{2})e^{-ikx_{1}}\
dx_{1}\quad((s,x_{2})\in Y).$

Finally, with $\displaystyle\tau>0$ and $\displaystyle\varphi_{0}$ defined by
(4.3), we set

$w^{k}=e^{\tau\varphi_{0}}u^{k},\quad\pi^{k}=e^{\tau\varphi_{0}}p^{k},\quad
h_{i}^{k}=e^{\tau\varphi_{0}}f_{i}^{k}\quad(i=1,2,3).$ (4.17)

Note that $\displaystyle h_{i}^{k}$ are the Fourier coefficients of the
functions $\displaystyle h_{i}$ defined by (4.9). Since
$\displaystyle\varphi_{0}$ only depends on $\displaystyle s$, and using (4.3),
(4.4), we have

$a_{k}(q)=\begin{bmatrix}(\pi^{k})_{|I_{0}}\\\
(\pi^{k})_{|I_{1}}\end{bmatrix}.$ (4.18)

Let us define the following conjugated operators:

$\displaystyle\displaystyle Q_{k,\varphi_{0}}$
$\displaystyle\displaystyle:=e^{\tau\varphi_{0}}(-\partial_{s}^{2}-\partial_{x_{2}}^{2}+k^{2})e^{-\tau\varphi_{0}}=-\partial_{s}^{2}-\partial_{x_{2}}^{2}+k^{2}+2\tau\varphi_{0}^{\prime}\partial_{s}-\tau^{2}(\varphi_{0}^{\prime})^{2}+\tau\varphi_{0}^{\prime\prime},$
$\displaystyle\displaystyle S_{k,\varphi_{0}}$
$\displaystyle\displaystyle:=e^{\tau\varphi_{0}}(-\partial_{s}^{2}+k^{2})e^{-\tau\varphi_{0}}=-\partial_{s}^{2}+k^{2}+2\tau\varphi_{0}^{\prime}\partial_{s}-\tau^{2}(\varphi_{0}^{\prime})^{2}+\tau\varphi_{0}^{\prime\prime}.$

Then, for $\displaystyle k\in\mathbb{Z}$, (3.18) transforms into

$\left\\{\begin{array}[]{l}Q_{k,\varphi_{0}}w_{1}^{k}+ik\pi^{k}=h_{1}^{k}\quad\text{in}\
Y,\\\
Q_{k,\varphi_{0}}w_{2}^{k}+\partial_{x_{2}}\pi^{k}=h_{2}^{k}\quad\text{in}\
Y,\\\ \operatorname{div}_{k}w^{k}=0\quad\text{in}\ Y,\\\
w_{1}^{k}=0\quad\text{on}\ I_{0}\cup I_{1},\\\ w_{2}^{k}=0\quad\text{on}\
I_{0},\\\ S_{k,\varphi}w_{2}^{k}=h_{3}^{k}+\pi^{k}\quad\text{on}\
I_{1},\end{array}\right.$ (4.19)

where

$\displaystyle\operatorname{div}_{k}\begin{bmatrix}f_{1}\\\
f_{2}\end{bmatrix}=ikf_{1}+\partial_{x_{2}}f_{2}.$

The relation (3.20) yields

$w^{k}=0\quad\text{and}\quad\pi^{k}=0\quad\text{if}\
s\notin\left[\frac{1}{3}S_{0},\frac{2}{3}S_{0}\right].$ (4.20)

######  Proposition 4.3.

If the solution of (4.19) satisfies (4.20), then there exist
$\displaystyle\lambda_{3}=\lambda_{3}(S_{0})>0$ and $\displaystyle C(S_{0})>0$
such that for any $\displaystyle\lambda\geqslant\lambda_{3}$ and
$\displaystyle k\in\mathrm{HF}_{\tau}$,

$\displaystyle\|\pi^{k}\|_{L^{2}(I_{0}\cup
I_{1})}\leqslant\frac{C}{(k^{2}+\tau^{2})^{1/4}}\left(\left\|h_{1}^{k}\right\|_{L^{2}(Y)}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}\right).$

###### Proof.

We multiply the first line (4.19) by $\displaystyle w_{1}^{k}$, the second
line (4.19) by $\displaystyle w_{2}^{k}$, and the last line (4.19) by
$\displaystyle w_{2}^{k}$. Integrating by parts and summing up yield

$\int_{Y}|\partial_{s}w^{k}|^{2}\ dy+\int_{Y}|\partial_{x_{2}}w^{k}|^{2}\
dy+k^{2}\int_{Y}|w^{k}|^{2}\
dy-\tau^{2}\int_{Y}(\varphi_{0}^{\prime})^{2}|w^{k}|^{2}\ dy\\\
+\int_{I_{1}}|\partial_{s}w_{2}^{k}|^{2}\ ds+k^{2}\int_{I_{1}}|w_{2}^{k}|^{2}\
ds-\tau^{2}\int_{I_{1}}(\varphi_{0}^{\prime})^{2}|w_{2}^{k}|^{2}\ ds\\\
=\operatorname{Re}\int_{Y}h_{1}^{k}\overline{w_{1}^{k}}\
dy+\operatorname{Re}\int_{Y}h_{2}^{k}\overline{w_{2}^{k}}\
dy+\operatorname{Re}\int_{I_{1}}h_{3}^{k}\overline{w^{k}_{2}}\ ds.$ (4.21)

Now, since $\displaystyle k\in\mathrm{HF}_{\tau}$, we have

$\displaystyle
k^{2}>\frac{\tau^{2}}{2}\inf|\partial_{x_{2}}\varphi|^{2}=\frac{\tau^{2}}{2}\lambda^{2}e^{-2s_{0}^{2}}.$

On the other hand,

$\displaystyle\sup_{[0,S_{0}]}|\varphi_{0}^{\prime}|\leqslant S_{0}.$

From the two previous relations, we deduce the existence of
$\displaystyle\lambda_{3}=\lambda_{3}(S_{0})>0$ such that for
$\displaystyle\lambda\geqslant\lambda_{3}$ and for $\displaystyle
k\in\mathrm{HF}_{\tau}$

$\displaystyle
k^{2}-\tau^{2}|\varphi_{0}^{\prime}|^{2}\geqslant\frac{1}{2}(\tau^{2}+k^{2}).$

Combining the above relation and (4.21) yields

$(\tau^{2}+k^{2})\left\|\begin{bmatrix}\partial_{s}w^{k},\partial_{x_{2}}w^{k}\end{bmatrix}\right\|_{L^{2}(Y)}^{2}+(\tau^{2}+k^{2})^{2}\left\|w^{k}\right\|_{L^{2}(Y)}^{2}+(\tau^{2}+k^{2})\left|\partial_{s}w_{2}^{k}\right|_{L^{2}(I_{1})}^{2}+(\tau^{2}+k^{2})^{2}\left|w_{2}^{k}\right|_{L^{2}(I_{1})}^{2}\\\
\lesssim\left\|h_{1}^{k}\right\|_{L^{2}(Y)}^{2}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}^{2}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}^{2}.$
(4.22)

Now, we write (4.19) under the form

$\left\\{\begin{array}[]{l}\left(-\partial_{s}^{2}-\partial_{x_{2}}^{2}+k^{2}\right)w_{1}^{k}+ik\pi^{k}=H_{1}^{k}\quad\text{in}\
Y,\\\
\left(-\partial_{s}^{2}-\partial_{x_{2}}^{2}+k^{2}\right)w_{2}^{k}+\partial_{x_{2}}\pi^{k}=H_{2}^{k}\quad\text{in}\
Y,\\\ \operatorname{div}_{k}w^{k}=0\quad\text{in}\ Y,\\\
w_{1}^{k}=0\quad\text{on}\ I_{0}\cup I_{1},\\\ w_{2}^{k}=0\quad\text{on}\
I_{0},\\\
\left(-\partial_{s}^{2}+k^{2}\right)w_{2}^{k}=H_{3}^{k}+\pi^{k}\quad\text{on}\
I_{1},\end{array}\right.$ (4.23)

where

$\displaystyle\displaystyle H_{1}^{k}$
$\displaystyle\displaystyle:=-2\tau\varphi_{0}^{\prime}\partial_{s}w_{1}^{k}+\tau^{2}(\varphi_{0}^{\prime})^{2}w_{1}^{k}-\tau\varphi_{0}^{\prime\prime}w_{1}^{k}+h_{1}^{k},$
$\displaystyle\displaystyle H_{2}^{k}$
$\displaystyle\displaystyle:=-2\tau\varphi_{0}^{\prime}\partial_{s}w_{2}^{k}+\tau^{2}(\varphi_{0}^{\prime})^{2}w_{2}^{k}-\tau\varphi_{0}^{\prime\prime}w_{2}^{k}+h_{2}^{k},$
$\displaystyle\displaystyle H_{3}^{k}$
$\displaystyle\displaystyle:=-2\tau\varphi_{0}^{\prime}\partial_{s}w_{2}^{k}+\tau^{2}(\varphi_{0}^{\prime})^{2}w_{2}^{k}-\tau\varphi_{0}^{\prime\prime}w_{2}^{k}++h_{3}^{k}.$

From (4.22), we deduce

$\left\|H_{1}^{k}\right\|_{L^{2}(Y)}+\left\|H_{2}^{k}\right\|_{L^{2}(Y)}+\left|H_{3}^{k}\right|_{L^{2}(I_{1})}\lesssim\left\|h_{1}^{k}\right\|_{L^{2}(Y)}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}.$
(4.24)

We multiply the first line (4.23) by
$\displaystyle-\partial_{s}^{2}w_{1}^{k}$, the second line (4.23) by
$\displaystyle-\partial_{s}^{2}w_{2}^{k}$, and the last line (4.23) by
$\displaystyle-\partial_{s}^{2}w_{2}^{k}$. Integrating by parts, summing up
and using (4.24) yield

$\left\|\partial_{s}^{2}w^{k}\right\|_{L^{2}(Y)}^{2}+\left\|\partial_{s}\partial_{x_{2}}w^{k}\right\|_{L^{2}(Y)}^{2}+k^{2}\left\|\partial_{s}w^{k}\right\|_{L^{2}(Y)}^{2}+\left|\partial_{s}^{2}w_{2}^{k}\right|_{L^{2}(I_{1})}^{2}+k^{2}\left|\partial_{s}w_{2}^{k}\right|_{L^{2}(I_{1})}^{2}\\\
\lesssim\left\|h_{1}^{k}\right\|_{L^{2}(Y)}^{2}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}^{2}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}^{2}.$
(4.25)

Next, we write (4.23) under the form

$\left\\{\begin{array}[]{l}\left(-\partial_{x_{2}}^{2}+k^{2}\right)w_{1}^{k}+ik\pi^{k}=\widetilde{H}_{1}^{k}\quad\text{in}\
Y,\\\
\left(-\partial_{x_{2}}^{2}+k^{2}\right)w_{2}^{k}+\partial_{x_{2}}\pi^{k}=\widetilde{H}_{2}^{k}\quad\text{in}\
Y,\\\ \operatorname{div}_{k}w^{k}=0\quad\text{in}\ Y,\\\
w_{1}^{k}=0\quad\text{on}\ I_{0}\cup I_{1},\\\ w_{2}^{k}=0\quad\text{on}\
I_{0},\\\ k^{2}w_{2}^{k}=\widetilde{H}_{3}^{k}+\pi^{k}\quad\text{on}\
I_{1},\end{array}\right.$ (4.26)

where

$\displaystyle\widetilde{H}_{1}^{k}:=H_{1}^{k}+\partial_{s}^{2}w_{1}^{k},\quad\widetilde{H}_{2}^{k}:=H_{2}^{k}+\partial_{s}^{2}w_{2}^{k},\quad\widetilde{H}_{3}^{k}:=H_{3}^{k}+\partial_{s}^{2}w_{1}^{k}.$

From (4.24) and (4.25), we deduce

$\left\|\widetilde{H}_{1}^{k}\right\|_{L^{2}(Y)}+\left\|\widetilde{H}_{2}^{k}\right\|_{L^{2}(Y)}+\left|\widetilde{H}_{3}^{k}\right|_{L^{2}(I_{1})}\lesssim\left\|h_{1}^{k}\right\|_{L^{2}(Y)}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}.$
(4.27)

The first two lines of (4.26) can be written as

$\nabla\left(e^{ikx_{1}}\pi^{k}\right)=e^{ikx_{1}}\begin{bmatrix}\widetilde{H}_{1}^{k}\\\
\widetilde{H}_{2}^{k}\end{bmatrix}+\Delta\left(e^{ikx_{1}}w^{k}\right)$ (4.28)

and thus

$\left\|\nabla\left(e^{ikx_{1}}\pi^{k}\right)\right\|_{H^{-1}(\Omega)}\leqslant\left\|e^{ikx_{1}}\begin{bmatrix}\widetilde{H}_{1}^{k}\\\
\widetilde{H}_{2}^{k}\end{bmatrix}\right\|_{H^{-1}(\Omega)}+\left\|\nabla\left(e^{ikx_{1}}w^{k}\right)\right\|_{L^{2}(\Omega)}$
(4.29)

and using that $\displaystyle k\neq 0$, we deduce from the above estimate that

$\left\|e^{ikx_{1}}\pi^{k}\right\|_{H^{-1}(\Omega)}\leqslant\left\|e^{ikx_{1}}\begin{bmatrix}\widetilde{H}_{1}^{k}\\\
\widetilde{H}_{2}^{k}\end{bmatrix}\right\|_{H^{-1}(\Omega)}+\left\|\nabla\left(e^{ikx_{1}}w^{k}\right)\right\|_{L^{2}(\Omega)}.$
(4.30)

Combining (4.29) and (4.30) with the Nečas inequality (see, for instance, [10,
p.231, Theorem IV.1.1]), we deduce that

$k\left\|e^{ikx_{1}}\pi^{k}\right\|_{L^{2}(\Omega)}\lesssim\left\|\widetilde{H}_{1}^{k}\right\|_{L^{2}(0,1)}+\left\|\widetilde{H}_{2}^{k}\right\|_{L^{2}(0,1)}+k\left\|\nabla\left(e^{ikx_{1}}w^{k}\right)\right\|_{L^{2}(\Omega)}$
(4.31)

and thus, with (4.27) and (4.22),

$k\left\|\pi^{k}\right\|_{L^{2}(Y)}\lesssim\left\|h_{1}^{k}\right\|_{L^{2}(Y)}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}.$
(4.32)

On the other hand, differentiating the divergence equation of system (4.26)
with respect to $\displaystyle x_{2}$ and using (4.22), (4.27) yield

$\displaystyle\|\partial_{x_{2}}^{2}w^{k}_{2}\|_{L^{2}(Y)}+\|\partial_{x_{2}}\pi^{k}\|_{L^{2}(Y)}\lesssim\left\|h_{1}^{k}\right\|_{L^{2}(Y)}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}.$

Then, combining the above relation with (4.32) and with a trace inequality, we
deduce

$\displaystyle\|\pi^{k}\|_{L^{2}(I_{0}\cup
I_{1})}\lesssim\frac{1}{k^{1/2}}\left(\left\|h_{1}^{k}\right\|_{L^{2}(Y)}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}\right).$

Using that $\displaystyle k\in\mathrm{HF}_{\tau}$, we deduce the result. ∎

###  4.4 Proof of 4.1

From 4.3 and from (4.18), for $\displaystyle k\in\mathrm{HF}_{\tau}$

$\displaystyle(\tau^{2}+k^{2})\left|a^{k}(q)\right|_{L^{2}(0,S_{0})}^{2}\lesssim
k^{2}\left(\left\|h_{1}^{k}\right\|_{L^{2}(Y)}^{2}+\left\|h_{2}^{k}\right\|_{L^{2}(Y)}^{2}+\left|h_{3}^{k}\right|_{L^{2}(I_{1})}^{2}\right)$

and thus, with the Parceval formula,

$\displaystyle\sum_{k\in\mathrm{HF}_{\tau}}(\tau^{2}+k^{2})|a_{k}(q)|_{L^{2}(0,S_{0})}^{2}\lesssim\left\|\partial_{x_{1}}h_{1}\right\|_{L^{2}(Z)}^{2}+\left\|\partial_{x_{1}}h_{2}\right\|_{L^{2}(Z)}^{2}+\left|\partial_{x_{1}}h_{3}\right|_{L^{2}(J_{1})}^{2},$

where we have used (4.17) and (4.9).

Combining this estimate with (4.16) we finally obtain the sought result.

∎

##  5 Proof of the spectral inequality

The proof of the spectral inequality, that is (3.12) is based on interpolation
estimates. More precisely, it will be a consequence of 5.3 stated below. In
order to show such a result, we first recall some interpolation inequalities
available in the literature and then we combine them with the global pressure
estimates, that is 4.1 to show 5.3. The last part of this section is devoted
to the proof of the spectral inequality from the interpolation inequality.

First, we need the following notation for this section:

$\mathcal{O}_{0}:=\left(s_{0}-\frac{S_{0}}{6},s_{0}+\frac{S_{0}}{6}\right)\times\omega_{0},\quad\mathcal{O}:=\left(s_{0}-\frac{S_{0}}{5},s_{0}+\frac{S_{0}}{5}\right)\times\omega,$
(5.1)
$\widetilde{Z}:=\left(s_{0}-\frac{S_{0}}{10},s_{0}+\frac{S_{0}}{10}\right)\times\Omega,\quad\widetilde{J}_{1}:=\left(s_{0}-\frac{S_{0}}{10},s_{0}+\frac{S_{0}}{10}\right)\times\Gamma_{1},$
(5.2)
$\widehat{Z}:=\left(s_{0}-\frac{S_{0}}{9},s_{0}+\frac{S_{0}}{9}\right)\times\Omega,\quad\widehat{J}_{1}:=\left(s_{0}-\frac{S_{0}}{9},s_{0}+\frac{S_{0}}{9}\right)\times\Gamma_{1}.$
(5.3)

Note that $\displaystyle\widetilde{Z}\subset\widehat{Z}$ and
$\displaystyle\widetilde{J}_{1}\subset\widehat{J}_{1}$.

###  5.1 Estimates on the velocity

The two components of the velocity satisfy different boundary conditions (see
(3.15)). We start by an estimate on the first component $\displaystyle U_{1}$
that satisfies homogeneous Dirichlet boundary conditions. For the proof of
this result, we refer to relation $\displaystyle(1)$ in Section 3 of [27].

######  Theorem 5.1.

There exist $\displaystyle C_{1}>0$ and $\displaystyle\mu_{1}\in(0,1)$ such
that for all $\displaystyle w\in H^{2}(Z)$ such that $\displaystyle
w_{|_{J_{0}\cup J_{1}}}=0$,

$\left\|w\right\|_{H^{1}\left(\widetilde{Z}\right)}\leqslant
C_{1}\left\|w\right\|_{H^{1}\left(\widehat{Z}\right)}^{1-\mu_{1}}\left(\left\|\Delta_{z}w\right\|_{L^{2}\left(\widehat{Z}\right)}+\left\|w\right\|_{L^{2}\left(\mathcal{O}_{0}\right)}\right)^{\mu_{1}}.$
(5.4)

Note that Theorem 5.1 is stated with an $\displaystyle H^{1}$ observation in
[27], but we can transform it into an $\displaystyle L^{2}$ observation as in
(5.4) by using a cut-off function and integrations by parts.

For the estimate of $\displaystyle U_{2}$, we note that it satisfies a Ventcel
boundary condition on $\displaystyle J_{1}$ and the Dirichlet boundary
condition on $\displaystyle J_{0}$. Hence, we use the following result, which
is basically a consequence of a Carleman estimate obtained in [11]. However
for the sake of completeness, we prove the next result in Appendix A.

######  Theorem 5.2.

There exist $\displaystyle C_{2}>0$ and $\displaystyle\mu_{2}\in(0,1)$ such
that

$\left\|w\right\|_{H^{1}\left(\widetilde{Z}\right)}+\left|w\right|_{H^{1}\left(\widetilde{J}_{1}\right)}\leqslant
C_{2}\left(\left\|w\right\|_{H^{1}\left(\widehat{Z}\right)}+\left|w\right|_{H^{1}\left(\widehat{J}_{1}\right)}\right)^{1-\mu_{2}}\\\
\times\left(\left\|\Delta_{z}w\right\|_{L^{2}\left(\widehat{Z}\right)}+\left|\left(\partial_{x_{2}}w\right)_{|_{J_{1}}}-\Delta_{x_{1},s}w_{|_{J_{1}}}\right|_{L^{2}(\widehat{J}_{1})}+\left\|w\right\|_{L^{2}(\mathcal{O}_{0})}\right)^{\mu_{2}}.$
(5.5)

for all $\displaystyle w\in H^{2}(Z)$ such that $\displaystyle
w_{|_{J_{0}}}=0$ and $\displaystyle w_{|_{J_{1}}}\in H^{2}(J_{1}).$

Note that both 5.1 and 5.2 hold for $\displaystyle\mu_{3}\in(0,1)$ such that
$\displaystyle\mu_{3}\leqslant\mu_{1}$ and
$\displaystyle\mu_{3}\leqslant\mu_{2}$ (with a modification of the constants
$\displaystyle C_{1}$ and $\displaystyle C_{2}$). Thus, with
$\displaystyle\mu_{3}=\min(\mu_{1},\mu_{2})$ and an adequate constant
$\displaystyle C_{3}$, we can apply 5.1 with $\displaystyle w=U_{1}$ and 5.2
with $\displaystyle w=U_{2}$, where $\displaystyle U$ satisfies (3.15) and we
deduce

$\left\|U\right\|_{H^{1}\left(\widetilde{Z}\right)}+\left|U_{2}\right|_{H^{1}\left(\widetilde{J}_{1}\right)}\leqslant
C_{3}\left(\left\|U\right\|_{H^{1}\left(\widehat{Z}\right)}+\left|U_{2}\right|_{H^{1}\left(\widehat{J}_{1}\right)}\right)^{1-\mu_{3}}\\\
\times\left(\left\|\nabla
P\right\|_{L^{2}\left(\widehat{Z}\right)}+\left|P_{|J_{1}}-m_{\mathcal{I}}(P)\right|_{L^{2}(\widehat{J}_{1})}+\left\|U\right\|_{L^{2}(\mathcal{O}_{0})}\right)^{\mu_{3}}.$
(5.6)

We have used here the fact that on $\displaystyle J_{1}$,

$\displaystyle\partial_{x_{2}}U_{2}=-\partial_{x_{1}}U_{1}=0.$

We are going now to combine the above estimate with the estimates of the
pressure terms obtained in Section 4.

###  5.2 Patching the estimates together

Combining the previous estimates, we can now prove the following result.

######  Theorem 5.3.

There exist $\displaystyle C>0$ and $\displaystyle\mu\in(0,1)$ such that for
any $\displaystyle\Lambda>0$ and for any
$\displaystyle(a_{j})_{j}\in\mathbb{C}^{\mathbb{N}}$, the function
$\displaystyle U$ defined by (3.13) satisfies

$\displaystyle\left\|U\right\|_{H^{1}\left(\widetilde{Z}\right)}+\left|U_{2}\right|_{H^{1}\left(\widetilde{J}_{1}\right)}\leqslant
C\left(\left\|U\right\|_{H^{2}\left(Z\right)}+\left|U_{2}\right|_{H^{2}\left(J_{1}\right)}\right)^{1-\mu}\left\|U\right\|_{H^{2}(\mathcal{O}_{0})}^{\mu}.$

###### Proof.

We start with the estimate (4.10), where we recall that $\displaystyle q$ is
given by (3.17), (4.4) and $\displaystyle h_{i}$, $\displaystyle i=1,2,3$ by
(4.9):

$\tau^{3}\|e^{\tau\varphi}\chi
P\|_{L^{2}(Z)}^{2}+\tau\|e^{\tau\varphi}\chi\nabla
P\|^{2}_{L^{2}(Z)}+\tau^{3}|e^{\tau\varphi_{0}}\chi(P-m_{\mathcal{I}}(P))|_{L^{2}(J_{1})}^{2}+\tau|e^{\tau\varphi_{0}}\chi\partial_{x_{1}}P|_{L^{2}(J_{1})}^{2}+\tau|e^{\tau\varphi_{0}}\chi\partial_{x_{2}}P|_{L^{2}(J_{1})}^{2}\\\
\leqslant
C\Big{(}\tau^{3}\|e^{\tau\varphi}P\|^{2}_{L^{2}(\mathcal{O}_{0})}+\tau\|e^{\tau\varphi}\nabla
P\|^{2}_{L^{2}(\mathcal{O}_{0})}+\tau\left\|e^{\tau\varphi_{0}}\partial_{x_{1}}f_{1}\right\|_{L^{2}(Z)}^{2}\\\
+\tau\left\|e^{\tau\varphi_{0}}\partial_{x_{1}}f_{2}\right\|_{L^{2}(Z)}^{2}+\tau\left|e^{\tau\varphi_{0}}\partial_{x_{1}}f_{3}\right|_{L^{2}(J_{1})}^{2}\Big{)}.$
(5.7)

Note that $\displaystyle f_{1},f_{2}$ (respectively $\displaystyle f_{3}$) are
supported in

$\displaystyle
Z_{8}:=\left(\operatorname{supp}\chi^{\prime}\right)\times\mathcal{I}\times(0,1)\quad(\text{respectively
in}\
J_{8}:=\left(\operatorname{supp}\chi^{\prime}\right)\times\mathcal{I}\times\\{1\\}),$

and since
$\displaystyle\operatorname{supp}\chi^{\prime}\subset\left[s_{0}-S_{0}/6,s_{0}+S_{0}/6\right]$

$\sup_{J_{8}}\varphi_{0}=\sup_{Z_{8}}\varphi_{0}\leqslant
e^{-\frac{S_{0}^{2}}{36}}.$ (5.8)

Hence, from (3.19)

$\displaystyle\displaystyle\left\|e^{\tau\varphi_{0}}\partial_{x_{1}}f_{1}\right\|_{L^{2}(Z)}+\left\|e^{\tau\varphi_{0}}\partial_{x_{1}}f_{2}\right\|_{L^{2}(Z)}+\left|e^{\tau\varphi_{0}}\partial_{x_{1}}f_{3}\right|_{L^{2}(J_{1})}\leqslant
Ce^{\tau
e^{-\frac{S_{0}^{2}}{36}}}\left(\|U\|_{H^{2}(Z)}+|U_{2}|_{H^{2}(J_{1})}\right).$
(5.9)

In $\displaystyle\widehat{Z}$ (respectively in
$\displaystyle\widehat{J}_{1}$), $\displaystyle\chi(s)\equiv 1$, and

$\inf_{\widehat{Z}}\varphi=\inf_{\widehat{Z}}\varphi_{0}=\inf_{\widehat{J}_{1}}\varphi_{0}=e^{-\frac{S_{0}^{2}}{81}}.$
(5.10)

Combining (5.7), (5.9) and (5.10), there exist
$\displaystyle\tau_{4},c_{1},c_{2}>0$ such that for all
$\displaystyle\tau\geqslant\tau_{4}$, we have

$\|\nabla
P\|^{2}_{L^{2}(\widehat{Z})}+|(P-m_{\mathcal{I}}(P))|_{L^{2}(\widehat{J}_{1})}^{2}\leqslant
e^{c_{1}\tau}\left(\left\|P\right\|^{2}_{L^{2}(\mathcal{O}_{0})}+\left\|\nabla
P\right\|^{2}_{L^{2}(\mathcal{O}_{0})}\right)+e^{-c_{2}\tau}\left(\|U\|_{H^{2}(Z)}+|U_{2}|_{H^{2}(J_{1})}\right).$
(5.11)

On the other hand, we deduce from (5.6) and a Young inequality that

$\left\|U\right\|_{H^{1}\left(\widetilde{Z}\right)}+\left|U_{2}\right|_{H^{1}\left(\widetilde{J}_{1}\right)}\leqslant
C_{4}e^{-\frac{\mu_{3}c_{2}}{2(1-\mu_{3})}\tau}\left(\left\|U\right\|_{H^{1}\left(\widehat{Z}\right)}+\left|U_{2}\right|_{H^{1}\left(\widehat{J}_{1}\right)}\right)\\\
+e^{\frac{c_{2}}{2}\tau}\left(\left\|\nabla
P\right\|_{L^{2}\left(\widehat{Z}\right)}+\left|P_{|J_{1}}-m_{\mathcal{I}}(P)\right|_{L^{2}(\widehat{J}_{1})}+\left\|U\right\|_{L^{2}(\mathcal{O}_{0})}\right)$

and combining this relation with (5.11), we deduce the existence of
$\displaystyle c_{3},c_{4}>0$ such that for all
$\displaystyle\tau\geqslant\tau_{4}$

$\displaystyle\left\|U\right\|_{H^{1}\left(\widetilde{Z}\right)}+\left|U_{2}\right|_{H^{1}\left(\widetilde{J}_{1}\right)}\lesssim
e^{-c_{3}\tau}\left(\left\|U\right\|_{H^{2}\left(Z\right)}+\left|U_{2}\right|_{H^{2}\left(J_{1}\right)}\right)\\\
+e^{c_{4}\tau}\left(\left\|P\right\|^{2}_{L^{2}(\mathcal{O}_{0})}+\left\|\nabla
P\right\|^{2}_{L^{2}(\mathcal{O}_{0})}+\left\|U\right\|_{L^{2}(\mathcal{O}_{0})}\right)$

Now, we can use $\displaystyle c_{P}$ in (3.13) so that for all $\displaystyle
s\in[0,S_{0}]$,

$\displaystyle\int_{\omega_{0}}P(s,x)\ dx=0$

and using the Poincaré-Wirtinger inequality, we deduce that

$\displaystyle\left\|P\right\|^{2}_{L^{2}(\mathcal{O}_{0})}\lesssim\left\|\nabla
P\right\|^{2}_{L^{2}(\mathcal{O}_{0})}\lesssim\left\|\Delta_{z}U\right\|_{L^{2}(\mathcal{O}_{0})}.$

We deduce that for some constants $\displaystyle c_{5},c_{6}>0$, for all
$\displaystyle\tau\geqslant\tau_{4}$,

$\displaystyle\left\|U\right\|_{H^{1}\left(\widetilde{Z}\right)}+\left|U_{2}\right|_{H^{1}\left(\widetilde{J}_{1}\right)}\lesssim
e^{-c_{5}\tau}\left(\left\|U\right\|_{H^{2}\left(Z\right)}+\left|U_{2}\right|_{H^{2}\left(J_{1}\right)}\right)\\\
+e^{c_{6}\tau}\left\|U\right\|_{H^{2}(\mathcal{O}_{0})}.$

Optimizing this inequality with respect to
$\displaystyle\tau\geqslant\tau_{4}$ (see, for instance, [11, Lemma 8.4])
allows us to conclude the proof of 5.3. ∎

###  5.3 From the interpolation inequality to the spectral inequality

Using 5.3, we are now in a position to prove 3.2. This inequality combined
with 2.1 yields the main result of the article (1.1).

###### Proof of 3.2.

From (3.8) and (3.1), we deduce that

$\displaystyle\left\|U\right\|_{H^{1}\left(\widetilde{Z}\right)}^{2}+\left|U_{2}\right|_{H^{1}\left(\widetilde{J}_{1}\right)}^{2}\geqslant\left\|U\right\|_{L^{2}\left(\widetilde{Z}\right)}^{2}+\left|U_{2}\right|_{L^{2}\left(\widetilde{J}_{1}\right)}^{2}\geqslant\int_{s_{0}-\frac{S_{0}}{10}}^{s_{0}+\frac{S_{0}}{10}}\sum_{\lambda_{j}\leqslant\Lambda}|a_{j}|^{2}\cosh(\sqrt{\lambda}_{j}s)^{2}\
ds\gtrsim\sum_{\lambda_{j}\leqslant\Lambda}|a_{j}|^{2}$

and

$\displaystyle\left\|U\right\|_{H^{2}\left(Z\right)}^{2}+\left|U_{2}\right|_{H^{2}\left(J_{1}\right)}^{2}\lesssim
e^{C\sqrt{\Lambda}}\sum_{\lambda_{j}\leqslant\Lambda}|a_{j}|^{2}$

Combining 5.3 with the previous relations, we deduce that

$\displaystyle\sum_{\lambda_{j}\leqslant\Lambda}|a_{j}|^{2}\lesssim
e^{C\sqrt{\Lambda}}\left\|U\right\|_{H^{2}(\mathcal{O}_{0})}^{2}.$

Using a cut-off function and integrations by parts, we deduce

$\displaystyle\sum_{\lambda_{j}\leqslant\Lambda}|a_{j}|^{2}\lesssim
e^{C\sqrt{\Lambda}}\left\|U\right\|_{L^{2}(\mathcal{O})}^{2}.$

and thus 3.2. ∎

##  Appendix A Proof of 5.2

###  A.1 A Carleman estimate

The proof of 5.2 is mainly based on a Carleman estimate obtained in [11] that
we recall here. We recall that $\displaystyle Z,J_{1},J_{0}$ are defined by
(3.14) whereas $\displaystyle\widetilde{Z}$ and
$\displaystyle\widetilde{J}_{1}$ are defined by (5.2). In what follows, we
consider $\displaystyle z^{0}\in\widetilde{J}_{1}$, an open neighborhood
$\displaystyle V$ of $\displaystyle z^{0}$ in $\displaystyle\overline{Z}$ and
a weight function $\displaystyle\varphi\in C^{\infty}(\overline{V})$. For any
$\displaystyle\sigma\in\mathbb{R}$, we define

$\displaystyle
p_{\varphi,\sigma}(z,\xi,\tau)=|\xi|^{2}-\tau^{2}|\nabla_{z}\varphi(z)|^{2}-\sigma^{2}+2i\tau\xi\cdot\nabla_{z}\varphi(z),\quad(z\in
V,\ \xi\in\mathbb{R}^{3},\ \tau\in\mathbb{R}).$

It is the principal symbol of the conjugated operator associated with
$\displaystyle-\Delta_{z}-\sigma^{2}$, that is, of the operator

$\displaystyle
P_{\varphi,\sigma}=-e^{\tau\varphi}\left(\Delta_{z}+\sigma^{2}\right)e^{-\tau\varphi}=-\Delta_{z}+2\tau\nabla_{z}\varphi\cdot\nabla_{z}-\tau^{2}|\nabla_{z}\varphi|^{2}+\tau(\Delta_{z}\varphi)-\sigma^{2}.$

We assume the following hypotheses on $\displaystyle\varphi$: sub-ellipticity
on $\displaystyle\overline{V}$, that is the existence of
$\displaystyle\tau_{0}>0$ such that for any $\displaystyle z\in V$,
$\displaystyle\xi\in\mathbb{R}^{3}$, $\displaystyle|\sigma|\geqslant 1$ and
$\displaystyle\tau\geqslant\tau_{0}|\sigma|$,

$p_{\varphi,\sigma}(z,\xi,\tau)=0\implies\frac{1}{2i}\left\\{\overline{p_{\varphi,\sigma}},p_{\varphi,\sigma}\right\\}(z,\xi,\tau)>0.$
(A.1)

and the two following conditions to handle Ventcel boundary conditions (see
[11, conditions (23) and (24)]):

$\nabla_{z}\varphi\neq 0\quad\text{in}\
\overline{V},\quad\text{and}\quad\sup_{\overline{V}\cap\widetilde{J}_{1}}\left|\nabla_{s,x_{1}}\varphi\right|\leqslant\nu_{0}\inf_{\overline{V}}\left|\partial_{x_{2}}\varphi\right|,$
(A.2)

for $\displaystyle\nu_{0}>0$ small enough. We recall that the Poisson bracket
is defined by

$\displaystyle\left\\{p^{(1)},p^{(2)}\right\\}=\sum_{j=1}^{3}\frac{p^{(1)}}{\partial\xi_{j}}\frac{p^{(2)}}{\partial
z_{j}}-\frac{p^{(2)}}{\partial\xi_{j}}\frac{p^{(1)}}{\partial z_{j}}$

where we set here $\displaystyle z_{1}=s$, $\displaystyle z_{2}=x_{1}$ and
$\displaystyle z_{3}=x_{2}$ to simplify.

Then we have the following result proved in [11]:

######  Theorem A.1.

Assume $\displaystyle z^{0}\in\widetilde{J}_{1}$ and $\displaystyle V$ is an
open neighborhood of $\displaystyle z^{0}$ in $\displaystyle\overline{Z}$.
Assume also that $\displaystyle\varphi\in C^{\infty}(\overline{V})$ satisfies
the conditions (A.1)) and (A.2) for $\displaystyle\nu_{0}$ small enough. Then,
there exist $\displaystyle\tau_{0}>0$ and $\displaystyle C>0$ such that for
all $\displaystyle|\sigma|\geqslant 1$, for all
$\displaystyle\tau\geqslant\tau_{0}|\sigma|$ and for all $\displaystyle w\in
C^{\infty}_{0}(V)$,

$\tau^{3}\left\|e^{\tau\varphi}w\right\|_{L^{2}(V)}^{2}+\tau\left\|e^{\tau\varphi}\nabla_{z}w\right\|_{L^{2}(V)}^{2}+\tau^{3}\left|e^{\tau\varphi}w_{|J_{1}}\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}^{2}+\tau\left|e^{\tau\varphi}\nabla_{s,x_{1}}w_{|J_{1}}\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}^{2}\\\
+\tau\left|e^{\tau\varphi}\partial_{x_{2}}w_{|J_{1}}\right|^{2}_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}\leqslant
C\left(\left\|e^{\tau\varphi}(-\Delta_{z}-\sigma^{2})w\right\|^{2}_{L^{2}(V)}+\tau\left|e^{\tau\varphi}(\partial_{x_{2}}w_{|J_{1}}-\Delta_{s,x_{1}}w_{|J_{1}}\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}^{2}\right).$

First to precise the above statement, by $\displaystyle w\in
C^{\infty}_{0}(V)$ we mean that $\displaystyle w$ is the restriction of a
$\displaystyle C^{\infty}$ function with compact support in $\displaystyle
V_{0}$ where $\displaystyle V_{0}$ is an open set of
$\displaystyle\mathbb{R}\times\mathcal{I}\times\mathbb{R}$ such that
$\displaystyle V_{0}\cap\overline{Z}=V$. Second, we use the above result in
the case $\displaystyle\sigma=1$, so that by taking $\displaystyle\tau_{0}$
large enough, we obtain that for all $\displaystyle\tau\geqslant\tau_{0}$ and
for all $\displaystyle w\in C^{\infty}_{0}(V)$,

$\tau^{3}\left\|e^{\tau\varphi}w\right\|_{L^{2}(V)}^{2}+\tau\left\|e^{\tau\varphi}\nabla_{z}w\right\|_{L^{2}(V)}^{2}+\tau^{3}\left|e^{\tau\varphi}w_{|J_{1}}\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}^{2}+\tau\left|e^{\tau\varphi}\nabla_{s,x_{1}}w_{|J_{1}}\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}^{2}\\\
+\tau\left|e^{\tau\varphi}\partial_{x_{2}}w_{|J_{1}}\right|^{2}_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}\leqslant
C\left(\left\|e^{\tau\varphi}\Delta_{z}w\right\|^{2}_{L^{2}(V)}+\tau\left|e^{\tau\varphi}\left(\partial_{x_{2}}w_{|J_{1}}-\Delta_{s,x_{1}}w_{|J_{1}}\right)\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}^{2}\right).$
(A.3)

###  A.2 Interpolation estimates for the Ventcel boundary condition

Using the Carleman inequality of the previous section, one can deduce, in a
classical way, an interpolation inequality. First let us define the weight
function that we are going to use.

We consider the following norms on
$\displaystyle\mathbb{R}\times\mathcal{I}\times\mathbb{R}$:

$\displaystyle\left|\begin{pmatrix}s,x_{1},x_{2}\end{pmatrix}\right|_{\lambda}:=\left(\frac{s^{2}}{\lambda^{2}}+\frac{x_{1}^{2}}{\lambda^{2}}+x_{2}^{2}\right)^{1/2}.$

We consider $\displaystyle z^{0}=(s^{*},x_{1}^{*},1)\in\widetilde{J}_{1}$,

$\displaystyle
V:=\left(s^{*}-\delta,s^{*}+\delta\right)\times\left(x_{1}^{*}-\delta,x_{1}^{*}+\delta\right)\times\left(1-\delta,1\right],$

with $\displaystyle\delta\in(0,1)$ small enough such that

$\displaystyle\left(s^{*}-\delta,s^{*}+\delta\right)\subset\left(s_{0}-\frac{S_{0}}{9},s_{0}+\frac{S_{0}}{9}\right)$

(see (5.3)). We also define $\displaystyle z^{*}=(s^{*},x_{1}^{*},0)$. Then we
define

$\displaystyle\psi(z):=\left|z-z^{*}\right|_{\lambda},\quad\varphi=e^{-\lambda\psi}.$

######  Lemma A.2.

There exists $\displaystyle\lambda_{0}>0$ such that for any
$\displaystyle\lambda\geqslant\lambda_{0}$, the weight function
$\displaystyle\varphi$ satisfies (A.1) and (A.2) on $\displaystyle V$ for some
$\displaystyle\tau_{0}=\tau_{0}(\lambda)$.

###### Proof.

We assume that $\displaystyle\lambda\geqslant 1$ in all what follows. First,
since $\displaystyle\nabla_{z}\psi\neq 0$ in $\displaystyle\overline{V}$, we
deduce the first point of (A.2). For the second point of (A.2), we first
notice that

$\inf_{\overline{V}\cap\widetilde{J}_{1}}\psi=1,\quad\sup_{\overline{V}}\psi=\left(\frac{2\delta^{2}}{\lambda^{2}}+1\right)^{1/2}\leqslant
2.$ (A.4)

We thus deduce

$\displaystyle\sup_{\overline{V}\cap\widetilde{J}_{1}}\left|\nabla_{s,x_{1}}\varphi\right|\leqslant\sup_{\overline{V}\cap\widetilde{J}_{1}}\varphi=e^{-\lambda}\quad\text{and}\quad\inf_{\overline{V}}\left|\partial_{x_{2}}\varphi\right|\geqslant\lambda\frac{1-\delta}{2}e^{-\left(2\delta^{2}+\lambda^{2}\right)^{1/2}}.$

Consequently there exists $\displaystyle\lambda_{1}$ such that the second
point of (A.2) holds for $\displaystyle\lambda\geqslant\lambda_{1}$.

For (A.1), we compute the Poisson bracket:

$\displaystyle\displaystyle\frac{1}{8\tau
i}\\{\overline{p_{\varphi,\sigma}},p_{\varphi,\sigma}\\}$
$\displaystyle\displaystyle=\tau^{2}\left(\nabla_{z}^{2}\varphi\right)\nabla_{z}\varphi\cdot\nabla_{z}\varphi+\left(\nabla_{z}^{2}\varphi\right)\xi\cdot\xi$
$\displaystyle\displaystyle=\tau^{2}\varphi^{3}\left(\lambda^{4}\left|\nabla_{z}\psi\right|^{4}-\lambda^{3}\left(\nabla_{z}^{2}\psi\right)(\nabla_{z}\psi)\cdot(\nabla_{z}\psi)\right)+\varphi\left(\lambda^{2}(\nabla_{z}\psi\cdot\xi)^{2}-\lambda\left(\nabla_{z}^{2}\psi\right)\xi\cdot\xi\right)$
$\displaystyle\displaystyle\geqslant\tau^{2}\lambda^{4}\varphi^{3}\left|\nabla_{z}\psi\right|^{4}-\tau^{2}\lambda^{3}\varphi^{3}\left|\nabla_{z}^{2}\psi\right|\left|\nabla_{z}\psi\right|^{2}-\lambda\varphi\left|\nabla_{z}^{2}\psi\right||\xi|^{2}.$

Now, if $\displaystyle p_{\varphi,\sigma}(z,\xi,\tau)=0$, then
$\displaystyle|\xi|^{2}=\tau^{2}\lambda^{2}\varphi^{2}|\nabla_{z}\psi(z)|^{2}+\sigma^{2}$
so that

$\displaystyle\frac{1}{8\tau
i}\\{\overline{p_{\varphi,\sigma}},p_{\varphi,\sigma}\\}\geqslant\tau^{2}\lambda^{4}\varphi^{3}\left|\nabla_{z}\psi\right|^{4}-2\tau^{2}\lambda^{3}\varphi^{3}\left|\nabla_{z}^{2}\psi\right|\left|\nabla_{z}\psi\right|^{2}-\lambda\varphi\left|\nabla_{z}^{2}\psi\right|\sigma^{2}.$

From (A.4), there exist positive constants independent of
$\displaystyle\lambda$ such that

$\displaystyle C_{1}\leqslant\left|\nabla_{z}\psi\right|\leqslant
C_{2},\quad\left|\nabla_{z}^{2}\psi\right|\leqslant C_{3}.$

In particular there exist $\displaystyle C>0$ and
$\displaystyle\lambda_{0}\geqslant\lambda_{1}$, such that for
$\displaystyle\lambda\geqslant\lambda_{0}$,

$\displaystyle\frac{1}{8\tau
i}\\{\overline{p_{\varphi,\sigma}},p_{\varphi,\sigma}\\}\geqslant
C\tau^{2}\lambda^{4}\varphi^{3}-\lambda\varphi\left|\nabla_{z}^{2}\psi\right|\sigma^{2}$

and there exists $\displaystyle\tau_{0}=\tau_{0}(\lambda)$ such that for
$\displaystyle\tau\geqslant\tau_{0}|\sigma|$,

$\displaystyle\frac{1}{8\tau
i}\\{\overline{p_{\varphi,\sigma}},p_{\varphi,\sigma}\\}\geqslant\frac{C}{2}\tau^{2}\lambda^{4}\varphi^{3}>0.$

∎

From now on, the value of $\displaystyle\lambda$ shall be kept fixed. We
define, for $\displaystyle\beta>0$,

$\displaystyle\widehat{Z}_{\beta}:=\\{z\in\widehat{Z}\ ;\
\operatorname{dist}(z,\widehat{J}_{1})>\beta\\}.$

######  Lemma A.3.

Assume $\displaystyle z^{0}\in\widetilde{J}_{1}$. There exist an open
neighborhood $\displaystyle\widetilde{V}$ of $\displaystyle z^{0}$ in
$\displaystyle\overline{Z}$, $\displaystyle\mu,\beta\in(0,1)$ and
$\displaystyle C>0$ such that for any $\displaystyle v\in H^{2}(Z)$ with
$\displaystyle v_{|_{J_{1}}}\in H^{2}(J_{1})$,

$\left\|v\right\|_{H^{1}(\widetilde{V})}+\left|v_{|J_{1}}\right|_{H^{1}\left(\widetilde{V}\cap\widetilde{J}_{1}\right)}\leqslant
C\left(\left\|v\right\|_{H^{1}\left(\widehat{Z}\right)}+\left|v_{|J_{1}}\right|_{H^{1}\left(\widehat{J}_{1}\right)}\right)^{1-\mu}\\\
\times\left(\left\|\Delta_{z}v\right\|_{L^{2}(\widehat{Z})}+\left|\partial_{x_{2}}v_{|J_{1}}-\Delta_{s,x_{1}}v_{|J_{1}}\right|_{L^{2}\left(\widehat{J}_{1}\right)}+\left\|v\right\|_{H^{1}\left(\widehat{Z}_{\beta}\right)}\right)^{\mu}.$
(A.5)

###### Proof.

Standard computation shows the existence of $\displaystyle r_{3}>1$ such that

$\displaystyle\left|z-z^{*}\right|_{\lambda}=r_{3}\quad z=(s,x_{1},1)\in
J_{1}\implies(s,x_{1})\in\left(s^{*}-\frac{\delta}{2},s^{*}+\frac{\delta}{2}\right)\times\left(x_{1}^{*}-\frac{\delta}{2},x_{1}^{*}+\frac{\delta}{2}\right).$

We consider $\displaystyle r_{2}\in(1,r_{3})$. We can also check the existence
of $\displaystyle r_{1}\in(0,\delta)$ such that

$\displaystyle\\{z=(s,x_{1},x_{2})\in Z\ ;\ 1-r_{1}\leqslant x_{2}\leqslant
1,\quad\left|z-z^{*}\right|_{\lambda}\leqslant r_{3}\\}\subset V.$

We consider two cut-off functions $\displaystyle\chi_{0},\chi_{1}\in
C^{\infty}(\mathbb{R}\times\mathcal{I}\times\mathbb{R})$ such that

$\chi_{0}(z)=\left\\{\begin{array}[]{lll}0&\text{if}&0<x_{2}<1-r_{1}\\\
1&\text{if}&1-r_{1}/2<x_{2}<1,\end{array}\right.\quad\chi_{1}(z)=\left\\{\begin{array}[]{lll}0&\text{if}&\left|z-z^{*}\right|_{\lambda}>r_{3}\\\
1&\text{if}&\left|z-z^{*}\right|_{\lambda}\leqslant r_{2}.\end{array}\right.$

Let us consider $\displaystyle v\in C^{\infty}(Z)$ and let us apply the
Carleman estimate (A.3) to $\displaystyle w=\chi_{0}\chi_{1}v\in
C^{\infty}_{0}(V)$. In the right-hand side of this estimate, we have

$\displaystyle\Delta_{z}\left(\chi_{0}\chi_{1}v\right)=\left(\chi_{0}\chi_{1}\right)\Delta_{z}v+2\nabla_{z}\left(\chi_{0}\chi_{1}\right)\cdot\nabla_{z}v+v\Delta_{z}\left(\chi_{0}\chi_{1}\right)$

Note that in $\displaystyle\operatorname{supp}\chi_{0}\chi_{1}$,
$\displaystyle\varphi\leqslant C_{3}:=e^{-\lambda(1-r_{1})}$. The two last
terms in the above relation are included in

$\displaystyle V\cap\left(\operatorname{supp}\nabla\chi_{0}\right)\subset
V\cap\left\\{x_{2}\in[1-r_{1},1-r_{1}/2]\right\\}$

and on this set, $\displaystyle\varphi\leqslant C_{3}$ or in

$\displaystyle V\cap\left(\operatorname{supp}\nabla\chi_{1}\right)\subset
V\cap\left\\{r_{2}\leqslant\left|z-z^{*}\right|_{\lambda}\leqslant
r_{3}\right\\}$

and on this set, $\displaystyle\varphi\leqslant C_{1}:=e^{-\lambda r_{2}}$.
Therefore,

$\left\|e^{\tau\varphi}\Delta_{z}w\right\|_{L^{2}(V)}\lesssim
e^{C_{3}\tau}\left\|\Delta_{z}v\right\|_{L^{2}(V)}+e^{C_{3}\tau}\left\|v\right\|_{H^{1}\left(V\cap\left\\{x_{2}\in[1-r_{1},1-r_{1}/2]\right\\}\right)}+e^{C_{1}\tau}\left\|v\right\|_{H^{1}\left(V\right)}$
(A.6)

and similarly,

$\left|e^{\tau\varphi}\left(\partial_{x_{2}}w_{|J_{1}}-\Delta_{s,x_{1}}w_{|J_{1}}\right)\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}\lesssim
e^{C_{3}\tau}\left|\partial_{x_{2}}v_{|J_{1}}-\Delta_{s,x_{1}}v_{|J_{1}}\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}+e^{C_{1}\tau}\left|v_{|J_{1}}\right|_{H^{1}\left(V\cap\widetilde{J}_{1}\right)}.$
(A.7)

There exists $\displaystyle r_{4}>0$ such that

$\displaystyle\left\\{z\in\mathbb{R}\times\mathcal{I}\times\mathbb{R}\ ;\
\left|z-z^{0}\right|\leqslant
r_{4}\right\\}\subset\left\\{z=(s,x_{1},x_{2})\in\mathbb{R}\times\mathcal{I}\times\mathbb{R}\
;\
1-\frac{r_{1}}{2}<x_{2},\quad\left|z-z^{*}\right|_{\lambda}<r_{2}\right\\}.$

Then on the set

$\displaystyle\widetilde{V}:=\left\\{z\in Z\ ;\
\left|z-z^{0}\right|<r_{4}\right\\}\subset V$

we have $\displaystyle\chi_{0}\chi_{1}=1$ and $\displaystyle\varphi\geqslant
C_{2}:=e^{-\lambda\sup_{\widetilde{V}}\psi}$, with $\displaystyle
C_{2}\in(C_{1},C_{3})$. Combining this with (A.3), (A.6) and (A.7), we deduce
that for all $\displaystyle\tau\geqslant\tau_{0}$,

$\left\|v\right\|_{H^{1}(\widetilde{V})}+\left|v_{|J_{1}}\right|_{H^{1}\left(\widetilde{V}\cap\widetilde{J}_{1}\right)}\\\
\lesssim
e^{\left(C_{3}-C_{2}\right)\tau}\left(\left\|\Delta_{z}v\right\|_{L^{2}(V)}+\left|\partial_{x_{2}}v_{|J_{1}}-\Delta_{s,x_{1}}v_{|J_{1}}\right|_{L^{2}\left(V\cap\widetilde{J}_{1}\right)}+\left\|v\right\|_{H^{1}\left(V\cap\left\\{x_{2}\in[1-r_{1},1-r_{1}/2]\right\\}\right)}\right)\\\
+e^{-\left(C_{2}-C_{1}\right)\tau}\left(\left\|v\right\|_{H^{1}\left(V\right)}+\left|v_{|J_{1}}\right|_{H^{1}\left(V\cap\widetilde{J}_{1}\right)}\right).$
(A.8)

Optimizing this inequality with respect to $\displaystyle\tau$ yields the
interpolation inequality for $\displaystyle v$ smooth. A density argument
permits to conclude the proof of A.3. ∎

###### Proof of Theorem 5.2.

By a compactness argument, one can deduce from Lemma A.3 an interpolation
result on a neighborhood of $\displaystyle\widetilde{J}_{1}$. Then we combine
this with classical interpolation estimates (see [27]) in the interior and at
the boundary $\displaystyle J_{0}$ where Dirichlet boundary condition hold to
conclude. A similar proof is done in [11] (see Lemma 8.3). ∎

.

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