数理解析研究所講究録
第 1645 巻 2009 年 80-89

80

Convergence rate toward planar stationary solution for thecompressible Navier-Stokes equation in half space
中村徹 (Tohru NAKAMURA)1
九州大学・数理西畑伸也 (Shinya NISHIBATA)2
東京工業大学・情報理工

1Faculty of Mathematics,
Kyushu University, Fukuoka 812-8581, Japan

2Department of Mathematical and Computing Sciences,
Tokyo Institute of Technology, Tokyo 152-8552, Japan

Abstract

The present paper concems a $largearrow$ time behavior of a solution to an isentropic model of the compressible Navier-Stokes equation in multi-dimensionalhalf space. Precisely, we obtain a convergence rate toward a planar stationary wave for an outflow problem, where fluid blows out from a boundary,under the assumption that an initial perturbation and a boundary strength
are sufficiently small. For a supersonic flow at spatial infinity, if the initialfor
perturbation belongs to the algebraically weighted Sobolev space$t^{-\alpha/2-(n-1)/4}$
convergence
rate
,
is
then the$:=[(n-1)/2]+2$ andin-norm. For a transonic flow, due to a degenerate property of the stationary solution, we require a restriction on a weight exponent to obtain an
algebraic convergence rate. Namely if the initial perturbation belongs to thewhereis a certainforalgebraically weighted Sobolev space$t^{-\alpha/4-(n-1)/4}$.
is
positive constant, then the convergence rate$H^{s}\cap L_{\alpha}^{2}$$\alpha\geq 0$$s$$L^{\infty}$$\alpha$$H_{\alpha}^{s}$1$\alpha\in[0,\alpha^{*})$$\alpha^{*}$

Introduction

We study an asymptotic behavior of a solution to the compressible Navier-Stokes
andforequation in the multi-dimensional half space$n=2,3$ :$\mathbb{R}_{+}^{n};=\mathbb{R}_{+}\cross \mathbb{R}^{n-1}$$\rho_{t}+div(\rho u)=0$$\mathbb{R}_{+}:=(0, \infty)$(l.la)(l.lb),$\rho\{u_{t}+(u\cdot\nabla)u\}=\mu_{1}\Delta u+(\mu_{1}+\mu_{2})\nabla(divu)-\nabla p(\rho)$,

where $x=(x_{1}, \ldots, x_{n})\in \mathbb{R}_{+}^{n}$ is a space variable, which is often abbreviated as$x=(x_{1}, x’)$ withand $x’$ $:=(x_{2}, \ldots, x_{n})\in \mathbb{R}^{n-1}$ . Unknown functions$u=u(t,x)=(u_{1}(t,x), \ldots, u_{n}(t, x))$ stand for fluid density and fluid$\rho=\rho(t, x)$ andvelocity, respectively. The function $p=p(\rho)$ means a pressure, which is explicitly. The constants and are
given byfor constants $K>0$ and$2\mu_{1}+n\mu_{2}\geq
0$ . We prescribe an initial$\mu_{1}>0$
andviscosity coefficients satisfying
condition$x_{1}\in \mathbb{R}_{+}$$p(\rho)$$:=K\rho^{\gamma}$$\gamma\geq 1$$(\rho, u)(0, x)=(\rho_{0}, u_{0})(x)$$\mu_{1}$$\mu_{2}$

(1.2)

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and an outflow boundary condition$u(t, 0, x’)=(u_{b}, 0, \ldots, 0)$ ,

(1.3)

where
is a negative constant. It is assumed that a spatial asymptotic state of theinitial data in a normal direction
is a constant. Precisely, a normal componentoftends to a certain constantand a tangential component
tends to :$u_{b}$$x_{1}$$u_{0}^{1}$$u_{0}=(u_{0}^{1}, \ldots, u_{0}^{n})$$u_{+}$$u_{0}’=(u_{0}^{2}, \ldots, u_{0}^{n})$$0$$\lim_{x_{1}arrow\infty}\rho_{0}(x)=\rho_{+}$,$\lim_{x_{1}arrow\infty}(u_{0}^{1}, u_{0}’)(x)=(u_{+}, 0)$

.

(1.4)

It is also assumed that the initial density is uniformly positive:$\inf_{x\in \mathbb{R}_{+}^{n}}\rho_{0}(x)>0$,$\rho_{+}>0$.

IFlrom the pioneering work [2] by Il’in and Oleinik, there have been many researches on the asymptotic stability of several kinds of nonlinear waves for the
initial value problem to scalar viscous conservation laws. Especially, Kawashima,
Matsumura and Nishihara in [6, 12] proved the asymptotic stability of viscous shockwaves and obtained the convergence rate by using a weighted energy method, of
which idea is also used to obtain the rate. The research by Liu, Matsumura andNishihara in [8] started the analysis on the asymptotic stability of stationary waves
for the initial and boundary value problem in one-dimensional half space.
For the isentropic model of the compressible Navier-Stokes equation, Matsumura
in [10] gave classification of the possible asymptotic states for a one-dimensional half
space problem and expected that one of asymptotic states for the outflow problem
is a stationary solution. This fact for the special case was verified by Kawashima,
Nishibata and Zhu in [7] under the smallness condition on the boundary strength andthe initial perturbation. The convergence rate for this system was firstly obtainedby Nakamura, Nishibata and Yuge in [14] by assuming that the initial perturbation
belongs to a certain weighted Sobolev space. Let us note that this type of convergence rate for the system is first obtained by Nishikawa and Nishibata in [16], wherea coupled system of Burgers Poisson equations is studied.
For the multi-dimensional problem for the isentropic model, Matsumura in [9]obtained a convergence rate $O(t^{-3/4})$ toward a constant state for a three-dimensionalcase under the smallness ofnorm of the initial data. The proof is based on timeweighted energy estimates of solutions. For the half space problem in, Kagei andKobayashi in [5] proved that the perturbation from a constant state decays with aconvergence rate
for the impermeable problem if the initial perturbationinis sufficiently small where $s\geq[n/2]+1$ . Kagei and Kawashima in[4] studied the outflow problem (1.1), (1.2) and (1.3) and showed that the planar
stationary wave is time asymptotically stable by using theenergy method where$H^{3}$$\mathbb{R}_{+}^{n}$$O(t^{arrow n/2})$$H^{s}\cap L^{1}$$H^{s}$$s\geq[n/2]+1$.

In the present paper, we study a convergence rate of the solution toward theplanar stationary wave
, which is a solution to 1.1), independent ofand , satisfying that is given by the form
. Therefore it satisfies$(\tilde{\rho}(x_{1}),\tilde{u}(x_{1}))$$x’$$\tilde{u}$$($$\tilde{u}=(\tilde{u}_{1},0, \ldots, 0)$

$t$

82

$(\tilde{\rho}\tilde{u}_{1})_{x_{1}}=0$(1.5a)(1.5b),$(\tilde{p}\tilde{u}_{1}^{2}+p(\tilde{\rho}))_{x_{1}}=\mu\tilde{u}_{1x_{1}x_{1}}$,is a positive constant defined by $:=2\mu_{1}+\mu_{2}$ . The stationary solution
is supposed to satisfy the boundary condition (1.3) and the spatial
asymptotic condition (1.4) as well as a positivity of the density:

where

$\mu$

$\mu$

$(\tilde{\rho}(x_{1}),\tilde{u}_{1}(x_{1}))$$\tilde{u}_{1}(0)=u_{b}$,$\lim_{x_{1}arrow\infty}(\tilde{\rho}(x_{1}),\tilde{u}_{1}(x_{1}))=(\rho_{+}, u_{+})$,$\inf_{x_{1}\in R+}\tilde{\rho}(x_{1})>0$

.

(1.6)

The solvability of the one-dimensional boundary value problem (1.5) and (1.6) is
discussed in the paper [7]. To summarize the existence result and the decay property
at theand the Mach numberof the stationary wave, we define sound speed
spatial asymptotic state:$M_{+}$$c_{+}$,$c_{+}:=\sqrt{p’(\rho_{+})}=\sqrt{\gamma K\rho_{+}^{\gamma-1}}$$M_{+}:= \frac{|u_{+}|}{c_{+}}$.

Moreover the quantity $:=|u_{b}-u_{+}|$ , which is called a boundary strength, plays anessential role in existence and stability analyses on the stationary wave.
$\delta$

Proposition 1.1 ([7]). There exists a positive constantary value problem (1.5) and (1.6) has a unique solution
conditions$u_{+}<0$ and $u_{b}<w_{c}u_{+}$$w_{c}$

such that the boundif and only if the$(\tilde{\rho},\tilde{u}_{1})$$M_{+}\geq 1_{f}$hold. Moreover the solution$(\tilde{\rho},\tilde{u}_{1})$(1.7)

satisfies the following decay estimates.

(i) If $M_{+}>1$ , there exist positive constants
solution
satisfies$c$and $C$ such that the stationary$(\tilde{\rho})\tilde{u}_{1})$$|\partial_{1}^{k}(\tilde{\rho}(x_{1})-\rho_{+},\tilde{u}_{1}(x_{1})-u_{+})|\leq C\delta e^{-cx_{1}}$for $k=0,1,2,$$\ldots$.

(1.8)(ii) If $M_{+}=1$ , there exist a positive constant $C$ such that the stationary solution$(\tilde{\rho},\tilde{u}_{1})$satisfies$| \partial_{1}^{k}(\tilde{\rho}(x_{1})-\rho_{+},\tilde{u}_{1}(x_{1})-u_{+})|\leq C\frac{\delta^{k+1}}{(1+\delta x_{1})^{k+1}}$

The constantdefined by$w_{c}$

for $k=0,1,2,$$\ldots$.

(1.9)

in (1.7) is determined as a root of $H(w_{c})=0$ , where $H$ is$H(w_{c});= \frac{\rho_{+}u_{+}}{\mu}(w_{c}-1)+\frac{K\rho_{+}^{\gamma}}{\mu u_{+}}(w_{c}^{-\gamma}-1)$

Namely, for the subsonic case $M+>1,$$w_{c}$.

is a root of the equation$\rho_{+}u_{+}^{2}(w_{c}-1)+K\rho_{+}^{\gamma}(w_{c}^{-\gamma}-1)=0$

is equal to 1.
satisfying $w_{c}>1$ . For the super sonic case $M_{+}=1,$
For the multi-dimensional half space problem, Kagei and Kawashima in [4]under smallproved the asynptotic stability of the planar stationary wave
. In theboundary
strengthandperturbation
theness assumptions on the initial$w_{c}$$(\tilde{\rho},\tilde{u})$$\delta$

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paper [13], a convergenoe rate of the solution toward the planar stationary wave isobtained by assuming that the initial perturbation decays in the normal direction
with the algebraic or the exponential rate. It is also required that the initial perturbation is sufficiently small inwith $n=2$ and 3. Here is a positive integer
defined by$H^{s}(\mathbb{R}_{+}^{n})$

$s$

$s$

$:=[ \frac{n-1}{2}]+2$,(1.10)

where $[x]$ denotes the greatest integer which does not exceed .
We summarize the results on convergence rates obtained in the paper [13]. In
Theorem 1.2, the convergence rate for a supersonic case $M_{+}>1$ is proved.
$x$

Theorem 1.2 ([13]). Let $n=2$ or 3, and be a positive integer defined by (1.10).
Suppose that the conditions $M_{+}>1_{f}(1.7)$ and 1
hold fora certain positive constant . Moreover, if the initial data satisfies, then the solution
to the initial boundary
for a certain constantvalue problem 1.1), 1.2) and (1.3) satisfies the decay estimate$s$$(\rho_{0}-\tilde{\rho}, u_{0}-\tilde{u})\Vert_{H^{s}}+\delta\leq\epsilon_{0}$$(\rho_{0}-\tilde{\rho},$$\epsilon_{0}$$($$u_{0}-\tilde{u})\in$$(\rho, u)$$\alpha\geq 0$$L_{\alpha}^{2}(\mathbb{R}_{+}^{n})$$($.

(1.11)$\Vert(\rho, u)(t)-(\tilde{\rho},\tilde{u})\Vert_{L}\infty\leq C(1+t)^{-\alpha/2-(n-1)/4}$

The convergence rate $(1+t)^{-\alpha/2}$ in 1.11) holds since the two characteristics ofthe corresponding Euler equation$($$\rho_{t}+(\rho u_{1})_{x_{1}}=0$,$(\rho u_{1})_{t}+(\rho u_{1}^{2}+p(\rho))_{x1}=0$$x_{1}arrow\infty$ .
On the other hand, the convergence rate $(1+$in (1.11) is obtained by using a dissipative property of the viscosity in theequation (l.lb). Therefore, the convergence rate (1.11) holds owing to both of thehyperbolicity and the parabolicity of the equations (1.1).
The next theorem shows the algebraic convergence for the transonic case $M_{+}=1$ .
In this case, owing to the degenerate property of the stationary wave, the convergence rate is worse than that for the supersonic case $M_{+}>1$ . Moreover we needa upper restriction on the exponent . Namely we assume that is smaller thana certain positive constantin (1.12) below. This kind of restriction is also required in the researches for scalar viscous conservation laws in [12, 17], in which theasymptotic stability of a degenerate traveling wave is proved.

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

are negative as$t)^{-(n-1)/4}$$\alpha$$\alpha$$\alpha_{*}$

Theorem 1.3 ([13]). Let $n=2$ or 3, and be a positive integer defined by (1.10).
Suppose that $M_{+}=1$ and (1.7) hold. Let,
be a constant satisfying
where
is a positive constant defined by$s$$\alpha\in[0, \alpha_{*})$$\alpha$$\alpha_{*}$$\alpha_{*}:=\frac{2}{a}(1+\sqrt{a+1})$,$a:=1+(n-1)( \frac{\mu_{1}+\mu_{2}}{2\mu_{1}+\mu_{2}})^{2}$

.

(112)

Then there exists a certain positive constant such that if I
, then the solution
to the initial boundary value problem (1.1), (1.2) and (1.3)
satisfies the estimate$\epsilon_{0}$$\epsilon_{0}$$(\rho_{0}-\tilde{\rho}, u_{0}-\tilde{u})\Vert_{H_{\alpha}^{\epsilon}}+\delta\leq$$(\rho, u)$$\Vert(\rho, u)(t)-(\tilde{\rho},\tilde{u})\Vert_{L^{\infty}}\leq C(1+t)^{-\alpha/4-(n-1)/4}$.

(1.13)

84
Remark 1.4. For the case of $M_{+}>1$ , we can also obtain the exponential convergence rate$\Vert(\rho, u)(t)-(\tilde{\rho},\tilde{u})\Vert_{L^{\infty}}\leq Ce^{-\alpha t}$

by assuming that the initial perturbation belongs to the exponentially weighted

space$L_{\alpha,\exp}^{2}(\mathbb{R}_{+}^{n}):=\{u\in L_{1oc}^{2}(\mathbb{R}_{+}^{n});e^{(\alpha/2)x_{1}}u\in L^{2}(\mathbb{R}_{+}^{n})\}$.

and $:=$. The operators
andNotations. Let
denote standard gradient and Palladian with respect to $x=(x_{1}, \ldots, x_{n})$ .
$:=$denote tangential gradientandThe operators$x’=(x_{2},\ldots,x_{n})$. The expression
is sometimes
and Laplacian with respect toand , the. For a non-negative integer , we denote byabbreviated tototality of all k-th order derivatives with respect to and , respectively. For nonandare defined byand , the operators
negative integers$:=$ Oi
and $1\leq p\leq\infty$ , the space. For a domain
and
. We
If
denotes the standard Lebesgue space equipped with the norm
. We also use the notations , $:=$
to $U$ if
sometimes abbreviate
denotes. For a non-negative integer
and.
We notesense with the normin thethe s-th order Sobolev space over$H^{0}=L^{2}$ and, the spacedenotes the. For a constant$:=\{u\in$space in the normal direction defined byalgebraically weighted
equipped with the norm$\nabla$$\partial_{t}:=\frac{\partial}{\theta t}$$\partial_{i}:=\frac{\partial}{\partial x:}$$\Delta$$:=(\partial_{1}, \ldots, \partial_{n})$$\sum_{i=1}^{n}\partial_{i}^{2}$$(\ , \ldots, \partial_{n})$$\partial_{x}/$$\Delta_{x’}:=\sum_{1=2}^{n}\partial_{i}^{2}$$\partial_{x’}u$$\nabla^{k}$$k$$u_{x}/$

$x’$

$x$

$k$

$i,$ $j$$T_{j,k}$$\theta_{x}^{;},\partial_{t}^{k}=\theta_{1}T_{j,k}$$\partial_{i,j,k}$$\partial_{x}^{k}$$T_{j,k}$$\partial_{i_{\nu}j,k}$$:=\dot{\theta}_{x’}\partial_{t}^{k}$$\Omega\subset \mathbb{R}_{+}^{n}$$\Vert\cdot\Vert_{L(\Omega)}p$$(\Omega)$$L^{p}(\Omega)$$L^{p}(\mathbb{R}^{n-1})$$L_{x}^{p_{1}}$$L_{x}^{p}$$\Omega=\mathbb{R}_{+}^{n}$$:=L^{P}(\mathbb{R}_{+})$$H^{s}=H^{s}(\mathbb{R}_{+}^{n})$$s,$$L^{2}$$\Vert\cdot\Vert_{H^{s}}$$\mathbb{R}_{+}^{n}$$\Vert\cdot\Vert$$\alpha\in \mathbb{R}$$:=\Vert\cdot\Vert_{L^{2}}$$L_{\alpha}^{2}(\mathbb{R}_{+}^{n})$$L^{2}$$L_{\alpha}^{2}(\mathbb{R}_{+}^{n})$$L_{1oc}^{2}(\mathbb{R}_{+}^{n});|u|_{\alpha}<\infty\}$$|u|_{\alpha}:= \Vert u\Vert_{L_{\alpha}^{2}}:=(\int_{R_{+}^{n}}(1+x_{1})^{\alpha}|u(x)|^{2}dx)^{1/2}$

The spacedefined by$H_{\alpha}^{s}(\mathbb{R}_{+}^{n})$$H_{\alpha}^{\epsilon}(\mathbb{R}_{+}^{n})$space corresponding todenotes the algebraic weighted$0\leqk\leqs\}$, equipped with the normfor$H^{\theta}$$L_{\alpha}^{2}$$:=\{u\in L_{\alpha}^{2};\nabla^{k}u\in L_{\alpha}^{2}$$\Vert u\Vert_{H_{\alpha}^{\delta}}:=(\sum_{k=0}^{\delta}|\nabla^{k}u|_{\alpha}^{2})^{1/2}$

2

.

.

A priori estimates for supersonic flow

In this section, we show key estimates of solutions, which are essential in the proofof the stability theorems. The proof is mainly based on the time and space weighted
spaces. To this end,
and weighted
a priori estimates of the perturbation in
we employ the perturbation$L^{2}$$H^{s}$$(\varphi,\psi)(t, x):=(\rho, u)(t, x)-(\tilde{\rho},\tilde{u})(x_{1})$. Owing to equations (1.1) and (1.5), the perturfrom the stationary solution
satisfies the system of equations
bation$(\tilde{\rho},\tilde{u})$$(\varphi, \psi)$$\varphi_{t}+u\cdot\nabla\varphi+\rho div\psi=f$,$\rho\{\psi_{t}+(u\cdot\nabla)\psi\}-L\psi+p’(\rho)\nabla\varphi=g$,(2.la)(2.lb)

85
where $f,$

$g$

and

$L\psi$are given by$f:=-div\tilde{u}\varphi-\nabla\overline{\rho}\cdot\psi$,,$g:=-\rho(\psi\cdot\nabla)\tilde{u}-\varphi(\tilde{u}\cdot\nabla)\tilde{u}-(p’(\rho)-p’(\tilde{\rho}))\nabla\tilde{\rho}$$L\psi:=\mu_{1}\triangle\psi+(\mu_{1}+\mu_{2})\nabla div\psi$

The initial and the boundary conditions for$(\varphi, \psi)$as.

are derived $hom(1.2)$ and (1.3)$(\varphi, \psi)(0, x)=(\varphi_{0}, \psi_{0})(x):=(\rho_{0}, u_{0})(x)-(\tilde{\rho},\tilde{u})(x_{1})$

,

(2.2)
(2.3)

$\psi(t, 0, x’)=0$ .

The perturbation is often abbreviated as$\Phi:=(\varphi, \psi)$,$\Phi_{0}:=(\varphi_{0}, \psi_{0})$To summarize the a priont estimate for$\Vert|u\Vert|_{m}^{2}:=\sum_{i-arrow 0}^{m}|[u]|_{i}^{2}$,$(\varphi,\psi)$.

, we employ the following notations:$|[u]|_{m}^{2}:= \sum_{k=0}^{[m/2]}\Vert\nabla^{m-2k}\partial_{t}^{k}u\Vert^{2}$and a time weighted norm $E(t)$ and a corresponding dissipative norm $D(t)$ definedby,$E(t)^{2}:= \sum_{j=0}^{s-1}(1+t)^{j}\Vert|\theta_{x}^{;},\Phi(t)\Vert|_{s-j}^{2}$$N(t):= \sup_{0\leq\tau\leq t}E(\tau)$$D(t)^{2}:= \sum_{j=0}^{\epsilon-1}(1+t)^{j}\hat{D}_{j}(t)^{2}$,,$\hat{D}_{j}(t)^{2}:=\sum_{i=1}^{\epsilon-j}|[\theta_{x}^{;},\Phi(t)]|_{i}^{2}+|[\theta_{x}^{j},\psi(t)]|_{s+1-j}^{2}+\Vert\partial_{x}^{;},\varphi(t, 0, \cdot)\Vert_{L_{x}^{2}}^{2},\cdot$

In addition, define spatial weighted norms$\tilde{E}_{\alpha}(t)^{2}:=E(t)^{2}+|\Phi(t)|_{\alpha}^{2}$,$\tilde{E}_{\alpha}(t)$and$\tilde{D}_{\alpha}(t)$by$\tilde{D}_{\alpha}(t)^{2}:=D(t)^{2}+\alpha|\Phi(t)|_{\alpha-1}^{2}+|\nabla\psi(t)|_{\alpha}^{2}$.

We show a uniform bound of, which is summarized in Proposition 2.2. To
this end, we employ function spaces as$\tilde{E}_{\alpha}(t)$$X(0, T):=\{(\varphi, \psi)\in C([0, T];H^{s});\nabla\varphi\in L^{2}(0, T;H^{\epsilon-1}), \nabla\psi\in L^{2}(0, T;H^{s})\}$,$X_{\alpha}(O, T):=\{(\varphi, \psi)\in X(0, T);(\varphi, \psi)\in C([0, T];L_{\alpha}^{2}), \nabla\psi\in L^{2}(0, T;L_{\alpha}^{2})\}$

for $T>0$ and.
The following lemma shows the existence of the solution to (2.1), (2.2) and (2.3)
locally in time, which can be proved by a standard iteration method with using theidea in [3].
$\alpha\geq 0$

Lemma 2.1. Suppose that the initial data satisfies
and a suitable compatibility condition. Then there exists a positive constantdependingonsuch that the problem (2.1), (2.2) and (2.3) has a unique solution$(\varphi, \psi)\in X(O, T)$ . Moreover,, it holds
if the initial data satisfies$(\varphi_{0}, \psi_{0})\in H^{s}(\mathbb{R}_{+}^{n})$$T$$\Vert(\varphi_{0}, \psi_{0})\Vert_{H^{\epsilon}}$$(\varphi_{0}, \psi_{0})\in L_{\alpha}^{2}(\mathbb{R}_{+}^{n})$$(\varphi, \psi)\in X_{\alpha}(0, T)$.

86
The following proposition gives the algebraically weighted a priori estimates
for the supersonic case $M_{+}>1$ . From the algebraically weighted estimates (2.4)
and (2.5), we see that the tangential derivatives of the solution verify better decay
estimates than the normal derivatives.

be a solution
Proposition 2.2. Suppose that $M_{+}>1$ holds. Let. Then there exist positiveto (2.1), (2.2) and (2.3) for certain $T>0$ and, then the solution
and $C$ independent of such that if
constants$t\in[0,T]$:satisfies the following estimates for$(\varphi, \psi)\in X_{\alpha}(O, T)$$\alpha\geq 0$$T$$\epsilon_{1}$$N(T)+\delta\leq\epsilon_{1}$$\Phi=(\varphi, \psi)$(2.4)$(1+t)^{\ell} \tilde{E}_{\alpha-\ell}(t)^{2}+\int_{0}^{t}(1+\tau)^{\ell}\tilde{D}_{\alpha-\ell}(\tau)^{2}d\tau\leq C(|\Phi_{0}|_{\alpha}^{2}+\Vert\Phi_{0}\Vert_{H^{\ell}}^{2})$for an arbitrary integer $\ell=0,$$[\alpha]$$\ldots,$

and$(1+t)^{\xi} \tilde{E}_{0}(t)^{2}+\int_{0}^{t}(1+\tau)^{\xi}\tilde{D}_{0}(\tau)^{2}d\tau\leq C(|\Phi_{0}|_{\alpha}^{2}+\Vert\Phi_{0}\Vert_{H^{\epsilon}}^{2})(1+t)^{(-\alpha}$

for an arbitrary$\xi>\alpha$

(2.5)

.

and
The proof of Proposition 2.2 is based on deriving the estimates in. To obtain these estimates, we utilize an interpolation inequality and the
Poincar\’e type inequality, which are summarized$L_{\alpha}^{2}(\mathbb{R}_{+}^{n})$$H^{\epsilon}(\mathbb{R}_{+}^{n})$Lemma 2.3. Let

be a solution to (2.1), (2.2) and (2.3).

$\Phi=(\varphi, \psi)$(i) Let $2<p\leq\infty$ and let$j$

and $m$ be integers satisfying$0\leq j+m\leq s$ ,Then the solution$\Phi$$\theta:=\frac{n}{m}(\frac{1}{2}-\frac{1}{p})\in(0,1)$.

satisfies.

$\Vert\partial_{x}^{i},\Phi(t)\Vert_{Lp}\leq C\Vert\dot{\theta}_{x},\Phi(t)\Vert^{1-\theta}\Vert\nabla^{m}\theta_{x}^{;},\Phi(t)\Vert^{\theta}\leq CE(t)(1+t)^{-j/2}$(2.6)

The inequality (2.6) also holds for the cases $p=2,$ $m=0,$ $\theta=0$ and $0\leq j\leq$$s-1$ .
(ii) Suppose that $M+>1$ holds. Let be a stationary solution to (1.5) and (1.6)
satisfying (1.8). Then satisfies$\tilde{u}$$\Phi$$\int_{R_{+}^{n}}|\nabla^{k}\tilde{u}||\dot{\theta}_{x’}\Phi(t)|^{2}dx\leq C\delta(\Vert\nabla\partial_{x}^{;},\Phi(t)\Vert^{2}+||\dot{\theta}_{x’}\varphi(t, 0, \cdot)\Vert_{L_{x’}^{2}}^{2})$for integers$k\geq 1$(2.7)

and $0\leq j\leq s-1$ .

Using the above lemma, we consider the derivation of the estimate of the per. To do this, we introduce an energy form , similarly asinturbationin [7]:$(\varphi,\psi)$$\mathcal{E}$$L_{\alpha}^{2}(\mathbb{R}_{+}^{n})$$\mathcal{E}:=K\tilde{\rho}^{\gamma-1}\omega(\frac{\tilde{\rho}}{\rho})+\frac{1}{2}|\psi|^{2}$,$\omega(r):=r-1-\int_{1}^{r}\eta^{-\gamma}d\eta$Under the smallness assumption on $N(T)$ , we haveenergy form is equivalent to the square of the perturbation.

$||\Phi(t)\Vert_{L}\infty\ll 1$$\mathcal{E}$$c(\varphi^{2}+|\psi|^{2})\leq \mathcal{E}\leq C(\varphi^{2}+|\psi|^{2})$.

$(\varphi, \psi)$. Hence, the

:
(2.8)

87
Moreover we have the uniform bounds of solutions as$|u(t, x)|\leq C$$0<c\leq\rho(t, x)\leq C_{\}}$

,

(2.9)

$-C\leq u_{1}(t, x)\leq-c<0$ ,owing to $u_{b}<0$ and $N(T)+\delta\ll 1$ . Using the time and space weighted energy
method, we obtain the energy inequality inframework.
$L^{2}$

Lemma 2.4. Suppose that the same conditions as in Proposition 2.2 hold. Then
there cists a positive constantsuch that if, it holds$e$$N(T)+\delta\leq\epsilon_{1}$$\epsilon_{1}$$(1+t)^{\xi}| \Phi(t)|_{\beta}^{2}+\int_{0}^{t}(1+\tau)^{\xi}(\beta|\Phi(\tau)|_{\beta-1}^{2}+|\nabla\psi(\tau)|_{\beta}^{2}+\Vert\varphi(\tau, 0, \cdot)\Vert_{L_{x}^{2}}^{2},)d\tau$$\leq C|\Phi_{0}|_{\beta}^{2}+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}|\Phi(\tau)|_{\beta}^{2}d\tau+C\delta\int_{0}^{t}(1+\tau)^{\xi}\Vert\nabla\varphi(\tau)\Vert^{2}d\tau$

for $t\in[0, T]$ and arbitrary constants$\beta\in[0, \alpha]$and$\xi\geq 0$(2.10).

Next we show the estimates for higher order derivatives. Precisely we derive
the time weighted energy estimate in, which is summarized in the next
proposition.
$H^{8}(\mathbb{R}_{+}^{n})$Proposition 2.5. Suppose that the same conditions as in Proposition 2.2 hold.
Then there exists a positive constant, it holds
such that if$N(T)+\delta\leq\epsilon_{1}$$\epsilon_{1}$$(1+t)^{\xi}E(t)^{2}+ \int_{0}^{t}(1+\tau)^{\xi}D(\tau)^{2}d\tau\leq C\Vert\Phi_{0}\Vert_{H^{\epsilon}}^{2}+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}\Vert|\Phi(\tau)\Vert|_{s}^{2}d\tau(2.11)$

for an arbitrary$\xi\geq 0$.

Here we give a brief outline of the proof. (For the details, the readers are referredto the paper [13]. It is divided into several steps. We firstly discuss the derivationof estimates for tangential and time derivatives
for $0\leq j+2k\leq s$ . By using$0\leqj+2k\leq s-1$ . Then we obtain
the parabolicity, we show estimates offorestimates of -derivatives of , i,e.,
for $0\leq i+j+2k\leq s-1$ . Finally
we get estimates of second order -derivatives of by substituting the previously
obtained estimates in the equation (2.lb). These computations give the desired
estimate (2. 11).
Next we discuss the derivation of the estimates (2.4) and (2.5). Adding (2.10)
to (2.11) and then letting suitably small, we have$)$$T_{j,k}\Phi$$\nabla T_{j,k}\psi$$x_{1}$$\partial_{i+1,j,k}\varphi$$\varphi$$x_{1}$$\psi$$\delta$$(1+t)^{\xi} \tilde{E}_{\beta}(t)^{2}+\int_{0}^{t}(1+\tau)^{\xi}(\beta|\Phi(\tau)|_{\beta-1}^{2}+\tilde{D}_{\beta}(\tau)^{2})d\tau$$\leq C(|\Phi_{0}|_{\beta}^{2}+\Vert\Phi_{0}\Vert_{H^{\epsilon}}^{2})+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}(|\Phi(\tau)|_{\beta}^{2}+\Vert|\Phi(\tau)\Vert|_{s}^{2})d\tau$Substituting the inequality$\Vert|\Phi(\tau)\Vert|_{s}^{2}=\Vert\Phi(\tau)\Vert^{2}+\sum_{i=1}^{\epsilon}|[\Phi(\tau)]|_{i}^{2}\leq|\Phi(\tau)|_{\beta}^{2}+\tilde{D}_{\beta}(\tau)^{2}$.

(2.12)

88in the second term on the right hand side of (2.12), we get$(1+t)^{\xi} \tilde{E}_{\beta}(t)^{2}+\int_{0}^{t}(1+\tau)^{\xi}(\beta|\Phi(\tau)|_{\beta-1}^{2}+\tilde{D}_{\beta}(\tau)^{2})d\tau$$\leq C(|\Phi_{0}|_{\beta}^{2}+\Vert\Phi_{0}\Vert_{H^{s}}^{2})+C\xi\int_{0}^{t}(1+\tau)^{\xi-1}(|\Phi(\tau)|_{\beta}^{2}+\tilde{D}_{\beta}(\tau)^{2})d\tau$.

Applying an induction with respect to and , of which idea is developed in [6] and[15], we obtain the desired estimates (2.4) and (2.5).
$\beta$$\xi$

Finally, Theorem 1.2 is proved by using the interpolation inequality in$L^{\infty}$

norm:$\Vert\Phi\Vert_{L^{\infty}}=\sup_{x_{1}\in R+}\Vert\Phi(x_{1}, \cdot)\Vert_{L_{x’}^{\infty}}$$\leq C\sup_{x_{1}\in R+}(\Vert\Phi(x_{1}, \cdot)\Vert_{L_{x}^{2}}^{1-\theta}\Vert\partial_{x}^{s-1}\Phi(x_{1}, \cdot)\Vert_{L_{x}^{2}}^{\theta},)$$\leq C\Vert(\Phi, \nabla\Phi)\Vert^{1-\theta}\Vert\partial_{x}^{s-1}(\Phi, \nabla\Phi)\Vert^{\theta}$for$\theta=\frac{n-1}{2(s-1)}$(2.13),and the Sobolev
which follows from the Gagliardo-Nirenberg inequality over
. Then substituting the decay estiinequalitymates$\mathbb{R}^{n-1}$$\Vert v\Vert_{L^{\infty}(R_{+})}\leq C\Vert v\Vert_{L^{2}(R_{+})}||v_{x}1\Vert_{L^{2}(R_{+})}$$\Vert(\Phi, \nabla\Phi)(t)\Vert\leq C(1+t)^{-\alpha/2}$,$\Vert\partial_{x}^{\epsilon-1}(\Phi, \nabla\Phi)(t)\Vert\leq C(1+t)^{-(\alpha+s-1)/2}$,

which are direct consequences of (2.5), in the inequality (2.13), we get the desired
decay estimate (1.11).

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