# Stress blow-up analysis when suspending rigid particles approach boundary in
3D Stokes flow

Haigang Li  School of Mathematical Sciences, Beijing Normal University,
Laboratory of MathematiCs and Complex Systems, Ministry of Education, Beijing
100875, China<EMAIL_ADDRESS>,  Longjuan Xu  Academy for Multidisciplinary
Studies, Capital Normal University, Beijing 100048, China.
<EMAIL_ADDRESS>and  PeiHao Zhang  School of Mathematical Sciences,
Beijing Normal University, Beijing 100875, China.
<EMAIL_ADDRESS>
###### Abstract.

The stress concentration is a common phenomenon in the study of fluid-solid
model. In this paper, we investigate the boundary gradient estimates and the
second order derivatives estimates for the Stokes flow when the rigid
particles approach the boundary of the matrix in dimension three. We classify
the effect on the blow-up rates of the stress from the prescribed various
boundary data: locally constant case and locally polynomial case. Our results
hold for general convex inclusions, including two important cases in practice,
spherical inclusions and ellipsoidal inclusions. The blow-up rates of the
Cauchy stress in the narrow region are also obtained. We establish the
corresponding estimates in higher dimensions greater than three.

##  1\. Introduction and main results

###  1.1. Background and Problem Formulation

In high-contrast composite materials, the physical fields in the narrow region
between two adjacent inclusions or between the inclusions and the matrix
boundary, such as the electric fields and the stress fields, may be
concentrated caused by this microstructure and become arbitrary large. In the
last two decades, there are many important progresses on the study of the
stress concentration phenomena for two close-to-touching particles in the
contexts of Laplace equation for electrostatic field, Lamé system for linear
elasticity and Stokes equations for viscous flow. In this paper, we
investigate the boundary estimates for the gradient of the solutions to the
incompressible Stokes flow in a bounded domain $D\subset\mathbb{R}^{d}\,(d\geq
3)$

$\mu\,\Delta{\bf u}=\nabla p,\quad\nabla\cdot{\bf u}=0,$

when the rigid particles suspending inside are very close to the matrix
boundary $\partial{D}$. The aim is to classify the effect on the blow-up rates
of the gradient and the Cauchy stress from various boundary data.

Based on the geometry of the particle and the domain and the given boundary
data, we construct a class of auxiliary functions to capture the main
singularity of the stress term and the pressure term. We first derive the
optimal blow-up rates of the stress, via establishing the pointwise upper
bounds in the neck region and the lower bounds at the midpoint of the shortest
line in between the particle and the boundary, and then proceed to estimate
the second order derivatives provided the domain and the boundary data are
sufficiently smooth. The advantage of this method is that when we deal with
the pressure term we do not need to solve the stream function. The simplified
model in 2D was considered in [39]. This is a continuation. However, Several
estimates have to be improved to asymptotic expansion and new difficulty in
the derivation of the optimality of the blow-up rate is overcome. Our results
hold for general convex particles, including the two important and useful
cases in practical application: spherical inclusions and ellipsoid inclusions
in dimension three. Our method also can be extended to deal with the problem
in higher dimensions greater than three. The corresponding upper bounds are
presented as well.

There is a long history on the study of the Stokes flow in presence of two
circular cylinders. More than one hundred years ago, Jeffrey developed in [30]
a separable solution method based on bipolar coordinates and then analyzed in
[31] the flow generated by two well-separated rotating circular cylinders. See
Wannier [47], Hillairet [28] and the references therein for the related works.
However, it is difficult to analyze the singular behavior of the solution when
the cylinders are close-to-touching, due to the high complexity of the
solution. Recently, Ammari, Kang, Kim, and Yu [3] investigated the stress
concentration in the two-dimensional Stokes flow, when inclusions are the two-
dimensional cross sections of circular cylinders of the same radii, by using
the bipolar coordinates to construct two vector-valued functions to derive an
asymptotic representation formula for the stress. The blow-up rate of
$|\nabla{\bf u}|$ is proved to be $\varepsilon^{-1/2}$, where $\varepsilon$ is
the distance between two cylinders. Laterly, the first two authors [38]
studied the general convex inclusions in dimensions two and three and proved
that the optimal blow-up rates of the stress are, respectively,
$\varepsilon^{-1/2}$ in dimension two and $(\varepsilon|\ln\varepsilon|)^{-1}$
in dimension three, by adapting the iteration method developed in the study of
linear elasticity problem [37, 12] to establish pointwise upper bounds and
lower bounds at the narrowest place between two inclusions. The estimates in
[3, 38] can be regarded as interior estimates. However, due to the effect
directly from the boundary data, the solutions of the Stokes flow may become
more irregular near the boundary. To clarify such singularity from the
geometry of boundary and given boundary data, we proceed in this paper to
study the stress concentration in the Stokes flow when particles approach the
boundary of the matrix in three dimension and higher dimensions. There are
some closely related works on the expansions of the energy norm, see Gérard-
Varet and Hillairet [24], Hillairet and Kelaï [29], and [25, 26] for more.

As mentioned at the beginning, the concentration phenomenon also occurs in the
electrostatic field and elastic stress composite materials. Since the well-
known numerical work of Babu ̆ska, Andersson, Smith and Levin [7], there have
been many important progresses in this field, see Bonnetier and Vogelious
[15], Li and Vogelious [42] and Li and Nirenberg [41]. It is of vital
importance in mathematics to consider the extreme cases when the parameter in
the inclusions degenerated to infinite, in order to investigate the
concentration degree depending on the small interparticle distance. In the
electrostatics case, when the particles are perfectly conductive, the blow-up
rate of the electric field, represented by the gradient of the solution, is
proved to be of order $\varepsilon^{-1/2}$ in dimension two and
$(\varepsilon|\ln\varepsilon|)^{-1}$ in three dimensions, see the work by
Ammari, Kang, and Lim [6], Bao, Li, and Yin [8]. For more literature related
to the electric field concentration, we refer the reader to the introduction
of [38] and [4, 5, 9, 10, 33, 34, 44, 49, 2, 16, 14, 15, 21, 22, 32], for
instance (this is far from a complete list). For the insulated conductivity
problem, see [19, 20, 9, 43, 48, 50]. In the context of linear elasticity,
when the Lamé parameters of inclusions degenerate to infinity, the optimal
blow-up rate of the gradient is proved to be $\varepsilon^{-1/2}$ in dimension
two [12, 35], and $(\varepsilon|\ln\varepsilon|)^{-1}$ in dimension three [13,
36]. Especially, for the boundary estimates when particles are close to the
boundary, see Bao, Ju and Li [11], and Li and Zhao [40]. We also refer the
reader to [17, 18, 27] for the corresponding nonlinear problem, and [45] for
the asymptotic behavior of fluid flows in domains perforated with a large
number of tiny holes.

Before we state our main results, we first fix our domain and notations. In
order to establish the estimate of the gradient and that of the second-order
partial derivatives, in this paper we assume that the domain is of class
$C^{3}$ and the boundary data $\bm{\varphi}\in C^{2,\alpha}(\partial
D;\mathbb{R}^{d})$, because we will use the $W^{2,\infty}$-estimates in [38,
Proposition 3.6], see Step 3 in the proof of Proposition 2.2 for the details.
Suppose $D\subset\mathbb{R}^{d}$ is a bounded open set with $C^{3}$ boundary,
and $D_{1}^{0}$ is a $C^{3}$ strictly convex subset of $D$, touching $\partial
D$, say, at the origin, and the $x_{1},\dots,x_{d-1}$-axes being in their
common tangent plane, after a translation and rotation of coordinates if
necessary. We use $x^{\prime}=(x_{1},\dots,x_{d-1})$. By a translation, set
$D_{1}^{\varepsilon}:=D_{1}^{0}+(0^{\prime},\varepsilon)$, for small constant
$\varepsilon>0$. For the sake of simplicity, we drop the superscript
$\varepsilon$ and denote

$D_{1}:=D_{1}^{\varepsilon}\,\quad\text{and}\quad\Omega:=D\setminus\overline{D}_{1}.$

Then $\overline{D}_{1}\subset D$, $\varepsilon:=\mbox{dist}(D_{1},\partial
D)$. We assume that the $C^{3}$ norms of $\partial D$ and $\partial D_{1}$ are
bounded by a constant independent of $\varepsilon$.

Let

$\Psi:=\Big{\\{}{\bm{\psi}}\in
C^{1}(\mathbb{R}^{d};\mathbb{R}^{d})~{}|~{}e({\bm{\psi}}):=\frac{1}{2}(\nabla{\bm{\psi}}+(\nabla{\bm{\psi}})^{\mathrm{T}})=0\Big{\\}},$

be the linear space of rigid displacements in $\mathbb{R}^{d}$. It is well
known that

$\\{\bm{e}_{i},~{}x_{k}\bm{e}_{j}-x_{j}\bm{e}_{k}~{}|~{}1\leq\,i\leq\,d,~{}1\leq\,j<k\leq\,d\\}$

is a basis of $\Psi$, where $e_{1},\dots,e_{d}$ is the canonical basis of
$\mathbb{R}^{d}$. Consider the following Dirichlet problem, describing one
rigid particle suspending in the Stokes flow,

$\displaystyle\begin{cases}\mu\Delta{\bf u}=\nabla
p,\quad\quad~{}~{}\nabla\cdot{\bf
u}=0,&\hbox{in}~{}~{}\Omega:=D\setminus\overline{D}_{1},\\\ {\bf u}|_{+}={\bf
u}|_{-},&\hbox{on}~{}~{}\partial{D_{1}},\\\ e({\bf
u})=0,&\hbox{in}~{}~{}D_{1},\\\ \int_{\partial{D}_{1}}\frac{\partial{\bf
u}}{\partial\nu}\Big{|}_{+}\cdot{\bm{\psi}}_{\alpha}-\int_{\partial{D}_{1}}p\,{\bm{\psi}}_{\alpha}\cdot\nu=0,&\alpha=1,2,\dots,\frac{d(d+1)}{2},\\\
{\bf u}={\bm{\varphi}},&\hbox{on}~{}~{}\partial{D},\end{cases}$ (1.1)

where $\mu>0$, ${\bm{\psi}}_{\alpha}\in\Psi$, $\frac{\partial{\bf
u}}{\partial\nu}\big{|}_{+}:=\mu(\nabla{\bf u}+(\nabla{\bf
u})^{\mathrm{T}})\nu$, $\nu$ is the unit outward normal vector of $D_{1}$, and
the subscript $+$ indicates the limit from inside the domain. Furthermore, by
using the Gauss theorem and $\nabla\cdot{\bf u}=0$ in $\Omega$, we know the
prescribed velocity field ${\bm{\varphi}}$ satisfies the compatibility
condition:

$\int_{\partial{D}}{\bm{\varphi}}\cdot\nu\,=0,$ (1.2)

which ensures the existence and uniqueness of the solution. See [1, 46, 23].

In order to formulate our results precisely, we assume that near the origin
the boundaries $\partial D_{1}$ and $\partial D$ are expressed by the graphs
of two functions, respectively. Namely, for some $0<R<1$,

$x_{d}=\varepsilon+h_{1}(x^{\prime})\quad\text{and}\quad
x_{d}=h(x^{\prime}),\quad\text{for}~{}|x^{\prime}|\leq 2R,$

where $h_{1}$ and $h$ are $C^{3}$ functions, satisfying

$\displaystyle
h(x^{\prime})<\varepsilon+h_{1}(x^{\prime}),\quad\mbox{for}~{}~{}|x^{\prime}|\leq
2R,$ $\displaystyle h_{1}(0)$
$\displaystyle=h(0)=0,\quad\nabla_{x^{\prime}}h_{1}(0)=\nabla_{x^{\prime}}h(0)=0,$
$\displaystyle h_{1}(x^{\prime})=\kappa_{1}|x^{\prime}|^{2}+$ $\displaystyle
O(|x^{\prime}|^{3}),\quad
h(x^{\prime})=\kappa|x^{\prime}|^{2}+O(|x^{\prime}|^{3}),\quad\mbox{for}~{}~{}|x^{\prime}|<2R,$

where we assume that the constants $\kappa_{1}>\kappa$. For $0\leq r\leq 2R$,
we denote the neck region

$\Omega_{r}:=\left\\{(x^{\prime},x_{d})\in\Omega:~{}h(x^{\prime})<x_{d}<\varepsilon+h_{1}(x^{\prime}),~{}|x^{\prime}|<r\right\\}.$
(1.3)

Throughout this paper, $C$ denotes a universal constant, which means the value
of $C$ may vary from line to line, depends only on $d,\mu,\kappa_{1},\kappa$,
and the upper bounds of the $C^{3}$ norm of $\partial{D}$, $\partial{D}_{1}$,
but independent of $\varepsilon$.

###  1.2. Upper Bounds of $|\nabla{\bf u}|$ and $|p|$ in 3D

Let $({\bf u},p)$ be a pair of solution to (1.1). We introduce the Cauchy
stress tensor

$\sigma[{\bf u},p]=2\mu e({\bf u})-p\mathbb{I},$ (1.4)

where $\mathbb{I}$ is the identity matrix. Then we reformulate (1.1) as

$\displaystyle\begin{cases}\nabla\cdot\sigma[{\bf
u},p]=0,~{}~{}\nabla\cdot{\bf u}=0,&\hbox{in}~{}~{}\Omega,\\\ {\bf
u}|_{+}={\bf u}|_{-},&\hbox{on}~{}~{}\partial{D_{1}},\\\ e({\bf
u})=0,&\hbox{in}~{}~{}D_{1},\\\ \int_{\partial{D}_{1}}\sigma[{\bf
u},p]\cdot{\bm{\psi}}_{\alpha}\nu=0,&\alpha=1,\dots,6,\\\ {\bf
u}={\bm{\varphi}},&\hbox{on}~{}~{}\partial{D}.\end{cases}$ (1.5)

As in [11, 39], by the third line in (1.5), we decompose $\bf u$ in $D_{1}$ as

${\bf
u}=\sum_{\alpha=1}^{6}C^{\alpha}{\bm{\psi}}_{\alpha},\quad\mbox{where}~{}{\bm{\psi}}_{\alpha}\in\Psi,\quad\mbox{in}~{}{D}_{1},$

where $C^{\alpha}$, $\alpha=1,2,\dots,6$, are some free constants to be
determined by the fourth line in (1.5) later. By virtue of the continuity of
the transmission condition on $\partial{D}_{1}$, then $({\bf u},p)$ in
$\Omega$ can be split as follows:

$\displaystyle{\bf u}(x)$ $\displaystyle=\sum_{\alpha=1}^{6}C^{\alpha}{\bf
u}_{\alpha}(x)+{\bf u}_{0}(x)\quad\mbox{and}\quad
p(x)=\sum_{\alpha=1}^{6}C^{\alpha}p_{\alpha}(x)+p_{0}(x),$ (1.6)

where ${\bf u}_{\alpha},{\bf
u}_{0}\in{C}^{2}(\Omega;\mathbb{R}^{3}),~{}p_{\alpha},p_{0}\in{C}^{1}(\Omega)$
satisfy, respectively,

$\begin{cases}\nabla\cdot\sigma[{\bf
u}_{\alpha},p_{\alpha}]=0,\quad\nabla\cdot{\bf
u}_{\alpha}=0&\mathrm{in}~{}\Omega,\\\ {\bf
u}_{\alpha}={\bm{\psi}}_{\alpha}&\mathrm{on}~{}\partial{D}_{1},\\\ {\bf
u}_{\alpha}=0&\mathrm{on}~{}\partial{D},\end{cases}\quad\alpha=1,2,\dots,6,$
(1.7)

and

$\begin{cases}\nabla\cdot\sigma[{\bf u}_{0},p_{0}]=0,\quad\nabla\cdot{\bf
u}_{0}=0&\mathrm{in}~{}\Omega,\\\ {\bf
u}_{0}=0&\mathrm{on}~{}\partial{{D}_{1}},\\\ {\bf
u}_{0}={\bm{\varphi}}&\mathrm{on}~{}\partial{D}.\end{cases}$ (1.8)

Thus,

$\nabla{\bf u}=\sum_{\alpha=1}^{6}C^{\alpha}\nabla{\bf u}_{\alpha}+\nabla{\bf
u}_{0}\quad\mathrm{in}~{}\Omega.$ (1.9)

For ${\bf u}_{\alpha}$, $\alpha=1,2,\dots,6$, we will construct auxiliary
functions ${\bf v}_{\alpha}$ having the same boundary conditions as ${\bf
u}_{\alpha}$ and the associated auxiliary functions ${\bar{p}}_{\alpha}$, and
then apply the iteration approach as in [12, 13, 38, 39] to prove $\nabla{\bf
v}_{\alpha}$ being the main singular terms of $\nabla{\bf u}_{\alpha}$, see
Proposition 2.2 below. However, in order to investigate the effect from the
boundary data ${\bm{\varphi}}$, from (1.8), it is also important to derive the
estimate of $|\nabla{\bf u}_{0}|$. Let

${\bf\Phi}:=\left\\{\bm{\varphi}\in C^{2,\alpha}(\partial
D;\mathbb{R}^{3})~{}|~{}\bm{\varphi}~{}\mbox{satisfies}~{}\eqref{compatibility}\right\\},$

and

$\Gamma_{2R}=\left\\{(x^{\prime},x_{3})\in\Omega:~{}x_{3}=h(x^{\prime}),~{}|x^{\prime}|<2R\right\\}\subset\partial{D}.$

Since the gap is narrowest near the origin, the stress may concentrate there.
According to the Taylor expansion of ${\bm{\varphi}}$ at the origin, we
consider two kinds of boundary data: being locally constant on $\Gamma_{2R}$
and being locally polynomial on $\Gamma_{2R}$. Specifically,

${\bf\Phi}_{1}:=\left\\{{\bm{\varphi}}\in{\bf\Phi}~{}|~{}{\bm{\varphi}}=\begin{pmatrix}x_{i}^{l_{1}}\\\
0\\\ 0\end{pmatrix},~{}\mbox{or}~{}{\bm{\varphi}}=\begin{pmatrix}0\\\
x_{i}^{l_{1}}\\\ 0\end{pmatrix},~{}i=1,2,~{}\mbox{on}~{}\Gamma_{2R}\right\\},$
(1.10)
${\bf\Phi}_{2}:=\left\\{\bm{\varphi}\in{\bf\Phi}~{}|~{}{\bm{\varphi}}=\begin{pmatrix}0\\\
0\\\ x_{i}^{l_{2}}\end{pmatrix},~{}i=1,2,~{}\mbox{on}~{}\Gamma_{2R}\right\\},$
(1.11)

and

${\bf\Phi}_{3}:=\left\\{\bm{\varphi}\in{\bf\Phi}~{}|~{}{\bm{\varphi}}=\begin{pmatrix}x_{3}^{l_{3}}\\\
0\\\ 0\end{pmatrix},~{}\mbox{or}~{}{\bm{\varphi}}=\begin{pmatrix}0\\\
x_{3}^{l_{3}}\\\
0\end{pmatrix},~{}\mbox{or}~{}{\bm{\varphi}}=\begin{pmatrix}0\\\ 0\\\
x_{3}^{l_{3}}\end{pmatrix},~{}\mbox{on}~{}\Gamma_{2R}\right\\},$ (1.12)

where $l_{1}=0$ and $l_{2}=0$ corresponds to the constant case, and
$l_{1},l_{2},l_{3}\in\mathbb{N}^{+}$ corresponds to the polynomial case. We
have to construct an auxiliary function ${\bf v}_{0}$ with the same boundary
condition as ${\bf u}_{0}$ for each $\bm{\varphi}\in{\bf\Phi}_{i}$, $i=1,2,3$,
to capture the main singularity of $|\nabla{\bf u}_{0}|$. The constructions
become more complicated and several new difficulties need to overcome. See
Propositions 3.1–3.3.

To estimate $C^{\alpha}$ in (1.6), $\alpha=1,2,\dots,6$, from the fourth line
of (1.5) and (1.9), we need to solve the linear system

$\sum\limits_{\alpha=1}^{6}C^{\alpha}a_{\alpha\beta}=Q_{\beta}[{\bm{\varphi}}],$
(1.13)

where

$\displaystyle a_{\alpha\beta}=-\int_{\partial
D_{1}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf u}_{\alpha},p_{\alpha}]\nu,\quad
Q_{\beta}[{\bm{\varphi}}]=\int_{\partial
D_{1}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf u}_{0},p_{0}]\nu.$ (1.14)

To this end, here we need to derive their explicit asymptotic expansions of
$a_{\alpha\beta}$ and $Q_{\beta}[\bm{\varphi}]$, instead of upper and lower
bounds estimates as before. See Lemmas 4.2–4.6 for details.

Denote

$\delta(x^{\prime})=\varepsilon+(\kappa_{1}-\kappa)|x^{\prime}|^{2}.$ (1.15)

Then we have the following main results on the upper bounds.

######  Theorem 1.1.

Assume that $D,D_{1},\Omega$, and $\varepsilon$ are defined as in Section 1.1.
For $\bm{\varphi}\in{\bf\Phi}_{i}$, $i=1,2,3$, let ${\bf u}\in
H^{1}(D;\mathbb{R}^{3})\cap C^{1}(\bar{\Omega};\mathbb{R}^{3})$ and $p\in
L^{2}(D)\cap C^{0}(\bar{\Omega})$ be the solution to (1.5). Then for
sufficiently small $0<\varepsilon<1$, the following assertions hold:

(i) for $\bm{\varphi}\in{\bf\Phi}_{1}$,

$\displaystyle|\nabla{{\bf
u}}(x)|\leq\frac{C(1+|\ln\varepsilon||x^{\prime}|)}{|\ln\varepsilon|\delta(x^{\prime})},~{}x\in\Omega_{R},$
(1.16) $\displaystyle|\nabla^{2}{{\bf u}}(x)|+|\nabla
p(x)|\leq\frac{C(1+|\ln\varepsilon|\sqrt{\delta(x^{\prime})})}{|\ln\varepsilon|\delta^{2}(x^{\prime})},~{}x\in\Omega_{R},$
(1.17)

and

$\displaystyle\inf_{c\in\mathbb{R}}\|p+c\|_{C^{0}(\bar{\Omega}_{R})}\leq\begin{cases}\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}},&l_{1}=0,~{}l_{1}\geq
2,\\\ \frac{C}{\varepsilon^{3/2}},&l_{1}=1,\end{cases}$

where $C$ is a universal constant and $\delta(x^{\prime})$ is defined in
(1.15);

(ii) for $\bm{\varphi}\in{\bf\Phi}_{2}$,

$\displaystyle|\nabla{{\bf
u}}(x)|\leq\begin{cases}\frac{C(1+|\ln\varepsilon||x^{\prime}|)}{|\ln\varepsilon|\delta(x^{\prime})},&l_{2}=0,~{}l_{2}\geq
2,\\\ \frac{C}{\delta(x^{\prime})},&l_{2}=1,\end{cases}~{}x\in\Omega_{R},$
$\displaystyle|\nabla^{2}{{\bf u}}(x)|+|\nabla
p(x)|\leq\begin{cases}\frac{C(1+|\ln\varepsilon|\sqrt{\delta(x^{\prime})})}{|\ln\varepsilon|\delta^{2}(x^{\prime})},&l_{2}=0,~{}l_{2}\geq
2,\\\ \frac{C}{\delta^{2}(x^{\prime})},&l_{2}=1,\end{cases}~{}x\in\Omega_{R},$

and

$\displaystyle\inf_{c\in\mathbb{R}}\|p+c\|_{C^{0}(\bar{\Omega}_{R})}\leq\begin{cases}\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}},&l_{2}=0,\\\
\frac{C}{\varepsilon^{3/2}},&l_{2}\geq 1;\end{cases}$

(iii) for $\bm{\varphi}\in{\bf\Phi}_{3}$, (1.16) and (1.17) also hold, and

$\displaystyle\inf_{c\in\mathbb{R}}\|p+c\|_{C^{0}(\bar{\Omega}_{R})}\leq\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}};$

and outside the narrow region, we have

$\|\nabla^{k_{1}}{\bf
u}\|_{L^{\infty}(\Omega\setminus\Omega_{R})}+\|\nabla^{k_{2}}p\|_{L^{\infty}(\Omega\setminus\Omega_{R})}\leq
C\|\bm{\varphi}\|_{C^{2,\alpha}(\partial D)},\quad k_{1}=1,2,~{}k_{2}=0,1.$

######  Remark 1.2.

In particular, we have

$\|\nabla{{\bf
u}}\|_{L^{\infty}(\Omega)}\leq\begin{cases}\frac{1}{\varepsilon|\ln\varepsilon|},&\bm{\varphi}\in{\bf\Phi}_{i},~{}i=1,3;~{}\mbox{or}~{}\bm{\varphi}\in{\bf\Phi}_{2},~{}l_{2}=0,~{}l_{2}\geq
2,\\\
\frac{C}{\varepsilon},&\bm{\varphi}\in{\bf\Phi}_{2},~{}l_{2}=1.\end{cases}$

Although the above results for $\bm{\varphi}\in{\bf\Phi}_{1}$ and
$\bm{\varphi}\in{\bf\Phi}_{3}$ look the same, the construction of the
auxiliary functions are totally different, see Section 2.

######  Remark 1.3.

Our proof also works for more general convex particles only with a minor
modification. For example, when

$h_{1}(x)=\kappa_{1}x_{1}^{2}+\kappa_{2}x_{2}^{2},\quad
h(x)=\kappa|x^{\prime}|^{2},\quad\kappa_{1},\kappa_{2}>\kappa,\quad\mbox{in}~{}\Omega_{2R},$

then
$\delta(x^{\prime})=\varepsilon+(\kappa_{1}-\kappa)x_{1}^{2}+(\kappa_{2}-\kappa)x_{2}^{2}$.
The details are given in Section 6. The argument in the proof of Theorem 1.1
works well for the case of $m$-convex inclusions, say,
$h_{1}(x)=\kappa_{1}|x^{\prime}|^{m}$ and $h(x)=\kappa|x^{\prime}|^{m}$ with
$m\geq 3$, by modifying the auxiliary functions in Section 2 slightly. We
leave the details to the interested readers.

###  1.3. Lower Bounds of $|\nabla{\bf u}|$ in 3D

Notice that the upper bounds of $|\nabla{\bf u}|$ established in Theorem 1.1
achieve their maximum at the shortest line $\\{|x^{\prime}|=0\\}\cap\Omega$.
We shall prove the lower bounds of $|\nabla{\bf u}|$ at the midpoint of the
shortest line in order to show the optimality of the blow-up rate. Assume that
$({\bf u}_{0}^{*},p_{0}^{*})$ and $({\bf u}_{\beta}^{*},p_{\beta}^{*})$,
$\beta=1,\dots,6$, satisfy

$\displaystyle\begin{cases}\nabla\cdot\sigma[{\bf
u}_{0}^{*},p_{0}^{*}]=0,&\hbox{in}\ \Omega^{0},\\\ \nabla\cdot{\bf
u}_{0}^{*}=0,&\hbox{in}\ \Omega^{0}\\\ {\bf u}_{0}^{*}=0,&\hbox{on}\ \partial
D_{1}^{0},\\\ {\bf u}_{0}^{*}={\bm{\varphi}},&\hbox{on}\
\partial{D},\end{cases}\quad\begin{cases}\nabla\cdot\sigma[{\bf
u}_{\beta}^{*},p_{\beta}^{*}]=0,&\mathrm{in}~{}\Omega^{0},\\\ \nabla\cdot{\bf
u}_{\beta}^{*}=0,&\mathrm{in}~{}\Omega^{0},\\\ {\bf
u}_{\beta}^{*}={\bm{\psi}}_{\beta},&\mathrm{on}~{}\partial{D}_{1}^{0}\setminus\\{0\\},\\\
{\bf u}_{\beta}^{*}=0,&\mathrm{on}~{}\partial{D},\end{cases}$

respectively.

For the locally constant boundary data ${\bm{\varphi}}\in{\bf\Phi}_{1}$ with
$l_{1}=0$, it follows from Lemma 4.4 that $Q_{\beta}[{\bm{\varphi}}]$, defined
by (1.14), goes to infinity as $\varepsilon\rightarrow 0$, $\beta=1$ or
$\beta=5$. For this, we define a new functional by

$\displaystyle
Q_{1,\beta}[{\bm{\varphi}}]:=Q_{\beta}[{\bm{\varphi}}]-a_{1\beta}=\int_{\partial
D_{1}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf u}_{0}+{\bf
u}_{1},p_{0}+p_{1}]\nu,\quad\beta=1,\dots,6,$ (1.18)

where $a_{1\beta}$ is defined by (1.14). Then
$|Q_{1,\beta}[{\bm{\varphi}}]|\leq C$ since ${\bf u}_{0}+{\bf u}_{1}$ takes
the same data on $\partial D_{1}$ and $\partial D$. Moreover, we shall prove
in Lemma 5.1 that $Q_{1,\beta}[{\bm{\varphi}}]$ converges to

$\displaystyle Q^{*}_{1,\beta}[{\bm{\varphi}}]:=\int_{\partial
D_{1}^{0}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf u}_{0}^{*}+{\bf
u}_{1}^{*},p_{0}^{*}+p_{1}^{*}]\nu,\quad\beta=1,\dots,6.$ (1.19)

Assume that ${\bm{\varphi}}\in C^{1,\alpha}(\partial D;\mathbb{R}^{3})$
satisfies the following condition:

$\displaystyle({\mathscr{H}}):$
$\displaystyle~{}Q_{1,1}^{*}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q_{1,5}^{*}[{\bm{\varphi}}]-\frac{a^{*}_{14}-(\kappa_{1}+\kappa)a^{*}_{54}}{a^{*}_{44}}Q^{*}_{1,4}[{\bm{\varphi}}]\neq
0$
$\displaystyle~{}\mbox{for~{}some}~{}{\bm{\varphi}}\in{\bf\Phi}_{1}~{}\mbox{with~{}}l_{1}=0,$

where

$a^{*}_{\alpha\beta}:=\int_{\partial
D_{1}^{0}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf
u}_{\alpha}^{*},p_{\alpha}^{*}]\nu,\quad\alpha,\beta=1,\dots,6.$ (1.20)

Then we have the lower bound as follows.

######  Theorem 1.4.

Assume that $D,D_{1},\Omega$, and $\varepsilon$ are defined as in Section 1.1.
Let ${\bf u}\in H^{1}(D;\mathbb{R}^{3})\cap
C^{1}(\bar{\Omega};\mathbb{R}^{3})$ and $p\in L^{2}(D)\cap
C^{0}(\bar{\Omega})$ be the solution to (1.5). If ${\bm{\varphi}}$ satisfies
$({\mathscr{H}})$, then for sufficiently small $\varepsilon>0$, we have

$\displaystyle|\nabla{\bf
u}(0,0,\varepsilon/2)|\geq\frac{1}{\varepsilon|\ln\varepsilon|}.$

######  Remark 1.5.

Similarly, for ${\bm{\varphi}}\in{\bf\Phi}_{2}$ with $l_{2}=0$, we define for
$\beta=1,\dots,6$,

$\displaystyle Q_{2,\beta}[{\bm{\varphi}}]:=\int_{\partial
D_{1}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf u}_{0}+{\bf
u}_{2},p_{0}+p_{2}]\nu,~{}Q^{*}_{2,\beta}[{\bm{\varphi}}]:=\int_{\partial
D_{1}^{0}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf u}_{0}^{*}+{\bf
u}_{2}^{*},p_{0}^{*}+p_{2}^{*}]\nu.$

While, for the polynomial cases, $l_{1}\geq 1$ and $l_{2}\geq 2$, we define

$\displaystyle Q^{*}_{\beta}[{\bm{\varphi}}]:=\int_{\partial
D_{1}^{0}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf
u}_{0}^{*},p_{0}^{*}]\nu,\quad\beta=1,\dots,6.$

By a similar proof as that of Lemma 5.1, we have

$Q_{2,\beta}[{\bm{\varphi}}]\rightarrow
Q^{*}_{2,\beta}[{\bm{\varphi}}]~{}\mbox{and}~{}Q_{\beta}[{\bm{\varphi}}]\rightarrow
Q^{*}_{\beta}[{\bm{\varphi}}]\quad\mbox{as}~{}\varepsilon\rightarrow 0.$

Then if ${\bm{\varphi}}\in C^{1,\alpha}(\partial D;\mathbb{R}^{3})$ satisfies
one of the following conditions

$\displaystyle({\mathscr{H}}_{2}):$
$\displaystyle~{}Q_{2,1}^{*}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q_{2,5}^{*}[{\bm{\varphi}}]-\frac{a^{*}_{14}-(\kappa_{1}+\kappa)a^{*}_{54}}{a^{*}_{44}}Q^{*}_{2,4}[{\bm{\varphi}}]\neq
0$
$\displaystyle~{}\mbox{for~{}some}~{}{\bm{\varphi}}\in{\bf\Phi}_{2}~{}\mbox{with~{}}l_{2}=0,$

or

$\displaystyle({\mathscr{H}}_{3}):$
$\displaystyle~{}Q_{1}^{*}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q_{5}^{*}[{\bm{\varphi}}]-\frac{a^{*}_{14}-(\kappa_{1}+\kappa)a^{*}_{54}}{a^{*}_{44}}Q^{*}_{4}[{\bm{\varphi}}]\neq
0\qquad$
$\displaystyle~{}\mbox{for~{}some}~{}{\bm{\varphi}}\in{\bf\Phi}_{j}~{}\mbox{with~{}}l_{1}\geq
1,~{}l_{2}\geq 2,~{}j=1,2,$

the result in Theorem 1.4 still holds true.

######  Remark 1.6.

From the perspective of practical application, it is more important to
investigate the hydrodynamic force and the hydrodynamic torque acting on the
rigid particle $D_{1}$ defined by

${\bf F}=-\int_{\partial D_{1}}\sigma[{\bf u},p]\nu,\quad{\bf
T}=\int_{\partial D_{1}}(x-x_{D_{1}})\times\sigma[{\bf u},p]\nu,$

where $\nu$ is the unit outer normal to $\partial D_{1}$ and $x_{D_{1}}$ is
the center of the mass of $D_{1}$. Using the fourth line of (1.1), here we
have ${\bf F}=0$. While, as a consequence of Theorem 1.1, for given boundary
data $\bm{\varphi}\in{\bf\Phi}_{i}$, $i=1,2,3$, we have

$|{\bf T}|\leq C,$

where the constant $C$ is independent of $\varepsilon$.

###  1.4. Upper bounds of $|\nabla{\bf u}|$ and $p$ in dimensions $d\geq 4$,
and upper bounds of Cauchy stress tensor $\sigma[{\bf u},p]$

As mentioned before, our method works in any dimension. Next we will present
the upper bounds of $|\nabla{\bf u}|$ and $p$ in dimensions $d\geq 4$ when the
boundary data $\bm{\varphi}=x_{1}^{l}e_{1}$ on $\Gamma_{2R}$. Other cases can
be handled similarly, we leave it to interested readers.

######  Theorem 1.7.

Assume that $D,D_{1},\Omega$, and $\varepsilon$ are defined as in Section 1.1.
For $\bm{\varphi}=x_{1}^{l}e_{1}$, let ${\bf u}\in H^{1}(D;\mathbb{R}^{d})\cap
C^{1}(\bar{\Omega};\mathbb{R}^{d})$ and $p\in L^{2}(D)\cap
C^{0}(\bar{\Omega})$ be the solution to (1.5). Then for sufficiently small
$0<\varepsilon<1$, we have

$\displaystyle|\nabla{{\bf
u}}(x)|\leq\frac{C(\varepsilon+|x^{\prime}|)}{\delta^{2}(x^{\prime})},\quad
x\in\Omega_{R};\quad\inf_{c\in\mathbb{R}}\|p+c\|_{C^{0}(\bar{\Omega}_{R})}\leq\frac{C}{\varepsilon^{2}},$
(1.21)

and $\|\nabla{\bf
u}\|_{L^{\infty}(\Omega\setminus\Omega_{R})}+\|p\|_{L^{\infty}(\Omega\setminus\Omega_{R})}\leq
C\|\bm{\varphi}\|_{C^{2,\alpha}(\partial D)}$, where $C$ is a universal
constant and $\delta(x^{\prime})$ is defined in (1.15).

Recalling the definition of $\sigma[{\bf u},p]$ in (1.4), as a consequence of
Theorems 1.1 and 1.7, we have the estimates for the Cauchy stress tensor
$\sigma[{\bf u},p]$ as follows.

######  Corollary 1.8.

Assume that $D,D_{1},\Omega$, and $\varepsilon$ are defined as in Section 1.1.
Let ${\bf u}\in H^{1}(D;\mathbb{R}^{d})\cap
C^{1}(\bar{\Omega};\mathbb{R}^{d})$ and $p\in L^{2}(D)\cap
C^{0}(\bar{\Omega})$ be the solution to (1.5) with $d\geq 3$. Then for
sufficiently small $0<\varepsilon<1$, we have the following assertions:

(i) When $d=3$, it holds that,

  1. (1)

if $\bm{\varphi}\in{\bf\Phi}_{1}$, then

$\displaystyle\inf_{c\in\mathbb{R}}\|\sigma[{\bf
u},p+c]\|_{C^{0}(\bar{\Omega}_{R})}\leq\begin{cases}\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}},&l_{1}=0,~{}l_{1}\geq
2,\\\ \frac{C}{\varepsilon^{3/2}},&l_{1}=1;\end{cases}$

  2. (2)

if $\bm{\varphi}\in{\bf\Phi}_{2}$, then

$\displaystyle\inf_{c\in\mathbb{R}}\|\sigma[{\bf
u},p+c]\|_{C^{0}(\bar{\Omega}_{R})}\leq\begin{cases}\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}},&l_{2}=0,\\\
\frac{C}{\varepsilon^{3/2}},&l_{2}\geq 1;\end{cases}$

  3. (3)

if $\bm{\varphi}\in{\bf\Phi}_{3}$, then

$\displaystyle\inf_{c\in\mathbb{R}}\|\sigma[{\bf
u},p+c]\|_{C^{0}(\bar{\Omega}_{R})}\leq\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}}.$

(ii) When $d\geq 4$, for example, if $\bm{\varphi}=x_{1}^{l}e_{1}$ on
$\Gamma_{2R}$, we have

$\displaystyle\inf_{c\in\mathbb{R}}\|\sigma[{\bf
u},p+c]\|_{C^{0}(\bar{\Omega}_{R})}\leq\frac{C}{\varepsilon^{2}}.$

######  Remark 1.9.

From the proof of Theorem 1.1 in Section 5, one can see that it is quite
challenging in dimension three to fix the lower bounds of $p$ and $\sigma[{\bf
u},p]$ by using our argument in this paper. For example, if
$\bm{\varphi}\in{\bf\Phi}_{1}$ and $l_{1}=0$ in dimension three, then it
follows from (5.1) that the largest term
$\frac{1}{|\ln\varepsilon|\varepsilon^{3/2}}$ comes from
$|C^{1}-1||p_{1}(x)-(q_{1})_{\Omega_{R}}|$,
$|C^{2}||p_{2}-(q_{2})_{\Omega_{R}}|$, and
$\sum_{\alpha=5}^{6}|C^{\alpha}||p_{\alpha}-(q_{\alpha})_{\Omega_{R}}|$.
However, it is difficult to determine the lower bounds of these free constants
$C^{1}-1$, $C^{2}$, $C^{5}$, and $C^{6}$. We leave this problem to interested
readers and look forward to new idea and technique to prove the lower bounds
and the optimal blow-up rates of the Cauchy stress tensor.

Our paper is organized as follows. In Section 2, we construct the auxiliary
functions $\bf v_{\alpha}$ and the associated $p_{\alpha}$ in dimension three,
present the required properties in Lemma 2.1 in order to apply the framework
for gradient estimates established in [38], and obtain the estimates of
$|\nabla\bf u_{\alpha}|$ and $p_{\alpha}$ in Proposition 2.2. Several kinds of
prescribed boundary data ${\bm{\varphi}}$ are considered and the similar
estimates for $|\nabla{\bf u}_{0}|$ and $p_{0}$ are also obtained in Section
3. By virtue of these precise estimates and asymptotic fomulas for
$(\nabla{\bf u}_{\alpha},p_{\alpha})$ and $(\nabla{\bf u}_{0},p_{0})$, Section
4 is devoted to the estimates of $C^{\alpha}$. Then in Section 5 we prove
Theorem 1.1 for the upper bounds, and Theorem 1.4 for the lower bounds in 3D.
We study the ellipsoid inclusions case, with two different principle
curvatures near the origin in Section 6. Finally, the proof of Theorem 1.7 for
the upper bounds in dimensions greater than three is given in Section 7.

##  2\. Estimates of $(\nabla{\bf u}_{\alpha},p_{\alpha})$ in 3D

In this section, we prove the estimates for $(\nabla{\bf
u}_{\alpha},p_{\alpha})$, $\alpha=1,2,\dots,6$. By the framework for the
gradient estimates developed in [38], we reduce the estimates for $(\nabla{\bf
u}_{\alpha},p_{\alpha})$ to the constructions of the appropriate auxiliary
functions $(\nabla{\bf v}_{\alpha},\bar{p}_{\alpha})$, which will be proved to
be the main singular terms for each part. To express our idea clearly and
simplify the calculation, without loss of generality, we assume that $h_{1}$
and $h$ are quadratic, say, $h_{1}(x^{\prime})=\kappa_{1}|x^{\prime}|^{2}$,
$h_{2}(x^{\prime})=\kappa|x^{\prime}|^{2}$ for $|x^{\prime}|\leq 2R$, where
$\kappa_{1}$, $\kappa$ are two constants and $\kappa_{0}=\kappa_{1}-\kappa>0$.

We first define a scalar Keller-type auxiliary function $k(x)\in
C^{2}(\mathbb{R}^{d})$, satisfying $k(x)=\frac{1}{2}$ on $\partial D_{1}$,
$k(x)=-\frac{1}{2}$ on $\partial D$, especially,

$k(x)=\frac{x_{3}-h(x^{\prime})}{\delta(x^{\prime})}-\frac{1}{2}\quad\hbox{in}\
\Omega_{2R},$ (2.1)

and $\|k(x)\|_{C^{2}(\mathbb{R}^{d}\backslash\Omega_{R})}\leq C$. Clearly,

$\displaystyle\partial_{x_{j}}k(x)=-\frac{(\kappa_{1}+\kappa)x_{j}}{\delta(x^{\prime})}-\frac{2(\kappa_{1}-\kappa)x_{j}}{\delta(x^{\prime})}k(x),~{}\,j=1,2,\quad\partial_{x_{3}}k(x)=\frac{1}{\delta(x^{\prime})},\quad\hbox{in}\
\Omega_{2R}.$

For $\alpha=1,2$, for convenience, we define

$\displaystyle\begin{split}F_{\alpha}(x)&:=-\frac{12}{5}\frac{(\kappa_{1}+\kappa)x_{\alpha}^{2}}{\delta(x^{\prime})}+\frac{3}{5}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa},\quad
G(x):=-\frac{12}{5}\frac{(\kappa_{1}+\kappa)x_{1}x_{2}}{\delta(x^{\prime})},\\\
H_{\alpha}(x)&:=(\kappa_{1}-\kappa)x_{\alpha}+2(\kappa_{1}+\kappa)x_{\alpha}k(x).\end{split}$
(2.2)

Now we construct ${\bf v}_{\alpha}\in C^{2}(\Omega;\mathbb{R}^{3})$, such that
${\bf v}_{\alpha}={\bf u}_{\alpha}={\bm{\psi}}_{\alpha}$ on $\partial{D}_{1}$
and ${\bf v}_{\alpha}={\bf u}_{\alpha}=0$ on $\partial{D}$, especially,

$\displaystyle{\bf
v}_{\alpha}=\bm{\psi}_{\alpha}\big{(}k(x)+\frac{1}{2}\big{)}+{\bf
E}_{\alpha}(x)\Big{(}k^{2}(x)-\frac{1}{4}\Big{)},\quad\alpha=1,2,\quad\mbox{in}~{}\Omega_{2R},$
(2.3)

and $\|{\bf v}_{\alpha}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq\,C$, where

${\bf
E}_{1}(x)=\Big{(}F_{1}(x),G(x),H_{1}(x)-\delta(x^{\prime})\partial_{x_{1}}k(x)F_{1}(x)-\delta(x^{\prime})\partial_{x_{2}}k(x)G(x)\Big{)}^{\mathrm{T}},$

and

${\bf
E}_{2}(x)=\Big{(}G(x),F_{2}(x),H_{2}(x)-\delta(x^{\prime})\partial_{x_{1}}k(x)G(x)-\delta(x^{\prime})\partial_{x_{2}}k(x)F_{2}(x)\Big{)}^{\mathrm{T}}.$

To cancel out the biggest term in $\Delta{\bf v}_{\alpha}$, $\alpha=1,2$, and
to make the technique for gradient estimates developed in [38] work well, the
associated pressure $\bar{p}_{\alpha}\in C^{1}(\Omega)$ are chosen to be

$\bar{p}_{\alpha}=\frac{6\mu}{5}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa}\frac{x_{\alpha}}{\delta^{2}(x^{\prime})}+\mu\partial_{x_{3}}{\bf
v}_{\alpha}^{(3)},\quad\alpha=1,2,\quad\mbox{in}~{}\Omega_{2R},$ (2.4)

and $\|\bar{p}_{\alpha}\|_{C^{1}(\Omega\setminus\Omega_{R})}\leq C$.

For $\alpha=3$, similarly, we seek ${\bf v}_{3}\in
C^{2}(\Omega;\mathbb{R}^{3})$ satisfying,

$\displaystyle{\bf v}_{3}=\bm{\psi}_{3}\big{(}k(x)+\frac{1}{2}\big{)}+{\bf
E}_{3}(x)\Big{(}k^{2}(x)-\frac{1}{4}\Big{)},\quad\mbox{in}~{}\Omega_{2R},$
(2.5)

and $\|{\bf v}_{3}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq\,C$, where

$\displaystyle{\bf
E}_{3}(x)=\Big{(}\frac{3x_{1}}{\delta(x^{\prime})},\frac{3x_{2}}{\delta(x^{\prime})},-2k(x)+\frac{3x_{1}}{\delta(x^{\prime})}\tilde{H}_{1}(x)+\frac{3x_{2}}{\delta(x^{\prime})}\tilde{H}_{2}(x)\Big{)}^{\mathrm{T}},$

where

$\displaystyle\tilde{H}_{\alpha}=(\kappa_{1}+\kappa)x_{\alpha}+2(\kappa_{1}-\kappa)x_{\alpha}k(x),\quad\alpha=1,2,$
(2.6)

and the associated $\bar{p}_{3}\in C^{1}(\Omega)$ satisfying, in
$\Omega_{2R}$,

$\bar{p}_{3}=-\frac{3\mu}{2(\kappa_{1}-\kappa)\delta^{2}(x^{\prime})}+\mu\partial_{x_{3}}{\bf
v}_{3}^{(3)}.$

For $\alpha=4$, the construction of ${\bf v}_{4}\in
C^{2}(\Omega;\mathbb{R}^{3})$ is easy and as follows:

$\displaystyle{\bf
v}_{4}=\bm{\psi}_{4}\big{(}k(x)+\frac{1}{2}\big{)}\quad\mbox{in}~{}\Omega_{2R},$
(2.7)

and here we can directly take $\bar{p}_{4}=0$.

For $\alpha=5,6$, we define

$\displaystyle
F_{\alpha}(x)=\frac{12}{5}\frac{x_{\alpha-4}^{2}}{\delta(x^{\prime})}-2k(x)x_{3}-\frac{3x_{3}^{2}}{\delta(x^{\prime})}+\frac{3}{5(\kappa_{1}-\kappa)},\quad\tilde{G}(x)=-\frac{12}{5}\frac{x_{1}x_{2}}{\delta(x^{\prime})}.$

The auxiliary function ${\bf v}_{\alpha}\in C^{2}(\Omega;\mathbb{R}^{3})$
satisfies

$\displaystyle{\bf
v}_{\alpha}=\bm{\psi}_{\alpha}\big{(}k(x)+\frac{1}{2}\big{)}+{\bf
E}_{\alpha}(x)\Big{(}k^{2}(x)-\frac{1}{4}\Big{)},\quad\alpha=5,6,\quad\mbox{in}~{}\Omega_{2R},$
(2.8)

and $\|{\bf v}_{\alpha}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq\,C$, where

$\displaystyle{\bf
E}_{5}(x)=\left(F_{5}(x),\tilde{G}(x),2k(x)x_{1}+\tilde{H}_{1}(x)\Big{(}F_{5}(x)+2k(x)x_{3}\Big{)}-\delta(x^{\prime})\partial_{x_{2}}k(x)\tilde{G}(x)\right)^{\mathrm{T}},$

and

$\displaystyle{\bf
E}_{6}(x)=\left(\tilde{G}(x),F_{6}(x),2k(x)x_{2}+\tilde{H}_{2}(x)\Big{(}F_{6}(x)+2k(x)x_{3}\Big{)}-\delta(x^{\prime})\partial_{x_{1}}k(x)\tilde{G}(x)\right)^{\mathrm{T}},$

where $\tilde{H}_{1}(x)$ and $\tilde{H}_{2}(x)$ are defined in (2.6). Here the
associated pressure $\bar{p}_{\alpha}\in C^{1}(\Omega)$ in $\Omega_{2R}$, are
chosen to be

$\bar{p}_{\alpha}=\frac{6\mu}{5(\kappa_{1}-\kappa)}\frac{x_{\alpha-4}}{\delta^{2}(x^{\prime})}+\mu\partial_{x_{3}}{\bf
v}_{\alpha}^{(3)},\quad\alpha=5,6,\quad\mbox{in}~{}\Omega_{2R}.$

By direct calculations, these auxiliary functions have the following
properties, which is necessary to apply the technique in [37, 38] to estimate
$(\nabla{\bf u_{\alpha}},p_{\alpha})$.

######  Lemma 2.1.

Let ${\bf v}_{\alpha}$ and $\bar{p}_{\alpha}$ are defined as above,
$\alpha=1,\dots,6$. Then in $\Omega_{2R}$, the following assertions hold.

(1) For $\alpha=1,\dots,6$, $\nabla\cdot{\bf v}_{\alpha}=0$;

(2)

$\displaystyle|\nabla{\bf
v}_{\alpha}(x)|\leq\begin{cases}\frac{C}{\delta(x^{\prime})},&\alpha=1,2,5,6,\\\
C\big{(}\frac{1}{\delta(x^{\prime})}+\frac{|x^{\prime}|}{\delta^{2}(x^{\prime})}\big{)},&\alpha=3,\\\
C\big{(}\frac{|x^{\prime}|}{\delta(x^{\prime})}+1\big{)},&\alpha=4;\end{cases}$

and

(3)

$\displaystyle|\mu\Delta{\bf
v}_{\alpha}-\nabla\bar{p}_{\alpha}|\leq\begin{cases}\frac{C}{\delta(x^{\prime})},&\alpha=1,2,5,6,\\\
\frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})},&\alpha=3,\\\
\frac{C|x^{\prime}|}{\delta(x^{\prime})},&\alpha=4.\end{cases}$

###### Proof.

We only consider the case of $\alpha=1$ for instance, since other cases are
similar. Recalling the definition of ${\bf v}_{1}$ and $\bar{p}_{1}$ in (2.3)
and (2.4), by direct calculations, we have, for $x\in\Omega_{2R}$,

$\displaystyle\partial_{x_{1}}{\bf v}_{1}^{(1)}$
$\displaystyle=\partial_{x_{1}}k(x)-\frac{24}{5}\frac{(\kappa_{1}+\kappa)x_{1}\big{(}\delta(x^{\prime})-(\kappa_{1}-\kappa)x_{1}^{2}\big{)}}{\delta^{2}(x^{\prime})}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}$
$\displaystyle\quad+2k(x)\partial_{x_{1}}k(x)F_{1}(x),$ (2.9)
$\displaystyle\partial_{x_{2}}{\bf v}_{1}^{(1)}$
$\displaystyle=\partial_{x_{2}}k(x)+\frac{24(\kappa_{1}+\kappa)(\kappa_{1}-\kappa)}{5}\frac{x_{1}^{2}x_{2}}{\delta^{2}(x^{\prime})}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}+2k(x)\partial_{x_{2}}k(x)F_{1}(x),$
(2.10) $\displaystyle\partial_{x_{3}}{\bf v}_{1}^{(1)}$
$\displaystyle=\frac{1}{\delta(x^{\prime})}+\frac{2k(x)}{\delta(x_{1})}F_{1}(x),$
(2.11) $\displaystyle\partial_{x_{i}}{\bf v}_{1}^{(2)}$
$\displaystyle=-\frac{12}{5}\frac{(\kappa_{1}+\kappa)(x_{j}\delta(x^{\prime})-2(\kappa_{1}-\kappa)x_{1}x_{2}x_{i})}{5\delta^{2}(x^{\prime})}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}$
$\displaystyle\quad+2k(x)\partial_{x_{i}}k(x)G(x),\quad i,j=1,2,~{}i\neq j,$
$\displaystyle\partial_{x_{3}}{\bf v}_{1}^{(2)}$
$\displaystyle=\frac{2k(x)}{\delta(x^{\prime})}G(x),\quad|\partial_{x_{j}}{\bf
v}_{1}^{(3)}|\leq C,\quad j=1,2,$ $\displaystyle\partial_{x_{3}}{\bf
v}_{1}^{(3)}$
$\displaystyle=\frac{2}{\delta(x^{\prime})}\bigg{(}(\kappa_{1}+\kappa)x_{1}+(\kappa_{1}-\kappa)\big{(}x_{1}F_{1}(x)+x_{2}G(x)\big{)}\bigg{)}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}$
$\displaystyle\quad+\frac{2k(x)}{\delta(x^{\prime})}{\bf E}_{1}^{(3)}(x),$
(2.12)

where $F_{1}(x),G(x)$, and ${\bf E}_{1}(x)$ are defined in (2.2). These
estimates imply that

$\displaystyle\nabla\cdot{\bf v}_{1}=0,\quad|\nabla{\bf
v}_{1}|\leq\frac{C}{\delta(x^{\prime})}\quad\mbox{in}~{}\Omega_{2R}.$ (2.13)

Furthermore, for the second-order partial derivatives, we have, for $i=1,2$,

$\displaystyle|\partial_{x_{i}x_{i}}{\bf v}_{1}^{(1)}|$
$\displaystyle\leq\frac{C}{\delta(x^{\prime})},\quad\partial_{x_{3}x_{3}}{\bf
v}_{1}^{(1)}=\frac{2}{\delta^{2}(x^{\prime})}F_{1}(x),$ (2.14)
$\displaystyle|\partial_{x_{i}x_{i}}{\bf v}_{1}^{(2)}|$
$\displaystyle\leq\frac{C|x^{\prime}|}{\delta(x^{\prime})},\quad\partial_{x_{3}x_{3}}{\bf
v}_{1}^{(2)}=\frac{2}{\delta^{2}(x^{\prime})}G(x),$ (2.15)
$\displaystyle|\partial_{x_{i}x_{i}}{\bf v}_{1}^{(3)}|$
$\displaystyle\leq\frac{C|x^{\prime}|}{\delta(x^{\prime})},\quad|\partial_{x_{3}x_{3}}{\bf
v}_{1}^{(3)}|\leq\frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})},\quad|\partial_{x_{i}x_{3}}{\bf
v}_{1}^{(3)}|\leq\frac{C}{\delta(x^{\prime})},$ (2.16)
$\displaystyle\partial_{x_{1}}\bar{p}_{1}$
$\displaystyle=\frac{2\mu}{\delta^{2}(x_{1})}F_{1}(x)+\mu\partial_{x_{1}x_{3}}{\bf
v}_{1}^{(3)},\quad\partial_{x_{2}}\bar{p}_{1}=\frac{2\mu}{\delta^{(}x^{\prime})}G(x)+\mu\partial_{x_{2}x_{3}}{\bf
v}_{1}^{(3)}.$ (2.17)

It is easy to see from the definition of $\bar{p}_{1}$ in (2.4) that

$\mu\partial_{x_{3}x_{3}}{\bf v}_{1}^{(3)}-\partial_{x_{3}}\bar{p}_{1}=0.$

Moreover, it follows from (2.14)–(2.17) that

$\mu\partial_{x_{3}x_{3}}{\bf
v}_{1}^{(1)}-\partial_{x_{1}}\bar{p}_{1}=-\mu\partial_{x_{1}x_{3}}{\bf
v}_{1}^{(3)},\quad\mu\partial_{x_{3}x_{3}}{\bf
v}_{1}^{(2)}-\partial_{x_{2}}\bar{p}_{1}=-\mu\partial_{x_{2}x_{3}}{\bf
v}_{1}^{(3)},$

and

$\displaystyle\left|\mu\Delta{\bf
v}_{1}-\nabla\bar{p}_{1}\right|\leq\frac{C}{\delta(x^{\prime})}.$ (2.18)

Lemma 2.1 is proved. ∎

For $|x^{\prime}|\leq R$, we define a constant independently of $\varepsilon$,

$(q_{\alpha})_{\Omega_{R}}=\frac{1}{|\Omega_{R}|}\int_{\Omega_{R}}q_{\alpha}(y)\mathrm{d}y,\quad\alpha=1,\dots,6,$
(2.19)

and denote

$\Omega_{\delta}(x^{\prime}):=\left\\{(y^{\prime},y_{3})\in\mathbb{R}^{3}\big{|}h(y^{\prime})<y_{3}<\varepsilon+h_{1}(y^{\prime}),\,|y^{\prime}-x^{\prime}|<\delta\right\\},$
(2.20)

where $\delta=\delta(x^{\prime})$.

######  Proposition 2.2.

Let ${\bf u}_{\alpha}\in{C}^{2}(\Omega;\mathbb{R}^{3})$,
$p_{\alpha}\in{C}^{1}(\Omega)$ be the solution to (1.7) with
$\alpha=1,\dots,6$. Then in $\Omega_{R}$,

$\|\nabla({\bf u}_{\alpha}-{\bf
v}_{\alpha})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\begin{cases}C,&\alpha=1,2,5,6,\\\
\frac{C}{\sqrt{\delta(x^{\prime})}},&\alpha=3,\\\
C\sqrt{\delta(x^{\prime})},&\alpha=4,\end{cases}$

and

$\|\nabla^{2}({\bf u}_{\alpha}-{\bf
v}_{\alpha})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}+\|\nabla
q_{\alpha}\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\begin{cases}\frac{C}{\delta(x^{\prime})},&\alpha=1,2,5,6,\\\
\frac{C}{\delta^{3/2}(x^{\prime})},&\alpha=3,\\\
\frac{C}{\sqrt{\delta(x^{\prime})}},&\alpha=4.\end{cases}$

Consequently, in $\Omega_{R}$, we have

$\displaystyle|\nabla{\bf
u}_{\alpha}(x)|\leq\begin{cases}\frac{C}{\delta(x^{\prime})},&\alpha=1,2,5,6,\\\
C\left(\frac{1}{\delta(x^{\prime})}+\frac{|x^{\prime}|}{\delta^{2}(x^{\prime})}\right),&\alpha=3,\\\
C\left(\frac{|x^{\prime}|}{\delta(x^{\prime})}+1\right),&\alpha=4,\end{cases}$
$|p_{\alpha}(x)-(q_{\alpha})_{\Omega_{R}}|\leq\begin{cases}\frac{C}{\varepsilon^{3/2}},&\alpha=1,2,5,6,\\\
\frac{C}{\varepsilon^{2}},&\alpha=3,\\\
\frac{C}{\sqrt{\varepsilon}},&\alpha=4,\end{cases}$

and

$\displaystyle|\nabla^{2}{\bf u}_{\alpha}(x)|+|\nabla
p_{\alpha}(x)|\leq\begin{cases}\frac{C}{\delta^{2}(x^{\prime})},&\alpha=1,2,5,6,\\\
C\left(\frac{1}{\delta^{2}(x^{\prime})}+\frac{|x^{\prime}|}{\delta^{3}(x^{\prime})}\right),&\alpha=3,\\\
\frac{C}{\delta(x^{\prime})},&\alpha=4.\end{cases}$

###### Proof.

We only consider the case $\alpha=1$ since the proof of the case
$\alpha=2,\dots,6$ is similar. For simplicity, we denote ${\bf v}:={\bf
v}_{1}$, $\bar{p}:=\bar{p}_{1}$, ${\bf u}:={\bf u}_{1}$, $p:=p_{1}$, and set
${\bf w}={\bf u}-{\bf v}$, $q=p-\bar{p}$. Then it follows from (1.7) that
$({\bf w},q)$ satisfies

$\displaystyle\begin{cases}-\mu\,\Delta{\bf w}+\nabla q={\bf
f}:=\mu\,\Delta{\bf v}-\nabla\bar{p}\quad&\mathrm{in}\,\Omega,\\\
\nabla\cdot{\bf w}=0\quad&\mathrm{in}\,\Omega_{2R},\\\ \nabla\cdot{\bf
w}=-\nabla\cdot{\bf v}\quad&\mathrm{in}\,\Omega\setminus\Omega_{R},\\\ {\bf
w}=0\quad&\mathrm{on}\,\partial\Omega,\end{cases}$ (2.21)

where $\Omega_{R}$ is defined in (1.3). We shall divide the proof into three
steps.

Step 1. Global boundedness of $|\nabla{\bf w}|$:

$\displaystyle\int_{\Omega}|\nabla{\bf w}|^{2}\mathrm{d}x\leq\,C.$ (2.22)

We first recall that if

$\displaystyle\Big{|}\int_{\Omega_{R}}\sum_{j=1}^{3}{\bf f}^{(j)}{\bf
w}^{(j)}\mathrm{d}x\Big{|}\leq\,C\left(\int_{\Omega}|\nabla{\bf
w}|^{2}\mathrm{d}x\right)^{1/2},$ (2.23)

then (2.22) follows. See [38, Lemma 3.7] for details. To prove (2.23), using
the Sobolev trace embedding theorem, we have

$\displaystyle\int_{\begin{subarray}{c}|x^{\prime}|=R,\\\
h(x^{\prime})<x_{3}<\varepsilon+h_{1}(x^{\prime})\end{subarray}}|{\bf
w}|\mathrm{d}x_{3}$ $\displaystyle\leq C\left(\int_{\Omega}|\nabla{\bf
w}|^{2}\mathrm{d}x\right)^{1/2}.$ (2.24)

By using (2.18), the integration by parts with respect to $x^{\prime}$, and
(2.24), we obtain

$\displaystyle\left|\int_{\Omega_{R}}{\bf f}^{(1)}{\bf
w}^{(1)}\mathrm{d}x\right|$ $\displaystyle=\left|\int_{\Omega_{R}}\mu{\bf
w}^{(1)}\big{(}\partial_{x_{1}x_{1}}{\bf v}^{(1)}+\partial_{x_{2}x_{2}}{\bf
v}^{(1)}-\partial_{x_{1}x_{3}}{\bf v}^{(3)}\big{)}\mathrm{d}x\right|$
$\displaystyle\leq\,C\int_{\Omega_{R}}|\nabla_{x^{\prime}}{\bf
w}^{(1)}|\big{(}|\partial_{x_{1}}{\bf v}^{(1)}|+|\partial_{x_{2}}{\bf
v}^{(1)}|+|\partial_{x_{3}}{\bf v}^{(3)}|\big{)}\mathrm{d}x$
$\displaystyle\quad+C\,\int_{\begin{subarray}{c}|x^{\prime}|=R,\\\
h(x^{\prime})<x_{3}<\varepsilon+h_{1}(x^{\prime})\end{subarray}}|{\bf
w}^{(1)}|\mathrm{d}x_{3}$ $\displaystyle\leq
C\int_{\Omega_{R}}|\nabla_{x^{\prime}}{\bf
w}^{(1)}|\big{(}|\partial_{x_{1}}{\bf v}^{(1)}|+|\partial_{x_{2}}{\bf
v}^{(1)}|+|\partial_{x_{3}}{\bf v}^{(3)}|\big{)}\mathrm{d}x$
$\displaystyle\quad+C\left(\int_{\Omega}|\nabla{\bf
w}|^{2}\mathrm{d}x\right)^{1/2}=:\mathrm{I}_{11}+\mathrm{I}_{12}.$

From (2.9), (2.10), and (2.12), it follows that

$|\partial_{x_{1}}{\bf v}^{(1)}|,|\partial_{x_{2}}{\bf
v}^{(1)}|,|\partial_{x_{3}}{\bf
v}^{(3)}|\leq\frac{C|x^{\prime}|}{\delta(x^{\prime})}\quad\mbox{
in}~{}\Omega_{2R}.$

Combining with Hölder’s inequality, we have

$\displaystyle|\mathrm{I}_{11}|$ $\displaystyle\leq
C\left(\int_{\Omega_{R}}\big{(}|\partial_{x_{1}}{\bf
v}^{(1)}|^{2}+|\partial_{x_{2}}{\bf v}^{(1)}|^{2}+|\partial_{x_{3}}{\bf
v}^{(3)}|^{2}\big{)}\mathrm{d}x\right)^{1/2}\left(\int_{\Omega}|\nabla_{x^{\prime}}{\bf
w}^{(1)}|^{2}\mathrm{d}x\right)^{1/2}$ $\displaystyle\leq
C\left(\int_{\Omega}|\nabla{\bf w}|^{2}\mathrm{d}x\right)^{1/2}.$

Thus, we conclude that

$\displaystyle\left|\int_{\Omega_{R}}{\bf f}^{(1)}{\bf
w}^{(1)}\mathrm{d}x\right|\leq C\left(\int_{\Omega}|\nabla{\bf
w}|^{2}\mathrm{d}x\right)^{1/2}.$

Similarly, using (2.15), (2.16), (2.18), and (2.24), we obtain

$\displaystyle\left|\int_{\Omega_{R}}{\bf f}^{(2)}{\bf
w}^{(2)}\mathrm{d}x\right|\leq C\left(\int_{\Omega}|\nabla{\bf
w}|^{2}\mathrm{d}x\right)^{1/2},$

and

$\displaystyle\left|\int_{\Omega_{R}}{\bf f}^{(3)}{\bf
w}^{(3)}\mathrm{d}x\right|=\left|\mu\,\int_{\Omega_{R}}{\bf
w}^{(3)}\big{(}\partial_{x_{1}x_{1}}{\bf v}^{(3)}+\partial_{x_{2}x_{2}}{\bf
v}^{(3)}\big{)}\mathrm{d}x\right|$ $\displaystyle\leq$
$\displaystyle\,C\int_{\Omega_{R}}|\nabla_{x^{\prime}}{\bf
w}^{(3)}|\big{(}|\partial_{x_{1}}{\bf v}^{(3)}|+|\partial_{x_{2}}{\bf
v}^{(3)}|\big{)}\mathrm{d}x+C\int_{\begin{subarray}{c}|x^{\prime}|=R,\\\
h(x_{1})<x_{3}<\varepsilon+h_{1}(x^{\prime})\end{subarray}}|{\bf
w}^{(3)}|\mathrm{d}x_{3}$ $\displaystyle\leq$
$\displaystyle\,C\left(\int_{\Omega}|\nabla{\bf
w}|^{2}\mathrm{d}x\right)^{1/2}.$

Therefore, we have (2.23).

Step 2. Local estimate of $|\nabla{\bf w}|$:

$\displaystyle\int_{\Omega_{\delta}(z^{\prime})}|\nabla{\bf
w}|^{2}\mathrm{d}x\leq C\delta^{3}(z^{\prime}),$ (2.25)

where $\delta:=\delta(z^{\prime})$, $|z^{\prime}|\leq R$, and
$\Omega_{\delta}(z^{\prime})$ is defined by (2.20). Recall that

$\displaystyle\int_{\Omega_{t}(z^{\prime})}|\nabla{\bf
w}|^{2}\mathrm{d}x\leq\,$
$\displaystyle\frac{C\delta^{2}(z^{\prime})}{(s-t)^{2}}\int_{\Omega_{s}(z^{\prime})}|\nabla{\bf
w}|^{2}\mathrm{d}x+C\left((s-t)^{2}+\delta^{2}(z^{\prime})\right)\int_{\Omega_{s}(z^{\prime})}|{\bf
f}|^{2}\mathrm{d}x.$ (2.26)

See [38, Lemma 3.10]. Denote

$F(t):=\int_{\Omega_{t}(z^{\prime})}|\nabla{\bf w}|^{2}\mathrm{d}x.$

From (2.18), it follows that

$\displaystyle\int_{\Omega_{s}(z^{\prime})}|{\bf
f}|^{2}\mathrm{d}x\leq\frac{Cs^{2}}{\delta(z^{\prime})}.$

Substituting it into (2.26), we have

$\displaystyle
F(t)\leq\left(\frac{c_{0}\delta(z^{\prime})}{s-t}\right)^{2}F(s)+C\left((s-t)^{2}+\delta^{2}(z^{\prime}\right)\frac{s^{2}}{\delta(z^{\prime})},$
(2.27)

where $c_{0}$ is a constant to be fixed. Let
$k_{0}=\left[\frac{1}{4c_{0}\sqrt{\delta(z^{\prime})}}\right]+1$ and
$t_{i}=\delta(z^{\prime})+2c_{0}i\delta(z^{\prime}),i=0,1,2,\dots,k_{0}$. Then
we take $s=t_{i+1}$ and $t=t_{i}$ in (2.27) to get

$\displaystyle
F(t_{i})\leq\frac{1}{4}F(t_{i+1})+C(i+1)\delta^{3}(z^{\prime}).$

After $k_{0}$ iterations, and using (2.22), we obtain, for sufficiently small
$\varepsilon$ and $|z^{\prime}|$,

$\displaystyle
F(t_{0})\leq\left(\frac{1}{4}\right)^{k}F(t_{k})+C\delta^{3}(z^{\prime})\sum\limits_{l=0}^{k-1}\left(\frac{1}{4}\right)^{l}(l+1)\leq
C\delta^{3}(z^{\prime}).$

This gives (2.25).

Step 3. Local $W^{1,\infty}$ and $W^{2,\infty}$ estimates. With (2.18) and
(2.25), we are ready to get the desired estimates. We obtain from [38,
Proposition 3.6], (2.18), and (2.25) that

$\displaystyle\|\nabla{\bf
w}\|_{L^{\infty}(\Omega_{\delta/2}(z^{\prime}))}\leq\frac{C}{\delta^{3/2}(z^{\prime})}\|\nabla{\bf
w}\|_{L^{2}(\Omega_{\delta}(z^{\prime}))}+C\delta(z^{\prime})\|{\bf
f}\|_{L^{\infty}(\Omega_{\delta}(z^{\prime}))}\leq C,$

and

$\displaystyle\|\nabla^{2}{\bf
w}\|_{L^{\infty}(\Omega_{\delta/2}(z^{\prime}))}+\|\nabla
q\|_{L^{\infty}(\Omega_{\delta/2}(z^{\prime}))}$
$\displaystyle\leq\frac{C}{\delta^{5/2}(z^{\prime})}\|\nabla{\bf
w}\|_{L^{2}(\Omega_{\delta}(z^{\prime}))}+C\|{\bf
f}\|_{L^{\infty}(\Omega_{\delta}(z^{\prime}))}+C\delta(z^{\prime})\|\nabla{\bf
f}\|_{L^{\infty}(\Omega_{\delta}(z^{\prime}))}\leq\frac{C}{\delta(z^{\prime})}.$
(2.28)

Recalling ${\bf w}={\bf u}-{\bf v}$, it follows from (2.13) that

$\displaystyle|\nabla{\bf u}|\leq|\nabla{\bf v}(z)|+|\nabla{\bf
w}(z)|\leq\frac{C}{\delta(z^{\prime})},\quad z\in\Omega_{R},$

and using (2.14)–(2.16), we have

$\displaystyle|\nabla^{2}{\bf
u}|\leq\frac{C}{\delta(z^{\prime})}+\frac{C|z^{\prime}|}{\delta^{2}(z^{\prime})},\quad
z\in\Omega_{R}.$

By $q=p-\bar{p}$, (2.4), the mean value theorem, and the estimate of $|\nabla
q|$ in (2), we have

$\displaystyle|p(z)-(q)_{\Omega_{R}}|\leq|q-(q)_{\Omega_{R}}|+|\bar{p}|\leq\frac{C}{\varepsilon}+\frac{C|z^{\prime}|}{\delta^{2}(z^{\prime})}\leq\frac{C}{\varepsilon^{3/2}},\quad
z\in\Omega_{R}.$

Thus, we finish the proof of Proposition 2.2. ∎

##  3\. Estimates of $(\nabla{\bf u}_{0},p_{0})$ in 3D

In this section we estimate $|\nabla{\bf u}_{0}|$ and $p_{0}$ satisfying
(1.8), according to the given different kinds of boundary data $\bm{\varphi}$.
Define

$(q_{0}^{\alpha})_{\Omega_{R}}=\frac{1}{|\Omega_{R}|}\int_{\Omega_{R}}q_{0}^{\alpha}(x)\mathrm{d}x,\quad\alpha=1,2,3,$

which is a constant independent of $\varepsilon$.

(a) For $\bm{\varphi}\in{\bf\Phi}_{1}$. Here we only present the details for
the case ${\bm{\varphi}}=(x_{1}^{l_{1}},0,0)^{\mathrm{T}}$ on $\Gamma_{2R}$,
the other cases are similar.

(a1) If $l_{1}=0$, for the locally constant boundary value
$\bm{\varphi}=(1,0,0)^{\mathrm{T}}$, we choose ${\bf v}_{0}^{1}\in
C^{2}(\Omega;\mathbb{R}^{3})$ satisfying

${\bf v}_{0}^{1}+{\bf
v}_{1}=(1,0,0)^{\mathrm{T}},\quad\mbox{in}~{}\Omega_{2R},$ (3.1)

where ${\bf v}_{1}$ is defined by (2.3). We construct $\bar{p}_{0}^{1}\in
C^{1}(\Omega)$ such that

$\bar{p}_{0}^{1}=-\bar{p}_{1}=-\frac{6\mu}{5}\frac{(\kappa_{1}+\kappa)}{(\kappa_{1}-\kappa)}\frac{x_{1}}{\delta^{2}(x^{\prime})}+\mu\partial_{x_{3}}({\bf
v}_{0}^{1})^{(3)},\quad\mbox{in}~{}\Omega_{2R}.$

Clearly, $\nabla\cdot{\bf v}_{0}^{1}=0$ in $\Omega_{2R}$, and the properties
of ${\bf v}_{1}$ is also useful here.

(a2) If $l_{1}\geq 1$, for the polynomial cases, we seek ${\bf v}_{0}^{1}\in
C^{2}(\Omega;\mathbb{R}^{3})$ satisfying, in $\Omega_{2R}$,

$\displaystyle{\bf
v}_{0}^{1}=\bm{\varphi}\big{(}\frac{1}{2}-k(x)\big{)}+\big{(}{\bf
E}_{0}^{1(1)},0,{\bf
E}_{0}^{1(3)}\big{)}^{\mathrm{T}}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)},$

where

$\displaystyle{\bf E}_{0}^{1(1)}$
$\displaystyle=\frac{32(\kappa_{1}-\kappa)}{l_{1}+2}\frac{x_{1}^{l_{1}+2}}{\delta(x^{\prime})}k(x)+\frac{12\kappa}{l_{1}+2}\frac{x_{1}^{l_{1}+2}}{\delta(x^{\prime})}-8x_{1}^{l_{1}}k(x)+3x_{1}^{l_{1}},$
$\displaystyle{\bf E}_{0}^{1(3)}$
$\displaystyle=(2k(x)-1)k(x)\delta(x^{\prime})l_{1}x_{1}^{l_{1}-1}-2x_{1}^{l_{1}+1}k(x)\Big{(}(\kappa_{1}+\kappa)+2(\kappa_{1}-\kappa)k(x)\Big{)}$
$\displaystyle\quad-\delta(x^{\prime})\partial_{x_{1}}k(x){\bf E}_{0}^{1(1)}.$

Then $\nabla\cdot{\bf v}_{0}^{1}=0$ in $\Omega_{2R}$, and the biggest term is
$\partial_{x_{3}x_{3}}({\bf
v}_{0}^{1})^{(1)}\sim\frac{|x^{\prime}|^{l_{1}}}{\delta^{2}(x\prime)}$. It is
a good term. So we can directly choose $\bar{p}_{0}^{1}=0$ in $\Omega$. A
direct calculation gives,

$\displaystyle\left|\mu\Delta{\bf
v}_{0}^{1}-\nabla\bar{p}_{0}^{1}\right|=\left|\mu\Delta{\bf
v}_{0}^{1}\right|\leq\begin{cases}\frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})},&\mbox{if}~{}l_{1}=1,\\\
\frac{C}{\delta(x^{\prime})},&\mbox{if}~{}l_{1}\geq
2,\end{cases}\quad\mbox{in}~{}\Omega_{2R}.$ (3.2)

Denote

$\displaystyle\rho_{1}^{l_{1}}=\begin{cases}\frac{1}{\sqrt{\delta(x^{\prime})}},&\quad
l_{1}=1;\\\ 1,&\quad l_{1}\neq 1.\end{cases}$

Then applying the energy method and iteration technique as in the proof of
Proposition 2.2, we derive the following result.

######  Proposition 3.1.

Let ${\bf u}_{0}\in{C}^{2}(\Omega;\mathbb{R}^{3}),~{}p_{0}\in{C}^{1}(\Omega)$
be the solution to (1.8). Then in $\Omega_{R}$, we have

$\|\nabla({\bf u}_{0}-{\bf
v}_{0}^{1})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq
C\rho_{1}^{l_{1}},$

and

$\|\nabla^{2}({\bf u}_{0}-{\bf
v}_{0}^{1})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}+\|\nabla
q_{0}^{1}\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq
C\begin{cases}\frac{C}{\delta^{3/2}(x^{\prime})},&~{}l_{1}=1,\\\
\frac{1}{\delta(x^{\prime})},&~{}l_{1}\neq 1.\end{cases}$

Consequently, for $x\in\Omega_{R}$,

$\displaystyle|\nabla{\bf u}_{0}(x)|\leq
C\rho_{1}^{l_{1}}+\frac{C|x^{\prime}|^{l_{1}}}{\delta(x^{\prime})},\quad|p_{0}(x)-(q_{0}^{1})_{\Omega_{R}}|\leq\begin{cases}\frac{C}{\varepsilon^{3/2}},&l_{1}=0,1,\\\
\frac{1}{\varepsilon},&~{}l_{1}\geq 2,\end{cases}$

and

$\displaystyle|\nabla^{2}{\bf u}_{0}(x)|+|\nabla
p_{0}(x)|\leq\begin{cases}\frac{C}{\delta^{2}(x^{\prime})},&l_{1}=0,\\\
\frac{C}{\delta^{3/2}(x^{\prime})},&l_{1}=1,\\\
\frac{1}{\delta(x^{\prime})},&~{}l_{1}\geq 2.\end{cases}$

###### Proof.

The proof is an adaptation of the proof of Proposition 2.2. One can see from
(3.1) that the case of $l_{1}=0$ is essentially the same as that of
Proposition 2.2 with $\alpha=1$. Moreover, the proof for $l_{1}\geq 2$ is also
similar to the case of $\alpha=1$ in Proposition 2.2, since the estimate (3.2)
when $l_{1}\geq 2$ is the same as that in (2.18). Thus, we will only work on
the case of $l_{1}=1$ in the following. As before, we denote ${\bf v}:={\bf
v}_{0}^{1}$, $\bar{p}:=\bar{p}_{0}$, ${\bf u}:={\bf u}_{0}$, $p:=p_{0}$, and
set ${\bf w}={\bf u}-{\bf v}$, $q=p-\bar{p}$. Then it follows from (1.8) that
$({\bf w},q)$ satisfies (2.21).

A direct calculation yields

$|\nabla{\bf v}|\leq\frac{C|x^{\prime}|^{l_{1}}}{\delta(x^{\prime})}.$

Then using the argument in the proof of Proposition 2.2 (see Step 1 for the
details), we find that (2.23) holds true. By (3.2), the estimate (2.27)
becomes

$\displaystyle
F(t)\leq\left(\frac{c_{1}\delta(z^{\prime})}{s-t}\right)^{2}F(s)+C\left((s-t)^{2}+\delta^{2}(z^{\prime}\right)\frac{s^{2}}{\delta^{3}(z^{\prime})}(s+|z^{\prime}|^{2}),$

where $c_{1}$ is a constant to be fixed. Then applying the iteration technique
as used in the proof of (2.25), we obtain

$\displaystyle\int_{\Omega_{\delta}(z^{\prime})}|\nabla{\bf
w}|^{2}\mathrm{d}x\leq C\delta^{2}(z^{\prime}),$

where $\delta:=\delta(z^{\prime})$, $|z^{\prime}|\leq R$, and
$\Omega_{\delta}(z^{\prime})$ is defined in (2.20). This together with [38,
Proposition 3.6] and (3.2) gives

$\displaystyle\|\nabla{\bf
w}\|_{L^{\infty}(\Omega_{\delta/2}(z^{\prime}))}\leq\frac{C}{\delta^{3/2}(z^{\prime})}\|\nabla{\bf
w}\|_{L^{2}(\Omega_{\delta}(z^{\prime}))}+C\delta(z^{\prime})\|{\bf
f}\|_{L^{\infty}(\Omega_{\delta}(z^{\prime}))}\leq\frac{C}{\sqrt{\delta(z^{\prime})}},$

and

$\displaystyle\|\nabla^{2}{\bf
w}\|_{L^{\infty}(\Omega_{\delta/2}(z^{\prime}))}+\|\nabla
q\|_{L^{\infty}(\Omega_{\delta/2}(z^{\prime}))}$
$\displaystyle\leq\frac{C}{\delta^{5/2}(z^{\prime})}\|\nabla{\bf
w}\|_{L^{2}(\Omega_{\delta}(z^{\prime}))}+C\|{\bf
f}\|_{L^{\infty}(\Omega_{\delta}(z^{\prime}))}+C\delta(z^{\prime})\|\nabla{\bf
f}\|_{L^{\infty}(\Omega_{\delta}(z^{\prime}))}\leq\frac{C}{\delta^{3/2}(z^{\prime})}.$

The proof is finished. ∎

(b) For $\bm{\varphi}\in{\bf\Phi}_{2}$. Here we only consider the case $i=1$
in (1.11), since the case $i=2$ is the same.

(b1) If $l_{2}=0$, for the locally constant
$\bm{\varphi}=(0,0,1)^{\mathrm{T}}$, then we choose ${\bf v}_{0}^{2}\in
C^{2}(\Omega;\mathbb{R}^{3})$ satisfying

${\bf v}_{0}^{2}+{\bf
v}_{3}=(0,0,1)^{\mathrm{T}}\quad\mbox{in}~{}\Omega_{2R}.$

We construct $\bar{p}_{0}^{2}\in C^{1}(\Omega)$ such that, in $\Omega_{2R}$,

$\bar{p}_{0}^{2}=-\frac{3\mu}{2(\kappa_{1}-\kappa)\delta^{2}(x^{\prime})}+\mu\partial_{x_{3}}({\bf
v}_{0}^{2})^{(3)}.$

(b2) If $l_{2}=1$, we construct ${\bf v}_{0}^{2}\in
C^{2}(\Omega;\mathbb{R}^{3})$ satisfying, in $\Omega_{2R}$,

$\displaystyle{\bf
v}_{0}^{2}=\bm{\varphi}\big{(}\frac{1}{2}-k(x)\big{)}+\big{(}{\bf
E}_{0}^{2(1)},{\bf E}_{0}^{2(2)},{\bf
E}_{0}^{2(3)}\big{)}^{\mathrm{T}}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}\quad\mbox{in}~{}\Omega_{2R},$

where

$\displaystyle{\bf E}_{0}^{2(1)}=$
$\displaystyle-\frac{12}{5}\frac{x_{1}^{2}}{\delta(x^{\prime})}+\frac{3}{5(\kappa_{1}-\kappa)},\quad{\bf
E}_{0}^{2(2)}=-\frac{12}{5}\frac{x_{1}x_{2}}{\delta(x^{\prime})},$
$\displaystyle{\bf E}_{0}^{2(3)}=$ $\displaystyle
2k(x)x_{1}-\delta(x^{\prime})\partial_{x_{1}}k(x){\bf
E}_{0}^{2(1)}-\delta(x^{\prime})\partial_{x_{2}}k(x){\bf E}_{0}^{2(2)}.$

(b3) If $l_{2}\geq 2$, we seek ${\bf v}_{0}^{2}\in
C^{2}(\Omega;\mathbb{R}^{3})$ satisfying, in $\Omega_{2R}$,

$\displaystyle{\bf
v}_{0}^{2}=\bm{\varphi}\big{(}\frac{1}{2}-k(x)\big{)}+\big{(}{\bf
E}_{0}^{2(1)},0,{\bf
E}_{0}^{2(3)}\big{)}^{\mathrm{T}}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}\quad\mbox{in}~{}\Omega_{2R}.$

Here

${\bf
E}_{0}^{2(1)}=-\frac{6}{l_{2}+1}\frac{x_{1}^{l_{2}+1}}{\delta(x^{\prime})},\quad{\bf
E}_{0}^{2(3)}=2k(x)x_{1}-\delta(x^{\prime})\partial_{x_{1}}k(x){\bf
E}_{0}^{2(1)}.$

It also holds that $\nabla\cdot{\bf v}_{0}^{2}=0$ in $\Omega_{2R}$ and

$\displaystyle|\nabla{\bf
v}_{0}^{2}|\leq\begin{cases}\frac{C}{\delta(x^{\prime})}+\frac{|x^{\prime}|}{\delta^{2}(x^{\prime})},&\quad
l_{2}=0,\\\ \frac{C|x^{\prime}|^{l_{2}-1}}{\delta(x^{\prime})},&\quad
l_{2}\geq 1.\end{cases}$

In order to control the biggest term in $\Delta{\bf v}_{0}^{2}$, we take

$\displaystyle\bar{p}_{0}^{2}=\begin{cases}\frac{6\mu}{5(\kappa_{1}-\kappa)}\frac{x_{1}}{\delta^{2}(x^{\prime})}+\mu\partial_{x_{3}}({\bf
v}_{0}^{2})^{(3)},&\quad l_{2}=1;\\\ 0,&\quad l_{2}\geq 2.\end{cases}$

Furthermore, we deduce, in $\Omega_{2R}$,

$\displaystyle\left|\mu\Delta{\bf
v}_{0}^{2}-\nabla\bar{p}_{0}^{2}\right|\leq\begin{cases}\frac{C}{\delta(x^{\prime})},&\quad
l_{2}=1,~{}l_{2}\geq 3,\\\ \frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})},&\quad
l_{2}=0,2.\end{cases}$

Let

$\displaystyle\rho_{2}^{l_{2}}=\begin{cases}\frac{1}{\sqrt{\delta(x^{\prime})}},&\quad
l_{2}=0,2;\\\ 1,&\quad l_{2}=1,~{}l_{2}\geq 3.\end{cases}$

Then apply the same argument as used in the proof of Proposition 3.1, we
derive Proposition 3.2 as follows.

######  Proposition 3.2.

Let ${\bf u}_{0}\in{C}^{2}(\Omega;\mathbb{R}^{3}),~{}p_{0}\in{C}^{1}(\Omega)$
be the solution to (1.8). Then in $\Omega_{R}$,

$\displaystyle\|\nabla({\bf u}_{0}-{\bf
v}_{0}^{2})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq
C\rho_{2}^{l_{2}},$

and

$\displaystyle\|\nabla^{2}({\bf u}_{0}-{\bf
v}_{0}^{2})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}+\|\nabla
q_{0}^{2}\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\frac{C}{\delta(x^{\prime})}+\frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})}.$

Consequently, for $x\in\Omega_{R}$,

$\displaystyle|\nabla{\bf
u}_{0}(x)|\leq\begin{cases}\frac{C}{\delta(x^{\prime})}+\frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})},&\quad
l_{2}=0,\\\ \frac{C}{\delta(x^{\prime})},&\quad l_{2}=1,\\\
\frac{C}{\sqrt{\delta(x^{\prime})}}+\frac{C|x^{\prime}|}{\delta(x^{\prime})},&\quad
l_{2}=2,\\\ C,&\quad l_{2}\geq 3,\end{cases}$
$\displaystyle|p_{0}(x)-(q_{0}^{2})_{\Omega_{R}}|\leq\begin{cases}\frac{C}{\varepsilon^{2}},&\quad
l_{2}=0,\\\ \frac{C}{\varepsilon^{3/2}},&\quad l_{2}\geq 1,\end{cases}$

and

$\displaystyle|\nabla^{2}{\bf u}_{0}(x)|+|\nabla
p_{0}(x)|\leq\begin{cases}\frac{C}{\delta^{2}(x^{\prime})}+\frac{C|x^{\prime}|}{\delta^{3}(x^{\prime})},&\quad
l_{2}=0,\\\ \frac{C}{\delta^{2}(x^{\prime})},&\quad l_{2}=1,\\\
\frac{C}{\delta(x^{\prime})}+\frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})},&\quad
l_{2}\geq 2.\end{cases}$

(c) If $\bm{\varphi}\in{\bf\Phi}_{3}$ with $j=1$ in (1.12), we seek ${\bf
v}_{0}^{3}\in C^{2}(\Omega;\mathbb{R}^{3})$ satisfying, in $\Omega_{2R}$,

$\displaystyle{\bf
v}_{0}^{3}=\bm{\varphi}\big{(}\frac{1}{2}-k(x)\big{)}+\begin{pmatrix}2x_{3}^{l_{3}}k(x)+\frac{6}{l_{3}+1}\frac{x_{3}^{l_{3}+1}}{\delta(x^{\prime})}\\\
\\\ 0\\\ \\\ {\bf
E}_{0}^{3(3)}\end{pmatrix}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}\quad\mbox{in}~{}\Omega_{2R}.$

Here

${\bf
E}_{0}^{3(3)}=\big{(}(\kappa_{1}+\kappa)x_{1}+2(\kappa_{1}-\kappa)x_{1}k(x)\big{)}\frac{6}{l_{3}+1}\frac{x_{3}^{l_{3}+1}}{\delta(x^{\prime})}.$

Using a direct computation, we have $\nabla\cdot{\bf v}_{0}^{3}=0$ in
$\Omega_{2R}$, $|\nabla{\bf v}_{0}^{3}|\leq C$, and $|\Delta{\bf
v}_{0}^{3}|\leq\frac{C}{\delta(x^{\prime})}$, so that we choose
${\bar{p}}_{0}^{3}=0$. Also, in $\Omega_{2R}$,

$\displaystyle\left|\mu\Delta{\bf
v}_{0}^{3}-\nabla\bar{p}_{0}^{3}\right|\leq\frac{C}{\delta(x^{\prime})}.$

######  Proposition 3.3.

Let ${\bf u}_{0}\in{C}^{2}(\Omega;\mathbb{R}^{3}),~{}p_{0}\in{C}^{1}(\Omega)$
be the solution to (1.8). Then there holds

$\|\nabla({\bf u}_{0}-{\bf
v}_{0}^{3})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq C,$

and

$\|\nabla^{2}({\bf u}_{0}-{\bf
v}_{0}^{3})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}+\|\nabla
q_{0}^{3}\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\frac{C}{\delta(x^{\prime})},~{}~{}\,x\in\Omega_{R}.$

Consequently, for $x\in\Omega_{R}$,

$\displaystyle|\nabla{\bf u}_{0}(x)|\leq
C,\quad|p_{0}(x)-(q_{0}^{3})_{\Omega_{R}}|\leq\frac{C}{\varepsilon},\quad|\nabla
p_{0}(x)|+|\nabla^{2}{\bf u}_{0}(x)|\leq\frac{C}{\delta(x^{\prime})}.$

The auxiliary functions for $j=2,3$ in (1.12) are constructed similarly, we
left it to the interested reader.

##  4\. Estimates of $C^{\alpha}$

In this section, we proceed to estimate $C^{\alpha}$, by using the estimates
of $(\nabla{\bf u}_{\alpha},p_{\alpha})$, $\alpha=1,\dots,6$, and $(\nabla{\bf
u}_{0},p_{0})$. We have the following Proposition.

######  Proposition 4.1.

(1) Assume that ${\bm{\varphi}}\in{\bf\Phi}_{1}$, where ${\bf\Phi}_{1}$ is
defined in (1.10). Then if $l_{1}=0$, we have

$\displaystyle|C^{1}-1|,|C^{\alpha}|\leq\frac{C}{|\ln\varepsilon|},~{}\alpha=2,5,6,\quad|C^{3}|\leq
C\varepsilon,\quad|C^{4}|\leq C;$

if $l_{1}\geq 1$, we have

$\displaystyle|C^{\alpha}|\leq\frac{C}{|\ln\varepsilon|},~{}\alpha=1,2,5,6,\quad|C^{3}|\leq
C\varepsilon,\quad|C^{4}|\leq C.$

(2) Assume that ${\bm{\varphi}}\in{\bf\Phi}_{2}$, where ${\bf\Phi}_{2}$ is
defined in (1.11). Then if $l_{2}=0$, we have

$\displaystyle|C^{\alpha}|\leq\frac{C}{|\ln\varepsilon|},~{}\alpha=1,2,5,6,\quad|C^{3}-1|\leq
C\varepsilon,\quad|C^{4}|\leq C;$

if $l_{2}=1$, we have

$\displaystyle|C^{\alpha}|\leq
C,~{}\alpha=1,4,5,\quad|C^{\alpha}|\leq\frac{C}{|\ln\varepsilon|},~{}\alpha=2,6,\quad|C^{3}|\leq
C\varepsilon;$

if $l_{2}\geq 2$, we have

$\displaystyle|C^{\alpha}|\leq\frac{C}{|\ln\varepsilon|},~{}\alpha=1,2,5,6,\quad|C^{3}|\leq
C\varepsilon,\quad|C^{4}|\leq C.$

(3) Assume that ${\bm{\varphi}}\in{\bf\Phi}_{3}$, where ${\bf\Phi}_{3}$ is
defined in (1.12). Then we have

$\displaystyle|C^{\alpha}|\leq\frac{C}{|\ln\varepsilon|},~{}\alpha=1,2,5,6,\quad|C^{3}|\leq
C\varepsilon,\quad|C^{4}|\leq C.$

From (1.13), to prove Proposition 4.1, we need to establish the following
estimates and asymptotic expansions of $a_{\alpha\beta}$ and
$Q_{\beta}[{\bm{\varphi}}]$.

###  4.1. Estimates of $a_{\alpha\beta}$.

By (1.14), using the integration by parts, we have

$\displaystyle a_{\alpha\beta}=\int_{\Omega}\left(2\mu e({\bf
u}_{\alpha}),e({\bf u}_{\beta})\right)\mathrm{d}x,\quad
Q_{\beta}[{\bm{\varphi}}]=-\int_{\Omega}\left(2\mu e({\bf u}_{0}),e({\bf
u}_{\beta})\right)\mathrm{d}x.$ (4.1)

We will make use of the estimates of $|\nabla{\bf u}_{\alpha}|$ in Proposition
2.2 to derive the following asymptotic expansions and estimates for
$a_{\alpha\beta}$. Set

$K(\kappa_{1},\kappa)=\frac{9}{25}\Big{(}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa}\Big{)}^{2}+\frac{72}{25}\frac{\kappa_{1}\kappa}{(\kappa_{1}-\kappa)^{2}}+\frac{6}{25}.$

######  Lemma 4.2.

We have

$\displaystyle a_{ii}$
$\displaystyle=\frac{\mu\pi}{(\kappa_{1}-\kappa)}\left(1+\Big{(}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa}\Big{)}^{2}K(\kappa_{1},\kappa)\right)|\ln\varepsilon|+O(1),\quad
i=1,2;$ $\displaystyle a_{33}$
$\displaystyle=\frac{\mu\pi}{24(\kappa_{1}-\kappa)^{2}}\frac{1}{\varepsilon}+O(|\ln\varepsilon|),\quad\frac{1}{C}\leq
a_{44}\leq\ C;$ $\displaystyle a_{ii}$
$\displaystyle=\frac{\mu\pi\,}{(\kappa_{1}-\kappa)^{3}}K(\kappa_{1},\kappa)|\ln\varepsilon|+O(1),\quad
i=5,6;$ $\displaystyle a_{ij}$
$\displaystyle=\frac{\mu\pi(\kappa_{1}+\kappa)}{(\kappa_{1}-\kappa)^{3}}K(\kappa_{1},\kappa)|\ln\varepsilon|+O(1),\quad(i,j)=(1,5)~{}\mbox{and}~{}(2,6).$

###### Proof.

We follow the technique used in [39]. By means of (4.1), Proposition 2.2, and
(2.13), we have

$\displaystyle a_{11}$ $\displaystyle=\int_{\Omega_{R}}\left(2\mu e({\bf
u}_{1}),e({\bf u}_{1})\right)\mathrm{d}x+O(1)$
$\displaystyle=\int_{\Omega_{R}}\left(2\mu e({\bf v}_{1}),e({\bf
v}_{1})\right)\mathrm{d}x+\int_{\Omega_{R}}\left(2\mu e({\bf v}_{1}),e({\bf
w}_{1})\right)\mathrm{d}x$ $\displaystyle\quad+\int_{\Omega_{R}}\left(2\mu
e({\bf w}_{1}),e({\bf v}_{1})\right)\mathrm{d}x+\int_{\Omega_{R}}\left(2\mu
e({\bf w}_{1}),e({\bf w}_{1})\right)\mathrm{d}x+O(1)$
$\displaystyle=\int_{\Omega_{R}}\left(2\mu e({\bf v}_{1}),e({\bf
v}_{1})\right)\mathrm{d}x+O(1),$

where ${\bf w}_{1}={\bf u}_{1}-{\bf v}_{1}$. From (2.9)–(2.12), we find that
the biggest term in $e({\bf v}_{1})$ is $\partial_{x_{3}}{\bf v}_{1}^{(1)}$
and $\partial_{x_{3}}{\bf v}_{1}^{(2)}$. Then we have

$\displaystyle a_{11}$
$\displaystyle=\mu\int_{\Omega_{R}}\left(\frac{1}{\delta(x^{\prime})}+\frac{6(\kappa_{1}+\kappa)}{5}\frac{k(x)}{\delta(x^{\prime})}\Big{(}-\frac{5x_{1}^{2}}{\delta(x^{\prime})}+\frac{1}{\kappa_{1}-\kappa}\Big{)}\right)^{2}$
$\displaystyle\quad+\left(\frac{24(\kappa_{1}+\kappa)}{5}\frac{x_{1}x_{2}}{\delta^{2}(x^{\prime})}k(x)\right)^{2}\mathrm{d}x+O(1)$
$\displaystyle=\frac{\mu\pi}{(\kappa_{1}-\kappa)}\left(1+\Big{(}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa}\Big{)}^{2}K(\kappa_{1},\kappa)\right)|\ln\varepsilon|+O(1).$

The proof of the estimates of $a_{22}$, $a_{33}$, $a_{55}$, $a_{66}$,
$a_{15}$, and $a_{26}$ follows from the same argument, so we omit the details
here. For the estimate of $a_{44}$,

$\displaystyle a_{44}\leq C\int_{\Omega}|\nabla{\bf u}_{4}|^{2}\mathrm{d}x\leq
C\int_{\Omega_{R}}\frac{|x^{\prime}|^{2}}{\delta^{2}(x^{\prime})}\,\mathrm{d}x+C\leq
C,$

and

$\displaystyle
a_{44}\geq\int_{\Omega_{R}\setminus\Omega_{R/2}}|\partial_{x_{3}}{\bf
v}_{4}^{(2)}|^{2}\geq\int_{\Omega_{R}\setminus\Omega_{R/2}}\frac{|x^{\prime}|^{2}}{\delta^{2}(x^{\prime})}\mathrm{d}x\geq\frac{1}{C}.$

Consequently, the lemma is proved. ∎

######  Lemma 4.3.

For other cases, we have

$\displaystyle|a_{\alpha\beta}|=|a_{\beta\alpha}|\leq
C,\quad\alpha,\beta=1,\dots,6.$

###### Proof.

By using (4.1) and Proposition 2.2, we have

$\displaystyle a_{12}$ $\displaystyle=\int_{\Omega_{R}}\left(2\mu e({\bf
u}_{1}),e({\bf u}_{2})\right)\mathrm{d}x+C$
$\displaystyle=\int_{\Omega_{R}}\left(2\mu e({\bf v}_{1}),e({\bf
v}_{2})\right)\mathrm{d}x+\int_{\Omega_{R}}\left(2\mu e({\bf v}_{1}),e({\bf
w}_{2})\right)\mathrm{d}x$ $\displaystyle\quad+\int_{\Omega_{R}}\left(2\mu
e({\bf w}_{1}),e({\bf v}_{2})\right)\mathrm{d}x+\int_{\Omega_{R}}\left(2\mu
e({\bf w}_{1}),e({\bf w}_{2})\right)\mathrm{d}x+C$
$\displaystyle=\int_{\Omega_{R}}\left(2\mu e({\bf v}_{1}),e({\bf
v}_{2})\right)\mathrm{d}x+C.$

Recalling the definitions of ${\bf v}_{\alpha}$ in (2.3), $\alpha=1,2$, a
direct calculation yields that the the biggest terms in $(2\mu e({\bf
v}_{1}),e({\bf v}_{2}))$ are $\partial_{x_{3}}{\bf
v}_{1}^{(1)}\cdot\partial_{x_{3}}{\bf v}_{2}^{(1)}$ and $\partial_{x_{3}}{\bf
v}_{1}^{(2)}\cdot\partial_{x_{3}}{\bf v}_{2}^{(2)}$. Note that

$\displaystyle\partial_{x_{3}}{\bf v}_{1}^{(1)}\cdot\partial_{x_{3}}{\bf
v}_{2}^{(1)}=\frac{2k(x)}{\delta(x^{\prime})}G(x)\left(\frac{1}{\delta(x^{\prime})}+\frac{2k(x)}{\delta(x_{1})}F_{1}(x)\right),$

which is an odd function with respect to $x_{j}$, $j=1,2$, where $F_{1}(x)$
and $G(x)$ are defined in (2.2). Similarly, $\partial_{x_{3}}{\bf
v}_{1}^{(2)}\cdot\partial_{x_{3}}{\bf v}_{2}^{(2)}$ is also an odd function
with respect to $x_{j}$, $j=1,2$. The integral of the rest terms is bounded.
Thus, we derive

$|a_{12}|\leq C.$

The rest of the terms is proved similarly, so we omit the details here. ∎

######  Lemma 4.4.

Assume ${\bm{\varphi}}\in{\bf\Phi}_{1}$, where ${\bf\Phi}_{1}$ is defined in
(1.10). If $l_{1}=0$, then we have

$\displaystyle Q_{1}[\bm{\varphi}]$
$\displaystyle=\frac{\mu\pi}{(\kappa_{1}-\kappa)}\left(1+\Big{(}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa}\Big{)}^{2}K(\kappa_{1},\kappa)\right)|\ln\varepsilon|+O(1),$
$\displaystyle Q_{5}[\bm{\varphi}]$
$\displaystyle=\frac{\mu\pi(\kappa_{1}+\kappa)}{(\kappa_{1}-\kappa)^{3}}K(\kappa_{1},\kappa)|\ln\varepsilon|+O(1),\quad|Q_{\beta}[\bm{\varphi}]|\leq
C,\quad\beta=2,3,4,6.$

If $l_{1}\geq 1$, then we have

$\displaystyle|Q_{\beta}[\bm{\varphi}]|$ $\displaystyle\leq
C,\quad\beta=1,2,4,5,6,$

and

$\displaystyle|Q_{3}[\bm{\varphi}]|\leq\begin{cases}C|\ln\varepsilon|,&\quad
l_{1}=1,\\\ C,&\quad l_{1}\geq 2.\end{cases}$

###### Proof.

If $l_{1}=0$, then by using the definition of $Q_{\beta}[\bm{\varphi}]$ in
(1.14), we find that the estimates of $|Q_{\beta}[\bm{\varphi}]|$ are the same
as that of $|a_{1\beta}|$. Next we consider the case of $l_{1}\geq 1$.

If $l_{1}=1$, then by using Propositions 2.2 and 3.1, we have

$\displaystyle|Q_{1}[\bm{\varphi}]|$
$\displaystyle=\left|\int_{\Omega}\left(2\mu e({\bf u}_{0}),e({\bf
u}_{1})\right)\mathrm{d}x\right|\leq\left|\int_{\Omega_{R}}\left(2\mu e({\bf
u}_{0}),e({\bf u}_{1})\right)\mathrm{d}x\right|+C$ $\displaystyle\leq
C\int_{\Omega_{R}}\frac{1}{\delta(x^{\prime})}\left(\frac{1}{\sqrt{\delta(x^{\prime})}}+\frac{|x^{\prime}|}{\delta(x^{\prime})}\right)\mathrm{d}x+C\leq
C.$ (4.2)

If $l_{1}\geq 2$, similar to (4.1), we have

$\displaystyle Q_{1}[\bm{\varphi}]|$
$\displaystyle=\left|\int_{\Omega}\left(2\mu e({\bf u}_{0}),e({\bf
u}_{1})\right)\mathrm{d}x\right|\leq
C\int_{\Omega_{R}}\frac{\mathrm{d}x}{\delta(x^{\prime})}\ +C\leq C.$

The estimates of $Q_{2}[\bm{\varphi}]$, $Q_{5}[\bm{\varphi}]$, and
$Q_{6}[\bm{\varphi}]$ in the case of $l_{1}\geq 1$ are the same as that of
$Q_{1}[\bm{\varphi}]$. We thus omit the details.

In order to estimate $Q_{3}[\bm{\varphi}]$, we first have from Propositions
2.2 and 3.1 that, for $l_{1}=1$,

$\displaystyle|Q_{3}[\bm{\varphi}]|$
$\displaystyle\leq\left|\int_{\Omega_{R}}\left(2\mu e({\bf u}_{0}),e({\bf
u}_{3})\right)\mathrm{d}x\right|+C$ $\displaystyle\leq C\int_{|x^{\prime}|\leq
R}\frac{1}{\sqrt{\delta(x^{\prime})}}\left(1+\frac{|x^{\prime}|}{\delta(x^{\prime})}\right)\mathrm{d}x+C\leq
C|\ln\varepsilon|.$

If $l_{1}\geq 2$, using the above process, we have

$|Q_{3}[\bm{\varphi}]|\leq C.$

The process of estimating $|Q_{4}[\bm{\varphi}]|$ is easy by using
Propositions 2.2 and 3.1, we omit the proof here. We thus prove the lemma. ∎

######  Lemma 4.5.

Assume ${\bm{\varphi}}\in{\bf\Phi}_{2}$, where ${\bf\Phi}_{2}$ is defined in
(1.11). The following assertions hold.

(a) If $l_{2}=0$, then

$\displaystyle
Q_{3}[\bm{\varphi}]=\frac{\mu\pi}{24(\kappa_{1}-\kappa)^{2}}\frac{1}{\varepsilon}+O(|\ln\varepsilon|);\quad|Q_{\beta}[\bm{\varphi}]|\leq
C,\quad\beta=1,2,4,5,6.$

(b) If $l_{2}\geq 1$, then

$\displaystyle|Q_{\beta}[\bm{\varphi}]|\leq\begin{cases}C|\ln\varepsilon|,&l_{2}=1,\\\
C,&l_{2}\geq 2,\end{cases}\quad\beta=1,5;\quad|Q_{\beta}[\bm{\varphi}]|\leq
C,\quad l_{2}\geq 1,\quad\beta=2,3,4,6.$

###### Proof.

The proof is similar to Lemma 4.4. If $l_{2}=0$, then the estimates of
$|Q_{\beta}[\bm{\varphi}]|$ is simiar to that of $|a_{3\beta}|$. For
$l_{2}\geq 1$, we only consider the case of $|Q_{1}[\bm{\varphi}]|$, since
other cases are similar. If $l_{2}=1$, then combining Propositions 2.2 and
3.2, we have

$\displaystyle|Q_{1}[\bm{\varphi}]|$
$\displaystyle=\left|\int_{\Omega}\left(2\mu e({\bf u}_{0}),e({\bf
u}_{1})\right)\mathrm{d}x\right|\leq\left|\int_{\Omega_{R}}\left(2\mu e({\bf
u}_{0}),e({\bf u}_{1})\right)\mathrm{d}x\right|+C$ $\displaystyle\leq
C\int_{\Omega_{R}}\frac{1}{\delta^{2}(x^{\prime})}\mathrm{d}x+C\leq
C|\ln\varepsilon|.$

If $l_{2}=2$,

$\displaystyle|Q_{1}[\bm{\varphi}]|$
$\displaystyle=\left|\int_{\Omega}\left(2\mu e({\bf u}_{0}),e({\bf
u}_{1})\right)\mathrm{d}x\right|\leq
C\int_{\Omega_{R}}\frac{1}{\delta(x^{\prime})}\left(\frac{1}{\sqrt{\delta(x^{\prime})}}+\frac{|x^{\prime}|}{\delta(x^{\prime})}\right)\mathrm{d}x+C\leq
C.$

Similarly, if $l_{2}\geq 3$,

$|Q_{1}[\bm{\varphi}]|\leq C.$

We thus complete the proof of the estimate of $|Q_{1}[\bm{\varphi}]|$. ∎

Similar to Lemma 4.5, we have the following result.

######  Lemma 4.6.

If ${\bm{\varphi}}\in{\bf\Phi}_{3}$, where ${\bf\Phi}_{3}$ is defined in
(1.12), then we have

$\displaystyle|Q_{\beta}[\bm{\varphi}]|$ $\displaystyle\leq
C,\quad\beta=1,\dots,6.$

Now let us prove Proposition 4.1.

###### Proof of Proposition 4.1..

Denote

$X=(C^{1},\dots,C^{6})^{\mathrm{T}},\quad
P=(Q_{1}[{\bm{\varphi}}],\dots,Q_{6}[{\bm{\varphi}}]),\quad\mathbb{A}=(a_{\alpha\beta})_{6\times
6}.$

Denote

$a:=\frac{1}{\kappa_{1}-\kappa},~{}b:=\frac{(\kappa_{1}+\kappa)^{2}}{(\kappa_{1}-\kappa)^{3}}K(\kappa_{1},\kappa).$

Then, in view of Lemma 4.2, it follows that

$\displaystyle
a_{11}a_{22}a_{55}a_{66}+a_{15}^{2}a_{26}^{2}-a_{15}a_{22}a_{51}a_{66}-a_{11}a_{26}a_{55}a_{62}$
$\displaystyle=\frac{(\mu\pi)^{4}}{(\kappa_{1}+\kappa)^{4}}\Big{(}(a+b)^{2}b^{2}+b^{4}-2(a+b)b^{3}\Big{)}|\ln\varepsilon|^{4}+O(|\ln\varepsilon|^{3})$
$\displaystyle=\frac{(\mu\pi)^{4}}{(\kappa_{1}+\kappa)^{4}}a^{2}b^{2}|\ln\varepsilon|^{4}+O(|\ln\varepsilon|^{3}).$
(4.3)

Combining with the fact that $a,b\textgreater 0$, we derive

$\displaystyle\det\mathbb{A}$
$\displaystyle=a_{33}a_{44}\Big{(}a_{11}a_{22}a_{55}a_{66}+a_{15}^{2}a_{26}^{2}-a_{15}a_{22}a_{51}a_{66}-a_{11}a_{26}a_{55}a_{62}\Big{)}+O\Big{(}\frac{|\ln\varepsilon|^{3}}{\varepsilon}\Big{)}$
$\displaystyle\geq\frac{|\ln\varepsilon|^{4}}{C\varepsilon}.$ (4.4)

Thus, $\mathbb{A}$ is invertible, and

$C^{\alpha}=\frac{\det\mathbb{A}_{\alpha}}{\det\mathbb{A}},$

where $\mathbb{A}_{\alpha}$ is a matrix after replacing the $\alpha$-th column
of $\mathbb{A}$ with $P$. Next we only prove the case of
${\bm{\varphi}}\in{\bf\Phi}_{1}$, since others are similar.

(a) If $l_{1}=0$, denote by $\mbox{cof}(\mathbb{A})_{ij}$ the cofactor of
$\mathbb{A}$. By making use of (1.13) and Cramer’s rule, we have

$\displaystyle C^{1}=\frac{1}{\det\mathbb{A}}\Big{(}$
$\displaystyle\mbox{cof}(\mathbb{A})_{11}Q_{1}[{\bm{\varphi}}]-\mbox{cof}(\mathbb{A})_{21}Q_{2}[{\bm{\varphi}}]+\mbox{cof}(\mathbb{A})_{31}Q_{3}[{\bm{\varphi}}]-\mbox{cof}(\mathbb{A})_{41}Q_{4}[{\bm{\varphi}}]$
$\displaystyle+\mbox{cof}(\mathbb{A})_{51}Q_{5}[{\bm{\varphi}}]-\mbox{cof}(\mathbb{A})_{61}Q_{6}[{\bm{\varphi}}]\Big{)}.$
(4.5)

From Lemma 4.4, (4.1), and (4.1), it follows that

$\displaystyle\mbox{cof}(\mathbb{A})_{11}=a_{22}a_{33}a_{44}a_{55}a_{66}-a_{26}a_{33}a_{44}a_{55}a_{62}+O(|\ln\varepsilon|^{2}\varepsilon^{-1}),$

and thus

$\displaystyle\frac{\mbox{cof}(\mathbb{A})_{11}}{\det\mathbb{A}}=\frac{\kappa_{1}-\kappa}{\mu\pi}\frac{1}{|\ln\varepsilon|}+O(|\ln\varepsilon|^{-2}).$
(4.6)

Similarly,

$\displaystyle|\mbox{cof}(\mathbb{A})_{21}|,|\mbox{cof}(\mathbb{A})_{61}|\leq
C|\ln\varepsilon|^{2}\varepsilon^{-1},~{}|\mbox{cof}(\mathbb{A})_{31}|\leq|\ln\varepsilon|^{3},~{}|\mbox{cof}(\mathbb{A})_{41}|\leq
C|\ln\varepsilon|^{3}\varepsilon^{-1},$ (4.7)

and

$\frac{\mbox{cof}(\mathbb{A})_{51}}{\det\mathbb{A}}=\frac{(\kappa_{1}-\kappa)(\kappa_{1}+\kappa)}{\mu\pi}\frac{1}{|\ln\varepsilon|}+O(|\ln\varepsilon|^{-2}).$
(4.8)

Substituting (4.6)–(4.8) into (4.1), we have

$\displaystyle
C^{1}=\frac{\kappa_{1}-\kappa}{\mu\pi}\left(Q_{1}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q_{5}[{\bm{\varphi}}]\right)\frac{1}{|\ln\varepsilon|}+O(|\ln\varepsilon|^{-1}).$
(4.9)

By using Lemma 4.4, we have

$Q_{1}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q_{5}[{\bm{\varphi}}]=\frac{\mu\pi}{\kappa_{1}-\kappa}|\ln\varepsilon|+O(1).$

This together with (4.9) implies that

$C^{1}=1+O(|\ln\varepsilon|^{-1}).$

Similary, using Lemmas 4.2–4.4, and (4.1), we obtain

$\displaystyle|C^{2}|$
$\displaystyle=\frac{1}{\det\mathbb{A}}\Big{(}-\mbox{cof}(\mathbb{A})_{12}Q_{1}[{\bm{\varphi}}]+\mbox{cof}(\mathbb{A})_{22}Q_{2}[{\bm{\varphi}}]-\mbox{cof}(\mathbb{A})_{32}Q_{3}[{\bm{\varphi}}]$
$\displaystyle\qquad\qquad\quad+\mbox{cof}(\mathbb{A})_{42}Q_{4}[{\bm{\varphi}}]-\mbox{cof}(\mathbb{A})_{52}Q_{5}[{\bm{\varphi}}]+\mbox{cof}(\mathbb{A})_{62}Q_{6}[{\bm{\varphi}}]\Big{)}$
$\displaystyle\leq\frac{C\varepsilon}{|\ln\varepsilon|^{4}}\Big{(}\frac{|\ln\varepsilon|^{3}}{\varepsilon}+|\ln\varepsilon|^{3}\Big{)}\leq\frac{C}{|\ln\varepsilon|}.$

The estimates of $|C^{\alpha}|$, $\alpha=3,\dots,6$, are proved similarly, we
thus omit the details.

(b) For $l_{1}\geq 1$, we only prove the estimate of $|C^{1}|$ for instance,
since other terms are proved in the same way. It follows from (4.1), Cramer’s
rule, (4.1), Lemmas 4.2–4.4 that

$\displaystyle|C^{1}|\leq\frac{C\varepsilon}{|\ln\varepsilon|^{4}}\Big{(}\frac{|\ln\varepsilon|^{3}}{\varepsilon}+|\ln\varepsilon|^{4}\Big{)}\leq\frac{C}{|\ln\varepsilon|}.$

The proof of Proposition 4.1 is finished. ∎

##  5\. Proof of Upper bounds and Lower bounds

In this section we prove the upper bounds of $(\nabla{\bf u},p)$ in Subsection
5.1 and the lower bounds in Subsection 5.2.

###  5.1. Upper bounds: Proof of Theorem 1.1

In this Subsection, we complete the proof of Theorem 1.1 by using Propositions
2.2, 3.1, and 4.1.

###### Proof of Theorem 1.1..

We only prove the case of ${\bm{\varphi}}\in{\bf\Phi}_{1}$ for instance, since
the case of ${\bm{\varphi}}\in{\bf\Phi}_{3}$ is the same.

(a1) If $l_{1}=0$, then we note that ${\bf u}_{1}+{\bf u}_{0}$ takes the same
value on the top and bottom boundaries of $\Omega_{2R}$. By using (3.1), and
the proof of Propositions 2.2 and 3.1, we find that in $\Omega_{R}$,

$|\nabla^{k_{1}}({\bf u}_{1}+{\bf u}_{0})|+|\nabla^{k_{2}}(p_{1}+p_{0})|\leq
C,\quad k_{1}=1,2,~{}k_{2}=0,1.$

This together with (1.6) yields

$|\nabla^{k_{1}}{\bf u}(x)|\leq|C^{1}-1||\nabla^{k_{1}}{\bf
u}_{1}(x)|+\sum_{\alpha=2}^{6}|C^{\alpha}||\nabla^{k_{1}}{\bf
u}_{\alpha}(x)|+C,$

and

$\nabla^{k_{2}}p(x)=(C^{1}-1)\nabla^{k_{2}}p_{1}(x)+\sum_{\alpha=2}^{6}C^{\alpha}\nabla^{k_{2}}p_{\alpha}(x)+\nabla^{k_{2}}(p_{1}+p_{0}).$
(5.1)

By applying Propositions 2.2, 3.1, and 4.1, we have

$\displaystyle|\nabla{\bf u}(x)|$
$\displaystyle\leq\frac{C}{|\ln\varepsilon|\delta(x^{\prime})}+C\varepsilon\left(\frac{1}{\delta(x^{\prime})}+\frac{|x^{\prime}|}{\delta^{2}(x^{\prime})}\right)+C\left(\frac{|x^{\prime}|}{\delta(x^{\prime})}+1\right)$
$\displaystyle\leq\frac{C(1+|\ln\varepsilon||x^{\prime}|)}{|\ln\varepsilon|\delta(x^{\prime})},$

and

$\displaystyle|\nabla^{2}{\bf
u}(x)|\leq\frac{C}{|\ln\varepsilon|\delta^{2}(x^{\prime})}+C\varepsilon\left(\frac{1}{\delta^{2}(x^{\prime})}+\frac{|x^{\prime}|}{\delta^{3}(x^{\prime})}\right)+\frac{C}{\delta(x^{\prime})}\leq\frac{C(1+|\ln\varepsilon||x^{\prime}|)}{|\ln\varepsilon|\delta^{2}(x^{\prime})}.$

In order to estimate $|p|$, we first denote

$q(x):=\sum_{\alpha=1}^{6}C^{\alpha}q_{\alpha}(x)+q_{0}(x),\quad(q)_{\Omega_{R}}:=\frac{1}{|\Omega_{R}|}\int_{\Omega_{R}}q(x)\mathrm{d}x,$

where $q_{\alpha}=p_{\alpha}-\bar{p}_{\alpha}$ and $q_{0}=p_{0}-\bar{p}_{0}$.
Then by using (5.1), Propositions 2.2, 3.1, and 4.1 again, we have

$\displaystyle|p(x)-(q)_{\Omega_{R}}|$
$\displaystyle\leq|C^{1}-1||p_{1}(x)-(q_{1})_{\Omega_{R}}|+\sum_{\alpha=2}^{6}|C^{\alpha}||p_{\alpha}-(q_{\alpha})_{\Omega_{R}}|+C$
$\displaystyle\leq\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}},$ (5.2)

and

$\displaystyle|\nabla p(x)|\leq|C^{1}-1||\nabla
p_{1}(x)|+\sum_{\alpha=2}^{6}|C^{\alpha}||\nabla
p_{\alpha}|+C\leq\frac{C(1+|\ln\varepsilon||x^{\prime}|)}{|\ln\varepsilon|\delta^{2}(x^{\prime})}.$

(a2) If $l_{1}\geq 1$, using (1.9), Propositions 2.2, 3.1, and 4.1 again, we
obtain

$\displaystyle|\nabla{\bf u}(x)|\leq\sum_{\alpha=1}^{6}|C^{\alpha}||\nabla{\bf
u}_{\alpha}(x)|+|\nabla{\bf
u}_{0}(x)|\leq\frac{C(1+|\ln\varepsilon||x^{\prime}|)}{|\ln\varepsilon|\delta(x^{\prime})},$
$\displaystyle|p(x)-(q)_{\Omega_{R}}|\leq\begin{cases}\frac{C}{\varepsilon^{3/2}},&l_{1}=1,\\\
\frac{C}{|\ln\varepsilon|\varepsilon^{3/2}},&l_{1}\geq 2,\end{cases}$

and

$\displaystyle|\nabla^{2}{\bf u}(x)|+|\nabla
p(x)|\leq\frac{C(1+|\ln\varepsilon|\sqrt{\delta(x^{\prime})})}{|\ln\varepsilon|\delta^{2}(x^{\prime})}.$

∎

###  5.2. Lower bounds: Proof of Theorem 1.4

In this section, we are devoted to the proof of Theorem 1.4. Before proving
the main result, we give a lemma whose proof is a slight modification of that
in [39, Lemmas 7.2 and 7.3], and we omit the details here.

######  Lemma 5.1.

For $\beta=1,4,5$, as $\varepsilon\to 0$, we have

$Q_{1,\beta}[{\bm{\varphi}}]\to Q^{*}_{1,\beta}[{\bm{\varphi}}],$

where $Q_{1,\beta}[{\bm{\varphi}}]$ and $Q^{*}_{1,\beta}[{\bm{\varphi}}]$ are
defined in (1.18) and (1.19), respectively.

Using (1.14) and the integration by parts, we have

$\displaystyle a_{\alpha\beta}=\int_{\partial
D_{1}}{\bm{\psi}}_{\beta}\cdot\sigma[{\bf
u}_{\alpha}-{\bm{\psi}}_{\alpha},p_{\alpha}]\nu=\int_{\partial
D}{\bm{\psi}}_{\alpha}\cdot\sigma[{\bf u}_{\beta},p_{\beta}]\nu.$

Similarly,

$\displaystyle a^{*}_{\alpha\beta}=\int_{\partial
D}{\bm{\psi}}_{\alpha}\cdot\sigma[{\bf u}_{\beta}^{*},p_{\beta}^{*}]\nu,$

where $a^{*}_{\alpha\beta}$ is defined in (1.20). Following the argument in
the proof of [39, Lemma 7.1], we get

$\displaystyle|{\bf u}_{4}-{\bf u}_{4}^{*}|\leq
C\varepsilon^{1/2}\quad\mbox{in}~{}V\setminus\mathcal{C}_{\varepsilon^{1/4}},$

where $V:=D\setminus\overline{D_{1}\cup D_{1}^{0}}$ and

$\mathcal{C}_{r}:=\left\\{x\in\mathbb{R}^{3}\big{|}|x^{\prime}|<r,~{}0\leq
x_{3}\leq\varepsilon+\kappa_{1}r^{2}\right\\},\quad r<R.$

######  Lemma 5.2.

For $\alpha=1,\dots,6$, as $\varepsilon\to 0$, we have

$a_{\alpha 4}\to a^{*}_{\alpha 4}.$

###### Proof of Theorem 1.4..

We have from the definition of $k(x)$ in (2.1) that

$k(0,0,\varepsilon/2)=0.$

It follows from (2.11) and (3.1) that

$\partial_{x_{3}}{\bf
v}_{1}^{(1)}(0,0,\varepsilon/2)=\frac{1}{\varepsilon},\quad\partial_{x_{3}}({\bf
v}_{0}^{1})^{(1)}(0,0,\varepsilon/2)=-\frac{1}{\varepsilon}.$

Moreover, from Propositions 2.2 and 3.1, it follows that

$\displaystyle|\partial_{x_{3}}{\bf
u}_{1}^{(1)}(0,0,\varepsilon/2)+\partial_{x_{3}}{\bf
u}_{0}^{(1)}(0,0,\varepsilon/2)|$ $\displaystyle\leq|\partial_{x_{3}}{\bf
v}_{1}^{(1)}(0,0,\varepsilon/2)+\partial_{x_{3}}({\bf
v}_{0}^{1})^{(1)}(0,0,\varepsilon/2)|$
$\displaystyle\quad+|\partial_{x_{3}}{\bf
w}_{1}^{(1)}(0,0,\varepsilon/2)+\partial_{x_{3}}{\bf
w}_{0}^{(1)}(0,0,\varepsilon/2)|\leq C.$ (5.3)

On the other hand, in view of Propositions 2.2 and 4.1, (2.3), (2.5), (2.7),
and (2.8), we obtain

$\left|\sum_{\alpha=2}^{6}C^{\alpha}\partial_{x_{3}}{\bf
u}_{\alpha}^{(1)}(0,0,\varepsilon/2)\right|\leq C.$

This, in combination with (1.9) and (5.2), leads to

$\displaystyle|\nabla{\bf u}(0,0,\varepsilon/2)|$
$\displaystyle=\left|\sum_{\alpha=1}^{6}C^{\alpha}\nabla{\bf
u}_{\alpha}(0,0,\varepsilon/2)+\nabla{\bf u}_{0}(0,0,\varepsilon/2)\right|$
$\displaystyle\geq\left|\sum_{\alpha=1}^{6}C^{\alpha}\partial_{x_{3}}{\bf
u}_{\alpha}^{(1)}(0,0,\varepsilon/2)+\partial_{x_{3}}{\bf
u}_{0}^{(1)}(0,0,\varepsilon/2)\right|$
$\displaystyle\geq|C^{1}\partial_{x_{3}}{\bf
u}_{1}^{(1)}(0,0,\varepsilon/2)+\partial_{x_{3}}{\bf
u}_{0}^{(1)}(0,0,\varepsilon/2)|-C$
$\displaystyle\geq|(C^{1}-1)\partial_{x_{3}}{\bf
u}_{1}^{(1)}(0,0,\varepsilon/2)|-C\geq\frac{|C^{1}-1|}{\varepsilon}-C.$ (5.4)

From (1.18) and (1.13), it follows that

$a_{1\beta}(C^{1}-1)+\sum_{\alpha=2}^{6}a_{\alpha\beta}C^{\alpha}=Q_{1,\beta}[{\bm{\varphi}}],$

where $Q_{1,\beta}[{\bm{\varphi}}]$ is defined in (1.18). By using Cramer’s
rule, (4.1), and (4.6)–(4.8), we obtain

$\displaystyle C^{1}-1$
$\displaystyle=\frac{1}{\det\mathbb{A}}\Big{(}\mbox{cof}(\mathbb{A})_{11}Q_{1,1}[{\bm{\varphi}}]-\mbox{cof}(\mathbb{A})_{21}Q_{1,2}[{\bm{\varphi}}]+\mbox{cof}(\mathbb{A})_{31}Q_{1,3}[{\bm{\varphi}}]$
$\displaystyle\qquad\qquad-\mbox{cof}(\mathbb{A})_{41}Q_{1,4}[{\bm{\varphi}}]+\mbox{cof}(\mathbb{A})_{51}Q_{1,5}[{\bm{\varphi}}]-\mbox{cof}(\mathbb{A})_{61}Q_{1,6}[{\bm{\varphi}}]\Big{)}$
$\displaystyle=\frac{\kappa_{1}-\kappa}{\mu\pi}\left(Q_{1,1}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q_{1,5}[{\bm{\varphi}}]-\frac{a_{14}-(\kappa_{1}+\kappa)a_{54}}{a_{44}}Q_{1,4}[{\bm{\varphi}}]\right)\frac{1}{|\ln\varepsilon|}$
$\displaystyle\quad+O(|\ln\varepsilon|^{-2}).$ (5.5)

If
$Q^{*}_{1,1}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q^{*}_{1,5}[{\bm{\varphi}}]-\frac{a^{*}_{14}-(\kappa_{1}+\kappa)a^{*}_{54}}{a^{*}_{44}}Q^{*}_{1,4}[{\bm{\varphi}}]\neq
0$, then by using Lemmas 5.1 and 5.2, there exists a small enough constant
$\varepsilon_{0}>0$ such that for $0<\varepsilon<\varepsilon_{0}$,

$\displaystyle|Q_{1,1}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q_{1,5}[{\bm{\varphi}}]-\frac{a_{14}-(\kappa_{1}+\kappa)a_{54}}{a_{44}}Q_{1,4}[{\bm{\varphi}}]|$
$\displaystyle\geq\frac{1}{2}|Q^{*}_{1,1}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q^{*}_{1,5}[{\bm{\varphi}}]-\frac{a^{*}_{14}-(\kappa_{1}+\kappa)a^{*}_{54}}{a^{*}_{44}}Q^{*}_{1,4}[{\bm{\varphi}}]|>0.$

Thus, from (5.2), we have

$\displaystyle|C^{1}-1|\geq\frac{|Q^{*}_{1,1}[{\bm{\varphi}}]-(\kappa_{1}+\kappa)Q^{*}_{1,5}[{\bm{\varphi}}]-\frac{a^{*}_{14}-(\kappa_{1}+\kappa)a^{*}_{54}}{a^{*}_{44}}Q^{*}_{1,4}[{\bm{\varphi}}]|}{C}\frac{1}{|\ln\varepsilon|}.$

In view of (5.2), we have

$\displaystyle|\nabla{\bf
u}(0,0,\varepsilon/2)|\geq\frac{1}{\varepsilon|\ln\varepsilon|}.$

The proof of Theorem 1.4 is complete. ∎

##  6\. Ellipsoid suspending particle case

In this section, we show that the boundary gradient estimates in Theorem 1.1
hold also for the ellipsoid inclusion case. We assume that $D_{1}$ is an
ellipsoid, and near the origin, the part of $\partial D_{1}$ can be
represented by

$x_{3}=\varepsilon+h_{1}(x^{\prime}),\quad\mbox{where}~{}h_{1}(x^{\prime})=\kappa_{1}x_{1}^{2}+\kappa_{2}x_{2}^{2}.$

We replace $\delta(x^{\prime})$ by

$\delta(x^{\prime})=\varepsilon+(\kappa_{1}-\kappa)x_{1}^{2}+(\kappa_{2}-\kappa)x_{2}^{2}.$
(6.1)

The proof is similar to the previous ones, we will only list the main
differences in the following.

From the argument in Subsection 1.2, it follows that the key point is to
construct auxiliary functions with the same boundary conditions as ${\bf
u}_{\alpha}$ and ${\bf u}_{0}$ defined in (1.7) and (1.8), respectively. For
this, we seek ${\bf v}_{1}\in C^{2}(\Omega;\mathbb{R}^{3})$, such that ${\bf
v}_{1}={\bf u}_{1}={\bm{\psi}}_{1}$ on $\partial{D}_{1}$ and ${\bf v}_{1}={\bf
u}_{1}=0$ on $\partial{D}$,

$\displaystyle{\bf
v}_{1}=\bm{\psi}_{1}\big{(}k(x)+\frac{1}{2}\big{)}+\big{(}{\bf
E}_{1}^{(1)},{\bf E}_{1}^{(2)},{\bf
E}_{1}^{(3)}\big{)}^{\mathrm{T}}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}\quad\mbox{in}~{}\Omega_{2R},$

where

$\displaystyle{\bf E}_{1}^{(1)}$
$\displaystyle=-\frac{12(\kappa_{1}^{2}-\kappa^{2})}{(3\kappa_{1}+2\kappa_{2}-5\kappa)}\frac{x_{1}^{2}}{\delta(x^{\prime})}+\frac{3(\kappa_{1}+\kappa)}{3\kappa_{1}+2\kappa_{2}-5\kappa},$
$\displaystyle{\bf E}_{1}^{(2)}$
$\displaystyle=-\frac{12(\kappa_{1}+\kappa)(\kappa_{2}-\kappa)}{(3\kappa_{1}+2\kappa_{2}-5\kappa)}\frac{x_{1}x_{2}}{\delta(x^{\prime})},$
$\displaystyle{\bf E}_{1}^{(3)}$
$\displaystyle=-\delta(x^{\prime})\partial_{x_{1}}k(x){\bf
E}_{1}^{(1)}+\delta(x^{\prime})\partial_{x_{2}}k(x){\bf
E}_{1}^{(2)}+(\kappa_{1}-\kappa)x_{1}+2(\kappa_{1}+\kappa)x_{1}k(x),$

$k(x)$ is defined in (2.1) with $\delta(x^{\prime})$ replaced by (6.1), and
$\|{\bf v}_{1}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq\,C$. We choose
$\bar{p}_{1}\in C^{1}(\Omega)$ such that

$\bar{p}_{1}=\frac{6(\kappa_{1}+\kappa)}{(3\kappa_{1}+2\kappa_{2}-5\kappa)}\frac{\mu
x_{1}}{\delta^{2}(x^{\prime})}+\mu\partial_{x_{3}}({\bf
v}_{1})^{(3)}\quad\mbox{in}~{}\Omega_{2R}.$

It is easy to see that if $\kappa_{2}=\kappa_{1}$, then ${\bf v}_{1}$ and
$\bar{p}_{1}$ here are the same as that in (2.3) and (2.4) in Section 2,
respectively. The construction for the case of $\alpha=2,3,5,6$ is a slight
modification as before. Thus, using the iteration technique presented in
Section 2, Proposition 2.2 also holds except for $\alpha=4$.

Next we construct ${\bf v}_{4}\in C^{2}(\Omega;\mathbb{R}^{3})$ satisfying,

$\displaystyle{\bf
v}_{4}=\bm{\psi}_{4}\big{(}k(x)+\frac{1}{2}\big{)}+\big{(}-3x_{2},3x_{1},H_{4}\big{)}^{\mathrm{T}}\Big{(}k^{2}(x)-\frac{1}{4}\Big{)}\quad\mbox{in}~{}\Omega_{2R},$

where

$H_{4}=-(\kappa_{2}-\kappa_{1})(2k(x)+1)x_{1}x_{2}+\delta(x^{\prime})\Big{(}x_{2}\partial_{x_{1}}k(x)-x_{1}\partial_{x_{2}}k(x)\Big{)}.$

We choose $\bar{p}_{4}=0$. By a direct calculation, we obtain in
$\Omega_{2R}$,

$\displaystyle\nabla\cdot{\bf v}_{4}=0,\quad|\nabla{\bf
v}_{4}|\leq\frac{|x^{\prime}|}{\delta(x^{\prime})},\quad|\mu\Delta{\bf
v}_{4}-\nabla\bar{p}_{4}|\leq\frac{C|x^{\prime}|}{\delta^{2}(x^{\prime})}.$

Then by applying the energy method and the iteration technique, instead of the
result of $\alpha=4$ in Proposition 2.2, we have the following result.

######  Proposition 6.1.

Let ${\bf u}_{4}\in{C}^{2}(\Omega;\mathbb{R}^{3}),~{}p_{4}\in{C}^{1}(\Omega)$
be the solution to (1.7). Then we have for $x\in\Omega_{R}$,

$\|\nabla({\bf u}_{4}-{\bf
v}_{4})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\frac{C}{\sqrt{\delta(x^{\prime})}},$

and

$\|\nabla^{2}({\bf u}_{4}-{\bf
v}_{4})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}+\|q_{4}-(q_{4})_{\Omega_{R}}\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\frac{C}{\delta^{3/2}(x^{\prime})}.$

Consequently, in $\Omega_{R}$,

$\displaystyle|\nabla{\bf
u}_{4}(x)|\leq\frac{C}{\sqrt{\delta(x^{\prime})}},~{}|p_{4}(x)-(q_{4})_{\Omega_{R}}|\leq\frac{C}{\varepsilon^{3/2}},~{}|\nabla
p_{4}(x)|+|\nabla^{2}{\bf u}_{4}(x)|\leq\frac{C}{\delta^{3/2}(x^{\prime})}.$

Therefore, with the preparations above, by mimicking the process in the
previous sections, the boundary gradient estimates in Theorem 1.1 hold for the
ellipsoid inclusion and the corresponding estimates for $|p|$ can also be
obtained. We omit the details here.

##  7\. Proof of Theorem 1.7

This section is devoted to the proof of Theorem 1.7. From (1.6), one can see
that it suffices to estimate $C^{\alpha}$, $(\nabla{\bf
u}_{\alpha},p_{\alpha})$, and $(\nabla{\bf u}_{0},p_{0})$, where
$\alpha=1,\dots,\frac{d(d+1)}{2}$. To derive the estimates of $(\nabla{\bf
u}_{\alpha},p_{\alpha})$, we construct ${\bf v}_{\alpha}\in
C^{2}(\Omega;\mathbb{R}^{d})$, such that ${\bf v}_{\alpha}={\bf
u}_{\alpha}={\bm{\psi}}_{\alpha}$ on $\partial{D}_{1}$ and ${\bf
v}_{\alpha}={\bf u}_{\alpha}=0$ on $\partial{D}$, especially,

$\displaystyle{\bf
v}_{\alpha}=\bm{\psi}_{\alpha}\big{(}k(x)+\frac{1}{2}\big{)}+{\bf
E}_{\alpha}(x)\Big{(}k^{2}(x)-\frac{1}{4}\Big{)},\quad\alpha=1,\dots,d-1,\quad\mbox{in}~{}\Omega_{2R},$

and $\|{\bf v}_{\alpha}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq\,C$, where
$k(x)$ is defined in (2.1) with $x_{d}$ in place of $x_{3}$,

$\displaystyle{\bf E}_{\alpha}(x)$
$\displaystyle=F_{\alpha}(x)e_{\alpha}+\sum_{i\neq\alpha,d}G_{i}(x)e_{i}$
$\displaystyle\quad+\big{(}H_{\alpha}(x)-\sum_{i\neq\alpha,d}\delta(x^{\prime})\partial_{x_{i}}k(x)G_{i}(x)-\delta(x^{\prime})\partial_{x_{\alpha}}k(x)F_{\alpha}(x)\big{)}e_{d},$

here $\delta(x^{\prime})$ is defined in (1.15),

$\displaystyle\begin{split}F_{\alpha}(x)&:=\frac{12}{1-2d}\frac{(\kappa_{1}+\kappa)x_{\alpha}^{2}}{\delta(x^{\prime})}-\frac{3}{1-2d}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa},\quad
G_{i}(x):=\frac{12}{1-2d}\frac{(\kappa_{1}+\kappa)x_{\alpha}x_{i}}{\delta(x^{\prime})},\\\
H_{\alpha}(x)&:=(\kappa_{1}-\kappa)x_{\alpha}+2(\kappa_{1}+\kappa)x_{\alpha}k(x).\end{split}$

We choose $\bar{p}_{\alpha}\in C^{1}(\Omega)$ such that

$\bar{p}_{\alpha}=\frac{6\mu}{2d-1}\frac{\kappa_{1}+\kappa}{\kappa_{1}-\kappa}\frac{x_{\alpha}}{\delta^{2}(x^{\prime})}+\mu\partial_{x_{d}}{\bf
v}_{\alpha}^{(d)},\quad\alpha=1,\dots,d-1,\quad\mbox{in}~{}\Omega_{2R},$

and $\|\bar{p}_{\alpha}\|_{C^{1}(\Omega\setminus\Omega_{R})}\leq C$.

For $\alpha=d$, we seek

$\displaystyle{\bf v}_{d}=\bm{\psi}_{d}\big{(}k(x)+\frac{1}{2}\big{)}+{\bf
E}_{d}(x)\Big{(}k^{2}(x)-\frac{1}{4}\Big{)},\quad\mbox{in}~{}\Omega_{2R},$

and $\|{\bf v}_{d}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq\,C$, where

$\displaystyle{\bf
E}_{d}(x)=\sum_{i=1}^{d-1}\frac{6x_{i}}{(d-1)\delta(x^{\prime})}e_{i}+\big{(}-2k(x)+\sum_{i=1}^{d-1}\frac{6x_{i}}{(d-1)\delta(x^{\prime})}\tilde{H}_{i}(x)\big{)}e_{d},$

with

$\displaystyle\tilde{H}_{i}=(\kappa_{1}+\kappa)x_{i}+2(\kappa_{1}-\kappa)x_{i}k(x),$

and the associated $\bar{p}_{d}\in C^{1}(\Omega)$ satisfying, in
$\Omega_{2R}$,

$\bar{p}_{d}=-\frac{3\mu}{(d-1)(\kappa_{1}-\kappa)\delta^{2}(x^{\prime})}+\mu\partial_{x_{d}}{\bf
v}_{d}^{(d)}.$

For $\alpha=d+1,\dots,\frac{d(d-1)}{2}$, the construction of ${\bf v}_{4}\in
C^{2}(\Omega;\mathbb{R}^{d})$ is as follows:

$\displaystyle{\bf
v}_{\alpha}=\bm{\psi}_{\alpha}\big{(}k(x)+\frac{1}{2}\big{)}\quad\mbox{in}~{}\Omega_{2R},$

and here we can directly take $\bar{p}_{\alpha}=0$.

Finally, for $\alpha=\frac{d(d-1)}{2}+1,\dots,\frac{d(d+1)}{2},$ we define the
auxiliary function ${\bf v}_{\alpha}\in C^{2}(\Omega;\mathbb{R}^{d})$
satisfying

$\displaystyle{\bf
v}_{\alpha}=\bm{\psi}_{\alpha}\big{(}k(x)+\frac{1}{2}\big{)}+{\bf
E}_{\alpha}(x)\Big{(}k^{2}(x)-\frac{1}{4}\Big{)},\quad\mbox{in}~{}\Omega_{2R},$

and $\|{\bf v}_{\alpha}\|_{C^{2}(\Omega\setminus\Omega_{R})}\leq\,C$, where

$\displaystyle{\bf E}_{\alpha}(x)$
$\displaystyle=F_{\alpha}e_{\alpha}+\sum_{i\neq\alpha,d}\tilde{G}_{i}(x)e_{i}+\big{(}2k(x)x_{1}+\tilde{H}_{\frac{d(d-1)}{2}-1}(x)\Big{(}F_{\alpha}(x)+2k(x)x_{d}\Big{)}$
$\displaystyle\quad-\sum_{i\neq\alpha,d}\delta(x^{\prime})\partial_{x_{i}}k(x)\tilde{G}_{i}(x)\big{)}e_{d},$

with

$\displaystyle
F_{\alpha}(x)=\frac{12}{2d-1}\frac{x_{\alpha-\frac{d(d-1)}{2}-1}^{2}}{\delta(x^{\prime})}-2k(x)x_{3}-\frac{3x_{3}^{2}}{\delta(x^{\prime})}+\frac{3}{(2d-1)(\kappa_{1}-\kappa)},$

and

$\displaystyle\tilde{G}_{i}(x)=\frac{12}{1-2d}\frac{x_{i}x_{\alpha-\frac{d(d-1)}{2}-1}^{2}}{\delta(x^{\prime})}.$

Then we choose $\bar{p}_{\alpha}\in C^{1}(\Omega)$ in $\Omega_{2R}$ such that

$\bar{p}_{\alpha}=\frac{6\mu}{(2d-1)(\kappa_{1}-\kappa)}\frac{x_{\alpha-\frac{d(d-1)}{2}-1}}{\delta^{2}(x^{\prime})}+\mu\partial_{x_{d}}{\bf
v}_{\alpha}^{(d)},\quad\mbox{in}~{}\Omega_{2R}.$

Then we can obtain the following result by repeating the argument as in the
proof of Proposition 2.2.

######  Proposition 7.1.

Let ${\bf u}_{\alpha}\in{C}^{2}(\Omega;\mathbb{R}^{d})$,
$p_{\alpha}\in{C}^{1}(\Omega)$ be the solution to (1.7) with
$\alpha=1,\dots,\frac{d(d+1)}{2}$. Then in $\Omega_{R}$,

$\|\nabla({\bf u}_{\alpha}-{\bf
v}_{\alpha})\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\begin{cases}C,&\alpha=1,\dots,d-1,\frac{d(d-1)}{2}+2,\dots,\frac{d(d+1)}{2},\\\
\frac{C}{\sqrt{\delta(x^{\prime})}},&\alpha=d,\\\
C\sqrt{\delta(x^{\prime})},&\alpha=d+1,\dots,\frac{d(d-1)}{2}+1,\end{cases}$

and

$\|\nabla
q_{\alpha}\|_{L^{\infty}(\Omega_{\delta/2}(x^{\prime}))}\leq\begin{cases}\frac{C}{\delta(x^{\prime})},&\alpha=1,\dots,d-1,\frac{d(d-1)}{2}+2,\dots,\frac{d(d+1)}{2},\\\
\frac{C}{\delta^{3/2}(x^{\prime})},&\alpha=d,\\\
\frac{C}{\sqrt{\delta(x^{\prime})}},&\alpha=d+1,\dots,\frac{d(d-1)}{2}+1.\end{cases}$

Consequently, in $\Omega_{R}$,

$\displaystyle|\nabla{\bf
u}_{\alpha}(x)|\leq\begin{cases}\frac{C}{\delta(x^{\prime})},&\alpha=1,\dots,d-1,\frac{d(d-1)}{2}+2,\dots,\frac{d(d+1)}{2},\\\
C\left(\frac{1}{\delta(x^{\prime})}+\frac{|x^{\prime}|}{\delta^{2}(x^{\prime})}\right),&\alpha=d,\\\
C\left(\frac{|x^{\prime}|}{\delta(x^{\prime})}+1\right),&\alpha=d+1,\dots,\frac{d(d-1)}{2}+1,\end{cases}$

and

$|p_{\alpha}(x)-(q_{\alpha})_{\Omega_{R}}|\leq\begin{cases}\frac{C}{\varepsilon^{3/2}},&\alpha=1,\dots,d-1,\frac{d(d-1)}{2}+2,\dots,\frac{d(d+1)}{2},\\\
\frac{C}{\varepsilon^{2}},&\alpha=d,\\\
\frac{C}{\sqrt{\varepsilon}},&\alpha=d+1,\dots,\frac{d(d-1)}{2}+1,\end{cases}$

where $(q_{\alpha})_{\Omega_{R}}$ is defined in (2.19) with
$\alpha=1,\dots,\frac{d(d+1)}{2}$.

For the case of $\bm{\varphi}=x_{1}^{l}e_{1}$, Proposition 3.1 also holds for
$d\geq 4$. Now we are ready to complete the proof of Theorem 1.7.

###### Proof of Theorem 1.7.

Denote

$q(x):=\sum_{\alpha=1}^{\frac{d(d+1)}{2}}C^{\alpha}q_{\alpha}(x)+q_{0}(x),\quad(q)_{\Omega_{R}}:=\frac{1}{|\Omega_{R}|}\int_{\Omega_{R}}q(x)\mathrm{d}x,$

where $q_{\alpha}=p_{\alpha}-\bar{p}_{\alpha}$ and $q_{0}=p_{0}-\bar{p}_{0}$.
By (1.6), the boundedness of $C^{\alpha}$, Propositions 7.1 and 3.1, we have
(1.21). ∎

Acknowledgements. H.G. Li was partially supported by NSF of China (11971061).

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