数理解析研究所講究録
第 1885 巻 2014 年 105-115

105

Large-time existence of the spherically symmetricflow of a self-gravitating viscous gas
Morimichi Umehara*
University of Miyazaki

1

Introduction

We consider the spherically symmetric motion of a viscous and heat-conductive gas overa central rigid core (sphere). The gas is bounded by a free-surface, and let the boundaries be thermally-insulated. In the Eulerian coordinate such a motion is described by
the following system of equations: for $:=|x|,$$r$$x\in \mathbb{R}$(1.1)$\{\begin{array}{l}\frac{D\rho}{Dt}=-\rho\frac{(r^{2}v)_{r}}{r^{2}},\rho\frac{Dv}{Dt}=(-p+\zeta\frac{(r^{2}v)_{r}}{r^{2}})_{r}+\rho(f_{c}+f_{g}) ,\rho\frac{De}{Dt}=(-p+\zeta\frac{(r^{2}v)_{r}}{r^{2}})\frac{(r^{2}v)_{r}}{r^{2}}-4\mu(\frac{(v^{2})_{r}}{r}+\frac{v^{2}}{r^{2}})-\frac{(r^{2}q)_{r}}{r^{2}}\end{array}$$:=\{r\in \mathbb{R}|R_{c}<r<R(t)\}$ with the radius of thein, wherecentral core $R_{c}>0$ . The boundary conditions are$\mathfrak{D}$$:= \bigcup_{t>0}(\mathfrak{D}_{t}\cross\{t\})$$\mathfrak{D}_{t}$(1.2)$\{\begin{array}{ll}\frac{dR(t)}{dt}=v(R(t), t) for t>0,-p+\zeta\frac{(r^{2}v)_{r}}{r^{2}}-4\mu\frac{v}{r}=-p_{e} on \Gamma_{t}, t>0q=0 on \Gamma_{t}\cup\Sigma, t>0v=0 on \Sigma, t>0\end{array}$with $\Gamma_{t};=\{(r, t)\in \mathbb{R}^{2}|r=R(t)\},$ $:=\{(r, t)\in \mathbb{R}^{2}|r=R_{c}\}$ . We seek to find, for any$t>0$ , the density $\rho=\rho(r, t)$ , the velocity $v=v(r, t)$ (the velocity vector in)isand the absolute temperature
together with the free-surface $R(t)$ satisfying(1.1) and (1.2) for given initial conditions$\Sigma$$\mathbb{R}^{3}$$v \frac{x}{r}$$\theta=\theta(r, t)$$R(O)=R_{0}, (\rho, v, \theta)|_{t=0}=(\rho_{0}, v_{0}, \theta_{0})=(\rho_{0}, v_{0}, \theta_{0})(r), r\in\overline{\mathfrak{D}_{0}}$

.

(1.3)

*Department of Applied Physics, Faculty of Engineering, University of Miyazaki, 1-1 Gakuen
Kibanadai Nishi, Miyazaki 889-2192, JAPAN; Fax: $+81985587378;E$ -mail: umehara@cc.miyazaki. ac.jp
$u$

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is the material derivative; and are coefficients of the shear andHerethe bulk viscosity, respectively, satisfying $\mu>0$ and $3\zeta-4\mu\geq 0;p_{e}$ is the externalpressure assumed to be a non-negative constant. In this paper we consider only thecase that the gas is ideal, i.e., the pressure $p=p(\rho, \theta)$ , and the internal energy perunit mass $e=e(\rho, \theta)$ are given by$\frac{D}{Dt}=\frac{\partial}{\partial t}+v\frac{\partial}{\partial r}$$\zeta$

$\mu$

(1.4)

,

$p(\rho, \theta)=R\rho\theta, e(\rho, \theta)=c_{\gamma}\theta$

where $R,$ are the perfect gas constant, the specific heat capacity at constant volume,
respectively, and that the heat-flux per unit area is given by the Newton-Fourier’s law,with the heat-conductivity . For, i. e., $q=q(r, t)$ has the form
and are all positive constants. The self-gravitation
simplicity we assume thatof the gas per unit mass $f_{g}=f_{g}(r, t)$ and an attractive force due to the gravity of thecore $f_{c}=f_{c}(r)$ are given by, respectively,$c_{v}$$q=-\kappa\theta_{r}$$q \frac{x}{r}=-\kappa\nabla\theta$$\mu,$$\zeta,$$\kappa$$\kappa$$c_{\gamma}$(1.5)$f_{g}(r, t)=- \frac{G_{0}}{r^{2}}\int_{R_{c}}^{r}4\pi s^{2}\rho(s, t)ds, f_{c}(r)=-\frac{G_{0}M_{0}}{r^{2}}$and mass of the corewith the Newtonian gravitational constantIn order to investigate our system of equations we introduce the Lagrangian-mass$:= \lim_{\tauarrow+0}X(\tau;r, t)$ ,$(CD(t)\cross\{t\})$ lettransformation. First, for eachwhere $X(\tau;r, t)$ is the solution of the following initial value problem$M_{0}.$$G_{0}$$\xi(r, t)$$(r, t) \in\bigcup_{t>0}$$\{\begin{array}{ll}\frac{dX(\tau;r,t)}{d\tau}=v(X(\tau;r, t), \tau) , \tau\in(0, t) ,X(t;r, t)=r.

We see form (1.6) that a implicit function$\xi=\xi(r, t)$ satisfies$r=\tilde{r}(\xi, t)$\end{array}$(1.6)

determined by the equation.

$\tilde{r}(\xi, t)=\xi+\int_{0}^{t}v(\tilde{r}(\xi, \tau), \tau)d_{\mathcal{T}}$(1.7)

can be solved inversely as $\xi=$. Equationis strictly positive function. From these change of variables theprovided thatfunction $f=f(r, t)$ defined on is transformed into $f=f(x, t)$ $:=f(\tilde{r}(\eta(x), t), t)$ , and$:=f_{0}(\eta(x))$ . Putting $(v, v_{0})=$
into
the function $f_{0}=f_{0}(x)$ defined on$(u, u_{0})=(u(x, t), u_{0}(x))$ $:=(\dot{v}(x, t),\dot{v}_{0}(x))$ , and$(v(x, t), v_{0}(x))$, we deduce the following equations
omitting the every
Second, let$\eta(x)$$x=\tilde{x}(\xi)$$:= \int_{R}^{\xi}\rho_{0}(s)s^{2}ds$$x=\tilde{x}(\xi)$$\rho_{0}$$\overline{\mathfrak{D}}$$\overline{\mathfrak{D}_{0}}$$\dot{f}_{0}=\dot{f}_{0}(x)$$:=(1/\dot{\rho}(x, t), 1/\dot{\rho}_{0}(x)),$$”\cup,,$$\{\begin{array}{l}v_{t}=(r^{2}u)_{x},u_{t}=r^{2}(-R\frac{\theta}{v}+\zeta\frac{(r^{2}u)_{x}}{v})_{x}-G\frac{M_{c}+x}{r^{2}},c_{\eta}\theta_{t}=(-R\frac{\theta}{v}+\zeta\frac{(r^{2}u)_{x}}{v})(r^{2}u)_{x}-4\mu(ru^{2})_{x}+(\frac{r^{4}\kappa}{v}\theta_{x})_{x}\end{array}$

(1.8)

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in

$\Omega\cross(0, \infty)$

with$:=(0, M)$ , the boundary conditions$\Omega$$\{\begin{array}{l}(-R\frac{\theta}{v}+\zeta\frac{(r^{2}u)_{x}}{v}-4\mu\frac{u}{r}, \theta_{x})|_{x=M}=(-p_{e}, 0) ,(u, \theta_{x})|_{x=0}=(0,0) ,\end{array}$(1.9)

and the initial conditions$(v, u, \theta)|_{t=0}=(v_{0}, u_{0}, \theta_{0})$Here $M:=\tilde{x}(R_{0}),$$M_{c}:=M_{\Delta}\overline{4}\pi’ G:=4\pi G_{0}$.

(1.10)

and$r=r(x, t)=(R_{c}^{3}+3 \int_{0}^{x}v(s, t)ds)A$

which satisfies
and $r_{t}=u.$
We impose on the initial data the compatibility conditions$r_{x}= \frac{v}{r^{2}}$$\{\begin{array}{l}(-R\frac{\theta_{0}}{v_{0}}+\zeta\frac{(r_{0^{2}}u_{0})’}{v_{0}}-4\mu\frac{u_{0}1}{r_{0}})x=M=-p_{e},u_{0}(0)=\theta_{0}’(0)=\theta_{0}’(M)=0,\end{array}$where$r_{0};=\overline{r}[v_{0}]$(1.11)

with, for an arbitrary function $V=V(x)$ ,$\overline{r}[V]:=(R_{c}^{3}+3\int_{0}^{x}V(s)ds)A$

The temporally global solvability of the initial-boundary value problem $(1.8)-(1.10)$
has already been investigated in [22]. Namely, by letting $Q_{T}:=\Omega\cross(0, T)$ one has
Theorem 1 (Umehara-Tani) Let $\alpha\in(0,1)$ . Assume that the initial data$(v_{0}, u_{0}, \theta_{0})\in C^{I+\alpha}(\overline{\Omega})\cross(C^{2+\alpha}(\overline{\Omega}))^{2}$satisfy (1.11) andunique solution$v_{0},$

on , and that $p_{e}>0,3\zeta-4\mu>0$ . Then there exists aof the initial-boundary value problem $(1.8)-(1.10)$ such that$\theta_{0}>0$$(v, u, \theta)$$\overline{\Omega}$$(v, v_{x}, v_{t}, u, \theta)\in(C_{x,t}^{\alpha,\frac{\alpha}{2}}(\overline{Q_{T}}))^{3}\cross(C_{x,t}^{2+\alpha,1+\frac{\alpha}{2}}(\overline{Q_{T}}))^{2}$for any positive number$T$

and the inequalities$\{\begin{array}{l}|v, v_{x}, v_{t}|_{Q_{T}}^{(\alpha)}+|u, \theta|_{Q_{T}}^{(2+\alpha)}\leq C,v, \theta\geq C^{-1} in \overline{Q_{T}}\end{array}$hold, where$C$

is a positive constant dependent on the initial data and $T.$

(1.12)

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We note that in [22] radiating and reactive gases were also discussed. One easily
sees that this theorem gives the solvability of the original problem formulated in the
Eulerian coordinate. That is to say, one can construct the solution $R(t),$of the initial-boundary value problem $(1.1)-(1.3)$ with (1.4) and (1.5) such that, for$(\rho, v, \theta)$$\mathfrak{D}_{T}:=\bigcup_{t\in(0,T)}(\mathfrak{D}^{(t)}\cross\{t\})$,$R(t)\in C^{2+\frac{\alpha}{2}}([0, T]), (\rho, \rho_{r}, \rho_{t}, v, \theta)\in\backslash (C_{y,t}^{\alpha,\frac{\alpha}{2}}(\overline{\mathfrak{D}_{T}}))^{3}\cross(C_{y,t}^{2+\alpha,1+\frac{\alpha}{2}}(\overline{\mathfrak{D}_{T}}))^{2}$for any positive number by imposing suitable initial data corresponding to the one
imposed in Theorem 1. For details, see the manner found in [23].
System of equations related to (1.8), (1.9) have been studied as some stellar $mo$ delsappearing in the astrophysical arguments (see, for example, [1, 2,9]). One may say thatmathematical studies concerning problems of compressible viscous fluid (especially,
ideal gas case) in the spherically symmetric or one-dimensional framework have been
based on the spirit of the works [8] and [7] due to Kazhikhov and Shelukhin, andKazhikhov, respectively. Next in [11] Nagasawa developed the work [8] to the one
applicable to the free-boundary case. Since in the astrophysical context it may not
be reasonable to take some restrictions on the size of initial data, the works underlarge initial data mentioned above are also fundamental for our problem. Studies of
temporally global behaviour of the viscous gases related to our problem are found,for example, in [24] discussed in a fixed annulus domain; in $[12]$ in an unboundeddomain around the core; in [5, 6] including cases of both radiating and reacting gases;). In thein [10, 13, 14] with contacting outer vacuume (in the sense of $p_{e}=0$ in
paper [3] the model closely-similar to ours was discussed. However, some statements
and proofs in [3] seem not to be clear for the present author; for example, it looks
rather hard to make hold the following estimate (Lemma 9 in [3]):$T$$(1.9)^{1}$$\int_{0}^{t}\int_{0}^{M}\theta v_{x}^{2}dxd\tau\leq C$

for any $t>0$(1.13)

with some constant $C>0$ independent of time (which plays an important roll for the
rest of estimates in that paper). The alternative to (1.13) is found in [21]. Although,wasrecently the system of equations (1.8), (1.9) with $\mu=0,$ $\zeta=\zeta(v),$investigated by Ducomet and Ne\v{c}asov\’a in [4], the asymptotic behaviour of the solution
was not discussed in [4]. We also note that although the situations discussed in [5, 6]
seem rather general, the pure free-boundary case, i. e., imposing Neumann boundary
condition for the temperature on both two edges of the medium was not included.
$\kappa=\kappa(v, \theta)$

2Statements of theorems

Theorem 1 guarantees that the classical solution exists uniquely up to an arbitrary(large) fixed time . However, since the estimates of the solution (1.12) is dependent
on time , those are not enough to investigate large-time behaviour of the solution.
On the other hand, from physical point of view it is expected that the solution of
$T$

$T$

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our problem $(1.8)-(1.10)$ converges, in some sense, to the corresponding stationary
solution as time tends to infinity. In the paper [25], for a spherically symmetric flowof a barotropic viscous fluid the stationary problem was discussed, and then solved
uniquely under a certain restricted condition on several physical parameters. In thepresent paper we obtain the condition similar to the one in [25] guaranteeing uniquesolvability of our stationary problem (see Proposition 1).
Before stating our result specifically, we briefly mention a theorem, obtained in [23],on the global behaviour of a one-dimensional gas flow having similar situation with ours.
In this case, the equation of motion is given by in$:=(0,1)$ ,with$\Omega_{1}\cross(0, \infty)$$\Omega_{1}$$u_{t}=(-R \frac{\theta}{v}+\mu\frac{u_{x}}{v})_{x}-G(x-\frac{\int_{0}^{M}sv(s,t)d_{\mathcal{S}}}{\int_{0}^{M}v(s,t)ds})$

in the Lagrangian-mass coordinate. Let$g(a):=[ \frac{5}{12}-a^{2}+\frac{(a^{2}-\frac{1}{4})^{2}}{a}\log(1+\frac{1}{a-\frac{1}{2}})]\frac{1}{a}\log(1+\frac{1}{a-\frac{1}{2}})$,$h(b):=\sqrt{2b+\frac{1}{4}}.$

The following condition confirms that the classical solution does not blow up at least
under a certain condition. Namely, if the condition(2.1)$( \frac{R}{c_{v}})^{2}(goh)(\frac{p_{e}}{G})<1$

is satisfied, then there exists a positive constant $C$ independent of such that theinequality (1.12) holds for the solution of the-one-dimensional problem. The functions$g(a)$ and $h(b)$ are derived from a calculation of the integrals containing $f(x)$ $:=p_{e}+$$\frac{G}{2}x(1-x)$ :$T$$\int_{0}^{1}\frac{x^{2}(1-x^{2})}{f(x)}dx=\frac{2}{G}[\frac{5}{12}-a^{2}+\frac{(a^{2}-\frac{1}{4})^{2}}{a}\log(1+\frac{1}{a-\frac{1}{2}})],$$\int_{0}^{1}\frac{1}{f(x)}dx=\frac{2}{G}[\frac{1}{a}\log(1+\frac{1}{a-\frac{1}{2}})]$

with$a=\sqrt{2\frac{p_{e}}{G}+\frac{1}{4}}.$For details, see [23]. We also note that the function $g(a)$ is monotone decreasing
and convex downward with respect towith $\lim_{aarrow\frac{1}{2}+0}g(a)=+\infty$ and$\lim_{aarrow+\infty}g(a)=0$ .
In the case that the condition (2.1) is satisfied, we proceed tofurther investigation concerning the temporally asymptotic behaviour of the flow. In
fact, the solution $(v, u- \int_{0}^{1}udx, \theta)$ of the one-dimensional problem converges to asteady statein the sense ofas $tarrow+\infty$ . Moreover, there exist
constants and independent of such that for any $t>0$ the inequality$a \in(\frac{1}{2}, +\infty)$$(\tilde{v}, 0,\overline{\theta})$

$C$

$c$

$H^{1}(\Omega_{1})\cap C(\overline{\Omega}_{1})$$t$$\Vert(v-\tilde{v}, u-\int_{0}^{1}udx, \theta-\overline{\theta})(t)\Vert\leq Ce^{-ct}$

110
holds. Here$\tilde{v}=\tilde{v}(x):=\frac{R\overline{\theta}}{f(x)}, \overline{\theta}:=\frac{E_{0}}{c_{\eta}+R},$$E_{0}:= \int_{0}^{1}[\frac{1}{2}(u_{0}-\int_{0}^{1}u_{0}dx)^{2}+c_{v}\theta_{0}(x)+f(x)v_{0}(x)]dx.$

Let us get back to our problem describing motion of a three-dimensional sphericallysymmetric gas flow. Our stationary problem is: To find the solution $(V, U, \Theta)=$$(V, U, \Theta)(x)$ of the boundary value problem$\{\begin{array}{ll}(\overline{r}[V]^{2}U)’=0 in \Omega,(R\frac{\Theta}{V})’=-G\frac{M_{c}+x}{\overline{r}[V]^{4}} in \Omega,-4\mu(\overline{r}[V]U^{2})’+(\frac{\overline{r}[V]^{4}\kappa}{V}\Theta’)’=0 in \Omega,U(0)=\Theta’(0)=\Theta’(M)=0, (R\frac{\Theta}{V}+4\mu\frac{U}{\overline{r}[V]})(M)=p_{e}.

We easily deduce that, if the solution exists, its form is limited toan any positive constant . Heremeans the solution
parameter , of the following boundary value problem$\overline{\Theta}$$V(x;\overline{\Theta})$\end{array}$$(V(x;\overline{\Theta}), 0,\overline{\Theta})$

with

$V$

, for each (given)

.

(2.2)

$\overline{\Theta}$$( \frac{R\overline{\Theta}}{V})’=-G\frac{M_{c}+x}{\overline{r}[V]^{4}}$$in$$\Omega,$$\frac{R\overline{\Theta}}{V}x=M=p_{e}$Concerning the existence and uniqueness of the problem (2.2), we obtain the following
result.
Proposition 1 Assume that the condition(2.3)$p_{e}- \frac{GM(M_{c}+M/2)}{R_{c^{7}}}\cdot\frac{4RM\overline{\Theta}}{p_{e}}>0$

is satisfied. Then there exists a unique solution
problem (2.2).

For proof of Proposition 1, see [20].
Here let us consider a function$\tilde{v}=\tilde{v}(x)$$V\in C^{1}(\overline{\Omega})$of the boundary value

and a positive constant$\{\begin{array}{l}\frac{R\overline{\theta}}{\tilde{v}}=p_{e}+\int_{x}^{M}G\frac{M_{c}+s}{\overline{r}[\tilde{v}]^{4}}ds for x\in\overline{\Omega},\int_{0}^{M}(c_{v}\overline{\theta}+p_{e}\tilde{v}-G\frac{M_{c}+x}{\overline{r}[\tilde{v}]})dx=E_{0}\end{array}$$\overline{\theta}$

satisfying

(2.4)

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with
$E_{0}:= \int_{0}^{M}(\frac{1}{2}u_{0^{2}}+c_{v}\theta_{0}+p_{e}v_{0}-G\frac{M_{c}+x}{\overline{r}[v_{0}]})dx.$

Our main result isand the initial data
Theorem 2 Let be an arbitrary positive number, andandsatisfy the hypotheses of Theorem 1. Assume that there exist a function
a positive constant satisfying the equalities (2.4), and that the condition$T$$\alpha,$$\mu,$$\zeta$$\tilde{v}\in C^{1}(\overline{\Omega})$$\overline{\theta}$(2.5)$p_{e}- \frac{R}{c_{v}}\cdot\frac{GM(M_{c}+M/2)}{R_{c^{4}}}(1+\frac{4C_{0}}{R_{c}^{3}})>0$

is satisfied with
$C_{0}:= \frac{1}{p_{e}}[E_{0}+\frac{GM(M_{c}+M/2)}{R_{c}}]$Then there exists a positive constant independent of such that the inequality (1.12)
holds for the solutionof the initial-boundary value problem $(1.8)-(1.10)$ . Moreover, the solution
as $tarrow+\infty$ in the sense of
converges to the state
.
$C$

$T$

$(v, u, \theta)$$(v, u, \theta)$$(\tilde{v}, 0,\overline{\theta})$$H^{1}(\Omega)\cap C(\overline{\Omega})$

Remark 2.1 It is easily seen that if one can find (together with ) satisfying (2.4),
it also be a special solutionof the stationary problem (2.2). On the other hand,
we see that the stationary solution of (2.2), whose unique existence is guranteed by
Proposition 1, satisfies$\overline{\theta}$$\tilde{v}$$V(x;\overline{\theta})$(2.6)$MR \overline{\Theta}=\int_{0}^{M}(p_{e}V+G\frac{M_{c}+x}{\overline{r}^{4}}\int_{0}^{x}Vds)dx$

with$\overline{r}$which denotes$\overline{r}[V]$. Here let us assume that the solution$V(x;\overline{\Theta})$also satisfies(2.7)$Mc_{v} \overline{\Theta}+\int_{0}^{M}(p_{e}V-G\frac{M_{c}+x}{\overline{r}})dx=K_{0}$for an any constantthe equation$K_{0}$. Combining (2.6) with (2.7) and putting$g(\overline{\Theta}, \lambda)=0$

on$\{(\overline{\Theta}, \lambda)\in \mathbb{R}^{2}|\Theta>0, \lambda\geq 0\}$$\lambda$$:= \frac{1}{R_{c}}$, we consider, where$g(\overline{\Theta}, \lambda):=M(c_{\gamma}+R)\overline{\Theta}$

一

with

$\hat{r}$$\{K_{0}+\int_{0}^{M}(\lambda G\frac{M_{c}+x}{\hat{r}}+\lambda^{4}G\frac{M_{c}+x}{\hat{r}^{4}}\int_{0}^{x}Vds)dx\}$$:=(1+3 \lambda^{3}\int_{0}^{x}V(s;\overline{\Theta})ds)^{1/3}$

By noting that for$\overline{\Theta}_{0}:=\frac{1}{M}\frac{K_{0}}{c_{v}+R}$$g( \overline{\Theta}_{0},0)=0, \frac{\partial g}{\partial\overline{\Theta}}(\overline{\Theta}_{0},0)=M(c_{v}+R)>0,$

there exists a constantnear
for suitably small $\lambda>0$ . Namely, we can say
is sufficiently large, fi andthat in the situation both that (2.3) is satisfied and that
(2.4)
satisfying
are both determined.
$\overline{\Theta}>0$$\overline{\Theta}_{0}$$R_{c}$$\overline{\theta}$

112
Remark 2.2 Since an any constant$\overline{\Theta}$

satisfying (2.7) has the estimate$M \overline{\Theta}\leq\frac{1}{c_{v}}[K_{0}+\frac{GM(M_{c}+M/2)}{R_{c}}],$

the inequality (2.3) is holded as far as we are in the situation of (2.5).

Proof of Theorem 1 was achieved by continuation of the temporally local solution
with the help of suitable a priori estimates of the solution. We note that the fundamental theorem about local solvability of problems around ours had been alreadyestablished, for example, in $[15-17,19]$ , which are applicable to our problem without
essential modifications. Concerning a priori estimates of the solution, we have (1.12).
To prove Theorem 2 we first establish the following proposition. For the norm$\Vert u\Vert_{E(Q_{T})}:=\Vert u\Vert_{E_{1}(Q_{T})}+\Vert u_{x}\Vert_{E_{1}(Q_{T})}+\Vert u_{xx}\Vert_{E_{1}(Q_{T})}+\Vert u_{t}\Vert_{E_{1}(Q_{T})},$$\Vert u\Vert_{E_{1}(Q_{T})}:=(\sup_{t\in[0,T]}\Vert u(t)\Vert^{2}+\int_{0}^{T}\Vert u_{x}(t)\Vert^{2}dt)^{\frac{1}{2}}$we obtain
Proposition 2 For generalized derivatives$(v_{xt}, v_{tt}, u_{xxx}, u_{xt}, \theta_{xxx}, \theta_{xt})\in(L^{2}(0, T;L^{2}(\Omega)))^{6}$of the solutionof such that$(v, u, \theta)$of the problem $(1.8)-(1.10)$ , there exists a constant$C$

independent$T$$\{\begin{array}{l}\sup\Vert(v, v_{x}, v_{t}, v_{xt})(t)\Vert\leq C,t\in[0,T]\Vert u, \theta-\overline{\theta}\Vert_{E(Q\tau)}+\int_{0}^{T}\Vert(v-\tilde{v},v_{x}-\tilde{v}_{x}, v_{tt})(t)\Vert^{2}dt\leq C,C^{-1}\leq v, \theta\leq C, |u|\leq C in \overline{Q_{T}}.\end{array}$Moreover,$\theta_{x}|(t)+\Vert(v-\tilde{v})(t)\Vert_{H^{1}(\Omega)\cap C(\overline{\Omega})}x\in$$su_{\frac{p}{\Omega}}|u_{x},$$+\Vert(u, \theta-\overline{\theta})(t)\Vert_{H^{2}(\Omega)\cap C(\overline{\Omega})}+\Vert(u_{t}, \theta_{t})(t)\Vertarrow 0$

as $tarrow+\infty.$

For $pro$ of of Proposition 2, see [21]. By virtue of Proposition 2 we obtain the H\"older
estimates of the solution 1.12 uniformly in time.
$($$)^{}$

113

Notations3

We introduce some function spaces. Let
be a domain, $m$ a non-negative integer,$0<\beta<1$ , and $0<\sigma\leq 1$ . We denote by
the standard -Sobolev space oforder $m$ equipped with the norm
. For the case $m=0$ , we simply denote
$u(x,
t)$
. For
, an
-valued function ofmeans
The space of continuous and bounded functions defined on is denoted by. We
use
as the space of functions $u=u(x)$ defined on which have boundedderivatives up to order $m$ on , and whose m-th order derivatives are uniformly H\"older
continuous in with exponent . Its norm is defined by$\mathfrak{S}\subset \mathbb{R}$$L^{2}$$H^{m}(\mathfrak{S})$$\Vert_{H^{m}(\mathfrak{S})}$$\Vert$$\Vert$$t,$$H^{m}(\mathfrak{S})$$\Vert$$\Vert u(t)\Vert_{H^{m}(\mathfrak{S})}$$\Vert u(\cdot, t)\Vert_{H^{m}(\mathfrak{S})}.$$\overline{\mathfrak{S}}$$C(\overline{\mathfrak{S}})$$\overline{\mathfrak{S}}$$C^{m+\beta}(\overline{\mathfrak{S}})$$\overline{\mathfrak{S}}$$\overline{\mathfrak{S}}$$\beta$,$|u|_{\mathfrak{S}}^{(m+\beta)};= \sum_{k=0}^{m}sux\in^{\frac{p}{\mathfrak{S}}}|D^{k}u(x)|+\sup_{x’\neq x’}\frac{|D^{m}u(x")-D^{m}u(x’)|}{|x"-x|^{\beta}}x,,x’\in\overline{\mathfrak{S}}$

Let

$T$

with

$D= \frac{d}{dx}.$

be a positive number. For a function $u=u(x, t)$ defined on QT we say that
if is continuous inand has a finite norm

$u\in C_{x,t}^{\beta,\frac{\beta}{2}}(\overline{Q_{T}})$$\overline{Q_{T}}$$u$$|u|_{Q_{T}}^{(\beta)}:=|u|_{Q_{T}}^{(0)}+[u]_{x,Q_{T}}^{(\beta)}+[u]_{t,Q_{T}}^{(_{2}^{g})},$

where$|u|_{Q_{T}}^{(0)}:=su|u(x, t)|(x,t)\in^{\frac{p}{Q_{T}}},$$[u]_{x,Q_{T}}^{(\sigma)}:= \sup_{x\neq x’},\frac{|u(x",t)-u(x’,t)|}{|x’-x’|^{\sigma}}(x’,t),(x’,t)\in\overline{Q_{T}},,$$[u]_{t,Q_{T}}^{(\sigma)}:= \sup_{t\neq t’}\frac{|u(x,t")-u(x,t’)|}{|t"-t’|^{\sigma}}(x,t’),(x,t")\in\overline{Q_{T}}.$

We also say thatand has a finite norm

$u\in C_{x,t}^{2+\beta,1+\frac{\beta}{2}}(\overline{Q_{T}})$

if

$u$

is continuous in$\overline{Q_{T}}$together with

$u_{x},$$u_{xx},$$u_{t},$$|u|_{Q_{T}}^{(2+\beta)}:=|u|_{Q_{T}}^{(0)}+|u_{x}|_{Q_{T}}^{(0)}+[u_{x}]_{t}^{(\frac{1+\beta}{Q_{T}2},)}+|u_{xx}|_{Q_{T}}^{(\beta)}+|u_{t}|_{Q_{T}}^{(\beta)}.$Remark 3.1 For the specific definitions ofof functions defined on a noncylindrical domain, the spaces
and, see [18], Chapter VI.

$C_{x,t}^{\beta,\frac{\beta}{2}}(\overline{\mathfrak{D}_{T}})$$\overline{\mathfrak{D}_{T}}$$C_{x,t}^{2+\beta,1+\frac{\beta}{2}}(\overline{\mathfrak{D}_{T}})$

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York, 1957.

[2] Ducomet, B., Hydrodynamical models of gaseous stars, Rev. Math. Phys., 8(1996), 957-1000.

114[3] Ducomet, B., Some asymptotics for a reactive Navier-Stokes-Posson system, Math.
Mod. Meth. Appl. Sci., 9 (1999), 1039-1076.
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