#  Long-time instability of planar Poiseuille-type flow in compressible fluid

Andrew Yang  Department of Mathematics, City University of Hong Kong, Hong
Kong
<EMAIL_ADDRESS>Zhu Zhang  Department of Applied Mathematics, The
Hong Kong Polytechnic University, Hong Kong
<EMAIL_ADDRESS>
###### Abstract

It is well-known that at the high Reynolds number, the linearized Navier-
Stokes equations around the inviscid stable shear profile admit growing mode
solutions due to the destabilizing effect of the viscosity. This phenomenon,
called Tollmien-Schlichting instability, has been rigorously justified by
Grenier-Guo-Nguyen [Adv. Math. 292 (2016); Duke J. Math. 165 (2016)] for
Poiseuille flows and boundary layers in the incompressible fluid. To reveal
this intrinsic instability mechanism in the compressible setting, in this
paper, we study the long-time instability of the Poiseuille flow in a channel.
Note that this instability arises in a low-frequency regime instead of a high-
frequency regime for the Prandtl boundary layer. The proof is based on the
quasi-compressible-Stokes iteration introduced by Yang-Zhang in [50] and
subtle analysis of the dispersion relation for the instability. Note that we
do not require symmetric conditions on the background shear flow or
perturbations.

††journal: Elsevier journal

##  1 Introduction

Stability/instability properties of laminar flows at the high Reynolds number
is the fundamental problem in Hydrodynamic Stability Theory. Significant
progress was made by pioneering works of Kelvin [29], Orr [42], Rayleigh [44],
and Heisenberg [24], among many others. The mathematical theory in this area
has been rapidly developed in this decade.

For inviscid flow when the Reynolds number is infinity, the main stabilizing
mechanism is the inviscid damping. That is, due to the mixing between
vorticity and the background flow, the velocity field asymptotically converges
to a laminar flow that is closed to its background. This phenomenon has been
rigorously justified for Couette flow in linear/nonlinear settings, and for
perturbations in high-regularity spaces, cf. [4, 26, 51, 52]. On the other
hand, if the perturbation is less smooth, some instabilities arise that
prevent the solution from converging to the shear flow. In this direction,
Lin-Zeng [35] constructed the time-periodic solution close to the Couette flow
in $H^{\frac{3}{2}-}$-space. Nonlinear instability in Gevrey $2^{-}$ space of
Couette flow was shown by Deng-Masmoudi [13]. For shear flows other than the
Couette, the dynamics is more complicated even in the linear level [2, 20, 26,
34, 48, 49], and the stability in full nonlinear setting is shown recently by
Ionescu-Jia [25] and Masmoudi-Zhao [40]. We also refer to [19, 31] for
nonlinear instability results of generic shear flows.

In the high Reynolds number regime, the role of viscosity is mixed. On the one
hand, the viscosity can help stabilize the shear flow with the aid of an
enhanced dissipation mechanism. Such a mechanism can also be used to
characterize the critical size of perturbations for the nonlinear stability of
the corresponding shear flows. We refer to the list of references [3, 5, 6, 7,
9, 10, 11, 16, 20, 39, 40, 45, 49] for the research in this direction. On the
other hand, small viscosity may destabilize the flow. This phenomenon, called
Tollmien-Schlichting instability, cf. [18, 46, 47], typically occurs in the
laminar flow with boundary layers. Precisely, for the linearized
incompressible Navier-Stokes equations around a Prandtl boundary layer, at
high tangential frequency, there exists an unstable eigenmode corresponding to
Gevrey function space with index $\frac{3}{2}$. Substantial progress on this
topic was made by Grenier-Guo-Nguyen [22] in which they justified this
phenomenon with mathematical rigor. In addition, the validity of Prandtl
expansion with critical Gevrey index $\frac{3}{2}$ was obtained by Gérard-
Varet-Maekawa-Masmoudi in [23]. For plane Poiseuille flow, this mechanism
induces a long-wave type of instability, cf. [21].

One of the powerful analytic tools in above studies for incompressible flow is
the Orr-Sommerfeld equation, which describes the time evolution of the Fourier
mode behavior of the stream function. In the absence of viscosity, the Orr-
Sommerfeld equation is the Rayleigh equation for inviscid flow. Thus, it can
be regarded as a singular perturbation of the Rayleigh equation at high
Reynolds numbers. The derivation of this famous equation relies on the stream
function representation of the velocity field so that it is effective in the
two-dimensional incompressible setting. However, for compressible flow,
eigenvalue problems for stability can not be reduced to a simple single
equation like the Orr-Sommerfeld. Therefore, despite many works from the
physical perspectives [8, 14, 32, 33, 37, 43], mathematical results on
hydrodynamic stability in compressible fluids are very few. The works that we
are aware of in this direction are [1, 53] for the Couette flow in two and
three-dimensional spaces (without boundary) respectively. See also [27, 28,
41] for stability of Poiseuille flow in different regimes.

In order to capture the Orr-Sommerfeld structure in the compressible setting,
in [50], the authors introduced a quasi-compressible system so that the Orr-
Sommerfeld type equation can be derived and used to approximate the
compressible Navier-Stokes for isentropic flow. This newly derived formulation
allows one to use the “incompressible techniques” to treat problems for
compressible flows, and it has been proved helpful in the subsonic regime
through the quasi-compressible and Stokes iteration to investigate both the
Tollmien-Schlichting instability and validity of Prandtl ansatz in suitable
function spaces in various physical situations. We also refer to the recent
work [38] for the whole subsonic regime.

As a further exploration of the compressible flow, particularly in a channel,
we will study the long-time instability phenomena of the planar Poiseuille
flow in this paper. Note that although this kind of instability phenomena has
been studied in [21] for Poiseuille flows in the incompressible fluid, there
is no corresponding work for compressible models. Motivated by this, we
consider the two-dimensional compressible Navier-Stokes equations for
isentropic flow in a channel $\Omega=\left\\{(x,y)\mid
x\in\mathbb{R},y\in[-1,1]\right\\}:$

$\left\\{\begin{aligned} &\partial_{t}\rho+\nabla\cdot(\rho\vec{U})=0,\\\
&\rho\partial_{t}\vec{U}+\rho\vec{U}\cdot\nabla\vec{U}+\nabla
P(\rho)=\mu\epsilon\Delta\vec{U}+\lambda\epsilon\nabla(\nabla\cdot\vec{U})+\rho\vec{F},\\\
&\vec{U}|_{y=\pm 1}=\vec{0}.\end{aligned}\right.$ (1.1)

Here, $\rho$, $\vec{U}=(u,v)$ and $P(\rho)$ represent the density, velocity
field and pressure of the fluid respectively. The vector field $\vec{F}$ is a
given external force. The constants $\mu>0,\lambda\geq 0$ are the rescaled
shear and bulk viscosity, and $0<\epsilon\ll 1$ is a small parameter which is
proportional to the reciprocal of the Reynolds number. Without loss of
generality, the constant $\mu$ is set to 1 throughout the paper.

As well known, the planar Poiseuille flow is a typical steady solution profile
to the Navier-Stokes equations when the tangential derivative of the pressure
function is balanced by the normal diffusion of the velocity field. In the
compressible setting under consideration, we can fix a class of Poiseuille-
type profile as

$\displaystyle(\rho_{s},\vec{\mathbf{U}}_{s})\overset{\mbox{\tiny{def}}}{=}(1,U_{s}(y),0),~{}y\in(-1,1),$
(1.2)

which is a steady solution to (1.1) with a small constant force
$\vec{F}=(-2U_{s}^{\prime\prime}\mu,0)$. Here we assume that $U_{s}$ satisfies

$\displaystyle U_{s}(\pm
1)=0,~{}U_{s}^{\prime}(-1)>0,~{}U_{s}^{\prime}(1)<0,\text{ and
}U_{s}^{\prime\prime}(y)<0,~{}\text{ for }y\in(-1,1).$ (1.3)

A typical example of this class of profile is the plane Poiseuille flow
$U_{s}(y)=1-y^{2}$.

To study the stability/instability of the Poiseuille flow, we consider a small
perturbation of the Poiseuille profile

$\displaystyle\rho=1+\tilde{\rho},~{}u=U_{s}(y)+\tilde{u},~{}v=\tilde{v}.$

By linearization, we have the following system for $(\rho,\vec{U})$

$\left\\{\begin{aligned}
&\partial_{t}\rho+U_{s}\partial_{x}\rho+\nabla\cdot\vec{U}=0,~{}t>0,~{}(x,y)\in\mathbb{R}\times{(-1,1)},\\\
&\partial_{t}\vec{U}+U_{s}\partial_{x}\vec{U}+m^{-2}\nabla\rho+v\partial_{y}U_{s}\vec{e}_{1}-{\epsilon}\Delta\vec{U}-\lambda\epsilon\nabla(\nabla\cdot\vec{U})-\rho\vec{F}=0,\\\
&\vec{U}|_{y=\pm 1}=\vec{0}.\end{aligned}\right.$ (1.4)

Here we have dropped tilde for the simplicity of notations. Note that in our
setting, $m=\frac{1}{\sqrt{P^{\prime}(1)}}$ is the Mach number because the
maximum magnitude of the Poiseuille profile is $1$ and $\sqrt{P^{\prime}(1)}$
is the background sound speed.

For stability analysis, we look for solution $(\rho,u,v)$ to the linearized
compressible Navier-Stokes system (1.4) in the following form

$\displaystyle(\rho,u,v)=e^{ik(x-ct)}(\hat{\rho},\hat{u},\hat{v})(y),$ (1.5)

where $k>0$ is frequency and $c\in\mathbb{C}$ is the spectral parameter.
Substituting (1.5) into (1.4), we obtain the following eigenvalue problem for
the profile function $(\hat{\rho},\hat{u},\hat{v})$

$\left\\{\begin{aligned}
&ik(U_{s}-c)\hat{\rho}+\text{div}_{k}(\hat{u},\hat{v})=0,~{}~{}y\in(-1,1),\\\
&-{\epsilon}\Delta_{k}\hat{u}-\lambda
ik{\epsilon}\text{div}_{k}(\hat{u},\hat{v})+ik(U_{s}-c)\hat{u}+(ikm^{-2}+{\epsilon}U_{s}^{\prime\prime})\hat{\rho}+\hat{v}U_{s}^{\prime}=0,\\\
&-{\epsilon}\Delta_{k}\hat{v}-\lambda{\epsilon}\partial_{y}\text{div}_{k}(\hat{u},\hat{v})+ik(U_{s}-c)\hat{v}+m^{-2}\partial_{y}\hat{\rho}=0,\end{aligned}\right.$
(1.6)

with no-slip boundary conditions on two boundaries $y=\pm 1$

$\displaystyle\hat{u}|_{y=\pm 1}=\hat{v}|_{y=\pm 1}=0.$ (1.7)

Here, in (1.6), $\Delta_{k}=(\partial_{y}^{2}-k^{2})$ and
$\text{div}_{k}(u,v)=iku+\partial_{y}v$ denote the Fourier transform of
Laplacian and divergence operators respectively. For convenience, we denote by
$\mathcal{L}(\hat{\rho},\hat{u},\hat{v})$ the linear operator (1.6).

The stability analysis relies on the solvability of the eigenvalue problem
(1.6) with boundary conditions (1.7). In fact, if for some complex number
$c\in\mathbb{C}$ with $\text{Im}c>0$ and frequency $k\in\mathbb{R}$, the ODE
system (1.6) with (1.7) has a non-zero solution, then the Poiseuille flow
(1.2) is spectral unstable. Otherwise, it is spectral stable.

In this paper we concentrate on the case when the background profile is the
planar shear flow that satisfies (1.3), and the main result is stated in the
following theorem.

######  Theorem 1.1.

Let Mach number $m\in\left(0,\frac{1}{\sqrt{3}}\right)$. For sufficiently
small $0<\epsilon\ll 1$, there exist frequency $k\sim\epsilon^{\frac{1}{7}}$
and wave number $c\in\mathbb{C}$ with $Imc\sim\epsilon^{\frac{2}{7}}$, such
that the linearized compressible Navier-Stokes system admits a solution in the
form of

$(\rho,u,v)(t,x,y)=e^{ik(x-ct)}(\hat{\rho}.\hat{u},\hat{v})(y).$

We have several remarks on the above result.

######  Remark 1.1.

Unlike the case of boundary layer profile [22, 50], for the planar Poiseuille
flow, the unstable mode localizes at low frequency
$k\sim\epsilon^{\frac{1}{7}}$ rather than in a high-frequency regime, and it
grows exponentially in time like $e^{tkImc}\sim e^{\epsilon^{\frac{3}{7}}t}$.
Note that the slow growth is consistent with the linear inviscid stability for
Poiseuille flow, see Lees-Lin’s criterion [30].

######  Remark 1.2.

In the work [27] by Kagei-Nishida, the instability of planar Poiseuille flows
is shown in the high Mach number regime, which is different from the regime
studied in this paper. Even though instabilities investigated in both papers
occur at low frequencies, the mechanisms are different. In [27], the unstable
eigenmode bifurcates from the zero eigenvalue of hyperbolic part of linearized
operator at zero frequency, while in this paper, the instability arises from
the interaction between the inviscid perturbation and boundary layers.

######  Remark 1.3.

The dispersion relation is derived from full boundary conditions (1.7),
instead of building upon symmetry assumptions on background flows and
perturbations as in [21]. Thus, the analysis can be used for more generic
shear flows without symmetry.

######  Remark 1.4.

The above theorem is proved in the subsonic regime, covering the
incompressible case when the Mach number is $0$. Note that the subsonic
condition is essential because the ellipticity of compressible Orr-Sommerfeld
equation does not hold when the Mach number $m\geq 1$. Hence, how to establish
mathematical theory for the cases in transonic and sonic regimes remains very
challenging and unsolved.

Now we briefly present the ideas and strategies for the proof of main theorem.
To match the full boundary conditions in (1.7), we need to construct four
independent solutions to the eigenvalue problem (1.6). As in the case of
boundary layer profiles [22, 50] and the case of Poiseuille flow under
incompressible perturbations [21], the first step is to construct approximate
solutions consisting of the inviscid and viscous parts. The inviscid component
is constructed by using the Rayleigh approximate operator in the low-frequency
regime, usually called the slow mode. In constrast, the viscous component
takes effect in a region close to the two boundaries $y=\pm 1$ to capture
interactions between the viscosity and no-slip boundary conditions.

The next step is to construct four exact solutions to (1.6) near these
approximations or equivalently to solve the remainder system (3.1). The
difficulty in analysis comes from the lack of stream function expression in
(1.6), so the resolvent problem (3.5) can not be reduced to the Orr-Sommerfeld
equation as in the incompressible case. To overcome this difficulty, we apply
the quasi-compressible-Stokes iteration introduced in [50] for studying
boundary layers to the study of flows in the channel. Roughly speaking, we
introduce the quasi-compressible operator, where the intrinsic effective
stream function can be defined by considering the compressibility of fluids.
The Orr-Sommerfeld type equation can be derived accordingly. On the other
hand, to recover the loss of regularity due to the quasi-compressible
approximation, the Stokes operator is used to capture the elliptic structure
of the system. The convergence of iteration scheme will be presented in
Section 3.3. The proof is based on energy estimates, which is different from
the Green function approach used in [21]. The concavity of Poiseuille flow
plays an important role in analysis.

Finally, in Section 4, we solve the dispersion relation for the instability of
Poiseuille flow. We remark that the dispersion relation is more complicated
than the one studied in [50] for the instability of boundary layer, because,
for latter case, one can choose the slow and fast modes decaying at infinity
so that far-field conditions can be automatically satisfied. In this paper, to
remove the symmetry assumptions made in [21], the dispersion relation is
derived from full boundary conditions (1.7): we look for the solution to
eigenvalue problem (1.6) in the following form

$\displaystyle(\rho,u,v)=(\rho_{+}^{s},u_{+}^{s},v_{+}^{s})+\alpha_{1}(\rho_{-}^{s},u_{-}^{s},v_{-}^{s})+\alpha_{2}(\rho_{-}^{f},u_{-}^{f},v_{-}^{f})+\alpha_{3}(\rho_{+}^{f},u_{+}^{f},v_{+}^{f}),$

where $(\rho_{\pm}^{s},u_{\pm}^{s},v_{\pm}^{s})$ are slow modes and
$(\rho_{\pm}^{f},u_{\pm}^{f},v_{\pm}^{f})$ are fast modes that involve
boundary layer structures near $y=\pm 1$ respectively, and constants
$\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ are chosen such that the following
three boundary conditions are fulfilled: $v(-1)=v(1)=u(-1)=0$. In this way,
the zero point problem for the $4\times 4$ matrix of dispersion relation is
reduced into a single equation for matching the last boundary condition
$u(1)=0.$ Inspired by [12, 15], we solve this by applying the Rouché theorem.

The rest of the paper is organized as follows. In the next section, we will
construct the approximate solutions consisting of the inviscid modes and
boundary layers corresponding to the inviscid and viscous effects
respectively. Then, in the third section, the remainder of these approximate
solutions are estimated by studying the resolvent problem. The quasi-
compressible and Stokes iteration is used there. The dispersion relation for
the instability is solved in Section 4. Finally, we list several useful
estimates in the Appendices.

Regime of parameters: For the sake of readers’ convenience, we list the
regimes of the parameters considered in this paper. Let $\epsilon$ be the
viscosity coefficient which is suffciently small. We set the tangential
frequency $k=T_{0}\epsilon^{\frac{1}{7}}$ where the constant $T_{0}\gg 1$ is
sufficiently large and of $O(1)$ with respect to $\epsilon.$ We also define
the rescaled frequency
$n\overset{\mbox{\tiny{def}}}{=}k/\epsilon=T_{0}\epsilon^{-\frac{6}{7}}$. The
wave speed $c=\text{Re}c+i\text{Im}c$ and satisfies

$\displaystyle\text{Re}c\approx T_{0}^{2}\epsilon^{\frac{2}{7}},\text{ and
}\text{Im}c\approx T_{0}^{-\frac{3}{2}}\epsilon^{\frac{2}{7}}.$ (1.8)

The precise range of $c$ is given in (4.29).

Notations: Throughout the paper, $C$ denotes a positive constant independent
of $\epsilon$ and may vary from line to line. $A\lesssim B$ and $A=O(1)B$ mean
that there exists a positive constant $C$, not depending on $\epsilon$, such
that $A\leq CB$. Similarly, $A\gtrsim B$ means $A\geq CB$ for some positive
constant $C$. $A=O(1)\epsilon^{\infty}$ means $A\leq C_{N}\epsilon^{N}$ for
any positive integer $N$. We use the notation $A\approx B$ if $A\lesssim B$
and $A\gtrsim B.$ We denote by $\|\cdot\|_{L^{2}}$ and
$\|\cdot\|_{L^{\infty}}$ respectively the standard $L^{2}$ and $L^{\infty}$
norm in the interval $(-1,1).$ For any non-negative integer $s$ and
$p\in[1,\infty]$, the standard Sobolev space $W^{s,p}=\\{f\in
L^{p}(-1,1)\mid\partial_{y}^{j}f\in L^{p}(-1,1),~{}j=1,2,\cdots,s\\}$ are
used. In particular, $H^{s}\overset{\mbox{\tiny{def}}}{=}W^{s,2}$.

##  2 Approximate solutions

The approximate solutions consisting of inviscid approximations and boundary
layers will be given in the following two subsections. The estimates on errors
generated by these approximations will be given in Section 2.3.

###  2.1 Inviscid solutions

We start from the following inviscid system corresponding to (1.6)

$\left\\{\begin{aligned}
&ik(U_{s}-c)\rho+\text{div}_{k}(u,v)=0,~{}~{}y\in(-1,1),\\\
&ik(U_{s}-c)u+m^{-2}ik\rho+vU_{s}^{\prime}=0,\\\
&ik(U_{s}-c)v+m^{-2}\partial_{y}\rho=0.\end{aligned}\right.$ (2.1)

As in [30, 50], the solution $(\rho,u,v)$ to the inviscid system (LABEL:2.1.1)
can be represented by using a new function $\varphi=\frac{i}{k}v$. In fact,
the first equation in (LABEL:2.1.1) gives

$\displaystyle u=\partial_{y}\varphi-(U_{s}-c)\rho.$ (2.2)

Substituting this into the second equation of (LABEL:2.1.1), we can obtain the
following relation between $\rho$ and $\varphi$:

$\displaystyle-m^{-2}A(y)\rho=(U_{s}-c)\partial_{y}\varphi-\varphi
U_{s}^{\prime},$

where

$\displaystyle A(y)\overset{\mbox{\tiny{def}}}{=}1-m^{2}(U_{s}-c)^{2}.$ (2.3)

Note that for the Poiseuille flow in the subsonic regime, that is,
$m\in(0,1)$, $A(y)$ is invertible when the module $|c|$ is sufficiently small.
Thus, we can represent $\rho$ in terms of $\varphi$ as follows:

$\displaystyle\rho=-m^{2}A^{-1}(y)\left[(U_{s}-c)\partial_{y}\varphi-\varphi
U_{s}^{\prime}\right].$ (2.4)

Plugging (2.4) into the third line of (LABEL:2.1.1), we derive the following
Lees-Lin equation (Rayleigh’s type) for $\varphi$, cf. [30, 50]:

$\displaystyle\text{Ray}_{\text{CNS}}(\varphi)\overset{\mbox{\tiny{def}}}{=}\partial_{y}\left\\{A^{-1}\left[(U_{s}-c)\partial_{y}\varphi-
U_{s}^{\prime}\varphi\right]\right\\}-k^{2}(U_{s}-c)\varphi=0,~{}~{}y\in(-1,1).$
(2.5)

Compared with the classical Lees-Lin equation around the Prandtl boundary
layer profile studied in [50], here we need to analyze (2.5) in a finite
interval $(-1,1)$ so that the boundary conditions for both $y=1$ and $y=-1$
need to be taken into account. For this, we construct two independent
approximate solutions to the Lees-Lin equation (2.5) at low frequency $k\ll
1$. We start from $k=0$. In this case, the equation (2.5) reduces to

$\displaystyle\partial_{y}\left\\{A^{-1}\left[(U_{s}-c)\partial_{y}\varphi-
U_{s}^{\prime}\varphi\right]\right\\}=0,~{}y\in(-1,1),$

and it admits following two independent solutions

$\displaystyle\varphi_{+}(y)$ $\displaystyle=U_{s}(y)-c,$ (2.6)
$\displaystyle\varphi_{-}(y)$
$\displaystyle=\left(U_{s}(y)-c\right)\int_{0}^{y}\frac{1}{(U_{s}(x)-c)^{2}}{\rm
d}x-m^{2}\left(U_{s}(y)-c\right)y.$ (2.7)

Based on $\varphi_{\pm}$, we construct two approximate solutions to (2.5) for
$k\ll 1$.

Define the approximate Green’s function:

$G(x,y)=-(U_{s}(x)-c)^{-1}\begin{cases}\varphi_{+}(y)\varphi_{-}(x),~{}x<y,\\\
\varphi_{+}(x)\varphi_{-}(y),~{}x>y,\end{cases}$

and two correctors at $k^{2}$-order:

$\varphi_{\pm,k}(y)\overset{\mbox{\tiny{def}}}{=}\int^{1}_{-1}G(x,y)(U_{s}(x)-c)\varphi_{\pm}(x){\rm
d}x.$ (2.8)

Set

$\varphi^{s}_{\pm}=\varphi_{\pm}+k^{2}\varphi_{\pm,k}.$ (2.9)

Plugging (2.9) into (2.5) gives

$\text{Ray}_{\text{CNS}}(\varphi^{s}_{\pm})=-k^{4}(U_{s}-c)\varphi_{\pm,k}.$
(2.10)

Thus, $\varphi^{s}_{\pm}$ solves the Lees-Lin equation (2.1.5) up to
$O(k^{4})$.

For later use, in the following lemma we summarize the boundary values of
$\varphi^{s}_{\pm}$ at $y=\pm 1$.

######  Lemma 2.1.

There exists a positive constant $\gamma_{1}$, such that for $k\leq\gamma_{1}$
and $|c|\leq\gamma_{1}$, the boundary values of $\varphi^{s}_{\pm}$ have the
following asymptotic behavior:

$\displaystyle\varphi^{s}_{+}(-1)$
$\displaystyle=-c+k^{2}\frac{1}{U^{\prime}_{s}(-1)}\int^{1}_{-1}U^{2}_{s}(x){\rm
d}x+O(1)k^{2}|c\log{Imc}|,$ (2.11)
$\displaystyle\partial_{y}\varphi^{s}_{+}(-1)$
$\displaystyle=U^{\prime}_{s}(-1)+O(1)k^{2}|\log{Imc}|,$ (2.12)
$\displaystyle\varphi^{s}_{+}(1)$ $\displaystyle=-c\left(1+O(1)k^{2}\right),$
(2.13) $\displaystyle\partial_{y}\varphi^{s}_{+}(1)$
$\displaystyle=U^{\prime}_{s}(1)+O(1)k^{2},$ (2.14)
$\displaystyle\varphi^{s}_{-}(\pm 1)$
$\displaystyle=-\frac{1}{U^{\prime}_{s}(\pm
1)}+O(1)\bigg{(}|c\log{Imc}|+k^{2}\bigg{)},$ (2.15)
$\displaystyle\partial_{y}\varphi^{s}_{-}(\pm 1)$
$\displaystyle=O(1)|\log{Imc}|.$ (2.16)

###### Proof.

Consider $\varphi_{+}^{s}$ first. By the definition of $\varphi_{+}^{s}$ in
(2.9) and using (2.8), we obtain the following explicit formula:

$\varphi^{s}_{+}(y)=\varphi_{+}(y)-k^{2}\varphi_{+}(y)\bigg{(}\int^{y}_{-1}\varphi_{+}(x)\varphi_{-}(x){\rm
d}x\bigg{)}-k^{2}\varphi_{-}(y)\bigg{(}\int^{1}_{y}\varphi_{+}^{2}(x){\rm
d}x\bigg{)}.$ (2.17)

Evaluating (2.17) at $y=-1$ and using the boundary value $\varphi_{-}(-1)$
given in (5.1), it holds that

$\displaystyle\varphi^{s}_{+}(-1)$
$\displaystyle=-c-k^{2}\varphi_{-}(-1)\bigg{(}\int^{1}_{-1}(U_{s}(x)-c)^{2}{\rm
d}x\bigg{)}$
$\displaystyle=-c-k^{2}\bigg{(}\frac{-1}{U^{\prime}_{s}(-1)}\int^{1}_{-1}U^{2}_{s}(x){\rm
d}x+O(1)|c\log\text{Im }c|\bigg{)}$
$\displaystyle=-c+\frac{k^{2}}{U^{\prime}_{s}(-1)}\int^{1}_{-1}U^{2}_{s}(x){\rm
d}x+O(1)k^{2}|c\log\text{Im }c|,$

which is (2.11). Similarly, we evaluate $\varphi_{+}^{s}(y)$ at $y=1$ to
obtain

$\displaystyle\varphi^{s}_{+}(1)$
$\displaystyle=-c+ck^{2}\bigg{[}\int^{1}_{-1}(U_{s}(x)-c)\varphi_{-}(x){\rm
d}x\bigg{]}$ $\displaystyle=-c\left(1+O(1)k^{2}\right).$

Here in the last line we have used the inequality
$\left|\int_{-1}^{1}(U_{s}-c)\varphi_{-}(x){\rm d}x\right|\leq
C\|\varphi_{-}\|_{L^{\infty}}\leq C,$ with the aid of (5.9). Thus, (2.13)
follows.

Next we compute boundary values of $\partial_{y}\varphi_{+}^{s}$ at $y=\pm 1$.
Differentiate (2.17) to obtain

$\displaystyle\partial_{y}\varphi^{s}_{+}(y)=U^{\prime}_{s}(y)-k^{2}U^{\prime}_{s}(y)\int^{y}_{-1}(U_{s}(x)-c)\varphi_{-}(x){\rm
d}x-k^{2}\varphi_{-}^{\prime}(y)\int^{1}_{y}(U_{s}(x)-c)^{2}{\rm d}x.$ (2.18)

Evaluating (2.18) at $y=-1$ and using (5.2) in Lemma 5.1, we deduce that

$\displaystyle\partial_{y}\varphi^{s}_{+}(-1)$
$\displaystyle=U^{\prime}_{s}(-1)-k^{2}\varphi^{\prime}_{-}(-1)\bigg{(}\int^{1}_{-1}(U_{s}(x)-c)^{2}{\rm
d}x\bigg{)}$ $\displaystyle=U^{\prime}_{s}(-1)+O(1)k^{2}|\log\text{Im}c|,$

which yields (2.12). Similarly, we can compute

$\displaystyle\partial_{y}\varphi^{s}_{+}(1)$
$\displaystyle=U^{\prime}_{s}(1)-k^{2}U^{\prime}_{s}(1)\left(\int^{1}_{-1}(U_{s}(x)-c)\varphi_{-}(x){\rm
d}x\right)$ $\displaystyle=U^{\prime}_{s}(1)-O(1)k^{2},$

which is (2.14). Therefore, we have completed estimates on boundary data of
$\varphi^{s}_{+}$.

Now we turn to consider $\varphi^{s}_{-}$. By definition (2.9), it holds that

$\displaystyle\varphi^{s}_{-}(y)=\varphi_{-}(y)-k^{2}\bigg{(}\int^{y}_{-1}\varphi^{2}_{-}(x){\rm
d}x\bigg{)}\varphi_{+}(y)-k^{2}\bigg{(}\int^{1}_{y}\varphi_{+}(x)\varphi_{-}(x){\rm
d}x\bigg{)}\varphi_{-}(y).$ (2.19)

Then evaluating $\varphi_{-}^{s}$ at $y=-1$ and using (5.1) again, we have

$\displaystyle\varphi^{s}_{-}(-1)$
$\displaystyle=\varphi_{-}(-1)\left(1-k^{2}\int^{1}_{-1}\varphi_{+}(x)\varphi_{-}(x){\rm
d}x\right)$
$\displaystyle=\bigg{(}\frac{-1}{U^{\prime}_{s}(-1)}+O(1)|c\log\text{Im}c|\bigg{)}\left(1+O(1)k^{2}\right)$
$\displaystyle=\frac{-1}{U^{\prime}_{s}(-1)}+O(1)\left(|c\log{\text{Im}c}|+k^{2}\right).$
(2.20)

Similarly, the value of $\varphi_{-}^{s}(1)$ can be computed as follows.

$\displaystyle\varphi^{s}_{-}(1)$
$\displaystyle=\varphi_{-}(1)-k^{2}\bigg{(}\int^{1}_{-1}\varphi^{2}_{-}(x){\rm
d}x\bigg{)}\varphi_{+}(1)$
$\displaystyle=\frac{-1}{U_{s}^{\prime}(1)}+O(1)\left(|c\log\text{Im}c|+k^{2}\right).$
(2.21)

Combining (2.20) and (2.21) together yields (2.15).

Finally we estimate the boundary values of derivative
$\partial_{y}\varphi^{s}_{-}$. Differentiating (2.19), we obtain

$\displaystyle\partial_{y}\varphi^{s}_{-}(y)=\varphi^{\prime}_{-}(y)-k^{2}\bigg{(}\int^{y}_{-1}\varphi^{2}_{-}(x){\rm
d}x\bigg{)}\varphi^{\prime}_{+}(y)-k^{2}\bigg{(}\int^{1}_{y}\varphi_{+}(x)\varphi_{-}(x){\rm
d}x\bigg{)}\varphi^{\prime}_{-}(y).$ (2.22)

Evaluating (2.22) at $y=-1$ and using Lemma 5.1, we have

$\displaystyle\partial_{y}\varphi^{s}_{-}(-1)$
$\displaystyle=\varphi^{\prime}_{-}(-1)\bigg{(}1-k^{2}\int^{1}_{-1}\varphi_{-}(x)\varphi_{+}(x){\rm
d}x\bigg{)}=O(1)|\log\text{Im}c|.$

Similarly, we have

$\displaystyle\partial_{y}\varphi^{s}_{-}(1)$
$\displaystyle=\varphi^{\prime}_{-}(1)-k^{2}\bigg{(}\int^{1}_{-1}\varphi^{2}_{-}(x){\rm
d}x\bigg{)}U^{\prime}_{s}(1)=O(1)|\log\text{Im}c|.$

Combining these two equalities gives (2.16). Therefore, we have shown
(2.11)-(2.16). The proof of Lemma 2.1 is completed. ∎

The following estimates on $\varphi_{\pm}^{s}$ can be derived from explicit
formula in (2.6)-(2.9) and bounds on $\varphi_{-}$ given in Lemma 5.2.

######  Lemma 2.2.

Let parameters $k=T_{0}\epsilon^{\frac{1}{7}}$ and $c$ satisfies (1.8). The
approximate Lees-Lin solutions $\varphi_{\pm}^{s}$ satisfy the following
uniform estimates with respect to $\epsilon$.

$\displaystyle\|\varphi_{+}^{s}\|_{W^{2,\infty}}+\epsilon^{\frac{1}{7}}\|\partial_{y}^{3}\varphi_{+}^{s}\|_{L^{2}}+\epsilon^{\frac{2}{7}}\|\partial_{y}^{3}\varphi_{+}^{s}\|_{L^{\infty}}\leq
C,$ (2.23)
$\displaystyle\|\varphi_{-}^{s}\|_{L^{\infty}}+\frac{\|\partial_{y}\varphi_{-}^{s}\|_{L^{\infty}}}{\left|\log\epsilon\right|}+\|\partial_{y}\varphi_{-}^{s}\|_{L^{2}}+\sum_{j=2}^{4}\epsilon^{\frac{2j-3}{7}}\|\partial_{y}^{j}\varphi_{-}^{s}\|_{L^{2}}+\sum_{j=2}^{4}\epsilon^{\frac{2(j-1)}{7}}\|\partial_{y}^{j}\varphi_{-}^{s}\|_{L^{\infty}}\leq
C.$ (2.24)

###### Proof.

First we estimate $\varphi_{+}^{s}$. Differentiating (2.17) yields

$\displaystyle\partial_{y}\varphi_{+}^{s}$
$\displaystyle=\varphi_{+}^{\prime}-k^{2}\varphi_{+}^{\prime}\left(\int_{-1}^{y}\varphi_{+}\varphi_{-}{\rm
d}x\right)-k^{2}\varphi^{\prime}_{-}\left(\int_{y}^{1}\varphi_{+}^{2}{\rm
d}x\right),$ (2.25) $\displaystyle\partial_{y}^{2}\varphi_{+}^{s}$
$\displaystyle=\varphi^{\prime\prime}_{+}-k^{2}\varphi_{+}^{\prime\prime}\left(\int_{-1}^{y}\varphi_{+}\varphi_{-}{\rm
d}x\right)-k^{2}\varphi_{-}^{\prime\prime}\left(\int_{y}^{1}\varphi_{+}^{2}{\rm
d}x\right)-k^{2}\varphi^{\prime}_{+}\varphi_{+}\varphi_{-}+k^{2}\varphi_{-}^{\prime}\varphi_{+}^{2},$
$\displaystyle\partial_{y}^{3}\varphi_{+}^{s}$
$\displaystyle=\varphi_{+}^{\prime\prime\prime}-k^{2}\varphi_{+}^{\prime\prime\prime}\left(\int_{-1}^{y}\varphi_{+}\varphi_{-}{\rm
d}x\right)-k^{2}\varphi_{-}^{\prime\prime\prime}\left(\int_{y}^{1}\varphi_{+}^{2}{\rm
d}x\right)$
$\displaystyle\quad+2k^{2}\varphi_{-}^{\prime\prime}\varphi_{+}^{2}-2k^{2}\varphi_{+}^{\prime\prime}\varphi_{+}\varphi_{-}+k^{2}\varphi_{+}\varphi_{+}^{\prime}\varphi_{-}^{\prime}-k^{2}(\varphi_{+}^{\prime})^{2}\varphi_{-}.$

By the explicit formula (2.25), using bounds
$\|\partial_{y}^{j}\varphi_{+}\|_{L^{\infty}}\leq C$, for $j=0,1,2,3,$ and
(5.9) in Lemma 5.2, we can obtain, for $k\approx\epsilon^{\frac{1}{7}}$ and
$\text{Im}c\approx\epsilon^{\frac{2}{7}}$, that

$\displaystyle\|\varphi_{+}^{s}\|_{W^{2,\infty}}$ $\displaystyle\leq
C\left(1+k^{2}\|\varphi_{-}\|_{W^{2,\infty}}\right)\leq
C\left(1+k^{2}|\text{Im}c|^{-1}\right)\leq C,$
$\displaystyle\|\partial_{y}^{3}\varphi_{+}^{s}\|_{L^{2}}$ $\displaystyle\leq
C\left(1+k^{2}\|\varphi_{-}\|_{H^{3}}\right)\leq
C\left(1+k^{2}|\text{Im}c|^{-\frac{3}{2}}\right)\leq
C\epsilon^{-\frac{1}{7}},$
$\displaystyle\|\partial_{y}^{3}\varphi_{+}^{s}\|_{L^{\infty}}$
$\displaystyle\leq C\left(1+k^{2}\|\varphi_{-}\|_{W^{3,\infty}}\right)\leq
C\left(1+k^{2}|\text{Im}c|^{-2}\right)\leq C\epsilon^{-\frac{2}{7}}.$

Combining these inequalities together yields (2.23). Similarly, for
$\varphi_{-}^{s}$, we differentiate (2.19) up to the fourth order and deduce
that

$\displaystyle\partial_{y}\varphi_{-}^{s}$
$\displaystyle=\varphi_{-}^{\prime}-k^{2}\left(\int_{-1}^{y}\varphi_{-}^{2}{\rm
d}x\right)\varphi_{+}^{\prime}-k^{2}\left(\int_{y}^{1}\varphi_{+}\varphi_{-}{\rm
d}x\right)\varphi_{-}^{\prime},$ (2.26)
$\displaystyle\partial_{y}^{2}\varphi_{-}^{s}$
$\displaystyle=\varphi^{\prime\prime}_{-}-k^{2}\varphi_{+}^{\prime\prime}\left(\int_{-1}^{y}\varphi_{-}^{2}{\rm
d}x\right)-k^{2}\varphi_{-}^{\prime\prime}\left(\int_{y}^{1}\varphi_{+}\varphi_{-}{\rm
d}x\right)-k^{2}\varphi^{\prime}_{+}\varphi_{-}^{2}+k^{2}\varphi_{-}^{\prime}\varphi_{+}\varphi_{-},$
$\displaystyle\partial_{y}^{3}\varphi_{-}^{s}$
$\displaystyle=\varphi_{-}^{\prime\prime\prime}-k^{2}\varphi_{+}^{\prime\prime\prime}\left(\int_{-1}^{y}\varphi_{-}^{2}{\rm
d}x\right)-k^{2}\varphi_{-}^{\prime\prime\prime}\left(\int_{y}^{1}\varphi_{+}\varphi_{-}{\rm
d}x\right)$
$\displaystyle\quad-2k^{2}\varphi_{+}^{\prime\prime}\varphi_{-}^{2}+2k^{2}\varphi_{-}^{\prime\prime}\varphi_{+}\varphi_{-}+k^{2}\varphi_{+}(\varphi_{-}^{\prime})^{2}-k^{2}\varphi_{-}^{\prime}\varphi_{+}^{\prime}\varphi_{-},$
$\displaystyle\partial_{y}^{4}\varphi_{-}^{s}$
$\displaystyle=\partial_{y}^{4}\varphi_{-}-k^{2}\partial_{y}^{4}\varphi_{+}\left(\int_{-1}^{y}\varphi_{-}^{2}{\rm
d}x\right)-k^{2}\partial_{y}^{4}\varphi_{-}\left(\int_{y}^{1}\varphi_{+}\varphi_{-}{\rm
d}x\right)-3k^{2}\varphi_{+}^{\prime\prime\prime}\varphi_{-}^{2}+3k^{2}\varphi_{-}^{\prime\prime\prime}\varphi_{+}\varphi_{-}$
$\displaystyle\quad-5k^{2}\varphi_{+}^{\prime\prime}\varphi_{-}^{\prime}\varphi_{-}+4k^{2}\varphi_{-}^{\prime\prime}\varphi_{-}^{\prime}\varphi_{+}+k^{2}\varphi_{-}^{\prime\prime}\varphi_{+}^{\prime}\varphi_{-}.$

Then from the explicit formula in (2.26), it holds, for $j=0,1,2,3,4$, that

$\displaystyle\|\partial_{y}^{j}\varphi_{-}^{s}\|_{L^{p}}\leq
C\left(\|\varphi_{-}\|_{W^{j,p}}+1\right),~{}p=2~{}\text{or}~{}\infty.$ (2.27)

The estimate (2.24) follows from (2.27) and (5.9). Therefore, the proof of
Lemma 2.2 is completed. ∎

Based on the approximate solutions $\varphi^{s}_{\pm}$ to the Lees-Lin
equation (2.5), we can define the following two inviscid modes:
$\vec{\Xi}^{s}_{\pm,\text{app}}=(\rho^{s}_{\pm,,\text{app}},u^{s}_{\pm,,\text{app}},v^{s}_{\pm,,\text{app}})$,
where

$\displaystyle v^{s}_{\pm,\text{app}}$ $\displaystyle=-ik\varphi^{s}_{\pm},$
(2.28) $\displaystyle\rho^{s}_{\pm,\text{app}}$
$\displaystyle=-m^{2}A^{-1}(y)\bigg{[}(U_{s}-c)\partial_{y}\varphi^{s}_{\pm}-\varphi^{s}_{\pm}U^{\prime}_{s}\bigg{]},$
(2.29) $\displaystyle u^{s}_{\pm,\text{app}}$
$\displaystyle=\partial_{y}\varphi^{s}_{\pm}-(U_{s}-c)\rho^{s}_{\pm}.$ (2.30)

From (2.10), we can see that $\vec{\Xi}_{\pm,\text{app}}^{s}$ solve the
inviscid system (LABEL:2.1.1) up to $k^{4}$.

From Lemma 2.1, the following asymptotic expansions of boundary data for
$u_{\pm,\text{app}}^{s}$ and $v_{\pm,\text{app}}^{s}$ are obtained.

######  Lemma 2.3.

Under the same assumption on $k$ and $c$ as in Lemma 2.1, it holds that

$\displaystyle v_{+,\text{app}}^{s}(-1)$
$\displaystyle=ik\left(c-\frac{k^{2}}{U_{s}^{\prime}(-1)}\int_{-1}^{1}U_{s}^{2}(x){\rm
d}x\right)+O(1)k^{3}|c\log\text{Im}c|,$ (2.31) $\displaystyle
v_{+,\text{app}}^{s}(1)$ $\displaystyle=ikc+O(1)k^{3}|c|,$ (2.32)
$\displaystyle v_{-,\text{app}}^{s}(\pm 1)$
$\displaystyle=\frac{ik}{U_{s}^{\prime}(\pm
1)}+O(1)k\left(|c|+k^{2}\right)|\log\text{Im}c|,$ (2.33) $\displaystyle
u_{+,\text{app}}^{s}(\pm 1)$ $\displaystyle=U_{s}^{\prime}(\pm
1)+O(1)\left(|c|+k^{2}\right)|\log\text{Im}c|,$ (2.34) $\displaystyle
u_{-,\text{app}}^{s}(\pm 1)$ $\displaystyle=O(1)|\log\text{Im}c|.$ (2.35)

###### Proof.

With Lemma 2.1 for the boundary values of $\varphi_{\pm}^{s}$, the expansions
(2.31)-(2.33) for $v_{\pm,\text{app}}^{s}$ directly follow from (2.28). Next
we show (2.34) and (2.35). From Lemma 2.1, the relation (2.29) and boundary
conditions $U_{s}(\pm 1)=0$, we have

$\displaystyle\rho_{+,\text{app}}^{s}(\pm 1)$ $\displaystyle=m^{2}A^{-1}(\pm
1)\left[-c\partial_{y}\varphi_{+}^{s}(\pm 1)-\varphi_{+}^{s}(\pm
1)U_{s}^{\prime}(\pm 1)\right]$ $\displaystyle=O(1)\left(k^{2}+|c|\right),$
(2.36)

and

$\displaystyle\rho_{-,\text{app}}^{s}(\pm 1)$ $\displaystyle=m^{2}A^{-1}(\pm
1)\left[-c\partial_{y}\varphi_{-}^{s}(\pm 1)-\varphi_{-}^{s}(\pm
1)U_{s}^{\prime}(\pm 1)\right]=O(1).$ (2.37)

Thus from (2.30) for $u_{+,\text{app}}^{s}$ and the boundary values (2.12),
(2.14) and (2.36), we obtain

$\displaystyle u_{+,\text{app}}^{s}(\pm 1)$
$\displaystyle=\partial_{y}\varphi_{+}^{s}(\pm
1)+O(1)|c|\left|\rho_{+,\text{app}}^{s}(\pm 1)\right|$
$\displaystyle=U_{s}^{\prime}(\pm 1)+\left(k^{2}+|c|\right)|\log\text{Im}c|,$

which implies (2.34). Similarly, from (2.30) for $u_{-,\text{app}}^{s}$,
(2.16) and (2.37), there holds

$\displaystyle u_{-,\text{app}}^{s}(\pm 1)$
$\displaystyle=\partial_{y}\varphi_{-}^{s}(\pm
1)+O(1)|c|\left|\rho_{-,\text{app}}^{s}(\pm 1)\right|=O(1)|\log\text{Im}c|.$

This yields (2.35). Therefore, the proof of Lemma 2.3 is completed. ∎

Using Lemma 2.2 and (2.28)-(2.30), we can obtain the following estimates on
$\vec{\Xi}_{\pm,\text{app}}^{s}.$

######  Corollary 2.1.

Under the same assumptions on $k$ and $c$ as in Lemma 2.2, the inviscid modes
$\vec{\Xi}^{s}_{\pm,\text{app}}=(\rho^{s}_{\pm,,\text{app}},u^{s}_{\pm,,\text{app}},v^{s}_{\pm,,\text{app}})$
have the following bounds:

$\displaystyle\|v_{+,\text{app}}^{s}\|_{W^{2,\infty}}+\left\|\left(m^{-2}\rho_{+,\text{app}}^{s},u^{s}_{+,\text{app}}\right)\right\|_{W^{1,\infty}}+\epsilon^{\frac{1}{7}}\left\|\left(m^{-2}\partial_{y}^{2}\rho_{+,\text{app}}^{s},\partial_{y}^{2}u^{s}_{+,\text{app}}\right)\right\|_{L^{2}}\leq
C,$ (2.38)

and

$\displaystyle\|v_{-,\text{app}}^{s}\|_{H^{2}}$
$\displaystyle+\epsilon^{\frac{2}{7}}\|\partial_{y}^{3}v_{-,\text{app}}^{s}\|_{L^{2}}+\left\|\left(m^{-2}\rho_{-,\text{app}}^{s},u^{s}_{-,\text{app}}\right)\right\|_{L^{2}}$
$\displaystyle+\sum_{j=1}^{3}\epsilon^{\frac{2j-1}{7}}\left\|\left(m^{-2}\partial_{y}^{j}\rho_{-,\text{app}}^{s},\partial_{y}^{j}u^{s}_{-,\text{app}}\right)\right\|_{L^{2}}\leq
C.$ (2.39)

######  Remark 2.1.

Note that in the above estimates, compared with
$\vec{\Xi}_{+,\text{app}}^{s},$ we need bound on one more derivative of
$\vec{\Xi}_{-,\text{app}}^{s}$ in order to get better estimates on boundary
values of remainder for $\vec{\Xi}^{s}_{-\text{app}}$. See Section 4 for
details.

###### Proof.

The bounds for $v_{\pm,\text{app}}^{s}$ are straightforward. That is, by using
(2.28) and bounds for $\varphi_{\pm}^{s}$ in Lemma 2.2, we obtain

$\displaystyle\|v_{+,\text{app}}^{s}\|_{W^{2,\infty}}\leq
C\|\varphi_{+}^{s}\|_{W^{2,\infty}}\leq C,$ (2.40)
$\displaystyle\|v_{-,\text{app}}^{s}\|_{H^{2}}+\epsilon^{\frac{2}{7}}\|\partial_{y}^{3}v_{-,\text{app}}^{s}\|_{L^{2}}\leq\|\varphi_{-}^{s}\|_{H^{2}}+\epsilon^{\frac{3}{7}}\|\partial_{y}^{3}\varphi_{-}^{s}\|_{L^{2}}\leq
C.$ (2.41)

For the density and tangential velocity component, we have from the relation
(2.29) and (2.30) that

$m^{-2}\rho_{\pm,\text{app}}^{s},~{}u_{\pm,\text{app}}^{s}\approx|\partial_{y}\varphi_{\pm}^{s}|+|\varphi_{\pm}^{s}|.$

Thus

$\displaystyle\left\|\left(m^{-2}\rho_{+,\text{app}}^{s},u_{+,\text{app}}^{s}\right)\right\|_{W^{1,\infty}}\leq
C\|\varphi_{+}^{s}\|_{W^{2,\infty}}\leq C,$

and

$\displaystyle\left\|\left(m^{-2}\partial_{y}^{2}\rho_{+,\text{app}}^{s},\partial_{y}^{2}u_{+,\text{app}}^{s}\right)\right\|_{L^{2}}\leq
C\|\varphi_{+}^{s}\|_{H^{3}}\leq C\epsilon^{-\frac{1}{7}}.$

Combining above estimates with (2.40) yields the bound (2.38) for
$\vec{\Xi}_{+,\text{app}}^{s}$. Similarly, for $\rho_{-,\text{app}}^{s}$ and
$u_{-,\text{app}}^{s}$, we have

$\displaystyle\left\|\left(m^{-2}\rho_{-,\text{app}}^{s},u^{s}_{-,\text{app}}\right)\right\|_{L^{2}}$
$\displaystyle+\sum_{j=1}^{3}\epsilon^{\frac{2j-1}{7}}\left\|\left(m^{-2}\partial_{y}^{j}\rho_{-,\text{app}}^{s},\partial_{y}^{j}u^{s}_{-,\text{app}}\right)\right\|_{L^{2}}$
$\displaystyle\leq
C\|\varphi_{-}^{s}\|_{H^{1}}+C\sum_{j=2}^{4}\epsilon^{\frac{2j-3}{7}}\|\partial_{y}^{j}\varphi_{-}^{s}\|_{L^{2}}\leq
C.$

Here we have used (2.24) in the last inequality. Combining this with (2.41)
for $v_{-,\text{app}}^{s}$ gives (2.39). Therefore, the proof of Corollary 2.1
is completed. ∎

###  2.2 Boundary layers

To match the exact boundary conditions in (1.7), we need to construct two
boundary layers near $y=\pm 1.$ Firstly, consider the boundary layer on the
bottom $y=-1$. For this, we introduce the fast variable

$z=\frac{1+y}{\delta}$

where $\delta>0$ is the scale of boundary layer to be determined later. The
approximate solution to (1.6) is in the following form

$\displaystyle\Xi_{-,\text{app}}^{f}=(\underline{\rho}^{-},\underline{u}^{-},ik\delta\underline{v}^{-})(z).$
(2.42)

Substituting the ansatz (2.42) into (1.6), we obtain the following system in
the new variable $z$:

$\left\\{\begin{aligned}
&(U_{s}-c)\underline{\rho}^{-}+\underline{u}^{-}+\partial_{z}\underline{v}^{-}=0,\\\
&-\partial^{2}_{z}\underline{u}^{-}+\frac{ik\delta^{2}}{\epsilon}(U_{s}-c)\underline{u}^{-}+\frac{ik\delta^{3}}{\epsilon}\underline{v}^{-}U^{\prime}_{s}\\\
&\qquad+\frac{\delta^{2}}{\epsilon}(ikm^{-2}+\epsilon
U^{\prime\prime})\underline{\rho}^{-}+(\lambda+1)k^{2}\delta^{2}\underline{u}^{-}+\lambda
k^{2}\delta^{2}\partial_{z}\underline{v}^{-}=0,\\\
&-\partial^{2}_{z}\underline{v}^{-}+\frac{ik\delta^{2}}{\epsilon}(U_{s}-c)\underline{v}^{-}+\frac{m^{-2}}{ik\epsilon}\partial_{z}\underline{\rho}^{-}+k^{2}\delta^{2}\underline{v}^{-}-\lambda\partial_{z}(\underline{u}^{-}+\partial_{z}\underline{v}^{-})=0.\end{aligned}\right.$
(2.43)

To obtain the leading order profile
$(\underline{\rho}^{-},\underline{u}^{-},\underline{v}^{-})$ near $y=-1$, we
expand $U_{s}(y)$ as follows:

$\displaystyle U_{s}(y)-c$
$\displaystyle=U_{s}(-1)-c+U_{s}^{\prime}(-1)(1+y)+O(1)(1+y)^{2}$
$\displaystyle=U^{\prime}_{s}(-1)\delta z-c+O(1)|\delta z|^{2}$
$\displaystyle=\delta U^{\prime}_{s}(-1)(z+z_{0})+O(1)|\delta z|^{2},$ (2.44)

where the boundary condition $U_{s}(-1)=0$ of Poiseuille flow has been used.
Here,

$\displaystyle z_{0}=\frac{-c}{U^{\prime}_{s}(-1)\delta}.$ (2.45)

We can also expand $U_{s}^{\prime}(y)$ at $y=-1$ as

$\displaystyle U_{s}^{\prime}(y)$
$\displaystyle=U_{s}^{\prime}(-1)+O(1)|\delta z|.$ (2.46)

Now we plug (2.44) and (2.46) into (2.43) and derive that the scale

$\displaystyle\delta=\frac{e^{-\frac{1}{6}\pi
i}}{\left[U_{s}^{\prime}(-1)n\right]^{\frac{1}{3}}},\text{ where
}n=\frac{k}{\epsilon},$ (2.47)

to balance the first three terms in the second equation in (2.43). Then by
taking the leading order of (2.43), the following system for boundary layer
profile $(\underline{\rho}^{-},\underline{u}^{-},\underline{v}^{-})$ is
derived.

$\displaystyle\underline{\rho}^{-}=0,~{}\underline{u}^{-}+\partial_{z}\underline{v}^{-}=0,~{}-\partial^{2}_{z}\underline{u}^{-}+(z+z_{0})\underline{u}^{-}+\underline{v}^{-}=0.$
(2.48)

From the above system, we can obtain the following equation for
$\underline{v}^{-}$.

$\displaystyle-\partial^{4}_{z}\underline{v}^{-}+(z+z_{0})\partial^{2}_{z}\underline{v}^{-}=0.$
(2.49)

As in [22, 23, 50], to solve (2.49) we use the function $\text{Ai}(z)$ which
is the solution to the Airy equation:

$\partial^{2}_{z}\text{Ai}-z\text{Ai}=0.$

The first and second order primitive functions of $\text{Ai}(z)$ are denoted
by $\text{Ai}(1,z)$ and $\text{Ai}(2,z)$ respectively. They satisfy

$\partial_{z}\text{Ai}(2,z)=\text{Ai}(1,z),\text{ and
}\partial_{z}\text{Ai}(1,z)=\text{Ai}(z).$

Moreover, $\text{Ai}(z)$, $\text{Ai}(1,z)$ and $\text{Ai}(2,z)$ decay at
infinity along the straight line $e^{\frac{1}{6}\pi i}\mathbb{R}_{+}$. Set

$\displaystyle\underline{v}^{-}=\frac{\text{Ai}(2,z+z_{0})}{\text{Ai}(2,z_{0})},~{}\underline{u}^{-}=-\frac{\text{Ai}(1,z+z_{0})}{\text{Ai}(2,z_{0})}.$

It is straightforward to check that $(\underline{u}^{-},\underline{v}^{-})$
satisfies (2.48). In view of (2.42), we define the boundary layer profile at
$y=-1$ as follows:

$\displaystyle\vec{\Xi}^{f}_{-,\text{app}}=(\rho^{f}_{-},u^{f}_{-},v^{f}_{-})=\left(0,\underline{u}^{-},ik\delta\underline{v}^{-}\right)(z),~{}~{}z=e^{\frac{1}{6}\pi
i}(1+y)\left[U_{s}^{\prime}(-1)n\right]^{\frac{1}{3}}.$ (2.50)

The boundary layer at $y=1$ can be constructed similarly. Set the fast
variable

$\tilde{z}=\frac{1-y}{\tilde{\delta}},$

where the scale

$\displaystyle\tilde{\delta}=\frac{e^{-\frac{1}{6}\pi
i}}{\left[-U_{s}^{\prime}(1)n\right]^{\frac{1}{3}}}$ (2.51)

of boundary layer can be derived in the same way as (2.47). Following
(2.42)-(2.50), we can construct the boundary layer profile near $y=1$:

$\displaystyle\vec{\Xi}^{f}_{+,\text{app}}=(\rho^{f}_{+},u^{f}_{+},v^{f}_{+})=\left(0,\underline{u}^{+},ik\tilde{\delta}\underline{v}^{+}\right)(\tilde{z}),$
(2.52)

where

$\displaystyle\underline{v}^{+}=\frac{\text{Ai}(2,\tilde{z}+\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})},~{}~{}\underline{u}^{+}=\frac{\text{Ai}(1,\tilde{z}+\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})},~{}~{}\text{with
}\tilde{z}_{0}=\frac{c}{U_{s}^{\prime}(1)\tilde{\delta}}.$ (2.53)

In the next Lemma, we summarize some pointwise estimates on
$\vec{\Xi}_{\pm,\text{app}}^{f}$. The proof follows from Lemma 3.9 in [36].
Therefore, we omit the details for brevity.

######  Lemma 2.4.

For $j=0,1,2$ and $l\geq 0$, there exists a positive constant $\tau_{1}$, such
that the boundary layers $\vec{\Xi}_{\pm,\text{app}}^{f}$ satisfy the
following pointwise bounds:

$\displaystyle\left|(1\pm y)^{l}\partial_{y}^{j}v_{\pm,\text{app}}^{f}\right|$
$\displaystyle\leq C|n|^{\frac{2(j-l)-3}{6}}e^{-\tau_{1}n^{\frac{1}{3}}(1\pm
y)},$ (2.54) $\displaystyle\left|(1\pm
y)^{l}\partial_{y}^{j}u_{\pm,\text{app}}^{f}\right|$ $\displaystyle\leq
C|n|^{\frac{j-l}{3}}e^{-\tau_{1}n^{\frac{1}{3}}(1\pm y)},$ (2.55)

for any $y\in(-1,1).$ Here the constant $C$ is independent of $\epsilon$.

By using explicit formula (2.50), (2.52) and pointwise estimates (2.54),
(2.55), we have the following lemma about the asymptotic behavior of boundary
values for $(u_{\pm,\text{app}}^{f},v_{\pm,\text{app}}^{f})$.

######  Lemma 2.5.

For sufficiently small $\epsilon$, boundary values of
$(u_{\pm,\text{app}}^{f},v_{\pm,\text{app}}^{f})$ have the following
expansions:

$\displaystyle
u_{+,\text{app}}^{f}(1)=\frac{\text{Ai}(1,\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})},~{}v_{+,\text{app}}^{f}(1)=ik\tilde{\delta},~{}~{}u_{+,\text{app}}^{f}(-1),v_{+,\text{app}}^{f}(-1)=O(1)\epsilon^{\infty},$
(2.56) $\displaystyle
u_{-,\text{app}}^{f}(-1)=-\frac{\text{Ai}(1,z_{0})}{\text{Ai}(2,z_{0})},~{}v_{-,\text{app}}^{f}(-1)=ik\delta,~{}~{}u_{-,\text{app}}^{f}(1),v^{f}_{-,\text{app}}(1)=O(1)\epsilon^{\infty}.$
(2.57)

###### Proof.

We only show (2.56). Boundary conditions for
$(u_{+,\text{app}}^{f},v_{+,\text{app}}^{f})$ at $y=1$ directly follow from
the explicit formula (2.52) and (2.53). From pointwise estimates (2.54) and
(2.55) for $j=l=0,$ we can obtain

$\displaystyle\left|v_{+,\text{app}}^{f}(-1)\right|$ $\displaystyle\leq
Cn^{-\frac{1}{2}}e^{-2\tau_{1}n^{\frac{1}{3}}}\leq
Ce^{-2\tau_{1}\epsilon^{-\frac{2}{7}}},$
$\displaystyle\left|u_{+,\text{app}}^{f}(-1)\right|$ $\displaystyle\leq
Ce^{-2\tau_{1}n^{\frac{1}{3}}}\leq Ce^{-\tau_{1}\epsilon^{-\frac{2}{7}}}.$

The proof of (2.57) is similar. ∎

###  2.3 Error estimates

Based on the approximate slow and fast modes constructed in the previous two
subsections, we now summarize the formulation and related estimates on errors
generated by the approximations. Recall inviscid modes
$\vec{\Xi}^{s}_{\pm,\text{app}}$ defined in (2.28)-(2.30), and viscous modes
$\vec{\Xi}^{f}_{\pm,\text{app}}$ given by (2.42) and (2.52) respectively.
Plugging these solutions into (1.6), we obtain the following error functions:

$\displaystyle\vec{E}_{\pm}^{s}\overset{\mbox{\tiny{def}}}{=}\mathcal{L}(\vec{\Xi}^{s}_{\pm,\text{app}})=\bigg{(}0,$
$\displaystyle-\epsilon\Delta_{k}u_{\pm,\text{app}}^{s}-\lambda\epsilon
ik\text{div}_{k}(u_{\pm,\text{app}}^{s},v_{\pm,\text{app}}^{s})+\epsilon
U_{s}^{\prime\prime}\rho_{\pm,\text{app}}^{s},$
$\displaystyle-\epsilon\Delta_{k}v_{\pm,\text{app}}^{s}-\lambda\epsilon\partial_{y}\text{div}_{k}(u_{\pm,\text{app}}^{s},v_{\pm,\text{app}}^{s})-k^{4}(U_{s}-c)\varphi_{\pm,k}\bigg{)},$
(2.58)

where $\varphi_{\pm,k}$ are defined in (2.8), and

$\displaystyle\vec{E}_{\pm}^{f}\overset{\mbox{\tiny{def}}}{=}\mathcal{L}(\vec{\Xi}^{f}_{\pm,\text{app}})=\bigg{(}0,~{}$
$\displaystyle k^{2}\epsilon
u_{\pm,\text{app}}^{f}+ik\left[U_{s}(y)-U_{s}^{\prime}(\pm 1)(1\pm
y)\right]u_{\pm,\text{app}}^{f}$
$\displaystyle+\left(U_{s}^{\prime}(y)-U_{s}^{\prime}(-1)\right)v_{\pm,\text{app}}^{f},~{}-\epsilon\Delta_{k}v_{\pm,\text{app}}^{f}+ik(U_{s}-c)v_{\pm,\text{app}}^{f}\bigg{)}.$
(2.59)

Based on bounds in Corollary 2.1 and Lemma 2.4, we have the following
estimates on error terms $\vec{E}^{s}_{\pm}$ and $\vec{E}^{f}_{\pm}$.

######  Proposition 2.1.

Decompose the error function $\vec{E}_{+}^{s}$ as

$\displaystyle\vec{E}_{+}^{s}=\vec{E}_{+,1}^{s}+\vec{E}_{+,2}^{s},$ (2.60)

where $\vec{E}_{+,2}^{s}=(0,0,-k^{4}(U_{s}-c)\varphi_{+,k}).$ Then it holds
that

$\displaystyle\|\vec{E}_{+,1}^{s}\|_{L^{2}}\leq
C\epsilon^{\frac{6}{7}},~{}\|\vec{E}_{+,2}^{s}\|_{H^{1}}\leq
C\epsilon^{\frac{4}{7}}.$ (2.61)

The error function $\vec{E}_{-}^{s}$ satifies the following uniform estimate:

$\displaystyle\|\vec{E}_{-}^{s}\|_{L^{2}}+\epsilon^{\frac{2}{7}}\|\partial_{y}\vec{E}_{-}^{s}\|_{L^{2}}\leq
C\epsilon^{\frac{4}{7}}.$ (2.62)

For the error functions $\vec{E}_{\pm}^{f}$ generated by the boundary layers,
it holds that

$\displaystyle\|\vec{E}_{\pm}^{f}\|_{L^{2}}\leq C\epsilon^{\frac{6}{7}}.$
(2.63)

###### Proof.

We start with the error function $\vec{E}_{+}^{s}$ defined in (2.3). By using
(2.38) in Corollary 2.1, we obtain

$\displaystyle\|\vec{E}_{+,1}^{s}\|_{L^{2}}\leq
C\epsilon\left\|\left(u_{+,\text{app}}^{s},v_{+,\text{app}}^{s}\right)\right\|_{H^{2}}+\epsilon\|\varrho_{+,\text{app}}^{s}\|_{L^{2}}\leq
C\epsilon^{\frac{6}{7}},$ (2.64)

and

$\displaystyle\|\vec{E}_{+,2}^{s}\|_{H^{1}}\leq
k^{4}\left(1+\|\varphi_{-}\|_{H^{1}}\right)\leq\epsilon^{\frac{4}{7}}.$ (2.65)

Then combining (2.64) and (2.65) yields (2.61).

For $\vec{E}_{-}^{s}$, we use (2.39) to obtain that

$\displaystyle\|\vec{E}_{-}^{s}\|_{L^{2}}\leq
C\epsilon\left\|\left(u_{-,\text{app}}^{s},v_{-,\text{app}}^{s}\right)\right\|_{H^{2}}+\epsilon\|\varrho_{-,\text{app}}^{s}\|_{L^{2}}+k^{4}\|\varphi_{-}\|_{L^{2}}\leq
C\epsilon^{\frac{4}{7}},$ (2.66)

and

$\displaystyle\|\partial_{y}\vec{E}_{-}^{s}\|_{L^{2}}\leq
C\epsilon\left\|\left(u_{-,\text{app}}^{s},v_{-,\text{app}}^{s}\right)\right\|_{H^{3}}+C\epsilon\|\varrho_{-,\text{app}}^{s}\|_{H^{1}}+Ck^{4}\|\varphi_{-}\|_{H^{1}}\leq
C\epsilon^{\frac{2}{7}}.$ (2.67)

Thus the inequality (2.62) follows from (2.66) and (2.67).

Finally for $\vec{E}_{\pm}^{f}$, using pointwise estimates (2.54) and (2.55)
gives

$\displaystyle\|\vec{E}_{\pm}^{f}\|_{L^{2}}\leq$ $\displaystyle
Ck^{2}\epsilon\|u_{\pm,\text{app}}^{f}\|_{L^{2}}+Ck\|(1\pm
y)^{2}u_{\pm,\text{app}}^{f}\|_{L^{2}}+C\|(1\pm
y)v_{\pm,\text{app}}^{f}\|_{L^{2}}$
$\displaystyle+C\left(\epsilon\|v_{\pm,\text{app}}^{f}\|_{H^{2}}+k\|(1\pm
y)v_{\pm,\text{app}}^{f}\|_{L^{2}}+|c|\|v_{\pm,\text{app}}^{f}\|_{L^{2}}\right)$
$\displaystyle\leq$ $\displaystyle C\epsilon^{\frac{6}{7}},$

which is (2.63). The proof of Proposition 2.1 is completed. ∎

##  3 Control of remainder

In the last section, we have constructed four independent approximate
solutions to the eigenvalue problem (1.6). To correct errors generated by the
approximation, we need to solve the following remainder system

$\left\\{\begin{aligned} &ik(U_{s}-c)\rho+\text{div}_{k}(u,v)=0,\quad
y\in(-1,1),\\\ &-\epsilon\Delta_{k}u-\lambda
ik\epsilon\text{div}_{k}(u,v)+ik(U_{s}-c)u+(ikm^{-2}+\epsilon
U^{\prime\prime}_{s})\rho+vU^{\prime}_{s}=f_{u},\\\
&-\epsilon\Delta_{k}v-\lambda\epsilon\partial_{y}\text{div}_{k}(u,v)+ik(U_{s}-c)v+m^{-2}\partial_{y}\rho=f_{v},\\\
&v|_{y=\pm 1}=0,\end{aligned}\right.$ (3.1)

where $(f_{u},f_{v})$ are in homogenuous source terms. Notice that in (3.1),
we only impose the boundary condition for normal velocity $v$. The system
(3.1) is studied in the regime of parameters: $k\approx\epsilon^{\frac{1}{7}}$
and $c$ satisfies (1.8).

The main result in this section is the following proposition.

######  Proposition 3.1.

Let the Mach number $m\in\left(0,\frac{1}{\sqrt{3}}\right)$. If
$f_{u},f_{v}\in L^{2}(-1,1)$, then remainder system (3.1) admits a solution
$(\rho,u,v)\in H^{1}(-1,1)\times\left(H^{2}(-1,1)\right)^{2}$. Moreover, the
solution satisfies following uniform-in-$\epsilon$ estimates

$\displaystyle\epsilon^{\frac{5}{7}}\|(m^{-1}\rho,u,v)\|_{L^{2}}+\epsilon^{\frac{5}{7}}\|(m^{-2}\partial_{y}\rho,\partial_{y}u,\partial_{y}v)\|_{L^{2}}+\epsilon^{\frac{9}{7}}\|(\partial_{y}^{2}u,\partial_{y}^{2}v)\|_{L^{2}}\leq
C\|(f_{u},f_{v})\|_{L^{2}}.$ (3.2)

For $f_{u}$, $f_{v}\in H^{1}(-1,1)$, we define the operator

$\displaystyle\Omega(f_{u},f_{v})=-f_{v}+\frac{1}{ik}\partial_{y}(A^{-1}f_{u}).$
(3.3)

Then the solution $(\rho,u,v)$ satisfies the following improved estimates:

$\displaystyle\epsilon^{\frac{2}{7}}\|(m^{-1}\rho,u,v)\|_{L^{2}}+\epsilon^{\frac{2}{7}}\|(m^{-2}\partial_{y}\rho,\partial_{y}u,\partial_{y}v)\|_{L^{2}}+\epsilon^{\frac{4}{7}}\|(\partial_{y}^{2}u,\partial_{y}^{2}v)\|_{L^{2}}$
$\displaystyle\qquad\leq
C\|\Omega(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{\frac{1}{7}}\|(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{\frac{2}{7}}\|\text{div}_{k}(f_{u},f_{v})\|_{L^{2}}.$
(3.4)

Furthermore, the boundary values $u(\pm 1;c)$ are analytic in $c$.

###  3.1 Quasi-compressible approximation

To solve (3.1), we introduce the following quasi-compressible approximate
problem:

$\left\\{\begin{aligned}
&ik(U_{s}-c)\varrho+\text{div}_{k}(\mathfrak{u},\mathfrak{v})=0,~{}y\in(-1,1),\\\
&-\epsilon\Delta_{k}(\mathfrak{u}+(U_{s}-c)\varrho)+ik(U_{s}-c)\mathfrak{u}+\mathfrak{v}U^{\prime}_{s}+ikm^{-2}\varrho=s_{1},\\\
&-\epsilon\Delta_{k}\mathfrak{v}+ik(U_{s}-c)\mathfrak{v}+m^{-2}\partial_{y}\varrho=s_{2},\\\
&\mathfrak{v}|_{y=\pm 1}=0.\end{aligned}\right.$ (3.5)

Here $(s_{1},s_{2})$ is a given source term. We denote the operator on the
left hand side of (3.5) by $\mathfrak{Q}$.

As in [50], we can decouple $(\varrho,\mathfrak{u},\mathfrak{v})$ and
reformulate the system (3.5) into an Orr-Sommerfeld type equation. In fact, by
the first equation in (3.5), it is natural to define the effective stream
function $\Phi$, which satisfies:

$\displaystyle\partial_{y}\Phi=\mathfrak{u}+(U_{s}-c)\varrho,~{}-ik\Phi=\mathfrak{v},~{}~{}\Phi|_{y=\pm
1}=0.$ (3.6)

Then plugging (3.6) into the second equation of (3.5), we can express
$\varrho$ in terms of $\Phi$ as

$\displaystyle
m^{-2}\varrho=-A^{-1}(y)\bigg{[}\frac{i}{n}\Delta_{k}\partial_{y}\Phi+(U_{s}-c)\partial_{y}\Phi-
U_{s}^{\prime}\Phi-(ik)^{-1}s_{1}\bigg{]},$ (3.7)

where the function $A(y)$ is defined in (2.3). Substituting (3.7) into the
third equation, we derive the following equation for $\Phi$:

$\text{OS}_{\text{CNS}}(\Phi)=\frac{i}{n}\Lambda(\Delta_{k}\Phi)+(U_{s}-c)\Lambda(\Phi)-\partial_{y}(A^{-1}U_{s}^{\prime})\Phi=\Omega(s_{1},s_{2}),~{}y\in(-1,1).$
(3.8)

Here $n=\frac{k}{\epsilon}$ is the rescaled frequency, and

$\displaystyle\Lambda:H^{2}(-1,1)\cap H^{1}_{0}(-1,1)\rightarrow L^{2}(-1,1),$
$\displaystyle\Lambda(\Phi)=\partial_{y}(A^{-1}\partial_{y}\Phi)-k^{2}\Phi$

is the modified vorticity, and the operator $\Omega$ is defined in (3.3). To
solve (3.8), the following boundary conditions are imposed:

$\Lambda(\Phi)|_{y=\pm 1}=\Phi|_{y=\pm 1}=0.$

If $\Phi$ is a solution to (3.8), then $(\varrho,\mathfrak{u},\mathfrak{v})$
is a solution to the quasi-compressible system (3.5).

According to the above argument, it is suffcient to study the following Orr-
Sommerfeld type equation.

$\left\\{\begin{aligned} &\text{OS}_{\text{CNS}}(\Phi)=h,~{}y\in(-1,1),\\\
&\Phi|_{y=\pm 1}=\Lambda(\Phi)|_{y=\pm 1}=0.\end{aligned}\right.$ (3.9)

Here $h\in L^{2}(-1,1)$ is a given source. Once the solution $\Phi$ is
obtained, the fluid quantities $(\varrho,\mathfrak{u},\mathfrak{v})$ can be
recovered from (3.6) and (3.7).

For the solvability of (3.9), we have the following lemma.

######  Lemma 3.1.

Let the Mach number $m\in\left(0,\frac{1}{\sqrt{3}}\right)$. There exists a
unique solution $\Phi\in H^{4}(-1,1)\cap H^{1}_{0}(-1,1)$ to the Orr-
Sommerfeld equation (3.9). Moreover, the solution satisfies

$\displaystyle\|\partial_{y}\Phi,k\Phi\|_{L^{2}}+\|\Lambda\Phi\|_{L^{2}}$
$\displaystyle\leq C\epsilon^{-\frac{2}{7}}\|h\|_{L^{2}},$ (3.10)
$\displaystyle\|\partial_{y}\Lambda(\Phi),k\Lambda(\Phi)\|_{L^{2}}$
$\displaystyle\leq C\epsilon^{-\frac{4}{7}}\|h\|_{L^{2}}.$ (3.11)

###### Proof.

We only show the a priori estimates (3.10) and (3.11). The existence and
uniqueness parts can be proved in the same way as in Lemma 3.5 of [50]. Set
the following weight function.

$\displaystyle w(y)=-[\partial_{y}(A^{-1}\partial_{y}U_{s})]^{-1}.$ (3.12)

Multiplying (3.9) by $-w\overline{\Lambda(\Phi)}$ and integrating the
resultant equation over $(-1,1)$, we have:

$\displaystyle\underbrace{-\frac{i}{n}\int^{1}_{-1}w\Lambda(\Delta_{k}\Phi)\overline{\Lambda(\Phi)}{\rm
d}y}_{I_{1}}$
$\displaystyle+\underbrace{\int^{1}_{-1}-(U_{s}-c)w|\Lambda(\Phi)|^{2}{\rm
d}y}_{I_{2}}$
$\displaystyle+\underbrace{\int^{1}_{-1}-\Phi\overline{\Lambda(\Phi)}{\rm
d}y}_{I_{3}}+\underbrace{\int^{1}_{-1}hw\overline{\Lambda(\Phi)}{\rm
d}y}_{I_{4}}=0.$ (3.13)

Now we estimate $I_{i}$, $i=1,2,3,4$, term by term. First we consider $I_{1}$.
Denote
$[\Delta_{k},\Lambda]=\Delta_{k}(\Lambda(\Phi))-\Lambda(\Delta_{k}(\Phi))$ and
write $I_{1}$ as

$\displaystyle I_{1}$
$\displaystyle=-\frac{i}{n}\int^{1}_{-1}w\Delta_{k}\left(\Lambda(\Phi)\right)\overline{\Lambda(\Phi)}{\rm
d}y+\frac{i}{n}\int^{1}_{-1}w[\Delta_{k},\Lambda](\Phi)\overline{\Lambda(\Phi)}{\rm
d}y=I_{11}+I_{12}.$ (3.14)

For the first term $I_{11}$, integrating by parts and using boundary
conditions $\Lambda(\Phi)|_{y=\pm 1}=0$ give

$\displaystyle
I_{11}=\frac{i}{n}\int^{1}_{-1}w(|\partial_{y}\Lambda(\Phi)|^{2}+k^{2}|\Lambda(\Phi)|^{2}){\rm
d}y+\frac{i}{n}\int^{1}_{-1}w^{\prime}\partial_{y}\Lambda(\Phi)\overline{\Lambda(\Phi)}{\rm
d}y,$ (3.15)

in which the last integral is bounded by

$\displaystyle\bigg{|}\frac{i}{n}\int^{1}_{-1}w^{\prime}\partial_{y}\Lambda(\Phi)\overline{\Lambda(\Phi)}dy\bigg{|}\leq
C\epsilon^{\frac{6}{7}}\|\partial_{y}\Lambda(\Phi)\|_{L^{2}}\|\Lambda(\Phi)\|_{L^{2}}.$
(3.16)

Concerning $I_{12}$, we first represent the commutator
$[\Delta_{k},\Lambda]\Phi$ in terms of $\Lambda(\Phi)$, $\partial_{y}\Phi$ and
$\Phi$:

$\displaystyle[\Delta_{k},\Lambda]\Phi$
$\displaystyle=\partial^{3}_{y}(A^{-1}\partial_{y}\Phi)-\partial_{y}(A^{-1}\partial^{3}_{y}\Phi)$
$\displaystyle=2\partial_{y}(A^{-1})\partial^{3}_{y}\Phi+3\partial^{2}_{y}(A^{-1})\partial^{2}_{y}\Phi+\partial^{3}_{y}(A^{-1})\partial_{y}\Phi.$
(3.17)

Then by using the relations

$\displaystyle\partial^{2}_{y}\Phi$
$\displaystyle=A\Lambda(\Phi)+A^{-1}\partial_{y}A\partial_{y}\Phi+k^{2}A\Phi,$
(3.18) $\displaystyle\partial^{3}_{y}\Phi$
$\displaystyle=A\partial_{y}\Lambda(\Phi)+2\Lambda(\Phi)\partial_{y}A+\partial_{y}\Phi(A^{-1}\partial^{2}_{y}A+k^{2}A)+2k^{2}\Phi\partial_{y}A,$
(3.19)

we obtain
$|\partial_{y}^{2}\Phi|\approx|\Lambda(\Phi)|+\left|\left(\partial_{y}\Phi,k\Phi\right)\right|$,
and
$|\partial_{y}^{3}\Phi|\approx\left|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right|+\left|\left(\partial_{y}\Phi,k\Phi\right)\right|$.
Plugging this into commutator (3.17) and taking $L^{2}$ norm, we obtain

$\left\|[\Delta_{k},\Lambda]\Phi\right\|_{L^{2}}\leq
C\left(\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\right).$

Thus, $I_{12}$ is bounded as

$\displaystyle|I_{12}|$
$\displaystyle\leq\frac{C}{n}\|[\Delta_{k},\Lambda]\Phi\|_{L^{2}}\|\Lambda(\Phi)\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{\frac{6}{7}}\|\Lambda(\Phi)\|_{L^{2}}\left(\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\right).$
(3.20)

By substituting estimates (3.15), (3.16) for $I_{11}$ and (3.20) for $I_{12}$
into (3.14), then taking real and imaginary parts of the result respectively,
we deduce that

$\displaystyle|\text{Re}I_{1}|\leq$
$\displaystyle\frac{C\|\text{Im}w\|_{L^{\infty}}}{n}\left\|\partial_{y}\Lambda(\Phi)\right\|^{2}_{L^{2}}+C\epsilon^{\frac{6}{7}}\|\Lambda(\Phi)\|_{L^{2}}\left(\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\right)$
$\displaystyle\leq$ $\displaystyle
C\epsilon^{\frac{8}{7}}\left\|\partial_{y}\Lambda(\Phi)\right\|^{2}_{L^{2}}+C\epsilon^{\frac{6}{7}}\|\Lambda(\Phi)\|_{L^{2}}\left(\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\right),$
(3.21)

and

$\displaystyle\text{Im}I_{1}\geq$
$\displaystyle\frac{1}{n}\int_{-1}^{1}\text{Re}w\left|\left(\partial_{y}\Lambda(\Phi),k\Lambda(\Phi)\right)\right|^{2}{\rm
d}y-C\epsilon^{\frac{6}{7}}\|\Lambda(\Phi)\|_{L^{2}}\bigg{(}\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\bigg{)}$
$\displaystyle\gtrsim$
$\displaystyle\epsilon^{\frac{6}{7}}\left\|\left(\partial_{y}\Lambda(\Phi),k\Lambda(\Phi)\right)\right\|^{2}-C\epsilon^{\frac{6}{7}}\|\Lambda(\Phi)\|_{L^{2}}\bigg{(}\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\bigg{)}.$
(3.22)

In (3.21) and (3.22), we have used the fact that
$|\text{Im}w|\lesssim|c|\lesssim\epsilon^{\frac{2}{7}}$ and $\text{Re}w\gtrsim
1$ respectively.

Next we estimate $I_{2}$. By using (5.13) and (5.14), we can obtain

$\left|\text{Re}\bigg{(}(U_{s}-c)w\bigg{)}\right|\lesssim 1,$

and

$\text{Im}\bigg{(}-(U_{s}-c)w\bigg{)}=\left(w_{0}-U_{s}w_{1}\right)\text{Im}c+O(1)|c|^{2}\gtrsim{\gamma}_{0}\text{Im}c+O(1)|c|^{2}\gtrsim\frac{\gamma_{0}}{2}\epsilon^{\frac{2}{7}}.$

Therefore, by taking real and imaginary part of $I_{2}$ respectively, we
deduce that:

$\displaystyle|\text{Re}I_{2}|$
$\displaystyle\leq\left\|\text{Re}\bigg{(}(U_{s}-c)w\bigg{)}\right\|_{L^{\infty}}\|\Lambda(\Phi)\|^{2}_{L^{2}}\leq
C\|\Lambda(\Phi)\|_{L^{2}},$ (3.23)

and

$\displaystyle\text{Im}I_{2}$
$\displaystyle\geq\int_{-1}^{1}\text{Im}\bigg{(}-(U_{s}-c)w\bigg{)}\left|\Lambda(\Phi)\right|^{2}{\rm
d}y\geq\frac{\gamma_{0}}{2}\epsilon^{\frac{2}{7}}\|\Lambda(\Phi)\|_{L^{2}}^{2}.$
(3.24)

For $I_{3}$, integrating by parts and using the boundary conditions
$\Phi|_{y=\pm 1}=0$ give

$\displaystyle
I_{3}=\int_{-1}^{1}\bar{A}^{-1}|\partial_{y}\Phi|^{2}+k^{2}|\Phi|^{2}{\rm
d}y.$

Then we have

$\bar{A}^{-1}=(1-m^{2}U_{s}^{2})^{-2}\left(1-m^{2}U_{s}^{2}-2m^{2}U_{s}\bar{c}+O(1)|c|^{2}\right),$

which implies that $\text{Re}\left(\bar{A}^{-1}\right)\sim 1$ and
$\text{Im}\left(\bar{A}^{-1}\right)\gtrsim\left(m^{2}\text{Im}c\right)U_{s}-|c|^{2}$.
Thus, we obtain

$\displaystyle\text{Re}I_{3}$
$\displaystyle=\int_{-1}^{1}\text{Re}\left(\bar{A}^{-1}\right)|\partial_{y}\Phi|^{2}+k^{2}|\Phi|^{2}{\rm
d}y\gtrsim\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}^{2},$ (3.25)

and

$\displaystyle\text{Im}I_{3}$
$\displaystyle=\int_{-1}^{1}\text{Im}\left(\bar{A}^{-1}\right)|\partial_{y}\Phi|^{2}{\rm
d}y\gtrsim m^{2}\text{Im}c\left(\int_{-1}^{1}U_{s}|\partial_{y}\Phi|^{2}{\rm
d}y\right)-|c|^{2}\|\partial_{y}\Phi\|_{L^{2}}^{2}$
$\displaystyle\gtrsim-\epsilon^{\frac{4}{7}}\|\partial_{y}\Phi\|_{L^{2}}^{2}.$
(3.26)

Finally, by Cauchy-Schwarz inequality, it holds that:

$|I_{4}|\lesssim\|h\|_{L^{2}}\|\Lambda(\Phi)\|_{L^{2}}.$

Thus, we have completed the estimates for $I_{1}$ to $I_{4}$. By plugging them
into (3.1) and taking real and imaginary parts respectively, we obtain

$\displaystyle\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}^{2}\leq$ $\displaystyle
C\|\Lambda(\Phi)\|_{L^{2}}^{2}+C\epsilon^{\frac{8}{7}}\left\|\partial_{y}\Lambda(\Phi)\right\|_{L^{2}}^{2}+C\|h\|_{L^{2}}^{2}$
$\displaystyle+C\epsilon^{\frac{6}{7}}\|\Lambda(\Phi)\|_{L^{2}}\left(\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\right)$
$\displaystyle\leq$ $\displaystyle
C\|\Lambda(\Phi)\|_{L^{2}}^{2}+C\epsilon^{\frac{8}{7}}\left\|\partial_{y}\Lambda(\Phi)\right\|_{L^{2}}^{2}+C\|h\|_{L^{2}}^{2},$
(3.27)

and

$\displaystyle\epsilon^{\frac{4}{7}}\left\|\left(\partial_{y}\Lambda(\Phi),k\Lambda(\Phi)\right)\right\|_{L^{2}}^{2}+\|\Lambda(\Phi)\|_{L^{2}}^{2}$
$\displaystyle\quad\leq\epsilon^{\frac{4}{7}}\|\Lambda(\Phi)\|_{L^{2}}\left(\left\|\left(\partial_{y}\Lambda(\Phi),\Lambda(\Phi)\right)\right\|_{L^{2}}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\right)$
$\displaystyle\qquad+\epsilon^{\frac{2}{7}}\|\partial_{y}\Phi\|_{L^{2}}^{2}+\epsilon^{-\frac{2}{7}}\|h\|_{L^{2}}\|\Lambda(\Phi)\|_{L^{2}}.$
$\displaystyle\quad\leq
o(1)\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}^{2}+\epsilon^{-\frac{4}{7}}\|h\|_{L^{2}}^{2}.$
(3.28)

Set

$\displaystyle\mathfrak{B}:=\epsilon^{\frac{4}{7}}\left\|\left(\partial_{y}\Lambda(\Phi),k\Lambda(\Phi)\right)\right\|_{L^{2}}^{2}+\|\Lambda(\Phi)\|_{L^{2}}^{2}+\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}^{2}.$

Combining (3.1) and (3.1) suitably, we have:

$\displaystyle\mathfrak{B}\leq
o(1)\mathfrak{B}+\epsilon^{-\frac{4}{7}}\|h\|_{L^{2}}.$

For sufficiently small $\epsilon\ll 1$, it holds that

$\mathfrak{B}\leq C\epsilon^{-\frac{4}{7}}\|h\|_{L^{2}}^{2}.$

This implies inequalities (3.10) and (3.11). The proof of lemma 3.1 is
completed. ∎

With the solution $\Phi$ to Orr-Sommerfeld type equation (3.8), we can recover
fluid quantities $(\varrho,\mathfrak{u},\mathfrak{v})$ in the following
corollary.

######  Corollary 3.1.

For any $s_{1},s_{2}\in H^{1}(-1,1)$, there exists a solution
$(\varrho,\mathfrak{u},\mathfrak{v})\in H^{2}(-1,1)$ to the quasi-compressible
system (3.5). The solution satisfies the following estimates:

$\displaystyle\|\mathfrak{u}\|_{H^{1}}$
$\displaystyle+\|(m^{-2}\varrho,\mathfrak{v})\|_{H^{2}}+\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathfrak{u},\mathfrak{v})\|_{H^{1}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|s_{1}\|_{L^{2}}+C\|s_{2}\|_{L^{2}}+C\|\text{div}_{k}(s_{1},s_{2})\|_{L^{2}},$
(3.29)

and

$\displaystyle\|\partial^{2}_{y}\mathfrak{u}\|_{L^{2}}\leq
C\epsilon^{-\frac{4}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|s_{1}\|_{L^{2}}+C\|s_{2}\|_{L^{2}}+C\|\text{div}_{k}(s_{1},s_{2})\|_{L^{2}}.$
(3.30)

###### Proof.

Let $\Phi$ be the solution to (3.8). From (3.10), (3.11), (3.18) and (3.19),
we obtain

$\displaystyle\|\partial^{2}_{y}\Phi\|_{L^{2}}\leq$ $\displaystyle
C\|\Lambda(\Phi)\|_{L^{2}}+C\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}},$ (3.31)

and

$\displaystyle\|\partial^{3}_{y}\Phi\|_{L^{2}}$ $\displaystyle\leq
C\|\partial_{y}\Lambda(\Phi)\|_{L^{2}}+C\|\Lambda(\Phi)\|_{L^{2}}+C\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\leq
C\epsilon^{-\frac{4}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}.$ (3.32)

From $\mathfrak{v}=-ik\Phi$, it holds that

$\displaystyle\|\mathfrak{v}\|_{H^{2}}$ $\displaystyle\leq
C\|\partial^{2}_{y}\Phi\|_{L^{2}}+C\|(\partial_{y}\Phi,k\Phi)\|_{L^{2}}\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}.$

This gives (3.1) for $\mathfrak{v}$. For the density $\varrho$, by using the
explicit formula (3.7) and the elementary inequality

$\displaystyle\|\Phi\|_{L^{\infty}}\leq\int^{1}_{-1}|\Phi^{\prime}(x)|{\rm
d}x\leq\sqrt{2}\|\Phi^{\prime}\|_{L^{2}},\text{ for any }\Phi\in
H^{1}_{0}(-1,1),$ (3.33)

we have

$\displaystyle m^{-2}\|\varrho\|_{L^{2}}$
$\displaystyle\leq\frac{C}{n}\|\partial^{3}_{y}\Phi\|_{L^{2}}+C(1+\frac{k^{2}}{n})\|\partial_{y}\Phi\|_{L^{2}}+C\|\Phi\|_{L^{\infty}}\|\partial_{y}U_{s}\|_{L^{2}}+\frac{C}{k}\|s_{1}\|_{L^{2}}$
$\displaystyle\leq\frac{C}{n}\|\partial^{3}_{y}\Phi\|_{L^{2}}+C\|\partial_{y}\Phi\|_{L^{2}}+\frac{C}{k}\|s_{1}\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|s_{1}\|_{L^{2}}.$
(3.34)

Here we have used (3.31) and (3.32) in the last inequality of (3.34). For the
derivatives of $\varrho$, we firstly observe that

$\displaystyle-m^{-2}\partial_{y}\varrho$
$\displaystyle=\text{OS}_{\text{CNS}}(\Phi)+k^{2}\left[\frac{i}{n}\Delta_{k}\Phi+(U_{s}-c)\Phi\right]-\frac{1}{ik}\partial_{y}(A^{-1}s_{1})$
$\displaystyle=-s_{2}+k^{2}\left[\frac{i}{n}\Delta_{k}\Phi+(U_{s}-c)\Phi\right].$

Then taking $L^{2}$-norm on both side of the above equation and using (3.31)
give

$\displaystyle m^{-2}\|\partial_{y}\varrho\|_{L^{2}}$ $\displaystyle\leq
C\|s_{2}\|_{L^{2}}+C\|\partial^{2}_{y}\Phi\|_{L^{2}}+Ck\|\Phi\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\|s_{2}\|_{L^{2}}.$
(3.35)

Moreover, by taking divergence of the last two equations in (3.5), we can
derive the following equation:

$-m^{-2}\Delta_{k}\varrho=-\text{div}_{k}(s_{1},s_{2})+k^{2}(U_{s}-c)^{2}\varrho+2k^{2}\Phi
U^{\prime}_{s}.$

Thus taking $L^{2}$-norm and using (3.31) and (3.34) yield that

$\displaystyle m^{-2}\|\partial^{2}_{y}\varrho\|_{L^{2}}$ $\displaystyle\leq
C\|\text{div}_{k}(s_{1},s_{2})\|_{L^{2}}+C(1+m^{-2})\|\varrho\|_{L^{2}}+Ck\|\Phi\|_{L^{2}}$
$\displaystyle\leq
C\|\text{div}_{k}(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|s_{1}\|_{L^{2}}.$
(3.36)

Combining (3.34), (3.35) and (3.36) together yields (3.1) for $\varrho$.
Concerning about the divergence field
$\text{div}_{k}(\mathfrak{u},\mathfrak{v})$, we have from the continuity
equation $\eqref{3.1.1}_{1}$ that

$\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathfrak{u},\mathfrak{v})\|_{H^{1}}\leq
C\|\varrho\|_{H^{1}}\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|s_{1}\|_{L^{2}}+C\|s_{2}\|_{L^{2}}.$

Finally, we estimate $\mathfrak{u}$. From (3.6), it holds that
$|\mathfrak{u}|\sim|\partial_{y}\Phi|+|\varrho|$. Thus, by using (3.31),
(3.32) and (3.34)-(3.36), we obtain

$\displaystyle\|\mathfrak{u}\|_{H^{1}}\leq
C\|\partial_{y}\Phi\|_{H^{1}}+C\|\varrho\|_{H^{1}}\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|s_{1}\|_{L^{2}}+C\|s_{2}\|_{L^{2}},$

and

$\displaystyle\|\partial^{2}_{y}\mathfrak{u}\|_{L^{2}}$ $\displaystyle\leq
C\|\partial^{3}_{y}\Phi\|_{L^{2}}+C\|\varrho\|_{H^{2}}\leq
C\epsilon^{-\frac{4}{7}}\|\Omega(s_{1},s_{2})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|s_{1}\|_{L^{2}}+C\|s_{2}\|_{L^{2}}+C\|\text{div}_{k}(s_{1},s_{2})\|_{L^{2}}.$

Therefore, the estimates (3.1) and (3.30) for $\mathfrak{u}$ are obtained.
This completes the proof of the corollary. ∎

It is straightforward to see that the quasi-compressible operator
$\mathfrak{Q}$ generates the error

$\displaystyle\vec{E}_{\mathfrak{Q}}(\varrho,\mathfrak{u},\mathfrak{v})$
$\displaystyle\overset{\mbox{\tiny{def}}}{=}\mathcal{L}(\varrho,\mathfrak{u},\mathfrak{v})-\mathfrak{Q}(\varrho,\mathfrak{u},\mathfrak{v})$
$\displaystyle=\bigg{(}0,\epsilon\Delta_{k}\left((U_{s}-c)\varrho\right)-\lambda\epsilon
ik\text{div}_{k}(\mathfrak{u},\mathfrak{v})+\epsilon
U_{s}^{\prime\prime}\varrho,-\epsilon\lambda\partial_{y}\text{div}_{k}(\mathfrak{u},\mathfrak{v})\bigg{)},$
(3.37)

which is not in $H^{1}(-1,1)$. This fact prevents us from iterating
$\mathfrak{Q}$ directly to solve (3.1). In the next subsection, we will
introduce the Stokes operator to recover the regularity.

###  3.2 Stokes approximation

To smooth out the error term (3.1) generated by the quasi-compressible system
(3.5) , we introduce the following Stokes approximation:

$\left\\{\begin{aligned}
&ik(U_{s}-c)\mathcal{P}+\text{div}_{k}(\mathcal{U},\mathcal{V})=q_{0},~{}y\in(-1,1),\\\
&-\epsilon\Delta_{k}\mathcal{U}-\lambda
ik\epsilon\text{div}_{k}(\mathcal{U},\mathcal{V})+ik(U_{s}-c)\mathcal{U}+(ikm^{-2}+\epsilon
U^{\prime\prime}_{s})\mathcal{P}=q_{1},\\\
&-\epsilon\Delta_{k}\mathcal{V}-\lambda\epsilon\partial_{y}\text{div}_{k}(\mathcal{U},\mathcal{V})+ik(U_{s}-c)\mathcal{V}+m^{-2}\partial_{y}\mathcal{P}=q_{2},\\\
&\partial_{y}\mathcal{U}|_{y=\pm 1}=\mathcal{V}|_{y=\pm
1}=0.\end{aligned}\right.$ (3.38)

Here $(q_{0},q_{1},q_{2})$ is any given inhomogenuous source term. Compared
with the original resolvent problem (3.1), we remove the stretching term
$\mathcal{V}U_{s}^{\prime}$ in the Stokes system (3.38). The solvability of
Stokes system is summarized as follows.

######  Lemma 3.2.

The Stokes system (3.38) admits a unique solution
$(\mathcal{P},\mathcal{U},\mathcal{V})\in H^{1}(-1,1)\times(H^{2}(-1,1))^{2}$.
Moreover, $(\mathcal{P},\mathcal{U},\mathcal{V})$ satisfies the following
estimates:

$\displaystyle\|(m^{-1}\mathcal{P},\mathcal{U},\mathcal{V})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{3}{7}}\|(m^{-1}q_{0},q_{1},q_{2})\|_{L^{2}},$ (3.39)
$\displaystyle\|(\partial_{y}\mathcal{U},k\mathcal{U})\|_{L^{2}}+\|(\partial_{y}\mathcal{V},k\mathcal{V})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{5}{7}}\|(m^{-1}q_{0},q_{1},q_{2})\|_{L^{2}},$ (3.40)
$\displaystyle\|div_{k}(\mathcal{U},\mathcal{V})\|_{H^{1}}+m^{-2}\|\partial_{y}\mathcal{P}\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\|(m^{-2}q_{0},q_{1},q_{2})\|_{L^{2}}+C\|\partial_{y}q_{0}\|_{L^{2}},$
(3.41)
$\displaystyle\|(\Delta_{k}\mathcal{U},\Delta_{k}\mathcal{V})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{9}{7}}\|(m^{-1}q_{0},q_{1},q_{2})\|_{L^{2}}+C\|\partial_{y}q_{0}\|_{L^{2}}.$
(3.42)

######  Remark 3.1.

In view of (3.39)-(3.41), the divergence
$\text{div}_{k}(\mathcal{U},\mathcal{V})$ and density $\mathcal{P}$ have
stronger estimates than other components. This indicates the absence of
boundary layers for these two components.

###### Proof.

We follow the proof in [50]. Multiplying the second and third equations by
$-\bar{\mathcal{U}}$ and $-\bar{\mathcal{V}}$ respectively and integrating by
parts, we obtain:

$\displaystyle\epsilon\bigg{(}\|\partial_{y}\mathcal{U},k\mathcal{U}\|^{2}_{L^{2}}+\|\partial_{y}\mathcal{V},k\mathcal{V}\|^{2}_{L^{2}}\bigg{)}+\lambda\epsilon\|\text{div}_{k}(\mathcal{U},\mathcal{V})\|^{2}_{L^{2}}$
$\displaystyle+\underbrace{ik\int^{1}_{-1}(U_{s}-c)\bigg{(}|\mathcal{U}|^{2}+|\mathcal{V}|^{2}\bigg{)}{\rm
d}y}_{J_{1}}+\underbrace{m^{-2}\int^{1}_{-1}-\mathcal{P}\overline{\text{div}_{k}(\mathcal{U},\mathcal{V})}{\rm
d}y}_{J_{2}}$ $\displaystyle\qquad=\underbrace{-\int^{1}_{-1}(q_{1}+\epsilon
U_{s}^{\prime\prime}\mathcal{P})\bar{\mathcal{U}}+q_{2}\bar{\mathcal{V}}{\rm
d}y}_{J_{3}}.$ (3.43)

The real part of $J_{1}$ gives

$\displaystyle\text{Re}J_{1}=k\text{Im}c\left\|(\mathcal{U},\mathcal{V})\right\|_{L^{2}}^{2}\gtrsim\epsilon^{\frac{3}{7}}\|(\mathcal{U},\mathcal{V})\|_{L^{2}}^{2}.$
(3.44)

For $J_{2}$, by using the continuity equation $\eqref{A2.1}_{1}$, we have

$\overline{\text{div}_{k}(\mathcal{U},\mathcal{V})}=ik(U_{s}-\bar{c})\bar{\mathcal{P}}+\bar{q_{0}}.$

Plugging this into $J_{2}$ and taking real part give

$\displaystyle\text{Re}J_{2}$
$\displaystyle=-ikm^{-2}\int^{1}_{-1}(U_{s}-\bar{c})|\mathcal{P}|^{2}{\rm
d}y-m^{-2}\int^{1}_{-1}\mathcal{P}\bar{q_{0}}{\rm d}y$ $\displaystyle\geq
k\text{Im}c\|m^{-1}\mathcal{P}\|^{2}_{L^{2}}-m^{-2}\|\mathcal{P}\|_{L^{2}}\|q_{0}\|_{L^{2}}$
$\displaystyle\gtrsim\epsilon^{\frac{3}{7}}\|m^{-1}\mathcal{P}\|_{L^{2}}^{2}-m^{-2}\|\mathcal{P}\|_{L^{2}}\|q_{0}\|_{L^{2}}.$
(3.45)

By Cauchy-Schwarz inequality, $J_{3}$ is bounded by

$\displaystyle|J_{3}|\lesssim\|(\mathcal{U},\mathcal{V})\|_{L^{2}}\|(q_{1},q_{2})\|_{L^{2}}+\epsilon\bigg{(}\|\mathcal{P}\|^{2}_{L^{2}}+\|\mathcal{U}\|^{2}_{L^{2}}\bigg{)}.$
(3.46)

Therefore, by substituting (3.44), (3.45), (3.46) into (3.43) and then taking
the real part, we deduce that

$\displaystyle\epsilon\bigg{(}\|\partial_{y}\mathcal{U},k\mathcal{U}\|^{2}_{L^{2}}$
$\displaystyle+\|\partial_{y}\mathcal{V},k\mathcal{V}\|^{2}_{L^{2}}\bigg{)}+\lambda\epsilon\|\text{div}_{k}(\mathcal{U},\mathcal{V})\|^{2}_{L^{2}}+\epsilon^{\frac{3}{7}}\|(m^{-1}\mathcal{P},\mathcal{U},\mathcal{V})\|_{L^{2}}^{2}$
$\displaystyle\leq
C\|(m^{-1}\mathcal{P},\mathcal{U},\mathcal{V})\|_{L^{2}}\|(m^{-1}q_{0},q_{1},q_{2})\|_{L^{2}}$
$\displaystyle\leq
o(1)\epsilon^{\frac{3}{7}}\|(m^{-1}\mathcal{P},\mathcal{U},\mathcal{V})\|_{L^{2}}^{2}+C\epsilon^{-\frac{3}{7}}\|m^{-1}q_{0},q_{1},q_{2})\|_{L^{2}}^{2}.$

Absorbing the first term on the right hand side to the left gives (3.39) and
(3.40).

Next we estimate the density and divergence part of the solution. For this, we
denote the vorticity by

$\omega=\partial_{y}\mathcal{U}-ik\mathcal{V}$

and the divergence of $(\mathcal{U},\mathcal{V})$ by
$\mathcal{D}=\text{div}_{k}(\mathcal{U},\mathcal{V})$. The diffusion in (3.38)
can be expressed in terms of $\mathcal{D}$ and $\omega$:

$\Delta_{k}\mathcal{U}=\partial_{y}\omega+ik\mathcal{D},~{}\Delta_{k}\mathcal{V}=-ik\omega+\partial_{y}\mathcal{D}.$

Thus we can rewrite the momentum equations in (3.38) as

$\left\\{\begin{aligned}
&ikm^{-2}\mathcal{P}=\epsilon\partial_{y}\omega+\epsilon(1+\lambda)ik\mathcal{D}-ik(U_{s}-c)\mathcal{U}-q_{1}-\epsilon
U^{\prime\prime}_{s}\mathcal{P},\\\ &m^{-2}\partial_{y}\mathcal{P}=-\epsilon
ik\omega+\epsilon(1+\lambda)\partial_{y}\mathcal{D}-ik(U_{s}-c)\mathcal{V}-q_{2}.\end{aligned}\right.$
(3.47)

By taking inner product of the first and second equations with
$\overline{ik{\mathcal{P}}}$ and $\partial_{y}\bar{\mathcal{P}}$ respectively,
we obtain:

$\displaystyle m^{-2}$
$\displaystyle\|(\partial_{y}\mathcal{P},k\mathcal{P}\|^{2}_{L^{2}}=\underbrace{\epsilon\int^{1}_{-1}\partial_{y}\omega\overline{ik{\mathcal{P}}}-ik\omega\partial_{y}\bar{\mathcal{P}}{\rm
d}y}_{J_{4}}+\underbrace{\epsilon(1+\lambda)\int^{1}_{-1}k^{2}\mathcal{D}\bar{\mathcal{P}}+\partial_{y}\mathcal{D}\partial_{y}\bar{\mathcal{P}}{\rm
d}y}_{J_{5}}$ $\displaystyle+\underbrace{\int^{1}_{-1}(q_{1}+\epsilon
U^{\prime\prime}_{s}\mathcal{P})ik\bar{\mathcal{P}}-q_{2}\partial_{y}\bar{\mathcal{P}}{\rm
d}y}_{J_{6}}+\underbrace{\int^{1}_{-1}ik(U_{s}-c)(ik\bar{\mathcal{P}}\mathcal{U}-\mathcal{V}\partial_{y}\bar{\mathcal{P}}){\rm
d}y}_{J_{7}}.$ (3.48)

Integrating by parts and using the boundary condition $\omega|_{y=\pm 1}=0$
give

$\displaystyle
J_{4}=-ik\epsilon\bar{\mathcal{P}}\omega\big{|}_{y=-1}^{y=1}=0.$ (3.49)

From the continuity equation $\eqref{A2.1}_{1}$, it holds that
$\mathcal{D}=-ik(U_{s}-c)\mathcal{P}+q_{0},$ and
$\partial_{y}\mathcal{D}=-ik(U_{s}-c)\partial_{y}\mathcal{P}-ikU^{\prime}_{s}\mathcal{P}+\partial_{y}q_{0}.$
Plugging these into $J_{5}$, we obtain

$\displaystyle J_{5}=$
$\displaystyle-\epsilon(1+\lambda)\int^{1}_{-1}ik(U_{s}-c)(|\partial_{y}\mathcal{P}|^{2}+k^{2}|\mathcal{P}|^{2}){\rm
d}y$ $\displaystyle-
ik\epsilon(1+\lambda)\int^{1}_{-1}U^{\prime}_{s}\mathcal{P}\partial_{y}\mathcal{P}{\rm
d}y+\epsilon(1+\lambda)\int^{1}_{-1}\left(\partial_{y}\bar{\mathcal{P}}\partial_{y}q_{0}+k^{2}q_{0}\bar{\mathcal{P}}\right){\rm
d}y.$ (3.50)

By Cauchy-Schwarz and Young’s inequalities, the last two terms on the right
hand side of $J_{5}$ are bounded by

$\bigg{|}ik\epsilon(1+\lambda)\int^{1}_{-1}U^{\prime}_{s}\mathcal{P}\partial_{y}\mathcal{P}dy\bigg{|}\leq
C\epsilon\|(\partial_{y}\mathcal{P},k\mathcal{P})\|^{2}_{L^{2}},$

and

$\displaystyle\bigg{|}\epsilon(1+\lambda)\int^{1}_{-1}\partial_{y}\bar{\mathcal{P}}\partial_{y}q_{0}+k^{2}q_{0}\bar{\mathcal{P}}dy\bigg{|}\leq$
$\displaystyle
C\epsilon\|(\partial_{y}\mathcal{P},k\mathcal{P})\|_{L^{2}}\|(\partial_{y}q_{0},kq_{0})\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{m^{-2}}{8}\|(\partial_{y}\mathcal{P},k\mathcal{P})\|^{2}_{L^{2}}+Cm^{2}\epsilon^{2}\|(\partial_{y}q_{0},kq_{0})\|^{2}_{L^{2}}.$

Therefore, by taking real part in (3.50), we deduce

$\displaystyle\text{Re}J_{5}$ $\displaystyle\leq-\epsilon
k\text{Im}c(1+\lambda)\|(\partial_{y}\mathcal{P},k\mathcal{P})\|^{2}_{L^{2}}+C\bigg{(}\epsilon+\frac{m^{-2}}{8}\bigg{)}\|(\partial_{y}\mathcal{P},k\mathcal{P})\|^{2}_{L^{2}}+Cm^{2}\epsilon^{2}\|(\partial_{y}q_{0},kq_{0})\|^{2}_{L^{2}}$
$\displaystyle\leq\frac{m^{-2}}{4}\|(\partial_{y}\mathcal{P},k\mathcal{P})\|^{2}_{L^{2}}+Cm^{2}\epsilon^{2}\|(\partial_{y}q_{0},kq_{0})\|^{2}_{L^{2}}.$
(3.51)

Moreover, by Cauchy-Schwarz and Young’s inequalities again, it holds that

$\displaystyle|J_{6}|+|J_{7}|$
$\displaystyle\leq\bigg{(}\|q_{1},q_{2})\|_{L^{2}}+k\|(\mathcal{U},\mathcal{V})\|_{L^{2}}\bigg{)}\|(\partial_{y}\mathcal{P},k\mathcal{P})\|_{L^{2}}+C\epsilon^{\frac{6}{7}}\|k\mathcal{P}\|^{2}_{L^{2}}$
$\displaystyle\leq\frac{m^{-2}}{4}\|(\partial_{y}\mathcal{P},k\mathcal{P})\|^{2}_{L^{2}}+Cm^{2}\bigg{(}\|(q_{1},q_{2})\|^{2}_{L^{2}}+k^{2}\|(\mathcal{U},\mathcal{V})\|^{2}_{L^{2}}\bigg{)}.$
(3.52)

By combining the estimates (3.49), (3.51) and (3.52) for $J_{4}$ to $J_{7}$,
and taking the real part of (3.48), we obtain that

$\displaystyle m^{-2}\|(\partial_{y}\mathcal{P},k\mathcal{P})\|_{L^{2}}$
$\displaystyle\leq
C\|(q_{1},q_{2})\|_{L^{2}}+C\epsilon\|(\partial_{y}q_{0},kq_{0})\|_{L^{2}}+Ck\|(\mathcal{U},\mathcal{V})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\|(m^{-1}q_{0},q_{1},q_{2})\|_{L^{2}}+C\epsilon\|\partial_{y}q_{0}\|_{L^{2}}.$
(3.53)

Here we have used (3.39) for $(\mathcal{U},\mathcal{V})$ in the last line of
(3.2). For the divergence, it holds from $\eqref{A2.1}_{1}$ and (3.2) that

$\displaystyle\|\text{div}_{k}(\mathcal{U},\mathcal{V})\|_{H^{1}}$
$\displaystyle\leq
C\|(\partial_{y}\mathcal{P},k\mathcal{P})\|_{L^{2}}+C\|q_{0}\|_{H^{1}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\|(m^{-1}q_{0},q_{1},q_{2})\|_{L^{2}}+C\|\partial_{y}q_{0}\|_{L^{2}}.$
(3.54)

Putting bounds (3.2) and (3.2) together gives (3.41).

Finally, we estimate second order derivatives of $(\mathcal{U},\mathcal{V})$.
By taking inner product of the first and second equations of (3.47) with
$\partial_{y}\bar{\omega}$ and $\overline{ik\omega}$ respectively, we can
deduce

$\displaystyle\epsilon\|(\partial_{y}\omega,k\omega)\|^{2}_{L^{2}}=$
$\displaystyle\int^{1}_{-1}(q_{1}+\epsilon
U^{\prime\prime}_{s}\mathcal{P})\partial_{y}\bar{\omega}+q_{2}\overline{ik\omega}{\rm
d}y+\int^{1}_{-1}ik(U_{s}-c)(\mathcal{U}\partial_{y}\bar{\omega}+\mathcal{V}\overline{ik{\omega}}){\rm
d}y$ $\displaystyle\leq$ $\displaystyle
C\bigg{(}\|(q_{1},q_{2})\|_{L^{2}}+\epsilon\|\mathcal{P}\|_{L^{2}}+k\|(\mathcal{U},\mathcal{V})\|_{L^{2}}\bigg{)}\|(\partial_{y}\omega,k\omega)\|_{L^{2}}.$
(3.55)

Thus by using (3.39) and (3.55), we have

$\displaystyle\|(\partial_{y}\omega,k\omega)\|_{L^{2}}$ $\displaystyle\leq
C\epsilon^{-1}\|(q_{1},q_{2})\|_{L^{2}}+C\epsilon^{-\frac{6}{7}}\|(m^{-1}\mathcal{P},\mathcal{U},\mathcal{V})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{9}{7}}\|(m^{-1}q_{0},q_{2},q_{2})\|_{L^{2}}.$ (3.56)

The estimate (3.42) follows from (3.2) and (3.56). Therefore, the proof of the
lemma is completed. ∎

Note that the error generated by $\mathfrak{S}$ is

$\displaystyle\vec{E}_{\mathfrak{S}}(\mathcal{P},\mathcal{U},\mathcal{V})\overset{\mbox{\tiny{def}}}{=}\mathcal{L}(\mathcal{P},\mathcal{U},\mathcal{V})-\mathfrak{S}(\mathcal{P},\mathcal{U},\mathcal{V})=\left(0,\mathcal{V}U_{s}^{\prime},0\right).$
(3.57)

By Lemma 3.2, we have $\vec{E}_{\mathfrak{S}}\in H^{2}(-1,1)$.

###  3.3 Solvability of resolvent problem

In this subsection, we solve the resolvent problem (3.1) by alternatively
iterating the above Quasi-compressible $\mathfrak{Q}$ and Stokes operators
$\mathfrak{S}.$ First of all, we construct the solution $(\rho,u,v)$ to (3.1)
for any given inhomogeneous source terms $f_{u},f_{v}\in L^{2}$, and show the
estimate (3.2). At the zeroth step, we introduce the Stokes solution
$\vec{\Xi}_{0}=(\mathcal{P}_{0},\mathcal{U}_{0},\mathcal{V}_{0})$, which
solves the following system:

$\displaystyle\mathfrak{S}(\mathcal{P}_{0},\mathcal{U}_{0},\mathcal{V}_{0})=(0,f_{u},f_{v}).$
(3.58)

Recall the error operator $\vec{E}_{\mathfrak{S}}$ defined in (3.57). The
error generated at this step is

$\displaystyle\vec{E}_{\mathfrak{S}}(\mathcal{P}_{0},\mathcal{U}_{0},\mathcal{V}_{0})=\left(0,\mathcal{V}_{0}U_{s}^{\prime},0\right).$
(3.59)

Since
$\vec{E}_{\mathfrak{S}}(\mathcal{P}_{0},\mathcal{U}_{0},\mathcal{V}_{0})\in
H^{2}(-1,1)$, we can introduce the solution
$(\varrho_{1},\mathfrak{u}_{1},\mathfrak{v}_{1})$ to the quasi-compressible
system:

$\displaystyle\mathfrak{Q}(\varrho_{1},\mathfrak{u}_{1},\mathfrak{v}_{1})=-\vec{E}_{\mathfrak{S}}(\mathcal{P}_{0},\mathcal{U}_{0},\mathcal{V}_{0}).$
(3.60)

The error term generated by $\mathfrak{Q}$ is

$\displaystyle\vec{E}_{\mathfrak{Q}}(\varrho_{1},\mathfrak{u}_{1},\mathfrak{v}_{1})=\left(0,\epsilon\Delta_{k}\left[(U_{s}-c)\varrho_{1}\right]-\epsilon\lambda
ik\text{div}_{k}(\mathfrak{u}_{1},\mathfrak{v}_{1})+\epsilon
U_{s}^{\prime\prime}\varrho_{1},\epsilon\lambda\partial_{y}\text{div}_{k}(\mathfrak{u}_{1},\mathfrak{v}_{1})\right).$
(3.61)

In view of Corollary 3.1, the right hand side of (3.61) is in $L^{2}(-1,1)$
instead of $H^{1}(-1,1)$, which forbids us to iterate $\mathfrak{Q}$ directly.
In order to recover the regularity, we introduce
$(\mathcal{P}_{1},\mathcal{U}_{1},\mathcal{V}_{1})$ as the solution to Stokes
approximation

$\displaystyle\mathfrak{S}(\mathcal{P}_{1},\mathcal{U}_{1},\mathcal{V}_{1})=-\vec{E}_{\mathfrak{Q}}.$
(3.62)

Now we define the following corrector $\vec{\Xi}_{1}$ at the Step 1:

$\displaystyle\vec{\Xi}_{1}=(\rho_{1},u_{1},v_{1})\overset{\mbox{\tiny{def}}}{=}(\varrho_{1},\mathfrak{u}_{1},\mathfrak{v}_{1})+(\mathcal{P}_{1},\mathcal{U}_{1},\mathcal{V}_{1}).$
(3.63)

One can check that the error term generated at this step is

$\displaystyle\vec{\mathcal{E}}_{1}$
$\displaystyle\overset{\mbox{\tiny{def}}}{=}\mathcal{L}(\vec{\Xi}_{0}+\vec{\Xi}_{1})-(0,f_{u},f_{v})$
$\displaystyle=\vec{E}_{\mathfrak{S}}(\mathcal{P}_{1},\mathcal{U}_{1},\mathcal{V}_{1})=(0,\mathcal{V}_{1}U_{s}^{\prime},0).$
(3.64)

Approximate solutions up to any order can be constructed by induction. Suppose
that at $N$-th step, we have the corrector

$\displaystyle\vec{\Xi}_{N}=(\rho_{N},u_{N},v_{N})\overset{\mbox{\tiny{def}}}{=}(\varrho_{N},\mathfrak{u}_{N},\mathfrak{v}_{N})+(\mathcal{P}_{N},\mathcal{U}_{N},\mathcal{V}_{N}),$
(3.65)

and the error

$\displaystyle\vec{\mathcal{E}}_{N}\overset{\mbox{\tiny{def}}}{=}\mathcal{L}\left(\sum_{j=0}^{N}\vec{\Xi}_{j}\right)-(0,f_{u},f_{v})=(0,\mathcal{V}_{N}U_{s}^{\prime},0).$
(3.66)

Then we define the $(N+1)$-th step corrector as

$\displaystyle\vec{\Xi}_{N+1}=(\rho_{N+1},u_{N+1},v_{N+1})\overset{\mbox{\tiny{def}}}{=}(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})+(\mathcal{P}_{N+1},\mathcal{U}_{N+1},\mathcal{V}_{N+1}),$
(3.67)

where $(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})$ is the solution
to quasi-compressible system

$\displaystyle\mathfrak{Q}(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})=-\vec{\mathcal{E}}_{N},$
(3.68)

that cancels the error $\vec{\mathcal{E}}_{N}$ (3.66) generated from $N$-th
step, and $(\mathcal{P}_{N+1},\mathcal{U}_{N+1},\mathcal{V}_{N+1})$ solves the
Stokes system

$\displaystyle\mathfrak{S}(\mathcal{P}_{N+1},\mathcal{U}_{N+1},\mathcal{V}_{N+1})=-\vec{E}_{\mathfrak{Q}}(\varrho_{N+1},\mathfrak{u}_{N+1},v_{N+1}),$
(3.69)

with the error operator $\vec{E}_{\mathfrak{Q}}$ being defined in (3.1). One
can check that the new error generated at $(N+1)$-th step is

$\displaystyle\vec{\mathcal{E}}_{N+1}$
$\displaystyle\overset{\mbox{\tiny{def}}}{=}\mathcal{L}\left(\sum_{j=0}^{N+1}\vec{\Xi}_{j}\right)-(0,f_{u},f_{v})$
$\displaystyle=\vec{E}_{\mathfrak{S}}(\mathcal{P}_{N+1},\mathcal{U}_{N+1},\mathcal{V}_{N+1})=(0,\mathcal{V}_{N+1}U_{s}^{\prime},0).$
(3.70)

Thus, the approximate solutions up to any order have been constructed.
Finally, if $\vec{\Xi}=\sum_{j=0}^{\infty}\vec{\Xi}_{j}$ converges, then this
series defines a solution to the resolvent problem (3.1).

Next, if both $f_{u}$ and $f_{v}$ have one order regularity, that is:
$(f_{u},f_{v})\in H^{1}(-1,1)$, then at the zeroth step we can introduce
$(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})$ which solves quasi-
compressible system

$\displaystyle\mathfrak{Q}(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})=(0,f_{u},f_{v}).$
(3.71)

The error generated by $(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})$ is
given by

$\displaystyle\vec{\mathcal{E}}_{-1}$
$\displaystyle\overset{\mbox{\tiny{def}}}{=}\vec{E}_{\mathfrak{Q}}(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})$
$\displaystyle=\left(0,\epsilon\Delta_{k}\left[(U_{s}-c)\varrho_{0}\right]-\epsilon\lambda
ik\text{div}_{k}(\mathfrak{u}_{0},\mathfrak{v}_{0})+\epsilon
U_{s}^{\prime\prime}\varrho_{0},-\epsilon\lambda\partial_{y}\text{div}_{k}(\mathfrak{u}_{0},\mathfrak{v}_{0})\right).$
(3.72)

Since $\vec{\mathcal{E}}_{-1}\in L^{2}(-1,1)$, we can repeat the procedure
(3.59)-(3.3) to construct a solution
$\vec{\Upsilon}=(\tilde{\rho},\tilde{u},\tilde{v})$ to the resolvent problem
$\mathcal{L}(\vec{\Upsilon})=-\vec{\mathcal{E}}_{-1}$. Finally, the solution
to the original resolvent problem (3.1) is given by
$\vec{\Xi}\overset{\mbox{\tiny{def}}}{=}\vec{\Upsilon}+(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})$.

We are now in the position to prove the convergence of iteration and
solvability of the resolvent problem.

Proof of Proposition 3.1: Recall (3.67) the definition of $(N+1)$-step
corrector. First we estimate
$\vec{\Xi}_{N+1}=(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})$ which
solves the quasi-compressible system (3.68) with source term
$s_{1}=\mathcal{V}_{N}U_{s}^{\prime}$ and $s_{2}=0$. To use Corollary 3.1, we
compute

$\displaystyle\Omega(s_{1},s_{2})=\frac{1}{ik}\partial_{y}\left(A^{-1}\mathcal{V}_{N}U_{s}^{\prime}\right)=\frac{1}{ik}\left(\partial_{y}(A^{-1}U_{s}^{\prime})\mathcal{V}_{N}+A^{-1}U_{s}^{\prime}\partial_{y}\mathcal{V}_{N}\right).$

By taking $L^{2}$-norm of $\Omega(s_{1},s_{2})$, using the elementary
inequality (3.33) and the fact
$\partial_{y}\mathcal{V}_{N}=\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})-ik\mathcal{U}_{N}$,
we deduce that

$\displaystyle\|\Omega(s_{1},s_{2})\|_{L^{2}}$ $\displaystyle\leq
Ck^{-1}\left(\|\partial_{y}(A^{-1}U_{s}^{\prime})\|_{L^{2}}\|\mathcal{V}_{N}\|_{L^{\infty}}+\|A^{-1}U_{s}^{\prime}\|_{L^{\infty}}\|\partial_{y}\mathcal{V}_{N}\|_{L^{2}}\right)$
$\displaystyle\leq Ck^{-1}\|\partial_{y}\mathcal{V}_{N}\|_{L^{2}}\leq
C\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+C\|\mathcal{U}_{N}\|_{L^{2}}.$
(3.73)

Similarly, it holds that

$\displaystyle\|s_{1}\|_{L^{2}}\leq C\|\mathcal{V}_{N}\|_{L^{\infty}}\leq
C\left(\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\epsilon^{\frac{1}{7}}\|\mathcal{U}_{N}\|_{L^{2}}\right),$
(3.74)

and

$\displaystyle\|\text{div}_{k}(s_{1},s_{2})\|_{L^{2}}\leq
Ck\|\mathcal{V}_{N}\|_{L^{\infty}}\leq
C\epsilon^{\frac{1}{7}}\left(\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\epsilon^{\frac{1}{7}}\|\mathcal{U}_{N}\|_{L^{2}}\right).$
(3.75)

Then by applying Corollary 3.1 to
$(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})$ and using the bounds
(3.73) to (3.75), we obtain

$\displaystyle\|\mathfrak{u}_{N+1}\|_{H^{1}}+\|(m^{-2}\varrho_{N+1},\mathfrak{v}_{N+1})\|_{H^{2}}+\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{H^{1}}$
$\displaystyle\quad\leq
C\epsilon^{-\frac{2}{7}}\left(\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\|\mathcal{U}_{N}\|_{L^{2}}\right),$
(3.76)

and

$\displaystyle\|\partial_{y}^{2}\mathfrak{u}_{N+1}\|_{L^{2}}\leq
C\epsilon^{-\frac{4}{7}}\left(\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\|\mathcal{U}_{N}\|_{L^{2}}\right).$
(3.77)

Next we estimate $(\mathcal{P}_{N+1},\mathcal{U}_{N+1},\mathcal{V}_{N+1})$
which solves the Stokes system (3.69). Recall the error operator
$\vec{E}_{\mathfrak{Q}}$ defined in (3.1). Then by using the estimate (3.3) we
obtain

$\displaystyle\|\vec{E}_{\mathfrak{Q}}(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{L^{2}}\leq$
$\displaystyle
C\epsilon\bigg{(}\|\varrho_{N+1}\|_{H^{2}}+\|\text{div}_{k}(\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{H^{1}}\bigg{)}$
$\displaystyle\leq$ $\displaystyle
C\epsilon\bigg{(}\|m^{-2}\varrho_{N+1}\|_{H^{2}}+k^{-1}\|\text{div}_{k}(\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{H^{1}}\bigg{)}$
$\displaystyle\leq$ $\displaystyle
C\epsilon^{\frac{5}{7}}\left(\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\|\mathcal{U}_{N}\|_{L^{2}}\right).$
(3.78)

By applying Lemma 3.2 to
$(\mathcal{P}_{N+1},\mathcal{U}_{N+1},\mathcal{V}_{N+1})$ and using the bound
(3.3), we deduce

$\displaystyle\|(m^{-1}\mathcal{P}_{N+1},\mathcal{U}_{N+1},\mathcal{V}_{N+1})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{3}{7}}\|\vec{E}_{\mathfrak{Q}}(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{\frac{2}{7}}\left(\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\|\mathcal{U}_{N}\|_{L^{2}}\right),$
(3.79)
$\displaystyle\epsilon^{-\frac{1}{7}}\left(\|\text{div}_{k}(\mathcal{U}_{N+1},\mathcal{V}_{N+1})\|_{H^{1}}+m^{-2}\|\partial_{y}\mathcal{P}_{N+1}\|_{L^{2}}\right)$
$\displaystyle\leq
C\epsilon^{-\frac{3}{7}}\|\vec{E}_{\mathfrak{Q}}(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{\frac{2}{7}}\left(\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\|\mathcal{U}_{N}\|_{L^{2}}\right),$
(3.80)
$\displaystyle\|\partial_{y}\mathcal{U}_{N+1},k\mathcal{U}_{N+1}\|_{L^{2}}+\|\partial_{y}\mathcal{V}_{N+1},k\mathcal{V}_{N+1}\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{5}{7}}\|\vec{E}_{\mathfrak{Q}}(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{L^{2}}$
$\displaystyle\leq
C\left(\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\|\mathcal{U}_{N}\|_{L^{2}}\right),$
(3.81)
$\displaystyle\|(\Delta_{k}\mathcal{U}_{N+1},\Delta_{k}\mathcal{V}_{N+1})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{9}{7}}\|\vec{E}_{\mathfrak{Q}}(\varrho_{N+1},\mathfrak{u}_{N+1},\mathfrak{v}_{N+1})\|_{L^{2}}$
$\displaystyle\leq
C\epsilon^{-\frac{4}{7}}\left(\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\|\mathcal{U}_{N}\|_{L^{2}}\right).$
(3.82)

Now for $N=0,1,2,\cdots$, we define

$\displaystyle
E_{N}\overset{\mbox{\tiny{def}}}{=}~{}\|(m^{-1}\mathcal{P}_{N},\mathcal{U}_{N},\mathcal{V}_{N})\|_{L^{2}}+\epsilon^{-\frac{1}{7}}\left(\|\text{div}_{k}(\mathcal{U}_{N},\mathcal{V}_{N})\|_{H^{1}}+m^{-2}\|\partial_{y}\mathcal{P}_{N}\|_{L^{2}}\right).$
(3.83)

From (3.79) and (3.80), it holds that $E_{N+1}\leq
C\epsilon^{\frac{2}{7}}E_{N}$. Then by taking $\epsilon\ll 1$ suffciently
small, we have

$\displaystyle\sum_{j=0}^{\infty}E_{j}\leq\sum_{j=0}^{\infty}\left(\frac{1}{2}\right)^{j}E_{0}\leq
CE_{0}.$ (3.84)

For other components, by using the estimates (3.3), (3.77), (3.81) and (3.82),
we obtain that

$\displaystyle\sum_{j=1}^{\infty}\|\mathfrak{u}_{j}\|_{H^{1}}+\|(m^{-2}\varrho_{j},\mathfrak{v}_{j})\|_{H^{2}}+\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathfrak{u}_{j},\mathfrak{v}_{j})\|_{H^{1}}\leq
C\epsilon^{-\frac{2}{7}}\left(\sum_{j=0}^{\infty}E_{j}\right)\leq
C\epsilon^{-\frac{2}{7}}E_{0},$ (3.85)
$\displaystyle\sum_{j=1}^{\infty}\|\partial_{y}^{2}\mathfrak{u}_{j}\|_{L^{2}}\leq
C\epsilon^{-\frac{4}{7}}\left(\sum_{j=0}^{\infty}E_{j}\right)\leq
C\epsilon^{-\frac{4}{7}}E_{0},$ (3.86)
$\displaystyle\sum_{j=1}^{\infty}\|(\partial_{y}\mathcal{U}_{j},\partial_{y}\mathcal{V}_{j})\|_{L^{2}}\leq
C\left(\sum_{j=0}^{\infty}E_{j}\right)\leq CE_{0},$ (3.87)
$\displaystyle\sum_{j=1}^{\infty}\|(\partial_{y}^{2}\mathcal{U}_{j},\partial_{y}^{2}\mathcal{V}_{j})\|_{L^{2}}\leq
C\epsilon^{-\frac{4}{7}}\left(\sum_{j=0}^{\infty}E_{j}\right)\leq
C\epsilon^{-\frac{4}{7}}E_{0}.$ (3.88)

Next we estimate
$\vec{\Xi}_{0}=(\mathcal{P}_{0},\mathcal{U}_{0},\mathcal{V}_{0})$. Since
$\vec{\Xi}_{0}$ solves the Stokes system (3.58), we can apply Lemma 3.2 to
$\vec{\Xi}_{0}$ with $q_{0}=0$, $q_{1}=f_{u}$, and $q_{2}=f_{v}$. Thus, it
holds that

$\displaystyle E_{0}$ $\displaystyle\leq
C\epsilon^{-\frac{3}{7}}\|(f_{u},f_{v})\|_{L^{2}},$ (3.89)
$\displaystyle\|\partial_{y}\mathcal{U}_{0},k\mathcal{U}_{0}\|_{L^{2}}+\|\partial_{y}\mathcal{V}_{0},k\mathcal{V}_{0}\|_{L^{2}}$
$\displaystyle\leq C\epsilon^{-\frac{5}{7}}\|(f_{u},f_{v})\|_{L^{2}},$ (3.90)
$\displaystyle\|(\partial_{y}^{2}\mathcal{U}_{0},\partial_{y}^{2}\mathcal{V}_{0})\|_{L^{2}}$
$\displaystyle\leq C\epsilon^{-\frac{9}{7}}\|(f_{u},f_{v})\|_{L^{2}}.$ (3.91)

Combining estimates (3.84)-(3.91) together gives

$\displaystyle\|(m^{-1}\rho,u,v)\|_{L^{2}}$
$\displaystyle\leq\sum_{j=0}^{\infty}\|(m^{-1}\mathcal{P}_{j},\mathcal{U}_{j},\mathcal{V}_{j})\|_{L^{2}}+\sum_{j=1}^{\infty}\|(m^{-1}\varrho_{j},\mathfrak{u}_{j},\mathfrak{v}_{j})\|_{L^{2}}$
$\displaystyle\leq C\left(1+\epsilon^{-\frac{2}{7}}\right)E_{0}\leq
C\epsilon^{-\frac{5}{7}}\|(f_{u},f_{v})\|_{L^{2}},$ (3.92)
$\displaystyle\|m^{-2}\partial_{y}\rho\|_{L^{2}}+\|\text{div}_{k}(u,v)\|_{L^{2}}$
$\displaystyle\leq\epsilon^{\frac{1}{7}}\left(\sum_{j=0}^{\infty}E_{j}\right)+\sum_{j=1}^{\infty}\left(\|m^{-2}\partial_{y}\varrho_{j}\|_{L^{2}}+\|\text{div}_{k}(\mathfrak{u}_{j},\mathfrak{v}_{j})\|_{L^{2}}\right)$
$\displaystyle\leq C(1+\epsilon^{-\frac{2}{7}})E_{0}\leq
C\epsilon^{-\frac{5}{7}}\|(f_{u},f_{v})\|_{L^{2}},$ (3.93)
$\displaystyle\|(\partial_{y}u,\partial_{y}v)\|_{L^{2}}$
$\displaystyle\leq\sum_{j=0}^{\infty}\|(\partial_{y}\mathcal{U}_{j},\partial_{y}\mathcal{V}_{j})\|_{L^{2}}+\sum_{j=1}^{\infty}\|(\partial_{y}\mathfrak{u}_{j},\partial_{y}\mathfrak{v}_{j})\|_{L^{2}}$
$\displaystyle\leq
C\left(1+\epsilon^{-\frac{2}{7}}\right)E_{0}+C\epsilon^{-\frac{5}{7}}\|(f_{u},f_{v})\|_{L^{2}}\leq
C\epsilon^{-\frac{5}{7}}\|(f_{u},f_{v})\|_{L^{2}},$ (3.94)
$\displaystyle\|(\partial_{y}^{2}u,\partial_{y}^{2}v)\|_{L^{2}}$
$\displaystyle\leq\sum_{j=0}^{\infty}\|(\partial_{y}^{2}\mathcal{U}_{j},\partial_{y}^{2}\mathcal{V}_{j})\|_{L^{2}}+\sum_{j=1}^{\infty}\|(\partial_{y}^{2}\mathfrak{u}_{j},\partial_{y}^{2}\mathfrak{v}_{j})\|_{L^{2}}$
$\displaystyle\leq
C\left(1+\epsilon^{-\frac{4}{7}}\right)E_{0}+C\epsilon^{-\frac{9}{7}}\|(f_{u},f_{v})\|_{L^{2}}\leq
C\epsilon^{-\frac{9}{7}}\|(f_{u},f_{v})\|_{L^{2}}.$ (3.95)

By putting (3.92)-(3.95) together, the inequality (3.2) is obtained.

Finally, we prove the improved estimate (3.4). Suppose that $f_{u}$ and
$f_{v}\in H^{1}(-1,1)$. As mentioned before, we look for a solution in the
form of

$(\rho,u,v)=(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})+(\tilde{\rho},\tilde{u},\tilde{v}),$

where $(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})$ solves the quasi-
compressible system (3.71) that generates an error $\vec{\mathcal{E}}_{-1}$
defined in (3.3), and $(\tilde{\rho},\tilde{u},\tilde{v})$ solves the
resolvent problem
$\mathcal{L}(\tilde{\rho},\tilde{u},\tilde{v})=-\vec{\mathcal{E}}_{-1}$. Now
we estimate $(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})$. Use Corollary
3.1 to have

$\displaystyle\|\mathfrak{u}_{0}\|_{H^{1}}$
$\displaystyle+\|(m^{-2}\varrho_{0},\mathfrak{v}_{0})\|_{H^{2}}+\epsilon^{-\frac{1}{7}}\|\text{div}_{k}(\mathfrak{u}_{0},\mathfrak{v}_{0})\|_{H^{1}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|(f_{u},f_{v})\|_{L^{2}}+C\|\text{div}_{k}(f_{u},f_{v})\|_{L^{2}},$
(3.96)

and

$\displaystyle\|\partial^{2}_{y}\mathfrak{u}_{0}\|_{L^{2}}\leq
C\epsilon^{-\frac{4}{7}}\|\Omega(f_{u},f_{v})\|_{L^{2}}+\epsilon^{-\frac{1}{7}}\|(f_{u},f_{v})\|_{L^{2}}+C\|\text{div}_{k}(f_{u},f_{v})\|_{L^{2}}.$
(3.97)

Moreover, to bound $(\tilde{\rho},\tilde{u},\tilde{v})$, we first observe that

$\displaystyle\|\vec{\mathcal{E}}_{-1}\|_{L^{2}}$
$\displaystyle\leq\|\vec{E}_{\mathfrak{Q}}(\varrho_{0},\mathfrak{u}_{0},\mathfrak{v}_{0})\|_{L^{2}}\leq
C\epsilon\|\varrho_{0}\|_{H^{2}}+C\epsilon\|\text{div}_{k}(\mathfrak{u}_{0},\mathfrak{v}_{0})\|_{H^{1}}$
$\displaystyle\leq
C\epsilon\left(\epsilon^{-\frac{2}{7}}\|\Omega(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|(f_{u},f_{v})\|_{L^{2}}+C\|\text{div}_{k}(f_{u},f_{v})\|_{L^{2}}\right),$
(3.98)

where we have used (3.3) in the last inequality. Then by applying (3.2) to
$(\tilde{\rho},\tilde{u},\tilde{v})$, and using the bound (3.98) on source
$\vec{\mathcal{E}}_{-1}$, we obtain that

$\displaystyle\|(m^{-1}\tilde{\rho},\tilde{u},\tilde{v})\|_{L^{2}}+\|(m^{-2}\partial_{y}\tilde{\rho},\partial_{y}\tilde{u},\partial_{y}\tilde{v})\|_{L^{2}}+\epsilon^{\frac{4}{7}}\|(\partial_{y}^{2}\tilde{\mathfrak{u}},\partial_{y}^{2}\tilde{\mathfrak{v}})\|_{L^{2}}$
$\displaystyle\qquad\leq
C\epsilon^{-\frac{5}{7}}\|\vec{\mathcal{E}}_{-1}\|_{L^{2}}\leq
C\|\Omega(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{\frac{1}{7}}\|(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{\frac{2}{7}}\|\text{div}_{k}(f_{u},f_{v})\|_{L^{2}}.$
(3.99)

Putting (3.3), (3.97) and (3.3) together gives

$\displaystyle\|(m^{-1}\rho,u,v)\|_{L^{2}}+\|(m^{-2}\partial_{y}\rho,\partial_{y}u,\partial_{y}v)\|_{L^{2}}$
$\displaystyle\qquad\leq
C\epsilon^{-\frac{2}{7}}\|\Omega(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{-\frac{1}{7}}\|(f_{u},f_{v})\|_{L^{2}}+C\|\text{div}_{k}(f_{u},f_{v})\|_{L^{2}},$
(3.100)

and

$\displaystyle\|(\partial_{y}^{2}u,\partial_{y}^{2}v)\|_{L^{2}}\leq
C\epsilon^{-\frac{4}{7}}\|\Omega(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{-\frac{3}{7}}\|(f_{u},f_{v})\|_{L^{2}}+C\epsilon^{-\frac{2}{7}}\|\text{div}_{k}(f_{u},f_{v})\|_{L^{2}}.$
(3.101)

Combining (3.100) and (3.101) together implies (3.4). The analyticity can be
proved as in [50]. We omit the details for brevity. Therefore, the proof of
the proposition is completed. ∎

##  4 Dispersion relation

Recall in Section 2.1 and 2.2 that we have four independent approximate
solutions:

$\vec{\Xi}^{s}_{\pm,\text{app}}\text{ and }\vec{\Xi}^{f}_{\pm,\text{app}}.$

Here $\vec{\Xi}^{s}_{\pm,\text{app}}$ are inviscid modes which are determined
in terms of solutions $\varphi^{s}_{\pm}$ to the Lees-Lin equation (2.5), and
$\vec{\Xi}^{f}_{\pm,\text{app}}$ are viscous modes which are boundary layers
at $y=\pm 1$. Using Proposition 3.1, we can construct four exact independent
solutions to (1.6) near $\vec{\Xi}^{s}_{\pm,\text{app}}$ and
$\vec{\Xi}^{f}_{\pm,\text{app}}.$

######  Proposition 4.1.

The eigenvalue problem (1.6) admits four solutions, that is,
$\vec{\Xi}^{s}_{\pm}=(\rho_{\pm}^{s},u_{\pm}^{s},v_{\pm}^{s})$ and
$\vec{\Xi}^{f}_{\pm}=(\rho_{\pm}^{f},u_{\pm}^{f},v_{\pm}^{f})$. Moreover,
their boundary values satisfy the following asymptotic properties:

$\displaystyle v_{+}^{s}(-1)$
$\displaystyle=ik\left(c-\tau{k^{2}}\right)+O(1)\epsilon^{\frac{5}{7}}|\log\epsilon|,\text{
with }\tau=\frac{1}{U_{s}^{\prime}(-1)}\int_{-1}^{1}U_{s}^{2}(x){\rm d}x,$
(4.1) $\displaystyle v_{+}^{s}(1)$
$\displaystyle=ikc+O(1)\epsilon^{\frac{5}{7}},$ (4.2) $\displaystyle
v_{-}^{s}(\pm 1)$ $\displaystyle=\frac{ik}{U_{s}^{\prime}(\pm
1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|,$ (4.3) $\displaystyle
u_{+}^{s}(\pm 1)$ $\displaystyle=U_{s}^{\prime}(\pm
1)+O(1)\epsilon^{\frac{1}{7}},$ (4.4) $\displaystyle u_{-}^{s}(\pm 1)$
$\displaystyle=O(1)|\log\epsilon|,$ (4.5) $\displaystyle u_{+}^{f}(1)$
$\displaystyle=\frac{\text{Ai}(1,\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})}+O(1)\epsilon^{\frac{1}{7}},~{}v_{+}^{f}(1)=ik\tilde{\delta},~{}v_{\pm}^{f}(-1)=O(1)\epsilon^{\infty},~{}u_{\pm}^{f}(-1)=O(1)\epsilon^{\frac{1}{7}},$
(4.6) $\displaystyle u_{-}^{f}(-1)$
$\displaystyle=-\frac{\text{Ai}(1,z_{0})}{\text{Ai}(2,z_{0})}+O(1)\epsilon^{\frac{1}{7}},~{}v_{-}^{f}(-1)=ik\delta,~{}v_{-}^{f}(1)=O(1)\epsilon^{\infty},~{}u_{-}^{f}(1)=O(1)\epsilon^{\frac{1}{7}}.$
(4.7)

Here $z_{0}$ and $\tilde{z}_{0}$ are defined in (2.45) and (2.53)
respectively.

###### Proof.

First we construct $\vec{\Xi}_{+}^{s}$. We look for the solution in form of

$\displaystyle\vec{\Xi}_{+}^{s}=\vec{\Xi}_{+,\text{app}}^{s}+\vec{\Xi}_{+,r}^{s},$

where the approximate solution $\vec{\Xi}_{+,\text{app}}^{s}$ is defined in
(2.28)-(2.30). Recall the error function $\vec{E}_{+}^{s}$ in (2.3) and
decomposition $\vec{E}_{+}^{s}=\vec{E}_{+,1}^{s}+\vec{E}_{+,2}^{s}$ in (2.60).
Then we decompose the remainder accordingly into:
$\vec{\Xi}_{+,r}^{s}=\vec{\Xi}_{+,r,1}^{s}+\vec{\Xi}_{+,r,2}^{s}$. Here,
$\vec{\Xi}_{+,r,j}^{s}=(\rho_{+,r,j}^{s},u_{+,r,j}^{s},v_{+,r,j}^{s})$
satisfies

$\displaystyle\mathcal{L}(\vec{\Xi}_{+,r,j}^{s})=-\vec{E}_{+,j}^{s},~{}~{}v_{+,r,j}^{s}\big{|}_{y=\pm
1}=0,~{}j=1,2.$ (4.8)

The solvability of $\vec{\Xi}_{+,r,j}^{s}$ is guaranteed by Proposition 3.1.
By using (2.61) and (3.2), we obtain

$\displaystyle\left|u_{+,r,1}^{s}(\pm
1)\right|\leq\|u_{+,r,1}^{s}\|_{H^{1}}\leq
C\epsilon^{-\frac{5}{7}}\|\vec{E}_{+,1}^{s}\|_{L^{2}}\leq
C\epsilon^{\frac{1}{7}},$ (4.9)

and

$\displaystyle\left|u_{+,r,2}^{s}(\pm 1)\right|\leq\|u_{+,r,2}^{s}\|_{H^{1}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\left(\|\Omega(\vec{E}_{+,2}^{s})\|_{L^{2}}+\epsilon^{\frac{1}{7}}\|\vec{E}_{+,2}^{s}\|_{L^{2}}+\epsilon^{\frac{2}{7}}\|\text{div}_{k}(\vec{E}_{+,2}^{s})\|_{L^{2}}\right)$
$\displaystyle\leq C\epsilon^{-\frac{2}{7}}\|\vec{E}_{+,2}^{s}\|_{H^{1}}\leq
C\epsilon^{\frac{2}{7}}.$ (4.10)

From (2.31), (2.32), (2.34), (4.8), (4.9) and (4.10), it holds that

$\displaystyle v_{+}^{s}(-1)$
$\displaystyle=v_{+,\text{app}}^{s}(-1)=ik(c-\tau
k^{2})+O(1)\epsilon^{\frac{5}{7}}|\log\epsilon|,$ $\displaystyle v_{+}^{s}(1)$
$\displaystyle=v_{+,\text{app}}^{s}(1)=ikc+O(1)\epsilon^{\frac{5}{7}},$
$\displaystyle u_{+}^{s}(\pm 1)$ $\displaystyle=u_{+,\text{app}}^{s}(\pm
1)+u_{+,r,1}^{s}(\pm 1)+u_{+,r,2}^{s}(\pm 1)=U_{s}^{\prime}(\pm
1)+O(1)\epsilon^{\frac{1}{7}}.$

Thus we have shown the asymptotic properties given in (4.1), (4.2) and (4.4)
for the boundary data of $\vec{\Xi}_{+}^{s}$.

For $\vec{\Xi}_{-}^{s}$, we recall the approximate solution
$\vec{\Xi}_{-,\text{app}}^{s}$ defined in (2.28)-(2.30). This approximation
generates an error $\vec{E}_{-}^{s}$ which is given in (2.3). To get rid of
the error, it is natural to seek the solution $\vec{\Xi}_{-}^{s}$ in the
following form

$\displaystyle\vec{\Xi}_{-}^{s}=\vec{\Xi}_{-,\text{app}}^{s}+\vec{\Xi}_{-,r}^{s},$

where the remainder
$\vec{\Xi}_{-,r}^{s}=(\rho_{-,r}^{s},u_{-,r}^{s},v_{-,r}^{s})$ satisfies

$\displaystyle\mathcal{L}(\vec{\Xi}_{-,r}^{s})=-\vec{E}_{-}^{s},~{}~{}v_{-}^{s}\big{|}_{y=\pm
1}=0.$ (4.11)

By using (3.4) and (2.62), we obtain that

$\displaystyle\left|u_{-,r}^{s}(\pm 1)\right|\leq C\|u_{-,r}^{s}\|_{H^{1}}$
$\displaystyle\leq
C\epsilon^{-\frac{2}{7}}\left(\|\Omega(\vec{E}_{-}^{s})\|_{L^{2}}+\epsilon^{\frac{1}{7}}\|\vec{E}_{-}^{s}\|_{L^{2}}+\epsilon^{\frac{2}{7}}\|\text{div}_{k}(\vec{E}_{-}^{s})\|_{L^{2}}\right)$
$\displaystyle\leq C\epsilon^{-\frac{2}{7}}\|\vec{E}_{-}^{s}\|_{H^{1}}\leq C.$
(4.12)

Then by (2.33), (4.11) and (4.12), we have

$\displaystyle v_{-}^{s}(\pm 1)=v_{-,\text{app}}^{s}(\pm
1)=\frac{ik}{U_{s}^{\prime}(\pm 1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|,$

and

$\displaystyle u_{-}^{s}(\pm 1)=u_{-,\text{app}}^{s}(\pm 1)+u_{-,r}^{s}(\pm
1)=O(1)|\log\epsilon|.$

This completes the proof of (4.3) and (4.5).

Finally we construct the viscous modes $\vec{\Xi}_{\pm}^{f}$ near the boundary
layer profiles $\vec{\Xi}_{\pm,\text{app}}^{f}$ defined in (2.50) and (2.52)
respectively. Without loss of generality, we consider $\vec{\Xi}_{+}^{f}$.
Recall the error function $\vec{E}_{+}^{f}$ given in (2.3). We look for the
solution in form of

$\displaystyle\vec{\Xi}_{+}^{f}=\vec{\Xi}_{+,\text{app}}^{f}+\vec{\Xi}_{+,r}^{f},$

where $\vec{\Xi}_{+,r}^{f}$ solves the following resolvent problem:

$\displaystyle\mathcal{L}(\vec{\Xi}_{+,r}^{f})=-\vec{E}_{+}^{f},~{}~{}v_{+,r}^{f}\big{|}_{y=\pm
1}=0.$ (4.13)

Using (2.63) and (3.2), we can obtain that

$\displaystyle|u_{+,r}^{f}(\pm 1)|\leq C\|u_{+,r}^{f}\|_{H^{1}}\leq
C\epsilon^{-\frac{5}{7}}\|\vec{E}_{+}^{f}\|_{L^{2}}\leq
C\epsilon^{\frac{1}{7}}.$ (4.14)

Then from (2.56), (4.13) and (4.14), it holds that

$\displaystyle
v_{+}^{f}(1)=v_{+,\text{app}}^{f}(1)=ik\tilde{\delta},~{}v_{+}^{f}(-1)=v_{+,\text{app}}^{f}(-1)=O(1)\epsilon^{\infty},$
$\displaystyle
u_{+}^{f}(1)=u_{+,\text{app}}^{f}(1)+u_{+,r}^{f}(1)=\frac{\text{Ai}(1,\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})}+O(1)\epsilon^{\frac{1}{7}},$
$\displaystyle
u_{+}^{f}(-1)=u_{+,\text{app}}^{f}(-1)+u_{+,r}^{f}(-1)=O(1)\epsilon^{\frac{1}{7}}.$

Combining these estimates together yields (4.6). The asymptotic expansion
(4.7) for $\vec{\Xi}_{-}^{f}$ can be proved similarly. Therefore, the proof of
the proposition is completed. ∎

To match exact boundary conditions at $y=\pm 1$, we construct a solution
$\vec{\Xi}$ to the eigenvalue problem (1.6) by the following linear
combination of $\vec{\Xi}^{s}_{\pm}$ and $\vec{\Xi}^{f}_{\pm}:$

$\displaystyle\vec{\Xi}=(\rho,u,v)\overset{\mbox{\tiny{def}}}{=}\vec{\Xi}^{s}_{+}+\alpha_{1}(c)\vec{\Xi}^{s}_{-}+\alpha_{2}(c)\vec{\Xi}^{f}_{-}+\alpha_{3}(c)\vec{\Xi}^{f}_{+},$
(4.15)

where $\alpha_{1},\alpha_{2},\alpha_{3}$ are constants. Now we choose
$\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ so that the following three
boundary conditions are satisfied:

$\displaystyle v(-1)$
$\displaystyle=v^{s}_{+}(-1)+\alpha_{1}v^{s}_{-}(-1)+\alpha_{2}v^{f}_{-}(-1)+\alpha_{3}v^{f}_{+}(-1)=0,$
(4.16) $\displaystyle u(-1)$
$\displaystyle=u^{s}_{+}(-1)+\alpha_{1}u^{s}_{-}(-1)+\alpha_{2}u^{f}_{-}(-1)+\alpha_{3}u^{f}_{+}(-1)=0,$
(4.17) $\displaystyle v(1)$
$\displaystyle=v^{s}_{+}(1)+\alpha_{1}v^{s}_{-}(1)+\alpha_{2}v^{f}_{-}(1)+\alpha_{3}v^{f}_{+}(1)=0.$
(4.18)

This reduces to solve the following algebraic system:

$\mathcal{M}(\alpha_{1},\alpha_{2},\alpha_{3})^{T}=-\bigg{(}v^{s}_{+}(-1),u^{s}_{+}(-1),v^{s}_{+}(1)\bigg{)}.$

where the coefficient matrix $\mathcal{M}$ is given by:

$\mathcal{M}=\left(\begin{aligned}
v^{s}_{-}(-1)\quad&&v^{f}_{-}(-1)\quad&&v^{f}_{+}(-1)\\\
u^{s}_{-}(-1)\quad&&u^{f}_{-}(-1)\quad&&u^{f}_{+}(-1)\\\
v^{s}_{-}(1)\quad&&v^{f}_{-}(1)\quad&&v^{f}_{+}(1)\end{aligned}\right).$

The following lemma gives the invertibility of $\mathcal{M}$ and asymptotic
formula of coefficients $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ for
$\epsilon\ll 1$. Denote
$\beta=\frac{U_{s}^{\prime}(-1)}{\left|U_{s}^{\prime}(1)\right|}$.

######  Lemma 4.1.

For any $\epsilon\ll 1$, the matrix $\mathcal{M}$ is invertible. Moreover, the
coefficients $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ in (4.16)-(4.18) have
the following asymptotic expansions:

$\displaystyle\alpha_{1}$
$\displaystyle=-\delta\left[U_{s}^{\prime}(-1)\right]^{2}\frac{\text{Ai}(2,z_{0})}{\text{Ai}(1,z_{0})}-U_{s}^{\prime}(-1)(c-\tau
k^{2})+O(1)\epsilon^{\frac{3}{7}},$ (4.19) $\displaystyle\alpha_{2}$
$\displaystyle=U_{s}^{\prime}(-1)\frac{\text{Ai}(2,z_{0})}{\text{Ai}(1,z_{0})}+O(1)\epsilon^{\frac{1}{7}},$
(4.20) $\displaystyle\alpha_{3}$
$\displaystyle=-\delta^{-1}\left[(1+\beta)\beta^{-\frac{1}{3}}c-\tau\beta^{\frac{2}{3}}k^{2}\right]-U_{s}^{\prime}(-1)\beta^{\frac{2}{3}}\frac{\text{Ai}(2,z_{0})}{\text{Ai}(1,z_{0})}+O(1)\epsilon^{\frac{1}{7}}.$
(4.21)

###### Proof.

By using asymptotic expansions of boundary values in Proposition 4.1, we have

$\mathcal{M}=\left(\begin{aligned}
\frac{ik}{U_{s}^{\prime}(-1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|\quad&&ik\delta\quad&&O(1)\epsilon^{\infty}\\\
O(1)|\log\epsilon|\quad&&-\frac{\text{Ai}(1,z_{0})}{\text{Ai}(2,z_{0})}+O(1)\epsilon^{\frac{1}{7}}\quad&&O(1)\epsilon^{\frac{1}{7}}\\\
\frac{ik}{U_{s}^{\prime}(1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|\quad&&O(1)\epsilon^{\infty}\quad&&ik\tilde{\delta}\end{aligned}\right).$

Then we compute the determinant of $\mathcal{M}$ as follows.

$\displaystyle\det\mathcal{M}=$ $\displaystyle
ik\tilde{\delta}\cdot\det\left(\begin{aligned}
\frac{ik}{U_{s}^{\prime}(-1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|\quad&&ik\delta\\\
O(1)|\log\epsilon|\quad&&-\frac{\text{Ai}(1,z_{0})}{\text{Ai}(2,z_{0})}+O(1)\epsilon^{\frac{1}{7}}\end{aligned}\right)$
$\displaystyle+O(1)\epsilon^{\frac{1}{7}}\cdot\det\left(\begin{aligned}
\frac{ik}{U_{s}^{\prime}(-1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|\quad&&ik\delta\\\
\frac{ik}{U_{s}^{\prime}(1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|\quad&&O(1)\epsilon^{\infty}\end{aligned}\right)+O(1)\epsilon^{\infty}$
$\displaystyle=$
$\displaystyle\frac{k^{2}\tilde{\delta}}{U_{s}^{\prime}(-1)}\left(\frac{\text{Ai}(1,z_{0})}{\text{Ai}(2,z_{0})}-O(1)\epsilon^{\frac{1}{7}}|\log\epsilon|\right)\sim
k^{2}\tilde{\delta}\neq 0.$

Therefore, the matrix $\mathcal{M}$ is invertible and we can solve

$(\alpha_{1},\alpha_{2},\alpha_{3})=-\mathcal{M}^{-1}\bigg{(}v^{s}_{+}(-1),u^{s}_{+}(-1),v^{s}_{+}(1)\bigg{)}.$

To study the asymptotic properties of $\alpha_{1},\alpha_{2}$ and
$\alpha_{3}$, we substitute (4.1)-(4.7) into (4.16)-(4.18) and obtain that

$\displaystyle\left(\frac{ik}{U_{s}^{\prime}(-1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|\right)\alpha_{1}+ik\delta\alpha_{2}+O(1)\epsilon^{\infty}\alpha_{3}=-\left[ik\left(c-\tau
k^{2}\right)+O(1)\epsilon^{\frac{5}{7}}|\log\epsilon|\right],$ (4.22)
$\displaystyle
O(1)|\log\epsilon|\alpha_{1}+\left(-\frac{\text{Ai}(1,z_{0})}{\text{Ai}(2,z_{0})}+O(1)\epsilon^{\frac{1}{7}}\right)\alpha_{2}+O(1)\epsilon^{\frac{1}{7}}\alpha_{3}=-U_{s}^{\prime}(-1)+O(1)\epsilon^{\frac{1}{7}},$
(4.23)
$\displaystyle\left(\frac{ik}{U_{s}^{\prime}(1)}+O(1)\epsilon^{\frac{3}{7}}|\log\epsilon|\right)\alpha_{1}+O(1)\epsilon^{\infty}\alpha_{2}+ik\tilde{\delta}\alpha_{3}=-\left(ikc+O(1)\epsilon^{\frac{5}{7}}\right).$
(4.24)

Recall that $k\sim\epsilon^{\frac{1}{7}}$,
$\delta\sim|c|\sim\epsilon^{\frac{2}{7}}$. From (4.22)-(4.24), we can see that
$\alpha_{1}\sim\epsilon^{\frac{2}{7}}$, $\alpha_{2}\sim O(1)$ and
$\alpha_{3}\sim O(1)$. Then by taking the leading order of both sides of
equations in (4.22)-(4.24), and using
$\tilde{\delta}=\beta^{\frac{1}{3}}\delta$, we obtain

$\displaystyle\alpha_{1}$
$\displaystyle=-\delta[U_{s}^{\prime}(-1)]^{2}\frac{\text{Ai}(2,z_{0})}{\text{Ai}(1,z_{0})}-U_{s}^{\prime}(-1)(c-\tau
k^{2})+O(1)\epsilon^{\frac{3}{7}},$ (4.25) $\displaystyle\alpha_{2}$
$\displaystyle=U_{s}^{\prime}(-1)\frac{\text{Ai}(2,z_{0})}{\text{Ai}(1,z_{0})}+O(1)\epsilon^{\frac{1}{7}},$
(4.26) $\displaystyle\alpha_{3}$
$\displaystyle=-\delta^{-1}\left[(1+\beta)\beta^{-\frac{1}{3}}c-\tau\beta^{\frac{2}{3}}k^{2}\right]-U_{s}^{\prime}(-1)\beta^{\frac{2}{3}}\frac{\text{Ai}(2,z_{0})}{\text{Ai}(1,z_{0})}+O(1)\epsilon^{\frac{1}{7}}.$
(4.27)

This completes the proof of the lemma. ∎

To match the last boundary condition, that is, $u(y;c)|_{y=1}=0$, we introduce
the function:

$\displaystyle\text{I}(c)\overset{\mbox{\tiny{def}}}{=}u(1;c)=u^{s}_{+}(1;c)+\alpha_{1}(c)u^{s}_{-}(1;c)+\alpha_{2}(c)u^{f}_{-}(1;c)+\alpha_{3}(c)u^{f}_{+}(1;c),$

where constants $\alpha_{1},\alpha_{2}$ and $\alpha_{3}$ are determined in
Lemma 4.1. It is obvious that if $c$ is a zero point of $\text{I}(c)$, then

$\displaystyle\vec{\Xi}(y;c)=\vec{\Xi}^{s}_{+}(y;c)+\alpha_{1}\vec{\Xi}^{s}_{-}(y;c)+\alpha_{2}\vec{\Xi}^{f}_{-}(y;c)+\alpha_{3}\vec{\Xi}^{f}_{+}(y;c)$
(4.28)

defines a solution to the eigenvalue problem (1.6) with desired boundary
conditions (1.7).

We are now ready to prove the main result in this paper.

Proof of Theorem 1.1: Let $k=T_{0}\epsilon^{\frac{1}{7}}$ where the constant
$T_{0}>1$ is a sufficiently large. We look for the zero point of $\text{I}(c)$
in the disk

$\displaystyle D=\bigg{\\{}c\in\mathbb{C}~{}\bigg{|}|c-c_{0}|\leq
T^{-2}_{0}\epsilon^{\frac{2}{7}}\bigg{\\}}$ (4.29)

centered at

$\displaystyle c_{0}=\frac{\tau\beta
T_{0}^{2}}{1+\beta}\epsilon^{\frac{2}{7}}+T_{0}^{-\frac{3}{2}}e^{\frac{1}{4}\pi
i}\frac{\left|U_{s}^{\prime}(1)\right|^{\frac{1}{2}}(1+\beta^{\frac{3}{2}})}{\tau^{\frac{1}{2}}\beta^{\frac{1}{2}}(1+\beta)^{\frac{1}{2}}}\epsilon^{\frac{2}{7}}.$
(4.30)

From Proposition 4.1 and (4.19)-(4.21) in Lemma 4.1, we have

$\displaystyle\text{I}(c)=$ $\displaystyle
U_{s}^{\prime}(1)-U_{s}^{\prime}(-1)\beta^{\frac{2}{3}}\frac{\text{Ai}(2,z_{0})}{\text{Ai}(1,z_{0})}\frac{\text{Ai}(1,\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})}$
$\displaystyle-\delta^{-1}\left[(1+\beta)\beta^{-\frac{1}{3}}c-\tau\beta^{\frac{2}{3}}k^{2}\right]\frac{\text{Ai}(1,\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})}+O(1)\epsilon^{\frac{1}{7}},$
(4.31)

where$z_{0}$ and $\tilde{z}_{0}$ are defined in (2.45) and (2.53)
respectively. Next we compute the leading order terms of
$\frac{\text{Ai}(1,\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})}$ and
$\frac{\text{Ai}(2,{z}_{0})}{\text{Ai}(1,{z}_{0})}$ as follows. By taking
$T_{0}>1$ sufficiently large, we can find a positive constant $b_{0}$
independent of $\epsilon$ and $T_{0}$, such that for any $c\in D$, it holds

$\displaystyle\text{Im}c\geq
b_{0}T_{0}^{-\frac{3}{2}}\epsilon^{\frac{2}{7}},~{}\arg
c\in\left(0,b_{0}T_{0}^{-\frac{7}{2}}\right),~{}|c|=\frac{\tau\beta
T_{0}^{2}}{1+\beta}\epsilon^{\frac{2}{7}}\left(1+O(1)T_{0}^{-\frac{7}{2}}\right).$
(4.32)

From (2.51), we have

$\displaystyle\tilde{z}_{0}=-\tilde{\delta}^{-1}c=-e^{\frac{1}{6}\pi
i}\left|U_{s}^{\prime}(1)\right|^{\frac{1}{3}}T_{0}^{\frac{1}{3}}\epsilon^{-\frac{2}{7}}c.$

If $c$ is in the disk $D$, then from (4.32) we can obtain

$\displaystyle|\tilde{z}_{0}|=\frac{\tau\beta\left|U_{s}^{\prime}(1)\right|^{\frac{1}{3}}}{1+\beta}T_{0}^{\frac{7}{3}}\left(1+O(1)T_{0}^{-\frac{7}{2}}\right),~{}-\frac{5}{6}\pi<\arg
z_{0}<-\frac{5}{6}\pi+b_{0}T_{0}^{-\frac{7}{2}}.$

Then from the asymptotic behavior of the Airy profile (cf. [22, 23]), it holds
that

$\displaystyle\frac{\text{Ai}(1,\tilde{z}_{0})}{\text{Ai}(2,\tilde{z}_{0})}=-\tilde{z}_{0}^{\frac{1}{2}}+O(1)|\tilde{z}_{0}|^{-1}=-e^{-\frac{5}{12}\pi
i}\frac{\tau^{\frac{1}{2}}\beta^{\frac{1}{2}}\left|U_{s}^{\prime}(1)\right|^{\frac{1}{6}}}{(1+\beta)^{\frac{1}{2}}}T_{0}^{\frac{7}{6}}+O(1)T_{0}^{-\frac{7}{3}}.$
(4.33)

Similarly, we have

$\displaystyle\frac{\text{Ai}(2,{z}_{0})}{\text{Ai}(1,{z}_{0})}=\frac{-e^{\frac{5}{12}\pi
i}(1+\beta)^{\frac{1}{2}}}{\tau^{\frac{1}{2}}\beta^{\frac{1}{2}}\left[U_{s}^{\prime}(-1)\right]^{\frac{1}{6}}}T_{0}^{-\frac{7}{6}}\left(1+O(1)T_{0}^{-\frac{7}{2}}\right).$
(4.34)

Plugging the asymptotic expansion (4.33) and (4.34) into (4.31), we deduce
that

$\displaystyle\text{I}(c)=\text{I}_{\text{lin}}(c)+O(1)\left(T_{0}^{-\frac{7}{2}}+\epsilon^{\frac{1}{7}}\right),$

where the linear function

$\displaystyle\text{I}_{\text{lin}}(c)=U_{s}^{\prime}(1)-U_{s}^{\prime}(-1)\beta^{\frac{1}{2}}+e^{-\frac{1}{4}\pi
i}\frac{\tau^{\frac{1}{2}}\left|U_{s}^{\prime}(1)\right|^{\frac{1}{2}}\beta^{\frac{5}{6}}}{(1+\beta)^{\frac{1}{2}}}T_{0}^{\frac{3}{2}}\left[(1+\beta)\beta^{-\frac{1}{3}}c-\tau\beta^{\frac{2}{3}}k^{2}\right].$

By using the same argument as in [50], one can show that the function
$\text{I}(c)$ is analytic in the disk $D$. Moreover, one can check that
$c_{0}$ given in (4.30) is the unique zero point of linear function
$\text{I}_{\text{lin}}(c)$. On the circle $\partial D$, we have

$\displaystyle\left|\text{I}_{\text{lin}}(c)\right|=\tau^{\frac{1}{2}}\left|U_{s}^{\prime}(1)\right|^{\frac{1}{2}}\beta^{\frac{1}{2}}(1+\beta)^{\frac{1}{2}}T_{0}^{-\frac{1}{2}},$

and

$\displaystyle\left|\text{I}(c)-\text{I}_{\text{lin}}(c)\right|\leq
C\left(T_{0}^{-\frac{7}{2}}+\epsilon^{\frac{1}{7}}\right)\leq\frac{1}{2}\tau^{\frac{1}{2}}\left|U_{s}^{\prime}(1)\right|^{\frac{1}{2}}\beta^{\frac{1}{2}}(1+\beta)^{\frac{1}{2}}T_{0}^{-\frac{1}{2}}\leq\frac{1}{2}\left|\text{I}_{\text{lin}}(c)\right|.$

By Rouché’s Theorem, the function $\text{I}(c)$ and $\text{I}_{\text{lin}}(c)$
have the same number of zero point in $D$. Thus, there exists a unique $c\in
D$, such that (4.28) defines a solution of eigenvalue problem (1.6) with
boundary conditions (1.7). Therefore, the proof of Theorem 1.1 is completed.

##  5 Appendices

First of all, we give in the following lemma the boundary values of
$\varphi_{-}$ defined in (2.7).

######  Lemma 5.1.

For sufficiently small $|c|$, the boundary values of $\varphi_{-}(y)$ satisfy:

$\displaystyle\varphi_{-}(\pm 1)$ $\displaystyle=\frac{-1}{U^{\prime}_{s}(\pm
1)}+O(1)|c\log{Imc}|,$ (5.1) $\displaystyle\varphi^{\prime}_{-}(\pm 1)$
$\displaystyle=O(1)|\log{Imc}|.$ (5.2)

###### Proof.

We only show (5.1) and (5.2) for $y=1$, since the boundary value at $y=-1$ can
be obtained in the same way. Evaluating (2.7) at $y=1$, we have

$\displaystyle\varphi_{-}(1)$
$\displaystyle=(U_{s}(1)-c)\int^{1}_{0}\frac{1}{(U_{s}(x)-c)^{2}}{\rm
d}x-m^{2}(U_{s}(1)-c)$
$\displaystyle=-c\int^{1}_{0}\frac{1}{(U_{s}(x)-c)^{2}}{\rm d}x+m^{2}c$
$\displaystyle=-c\int^{1}_{\frac{1}{2}}\frac{1}{(U_{s}(x)-c)^{2}}{\rm
d}x-c\int^{\frac{1}{2}}_{0}\frac{1}{(U_{s}(x)-c)^{2}}{\rm d}x+O(1)|c|.$ (5.3)

For $x\in\left(0,\frac{1}{2}\right)$, we have
$U_{s}(x)=1-x^{2}\geq\frac{3}{4}$. Then the second integral in (5.3) satisfies

$\bigg{|}\int^{\frac{1}{2}}_{0}\frac{c}{(U_{s}(x)-c)^{2}}{\rm
d}x\bigg{|}\leq\frac{|c|}{2\left(\frac{9}{16}-|c|^{2}\right)}\\\ \leq
O(1)|c|,\text{ for }|c|\ll 1.$ (5.4)

When $x\in\left(\frac{1}{2},1\right)$, $U$ is not denegerate because that
$|U^{\prime}_{s}(x)|=2x\geq 1$. Thus, we can compute the first integral in
(5.3) as

$\displaystyle-\int^{1}_{\frac{1}{2}}\frac{c}{(U_{s}(x)-c)^{2}}{\rm
d}x=\int^{1}_{\frac{1}{2}}\frac{{\rm d}}{{\rm
d}x}\bigg{(}\frac{c}{U_{s}(x)-c}\bigg{)}\frac{1}{U^{\prime}_{s}(x)}{\rm d}x$
$\displaystyle\qquad=\frac{c}{U^{\prime}_{s}(1)(U_{s}(1)-c)}-\frac{c}{U^{\prime}_{s}\left(\frac{1}{2}\right)\left(U_{s}\left(\frac{1}{2}\right)-c\right)}-\int^{1}_{\frac{1}{2}}\frac{c}{U_{s}(x)-c}\bigg{(}\frac{1}{U^{\prime}_{s}(x)}\bigg{)}^{\prime}{\rm
d}x$
$\displaystyle\qquad=-\frac{1}{U^{\prime}_{s}(1)}+c\int^{1}_{\frac{1}{2}}\frac{{\rm
d}}{{\rm
d}x}(\log(U_{s}(x)-c))\frac{U^{\prime\prime}_{s}(x)}{(U^{\prime}_{s}(x))^{3}}+O(1)|c|$
$\displaystyle\qquad=-\frac{1}{U^{\prime}_{s}(1)}+c\log(U_{s}(y)-c)\frac{U^{\prime\prime}_{s}(y)}{(U^{\prime}_{s}(y))^{3}}\bigg{|}_{y=\frac{1}{2}}^{y=1}-c\int^{1}_{\frac{1}{2}}\log(U_{s}(x)-c)\left(\frac{U^{\prime\prime}_{s}}{(U^{\prime}_{s})^{3}}\right)^{\prime}{\rm
d}x+O(1)|c|$
$\displaystyle\qquad=-\frac{1}{U^{\prime}_{s}(1)}+O(1)|c\log\text{Im}c|.$
(5.5)

Plugging (5.4) and (5.5) into (5.3), we obtain:

$\varphi_{-}(1)=-\frac{1}{U^{\prime}_{s}(1)}+O(1)|c\log\text{Im}c|,$

which is (5.1). Next we show (5.2). Differentiating (2.7) yields

$\displaystyle\varphi_{-}^{\prime}(y)=\frac{U_{s}^{\prime}(y)\varphi_{-}(y)+A(y)}{U_{s}(y)-c}.$
(5.6)

Evaluaing both sides of (5.6) at $y=1$ gives

$\displaystyle\varphi^{\prime}_{-}(1)$
$\displaystyle=\frac{U^{\prime}_{s}(1)\varphi_{-}(1)+1}{U_{s}(1)-c}-m^{2}(U_{s}(1)-c)$
$\displaystyle=\frac{O(1)|c\log\text{Im}c|}{-c}+m^{2}c$
$\displaystyle=O(1)|\log\text{Im}c|,$

which is (5.2). The proof of Lemma 5.1 is completed. ∎

The following lemma concerns some bounds on $\varphi_{-}.$

######  Lemma 5.2.

There exists a positive constant $\gamma,$ such that if $c$ lies in the half
disk $\\{\text{Im}c>0,|c|\leq\gamma\\}$, then $\varphi_{-}$ satisfies the
following estimates.

  * (1)

If $y\in\left(-\frac{1}{2},\frac{1}{2}\right)$, we have

$\displaystyle\left|\partial_{y}^{k}\varphi_{-}(y)\right|\leq
C,~{}k=0,1,2,3,4.$ (5.7)

  * (2)

If $y\in(-1,1)\setminus\left(-\frac{1}{2},\frac{1}{2}\right),$ we have the
following pointwise estimate

$\displaystyle\left|\varphi_{-}(y)\right|\leq
C,~{}\varphi_{-}^{\prime}(y)=\frac{-U_{s}^{\prime\prime}(y)}{\left(U_{s}^{\prime}(y)\right)^{2}}\log\left(U_{s}(y)-c\right)+O(1),$
(5.8)
$\displaystyle\varphi_{-}^{\prime\prime}(y)=-\frac{U_{s}^{\prime\prime}(y)}{U_{s}^{\prime}(y)\left(U_{s}(y)-c\right)}-\frac{\left(U_{s}^{\prime\prime}(y)\right)^{2}}{\left(U_{s}^{\prime}(y)\right)^{3}}\log\left(U_{s}(y)-c\right)+O(1),$
$\displaystyle\varphi_{-}^{\prime\prime\prime}(y)=\frac{U_{s}^{\prime\prime}(y)}{\left(U_{s}(y)-c\right)^{2}}+O(1),~{}\partial_{y}^{4}\varphi_{-}=-\frac{3U_{s}^{\prime}U_{s}^{\prime\prime}}{(U_{s}-c)^{3}}.$

  * (3)

Furthermore, it holds that

$\displaystyle\|\varphi_{-}\|_{L^{\infty}}\leq
C,~{}\|\varphi_{-}^{\prime}\|_{L^{\infty}}\leq
C\left|\log\text{Im}c\right|,~{}\|\varphi_{-}^{\prime}\|_{L^{2}}\leq C,$ (5.9)
$\displaystyle\|\partial_{y}^{j}\varphi_{-}\|_{L^{\infty}}\leq
C\left|\text{Im}c\right|^{-j+1},~{}j=2,3,4,$
$\displaystyle\|\partial_{y}^{j}\varphi_{-}\|_{L^{2}}\leq
C\left|\text{Im}c\right|^{-j+\frac{3}{2}},~{}j=1,2,3,4.$

###### Proof.

By differentiating (2.7) up to the fourth order, we obtain

$\displaystyle\varphi_{-}^{\prime}$
$\displaystyle=U_{s}^{\prime}\int_{0}^{y}\frac{1}{(U_{s}-c)^{2}}{\rm
d}x+\frac{1}{U_{s}-c}-m^{2}\left(yU_{s}^{\prime}+U_{s}-c\right),$ (5.10)
$\displaystyle\varphi_{-}^{\prime\prime}$
$\displaystyle=U_{s}^{\prime\prime}\int_{0}^{y}\frac{1}{(U_{s}-c)^{2}}{\rm
d}x-m^{2}\left(yU_{s}^{\prime\prime}+2U_{s}^{\prime}\right),$
$\displaystyle\varphi_{-}^{\prime\prime\prime}$
$\displaystyle=\frac{U_{s}^{\prime\prime}}{(U_{s}-c)^{2}}-3m^{2}U_{s}^{\prime\prime},~{}\partial_{y}^{4}\varphi_{-}=-\frac{2U_{s}^{\prime}U_{s}^{\prime\prime}}{(U_{s}-c)^{3}}.$

Here we have used $\partial_{y}^{j}U_{s}=0$ when $j=3,4$. For
$y\in\left(-\frac{1}{2},\frac{1}{2}\right)$, $|U_{s}-c|\gtrsim 1$. Thus (5.7)
follows from the explicit formula (5.10). For
$y\in(-1,1)\setminus\left(-\frac{1}{2},\frac{1}{2}\right)$, as (5.5) we can
obtain the following pointwise estimate

$\displaystyle\int_{0}^{y}\frac{1}{(U_{s}-c)^{2}}{\rm
d}x=-\frac{1}{U_{s}^{\prime}(y)(U_{s}(y)-c)}-\log(U_{s}(y)-c)\frac{U_{s}^{\prime\prime}(y)}{(U_{s}^{\prime}(y))^{3}}+O(1).$
(5.11)

Substituting (5.11) into the explicit formula (5.10), we obtain pointwise
estimates (5.8). The bounds (5.9) on $L^{2}$ and $L^{\infty}$-norms can be
obtained in the same way as in [36] and we omit the details for brevity. The
proof of Lemma 5.2 is completed. ∎

Finally, we have the following lemma concerning about the weight function
$w(y)$ defined in (3.12). Set

$\displaystyle\mathcal{W}(y)=-(1-m^{2}U_{s}^{2})U_{s}^{\prime\prime}-2m^{2}U_{s}|U_{s}^{\prime}|^{2}.$
(5.12)

######  Lemma 5.3.

Let the Mach number $m\in\left(0,\frac{1}{\sqrt{3}}\right)$. For sufficiently
small $|c|\ll 1$, the weight function $w$ has the following expansion

$\displaystyle w(y)=w_{0}(y)+cw_{1}(y)+O(1)|c|^{2},$ (5.13)

where

$\displaystyle\ w_{0}$ $\displaystyle=(1-m^{2}U_{s}^{2})^{2}\mathcal{W}^{-1},$
$\displaystyle w_{1}$
$\displaystyle=4m^{2}U_{s}(1-m^{2}U_{s}^{2})\mathcal{W}^{-1}-2m^{2}(1-m^{2}U_{s}^{2})^{2}\left(|U_{s}^{\prime\prime}|U_{s}+|U_{s}^{\prime}|^{2}\right).$

Moreover, there exists a positive constant $\gamma_{0}$, such that

$\displaystyle w_{0}-U_{s}w_{1}\geq\gamma_{0}.$ (5.14)

###### Proof.

We show that $\mathcal{W}(y)$ has a strictly positive lower bound, hence
$w_{0}$ and $w_{1}$ are well-defined. Then the expansion (5.13) follows from a
straightforward computation. We take the plane Poiseuille flow
$U_{s}(y)=1-y^{2}$ as an example. Then

$\displaystyle\mathcal{W}(y)=2(1-m^{2})+6m^{2}y^{2}\left(y^{2}-\frac{2}{3}\right).$

Therefore,
$\min_{y\in[-1,1]}\mathcal{W}(y)=\mathcal{W}(y)|_{y=\pm\frac{1}{\sqrt{3}}}=2\left(1-\frac{4m^{2}}{3}\right),$
which is positive when the Mach number
$m\in\left(0,\frac{\sqrt{3}}{2}\right)$. For (5.14), following the same
computation as (3.29) in [50], one has $w_{0}-U_{s}w_{1}\sim
1-3m^{2}U_{s}^{2}$. Therefore, $w_{0}-U_{s}w_{1}$ has a strictly positive
lower bound when the Mach number $m\in\left(0,\frac{1}{\sqrt{3}}\right)$. The
proof of Lemma 5.3 is completed. ∎

Acknowledgment. The research of Yang was supported by the Hong Kong PhD
Fellowship. The research of Zhang was supported by the Start-up Fund
(P0043862), and by the Research Centre for Nonlinear Analysis of The Hong Kong
Polytechnic University.

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