数理解析研究所講究録
第 1782 巻 2012 年 17-27

17Lower bound of
decay of the Navier-Stokes
flow in the half space$L^{2}$$R_{+}^{n}$

東北大学大学院理学研究科 岡部 考宏 (Takahiro Okabe)

Mathematical institute, Tohoku University,e-mail: sa6m08@math.tohoku.ac.jp

1

Introduction

In this paper, we consider an asymptotic behavior inof the Navier-Stokes equations in the half-space:$L^{2}$

of weak solutions$\mathbb{R}_{+}^{n}$(N-S)$\{\begin{array}{l}\frac{\partial u}{\partial t}-\triangle u+u\cdot\nabla u+\nabla p=0 in \mathbb{R}_{+}^{n}\cross(0, \infty)divu=0 in \mathbb{R}_{+}^{n}\cross(0, \infty)u=0 on \partial \mathbb{R}_{+}^{n}\cross(0, \infty)u(0)=a in\mathbb{R}_{+}^{n},\end{array}$where $n\geq 3,$ $\mathbb{R}_{+}^{n};=\{x=(x_{1}, \ldots, x_{n})\in \mathbb{R}^{n};x_{n}>0\}$ denotes the upper
half-space. Here, $u=u(x, t)=(u^{1}(x, t), \ldots, u^{n}(x, t))$ and $p=p(x, t)$ denotethe unknown velocity vector and pressure of the fluid at point $(x, t)\in$, respectively, while $a=a(x)=(a^{1}(x), \ldots, a^{n}(x))$ is the given
initial velocity.
In his celebrated paper [7], Leray proposed the problem whether or not
weak solutions of (N-S) tend to zero inas the time goes to infinity.
Masuda [8] first gave a partial answer to Leray’s problem and clarified thatthe energy inequality of strong type plays an important role indecay ofweak solutions. Here, we mean by the energy inequality of strong type:$\mathbb{R}_{+}^{n}\cross(0, \infty)$$L^{2}$$L^{2}$$\Vert u(t)\Vert_{2}^{2}+2\int_{s}^{t}\Vert\nabla u(\tau)\Vert_{2}^{2}d\tau\leq\Vert u(s)\Vert_{2}^{2}$(1.1)

for almost all $s\geq 0$ , including $s=0$ . and for all $t\geq s$ . Leray called a weaksolution with (1.1) a turbulent solution. Later on, exact algebraic decayrate of energy decay was shown by Schonbek [12], Kajikiya-Miyakawa [5]and Wiegner [21]. For example, by the asymptotic expansion of heat kernel,

18
Fujigaki-Miyakawa showed that there exists a turbulent solution of (N-S)
such that
(1.2)
,
as
$\Vert u(t)\Vert_{2}=O(t^{-\frac{l+2}{4}})$$tarrow\infty$

satisfies $\int_{\mathbb{R}^{n}}(1+|x|)|a(x)|dx<0$ . They also
if initial data
found the necessary and sufficient condition that decay rate $t^{-(n+2)/4}$ is
optimal. Furthermore, it is well known that the decay rate as in (1.2) is
one of the nonlinear Duhamel term. Indeed, Kajikiya-Miyakawa [5] andBorchers-Miyakawa [1] proved that the decay rate of the difference between
for thethe nonlinear Navier-Stokes flow and the linear Stokes flow in.
andcase of$a\in L_{\sigma}^{2}(\mathbb{R}^{n})$$L^{2}$$\mathbb{R}^{n}$$\mathbb{R}_{+}^{n}$. If $1\leq r<2n/(n+2)$ .
Proposition 1.1 ([5],[1]). Let $1\leq r<2$ andthen every weak solution $u(t)$ of (N-S) with (1.1) satisfies$a\in L_{\sigma}^{2}$$\Vert u(t)-e^{-tA}a\Vert_{2}=O(t^{-\frac{\iota+2}{4}})$where$e^{tA}$is the Stokes semigroup and

$A$

as

$tarrow\infty$

(1.3)

,

is the Stokes operator.

Form this proposition, it is easy to see that if we require the slower decay
for the nonlinear Navier-Stokes flow, the linear Stokes flow is dominant andshould be investigated. Here we note that our aim this article is to establish
the lower bound of the energy decay for the Navier-Stokes flow in the half

space, i.e.,$\Vert u(t)\Vert_{2}\geq Ct^{-\alpha}$,$t\gg 1$

(1.4),

where $n/4\leq\alpha<(n+2)/4$ .
Our original motivation and background is based on the energy concentration phenomenon in the frequency space in order to investigate the asymp. For thistotic profile of the Naiver Stokes flow in the whole space
purpose, we consider the following asymptotic behavior:$\mathbb{R}^{n}$(1.5)$\lim_{tarrow\infty}\frac{\Vert E_{\lambda}u(t)\Vert_{2}}{\Vert u(t)||_{2}}=1$

associatedis a family of projection operators on
wherewith the spectral decomposition of the Stokes operator . Furthermore, forcan be regarded as a low frequency
the case of the whole spacecomponent of in the frequency space. Indeed, we introduce a cut-off:
function$L_{\sigma}^{2}(\mathbb{R}^{n})$$\{E_{\lambda}\}_{\lambda\geq 0}$$A$$\mathbb{R}^{n},$$E_{\lambda}u$$u$$\chi_{R}$$\chi_{R}(\xi):=\{\begin{array}{l}1 |\xi|\leq R0 |\xi|>R.\end{array}$

Then by the Fourier transform, we see thatへ$\hat{E_{\lambda}u}(\xi)=\chi_{\sqrt{\lambda}}(\xi)u$ $(\xi)$.

19Hence, (1.5) meas that the partial energy of the lower frequency componentof $u(t)$ up to
becomes dominant over the whole energy of $u(t)$ as $tarrow$. So it is an interesting problem to clarified that whether concentration
phenomenon (1.5) does occur or not, that what initial data causes (1.5) andthat what energy concentrates.
To prove (1.5), we found out the following inequality:$\sqrt{\lambda}$$\infty$$\lambda$$1- \frac{\Vert E_{\lambda}u(t)\Vert_{2}^{2}}{\Vert u(t)\Vert_{2}^{2}}\leq\frac{1}{\lambda}\frac{\Vert\nabla u(t)||_{2}^{2}}{||u(t)\Vert_{2}^{2}}$

,

(1.6)

for all $t>0$ and all $\lambda>0$ . Here, it is well-known thatdecayswith the same rate as one ofestimate of the Stokes semigroup,
if initial datawith some $1\leq r<2$ . Hence, in order to provethe convergence of the L.H.S. in (1.6), it suffices to derive the lower boundof the decay ofand to compare with each rate. However, the fastest$t^{-(n+2)/4}$
decay ofisviaestimate for $r=1$ . This is why
we need such a slow decay (1.4).
In this direction, the author established precise behavior of solutions of
the lower bound in. We note that to derive such a slow decay,
the analysis on the linear Stokes flow is essential. By the Fourier splitting
method, the behavior at $t=\infty$ of the Stokes flow is controlled by the lower
frequency component of initial data. Indeed, introducing a classfor initial data, defined by$\Vert\nabla u(t)\Vert_{2}$$e^{-tA}$$L^{p}-L^{q}$$a\in L^{r}\cap L_{\sigma}^{2}$$\Vert u(t)|_{2}$$L^{p}-L^{q}$$\Vert\nabla u(t)\Vert_{2}$$L^{2}(\mathbb{R}^{n})$$K_{\alpha,\delta}^{m}(\mathbb{R}^{n})$he [10] proved that if
solution $u(t)$ of (N-S) satisfies$a\in K_{\alpha,\delta}^{m}(\mathbb{R}^{n})\cap L^{r}(\mathbb{R}^{n})$

,

,
(1.7)
with $1<r<2$ , then the weak

$K_{\alpha,\delta}^{m}(\mathbb{R}^{n}):=\{\phi\in L^{2}(\mathbb{R}^{n});|\hat{\phi}(\xi)|\geq\alpha|\xi|^{m}, |\xi I \leq\delta\}$$\Vert u(t)\Vert_{L^{2}(N^{1})}\geq C(1+t)^{-\frac{n+2m}{4}}$$m\geq 0,$,$\alpha,$$\delta>0$(1.8)

for $n=2,3,4$ . We note that the set
has a different character ofthe initial profile from that of [13, 14] and [9], and that in particular, our$K_{\alpha,\delta}^{m}(\mathbb{R}^{n})$characterization covers the results of [12, 13, 14], when $0\leq m<1$ .
Rom this observation, to derive energy concentration (1.5), the slow decay ofis essential. Here, we notice that the method to derive thelower bound depends heavily on the Fourier transform. Hence it is interesting problem to establish the lower bound of the energy decay in other$\Vert u(t)\Vert_{2}$domains where the Fourier transform does not work well.
Next we consider the Navier-Stokes flow in the half-space. In the halfspace, there are many results for the upper bound of the temporal decayof the Stokes flow and the Navier-Stokes flow. See, for instance, Borchers$\mathbb{R}_{+}^{n}$20
and Miyakawa [1], Fujigaki and Miyakawa [3, 4]. However, up to now, it
seems that there are few results for the lower bound of the energy decay. Insuch a situation, [3, 4] obtained the same lower bound as (1.8) under somecondition on initial data. Especially, in [4], it was clarified that the strong
if and only if the Stokes
solution $u(t)$ of (N-S) satisfies. As is mentioned in [4], it seems to be an
flow $v(t)$ satisfies
interesting problem to characterize a class of the initial data which exhibits. In the present article,
a lower bound of the Stokes flow in the half-spacefocusing on the profile of initial data, we investigate the lower bound such as(1.8) for weak solutions of (N-S) which satisfy the energy inequality of strong. Our rate as in (1.8) improves the rate given
type (1.1) in the half-spaceby [3] like ?? . Furthermore, we give a positive answer to the question of[4] for the slow decay of the Stokes flow by the concrete characterization ofwhich is similar to (1.7).
the initial data in
To study on the asymptotic behavior of the Navier-Stokes flow in thehalf-space, we first consider the Stokes flow and establish the estimate frombelow in terms of the explicit solution formula given by Ukai [20]. In the, a number of decay properties of lower bounds relies heavily
whole spaceon the Fourier transform. However, in order to overcome such difficulty, we
split the variables of the initial data with the following form:$\Vert u(t)\Vert_{2}\geq Ct^{-n/4}$$\Vert v(t)\Vert_{2}\geq Ct^{-n/4}$$\mathbb{R}_{+}^{n}$$\mathbb{R}_{+}^{n}$

$($

$)$

$\mathbb{R}_{+}^{n}$$\mathbb{R}^{n}$$a$$a(x)=a’(x’)\eta(x_{n})$ ,where $x=(x’, x_{n})\in \mathbb{R}^{n}$ and $x’$ $:=(x_{1}, \ldots, x_{n-1})\in \mathbb{R}^{n-1}$ . Moreover, undersome restriction on and , we notice that the property of is dominant
to the slow decay of the Stokes flow. By this form, the problem is reduced. Conversely, we seeto that on the lower dimensional whole spacethat the 2-dimensional initial data can be embedded in the 3-dimensional
half-space
and also the whole space , where the slow decay properties
are preserved. In the same manner, for every $n\in N$ , we find out a hierfor the decay of the lower bounds
andarchy structure betweenof solutions with respect to the initial data. On the other hand, insteadof
as in (1.7), we introduce a more general profile on the lower
ffequency part on initial data such as
$a’$

$a’$

$\eta$$\mathbb{R}^{n-1}$$\mathbb{R}^{3}$$\mathbb{R}_{+}^{3}$$\mathbb{R}^{n}$$\mathbb{R}^{n+1}$$K_{\alpha,\delta}^{m}(\mathbb{R}^{n})$$T_{\alpha,\gamma,\delta}^{m}(\mathbb{R}^{n}):=\{\phi\in L^{2}(\mathbb{R}^{n});|\hat{\phi}(\xi)|\geq\alpha|\xi_{n}|^{m}, |\xi_{n}|\leq\gamma, |\xi’|\leq\delta\}$

,

(1.9)

$:=(\xi_{1}, \ldots, \xi_{n-1})\in$$\delta>0$ , whereandfor $m\geq 0,$
can be characterized. It should be noted that the classin terms of the estimate from below of the low frequency $\xi=(\xi’, \xi_{n})$ in
the -direction. It turns out that such a profile of initial data only in one$\alpha,$$\mathbb{R}^{n-1}$$\xi_{n}$$\gamma,$$\xi=(\xi’, \xi_{n})\in \mathbb{R}^{n}$$T_{\alpha,\gamma,\delta}^{m}(\mathbb{R}^{n})$$\xi’$

21
direction todominates the asymptotic behavior in time from below of
the Stokes flow. We also note that by making use of, we canimprove the previous result in [10] for the whole space. By the virtue ofUkai solution formula of the Stokes flow, the profile of initial data can bedirectly applicable to the exact exponent of the decay in (1.8). If we take$m=0$ in (1.8) and (1.9), then we obtain such a lower bound as:$\xi_{n}$$T_{\alpha,\gamma,\delta}^{m}(\mathbb{R}^{n})$$\mathbb{R}^{n}$$s$$\Vert u(t)\Vert_{2}\geq Ct^{-\frac{l1}{4}}$$t\gg 1$.

(1.10)へ

In addition, if $|a’(\xi’)|$ $\leq M$ for near $\xi’=0$ , it is easy to see that$\Vert u(t)\Vert_{2}\leq C(1+t)^{-\frac{l}{4}}$.

(1.11)

Therefore, (1.11) gives the optimal decay rate of the energy of the NavierStokes flow in the half-spacefor such a initial data. Indeed, we construct an initial data which causes both (1.10) and (1.11), as an example in$\mathbb{R}_{+}^{n}$$T_{\alpha,\gamma,\delta}^{0}(\mathbb{R}^{n})$2.

Results
We consider the following assumption on initial data:

Assumption. (A 1) $a(x)=(a^{1}(x), \ldots, a^{n-1}(x), 0)=:(a’(x), 0)$$(A2)a’(x)=a”(x’)\eta(x_{n})$(A 3) $\eta(-x_{n})=-\eta(x_{n})$ and $|\hat{\eta}(\xi_{n})|\geq Cnear\xi_{n}=0$$(A4)a”\in T_{\alpha,\gamma,\delta}^{m}(\mathbb{R}^{n})$. $i.e..|$へ$a^{\prime l}(\xi’)|\geq C|\xi_{n-1}|^{m}$near $\xi’=0$

Now our results read:
Theorem 2.1. Let $n\geq 3$ , and let$r$

and $m$ sa tisfy either (i) or (ii):(i) $1<r\leq 2n/(n+2),$ $0\leq m<1$ ,(ii) $2n/(n+2)<r<2n/(n+1),$ $0\leq m<2n/r-n-1$ .

If
sa $tisBes$ the assumptions $(Al),$ $(A2),$ $(A3)$ and $(A4)$$\delta>0$ , thenfor some
there exist $T>1$ and a constant $C>0$ suchth at every weak solution $u(t)$ of (N-S) with (1.1) fulfills the estima $te$,$a\in L^{r}(\mathbb{R}_{+}^{n})\cap L_{\sigma}^{2}(\mathbb{R}_{+}^{n})$$\alpha,$$\gamma,$$\Vert u(t)\Vert_{2}\geq Ct^{-\frac{21+2m}{4}}$for all $t\geq T$ .

(2.1)

22
Remark 2.1. (i) We note that (2.1) improves the result in [4] when$0\leq$$m<1$ .

(ii) The estimate (2.1) inspires us that the optimal decay rate for such an
initial data seems to be $n/4$ . Indeed, by taking $m=0$ in (2.1), we obtain$Ct^{-\frac{\iota}{4}}\leq\Vert u(t)\Vert_{2}\leq C_{r}(1+t)^{-\frac{\iota}{2}(\frac{1}{r}-\frac{1}{2})}$

for$a\in L^{r}(\mathbb{R}_{+}^{n})\cap L^{2}(\mathbb{R}_{+}^{n}),$

,

(2.2)

$t>T$ ,

$1<r<2$ . Letting $rarrow 1$ in (2.2) formally, we

may expect an exact estimate both from below and above such that
$Ct^{-\frac{}{4}}\leq\Vert u(t)\Vert_{2}\leq C(1+t)^{-\frac{l}{4}}$,$t\geq T$.

However, up to now, we do not establish any uniform estimate with respectin (2.2).
to $1<r<2$ on the constantfor near $\xi’=0$ and(iii) In addition to the case $m=0$ , if
for near $\xi_{n}=0$ then we obtain the optimal decay rate $n/4$ for$C_{r}$$|\hat{a’’}(\xi^{l})|\leq M$$|\eta^{\hat{*}}(\xi_{n})|\leq M$such an initial data, since it holds that
$Ct^{-\frac{\iota}{4}}\leq\Vert u(t)\Vert_{2}\leq C(1+t)^{-\frac{l}{4}}$3,$t\geq T$

Stokes flow in the half-space.

$\mathbb{R}_{+}^{n}$

To prove our main theorem, it is essential to investigate the energy decayof the linear Stokes flow in the half-space. For this purpose, we first introduce some specific properties of solutions, $v=(v^{l}, v^{n}),$ $v^{l}=(v^{1}, \ldots, v^{n-1})$ ,of the Stokes equations:(S)$\{\begin{array}{l}\frac{\partial v}{\partial t}-\Delta v+\nabla p=0 in \mathbb{R}_{+}^{n}\cross( 0, oo)divv=0 in \mathbb{R}_{+}^{n}\cross(0, \infty)v=0 on \partial \mathbb{R}_{+}^{n}\cross(0, \infty)v(0)=a in \mathbb{R}_{+}^{n}.\end{array}$

Ukai [20] gave a explicit solution formula for (S). To state Ukai formula
we prepare some notations. Let $R=(R’, R_{n})$ with $R’=(R_{1}, \ldots, R_{n-1})$,
andand $S=(S_{1}, \ldots, S_{n-1})$ denote the Riesz transform over
) is a bounded linear operator on(resp.
respectively. Each
(resp.
), $1<r<\infty$ . For a function $f(x’, x_{n})$ , we umderstand thatis regarded as
acts as a convolution with respect to the variables ‘, so$1<r<\infty$ . Let $B=B_{r}=$anda bounded operator on both
. Itwith domain $D(B)$-A be the Laplacian on$s$$\mathbb{R}^{n}$$L^{r}(\mathbb{R}^{n})$$S_{j}$$R_{j}$$\mathbb{R}^{n-1}$$L^{r}(\mathbb{R}^{n-1})$$x$$S_{j}$$L^{r}(\mathbb{R}^{n})$$\mathbb{R}_{+}^{n}$$S_{j}$$L^{r}(\mathbb{R}_{+}^{n}),$$:=W^{2,r}(\mathbb{R}_{+}^{n})\cap W_{0}^{1,r}(\mathbb{R}_{+}^{n})$23
is well known $that-B$ generates a bounded analytic semigroupon$L^{r}(\mathbb{R}_{+}^{n}),$$1<r<\infty$ .

$\{e^{-tB}\}_{t\geq 0}$

More precisely, we have$e^{-tB}f=e^{t\triangle}f^{*}|_{\mathbb{R}_{+}^{?l}}$,for$f\in L^{r}(\mathbb{R}_{+}^{n})$whereis the usual heat operator onwith respect to variable , i.e.,$e^{t\triangle}$$\mathbb{R}^{n}$and$f^{*}$,$1<r<\infty$ ,denotes the odd extension$x_{n}$$f^{*}(x’, x_{n}):=\{\begin{array}{ll}f(x’, x_{n}), x_{n}>0,-f(x^{l}, -x_{n}), x_{n}<0.\end{array}$

The solution formula of Ukai [20] is now read:
Proposition 3.1 (Ukai [20]). For$v=(v’, v^{n})$ of $(S)$ is expressed as$v^{n}(t)=Ue^{-tB}[a^{n}+S\cdot a’]$where

$a\in L_{\sigma}^{r}(\mathbb{R}_{+}^{n}),$,$1<r<\infty$ ,the solution$v’(t)=e^{-tB}[a’-Sa^{n}]+Sv^{n}$is the bounded operator on, indeed, $Uf=R’\cdot S(R’\cdot S-R_{n})ef|_{\mathbb{R}}\dotplus^{l}$ ’which is also expressed with the Fourier transform on
as$U$$L^{r}(\mathbb{R}_{+}^{n})$$\mathbb{R}^{n-1}$$\hat{Uf}(\xi’, x_{n})=|\xi’|\int_{0^{e^{-|\xi’|(x_{l}-y)}}}^{x_{l}\prime},f(\xi’, y)dy$

Here, $ef$ denotes the zero extension of

$f$

from

$\mathbb{R}_{+}^{n}$over$\mathbb{R}^{n}$

.
:

(3.1)

$ef(x’, x_{n})=\{\begin{array}{ll}f(x’, x_{n}) x_{n}>00 x_{n}<0.\end{array}$Remark 3.1. In this paper, we use the Fourier transform with the following
form:$\hat{f}(\xi)$$:=(2 \pi)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}f(x)dx$,$i$

Furthermore, we note that the symbols of Riesz$\sigma(R_{j})=-i\xi_{j}/|\xi|$$\sigma(S_{j})=-i\xi_{j}/|\xi’|$,,$j=1,$$j=1,$$s$operator$\ldots,$$\ldots,$$:=\sqrt{-1}$$n$.

$R_{j}$

and$S_{j}$

are,$n-1$ ,

which have opposite signs of ones in [20, 1].
With Ukai solution formula for the linear Stokes flow in the half-space,
we can directly calculate the Stokes flow if the initial data is given. Indeed,
we have the following Theorem for the lower bound of the energy decay for
the Stokes flow in the half-space:$s$24
Theorem 3.1 (The half-space). Let $n\geq 3$ and put $v(t)=e^{-tA}a$ . If
sa tisfies assumptions $(Al),$ $(A2),$ $(A3)$ and $(A4)$ then the Stokes
flow $v(t)$ satisfies(3.2)
for $t\geq 1$$a\in L_{\sigma}^{2}(\mathbb{R}_{+}^{n})$$\Vert v(t)\Vert_{2}\geq Ct^{-\frac{n+2m}{4}}$where $C=C(n, m, \alpha,\gamma, \delta)>0$ .

Since we focus on the initial data with splitting variables $a(x)=a”(x’)\eta(x_{n})$ ,the following lemma for the Stokes flow in the whole space plays an important role for$a^{l\prime}(x’)$

Lemma 3.1 (The whole space). Let $n\geq 2$ and put $v(t)=e^{-tA}a$ with thefor some. Ifon
Stokes semigroup$\delta>0$ , then $v(t)$ satisfies$m\geq 0$ and$e^{-tA}$$\alpha,$$L_{\sigma}^{2}(\mathbb{R}^{n})$$a\in L_{\sigma}^{2}(\mathbb{R}^{n})\cap T_{\alpha,\gamma,\delta}^{m}(\mathbb{R}^{n})$$\gamma,$$\Vert v(t)\Vert_{2}\geq Ct^{-\frac{\iota+2m}{4}}$for $t\geq 1$ ,(3.3)

where $C=C(n, m, \alpha, \gamma, \delta)>0$ .

Proof. By Plancherel theorem and Fubini’s theorem, we have$s$$\Vert v(t)\Vert_{2}^{2}=\Vert\hat{v}(t)\Vert_{2}^{2}\geq\int_{|\xi_{1}|\leq\gamma,|\xi’|\leq\delta}e^{-2t|\xi|^{2}}|\hat{a}(\xi)|^{2}d\xi$$\geq\alpha^{2}\int_{|\xi_{l}|\leq\gamma,|\xi’|\leq\delta}e^{-2t|\xi|^{2}}|\xi_{n}|^{2m}d\xi$$= \alpha^{2}(\int_{|\xi_{n}|\leq\gamma}e^{-2t\xi_{1}^{2}},|\xi_{n}|^{2m}d\xi_{n})(\int_{|\xi’|\leq\delta}e^{-2t|\xi’|^{2}}d\xi’)$$=:\alpha^{2}I_{1}\cdot I_{2}$,for all $t\geq 0$ . By changing variables we have$I_{1}=2 \int_{0}^{\gamma}e^{-2t\xi_{l}^{2}}\cdot\xi_{n}^{2m}d\xi_{n}$$=2 \int_{0}^{\sqrt{t}\gamma}e^{-2\rho^{2}}(\frac{\rho}{\sqrt{t}})^{2m}\frac{d\rho}{\sqrt{t}}$$\geq 2t^{-\frac{2m+1}{2}}\int_{0}^{\gamma}e^{-2\rho^{2}}\rho^{2m}d\rho$

for all $t\geq 1$ . Similarly by polar coordinates$\xi’=\rho\omega\in \mathbb{R}^{n-1}$$I_{2}=(n-1) \omega_{n-1}\int_{0}^{\delta}e^{-2t\rho^{2}}\rho^{n-2}d\rho$$=(n-1) \omega_{n-1}\int_{0}^{\sqrt{t}\delta}e^{-2\rho^{2}}(\frac{\rho}{\sqrt{t}})^{n-2}\frac{d\rho}{\sqrt{t}}$,$\geq(n-1)\omega_{n-1}t^{-\frac{n-1}{2}}\int_{0}^{\delta}e^{-2\rho^{2}}\rho^{n-2}d\rho$, we have

25
for all $t\geq 1$ , whereis the volume of the unit ball in
Therefore, we obtain (3.3) with a constant$\omega_{n-1}$$\mathbb{R}^{n-1}$.

$C^{2}=2 \alpha^{2}(n-1)\omega_{n-1}(\int_{0}^{\gamma}e^{-2\rho^{2}}\rho^{2m}d\rho)(\int_{0}^{\delta}e^{-2\rho^{2}}\rho^{n-2}d\rho)$.

This completes the proof of Lemma 3.1$\square$Remark 3.2. We note that Lemma 3.1 still holds, if we replace $a\in$$L_{\sigma}^{2}(\mathbb{R}^{n})\cap T_{\alpha,\gamma,\delta}^{m}(\mathbb{R}^{n})$and$e^{-tA}$by$a\in T_{\alpha,\gamma,\delta}^{m}(\mathbb{R}^{n})$and$e^{t\triangle}$

respectively.

Finally, with this lemma and theorem for the linear Stokes flow, we obtainmain theorem for the nonlinear Navier-Stokes flow in the half-space.

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