数理解析研究所講究録 1425 巻 2005 年 54-64

54

Stability of 1-dimensional stationary solution
to the compressible Navier-Stokes equations
on the half space
隠居 良行 (Yoshiyuki KAGEI)

Faculty of Mathem atics, Kyushu University
Fukuoka 812-8581, JAPAN1. Introduction
This article is concerned with the compressible Navier-Stokes equation;on the half space$\mathrm{R}_{+}^{n}(n\geq 2)$$\partial_{t}\rho+\mathrm{d}\mathrm{i}\mathrm{v}$(1.1)$\partial_{t}$(pu)$(pu)=0$ ,$(\rho u\otimes u)+$$+\mathrm{d}\mathrm{i}\mathrm{v}$$p(\rho)=K\rho^{\gamma}$

Vp $(\rho)=$ pbu,$+(\mu+\mu’)\nabla \mathrm{d}\mathrm{i}\mathrm{v}u$.

Here $\mathrm{R}_{+}^{n}=\{x=(x_{1}, x’):x’=(x_{2)}\cdots, x_{n})\in \mathrm{R}^{n-1}, x_{1}>0\};\rho=\rho(x, t)$
and $u=$ $(u^{1}(x,t)$ ,, $u^{n}(x, t))$ denote the unknown density and velocity$K$
respectively; , , and 7 are constants satisfying $\mu>0$ ,, $K>0$and $\gamma>1$ . We consider (1.1) under the initial and boundary conditions$\cdots$$\mu$$\frac{2}{n}\mu+\mu’\geq 0$$\mu’$$u|_{x_{1}=0}=(u_{b}^{1},0, \cdots, 0)$(1.2)$\rhoarrow\rho_{+}$,,$uarrow(u_{+}^{1},0, \cdots, 0)$$(\rho, u)|‘=0=(\rho_{0}, u_{0})$$(x_{1}arrow\infty)$,,

where ,
andare given constants satisfying $\rho_{+}>0$ and.
Kawashima, Nishibata and Zhu [4] investigated the conditions for,andunder which planar stationary motions occur. Namely, they
showed that under suitable conditions for,andthere exists a
stationary solutionof problem (1.1)-(1.2) in the form,
. Furthermore, it was shown in [4] thatis asymptotically stable with respect to small one-dimensional perturbations; i.e.
$\rho_{+}$$u_{+}^{1}$$u_{b}^{1}<0$$u_{b}^{1}$$\rho_{+}$$u_{+}^{1}$$u_{b}^{1}$$\rho_{+}$$(\tilde{\rho},\tilde{u})$$\overline{u}=$$(\overline{u}^{1}(x_{1}), 0, \cdots, 0)$$u_{+}^{1}$$u_{b}^{1}$$\tilde{\rho}=\overline{\rho}(x_{1})$$(\tilde{\rho},\overline{u})$55, $u-\overline{u}=(u^{1}(x_{1}, t)-$perturbations in the form
is sufficiently small.
, 0,, 0), provided thatis
In this article we will give a summary of the results in [3], where
shown to be asymptotically stable with respect to multi-dimensional perturis sufficiently small. Here, provided thatbations small inis an integer satisfying $s\geq[n/2]+1$ .
$\rho-\overline{\rho}=\rho(x_{1\}}t)-\tilde{\rho}(x_{1})$$\overline{u}^{1}(x_{1})$$|u_{+}^{1}-u_{b}^{1}|$$\cdots$$(\overline{\rho}, \overline{u})$$|u_{+}^{1}-u_{b}^{1}|$$H^{s}(\mathrm{R}_{+}^{n})$$s$2. Stability Result
We first consider the one-dimensional stationary problem whose solu. We look for a smoothtions represent planar stationary motions inandof (1.1)-(1.2) of the form
stationary solution
is written as. Then the problem for$\mathrm{R}_{+}^{n}$$\tilde{\rho}=\overline{\rho}(x_{1})>0$$(\tilde{\rho},\tilde{u})$$\tilde{u}=$$(\overline{\rho},\overline{u}^{1})$$(\overline{u}^{1}(x_{1}), 0, \cdots, 0)$$(\overline{\rho}\overline{u}^{1})_{x_{1}}=0$$(x_{1}>0)2$$(\overline{\rho}(\overline{u}^{1})^{2})_{x_{1}}+p(\tilde{\rho})_{x_{1}}=(2\mu+\mu’)\overline{u}_{x_{1}x_{1}}^{1}$(2.1)$\overline{u}|_{x_{1}=0}=u_{b}^{1}$$\overline{\rho}arrow\rho_{+}$,$(x_{1}>0)$,,$\overline{u}^{1}arrow u_{+}^{1}$$(x_{1}arrow\infty)$,

stands for differentiation in .
Kawashima, Nishibata and Zhu [4] investigated problem (2.1) and gave
a necessary and sufficient condition for the existence of solutions. Following[4], we introduce the Mach number at infinity defined bywhere subscript$x_{1}$$x_{1}$$M_{+} \equiv\frac{|u_{+}|}{\sqrt{p^{t}(\rho_{+})}}$We also set$\delta$$\equiv|u_{+}^{1}-u_{b}^{1}|$.

,which measures the strength of the stationary solution.

Proposition 2,1.([4]) Let $u_{+}^{1}<0$ . Then problem (2.1) has a smooth solution
is a certain positive, whereif and only if $M_{+}\geq 1$ andis monotonis monotonic, in particular,
number. The solution
has the following decay
ically increasing when $M_{+}=1$ . Furthermore,.
properties as(i) If $M_{+}>1$ , then for any nonnegative integer there exists a constant$C>0$ stich that$w_{\mathrm{c}}u_{+}>u_{b}$$(\overline{\rho},\overline{u}^{1})$$w_{\mathrm{c}}$$\overline{u}^{1}(x_{1})$$(\overline{\rho},\overline{u}^{1})$$(\tilde{\rho}, \tilde{u}^{1})$$x_{1}arrow\infty$$k$$|\partial_{x_{1}}^{k}(\overline{\rho}-\rho_{+},\overline{u}^{1}-u_{+}^{1})|\leq C\delta e^{-\sigma x_{1}}$for some positive constant .
$\sigma$

58(ii) If $M_{+}=1_{f}$ then for any nonnegative integer$C>0$ such that$k$

there exists a constant$| \partial_{x_{1}}^{k}(\tilde{\rho}-\rho_{+},\overline{u}^{1}-u_{+}^{1})|\leq C\frac{\delta^{k+1}}{(1+\delta x_{1})^{k+1}}$.

, withOur interest is the stability properties ofrespect to multi-dimensional perturbations. To state our stability result we, we define the,introduce function spaces. For $0<T\leq\infty$ andBanach space,$(\overline{\rho},\overline{u}),\overline{u}=(\overline{u}^{1},0, \cdots, 0)$$\sigma\in \mathrm{Z}$$\sigma\geq 0$$Z^{\sigma}(T)=X^{\sigma}(T)\rangle\langle Y^{\sigma}(T)^{n}$

where
.

$X^{\sigma}(T)=C^{j}([0, T])H^{\sigma-2j})j=01 \frac{\sigma}{\mathrm{n}’\sim)}\mathrm{J}$and$Y^{\sigma}(T)=X^{\sigma}(T) \cap H^{j}(0, T,\overline{H}^{\sigma+1-2j})[\frac{\sigma+1}{j=0\cap^{2}}].$

Here$\overline{H}^{m}=H^{m}\cap H_{0}^{1}$$Z^{\sigma}(T)$is define by$\mathrm{d}$when $m\geq 1$ and$\overline{H}^{m}=L^{2}$.

when $m=0$ . The norm offor $U=\{\phi$ , ), where$||U||_{Z^{\sigma}(T)}=||\phi||_{X^{\sigma}(T)}+||\psi||_{Y^{\sigma}(T)}$$|| \phi||_{X^{\sigma}(T)}=\sup_{0\leq t\leq T}|[\phi(t)]|_{\sigma}$,$\psi$$|| \psi||_{Y^{\sigma}(T\rangle}=(||\psi||_{X^{\sigma}(T)}^{2}+\oint_{0}^{T}|[\psi(t)]|_{\sigma+1}^{2}dt)^{1/2}$

with$|[ \phi(t)]|_{\sigma,k}=(\sum_{j=0}^{k}||\partial_{t}^{j}\phi(t)||_{H^{\sigma-2j)^{1/2}}}^{2},$We simply denote by$Z^{\sigma}$,$X^{\sigma}$

and$Y^{\sigma}$$|[\phi(t)]|_{\sigma}=|[\phi(t)]|_{\sigma,[\frac{\sigma}{2}]}$.

when $T=\infty$ .

Theorem 2.2. Let be an integer satisfying $s\geq[n/2]+1$ and let
be the solution of (2.1). Then there exists a positive number
such that,there
is stable with respect to perturbations small
ifinin the following sense: there exist
and $C>0$ such that$(\rho(0)-\tilde{\rho},u(0)-\tilde{u})\inH^{s}$and satisfies a suitable
if the initial perturbationcompatibility condition, then perturbations.
,exists inand it satisfies$s$$(\tilde{\rho},\tilde{u})$$\delta_{0}$$|u_{b}^{1}-u_{+}^{1}|<\delta_{0}$$H^{s}(\mathrm{R}_{+}^{n})$$(\tilde{\rho},\overline{u})$$\epsilon_{0}>0$$(\rho(t)-\overline{\rho}, u(t)-$$\mathrm{i})$$||(\rho(t)-\tilde{\rho}, u(t)-\overline{u})||_{H^{\mathrm{s}}}\leq C||(\rho(0)-\overline{\rho}, u(0)-\overline{u})||_{H^{s}}$$Z^{s}$

57

for all$t\geq 0$

and,$\lim_{tarrow\infty}||\partial_{x}(\rho(t)-\overline{\rho}, u(t)-\tilde{u})||_{H^{\epsilon-1}}=0$

provided that$||(\rho(0)-\overline{\rho}, u(0)-\overline{u})|_{1}^{1_{H^{s}}}\leq\in 0$. In particular,.

$\lim_{tarrow\infty}||(\rho(t)-\overline{\rho}, u(t)-\overline{u})||_{\infty}=0$

was firstly investigated in [4] and they(i) The stability ofis stable with respect to small
proved Theorem 2.1 for $n=1$ , i.e.,
perturbations in the form $\rho-\tilde{\rho}=\rho(x_{1}, t)-\tilde{\rho}(x_{1}))u-\tilde{u}=(u^{1}(x_{1}, t)-$
, 0).
, 0,
(ii) We here consider large time behavior of solutions of (1.1)-(1.2) only
given in Proposition 2.1. As is easily
andunder the conditions for ,imagined, if one of these conditions would be disturbed, then complicatedphenomena might occur. In fact, Matsumura [5] proposed a classificationof all possible time asymptotic states in terms of boundary data for onedimensional problem. Some parts of this classification were already proved
rigorously. See [5].
Remarks,$(\tilde{\rho},\overline{u})$$(\overline{\rho},\overline{u})$$\tilde{u}^{1}(x_{1})$$\cdots$$u_{+}^{1}$$u_{b}^{1}$$\rho_{+}$

3. Outline of the Proof(

Let us rewrite the problem into the one for perturbations. We set $(\phi, \psi)=$, $u-u\gamma$ . Then problem (1.1)-(1.2) is transformed into$\rho-\overline{\rho}$$\partial_{t}\phi+u$.

,$\nabla\phi+\rho \mathrm{d}\mathrm{i}\mathrm{v}\psi=F$$\rho(\partial_{t}\psi+u\cdot\nabla\psi)+L\psi+p’(\rho)\nabla\phi=G$

(3.1)

.

$\psi|_{x_{1}=0}=0_{7}$$(\phi)\psi)arrow(0,0)$,$(x_{1}arrow\infty)$,$(\phi, \psi)|_{t=0}=(\phi_{0},\psi_{0})$where$L\psi=-\mu\Delta\psi-$$F=-\psi$.

$(\mathrm{p}\mathrm{a}+\mu’)\nabla \mathrm{d}\mathrm{i}\mathrm{v}\psi$$\nabla\tilde{\rho}-\phi \mathrm{d}\mathrm{i}\mathrm{v}\overline{u}$,,$G=-(\rho\psi+\phi\overline{u})\cdot\nabla\tilde{u}-(p’(\rho)-p’(\gamma\rho)\nabla\tilde{\rho}$.

The proof of Theorem 2.1 is thus reduced to showing the global existence, where is an integer satisfyingof (3.1) in the classof solution$(\phi, \psi)$$s\geq[n/2]+1$ .

$Z^{s}$$s$

58

Let us firstly consider the local existence of solutions. The local existence
can be proved by applying the result in [2], In fact, problem (3.1) is a
hyperbolic-parabolic system satisfying the assumptions in [2] that guarantees
the local solvability infor satisfying $s\geq[n/2]+1$ . Therefore, we obtainthe following$H^{s}$$s$

Proposition 3.1. Let s be an integer satisfying s $\geq s_{0}=$ [ ] +1. Assume
that the initial value
satisfies the following conditions.
$(\phi_{0}, \psi_{0})$(a)$(\phi_{0}, \psi_{0})\in H^{s}$

tion,

(b)

$t$

here

and

$\hat{s}=[\frac{s-1}{2}]$$\inf_{x}\rho o(x)\geq-\frac{1}{4}$satisfies the ’s-th order compatibility condi-$(\phi_{0_{7}}\psi_{0})$

infxi

.
$\tilde{\rho}(x_{1})$.

anddepending on
Then there exists a positive number$(\phi,\psi)\inZ^{s}(T_{0})$solution(3.1)
aproblemsuch thathas unique. Furthemore,satisfying,?
for all$>0$$con$there exist
depending on ,stants $C>0$ andandsuch that$||(\phi_{0}, \psi_{0})||_{H^{s}}$$T_{0}$$\inf_{x_{1}}\overline{\rho}(x_{1})$$\phi(x$$)$$(x, t)\in \mathrm{R}_{+}^{n}\rangle\langle[0, T_{0}]$$\geq-\frac{1}{2}\inf_{x_{1}}\tilde{\rho}(x_{1})$$s$$\gamma$$||(\phi_{0},\psi 0)||_{H^{\epsilon}}$$\inf_{x_{1}}\overline{\rho}(x_{1})$.

$||(\phi, \psi)||_{Z^{\theta}(T_{0}\rangle}^{2}\leq C\{1+||(\phi_{0}, \psi_{0})||_{H^{\theta}}^{2}\}^{\gamma}||(\phi_{0}, \psi_{0})||_{H^{s}}^{2}$

We next derive a priori estimates to show the global existence of solution.
We defineandby$E_{\sigma}(t)$$D_{\sigma}(t)$$E_{\sigma}(t)=( \sup_{0\leq\tau\leq t}\{|[\psi(\tau)]|_{\sigma}^{2}+||\phi(\tau)||_{H^{\sigma}}^{2}+|[\partial_{\tau}\phi(\tau)]|_{\sigma-1}^{2}\})^{1/2}$

and
for $\sigma=0$ ,$(I_{0}^{t}||\partial_{x}\psi||_{2}^{2}+||\phi|_{x_{1}=0}||_{L^{2}(\mathrm{R}^{n-1})}^{2}d\tau)^{1/2}$$D_{\sigma}(t)=\{$$( \oint_{0}^{t}||\partial_{x}\psi||_{H^{\sigma}}^{2}+||\phi|_{x_{1}=0}||_{L^{2}(\mathrm{R}^{n-1})}^{2}$$+||\partial_{x}\phi||_{H^{\sigma-1}}^{2}+|[\partial_{\tau}\phi]|_{\sigma-1}^{2}+|[\partial_{\tau}\psi]|_{\sigma-1}^{2}d\tau)^{1/2}$($\phi_{0}$In what follows we will denote the solution, Vo) by
$U=(\phi,\psi)$,$(\phi,\psi)$$U_{0}=(\phi_{0},\psi_{0})$for$\sigma\geq 1$.

and the initial value.

Theorem 2.2 follows from Proposition 3.1 and the following a priori estimate.

59

Proposition 3.2. Let $U=(\phi, \psi)$ be a solution of (3.1) on $[0, T]$ . Assume
and $C>0$ ,that $E_{s}(t)<1$ for all $t\in[0, T]$ . Then there exist constants
which are independent of $T>0$ , such that$\epsilon 0>0$$E_{s}(t)^{2}+D_{s}(t)^{2}\leq C||U_{0}||_{H^{s}}^{2}$for all $t\in[0, T]$ , provided that$||U_{0}||_{H^{s}}<\epsilon 0$.

Outline of the proof of Proposition 3.2
As in the one-dimensional problem studied in [4] , the point in the proof.
norm ofof Proposition 3.2 is to derive a suitable bound for the
Due to the fact that the stationary solution has no shear components, one
bound in the same way as in the one-dimensional case in
can obtain the[4].
$L^{2}$$(\phi, \psi)$$L^{2}$

Proposition 3.3. There eists a constant M $>0$ such that if(3.2)$E_{s}(t)\leq M$

for all $t\in[0, T]$ , then$E_{0}(t)^{2}+D_{0}(t)^{2}\leq C\{||U_{0}||_{2}^{2}+R_{0}(t)^{2}\}$

uniformly in $t\in[0$ , ? , where $C>0$ is independent of
$]$

$T$

,
and
.

$R_{0}(t)^{2}=- \int_{0}^{t}\{(\rho\psi\cdot\nabla\overline{u}, \psi)+((p(\rho)-p(\rho\gamma-p’(\rho))\phi, \mathrm{d}\mathrm{i}\mathrm{v}\overline{u})+(\frac{1}{\frac{}{\rho}}\phi L\overline{u}, \psi)\}d\tau$Proof. As in [4], we introduce an energy functional based on the energyfunction defined by$\rho \mathcal{E}=\rho\{\frac{1}{2}|u|^{2}+\Phi(\rho)\}$Note that$\Phi(\rho)$,.

$\Phi(\rho)=\int^{\rho}\frac{p(\zeta)}{\zeta^{2}}d\langle$

is a strictly convex function of . We then define$\frac{1}{p}$$\rho\overline{\mathcal{E}}=\rho\{\frac{1}{2}|\psi|^{2}+\Psi(\rho,\overline{\rho})\}$

,

where
$\Psi(\rho,\tilde{\rho})$$(\begin{array}{l}11\overline{\rho}\rho-=\end{array})$$=$$\Phi(\rho)-\Phi(\overline{\rho})-\partial_{\frac{1}{\rho}}\Phi(\rho\gamma$$=$$\mathit{1}_{\tilde{\rho}}^{p}.\frac{p(\zeta)-p(\overline{\rho})}{\zeta^{2}}d\zeta$.

eo

is equivalent toAs shown in [4],and hence, there are positive constants and$|\rho-\rho\neg^{2}$$\rho\Psi(\rho,\tilde{\rho})$$c\mathit{0}$(3.3)$c_{1}$$c_{0}^{1}|U|\leq\rho\overline{\mathcal{E}}\leq c_{0}|U|$for suitably small
such that$|\rho-\rho\neg$

,

,

.
with
where $U=(\phi, \psi)$ ,Since
we can find a number $M>0$ such that if $E_{s}(t)\leq M$ ,for all $t\in[0, T]$ .
thenand
A direct calculation shows$|\phi|\leq c_{1}$$\phi=\rho-\tilde{\rho}$$H^{s}arrow\neq\neq L^{\infty}$$\inf_{x}\phi(x, t)\geq-\frac{1}{4}\inf_{x_{1}}\tilde{\rho}(x_{1})$$||\phi(t)||_{\infty}\leq c_{1}$$= \mu \mathrm{d}\mathrm{i}\mathrm{v}(\frac{1}{2}|\nabla\psi|^{2})+(\mu+\mu’)\mathrm{d}\mathrm{i}\mathrm{v}(\psi \mathrm{d}\mathrm{i}\mathrm{v}\psi)$$\partial_{t}(\rho \mathcal{E})+\mathrm{d}\mathrm{i}\mathrm{v}(\rho u\mathcal{E}+(p(\rho)-p(\rho\gamma)\psi)$,$-\mu|\nabla\psi|^{2}-(\mu+\mu’)(\mathrm{d}\mathrm{i}\mathrm{v}\psi)^{2}+\mathcal{R}_{0}$where$\mathcal{R}_{0}=\mathcal{R}_{0}(x, t)$is the function defined by.

$\mathcal{R}_{0}=-\rho(\psi\cdot\nabla\overline{u})$$\psi$$-(p( \rho)-p(\overline{\rho})-p’(\rho\gamma\phi)\mathrm{d}\mathrm{i}\mathrm{v}\overline{u}-\frac{1}{\tilde{\rho}}\phi\psi$.

$L\overline{u}$.

Proposition 3.3 now follows from this identity and (3.3). This completes theproof.

To estimate higher order derivatives, we rewrite (3.1) as$\partial_{t}\phi+u\cdot\nabla\phi+\rho_{+}\mathrm{d}\mathrm{i}\mathrm{v}\psi=f)$$\partial_{t}\psi+\frac{1}{\rho+}L\psi+\frac{p’(\rho_{+})}{\rho+}\nabla\phi=g$

I

(3.4)

$|_{x_{1}=0}=0$,$(\phi,\psi)arrow(0,0)$($(\phi,\psi)|_{t=0}=$

where$L\psi=-\underline{\mu}\Delta\psi-(\mu+\mu’)\nabla \mathrm{d}\mathrm{i}\mathrm{v}\psi$$\hat{f}=-\phi \mathrm{d}\mathrm{i}\mathrm{v}\psi$,$f=-(\overline{\rho}-\rho_{+})\mathrm{d}\mathrm{i}\mathrm{v}\psi$$\tilde{g}=\overline{g}^{(1)}+\overline{g}^{(2)}+\overline{g}^{(3)}$,,$(x_{1}arrow\infty)$$\phi_{0)}$,Vo)$f=\hat{f}+\overline{f}$and $g=-\tilde{u}$ .
, and$-\psi\cdot\nabla\overline{\rho}-\phi \mathrm{d}\mathrm{i}\mathrm{v}\overline{u}$. Here,$\nabla\psi+\hat{g}+\overline{g}$$\hat{g}=\hat{g}^{(1\}}+\hat{g}2\rangle$ $+\hat{g}^{\{3)}$with$\hat{g}^{\{1\}}=\hat{P}(\rho, \rho_{+})\phi\nabla\phi$,,$\hat{g}^{\langle 2)}=\frac{1}{\rho\rho+}\phi L\psi,\hat{g}^{\langle 3)}=-\psi\cdot\nabla\psi$,$\hat{g}^{(1)}=\hat{P}(\rho_{\mathrm{J}}\rho_{+})(\overline{\rho}-\rho_{+})\nabla\phi+\hat{P}(\rho,\tilde{\rho})\phi\nabla\overline{\rho}$$\triangleleft 2)g=\frac{1}{\rho\overline{\rho}}(L\overline{u})\phi+\frac{1}{\rho\rho+}(\overline{\rho}-\rho_{+})L\psi$$\overline{g}^{(3)}=-\psi$.

$\nabla\overline{u}$,,$P( \rho_{1}, \rho_{2})=\int_{0}^{1}p^{\prime/}(\rho_{2}+\theta(\rho_{1}-\rho_{2}))d\theta$,.

$\hat{P}(\rho_{1}, \rho_{2})=\frac{p’\{\rho_{1})}{\rho_{1}\rho_{2}}-\frac{P(\rho_{1},\rho_{2})}{\rho_{2}}$

Gl

$0$

Before proceeding further, we introduce some notations. We definebydr$N_{\sigma}(t)^{2}$$=$$N_{\sigma}\geq$$\int_{0}^{t}|[\hat{f}]|_{\sigma}^{2}+|[\hat{g}]|_{\sigma-1}^{2}+|[\psi\cdot\nabla\phi]|_{\sigma-1}^{2}$$+ \sum_{1\leq 2j+|\alpha’|\leq\sigma}\int_{0}^{t}|(\partial_{\tau}^{j}\partial_{x’}^{\alpha’}\hat{g}, \partial_{\tau}^{j}\partial_{x}^{\alpha’},\psi)|d\tau$$+ \sum_{1\leq 2j+|\alpha\}\leq\sigma}\int_{0}^{t}|(\mathrm{d}\mathrm{i}\mathrm{v}\psi, |\partial_{\tau}^{j}\partial_{x}^{\alpha}\phi|^{2})|d\tau$$+ \sum_{2j+|\alpha|\leq\sigma}\int_{0}^{t}||[\partial_{\tau}^{j}\partial_{x}^{\alpha},\psi\cdot\nabla]\phi||_{2}^{2}$

where $[C, D]$ denotes the commutator of$[C, D]$We also define$R_{\sigma}\geq 0(\sigma\geq 1)$$R_{\sigma}(t)^{2}$

$=$

$C$

,

$d\tau$,and $D$$=CD-DC$ .

by$R_{\sigma-1}(t \rangle^{2}\dashv-\int_{0}^{t}|[\tilde{f}]|_{\sigma}^{2}+|[\neg g|_{\sigma-1}^{2}+|[\overline{u}\cdot\nabla\phi]|_{\sigma-1}^{2}d\tau$$+ \sum_{1\leq 2j+|\alpha’|\leq\sigma}\oint_{0}^{t}|(\partial_{\tau}^{j}\partial_{x}^{\alpha’},\overline{g}, \partial_{\tau}^{j}\partial_{x^{l}}^{\alpha’}\psi)|d\tau$$+ \sum_{1\leq 2j+|\alpha|\leq\sigma}\int_{0}^{t}|(\mathrm{d}\mathrm{i}\mathrm{v}\tilde{u}, |\partial_{\tau}^{j}\partial_{x}^{\alpha}\phi|^{2})|d\tau$$+ \sum_{2\mathrm{i}+|\alpha|+\ell\leq\sigma-1}\int_{0}^{t}||[\partial_{\tau}^{J}\partial_{x}^{\alpha}\partial_{x_{1}}^{\ell+1},\tilde{u}\cdot\nabla]\phi||_{2}^{2}$,$d\tau$,Proposition 3.4. Let $1\leq\sigma\leq s$ . Assume that (3.2) holds. Then there
exists a constant $C>0$ such that.

$E_{\sigma}(t)^{2}+D_{\sigma}(t)^{2}\leq C\{||U_{0}||_{H^{s}}^{2}+R_{\sigma}(t)^{2}+N_{\sigma}(t)^{2}\}$

To prove Proposition 3.4 we introduce a notation$|v|_{k}=( \sum_{|\alpha|=k}||\partial_{x}^{\alpha}v||_{2}^{2})1/2$

We also define$T_{j,\alpha’}$by$T_{j,\alpha’}v=\partial_{t}^{j}\partial_{x}^{\alpha^{J}},v$.

62

Proposition 3.4 follows from the following inequalities.
Proposition 3.5. Let a be a nonnegative integer satisfying a(i) Let$||T_{j,\alpha}$

$wh$

ere

and

$j$

$\alpha’$$\leq s$.

satisfy $2j+|\alpha’|=\sigma$ . Then

, $U(t)||_{2}^{2}+ \int_{0}^{t}||L^{1/2}T_{j,\alpha’}\psi||_{2}^{2}d\tau\leq C\{||U_{0}||_{H^{s}}^{2}+R_{\sigma}(t)^{2}+N_{\sigma}(t^{2})\})$$||L^{1/2}\psi||_{2}^{2}=\mu||\nabla\psi||_{2}^{2}+(\mu+\mu’)||\mathrm{d}\mathrm{i}\mathrm{v}\psi||_{2}^{2}$(ii) Letand$j$$\alpha’$.

satisfy $2j+|\alpha’|=\sigma-1$ . Then$||L^{1/2}T_{j,\alpha’} \psi(t)||_{2}^{2}+\oint_{0}^{t}||T_{j+1,\alpha’}\psi||_{2}^{2}d\tau\leq\eta D_{\sigma}(t)^{2}+C_{\eta}N_{\sigma}(t)^{2}$for any $\eta>0$ . Here and in what follows$N_{\sigma}(t)^{2}$denotes$N_{\sigma}(t)^{2}=||U_{0}||_{H^{\mathrm{s}}}^{2}+E_{\sigma-1}(t)^{2}+D_{\sigma-1}(t)^{2}+R_{\sigma}(t)^{2}+N_{\sigma}(t^{2})$(iii) Let

and

$j$

$\alpha’$.

satisfy $2j+|\alpha’|+\ell=\sigma-1$ . Then$||T_{j,\alpha’} \partial_{x_{1}}^{l+1}\phi(t)||_{2}^{2}+\int_{0}^{t}||T_{j,\alpha’}\partial_{x_{1}}^{\ell+1}\phi||_{2}^{2}d\tau$$\leq$$\eta D_{\sigma}(t)^{2}+C_{\eta}\{N_{\sigma}(t)^{2}+\int_{0}^{t}||T_{j+1,\alpha’}\partial_{x_{1}}^{\ell}\psi||_{2}^{2}+||\partial_{x}\partial_{x’}T_{j,\alpha’}\partial_{x_{1}}^{\ell}\psi||_{2}^{2}d\tau\}$

for any y7 $>0$ .
(iv) Let
The

and

$j$

$\alpha’$satisfy $2j+|\alpha’|+\ell=\sigma-1$ and set$\frac{D\phi}{Dt}=\partial_{t}\phi+u\cdot\nabla\phi$.

$n$$f_{0}^{t}|T_{j,\alpha’} \frac{D\phi}{Dt}|_{\ell+1}^{2}d\tau\leq\eta D_{\sigma}(t)^{2}+C_{\eta}\{N_{\sigma}(t)^{2}+\int_{0}^{t}||T_{j+1,\alpha’}\partial_{x_{1}}^{l}\psi||_{2}^{2}+||\partial_{x}\partial_{x’}T_{j,\alpha’}\partial_{x_{1}}^{\ell}\psi||_{2}^{2}d\tau\}$

for any$\eta>0$(i) Let

$j$

.

and

$\alpha’$satisfy $2j+|\alpha’|+\ell=\sigma-1$ . Then$\oint_{0}^{t}|T_{j,\alpha^{J}}\psi|_{l+2}^{2}+|T_{j,\alpha’}\phi|_{l+1}^{2}d\tau$$\leq$$C \int_{0}^{t}\{|T_{j+1,\alpha’}\psi|_{f}^{2}+|T_{j,\alpha’}f|_{l+1}^{2}+|T_{j,\alpha’}\frac{D\phi}{Dt}|_{l+1}^{2}$$+|T_{j,\alpha’}(\overline{u}\cdot\nabla\psi)|_{\ell}^{2}+|T_{j,\alpha’}\hat{g}|_{l}^{2}+|T_{j,\alpha’}\tilde{g}|_{f}^{2}\}d\tau$(i) Let$j$and$\alpha’$satisfy $2j+1\leq\sigma$ . Then$|| \partial_{t}^{j+1}\phi(t)||_{2}^{2}+\int_{0}^{t}||\partial_{\tau}^{j+1}\phi||_{2}^{2}d\tau\leq\eta D_{\sigma}(t)^{2}+C_{\eta}N_{\sigma}(t)^{2}$.

83

for any $\eta>0$ .
Proof. Proposition 3.5 can be proved by the energy method as in [1, 6]. The
details can be found in [3].
It remains to estimatecase of Hardy’s ineq uality(3.5)$R_{\sigma}$

and$N_{\sigma}$. To estimate $R0$ we will use a special$|| \frac{1}{x_{1}}\int_{0}^{x_{1}}v(y)dy||_{L^{2}(0,\infty)}\leq C||v||_{L^{2}(0,\infty\rangle}$.

In a similar manner as in $[1, 4]$ , applying (3.5) and the decay estimatesin Proposition 2.1 together with the Gagliardo-Nirenberg inequality, one canshow that$R_{0}(t)^{2}\leq C\{\delta D_{0}(t)^{2}+E_{s}(t)D_{s}(t)^{2}\}$

Here we note that we also use the monotonicity of, one can show, as in [1], thatFor.

$\overline{u}^{1}(x_{1})$when $M_{+}=1$ .

$\sigma\geq 1$$R_{\sigma}(t)^{2}+N_{\sigma}(t)^{2}\leq C\{D_{\sigma-1}(t)^{2}+\delta D_{\sigma}(t)^{2}+E_{s}(t)D_{s}(t)^{2}\}$,

provided that $E_{s}(t)< \min\{M, 1\}$ . Therefore, it follows that if is sufficiently
small and $E_{s}(t)< \min\{M, 1\}$ then$\delta$$E_{s}(t)^{2}+D_{s}(t)^{2}\leq C\{||U_{0}||_{H^{\theta}}^{2}+E_{s}(t)D_{s}(t)^{2}\}$,and hence, we conclude that
$E_{s}(t)^{2}+D_{s}(t)^{2}\leq C||U_{0}||_{H^{s}}^{2}$

provided thatsition 3.2.

$||U_{0}||_{H^{s}}$,is sufficiently small. This completes the proof of Propo-

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