数理解析研究所講究録 1425 巻 2005 年 150-159

150
Large-time behavior of spherically symmetric flowfor viscous heat-conductive gas
東工大・情報理工 中村 徹 (Tohru NAKAMURA)
東工大・情報理工 西畑 伸也 (Shinya NISHIBATA)

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

1

Introduction

We consider the asymptotic behavior of a spherically symmetric solution to a polytropic
ideal model of a compressible viscous gas over an unbounded exterior domain; $|\xi|>1$ }, where $n(\geq 3)$ is a space dimension. The motion of the polytropic ideal
gas is governed by the system of equations$\Omega:=\{\xi\in$$\mathbb{R}^{n}$$\rho_{t}+\nabla$. $(\rho u)=0$ ,,$\rho\{u_{t}+(u\cdot\nabla)u\}=\mu_{1}\triangle u+(\mu_{1}+\mu_{2})\nabla(\nabla\cdot u)-\nabla P(\rho, \theta)+\rho f$$c_{V}\rho\{\theta_{t}+ (u.

\nabla)\theta\}=\kappa\triangle\theta-P(\rho, \theta)\nabla\cdot$$u+\mu_{2}(\nabla\cdot u)^{2}+2\mu_{1}D\cdot$ $D$(l.la)(l.lb)(l.la)

andin the Eulerian coordinate, where the mass density , the velocity $u=$$P(\rho,\theta)=R\rho\theta(R>0)$
the absolute temperature 0 are unknown functions. In addition,andare constants called theis the pressure; $f=f(\xi)$ is the external force;viscosity coefficients satisfying $\mu_{1}>0$ and $2\mu_{1}+$ np2
and $cv$ are positive
constants called the thermal conductivity and the specific heat at constant volume,
respectively; $D=D(u)$ is the deformation tensor,$(u_{1}, \ldots, u_{n})$$\rho$$\mu_{2}$$\mu_{1}$$>0,\cdot$$D\cdot$

and$D:= \sum_{\mathrm{i},j=1}^{n}D_{ij}^{2}$,.

$D_{\mathrm{i}j}:= \frac{1}{2}(\frac{\partial u_{\mathrm{i}}}{\partial\xi_{j}}+\frac{\partial u_{j}}{\partial\xi_{i}})$Our concern is the problem of which initial data
given by the spherically symmetric function:$\beta \mathrm{o}(\xi)=\hat{\rho}_{0}(r)$$\kappa$$u_{0}( \xi)=\frac{\xi}{r}\hat{u}_{0}(r)$,$(\rho, u, \theta)(\xi, 0)=(\rho_{0\}}u_{0}, \theta_{0})(\xi)$$\theta_{0}(\xi)=\hat{\theta}_{0}(r)$

is

(1.2)

,

where $r:=|\xi|$ and eachand is a sealer function. We assume that the external
force is also given by the spherically symmetric potential force,$\hat{\theta}_{0}$$\hat{\rho}0,\hat{r\iota}_{0}$$f$$f:=- \nabla U=-\frac{\xi}{r}U_{r}(r)$

.

(1.3)

Under the assumptions (1.2) and (1.3), the solution
to (1.1) becomes sphericallysymmetric since (1.1) is rotationally invariant (see [6]). Namely, a solution to (1.1) isin the form of$(\rho, u, \theta)$$\rho(\xi,t)=\hat{\rho}(r, t)$,$u(\xi, t)$$= \frac{\xi}{r}\hat{u}(r, t)$,$\theta(\xi, t)=\hat{\theta}(r,t)$

.

(1.1)

151
to express spherically symmetric
For simplicity, hereafter, we abbreviate the symbol
isfunctions. Therefore, the equations for the spherically symmetric solution”$”\wedge$$(\rho, u, \theta)$$\rho_{t}+\frac{(r^{n-1}\rho u)_{r}}{r^{n-1}}=0$(1.5a),$\rho(u_{t}+uu_{\Gamma})=\mu(\frac{(r^{n-1}u)_{r}}{r^{n-1}})_{\tau}-P(\rho, \theta)_{\tau}-\rho U_{r}$$c_{V} \rho(\theta_{t}+u\theta_{T})=\kappa\frac{(r^{n-1}\theta_{r})_{r}}{r^{n-1}}.-P(\rho, \theta)\frac{(r^{n-1}u)_{r}}{r^{n-1}}$$(1.5b),$\mu_{2}(\frac{(r^{n-1}u)_{r}}{r^{n-1}})^{2}+2\mu_{1}u_{r}^{2}$$+2(n-1) \mu_{1}\frac{u^{2}}{r^{2}}$, (1.5c)where $\mu.--2\mu_{1}+\mu_{2}$ is a positive constant. The initial and the boundary conditions areprescribed as$\rho(r\}0)=\rho_{0}(r)$,$u(r, 0)=u_{0}(r)$ ,$u(1, t)=0$ ,$\theta(r, 0)$$\theta_{r}(1, t)=0$$=\theta_{0}(r)$

(1.6)
(1.7)

,

.

It is also assumed that the initial data satisfies$\inf_{r\in[1,\infty)}\rho_{0}(r)>0$$\lim_{rarrow\infty}(\rho_{0}(r), u_{0}(r)$,,$\inf_{\tau\in|1,\infty)}\theta_{0}(r)>0$$\theta_{0}(r))=(\rho_{+}fu_{+}, \theta_{+})$

,

(1.8)

,

$\rho_{+}>0$,

are constants. Moreover, the initial data
andwhere ,
to be compatible with the boundary data (1.7):$\rho_{+}$$u_{+}$$\theta_{+}>0$

(1.9)

,

$(\rho_{0}, u_{0}, \theta_{0})$$\theta_{+}$$u_{0}(1)=0\}$$\{\mu(\frac{(r^{n-1}u_{0})_{r}}{r^{n-1}})\cdot \mathrm{r}$$\theta_{0r}(1)=0$

is supposed(l.lOa),$-P(\rho_{0}, \theta_{0})_{r}-\rho_{0}U_{r}\}|_{r=1}=0$(l.lOb).

to the problem (1.5), (1.6)
The aim of the present paper is to show that theand (1.7) converges to the corresponding stationary solution as time goes to infinity for
Sobolev space. The stationary solution
an arbitrary initial disturbance belonging to
is a solution to the equations (1.5) independent of time and satisfies
Therefore, thethe same boundary and spatial asymptotic conditions (1.7) and (1.9).
stationary solution verifies$\mathrm{s}\mathrm{o}1_{11}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$H^{1}$$t$$(\tilde{\rho}(r),\tilde{u}(r),$ $\theta(r))$$\frac{(r^{n-1}\tilde{\rho}\tilde{u})_{r}}{r^{n-1}}=0$(l.lla),$\tilde{\rho}\tilde{u}\tilde{u}_{r}=l^{l}(\frac{(r^{n-1}\overline{u})_{\tau^{\backslash }}}{r^{n-1}})\Gamma-P(\tilde{\rho},\overline{\theta})_{r}-\tilde{\rho}U_{r}$,(l.llb)$c_{V} \tilde{\rho}\tilde{u}\tilde{\theta}_{r}=\kappa\frac{(r^{n-1}\overline{\theta},.)_{r}}{r^{?l}-1}-P(\tilde{\rho},\tilde{\theta})\frac{(r^{n-1}\overline{u})_{r}}{r^{?\iota-1}}+\mu_{2}(\frac{(r^{n-1}\tilde{u})_{\varphi}}{r^{n-1}})^{2}+2\mu_{1}\overline{u}_{r}^{2}+2(n-1)\mu_{1}\frac{\overline{u}^{2}}{r^{2}}$,(l.llc)152$\tilde{u}(1)=0$,$\tilde{\theta}_{r}(1)=0$.

,(1.12)$\lim_{rarrow\infty}(\tilde{\rho}(r),\tilde{u}(r),\tilde{\theta}(r))=(\rho_{+}, u_{+}, \theta_{+})$

Solving the problem (1.11) with (1.12), we obtain the stationary solution explicitly as$\overline{\rho}(r)=\rho_{+}\exp(-\frac{1}{R\theta_{+}}U(r))$,$\tilde{u}(r)=0$,$\tilde{\theta}(r)=\theta_{+}$,(1.13)

where we have assumed that$\lim_{Tarrow\infty}U(r)=\mathrm{k}.\mathrm{m}\int^{T}\tauarrow\infty U_{r}(\eta)d\eta$(1.14)$+U(1)=0$

without loss of generality. Here, let us note that the stationary solution is a constant
if the external force is equal to zero, that is, , $(r)\equiv 0$ .
The main result in the present paper is stated in the next theorem, which meansthat the stationary solution (1.13) is time asymptotically stable. This stability th notedoes not need any smallness assumptions on the initial data. Moreover, due to the, $U_{T}(r)\geq 0$ , it can be arbitrary
condition (1.16), if the external force U. is attractive,
large. The typical example allowing this assumption is the case that the external force
is given by a gravitational force of the sphere.
$U$$(\rho_{+}, 0, \theta_{+})$$\mathrm{m}$$\mathrm{i}.\mathrm{e}.$Theorem 1.1. Suppose that the initial data$r^{\frac{n-1}{2}}(\rho_{0}-\tilde{\rho})$,$r^{\frac{n-1}{2}}u_{0}$,$r^{\frac{n-1}{2}}(\theta_{0}-\theta_{+})$,

satisfies$(\rho 0, u_{0}, \theta_{0})$$r^{\frac{n-1}{2}}(\rho_{0}-\tilde{\rho})_{r}$,$r^{\frac{n-1}{2}}u_{0r}$,$r^{\frac{n-1}{2}}\theta_{0r}\in L^{2}(1, \infty))$(1.15a)$\rho_{0}\in B^{1+\sigma}[1, \infty)$

for a certain constantexternal force$\sigma\in(0,1)$,(1.15b)$(u_{0}, \theta_{0})\in B^{2+\sigma}[1, \infty)$. Then there exists a constant$U_{r}\in C^{1}[1, \infty)\cap B^{\sigma}[1, \infty)$$\delta$$>0$ such thatsatisfies

if the(1.16)$-\delta\leq U_{r}(r)$for an arbitrary $r\geq 1$ , then the initial boundary value problem (1.5), (1.6) and (1.7)
has a unique solution$r^{\frac{\prime\iota-1}{2}}(\rho-\tilde{\rho})$,$r^{\frac{7(-)}{2}}u$,$(\rho, u, \ )$satisfying$r^{\frac{\prime\prime-1}{2}}(\theta-\theta_{+})$$\rho\in B^{1+\sigma,1+\sigma/2}([1, \infty)\cross$,$r^{\frac{\tau\iota-1}{2}}(\rho-\tilde{\rho})_{r}$$[0, T])$,,$r^{\frac{\tau\iota-1}{2}}u_{r}$,$(u, \theta)\in B^{2+\sigma,1+\sigma/2}([1, \infty)\mathrm{x}$for an arbitrary $T>0$ . Moreover, the solution$(\rho, u, \theta)$

stationary solution (1.13) as iime tends to infinity:$\lim_{tarrow\infty_{r}}\sup_{\in[1,\infty)}|(\rho(r, t)-\tilde{\rho}(r),u(r,t)$,(1.17a)(1.17b)$r^{\frac{n-1}{2}}\theta_{r}\in C([0, T];L^{2}(1, \infty))$$[0, T])$

converges to the corresponding, $\theta(r,t)-\theta_{+})|=0$ .

(1.18)

Remark 1.2. The same result as in Theorem 1.1 is proved for the case $P(\rho, \theta)=$in place of. It is slightly more general than the ideal
polytropic gas. If $>1$ , the stationary solution
is given by$P(\rho, \theta)=R\rho\theta$$R\rho^{\alpha}\theta(\alpha\geq 1)$$(\tilde{\rho},\tilde{u},\tilde{\theta})$$\alpha$$\tilde{\rho}(r)=(\rho_{+}^{\alpha-1}+\frac{\alpha-1}{\alpha R\theta_{+}}U(r))^{1/(\alpha-1)}$,$\tilde{u}(r)=0$,$\tilde{\theta}(r)=\theta_{+}$,

153
which is proved similarly as the derivation of (1 i3). The difference of the proofs of theappears in the derivation of the upper boundstabilities for the cases a $=1$ andof the specific volume.
$\alpha>1$

Remark 1.3. For the Dirichlet boundary condition on the temperature, the same result
is posed in place of $\theta,.(1, t)=0$ in (1.7).
as in Theorem 1.1 holds if$\theta(1, t)=\theta_{+}$

Related results. The asymptotic behavior of a solution to the compressible NavierStokes equation (1.1) is first consiclered by Matsumura and Nishida in [12], where they
thespherically symmetric, converges
prove the solution, which is notexterior domain ofcorresponding stationary solution as time tends to infinity inforce. For
theexternalandunder the smallness assumptions on the initial data
large initial data, Itaya in [6] shows a time global existence of the spherically symmetricaresolution to (1.1) on a bounded annulus domain. After this research, many
domainannulusboundedobtained for the spherically symmetric solution to (1.1) on ain [11] considers isothermal flow
data. For example,
with large
arbitraryand proves that the stationary solution is time asymptotically stable for an
external force. Here, he also obtains a exponential decay rate.
exterior domain, Jiang in
For the spherically symmetric problem in an
the external forces,(1.1)
without[7] shows the time global existence of the solution toPrecisely,completely.
. In the paper [7], the asymptotic state is noti.e.,
as $tarrow\infty$ , where is an arbitrarily fixed
it shows that, for $n=3$ ,to 2. Thus, the asymptotic behavior for this problem is
integer greater than or
and Yanagi inremained open. This open problem is solved byfor the[16] for the isentropic flow. In the present paper, we consider the same prob lem
ideal polytropic gas to solve this problem.
$\mathrm{t}_{\mathrm{J}}\mathrm{o}$$\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{a}1^{\mathrm{s}}\mathrm{i}1\mathrm{y}$$\mathrm{t},\mathrm{h}\mathrm{e}$$\mathrm{a}$$\mathbb{R}^{3}$$\mathrm{r}\mathrm{e}\mathrm{s}\iota\iota \mathrm{l}\mathrm{t}\mathrm{s}$$\mathrm{M}\mathrm{a}\mathrm{t}_{l}\mathrm{s}\mathrm{u}\mathrm{m}\iota\iota \mathrm{r}\mathrm{a}$$\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{l}|$$\iota\iota \mathrm{n}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}$$01_{3}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$$f\equiv 0$$j$$||u(t)||_{L^{2j}}arrow 0$$\mathrm{e}\mathrm{q}\iota \mathrm{l}\mathrm{a}\mathrm{l}$$\mathrm{N}\mathrm{i}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{b}\mathrm{a}\mathrm{t}_{1}\mathrm{a}$$\mathrm{N}\mathrm{a}\mathrm{l}\{\mathrm{a}\mathrm{m}\iota\iota \mathrm{r}\mathrm{a},$denotes the l-ih order Sobolev
Notation. For a non-negative integer $l\geq 0,$.
sense with the norm . . We note $H^{0}=L^{2}$ andspace over in thewhich have$(0, 1)$ Ba(Q) denotes the space of bounded functions over,Fordenotes theuniform Holder continuity with exponent . For an integer ,$i\in[0,k]$ . For
for an arbitrary integer
space of the functions satisfying
lder$[0, T]$ ,denotes the space of the uniform
a domain
and with respect to and ,the H\"older
continuous
denotes the space of the functions
respectively. For integers and 1,
and $j\in[0, l]$ .
for arbitrary integers,
satisfying$H^{f}(\Omega)$$||\cdot$$\Omega$$L^{2}$$||$$||_{l}$$||:=||\cdot$$||0$$\mathrm{t}_{1}\mathrm{h}\mathrm{e}$$\Omega$$\alpha\in$$k$$B^{k+\alpha}(\Omega)$$\alpha$$\partial_{x}^{i}u\in B^{\alpha}(\Omega)$$\mathrm{H}\dot{\mathrm{o}}$$B^{\alpha,\beta}(Q_{\mathit{1}’}’)$$Q_{\mathit{1}’}’\underline{\subseteq}[0, \infty)\mathrm{x}$$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t},\mathrm{s}$$\mathrm{f}\mathfrak{U}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{w}\mathrm{i}\mathrm{f}_{1}\mathrm{h}$$k$$\partial_{x}^{\dot{\mathrm{t}}}u$$\partial_{t}^{j}u\in B^{\alpha,\beta}(Q_{T})$$\alpha$$\beta$$x$$t$$B^{k+\alpha}’ f+\beta(Q\tau)$$\mathrm{i}\in[0, k^{\wedge}]$2

Local existence in the Lagrangian coordinate

2.1

The System in the Lagrangian coordinate

by employing
To prove Theorem 1.1, we derive the a priori estimate of the solution(1.5) inequationthethe energy method. To this end, it is convenient to transform
The transformatio
the Eulerian coordinate into that in the Lagrangian coordinate.

$\mathrm{n}$154from the Eulerian coordinate $(r, t)$ to the Lagrangian coordinate $(x,t)$ is
the transformation$x=l^{\Gamma}s^{n-1}\rho(s,t)ds$,$r_{x}= \frac{v}{r^{n-1}}$,$r_{t}=u$$\mathrm{e}\mathrm{x}’\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d}1\supset \mathrm{y}$

(2.1)

,

where $v:=1/\rho$ is the specific volume. Using $(2,1)$ , we deduce the system (1.5) to(2.2a)$v_{\mathrm{t}}=(r^{n-1}u)_{x)}$$u_{t}= \mu,r^{n-1}(\frac{(r^{n-1}u)_{x}}{v})_{x}-r^{r1-1}p(v, \theta)_{x}-U_{r}$(2.2b),$c_{V} \theta_{t}=\kappa(\frac{r^{2n-2}\theta_{x}}{v})_{x}-p(v, \theta)(r^{n-1}u)_{x}+\mu\frac{(r^{n-1}u)_{x}^{2}}{v}-2(n-1)\mu_{1}(r^{n-2}u^{2})_{x}$,(2.2c)

where $p(v, \theta).--Rv^{-1}\theta$ . The initial and the boundary conditions are$(v,u, \theta)(x, 0)=(v_{0}, u_{0}, \theta_{0})(x)$$r\iota(0, t)=0$

Owing to (1.8) and (1.9), the$\mathrm{i}\mathrm{n}1^{1}\mathrm{t}\mathrm{i}\mathrm{a}1$,$v_{0}.--1/\beta 0$$\theta_{x}(0, t)=0$

data$\inf_{x\in\{0,\infty)}v_{0}(x)>0$,$(v_{0},u_{0}, \theta_{0})$,.

satisfies$\inf_{x\in[0,\infty)}\theta_{0}(x)>0$$x\varliminf_{\infty}^{(v_{0},u_{0},\theta_{0})(x)=(v_{+},0,\theta_{+})}$

(2.3)
(2.4)

,

,

(2.5)

,

$v_{+}:=1/\rho_{+}$

.

(2.6)

Since the variable $r=r(x, t)$ is a function of $(x, t)$ , the stationary solution
depends on $(x,t)$ in the Lagrangian coordinate. Thus, let$\overline{v}_{0}(x):=1/\overline{\rho}(r_{0}(x))$,$r_{0}(x):=r(x, 0)$$\overline{v}(r).=1/\overline{\rho}(r)$.

It is also assumed that the initial data is compatible with the boundary data (2.4),
in thethat is, (1.10) holds. The stability theorem of the stationary solution
in
next
theorem.
isthheLagrangian coordinate stated$(\tilde{v},\tilde{u},\tilde{\theta})$

Theorem 2.1, Suppose that the initial data
$v_{0}-\mathrm{v}\mathrm{o}$,$u_{0}$,$\theta_{0}-\theta_{+}$,

satisfies$(v_{0}, u_{0}, \theta_{0})$$r_{0}^{n-1}(v_{0}-\tilde{v}_{0})_{x}$,$r_{0}^{n-1}u_{0x}$,$r_{0}^{n-1}\theta_{0x}\in L^{2}(0, \infty)$

.

(2.7)

Then there exists a constant $>0$ such that the external force
satisfiesand (L16), then the initial boundary value problem (2.2); (2.3) and (2.4) has
satisfying
a unique global solution$\delta$$U_{r}\in$$U_{r}$$\iota f$$C^{1}[1, \infty)$$(v, u, \theta)$$v-\tilde{v}$

,

$u$

,

$\theta-\theta_{+}$

Moreover the solution
tends to infinity:$\lim$,$r^{n-1}(v-\tilde{v})_{x}$$(v, u, \theta)$$\sup$$tarrow\infty_{x\in(0,\infty\}}$,$r^{n-1}u_{x}$,$r^{n-1}\theta_{x}\in C([0, \infty)_{\gamma}.L^{2}(0, \infty))$

converges to the stationary solution,.

$(\tilde{v}, 0, \theta_{+})$$|(v(x, t)-\tilde{v}(r(x, t))_{)}u(x,t)$ $\theta(x, t)-\theta_{+})|=0$

.

(2.8)

as time

$t$

(2.3)

155
The initial condition (2.7) holds owing to (1.15a). Because several coefficients in, we first consider the approximate problem(2.2) are unbounded over the domain$(0,m)$
for a positive integer $m$ . Following ideas inrestricted in the bounded intervalby$[2, 7]$ , we define the “cut-off-function”$(0, \infty)$$\phi_{m}\in C^{3}[0, \infty)$$1_{\}}$$\phi_{m}(x):=\{$$0\leq\phi_{\uparrow\eta}(x)\leq 1$,

for$0 \leq x\leq\frac{r\mathfrak{s}x}{2}$for$m\leq x$0‘(2.10),$(\mathrm{i}=1,2,3)$$| \partial_{x}^{i}\phi_{7n}(x)|\leq\frac{C}{m’}$

,

,

for

$\frac{m}{2}\leq x\leq m$,where $m$ is an arbitrary positive integer. We consider the equations for an unknown
in the bounded domain $(0, m)$ :function(2.11a),$(v_{m}, u_{m}, \theta_{m})$$v_{mt}=(r_{m}^{n-1}u_{m})_{x}$$u_{m\mathrm{t}}= \mu r_{m}^{n-1}(\frac{(r_{m}^{n-1}u_{m})_{x}}{v_{m}})_{x}-r_{m}^{n-1}p(v_{m}, \theta_{m})_{x}-U_{r}$,(2.11b)$c_{V} \theta_{m\ell}=\kappa(\frac{r_{m}^{2n-2}\theta_{mx}}{v_{m}})_{x}-p(v_{m}, \theta_{m})(r_{m}^{n-1}u_{m})_{x}+\mu\frac{(r_{m}^{n-1}u_{m})_{x}^{2}}{v_{m}}-2(n-1)\mu_{1}(r_{m}^{n-2}u_{m}^{2})_{x}$(2.11c)with the initial and the boundary conditions(X. 12)$(v_{m}, u_{m}, \theta_{m})(x, 0)=(v_{7n0}, u_{m0}, \theta_{m0})(x))$$u_{m}(0, t)=u_{m}(n\tau, t)=0$

where the initial data$(v_{m0}, u_{rn0}, \theta_{m0})$,$\theta_{mx}(0, t)=\theta_{mx}(m, t)=0$

is defined by using$\phi_{m}$as$v_{m0}(x).--(v_{0}(x)-\tilde{v}_{0}(x))\phi_{\mathfrak{m}}(x)+\tilde{v}_{0}(x)$$u_{ml\}}(x):=u_{0}(x)\phi_{n?}(x)$$r_{m}(x, t).--\{$$r_{m}$

and$1+n \int_{0}^{x}v_{m}(y, t)dy\}^{1/n}$

,

,

,

$\theta_{m0}(x):=(\theta_{0}(x)-\theta_{+})\phi_{m}(x)+\theta_{+}$In the equations (2.11), the functions(2.13),$r_{m0}$.

are given by$r_{m0}(x).-- \{1+n\int_{0}^{x}v_{n\uparrow 0}(y)dy\}^{1/\uparrow \mathrm{t}}$In order to state the local existence theorem precisely, we define the function

space:$X_{D,\overline{v},\underline{v},\overline{\theta},\underline{\theta}}^{m}(0, T):=$$\{(v, u, \theta)|(v-\tilde{v}, u, \theta-\theta_{+})\in C^{0}([0, T]).H^{1}(0, m)), u_{x}, \theta_{x}\in L^{2}(0, T;H^{1}(0m)\})$$||(v-\tilde{v}, u, \theta-\mathit{0}_{+})(t)||_{1,r,m}\leq D$$\underline{v}\leq v(x, t)\leq\overline{v}$

for constants

1,$\underline{\theta}\leq\theta(x, t)\leq\overline{\theta}$$D>0,\overline{v}>\underline{v}>0,\overline{\theta}>\underline{\theta}>0$,

for $t\in[0, T]$ ,for $(x, t)\in[0, m]\mathrm{x}[0, T]\}$and $T>0$ . The norm$||\cdot$$||\cdot r,m$

is given by$(v-\tilde{v}, u, \theta-\theta_{+})||_{1,r,m}:=||(v-\tilde{v}, u, \theta-\theta_{+}, r^{n-1}(v-\tilde{v})_{x}, r^{??-1}u_{x}, r^{n-1}\theta_{x})||_{L^{2}(0,m)}$.

(2.12) and (2.13) is
A time local solution to the initial boundary value problem (2.11),
established by employing the standard iteration method

158
and
Proposition 2,2. For arbitrary constants
data
theinitial
that
such
if
exists a positive constantandsatisfies
then the problem (2.11), (2.12) and (2.13) has a unique solution.

$\overline{\theta}>\underline{\theta}>0$$D>0_{f}\overline{v}>\underline{v}>0$$T=T(D,\overline{v}, \underline{v}\overline{\theta}, \underline{\theta})\}$$||(v_{m0}-\tilde{v},u_{m0}, \theta_{m0}-\theta+)||_{1,r,m}\leq D_{t}\underline{v}\leq v_{m0}(x)\leq\overline{v}$, there$(v_{m0}, u_{m0}, \theta_{m0})$$\underline{\theta}\leq\theta_{m0}(x)\leq$$(v_{m}, u_{m}, \theta_{m})\in$$\overline{\theta}_{f}$$X_{2D_{y}2\overline{v},\underline{v}/2,2\overline{\theta},\underline{\theta}/2}^{m}(0, T)$A priori estimate

2.2

in this subsection, we discuss the a$(0, \infty)$$\mathrm{P}^{\mathrm{r}\mathrm{i}_{\mathrm{o}\mathrm{r}1}^{1}}$estimate, uniformly in$\eta \mathit{1}$, in$H^{1}- 8\mathrm{o}\mathrm{b}\mathrm{o}1\mathrm{e}\mathrm{v}$. Hereafter in this subsection, for simplicity, we omit the subscript $m$ and denote
by $(v, u, \theta, r)$ .

$(v_{m}, u_{m}, \theta_{m}, r_{m})$We define the energy form$\mathcal{E}$as$\mathcal{E}:=\frac{1}{2}u^{2}+\Phi(v,\overline{v})+c_{V}\mathit{0}_{+}(\frac{\theta}{\theta_{+}}-1-\log\frac{\theta}{\theta_{+}})$$\Psi(v,\tilde{\tau)}):=p(\tilde{v}, \theta_{+})(\uparrow \mathit{1}-\tilde{TJ})-\varphi(V,\tilde{7J})$(2.14),(2.15),$\varphi(v,\overline{v}):=\int^{\tilde{v}}p(\xi, \theta_{+})d\xi=R\theta_{+}\log\frac{v}{\overline{v}}$(2.16).

The following lemmas are proved by using the standard energy method. For details,see [15].
T) to (2.11), (2.12) and (2.13),

Lemma 2.3. For a solution (v, u,$\theta)\in X_{D,\overline{v},\underline{v},\overline{\theta},\underline{\theta}}^{m}(0,$,$\oint_{0}^{m}\mathcal{E}(t)dx+c\int_{0}^{t}\int_{0}^{m}\frac{v}{\theta r^{2}}u^{2}+\frac{r^{2n-2}}{v\theta}u_{x}^{2}+\frac{r^{2n-2}}{v\theta^{2}}\theta_{x}^{2}dxd\tau\leq\int_{0}^{m}\mathcal{E}(0)dx$

where

$c$

is a positive constant depending only on the initial data.

Lemma 2.4. For a solution (v, u,$?)$T) to (2.11), (2.12) and (2.13),$\in X_{D,\overline{v}_{1}\underline{v},\overline{\theta},\underline{\theta}}^{m}(0,$,$\int_{0}^{m}\frac{u^{4}+(\theta-\theta_{+})^{2}}{r^{2n-2}}dx+\oint_{0}^{t}\int_{0}^{m}u^{2}u_{x}^{2}+\theta_{x}^{2}dxd\tau\leq C_{\tilde{v}_{J}\underline{v}}$where(2.17)

a positive constant depending only on ,$\overline{v}$$C_{\overline{v}_{f}\underline{v}}\iota s$

Lemma 2.3. For a solution (v, u,$\underline{v}$(2.18)and the initial data.

T) to (2.11), (2.12) and (2.13),$\theta)\in X_{D,\overline{v},\underline{v},\overline{\theta},\underline{\theta}}^{m}(0,$$\int_{0}^{m}\varphi_{x}^{2}dx+\int_{0}^{t}\int_{0}^{m}(1+\theta)\varphi_{x}^{2}dxd\tau\leq C_{\overline{v},\underline{v}}$,(2.19),(2.20).

(2.21)$\int_{0}^{\iota n}u_{x}^{2}dx+\int_{0}^{t}\oint_{0}^{m}.r^{2n-2}u_{xx}^{2}dxd\tau\leq C_{\overline{v},\underline{v}}(\overline{\theta}+1)$$\oint_{0}^{m}.\theta_{x}^{2}dx+\int_{0}^{t}\int_{0}^{m}r^{2\uparrow \mathrm{z}-2}\theta_{xx}^{2}dxd\tau\leq C_{\overline{v},\underline{v}}(\overline{\theta}+1)^{2}$

157
Using the estimates (2.20) and (2.21), we show the uniform positive bound of zz and
0 in the next lemma. Thus, we see that the estimates (2.20) and (2.21) are independentof$\overline{\theta}$.

Lemma 2.6. For a solution (v, u,T) to (2.11), (2.12) and (2.13),$\theta)\in X_{D,\overline{u},\underline{v},\overline{\theta},\underline{\theta}}^{m}(0,$$|u(x, t)|\leq C_{\overline{v},\underline{v}}$$0<ce$

for$(x,t)\in[0, n\mathrm{z}]\mathrm{x}$$[0, T]$(2.22),(2.23)$-C_{\overline{\mathrm{t}^{1}},\underline{v}}(1+t)\leq\theta(x,t)\leq C_{\overline{v},\underline{v}}$.

Lemma 2.7. For a solution$(v, u, \theta)\in X_{D,\overline{v}_{\mathrm{J}}\underline{v}_{1}\overline{\theta},\underline{\theta}}^{m}(0T)\}$to (2.11), (2.12) and (2.13),(2.24).

$\int_{0}^{m}r^{2n-2}\varphi_{x}^{2}+r^{2n-2}u_{x}^{2}+r^{2n-2}\theta_{x}^{2}dx\leq C_{\overline{v},\underline{v}}$Utilizing the estimates obtained in the above lemmas and letting
to the problem (2.2), (2.3) and (2.4) satisfying
the local solution$arrow\infty$$n\iota$, we have$(v, u, \mathit{0})$$v-\tilde{v}$

, ,
$u$

$\varphi_{x}$,$\theta-\theta_{+}$$r^{n-1}u_{x}$,,$r^{n-1}(v-\tilde{v})_{x}$$r^{n-1}\theta_{x}$,,$r^{n-1}u_{x}$$r^{n-1}u_{xx}$, $r^{n-1}\theta_{x}\in C([0,T];L^{2}(0, \infty))$ ,, $r^{2n-2}\theta_{xx}\in L^{2}(0, T;L^{2}(0, \infty))$Mor over we have the energy estimates over an unbounded domain$\int_{0}^{\infty}\mathcal{E}dx+J_{0}^{t}.J_{0}^{\infty}.\frac{v}{\theta r^{2}}u^{2}+\frac{r^{2n-2}}{v\theta}u_{x}^{2}+\frac{r^{2n-2}}{v\theta^{2}}\theta_{x}^{2}dx$dr$(0, \infty)$$\leq C$as(2.25a),$\int_{0}^{\infty}r^{2n-2}(v-\overline{v})_{x}^{2}+r^{2n-2}u_{x}^{2}+r^{2n-2}\theta_{x}^{2}dx$$+ \int_{0}^{t}\int_{0}^{\infty}\varphi_{x}^{2}+r^{2n-2}\tau\iota_{xx}^{2}+r^{4n-4}\theta_{xx}^{2}dxd\tau\leq C_{\overline{v},\underline{v}}$.

(2.25b)3

Asymptotic stability of the stationary solution in
Lagrangian coordinate

3.1

Positive bound of the specific volume

specific volume $v(x, t)$
In this subsection, we derive the pointwise positive bound of thetheuniformly in . We prove this bound by using the basic estimate (2.25a) andrepresentation formula of $v(x, t)$ .
be a solution to the problem (2.2), (2.3)
Proposition 3.1. Leton the initial dataand (2.4). Then there exist positive constants and $Cdepend,ing$ only
satisfies
such that if the external force$t$$(v, u, \theta)\in X_{D,\overline{v},\underline{v},\overline{\theta},\underline{\theta}}^{\infty}(0, T)$$c$$U_{r}$$-\delta_{D}\leq U_{r}(r)$,$\delta_{D}.--\frac{1}{2}(\sqrt{2}D+v_{+}e^{\frac{M}{R\theta+}})^{-1}$

,

(3.1)

158

where

$M\iota s$$v(x, t)$$s$a positive constant satisfying $\sup_{r\geq 1}U(r)\leq M$ , then the specific volume

atisfies$c\leq v(x, t)$$\leq C$for$(x, t)\in[0, \infty)\mathrm{x}$(3.2)$[0, T]$ .

To prove (3.2), we derive the representation formula of the specific volume $v(x, t)$ .
:To this end, we define the “cut-off-function”,
for1,$\leq x\leq(k+1)\epsilon$ ,
(3.3)
, for
$\eta(x).--\{$$\eta(x)$$0\leq x\leq k\epsilon$$k+1- \frac{x}{\epsilon}$0for$\epsilon$$>0$

and $k=1,2$ ,$\ldots$$k\epsilon$

for $(k+1)\epsilon\leq x$.

Lemma 3.2, Suppose that the same assumptions as in Proposition 3.1 hold. Then thespecific volume $v(x, t)$ is expressed by the fomula(3.4)$v(x, t)= \frac{v_{0}(x)+\frac{R}{\mu}f_{0}^{t}A_{\epsilon}(x,\tau)B_{\epsilon}(x,\tau)\theta(x,\tau)d\tau}{A_{\mathrm{b}}(x,t)B_{\epsilon}(x,t)}$for$x\in$$[$(&-1)\epsilon$k\epsilon)]$and $t\in[0, T]_{f}$ where$A_{\mathrm{a}}(x, t):=A_{\in}^{0}(t)A_{\epsilon}^{1}(x, t)A_{\epsilon}^{2}(x, t)$,$\mathrm{A}_{\epsilon}^{0}(t).--\exp(\frac{R}{\mu\epsilon}\oint_{0}^{t}\oint_{k^{\wedge}\epsilon}^{(k+1)\epsilon}\frac{\theta}{v}dxd\tau))A_{\epsilon}^{1}(x, t):=\exp(\frac{n-1}{\mu}\int_{0}^{t}\oint_{x}^{\infty}\frac{u^{2}}{r^{n}}\eta dxd\tau)$$A_{\epsilon}^{2}(x, t):= \exp(\frac{1}{\mu}\int_{0}^{t}\oint_{x}^{\infty}\frac{U_{r}}{r^{n-1}}\eta dxd\tau)$

,

,
(S.5)

.

$B_{\epsilon}(x_{;}t).-- \exp(\frac{1}{\mu}\oint_{x}^{\infty}(\frac{u}{r^{n-1}}-\frac{u_{0}}{r_{0}^{n-1}})\eta dx-\frac{1}{\epsilon}f_{k\epsilon}^{(k+1)\epsilon}.\log\frac{v}{?J_{0}}dx)$

Owing to the estimate (2.25a) with the aid of the Jensen inequality, we have theestimates(3.6),$c_{\epsilon} \leq\int_{a}^{a+\in}v(x, t)dx\leq C_{\epsilon}$$c_{\epsilon}\leq l^{a+b}\theta(x, t)dx\leq C_{\epsilon}$

for an arbitrary constant $a\geq 0$ . Substituting (3.6) in (3.4), we obtain the desired
estimate (3.2). For details, readers are referred to [15].
Substituting the pointwise estimate (3.2) in (2.25b), we obtain a uniform a priori
estimate:$\int_{0}^{\infty}(v-\overline{v})^{2}+u^{2}+(\theta-\theta_{+})^{2}+r^{2n-2}(v-\tilde{v})_{x}^{2}+r^{2n-2}u_{x}^{2}+r^{2n-2}\theta_{x}^{2}dx$. (3.7)$+ \oint_{0\acute{0}}^{t\cdot\infty}\frac{u^{2}}{r^{2}}+\varphi_{x}^{2}+r^{2n-2}u_{x}^{2}+r^{2n-2}\theta_{x}^{2}+r^{2n-2}u_{xx}^{2}+r^{4n-4}\theta_{xx}^{2}dxd\tau\leq C$

The estimate (3.7) yields the existence of the solution to the problem (2.2), $(2,3)$ and(2.4) globally in time by the standard continuation argument. Moreover, we see theconvergence (2.9) holds. Finally, by employing the Schauder theory for the parabolic
equations, we have the estimate of the solution in the H\"older space. By virtue of the
H\"older continuity of the solution, we translate Theorem 2.1 into that
the Euleriancoord inate. These procedures show the convergence (1.18)$|\ln$

159
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