数理解析研究所講究録
第 1985 巻 2016 年 60-79

60

Time-periodic problem for the compressible
Navier-Stokes-Korteweg system on$\mathbb{R}^{3}$$E$

1

Kazuyuki TSUDA
Graduate School of Mathematics,
Kyushu University,
Fukuoka 819-0395, JAPAN-mail: k-tsuda@math.kyusyu-u.ac.jp

Introduction

We consider time periodic problem for the following compressible Navier-Stokes-Korteweg
system in:$\mathbb{R}^{3}$$\{\begin{array}{ll}\partial_{t}\rho+divM=0, (1.1)\partial_{t}M+div(\frac{M\otimes M}{\rho})=div(S(\frac{M}{\rho})+\mathcal{K}(\rho))+\rho g, (1.2)\partial_{t}(\rho E)+div(ME)+div(P(\rho, \theta)\frac{M}{\rho}) \end{array}$$= \tilde{\alpha}\triangle\theta+div((S(\frac{M}{\rho})+\mathcal{K}(\rho))\frac{M}{\rho})+Mg$

.

(1.3)

Here $\rho=\rho(x, t)$ , $M=(M_{1}(x, t), M_{2}(x, t), M_{3}(x, t))$ and $E=E(x, t)>0$ denote theunknown density, momentum, and total energy respectively, at timeand position
denotes the absolute temperature of fluid satisfying$t\in \mathbb{R}$$x\in \mathbb{R}^{3};\theta$$E=C_{v} \theta+\frac{1}{2}\frac{|M|^{2}}{\rho^{2}},$

denotes the heat capacity at the constant volume, that is assumed to be apositive constant;anddenote the viscous stress tensor and the Korteweg stresstensor that are given by

where$C_{v}$$\mathcal{S}$$\mathcal{K}$$\{S(\frac{M}{\rho)\rho})=(\mu’div\frac{M}{\rho 1})\delta_{i,j}+2\mu d_{ij}9_{4}^{\frac{M}{\rho}})_{\partial}\mathcal{K}(=\frac{\kappa}{2}(\triangle\rho^{2}-\nabla\rho|^{2})\delta i,j_{\dot{\partial}x_{t}\partial x_{J}}-\kappa--4_{-},$

where$d_{ij}( \frac{M}{\rho})=\frac{1}{2}(\frac{\partial}{\partial x_{l}}(\frac{M}{\rho})_{j}+\frac{\partial}{\partial x_{J}}(\frac{M}{\rho})_{i});\mu$and

are the viscosity coeﬃcients that$\mu’$

are assumed to be constants satisfying$\mu>0, \frac{2}{3}\mu+\mu’\geq 0$

(1.4)

;

61
$P=P(\rho, \theta)$

is the pressure that is assumed to be a smooth function of$\rho$

and$\theta$satisfying$P_{\rho}(\rho_{*}, \theta_{*})>0, P_{\theta}(\rho_{*}, \theta_{*})>0,$

where
and
are given positive constants; and denote the capillary constant andthe heat conductivity coeﬃcient respectively, that are assumed to be positive constants;
and $9=g(x, t)$ is a given external force periodic in . We assume that $g=g(x, t)$ satisfies
the condition$9(x, t+T) = g(x, t) (x\in \mathbb{R}^{3}, t\in \mathbb{R})$(1.5)$\tilde{\alpha}$$\theta_{*}$$\kappa$$\rho_{*}$$t$for some constant $T>0.$
The system $(1.1)-(1.3)$ is known to be a model system for two phase flow with phasetransition between liquid and vapor in compressible fluid. In deriving $(1.1)-(1.3)$ , phasetransition boundary is regarded as a diﬀuse interface. So $(1.1)-(1.3)$ describes fluid stateby the changes of the density. $(Cf., [4, 6, 11] for the$ derivation $of (1.1)-(1.3).$ )As for the mathematical analysis for $(1.1)-(1.3)$ , most of literatures treated the systemin terms of the density , velocity $v=M/\rho$ and absolute temperature :$\theta$$\rho$$\{\begin{array}{ll}\partial_{t}\rho+div(\rho v)=0, (1.6)\rho(\partial_{t}v+(v\cdot\nabla)v)+\nabla P(\rho, \theta)=\mu\triangle v+(\mu+\mu’)\nabla divv+\kappa\rho\nabla\triangle\rho+\rho g, (1.7)\rho C_{v}(\theta_{t}+(v\cdot\nabla)\theta)+\theta P_{\theta}(\rho, \theta)divv=\tilde{\alpha}\triangle\theta+\Psi(v)+\tilde{\Phi}(\rho, v) , (1.8)\end{array}$where$\Psi(v)$and$\tilde{\Phi}(\rho, v)$

are given by$\{\begin{array}{l}\Psi(v)=\mu’(divv)^{2}+2\mu \mathbb{D}v:\mathbb{D}v, \mathbb{D}v=(d_{ij}(v))_{i,j=1}^{3},\tilde{\Phi}(\rho, v)=\kappa(\frac{|\nabla\rho|^{2}}{2}+\rho\triangle\rho)divv-\kappa(\nabla\rho\otimes\nabla\rho):\nabla v.\end{array}$Chen and Zhao ([3]) considered the stationary problem $(1.6)-(1.8)$ for of the form $g(x)=$. It was shown in [3] that if satisfies
around$g$$divg_{1}(x)+g_{2}(x)$$(\rho_{*}, 0, \theta_{*})$$g$$\sum_{k=1}^{3}\Vert(1+|x|)^{k+1}\nabla^{k}g\Vert_{L^{2}}+\sum_{k=0}^{1}\Vert(1+|x|)^{3+k}\nabla^{k}g\Vert_{L^{\infty}}$$+\Vert(1+|x|)^{2}g_{1}\Vert_{L}\infty+\Vert(1+|x|)^{-1}g_{2}\Vert_{L^{1}}\ll 1$

,

(1.9)

then there exists a stationary solution for problem $(1.6)-(1.8)$ in the weighted
space. The stability of the stationary solution was also considered in [3]. It was shown in
is asymptotically stable[3] that if satisfies (1.9), then the stationary solution
under suﬃciently small initial perturbations, and the perturbation satisfies$L^{\infty}\cap L^{2}$$g$$(\rho^{*}, v^{*}, \theta^{*})$$\Vert(\rho(t), v(t), \theta(t))-(\rho^{*}, v^{*}, \theta^{*})\Vert_{L^{\infty}}arrow 0$. Chen, Xiao and Zhao ([2]) and Cai, Tan and Xu ([1]) then considered time
periodic problem for the barotropic and non-barotropic system of $(1.6)-(1.8)$ , respectively,onwith $n\geq 5$ . They proved that there exists a time periodic solution
for a suﬃciently small $g\in C^{0}(\mathbb{R};H^{N-1}\cap L^{1})$ satisfying (1.5), wherearoundas$tarrow\infty$$(\rho_{per}, v_{per}, \theta_{per})$$\mathbb{R}^{n}$$(\rho_{*}, 0, \theta_{*})$

62
satisfying $N\geq n+2$ . Furthermore, the time periodic solution is stable undersuﬃciently small perturbations and it holds that
$N\in \mathbb{Z}$$\Vert(\rho(t)-\rho_{per}(t), v(t)-v_{per}(t), \theta(t)-\theta_{per}(t))\Vert_{L}\inftyarrow 0 (tarrow\infty)$.

In this paper we consider time periodic problem for $(1.1)-(1.3)$ instead of $(1.6)-(1.8)$ .
We will show the existence of a time periodic solution for $(1.1)-(1.3)$ aroundonwith. It will be proved that if satisfies (1.5) and$(\rho_{*}, 0, E_{*})$$\mathbb{R}^{3}$$E_{*}=C_{v}\theta_{*}$$g$$\Vert_{9}\Vert_{C([0,T];L^{1})}+\Vert(1+|x|^{3})g\Vert_{C([0,T];L)}\infty+\Vert(1+|x|^{2})g\Vert_{L^{2}(0,T;H^{s-1})}\ll 1$

for an integer $s\geq 2$ , then there exists a time periodic solution $(\rho_{per}-\rho_{*}, M_{per}, E_{per}-E_{*})\in$$C([O, T];H^{s})$ with period
for $(1.1)-(1.3)$ , and $(\rho_{per}-\rho_{*}, M_{per}, E_{per}-E_{*})$ satisfies theestimate$T$$\sup_{t\in[0,T]}\{\sum_{j=0}^{1}\Vert(1+|x|^{1+j})\partial_{x}^{j}(\rho_{per}-\rho_{*})(t)\Vert_{L^{\infty}}+\sum_{j=0}^{1}\Vert(1+|x|^{1+j})\partial_{x}^{j}M_{per}(t)\Vert_{L^{\infty}}$$+ \sum_{j=0}^{1}\Vert(1+|x|^{1+j})\partial_{x}^{j}(E_{per}-E_{*})(t)\Vert_{L}\infty\}$$\leq C(\Vert g\Vert_{C([0,T];L^{1})}+\Vert(1+|x|^{3})g\Vert_{C(0,T,L)}\infty+\Vert(1+|x|^{2})g\Vert_{L^{2}(0,T;H^{s-1})})$.

(1.10)

Furthermore, the time periodic solution
for $(1.1)-(1.3)$ is asymptoticallystable under suﬃciently small initial perturbations and the perturbation satisfies$(\rho_{per}, M_{per}, E_{per})$$\Vert(\rho(t), M(t), E(t))-(\rho_{per}(t), M_{per}(t), E_{per}(t))\Vert_{L}\inftyarrow 0 (tarrow\infty)$.

The precise statements of our results are given in Theorem 2.1 and Theorem 2.2 below.
The existence of time periodic solution is proved by using the time-T map for the
linearized semigroup at. We will employ a function space of hybrid type which,roughly speaking, consists of functions whose low frequency parts belong to a weighted
space and high frequency parts belong to a weighted -Sobolev space. For the
low frequency part we introduce a function space similar to that employed in the study ofthe stationary problem in [3], that is, a set of periodic functions with values in a weighted
space similar to (1.9). We investigate the spatial decay properties of the integral
kernel of the time-T map, and establish the estimates for the low frequency part by apotential theoretic method. Due to the conservation form of momentum and total energy
we can estimate the nonlinear terms for the low frequency part directly. As for the highfrequency part, we employ the weighted energy method to obtain the a priori estimates.
Note that by making use of the smoothing eﬀect for due to the termarising in
the Korteweg tensor, the derivative loss due to the term
does not occur for the highfrequency part and we can directly treat $(1.1)-(1.3)$ .
The asymptotic stability of the time periodic solution
is proved by
the energy method using the Hardy inequality as in [3, 7].
$(\rho_{*}, 0, E_{*})$$L^{\infty}\cap L^{2}$$L^{2}$$L^{\infty}\cap L^{2}$$\kappa\nabla\triangle\rho$$\rho$$v\cdot\nabla\rho$$(\rho_{per}, M_{per}, E_{per})$

63

Main results

2

To state our results, we define function spaces with spatial weight.
the weighted
For a nonnegative integer and $1\leq p\leq\infty$ , we denote bydefined by$\ell$$L_{\ell}^{p}$$L^{p}$

space$L_{\ell}^{p}=\{u\in L^{p};\Vert u\Vert_{L_{\ell}^{p}} :=\Vert(1+|x|)^{\ell}u\Vert_{L^{p}}<\infty\}.$

Let

$k$

and

$\ell$

be nonnegative integers. We define the weighted$L^{2}$-Sobolev space$H_{\ell}^{k}$by$H_{\ell}^{k}=\{u\in H^{k};\Vert u\Vert_{H_{p}^{k}}<+\infty\},$where$H_{l}^{k}$$=$$<+\infty\}$$\{u\in H^{k};\Vert u\Vert_{H_{\ell}^{k}}:=(\sum_{|\alpha|\leq k}\Vert\partial_{x}^{\alpha}u\Vert_{L_{\ell}^{2}}^{2})$We also introduce function spaces of -periodic functions in . We denote by
the set of all -periodic continuous functions with values in $X$ equipped with the norm
and we denote by
the set of all -periodic locally square integrable
functions with values in $X$ equipped with the norm$T$

$C_{per}(\mathbb{R};X)$

$t$

$T$

$L_{per}^{2}(\mathbb{R};X)$$\Vert\cdot\Vert_{C([0,T];X);}$$T$$\Vert\cdot\Vert_{L^{2}(0,T;X)}.$

Our result on the existence of a time periodic solution is stated as follows.
Theorem 2.1. Let
and$\mathcal{S}$be an integer satisfying $s\geq 2$ . Assume that $g(x, t)$ satisfies (1.5). Set$g\in C_{per}(\mathbb{R};L^{1}\cap L_{3}^{\infty})\cap L_{per}^{2}(\mathbb{R};H_{2}^{s-1})$$[g]_{s}=\Vert g\Vert_{C([0,T];L^{1}\cap L_{3}^{\infty})}+\Vert g\Vert_{L^{2}(0,T;H_{2}^{s-1})}.$, then the $sy\mathcal{S}tem(1.1)-(1.3)$
Then there exists a constant $\delta>0$ such that if
has a time periodic solution $u_{per}=T(\rho_{per}-\rho_{*}, M_{per}, E_{per}-E_{*})\in C_{per}(\mathbb{R};L^{\infty})$ withsatisfying$[g]_{s}\leq\delta$$\nabla u_{per}\in C_{per}(\mathbb{R};H^{s}\cross H^{s-1})$$\sup_{t\in[0,T]}(\Vert(1+|x|)u_{per}(t)\Vert_{L^{\infty}}+\Vert(1+|x|)^{2}\nabla u_{per}(t)\Vert_{L^{\infty}})\leq C[g]_{s}.$Our next issue to study the stability of the time periodic solution obtained in Theorem
2.1. Let $T(\rho_{per}, M_{per}, E_{per})$ be the time-periodic solution obtained in Theorem 2.1, let the$\tilde{M}=M-M_{per},$, whereperturbation be denoted by$E-E_{per}$ and let the initial perturbation be denoted by$\tilde{u}=T(\tilde{\rho},\tilde{M},\tilde{E})$$\tilde{E}=$$\tilde{\rho}=\rho-\rho_{per},$$\tilde{u}_{0}=\tilde{u}|_{t=0}=T(\rho(0)-\rho_{per}(0), M(0)-M_{per}(0), E(O)-E_{per}(0))$

We have the following stability result of the time periodic solution.

.

64
be an integer satisfying $s\geq 2$ . Assume that $g(x, t)$ satisfies (1.5) and. Let $T(\rho_{per}, M_{per}, E_{per})$ be the time-periodic solution. Then there exist constants
andobtained in Theorem 2.1 and letsuch that if

Theorem 2.2. Let$s$$g\in C_{per}(\mathbb{R};L^{1}\cap L_{3}^{\infty})\cap L_{per}^{2}(\mathbb{R};H_{2}^{s})$$\tilde{u}_{0}\in H^{s+1}\cross H^{s}$$\epsilon_{1}>0$$\epsilon_{2}>0$$[g]_{s+1}\leq\epsilon_{1}, \Vert\tilde{u}_{0}\Vert_{H^{s}+1\cross H^{8}}\leq\epsilon_{2},$then $u(t)$ exists globally in time and $u(t)$ satisfies$\tilde{u}\in C([0, \infty);H^{s+1}\cross H^{s})$$\Vert\tilde{u}(t)\Vert_{H^{s+1}\cross H^{S}}^{2}+$むオ,$\Vert\nabla\tilde{u}(\tau)\Vert_{H^{s+1}\cross H^{S}}^{2}d\tau\leq C\Vert\tilde{u}_{0}\Vert_{H^{\epsilon+1}\cross H^{\delta}}^{2}$$\Vert\tilde{u}(t)\Vert_{L^{\infty}}arrow 0(tarrow\infty)$$(t\in[0,$$\infty$.

Theorem 2.2 is proved as follows. We write $(1.1)-(1.3)$ into $(1.6)-(1.8)$ . Let
be the periodic solution given in Theorem 2.1. We setby$T(\rho_{per}, M_{per}, E_{per})$$v_{per},$and$\theta_{per}$$U_{per}$$v_{per}= \frac{M_{per}}{\rho_{per}}, \theta_{per}=\frac{1}{C_{v}}(E_{per}-\frac{|M_{per}|^{2}}{2\rho_{per}^{2}}) , U_{per}=T(\rho_{per}, v_{per}, \theta_{per})$

It follows from Theorem 3.1 that

$U_{per}$.

satisfies the estimate$\Vert^{T}(v_{per}, \theta_{per}-\theta_{*})\Vert_{C([0,T];L_{1}^{\infty})}\leq C[g]_{s+1}$$\Vert\nabla\{^{T}(v_{per}, \theta_{per}-\theta_{*})\}\Vert_{C([0,T];L_{2}^{\infty})}\leq C[g]_{s+1}$

(2.1)

,

.

(2.2)

Let the perturbation be denoted by $U=T(\phi, w, \theta)$ , where $\phi=\rho-\rho_{per},$ $w=v-v_{per},$. Then the perturbation $U=T(\phi, w, \theta)$ is governed by$\theta=$$\theta-\theta_{per}$$\{\begin{array}{l}\partial_{t}\phi+v_{per}\cdot\nabla\phi+\phi divv_{per}+\rho_{per}divw +w\cdot\nabla\rho_{per}=f^{1},\partial_{t}w-\frac{1}{\rho_{per}}\{\mu\triangle w+(\mu+\mu’)\nabla divw\}+B_{1}(U, U_{per})\nabla\phi-\kappa\nabla\triangle\phi+B_{2}(U, U_{per})\nabla\theta=f^{2},\partial_{t}\theta-\tilde{\alpha}B_{3}(U_{per})\Delta\theta+B_{4}(U, U_{per})divw=f^{3},\end{array}$where$f^{1}=-div(\phi w)$,$f^{2}=-(v_{per}\cdot\nabla)w-(w\cdot\nabla)(v_{per}+w)-(B_{1}(U, U_{per})-B_{1}(U_{per}))\nabla\rho_{per}$$-(B_{2}(U, U_{per})-B_{2}(U_{per}))\nabla\theta_{per}$$- \frac{\phi}{\rho_{per}(\rho_{per}+\phi)}\{\mu\triangle(v_{per}+w)+(\mu+\mu’)\nabla div(v_{per}+w$

(2.3)

65
$f^{3}=-(v_{per}\cdot\nabla)\theta-(w\cdot\nabla)(\theta_{per}+\theta)+\tilde{\alpha}(B_{3}(U, U_{per})-B_{3}(U_{per}))\triangle(\theta_{per}+\theta)$$+(B_{3}(U, U_{per})-B_{3}(U_{per}))(\Psi(v_{per})+\tilde{\Phi}(\rho_{per}, v_{per}))$$+B_{3}(U, U_{per})\{\Psi(v)-\Psi(v_{per})+\tilde{\Phi}(\rho, v)-\tilde{\Phi}(\rho_{per}, v_{per})\}$$-(B_{4}(U, U_{per})-B_{4}(U_{per}))divv_{per},$$B_{1}(U, U_{per})= \frac{P_{\rho}(\rho_{per}+\phi,\theta_{per}+\theta)}{\rho_{per}+\phi}, B_{2}(U, U_{per})=\frac{P_{\theta}(\rho_{per}+\phi,\theta_{per}+\theta)}{\rho_{per}+\phi},$$B_{3}(U, U_{per})= \frac{1}{C_{v}(\rho_{per}+\phi)}, B_{4}(U, U_{per})=\frac{(\theta_{per}+\theta)P_{\theta}(\rho_{per}+\phi,\theta_{per}+\theta)}{C_{v}(\rho_{per}+\phi)}$with$B_{1}(U_{per})= \frac{P_{\rho}(\rho_{per},\theta_{per})}{\rho_{per}}) B_{2}(U_{per})=\frac{P_{\theta}(\rho_{per},\theta_{per})}{\rho_{per}},$$B_{3}(U_{per})= \frac{1}{C_{v}\rho_{per}}, B_{4}(U_{per})=\frac{\theta_{per}P_{\theta}(\rho_{per},\theta_{per})}{C_{v}\rho_{per}}.$We consider the initial value problem for (2.3) under the initial condition$U|_{t=0}=U_{0}=^{T}(\phi_{0}, w_{0}, \theta_{0})$One can show that if
and$U(t)$
in time andsatisfies$[g]_{s+1}$$\Vert U_{0}\Vert_{H^{S}+1\cross H^{S}}$$U\in C([0, \infty);H^{s+1}\cross H^{s})$$\Vert U(t)\Vert_{H^{s+1}\cross H^{s}}^{2}+$むオ.

are suﬃciently small, then $U(t)$ exists globally,$\Vert\nabla U(\tau)\Vert_{H^{s+1}\cross H^{8}}^{2}d\tau\leq C\Vert U_{0}\Vert_{H^{s+1}\cross H^{s}}^{2}(t\in[0,$$\Vert U(t)\Vert_{L}\inftyarrow 0(tarrow\infty)$$\infty$.

These can be proved by similar methods as those in [3, 7], since the Hardy inequality
works well to deal with the linear terms including
due to the estimatesforin Theorem 3.1 and $(2.1)-(2.2)$ . We thus omit the details.
$T(\rho_{per}, v_{per}, \theta_{per})$$T(\rho_{per}, v_{per}, \theta_{per})$3

Outline of the proof of the main result

3.1

Formulation

To prove Theorem 3.1, we rewrite $(1.1)-(1.3)$ as follows. Let$\gamma_{1}=\sqrt{P_{\rho}(\rho_{*},\theta_{*})}, \gamma_{2}=\gamma_{1}\sqrt{\frac{C_{v}P(\rho_{*},\theta_{*})}{P_{\theta}(\rho_{*},\theta_{*})}}, \gamma_{3}=\frac{P_{\theta}(\rho_{*},\theta_{*})\gamma_{2}}{\gamma_{1}C_{v}}.$

66
and by $\phi=\rho-\rho_{*},$$(1.1)-(1.3)$ is rewritten as

We define$\phi,$$m$$\epsilon$$m= \frac{M}{\gamma_{1}}$and$\epsilon=(\rho_{*}+\phi)\frac{E-E}{\gamma_{2}}$$\partial_{t}u+Au=F(u, g)$, respectively. Then

(3.1)

,

where
$u=T(\phi, m, \epsilon) , A=(\begin{array}{lll}0 \gamma_{1}div 0\gamma_{1}\nabla-\kappa_{0}\nabla\triangle -\nu\triangle-\tilde{v}\nabla div \zeta\nabla 0 \zeta div -\alpha_{0}\triangle\end{array})$

(3.2)

,

$\nu=\frac{\mu}{\rho_{*}}, \tilde{\nu}=\frac{\mu+\mu’}{\rho_{*}}, \zeta=\frac{\gamma_{1}P(\rho_{*},\theta_{*})}{\gamma_{2}\rho_{*}}, \kappa_{0}=\frac{\kappa\rho_{*}}{\gamma_{1}}, \alpha_{0}=\frac{\tilde{\alpha}}{C_{v}\rho_{*}}$

and$F(u, g) = (\begin{array}{l}0F^{2}(u,g)F^{3}(u)\end{array})$

(3.3)

,

$F^{2}(u, g) = - \{\frac{\rho_{*}}{\gamma_{1}}div(m\otimes m)+\gamma_{1}div(P^{(1)}(\phi)\phi m\otimes m)$$+\rho_{*}\nu\triangle(P^{(1)}(\phi)\phi m)+\rho_{*}\tilde{v}\nabla div(P^{(1)}(\phi)\phi m)+\gamma_{3}\nabla(P^{(1)}(\phi)\phi\epsilon)$$+ \frac{1}{\gamma_{1}}\nabla(P^{(2)}(\phi)\phi^{2})-\frac{1}{\gamma_{1}}\nabla(P_{\theta}(\rho_{*}, \theta_{*})\frac{\gamma_{1}^{2}|m|^{2}}{2C_{v}(\rho_{*}+\phi)^{2}})$$+ \frac{1}{C_{v}^{2}\gamma_{1}}\nabla\{P^{(3)}(\theta)((\frac{\gamma_{1}^{2}|m|^{2}}{2(\rho_{*}+\phi)^{2}})^{2}-\frac{\gamma_{1}^{2}\gamma_{2}\epsilon|m|^{2}}{(\rho_{*}+\phi)^{3}}+\frac{\gamma_{2}^{2}\epsilon^{2}}{(\rho_{*}+\phi)^{2}})\}$$+ \frac{1}{C_{v}\gamma_{1}}\nabla\{P^{(4)}(\theta)(\frac{\gamma_{2}\phi\epsilon}{\rho_{*}+\phi}-\frac{\gamma_{1}^{2}|m|^{2}\phi}{2(\rho_{*}+\phi)^{2}})\}$$-\underline{1}div\Phi(\phi)-\underline{1}(\rho_{*}+\phi)g\}$

(3.4)

,

$\gamma_{1} \gamma_{1}$$\theta = \frac{1}{C_{v}}(E_{*}+\frac{\gamma_{2}\epsilon}{\rho_{*}+\phi}-\gamma_{1}^{2}\frac{|m|^{2}}{2(\rho_{*}+\phi)^{2}})$,$P^{(1)}( \phi) = \int_{0}^{1}f’(\rho_{*}+\tau\phi)d\tau, f(\tau)=\frac{1}{\tau}(\tau\in \mathbb{R})$$P^{(2)}(\phi, \theta)$$=$,$\int_{0}^{1}(1-\tau)P_{\rho\rho}(\rho_{*}+\tau\phi, \theta)d\tau,$$P^{(3)}( \theta) = \int_{0}^{1}(1-\tau)P_{\theta\theta}(\rho_{*}, \theta_{*}+\tau(\theta-\theta_{*}))d\tau,$$P^{(4)}( \theta) = \int_{0}^{1}P_{\rho\theta}(\rho_{*}, \theta_{*}+\tau(\theta-\theta_{*}))d\tau,$$\Phi(\phi) = \kappa\{\phi\triangle\phi I+(\nabla\phi)\cdot(\nabla\phi)I-\frac{|\nabla\phi|^{2}}{2}I-\nabla\phi\otimes\nabla\phi\},$$F^{3}(u) = - \{\frac{\gamma_{1}}{\rho_{*}}div(\epsilon m)+\gamma_{1}div(P^{(1)}(\phi)\phi\epsilon)-\alpha_{0}\rho_{*}\triangle(P^{(1)}(\phi)\phi\epsilon)$,

67$+ \frac{\alpha_{0}}{C_{v}\gamma_{2}}\triangle(\frac{\gamma_{1}^{2}|m|^{2}}{2(\rho_{*}+\phi)^{2}})+\frac{\gamma_{1}}{\gamma_{2}}div(P^{(1)}(\phi)\phi mP(\rho_{*}+\phi, \theta))$$+ \frac{\gamma_{1}}{\rho_{*}\gamma_{2}}div(mP^{(5)}(\phi, \theta)\phi)$$+ \frac{\gamma_{1}}{C_{v}\rho_{*}\gamma_{2}}div(mP^{(6)}(\theta)(\frac{\gamma_{2}\epsilon}{(\rho_{*}+\phi)}-\frac{\gamma_{1}^{2}|m|^{2}}{2(\rho_{*}+\phi)^{2}}))$$- \frac{\gamma_{1}}{\gamma_{2}}div((S(\frac{\gamma_{1}m}{\rho_{*}+\phi})+\mathcal{K}(\rho_{*}+\phi))\frac{m}{\rho_{*}+\phi})-\frac{1}{\gamma_{2}}mg\}$

,

(3.5)

$P^{(5)}( \phi, \theta) = \int_{0}^{1}P_{\rho}(\rho_{*}+\tau\phi, \theta)d\tau,$$P^{(6)}( \theta) = \int_{0}^{1}P_{\theta}(\rho_{*}, \theta_{*}+\tau(\theta-\theta_{*}))d\tau.$

Let us introduce a semigroup $S(t)=e^{-tA}$ generated by$A$;$S(t)=e^{-tA}=\mathcal{F}^{-1}e^{-t\hat{A}_{\xi}}\mathcal{F},$where$\hat{A}_{\xi}=(\begin{array}{llll} 0 i\gamma_{1}^{T}\xi 0i\gamma_{1}\xi +i\kappa_{0}|\xi|^{2}\xi \nu|\xi|^{2}I_{n}+\tilde{\nu}\xi^{T}\xi i\zeta\xi 0 i\zeta^{T}\xi \alpha_{0}|\xi|^{2}\end{array}) (\xi\in \mathbb{R}^{3})$.

Then $S(t)$ has the following properties.
Proposition 3.1. Let be a nonnegative integer satisfying $s\geq 2$ . Then $S(t)=e^{-tA}$ is acontraction semigroup on. In addition, for each $u\in H^{s}\cross H^{s-1}\cross H^{s-1}$
and all $T’>0,$ $S(t)$ satisfies$s$$H^{8}\cross H^{s-1}\cross H^{s-1}$$S(\cdot)u\in C([0, T H^{8}\cross H^{s-1}\cross H^{s-1}) , S(O)u=u$

and there hold the estimates$\Vert S(t)u\Vert_{H^{s}\cross H^{s-1}\cross H^{s-1}}\leq\Vert u\Vert_{H^{S}\cross H^{s-1}\cross H^{s-1}}$

for$u\in H^{S}\cross H^{s-1}\cross H^{s-1}$

We set an operator$\Gamma$(3.6)

and $t\geq 0.$

using the time-T map by$\Gamma[F]=S(t)(I-S(T))^{-1}\mathscr{S}(T)F+\mathscr{S}(t)F(t\in[0, T$

where
ノ ( ) $F$ $:=S(t-\tau)F(\tau)d\tau Jo^{t}.$
$t$

(3.7)

68
To solve the time periodic problem for (3.1), as in [9], we look for a fixed point$\Gamma[F(u,$ $g$

$u$

of

i.e.,
(3.8)

$u=\Gamma[F(u, g)](t\in[0, T$

where $u=T(\phi, m, \epsilon)$ and $F(u, g)$ is given by $(3.3)-(3.5)$ . From (3.7) and (3.8), it holds
that $u(T)=u(O)$ . Therefore, we will investigate properties of the map$\Gamma.$

We next introduce function spaces. We define a space$\mathscr{X}$by$\mathscr{X}=\{w\in L_{1}^{\infty}, \nabla w\in H^{1};\Vert w\Vert_{\mathscr{X}}<+\infty\},$where$\Vert w\Vert_{\mathscr{X}}:=\sum_{j=0}^{1}\Vert(1+|x|)^{1+j}\nabla^{j}w\Vert_{L^{\infty}}+\sum_{j=1}^{2}\Vert(1+|x|)^{j-1}\nabla^{j}w\Vert_{L^{2}}.$

Note that decays in the same order as the fundamental solution of the Laplace equation.
Accordingly, this space is a similar one to that introduced in the stationary problem [3].
Let be a nonnegative integer and let be a nonnegative integer satisfying $s\geq 2$ . We
bydefine the weighted -Sobolev space$w$$\ell$$s$$L^{2}$$\mathscr{Y}_{\ell}^{s}(a, b)$$\mathscr{Y}_{p}^{s}(a, b)=[C([a, b];H_{\ell}^{s+1})\cap L^{2}(a, b;;H_{\ell}^{s+2})]$$\cross[C([a, bH_{\ell}^{s})\cap L^{2}(a, b;H_{\ell}^{s+1})].$Let us introduce operators which decompose a function into its low and high frequency
are defined byandonparts. Operators$P_{1}$$P_{\infty}$$L^{2}$$P_{j}f=\mathcal{F}^{-1}\hat{\chi}_{j}\mathcal{F}[f] (f\in L^{2},j=1, \infty)$

,

where
$\hat{\chi}_{j}(\xi)\in C^{\infty}(\mathbb{R}^{n}) (j=1, \infty) , 0\leq\hat{\chi}_{j}\leq 1 (j=1, \infty)$,$\hat{\chi}_{1}(\xi)=\{\begin{array}{l}1 (|\xi|\leq r_{1}) ,0 (|\xi|\geq r_{\infty}) ,\end{array}$$\hat{\chi}_{\infty}(\xi)=1-\hat{\chi}_{1}(\xi) , 0<r_{1}<r_{\infty}.$

We fixholds for$0<r_{1}<r_{\infty}< \frac{2\gamma}{\nu+\tilde{\nu}}$

Let

$s$

in such a way that the estimate (3.10) in Lemma 3.7 below$|\xi|\leq r_{\infty}.$

be a nonnegative integer satisfying $s\geq 2$ . We define a solution spaceby$\mathscr{Z}^{s}(a, b)=\{u;P_{1}u\in C(a, b;\mathscr{X}) , P_{\infty}u\in\Psi_{2}(a, b$$\mathscr{Z}^{s}(a, b)$

69
and the norm is defined by$\Vert u\Vert_{\mathscr{Z}^{s}(a,b)}=\Vert P_{1}u\Vert_{C(a,b;\mathscr{X})}+\Vert P_{\infty}u\Vert_{\Psi_{2}(a,b)}.$Observe that
$P_{j}\Gamma[F(u, g)]=\Gamma[P_{j}F(u, g)](j=1, \infty)$

and

supp

$\hat{P_{1}F}(u, g)\subset\{|\xi|\leq r_{\infty}\},$

supp$\overline{P_{\infty}F}(u, g)\subset\{|\xi|\geq r_{1}\}.$

So we will investigate the restriction of to the space of functions whose Fourier transin subsection
and will then establish estimates forforms have support in
3.2. Likewise, the restriction of to the high frequency part will be investigated toin subsection 3.3.
establish estimates for$\Gamma$$\Gamma P_{1}$$\{|\xi|\leq r_{\infty}\}$$\Gamma$$\Gamma P_{\infty}$

In the remaining of this subsection we introduce some lemmas which will be used in
the proof of Theorem 2.1.

We will use the following lemma for the estimates for the integral kernels which willappear in the analysis of the low frequency part.
Lemma 3.2. [16, Theorem 2.3] Let$(x\in \mathbb{R}^{3})$, where$\hat{\Phi}_{\ell}\in C^{\infty}(\mathbb{R}^{3}-\{O\})$be a nonnegative integer and let
is a function satisfying$\ell$$\partial_{\xi}^{\alpha}\hat{\Phi}_{\ell}\in L^{1} (|\alpha|\leq\ell)$$E(x)=\mathscr{F}^{-1}\hat{\Phi}_{\ell}$,$|\partial_{\xi}^{\beta}\hat{\Phi}_{\ell}|\leq C|\xi|^{-2-|\beta|+\ell} (\xi\neq0, |\beta|\geq 0)$.

Then $E(x)(x\neq 0)$ satisfies the estimate$|E(x)|\leq C|x|^{-(1+\ell)}.$The following lemma is related to the estimates for the convolutions which appear in
the analysis of the low frequency part.

Lemma 3.3. (i) [17, Lemma 2.5] Let$E(x)(x\in \mathbb{R}^{3})$be a function satisfying$| \partial_{x}^{\alpha}E(x)|\leq\frac{C}{(1+|x|)^{|\alpha|+1}} (|\alpha|=0,1,2)$

Assume thatis a function satisfying
estimate for $|\alpha|=0$ , 1.
$f$$\Vert f\Vert_{L_{3}^{\infty}\cap L^{1}}<\infty$

.

(3.9)

. Then there holds the following$|[ \partial_{x}^{\alpha}E*f](x)|\leq\frac{C}{(1+|x|)^{|\alpha|+1}}\Vert f\Vert_{L_{3}^{\infty}\cap L^{1}}.$

70(ii) [17, Lemma 2.5] Let $E(x)(x\in \mathbb{R}^{3})$ be a junction satisfying(3.9). Assume thata function of the form:
for some $1\leq j\leq n$ satisfying
Then there holds the following estimate for $|\alpha|=0$ , 1.

$f$

is

$\Vert\partial_{x_{g}}f_{1}\Vert_{L_{3}}\infty+\Vert f_{1}\Vert_{L_{2}}\infty<\infty.$$f=\partial_{x_{g}}f_{1}$$|[ \partial_{x}^{\alpha}E*f](x)|\leq\frac{C}{(1+|x|)^{|\alpha|+1}}(\Vert\partial_{x_{J}}f_{1}\Vert_{L_{3}^{\infty}}+\Vert f_{1}\Vert_{L_{2}^{\infty}})$.

(iii) [14, Lemma 4.9] Let $E(x)(x\in \mathbb{R}^{3})$ be a junction satisfying$| \partial_{x}^{\alpha}E(x)|\leq\frac{C}{(1+|x|)^{|\alpha|+2}} (|\alpha|=0,1)$

Assume that
estimate for$f$

is a function satisfying, 1.

$\Vert f\Vert_{L_{3}}\infty<\infty$.

. Then there holds the following$|\alpha|=0$$|[ \partial_{x}^{\alpha}E*f](x)|\leq\frac{C\log|x|}{(1+|x|)^{|\alpha|+2}}\Vert f\Vert_{L_{3}^{\infty}}.$3.2

Estimate of$\Gamma$for the low frequency partspace
In this subsection we estimate for the low frequency part. We introduce a$u\in L^{2}$ satisfying
of
all
for
the
set
standsThe
part.
symbolfor the low frequency. (Cf.,
supp. For any nonnegative integer , we see that
bellow.)(ii)
Lemma 3.4
We next state some properties of$L^{2}$$\Gamma$$L_{(1)}^{2}$$k$$\hat{f}\subset\{|\xi|\leq r_{\infty}\}$$H^{k}\cap L_{(1)}^{2}=L_{(1)}^{2}$$P_{1}.$

Lemma 3.4. ([9, Lemma 4.3]) (i) Let be a nonnegative integer. Thentoandlinear operator fromsatisfies the estimates$k$$H^{k}$$L^{2}$$re\mathcal{S}ult$(ii) Let, for any $2\leq p\leq\infty,$$k$

is a bounded$P_{1}$$\Vert\nabla^{k}P_{1}f\Vert_{L^{2}}\leq C\Vert f\Vert_{L^{2}} (f\in L^{2})$As a$P_{1}$$P_{1}$

is bounded from$L^{2}$

to.

$L^{p}.$

be a nonnegative integer. Then there hold the estimates$\Vert\nabla^{k}f_{1}\Vert_{L^{2}}+\Vert f_{1}\Vert_{L^{p}}\leq C\Vert f_{1}\Vert_{L^{2}} (f\in L_{(1)}^{2})$,where $2\leq p\leq\infty.$

We derive the following inequalities for the weighted
part.

Lemma 3.5. Let
the estimate

$k$

and

$\ell$$L^{p}$

norm of the low frequency

be nonnegative integers and let $1\leq p\leq\infty$ . Then there holds$\Vert\nabla^{k}f_{1}\Vert_{L_{\ell}^{p}}\leq C\Vert f_{1}\Vert_{L_{\ell}^{p}}(f\in L_{(1)}^{2}\cap L_{\ell}^{p})$.

71
The proof of the estimate is given in [14, Lemma 4.3].
We define a space$\mathscr{X}_{(1)}$by$\mathscr{X}_{(1)}=\{u\in \mathscr{X};$We set operators

and$S_{1}(t)$$\mathscr{S}_{1}(t)$supp$\hat{u}\subset\{|\xi|\leq r_{\infty}$byオ$S_{1}(t)=S(t)|_{\mathscr{X}_{(1)}},$$\mathscr{S}_{1}(t)F_{1}=\int_{0}$$S_{1}(t-\tau)F_{1}(\tau)d\tau.$

Then we have the following

Proposition 3.6. (i)is a uniformly continuous semigroup on
and all $T’>0,$
for eachsatisfies$S_{1}(t)$$\mathscr{X}_{(1)}$. In addition,$S_{1}(t)$$u_{1}\in \mathscr{X}_{(1)}$$S_{1}(t)u_{1}\in C^{1}([0, T \mathscr{X}_{(1)})$,$\partial_{t}S_{1}(t)u_{1}=-A_{1}S_{1}(t)u_{1}(=-AS_{1}(t)u_{1}) , S_{1}(0)u_{1}=u_{1},$and there hold the estimates$\Vert\partial_{t}^{k}S_{1}(\cdot)u_{1}\Vert_{C([0,T’];\mathscr{X}_{(1)})}\leq C\Vert u_{1}\Vert_{\mathscr{X}_{(1)}}$for$u_{1}\in \mathscr{X}_{(1)},$$k=0$ , 1, where $T’>0$ is any given positive number and $C$ is a positive

constant depending on $T’.$(ii):$\mathscr{S}_{1}(t)$$L^{2}(0, T;\mathscr{X}_{(1)})arrow C([0, T];\mathscr{X}_{(1)})\cap H^{1}(0, T;\mathscr{X}_{(1)})$is a bounded linear operator for $t\in[0, T]$ satisfying
哉$\mathscr{S}$

1 $(t)F_{1}+A_{1}\mathscr{S}_{1}(t)F_{1}=F_{1}(t)$ ,$\mathscr{S}_{1}(0)F_{1}=0,$$\Vert \mathscr{S}_{1}(\cdot)F_{1}\Vert_{C([0,T];\mathscr{X}_{(1)})}\leq C\Vert F_{1}\Vert_{L^{2}(0,T;\mathscr{X}_{(1)})},$$\Vert\partial_{t}\mathscr{S}_{1}(\cdot)F_{1}\Vert_{L^{2}(0,T;\mathscr{X}_{(1)})}\leq C\Vert F_{1}\Vert_{L^{2}(0,T;\mathscr{X}_{(1)})}.$

for$F_{1}\in L^{2}(0, T;\mathscr{X}_{(1)})$, where $C$ is a positive constant depending on $T.$(iii) It holds that
$S_{1}(t)\mathscr{S}_{1}(t’)F_{1}=\mathscr{S}_{1}(t’)[S_{1}(t)F_{1}]$for any$t\geq 0,$$t’\in[0, T]$and$F_{1}\in L^{2}(0, T;\mathscr{X}_{(1)})$.

Proposition 3.6 can be proved in a similar manner to the proof of [14, Proposition 5.1];
and we omit the proof.

To estimate , we prepare some lemmas. The following lemma plays an importantrole to investigate the spatial decay properties of the time-T map.
$\Gamma$

72
Lemma 3.7. (i) Let$\hat{A}_{\xi}=(\begin{array}{lll}0 i\gamma_{1^{T}}\xi 0i\gamma_{1}\xi+i\kappa_{0}|\xi|^{2}\xi v|\xi|^{2}I_{3}+\tilde{\nu}\xi^{T}\xi i\zeta\xi 0 i\zeta^{T}\xi \alpha_{0}|\xi|^{2}\end{array}) (\xi\in \mathbb{R}^{3})$Then there exists
such that if$\lambda_{j}(\xi)(j=1,\cdots4)$ , whereconsists of$\delta_{0}>0$$0<r_{\infty}\leq\delta_{0}$.

, the set of all eigenvalues$of-\hat{A}_{\xi}$$\{\begin{array}{l}\lambda_{1}(\xi)=-\nu|\xi|^{2}+O(|\xi|^{3}) ,\lambda_{2}(\xi)=-\frac{\alpha 0\gamma^{2}}{\gamma_{1}^{2}+\zeta^{2}}|\xi|^{2}+O(|\xi|^{3}) ,\lambda_{3}(\xi)=i\sqrt{\gamma_{1}^{2}+\zeta^{2}}|\xi|-\frac{\nu+\tilde{\nu}}{2}|\xi|^{2}-\frac{\alpha_{O}\zeta^{2}}{2(\gamma_{1}^{2}+\zeta^{2})}|\xi|^{2}+O(|\xi|^{3}) ,\lambda_{4}(\xi)=\overline{\lambda}_{3} (complex conjugate).\end{array}$(ii) For$e^{-t\hat{A}_{\xi}}$$|\xi|\leq\delta_{0},$has the spectral resolution$e^{-t\hat{A}_{\xi}}= \sum_{j=1}^{4}e^{t\lambda_{J}(\xi)}\Pi_{j}(\xi)$

where$\Pi_{j}(\xi)$,is eigenprojections for $\lambda_{j}(\xi)(j=1, \cdots, 4)$ , and $\Pi_{j}(\xi)(j=1, \cdots, 4)$ satisfy$\Pi_{1}(\xi)=(\begin{array}{lll}0 0 00 I_{3_{|\xi|}^{-i^{T}1_{2}}} 00 0 0\end{array})+O(|\xi|)$,$\Pi_{2}(\xi)=(^{1-\frac{\gamma^{2}}{o_{L}^{2}\gamma_{1}++\zeta}}-\not\simeq\gamma_{1}^{2}\overline{\zeta^{2}} 000 1-\frac{0_{\zeta^{2}}^{+\zeta}2L_{2}}{\gamma_{1}^{2}+\zeta^{2}}-2\overline{\gamma}_{1)}+O(|\xi|)$,$\Pi_{3}(\xi)=\frac{1}{2}(-\frac{\frac{\gamma_{1}^{2}}{\gamma_{1}^{2}+\zeta^{2}i\gamma_{1}\xi}}{i\sqrt{\gamma_{1}^{2}+\zeta^{2}}|\xi|,\gamma_{1}+\zeta\not\simeq^{L_{2}}} -\frac{}{}-\frac{i\gamma_{1^{T}}\xi}{i\sqrt{\gamma_{1}^{2}+\zeta^{2}}|\xi|,i\sqrt{\gamma_{1}^{2}+\zeta^{2}}|\xi|\simeq^{T}|\xi|^{2}i\zeta^{T}\xi} -\frac{\frac{\gamma_{1}\zeta}{\gamma_{1}^{2}+\zeta^{2}i\zeta\xi}}{i\sqrt{+\zeta^{2}}|\xi|,\frac{\gamma_{1}^{2}\zeta^{2}}{\gamma_{1}^{2}+\zeta^{2}}}1+O(|\xi|)$$\Pi_{4}(\xi)=\frac{1}{2}(\frac{\frac{\gamma_{I}^{2}}{\gamma_{1}^{2}+\zeta^{2}i\gamma_{1}\xi}}{i\sqrt{\gamma}|\xi|,\frac{\gamma_{1}\zeta 21^{+\zeta^{2}}}{\gamma_{1}^{2}+\zeta^{2}}} \frac{}{}\frac{i\gamma_{1^{T}}\xi}{i\sqrt{\gamma_{1}^{2}+\zeta^{2}}|\xi|i\sqrt{\gamma_{1}^{2}+\zeta^{2}}|\xi|i\zeta^{T}\xi\simeq^{T}|\xi|^{2}} i\sqrt{\gamma}\frac{\zeta^{2}}{\gamma_{1}^{2}+\zeta^{2}}\frac{\gamma_{1}\zeta}{\gamma_{1}^{2}+\zeta^{2},i\zeta\xi 1^{+\zeta^{2}}2}|\xi|)+O(|\xi|)$,.

Furthermore, there exist a constant $C>0$ such that the estimates$\Vert\Pi_{j}(\xi)\Vert\leq C(j=1, \cdots, 4)$hold for$|\xi|\leq r_{\infty}.$(3.10)

73
Lemma 3.7 is proved by the analytic perturbation theory ([10]). We set$\xi=|\xi|\omega, \omega=\frac{\xi}{|\xi|}, -\hat{A}_{\xi}=r\tilde{A}_{\xi}, \tilde{A}_{\xi}=L_{1}+rL_{2}+r^{2}L_{3},$where $r=|\xi|,$$L_{1} = -i(\begin{array}{lll}0 \gamma_{1^{T}}\omega 0\gamma_{1}\omega 0 \zeta\omega 0 \zeta^{T}\omega 0\end{array}), L_{2}=-(\begin{array}{lll}0 0 00 vI_{3}+\omega^{T}\omega 00 0 \alpha_{0}\end{array})$and$L_{3}=-(\begin{array}{lll}0 0 0i\kappa_{0}\omega 0 00 0 0\end{array})$

Applying the reduction process ([10, Section I-2-3]), we can prove Lemma 3.7. See also[12, Lemma 3.1].
Hereafter we fix

Lemma 3.8. Let

for

$\xi$

(i)
(ii)

with

$0<r_{1}<r_{\infty}\leq\delta_{0}$so that (3.10) in Lemma 3.7 holds for

be a multi-index. Then the following estimates hold true uniformly
and $t\in[O, T].$$\alpha$$|\xi|\leq r_{\infty}$$((|\alpha|\geq 0, j=1,2),$$|\partial_{\xi}^{\alpha}\lambda_{j}|\leq C|\xi|^{2-|\alpha|}$$|(\partial_{\xi}^{\alpha}\Pi_{j})\hat{F}|\leq C|\xi|^{-|\alpha|}|\hat{F}|(|\alpha|\geq 0)$$|\partial_{\xi}^{\alpha}\lambda_{j}|\leq C|\xi|^{1-|\alpha|}$$((|\alpha|\geq 0, j=3,4)$

.

.

(iii)

$|\partial_{\xi}^{\alpha}(e^{\lambda_{\mathcal{J}}t})|\leq C|\xi|^{2-|\alpha|}(|\alpha|\geq 1, j=1,2)$

.

(iv)

$|\partial_{\xi}^{\alpha}(e^{\lambda_{2}t})|\leq C|\xi|^{1-|\alpha|}(|\alpha|\geq 1, j=3,4)$.

(v)$|\xi|\leq r_{\infty}.$$|(\partial_{\xi}^{\alpha}e^{-t\hat{A}_{\xi}})\hat{F}|\leq C|\xi|^{-|\alpha|}|\hat{F}|(|\alpha|\geq 1)$

.

(vi)

$|\partial_{\xi}^{\alpha}(I-e^{\lambda_{J}t})^{-1}|\leq C|\xi|^{-2-|\alpha|}(|\alpha|\geq 0, j=1,2)$

.

(vii)

$|\partial_{\xi}^{\alpha}(I-e^{\lambda_{\mathcal{J}}t})^{-1}|\leq C|\xi|^{-1-|\alpha|}(|\alpha|\geq 0, j=3,4)$.

Lemma 3.8 can be verified by direct computations based on Lemma 3.7.

We are now in a position to give an estimate of

Proposition 3.9. Let$T(\phi, m, \epsilon)$$s$$\Gamma$for the low frequency part.

be a nonnegative integer satisfying $s\geq 2$ . Assume that $u=$satisfies$\Vert u\Vert \mathscr{Z}(0,T)<<1.$

Then it holds that$\Vert\Gamma[P_{1}F(u, g)]\Vert_{C([0,T];\mathscr{X})}\leq C\Vert u\Vert_{\mathscr{Z}_{(0,T)}}^{2}+C(1+\Vert u\Vert_{\mathscr{Z}_{(0,T)}})[g]_{s}$uniformly for $u.$

74
Proof. We set$\Gamma_{1}[P_{1}F(u, g)] := S(t)(I-S(T))^{-1}\mathscr{S}(T)[P_{1}F(u, g$$\Gamma_{2}[P_{1}F(u,g)] := \mathscr{S}(t)[P_{1}F(u, g$

By using Lemma 3.8, one can easily obtain the required estimates for$\Vert\nabla^{k}\Gamma_{j}[P_{1}F(u, g)]\Vert_{L_{k-1}^{2}}$$(j, k=1,2)$ .

We estimatetion 3.6 we have$\Gamma_{j}$in the weighted$L^{\infty}$

space. As for the term$\Gamma_{1}[P_{1}F(u,$ $g$

by Proposi-$\Gamma_{1}[P_{1}F(u, g)] = S_{1}(t)(I-S_{1}(T))^{-1}\mathscr{S}_{1}(T)[P_{1}F(u, g)]$ア$=$$e^{-(T-\tau)\hat{A}_{\xi}}\hat{\chi}_{1}\hat{F}(\tau, u, g)d\tau\}$$\mathscr{F}^{-1}\{e^{-t\hat{A}}\epsilon(I-e^{-T\hat{A}}\epsilon)^{-1}\int_{0}$

ア
$=$

:

$\int_{0}$$E_{1}(t, \tau)*P_{1}F(\tau, u, g)d\tau$(3.11),

where$E_{1}(t, \tau)=\mathcal{F}^{-1}\{\hat{\chi}_{0}e^{-t\hat{A}_{\xi}}(I-e^{-T\hat{A}_{\xi}})^{-1}e^{-(T-\tau)\hat{A}_{\xi}}\},$$\chi_{0}$is a cut-oﬀ function defined by$\hat{\chi}_{0}\in C^{\infty}(\mathbb{R}^{n})$,By Lemma 3.7,

$0\leq\hat{\chi}_{0}\leq 1,$$e^{-t\hat{A}_{\’{e}}}$$\chi_{0}=\mathcal{F}^{-1}\hat{\chi}_{0}$$\hat{\chi}_{0}=1$

on

with

$\hat{\chi}_{0}$$\lambda_{j}$and supp$\{|\xi|\leq r_{\infty}\}$,and $\Pi_{j}(j=1, \cdots, 4)$ are the same ones in Lemma 3.7. Therefore, we see that$(I-e^{-T\hat{A}} \epsilon)^{-1}=\sum_{j=1}^{4}(I-e^{T\lambda_{j}})^{-1}\Pi_{j}$Let$\alpha$be a multi-index satisfying$|\alpha|\geq$.

$\int_{|\xi|\leq 2r_{\infty}}|\xi|^{-2}d\xi<\infty$(3.12)

O. It follows from Lemma 3.8 that$\sum_{j}|\partial_{x}^{\alpha}E_{1}(x)|\leq C\int_{|\xi|\leq 2r_{\infty}}|\xi|^{-2}d\xi(x\in \mathbb{R}^{3})$Since$\hat{\chi}_{0}\subset\{|\xi|\leq 2r_{\infty}\}.$

has the spectral resolution$e^{-t\hat{A}_{\xi}}= \sum_{j=1}^{4}e^{t\lambda_{J}(\xi)}\Pi_{j}(\xi)$wheresatisfying.

, we see that$\sum_{j}|\partial_{x}^{\alpha}E_{1}(x)|\leq C(x\in \mathbb{R}^{3})$where $C>0$ is a constant depending on$|\partial_{\xi}^{\beta}((i\xi)^{\alpha}\hat{\chi}_{0}(I-e^{\lambda_{J}T})^{-1}\Pi_{j})|$$\alpha,$$\leq$$T$(3.13),. By Lemma 3.8, we have

$C|\xi|^{-2+|\alpha|-|\beta|}$for $j=1$ , 2,$|\beta|\geq 0,$

75$|\partial_{\xi}^{\beta}((i\xi)^{\alpha}\hat{\chi}_{0}(I-e^{\lambda_{g}T})^{-1}\Pi_{j})|$$\leq$$C|\xi|^{-1+|\alpha|-|\beta|}$for $j=3$ , 4,$|\beta|\geq 0.$

It then follows from Lemma 3.2 and (3.12) that$|\partial_{x}^{\alpha}E_{1}(x)|\leq C|x|^{-(1+|\alpha|)}$

(314)

.

From (3.13) and (3.14), we obtain that(315)$|\partial_{x}^{\alpha}E_{1}(x)|\leq C(1+|x|)^{-(1+|\alpha|)}$uniformly forConcerning the estimate for the nonlinear term $P_{1}div(m\otimes m)$ in the estimate of$\Gamma_{1}[P_{1}F(u, g due to the$ conservation form, applying Lemma $3.3,$ Lemma $3.8, (3.11)$ and(3.15) with, we see that$x\in \mathbb{R}^{3}.$$|\alpha|\geq 1$$\Vert\Gamma_{1}[F_{1}(u)]\Vert_{\mathscr{Z}_{(1)}}(0,T)\leq C\Vert u\Vert_{\mathscr{Z}}^{2}(0,T)$(3.16)’

where $F_{1}(u)=T(0, P_{1}div(m\otimes m), 0)$ . Similarly to (3.16), the remaining terms can be
estimated. Hence, we obtain the desired estimate for . The estimate forcan beproved in a similar manner to the proof of the estimate for . This completes the proof.
$\Gamma_{2}$$\Gamma_{1}$$\Gamma_{1}$口

3.3

Estimate offor the high frequency part$\Gamma$In this subsection we establish an estimate for the high frequency part. The following
function spaces are introduced for the high frequency part. Let and be nonnegative
integers. The symbol
stands for the set of all $u\in H^{k}$ satisfying supp
and the spaceis defined by$\Gamma$$k$$\ell$$\hat{u}\subset\{|\xi|\geq r_{1}\}$$H_{(\infty)}^{k}$$H_{(\infty),\ell}^{k}$$H_{(\infty),\ell}^{k}=\{u\in H_{(\infty)}^{k};\Vert u\Vert_{H_{\ell}^{k}}<+\infty\}.$We prepare some lemmas for the high frequency part.
Lemma 3.10. [9, Lemma 4.4] (i) Let
linear operator on$k$be a nonnegative integer. Then

$P_{\infty}$

is a bounded$H^{k}.$(ii) There hold the inequalities$\Vert P_{\infty}f\Vert_{L^{2}} \leq C\Vert\nabla f\Vert_{L^{2}} (f\in H^{1})$,$\Vert f_{\infty}\Vert_{L^{2}} \leq C\Vert\nabla f_{\infty}\Vert_{L^{2}} (f_{\infty}\in H_{(\infty)}^{1})$Lemma 3.11. [14, Lemma 4.13] Let
depending only on such that$\ell\in \mathbb{N}$.

. Then there exists a positive constant$\ell$$\Vert P_{\infty}f\Vert_{L_{\ell}^{2}}\leq C\Vert\nabla f\Vert_{L_{\ell}^{2}}.$

$C$

76
Let

$s$

be a nonnegative integer satisfying $s\geq 2$ . By Proposition 3.1, we define anoperator$S_{\infty}(t)$by$S_{\infty}(t)u_{\infty}=S(t)u_{\infty}$for:$H_{(\infty)}^{s+1}\cross H_{(\infty)}^{s}arrow H_{(\infty)}^{s+1}\cross H_{(\infty)}^{s}(t\geq 0)$$u_{\infty}\in H_{(\infty)}^{s+1}\cross H_{(\infty)}^{s}$. We also define$\mathscr{S}_{\infty}(t):L^{2}(0, T;H_{(\infty)}^{s}\cross H_{(\infty)}^{s-1})arrow H_{(\infty)}^{s+1}\cross H_{(\infty)}^{s}(t\in[O, T])$by$\mathscr{S}_{\infty}(t)F_{\infty}=\int_{0}^{t}S_{\infty}(t-\tau)F_{\infty}(\tau)d\tau.$for.
We have the following properties for$F_{\infty}\in L^{2}(0, T;H_{(\infty)}^{s}\cross H_{(\infty)}^{s-1})$

Proposition 3.12. (i) It holds that

and$\mathscr{S}_{\infty}.$$S_{\infty}(\cdot)u_{0\infty}\in C([0, \infty);H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s})$$T(\phi_{0\infty}, m_{0\infty}, \epsilon_{0\infty})\in H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}$$S_{\infty}(t)$$S_{\infty}$

for each$u_{0\infty}=$and there exist constants $a>0$ and $C>0$ such thatsatisfies the estimate$\Vert S_{\infty}(t)u_{0\infty}\Vert_{H_{(\infty)_{)}2^{\cross H_{(\infty),2}^{\delta}}}^{s+1}}\leq Ce^{-at}\Vert u_{0\infty}\Vert_{H_{(\infty),2^{\cross H_{(\infty)_{)}2}^{s}}}^{s+1}}$

for all$t\geq 0$

and$u_{0\infty}\in H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}$$r_{H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}}(S_{\infty}(T))$$S_{\infty}(T)$. Furthermore,denotes the spectral radius of

has a bounded inverse $(I-S_{\infty}(T))^{-1}$ on$S_{\infty}(T)$for$\mathscr{S}_{\infty}(\cdot)F_{\infty}\in C([0, T];H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s})$$L^{2}(0, T;H_{(\infty),2}^{s}\cross H_{(\infty),2}^{s-1})$and$\mathscr{S}_{\infty}(t)$on$H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}$$\Vert(I-S_{\infty}(T))^{-1}u\Vert_{H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}}\leq C\Vert u\Vert_{H_{(\infty),2}^{s+1}\cross H_{(\infty)_{)}2}^{s}}$(ii) It holds that$r_{H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{\partial}}(S_{\infty}(T))<1$$H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}$, where; and $I-$

and $(I-S_{\infty}(T))^{-1}$ satisfies$u\in H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}.$for each$F_{\infty}=T(F_{\infty}^{1}, F_{\infty}^{2}, F_{\infty}^{3})\in$

satisfies the estimate$\Vert \mathscr{S}_{\infty}(t)[F_{\infty}]\Vert_{H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s}}\leq C\{\int_{0}^{t}e^{-a(t-\tau)}\Vert F_{\infty}\Vert_{H_{(\infty),2^{\cross H_{(\infty),2}^{s-1}}}^{s}}^{2}d\tau\}^{\frac{1}{2}}$for $t\in[O, T]$ and$F_{\infty}\in L^{2}(0, T;H_{(\infty),2}^{s}\cross H_{(\infty),2}^{s-1})$

with a positive constant$C$depending on$T.$Proposition 3.12 can be proved by the weighted -energy method. In fact, Proposition
3.12 is an immediate consequence of the following proposition.
$L^{2}$

Proposition 3.13. Let$s$be a nonnegative integer satisfying$\mathcal{S}\geq 2$$u_{0\infty}=^{T}(\phi_{0\infty}, m_{0\infty}, \epsilon_{0\infty})\in H_{(\infty),2}^{s+1}\cross H_{(\infty),2}^{s},$. Assume that77
$F_{\infty}=T(F_{\infty}^{1}, F_{\infty}^{2}, F_{\infty}^{3})\in L^{2}(0, T’;H_{(\infty),2}^{s}\cross H_{(\infty),2}^{s-1})$

for all $T’>$ O. Assume also that$u_{\infty}=T(\phi_{\infty}, m_{\infty}, \epsilon_{\infty})$satisfies(3.17)$\{\begin{array}{l}\partial_{t}u_{\infty}+Au_{\infty}=F_{\infty},u_{\infty}|_{t=0}=u_{0\infty}.\end{array}$and$\phi_{\infty}\in C([O, T H_{(\infty)}^{s+1})\cap L^{2}(0, T’;H_{(\infty)}^{s+2})$Then$u_{\infty}$,$T(m_{\infty}, \epsilon_{\infty})\in C([0, T H_{(\infty)}^{s})\cap L^{2}(0, T’;H_{(\infty)}^{s+1})$

satisfies$\phi_{\infty}\in C([O, T H_{(\infty),2}^{s+1})\cap L^{2}(0, T’;H_{(\infty),2}^{s+2})$,$T(m_{\infty}, \epsilon_{\infty})\in C([0, T H_{(\infty),2}^{s})\cap L^{2}(0, T’;H_{(\infty),2}^{s+1})$

for all $T’>0$ and there exists an energy functional$\mathcal{E}^{s}[u_{\infty}]$such that there holds theestimate$\frac{d}{dt}\mathcal{E}^{s}[u_{\infty}](t)+d(\Vert\phi_{\infty}(t)\Vert_{H_{2}^{\epsilon+2}}^{2}+\Vert m_{\infty}(t)\Vert_{H_{2}^{s+1}}^{2}+\Vert\epsilon_{\infty}(t)\Vert_{H_{2}^{s+1}}^{2})$(3.18)$\leq C\Vert F_{\infty}(t)\Vert_{H_{2}^{s}\cross H_{2}^{\epsilon-1}}^{2}$

on $(0, T’)$ for all $T’>0$ . Here is a positive constant;on but not on
is equivalent to
$d$

$T$

$T’;\mathcal{E}^{S}[u_{\infty}]$$C$

is a positive constant depending$\Vert u_{\infty}\Vert_{H_{2}^{s+1}\cross H_{2}^{s}}^{2},$$i.e,$$C^{-1}\Vert u_{\infty}\Vert_{H_{2}^{s+1}\cross H_{2}^{s}}^{2}\leq \mathcal{E}^{S}[u_{\infty}]\leq C\Vert u_{\infty}\Vert_{H_{2}^{s+1}\cross H_{2}^{s}}^{2}$and$\mathcal{E}^{s}[u_{\infty}](t)$;is absolutely continuous in $t\in[0, T’]$ for all $T’>0.$

Making use of the smoothing eﬀect of arising in the Korteweg stress tensor, we canprove Proposition 3.13 in a similar manner to the -energy method as in [1, 5], and weomit the details here.
$\rho$$L^{2}$

It follows from Proposition 3.12 that we obtain the following estimate of
frequency part.

Proposition 3.14. Let$T(\phi, m, \epsilon)$$s$$\Gamma$for the highbe a nonnegative integer satisfying $s\geq 2$ . Assume that $u=$satisfies$\Vert u\Vert_{\mathscr{Z}}(0,T)<<1.$

Then it holds that$\Vert\Gamma[P_{\infty}F(u, g 劣 (o,T) \leq C\Vert u\Vert_{\mathscr{Z}}^{2}(0,T)+C(1+\Vert u\Vert_{\mathscr{Z}}(0,T))[g]_{8}$uniformly for $u.$
By Proposition 3.9 and Proposition 3.14, as in [9], we obtain Theorem 3.1.

78

References[1] H. Cai, Z.Tan and Q. Xu, Time periodic solutions of the non-isentropic compressible
fluid models of Korteweg type, Kinet. Relat. Models., 8 (2015), pp. 29-51.

[2] Z. Chen, Q. Xiao, and H. Zhao, Time periodic solutions of compressible fluid modelsof Korteweg type, preprint, 2012, Analysis of PDEs.
[3] Z. Chen and H. Zhao, Existence and nonlinear stability of stationary solutions to thefull compressible Navier-Stokes-Korteweg system, J. Math. Pures Appl., 101 (2014),

pp. 330-371.
[4] J.E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch.
Rational Mech. Anal., 88 (1985), pp. 95-133.
[5] H. Hattori and D. N. Li, The existence of global solutions to a fluid dynamic model formaterials for Korteweg type, J. Partial Diﬀerential Equations, 9 (1996), pp. 323-342.

[6] M. Heida and J. M\’alek, On compressible Korteweg fluid-like materials, Internat. J.
Engrg. Sci., 48 (2010), pp. 1313-1324.
[7] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space, Commun. Math. Phys., 266 (2006),

pp. 401-430.
[8] Y. Kagei and T. Kobayashi, Asymptotic Behavior of Solutions of the Compressible
Navier-Stokes Equation on the Half Space, Arch. Rational Mech. Anal., 177 (2005),
pp. 231-330.
[9] Y. Kagei and K. Tsuda, Existence and stability of time periodic solution to thecompressible Navier-Stokes equation for time periodic external force with symmetry,
J. Diﬀerential Equations, 258 (2015), pp. 399-444.
[10] T.Kato, Perturbation Theory for Liner Operators, Classics math., Springer-Verlag,
Berlin, 1995, reprint of the 1980 edition.

[11] D.J. Korteweg, Sur la forme que prennent les \’equations du mouvement des fluides
si 1 on tient compte des forces capillaires caus\’ees par des variations de densit\’econsid\’erables mais continues et sur la th\’eorie de la capillarite dans 1 hypoth\‘ese
une variation continue de la densit\’e, Archives N\’eerlandaises des sciences exactes etnaturelles, Ser 2 (6) (1901), pp. 1-24.
’

’

$d$

’

[12] A. Matsumura and T. Nishida, The initial value problem for the equations of motionof compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. , 55
(1979), pp. 337-342.
$A$[13] M. Okita, On the convergence rates for the compressible Navier- Stokes equations
with potential force, Kyushu J. Math. 68 (2014), pp. 261-286.

79[14] K. Tsuda, On the existence and stability of a time periodic solution to the compressible Navier-Stokes equation on the whole space, to appear in Arch. Rational Mech.
Anal..

[15] K. Tsuda, Existence and stability of time periodic solution to the compressible
Navier-Stokes-Korteweg system on , to appear in J. Math. Fluid Mech..
$\mathbb{R}^{3}$[16] Y. Shibata and S. Shimizu, A decay property of the Fourier transform and its application to the Stokes problem, J. Math. Fluid Mech, 3 (2001), pp. 213–230.

[17] Y. Shibata and K. Tanaka, On the steady flow of compressible viscous fluid andits stability with respect to initial disturbance, J. Math. Soc. Japan, 55 (2003), pp.
797-826.