# Prandtl-Batchelor flows on an annulus

Mingwen Fei  School of Mathematics and Statistics, Anhui Normal University,
Wuhu 241002, China<EMAIL_ADDRESS>,  Chen Gao  Beijing International
Center for Mathematical Research, Peking University, Beijing,100871, China
<EMAIL_ADDRESS>,  Zhiwu Lin  School of Mathematics, Georgia Institute of
Technology, 30332, Atlanta, GA, USA<EMAIL_ADDRESS>and  Tao Tao  School
of Mathematics, Shandong University, jinan, Shandong, 250100, China
<EMAIL_ADDRESS>
###### Abstract.

For steady two-dimensional flows with a single eddy (i.e. nested closed
streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956)
found that in the limit of vanishing viscosity, the vorticity is constant in
an inner region separated from the boundary layer. In this paper, we consider
the generalized Prandtl-Batchelor theory on an annulus that is non-simply-
connected. First, we observe that in the vanishing viscosity if steady Navier-
Stokes solutions with nested closed streamlines on an annulus converge to
steady Euler flows which are rotating shear flows, then the vorticity of Euler
flows must be $a\ln r+b$ and the associated velocity must be
$(ar+\frac{b}{r}+cr\ln r,0)$ in polar coordinates. We call solutions of steady
Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then,
by constructing higher order approximate solutions of the Navier-Stokes
equations and establishing the validity of Prandtl boundary layer expansion,
we give a rigorous proof of the existence of Prandtl-Batchelor flows on an
annulus with the wall velocity slightly different from the rigid-rotation. In
particular, for the above boundary conditions, we prove that there is a
continuous curve (i.e. infinitely many) of solutions to the steady Navier-
Stokes equations when the viscosity is sufficiently small.

##  1\. Introduction

In his famous paper [35] on the birth of boundary layer theory, Prandtl (1904)
noted that the vorticity of steady two-dimensional laminar flows in the limit
of infinite Reynolds number becomes constant within a region of nested closed
streamlines (i.e. a single eddy). This important property was later
rediscovered by Batchelor (1956) in [1]. See also the conference abstract [6]
of Feynman and Lagerstrom (1956). This class of result is now usually referred
to as Prandtl-Batchelor theory and such laminar flows are called Prandtl-
Batchelor (PB) flows in the literature. Furthermore, for PB flows on a
circular disk, the formula of the limiting vorticity constant was given in
([1, 6, 44]) and is usually referred to as the Batchelor-Wood formula.
However, for PB flows on more general domain, it is very difficult to
determine the limiting vorticity constant and partial results were obtained in
([5, 6, 27, 39, 40, 44]). The Prandtl-Batchelor theory has important
applications in many studies involving laminar flows with small viscosity, for
example, the nonlinear critical layer theory near shear flows in [29]. It also
implies a selection mechanism of the Navier-Stokes equations in the inviscid
limit. Although the Euler equations have infinitely many steady solutions,
only those Euler solutions satisfying the PB theory can appear in the inviscid
limit. Moreover, the Prandtl-Batchelor theory can be applied to general 2D
advection diffusion equation of a passive scalar field $\theta(x,y)$

$u\cdot\nabla\theta-R^{-1}\Delta\theta=0,$

where $u(x,y)$ is a given steady incompressible flow, and $R^{-1}$ is the
diffusion coefficient. For example, homogenization of potential vorticity in
the ocean circulation theory ([34, 37, 38]) and flux expulsion in the self-
excited dynamo theory ([30, 31, 43]).

Despite the importance of PB theory and its wide applications, there is
relatively few mathematical works on this problem. In the existing “proof” of
PB theory stemming from [1, 35], the eddy structure of the laminar flows is a
priori assumed. Kim initiated a mathematical study of PB flows on a disk in a
series of works ([21, 22, 23, 24, 25, 26, 27]). In particular, when the
boundary velocity is slightly different from a constant, the well-posedness of
the Prandtl equations under the Batchelor-Wood condition was shown in [22] and
some asymptotic study of the boundary layer expansion was given in [21].
However, he didn’t prove the convergence of the boundary layer expansion in
the limit of infinite Reynolds number. Recently, by constructing higher order
approximate solutions of the Navier-Stokes equations and establishing the
validity of Prandtl boundary layer expansion, we give a rigorous proof of the
existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly
different from the rigid-rotation in [7].

In this paper, we extend the Prandtl-Batchelor theory to the annulus which is
a non-simply-connected domain. In this setting, we found that if the solutions
of steady Navier-Stokes equations with nested closed streamlines converge to
steady Euler flows which are rotating shear flows, then their vorticity must
be $a\ln r+b$ and the associated velocity must be $(ar+\frac{b}{r}+cr\ln r,0)$
in polar coordinates. We call the solutions of steady Navier-Stokes equations
satisfying the above property Prandtl-Batchelor flows on an annulus. This is
different from the disk case where the vorticity of Prandtl-Batchelor flows
must be constant in the limit of infinite Reynolds number. In the Appendix, we
give proof of the above property under the following assumptions: for
sufficiently small viscosity i) the steady flows of Navier-Stokes equations
have nested closed streamlines; ii) any interior domain of the annulus is
separated from the boundary layer uniformly for vanishing viscosity; iii)
inside the interior domain, steady Navier-Stokes solutions converge to a
steady Euler solution in $C^{2}$. However, given a domain and the boundary
data, it is hard to understand the streamline structures of the steady
solutions of Navier-Stokes equations and control the boundary layer, hence it
seems difficult to verify the above assumptions rigorously. Then, following
the argument of [7], by constructing higher order approximate solutions and
establishing the convergence of Prandtl boundary layer expansion, we construct
a class of Prandtl-Batchelor flows to the steady Navier-Stokes equations on an
annulus with both the wall velocities slightly different from the rigid-
rotation.

More precisely, we consider the steady Navier-Stokes equations in a two-
dimensional annulus centered on zero $A:=B_{1}(0)\backslash B_{r_{0}}(0)$

$\left\\{\begin{array}[]{ll}\mathbf{u}^{\varepsilon}\cdot\nabla\mathbf{u}^{\varepsilon}+\nabla
p^{\varepsilon}-\varepsilon^{2}\Delta\mathbf{u}^{\varepsilon}=0,\\\\[5.0pt]
\nabla\cdot\mathbf{u}^{\varepsilon}=0,\end{array}\right.$ (1.1)

with rotating boundary

$\displaystyle\mathbf{u}^{\varepsilon}\big{|}_{\partial B_{1}}=(\alpha+\eta
f(\theta))\mathbf{t},\quad\mathbf{u}^{\varepsilon}\big{|}_{\partial
B_{r_{0}}}=(\beta+\eta g(\theta))\mathbf{t},$ (1.2)

where $\varepsilon^{2}>0$ is reciprocal to Reynolds number, $0<r_{0}<1$,
$\alpha>0,\beta>0$, $\mathbf{u}^{\varepsilon}$ is the velocity,
$p^{\varepsilon}$ is the pressure, $\eta$ is a small number, $\mathbf{t}$ is
the unit tangential vector to $\partial B_{1}$ or $\partial B_{r_{0}}$,
$f(\theta),g(\theta)$ are $2\pi$-periodic smooth functions.

Let $u^{\varepsilon}(\theta,r)$ be the tangential component and
$v^{\varepsilon}(\theta,r)$ be the normal component of
$\mathbf{u}^{\varepsilon}$ in polar coordinates, then (1.1) reads

$\displaystyle\left\\{\begin{array}[]{lll}u^{\varepsilon}u^{\varepsilon}_{\theta}+rv^{\varepsilon}u^{\varepsilon}_{r}+u^{\varepsilon}v^{\varepsilon}+p^{\varepsilon}_{\theta}-\varepsilon^{2}\big{(}\frac{u^{\varepsilon}_{\theta\theta}}{r}+ru^{\varepsilon}_{rr}+u^{\varepsilon}_{r}+\frac{2}{r}v^{\varepsilon}_{\theta}-\frac{u^{\varepsilon}}{r}\big{)}=0,\\\\[5.0pt]
u^{\varepsilon}v^{\varepsilon}_{\theta}+rv^{\varepsilon}v^{\varepsilon}_{r}-(u^{\varepsilon})^{2}+rp^{\varepsilon}_{r}-\varepsilon^{2}\big{(}\frac{v^{\varepsilon}_{\theta\theta}}{r}+rv^{\varepsilon}_{rr}+v^{\varepsilon}_{r}-\frac{2}{r}u^{\varepsilon}_{\theta}-\frac{v^{\varepsilon}}{r}\big{)}=0,\\\\[5.0pt]
u^{\varepsilon}_{\theta}+rv^{\varepsilon}_{r}+v^{\varepsilon}=0,\\\\[5.0pt]
u^{\varepsilon}(\theta,1)=\alpha+\eta
f(\theta),~{}~{}v^{\varepsilon}(\theta,1)=0,\\\\[5.0pt]
u^{\varepsilon}(\theta,r_{0})=\beta+\eta
g(\theta),~{}~{}v^{\varepsilon}(\theta,r_{0})=0,\end{array}\right.$ (1.8)

where $(\theta,r)\in\Omega:=[0,2\pi]\times[r_{0},1]$. Formally, as
$\varepsilon\rightarrow 0$, we obtain the steady Euler equations for
$(u_{e},v_{e})$

$\displaystyle\left\\{\begin{aligned}
&u_{e}\partial_{\theta}u_{e}+v_{e}r\partial_{r}u_{e}+u_{e}v_{e}+\partial_{\theta}p_{e}=0,\\\
&u_{e}\partial_{\theta}v_{e}+v_{e}r\partial_{r}v_{e}-(u_{e})^{2}+r\partial_{r}p_{e}=0,\\\
&\partial_{\theta}u_{e}+\partial_{r}(rv_{e})=0.\end{aligned}\right.$ (1.9)

We will show the existence of solution $(u^{\varepsilon},v^{\varepsilon})$ to
(1.8) which converges to a solution of steady Euler equations (1.9) with
vorticity $a\ln r+b$ as $\varepsilon\rightarrow 0$. Following the argument of
[7], we first choose a steady Euler flow whose vorticity is $a\ln r+b$, then
construct a class of Prandtl-Batchelor flows
$(u^{\varepsilon},v^{\varepsilon})$ to (1.8) by perturbing this steady Euler
flow. Generally, there is a mismatch between the tangential velocities of the
Euler flow $u_{e}$ and the prescribed Navier-Stokes flows $u^{\varepsilon}$.
Due to the mismatch on the boundary, Prandtl in 1904 formally introduced the
boundary layer theory to correct this mismatch. However, the justification of
this formal boundary expansion is a challenging problem. The studies on steady
Prandtl equations and linearized steady Prandtl equations can be found in [3,
14, 33, 41, 42] and the validity of Prandtl boundary layer expansion has also
some important progresses, see [11, 15, 16, 17, 12, 8, 19, 9]. Moreover, the
stability in Sobolev space for some class shear flow of Prandtl type has been
studied in [3, 2].

In the current work, we assume that
$\int_{0}^{2\pi}f(\theta)d\theta=\int_{0}^{2\pi}g(\theta)d\theta=0$ and take
the leading order Euler flow $(u_{e}(\theta,r),v_{e}(\theta,r))$ is the
following shear flow

$\displaystyle
u_{e}(\theta,r)=u_{e,\delta}(r):=a_{0}r+\frac{b_{0}}{r}+\delta\tilde{u}_{e}(r),\
v_{e}(\theta,r)=0,$ (1.10)

here

$\displaystyle
a_{0}+b_{0}=\Big{(}\alpha^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}f^{2}(\theta)d\theta\Big{)}^{\frac{1}{2}},\
a_{0}r_{0}+\frac{b_{0}}{r_{0}}=\Big{(}\beta^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}g^{2}(\theta)d\theta\Big{)}^{\frac{1}{2}}$

and $\tilde{u}_{e}(r)=\tilde{a}_{0}r+\frac{\tilde{b}_{0}}{r}+\tilde{c}_{0}r\ln
r$ satisfies

$\displaystyle\tilde{u}_{e}(1)=\tilde{u}_{e}(r_{0})=0,\ \tilde{u}_{e}(r)\geq
0,\ \forall r\in[r_{0},1],$

where $(\tilde{a}_{0},\tilde{b}_{0},\tilde{c}_{0})$ are fixed numbers and we
take $\delta\in[0,1]$. There is an important basis for this choice—the
generalized Prandtl-Batchelor theory, see Appendix C. This theory shows that
if the Euler flow is rotating shear $u_{e}(r)\vec{e}_{\theta}$ in the
vanishing viscosity limit, then there holds

$u_{e}(r)=ar+\frac{b}{r}+cr\ln r,$

where $a,b,c$ are constants. While the Batchelor-Wood formula in Lemma 2.1
shows

$\displaystyle
u_{e}^{2}(1)=\alpha^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}f^{2}(\theta)d\theta,$

and

$\displaystyle
u_{e}^{2}(r_{0})=\beta^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}g^{2}(\theta)d\theta,$

which give some relationship of $a,b,c$. Moreover, the solvability of
linearized second order Euler equations makes $u_{e}(r)$ necessary to be this
form, see (2.141).

Next we introduce the steady Prandtl equations near $r=1$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e,\delta}(1)+u_{p}^{(0)}\big{)}\partial_{\theta}u_{p}^{(0)}+\big{(}v_{p}^{(1)}-v_{p}^{(1)}(\theta,0)\big{)}\partial_{Y}u_{p}^{(0)}-\partial_{YY}u_{p}^{(0)}=0,\\\\[5.0pt]
\partial_{\theta}u_{p}^{(0)}+\partial_{Y}v_{p}^{(1)}=0,\\\\[5.0pt]
u_{p}^{(0)}(\theta,Y)=u_{p}^{(0)}(\theta+2\pi,Y),\quad
v_{p}^{(0)}(\theta,Y)=v_{p}^{(0)}(\theta+2\pi,Y),\\\\[5.0pt]
u_{p}^{(0)}\big{|}_{Y=0}=\alpha+\eta
f(\theta)-u_{e,\delta}(1),\quad\lim\limits_{Y\rightarrow-\infty}u_{p}^{(0)}=\lim\limits_{Y\rightarrow-\infty}v_{p}^{(1)}=0\end{array}\right.$
(1.15)

and the steady Prandtl equations near $r=r_{0}$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e,\delta}(r_{0})+\hat{u}_{p}^{(0)}\big{)}\partial_{\theta}\hat{u}_{p}^{(0)}+\big{(}\hat{v}_{p}^{(1)}-\hat{v}_{p}^{(1)}(\theta,0)\big{)}r_{0}\partial_{Z}\hat{u}_{p}^{(0)}-r_{0}\partial_{ZZ}\hat{u}_{p}^{(0)}=0,\\\\[5.0pt]
\partial_{\theta}\hat{u}_{p}^{(0)}+r_{0}\partial_{Z}\hat{v}_{p}^{(1)}=0,\\\\[5.0pt]
\hat{u}_{p}^{(0)}(\theta,Z)=\hat{u}_{p}^{(0)}(\theta+2\pi,Z),\quad\hat{v}_{p}^{(0)}(\theta,Z)=\hat{v}_{p}^{(0)}(\theta+2\pi,Z),\\\\[5.0pt]
\hat{u}_{p}^{(0)}\big{|}_{Z=0}=\beta+\eta g(\theta)-u_{e,\delta}(r_{0}),\ \
\lim\limits_{Z\rightarrow+\infty}\hat{u}_{p}^{(0)}=\lim\limits_{Z\rightarrow+\infty}\hat{v}_{p}^{(1)}=0,\end{array}\right.$
(1.20)

where $Y=\frac{r-1}{\varepsilon}$ and $Z=\frac{r-r_{0}}{\varepsilon}$. The
above steady Prandtl equations will be derived by matched asymptotic expansion
and their solvability will be studied in the next section.

Now our main theorem is stated as follows.

######  Theorem 1.1.

Assume that $f(\theta),g(\theta)$ are $2\pi$-periodic smooth functions which
satisfy $\int_{0}^{2\pi}f(\theta)d\theta=\int_{0}^{2\pi}g(\theta)d\theta=0$,
then there exist $\varepsilon_{0}>0,\eta_{0}>0,C>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0}),\eta\in(0,\eta_{0}),\delta\in[0,1]$, the
Navier-Stokes equations (1.8) have a solution
$(u^{\varepsilon}(\theta,r),v^{\varepsilon}(\theta,r))$ which satisfies

$\displaystyle\Big{\|}u^{\varepsilon}(\theta,r)-u_{e,\delta}(r)-u_{p}^{(0)}\Big{(}\theta,\frac{r-1}{\varepsilon}\Big{)}-\hat{u}_{p}^{(0)}\Big{(}\theta,\frac{r-r_{0}}{\varepsilon}\Big{)}\Big{\|}_{L^{\infty}(\Omega)}\leq
C\varepsilon,$ $\displaystyle\|v^{\varepsilon}\|_{L^{\infty}(\Omega)}\leq
C\varepsilon,$

where $(u_{e,\delta}(r),0)$ is the shear flow in (1.10),
$(u_{p}^{(0)},v_{p}^{(1)})$ is the solution of steady Prandtl equations (1.15)
and $(\hat{u}_{p}^{(0)},\hat{v}_{p}^{(1)})$ is the solution of steady Prandtl
equations (1.20).

Moreover, for any $r_{0}<r_{1}<r_{2}<1$, there holds

$\displaystyle\lim_{\varepsilon\rightarrow
0}\|w^{\varepsilon}-(2\delta\tilde{c}_{0}\ln
r+2a_{0}+2\delta\tilde{a}_{0}+\delta\tilde{c}_{0})\|_{L^{\infty}(A_{r_{1},r_{2}})}=0,$

where $w^{\varepsilon}(\theta,r)$ is the vorticity of
$(u^{\varepsilon}(\theta,r),v^{\varepsilon}(\theta,r))$ and
$A_{r_{1},r_{2}}=B_{r_{2}}(0)\backslash B_{r_{1}}(0)$.

######  Remark 1.2.

1\. The condition
$\int_{0}^{2\pi}f(\theta)d\theta=\int_{0}^{2\pi}g(\theta)d\theta=0$ can be
dropped due to the fact,e.g.,

$\alpha+\eta
f(\theta)=\alpha+\frac{\eta}{2\pi}\int_{0}^{2\pi}f(\theta)d\theta+\eta\tilde{f}(\theta),$

where $\int_{0}^{2\pi}\tilde{f}(\theta)d\theta=0.$ Moreover, the smoothness of
$f(\theta),g(\theta)$ can be relaxed, but we don’t pursue this issue in this
manuscript.

2\. $\delta\in[0,1]$ can be relaxed to $\delta\in[\delta_{1},\delta_{2}]$ for
any $[\delta_{1},\delta_{2}]\subset[0,+\infty)$ such that
$u_{e}\left(r\right)$ defined by (1.10) is strictly positive.

3\. This theorem implies that there are infinity many solutions of the steady
Navier-Stokes equations when the viscosity is sufficiently small. Moreover, we
can prove that for any $\varepsilon$ sufficiently small, the solution
$(u^{\varepsilon}(\theta,r;\delta),u^{\varepsilon}(\theta,r;\delta))$ of (1.8)
is continuous dependence on $\delta$.

######  Remark 1.3.

In this remark, we give some explanations about the difference between the
disk and annulus. (1)For the disk case, the classical Prandtl-Batchelor theory
tells us that the Euler flow in the vanishing viscosity limit must be the
Couette flow $(ar,0)$, and the Batchelor-Wood formula(a necessary condition
for the solvability of steady Prandtl equations) determines the value of $a$.
Thus, in the disk case, the Euler flow in the vanishing viscosity limit was
uniquely determined by the boundary condition of the Navier-Stokes equations.
However, there is a new difficulty for justifying the vanishing viscosity
limit in this case: the coordinate singularity $\frac{1}{r}$ near $r=0$ in the
error equations which are written in the polar coordinates. For overcoming
this difficulty, we need to study the precise properties of the approximate
solution $(u^{a},v^{a})$. Fortunately, due to the specialty of Couette flow
$(ar,0)$, our approximate solution $(u^{a},v^{a})$ satisfies
$|\partial_{\theta}u_{e}^{a}+v_{e}^{a}|+|\partial_{\theta}v_{e}^{a}-u^{a}_{e}+ar|\leq
C\varepsilon\eta r$, this key property can cancel the coordinate singularity.
(2)For the annulus case, the classical Prandtl-Batchelor theory isn’t true,
and the generalized Prandtl-Batchelor theory(see Appendix C) tells us that in
the annulus the rotating shear flow in the vanishing viscosity limit must be
the form $(ar+\frac{b}{r}+cr\ln r,0)$. Similarly, the Batchelor-Wood formula
gives some relationship of $a,b,c$, and there are infinity many chooses of
$a,b,c$ which satisfy the Batchelor-Wood formula. Thus, in the annulus case,
there are infinity many Euler flows in the vanishing viscosity limit, and
using this fact we can prove that there are infinity many solutions of the
Navier-Stokes equations (1.1) with boundary condition (1.2) for sufficiently
small viscosity. Actually, for the given boundary conditions we construct a
continuous family of steady Navier-Stokes solutions. This implies that such
steady Navier-Stokes solutions are not isolated and when studying their
dynamical behaviors (e.g. stability) we need to consider the whole family.

Now we present a sketch of the proof and some key ideas. For convenience, we
use $u_{e}(r)$ to express $u_{e,\delta}(r)$.

Step 1: Construction of the approximate solution. We construct an approximate
solution $(u^{a},v^{a})$ by matched asymptotic expansion. The leading order of
$(u^{a},v^{a})$ is

$\displaystyle\Big{(}u_{e}(r)+u_{p}^{(0)}\Big{(}\theta,\frac{r-1}{\varepsilon}\Big{)}+\hat{u}_{p}^{(0)}\Big{(}\theta,\frac{r-r_{0}}{\varepsilon}\Big{)},0\Big{)}.$

The details of constructing the approximate solution will be given in Section
2. After the construction of the approximate solution, we derive the equations
(3.7) for the error $(u,v):=(u^{\varepsilon}-u^{a},v^{\varepsilon}-v^{a})$.
Notice that the nonlinear term can be easily handled by higher order
approximation, hence we only need to consider the linearized error equations

$\displaystyle\left\\{\begin{array}[]{lll}u^{a}u_{\theta}+uu^{a}_{\theta}+v^{a}ru_{r}+vru^{a}_{r}+v^{a}u+vu^{a}+p_{\theta}-\varepsilon^{2}\big{(}ru_{rr}+\frac{u_{\theta\theta}}{r}+2\frac{v_{\theta}}{r}+u_{r}-\frac{u}{r}\big{)}=F_{1},\\\\[5.0pt]
u^{a}v_{\theta}+uv^{a}_{\theta}+v^{a}rv_{r}+vrv^{a}_{r}-2uu^{a}+rp_{r}-\varepsilon^{2}\big{(}rv_{rr}+\frac{v_{\theta\theta}}{r}-2\frac{u_{\theta}}{r}+v_{r}-\frac{v}{r}\big{)}=F_{2},\\\\[5.0pt]
u_{\theta}+(rv)_{r}=0,\\\\[5.0pt] u(\theta+2\pi,r)=u(\theta,r),\
v(\theta+2\pi,r)=v(\theta,r),\\\\[5.0pt] u(\theta,1)=0,\
v(\theta,1)=0,\\\\[5.0pt] u(\theta,r_{0})=0,\
v(\theta,r_{0})=0.\end{array}\right.$ (1.27)

Step 2: Linear stability estimate for (1.27). Equations (1.27) are the
linearized Navier-Stokes equations around the approximate solution
$(u^{a},v^{a})$. For simply, we assume
$(u^{a},v^{a})\approx\Big{(}u_{e}(r)+u_{p}^{(0)}\big{(}\theta,\frac{r-1}{\varepsilon}\big{)},0\Big{)}$
because we can deal with the boundary layer profile near $r=r_{0}$ in the
similar way. Since $|u^{a}_{\theta}|=|u^{(0)}_{p\theta}|\lesssim\eta$, the
leading order of the above system can be simplified as

$\displaystyle\left\\{\begin{array}[]{lll}u^{a}u_{\theta}+vru^{a}_{r}+vu^{a}+p_{\theta}-\varepsilon^{2}\big{(}ru_{rr}+\frac{u_{\theta\theta}}{r}+2\frac{v_{\theta}}{r}+u_{r}-\frac{u}{r}\big{)}=F_{1},\\\\[5.0pt]
u^{a}v_{\theta}-2uu^{a}+rp_{r}-\varepsilon^{2}\big{(}rv_{rr}+\frac{v_{\theta\theta}}{r}-2\frac{u_{\theta}}{r}+v_{r}-\frac{v}{r}\big{)}=F_{2},\\\\[5.0pt]
u_{\theta}+(rv)_{r}=0,\\\\[5.0pt] u(\theta+2\pi,r)=u(\theta,r),\
v(\theta+2\pi,r)=v(\theta,r),\\\\[5.0pt] u(\theta,1)=0,\
v(\theta,1)=0,\\\\[5.0pt] u(\theta,r_{0})=0,\
v(\theta,r_{0})=0,\end{array}\right.$ (1.34)

where $u^{a}$ can be regarded as
$u_{e}(r)+u_{p}^{(0)}\big{(}\theta,\frac{r-1}{\varepsilon}\big{)}$.

The linear stability estimate consists of a basic energy estimate and
positivity estimate. In fact, it is easy to know that the basic energy
estimate is not good enough for closing the estimate because $\varepsilon$ is
small. The key point is the following observation which gives the important
positivity estimate: $u^{a}$ is strictly positive, we should make use of the
terms $u^{a}u_{\theta}$ and $u^{a}v_{\theta}$ to obtain a positive quantity
from the convective term. To do this, we choose
$\big{(}-\big{(}\frac{r^{2}v}{u^{a}}\big{)}_{r},\big{(}\frac{rv}{u^{a}}\big{)}_{\theta}\big{)}$
as the multiplier, the pressure is eliminated due to the divergence-free
condition and the diffusion term is dominated easily due to the small
viscosity $\varepsilon^{2}$. Here we only give a sketch of deriving the key
positive quantity as follows. Firstly one has

$\displaystyle-\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u^{a}u_{\theta}+vru^{a}_{r}+vu^{a}\big{)}\Big{(}\frac{r^{2}v}{u^{a}}\Big{)}_{r}d\theta
dr$
$\displaystyle\quad\quad+\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u^{a}v_{\theta}-2uu^{a}\big{)}\Big{(}\frac{rv}{u^{a}}\Big{)}_{\theta}d\theta
dr$ $\displaystyle=$
$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u^{a}u_{\theta}+vru^{a}_{r}+vu^{a}\big{)}\Big{(}\frac{ru_{\theta}}{u^{a}}-rv\big{(}\frac{r}{u^{a}}\big{)}_{r}\Big{)}d\theta
dr$
$\displaystyle\quad\quad+\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u^{a}v_{\theta}-2uu^{a}\big{)}\Big{(}\frac{rv_{\theta}}{u^{a}}+rv\big{(}\frac{1}{u^{a}}\big{)}_{\theta}\Big{)}d\theta
dr$ $\displaystyle\approx$
$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u^{a}u_{\theta}+vru^{a}_{r}+vu^{a}\big{)}\Big{(}\frac{ru_{\theta}}{u^{a}}-rv\big{(}\frac{r}{u^{a}}\big{)}_{r}\Big{)}d\theta
dr$
$\displaystyle\quad\quad+\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u^{a}v_{\theta}-2uu^{a}\big{)}\Big{(}\frac{rv_{\theta}}{u^{a}}\Big{)}d\theta
dr$ $\displaystyle\approx$
$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}ru^{2}_{\theta}+rv^{2}_{\theta}\big{)}d\theta
dr+\int_{r_{0}}^{1}\int_{0}^{2\pi}\frac{1}{u^{a}}\Big{(}ru^{a}_{rr}+u^{a}_{r}-\frac{u^{a}}{r}\Big{)}(rv)^{2}d\theta
dr,$

here the details of obtaining the last step will be given in subsection 3.2.

Since $u^{a}\approx u_{e}(r)+u^{(0)}_{p}(\theta,\frac{r-1}{\varepsilon})$, the
boundary layer profile $u^{(0)}_{p}(\theta,Y)$ is small in the Prandtl
variable $Y=\frac{r-1}{\varepsilon}$, we deduce that

$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\frac{1}{u^{a}}\Big{(}r\partial^{2}_{r}u^{(0)}_{p}+\partial_{r}u^{(0)}_{p}-\frac{u^{(0)}_{p}}{r}\Big{)}(rv)^{2}d\theta
dr$ $\displaystyle=$
$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\frac{1}{u^{a}}\Big{(}rY^{2}\partial^{2}_{Y}u^{(0)}_{p}+\varepsilon
Y^{2}\partial_{Y}u^{(0)}_{p}-\frac{(\varepsilon
Y)^{2}u^{(0)}_{p}}{r}\Big{)}\Big{(}\frac{rv}{r-1}\Big{)}^{2}d\theta dr$
$\displaystyle\lesssim$
$\displaystyle\eta\int_{r_{0}}^{1}\int_{0}^{2\pi}\Big{(}\frac{rv}{r-1}\Big{)}^{2}d\theta
dr\lesssim\eta\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}(rv)_{r}\big{)}^{2}d\theta
dr=\eta\int_{r_{0}}^{1}\int_{0}^{2\pi}u_{\theta}^{2}d\theta dr,$

where we have used the Hardy inequality.

Moreover, if the Euler flow is Taylor-Coutte flow, that is
$u_{e}(r)=ar+\frac{b}{r}$, then $ru_{err}+u_{er}-\frac{u_{e}}{r}=0$. So

$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}ru^{2}_{\theta}+rv^{2}_{\theta}\big{)}d\theta
dr+\int_{r_{0}}^{1}\int_{0}^{2\pi}\frac{1}{u^{a}}\Big{(}ru^{a}_{rr}+u^{a}_{r}-\frac{u^{a}}{r}\Big{)}(rv)^{2}d\theta
dr$
$\displaystyle\geq(1-C\eta)\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}ru^{2}_{\theta}+rv^{2}_{\theta}\big{)}d\theta
dr,$

where the constant $C$ is independent of $\eta$. So we obtain a positive
quantity $\|(u_{\theta},v_{\theta})\|_{2}$. When $u_{e}$ is strictly positive,
we can also obtain the same estimate which will be given in the text.

However, the positive quantity $\|(u_{\theta},v_{\theta})\|_{2}$ don’t contain
zero-frequency of $(u,v)$ which is needed to obtain the $L^{\infty}$ estimate
for the error $(u,v)$. Notice that $\int_{0}^{2\pi}v(\theta,r)d\theta=0$
because of the divergence-free condition and the boundary conditions, we can
dominate $\|v\|_{2}$ by $\|v_{\theta}\|_{2}$ using the Poincaré inequality.
However $u_{0}(r):=\frac{1}{2\pi}\int_{0}^{2\pi}u(\theta,r)d\theta\neq 0$, we
need to obtain the estimate of $|u_{0}(r)|$ from the basic energy estimate.

Now we give a sketch of the basic energy estimate. Choose $u_{0}$ as a
multiplier to the first equation in (1.34). The diffusion term is

$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}-\varepsilon^{2}\big{(}ru_{rr}+\frac{u_{\theta\theta}}{r}+2\frac{v_{\theta}}{r}+u_{r}-\frac{u}{r}\big{)}u_{0}(r)d\theta
dr=\varepsilon^{2}\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}r|u^{\prime}_{0}|^{2}+\frac{u_{0}^{2}}{r}\big{)}d\theta
dr.$

We decompose the approximate solution as $u^{a}\approx
u_{e}(r)+u^{(0)}_{p}(\theta,\frac{r-1}{\varepsilon})$. Due to the Euler flow
is radial, we deduce that the following quantity vanishes

$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u_{e}u_{\theta}+vru_{er}+vu_{e}+p_{\theta}\big{)}u_{0}d\theta
dr=0,$

where we used $\int_{0}^{2\pi}v(\theta,r)d\theta=0$. Moreover, the Prandtl
part can be handled by the Hardy inequality as above

$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\big{(}u_{p}^{(0)}u_{\theta}+vr\partial_{r}u_{p}^{(0)}+vu_{p}^{(0)}\big{)}u_{0}d\theta
dr$ $\displaystyle=$
$\displaystyle\int_{r_{0}}^{1}\int_{0}^{2\pi}\varepsilon\big{(}Yu_{p}^{(0)}u_{\theta}+\frac{vr}{r-1}Y^{2}\partial_{Y}u_{p}^{(0)}+vYu_{p}^{(0)}\big{)}\frac{u_{0}}{r-1}d\theta
dr\lesssim\eta\varepsilon\|(u_{\theta},v_{\theta})\|_{2}\|u^{\prime}_{0}\|_{2}.$

Thus, we obtain that
$\varepsilon^{2}\|u^{\prime}_{0}\|_{2}\lesssim\|(u_{\theta},v_{\theta})\|_{2}$.
One can see Lemma 3.2 for the basic energy estimate.

Combining the positivity estimate and basic energy estimate, we obtain the
linear stability of equations (1.34).

Step 3: $L^{\infty}$ estimate for (1.27). To close the nonlinearity, we need
the $L^{\infty}$ estimate. Using the anisotropic Sobolev embedding, we only
need to estimate $\|(u_{r\theta},v_{r\theta})\|_{2}$ which can be obtained by
studying the Stokes equations.

Finally, combining the linear stability estimate and $L^{\infty}$ estimate, we
establish the well-posedness of the error equation by contraction mapping
theorem.

The paper is organized as follows. In Section 2, we construct the approximate
solution by the matched asymptotic expansion method. In Section 3, we derive
the error equations and establish the linear stability estimate which consists
of the basic energy estimate and positivity estimate. In Section 4, we firstly
establish the “$H^{2}$ estimate” for a Stokes system, then obtain $L^{\infty}$
estimate by the anisotropic Sobolev embedding, finally complete the proof of
Theorem 1.1 by combining the linear stability estimate and $L^{\infty}$
estimate.

##  2\. Construction of approximate solutions

In this section, we construct an approximate solution of the Navier-Stokes
equations (1.8) by matched asymptotic expansion.

###  2.1. Euler expansions

Away from the boundary, we make the following formal expansions

$\displaystyle u^{\varepsilon}(\theta,r)=u_{e}^{(0)}(\theta,r)+\varepsilon
u_{e}^{(1)}(\theta,r)+\text{h.o.t.},$ $\displaystyle
v^{\varepsilon}(\theta,r)=v_{e}^{(0)}(\theta,r)+\varepsilon
v_{e}^{(1)}(\theta,r)+\text{h.o.t.},$ $\displaystyle
p^{\varepsilon}(\theta,r)=p_{e}^{(0)}(\theta,r)+\varepsilon
p_{e}^{(1)}(\theta,r)+\text{h.o.t.},$

here and in what follows, “h.o.t.” means higher order terms.

####  2.1.1. Equations for $(u_{e}^{(0)},v_{e}^{(0)},p_{e}^{(0)})$

By substituting the above expansions into (1.8) and collecting the
$\varepsilon$-zeroth order terms, we deduce that
$(u_{e}^{(0)},v_{e}^{(0)},p_{e}^{(0)})$ satisfies the following steady
nonlinear Euler equations

$\displaystyle\left\\{\begin{array}[]{ll}u_{e}^{(0)}\partial_{\theta}u_{e}^{(0)}+rv_{e}^{(0)}\partial_{r}u_{e}^{(0)}+u_{e}^{(0)}v_{e}^{(0)}+\partial_{\theta}p_{e}^{(0)}=0,\\\\[7.0pt]
u_{e}^{(0)}\partial_{\theta}v_{e}^{(0)}+rv_{e}^{(0)}\partial_{r}v_{e}^{(0)}-(u_{e}^{(0)})^{2}+r\partial_{r}p_{e}^{(0)}=0,\\\\[7.0pt]
\partial_{\theta}u_{e}^{(0)}+r\partial_{r}v_{e}^{(0)}+v_{e}^{(0)}=0.\end{array}\right.$
(2.4)

In our current work, we assume that the leading order Euler flow
$(u_{e}^{(0)},v_{e}^{(0)})$ is the shear flow in (1.10). Then equations (2.4)
lead to

$\displaystyle\partial_{\theta}p_{e}^{(0)}(\theta,r)=0,\quad\partial_{r}p_{e}^{(0)}(\theta,r)=\frac{1}{r}u_{e}^{2}.$

Thus we can write

$\displaystyle p_{e}^{(0)}(\theta,r)=p_{e}(r),\quad
p^{\prime}_{e}(r)=\frac{1}{r}u_{e}^{2}(r).$ (2.5)

####  2.1.2. Equations for $(u_{e}^{(1)},v_{e}^{(1)},p_{e}^{(1)})$

By collecting the $\varepsilon$-order terms, we deduce that
$(u_{e}^{(1)},v_{e}^{(1)},p_{e}^{(1)})$ satisfies the following linearized
Euler equations in $\Omega$

$\displaystyle\left\\{\begin{array}[]{ll}u_{e}(r)\partial_{\theta}u_{e}^{(1)}+rv_{e}^{(1)}u_{e}^{\prime}(r)+u_{e}(r)v_{e}^{(1)}+\partial_{\theta}p_{e}^{(1)}=0,\\\\[5.0pt]
u_{e}(r)\partial_{\theta}v_{e}^{(1)}-2u_{e}(r)u_{e}^{(1)}+r\partial_{r}p_{e}^{(1)}=0,\\\\[5.0pt]
\partial_{\theta}u_{e}^{(1)}+r\partial_{r}v_{e}^{(1)}+v_{e}^{(1)}=0,\end{array}\right.$
(2.9)

which are equipped with the boundary conditions

$\displaystyle v_{e}^{(1)}|_{r=1}=-v_{p}^{(1)}|_{Y=0},\quad
v_{e}^{(1)}|_{r=r_{0}}=-\hat{v}_{p}^{(1)}|_{Z=0},\quad
v_{e}^{(1)}(\theta,r)=v_{e}^{(1)}(\theta+2\pi,r),$ (2.10)

where $v_{p}^{(1)},\hat{v}_{p}^{(1)}$ is defined in (2.17) and (2.29)
respectively in next subsection.

###  2.2. Prandtl expansions near $\partial B_{1}$

We introduce the scaled variable $Y=\frac{r-1}{\varepsilon}\in(-\infty,0]$ and
make the following Prandtl expansions near $\partial B_{1}$

$\displaystyle\begin{aligned}
&u^{\varepsilon}=u_{e}^{(0)}(\theta,r)+u_{p}^{(0)}(\theta,Y)+\varepsilon\big{[}u_{e}^{(1)}(\theta,r)+u_{p}^{(1)}(\theta,Y)\big{]}+\varepsilon^{2}\big{[}u_{e}^{(2)}(\theta,r)+u_{p}^{(2)}(\theta,Y)\big{]}+\text{h.o.t.},\\\\[5.0pt]
&v^{\varepsilon}=v_{e}^{(0)}(\theta,r)+v_{p}^{(0)}(\theta,Y)+\varepsilon\big{[}v_{e}^{(1)}(\theta,r)+v_{p}^{(1)}(\theta,Y)\big{]}+\varepsilon^{2}\big{[}v_{e}^{(2)}(\theta,r)+v_{p}^{(2)}(\theta,Y)\big{]}+\text{h.o.t.},\\\\[5.0pt]
&p^{\varepsilon}=p_{e}^{(0)}(\theta,r)+p_{p}^{(0)}(\theta,Y)+\varepsilon\big{[}p_{e}^{(1)}(\theta,r)+p_{p}^{(1)}(\theta,Y)\big{]}+\varepsilon\big{[}p_{e}^{(2)}(\theta,r)+p_{p}^{(2)}(\theta,Y)\big{]}+\text{h.o.t.},\end{aligned}$
(2.11)

where as $Y\rightarrow-\infty$

$\displaystyle\partial_{\theta}^{l}\partial_{Y}^{m}v_{p}^{(i)}(\theta,Y)\rightarrow
0,\ \partial_{\theta}^{l}\partial_{Y}^{m}p_{p}^{(i)}(\theta,Y)\rightarrow 0,$
(2.12)

here $l,m\geq 0,i=0,1,\cdots,$ and satisfy the following boundary conditions

$\displaystyle u_{e}^{(0)}(\theta,1)+u_{p}^{(0)}(\theta,0)=\alpha+\eta
f(\theta),\ u_{e}^{(i)}(\theta,1)+u_{p}^{(i)}(\theta,0)=0,\ i\geq 1,$
$\displaystyle v_{e}^{(i)}(\theta,1)+v_{p}^{(i)}(\theta,0)=0,i\geq 0,\
\lim_{Y\rightarrow-\infty}(u_{p}^{(0)}(\theta,Y),\partial_{Y}u_{p}^{(1)}(\theta,Y))=(0,0).$

####  2.2.1. Equations for $(v_{p}^{(0)},p_{p}^{(0)})$

By substituting the above expansions into (1.8) and collecting the
$\frac{1}{\varepsilon}$ order terms, we get

$\displaystyle\partial_{Y}v_{p}^{(0)}(\theta,Y)=0,\ \
\partial_{Y}p_{p}^{(0)}(\theta,Y)=0,$

which together with (2.12) implies

$\displaystyle v_{p}^{(0)}=0,\ \ p_{p}^{(0)}=0.$

####  2.2.2. Equations for $(u_{p}^{(0)},v_{p}^{(1)},p_{p}^{(1)})$

By substituting the above expansions into (1.8) and collecting the
$\varepsilon$-zeroth order terms, we obtain the following steady Prandtl
equations for $(u_{p}^{(0)},v_{p}^{(1)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(1)+u_{p}^{(0)}\big{)}\partial_{\theta}u_{p}^{(0)}+\big{(}v_{p}^{(1)}-v_{p}^{(1)}(\theta,0)\big{)}\partial_{Y}u_{p}^{(0)}-\partial_{YY}u_{p}^{(0)}=0,\\\\[5.0pt]
\partial_{\theta}u_{p}^{(0)}+\partial_{Y}v_{p}^{(1)}=0,\\\\[5.0pt]
u_{p}^{(0)}(\theta,Y)=u_{p}^{(0)}(\theta+2\pi,Y),\quad
v_{p}^{(0)}(\theta,Y)=v_{p}^{(0)}(\theta+2\pi,Y),\\\\[5.0pt]
u_{p}^{(0)}\big{|}_{Y=0}=\alpha+\eta f(\theta)-u_{e}(1),\ \
\lim\limits_{Y\rightarrow-\infty}u_{p}^{(0)}=\lim\limits_{Y\rightarrow-\infty}v_{p}^{(1)}=0\end{array}\right.$
(2.17)

and the pressure $p_{p}^{(1)}$ satisfies

$\displaystyle\partial_{Y}p_{p}^{(1)}(\theta,Y)=(u_{p}^{(0)})^{2}(\theta,Y)+2u_{e}(1)u_{p}^{(0)}(\theta,Y),\quad\lim_{Y\rightarrow-\infty}p_{p}^{(1)}(\theta,Y)=0.$
(2.18)

####  2.2.3. Equations for $(u_{p}^{(1)},v_{p}^{(2)})$

By substituting the above expansions into the first and third equation of
(1.8) and collecting the $\varepsilon$-order terms, we obtain the following
linearized steady Prandtl equations for $(u_{p}^{(1)},v_{p}^{(2)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(1)+u_{p}^{(0)}\big{)}\partial_{\theta}u_{p}^{(1)}+\big{(}v_{e}^{(1)}(\theta,1)+v_{p}^{(1)}\big{)}\partial_{Y}u_{p}^{(1)}+(v_{p}^{(2)}-v_{p}^{(2)}(\theta,0))\partial_{Y}u_{p}^{(0)}\\\\[5.0pt]
\quad\quad\quad\quad+u_{p}^{(1)}\partial_{\theta}u_{p}^{(0)}-\partial_{YY}u_{p}^{(1)}=f_{1}(\theta,Y),\\\\[5.0pt]
\partial_{\theta}u_{p}^{(1)}+\partial_{Y}v_{p}^{(2)}+\partial_{Y}(Yv_{p}^{(1)})=0,\\\\[5.0pt]
u_{p}^{(1)}(\theta,Y)=u_{p}^{(1)}(\theta+2\pi,Y),\quad
v_{p}^{(2)}(\theta,Y)=v_{p}^{(2)}(\theta+2\pi,Y),\\\\[5.0pt]
u_{p}^{(1)}\big{|}_{Y=0}=-u_{e}^{(1)}\big{|}_{r=1},\ \
\lim\limits_{Y\rightarrow-\infty}\partial_{Y}u_{p}^{(1)}=\lim\limits_{Y\rightarrow-\infty}v_{p}^{(2)}=0\end{array}\right.$
(2.24)

and

$\displaystyle f_{1}(\theta,Y)=$
$\displaystyle-\partial_{\theta}p_{p}^{(1)}+Y\partial_{YY}u_{p}^{(0)}+\partial_{Y}u_{p}^{(0)}$
$\displaystyle-
u_{p}^{(0)}\big{(}\partial_{\theta}u_{e}^{(1)}(\theta,1)+v_{e}^{(1)}(\theta,1)+v_{p}^{(1)}\big{)}-(u^{\prime}_{e}(1)Y+u_{e}^{(1)}(\theta,1))\partial_{\theta}u_{p}^{(0)}$
$\displaystyle-(\partial_{r}v_{e}^{(1)}(\theta,1)+v_{e}^{(1)}(\theta,1))Y\partial_{Y}u_{p}^{(0)}-(u_{e}^{\prime}(1)+Y\partial_{Y}u_{p}^{(0)}+u_{e}(1))v_{p}^{(1)}.$

###  2.3. Prandtl expansions near $\partial B_{r_{0}}$

Similarly as above, we introduce the scaled variable
$Z=\frac{r-r_{0}}{\varepsilon}\in[0,+\infty)$ and make the following Prandtl
expansions near $\partial B_{r_{0}}$

$\displaystyle
u^{\varepsilon}=u_{e}^{(0)}(\theta,r)+\widehat{u}_{p}^{(0)}(\theta,Z)+\varepsilon\big{[}u_{e}^{(1)}(\theta,r)+\widehat{u}_{p}^{(1)}(\theta,Z)\big{]}+\varepsilon^{2}\big{[}u_{e}^{(2)}(\theta,r)+\widehat{u}_{p}^{(2)}(\theta,Z)\big{]}+\text{h.o.t.},$
$\displaystyle
v^{\varepsilon}=v_{e}^{(0)}(\theta,r)+\widehat{v}_{p}^{(0)}(\theta,Z)+\varepsilon\big{[}v_{e}^{(1)}(\theta,r)+\widehat{v}_{p}^{(1)}(\theta,Z)\big{]}+\varepsilon^{2}\big{[}v_{e}^{(2)}(\theta,r)+\widehat{v}_{p}^{(2)}(\theta,Z)\big{]}+\text{h.o.t.},$
$\displaystyle
p^{\varepsilon}=p_{e}^{(0)}(\theta,r)+\widehat{p}_{p}^{(0)}(\theta,Z)+\varepsilon\big{[}p_{e}^{(1)}(\theta,r)+\widehat{p}_{p}^{(1)}(\theta,Z)\big{]}+\varepsilon^{2}\big{[}p_{e}^{(1)}(\theta,r)+\widehat{p}_{p}^{(1)}(\theta,Z)\big{]}+\text{h.o.t.},$

where as $Z\rightarrow+\infty$

$\displaystyle\partial_{\theta}^{l}\partial_{Z}^{m}\widehat{v}_{p}^{(i)}(\theta,Z)\rightarrow
0,\
\partial_{\theta}^{l}\partial_{Z}^{m}\widehat{p}_{p}^{(i)}(\theta,Z)\rightarrow
0,$

here $l,m\geq 0,i=0,1,\cdots,$ and there hold the following boundary
conditions

$\displaystyle
u_{e}^{(0)}(\theta,r_{0})+\hat{u}_{p}^{(0)}(\theta,0)=\beta+\eta g(\theta),\
u_{e}^{(i)}(\theta,r_{0})+\hat{u}_{p}^{(i)}(\theta,0)=0,\ i\geq 1,$
$\displaystyle v_{e}^{(i)}(\theta,r_{0})+\hat{v}_{p}^{(i)}(\theta,0)=0,i\geq
0,\
\lim_{Z\rightarrow+\infty}(\hat{u}_{p}^{(0)}(\theta,Z),\partial_{Z}\hat{u}_{p}^{(1)}(\theta,Z))=(0,0).$

####  2.3.1. Equations for $(\hat{v}_{p}^{(0)},\hat{p}_{p}^{(0)})$

By substituting the above expansions into (1.8) and collecting the
$\frac{1}{\varepsilon}$ order terms, we get

$\displaystyle\partial_{Z}\hat{v}_{p}^{(0)}(\theta,Z)=0,\ \
\partial_{Z}\hat{p}_{p}^{(0)}(\theta,Z)=0,$

which together with (2.12) implies

$\displaystyle\hat{v}_{p}^{(0)}=0,\ \ \hat{p}_{p}^{(0)}=0.$

####  2.3.2. Equations for $(u_{p}^{(0)},v_{p}^{(1)},p_{p}^{(1)})$

By substituting the above expansions into (1.8) and collecting the
$\varepsilon$-zeroth order terms, we obtain the following steady Prandtl
equations for $(\hat{u}_{p}^{(0)},\hat{v}_{p}^{(1)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(r_{0})+\hat{u}_{p}^{(0)}\big{)}\partial_{\theta}\hat{u}_{p}^{(0)}+\big{(}\hat{v}_{p}^{(1)}-\hat{v}_{p}^{(1)}(\theta,0)\big{)}r_{0}\partial_{Z}\hat{u}_{p}^{(0)}-r_{0}\partial_{ZZ}\hat{u}_{p}^{(0)}=0,\\\\[5.0pt]
\partial_{\theta}\hat{u}_{p}^{(0)}+r_{0}\partial_{Z}\hat{v}_{p}^{(1)}=0,\\\\[5.0pt]
\hat{u}_{p}^{(0)}(\theta,Z)=\hat{u}_{p}^{(0)}(\theta+2\pi,Z),\quad\hat{v}_{p}^{(0)}(\theta,Z)=\hat{v}_{p}^{(0)}(\theta+2\pi,Z),\\\\[5.0pt]
\hat{u}_{p}^{(0)}\big{|}_{Z=0}=\beta+\eta g(\theta)-u_{e}(r_{0}),\ \
\lim\limits_{Z\rightarrow+\infty}\hat{u}_{p}^{(0)}=\lim\limits_{Z\rightarrow+\infty}\hat{v}_{p}^{(1)}=0\end{array}\right.$
(2.29)

and the pressure $\hat{p}_{p}^{(1)}$ satisfies

$\displaystyle
r_{0}\partial_{Z}\hat{p}_{p}^{(1)}(\theta,Z)=(\hat{u}_{p}^{(0)})^{2}(\theta,Z)+2u_{e}(r_{0})\hat{u}_{p}^{(0)}(\theta,Z),\quad\lim_{Z\rightarrow+\infty}\hat{p}_{p}^{(1)}(\theta,Z)=0.$
(2.30)

####  2.3.3. Equations for $(\hat{u}_{p}^{(1)},\hat{v}_{p}^{(2)})$

By substituting the above expansions into the first and third equation of
(1.8) and collecting the $\varepsilon$-order terms, we obtain the following
linearized steady Prandtl equations for
$(\hat{u}_{p}^{(1)},\hat{v}_{p}^{(2)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(r_{0})+\hat{u}_{p}^{(0)}\big{)}\partial_{\theta}\hat{u}_{p}^{(1)}+\big{(}v_{e}^{(1)}(\theta,r_{0})+\hat{v}_{p}^{(1)}\big{)}r_{0}\partial_{Z}\hat{u}_{p}^{(1)}+\hat{u}_{p}^{(1)}\partial_{\theta}\hat{u}_{p}^{(0)}\\\\[5.0pt]
\quad\quad\quad\quad\quad\quad\quad\quad\quad+(\hat{v}_{p}^{(2)}-\hat{v}_{p}^{(2)}(\theta,0))r_{0}\partial_{Z}\hat{u}_{p}^{(0)}-r_{0}\partial_{ZZ}\hat{u}_{p}^{(1)}=\hat{f}_{1}(\theta,Z),\\\\[5.0pt]
\partial_{\theta}\hat{u}_{p}^{(1)}+r_{0}\partial_{Z}\hat{v}_{p}^{(2)}+\partial_{Z}(Z\hat{v}_{p}^{(1)})=0,\\\\[5.0pt]
\hat{u}_{p}^{(1)}(\theta,Z)=\hat{u}_{p}^{(1)}(\theta+2\pi,Z),\quad\hat{v}_{p}^{(2)}(\theta,Z)=\hat{v}_{p}^{(2)}(\theta+2\pi,Z),\\\\[5.0pt]
\hat{u}_{p}^{(1)}\big{|}_{Z=0}=-u_{e}^{(1)}\big{|}_{r=r_{0}},\ \
\lim\limits_{Z\rightarrow+\infty}\partial_{Z}\hat{u}_{p}^{(1)}=\lim\limits_{Z\rightarrow+\infty}\hat{v}_{p}^{(2)}=0,\end{array}\right.$
(2.36)

where

$\displaystyle\hat{f}_{1}(\theta,Y)=$
$\displaystyle-\partial_{\theta}\hat{p}_{p}^{(1)}+Z\partial_{ZZ}\hat{u}_{p}^{(0)}+\partial_{Z}\hat{u}_{p}^{(0)}$
$\displaystyle-\hat{u}_{p}^{(0)}(\partial_{\theta}u_{e}^{(1)}(\theta,r_{0})+v_{e}^{(1)}(\theta,r_{0})+\hat{v}_{p}^{(1)})-(u^{\prime}_{e}(r_{0})Z+u_{e}^{(1)}(\theta,r_{0}))\partial_{\theta}\hat{u}_{p}^{(0)}$
$\displaystyle-(r_{0}\partial_{r}v_{e}^{(1)}(\theta,r_{0})+v_{e}^{(1)}(\theta,r_{0})+\hat{v}_{p}^{(1)})Z\partial_{Z}u_{p}^{(0)}-[u_{e}(r_{0})+r_{0}u^{\prime}_{e}(r_{0})]\hat{v}_{p}^{(1)}.$

Before performing the higher order expansions we then aim to solve the above
Euler equations and Prandtl equations.

###  2.4. Solvabilities of Euler equations and Prandtl equations

The order in which we solve the equation is as follows

$\displaystyle(u_{e}(r),0)\rightarrow(u_{p}^{(0)},v_{p}^{(1)})/(\hat{u}_{p}^{(0)},\hat{v}_{p}^{(1)})\rightarrow(u_{e}^{(1)},v_{e}^{(1)})\rightarrow(u_{p}^{(1)},v_{p}^{(2)})/(\hat{u}_{p}^{(1)},\hat{v}_{p}^{(2)}).$

####  2.4.1. Prandtl system and its solvability

We first give some necessary conditions for the solvability of Prandtl
equations, one can also refer to Lemma 2.1 in [7].

######  Lemma 2.1.

(Batchelor-Wood formula [21][22]) Let
$\int_{0}^{2\pi}f(\theta)d\theta=\int_{0}^{2\pi}g(\theta)d\theta=0$. If the
nonlinear Prandtle equations (2.17) have a solution
$(u_{p}^{(0)},v_{p}^{(1)})$ and (2.29) have a solution
$(\hat{u}_{p}^{(0)},\hat{v}_{p}^{(1)})$ which satisfy

$\displaystyle u_{e}(1)+u_{p}^{(0)}(\theta,Y)>0,\quad\forall Y\leq 0;\quad
u_{e}(r_{0})+\hat{u}_{p}^{(0)}(\theta,Z)>0,\quad\forall Z\geq 0,$
$\displaystyle\|u_{p}^{(0)}\|_{\infty}+\|\hat{u}_{p}^{(0)}\|_{\infty}\leq M,$

where $M>0$ is a constant, then there hold

$\displaystyle
u_{e}^{2}(1)=\alpha^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}f^{2}(\theta)d\theta,$
(2.37)

and

$\displaystyle
u_{e}^{2}(r_{0})=\beta^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}g^{2}(\theta)d\theta.$
(2.38)

###### Proof.

We only need to prove $(\ref{BW formula})$ since $(\ref{BW formula-1})$ can be
derived similarly. We introduce the von Mises variable

$\displaystyle\psi=\int_{0}^{Y}\big{(}u_{e}(1)+u_{p}^{(0)}(\theta,z)\big{)}dz,\
\ \mathcal{U}^{(0)}(\theta,\psi)=u_{e}(1)+u_{p}^{(0)}(\theta,Y).$

Then from (2.17), we deduce that $\mathcal{U}^{(0)}$ satisfies

$\displaystyle\left\\{\begin{array}[]{ll}2\mathcal{U}^{(0)}_{\theta}=\big{(}(\mathcal{U}^{(0)})^{2}\big{)}_{\psi\psi},\\\\[5.0pt]
\mathcal{U}^{(0)}(\theta,\psi)=\mathcal{U}^{(0)}(\theta+2\pi,\psi),\\\\[5.0pt]
\mathcal{U}^{(0)}\big{|}_{\psi=0}=\alpha+\eta f(\theta),\ \
\lim_{\psi\rightarrow-\infty}\mathcal{U}^{(0)}=u_{e}(1).\end{array}\right.$
(2.42)

Here we have used the facts:

$\displaystyle\partial_{\theta}u_{p}^{(0)}$
$\displaystyle=\mathcal{U}^{(0)}_{\theta}+\mathcal{U}^{(0)}_{\psi}\int_{0}^{Y}\partial_{\theta}u_{p}^{(0)}(\theta,z)\big{)}dz$
$\displaystyle=\mathcal{U}^{(0)}_{\theta}+\mathcal{U}^{(0)}_{\psi}\big{(}v_{p}^{(1)}(\theta,0)-v_{p}^{(1)}(\theta,Y)\big{)}$
$\displaystyle=\mathcal{U}^{(0)}_{\theta}-\mathcal{U}^{(0)}_{\psi}\big{(}v^{(1)}(\theta,1)+v_{p}^{(1)}(\theta,Y)\big{)},$
$\displaystyle\partial_{Y}u_{p}^{(0)}$
$\displaystyle=\mathcal{U}^{(0)}_{\psi}\mathcal{U}^{(0)}.$

Integrating the first equation in (2.42) from $0$ to $2\pi$ about $\theta$
leads to

$\displaystyle\frac{\partial^{2}}{\partial\psi^{2}}\int_{0}^{2\pi}(\mathcal{U}^{(0)})^{2}(\theta,\psi)d\theta=0.$

Notice that $\mathcal{U}^{(0)}$ is bounded at $\psi\rightarrow-\infty$, we
deduce that

$\displaystyle\frac{\partial}{\partial\psi}\int_{0}^{2\pi}(\mathcal{U}^{(0)})^{2}(\theta,\psi)d\theta=0.$

Therefore combining the boundary condition in (2.42), we deduce that

$\displaystyle u_{e}^{2}(1)=\frac{1}{2\pi}\int_{0}^{2\pi}\big{(}\alpha+\eta
f(\theta)\big{)}^{2}(\theta)d\theta$
$\displaystyle=\alpha^{2}+\frac{\alpha\eta}{\pi}\int_{0}^{2\pi}f(\theta)d\theta+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}f^{2}(\theta)d\theta$
$\displaystyle=\alpha^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}f^{2}(\theta)d\theta.$

Thus, we complete the proof of this lemma. MM $\square$

The following corollary is a direct result of Lemma 2.1.

######  Corollary 2.1.

If Lemma 2.1 holds, then $(a_{0},b_{0})$ must satisfy

$\displaystyle
a_{0}+b_{0}=\Big{(}\alpha^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}f^{2}(\theta)d\theta\Big{)}^{\frac{1}{2}},\
a_{0}r_{0}+\frac{b_{0}}{r_{0}}=\Big{(}\beta^{2}+\frac{\eta^{2}}{2\pi}\int_{0}^{2\pi}g^{2}(\theta)d\theta\Big{)}^{\frac{1}{2}}.$

Next we aim to solve the steady Prandtl equations (2.17), one can refer to
Corollary 2.2 in [7].

######  Proposition 2.2.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and any
$j,k,l\in\mathbb{N}\cup\\{0\\}$, the equations (2.42) have a unique solution
$\mathcal{U}^{(0)}$ which satisfies

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{|}\partial_{\theta}^{j}\partial_{\psi}^{k}\big{(}\mathcal{U}^{(0)}-u_{e}(1)\big{)}\Big{|}^{2}\big{<}\psi\big{>}^{2l}d\theta
d\psi\leq C(j,k,l)\eta^{2},$

here $\big{<}\psi\big{>}=\sqrt{1+\psi^{2}}$.

###### Proof.

We will use the contraction mapping theorem to prove the desired conclusions
and divide the proof into five steps.

Step 1: Derivation of equivalent equations.

Let $Q(\theta,\psi):=(\mathcal{U}^{(0)})^{2}(\theta,\psi)-u_{e}^{2}(1)$ and we
rewrite (2.42) as

$\displaystyle\left\\{\begin{array}[]{ll}Q_{\theta}=\mathcal{U}^{(0)}Q_{\psi\psi},\\\\[7.0pt]
Q(\theta,\psi)=Q(\theta+2\pi,\psi),\\\\[5.0pt]
Q\big{|}_{\psi=0}=\alpha^{2}+2\alpha\eta
f(\theta)+\eta^{2}f^{2}(\theta)-u_{e}^{2}(1),\ \
Q\big{|}_{\psi\rightarrow-\infty}=0.\end{array}\right.$ (2.46)
$\displaystyle\left\\{\begin{array}[]{ll}Q_{\theta}=\sqrt{Q+C}\cdot
Q_{\psi\psi},\\\ Q(\theta,\psi)=Q(\theta+2\pi,\psi),\\\ Q\big{|}_{\psi=0}=\eta
f(\theta),\ \ Q\big{|}_{\psi=-\infty}=0,\end{array}\right.$

here $|\eta|\ll 1$, $C>0$ is a constant.

Defining

$\displaystyle\mathcal{G}(Q)=Q-2u_{e}(1)\sqrt{Q+u_{e}^{2}(1)}+2u_{e}^{2}(1),$

then (2.46) is equivalent to

$\displaystyle\left\\{\begin{array}[]{ll}Q_{\theta}-u_{e}(1)Q_{\psi\psi}=(\mathcal{G}(Q))_{\theta},\\\\[7.0pt]
Q(\theta,\psi)=Q(\theta+2\pi,\psi),\\\\[5.0pt]
Q\big{|}_{\psi=0}=\alpha^{2}+2\alpha\eta
f(\theta)+\eta^{2}f^{2}(\theta)-u_{e}^{2}(1),\ \
Q\big{|}_{\psi\rightarrow-\infty}=0.\end{array}\right.$ (2.51)

Let $Q_{0}$ be the solution to

$\displaystyle\left\\{\begin{array}[]{ll}(Q_{0})_{\theta}=u_{e}(1)(Q_{0})_{\psi\psi},\\\\[7.0pt]
Q_{0}(\theta,\psi)=Q_{0}(\theta+2\pi,\psi),\\\\[5.0pt]
Q_{0}\big{|}_{\psi=0}=\alpha^{2}+2\alpha\eta
f(\theta)+\eta^{2}f^{2}(\theta)-u_{e}^{2}(1),\ \
Q_{0}\big{|}_{\psi\rightarrow-\infty}=0,\end{array}\right.$ (2.55)

which will be solved in Appendix A, then (2.51) is equivalent to

$\displaystyle\left\\{\begin{array}[]{ll}Q_{\theta}-u_{e}(1)Q_{\psi\psi}=(\mathcal{H}(Q))_{\theta},\\\\[7.0pt]
Q(\theta,\psi)=Q(\theta+2\pi,\psi),\\\\[5.0pt] Q\big{|}_{\psi=0}=0,\ \
Q\big{|}_{\psi\rightarrow-\infty}=0,\end{array}\right.$ (2.59)

where

$\displaystyle\mathcal{H}(Q)=Q+Q_{0}-2u_{e}(1)\sqrt{Q+Q_{0}+u_{e}^{2}(1)}+2u_{e}^{2}(1).$

Defining the linear operator $\mathcal{L}:\Lambda\longmapsto\Phi$ such that

$\displaystyle\Phi=\mathcal{L}\Lambda\Longleftrightarrow\left\\{\begin{array}[]{ll}\Phi_{\theta}-u_{e}(1)\Phi_{\psi\psi}=\Lambda_{\theta},\\\\[7.0pt]
\Phi(\theta,\psi)=\Phi(\theta+2\pi,\psi),\\\\[5.0pt] \Phi\big{|}_{\psi=0}=0,\
\ \Phi\big{|}_{\psi\rightarrow-\infty}=0,\end{array}\right.$ (2.63)

then (2.59) is equivalent to

$\displaystyle\left\\{\begin{array}[]{ll}Q=\mathcal{L}(\mathcal{H}(Q)),\\\\[7.0pt]
Q(\theta,\psi)=Q(\theta+2\pi,\psi),\\\\[5.0pt] Q\big{|}_{\psi=0}=0,\ \
Q\big{|}_{\psi=-\infty}=0.\end{array}\right.$ (2.67)

Defining the function space $X$ as follows

$\displaystyle X=\bigg{\\{}$ $\displaystyle
Q:Q(\theta,\psi)=Q(\theta+2\pi,\psi),\
Q\big{|}_{\psi=0}=Q\big{|}_{\psi\rightarrow-\infty}=0,$
$\displaystyle\quad\quad\|Q\|_{X}^{2}:=\sum\limits_{j+k\leq m,l\geq
0}\int_{-\infty}^{0}\int_{0}^{2\pi}\big{|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q\big{|}^{2}\big{<}\psi\big{>}^{l}d\theta
d\psi<+\infty\bigg{\\}}$

and a ball $B_{0}$ in $X$

$\displaystyle B_{0}=\big{\\{}Q\in X:\|Q\|_{X}\leq\nu\big{\\}},$

here $m$ is a positive integer and $\nu$ is a small number which will be
determined later.

In the next three steps we aim to verify that $\mathcal{L}\circ\mathcal{H}$ is
a contraction map from $B_{0}$ to $B_{0}$ with suitable small $\nu$.

Step 2: Boundedness of $\mathcal{L}$ in $X$. In this step, we prove that for
any $m\geq 0,l\geq 0$, there holds

$\displaystyle\sum_{j+k\leq
m}\big{\|}\partial^{j}_{\theta}\partial^{k}_{\psi}\Phi\big{<}\psi\big{>}^{l}\big{\|}_{2}\leq
C(m,l)\sum_{j+k\leq m,q\leq
l}\big{\|}\partial^{j}_{\theta}\partial^{k}_{\psi}\Lambda\big{<}\psi\big{>}^{q}\big{\|}_{2}.$
(2.68)

First, we prove (2.68) for $l=0$. Multiplying the equation in (2.63) by $\Phi$
and integrating with respect to $(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$,
we obtain for any $\lambda>0$

$\displaystyle u_{e}(1)\|\Phi_{\psi}\|_{2}\leq
C(\lambda)\|\Lambda\|_{2}+\lambda\|\Phi_{\theta}\|_{2}.$ (2.69)

Then, multiplying the equation in (2.63) by $\Phi_{\theta}$ and integrating
with respect to $(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$, one has

$\displaystyle\|\Phi_{\theta}\|_{2}\leq
C(\lambda)\|\Lambda_{\theta}\|_{2}+\lambda\|\Phi_{\theta}\|_{2}.$ (2.70)

Combining (2.69)-(2.70) and choosing small $\lambda>0$ we get

$\displaystyle\|(\Phi_{\psi},\Phi_{\theta})\|_{2}\leq
C\|(\Lambda,\Lambda_{\theta})\|_{2}.$ (2.71)

Integrating (2.63) with respect to $\theta\in(0,2\pi)$ gives

$\displaystyle\frac{d^{2}}{d\psi^{2}}\int_{0}^{2\pi}\Phi(\theta,\psi)d\theta=0,$

which and $\Phi|_{\psi=0}=\Phi|_{\psi\rightarrow-\infty}=0$ imply

$\displaystyle\int_{0}^{2\pi}\Phi(\theta,\psi)d\theta=0.$ (2.72)

Due to (2.72) and the Poincaré inequality we have

$\displaystyle\|\Phi\|_{2}\leq C\|\Phi_{\theta}\|_{2}\leq
C\|(\Lambda,\Lambda_{\theta})\|_{2}.$ (2.73)

For any $j\geq 0,k\geq 2$, from the equation (2.63), we deduce that

$\displaystyle\|\partial_{\theta}^{j}\partial_{\psi}^{k}\Phi\|_{2}\leq
C(\|\partial_{\theta}^{j}\partial_{\psi}^{k-2}\Phi_{\theta}\|_{2}+\|\partial_{\theta}^{j}\partial_{\psi}^{k-2}\Lambda_{\theta}\|_{2}).$
(2.74)

For any $j\geq 0$, applying $\partial^{j}_{\theta}$ to the equation in (2.63),
multiplying the resultant equation by $\partial^{j}_{\theta}\Phi$, integrating
with respect to $(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$ and using the
Young inequality one has

$\displaystyle\|\partial^{j}_{\theta}\partial_{\psi}\Phi\|_{2}\leq
C\|\partial^{j+1}_{\theta}\Lambda\|_{2}+C\|\partial^{j}_{\theta}\Phi\|_{2}.$
(2.75)

For any $j\geq 0$, multiplying the equation in (2.63) by
$\partial^{2j-1}_{\theta}\Phi$, integrating with respect to
$(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$, one can get

$\displaystyle\|\partial^{j}_{\theta}\Phi\|^{2}_{2}=\int_{-\infty}^{0}\int_{0}^{2\pi}\partial^{2j-1}_{\theta}\Phi\Lambda_{\theta}d\theta
d\psi\leq
C\|\partial^{j}_{\theta}\Phi\|_{2}\|\partial^{j}_{\theta}\Lambda\|_{2}.$
(2.76)

Based on (2.74), (2.75) and (2.76), we obtain that for any $m\geq 2$, there
holds

$\displaystyle\sum_{j+k=m}\|\partial^{j}_{\theta}\partial^{k}_{\psi}\Phi\|_{2}\leq
C\sum_{j+k\leq
m-1}\|\partial^{j}_{\theta}\partial^{k}_{\psi}\Phi\|_{2}+C\sum_{j+k\leq
m}\|\partial^{j}_{\theta}\partial^{k}_{\psi}\Lambda\|_{2}.$

Thus, by induction and (2.71), (2.73), we deduce that for any $m\geq 0$, there
holds

$\displaystyle\sum_{j+k\leq
m}\|\partial^{j}_{\theta}\partial^{k}_{\psi}\Phi\|_{2}\leq C(m)\sum_{j+k\leq
m}\|\partial^{j}_{\theta}\partial^{k}_{\psi}\Lambda\|_{2}.$ (2.77)

Next, we prove (2.68) for $m\leq 1,l\geq 1$. Multiplying the equation in
(2.63) by $\Phi\psi^{2l}$, integrating with respect to
$(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$, we have

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\Phi_{\psi}^{2}\psi^{2l}d\theta
d\psi$ $\displaystyle\leq
C\int_{-\infty}^{0}\int_{0}^{2\pi}\Phi^{2}\psi^{2(l-1)}d\theta d\psi$
$\displaystyle\quad+C(\lambda)\int_{-\infty}^{0}\int_{0}^{2\pi}\Lambda^{2}\psi^{2l}d\theta
d\psi+\lambda\int_{-\infty}^{0}\int_{0}^{2\pi}\Phi_{\theta}^{2}\psi^{2l}d\theta
d\psi.$ (2.78)

Multiplying the equation in (2.63) by $\Phi_{\theta}\psi^{2l}$, integrating
with respect to $(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$, we obtain

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\Phi_{\theta}^{2}\psi^{2l}d\theta
d\psi$ $\displaystyle\leq
C\int_{-\infty}^{0}\int_{0}^{2\pi}\Phi_{\psi}^{2}\psi^{2(l-1)}d\theta
d\psi+C\int_{-\infty}^{0}\int_{0}^{2\pi}\Lambda_{\theta}^{2}\psi^{2l}d\theta
d\psi.$ (2.79)

Combining (2.78)-(2.79), using the Poincaré inequality and choosing small
$\lambda>0$ we get

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\big{(}\Phi^{2}+\Phi_{\psi}^{2}+\Phi_{\theta}^{2}\big{)}\psi^{2l}d\theta
d\psi$ $\displaystyle\leq$
$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\big{(}\Phi^{2}+\Phi_{\psi}^{2}+\Phi_{\theta}^{2}\big{)}\psi^{2(l-1)}d\theta
d\psi+C\int_{-\infty}^{0}\int_{0}^{2\pi}\big{(}\Lambda^{2}\psi^{2l}+\Lambda_{\theta}^{2}\psi^{2l}\big{)}d\theta
d\psi.$

By induction on $l$ and using (2.77), we deduce that for any $l\geq 1$, there
holds

$\displaystyle\big{\|}\big{(}\Phi_{\psi},\Phi_{\theta},\Phi\big{)}\psi^{l}\big{\|}_{2}\leq
C(l)\sum_{q\leq
l}\big{\|}\big{(}\Lambda,\Lambda_{\theta}\big{)}\psi^{q}\big{\|}_{2}.$ (2.80)

Finally, we prove (2.68) any $m\geq 2,l\geq 1$. When $j+k=m,k\geq 2$, from the
equation (2.63), we deduce that

$\displaystyle\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}\Phi\psi^{l}\Big{\|}_{2}\leq
C\Big{(}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k-2}\partial_{\theta}\Phi\psi^{l}\Big{\|}_{2}+\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k-2}\partial_{\theta}\Lambda\psi^{l}\Big{\|}_{2}\Big{)}.$
(2.81)

Applying $\partial^{m-1}_{\theta}$ to the equation in (2.63), multiplying the
resultant equation by $\partial^{m-1}_{\theta}\Phi\psi^{2l}$, integrating with
respect to $(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$ and using the Young
inequality one has

$\displaystyle\Big{\|}\partial_{\theta}^{m-1}\partial_{\psi}\Phi\psi^{l}\Big{\|}_{2}\leq
C\Big{\|}\partial_{\theta}^{m}\Lambda\psi^{l}\Big{\|}_{2}+C\Big{\|}\partial_{\theta}^{m-1}\Phi\psi^{l}\Big{\|}_{2}+C(l)\Big{\|}\partial_{\theta}^{m-1}\Phi\psi^{l-1}\Big{\|}_{2}.$
(2.82)

Multiplying the equation in (2.63) by $\partial^{2m-1}_{\theta}\Phi\psi^{2l}$,
integrating with respect to $(\theta,\psi)\in(0,2\pi)\times(-\infty,0)$, one
can get

$\displaystyle\Big{\|}\partial^{m}_{\theta}\Phi\psi^{l}\Big{\|}_{2}\leq
C\Big{\|}\partial^{m}_{\theta}\Lambda\psi^{l}\Big{\|}_{2}+C(l)\Big{\|}\partial_{\theta}^{m-1}\partial_{\psi}\Phi\psi^{l-1}\Big{\|}_{2}.$
(2.83)

Thanks to (2.81), (2.82) and (2.83), we obtain that for any $m\geq 2,l\geq 1$,
there holds

$\displaystyle\sum_{j+k=m}\Big{\|}\partial^{j}_{\theta}\partial^{k}_{\psi}\Phi\psi^{l}\Big{\|}_{2}\leq
C(m,l)\sum_{j+k\leq m-1,q\leq
l}\Big{\|}\partial^{j}_{\theta}\partial^{k}_{\psi}\Phi\psi^{q}\Big{\|}_{2}+C(m,l)\sum_{j+k\leq
m}\Big{\|}\partial^{j}_{\theta}\partial^{k}_{\psi}\Lambda\psi^{l}\Big{\|}_{2}.$

Thus, by induction $m$ and using (2.80), we deduce that for any $m\geq 2,l\geq
1$, there holds

$\displaystyle\sum_{j+k\leq
m}\Big{\|}\partial^{j}_{\theta}\partial^{k}_{\psi}\Phi\psi^{l}\Big{\|}_{2}\leq
C(m,l)\sum_{j+k\leq m,q\leq
l}\Big{\|}\partial^{j}_{\theta}\partial^{k}_{\psi}\Lambda\psi^{q}\Big{\|}_{2}.$
(2.84)

This complete the proof of (2.68). Thus, we obtain

$\displaystyle\|\mathcal{L}\Lambda\|_{X}\leq C\|\Lambda\|_{X},\ \
\forall\Lambda\in X.$ (2.85)

Step 3: $\mathcal{L}\circ\mathcal{H}$ is a continuous map from $B_{0}$ to
$B_{0}$. In this section, we first prove that for any $m\geq 2,l\geq 0$, there
holds

$\displaystyle\sum_{j+k\leq
m}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}\mathcal{H}(Q)\big{<}\psi\big{>}^{l}\Big{\|}_{2}\leq
C(m,l)\Big{(}\sum_{j+k\leq
m}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q\big{<}\psi\big{>}^{l}\Big{\|}_{2}+\sum_{j+k\leq
m}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q_{0}\big{<}\psi\big{>}^{l}\Big{\|}_{2}\Big{)}^{2}.$
(2.86)

Set

$\displaystyle\tilde{H}(x)=x-2u_{e}(1)\sqrt{x+u_{e}^{2}(1)}+2u^{2}_{e}(1),|x|\ll
u_{e}^{2}(1),$

then $\mathcal{H}(Q)=\tilde{H}(Q+Q_{0}).$ Direct computation gives

$\displaystyle|\tilde{H}^{\prime}(x)|\leq C|x|,\quad|\tilde{H}^{(k)}(x)|\leq
C,\quad k\geq 2.$ (2.87)

Since

$\displaystyle\mathcal{H}(Q)=\big{(}\sqrt{Q+Q_{0}+u_{e}^{2}(1)}-u_{e}(1)\big{)}^{2}=\bigg{(}\frac{Q+Q_{0}}{\sqrt{Q+Q_{0}+u_{e}^{2}(1)}+u_{e}(1)}\bigg{)}^{2},$

it’s easy to get

$\displaystyle\mathcal{H}^{2}(Q)\big{<}\psi\big{>}^{l}\leq
C\|Q+Q_{0}\|_{L^{\infty}}^{2}\big{(}Q^{2}\big{<}\psi\big{>}^{l}+Q_{0}^{2}\big{<}\psi\big{>}^{l}\big{)},\quad
l\geq 0.$

Using (2.87), we deduce that

$\displaystyle\big{(}\partial_{\psi}\mathcal{H}(Q)\big{)}^{2}\big{<}\psi\big{>}^{l}\leq$
$\displaystyle
C\|Q+Q_{0}\|_{L^{\infty}}^{2}\big{(}Q_{\psi}^{2}\big{<}\psi\big{>}^{l}+(Q_{0})_{\psi}^{2}\big{<}\psi\big{>}^{l}\big{)},\quad
l\geq 0,$
$\displaystyle\big{(}\partial_{\theta}\mathcal{H}(Q)\big{)}^{2}\big{<}\psi\big{>}^{l}\leq$
$\displaystyle
C\|Q+Q_{0}\|_{L^{\infty}}^{2}\big{(}Q_{\theta}^{2}\big{<}\psi\big{>}^{l}+(Q_{0})_{\theta}^{2}\big{<}\psi\big{>}^{l}\big{)},\quad
l\geq 0.$

Thus, we obtain

$\displaystyle\sum_{j+k\leq
1}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}\mathcal{H}(Q)\big{<}\psi\big{>}^{l}\Big{\|}_{2}$
$\displaystyle\leq$ $\displaystyle
C\|Q+Q_{0}\|_{L^{\infty}}\Big{(}\sum_{j+k\leq
1}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q\big{<}\psi\big{>}^{l}\Big{\|}_{2}+\sum_{j+k\leq
1}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q_{0}\big{<}\psi\big{>}^{l}\Big{\|}_{2}\Big{)}.$
(2.88)

Since

$\displaystyle\partial_{\psi\theta}\mathcal{H}(Q)=\tilde{H}^{\prime}(Q+Q_{0})(\partial_{\psi\theta}Q+\partial_{\psi\theta}Q_{0})+\tilde{H}^{\prime\prime}(Q+Q_{0})(\partial_{\theta}Q+\partial_{\theta}Q_{0})(\partial_{\psi}Q+\partial_{\psi}Q_{0}),$

thus, using (2.87), we obtain

$\displaystyle\Big{\|}\partial_{\psi\theta}\mathcal{H}(Q)\big{<}\psi\big{>}^{l}\Big{\|}_{L^{2}}$
$\displaystyle\leq
C\|Q+Q_{0}\|_{L^{\infty}}\Big{(}\Big{\|}Q_{\psi\theta}\big{<}\psi\big{>}^{l}\Big{\|}_{L^{2}}+\Big{\|}(Q_{0})_{\psi\theta}\big{<}\psi\big{>}^{l}\Big{\|}_{L^{2}}\Big{)}$
$\displaystyle\quad+C\big{\|}Q_{\psi}+(Q_{0})_{\psi}\big{\|}_{L^{4}}\big{\|}(Q_{\theta}+(Q_{0})_{\theta})\big{<}\psi\big{>}^{l}\big{\|}_{L^{4}}$
$\displaystyle\leq
C\|Q+Q_{0}\|_{L^{\infty}}\Big{(}\Big{\|}Q_{\psi\theta}\big{<}\psi\big{>}^{l}\Big{\|}_{L^{2}}+\Big{\|}(Q_{0})_{\psi\theta}\big{<}\psi\big{>}^{l}\Big{\|}_{L^{2}}\Big{)}$
$\displaystyle\quad+C\big{\|}Q_{\psi}+(Q_{0})_{\psi}\big{\|}_{H^{1}}\big{\|}(Q_{\theta}+(Q_{0})_{\theta})\big{<}\psi\big{>}^{l}\big{\|}_{H^{1}}.$

Hence by Sobolev imbedding, we deduce that

$\displaystyle\Big{\|}\partial_{\psi\theta}\mathcal{H}(Q)\big{<}\psi\big{>}^{l}\Big{\|}_{2}\leq
C\Big{(}\sum_{j+k\leq
2}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q\big{<}\psi\big{>}^{l}\Big{\|}_{2}+\sum_{j+k\leq
2}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q_{0}\big{<}\psi\big{>}^{l}\Big{\|}_{2}\Big{)}^{2}.$

Same estimates hold for
$\Big{\|}\partial_{\theta\theta}\mathcal{H}(Q)\big{<}\psi\big{>}^{l}\Big{\|}_{2}$
and
$\Big{\|}\partial_{\psi\psi}\mathcal{H}(Q)\big{<}\psi\big{>}^{l}\Big{\|}_{2}$.
Thus, combing the estimate (2.4.1), we arrive at

$\displaystyle\sum_{j+k\leq
2}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}\mathcal{H}(Q)\big{<}\psi\big{>}^{l}\Big{\|}_{2}\leq
C\Big{(}\sum_{j+k\leq
2}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q\big{<}\psi\big{>}^{l}\Big{\|}_{2}+\sum_{j+k\leq
2}\Big{\|}\partial_{\theta}^{j}\partial_{\psi}^{k}Q_{0}\big{<}\psi\big{>}^{l}\Big{\|}_{2}\Big{)}^{2}.$

Using (2.87) and repeating the above arguments, we obtain (2.86).

Consequently, if we take $\nu=\|Q_{0}\|_{X}$ and $\eta$ small enough, then

$\displaystyle\|\mathcal{L}\mathcal{H}(Q)\|_{X}\leq
C\|\mathcal{H}(Q)\|_{X}\leq 4C\nu^{2}\leq 4C\eta\nu\leq\nu,\ \ \forall Q\in
B_{0},$

here we have used (5.6). Thus, $\mathcal{L}\mathcal{H}$ is a continuous map
from $B_{0}$ to $B_{0}$ for small $\eta$.

Step 4: $\mathcal{L}\circ\mathcal{H}$ is a contraction map in $B_{0}$.

Noting firstly that

$\displaystyle\mathcal{H}(Q_{1})-\mathcal{H}(Q_{2})$
$\displaystyle=(Q_{1}-Q_{2})\bigg{(}1-\frac{2u_{e}(1)}{\sqrt{Q_{1}+Q_{0}+u_{e}^{2}(1)}+\sqrt{Q_{2}+Q_{0}+u_{e}^{2}(1)}}\bigg{)}$

and

$\displaystyle
1-\frac{2u_{e}(1)}{\sqrt{Q_{1}+Q_{0}+u_{e}^{2}(1)}+\sqrt{Q_{2}+Q_{0}+u_{e}^{2}(1)}}$
$\displaystyle=\frac{Q_{1}+Q_{0}}{\big{(}\sqrt{Q_{1}+Q_{0}+u_{e}^{2}(1)}+\sqrt{Q_{2}+Q_{0}+u_{e}^{2}(1)}\big{)}\big{(}\sqrt{Q_{1}+Q_{0}+u_{e}^{2}(1)}+u_{e}(1)\big{)}}$
$\displaystyle\quad+\frac{Q_{2}+Q_{0}}{\big{(}\sqrt{Q_{1}+Q_{0}+u_{e}^{2}(1)}+\sqrt{Q_{2}+Q_{0}+u_{e}^{2}(1)}\big{)}\big{(}\sqrt{Q_{2}+Q_{0}+u_{e}^{2}(1)}+u_{e}(1)\big{)}}.$

With the help of the similar arguments in Step 3, we can obtain that there
exist $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$, there holds

$\displaystyle\big{\|}\mathcal{L}\mathcal{H}(Q_{1})-\mathcal{L}\mathcal{H}(Q_{2})\big{\|}_{X}$
$\displaystyle\leq C\big{\|}\mathcal{H}(Q_{1})-\mathcal{H}(Q_{2})\big{\|}_{X}$
$\displaystyle\leq
C\big{\|}Q_{1}-Q_{2}\big{\|}_{X}\big{(}\|Q_{0}\|_{X}+\|Q_{1}\|_{X}+\|Q_{2}\|_{X}\big{)}$
$\displaystyle\leq C\|Q_{0}\|_{X}\big{\|}Q_{1}-Q_{2}\big{\|}_{X}$
$\displaystyle\leq\frac{1}{2}\big{\|}Q_{1}-Q_{2}\big{\|}_{X},$

that is, $\mathcal{L}\mathcal{H}$ is a contraction map in $B_{0}$.

Step 5: Existence and uniqueness of the system (2.42). By the standard
contraction mapping principle, we know that there exists $\eta_{0}>0$ such
that for any $\eta\in(0,\eta_{0})$ and any $j,k,l\in\mathbb{N}\cup\\{0\\}$,
the equation (2.42) has a unique solution $\mathcal{U}^{(0)}$ which satisfies

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{|}\partial_{\theta}^{j}\partial_{\psi}^{k}\big{(}\mathcal{U}^{(0)}-u_{e}(1)\big{)}\Big{|}^{2}\big{<}\psi\big{>}^{2l}d\theta
d\psi\leq C(j,k,l)\eta^{2},$

this completes the proof of this proposition. MM $\square$

Return to the equations (2.17), we have the following result.

######  Proposition 2.3.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$, equation
(2.17) has a unique solution $(u_{p}^{(0)},v_{p}^{(1)})$ which satisfies

$\displaystyle\begin{aligned} &\sum_{j+k\leq
m}\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{|}\partial_{\theta}^{j}\partial_{Y}^{k}(u_{p}^{(0)},v_{p}^{(1)})\Big{|}^{2}\big{<}Y\big{>}^{2l}d\theta
dY\leq C(m,l)\eta^{2},\ \forall m,l\geq 0,\\\
&\int_{0}^{2\pi}v_{p}^{(1)}(\theta,Y)d\theta=0,\ \forall Y\leq
0.\end{aligned}$ (2.89)

Finally, solving (2.18), we obtain $p_{p}^{(1)}(\theta,Y)$ which decay very
fast as $Y\rightarrow-\infty$.

By the same argument, we can obtain the well-posedness of equations (2.29).

######  Proposition 2.4.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$, equation
(2.29) has a unique solution $(\hat{u}_{p}^{(0)},\hat{v}_{p}^{(1)})$ which
satisfies

$\displaystyle\begin{aligned} &\sum_{j+k\leq
m}\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{|}\partial_{\theta}^{j}\partial_{Y}^{k}(\hat{u}_{p}^{(0)},\hat{v}_{p}^{(1)})\Big{|}^{2}\big{<}Y\big{>}^{2l}d\theta
dY\leq C(m,l)\eta^{2},\ m,l\geq 0,\\\
&\int_{0}^{2\pi}\hat{v}_{p}^{(1)}(\theta,Z)d\theta=0,\ \forall Z\geq
0.\end{aligned}$ (2.90)

Similarly, we can obtain $\hat{p}_{p}^{(1)}$ by solving (2.30) and
$\hat{p}_{p}^{(1)}$ decay very fast as $Z\rightarrow+\infty$.

####  2.4.2. Linearized Euler equations for
$(u_{e}^{(1)},v_{e}^{(1)},p_{e}^{(1)})$ and their solvability

######  Proposition 2.5.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and
$\delta\in[0,1]$, the linearized Euler equation (2.9) has a solution
$(u_{e}^{(1)},v_{e}^{(1)},p_{e}^{(1)})$ which satisfies

$\displaystyle\|\partial^{k}_{\theta}\partial^{j}_{r}(u_{e}^{(1)},v_{e}^{(1)})\|_{2}\leq
C(\delta,k,j)\eta,\quad\forall j,k\geq 0.$ (2.91)

###### Proof.

Eliminating pressure $p_{e}^{(1)}$ in the equation (2.9), we obtain

$\displaystyle-
u_{e}(r)\Big{(}\partial_{rr}v_{e}^{(1)}+\frac{\partial_{\theta\theta}v_{e}^{(1)}}{r^{2}}+\frac{3\partial_{r}v_{e}^{(1)}}{r}\Big{)}+v_{e}^{(1)}\Big{(}u^{\prime\prime}_{e}(r)+\frac{u^{\prime}_{e}(r)}{r}-\frac{2u_{e}(r)}{r^{2}}\Big{)}=0.$

Notice that
$\triangle=\partial_{rr}+\frac{\partial_{\theta\theta}}{r^{2}}+\frac{\partial_{r}}{r}$,
we deduce that

$\displaystyle-
u_{e}(r)\triangle(rv_{e}^{(1)})+\Big{(}u^{\prime\prime}_{e}(r)+\frac{u^{\prime}_{e}(r)}{r}-\frac{u_{e}(r)}{r^{2}}\Big{)}(rv_{e}^{(1)})=0,$

hence we obtain the following equation for $rv_{e}^{(1)}$ in $\Omega$

$\displaystyle\left\\{\begin{array}[]{ll}-\triangle(rv_{e}^{(1)})+\frac{1}{u_{e}(r)}\Big{(}u^{\prime\prime}_{e}(r)+\frac{u^{\prime}_{e}(r)}{r}-\frac{u_{e}(r)}{r^{2}}\Big{)}(rv_{e}^{(1)})=0,\\\\[5.0pt]
rv_{e}^{(1)}|_{r=1}=-v_{p}^{(1)}|_{Y=0},\quad
rv_{e}^{(1)}|_{r=r_{0}}=-r_{0}\hat{v}_{p}^{(1)}|_{Z=0}.\end{array}\right.$
(2.94)

Firstly, we deduce that equations (2.94) a unique solution $rv_{e}^{(1)}$. In
fact, we assume that $f(\theta,r)$ solving the equations

$\displaystyle\left\\{\begin{array}[]{ll}-\triangle
f+\frac{1}{u_{e}(r)}\Big{(}u^{\prime\prime}_{e}(r)+\frac{u^{\prime}_{e}(r)}{r}-\frac{u_{e}(r)}{r^{2}}\Big{)}f=0,\\\\[5.0pt]
f|_{r=1}=0,\ f|_{r=r_{0}}=0,\end{array}\right.$

then multiple $rf$ and integrating in $\Omega$, we obtain

$\displaystyle\int_{0}^{2\pi}\int_{r_{0}}^{1}\Big{(}r(\partial_{r}f)^{2}+(\partial_{\theta}f)^{2}+\frac{1}{u_{e}(r)}\Big{(}ru^{\prime\prime}_{e}(r)+u^{\prime}_{e}(r)-\frac{u_{e}(r)}{r}\Big{)}f^{2}\Big{)}d\theta
dr=0.$

Let $g=\frac{f}{u_{e}}$, we arrive at

$\displaystyle\int_{0}^{2\pi}\int_{r_{0}}^{1}\Big{(}ru^{2}_{e}(r)(\partial_{r}g)^{2}+\frac{u_{e}^{2}(r)g^{2}_{\theta}}{r}-\frac{u_{e}^{2}(r)g^{2}}{r}\Big{)}d\theta
dr=0.$

Notice that $\int_{0}^{2\pi}g(\theta,r)d\theta=0,\ \forall r\in[r_{0},1]$, by
Póincare inequality, we deduce that $g=0$, hence $f=0$. Thus, equation (2.94)
has a unique solution.

Then, notice (2.89) and (2.90), we deduce that

$\displaystyle\|\partial^{k}_{\theta}\partial^{j}_{r}v_{e}^{(1)}\|_{2}\leq
C(\delta,k,j)\eta,\quad\forall k,j\geq 0.$

Moreover, due to
$\int_{0}^{2\pi}v_{p}^{(1)}(\theta,0)d\theta=\int_{0}^{2\pi}\hat{v}_{p}^{(1)}(\theta,0)d\theta=0$
and the divergence-free condition, we deduce that

$\displaystyle\int_{0}^{2\pi}v_{e}^{(1)}(\theta,r)d\theta=0,\quad\forall
r\in[r_{0},1].$ (2.97)

Then we can construct $u_{e}^{(1)}$ by solving the following equation

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\theta}u_{e}^{(1)}+\partial_{r}(rv_{e}^{(1)})=0,\\\\[5.0pt]
u_{e}^{(1)}(\theta,r)=u_{e}^{(1)}(\theta+2\pi,r).\end{array}\right.$

After obtaining $(u_{e}^{(1)},v_{e}^{(1)})$, we construct $p_{e}^{(1)}$ as
following

$\displaystyle
p_{e}^{(1)}(\theta,r):=\phi(r)-\int_{0}^{\theta}[u_{e}(r)\partial_{\theta^{\prime}}u_{e}^{(1)}+ru^{\prime}_{e}v_{e}^{(1)}+u_{e}v_{e}^{(1)}](\theta^{\prime},r)d\theta^{\prime},$

where $\phi(r)$ is a function which satisfies

$\displaystyle
r\partial_{r}\phi(r)+u_{e}(r)\partial_{\theta}v_{e}^{(1)}(0,r)-2u_{e}(r)u_{e}^{(1)}(0,r)=0.$

Combining the equations of $(u_{e}^{(1)},v_{e}^{(1)})$, it’s direct to obtain

$\displaystyle
u_{e}\partial_{\theta}v_{e}^{(1)}-2u_{e}u_{e}^{(1)}+r\partial_{r}p_{e}^{(1)}=0.$

Hence, $(u_{e}^{(1)},v_{e}^{(1)},p_{e}^{(1)})$ solves the equation (2.9) and
satisfies (2.91). MM $\square$

####  2.4.3. Linearized Prandtl equations for $(u_{p}^{(1)},v_{p}^{(2)})$,
$(\hat{u}_{p}^{(1)},\hat{v}_{p}^{(2)})$ and their solvabilities

In this subsubsection, we consider the solvabilities of (2.24) and (2.36). One
can also refer to Proposition 2.5 in [7].

######  Proposition 2.6.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and
$\delta\in[0,1]$, the equation (2.24) has a unique solution
$(u_{p}^{(1)},v_{p}^{(2)})$ which satisfies

$\displaystyle\begin{aligned} &\sum_{j+k\leq
m}\int_{-\infty}^{0}\int_{0}^{2\pi}\big{|}\partial_{\theta}^{j}\partial_{Y}^{k}\big{(}u_{p}^{(1)}-A_{1\infty},v_{p}^{(2)}\big{)}\big{|}^{2}\big{<}Y\big{>}^{2l}d\theta
dY\leq C(\delta,m,l)\eta^{2},\ \forall m,l\geq 0,\\\
&\int_{0}^{2\pi}v_{p}^{(2)}(\theta,Y)d\theta=0,\ \forall Y\leq
0,\end{aligned}$ (2.100)

where $A_{1\infty}:=\lim\limits_{Y\rightarrow-\infty}u_{p}^{(1)}(\theta,Y)$ is
a constant which satisfies $|A_{1\infty}|\leq C(\delta)\eta.$

###### Proof.

Let $\eta\in C_{c}^{\infty}((-\infty,0])$ satisfy

$\displaystyle\eta(0)=1,\quad\int_{0}^{+\infty}\eta(y)dy=0.$

For simplicity, we set

$\displaystyle\bar{u}:$
$\displaystyle=u_{e}(1)+u_{p}^{(0)},\quad\bar{v}:=v_{e}^{(1)}(\theta,1)+v_{p}^{(1)},$
$\displaystyle u:$
$\displaystyle=u_{p}^{(1)}+u_{e}^{(1)}(\theta,1)\eta(Y),\quad
v:=v_{p}^{(2)}-v_{p}^{(2)}(\theta,0)+Yv_{p}^{(1)}-\partial_{\theta}u_{e}^{(1)}(\theta,1)\int_{0}^{Y}\eta(z)dz.$

Then, the equations (2.24) reduce to

$\displaystyle\left\\{\begin{array}[]{ll}\bar{u}\partial_{\theta}u+\bar{v}\partial_{Y}u+u\partial_{\theta}\bar{u}+v\partial_{Y}\bar{u}-\partial_{YY}u=\tilde{f},\\\\[7.0pt]
\partial_{\theta}u+\partial_{Y}v=0,\\\\[5.0pt]
u(\theta,Y)=u(\theta+2\pi,Y),\quad v(\theta,Y)=v(\theta+2\pi,Y)\\\\[5.0pt]
u|_{Y=0}=v|_{Y=0}=0,\ \
\lim\limits_{Y\rightarrow-\infty}\partial_{Y}u=0,\end{array}\right.$ (2.105)

where $\tilde{f}(\theta,Y)$ is $2\pi$-periodic function and decays fast as
$Y\rightarrow-\infty$.

We can solve the equations (2.105) by considering the following approximate
system. Let $\gamma>0$ be a constant, we consider the following elliptic
equation

$\displaystyle\left\\{\begin{array}[]{ll}\bar{u}\partial_{\theta}u^{\gamma}+\bar{v}\partial_{Y}u^{\gamma}+\big{[}\int_{Y}^{0}\partial_{\theta}u^{\gamma}(\theta,z)dz\big{]}\partial_{Y}\bar{u}+u^{\gamma}\partial_{\theta}\bar{u}-\partial_{YY}u^{\gamma}-\gamma\partial_{\theta\theta}u^{\gamma}=\tilde{f},\\\\[7.0pt]
u^{\gamma}(\theta,Y)=u^{\gamma}(\theta+2\pi,Y),\\\\[5.0pt]
u^{\gamma}|_{Y=0}=0.\end{array}\right.$ (2.109)

We expect the solution of this equation is in
$\dot{H}^{1}_{0}=\\{u|\partial_{\theta}u\in L^{2},\partial_{Y}u\in
L^{2},u|_{Y=0}=0\\}$ rather than $H^{1}_{0}=\\{u|u\in
L^{2},\partial_{\theta}u\in L^{2},\partial_{Y}u\in L^{2},u|_{Y=0}=0\\}$. Now
we establish apriori estimate of equation (2.109). Multiplying the first
equation in (2.109) by $u^{\gamma}$ and integrating in
$(\theta,r)\in(0,2\pi)\times(-\infty,0)$, we obtain that

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{[}\bar{u}\partial_{\theta}u^{\gamma}+\bar{v}\partial_{Y}u^{\gamma}+\Big{(}\int_{Y}^{0}\partial_{\theta}u^{\gamma}(\theta,z)dz\Big{)}\partial_{Y}\bar{u}+u^{\gamma}\partial_{\theta}\bar{u}-\partial_{YY}u^{\gamma}-\gamma\partial_{\theta\theta}u^{\gamma}\Big{]}u^{\gamma}d\theta
dY$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\tilde{f}u^{\gamma}d\theta dY.$

It’s easy to get

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\big{[}-\partial_{YY}u^{\gamma}-\gamma\partial_{\theta\theta}u^{\gamma}\big{]}u^{\gamma}d\theta
dY=\|\partial_{Y}u^{\gamma}\|_{2}^{2}+\gamma\|\partial_{\theta}u^{\gamma}\|_{2}^{2}.$

Recall the estimates (2.89) and (2.91), we have

$\displaystyle\big{|}\partial^{j}_{\theta}\partial^{k}_{Y}(\bar{u}-u_{e}(1))\big{<}Y\big{>}^{l}\big{|}\leq
C(j,k,l)\eta,$
$\displaystyle\big{|}\partial^{j}_{\theta}\partial^{k}_{Y}(\bar{v}-v^{(1)}_{e}(\theta,1))\big{<}Y\big{>}^{l}\big{|}\leq
C(j,k,l)\eta,\quad\big{|}\partial^{j}_{\theta}v^{(1)}_{e}(\theta,1)\big{|}\leq
C(\delta,j)\eta,$

thus we can deduce that

$\displaystyle-\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{[}\bar{u}\partial_{\theta}u^{\gamma}+\bar{v}\partial_{Y}u^{\gamma}+\Big{(}\int_{Y}^{0}\partial_{\theta}u^{\gamma}(\theta,z)dz\Big{)}\partial_{Y}\bar{u}+u^{\gamma}\partial_{\theta}\bar{u}\Big{]}u^{\gamma}d\theta
dY+\int_{-\infty}^{0}\int_{0}^{2\pi}\tilde{f}u^{\gamma}d\theta dY$
$\displaystyle\leq$
$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\frac{1}{2}\big{[}\partial_{\theta}\bar{u}+\partial_{Y}\bar{v}\big{]}\big{(}u^{\gamma}\big{)}^{2}d\theta
dr+\|Y^{2}\bar{u}_{Y}\|_{\infty}\Big{\|}\frac{\int_{Y}^{0}\partial_{\theta}u^{\gamma}dz}{Y}\Big{\|}_{2}\Big{\|}\frac{u^{\gamma}}{Y}\Big{\|}_{2}$
$\displaystyle+\|Y^{2}\bar{u}_{\theta}\|_{\infty}\Big{\|}\frac{u^{\gamma}}{Y}\Big{\|}_{2}^{2}+\|Y\tilde{f}\|_{2}\Big{\|}\frac{u^{\gamma}}{Y}\Big{\|}_{2}$
$\displaystyle\leq$ $\displaystyle
C\eta\big{[}\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+\|\partial_{Y}u^{\gamma}\|_{2}^{2}\big{]}+C\|Y\tilde{f}\|_{2}\|\partial_{Y}u^{\gamma}\|_{2},$

where we use $\bar{u}_{\theta}+\bar{v}_{Y}=0$ and the Hardy inequality

$\Big{\|}\frac{\int_{Y}^{0}\partial_{\theta}u^{\gamma}dz}{Y}\Big{\|}_{2}\leq
C\|\partial_{\theta}u^{\gamma}\|_{2},\quad\Big{\|}\frac{u^{\gamma}}{Y}\Big{\|}_{2}\leq
C\|\partial_{Y}u^{\gamma}\|_{2}.$

By collecting the above estimates, we obtain

$\displaystyle\|\partial_{Y}u^{\gamma}\|_{2}^{2}+\gamma\|\partial_{\theta}u^{\gamma}\|_{2}^{2}\leq
C\eta\big{[}\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+\|\partial_{Y}u^{\gamma}\|_{2}^{2}\big{]}+C\|Y\tilde{f}\|_{2}\|\partial_{Y}u^{\gamma}\|_{2},$

where $C$ is independent on $\eta$ and $\gamma$. If $\eta$ is small enough, we
deduce that

$\displaystyle\|\partial_{Y}u^{\gamma}\|_{2}^{2}+\gamma\|\partial_{\theta}u^{\gamma}\|_{2}^{2}\leq
C\eta\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+C\|Y\tilde{f}\|_{2}^{2}.$ (2.110)

Next we multiply the first equation in (2.109) by
$\partial_{\theta}u^{\gamma}$ and integrate in
$(\theta,r)\in(0,2\pi)\times(-\infty,0)$, we arrive at

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{[}\bar{u}\partial_{\theta}u^{\gamma}+\bar{v}\partial_{Y}u^{\gamma}+\Big{(}\int_{Y}^{0}\partial_{\theta}u^{\gamma}(\theta,z)dz\Big{)}\partial_{Y}\bar{u}+u^{\gamma}\partial_{\theta}\bar{u}-\partial_{YY}u^{\gamma}-\gamma\partial_{\theta\theta}u^{\gamma}\Big{]}\partial_{\theta}u^{\gamma}d\theta
dY$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\tilde{f}\partial_{\theta}u^{\gamma}d\theta
dY.$

It’s direct to obtain

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\bar{u}\partial_{\theta}u^{\gamma}\partial_{\theta}u^{\gamma}d\theta
dY=\int_{-\infty}^{0}\int_{0}^{2\pi}\bar{u}\big{|}\partial_{\theta}u^{\gamma}\big{|}^{2}d\theta
dY\geq(\alpha-C\eta)\|\partial_{\theta}u^{\gamma}\|^{2}_{2}.$

The diffusion term can be computed as follows

$\displaystyle\int_{-\infty}^{0}\int_{0}^{2\pi}\big{[}-\partial_{YY}u^{\gamma}-\gamma\partial_{\theta\theta}u^{\gamma}\big{]}\partial_{\theta}u^{\gamma}d\theta
dY=\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{(}\frac{1}{2}\partial_{\theta}\big{(}\partial_{Y}u^{\gamma}\big{)}^{2}-\frac{\gamma}{2}\partial_{\theta}\big{(}\partial_{\theta}u^{\gamma}\big{)}^{2}\Big{)}d\theta
dY=0.$

Moreover, there holds

$\displaystyle-\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{[}\bar{v}\partial_{Y}u^{\gamma}+\Big{(}\int_{Y}^{0}\partial_{\theta}u^{\gamma}(\theta,z)dz\Big{)}\partial_{Y}\bar{u}+u^{\gamma}\partial_{\theta}\bar{u}\Big{]}\partial_{\theta}u^{\gamma}d\theta
dY+\int_{-\infty}^{0}\int_{0}^{2\pi}\tilde{f}\partial_{\theta}u^{\gamma}d\theta
dY$ $\displaystyle\leq$
$\displaystyle\|\bar{v}\|_{\infty}\|\partial_{Y}u^{\gamma}\|_{2}\|\partial_{\theta}u^{\gamma}\|_{2}+\|Y\bar{u}_{Y}\|_{\infty}\Big{\|}\frac{\int_{Y}^{0}\partial_{\theta}u^{\gamma}dz}{Y}\Big{\|}_{2}\|\partial_{\theta}u^{\gamma}\|_{2}$
$\displaystyle+\|Y\partial_{\theta}\bar{u}\|_{\infty}\Big{\|}\frac{u^{\gamma}}{Y}\Big{\|}_{2}\|\partial_{\theta}u^{\gamma}\|_{2}+\|\tilde{f}\|_{2}\|\partial_{\theta}u^{\gamma}\|_{2}$
$\displaystyle\leq$ $\displaystyle
C(\delta)\eta\big{[}\|\partial_{Y}u^{\gamma}\|^{2}_{2}+\|\partial_{\theta}u^{\gamma}\|^{2}_{2}\big{]}+\|\tilde{f}\|_{2}\|\partial_{\theta}u^{\gamma}\|_{2}.$

Thus, we obtain

$\displaystyle\alpha\|\partial_{\theta}u^{\gamma}\|_{2}^{2}\leq
C(\delta)\eta\big{[}\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+\|\partial_{Y}u^{\gamma}\|_{2}^{2}\big{]}+\|\tilde{f}\|_{2}\|\partial_{\theta}u^{\gamma}\|_{2}.$

We then choose $\eta_{0}$ small enough such that for any $\eta\in(0,\eta_{0})$
and $\delta\in[0,1]$, there holds

$\displaystyle\alpha\|\partial_{\theta}u^{\gamma}\|_{2}^{2}\leq
C(\delta)\eta\big{\|}\partial_{Y}u^{\gamma}\|_{2}^{2}+C\|\tilde{f}\|_{2}^{2}.$
(2.111)

It follows from (2.110) and (2.111) that we have

$\displaystyle\alpha\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+\|\partial_{Y}u^{\gamma}\|_{2}^{2}+\gamma\|\partial_{\theta}u^{\gamma}\|_{2}^{2}\leq
C\|Y\tilde{f}\|_{2}^{2}+C\|\tilde{f}\|_{2}^{2}.$ (2.112)

According to the first equation in (2.109), we deduce

$\displaystyle\|\partial_{YY}u^{\gamma}\|^{2}_{2}+2\gamma\|\partial_{\theta
Y}u^{\gamma}\|^{2}_{2}+\gamma^{2}\|\partial_{\theta\theta}u^{\gamma}\|^{2}_{2}=\Big{\|}\partial_{YY}u^{\gamma}+\gamma\partial_{\theta\theta}u^{\gamma}\Big{\|}_{2}^{2}$
$\displaystyle=$
$\displaystyle\bigg{\|}\bar{u}\partial_{\theta}u^{\gamma}+\bar{v}\partial_{Y}u^{\gamma}+\Big{(}\int_{Y}^{0}\partial_{\theta}u^{\gamma}(\theta,z)dz\Big{)}\partial_{Y}\bar{u}+u^{\gamma}\partial_{\theta}\bar{u}-\tilde{f}\bigg{\|}_{2}^{2}$
$\displaystyle\leq$ $\displaystyle
C(\delta)\big{[}\|\tilde{f}\|^{2}_{2}+\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+\|\partial_{Y}u^{\gamma}\|_{2}^{2}\big{]}\leq
C(\delta)\|Y\tilde{f}\|_{2}^{2}+C(\delta)\|\tilde{f}\|_{2}^{2}.$ (2.113)

Collecting the estimate (2.112) and (2.4.3), we obtain

$\displaystyle\alpha\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+\|\partial_{Y}u^{\gamma}\|_{2}^{2}+\|\partial_{YY}u^{\gamma}\|^{2}_{2}+\gamma\|\partial_{\theta}u^{\gamma}\|_{2}^{2}+2\gamma\|\partial_{\theta
Y}u^{\gamma}\|^{2}_{2}+\gamma^{2}\|\partial_{\theta\theta}u^{\gamma}\|^{2}_{2}\leq
C(\delta)\|\big{<}Y\big{>}\tilde{f}\|_{2}^{2}$

which shows the existence and uniqueness of solution for system (2.109) for
any $\gamma>0$ in $\dot{H}^{1}_{0}$. Moreover the solution is smooth if
$\tilde{f}$ is smooth. Set

$\displaystyle u:=\lim_{\gamma\rightarrow 0}u^{\gamma},\quad
v:=\int_{Y}^{0}\partial_{\theta}u(\theta,d)dz,$

then

$\displaystyle\alpha\|\partial_{\theta}u\|_{2}^{2}+\|\partial_{Y}u\|_{2}^{2}+\|\partial_{YY}u\|^{2}_{2}\leq
C(\delta)\|\big{<}Y\big{>}\tilde{f}\|^{2}_{2}$ (2.114)

and $(u,v)$ solves the system (2.105) except the boundary condition
$\lim\limits_{Y\rightarrow-\infty}\partial_{Y}u=0$.

Finally, we show that the derivatives of $u,v$ decay fast as
$Y\rightarrow-\infty$. Let $\psi=\int_{0}^{Y}\bar{u}(\theta,z)dz$ and

$\displaystyle\tilde{u}(\theta,\psi)=u(\theta,Y(\theta,\psi)),\
\tilde{v}(\theta,\psi)=v(\theta,Y(\theta,\psi)),\
F(\theta,\psi)=\frac{\tilde{f}(\theta,Y(\theta,\psi))}{\bar{u}(\theta,Y(\theta,\psi))},$

then there holds

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\theta}\tilde{u}-\partial_{\psi}(a(\theta,\psi)\partial_{\psi}\tilde{u})+b(\theta,\psi)\tilde{u}+c(\theta,\psi)\tilde{v}=F(\theta,\psi),\\\\[7.0pt]
\tilde{u}(\theta+2\pi,\psi)=\tilde{u}(\theta,\psi),\\\\[5.0pt]
\tilde{u}(\theta,0)=0,\
\lim_{\psi\rightarrow-\infty}\partial_{\psi}\tilde{u}(\theta,\psi)=0,\end{array}\right.$
(2.118)

where

$\displaystyle a(\theta,\psi)=\bar{u}(\theta,Y(\theta,\psi)),\
b(\theta,\psi)=\frac{\partial_{\theta}\bar{u}(\theta,Y(\theta,\psi))}{\bar{u}(\theta,Y(\theta,\psi))},\
c(\theta,\psi)=\frac{\partial_{Y}\bar{u}(\theta,Y(\theta,\psi))}{\bar{u}(\theta,Y(\theta,\psi))}.$

Notice that there exist $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$,
there holds

$\displaystyle\frac{\alpha}{2}\leq\bar{u}(\theta,Y)\leq\alpha,\
\forall(\theta,Y)\in[0,2\pi]\times(-\infty,0].$

Thus, we deduce that $\frac{\alpha}{2}\leq\frac{|\psi|}{|Y|}\leq\alpha$.

$\displaystyle\|\partial_{\theta}\tilde{u}\psi^{l}\|_{2}+\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l}\|_{2}+\|\partial_{\psi\psi}\tilde{u}\psi^{l}\|_{2}\leq
C(\delta,l)\|F\big{<}\psi\big{>}^{l+1}\|_{2},$ (2.119)

where

$\tilde{u}_{\neq}=\tilde{u}-u_{0}(\psi),\
u_{0}(\psi)=\frac{1}{2\pi}\int_{0}^{2\pi}\tilde{u}(\theta,\psi)d\theta.$

From (2.114), we deduce that

$\displaystyle\|\partial_{\theta}\tilde{u}\|_{2}+\|\partial_{\psi}\tilde{u}\|_{2}+\|\partial_{\psi\psi}\tilde{u}\|_{2}\leq
C(\delta)\|F\big{<}\psi\big{>}\|_{2},$

thus (2.119) holds for $l=0$.

For any $l\geq 1$, multiplying $\tilde{u}_{\neq}\psi^{2l}$ in (2.118) and
integrating in $[0,2\pi]\times(-\infty,0]$, we obtain

$\displaystyle\underbrace{\int_{0}^{2\pi}\int_{-\infty}^{0}\partial_{\theta}\tilde{u}\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta}_{I_{1}}-\underbrace{\int_{0}^{2\pi}\int_{-\infty}^{0}\partial_{\psi}(a(\theta,\psi)\partial_{\psi}\tilde{u})\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta}_{I_{2}}$ $\displaystyle=$
$\displaystyle\underbrace{\int_{0}^{2\pi}\int_{-\infty}^{0}[F(\theta,\psi)-b(\theta,\psi)\tilde{u}-c(\theta,\psi)\tilde{v}]\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta}_{I_{3}}.$

Obviously, $I_{1}=0$. Due to the fast decay of $b(\theta,\psi),c(\theta,\psi)$
as $\psi\rightarrow-\infty$, we deduce that

$\displaystyle|I_{3}|\leq$ $\displaystyle
C\|F\big{<}\psi\big{>}^{l+1}\|_{2}\|\partial_{\theta}\tilde{u}_{\neq}\psi^{l-1}\|_{2}+C(\delta)(\|\partial_{\theta}\tilde{u}\|^{2}_{2}+\|\partial_{\psi}\tilde{u}\|^{2}_{2})$
$\displaystyle\leq$ $\displaystyle
C(\delta)\|F\big{<}\psi\big{>}^{l+1}\|_{2}^{2}+C(\delta)\|\partial_{\theta}\tilde{u}\psi^{l-1}\|_{2}^{2}.$

Moreover, there holds

$\displaystyle
I_{2}=\underbrace{\int_{0}^{2\pi}\int_{-\infty}^{0}a(\theta,\psi)\partial_{\psi}\tilde{u}\partial_{\psi}\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta}_{I_{21}}+\underbrace{2l\int_{0}^{2\pi}\int_{-\infty}^{0}a(\theta,\psi)\partial_{\psi}\tilde{u}\tilde{u}_{\neq}\psi^{2l-1}d\psi
d\theta}_{I_{22}}.$

Notice that $a(\theta,\psi)=u_{e}(1)+u_{p}^{(0)}(\theta,Y(\theta,\psi))$, we
deduce that

$\displaystyle I_{21}=$ $\displaystyle
u_{e}(1)\int_{0}^{2\pi}\int_{-\infty}^{0}\partial_{\psi}\tilde{u}\partial_{\psi}\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta+\int_{0}^{2\pi}\int_{-\infty}^{0}u_{p}^{(0)}(\theta,Y(\theta,\psi))\partial_{\psi}\tilde{u}\partial_{\psi}\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta$ $\displaystyle=$ $\displaystyle
u_{e}(1)\int_{0}^{2\pi}\int_{-\infty}^{0}\partial_{\psi}\tilde{u}_{\neq}\partial_{\psi}\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta+\int_{0}^{2\pi}\int_{-\infty}^{0}u_{p}^{(0)}(\theta,Y(\theta,\psi))\partial_{\psi}\tilde{u}\partial_{\psi}\tilde{u}_{\neq}\psi^{2l}d\psi
d\theta$ $\displaystyle\geq$
$\displaystyle\frac{\alpha}{2}\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l}\|^{2}_{2}-C(\delta)\|\partial_{\psi}\tilde{u}\|^{2}_{2}.$

Moreover, by Hölder inequality and Póincare inequality, there holds

$\displaystyle|I_{22}|\leq
C(\delta,l)\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l}\|_{2}\|\partial_{\theta}\tilde{u}\psi^{l-1}\|_{2}.$

Thus, we obtain

$\displaystyle\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l}\|_{2}\leq
C(\delta,l)\|\partial_{\theta}\tilde{u}\psi^{l-1}\|_{2}+C(\delta)\|F\big{<}\psi\big{>}^{l+1}\|_{2}.$
(2.120)

Moreover, we can easily get

$\displaystyle\partial_{\theta}\tilde{u}-a(\theta,\psi)\partial_{\psi\psi}\tilde{u}=F(\theta,\psi)+\partial_{\psi}a(\theta,\psi)\partial_{\psi}\tilde{u}-b(\theta,\psi)\tilde{u}-c(\theta,\psi)\tilde{v},$

thus there holds

$\displaystyle\|[\partial_{\theta}\tilde{u}-a(\theta,\psi)\partial_{\psi\psi}\tilde{u}]\psi^{l}\|_{2}^{2}=\|[F(\theta,\psi)+\partial_{\psi}a(\theta,\psi)\partial_{\psi}\tilde{u}-b(\theta,\psi)\tilde{u}-c(\theta,\psi)\tilde{v}]\psi^{l}\|_{2}^{2}.$

The right side can be controlled by

$\displaystyle
C(\delta)\|F\big{<}\psi\big{>}^{l}\|_{2}^{2}+C(\delta)(\|\partial_{\theta}\tilde{u}\|^{2}_{2}+\|\partial_{\psi}\tilde{u}\|^{2}_{2}).$

Moreover, there holds

$\displaystyle\|[\partial_{\theta}\tilde{u}-a(\theta,\psi)\partial_{\psi\psi}\tilde{u}]\psi^{l}\|_{2}^{2}$
$\displaystyle=$
$\displaystyle\|\partial_{\theta}\tilde{u}\psi^{l}\|_{2}^{2}+\|a(\theta,\psi)\partial_{\psi\psi}\tilde{u}\psi^{l}\|_{2}^{2}-2\int_{0}^{2\pi}\int_{-\infty}^{0}a(\theta,\psi)\partial_{\theta}\tilde{u}\partial_{\psi\psi}\tilde{u}\psi^{2l}d\psi
d\theta$ $\displaystyle\geq$
$\displaystyle\|\partial_{\theta}\tilde{u}\psi^{l}\|_{2}^{2}+\frac{\alpha}{2}\|\partial_{\psi\psi}\tilde{u}\psi^{l}\|_{2}^{2}\underbrace{-2\int_{0}^{2\pi}\int_{-\infty}^{0}a(\theta,\psi)\partial_{\theta}\tilde{u}\partial_{\psi\psi}\tilde{u}\psi^{2l}d\psi
d\theta}_{I}.$

Integrating by parts, we deduce that

$\displaystyle I=$
$\displaystyle\underbrace{2\int_{0}^{2\pi}\int_{-\infty}^{0}\partial_{\psi}a(\theta,\psi)\partial_{\theta}\tilde{u}\partial_{\psi}\tilde{u}\psi^{2l}d\psi
d\theta}_{I_{1}}+\underbrace{4l\int_{0}^{2\pi}\int_{-\infty}^{0}a(\theta,\psi)\partial_{\theta}\tilde{u}\partial_{\psi}\tilde{u}\psi^{2l-1}d\psi
d\theta}_{I_{2}}$
$\displaystyle+\underbrace{2\int_{0}^{2\pi}\int_{-\infty}^{0}a(\theta,\psi)\partial_{\theta\psi}\tilde{u}\partial_{\psi}\tilde{u}\psi^{2l}d\psi
d\theta}_{I_{3}}.$

Obviously, there holds

$\displaystyle|I_{1}|+|I_{3}|\leq
C(\delta)(\|\partial_{\theta}\tilde{u}\|^{2}_{2}+\|\partial_{\psi}\tilde{u}\|^{2}_{2}).$

Moreover,

$\displaystyle|I_{2}|=$
$\displaystyle\Big{|}4lu_{e}(1)\int_{0}^{2\pi}\int_{-\infty}^{0}\partial_{\theta}\tilde{u}\psi^{2l-1}\partial_{\psi}\tilde{u}d\psi
d\theta+4l\int_{0}^{2\pi}\int_{-\infty}^{0}u_{p}^{(0)}(\theta,Y(\theta,\psi))\partial_{\theta}\tilde{u}\psi^{2l-1}\partial_{\psi}\tilde{u}d\psi
d\theta\Big{|}$ $\displaystyle=$
$\displaystyle\Big{|}4lu_{e}(1)\int_{0}^{2\pi}\int_{-\infty}^{0}\partial_{\theta}\tilde{u}\psi^{2l-1}\partial_{\psi}\tilde{u}_{\neq}d\psi
d\theta+4l\int_{0}^{2\pi}\int_{-\infty}^{0}u_{p}^{(0)}(\theta,Y(\theta,\psi))\partial_{\theta}\tilde{u}\psi^{2l-1}\partial_{\psi}\tilde{u}d\psi
d\theta\Big{|}$ $\displaystyle\leq$ $\displaystyle
C(l)\|\partial_{\theta}\tilde{u}\psi^{l}\|_{2}\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l-1}\|_{2}+C(\delta,l)(\|\partial_{\theta}\tilde{u}\|^{2}_{2}+\|\partial_{\psi}\tilde{u}\|^{2}_{2}).$

Thus, we obtain

$\displaystyle\|\partial_{\theta}\tilde{u}\psi^{l}\|_{2}+\|\partial_{\psi\psi}\tilde{u}\psi^{l}\|_{2}\leq
C(\delta,l)\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l-1}\|_{2}+C(\delta,l)\|F\big{<}\psi\big{>}^{l+1}\|_{2}.$
(2.121)

Combining the estimate (2.120) and (2.121), we obtain that for any $l\geq 1$,
there holds

$\displaystyle\|\partial_{\theta}\tilde{u}\psi^{l}\|_{2}+\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l}\|_{2}+\|\partial_{\psi\psi}\tilde{u}\psi^{l}\|_{2}\leq
C(\delta,l)(\|\partial_{\psi}\tilde{u}_{\neq}\psi^{l-1}\|_{2}+\|\partial_{\theta}\tilde{u}\psi^{l-1}\|_{2})+C(\delta,l)\|F\big{<}\psi\big{>}^{l}\|_{2}.$

Thus, by induction, we obtain (2.119).

Furthermore, by the Hardy inequality, we have

$\displaystyle\|\partial_{\psi}\tilde{u}\psi^{l-1}\|_{2}\leq
C\|\partial_{\psi\psi}\tilde{u}\psi^{l}\|_{2},$

hence there holds

$\displaystyle\|\partial_{\theta}\tilde{u}\psi^{l}\|_{2}+\|\partial_{\psi}\tilde{u}\psi^{l-1}\|_{2}+\|\partial_{\psi\psi}\tilde{u}\psi^{l}\|_{2}\leq
C(\delta)\|F\big{<}\psi\big{>}^{l+1}\|_{2}.$

Returning to the original variable, we obtain that for any $l\geq 1$, there
holds

$\displaystyle\big{\|}\big{<}Y\big{>}^{l}\partial_{YY}u\big{\|}_{2}^{2}+\|\big{<}Y\big{>}^{l-1}\partial_{\theta}u\|^{2}_{2}+\|\big{<}Y\big{>}^{l-1}\partial_{Y}u\|^{2}_{2}\leq
C(\delta,l)\|\big{<}Y\big{>}^{l+1}\tilde{f}\|^{2}_{2}.$

Furthermore, by induction, we obtain that for any
$m\in\mathbb{N}_{+},l\in\mathbb{N}$, there holds

$\displaystyle\sum_{j+k\leq
m}\Big{(}\big{\|}\big{<}Y\big{>}^{l}\partial^{j}_{\theta}\partial^{k}_{Y}\partial_{YY}u\big{\|}_{2}^{2}+\big{\|}\big{<}Y\big{>}^{l-1}\partial^{j}_{\theta}\partial^{k}_{Y}\partial_{\theta}u\big{\|}_{2}^{2}+\big{\|}\big{<}Y\big{>}^{l-1}\partial^{j}_{\theta}\partial^{k}_{Y}\partial_{Y}u\big{\|}_{2}^{2}\Big{)}$
$\displaystyle\leq C(\delta,m,l)\sum\limits_{j+k\leq
m}\big{\|}\big{<}Y\big{>}^{l+1}\partial^{j}_{\theta}\partial^{k}_{Y}\tilde{f}\big{\|}^{2}_{2}\leq
C(\delta,m,l)\eta^{2}.$

Noting that $\lim\limits_{Y\rightarrow-\infty}(u_{\theta},u_{Y})=0$ and
$A_{1\infty}:=\lim\limits_{Y\rightarrow-\infty}u(\theta,Y)$ is a constant
independent on $\theta$, then by the Hardy inequality we have for any $l\geq
2$

$\displaystyle\|Y^{l-2}(u-A_{1\infty})\|_{2}\leq
C(\delta,l)\|Y^{l-1}\partial_{Y}u\|_{2}\leq C(\delta,l)\eta^{2}.$

This completes the proof of this proposition. MM $\square$

Next, we construct the pressure $p_{p}^{(2)}(\theta,Y)$. Consider the equation

$\displaystyle\partial_{Y}p_{p}^{(2)}(\theta,Y)=g_{1}(\theta,Y),\quad\lim_{Y\rightarrow-\infty}p_{p}^{(2)}(\theta,Y)=0,$
(2.122)

where

$\displaystyle g_{1}(\theta,Y)=$
$\displaystyle-Y\partial_{Y}p_{p}^{(1)}+\partial_{YY}v_{p}^{(1)}-u_{e}(1)\partial_{\theta}v_{p}^{(1)}-u_{p}^{(0)}(\partial_{\theta}v_{e}^{(1)}(\theta,1)+\partial_{\theta}v_{p}^{(1)})$
$\displaystyle-\partial_{Y}v_{p}^{(1)}(v_{e}^{(1)}(\theta,1)+v_{p}^{(1)})-2(Yu^{\prime}_{e}(1)u_{p}^{(0)}+u_{e}(1)\tilde{u}_{p}^{(1)}+[u_{e}^{(1)}(\theta,1)+A_{1}]u_{p}^{(0)}+u_{p}^{(0)}\tilde{u}_{p}^{(1)},$

here and below,

$\tilde{u}_{p}^{(1)}=u_{p}^{(1)}-A_{1\infty}.$

$g_{1}(\theta,Y)$ can be obtained by replacing $u_{p}^{(1)}$ by
$\tilde{u}_{p}^{(1)}$ in the expansion (2.11) and putting the new expansion
into the second equation of (1.8), then collecting the $\varepsilon^{1}$-order
terms together. Notice that $g_{1}(\theta,Y)$ decay fast as
$Y\rightarrow-\infty$, we can get $p_{p}^{(2)}(\theta,Y)$ by solving (2.11)
and deduce that $p_{p}^{(2)}(\theta,Y)$ decay fast as $Y\rightarrow-\infty$.

By the same argument as above, we can obtain the well-posedness of equations
(2.36).

######  Proposition 2.7.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and
$\delta\in[0,1]$, the equation (2.36) has a unique solution
$(\hat{u}_{p}^{(1)},\hat{v}_{p}^{(2)})$ which satisfies

$\displaystyle\begin{aligned} &\sum_{j+k\leq
m}\int_{0}^{+\infty}\int_{0}^{2\pi}\big{|}\partial_{\theta}^{j}\partial_{Z}^{k}\big{(}\hat{u}_{p}^{(1)}-\hat{A}_{1\infty},v_{p}^{(2)}\big{)}\big{|}^{2}\big{<}Z\big{>}^{2l}d\theta
dZ\leq C(\delta,m,l)\eta^{2},\ \forall m,l\geq 0,\\\
&\int_{0}^{2\pi}\hat{v}_{p}^{(2)}(\theta,Z)d\theta=0,\ \forall Z\geq
0,\end{aligned}$ (2.123)

where
$\hat{A}_{1\infty}:=\lim\limits_{Z\rightarrow+\infty}\hat{u}_{p}^{(1)}(\theta,Z)$
is a constant which satisfies $|\hat{A}_{1\infty}|\leq C(\delta)\eta.$

Similarly, we can also construct $\hat{p}_{p}^{(2)}(\theta,Z)$ which decays
fast as $Z\rightarrow+\infty$.

####  2.4.4. Linearized Euler system for
$(u_{e}^{(2)},v_{e}^{(2)},p_{e}^{(2)})$ and its solvability

Let $r_{1}=\frac{1+2r_{0}}{3},r_{2}=\frac{2+r_{0}}{3}$ and $\chi(r)\in
C^{\infty}([r_{0},1])$ be an increasing smooth function such that

$\displaystyle\chi(r)=\left\\{\begin{array}[]{lll}0,\quad
r\in[r_{0},r_{1}],\\\\[5.0pt] 1,\quad r\in[r_{2},1]\end{array}\right.$

and

$\displaystyle\bar{u}_{e}^{(1)}(\theta,r):$
$\displaystyle=u_{e}^{(1)}(\theta,r)+\chi(r)A_{1\infty}+(1-\chi(r))\hat{A}_{1\infty},\quad\bar{v}_{e}^{(1)}(\theta,r)=v_{e}^{(1)}(\theta,r),$
$\displaystyle\bar{p}_{e}^{(1)}(\theta,r):$
$\displaystyle=p_{e}^{(1)}(\theta,r)+\int_{r_{0}}^{r}\frac{2u_{e}(s)}{s}[\chi(s)A_{1\infty}+(1-\chi(s))\hat{A}_{1\infty}]ds,$

then $(\bar{u}_{e}^{(1)},\bar{v}_{e}^{(1)},\bar{p}_{e}^{(1)})$ also satisfies
the linearized Euler equations (2.9) with the boundary condition (2.10).

Putting

$\displaystyle
u^{\varepsilon}(\theta,r)=u_{e}(r)+\varepsilon\bar{u}_{e}^{(1)}(\theta,r)+\varepsilon^{2}u_{e}^{(2)}(\theta,r)+\text{h.o.t.},$
$\displaystyle
v^{\varepsilon}(\theta,r)=\varepsilon\bar{v}_{e}^{(1)}(\theta,r)+\varepsilon^{2}v_{e}^{(2)}(\theta,r)+\text{h.o.t.},$
$\displaystyle
p^{\varepsilon}(\theta,r)=p_{e}(r)+\varepsilon\bar{p}_{e}^{(1)}(\theta,r)+\varepsilon^{2}p_{e}^{(2)}(\theta,r)+\text{h.o.t.}$

into the Navier-Stokes equation (1.8), we obtain the following linearized
Euler equations for $(u_{e}^{(2)},v_{e}^{(2)},p_{e}^{(2)})$

$\displaystyle\left\\{\begin{array}[]{ll}u_{e}(r)\partial_{\theta}u_{e}^{(2)}+rv_{e}^{(2)}u^{\prime}_{e}(r)+u_{e}(r)v_{e}^{(2)}+\partial_{\theta}p_{e}^{(2)}+\bar{u}_{e}^{(1)}\partial_{\theta}\bar{u}_{e}^{(1)}+\bar{v}_{e}^{(1)}r\partial_{r}\bar{u}_{e}^{(1)}+\bar{u}_{e}^{(1)}\bar{v}_{e}^{(1)}=2\delta\tilde{c}_{0},\\\\[5.0pt]
u_{e}(r)\partial_{\theta}v_{e}^{(2)}-2u_{e}(r)u_{e}^{(2)}+r\partial_{r}p_{e}^{(2)}+\bar{u}_{e}^{(1)}\partial_{\theta}\bar{v}_{e}^{(1)}+\bar{v}_{e}^{(1)}r\partial_{r}\bar{v}_{e}^{(1)}-(\bar{u}_{e}^{(1)})^{2}=0,\\\\[7.0pt]
\partial_{\theta}u_{e}^{(2)}+r\partial_{r}v_{e}^{(2)}+v_{e}^{(2)}=0,\end{array}\right.$
(2.127)

with the boundary conditions

$\displaystyle v_{e}^{(2)}|_{r=1}=-v_{p}^{(2)}|_{Y=0},\quad
v_{e}^{(2)}|_{r=r_{0}}=-\hat{v}_{p}^{(2)}|_{Z=0},\quad
v_{e}^{(2)}(\theta,r)=v_{e}^{(2)}(\theta+2\pi,r).$

Moreover, eliminate the pressure $p_{e}^{(2)}$ in (2.127) lead to

$\displaystyle-
u_{e}(r)\triangle(rv_{e}^{(2)})+\Big{(}u^{\prime\prime}_{e}(r)+\frac{u^{\prime}_{e}(r)}{r}-\frac{u_{e}(r)}{r^{2}}\Big{)}(rv_{e}^{(2)})$
$\displaystyle=r\partial_{r}\big{(}\bar{u}_{e}^{(1)}\partial_{\theta}\bar{u}_{e}^{(1)}+\bar{v}_{e}^{(1)}r\partial_{r}\bar{u}_{e}^{(1)}+\bar{u}_{e}^{(1)}\bar{v}_{e}^{(1)}\big{)}-\partial_{\theta}\big{(}\bar{u}_{e}^{(1)}\partial_{\theta}\bar{v}_{e}^{(1)}+\bar{v}_{e}^{(1)}r\partial_{r}\bar{v}_{e}^{(1)}-(\bar{u}_{e}^{(1)})^{2}\big{)}$

which and $\int_{0}^{2\pi}v_{e}^{(2)}d\theta=0$ imply

$\displaystyle\frac{d}{dr}\Big{(}\int_{0}^{2\pi}\big{(}\bar{u}_{e}^{(1)}\partial_{\theta}\bar{u}_{e}^{(1)}+\bar{v}_{e}^{(1)}r\partial_{r}\bar{u}_{e}^{(1)}+\bar{u}_{e}^{(1)}v_{e}^{(1)}\big{)}d\theta\Big{)}=0,\
\forall r\in[r_{0},1],$ (2.128)

which is a necessary condition for the solvability of (2.127).

######  Proposition 2.8.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and
$\delta\in[0,1]$, the linearized Euler system (2.127) has a solution
$(u_{e}^{(2)},v_{e}^{(2)},p_{e}^{(2)})$ which satisfies

$\displaystyle\|\partial^{k}_{\theta}\partial^{j}_{r}(u_{e}^{(2)},v_{e}^{(2)})\|_{2}\leq
C(\delta,j,k)\eta,\quad\forall j,k\geq 0.$ (2.129)

###### Proof.

Firstly, we need to check the necessary condition (2.128) for the solvability.
In fact, there holds

$\displaystyle\int_{0}^{2\pi}[\bar{u}_{e}^{(1)}\partial_{\theta}\bar{u}_{e}^{(1)}+\bar{v}_{e}^{(1)}r\partial_{r}\bar{u}_{e}^{(1)}+\bar{u}_{e}^{(1)}\bar{v}_{e}^{(1)}]d\theta=0,\
\forall r\in[r_{0},1].$ (2.130)

We set

$\displaystyle\bar{v}_{e}^{(1)}(\theta,r)=\sum_{n\neq
0}V_{n}(r)e^{in\theta},\quad\bar{u}_{e}^{(1)}(\theta,r)=a(r)-\sum_{n\neq
0}\frac{(rV_{n})^{\prime}(r)}{in}e^{in\theta},$

here we have used (2.97) and the third equation of (2.9).

Then, it’s easy to obtain that

$\displaystyle\bar{v}_{e}^{(1)}r\partial_{r}\bar{u}_{e}^{(1)}(\theta,r)+\bar{u}_{e}^{(1)}\bar{v}_{e}^{(1)}(\theta,r)$
$\displaystyle=(ra^{\prime}(r)+a(r))\bar{v}_{e}^{(1)}(\theta,r)-\bar{v}_{e}^{(1)}(\theta,r)\sum_{n\neq
0}\frac{e^{in\theta}}{in}[r(rV_{n})^{\prime\prime}+(rV_{n})^{\prime}].$
(2.131)

Moreover, due to
$-\triangle(r\bar{v}_{e}^{(1)})+U_{e}(r)(r\bar{v}_{e}^{(1)})=0$, where
$U_{e}(r)=\frac{1}{u_{e}(r)}\Big{(}u^{\prime\prime}_{e}(r)+\frac{u^{\prime}_{e}(r)}{r}-\frac{u_{e}(r)}{r^{2}}\Big{)},$
we deduce that

$\displaystyle\sum_{n\neq
0}e^{in\theta}[r^{2}(rV_{n})^{\prime\prime}+r(rV_{n})^{\prime}-(n^{2}+r^{2}U_{e}(r))rV_{n}]=0,$

hence there holds
$r(rV_{n})^{\prime\prime}+(rV_{n})^{\prime}=(n^{2}+r^{2}U_{e}(r))V_{n}$.

Thus, by (2.131) we deduce that

$\displaystyle\bar{v}_{e}^{(1)}r\partial_{r}\bar{u}_{e}^{(1)}(\theta,r)+\bar{u}_{e}^{(1)}\bar{v}_{e}^{(1)}(\theta,r)=$
$\displaystyle(ra^{\prime}(r)+a(r))\bar{v}_{e}^{(1)}(\theta,r)+\bar{v}_{e}^{(1)}(\theta,r)\Big{(}\sum_{n\neq
0}ine^{in\theta}V_{n}-r^{2}U_{e}(r)\sum_{n\neq
0}\frac{e^{in\theta}}{in}V_{n}\Big{)}$ $\displaystyle=$
$\displaystyle(ra^{\prime}(r)+a(r))\bar{v}_{e}^{(1)}(\theta,r)+\bar{v}_{e}^{(1)}(\theta,r)\partial_{\theta}\bar{v}_{e}^{(1)}(\theta,r)$
$\displaystyle\quad\quad\quad-\frac{r^{2}U_{e}(r)}{2}\partial_{\theta}\Big{(}\sum_{n\neq
0}\frac{e^{in\theta}}{in}V_{n}\Big{)}^{2},$ (2.132)

hence (2.130) can be obtained.

Then, we can construct $(u_{e}^{(2)},v_{e}^{(2)},p_{e}^{(2)})$ follows the
line of the construction of $(u_{e}^{(1)},v_{e}^{(1)},p_{e}^{(1)})$:
eliminating pressure $p_{e}^{(2)}$ in the equation (2.127) and obtain the
elliptic equation for $rv_{e}^{(2)}$; then obtaining $u_{e}^{(2)}$ from the
divergence-free condition; finally, construct pressure $p_{e}^{(2)}$ from the
first equation in (2.127). Thus, $(u_{e}^{(2)},v_{e}^{(2)},p_{e}^{(2)})$
satisfies the equation (2.127) and (2.129).
MM $\square$

Let $A_{1}(r)$ be a smooth function such that

$\displaystyle\left\\{\begin{array}[]{ll}rA^{\prime\prime}_{1}(r)+A^{\prime}_{1}(r)-\frac{A_{1}(r)}{r}\\\\[5.0pt]
\quad\quad\quad=\frac{1}{2\pi}\int_{0}^{2\pi}\Big{(}v^{(1)}_{e}\partial_{r}(ru^{(2)}_{e})+v^{(2)}_{e}\partial_{r}(r\bar{u}^{(1)}_{e})-\Big{(}r\partial_{rr}\bar{u}^{(1)}_{e}+\partial_{r}\bar{u}^{(1)}_{e}-\frac{\bar{u}^{(1)}_{e}}{r}\Big{)}\Big{)}d\theta,\\\\[10.0pt]
A_{1}(r_{0})=A_{1}(1)=0,\end{array}\right.$ (2.136)

then $\|\partial_{r}^{k}A_{1}(r)\|_{\infty}\leq C(\delta,k)\eta.$

Set

$\displaystyle\tilde{u}_{e}^{(1)}=\bar{u}_{e}^{(1)}+A_{1}(r),\quad\tilde{v}_{e}^{(1)}=\bar{v}_{e}^{(1)},\quad\tilde{p}_{e}^{(1)}=\bar{p}_{e}^{(1)}+\int_{r_{0}}^{r}\frac{2u_{e}(s)}{s}A_{1}(s)ds,$

then $(\tilde{u}_{e}^{(1)},\tilde{v}_{e}^{(1)},\tilde{p}_{e}^{(1)})$ is also a
solution of (2.9) with the boundary condition (2.10), and there holds

$\displaystyle\|\partial_{\theta}^{j}\partial_{r}^{k}(\tilde{u}^{(1)}_{e},\tilde{v}_{e}^{(1)})\|_{\infty}\leq
C(\delta,j,k)\eta,\ \forall k,j\geq 0.$ (2.137)

Thus, by the same argument as Proposition 2.8, we deduce that there exists
$\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and $\delta\in[0,1]$, the
following linearized Euler sysem

$\displaystyle\left\\{\begin{array}[]{ll}u_{e}(r)\partial_{\theta}u_{e}^{(2)}+rv_{e}^{(2)}u^{\prime}_{e}(r)+u_{e}(r)v_{e}^{(2)}+\partial_{\theta}p_{e}^{(2)}+\tilde{u}_{e}^{(1)}\partial_{\theta}\tilde{u}_{e}^{(1)}+\tilde{v}_{e}^{(1)}r\partial_{r}\tilde{u}_{e}^{(1)}+\tilde{u}_{e}^{(1)}\tilde{v}_{e}^{(1)}=2\delta\tilde{c}_{0},\\\\[5.0pt]
u_{e}(r)\partial_{\theta}v_{e}^{(2)}-2u_{e}(r)u_{e}^{(2)}+r\partial_{r}p_{e}^{(2)}+\tilde{u}_{e}^{(1)}\partial_{\theta}\tilde{v}_{e}^{(1)}+\tilde{v}_{e}^{(1)}r\partial_{r}\tilde{v}_{e}^{(1)}-(\tilde{u}_{e}^{(1)})^{2}=0,\\\\[7.0pt]
\partial_{\theta}u_{e}^{(2)}+r\partial_{r}v_{e}^{(2)}+v_{e}^{(2)}=0\end{array}\right.$
(2.141)

with the boundary conditions

$\displaystyle v_{e}^{(2)}|_{r=1}=-v_{p}^{(2)}|_{Y=0},\quad
v_{e}^{(2)}|_{r=r_{0}}=-\hat{v}_{p}^{(2)}|_{Z=0},\quad
v_{e}^{(2)}(\theta,r)=v_{e}^{(2)}(\theta+2\pi,r),$ (2.142)

has a solution $(\bar{u}_{e}^{(2)},\bar{v}_{e}^{(2)},\bar{p}_{e}^{(2)})$ which
satisfies

$\displaystyle\|\partial_{\theta}^{j}\partial_{r}^{k}(\bar{u}^{(2)}_{e},\bar{v}_{e}^{(2)})\|_{\infty}\leq
C(\delta,j,k)\eta.$ (2.143)

####  2.4.5. Linearized Prandtl equations for $(u_{p}^{(2)},v_{p}^{(3)})$,
$(\hat{u}_{p}^{(2)},\hat{v}_{p}^{(3)})$ and their solvabilities

Let

$\displaystyle
u^{\varepsilon}(\theta,r)=u_{e}(r)+u_{p}^{(0)}(\theta,Y)+\varepsilon\big{[}\tilde{u}_{e}^{(1)}(\theta,r)+\tilde{u}_{p}^{(1)}(\theta,Y)\big{]}+\varepsilon^{2}\big{[}\bar{u}_{e}^{(2)}(\theta,r)+u_{p}^{(2)}(\theta,Y)\big{]}+\text{h.o.t.},$
$\displaystyle
v^{\varepsilon}(\theta,r)=\varepsilon\big{[}\tilde{v}_{e}^{(1)}(\theta,r)+v_{p}^{(1)}(\theta,Y)\big{]}+\varepsilon^{2}\big{[}\bar{v}_{e}^{(2)}(\theta,r)+v_{p}^{(2)}(\theta,Y)\big{]}+\varepsilon^{3}[v_{e}^{(3)}(\theta,r)+v_{p}^{(3)}(\theta,Y)]+\text{h.o.t.},$
$\displaystyle
p^{\varepsilon}(\theta,r)=p_{e}(r)+\varepsilon\big{[}\tilde{p}_{e}^{(1)}(\theta,r)+p_{p}^{(1)}(\theta,Y)\big{]}+\varepsilon\big{[}\bar{p}_{e}^{(2)}(\theta,r)+p_{p}^{(2)}(\theta,Y)\big{]}+\varepsilon^{3}p_{p}^{(3)}(\theta,Y)+\text{h.o.t.},$

with the boundary conditions

$\displaystyle\bar{u}_{e}^{(2)}(\theta,1)+u_{p}^{(2)}(\theta,0)=0,\
v_{e}^{(3)}(\theta,1)+v_{p}^{(3)}(\theta,0)=0,\
\lim_{Y\rightarrow-\infty}(\partial_{Y}u_{p}^{(2)},v_{p}^{(3)})(\theta,Y)=(0,0),$

we obtain the following linearized steady Prandtl equations for
$(u_{p}^{(2)},v_{p}^{(3)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(1)+u_{p}^{(0)}\big{)}\partial_{\theta}u_{p}^{(2)}+\big{(}v_{e}^{(1)}(\theta,1)+v_{p}^{(1)}\big{)}\partial_{Y}u_{p}^{(2)}+(v_{p}^{(3)}-v_{p}^{(3)}(\theta,0))\partial_{Y}u_{p}^{(0)}\\\\[5.0pt]
\quad\quad\quad\quad+(u_{p}^{(2)}+\bar{u}_{e}^{(2)}(\theta,1))\partial_{\theta}u_{p}^{(0)}-\partial_{YY}u_{p}^{(2)}=f_{2}(\theta,Y),\\\\[5.0pt]
\partial_{\theta}u_{p}^{(2)}+\partial_{Y}v_{p}^{(3)}+\partial_{Y}(Yv_{p}^{(2)})=0,\\\\[5.0pt]
u_{p}^{(2)}(\theta,Y)=u_{p}^{(2)}(\theta+2\pi,Y),\quad
v_{p}^{(3)}(\theta,Y)=v_{p}^{(3)}(\theta+2\pi,Y),\\\\[5.0pt]
u_{p}^{(2)}\big{|}_{Y=0}=-\bar{u}_{e}^{(2)}\big{|}_{r=1},\ \
\lim\limits_{Y\rightarrow-\infty}\partial_{Y}u_{p}^{(2)}(\theta,Y)=\lim\limits_{Y\rightarrow-\infty}v_{p}^{(3)}(\theta,Y)=0\end{array}\right.$
(2.149)

here

$\displaystyle f_{2}(\theta,Y)=$
$\displaystyle-\partial_{\theta}p_{p}^{(2)}+Y\partial_{YY}u_{p}^{(1)}+\partial_{Y}u_{p}^{(1)}+\partial_{\theta\theta}u_{p}^{(0)}-u_{p}^{(0)}-\tilde{u}_{p}^{(1)}\partial_{\theta}u_{p}^{(1)}-v_{p}^{(2)}\partial_{Y}u_{p}^{(1)}-\sum_{i+j=2}v_{p}^{(i)}Y\partial_{Y}u_{p}^{(j)}$
$\displaystyle-\sum_{k=0}^{2}\sum_{i+j=2-k,(k,j)\neq(0,2)}\Big{(}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\partial_{\theta}\tilde{u}_{p}^{(j)}+\tilde{u}_{p}^{(j)}\frac{\partial_{r}^{k}\partial_{\theta}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\Big{)}+u_{e}^{(2)}(\theta,1)\partial_{\theta}u_{p}^{(0)}$
$\displaystyle-\sum_{k=0}^{1}\sum_{i+j=2-k}\Big{(}\frac{\partial_{r}^{k}\tilde{v}_{e}^{(i)}(\theta,1)}{k!}Y^{k+1}\partial_{Y}\tilde{u}_{p}^{(j)}+v_{p}^{(i)}\frac{\partial_{r}^{k}(r\partial_{r}\tilde{u}_{e}^{(j)})(\theta,1)}{k!}Y^{k}\Big{)}$
$\displaystyle-\sum_{k=0}^{2}\sum_{i+j=3-k,(k,j)\neq(0,2),(0,0)}\frac{\partial_{r}^{k}\tilde{v}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\partial_{Y}\tilde{u}_{p}^{(j)},$

where $\tilde{u}_{p}^{(0)}=u_{p}^{(0)},\ \tilde{u}_{e}^{(0)}=u_{e}(r),\
\tilde{u}_{e}^{(2)}=\bar{u}_{e}^{(2)},\
\tilde{v}_{e}^{(2)}=\bar{v}_{e}^{(2)}.$

Similarly, let

$\displaystyle
u^{\varepsilon}(\theta,r)=u_{e}(r)+\widehat{u}_{p}^{(0)}(\theta,Z)+\varepsilon\big{[}\tilde{u}_{e}^{(1)}(\theta,r)+(\widehat{u}_{p}^{(1)}(\theta,Z)-\hat{A}_{1})\big{]}+\varepsilon^{2}\big{[}\bar{u}_{e}^{(2)}(\theta,r)+\widehat{u}_{p}^{(2)}(\theta,Z)\big{]}+\text{h.o.t.},$
$\displaystyle
v^{\varepsilon}(\theta,r)=\varepsilon\big{[}\tilde{v}_{e}^{(1)}(\theta,r)+\widehat{v}_{p}^{(1)}(\theta,Z)\big{]}+\varepsilon^{2}\big{[}\bar{v}_{e}^{(2)}(\theta,r)+\widehat{v}_{p}^{(2)}(\theta,Z)\big{]}+\varepsilon^{3}[v_{e}^{(3)}(\theta,r)+\widehat{v}_{p}^{(3)}(\theta,Z)]+\text{h.o.t.},$
$\displaystyle
p^{\varepsilon}(\theta,r)=p_{e}(r)+\varepsilon\big{[}\tilde{p}_{e}^{(1)}(\theta,r)+\widehat{p}_{p}^{(1)}(\theta,Z)\big{]}+\varepsilon^{2}\big{[}\tilde{p}_{e}^{(2)}(\theta,r)+\widehat{p}_{p}^{(2)}(\theta,Z)\big{]}+\varepsilon\widehat{p}_{p}^{(3)}(\theta,Z)+\text{h.o.t.},$

with the following boundary conditions

$\displaystyle\bar{u}_{e}^{(2)}(\theta,r_{0})+\hat{u}_{p}^{(2)}(\theta,0)=0,\
v_{e}^{(3)}(\theta,r_{0})+\widehat{v}_{p}^{(3)}(\theta,0)=0,\
\lim_{Z\rightarrow+\infty}(\partial_{Z}\hat{u}_{p}^{(2)},\hat{v}_{p}^{(3)})(\theta,Z)=(0,0),$

we obtain the following linearized Prandlt equations for
$(\hat{u}_{p}^{(2)},\hat{v}_{p}^{(3)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(r_{0})+\hat{u}_{p}^{(0)}\big{)}\partial_{\theta}\hat{u}_{p}^{(2)}+\big{(}v_{e}^{(1)}(\theta,r_{0})+\hat{v}_{p}^{(1)}\big{)}r_{0}\partial_{Z}\hat{u}_{p}^{(2)}+(\hat{u}_{p}^{(2)}+\bar{u}_{e}^{(2)}(\theta,r_{0}))\partial_{\theta}\hat{u}_{p}^{(0)}\\\\[5.0pt]
\quad\quad\quad\quad\quad\quad\quad\quad\quad+(\hat{v}_{p}^{(3)}-\hat{v}_{p}^{(3)}(\theta,0))r_{0}\partial_{Z}\hat{u}_{p}^{(0)}-r_{0}\partial_{ZZ}\hat{u}_{p}^{(2)}=\hat{f}_{2}(\theta,Z),\\\\[5.0pt]
\partial_{\theta}\hat{u}_{p}^{(2)}+r_{0}\partial_{Z}\hat{v}_{p}^{(3)}+\partial_{Z}(Z\hat{v}_{p}^{(2)})=0,\\\\[5.0pt]
\hat{u}_{p}^{(2)}(\theta,Z)=\hat{u}_{p}^{(2)}(\theta+2\pi,Z),\quad\hat{v}_{p}^{(3)}(\theta,Z)=\hat{v}_{p}^{(3)}(\theta+2\pi,Z),\\\\[5.0pt]
\hat{u}_{p}^{(2)}\big{|}_{Z=0}=-u_{e}^{(2)}\big{|}_{r=r_{0}},\ \
\lim\limits_{Z\rightarrow+\infty}(\partial_{Z}\hat{u}_{p}^{(2)},\hat{v}_{p}^{(3)}(\theta,Z))=(0,0).\end{array}\right.$
(2.155)

The expressions of functions $\hat{f}_{2}(\theta,Z)$ are similar to
$f_{2}(\theta,Y)$, we omit the details here.

######  Proposition 2.9.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and
$\delta\in[0,1]$, equation (2.149) has a unique solution
$(u_{p}^{(2)},v_{p}^{(3)})$ and equation (2.155) has a unique solution
$(\hat{u}_{p}^{(2)},\hat{v}_{p}^{(3)})$ which satisfies

$\displaystyle\begin{aligned} &\sum_{j+k\leq
m}\int_{-\infty}^{0}\int_{0}^{2\pi}\Big{|}\partial_{\theta}^{j}\partial_{Y}^{k}(u_{p}^{(2)}-A_{2\infty},v_{p}^{(3)})\Big{|}^{2}\big{<}Y\big{>}^{2l}d\theta
dY\leq C(\delta,m,l)\eta^{2},\ \forall m,l\geq 0,\\\ &\sum_{j+k\leq
m}\int_{0}^{+\infty}\int_{0}^{2\pi}\Big{|}\partial_{\theta}^{j}\partial_{Z}^{k}(\hat{u}_{p}^{(2)}-\hat{A}_{2\infty},\hat{v}_{p}^{(3)})\Big{|}^{2}\big{<}Z\big{>}^{2l}d\theta
dZ\leq C(\delta,m,l)\eta^{2},\ \forall m,l\geq 0,\\\
&\int_{0}^{2\pi}v_{p}^{(3)}(\theta,Y)d\theta=\int_{0}^{2\pi}\hat{v}_{p}^{(3)}(\theta,Z)d\theta=0,\
\forall\ Y\leq 0,\ Z\geq 0\end{aligned}$ (2.156)

where

$\displaystyle
A_{2\infty}:=\lim_{Y\rightarrow-\infty}u_{p}^{(2)}(\theta,Y),\quad\hat{A}_{2\infty}:=\lim_{Z\rightarrow+\infty}\hat{u}_{p}^{(2)}(\theta,Z).$

and satisfies $|A_{2\infty}|+|\hat{A}_{2\infty}|\leq C(\delta)\eta$.

The proof is same with Proposition 2.6 by noticing that $f_{2}(\theta,Y)$
decay very fast as $Y\rightarrow-\infty$, we omit the details.

We construct the pressure $p_{p}^{(3)}(\theta,Y)$ by considering the equation

$\displaystyle\partial_{Y}p_{p}^{(3)}(\theta,Y)=g_{2}(\theta,Y),\quad\lim_{Y\rightarrow-\infty}p_{p}^{(3)}(\theta,Y)=0,$
(2.157)

where

$\displaystyle g_{2}(\theta,Y)=$
$\displaystyle\partial_{YY}v_{p}^{(2)}+Y\partial_{YY}v_{p}^{(1)}+\partial_{Y}v_{p}^{(1)}-2\partial_{\theta}u_{p}^{(0)}-Y\partial_{Y}p_{p}^{(2)}-\sum_{i+j=3}v_{p}^{(i)}\partial_{Y}v_{p}^{(j)}$
$\displaystyle-\sum_{i+j=2}\Big{(}\tilde{u}_{p}^{(i)}\partial_{\theta}v_{p}^{(j)}+v_{p}^{(i)}Y\partial_{Y}v_{p}^{(j)}-\tilde{u}_{p}^{(i)}\tilde{u}_{p}^{(j)}+\tilde{v}_{e}^{(i)}(\theta,1)Y\partial_{Y}v_{p}^{(j)}+v_{p}^{(i)}\partial_{r}\tilde{v}_{e}^{(j)}(\theta,1)\Big{)}$
$\displaystyle-\sum_{k=0}^{1}\sum_{i+j=2-k}\Big{(}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\partial_{\theta}v_{p}^{(j)}+\frac{\partial_{r}^{k}\partial_{\theta}\tilde{v}_{e}^{(j)}(\theta,1)}{k!}Y^{k}\partial_{\theta}\tilde{u}_{p}^{(i)}\Big{)}$
$\displaystyle-\sum_{k=0}^{2}\sum_{i+j=2-k}\Big{(}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\tilde{u}_{p}^{(j)}+\frac{\partial_{r}^{k}\tilde{u}_{e}^{(j)}(\theta,1)}{k!}Y^{k}\tilde{u}_{p}^{(i)}\Big{)},$

where $\tilde{u}_{p}^{(0)}=u_{p}^{(0)},\ \tilde{u}_{e}^{(0)}=u_{e}(r),\
\tilde{u}_{e}^{(2)}=\bar{u}_{e}^{(2)}+A_{2\infty},\
\tilde{v}_{e}^{(2)}=\bar{v}_{e}^{(2)}$, and here and below
$\tilde{u}_{p}^{(2)}=u_{p}^{(2)}-A_{2\infty}$. $g_{2}(\theta,Y)$ can be
derived by the same argument as $g_{1}(\theta,Y)$. Moreover, notice that
$g_{2}(\theta,Y)$ decay fast as $Y\rightarrow-\infty$, we can obtain
$p_{p}^{(3)}$ by solving (2.157) and $p_{p}^{(3)}$ also decay fast as
$Y\rightarrow-\infty$.

Similarly, we can construct $\hat{p}^{(3)}_{p}(\theta,Z)$ and
$\hat{p}^{(3)}_{p}(\theta,Z)$ also decay fast as $Z\rightarrow+\infty$.

####  2.4.6. Linearized Euler system for
$(u_{e}^{(3)},v_{e}^{(3)},p_{e}^{(3)})$ and its solvability

Let

$\displaystyle\tilde{u}_{e}^{(2)}(\theta,r):$
$\displaystyle=\bar{u}_{e}^{(2)}(\theta,r)+\chi(r)A_{2\infty}+(1-\chi(r))\hat{A}_{2\infty},\quad\tilde{v}_{e}^{(2)}(\theta,r)=\bar{v}_{e}^{(2)}(\theta,r),$
$\displaystyle\tilde{p}_{e}^{(2)}(\theta,r):$
$\displaystyle=\bar{p}_{e}^{(2)}(\theta,r)+\int_{r_{0}}^{r}\frac{2u_{e}(s)}{s}[\chi(s)A_{2\infty}+(1-\chi(s))\hat{A}_{2\infty}]ds,$

then $(\tilde{u}_{e}^{(2)},\tilde{v}_{e}^{(2)},\tilde{p}_{e}^{(2)})$ also
satisfies the linearized Euler equations (2.141) with the boundary conditions
(2.142), and there holds

$\displaystyle\|\partial_{\theta}^{j}\partial_{r}^{k}(\tilde{u}^{(2)}_{e},\tilde{v}_{e}^{(2)})\|_{\infty}\leq
C(\delta,j,k)\eta,\ \forall k,j\geq 0.$ (2.158)

Putting

$\displaystyle
u^{\varepsilon}(\theta,r)=u_{e}(r)+\sum_{i=1}^{3}\varepsilon^{i}\tilde{u}_{e}^{(i)}(\theta,r)+\text{h.o.t.},$
$\displaystyle
v^{\varepsilon}(\theta,r)=\sum_{i=1}^{3}\varepsilon^{i}\tilde{v}_{e}^{(i)}(\theta,r)+\text{h.o.t.},$
$\displaystyle
p^{\varepsilon}(\theta,r)=p_{e}(r)+\sum_{i=1}^{3}\varepsilon^{i}\tilde{p}_{e}^{(i)}(\theta,r)+\text{h.o.t.},$

into the Navier-Stokes equation (1.8), we find that
$(\tilde{u}_{e}^{(3)},\tilde{v}_{e}^{(3)},\tilde{p}_{e}^{(3)})$ satisfies the
following linearized Euler equations in $\Omega$

$\displaystyle\left\\{\begin{array}[]{ll}u_{e}(r)\partial_{\theta}\tilde{u}_{e}^{(3)}+r\tilde{v}_{e}^{(3)}u^{\prime}_{e}(r)+u_{e}(r)\tilde{v}_{e}^{(3)}+\partial_{\theta}\tilde{p}_{e}^{(3)}+f_{e}(\theta,r)=0,\\\\[5.0pt]
u_{e}(r)\partial_{\theta}\tilde{v}_{e}^{(3)}-2u_{e}(r)\tilde{u}_{e}^{(3)}+r\partial_{r}\tilde{p}_{e}^{(3)}+g_{e}(\theta,r)=0,\\\\[7.0pt]
\partial_{\theta}\tilde{u}_{e}^{(3)}+r\partial_{r}\tilde{v}_{e}^{(3)}+\tilde{v}_{e}^{(3)}=0,\end{array}\right.$
(2.162)

and equipped with the following boundary conditions

$\displaystyle\tilde{v}_{e}^{(3)}|_{r=1}=-v_{p}^{(3)}|_{Y=0},\quad\tilde{v}_{e}^{(3)}|_{r=r_{0}}=-\hat{v}_{p}^{(3)}|_{Z=0},\quad\tilde{v}_{e}^{(3)}(\theta,r)=\tilde{v}_{e}^{(3)}(\theta+2\pi,r),$

where

$\displaystyle f_{e}(\theta,r)=$
$\displaystyle\tilde{u}_{e}^{(1)}\partial_{\theta}\tilde{u}_{e}^{(2)}+\tilde{u}_{e}^{(2)}\partial_{\theta}\tilde{u}_{e}^{(1)}+\tilde{v}_{e}^{(1)}r\partial_{r}\tilde{u}_{e}^{(2)}+\tilde{v}_{e}^{(2)}r\partial_{r}\tilde{u}_{e}^{(1)}+\tilde{u}_{e}^{(1)}\tilde{v}_{e}^{(2)}+\tilde{u}_{e}^{(2)}\tilde{v}_{e}^{(1)}$
$\displaystyle-\Big{(}\frac{\partial_{\theta\theta}\tilde{u}_{e}^{(1)}}{r}+r\partial_{rr}\tilde{u}_{e}^{(1)}+\partial_{r}\tilde{u}_{e}^{(1)}-\frac{2}{r}\partial_{\theta}\tilde{v}_{e}^{(1)}-\frac{\tilde{u}_{e}^{(1)}}{r}\Big{)},$
$\displaystyle g_{e}(\theta,r)=$
$\displaystyle\tilde{u}_{e}^{(1)}\partial_{\theta}\tilde{v}_{e}^{(2)}+\tilde{u}_{e}^{(2)}\partial_{\theta}\tilde{v}_{e}^{(1)}+\tilde{v}_{e}^{(1)}r\partial_{r}\tilde{v}_{e}^{(2)}+\tilde{v}_{e}^{(2)}r\partial_{r}\tilde{v}_{e}^{(1)}-2\tilde{u}_{e}^{(1)}\tilde{u}_{e}^{(2)}$
$\displaystyle-\Big{(}\frac{\partial_{\theta\theta}\tilde{v}_{e}^{(1)}}{r}+r\partial_{rr}\tilde{v}_{e}^{(1)}+\partial_{r}\tilde{v}_{e}^{(1)}-\frac{2}{r}\partial_{\theta}\tilde{u}_{e}^{(1)}+\frac{\tilde{v}_{e}^{(1)}}{r}\Big{)}.$

Similarly to get (2.128), we can derive the necessary condition to the
solvability of (2.162):

$\displaystyle\frac{d}{dr}\int_{0}^{2\pi}f_{e}(\theta,r)d\theta=0.$ (2.163)

######  Proposition 2.10.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and
$\delta\in[0,1]$, the linearized Euler equation (2.162) has a solution
$(\tilde{u}_{e}^{(3)},\tilde{v}_{e}^{(3)},\tilde{p}_{e}^{(3)})$ which
satisfies

$\displaystyle\|\partial^{k}_{\theta}\partial^{j}_{r}(\tilde{u}_{e}^{(3)},\tilde{v}_{e}^{(3)})\|_{2}\leq
C(\delta,j,k)\eta,\ \forall j,k\geq 0.$ (2.164)

###### Proof.

We only need to check (2.163), in fact, we can arrive at
$\int_{0}^{2\pi}f_{e}(\theta,r)d\theta=0$, that is,

$\displaystyle\int_{0}^{2\pi}\Big{(}\tilde{v}^{(1)}_{e}\partial_{r}(r\tilde{u}^{(2)}_{e})+\tilde{v}^{(2)}_{e}\partial_{r}(r\tilde{u}^{(1)}_{e})-\Big{(}r\partial_{rr}\tilde{u}^{(1)}_{e}+\partial_{r}\tilde{u}^{(1)}_{e}-\frac{\tilde{u}^{(1)}_{e}}{r}\Big{)}\Big{)}d\theta=0,\
\forall r\in[r_{0},1].$ (2.165)

By (2.136), we only need to check that

$\displaystyle\int_{0}^{2\pi}\Big{(}v^{(1)}_{e}\partial_{r}(ru^{(2)}_{e})+v^{(2)}_{e}\partial_{r}(r\bar{u}^{(1)}_{e})\Big{)}(\theta,r)d\theta=\int_{0}^{2\pi}\Big{(}\tilde{v}^{(1)}_{e}\partial_{r}(r\tilde{u}^{(2)}_{e})+\tilde{v}^{(2)}_{e}\partial_{r}(r\tilde{u}^{(1)}_{e})\Big{)}(\theta,r)d\theta,\
\forall r\in[r_{0},1].$

Notice that $(\bar{u}_{e}^{(1)},v_{e}^{(1)})$ and
$(\tilde{u}_{e}^{(1)},\tilde{v}_{e}^{(1)})$ are both the solution of (2.9) and
$\tilde{v}_{e}^{(1)}=v_{e}^{(1)}$, the same argument as (2.4.4) gives

$\displaystyle\int_{0}^{2\pi}v^{(2)}_{e}\partial_{r}(r\bar{u}^{(1)}_{e})(\theta,r)d\theta=\int_{0}^{2\pi}v^{(2)}_{e}\partial_{\theta}v_{e}^{(1)}(\theta,r)d\theta,\
\forall r\in[r_{0},1],$
$\displaystyle\int_{0}^{2\pi}\tilde{v}^{(2)}_{e}\partial_{r}(r\tilde{u}^{(1)}_{e})(\theta,r)d\theta=\int_{0}^{2\pi}\tilde{v}^{(2)}_{e}\partial_{\theta}v_{e}^{(1)}(\theta,r)d\theta,\
\forall r\in[r_{0},1].$

Thus, we deduce that

$\displaystyle\int_{0}^{2\pi}[v^{(2)}_{e}\partial_{r}(r\bar{u}^{(1)}_{e})-\tilde{v}^{(2)}_{e}\partial_{r}(r\tilde{u}^{(1)}_{e})](\theta,r)d\theta=\int_{0}^{2\pi}[v^{(2)}_{e}-\tilde{v}^{(2)}_{e}]\partial_{\theta}v_{e}^{(1)}(\theta,r)d\theta,\
\forall r\in[r_{0},1].$ (2.166)

Similarly, using (2.127), (2.141) and the fact
$\tilde{v}_{e}^{(1)}=v_{e}^{(1)}$, we can deduce that

$\displaystyle-
ru_{e}(r)\triangle(rv_{e}^{(2)})=v_{e}^{(1)}(-r^{2}\triangle\bar{u}_{e}^{(1)}+\bar{u}_{e}^{(1)})-2v_{e}^{(1)}\partial_{\theta}v_{e}^{(1)},$
$\displaystyle-
ru_{e}(r)\triangle(r\tilde{v}_{e}^{(2)})=v_{e}^{(1)}(-r^{2}\triangle\tilde{u}_{e}^{(1)}+\tilde{u}_{e}^{(1)})-2v_{e}^{(1)}\partial_{\theta}v_{e}^{(1)}.$

Thus, there holds

$\displaystyle-
ru_{e}(r)\triangle(rv_{e}^{(2)}-r\tilde{v}_{e}^{(2)})=v_{e}^{(1)}(-r^{2}A^{\prime\prime}_{1}(r)+A_{1}(r)).$

The same argument as (2.4.4) gives

$\displaystyle\int_{0}^{2\pi}v^{(1)}_{e}\big{(}\partial_{r}(ru^{(2)}_{e})-\partial_{r}(r\tilde{u}^{(2)}_{e})\big{)}(\theta,r)d\theta=\int_{0}^{2\pi}v^{(1)}_{e}\partial_{\theta}\big{(}v^{(2)}_{e}-\tilde{v}^{(2)}_{e})(\theta,r)d\theta,\
\forall r\in[r_{0},1].$ (2.167)

With the help of (2.166) and (2.167), we obtain (2.165). MM $\square$

####  2.4.7. Linearized Prandtl equations for $(u_{p}^{(3)},v_{p}^{(4)})$,
$(\hat{u}_{p}^{(3)},\hat{v}_{p}^{(4)})$ and their solvabilities

Let

$\displaystyle
u^{\varepsilon}(\theta,r)=u_{e}(r)+u_{p}^{(0)}(\theta,Y)+\sum_{i=1}^{2}\varepsilon^{i}\big{[}\tilde{u}_{e}^{(i)}(\theta,r)+\tilde{u}_{p}^{(i)}(\theta,Y)\big{]}+\varepsilon^{3}[\tilde{u}_{e}^{(3)}(\theta,r)+u_{p}^{(3)}(\theta,Y)\big{]}+\text{h.o.t.},$
$\displaystyle
v^{\varepsilon}(\theta,r)=\sum_{i=1}^{3}\varepsilon^{i}\big{[}\tilde{v}_{e}^{(i)}(\theta,r)+v_{p}^{(i)}(\theta,Y)\big{]}+\varepsilon^{4}v_{p}^{(4)}(\theta,Z)+\text{h.o.t.},$
$\displaystyle
p^{\varepsilon}(\theta,r)=p_{e}(r)+\sum_{i=1}^{3}\varepsilon^{i}\big{[}\tilde{p}_{e}^{(i)}(\theta,r)+p_{p}^{(i)}(\theta,Y)\big{]}+\varepsilon^{4}p_{p}^{(4)}(\theta,Y)+\text{h.o.t.},$

with the boundary conditions

$\displaystyle\tilde{u}_{e}^{(3)}(\theta,1)+u_{p}^{(3)}(\theta,0)=0,\quad\lim_{Y\rightarrow\infty}\partial_{Y}u_{p}^{(3)}(\theta,Y)=v_{p}^{(4)}(\theta,0)=0,$

we obtain the following linearized Prandtl problem for
$(u_{p}^{(3)},v_{p}^{(4)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(1)+u_{p}^{(0)}\big{)}\partial_{\theta}u_{p}^{(3)}+\big{(}v_{e}^{(1)}(\theta,1)+v_{p}^{(1)}\big{)}\partial_{Y}u_{p}^{(3)}+(u_{p}^{(3)}+\tilde{u}_{e}^{(3)}(\theta,1))\partial_{\theta}u_{p}^{(0)}\\\\[5.0pt]
\quad\quad\quad\quad
v_{p}^{(4)}\partial_{Y}u_{p}^{(0)}-\partial_{YY}u_{p}^{(3)}=f_{3}(\theta,Y)\\\\[5.0pt]
\partial_{\theta}u_{p}^{(3)}+\partial_{Y}v_{p}^{(4)}+\partial_{Y}(Yv_{p}^{(3)})=0,\\\\[5.0pt]
u_{p}^{(3)}(\theta,Y)=u_{p}^{(3)}(\theta+2\pi,Y),\quad
v_{p}^{(4)}(\theta,Y)=v_{p}^{(4)}(\theta+2\pi,Y),\\\\[5.0pt]
u_{p}^{(3)}\big{|}_{Y=0}=-\tilde{u}_{e}^{(3)}\big{|}_{r=1},\ \
\lim\limits_{Y\rightarrow\infty}\partial_{Y}u_{p}^{(3)}(\theta,Y)=v_{p}^{(4)}(\theta,0)=0\end{array}\right.$
(2.173)

and

$\displaystyle\partial_{Y}p_{p}^{(4)}(\theta,Y)=g_{3}(\theta,Y),\
p_{p}^{(4)}(\theta,0)=0,$ (2.174)

where

$\displaystyle f_{3}(\theta,Y)=$
$\displaystyle-\partial_{\theta}p_{p}^{(3)}+Y\partial_{YY}u_{p}^{(3)}+\partial_{Y}u_{p}^{(3)}+\sum_{k=0}^{1}(-1)^{k}\frac{Y^{k}\partial_{\theta\theta}\tilde{u}_{p}^{(1-k)}}{k!}+2\partial_{\theta}v_{p}^{(2)}-2Y\partial_{\theta}v_{p}^{(1)}$
$\displaystyle-\sum_{k=0}^{1}(-1)^{k}\frac{Y^{k}\tilde{u}_{p}^{(1-k)}}{k!}-\sum_{i+j=3,1\leq
i\leq
2}\tilde{u}_{p}^{(i)}\partial_{\theta}\tilde{u}_{p}^{(j)}-\sum_{i+j=3}[v_{p}^{(i)}Y\partial_{Y}\tilde{u}_{p}^{(j)}+\tilde{u}_{p}^{(i)}v_{p}^{(j)}]$
$\displaystyle-\sum_{k=0}^{3}\sum_{i+j=3-k,(k,j)\neq(0,3)}\Big{(}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\partial_{\theta}\tilde{u}_{p}^{(j)}+\tilde{u}_{p}^{(i)}\frac{\partial_{r}^{k}\partial_{\theta}\tilde{u}_{e}^{(j)}(\theta,1)}{k!}Y^{k}\Big{)}$
$\displaystyle-\sum_{k=0}^{3}\sum_{i+j=3-k}\Big{(}\frac{\partial_{r}^{k}\tilde{v}_{e}^{(i)}(\theta,1)}{k!}Y^{k+1}\partial_{Y}\tilde{u}_{p}^{(j)}+v_{p}^{(i)}\frac{\partial_{r}^{k}(r\partial_{r}\tilde{u}_{e}^{(j)})(\theta,1)}{k!}Y^{k}\Big{)}$
$\displaystyle-\sum_{k=0}^{3}\sum_{i+j=4-k,(k,j)\neq(0,3),(0,0),i\leq
3}\frac{\partial_{r}^{k}\tilde{v}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\partial_{Y}\tilde{u}_{p}^{(j)}$
$\displaystyle-\sum_{k=0}^{3}\sum_{i+j=3-k}\Big{(}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}v_{p}^{(j)}+\tilde{u}_{p}^{(i)}\frac{\partial_{r}^{k}\tilde{v}_{e}^{(j)}(\theta,1)}{k!}Y^{k}\Big{)},$
$\displaystyle g_{3}(\theta,Y)=$
$\displaystyle\partial_{YY}v_{p}^{(3)}+Y\partial_{YY}v_{p}^{(2)}+\partial_{Y}v_{p}^{(2)}+\partial_{\theta\theta}v_{p}^{(1)}-\sum_{k=0}^{1}(-1)^{k}\frac{Y^{k}}{k!}\partial_{\theta}\tilde{u}_{p}^{(2-k)}-v_{p}^{(1)}-Y\partial_{Y}p_{p}^{(3)}$
$\displaystyle-\sum_{i+j=3}[\tilde{u}_{p}^{(i)}\partial_{\theta}v_{p}^{(j)}+v_{p}^{(i)}Y\partial_{Y}v_{p}^{(j)}-\tilde{u}_{p}^{(i)}\tilde{u}_{p}^{(j)}]-\sum_{i+j=4}v_{p}^{(i)}Y\partial_{Y}v_{p}^{(j)}-\sum_{k=0}^{2}\sum_{i+j=4-k}\frac{\partial_{r}^{k}\tilde{v}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\partial_{Y}v_{p}^{(j)}$
$\displaystyle-\sum_{k=0}^{3}\sum_{i+j=3-k}\Big{(}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\partial_{\theta}v_{p}^{(j)}+\tilde{u}_{p}^{(i)}\frac{\partial_{r}^{k}\partial_{\theta}\tilde{v}_{e}^{(j)}(\theta,1)}{k!}Y^{k}\Big{)}$
$\displaystyle+\sum_{k=0}^{3}\sum_{i+j=3-k}\Big{(}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(i)}(\theta,1)}{k!}Y^{k}\tilde{u}_{p}^{(j)}+\tilde{u}_{p}^{(i)}\frac{\partial_{r}^{k}\tilde{u}_{e}^{(j)}(\theta,1)}{k!}Y^{k}\Big{)}$
$\displaystyle-\sum_{k=0}^{1}\sum_{i+j=3-k}\Big{(}\frac{\partial_{r}^{k}\tilde{v}_{e}^{(i)}(\theta,1)}{k!}Y^{k+1}\partial_{Y}v_{p}^{(j)}+v_{p}^{(i)}\frac{\partial_{r}^{k}(r\partial_{r}\tilde{v}_{e}^{(j)})(\theta,1)}{k!}Y^{k}\Big{)}$

with $\tilde{u}_{p}^{(0)}=u_{p}^{(0)},\ \tilde{u}^{(0)}_{e}=u_{e}(r),\
\tilde{u}_{p}^{(3)}=u_{p}^{(3)}.$

Similarly, let

$\displaystyle
u^{\varepsilon}(\theta,r)=u_{e}(r)+\widehat{u}_{p}^{(0)}(\theta,Z)+\sum_{i=1}^{2}\varepsilon^{i}\big{[}\tilde{u}_{e}^{(i)}(\theta,r)+(\widehat{u}_{p}^{(i)}(\theta,Z)-\hat{A}_{i\infty})\big{]}+\varepsilon^{3}\big{[}\tilde{u}_{e}^{(3)}(\theta,r)+\widehat{u}_{p}^{(3)}(\theta,Z)\big{]}+\text{h.o.t.},$
$\displaystyle
v^{\varepsilon}(\theta,r)=\sum_{i=1}^{3}\varepsilon^{i}\big{[}\tilde{v}_{e}^{(i)}(\theta,r)+\widehat{v}_{p}^{(i)}(\theta,Z)\big{]}+\varepsilon^{4}\widehat{v}_{p}^{(4)}(\theta,Z)+\text{h.o.t.},$
$\displaystyle
p^{\varepsilon}(\theta,r)=p_{e}(r)+\sum_{i=1}^{3}\varepsilon\big{[}\tilde{p}_{e}^{(i)}(\theta,r)+\widehat{p}_{p}^{(i)}(\theta,Z)\big{]}+\varepsilon^{4}\widehat{p}_{p}^{(4)}(\theta,Z)+\text{h.o.t.},$

with the following boundary conditions

$\displaystyle\tilde{u}_{e}^{(3)}(\theta,r_{0})+\hat{u}_{p}^{(3)}(\theta,0)=0,\quad\lim_{Z\rightarrow+\infty}\partial_{Z}\hat{u}_{p}^{(3)}(\theta,Z)=\hat{v}_{p}^{(4)}(\theta,0)=0,$

we obtain the following linearized Prandlt equations for
$(\hat{u}_{p}^{(3)},\hat{v}_{p}^{(4)})$

$\displaystyle\left\\{\begin{array}[]{ll}\big{(}u_{e}(r_{0})+\hat{u}_{p}^{(0)}\big{)}\partial_{\theta}\hat{u}_{p}^{(3)}+\big{(}v_{e}^{(1)}(\theta,r_{0})+\hat{v}_{p}^{(1)}\big{)}r_{0}\partial_{Z}\hat{u}_{p}^{(3)}+\hat{u}_{p}^{(3)}\partial_{\theta}\hat{u}_{p}^{(0)}\\\\[5.0pt]
\quad\quad\quad\quad\quad\quad\quad\quad\quad+\hat{v}_{p}^{(4)}r_{0}\partial_{Z}\hat{u}_{p}^{(0)}-r_{0}\partial_{ZZ}\hat{u}_{p}^{(3)}=\hat{f}_{3}(\theta,Z),\\\\[5.0pt]
\partial_{\theta}\hat{u}_{p}^{(3)}+r_{0}\partial_{Z}\hat{v}_{p}^{(4)}+\partial_{Z}(Z\hat{v}_{p}^{(3)})=0,\\\\[5.0pt]
\hat{u}_{p}^{(3)}(\theta,Z)=\hat{u}_{p}^{(3)}(\theta+2\pi,Z),\quad\hat{v}_{p}^{(4)}(\theta,Z)=\hat{v}_{p}^{(4)}(\theta+2\pi,Z),\\\\[5.0pt]
\hat{u}_{p}^{(3)}\big{|}_{Z=0}=-\tilde{u}_{e}^{(3)}\big{|}_{r=r_{0}},\ \
\lim\limits_{Z\rightarrow+\infty}\partial_{Z}\hat{u}_{p}^{(3)}(\theta,Z)=\hat{v}_{p}^{(4)}(\theta,0)=0\end{array}\right.$
(2.180)

and the pressure $\hat{p}_{p}^{(4)}$ satisfies

$\displaystyle\partial_{Z}\hat{p}_{p}^{(4)}(\theta,Z)=\hat{g}_{3}(\theta,Z),\quad\hat{p}_{p}^{(4)}(\theta,0)=0.$
(2.181)

The functions $(\hat{f}_{3}(\theta,Z),\hat{g}_{3}(\theta,Z))$ are similar to
$(f_{3}(\theta,Y),g_{3}(\theta,Y))$, we omit the details here.

######  Proposition 2.11.

There exists $\eta_{0}>0$ such that for any $\eta\in(0,\eta_{0})$ and
$\delta\in[0,1]$, equations (2.173) have a unique solution
$(u_{p}^{(3)},v_{p}^{(4)})$ and equations (2.180) have a unique solution
$(\hat{u}_{p}^{(3)},\hat{v}_{p}^{(4)})$ which satisfy

$\displaystyle\begin{aligned} &\sum_{0<j+k\leq
m}\int_{-\infty}^{0}\int_{0}^{2\pi}\big{|}\partial_{\theta}^{j}\partial_{Y}^{k}(u_{p}^{(3)},v_{p}^{(4)})\big{|}^{2}\big{<}Y\big{>}^{2l}d\theta
dY\leq C(\delta,m,l)\eta^{2},\ \forall\ m,l\geq 0,\\\ &\sum_{0<j+k\leq
m}\int_{0}^{+\infty}\int_{0}^{2\pi}\big{|}\partial_{\theta}^{j}\partial_{Z}^{k}(\hat{u}_{p}^{(3)},\hat{v}_{p}^{(4)})\big{|}^{2}\big{<}Z\big{>}^{2l}d\theta
dZ\leq C(\delta,m,l)\eta^{2},\ \forall\ m,l\geq 0,\\\
&\int_{0}^{2\pi}v_{p}^{(4)}(\theta,Y)d\theta=\int_{0}^{2\pi}\hat{v}_{p}^{(4)}(\theta,Z)d\theta=0,\
\forall\ Y\leq 0,\ Z\geq 0\end{aligned}$ (2.182)

and

$\displaystyle\|(u_{p}^{(3)},v_{p}^{(4)})\|_{\infty}+\|(\hat{u}_{p}^{(3)},\hat{v}_{p}^{(4)})\|_{\infty}\leq
C(\delta)\eta.$

The proof is same with Proposition 2.6, we omit the details. Moreover, we can
construct $(p_{p}^{(4)},\hat{p}_{p}^{(4)})$ by solving the equations (2.174)
and (2.181).

###  2.5. Approximate solutions

In this subsection, we construct an approximate solution of Navier-Stokes
equations (1.8). Set

$\displaystyle\tilde{u}_{p}^{a}(\theta,r):=$
$\displaystyle\chi(r)\Big{(}u_{p}^{(0)}(\theta,Y)+\varepsilon\tilde{u}_{p}^{(1)}(\theta,Y)+\varepsilon^{2}\tilde{u}_{p}^{(2)}(\theta,Y)+\varepsilon^{3}u_{p}^{(3)}(\theta,Y)\Big{)}$
$\displaystyle+(1-\chi(r))\Big{(}\hat{u}_{p}^{(0)}(\theta,Z)+\varepsilon\big{[}\hat{u}_{p}^{(1)}(\theta,Z)-\hat{A}_{1\infty}\big{]}+\varepsilon^{2}[\hat{u}_{p}^{(2)}(\theta,Z)-\hat{A}_{2\infty}]+\varepsilon^{3}\hat{u}_{p}^{(3)}(\theta,Z)\Big{)}$
$\displaystyle:=$ $\displaystyle\chi(r)u_{p}^{a}+(1-\chi(r))\hat{u}_{p}^{a},$
$\displaystyle\tilde{v}_{p}^{a}(\theta,r):=$
$\displaystyle\chi(r)\Big{(}\sum_{i=1}^{4}\varepsilon^{i}v_{p}^{(i)}(\theta,Y)\Big{)}+(1-\chi(r))\Big{(}\sum_{i=1}^{4}\varepsilon^{i}\hat{v}_{p}^{(i)}(\theta,Z)\Big{)}$
$\displaystyle:=$ $\displaystyle\chi(r)v_{p}^{a}+(1-\chi(r))\hat{v}_{p}^{a},$
$\displaystyle\tilde{p}_{p}^{a}(\theta,r):=$
$\displaystyle\chi^{2}(r)\Big{(}\sum_{i=1}^{4}\varepsilon^{i}p_{p}^{(i)}(\theta,Y)\Big{)}+(1-\chi(r))^{2}\Big{(}\sum_{i=1}^{4}\varepsilon^{i}\hat{p}_{p}^{(i)}(\theta,Z)\Big{)}$
$\displaystyle:=$
$\displaystyle\chi^{2}(r)p_{p}^{a}+(1-\chi(r))^{2}\hat{p}_{p}^{a},$

and

$\displaystyle u_{e}^{a}(\theta,r):=$ $\displaystyle
u_{e}(r)+\sum_{i=1}^{3}\varepsilon^{i}\tilde{u}_{e}^{(i)}(\theta,r),$
$\displaystyle v_{e}^{a}(\theta,r):=$
$\displaystyle\sum_{i=1}^{3}\varepsilon^{i}\tilde{v}_{e}^{(i)}(\theta,r),$
$\displaystyle p_{e}^{a}(\theta,r):=$ $\displaystyle
p_{e}(r)+\sum_{i=1}^{3}\varepsilon^{i}\tilde{p}_{e}^{(i)}(\theta,r).$

We construct an approximate solution $(u^{a},v^{a},p^{a})$

$\displaystyle u^{a}(\theta,r):$
$\displaystyle=u_{e}^{a}(\theta,r)+\tilde{u}_{p}^{a}(\theta,r)+\varepsilon^{4}h(\theta,r),$
(2.183) $\displaystyle v^{a}(\theta,r):$
$\displaystyle=v_{e}^{a}(\theta,r)+\tilde{v}_{p}^{a}(\theta,r),$ (2.184)
$\displaystyle p^{a}(\theta,r):$
$\displaystyle=p_{e}^{a}(\theta,r)+\tilde{p}_{p}^{a}(\theta,r),$ (2.185)

where the corrector $h(\theta,r)$ will be given in Appendix A which satisfies

$\displaystyle h(\theta,1)=0,\quad
h(\theta,r_{0})=0,\quad\|\partial_{\theta}^{j}\partial_{r}^{k}h\|_{2}\leq
C(\delta,j,k)\varepsilon^{-k}$

and makes $(u^{a},v^{a})$ be divergence-free

$\displaystyle u^{a}_{\theta}+rv^{a}_{r}+v^{a}=0.$

Moreover, $(u^{a},v^{a})$ satisfies the following boundary conditions

$\displaystyle u^{a}(\theta+2\pi,r)$ $\displaystyle=u^{a}(\theta,r),\
v^{a}(\theta+2\pi,r)=v^{a}(\theta,r),$ $\displaystyle u^{a}(\theta,1)$
$\displaystyle=\alpha+\eta f(\theta),\ v^{a}(\theta,1)=0,$ $\displaystyle
u^{a}(\theta,r_{0})$ $\displaystyle=\beta+\eta g(\theta),\
v^{a}(\theta,r_{0})=0.$

By collecting the estimates (2.137), (2.158), (2.164), we deduce that

$\displaystyle\|u^{a}_{e}-u_{e}(r)\|_{\infty}+\|(\partial_{r}(u^{a}_{e}-u_{e}(r)),\partial_{\theta}u^{a}_{e})\|_{\infty}\leq
C(\delta)\varepsilon\eta,$
$\displaystyle\|v^{a}_{e}\|_{\infty}+\|(\partial_{r}v^{a}_{e},\partial_{\theta}v^{a}_{e})\|_{\infty}\leq
C(\delta)\varepsilon\eta,$ (2.186)

and collecting the estimates (2.89), (2.100), (2.156), (2.182) one has

$\displaystyle\|(Y^{j}\partial_{Y}^{k}u_{p}^{a},Y^{j}\partial_{\theta}^{k}u_{p}^{a})\|_{\infty}\leq
C(\delta)\eta,\ \
\|(Y^{j}\partial_{Y}^{k}v_{p}^{a},Y^{j}\partial_{\theta}^{k}v_{p}^{a})\|_{\infty}\leq
C(\delta)\varepsilon\eta,\ \ \forall j\leq 2,k\leq 2.$ (2.187)

Similarly, there holds

$\displaystyle\|(Z^{j}\partial_{Z}^{k}\hat{u}_{p}^{a},Z^{j}\partial_{\theta}^{k}\hat{u}_{p}^{a})\|_{\infty}\leq
C(\delta)\eta,\ \
\|(Z^{j}\partial_{Z}^{k}\hat{v}_{p}^{a},Z^{j}\partial_{\theta}^{k}\hat{v}_{p}^{a})\|_{\infty}\leq
C(\delta)\varepsilon\eta,\ \ \forall j\leq 2,k\leq 2.$

By (1.10), (2.183), (2.5) and (2.187), we deduce that there exists
$\varepsilon_{0}>0,\eta_{0}>0$ such that for any
$\varepsilon\in(0,\varepsilon_{0}),\eta\in(0,\eta_{0})$ and any
$\delta\in[0,1]$, there holds

$\displaystyle u^{a}(\theta,r)\geq d_{0}>0,\ \forall(\theta,r)\in\Omega,$
(2.188)

where $d_{0}$ is a fixed number.

Finally, set

$\displaystyle R_{u}^{a}:$
$\displaystyle=u^{a}u^{a}_{\theta}+v^{a}ru^{a}_{r}+u^{a}v^{a}+p^{a}_{\theta}-\varepsilon^{2}\big{(}ru^{a}_{rr}+\frac{u^{a}_{\theta\theta}}{r}+2\frac{v^{a}_{\theta}}{r}+u^{a}_{r}-\frac{u^{a}}{r}\big{)},$
$\displaystyle R_{v}^{a}:$
$\displaystyle=u^{a}v^{a}_{\theta}+v^{a}rv^{a}_{r}-(u^{a})^{2}+rp^{a}_{r}-\varepsilon^{2}\big{(}rv^{a}_{rr}+\frac{v^{a}_{\theta\theta}}{r}-2\frac{u^{a}_{\theta}}{r}+v^{a}_{r}-\frac{v^{a}}{r}\big{)},$

then there holds

$\displaystyle\|R_{u}^{a}\|_{2}+\|\partial_{\theta}R_{u}^{a}\|_{2}\leq
C(\delta)\varepsilon^{4},\quad\|R_{v}^{a}\|_{2}+\|\partial_{\theta}R_{v}^{a}\|_{2}\leq
C(\delta)\varepsilon^{4}$

and $(u^{a},v^{a},p^{a})$ satisfies

$\displaystyle\left\\{\begin{array}[]{lll}u^{a}u^{a}_{\theta}+v^{a}ru^{a}_{r}+u^{a}v^{a}+p^{a}_{\theta}-\varepsilon^{2}\big{(}ru^{a}_{rr}+\frac{u^{a}_{\theta\theta}}{r}+2\frac{v^{a}_{\theta}}{r}+u^{a}_{r}-\frac{u^{a}}{r}\big{)}=R_{u}^{a},\
(\theta,r)\in\Omega,\\\\[5.0pt]
u^{a}v^{a}_{\theta}+v^{a}rv^{a}_{r}-(u^{a})^{2}+rp^{a}_{r}-\varepsilon^{2}\big{(}rv^{a}_{rr}+\frac{v^{a}_{\theta\theta}}{r}-2\frac{u^{a}_{\theta}}{r}+v^{a}_{r}-\frac{v^{a}}{r}\big{)}=R_{v}^{a},\
(\theta,r)\in\Omega,\\\\[5.0pt] u^{a}_{\theta}+(rv^{a})_{r}=0,\
(\theta,r)\in\Omega,\\\\[5.0pt] u^{a}(\theta+2\pi,r)=u^{a}(\theta,r),\
v^{a}(\theta+2\pi,r)=v^{a}(\theta,r),\ (\theta,r)\in\Omega,\\\\[5.0pt]
u^{a}(\theta,1)=\alpha+f(\theta)\eta,\ v^{a}(\theta,1)=0,\
\theta\in[0,2\pi],\\\\[5.0pt] u^{a}(\theta,r_{0})=\beta+g(\theta)\eta,\
v^{a}(\theta,r_{0})=0,\ \theta\in[0,2\pi].\end{array}\right.$ (2.195)

##  3\. Linear stability estimates of error equations

In this section, we derive the error equations and establish the linear
stability estimate of the error equations.

###  3.1. Error equations

Set the error

$\displaystyle u:=u^{\varepsilon}-u^{a},\quad v:=v^{\varepsilon}-v^{a},\quad
p:=p^{\varepsilon}-p^{a},$

then there hold

$\displaystyle\left\\{\begin{array}[]{lll}u^{a}u_{\theta}+uu^{a}_{\theta}+uu_{\theta}+v^{a}ru_{r}+vru^{a}_{r}+vru_{r}+v^{a}u+vu^{a}+vu+p_{\theta}-\varepsilon^{2}\big{(}ru_{rr}+\frac{u_{\theta\theta}}{r}+2\frac{v_{\theta}}{r}+u_{r}-\frac{u}{r}\big{)}=R_{u}^{a},\\\\[5.0pt]
u^{a}v_{\theta}+uv^{a}_{\theta}+uv_{\theta}+v^{a}rv_{r}+vrv^{a}_{r}+vrv_{r}-(u^{2}+2uu^{a})+rp_{r}-\varepsilon^{2}\big{(}rv_{rr}+\frac{v_{\theta\theta}}{r}-2\frac{u_{\theta}}{r}+v_{r}-\frac{v}{r}\big{)}=R_{v}^{a},\\\\[5.0pt]
u_{\theta}+(rv)_{r}=0,\\\\[5.0pt] u(\theta+2\pi,r)=u(\theta,r),\
v(\theta+2\pi,r)=v(\theta,r),\\\\[5.0pt] u(\theta,1)=0,\
v(\theta,1)=0,\\\\[5.0pt] u(\theta,r_{0})=0,\
v(\theta,r_{0})=0.\end{array}\right.$ (3.7)

We set

$\displaystyle S_{u}:$
$\displaystyle=u^{a}u_{\theta}+v^{a}ru_{r}+uu^{a}_{\theta}+vru^{a}_{r}+v^{a}u+vu^{a},$
$\displaystyle S_{v}:$
$\displaystyle=u^{a}v_{\theta}+v^{a}rv_{r}+uv^{a}_{\theta}+vrv^{a}_{r}-2uu^{a},$

then the error equations become

$\displaystyle\left\\{\begin{array}[]{lll}-\varepsilon^{2}\big{(}ru_{rr}+\frac{u_{\theta\theta}}{r}+2\frac{v_{\theta}}{r}+u_{r}-\frac{u}{r}\big{)}+p_{\theta}+S_{u}=R_{u},\\\\[5.0pt]
-\varepsilon^{2}\big{(}rv_{rr}+\frac{v_{\theta\theta}}{r}-2\frac{u_{\theta}}{r}+v_{r}-\frac{v}{r}\big{)}+rp_{r}+S_{v}=R_{v},\\\\[5.0pt]
u_{\theta}+(rv)_{r}=0,\\\\[5.0pt] u(\theta,r)=u(\theta+2\pi,r),\
v(\theta,r)=v(\theta+2\pi,r),\\\\[5.0pt] u(\theta,1)=0,\