# Singular velocity of the Stokes and Navier-Stokes equations near boundary in
the half space

###### Abstract.

Local behaviors near boundary are analyzed for solutions of the Stokes and
Navier-Stoke equations in the half space with localized non-smooth boundary
data. We construct solutions of Stokes equations whose velocity field is not
bounded near boundary away from the support of boundary data, although
velocity and gradient velocity of solutions are locally square integrable.
This is an improvement compared to known results in the sense that velocity
field is unbounded itself, since previously constructed solutions were bounded
near boundary, although their normal derivatives are singular. We also
establish singular solutions and their derivatives that do not belong to
$L^{q}_{\rm{loc}}$ near boundary with $q>1$. For such examples, there
corresponding pressures turn out not to be locally integrable. Similar
construction via a perturbation argument is available to the Navier-Stokes
equations near boundary as well.

2020 Mathematics Subject Classification. primary 35Q30, secondary 35B65.

Keywords and phrases: Singualr velocity Stokes and Navier-Stokes equations,
local regularity near boundary

Tongkeun Chang 111Department of Mathematics, Yonsei University Seoul, 03722,
South Korea<EMAIL_ADDRESS>and Kyungkeun Kang 222Department of
Mathematics, Yonsei University Seoul, 03722, South Korea<EMAIL_ADDRESS>
##  1\. Introduction

In this paper, we study local regularity of the Stokes and the Navier-Stokes
equations near flat boundary. Let
$B^{+}_{r}:=\\{x=(x^{\prime},x_{n})\in{\mathbb{R}}^{n-1}\times{\mathbb{R}}:|x|<r,x_{n}>0\\}$,
$n\geq 2$. We consider the Stokes system near local boundaries.

(1.1) $u_{t}-\Delta u+\nabla\pi=0\qquad\mbox{div }u=0\quad\mbox{ in
}\,Q_{2}^{+}:=B^{+}_{2}\times(0,4),$

where $u$ indicates the velocity of the fluid and $\pi$ is the associated
pressure. We emphasize that no-slip boundary condition is given only on the
flat boundary, i.e.

(1.2) $u=0\quad\mbox{ on }\,\Sigma:=(B_{2}\cap\\{x_{n}=0\\})\times(0,4).$

Similar situation can be examined for the Navier-Stokes equations, namely

(1.3) $u_{t}-\Delta u+(u\cdot\nabla)u+\nabla\pi=0,\qquad\mbox{div
}u=0\quad\mbox{ in }\,Q_{2}^{+}$

together with the boundary condition (1.2).

Unlike the heat equation, due to non-local effect of incompressible fluid
flow, we cannot expect local smoothing property for (1.1) or (1.3) near
boundary. Indeed, the first author constructed a weak solution of (1.1) whose
normal derivatives near boundary are unbounded. To be more precise, the
singular solution, denoted by $u$, constructed in [11] is locally bounded weak
solutions in $Q_{1}^{+}$, namely,

(1.4)
$\left\|u\right\|_{L^{\infty}(Q_{1}^{+})}+\left\|u\right\|_{L^{2}(Q_{1}^{+})}<\infty,\qquad\left\|u\right\|_{L^{\infty}(Q_{1}^{+})}=\infty.$

In [12], such construction was extended to the Navier-Stokes equations as well
and various types of blow-up rates were established near boundary. We remark
that constructed singular solutions in [11] and [12] are due to nonlocal
effect of solutions in the half-space caused by compactly supportd singular
boundary data.

Meanwhile, in [15], Seregin and S̆verák constructed a Hölder continuous
solution up to boundary in the form of shear flow whose normal derivatives
are, however, unbounded near boundary in the half-space. It is remarkable that
such a solution satisfies homogeneous initial and boundary conditions and the
solution, however, is not of finite energy in the half space in contrast to
ones constructed in [11] and [12].

On the other hand, a certain type of local maximal regularity was proved near
boundary via a priori estimate. More precisely, in [14, Proposition 2] it was
shown in three dimensions that for given $p,q\in(1,2]$ the solution of the
Stokes equations (1.1)-(1.2) satisfies the following a priori estimate: For
any $r$ with $p\leq r<\infty$

(1.5)
$\left\|u_{t}\right\|_{L^{q}_{t}L^{r}_{x}(Q_{\frac{1}{4}}^{+})}+\left\|\nabla^{2}u\right\|_{L^{q}_{t}L^{r}_{x}(Q_{\frac{1}{4}}^{+})}+\left\|\nabla\pi\right\|_{L^{q}_{t}L^{r}_{x}(Q_{\frac{1}{4}}^{+})}\leq
C\left(\left\|u\right\|_{L^{q}_{t}W^{1,p}_{x}(Q_{1}^{+})}+\left\|\pi\right\|_{L^{q}_{t}L^{p}_{x}(Q_{1}^{+})}\right).$

Furthermore, due to parabolic embedding, it follows that

(1.6)
$\left\|u\right\|_{{\mathcal{C}}^{\frac{\alpha}{2}}_{t}{\mathcal{C}}^{\alpha}_{x}(Q_{\frac{1}{4}}^{+})}<\infty,\qquad
0<\alpha<2-\frac{2}{q}.$

It is worth to mention that the estimates (1.5) is optimal, since the authors
constructed in [8] an example that satisfies (1.5) but any higher
integrability of the example fails (see also [12]).

We remark very recently that local boundary regularity of solutions was
improved for the Stokes system, when boundary condition is given as Navier-
boundary condition, not no-slip boundary condition (see [10]).

So far, singular solutions that has been constrructed in above references are
bounded, in fact, Hölder continuous but the normal derivatives of velocity are
singular near boundary in the half space. One may wonder whether or not there
exists a singular solution that is unbounded itself near boundary. Our main
objective is to construct a weak solution of the Stokes and Navier-Stokes
equations sucht that velocity field is singular near boundary in the half
space (the definition of weak solutions is given in section 2). Furthermore,
we are also interested in establishing an example so that velocity fileld is
bounded but it does not belong to ${\mathcal{C}}^{\alpha}$, class of Hölder
continuous functions, with any exponent $\alpha>0$ near boundary. We remark
that since we are looking for solutions that are not Hölder continuous
functions itself, due to above a priori estimate (1.5), the associated
pressure is unlikely to belong to $L^{p}_{\rm loc}$, $p>1$, which turns out to
be the case in examples of our construction. Our main resutls will be
precisely stated in the next section.

##  2\. Main results

For convenience, we introduce a tensor $L_{ij}$ defined by

(2.1) $\displaystyle L_{ij}(x,t)$
$\displaystyle=D_{x_{j}}\int_{0}^{x_{n}}\int_{{\mathbb{R}}^{n-1}}D_{z_{n}}\Gamma(z,t)D_{x_{i}}N(x-z)dz,\quad
i,j=1,2,\cdots,n,$

where $\Gamma$ and $N$ are the $n$ dimensional Gaussian kernel and Newtonian
kernel defined by

(2.6)
$\displaystyle\Gamma(x,t)=\left\\{\begin{array}[]{cc}\vspace{2mm}\frac{1}{(4\pi
t)^{\frac{n}{2}}}e^{-\frac{|x|^{2}}{4t}}&\quad t>0,\\\ 0&\quad
t<0,\end{array}\right.\quad
N(x)=\left\\{\begin{array}[]{ll}\vspace{2mm}-\frac{1}{n(n-2)\omega_{n}}|x|^{-n+2}&\quad
n\geq 3,\\\ \frac{1}{2\pi}\ln|x|&\quad n=2.\end{array}\right.$

It is known that $L_{ij}$ satisfies following relations:

(2.7) $\displaystyle\sum_{i=1}^{n}L_{ii}=\frac{1}{2}D_{x_{n}}\Gamma,\quad
L_{ij}=L_{ji}\,\,1\leq i,j\leq n-1,\quad L_{in}=L_{ni}+B_{in}\,\,\mbox{ if
}\,i\neq n,$

where

(2.8)
$B_{in}(x,t)=\int_{{\mathbb{R}}^{n-1}}D_{x_{n}}\Gamma(x^{\prime}-z^{\prime},x_{n},t)D_{z_{i}}N(z^{\prime},0)dz^{\prime}.$

Furthermore, we remind an estimate of $L_{ij}$ defined in (2.1) (see (2.6) in
[16])

(2.9)
$|D^{l_{0}}_{x_{n}}D^{k_{0}}_{x^{\prime}}L_{ij}(x,t)|\leq\frac{c}{t^{\frac{1}{2}}(|x|^{2}+t)^{\frac{1}{2}n+\frac{1}{2}k_{0}}(x_{n}^{2}+t)^{\frac{1}{2}l_{0}}},$

where $1\leq i\leq n$ and $1\leq j\leq n-1$.

We recall the following Stokes equations in a half space:

(2.10) $\displaystyle\begin{split}&w_{t}-\Delta w+\nabla p=0,\qquad\mbox{div
}\,w=0\quad\mbox{ in }\,{\mathbb{R}}^{n}_{+}\times(0,\infty),\\\
&\qquad\quad\displaystyle w|_{t=0}=0,\qquad w|_{x_{n}=0}=g.\end{split}$

It is known that $w$ is represented by

(2.11) $\displaystyle w_{i}(x,t)$
$\displaystyle=\sum_{j=1}^{n}\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}K_{ij}(x^{\prime}-y^{\prime},x_{n},t-s)g_{j}(y^{\prime},s)dy^{\prime}ds,$

where the Poisson kernel $K$ of the Stokes equations in
${\mathbb{R}}^{n}_{+}\times(0,\infty)$ is given as follows (see [16]):

(2.12)
$\displaystyle\begin{split}K_{ij}(x^{\prime}-y^{\prime},x_{n},t)&=-2\delta_{ij}D_{x_{n}}\Gamma(x^{\prime}-y^{\prime},x_{n},t)+4L_{ij}(x^{\prime}-y^{\prime},x_{n},t)\\\
&\qquad+2\delta_{jn}\delta(t)D_{x_{i}}N(x^{\prime}-y^{\prime},x_{n}).\end{split}$

The formula of the associated pressure $p$ is also given as

(2.13)
$\displaystyle\begin{split}p(x,t)=&-\sum_{k=1}^{k=n}\int_{{\mathbb{R}}^{n-1}}D_{x_{k}}D_{x_{n}}N(x^{\prime}-y^{\prime},x_{n})g_{k}(y^{\prime},t)dy^{\prime}\\\
&+2\sum_{k=1}^{k=n}(D_{t}-\Delta^{\prime})\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}g_{k}(z^{\prime},s)\int_{{\mathbb{R}}^{n-1}}D_{y_{k}}N(y^{\prime},x_{n})\Gamma(x^{\prime}-y^{\prime}-z^{\prime},0,t-s)dy^{\prime}dz^{\prime}ds\\\
&+N*^{\prime}D_{t}g_{n}(x,t),\end{split}$

We specify a boundary data
$g:\mathbb{R}^{n-1}\times\mathbb{R}_{+}\rightarrow{\mathbb{R}}^{n}$ with only
non-zero $n-$th component defined as follows:

(2.14) $\displaystyle
g(y^{\prime},s)=\alpha(0,\cdots,0,g_{n}(y^{\prime},s))=\alpha(0,\cdots,0,g^{{\mathcal{S}}}_{n}(y^{\prime})g^{{\mathcal{T}}}_{n}(s)),$

where $\alpha>0$ is a parameter decided later for the Navier-Stokes equations.
Here $g_{n}^{\mathcal{S}}\geq 0$ and $g_{n}^{\mathcal{T}}\geq 0$ are assumed
to satisfy

(2.15) $\displaystyle g^{\mathcal{S}}_{n}\in C^{\infty}_{c}(A),\qquad{\rm
supp}\,\,g_{n}^{\mathcal{T}}\subset(0,1),\quad g^{\mathcal{T}}_{n}\in
L^{1}({\mathbb{R}})\cap C^{\infty}([0,1)),$

where the set $A\subset{\mathbb{R}}^{n-1}$ is defined by

(2.16) $\displaystyle
A=\\{y^{\prime}=(y_{1},y_{2},\cdots,y_{n-1})\in{\mathbb{R}}^{n-1}\,|\,1<|y^{\prime}|<2,\,-2<y_{i}<-1,\,\,1\leq
i\leq n-1\\}.$

Here we consider the case of non-smooth $g^{\mathcal{T}}_{n}$, which is
supposed to be three kinds of prototype types. More precisely, we assume the
following for $g^{\mathcal{T}}_{n}$:

######  Assumption 2.1.

We suppose that $g^{\mathcal{T}}_{n}(s)=\phi(1-s)$, where $\phi$ satisfies

(2.17) $\displaystyle\phi\geq 0,\quad{\rm
supp}\,\,\phi\subset(0,1),\quad\phi\in L^{1}({\mathbb{R}})\cap
C^{\infty}((0,1]),\quad\phi(1)=0.$

Assume further that either

(2.18) $\displaystyle\phi(s)=\left|\ln
s\right|\quad\mbox{or}\quad\phi(s)=\left|\ln s\right|^{-1},\qquad
s\in(0,\frac{1}{2}]$

or $\phi$ satisfies

(2.19) $\displaystyle\phi^{(k)}(s)\approx\phi^{(k)}(2s),\quad\qquad
s\in(0,\frac{1}{4}],\,\,\,\,k=0,\,\,1.$

Here, $A\approx B$ means that there exists a constant $c>1$ such that
$c^{-1}B\leq A\leq cB$. Furthermore, we also assume that

(2.20)
$\displaystyle\displaystyle\int_{\epsilon}^{1}s^{-\alpha}\phi(s)ds\approx\max(1,\epsilon^{1-\alpha}\phi(\epsilon)),\quad
0<\alpha,\quad\alpha\neq 1.$ (2.21)
$\displaystyle\displaystyle\int_{\epsilon}^{1}s^{-\alpha}\phi^{\prime}(s)ds\approx\epsilon^{1-\alpha}\phi^{\prime}(\epsilon),\quad
0<\alpha\quad\alpha\neq 1.$ (2.22)
$\displaystyle\displaystyle\int_{0}^{\epsilon}\phi(s)ds\approx\epsilon\phi(\epsilon).$

Some comments are prepared for the Assumption 2.1.

  * (i)

In case that $\frac{3}{4}<t<1$, if $\phi$ satisfies (2.19), then one can see
that

(2.23) $\displaystyle\phi^{(k)}(t-s)\approx\phi^{(k)}(1-s)\quad\mbox{ for
}\quad\frac{1}{2}<s<2t-1\quad k=0,1,$ (2.24)
$\displaystyle\phi^{(k)}(1-t)\approx\phi^{(k)}(1-s)\quad\mbox{ for }\quad
2t-1<s<t\quad k=0,1,$

since $\frac{1}{2}(1-s)\leq t-s\leq(1-s)$ for $\frac{1}{2}<s<2t-1$ and
$\frac{1}{2}(1-s)\leq 1-t\leq(1-s)$ for $2t-1<s<t$. Similarly, if
$1<t<\frac{5}{4}$, it is straightforward that

(2.25) $\displaystyle\phi^{(k)}(t-s)\approx\phi^{(k)}(1-s)\quad\mbox{ for
}\quad\frac{3}{4}<s<2-t\quad k=0,1,$ (2.26)
$\displaystyle\phi^{(k)}(t-1)\approx\phi^{(k)}(t-s)\quad\mbox{ for }\quad
2-t<s<1\quad k=0,1.$

This is also due to facts that $(1-s)\leq t-s\leq 2(1-s)$ and
$\frac{1}{2}(t-s)\leq t-1\leq(t-s)$ for $0<s<2-t$ and $2-t<s<1$, respectively.

  * (ii)

One can check that $\phi(s)=\left|\ln s\right|$ or $\phi(s)=\left|\ln
s\right|^{-1}$ satisfy (2.20)-(2.22). We note that $\phi(t)=t^{-a}$ with
$a\in(0,1)$, $a+\alpha\neq 1$ is another typical type that we will treat in
this paper.

  * (iii)

We note that $\phi(t)=t^{-a}\eta(t)$ is a prototype in Assumption 2.1, where
$\eta$ is cut off fuction with $\eta(t)=1$ for $t\in(0,\frac{1}{2})$ and
$\eta(t)=0$ for $t\geq\frac{3}{4}$.

As a notational convention, we denote $\rm{sgn}\,(a)=1$ if $a>0$ and
$\rm{sgn}\,(a)=-1$ if $a<0$.

Next lemma, some elementary properties of $\Phi$ are stated and, for clarity,
its proof will be given in section 3.

######  Lemma 2.2.

Let $0<t<\frac{1}{4}$ and $\phi$ satisfy the Assumption 2.1. Then, for
$0<x_{n}<\frac{1}{4}$ and $\sqrt{t}<x_{n}$,

(2.27) $\displaystyle\phi(t)e^{-\delta_{1}\frac{x_{n}^{2}}{t}}\leq
c\phi(x_{n}^{2}),\quad{\rm
sgn}(\phi^{\prime})\phi^{\prime}(t)e^{-\delta_{1}\frac{x_{n}^{2}}{t}}\leq
c\,{\rm sgn}(\phi^{\prime})\phi^{\prime}(x_{n}^{2}).$

In the case $\phi=t^{-a}$, there is a large constant $\epsilon_{1}\gg 1$ such
that if $\epsilon_{1}\sqrt{t}<x_{n}$, then there exists there a sufficiently
small $\epsilon$ depending on $\epsilon_{1}$ such that

(2.28) $\displaystyle
t^{-\frac{1}{2}}e^{-\delta_{1}\frac{x_{n}^{2}}{t}}\phi(t)\leq\epsilon
x_{n}\,{\rm sgn}(\phi^{\prime})\phi^{\prime}(x_{n}^{2}).$

From (2.11), (2.12), and $\eqref{1006-3}_{3}$, for $1\leq i\leq n$, we have

(2.29)
$\displaystyle\begin{split}w_{i}(x,t)&=4\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}L_{ni}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds\\\
&\quad+4(1-\delta_{in})\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}B_{in}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds\\\
&\quad+2\int_{{\mathbb{R}}^{n-1}}D_{x_{i}}N(x^{\prime}-y^{\prime},x_{n})g_{n}(y^{\prime},t)dy^{\prime}\\\
&:=w^{L}_{i}(x,t)+w^{B}_{i}(x,t)+w^{N}_{i}(x,t).\end{split}$

Form now on, for simplicity, we denote by $\Gamma^{\prime}$ the $n-1$
dimensional Gaussian kernel.

######  Definition 2.3.

As a notational convention, we denote $A\vee B:=\max\left\\{A,B\right\\}$.
Further, for a positive function $g$, we say that $f\approx\pm e^{-\delta g}$
if $\pm c^{-1}e^{-g(x)}\leq f(x)\leq\pm ce^{-\frac{1}{8}g(x)}$ for some
constant $c>0$.

######  Lemma 2.4.

Let $\Gamma^{\prime}$ and $N$ be $n-1$ dimensional Gaussian kernel and $n$
dimensional Newtonian kernel defined in (2.6). Let
$X^{\prime}\in{\mathbb{R}}^{n-1}$ with $|X^{\prime}|\geq 1$,
$0<x_{n}<\frac{1}{2}$ and $t>0$. Then, there exists $c>0$, independent of
$X^{\prime},x_{n}$ and $t$, such that

(2.30)
$\displaystyle\begin{split}\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(X^{\prime}-z^{\prime},t)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}&=D_{x_{n}}N(X^{\prime},x_{n})+I(X^{\prime},x_{n},t)+J_{1}(X^{\prime},x_{n},t)\quad
k=0,1,\\\
\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(X^{\prime}-z^{\prime},t)D_{z_{i}}N(z^{\prime},0)dz^{\prime}&=D_{X_{i}}N(X^{\prime},0)+J_{2}(X^{\prime},t),\end{split}$

where $i=1,2,\cdots,n-1$ and $I$, $J_{1}$ and $J_{2}$ satisfy

(2.31) $\displaystyle\begin{split}I(x^{\prime},x_{n},t)\approx
t^{-\frac{n-1}{2}}e^{-\delta\frac{|x^{\prime}|^{2}}{t}},\quad|J_{1}(X^{\prime},x_{n},t)|&\leq
cx_{n}t^{\frac{1}{2}},\qquad\left|J_{2}(X^{\prime},t)\right|\leq
ct^{\frac{1}{2}}\quad m\geq 1,\end{split}$

where ${}^{\prime\prime}\approx^{\prime\prime}$ and $\delta$ are defined in
Definition 2.3.

The proof of Lemma 2.4 will be provided in section 3.

######  Remark 2.5.

Let $i=1,2,\cdots,n-1$. Suppose that
$x^{\prime},y^{\prime}\in{\mathbb{R}}^{n-1}$ with $|x^{\prime}|<\frac{1}{2}$
and $y^{\prime}\in A$ and, for convenience, we denote
$X^{\prime}=x^{\prime}-y^{\prime}$. Then, it is direct that
$\frac{1}{2}<\left|X^{\prime}\right|\leq\frac{5}{2}$, because $\frac{1}{2}\leq
x_{i}-y_{i}$ and $|X^{\prime}|\leq(|x^{\prime}|+|y^{\prime}|)\leq\frac{5}{2}$,
which implies that there are positive constants $c_{1}$ and $c_{2}$ such that

$c_{1}\leq D_{x_{i}}N(x^{\prime}-y^{\prime},0)\leq c_{2},\qquad c_{1}x_{n}\leq
D_{x_{n}}N(x^{\prime}-y^{\prime},x_{n})\leq c_{2}x_{n}.$

Therefore, it follows from $\eqref{0515-1}$ and $\eqref{Jkl}$ that

(2.32)
$\displaystyle\left|\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t)D_{z_{i}}N(z^{\prime},0)dz^{\prime}\right|\leq
c,\qquad|x^{\prime}|\leq\frac{1}{2},\,\,\,y^{\prime}\in A,$ (2.33)
$\displaystyle\left|\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}\right|\leq
c\Big{(}x_{n}+t^{-\frac{n-1}{2}}e^{-\frac{1}{8t}}\Big{)},\qquad|x^{\prime}|\leq\frac{1}{2},\,\,\,y^{\prime}\in
A,$

where $\frac{3}{4}<t<\frac{5}{4}$, $0<s<t$ and $k\geq 0$. Furthermore, there
exist $c_{0}>0$ and $t_{0}\in(0,\frac{1}{4})$ such that if $0<t-s<t_{0}$, then

(2.34) $\displaystyle
c_{0}\leq\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{z_{i}}N(z^{\prime},0)dz^{\prime}\qquad|x^{\prime}|\leq\frac{1}{2},\,\,\,y^{\prime}\in
A,$ (2.35) $\displaystyle
c_{0}(x_{n}+t^{-\frac{n-1}{2}}e^{-\frac{1}{t}})\leq\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime},\qquad|x^{\prime}|\leq\frac{1}{2},\,\,\,y^{\prime}\in
A.$

Indeed, the estimate (2.35) is due to the computations that

$\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}$
$\geq
D_{x_{n}}N(X^{\prime},x_{n})+t^{-\frac{n-1}{2}}e^{-\frac{1}{t}}-\left|J_{1}(x^{\prime},x_{n},t-s)\right|\geq
c(x_{n}+t^{-\frac{n-1}{2}}e^{-\frac{1}{t}}).$

In conclusion, due to (2.32)-(2.35), we obtain

(2.36)
$\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{z_{i}}N(z^{\prime},0)dz^{\prime}\approx
c,\qquad|x^{\prime}|\leq\frac{1}{2},\,\,\,y^{\prime}\in A,$ (2.37)
$\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}\approx
x_{n}+t^{-\frac{n-1}{2}}e^{-\frac{1}{t}},\qquad|x^{\prime}|\leq\frac{1}{2},\,\,\,y^{\prime}\in
A.$

∎

We define for notational convention

${\mathcal{D}}_{+}=\left\\{(x,t)\in\mathbb{R}^{n}_{+}\times{\mathbb{R}}^{n}_{+}:|t-1|^{\frac{1}{2}}\geq
x_{n},\,\,\left|t-1\right|<t_{0},\,\,0<x_{n}<\frac{1}{2}\right\\},$
${\mathcal{D}}_{-}=\left\\{(x,t)\in\mathbb{R}^{n}_{+}\times{\mathbb{R}}^{n}_{+}:|t-1|^{\frac{1}{2}}<x_{n},\,\,\left|t-1\right|<t_{0},\,\,0<x_{n}<\frac{1}{2}\right\\},$
${\mathcal{D}}_{0}=\left\\{(x,t)\in\mathbb{R}^{n}_{+}\times{\mathbb{R}}^{n}_{+}:\frac{1}{2}<\frac{x_{n}}{|t-1|^{\frac{1}{2}}}<1,\,\,\left|t-1\right|<t_{0},\,\,0<x_{n}<\frac{1}{2}\right\\}.$

Here $t_{0}>0$ is given in Remark 2.5 and $t_{1}<t_{0}$ will be determined
later. We also introduce two different regions defined by

(2.38)
$\displaystyle\begin{split}\tilde{{\mathcal{D}}}_{+}&={\mathcal{D}}_{+}\cap\left\\{(x,t):x_{n}\leq\epsilon_{0}|t-1|^{\frac{1}{2}}\right\\},\\\
\tilde{{\mathcal{D}}}_{-}&={\mathcal{D}}_{-}\cap\left\\{(x,t):\epsilon_{1}|t-1|^{\frac{1}{2}}\leq
x_{n}\right\\},\end{split}$

where $\epsilon_{0}$ and $\epsilon_{1}$ be positive constants with
$0<\epsilon_{0}\ll 1\ll\epsilon_{1}<\infty$, which will be specified later.

######  Theorem 2.6.

Let $w$ be the solution of the Stokes equations (2.10) in the half-space with
boundary data $g$ given in (2.14) and (2.15). Suppose that $\phi$ satisfies
Assumption 2.1. Then, there is $0<t_{1}<t_{0}$ such that if
$\left|t-1\right|<t_{1}$, $a\neq\frac{1}{2}$, $|x^{\prime}|<\frac{1}{2}$ and
$0<x_{n}<\frac{1}{2}$ such that we prove the following: Let
$i=1,2,\cdots,n-1$.

(2.41)
$\displaystyle\left|w_{i}(x,t)\right|\lesssim\left\\{\begin{array}[]{ll}\vspace{2mm}1&\mbox{if}\,\,\phi(t)=\left|\ln
t\right|^{-1}\\\ \phi(|1-t|)&\mbox{otherwise}.\end{array}\right.$

In case that $\phi(t)\neq\left|\ln t\right|^{-1}$, there exist $\epsilon_{0}$
and $\epsilon_{1}$, depending on $t_{1}$, in (2.38) such that

(2.42) $\displaystyle w_{i}(x,t)\geq c\phi(1-t)\quad$
$\displaystyle\mbox{if}\,\,\,\,(x,t)\in\tilde{{\mathcal{D}}}_{-},\quad t<1.$
(2.43) $\displaystyle
w_{i}(x,t)\leq-c\phi(t-1)\quad\mbox{if}\,\,\,\,(x,t)\in\tilde{{\mathcal{D}}}_{-}\cup{\mathcal{D}}_{0},\quad
1<t<1+t_{0}.$ (2.44) $\displaystyle\begin{split}|D_{x_{n}}w_{i}(x,t)|\leq
c\left\\{\begin{array}[]{cl}\vspace{2mm}\left((t-1)\vee
x_{n}^{2}\right)^{-\frac{1}{2}}\phi\left((t-1)\vee x_{n}^{2}\right)),&\mbox{
if }\,\,1<t<\frac{5}{4},\\\ \left((1-t)\vee
x_{n}^{2}\right)^{\frac{1}{2}}|\phi^{\prime}\left((1-t)\vee
x_{n}^{2}\right))|,&\mbox{ if
}\,\,\frac{3}{4}<t<1.\end{array}\right.\end{split}$ (2.45) $\displaystyle
D_{x_{n}}w_{i}(x,t)$
$\displaystyle\approx-(1-t)^{\frac{1}{2}}\phi^{\prime}(1-t){\bf
I}_{\tilde{{\mathcal{D}}_{+}}}-x_{n}\phi^{\prime}(x_{n}^{2}){\bf
I}_{\tilde{{\mathcal{D}}_{-}}},\quad 1-t_{1}<t<1.$ (2.46) $\displaystyle
D_{x_{n}}w_{i}(x,t)\approx-(t-1)^{-\frac{1}{2}}\phi(t-1)\quad\mbox{if}\,\,\,\,(x,t)\in\tilde{{\mathcal{D}}}_{+},\quad
1<t<1+t_{0}.$

If $1<t<1+t_{1}$ and $(x,t)\in\tilde{{\mathcal{D}}}_{-}$ then

(2.47)
$\displaystyle\begin{split}D_{x_{n}}w_{i}(x,t)\approx\left\\{\begin{array}[]{cl}\vspace{2mm}-x_{n}\phi^{\prime}(x_{n}^{2}),&\mbox{
if }\,\,\phi=t^{-a},\\\
-x_{n}\phi^{\prime}(x_{n}^{2})+(t-1)^{-\frac{3}{2}}x_{n}^{2}e^{-\frac{x_{n}^{2}}{4(t-1)}}\phi(t-1),&\mbox{
otherwise}.\end{array}\right.\end{split}$

(2.48)
$\displaystyle\begin{split}p(x,t)\approx\left\\{\begin{array}[]{cl}\vspace{2mm}-\phi^{\prime}(1-t),&\mbox{
if }\,\,1-t_{1}<t<1,\\\ x_{n}(t-1)^{-\frac{1}{2}}\phi(t-1)\pm 1,&\mbox{ if
}\,\,1<t<1+t_{1}.\end{array}\right.\end{split}$

######  Remark 2.7.

The results of Theorem 2.6 imply the following regularity of the solution for
$\left|t-1\right|<t_{1}$ and $|x|<\frac{1}{2}$ near boundary: Let
$Q:=B^{+}_{\frac{1}{2}}\times(1-t_{1},1+t_{1})$ and $i=1,2,\cdots,n-1$.

  * (1)

(The case $\phi(t)=|\ln(t)|)$ For any $p_{1}<\infty$ and $p_{2}<3$

(2.49) $\displaystyle\begin{split}&w_{i}\in L^{p_{1}}(Q),\quad
D_{x_{n}}w_{i}\in L^{p_{2}}(Q),\quad p\in L^{1}_{weak}(Q)\quad\mbox{ but}\\\
&w_{i}\notin L^{\infty}(Q),\quad D_{x_{n}}w_{i}\notin L^{3}(Q),\quad p\notin
L^{1}(Q).\end{split}$

  * (2)

(The case $\phi(t)=|\ln(t)|^{-1}$) For any $p_{3}>3$, $p_{4}>1$ and
$\alpha\in(0,1)$

(2.50) $\displaystyle\begin{split}&w_{i}\in L^{\infty}(Q),\quad
D_{x_{n}}w_{i}\in L^{3}(Q),\quad p\in L^{1}(Q),\quad\mbox{ but}\\\
&w_{i}\notin{\mathcal{C}}^{\alpha}(Q),\quad D_{x_{n}}w_{i}\notin
L^{p_{3}}(Q),\quad p\notin L^{p_{4}}(Q).\end{split}$

  * (3)

(The case $\phi(t)=t^{-a},\,\,0<a<1$). For any $p_{5}<\frac{3}{2a}$,
$p_{6}<\frac{3}{1+2a}$ and $p_{7}<\frac{1}{1+a}$

(2.51) $\displaystyle\begin{split}&w_{i}\in L^{p_{5}}(Q),\quad
D_{x_{n}}w_{i}\in L^{p_{6}}(Q),\quad p\in L^{p_{7}}(Q),\quad\mbox{ but}\\\
&w_{i}\notin L^{\frac{3}{2a}}(Q),\quad\,D_{x_{n}}w_{i}\notin
L^{\frac{3}{1+2a}}(Q),\quad\,p\notin L^{\frac{1}{1+a}}(Q).\end{split}$

The case that $\phi(t)=t^{-a},\,\,0<a<1$ is most interesting and, combining
Theorem 2.6 and Remark 2.7, we make a table to show asymptotic behavior of the
solution near boundary.

Region$w_{i}$$D_{x_{n}}w_{i}$$p$$t<1$$t>1$$\small{\tilde{D}_{+}}$$\small{\tilde{D}_{-}}$$\small{\tilde{D}_{+}}$$\small{\tilde{D}_{-}}$$\small{\tilde{D}_{0}}$$\approx-(1-t)^{\frac{1}{2}}\phi^{\prime}(1-t)\approx(1-t)^{-\frac{1}{2}-a}$$\approx\phi(1-t)$$\approx-
x_{n}\phi^{\prime}(x_{n}^{2})\approx x_{n}^{-1-2a}$
$\approx-\phi^{\prime}(1-t)\approx(1-t)^{-1-a}$
$\approx-(t-1)^{-\frac{1}{2}}\phi(t-1)\approx-(t-1)^{-\frac{1}{2}-a}$
$\approx\phi(t-1)$ $\approx-x_{n}\phi^{\prime}(x_{n}^{2})\approx
x_{n}^{-1-2a}$ $\approx-x_{n}(t-1)^{-\frac{1}{2}}\phi(t-1)\approx(t-1)^{-a}$
$\approx\phi(t-1)$ $\approx-
x_{n}(t-1)^{-\frac{1}{2}}\phi(t-1)\approx(t-1)^{-a}$ Integrable property $\in
L^{p},\,\,p<\frac{1}{a}$ $\mbox{but}\,\,\notin L^{\frac{1}{a}}$ $\in
L^{p},\,\,p<\frac{3}{1+2a}\quad\mbox{but}\,\,\notin L^{\frac{3}{1+2a}}$ $\in
L^{p},\,\,p<\frac{1}{1+a}\quad\mbox{but}\,\,\notin L^{\frac{1}{1+a}}$
$\big{(}\phi(t)=t^{-a},\,\,0<a<1\big{)}$

######  Remark 2.8.

When it comes to global estimate of the solution $w$ in the half-space, it is
worth mentioning that $w$ belongs to the same spaces as shown in Remark 2.7.
For example, in the case $\phi=|\ln t|^{-1}$, since the boundary data
$g_{n}(t)$ in (2.14) is Dini-continuity for all $0<t<\infty$, it follows due
to [1] that $w$ is boudned in the half space, i.e. $w\in
L^{\infty}({\mathbb{R}}^{n}_{+}\times(0,\infty))$ (see also [2] and [3]).
Moreover, since $\phi\in L^{\infty}\cap\dot{B}^{\frac{1}{3}}_{3}$, using the
results in [4] and [5], one can see that

(2.52) $\displaystyle\|\nabla
w\|_{L^{3}({\mathbb{R}}^{n}_{+}\times(0,\infty))}$ $\displaystyle\leq
c\left(\|\phi\|_{L^{3}(0,\infty)}+\|\phi\|_{\dot{B}^{\frac{1}{3}}_{3}(0,\infty)}\right)<\infty.$

The other cases such as $\phi(t)=\left|\ln t\right|$ or $t^{-a}$ can be
treated similarly, and thus we skip their details. Our results can be
understood that temporal singular behaviors caused by compactly supported
boundary data are immediately transfered to all over the place and, in
addition, unlike the interior local smoothing in spatial variables is not
available near boundary. We remark that spatial singular behavior of compactly
supported boundary data is not delivered to other places and local smoothing
effect is available in any region away from the data (see [13]).

######  Remark 2.9.

We remark that authors showed in [7] that there is a very weak solution of
(1.1) or (1.3) so that $\nabla w$ is not squre integrable, i. e.,
$\left\|\nabla w\right\|_{L^{2}(Q_{1}^{+})}=\infty$, although
$\left\|w\right\|_{L^{2}(Q_{1}^{+})}<\infty$. It turns out that it is a
special case almost close to $a=\frac{1}{4}$ and thus it is observable that
$w$ is not bounded and $\pi$ is not integrable in $Q_{1}^{+}$.

Let $\phi_{1}\in C_{c}^{\infty}({\mathbb{R}}^{n})$ be a cut-off function
satisfying $\phi_{1}\geq 0$, ${\rm supp}\,\phi_{1}\subset
B_{\frac{1}{\sqrt{2}}}$ and $\phi_{1}\equiv 1$ in $B_{\frac{3}{8}}$. Also, let
$\phi_{2}\in C_{c}^{\infty}(-\infty,\infty)$ be a cut-off function satisfying
$\phi_{2}\geq 0$, ${\rm supp}\,\phi_{2}\subset(-3,3)$ and $\phi_{2}\equiv 1$
in $(-2,2)$. Let $\phi(x,t)=\phi_{1}(x)\phi_{2}(t)$. Let $U=w\phi$ and
$\Pi=\pi\phi$ such that $U|_{Q^{+}_{\frac{1}{4}}}=w$ and
$\Pi|_{Q^{+}_{\frac{1}{4}}}=\pi$.

We consider the following perturbed Navier-Stokes equations in
${\mathbb{R}}^{n}_{+}\times(0,\infty)$:

(2.53) $v_{t}-\Delta v+\nabla q+{\rm div}\,\left(v\otimes v+v\otimes
U+U\otimes v\right)=-{\rm div}\,(U\otimes U),\quad{\rm div}\,v=0$

with homogeneous initial and boundary data, i.e.

(2.54) $v(x,0)=0\,\,\mbox{in}\,\,\mathbb{R}^{n}_{+},\qquad
v(x,t)=0\,\,\mbox{on}\,\,\\{x_{n}=0\\}\times[0,\infty).$

######  Proposition 2.10.

Let $0<a<\frac{3}{2(n+2)}$ and $\frac{n+2}{2}<q<\frac{3}{4a}$. Suppose
$\|w\|_{L^{2q}}<c\alpha$ for suffinciently small $\alpha>0$, where $\alpha>0$
is defined in (2.14). Then, there is a solution $v$ of (2.53)-(2.54) such that

(2.55) $\displaystyle\|\nabla
v\|_{L^{q}({\mathbb{R}}^{n}_{+}\times(0,1))}+\|v\|_{L^{2q}({\mathbb{R}}^{n}_{+}\times(0,1))}\leq
c(\|w\|_{L^{q}},\|w\|_{L^{2q}}).$

In case that $0<a<\frac{3}{4(n+2)}$ , if $\|w\|_{L^{2q}}<c\alpha$ with
$n+2<q<\frac{3}{4a}$ for suffinciently small $\alpha>0$, where $\alpha>0$ is
defined in (2.14), then $v$ satisfies

(2.56) $\displaystyle\|v\|_{L^{\infty}({\mathbb{R}}^{n}_{+}\times(0,1))}\leq
c(\|w\|_{L^{q}},\|w\|_{L^{2q}})).$

The proof of Proposition 2.10 will be given in section 5.

We then set $u:=v+U$ and $p=\pi+q$, which becomes a weak solution of the
Navier-Stokes equations in $\tilde{Q}_{+}=B_{\frac{1}{2}}^{+}\times(0,2)$,
namely

(2.57) $u_{t}-\Delta u+\nabla p=-{\rm div}\,\left(u\otimes u\right),\qquad{\rm
div}\,u=0\quad Q^{+}_{1},$

with boundary data $u(x,t)=0$ on $\\{x_{n}=0\\}$ and homogeneous initial data
$u(x,0)=0$.

With the aid of Proposition 2.10, we can construct a solution of the Navier-
Stokes equations such that its behaviors near boundary are more or less the
same as those of the Stokes equations mentioned in Theorem 2.6. Since its
verification is straightforward by combining Theorem 2.6 and Proposition 2.10,
we just state it without providing its details.

######  Theorem 2.11.

Suppose that $\phi$ satisfies Assumption 2.1 with an additional restriction
that $0<a<\frac{3}{2(n+2)}$ for $\phi(t)=t^{-a}$. Then, there exists a
solution of the Navier-Stokes equations, $u$, in (2.57) such that it satisfies
the same regularity properties mentioned in Remark 2.7. Furthermore, if
$0<a<\frac{3}{4(n+2)}$, then $u$ satisfies pointwise estimates (2.41), (2.42)
and (2.43).

######  Remark 2.12.

The pointwise estimate of $u$ in Theorem 2.11 is the combination of (2.56) for
$v$ in Proposition 2.10 and (2.41)-(2.43) for $w$ in Theorem 2.6. However, the
pointwise estimate of $\nabla u$ is not clear because of the lack of that of
$\nabla v$, and instead, $L^{q}$ estimate is only available up to
$q<\frac{3}{4a}$ as shown in (2.55). Thus, we leave it as an open question on
construction of solutions of the Navier-Stokes equations whose normal
derivatives satisfies pointwise estimates near boundary given in Theorem 2.6.

This paper is organized as follows: In section 3, proofs of some lemmas are
provided. Section 4 is devoted to proving Theorem 2.6. We present the proof of
Proposition 2.10 in Section 5.

##  3\. Preliminaries

In this section, we mainly provide proofs of lemmas, which are useful for our
analysis. Before we do that, we first define the notion of weak solutions of
the Stokes equations in the half space.

######  Definition 3.1.

Let $\phi$ satisfy Assumption 2.1. We say that $w\in
L^{1}_{\rm{loc}}(0,\infty;W^{1,1}_{\rm{loc}}({\mathbb{R}}^{n}_{+}))$ is a weak
solution of the Stokes equations (2.10) in the half-space with boundary data
$g$ given in (2.14) and (2.15), if the following equality holds:

$\displaystyle\int^{T}_{0}\int_{{\mathbb{R}}^{n}_{+}}w\cdot\Phi_{t}dxdt+\int^{T}_{0}\int_{{\mathbb{R}}^{n}_{+}}w\cdot\Delta\Phi
dxdt+\int^{T}_{0}\int_{\partial{\mathbb{R}}^{n}_{+}}g_{n}\cdot D_{x_{n}}\Phi
dx^{\prime}dt=0$

for each $\Phi\in C^{2}_{c}(\overline{{\mathbb{R}}^{n}_{+}}\times[0,T])$ with

(3.1) $\displaystyle\mbox{\rm div
}\Phi=0,\quad\Phi\big{|}_{\partial{\mathbb{R}}^{n}_{+}\times(0,T)}=0,\quad\Phi(\cdot,T)=0.$

In addition, for each $\Psi\in C^{1}_{c}(\overline{{\mathbb{R}}^{n}_{+}})$
with $\Psi\big{|}_{\partial{\mathbb{R}}^{n}_{+}}=0$

(3.2) $\int_{{\mathbb{R}}^{n}_{+}}u(x,t)\cdot\nabla\Psi(x)dx=0\quad\mbox{ for
all}\quad 0<t<T.$

For notational convenition, we denote for fixed $\alpha\geq 0$ and $k=0,1$

(3.3) $\displaystyle\Psi_{\phi,k}(t):$
$\displaystyle=\int_{0}^{\min(1,t)}(t-s)^{-\alpha}\phi^{(k)}(1-s)ds,$ (3.4)
$\displaystyle{\tilde{\Psi}}_{\phi,k}(t):$
$\displaystyle=\int_{0}^{\min(1,t)}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{(k)}(1-s)ds.$

The following is a key lemma to obtain main results.

######  Lemma 3.2.

Let $0<\alpha$, $\alpha\neq 1$, $\alpha+a\neq 1$ and
$\frac{3}{4}<t<\frac{5}{4}$. Suppose that $\phi$ satisfies Assumption 2.1.
Then, $\Psi_{\phi,k}(t)$ and ${\tilde{\Psi}}_{\phi,k}(t)$ in (3.3)-(3.4)
sastisfy following:

  * (i)

Let $1<t<\frac{5}{4}$. Then,

(3.7)
$\displaystyle\Psi_{\phi,0}(t)\approx\left\\{\begin{array}[]{cl}\vspace{2mm}(t-1)^{1-\alpha}\phi(t-1)&\mbox{
if }\,\,\alpha+a>1,\\\ 1&\mbox{ otherwise }.\end{array}\right.$ (3.8)
$\Psi_{\phi,1}(t)\approx(t-1)^{1-\alpha}\phi^{\prime}(t-1).$ (3.11)
$\displaystyle{\tilde{\Psi}}_{\phi,0}(t)\approx\left\\{\begin{array}[]{cl}\vspace{2mm}\left((t-1)\vee
x_{n}^{2}\right)^{1-\alpha}\phi\left((t-1)\vee x_{n}^{2}\right))&\mbox{ if
}\,\,\alpha+a>1,\\\ 1&\mbox{ otherwise }.\end{array}\right.$

(3.12) $\displaystyle{\tilde{\Psi}}_{\phi,1}(t)$
$\displaystyle\approx((t-1)\vee x_{n}^{2})^{1-\alpha}\phi^{\prime}((t-1)\vee
x_{n}^{2}).$

  * (ii)

Let $\frac{3}{4}<t<1$. Suppose that $0<\alpha<1$. Then,

(3.15)
$\displaystyle\Psi_{\phi,0}(t)\approx\left\\{\begin{array}[]{cl}\vspace{2mm}(1-t)^{1-\alpha}\phi(1-t)&\mbox{
if }\,\,\alpha+a>1,\\\ 1&\mbox{ otherwise }.\end{array}\right.$

(3.16) $\Psi_{\phi,1}(t)\approx(1-t)^{1-\alpha}\phi^{\prime}(1-t).$

(3.20)
$\displaystyle{\tilde{\Psi}}_{\phi,0}(t)\approx\left\\{\begin{array}[]{cl}\vspace{2mm}\max(1,x_{n}^{-2\alpha+2}\phi(x_{n}^{2}))+\phi(1-t)(1-t)^{2-\alpha}x_{n}^{-2}e^{-\delta\frac{x_{n}^{2}}{1-t}}&\mbox{
if }\,\,1-t<x_{n}^{2},\quad\alpha>0,\\\
\vspace{2mm}\phi(1-t)x_{n}^{2-2\alpha}&\mbox{ if
}\,\,x_{n}^{2}<1-t,\quad\alpha>1,\\\ \max(1,(1-t)^{-\alpha+1}\phi(1-t))&\mbox{
if }\,\,x_{n}^{2}<1-t,\quad\alpha<1.\end{array}\right.$

(3.24)
$\displaystyle{\tilde{\Psi}}_{\phi,1}(t)\approx\left\\{\begin{array}[]{cl}\vspace{2mm}x_{n}^{-2\alpha+2}\phi^{\prime}(x_{n}^{2})+\phi^{\prime}(1-t)(1-t)^{2-\alpha}x_{n}^{-2}e^{-\delta\frac{x_{n}^{2}}{1-t}}&\mbox{
if }\,\,1-t<x_{n}^{2},\quad\alpha>0,\\\
\vspace{2mm}\phi^{\prime}(1-t)x_{n}^{2-2\alpha}&\mbox{ if
}\,\,x_{n}^{2}<1-t,\quad\alpha>1,\\\
(1-t)^{-\alpha+1}\phi^{\prime}(1-t)&\mbox{ if
}\,\,x_{n}^{2}<1-t,\quad\alpha<1.\end{array}\right.$

###### Proof.

We first treat the cases $\Psi_{\phi,0}$ and $\tilde{\Psi}_{\phi,0}$. We first
remind that

(3.25) $\displaystyle\begin{split}&\frac{1}{2}(1-s)\leq
t-s\leq(1-s),\qquad\frac{3}{4}<t<1\qquad\mbox{for}\quad 0<s<2t-1,\\\
&\frac{1}{2}(1-s)\leq 1-t\leq(1-s),\qquad\frac{3}{4}<t<1\qquad\mbox{for}\quad
2t-1<s<t,\\\ &(1-s)\leq t-s\leq 2(1-s),\qquad
1<t<\frac{5}{4}\qquad\mbox{for}\quad 0<s<2-t,\\\ &\frac{1}{2}(t-s)\leq
t-1\leq(t-s),\qquad 1<t<\frac{5}{4}\qquad\mbox{for}\quad 2-t<s<1.\end{split}$

$\bullet$ (The case $1<t<\frac{5}{4}$) We split the integral over $[0,1]$ into
$[0,2-t]$ and $[2-t,1]$. Then, from $\eqref{eq1113-1}_{3,4}$ and the change of
variables, we have

(3.26)
$\displaystyle\begin{split}\Psi_{\phi,0}(t)&\approx\int_{0}^{2-t}(1-s)^{-\alpha}\phi(1-s)ds+\int_{2-t}^{1}(t-1)^{-\alpha}\phi(1-s)ds,\\\
&=\int^{1}_{t-1}s^{-\alpha}\phi(s)ds+(t-1)^{-\alpha}\int_{0}^{t-1}\phi(s)ds.\end{split}$

Applying (2.20) and (2.22) in (3.26), we obtain (3.7).

Using $\eqref{eq1113-1}_{4}$, we have $e^{-\frac{x_{n}^{2}}{2(t-1)}}\leq
e^{-\frac{x_{n}^{2}}{2(t-s)}}\leq e^{-\frac{2x_{n}^{2}}{(t-1)}}$ for
$2-t<s<1$, namely $e^{-\frac{x_{n}^{2}}{4(t-1)}}\leq
e^{-\frac{x_{n}^{2}}{4(t-s)}}\leq e^{-\frac{x_{n}^{2}}{(t-1)}}$. Thus, we note
that $e^{-\frac{x_{n}^{2}}{4(t-s)}}\approx e^{-\delta\frac{x_{n}^{2}}{t-1}}$
for $2-t<s<1$ (see Notation 2.3). Similarly, we split the following integral:

(3.27)
$\displaystyle\begin{split}{\tilde{\Psi}}_{\phi,0}(t)&\approx\int_{t-1}^{1}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds+(t-1)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{t-1}}\int_{0}^{t-1}\phi(s)ds.\end{split}$

Due to (2.22), we have

(3.28)
$\displaystyle\begin{split}(t-1)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{t-1}}\int_{0}^{t-1}\phi(s)ds&\approx(t-1)^{-\alpha+1}\phi(t-1)e^{-\delta_{1}\frac{x_{n}^{2}}{t-1}}.\end{split}$

For the first term of the right-hand side of (3.27), we separate the cases
that $x_{n}^{2}\leq t-1$ and $x_{n}^{2}>t-1$. In the case that $x_{n}^{2}\leq
t-1$, since $e^{-\frac{x_{n}^{2}}{4s}}\approx 1$ for $x_{n}^{2}\leq t-1\leq
s<1$, we can see via (2.20) that

(3.29)
$\displaystyle\int_{t-1}^{1}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds$
$\displaystyle\approx\int^{1}_{t-1}s^{-\alpha}\phi(s)ds\approx\max(1,(t-1)^{-\alpha+1}\phi(t-1)).$

Thus, if $x_{n}^{2}\leq t-1$, it follows from (3.27), (3.28) and (3.29) that

(3.30)
$\displaystyle\begin{split}{\tilde{\Psi}}_{\phi,0}(t)\approx\max(1,(t-1)^{-\alpha+1}\phi(t-1)).\end{split}$

Secondly, we treat the case that $t-1<x_{n}^{2}$. As in (3.29), we have

(3.31)
$\displaystyle\begin{split}\int_{t-1}^{1}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds&=\int_{t-1}^{x_{n}^{2}}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds+\int_{x_{n}^{2}}^{1}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds.\end{split}$

Since $e^{-\frac{x_{n}^{2}}{4s}}\approx 1$ for $x_{n}^{2}<s<1$, it follows
from (2.20) that

(3.32)
$\displaystyle\begin{split}\int_{x_{n}^{2}}^{1}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds\approx\int^{1}_{x_{n}^{2}}s^{-\alpha}\phi(s)ds\approx\max(1,x_{n}^{-2\alpha+2}\phi(x_{n}^{2})).\end{split}$

This implies that

(3.33)
$\displaystyle\max(1,x_{n}^{-2\alpha+2}\phi(x_{n}^{2}))\lesssim\int_{t-1}^{1}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds.$

On the other hand, noting that $e^{-b}\leq c_{m}b^{-m}$ for any $b>0$ and
$m\in{\mathbb{N}}$, we have

(3.34)
$\displaystyle\begin{split}\int_{t-1}^{x_{n}^{2}}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds&\leq
c\int_{t-1}^{x_{n}^{2}}s^{-\alpha}(\frac{x_{n}^{2}}{4s})^{-m}\phi(s)ds\leq
cx_{n}^{-2m}\int_{t-1}^{x_{n}^{2}}s^{-\alpha+m}\phi(s)ds\\\ &\leq
cx_{n}^{-2m}x_{n}^{-2\alpha+2m+2}\phi(x_{n}^{2})\leq
cx_{n}^{-2\alpha+2}\phi(x_{n}^{2}).\end{split}$

Estimates (3.31) -(3.34) implies that for $1<t$ and
$t-1<x_{n}^{2}<\frac{1}{2}$

(3.35)
$\displaystyle\int_{t-1}^{1}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi(s)ds$
$\displaystyle\approx\max(1,x_{n}^{-2\alpha+2}\phi(x_{n}^{2})).$

Adding up (3.27), (3.28), (3.30) and (3.35), we get (3.11).

$\bullet$ (The case $\frac{3}{4}<t<1$) Noting that $\frac{1}{2}<2t-1<t$ and
using $\eqref{eq1113-1}_{1}$, (2.23), (2.24), (2.20) and change of variables,
it follows that

(3.36)
$\displaystyle\begin{split}\Psi_{\phi,0}(t)&\approx\int_{0}^{2t-1}(1-s)^{-\alpha}\phi(1-s)ds+\int_{2t-1}^{t}(t-s)^{-\alpha}\phi(1-t)ds\\\
&=\int^{1}_{2-2t}s^{-\alpha}\phi(s)ds+\phi(1-t)\int_{0}^{1-t}s^{-\alpha}ds\\\
&\approx\max(1,(1-t)^{1-\alpha}\phi(1-t)).\end{split}$

Hence, we obtain (3.15). It remains to show (3.20). We note first via
$\eqref{eq1113-1}_{1,2}$, (2.23) and (2.24) that

(3.37)
$\displaystyle\begin{split}{\tilde{\Psi}}_{\phi,0}(t)&\approx\int_{0}^{2t-1}(1-s)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{1-s}}\phi(1-s)ds+\int_{2t-1}^{t}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi(1-t)ds\\\
&:=I_{1}+I_{2}.\end{split}$

In case that $x_{n}^{2}<1-t$, since $e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\approx
1$ for $1-t<s<1$, it follows due to (2.20) that

(3.38)
$\displaystyle\begin{split}I_{1}=\int_{2-2t}^{1}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi(s)ds&\approx\int_{2-2t}^{1}s^{-\alpha}\phi(s)ds\approx\max(1,(1-t)^{-\alpha+1}\phi(1-t)).\end{split}$

For $I_{2}$, using the change of variables, we have

(3.39)
$\displaystyle\begin{split}I_{2}&=\phi(1-t)\int_{0}^{1-t}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}ds=\phi(1-t)x_{n}^{2-2\alpha}\int^{\infty}_{\frac{x_{n}^{2}}{1-t}}s^{-2+\alpha}e^{-\frac{s}{4}}ds\\\
&\approx\left\\{\begin{array}[]{cl}\vspace{2mm}\phi(1-t)x_{n}^{2-2\alpha}&\mbox{
if }\,\,\alpha>1,\\\ \phi(1-t)(1-t)^{1-\alpha}&\mbox{ if
}\,\,\alpha<1.\end{array}\right.\end{split}$

Summing up (3.37), (3.38) and (3.39), we obtain

$\displaystyle{\tilde{\Psi}}_{\phi,0}(t)$
$\displaystyle\approx\left\\{\begin{array}[]{cl}\vspace{2mm}\phi(1-t)x_{n}^{2-2\alpha}+\max(1,(1-t)^{-\alpha+1}\phi(1-t))&\mbox{
if }\,\,\alpha>1,\\\ \max(1,(1-t)^{-\alpha+1}\phi(1-t))&\mbox{ if
}\,\,\alpha<1\end{array}\right.$
$\displaystyle\approx\left\\{\begin{array}[]{cl}\vspace{2mm}\phi(1-t)x_{n}^{2-2\alpha}&\mbox{
if }\,\,\alpha>1,\\\ \max(1,(1-t)^{-\alpha+1}\phi(1-t))&\mbox{ if
}\,\,\alpha<1.\end{array}\right.$

Thus, we have (3.20) for the case that $x_{n}^{2}<1-t$.

On the other hand, in case that $1-t<x_{n}^{2}$, we split $I_{1}$ as follows:

(3.40) $\displaystyle I_{1}$
$\displaystyle=\int_{2-2t}^{x_{n}^{2}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi(s)ds+\int_{x_{n}^{2}}^{1}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi(s)ds:=I_{11}+I_{12}.$

It is straightforwad from (2.20) that

(3.41)
$\displaystyle\begin{split}I_{12}\approx\int^{1}_{x_{n}^{2}}s^{-\alpha}\phi(s)ds\approx\max(1,x_{n}^{-2\alpha+2}\phi(x_{n}^{2})).\end{split}$

With the same estimate of (3.34), we get

(3.42)
$\displaystyle\begin{split}I_{11}=\int_{2-2t}^{x_{n}^{2}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi(s)ds&\leq
cx_{n}^{-2\alpha+2}\phi(x_{n}^{2}).\end{split}$

Adding up (3.40), (3.41) and (3.42), we obtain for $1-t<x_{n}^{2}$

(3.43) $\displaystyle
I_{1}=\int_{0}^{2t-1}(1-s)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{1-s}}\phi(1-s)ds\approx\max(1,x_{n}^{-2\alpha+2}\phi(x_{n}^{2})).$

Direct computations via the change of variables give

(3.44)
$\displaystyle\begin{split}I_{2}&=\phi(1-t)x_{n}^{2-2\alpha}\int^{\infty}_{\frac{x_{n}^{2}}{1-t}}s^{-2+\alpha}e^{-\frac{s}{4}}ds\approx\phi(1-t)(1-t)^{2-\alpha}x_{n}^{-2}e^{-\delta\frac{x_{n}^{2}}{1-t}}.\end{split}$

Hence, from (3.37), (3.44), (3.43) and Lemma 2.2, for $1-t\leq x_{n}^{2}$,

$\displaystyle{\tilde{\Psi}}_{\phi,0}(t)$
$\displaystyle\approx\max(1,x_{n}^{-2\alpha+2}\phi(x_{n}^{2}))+\phi(1-t)(1-t)^{2-\alpha}x_{n}^{-2}e^{-\delta\frac{x_{n}^{2}}{1-t}}.$

We obtain (3.20).

Next, we prove the estimate (3.12) of ${\tilde{\Psi}}_{\phi,1}(t)$. Again we
separately consdier the cases that $t<1$ and $t>1$.
$\bullet$ (The case $1<t<\frac{5}{4}$) For convenience, we define

(3.45) $\displaystyle{\tilde{\Psi}}_{\phi,k}(t;\tau):=$
$\displaystyle\int_{0}^{\tau}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{(k)}(1-s)ds.\qquad
k=0,1.$

We then split the integral, ${\tilde{\Psi}}_{\phi,1}(t;2-t)$, as follows:

(3.46)
$\displaystyle\begin{split}{\tilde{\Psi}}_{\phi,1}(t;2-t)&=\int_{0}^{\frac{3}{4}}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds+\int_{\frac{3}{4}}^{2-t}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds.\end{split}$

Since $\phi$ is smooth in $[\frac{1}{2},1]$, from direct caculation, we have
for $0<x_{n}<\frac{1}{2}$

(3.47)
$\displaystyle|\int_{0}^{\frac{3}{4}}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds|\leq
A_{0}<\infty.$

For the second term in (3.46), using (2.25) and (2.26), we have

(3.48)
$\displaystyle\int_{\frac{1}{4}}^{2-t}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)$
$\displaystyle\approx\int_{\frac{1}{4}}^{2-t}(1-s)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{1-s}}\phi^{\prime}(1-s)=\int_{t-1}^{\frac{1}{4}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s).$

In the case that $x_{n}^{2}\leq t-1$, since
$e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\approx 1$ for $x_{n}^{2}\leq t-1\leq s<1$,
we can see via (2.21) that

(3.49)
$\displaystyle\begin{split}\int_{t-1}^{\frac{1}{4}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s)ds\approx\int_{t-1}^{\frac{1}{4}}s^{-\alpha}\phi^{\prime}(s)ds\approx(t-1)^{-\alpha+1}\phi^{\prime}(t-1).\end{split}$

Adding up (3.46), (3.47), (3.48) and (3.49), it follows that

(3.50)
$\displaystyle\begin{split}{\tilde{\Psi}}_{\phi,1}(t;2-t)\approx(t-1)^{-\alpha+1}\phi^{\prime}(t-1),\end{split}$

and thus we obtain (3.12) for $x_{n}^{2}<t-1$.

Secondly, if $t-1<x_{n}^{2}$, we split the integral (3.49) as follows:

(3.51)
$\displaystyle\begin{split}\int_{t-1}^{\frac{1}{4}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s)ds&=\int_{t-1}^{x_{n}^{2}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s)ds+\int_{x_{n}^{2}}^{\frac{1}{4}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s)ds.\end{split}$

Since $e^{-\frac{x_{n}^{2}}{s}}\approx 1$ for $x_{n}^{2}<s<1$, it follows from
(2.21) that

(3.52)
$\displaystyle\begin{split}\int^{\frac{1}{4}}_{x_{n}^{2}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s)ds\approx\int^{\frac{1}{4}}_{x_{n}^{2}}s^{-\alpha}\phi^{\prime}(s)ds\approx
x_{n}^{-2\alpha+2}\phi^{\prime}(x_{n}^{2}).\end{split}$

Recalling that ${\rm sgn}(\phi^{\prime})\phi^{\prime}(s)>0$ for
$0<s<\frac{1}{2}$ and $e^{-b}\leq c_{m}b^{-m}$ for any $b>0$ and
$m\in{\mathbb{N}}$, we have

(3.53) $\displaystyle\begin{split}{\rm
sgn}(\phi^{\prime})\int_{t-1}^{x_{n}^{2}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s)ds&\leq
c\,{\rm
sgn}(\phi^{\prime})\int_{t-1}^{x_{n}^{2}}s^{-\alpha}(\frac{x_{n}^{2}}{\delta_{1}s})^{-m}\phi^{\prime}(s)ds\\\
&\leq c\,{\rm
sgn}(\phi^{\prime})x_{n}^{-2m}\int_{t-1}^{x_{n}^{2}}s^{-\alpha+m}\phi^{\prime}(s)ds\\\
&\leq cx_{n}^{-2m}x_{n}^{-2\alpha+2m+2}{\rm
sgn}(\phi^{\prime})\phi^{\prime}(x_{n}^{2})\\\ &\leq cx_{n}^{-2\alpha+2}{\rm
sgn}(\phi^{\prime})\phi^{\prime}(x_{n}^{2}).\end{split}$

Therefore, we deduce from estimates (3.51) -(3.53) that

(3.54)
$\displaystyle\int_{t-1}^{\frac{1}{2}}s^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi^{\prime}(s)ds$
$\displaystyle\approx x_{n}^{-2\alpha+2}\phi^{\prime}(x_{n}^{2}).$

Adding up estimates (3.46), (3.47) , (3.48) and (3.54), it follows in case
$t-1<x_{n}^{2}$ that

(3.55) $\displaystyle{\tilde{\Psi}}_{\phi,1}(t;2-t)\approx
x_{n}^{-2\alpha+2}\phi^{\prime}(x_{n}^{2}),$

which implies (3.12) for $t-1<x_{n}^{2}$.

$\bullet$ (The case $\frac{3}{4}<t<1$) Noting that $\frac{1}{2}<2t-1<t$.
Hence, we have

(3.56) $\displaystyle{\tilde{\Psi}}_{\phi,1}(t)$
$\displaystyle=\int_{0}^{\frac{1}{2}}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds+\int_{\frac{1}{2}}^{t}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds.$

Since $\phi$ is smooth in $[\frac{1}{2},1]$, it is direct for
$0<x_{n}<\frac{1}{2}$ that

(3.57)
$\displaystyle|\int_{0}^{\frac{1}{2}}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds|\leq
B_{0}<\infty.$

Using $\eqref{eq1113-1}_{1,2}$, it follows that

(3.58)
$\displaystyle\begin{split}&\int_{\frac{1}{2}}^{t}(t-s)^{-\alpha}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds\\\
\approx&\int_{\frac{1}{2}}^{2t-1}(1-s)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{1-s}}\phi^{\prime}(1-s)ds+\int_{2t-1}^{t}(t-s)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{t-s}}\phi^{\prime}(1-t)ds\\\
:=&I_{1}+I_{2}.\end{split}$

In case that $x_{n}^{2}\leq 1-t$, direct computations via the change of
variables give

(3.59)
$\displaystyle\begin{split}I_{2}&=\phi^{\prime}(1-t)x_{n}^{2-2\alpha}\int^{\infty}_{\frac{x_{n}^{2}}{1-t}}s^{-2+\alpha}e^{-\delta_{1}s}ds\approx\left\\{\begin{array}[]{cl}\vspace{2mm}\phi^{\prime}(1-t)x_{n}^{2-2\alpha}&\mbox{if}\,\,\,\alpha>1,\\\
\phi^{\prime}(1-t)(1-t)^{1-\alpha}&\mbox{if}\,\,\,\alpha<1.\end{array}\right.\end{split}$

Since $e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\approx 1$ for $1-t\leq s$, it
follows from (2.20) that

(3.60)
$\displaystyle\begin{split}I_{1}&\approx\int_{2-2t}^{1}s^{-\alpha}\phi^{\prime}(s)ds\approx(1-t)^{-\alpha+1}\phi^{\prime}(1-t).\end{split}$

From (3.56) (3.57), (LABEL:eq1017-10-1), (3.59) and (3.60), we have

$\displaystyle{\tilde{\Psi}}_{\phi,1}(t)$
$\displaystyle\approx\left\\{\begin{array}[]{cl}\vspace{2mm}\phi(1-t)x_{n}^{2-2\alpha}&\mbox{if}\,\,\,\alpha>1,\\\
(1-t)^{-\alpha+1}\phi(1-t)&\mbox{if}\,\,\,\alpha<1.\end{array}\right.$

We obtain (3.24) for $x_{n}^{2}<1-t$.

On the other hand, in case that $1-t<x_{n}^{2}$, we split $I_{1}$ as follows:

(3.61) $\displaystyle I_{1}$
$\displaystyle=\int_{2-2t}^{x_{n}^{2}}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi^{\prime}(s)ds+\int_{x_{n}^{2}}^{\frac{1}{2}}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi^{\prime}(s)ds:=I_{11}+I_{12}.$

It is straightforwad from (3.52) that

(3.62)
$\displaystyle\begin{split}I_{12}\approx\int^{\frac{1}{2}}_{x_{n}^{2}}s^{-\alpha}\phi^{\prime}(s)ds\approx
x_{n}^{-2\alpha+2}\phi^{\prime}(x_{n}^{2}).\end{split}$

With the same estimate of (3.53), we have

(3.63) $\displaystyle\begin{split}{\rm
sgn}(\phi^{\prime})\int_{2-2t}^{x_{n}^{2}}s^{-\alpha}e^{-\frac{x_{n}^{2}}{4s}}\phi^{\prime}(s)ds&\leq
c\,{\rm
sgn}(\phi^{\prime})x_{n}^{-2\alpha+2}\phi^{\prime}(x_{n}^{2}).\end{split}$

Adding up (3.61) and (3.63), we obtain for $1-t<x_{n}^{2}$

(3.64) $\displaystyle
I_{1}\approx\int_{\frac{1}{2}}^{2t-1}(1-s)^{-\alpha}e^{-\delta_{1}\frac{x_{n}^{2}}{1-s}}\phi^{\prime}(1-s)ds\approx
x_{n}^{-2\alpha+2}\phi^{\prime}(x_{n}^{2}).$

Direct computations via the change of variables give

(3.65)
$\displaystyle\begin{split}I_{2}&\approx\phi^{\prime}(1-t)x_{n}^{2-2\alpha}\int^{\infty}_{\frac{x_{n}^{2}}{1-t}}s^{-2+\alpha}e^{-\delta_{1}s}ds\approx\phi^{\prime}(1-t)(1-t)^{2-\alpha}x_{n}^{-2}e^{-\delta\frac{x_{n}^{2}}{1-t}}.\end{split}$

Hence, from (3.56), (3.57), (LABEL:eq1017-10-1), (3.64), (3.65) and Lemma 2.2,
we obtain (3.24) for $1-t\leq x_{n}^{2}$. Following similar way of proving
(3.7) and (3.15), one can also obtain that

(3.66)
$\Psi_{\phi,1}(t)\approx\left|t-1\right|^{1-\alpha}\phi^{\prime}(\left|t-1\right|),\qquad\frac{3}{4}<t<\frac{5}{4}.$

which implies (3.8) and (3.16). Since its proof is tedious repetition, the
details are omitted. ∎

Next, we provide the proofs of Lemma 2.2 and Lemma 2.4.

Proof of Lemma 2.2 Let $k\in{\mathbb{N}}\cup\\{0\\}$ be an integer with
$2^{k}\leq\frac{x_{n}^{2}}{t}<2^{k+1}$. Due to the Assumption 2.1, it follows
that

(3.67) $c^{-k-1}\phi(t)\leq\phi(2^{k+1}t)\leq\phi(x_{n}^{2}).$

We choose a positve integer $m$ such that $c\leq 2^{m}$, where $c$ is the
constant in (3.67). Noting that $e^{-x}\leq c_{m}x^{-m}$, we obtain

$e^{-\delta_{1}\frac{x_{n}^{2}}{t}}\phi(t)\leq
e^{-\delta_{1}2^{k}}c^{k+1}\phi(x_{n}^{2})\leq
c_{m}(\delta_{1}2^{k})^{-m}c^{k+1}\phi(x_{n}^{2})\leq
c_{m}c\delta_{1}^{-m}(\frac{c}{2^{m}})^{k}\phi(x_{n}^{2})\leq
c_{m}c\delta_{1}^{-m}\phi(x_{n}^{2}).$

Since $cc_{m}$ is an absolute constant, the first inequality in (2.27) is
proved. For the second inquality, it follows from the following observation:

$c^{-k-1}\,{\rm sgn}(\phi^{\prime}(t))\,\phi^{\prime}(t)\leq\,{\rm
sgn}(\phi^{\prime}(2^{k+1}t))\phi^{\prime}(2^{k+1}t)\leq{\rm
sgn}(\phi^{\prime}(x_{n}^{2}))\phi(x_{n}^{2}).$

Similar computations as above deduce the second inequality, and thus we skip
its details. In the case that $\phi=t^{-a}$, it is starightforward that
$t^{-1}\phi(t)\approx{\rm sgn}(\phi^{\prime})\phi^{\prime}(t)$ for
$0<t<\frac{1}{2}$. Again, the similar arguments as (2.27) implies (2.28), and
the details are omitted. ∎

Proof of Lemma 2.4 The second equality of (2.30) was shown in Lemma 3.2 in
[8]. Thus it suffices to prove the first equality of (2.30). Let
$x^{\prime}\in{\mathbb{R}}^{n-1}$ with $\left|x^{\prime}\right|\geq 1$. We
divide ${\mathbb{R}}^{n-1}$ by three disjoint sets $D_{1},D_{2}$ and $D_{3}$,
which are defined by

$D_{1}=\left\\{z^{\prime}\in{\mathbb{R}}^{n-1}:|x^{\prime}-z^{\prime}|\leq\frac{1}{10}|x^{\prime}|\right\\},$
$D_{2}=\left\\{z^{\prime}\in{\mathbb{R}}^{n-1}:|z^{\prime}|\leq\frac{1}{10}|x^{\prime}|\right\\},\qquad
D_{3}={\mathbb{R}}^{n-1}\setminus(D_{1}\cup D_{2}).$

We then split the following integral into three terms as follows:

(3.68)
$\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-z^{\prime},t)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}=\int_{D_{1}}\cdots+\int_{D_{2}}\cdots+\int_{D_{3}}\cdots:=J_{1}+J_{2}+J_{3}.$

We first estimtate $J_{2}$.

$\displaystyle J_{2}$
$\displaystyle=\int_{|z^{\prime}|\leq\frac{1}{10}|x^{\prime}|}D_{x_{n}}N(z^{\prime},x_{n})\Gamma^{\prime}(x^{\prime}-z^{\prime},t)\big{)}dz^{\prime}$
$\displaystyle\approx
ct^{-\frac{n-1}{2}}e^{-\delta\frac{|x^{\prime}|^{2}}{t}}\int_{|z^{\prime}|\leq\frac{1}{10}|x^{\prime}|}\frac{x_{n}}{(|z^{\prime}|^{2}+x_{n}^{2})^{\frac{n}{2}}}dz^{\prime}$
$\displaystyle\approx
ct^{-\frac{n-1}{2}}e^{-\delta\frac{|x^{\prime}|^{2}}{t}}\int_{|z^{\prime}|\leq\frac{|x^{\prime}|}{10x_{n}}}\frac{1}{(|z^{\prime}|^{2}+1)^{\frac{n}{2}}}dz^{\prime}$
(3.69) $\displaystyle\approx
ct^{-\frac{n-1}{2}}e^{-\delta\frac{|x^{\prime}|^{2}}{t}},$

where the mean value theorem and
$e^{-\frac{|x^{\prime}|^{2}}{2t}}\lesssim(\frac{t^{\frac{1}{2}}}{\left|x^{\prime}\right|})^{n+1}$
are used.

On the other hand, the term $J_{3}$ is controlled as follows:

$\displaystyle|J_{3}|$ $\displaystyle\leq
c\frac{x_{n}}{(|x^{\prime}|^{2}+x_{n}^{2})^{\frac{n}{2}}}t^{-\frac{n-1}{2}}\int_{\\{|z^{\prime}|\geq\frac{1}{10}|x^{\prime}|\\}}e^{-\frac{|z^{\prime}|^{2}}{t}}dz^{\prime}$
$\displaystyle\leq
c\frac{x_{n}}{(|x^{\prime}|^{2}+x_{n}^{2})^{\frac{n}{2}}}\int_{\\{|z^{\prime}|\geq\frac{|x^{\prime}|}{10\sqrt{t}}\\}}e^{-|z^{\prime}|^{2}}dz^{\prime}$
(3.70) $\displaystyle\leq
c\frac{x_{n}}{(|x^{\prime}|^{2}+x_{n}^{2})^{\frac{n}{2}}}e^{-\frac{|x^{\prime}|^{2}}{t}}\leq
cx_{n}t^{\frac{1}{2}},$

where we used that
$e^{-\frac{|x^{\prime}|^{2}}{2t}}\lesssim\frac{t^{\frac{m}{2}}}{\left|x^{\prime}\right|^{m}}$
for $m\geq 0$ (in fact, we take $m=\frac{1}{2}$).

Meanwhile, we decompose $J_{1}$ in the following way.

$\displaystyle J_{1}$
$\displaystyle=\int_{D_{1}}\Gamma^{\prime}(x^{\prime}-z^{\prime},t)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}$
$\displaystyle=\int_{\\{|z^{\prime}|\leq\frac{1}{10}\frac{|x^{\prime}|}{\sqrt{t}}\\}}\Gamma^{\prime}(z^{\prime},1)\Big{(}D_{x_{n}}N(x^{\prime}-\sqrt{t}z^{\prime},x_{n})-D_{x_{n}}N(x^{\prime},x_{n})\Big{)}dz^{\prime}$
$\displaystyle\quad+D_{x_{n}}N(x^{\prime},x_{n})\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(z^{\prime},1)dz^{\prime}-D_{x_{n}}N(x^{\prime},x_{n})\int_{\\{|z^{\prime}|\geq\frac{1}{10}\frac{|x^{\prime}|}{\sqrt{t}}\\}}\Gamma^{\prime}(z^{\prime},1)dz^{\prime}$
$\displaystyle=J_{11}+D_{x_{n}}N(x^{\prime},x_{n})+J_{12},$

where we used
$\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(z^{\prime},1)dz^{\prime}=1$. Since
$\frac{|x^{\prime}|^{2}}{t}\geq 1$, we observe that

(3.71)
$\begin{split}|J_{11}(x^{\prime},t)|\leq&c\frac{x_{n}}{(|x^{\prime}|^{2}+x_{n}^{2})^{\frac{n+1}{2}}}t^{\frac{1}{2}}\int_{\\{|z^{\prime}|\leq\frac{1}{10}\frac{|x^{\prime}|}{\sqrt{t}}\\}}e^{-|z^{\prime}|^{2}}|z^{\prime}|dz^{\prime}\\\
\leq&c\frac{x_{n}}{(|x^{\prime}|^{2}+x_{n}^{2})^{\frac{n+1}{2}}}t^{\frac{1}{2}}\leq
cx_{n}t^{\frac{1}{2}}\end{split}$

and

(3.72)
$\begin{split}|J_{12}(x^{\prime},t)|\leq&c\frac{x_{n}}{(|x^{\prime}|^{2}+x_{n}^{2})^{\frac{n}{2}}}\int_{\\{|z^{\prime}|\geq\frac{1}{10}\frac{|x^{\prime}|}{\sqrt{t}}\\}}e^{-|z^{\prime}|^{2}}dz^{\prime}\\\
\leq&c\frac{x_{n}}{(|x^{\prime}|^{2}+x_{n}^{2})^{\frac{n}{2}}}e^{-\frac{|x^{\prime}|^{2}}{2t}}\leq
cx_{n}t^{\frac{1}{2}}.\end{split}$

Adding up (3) -(3.72), we deduce (2.30)-$\eqref{Jkl}$. ∎

##  4\. Proof of Theorem 2.6

###  4.1. Estimate of $w$

We remind the decomposition of $w$, which given as $w=w^{L}+w^{N}+w^{B}$ in
(2.29). Let $x^{\prime}\in B^{\prime}_{1}\subset\mathbb{R}^{n-1}$. Noting that
$|x^{\prime}-y^{\prime}|\geq\frac{1}{2}$ for $y^{\prime}\in A$ and recalling
boundary conditions in (2.14)-(2.15) with Assumption 2.1, it follows for
$\frac{3}{4}<t<\frac{5}{4}$ that

(4.1)
$\displaystyle\begin{split}w^{L}_{i}(x,t)&=\int_{0}^{t}\int_{A}L_{ni}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds\\\
&\leq c\int_{0}^{t}(t-s)^{-\frac{1}{2}}\phi(1-s)ds\\\
&\approx\max(1,\left|t-1\right|^{\frac{1}{2}}\phi(|t-1|)),\end{split}$

where we used (2.9).

On the other hand, we note that $w^{N}_{i}$ vanishes for $t>1$. Furthermore,
it follows from (2.29) that

(4.2) $\displaystyle
w^{N}_{i}(x,t)=2\int_{A}D_{x_{i}}N(x^{\prime}-y^{\prime},x_{n})g_{n}(y^{\prime},t)dy^{\prime}=2T_{i}g^{\mathcal{S}}_{n}(x^{\prime})\phi(1-t),$

where

$T_{i}g^{\mathcal{S}}_{n}(x^{\prime})=\int_{A}D_{x_{i}}N(x^{\prime}-y^{\prime},x_{n})g^{\mathcal{S}}_{n}(y^{\prime})dy^{\prime}.$

Since $T_{i}g^{\mathcal{S}}_{n}(x^{\prime})$ is positive and finite, we have
$T_{i}g^{\mathcal{S}}_{n}(x^{\prime})\approx 1$ for $|x^{\prime}|<\frac{1}{2}$
and $x_{n}\in(0,\frac{1}{2})$, which implies via (4.2) that

(4.3) $\displaystyle w^{N}_{i}(x,t)\approx\phi(1-t){\bf
I}_{\left\\{t<1\right\\}}.$

It remains to estimate $w_{i}^{\mathcal{B}}$. Since $w_{i}^{\mathcal{B}}=0$
for $i=n$, we note that $|x^{\prime}-y^{\prime}|\geq 1$ for
$|x^{\prime}|<\frac{1}{2}$ and $y^{\prime}\in A$. Using (2.32) and (2.34), for
$i\neq n$, we have for $\frac{3}{4}<t<\frac{5}{4}$

(4.4)
$\displaystyle\begin{split}w_{i}^{\mathcal{B}}(x,t)&=\int_{0}^{t}D_{x_{n}}\Gamma_{1}(x_{n},t-s)g_{n}^{\mathcal{T}}(s)\int_{A}\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t)D_{z_{i}}N(z^{\prime},0)dz^{\prime}g^{\mathcal{S}}_{n}(y^{\prime})dy^{\prime}ds\\\
&\approx\int_{0}^{t}D_{x_{n}}\Gamma_{1}(x_{n},t-s)g_{n}^{\mathcal{T}}(s)ds.\end{split}$

We separately treat (4.4) for the cases that $t<1$ and for $t>1$.

$\bullet$ (The case $1-t_{0}<t<1$) Using (4.4) and (3.20), we observe that

(4.5)
$\displaystyle\begin{split}w_{i}^{\mathcal{B}}(x,t)&\approx-\int_{0}^{t}\frac{x_{n}}{(t-s)^{\frac{3}{2}}}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi(1-s)ds\\\
&\approx-\left\\{\begin{array}[]{ll}\vspace{2mm}\phi(1-t)&\mbox{if}\,\,x_{n}<(1-t)^{\frac{1}{2}},\\\
\phi(x_{n}^{2})+\phi(1-t)(1-t)^{\frac{1}{2}}x_{n}^{-1}e^{-\delta\frac{x_{n}^{2}}{1-t}}&\mbox{if}\,\,(1-t)^{\frac{1}{2}}<x_{n}.\end{array}\right.\end{split}$

Therefore, we have

(4.6)
$\displaystyle\begin{split}\left|w_{i}^{\mathcal{B}}(x,t)\right|\lesssim\left\\{\begin{array}[]{ll}\vspace{2mm}1&\mbox{if}\,\,\phi(t)=\left|\ln
t\right|^{-1}\\\ \phi(1-t)&\mbox{otherwise}.\end{array}\right.\end{split}$

Adding up (4.1), (4.3) and (4.6), we obtain

(4.7)
$\displaystyle\begin{split}\left|w_{i}(x,t)\right|\lesssim\left\\{\begin{array}[]{ll}\vspace{2mm}1&\mbox{if}\,\,\phi(t)=\left|\ln
t\right|^{-1}\\\ \phi(1-t)&\mbox{otherwise}.\end{array}\right.\end{split}$

We complete the proof of (2.41) for $t<1$.

Let $\phi\neq\left|\ln t\right|^{-1}$ and we choose a sufficiently large
$\epsilon_{1}\gg 1$. We then show that there exists a sufficiently small
$\epsilon$ depending on $\epsilon_{1}$ such that
$w^{B}(x,t)>-\epsilon\phi(1-t)$ in
$\tilde{{\mathcal{D}}}_{-}=\left\\{(x,t)\in{\mathcal{D}}_{-}:\epsilon_{1}\sqrt{1-t}<x_{n}<\frac{1}{2}\right\\}$.
Indeed, recalling the second estimate in (4.5), it follows that

(4.8)
$w_{i}^{\mathcal{B}}(x,t)\gtrapprox-\phi(\epsilon_{1}^{2}(1-t))-\phi(1-t)\epsilon_{1}^{-1}e^{-\delta\epsilon_{1}^{2}}\geq-\epsilon\phi(1-t).$

In a similar way, we can see via (4.1) that

(4.9) $|w^{L}(x,t)|<\epsilon\phi(1-t),\qquad\mbox{ for
}\,\,(x,t)\in\tilde{{\mathcal{D}}}_{-}.$

Summing up estimates (4.3), (4.8) and (4.9), we conclude in
$\tilde{{\mathcal{D}}}_{-}$ that

(4.10) $\displaystyle w_{i}(x,t)\geq
w^{N}_{i}(x,t)+w^{B}_{i}(x,t)-\left|w^{L}_{i}(x,t)\right|\geq(c-2\epsilon)\phi(1-t)\geq\frac{c}{2}\phi(1-t).$

We complete the proof of (2.42) for $t<1$.

$\bullet$ (The case $1<t<1+t_{0}$) With the aid of (4.5) and (3.11), we have

(4.11)
$\displaystyle\begin{split}w_{i}^{\mathcal{B}}(x,t)&\approx-\int_{0}^{1}\frac{x_{n}}{(t-s)^{\frac{3}{2}}}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi(1-s)ds\\\
&\approx-\left\\{\begin{array}[]{ll}\vspace{2mm}x_{n}(t-1)^{-\frac{1}{2}}\phi(t-1)&\mbox{if}\,\,x_{n}^{2}\leq
t-1,\\\ \max(1,\phi(x_{n}^{2}))&\mbox{if}\,\,t-1\leq
x_{n}^{2}.\end{array}\right.\end{split}$

Owing to (4.1), (4.3) and (4.11), it follows that

(4.12)
$\displaystyle\begin{split}\left|w_{i}(x,t)\right|\lesssim\left\\{\begin{array}[]{ll}\vspace{2mm}1&\mbox{if}\,\,\phi(t)=\left|\ln
t\right|^{-1}\\\ \phi(1-t)&\mbox{otherwise},\end{array}\right.\end{split}$

which implies (2.41) for $t\in(1,\frac{5}{4})$ and thus together with (4.7),
the proof of (2.41) is completed. Moreover, if $\phi\neq\left|\ln
t\right|^{-1}$, by choosing sufficiently small $t_{0}$ such that for
$\epsilon_{1}\sqrt{t-1}<x_{n}$ and $\sqrt{t-1}<t_{0}$, we have

(4.13) $\displaystyle
w^{B}_{i}(x,t)+w^{L}_{i}(x,t)\leq\big{(}-c+c_{1}\sqrt{t-1}\big{)}\phi(t-1)\leq-\frac{c}{2}\phi(t-1).$

On the other hand, from (4.1) and (4.11), for $1<t<1+t_{0}$ and
$\frac{1}{2}\sqrt{t-1}<x_{n}<\sqrt{t-1}$, we have

$\displaystyle
w_{i}(x,t)\leq(-\frac{c}{2}+\epsilon)\phi(t-1)\leq-\frac{c}{4}\phi(t-1).$

This completes the proof of (2.43). ∎

###  4.2. Estimate of $\nabla w$

We note for $1\leq i\leq n-1$ that

(4.14) $\displaystyle D_{x_{n}}w^{L}_{i}(x,t)$
$\displaystyle=-\sum_{k=1}^{n-1}D_{x_{k}}w^{L}_{ik}(x,t)+\frac{1}{2}D_{x_{i}}f(x,t),$

where

$\displaystyle f(x,t)$
$\displaystyle=\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}D_{x_{n}}\Gamma(x^{\prime}-z^{\prime},x_{n},t-s)g_{n}(z^{\prime},s)dz^{\prime}ds,$
$\displaystyle w^{L}_{ik}(x,t)$
$\displaystyle=\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}L_{ik}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds,\quad
k=1,2,\cdots,n-1.$

Similar estimates as in (4.1) imply for $\frac{3}{4}<t<\frac{5}{4}$ that

(4.15) $\displaystyle D_{x_{n}}w^{L}_{i}(x,t)$ $\displaystyle\leq
c\max(1,|t-1|^{\frac{1}{2}}\phi(|t-1|)).$

It is also straightforward as in (4.3) that

(4.16) $\displaystyle|D_{x}w^{N}_{i}(x,t)|\leq c\phi(\left|t-1\right|){\bf
I}_{\left\\{t<1\right\\}}.$

We can easily see that tangential derivatives of $w^{\mathcal{B}}_{i}$ satisfy
essentially the same estimates as those of $w^{\mathcal{B}}_{i}$, namely
$|D_{x^{\prime}}w_{i}^{\mathcal{B}}(x,t)|$ satisfies (4.6). Therefore, it
remains to estimate $D_{x_{n}}w_{i}^{\mathcal{B}}$. As before, we treat the
cases that $t<1$ and $t>1$ seperately.

$\bullet$ (The case $1-t_{0}<t<1$) We recall that

$\displaystyle B_{in}(x^{\prime},x_{n},t)=D_{x_{n}}A_{i}(x^{\prime},x_{n},t),$

where

(4.17) $\displaystyle
A_{i}(x,t)=\int_{{\mathbb{R}}^{n-1}}\Gamma(x^{\prime}-z^{\prime},x_{n},t)D_{z_{i}}N(z^{\prime},0)dz^{\prime},\qquad
i=1,2,\cdots,n-1.$

Using that $(D_{t}-\Delta)A_{i}=0$ for
$(x,t)\in{\mathbb{R}}^{n}_{+}\times(0,\infty)$ and
$g_{n}^{\mathcal{T}}(y^{\prime},0)=0$, we have

(4.18)
$\displaystyle\begin{split}D_{x_{n}}w_{i}^{\mathcal{B}}(x,t)&=\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}D_{x_{n}}B_{in}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}^{\mathcal{T}}(y^{\prime},s)dy^{\prime}ds\\\
&=\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}(D_{t}-\Delta^{\prime}_{x})A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}^{\mathcal{T}}(y^{\prime},s)dy^{\prime}ds\\\
&=\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)(D_{s}-\Delta^{\prime}_{y})g_{n}^{\mathcal{T}}(y^{\prime},s)dy^{\prime}ds.\end{split}$

Due to the estimates (2.32), and (3.20), the second term is estimated as
follows:

(4.19)
$\displaystyle\begin{split}&\left|\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)\Delta^{\prime}_{y}g_{n}^{\mathcal{T}}(y^{\prime},s)dy^{\prime}ds\right|\\\
\lesssim&\int_{0}^{t}\frac{1}{(t-s)^{\frac{1}{2}}}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi(1-s)ds\\\
\approx&\left\\{\begin{array}[]{cl}\max(1,x_{n}\phi(x_{n}^{2}))+\phi(1-t)(1-t)^{\frac{3}{2}}x_{n}^{-2}e^{-\delta_{1}\frac{x_{n}^{2}}{1-t}}&\mbox{if}\,\,1-t<x_{n}^{2},\vspace{2mm}\\\
\max(1,(1-t)^{\frac{1}{2}}\phi(1-t))&\mbox{if}\,\,x_{n}^{2}<1-t.\end{array}\right.\end{split}$

On the other hand, due to (2.32), (2.34) and (3.24), we compute for the first
term as follows:

(4.20)
$\displaystyle\begin{split}&\int_{0}^{t}\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)D_{s}g_{n}^{\mathcal{T}}(y^{\prime},s)dy^{\prime}ds\\\
\approx&-\int_{0}^{t}\frac{1}{(t-s)^{\frac{1}{2}}}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds\\\
\approx&-\left\\{\begin{array}[]{cl}\vspace{2mm}x_{n}\phi^{\prime}(x_{n}^{2})+\phi^{\prime}(1-t)(1-t)^{\frac{3}{2}}x_{n}^{-2}e^{-\delta_{1}\frac{x_{n}^{2}}{1-t}}&\mbox{if}\,\,1-t<x_{n}^{2},\\\
(1-t)^{\frac{1}{2}}\phi^{\prime}(1-t)&\mbox{if}\,\,x_{n}^{2}<1-t.\end{array}\right.\end{split}$

Thus, using Lemma 2.2, it follows that

(4.21) $\displaystyle|D_{x_{n}}w^{B}_{i}(x,t)|$
$\displaystyle\leq-c(1-t)^{\frac{1}{2}}\phi^{\prime}(1-t){\bf
I}_{{\mathcal{D}}_{+}}+x_{n}\phi^{\prime}(x_{n}^{2}){\bf
I}_{{\mathcal{D}}_{-}}.$

Hence, we obtain the second part in (2.44). In addition, we also note that

(4.22)
$\displaystyle\begin{split}D_{x_{n}}w_{i}(x,t)&\approx\left\\{\begin{array}[]{cl}\vspace{2mm}-(1-t)^{\frac{1}{2}}\phi^{\prime}(1-t)&\mbox{
if }\,\,0<1-t<\min\left\\{t_{0},t_{1}\right\\}\mbox{ and
}(x,t)\in\tilde{{{\mathcal{D}}}_{+}}\\\ -x_{n}\phi^{\prime}(x_{n}^{2})&\mbox{
if }\,\,0<1-t<\min\left\\{t_{0},t_{1}\right\\}\mbox{ and
}(x,t)\in\tilde{{{\mathcal{D}}}_{-}}.\end{array}\right.\end{split}$

This completes the proof of (2.45).
$\bullet$ (The case $1<t<1+t_{0}$)
We first recall that

(4.23)
$\displaystyle\begin{split}D_{x_{n}}w_{i}^{\mathcal{B}}(x,t)&=\int_{0}^{1}\int_{{\mathbb{R}}^{n-1}}D^{2}_{x_{n}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds,\end{split}$

where $A_{i}$ is defined in (4.17). Since
$|D^{2}_{x_{n}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)|\leq
c(t-s)^{-\frac{3}{2}}e^{-\frac{x_{n}^{2}}{4(t-s)}}$, it follows from (3.11)
that

$\displaystyle|D_{x_{n}}w_{i}^{\mathcal{B}}(x,t)|\leq
c\int_{0}^{1}(t-s)^{-\frac{3}{2}}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi(1-s)ds\leq
c\left((t-1)\vee x_{n}^{2}\right)^{-\frac{1}{2}}\phi\left((t-1)\vee
x_{n}^{2}\right)).$

We complete the proof of the first part in (2.44).

In case that $x_{n}\leq\epsilon_{0}\sqrt{t-1}$, it is direct that for $0<s<1$,

$D^{2}_{x_{n}}\Gamma_{1}(x_{n},t-s)\approx-(t-s)^{-\frac{3}{2}}e^{-\delta_{1}\frac{x_{n}^{2}}{t-s}}\approx-(t-s)^{-\frac{3}{2}}.$

Hence, it follows from (2.32), (2.34) and (3.7) that

(4.24)
$\displaystyle\begin{split}D_{x_{n}}w_{i}^{\mathcal{B}}(x,t)\approx-(t-1)^{-\frac{1}{2}}\phi(t-1).\end{split}$

Combining (4.15) and (4.24), for $x_{n}\leq\epsilon_{0}\sqrt{t-1}$, we have

$\displaystyle\begin{split}D_{x_{n}}w_{i}(x,t)\approx-(t-1)^{-\frac{1}{2}}\phi(t-1),\end{split}$

which completes the proof of (2.46) for $x_{n}\leq\epsilon_{0}\sqrt{t-1}$.

Next, we consider the case that $\epsilon_{1}\sqrt{t-1}<x_{n}$. Dividing time
interval into two parts such as $0<s<2-t$ and $2-t<s<1$, we split the integral

(4.25)
$\displaystyle\begin{split}D_{x_{n}}w_{i}^{\mathcal{B}}(x,t)=&\int_{0}^{2-t}\int_{{\mathbb{R}}^{n-1}}D^{2}_{x_{n}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds\\\
&+\int_{2-t}^{1}\int_{{\mathbb{R}}^{n-1}}D^{2}_{x_{n}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds\\\
:=&J_{1}+J_{2}.\end{split}$

Noting that
$D^{2}_{x_{n}}\Gamma_{1}(x_{n},t-s)\approx(t-1)^{-\frac{5}{2}}x_{n}^{2}e^{-\frac{x_{n}^{2}}{4(t-1)}}$
for $2-t<s$, we obtain

(4.26)
$\displaystyle\begin{split}J_{2}&\approx(t-1)^{-\frac{5}{2}}x_{n}^{2}e^{-\frac{x_{n}^{2}}{4(t-1)}}\int_{2-t}^{1}\phi(1-s)ds\approx(t-1)^{-\frac{3}{2}}x_{n}^{2}e^{-\frac{x_{n}^{2}}{4(t-1)}}\phi(t-1).\end{split}$

Using that $(D_{t}-\Delta)A_{i}=0$ for
$(x,t)\in{\mathbb{R}}^{n}_{+}\times(0,\infty)$. Since $\phi(1)=0$ and
integrating $J_{1}$ by parts, we get

(4.27)
$\displaystyle\begin{split}J_{1}=&\int_{0}^{2-t}\int_{{\mathbb{R}}^{n-1}}(D_{t}-\Delta^{\prime}_{x})A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)g_{n}(y^{\prime},s)dy^{\prime}ds\\\
=&\int_{0}^{2-t}\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)D_{s}g_{n}(y^{\prime},s)dy^{\prime}ds\\\
&-\int_{0}^{2-t}\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)\Delta^{\prime}_{y}g_{n}(y^{\prime},s)dy^{\prime}ds\\\
&-\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},2t-2)\phi(t-1)g^{\mathcal{S}}_{n}(y^{\prime})dy^{\prime}\\\
:=&J_{11}+J_{12}+J_{13}.\end{split}$

It is straightforward from (2.32) and (2.34) that $J_{13}$ satisfies

(4.28) $\displaystyle
J_{13}=-\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},2t-2)\phi(t-1)g_{n}(y^{\prime})dy^{\prime}$
$\displaystyle\approx-\frac{1}{(t-1)^{\frac{1}{2}}}e^{-\frac{x_{n}^{2}}{4(t-1)}}\phi(t-1).$

From the proof of (3.11) , the second term $J_{12}$ is controlled as follows:

(4.29)
$\displaystyle\begin{split}\left|J_{12}\right|&\leq\left|\int_{0}^{2-t}\int_{{\mathbb{R}}^{n-1}}A_{i}(x^{\prime}-y^{\prime},x_{n},t-s)\Delta^{\prime}_{y}g_{n}(y^{\prime},s)dy^{\prime}ds\right|\\\
&\leq
c\int_{0}^{2-t}\frac{1}{(t-s)^{\frac{1}{2}}}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi(1-s)ds\\\
&\leq
c\int_{0}^{2-t}\frac{1}{(1-s)^{\frac{1}{2}}}e^{-\delta_{1}\frac{x_{n}^{2}}{1-s}}\phi(1-s)ds\\\
&=c\int_{t-1}^{1}\frac{1}{s^{\frac{1}{2}}}e^{-\delta_{1}\frac{x_{n}^{2}}{s}}\phi(s)ds\\\
&\leq c\max(1,x_{n}\phi(x_{n}^{2})).\end{split}$

Due to (3.12), we have

(4.30) $\displaystyle J_{11}$
$\displaystyle\approx-\int_{0}^{2-t}\frac{1}{(t-s)^{\frac{1}{2}}}e^{-\frac{x_{n}^{2}}{4(t-s)}}\phi^{\prime}(1-s)ds\approx-
x_{n}\phi^{\prime}(x_{n}^{2}).$

We note for $\epsilon_{1}\sqrt{t-1}<x_{n}$ with sufficiently large
$\epsilon_{1}>0$ that

$\displaystyle(t-1)^{-\frac{3}{2}}x_{n}^{2}e^{-\frac{x_{n}^{2}}{4(t-1)}}\phi(t-1)-(t-1)^{-\frac{1}{2}}e^{-\frac{x_{n}^{2}}{4(t-1)}}\phi(t-1)$
$\displaystyle\approx(t-1)^{-\frac{3}{2}}x_{n}^{2}e^{-\frac{x_{n}^{2}}{4(t-1)}}\phi(t-1),$
$\displaystyle\max(1,x_{n}\phi(x_{n}^{2}))-x_{n}\phi^{\prime}(x_{n}^{2})$
$\displaystyle\approx-x_{n}\phi^{\prime}(x_{n}^{2}).$

From (4.25) to (4.30) , for $t-1<t_{0}$ for small $t_{0}$, we have

(4.31) $\displaystyle\begin{split}D_{x_{n}}w_{i}^{\mathcal{B}}(x,t)&\approx-
x_{n}\phi^{\prime}(x_{n}^{2})+(t-1)^{-\frac{3}{2}}x_{n}^{2}e^{-\frac{x_{n}^{2}}{4(t-1)}}\phi(t-1).\end{split}$

With the aid of (4.15), (4.31) and Lemma 2.2, we complete the proofs of (2.47)
for $\epsilon_{1}\sqrt{t-1}\leq x_{n}$. ∎

###  4.3. Estimate of pressure

For convenience, we denote

$\displaystyle{\mathcal{U}}g_{n}(x^{\prime},0,t)=\int_{0}^{t}\int_{A}\Gamma(x^{\prime}-y^{\prime},0,t-s)g_{n}(y^{\prime},s)dy^{\prime}ds.$

Recalling that $g_{n}(x^{\prime},0)=0$ and (2.13), we decompse
$p=p_{1}+p_{2}+p_{3}$, where

(4.32)
$\displaystyle\begin{split}p_{1}(x,t)&=-D_{x_{n}}D_{x_{n}}N*^{\prime}g_{n}(x,t),\\\
p_{2}(x,t)&=2(D_{t}-\Delta^{\prime})D_{x_{n}}N*^{\prime}{\mathcal{U}}g_{n}(x,t)\\\
&=2D_{t}D_{x_{n}}N*^{\prime}{\mathcal{U}}g_{n}(x,t)-2\Delta^{\prime}D_{x_{n}}N*^{\prime}{\mathcal{U}}g_{n}(x,t)\\\
&:=p_{21}+p_{22},\\\ p_{3}(x,t)&=N*^{\prime}D_{t}g_{n}(x,t).\end{split}$

Since $|x^{\prime}-y^{\prime}|\geq 1$ for $|x^{\prime}|<\frac{1}{2}$ and
$y^{\prime}\in A$, we can estimate $p_{1}$ and $p_{3}$ as follows:

(4.33) $\displaystyle\begin{split}|p_{1}(x,t)|&\leq c\phi(1-t){\bf
I}_{\left\\{0<t<1\right\\}},\\\ p_{3}(x,t)&\approx-\phi^{\prime}(1-t){\bf
I}_{\left\\{0<t<1\right\\}}.\end{split}$

It remains to estimate $p_{2}$. Firstly, for $p_{22}$ we note that

(4.34)
$\displaystyle\begin{split}p_{22}(x,t)&=-2\Delta^{\prime}D_{x_{n}}N*^{\prime}{\mathcal{U}}g_{n}(x,t)=-2D_{x_{n}}N*^{\prime}{\mathcal{U}}\Delta^{\prime}g_{n}(x,t)\\\
&=-2\int_{0}^{t}\int_{A}(t-s)^{-\frac{1}{2}}\Delta^{\prime}g_{n}(y^{\prime},s)\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}dy^{\prime}ds.\end{split}$

From (4.34) and $\eqref{240116-1}$, we have

(4.35) $\displaystyle\begin{split}\left|p_{22}(x,t)\right|&\leq
c\int_{0}^{t}(t-s)^{-\frac{1}{2}}\big{(}x_{n}+(t-s)^{-\frac{n-1}{2}}e^{-\frac{1}{(t-s)}}\big{)}g^{\mathcal{T}}_{n}(s)ds\\\
&\leq c\big{(}1+x_{n}\max(1,|t-1|^{\frac{1}{2}}\phi(|t-1|))\big{)}\\\ &\leq
c\max(1,x_{n}|t-1|^{\frac{1}{2}}\phi(|t-1|).\end{split}$

$\bullet$ (The case $\frac{3}{4}<t<1$)
Since $g_{n}(y^{\prime},0)=0$, by intergration by parts and (2.35), we have

(4.36)
$\displaystyle\begin{split}p_{21}(x,t)&=\int_{0}^{t}\int_{A}(t-s)^{-\frac{1}{2}}\phi^{\prime}(1-s)\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}g_{n}(y^{\prime})dy^{\prime}ds\\\
&\approx\int_{0}^{t}(t-s)^{-\frac{1}{2}}\phi^{\prime}(1-s)\big{(}x_{n}+(t-s)^{-\frac{n-1}{2}}e^{-\delta\frac{1}{(t-s)}}\big{)}ds\\\
&\approx\max(1,x_{n}(1-t)^{\frac{1}{2}}\phi^{\prime}(1-t)).\end{split}$

Therefore, adding up (4.33), (4.35) and (4.36), we otain for
$x_{n}<\frac{1}{2}$ and $1-t<t_{1}$,

(4.37) $\displaystyle p(x,t)\approx-c\phi^{\prime}(1-t).$

We prove the first part of (2.48) for $t<1$.

$\bullet$ (The case $1<t$) Since $p_{1}(x,t)=p_{3}(x,t)=0$ for $t>1$, Lemma
2.4 implies that

(4.38)
$\displaystyle\begin{split}p_{22}&=-2\int_{0}^{1}\int_{{\mathbb{R}}^{n-1}}(t-s)^{-\frac{1}{2}}\Delta^{\prime}g_{n}(y^{\prime},s)\big{(}D_{x_{n}}N(x^{\prime}-y^{\prime},x_{n})+(I+J_{1})(x^{\prime}-y^{\prime},x_{n},t)\big{)}dy^{\prime}ds\\\
&\leq
c\int_{0}^{1}\big{(}x_{n}(t-s)^{-\frac{1}{2}}+(t-s)^{-\frac{n}{2}}e^{-\frac{1}{8(t-s)}}\big{)}\phi(1-s)ds\\\
&\leq c\big{(}1+x_{n}\max(1,(t-1)^{\frac{1}{2}}\phi(t-1))\big{)}\end{split}$

and

(4.39)
$\displaystyle\begin{split}p_{21}&=\int_{0}^{1}\int_{{\mathbb{R}}^{n-1}}(t-s)^{-\frac{3}{2}}\phi(1-s)\int_{{\mathbb{R}}^{n-1}}\Gamma^{\prime}(x^{\prime}-y^{\prime}-z^{\prime},t-s)D_{x_{n}}N(z^{\prime},x_{n})dz^{\prime}g_{n}(y^{\prime})dy^{\prime}ds\\\
&\approx\int_{0}^{1}\big{(}x_{n}(t-s)^{-\frac{3}{2}}+(t-s)^{-\frac{n+1}{2}}e^{-\frac{1}{(t-s)}}\big{)}\phi(1-s)ds\\\
&\approx x_{n}(t-1)^{-\frac{1}{2}}\phi(t-1)+1.\end{split}$

Hence, we have

(4.40) $\displaystyle c_{1}x_{n}(t-1)^{-\frac{1}{2}}\phi(t-1)-c_{2}\leq
p_{21}(x,t)\leq c_{3}x_{n}(t-1)^{-\frac{1}{2}}\phi(t-1)+c_{4},$

which implies the second part of (2.48) for $t>1$, and thus we complete the
proof of (2.48). ∎

##  5\. Proof of Proposition 2.10

To prove Propositiion 2.10, we consider the following equations

(5.4) $\displaystyle\left\\{\begin{array}[]{l}\vspace{2mm}W_{t}-\Delta
W+\nabla Q=f\quad{\mathbb{R}}^{n}_{+}\times(0,T),\\\ \vspace{2mm}{\rm
div}\,W=0\quad{\mathbb{R}}^{n}_{+}\times(0,T),\\\ W|_{t=0}=0,\quad
W|_{x_{n}=0}=0,\end{array}\right.$

where $0<T\leq\infty$. We recall some known results for the system (5.4).

######  Proposition 5.1.

([7, Proposition 2.3]) Let $1<p,q<\infty$. Suppose that $f={\rm
div}\,{\mathcal{F}}$ with $F\in L^{q}(0,\infty;L^{p}({\mathbb{R}}^{n}_{+}))$,
${\mathcal{F}}|_{x_{n}=0}=0$. Then, the solution $W$ of (5.4) satisfies

(5.5) $\displaystyle\|\nabla
W\|_{L^{q}(0,\infty;L^{p}({\mathbb{R}}^{n}_{+}))}\leq
c\|{\mathcal{F}}\|_{L^{q}(0,\infty;L^{p}({\mathbb{R}}^{n}_{+}))}.$

######  Proposition 5.2.

([6, Theorem 1.2]) Let $1<p,q<\infty$. Assume that $f={\rm div}{\mathcal{F}}$
with ${\mathcal{F}}\in L^{q_{1}}(0,\infty;L^{p_{1}}({\mathbb{R}}^{n}_{+}))$,
where $p_{1},q_{1}$ satisfy $n/p_{1}+2/q_{1}=n/p+2/q+1$ with $q_{1}\leq q$ and
$p_{1}\leq p$. Then, there is the unique very weak solution $W\in
L^{q}(0,\infty;L^{p}({\mathbb{R}}^{n}_{+}))$ of the Stokes equations (5.4)
such that $W$ satisfies

$\|W\|_{L^{q}(0,\infty;L^{p}({\mathbb{R}}^{n}_{+}))}\leq
c\|{\mathcal{F}}\|_{L^{q_{1}}(0,\infty;L^{p_{1}}({\mathbb{R}}^{n}_{+}))}.$

######  Proposition 5.3.

[3, Theorem 2.2] Let $n+2<p<\infty$. Let $f={\rm div}\,{\mathcal{F}}$ with
${\mathcal{F}}\in L^{p}({\mathbb{R}}^{n}_{+}\times(0,T))$. Then, there exists
unique very weak solution, $w\in L^{\infty}({\mathbb{R}}^{n}_{+}\times(0,T))$,
of the Stokes equations (5.4) such that

$\|W\|_{L^{\infty}({\mathbb{R}}^{n}_{+}\times(0,T))}\leq
cT^{\frac{1}{2}(1-\frac{n+2}{p})}\|{\mathcal{F}}\|_{L^{p}({\mathbb{R}}^{n}_{+}\times(0,T))}.$

######  Proposition 5.4.

[9, Proposition 3.4] Suppose that $f\in
L^{q_{1}}(0,1;L^{p_{1}}({\mathbb{R}}^{n}_{+}))$ with $1<p_{1},q_{1}<\infty$.
If $p$ and $q$ satisfy $p>\frac{n}{n-1}p_{1}$ and
$\frac{2}{q_{1}}+\frac{n}{p_{1}}-1\leq\frac{2}{q}+\frac{n}{p}$, then

(5.6)
$\|Q\|_{L^{q}(0,1;L^{p}({\mathbb{R}}^{n}_{+}))}<c\|f\|_{L^{q_{1}}(0,1;L^{p_{1}}({\mathbb{R}}^{n}_{+}))}.$

Our aim is to establish the existence of solution $v$ for (2.53) satisfying
$v\in L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,\infty))$ and $\nabla v\in
L^{q}({\mathbb{R}}^{n}_{+}\times(0,\infty))$ for fixed $q>\frac{n+2}{2}$. In
order to do that, we consider the iterative scheme for (2.53), which is given
as follows: For a positive integer $m\geq 1$

$\displaystyle v^{m+1}_{t}-\Delta v^{m+1}+\nabla q^{m+1}=-{\rm
div}\,\left(v^{m}\otimes v^{m}+v^{m}\otimes U+U\otimes v^{m}+U\otimes
U\right),$ $\displaystyle\qquad\qquad\qquad\qquad\qquad{\rm div}\,v^{m+1}=0$

with homogeneous initial and boundary data, i.e. $v^{m+1}(x,0)=0$ and
$v^{m+1}(x,t)=0$ on $\\{x_{n}=0\\}$.

Let $q>\frac{n+2}{2}$. We set $v^{1}=0$. We then have, due to Proposition 5.1
and Proposition 5.2,

(5.7) $\displaystyle\begin{split}\|\nabla
v^{2}\|_{L^{q}({\mathbb{R}}^{n}_{+}\times(0,T))}&\leq
c\||U|^{2}\|_{L^{q}({\mathbb{R}}^{n}_{+}\times(0,T))}\leq
c\|U\|^{2}_{L^{2q}({\mathbb{R}}^{n}_{+}\times(0,T))}\leq
c\|w\|^{2}_{L^{2q}},\\\
\|v^{2}\|_{L^{2q}({\mathbb{R}}^{n}_{+}\times(0,T))}&\leq
c\||U|^{2}\|_{L^{q_{1}}(0,T;L^{q}({\mathbb{R}}^{n}_{+}))}\leq
cT^{\frac{1}{2}(1-\frac{n+2}{2q})}\||U|^{2}\|_{L^{q}({\mathbb{R}}^{n}_{+}\times(0,T))}\\\
&\leq cT^{\frac{1}{2}(1-\frac{n+2}{2q})}\|w\|^{2}_{L^{2q}},\end{split}$

where $\frac{2}{q_{1}}+\frac{n}{q}=\frac{n+2}{2q}+1$. We have
$A:=\|w\|_{L^{2q}}+\|\nabla w\|_{L^{q}}\leq c\alpha$, where $\alpha>0$ is
defined in (2.14). We take $T=1$. Taking $\alpha>0$ small such that
$A<\frac{1}{8c}$, where $c$ is the constant in (5.7) such that

$\displaystyle\|\nabla
v^{2}\|_{L^{q}({\mathbb{R}}^{n}_{+}\times(0,1))}+\|v^{2}\|_{L^{2q}({\mathbb{R}}^{n}_{+}\times(0,1))}\leq
cA^{2}<A.$

Then, iterative arguments show that

(5.8)
$\displaystyle\begin{split}\|v^{m+1}\|_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}&\leq
c\||v^{m}|^{2}+|v^{m}U|+|U|^{2}\|_{L^{q}(({\mathbb{R}}^{n}_{+}\times(0,1))}\\\
&\leq
2c\left(\|v^{m}\|^{2}_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}+\|w\|^{2}_{L^{2q}}\right)\leq
4cA^{2}<A,\\\ \|\nabla v^{m+1}\|_{L^{q}({\mathbb{R}}^{n}_{+}\times(0,1))}&\leq
c\big{(}\||v^{m}|^{2}+|v^{m}U|+|U|^{2}\|_{L^{q}(({\mathbb{R}}^{n}_{+}\times(0,1))}\big{)}\\\
&\quad\leq
2c\left(\|v^{m}\|^{2}_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}+\|w\|^{2}_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}\right)\\\
&\quad\leq 4cA^{2}<A.\end{split}$

We denote $V^{m+1}:=v^{m+1}-v^{m}$ and $Q^{m+1}:=q^{m+1}-q^{m}$ for $m\geq 1$.
We then see that $(V^{m+1},Q^{m+1})$ solves

$V^{m+1}_{t}-\Delta V^{m+1}+\nabla Q^{m+1}=-{\rm div}\,\left(V^{m}\otimes
v^{m}+v^{m-1}\otimes V^{m}+V^{m}\otimes U+U\otimes V^{m}\right),$ ${\rm
div}\,V^{m+1}=0$

with homogeneous initial and boundary data, i.e. $V^{m+1}(x,0)=0$ and
$V^{m+1}(x,t)=0$ on $\\{x_{n}=0\\}$. It follows from (5.8) that

$\displaystyle\|\nabla
V^{m+1}\|_{L^{q}(({\mathbb{R}}^{n}_{+}\times(0,1))}+\|V^{m+1}\|_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}$
$\displaystyle\leq
c\left\||V^{m}v^{m}|+|V^{m}v^{m-1}|+|V^{m}U|\right\|_{L^{q}(({\mathbb{R}}^{n}_{+}\times(0,1))}$
$\displaystyle\leq
3cA\left\|V^{m}\right\|_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}<\frac{1}{2}\left\|V^{m}\right\|_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}.$

Therefore, $(v^{m},\nabla v^{m})$ converges to $(v,\nabla v)$ in
$L^{2q}({\mathbb{R}}^{n}_{+}\times(0,1))\times
L^{q}({\mathbb{R}}^{n}_{+}\times(0,1))$ such that $v$ solves in the sense of
distributions

$v_{t}-\Delta v+\nabla q=-{\rm div}\,\left(v\otimes v+v\otimes U+v\otimes
U+U\otimes U\right),$ ${\rm div}\,v=0$

with homogeneous initial and boundary data, i.e. $v(x,0)=0$ and $v(x,t)=0$ on
$\\{x_{n}=0\\}$. In addition, via (5.8), we have

$\displaystyle\|\nabla
v\|_{L^{q}(({\mathbb{R}}^{n}_{+}\times(0,1))},\quad\|v\|_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}\leq
A.$

Furthermore, if $q>n+2$ and $\|w\|^{2}_{L^{2q}}\leq c\alpha$, it follows from
Proposition 5.3 that

(5.9)
$\displaystyle\begin{split}\|v\|_{L^{\infty}({\mathbb{R}}^{n}_{+}\times(0,2))}&\leq
c\||v|^{2}+|vU|+|U|^{2}\|_{L^{q}({\mathbb{R}}^{n}_{+}\times(0,1))}\\\ &\leq
2c\left(\|v\|^{2}_{L^{2q}(({\mathbb{R}}^{n}_{+}\times(0,1))}+\|w\|^{2}_{L^{2q}}\right)\leq
4cA^{2}<A.\end{split}$

This complete the proof of Proposition 2.10. ∎

## Acknowledgement

T. Chang is supported by RS-2023-00244630.

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