123

数理解析研究所講究録
第 734 巻 1990 年 123-142

The initial value problem for the equations of motion ofgeneral fluids with general slip boundary condition

Atusi TANI

温之)

(谷

Department of Mathematics, Keio University\S 1.

Introduction and Main Theorem

In this $com[!unication$ we are concerned with the initial-boundary
value problem for compressible viscous isotropic Newtonian fluids (say,general fluids) which happen to slip on the solid boundary.

The motion of general fluids filled in a bounded domain$\Omega\subset R^{3}$is governed by the so-called compressible $Navier-Stokes$ equations:$\frac{Dp}{Dt}=-\rho$$($$\nabla\cdot v$,(1)$\ovalbox{\tt\small REJECT}_{\rho\theta\frac{DS}{Dt}}---\rho\frac{LDv}{Dt}\nabla\cdot(\kappa=_{\nabla\theta^{\nabla\cdot P})+}+,\rho f\mu(\nabla\cdot v)^{2}+2_{l}\iota D(v):D(v)x\in\Omega,t\rangle 0,$.
Here$p=\rho$( $x$ , t)is the density,

the velocity vector field,$\theta=\theta$$v=v(x , t)=(v_{1} , v_{2} , v_{3})$( $x$ , t)is the absolute temperature,$f=f(x, \iota")$

is a vector field of external forces, $P=(-p+$$+l\ell’\nabla\cdot v)I_{3}+2$

SC $D(v)$is the stress tensor,$D(v)$

is the velocity

deformation tensor with the elements$D_{ij}=- \frac{1}{2}$$( \frac{\partial v_{i}}{\partial x_{j}}+-\frac{\partial v_{j}}{\partial x_{i}})$,$D(v);D(v)=D_{Jk}D_{jk}$ ,

–1–

is

124
is a pressure,$p=p$ $(\rho , 6!|)$

is an entropy,$c\backslash =S(\rho \theta)$‘$\chi\ell$

$,$

$/l$

,

$\kappa$are, respectively, coefficient of viscosity, second coefficient of vis-

cosity, coefficient of heat conductivity, which are all assumed to beconstants satisfyingand$I_{3}$$\mu\rangle$$0,2_{l’}+3_{l\ell}’\geq 0$,$\kappa\rangle$$0,$$D/Dt=\partial/\partial t+v\cdot\nabla$

is an identity matrix of degree 3.

Here and in what follows we use the well-known notation of vector

And we should refer toanalysis and the summation convention.

$[7,8]$

for the notation not stated here explicitely.
We have already studied the initial-boundary value problem for(1)

in the perfect slip case

re

and

$K\equiv 0$$e^{---\iota}$orin [10].

$\kappa_{e}\langle 1$

Here we consider the general slip boundary condition, which isformuiated as follows:$u\cdot n=0$,,$\tau=KPn\cdot\tau$1)or equivalently,
(2)

$v$

where

$n$

and

$\tau$$n=0$ ,

$v=K_{L}^{\overline{!}}Pn-(Pr\iota\cdot n)_{n}$are a unit inward normal and a unit tangential

vector, respectively, such that

positive function defined on

and$nX\tau=1$$\Gamma_{\Gamma-}^{-}f\cdot x[0, T ]$’1’ is any, but fixed, positive number).

1 $f/1K$],

and using the same letter($K$$l^{\urcorner}$

is a boundary of$1/(1+l\ell K)$, wededuce from (2)$v\cdot n^{--}O$,$\frac{l}{g_{l}}(\backslash 1-K)^{!}||Pn-(Pn\cdot n)_{n}\rceil-Kv=0$ ,Similarly, the boundary condition for$-\kappa\nabla\theta\cdot n^{--}\kappa_{p}(\theta_{e}-\theta)+g$

implies that, using the same letters$\kappa_{e}/(\kappa+\kappa_{e})$and$g/(\kappa+\kappa_{e})$,$\kappa_{e}$$l\rangle$]$K\geqq 0$.

$\theta$$\kappa_{e}\geqq 0$

and, respectively,–2–$\Omega$,

Dividing both sides of (2) byin place of$K$

is assumed to be a$g$in place of125$(1-\kappa_{e})\nabla\theta\cdot n-\kappa_{e}(\theta-\theta_{e})=g$,$1\rangle$$\kappa_{e}\geqq 0$.

The aim of this paper is to establish the unique solvability,

local in time, of the initial-boundary value probiem (1) with theinitial condition(3)$(p, p;, \theta)|_{t=0}=(\rho_{0}, v_{0}, \theta_{0})(x)$ ,and the boundary conditions(4)$\{\begin{array}{l}v\cdot n=0,\frac{1}{;l}(l-I\{)[Pn-(Pn\cdot n)_{n}]-Kv=Q(1-\kappa_{e})\nabla\theta\cdot\tau\iota-\kappa_{e}(\theta-\theta_{e})--g\end{array}$

where$(K, \kappa_{\rho})=(K , \kappa_{e})(x, t)$,$1\geq K,$$\kappa_{e}\geq 0$.

The following is our main theorem:

Theorem.

Let$\tau^{\tau}$bounded doma $ininR^{3}$be an arbitrary positive number andwi th boundary$\Gamma$of class$C^{2+\alpha}$

,

be a

$\Omega$$\alpha F_{-}^{-}(0,1)$.

Furthermore, we assume that
(i)

,

,

$\rho_{0}’\leqq\rho_{0}(x)-\leq-\rho_{0}’’$$(p_{0}, \iota_{0}’, \theta_{0})f_{-}- C^{1+\alpha}(\overline{\overline{\Omega}})\backslash xC^{2+\alpha}(\overline{\Omega})xC^{z+a}(\overline{\Omega})$$\theta_{0}’\leqq\theta_{0}(x)\leqq\theta_{0}’’$($\rho_{0}’,$$p_{0}$“,$\theta_{0}’$ard$\theta_{o}’’$are posit ive constants);$(||)(K, \kappa_{e})=(K, \kappa_{e})(x, t)t_{-}^{-}- C_{xt}^{1+a}(1+a)/2(I_{?}^{Y})$,$0\leqq K,$$(||i)(\theta_{e}, g)=(\theta_{e}, g)(x, t)\in c_{\chi}^{1};_{t^{a,}}(1+a)f2(\Gamma_{T})$,moreover,$(\theta_{e-}, g)\in c_{\chi}^{2};_{t^{a,1+aj2}}(l_{\tau^{\backslash }}\urcorner’)$

$\mu$

,

$\geqq 0$

’

$ll$

,

and

$1l\rangle$ $0$,$\kappa$$\kappa\rangle$$0$;$I_{T}^{\tau\prime}=\cup\{x\in\Gamma|\kappa_{e}(x, t)=1;0\leqq t\leq T\}$$(i_{Y})f=f(x, t)\in C_{x,tT^{-=}}^{a,a/2}(\overline{Q}--\overline{\Omega}x[0,7’])$(Y)$\kappa_{e}\leqq 1$;;

are constants satisfying the relations 2 $\mu+3\mu’\geqq$
and$(p, S)=(p, S)(\rho, \theta)\in$$\llcorner^{-}-C^{2+a}(!-\beta\rho_{0}’, \mathcal{B}^{-1}0_{0|X_{\llcorner}^{i}\beta\theta_{0}’}^{\prime\prime^{1}}, \beta^{-1}\theta_{0}’’j)$–3–

for some posit ive

126

constant$(|’|)$$i$$\beta$\langlesuch that$1$$S_{\theta}(---\partial S/\partial\theta)\rangle$$0$;

the compatibility conditions between the system (1) and the in-tial and the boundary conditions (3) , (4) are valid.
Then there exists a unique solution

wh ch belongs to$i$$(\rho, v 0)$of (1) , (3), (4) ,$\lfloor B^{1+a}(\overline{Q}_{\Gamma}, );)\{\beta\rho_{0}’\leqq\rho(x, t)\leq\beta^{-1}\rho_{0}’’\}$]$x$$\gamma_{\iota}c_{X,}^{\ell};_{t^{\alpha,1+a/\angle}}’(\overline{Q}_{I’}.)x_{1_{-}^{-}}c_{\chi}^{1};_{t^{a,1+a/2}}(\overline{Q}_{T’})r|\{\beta\theta_{0}’\leqq\theta(\chi, t)\leqq\beta^{-1}\theta_{0}’’\}_{\lrcorner}^{\urcorner}$for some\S 2.

$T’\in(0, T$ ].

Outline of the Proof of Theorem.rst of all, we introduce the characteristic transformation

ff,$F^{\cdot}1’$$x-\xi^{-}--X(0;x, t)$ ,

wh re
$e$

$X$

(

;

$\tau$, t)$x$$(0-\leq\tau\leqq t , x\in\overline{\Omega})$$;t$

is the solutionof the system of equations(5)

If$\frac{\prime l}{dc}X(\tau ; x, t)=v(X (\tau ; x, t),$$v$$\tau$),$X(t ; x, t)=x$ .

is suitably smooth, then (5) has a unique solution curve byvirtue of the basic theorem of ordinary differential equations.
gives us the relation between
(6)

and

$x$

:

$x-X(t ; \xi, 0)=\xi+I_{0}’\prime u(i_{X}^{\zeta}’\tau)d\tau--- X_{u}(\xi, t)$

where$u(\xi, t)=\iota)(X(t ; \zeta\vee , 0),$$t$

is an $one-to$ -one mapping from,).

According to the boundary condition$\Pi^{x_{\xi}}$

$\xi$

It

$\overline{\overline{Q}}_{J}’$$v\cdot n=0$

. ontoon$\overline{Q}_{T}$$1_{T}^{\urcorner}$, it is clear that.

In a similar way to that in [10] , we use this transformatioin

only for the first equation in (1), whence the unique solution of (1)is given by

–4–

127

$(7\rangle \rho(\chi, t)=\Pi_{x}’\cdot\rho_{0}(\xi)\exp^{\ulcorner}\lfloor-\downarrow_{0}^{t}\urcorner\nabla_{l4}\cdot u(\xi, \tau)d\tau_{1}^{-|}\urcorner$

provi ded thatis the inverse rnano $ing$ of$h^{\tau_{p}}.re\Pi_{x}^{\xi}$–t

is given.

$u\in C_{x,t}^{2+a.1+a/2}(\overline{Q}_{T})$$(\partial X_{4}/\partial\xi)^{-- 1}$,$\Gamma f_{\epsilon}^{x}$$\nabla_{\epsilon}=(\frac{\partial}{\partial\xi 1}, \frac{\partial}{\partial\xi_{2}’} \frac{\partial}{\partial\zeta_{*}\backslash \prime})$,$\nabla_{u}-G\nabla_{F}.’G=(g_{jk})=$.

Hence the problem (1), (3) , (4) can be reduced to the foli owing initialboundary value problems with respect to$\sigma=\theta-\theta_{0}$$w=v-v_{0}$and with respect to:$Q_{7}\cdot$

(8)

(9)

$\{\begin{array}{l}\frac{\partial u)}{\partial t}=A(x,t,w,\nabla)w+\Phi(x,t,u\prime.\sigma)u,|_{\ell=0}-- 0on\Omega B(x,t.\nabla)\iota v^{-}--B(x,t.\nabla)v_{0}on\end{array}$$\Gamma_{T}in$$\{\begin{array}{l}\frac{\partial\sigma}{\partial t}=\Lambda’(x,t,w,\sigma)\Delta\sigma+\Psi(x,t,w,\sigma)inQ_{T}\sigma|_{t=0}=0on\Omega(1-\kappa_{e})\nabla\sigma\cdot p\iota-\kappa_{e}\sigma^{-}- g-\kappa_{e}\theta_{e}-(1-\kappa_{e})\nabla\theta_{o7}\iota+\kappa_{e-}\theta_{\zeta t}on \Gamma_{T}\end{array}$where the principal parts$\Psi$,, the boundary operator

$A$

and

$A’$

, the iower order terms$B=(B_{Jk})_{j,k=1,2,3}$$A=( \frac{\chi\iota+\mu’}{p}\nabla_{j}\nabla_{k}.-\}-\frac{ll}{\rho}\nabla_{p}^{2})_{J\cdot=1}k$andare given by the formulae:$A’= \frac{\kappa}{\rho\theta S_{\theta}}$,$\Phi=-\div\nabla p+f-(v\cdot\nabla)_{tJ}-AUo$–5–$\Phi$,

128$\Psi=\frac{1}{p\theta S_{\theta}}\lfloor_{-}^{-}u’(\nabla\cdot v)^{2}+2\mu D(v);D(v)+p^{2}\theta S_{\rho}\nabla\cdot v]-(v \nabla)\theta-$with$\frac{\kappa}{\rho\theta S_{\theta}}\Delta\theta_{0}$$(w+\{;_{0}\sigma+\theta_{0})$$\rho$

and$(v, \theta)$

replaced by (7) and, respect ively,$B_{Jk}=\{\begin{array}{l}n_{k}(j=1_{\prime}h--- 1,2,3)(1-K)(n_{k}\delta_{j- 1}\iota\star- n\iota\delta_{j\cdots 1}k^{-2n_{J- 1}}n_{k}n\iota)\nabla\iota-K\delta_{j- 1}k(j--2,3,k^{--}1,2,3)\end{array}$($\delta_{jk}$

$is$

2. 1

Kronecker‘$s$delta).

Linearized problem of (8) and (9)First of all, we consider the following linearized problem of (8):in

$Q_{T}$,$\{\begin{array}{l}\frac{\partial w}{\partial t}--A(x,t,w’,\nabla)w+\Phi(x,t,w,\sigma’)w|_{t=0}--0on\Omega B(x,t.\nabla)w^{---}B(x,t.\nabla)v_{0}on\Gamma_{T}\end{array}$

(10)

Here

$(w’ , \sigma’)$is a given function beionging to the class,$\mathscr{J}_{\tau-}^{-}\{(w, \sigma)\in c_{\chi}^{2}:_{t^{a,1+a/2}}(\overline{Q}_{T})|(w, \sigma)|_{t=0}=0$$\sum_{|s|=2}|D_{A}^{s}(w, \sigma)|x^{(\alpha)}\overline{Q}T\langle M_{2}\}$

where$\det$$M_{1}$

is a positive number,$M_{z}$$||u|| \equiv\sum_{2r+|s|=0}\overline{Q^{(m)}}_{T}m\Vert D_{t}^{r}D_{\chi}^{s}u\Vert_{\frac{(}{Q}}^{0}r^{)}$

–6–

,

,

is an arbitrary positive number,ermi ned later,$\Vert(w, \sigma)\Vert_{\frac{(}{Q}\langle}^{2)}TM_{1}$,$||u||_{\overline{Q^{(0)}}_{T}}\equiv$

129$\equiv$-

strp $\{|u(x, t)| ; (x, t)\in\overline{Q}_{T}\}$

$u$

and

$|u|_{x,\overline{Q}_{T}}^{(\alpha)} \equiv\sup\{|u(x, t)-$$(x’ , t)||x-x’|^{-a}$ ; $(x, t),$ $(x’, t)\in\overline{Q}_{T},$ $x\neq x’$ }.

Then the following fact holds.

The system of differential equations (10) is uniformly

Lemma 1.

parabolic in the sense of Petrowsky with modulo of parabolicity

if we take$T$

$M_{1}T\langle\theta_{0}’,$in such a way that$0\langle M_{3^{-}}^{-}--(M_{1}+\Vert v_{0}\Vert^{\{1)})T/[1-(M_{1}-\vdash\Vert v_{0}i|^{(1)})T]\langle M_{0},$$\overline{\Omega}$

where

Proof.

$M_{3}T\langle 1$,$\overline{\Omega}$

is a positive root of the equation$M_{0}$$\delta$$1-3x-6x^{2}-6x^{3}=0$

.

Since

$\det LA$ $(x, t, w’ ; i \xi)-\lambda I_{3}\rfloor=(\lambda+\frac{|\xi|^{2}}{a_{1}})^{2}(\lambda+\frac{|\xi|^{2}}{a_{3}})$$a_{3}=\rho/(2_{ll}-\vdash/\iota’))$,$(a_{1}=a_{2}=p/ll$ ,and the estimates(11)$\{\begin{array}{l}||u’||_{\overline{Q^{(0)}}}\leq M_{1}T\Sigma|S|=1||D_{\chi}^{s}u’||_{\frac{t}{Q}\underline{\leq}\Lambda M_{3}}^{0}T\Sigma|S|=2||D_{x}^{s}u’||^{0)}\underline{\backslash }T\backslash |g_{jk}-\delta_{jk}|\leqq--\frac{M_{3}-\vdash 4M_{3}+6M_{3}^{3}}{1-3M_{3}-6M_{3}^{2}-6\Lambda I_{3}^{3}}=- C_{1}(M_{1},T)\end{array}$$(j, k^{--}1,2,3)$

follow from (6) for
take$\delta$$u’=\Pi_{\epsilon}^{x}w’$,$(w’, \sigma’)\in \mathscr{J}_{T}$,it is sufficient toin such a way that$\delta=\mu p_{0}^{2-1}\exp[-3(1-3C_{1})TM_{3\overline{\lrcorner^{\}}}}$,

by virtue of (7).

$\square$

–7–

130

The following compl ement ng condi ion in the case of
$t$

$i$

There exists a positive constantsuch that for any,${\rm Re}\nu\underline{\geq_{-}}-\delta’\xi^{\prime 2}$

the row vectors of the matri

$B$

$x$

adi ugate matri

$x$

of

$A$

smaller than$\delta’$$\delta$satisfying$\nu\in C^{1}$$\xi’4+|\nu|^{2}\rangle$ $0$,$(x, t ; i\xi)\hat{A}(x, t , w’ ; i\xi, \nu)$fi xed) are linearly,$M= \prod_{r=1}3(\zeta_{3}-\xi_{3}^{+(r)}(\xi’, \nu))$roots inand$\xi’=(\xi_{1}, \xi_{2})\in R^{2}$(12)$((x, t)\in l_{T}^{\urcorner}$,is essential throughout our investigation.

$\Gamma=\{x\in R^{3}|x_{3}=0\}$

Lemma 2.

$\Omega=R_{+}^{3}$$i$, wherendependent modulo

is an$\hat{A}(x, t , w’ ; i\xi, \nu)$$(x, t, w’ ; i\xi)-\nu I_{3}$and$\xi_{3}$of$\det[A (x, t, w’ ; i\xi)-\nu I_{3}]=0$.

Let$\sum_{S=1}\alpha\xi_{3}3(S)S-1$’$\xi_{3}^{+tr)}$$s$

are thewi th positive imagin-ary parts.

$\underline{Proof}$$B$be the remainder term when we divide$(x, t ;i\xi)\hat{A}(x, t, w’ ;i\xi, \nu)$

by

.

$M$

Then after some lengthy

calculations we have
(13)

$X$

$\det oe^{t3)}=-a_{1}^{-3}a_{3}^{-3}J_{11}^{2}J_{31}(\xi_{3}^{+(1)}+\xi_{3}^{+(3)})(a\xi_{3}^{+(1)}+\xi_{3}^{+(3)})x$[( -K)$1.$where$i\xi_{3}^{+(1)}$-K\rceil 「$a=a_{3}/a_{1}=n/(2_{l}z+\mu’)$ ,Since$Im\xi_{3}^{+(r)}\rangle$$itis$ obvious that

Moreover,to {$f_{pq}=\xi_{3}^{+t\Phi)}-\xi_{3}^{-t4)}$$0(r=1,2,3)$$|\det\alpha^{(3)}|\rangle$ $0$.

follows from the assumption (12),.

$\square$we can extend the domain of def nit ion (12) of$(\zeta’\equiv\xi’+i\eta’, q)\in C^{2}XC^{1}|\xi^{\prime 4}+|q|^{2}\rangle$ $0,$$|\eta’|\leqq\beta’’(\xi’4+|q|^{2})^{1/2}\}$$\det\alpha^{(3)}$$i$${\rm Re} q\geqq-\beta’|{\rm Im} q|$

for some posit ive constants–8–$\beta’$,and$\beta’’$,131

so that there(14)

is estimated from below$\det\alpha(3)$$|\det\alpha^{t3)}|(\zeta’, q-\delta’\zeta’)\geqq 2$$\geqq C_{2}(\xi’+|q4|^{2})^{\frac{3}{2}}[K+(1-K)(\xi’’4-\vdash|q|^{2})^{\frac{1}{2}}]^{2}$,hence we have the estimates of the inverse matrix $(\alpha_{3^{(j,k)}})_{jk=1,2,\hat{o}}$of$\alpha(3)$:(15) $|\alpha_{s^{(j,k)}-}|(\zeta’’, q-\delta’\zeta’)-\leq_{-}2$1(j–1,2,3;

$k=1$

),

$\leq C_{J}’(\xi’\vdash|q4|^{2})^{-\frac{1}{2}\dagger}$$($$[K+(l-K)(\xi’+|q4|^{2})^{\frac{1}{4}}]^{-1}(j=1,2,3; k=2,3)$

From these estimates, we can construct the Poisson kernel

and the Green matrix$H_{0}$

in the half space.

$H_{1}$$R_{+}^{3}:$$\hat{H}_{1}(y, \nu)=(2\pi i)^{-1}ri^{\hat{A}}7_{+}(x, t, w’ ; iy’, j\xi_{3}, \nu)\alpha_{3}(g’, \xi_{3}, \nu)x$$Xe^{iy_{3^{\xi}3}}/M(y’, \xi_{3}, \nu)d\xi_{3}$$\epsilon+i\infty$$H_{1}(g, \tau)=-i(2\pi)^{-3}$.

,$\Lambda$$R^{2}e^{i}(y’\epsilon_{d\xi’\uparrow_{1}e^{\tau v}H_{1}(\xi’}’)\prime c\epsilon-i\infty$$J_{3}r,$$\nu$) $d\nu$$(\epsilon\rangle-\delta’\xi’2)$$H_{0}(\iota J\tau ; \xi, \tau_{0})=Z_{0}(\uparrow j-\xi, \tau-\tau_{0} ; x, t ; w’, \sigma’)-$$-]_{\tau}^{\tau_{0}}d \tau’\int_{R}{}_{2}H_{1}(y-\eta’, \tau-\tau’)B(x,$$t$;$.\swarrow_{J}’(\eta-\xi 0 \tau’-\tau_{0} ; x, t ; w’, \sigma’)|\eta_{3}=0d\eta’$–9–$\nabla_{\eta}\grave{J}$

.

,

132

where

acontour enclos ng all$is$$7_{+}$$\xi_{3}^{+(r)}(r=1,2,3)$

$i$

and

is

$Z_{0}$

the fundamental solution of the system of equations$\frac{\partial W}{\partial t}=A$$(x, t, w’ ; \nabla_{y})W$.

Then, tracing the proof of Lemma 3. 14 in [7] , we obtain

Lemma 3$|D_{\tau}^{r}D_{y}^{s}(H_{1})_{jk}| \leqq C_{4}\tau^{-(2r+|S|+4)/2}\exp[- d\frac{|y|^{2}}{\tau}\rfloor\{\begin{array}{l}[K+(1-A^{r})\tau^{-\frac{1}{2}}|^{-1}(j--- 1,2,3,k--- 1)1(j=1,2,3,h^{--}2,3)\end{array}$$|D_{r}^{r}D_{y}^{s}H_{0}|_{-}\leq- C_{4}(\tau-\tau_{0})^{-\{2r+|S|+3)/2}\exp\lfloor-d\underline{|y-\xi|^{2}}]$.

$\tau-\tau_{0}$In the present problem just unlike the previous one [10] , it isnecessary to introduce two systems of covering

{

Let

{

of

$\Omega_{k}(t)\rangle$

depending on the time variable$\overline{\Omega}$

and$\{\Omega_{k}(t)\}$$(|)a)_{k}(t)\subset\Omega_{k}(t)L\overline{\Omega}$(ii)

for any

dist$x\overline{\epsilon}\overline{\Omega}$,$\lambda\rangle$

$0$

and

.
We construct

as follows (cf. [7]):$\bigcup_{k}\omega_{\text{鳶}}(t)=\bigcup_{k}\Omega_{k}(t)=\overline{\Omega}$, there exists$(x,\overline{\Omega}-\omega_{k}(t))\geq\beta J\lambda$(iii) for any$t$be an arbitrary small positive number.

$\lambda$$\omega_{k}(t)\rangle$$\{\omega_{k}(t)\}$$\omega_{k}(t)$for some, there exists a number;

such that$\beta_{1}\rangle$ $0$$N_{0}$

and$x\in\omega_{k}(t)$;independent of$\lambda$

such

that
$N_{0}+1$$k^{--1}(1_{-}\Omega_{k}(t)=\phi$$(\{r-1)$

if$\Omega_{k}(t)\cap\Gamma=\phi$$\omega_{k’}(t)$and;(in this case,$\Omega_{k’}(t)$we shall denotek—k), then

are the cubes with the same center andwith the length of their edges, in a parallel direction with axes,
equal to$\lambda/2$

and$\lambda$, respectively (indeed,-10–$\Omega_{k}\cdot(t)$

and

133
do not depend on$\omega_{k}\cdot(t)$$(i\uparrow-2)$

if$\omega_{k}(t)\cap\Gamma\neq\phi$t), then we constructand$\omega_{k}(t)$means of the 1 ocal rectangular coordinate system
origin at some pointatas the$\xi_{k}\in I\urcorner$$\underline{p}/3$

tangential plane atFor$\xi_{k}’-\in\Gamma-7(t)$$\xi_{k}\in I^{\urcorner}$,$i\cdot e$.

Letwith the$y_{1}-$,$g_{2}$$\Gamma$-axis in the

7 $(t)=\{x\in I^{\urcorner}|K(x , t)=\dagger\rangle$ .

(in this case, let us denoteby the local rectangular coordinate system(16)$\{y\}$

by, we take the inner normal to-axis and place the$\xi_{k}$$\Omega_{k}(t)$$k=k’$ ),we define$\{y\}$$\{\begin{array}{l}\omega_{k^{\wedge}}(t)- y_{j}|\leqq\frac{1}{2}\beta_{2}\lambda(j=1,2),0_{-}<\simy_{3}-F(\iota y’,\xi_{k}\sim)\leqq\beta_{2}\lambda\}\Omega_{k}’(t)--II_{x}^{y}\{|y_{j}|\leqq\beta_{2}\lambda(i--- 1,2),0=’\backslash _{-}y_{3}-\Gamma(g’\xi_{k}\cdot\prime)\leq 2\beta_{2}\lambda\}\end{array}$where the equation

boundary$\Gamma$$/\ell 3=F$ $(y’ ; \xi_{k}^{\sim})$in the neighborhood of the point

positi ve constant independent of$\bigcup_{k}\sim(\omega_{k}\sim(t)\cap\Gamma)$$\{\omega_{k}.(t)\}$and$\lambda$.

andconstant$\beta_{3}(\leqq\beta_{z})$$\Omega_{k}^{\sim}(t)$$\beta_{2}$is ais covered by$\overline{\Omega}$

constructed above.

Otherwise (in this case, we shall denote$\omega_{k}^{\prime\wedge}(t)$and$\xi_{k^{*}}$

If 7 $(t)$ is covered by, then it is clear that$\{\Omega_{k}(t)\}$represent the$(/\iota ‘ = (g_{1} , y_{2}))$$k=k”’$ ),we defineby the same way as (16) with anoter positive

also independent ofso that$\lambda$

7 $(t)- \bigcup_{k}.\sim(\omega_{k}arrow(t)r)I’)c^{-}\bigcup_{k}\sim’(\Omega_{k}\sim(t)\cap\Gamma)c_{-7}(t)$ .

Now we introduce two families of smooth functions
and$\{\eta_{k}(x)\}$assoc ated with the cover ngs
$i$

$i$

$\zeta_{k}(x)=\{\begin{array}{l}1ifx\in\omega_{k}(t)0ifx^{\xi-}.\overline{\Omega}-\Omega_{k}(t)\end{array}$$|D_{x}^{s}\zeta_{k}(x)|\leq C_{5}\lambda^{-Is1}$,$\{\omega_{k}(t)\}$$0\leqq\zeta_{k}(x)\leqq 1$$\{\zeta_{k}(x)\}$,,$\eta_{k}(x)=\zeta_{k}(x)/\sum_{k^{\lrcorner}}\neg\zeta_{k}(x)^{2}$.

$\{\Omega_{k}(t )\}$:

Then similarly to$\frac{\partial w}{\partial t}=A$(10)$\tau,$$[7,8]$ ,the regularizer$(x, t, w’ ; \nabla)w+\Phi$$i$$\backslash$on$B$ $(x, t ; \nabla)w=\varphi$$(\forall\tau\geqq 0, O\langle\forall h\leqq T-\tau)$$Q_{\tau\tau+h^{-}}--\Omega x(\tau, \tau+h)$on$w|_{t=\tau}=0$$\tau+h$ii

of the problem$R$$\Omega$$\Gamma_{\tau},$,,$r+h\equiv\Gamma x(\tau, \tau+h)$can be constructed and has the following prop-

erties.

Lemma 4.

Assume thatare sufficiently small).
provi ded$\Phi\in C\circ"$$\Gamma\in C^{2+a}$Then$R_{\lambda}$.

and$h=\chi\lambda^{2}(\chi(\rangle 0)$

and$\lambda$$\Phi\in C_{x,t}^{2+a,1+a/2}\circ(Q-\Omega_{k}x[\tau\tau,\tau+h^{-} \tau+h])$$a/2(\overline{Q}_{\tau}, \tau+h:\overline{\Omega}x\lfloor\tau, \tau+h^{-}-)$.

Furthermore the following

estimates hold:$(2 r-\vdash|s|\leq 2)$$|D_{t}^{r}D_{\chi}^{s}R_{\dot{k}}\Phi|\leqq C_{t\grave{\}}}(t-\tau)^{(\gamma,S|+\alpha)/2}2-2-|\Vert\Phi\Vert_{Q_{\tau,\tau*h}^{(\alpha)}}-$,$(2 r+|s|=2)$ ,$|\triangle_{x}^{x}D_{t}^{r}D_{x}^{s}R_{k}\Phi|\leqq C_{6}|x-x’|^{a}\Vert\Phi\Vert_{Q_{\tau\cdot\tau\star h}^{(a)}}-$’$|_{i} \bigwedge_{-t}.t$$D_{t}^{r}D_{\chi}^{s}R_{k}\Phi|\leqq C_{6}|t-t’|\langle 2-2r-\{s\}+a$)$/\wedge 0\Vert\Phi\Vert_{Q_{\tau\tau*h}^{--}}^{ta,j}$

where$R_{k}$,$\Phi=d\tau’1^{\backslash t_{\tau}}$I$Z_{0}(x-\overline{x}, t-\tau ; \xi_{k}’, \tau ; w’, \sigma’)x$$\Omega_{k}$,$x\zeta_{k}\cdot(\overline{x}\backslash ’\tau)\Phi(\overline{x}, \tau’)d\overline{x}$

–12–

,

$(0\langle 2r+|s|\leq 2)$

,

135

$R_{k^{r}}\Phi=\Pi_{\chi}^{z}\overline{R}_{k}\Phi$,$R_{k}\sim’.\Phi=\Pi_{x}^{z}\overline{R}_{k}\sim’\Phi$$\overline{R}_{k}\Phi=I^{d\tau’}\backslash \backslash H_{0}^{tk}(y\iota^{\backslash } t :,,\tilde{z}, \tau’)\overline{\zeta}_{k}\infty(\wedge’-)\overline{\Phi}(z, \tau’)dz$)$\overline{R}_{k}\Phi=]^{t_{\tau}\backslash }d\tau’\downarrow_{K_{2}}H_{0^{(}}$$(\zeta_{k^{r\prime}}’(\wedge)=II_{z}^{\chi}\zeta_{k}\sim(x)-$,$(y, t ; 2, \tau’)\overline{\zeta}_{k^{w}}(z)\overline{\Phi}(z, \tau’)dz$$\overline{\Phi}(z, \tau)=F_{z}^{\chi}\Phi(x, \tau)$,$K_{1}=\Pi_{z}^{\chi}t\Omega_{k}\prime\prime$,,’$K_{2}=F_{z}^{\chi}\Omega_{k}-$$\bigwedge_{p_{-t}^{\iota}}t."=/1_{x,f^{\wedge}}^{xt}-$$H_{0^{(k)}}\sim$

and

and

,

$\Delta_{\chi,t}^{\chi,f}g(x, t)=g(x, t)-g(x’, i’)$ ,,$z_{j}-- y_{j}(j=1,2)$ ,$H_{0}^{tk)}\sim$’$z_{3^{-J_{3}}}^{-};-F(\iota j’;\xi_{\sim p ,k}^{arrow}))$

are the Green matrix for$A=\Pi_{y}^{\chi}A$ $(\xi_{k}\sim, \tau, w’ ; \nabla_{x})$’$L^{\prime t_{x}^{x}}--/\backslash _{xt}^{x_{l},t}\lrcorner$

wi th$\nabla_{y}$$A=\Pi_{y}^{\chi}A(_{s}c_{k^{\nu}}\tau, w’ ; \nabla_{\chi})$replaced by$\nabla_{z}$$in(10)$..

$\tau+h$respectively.

$\underline{I,emma}5$.

Under the same assumptions as those in Lemma 4,$\in c_{\chi};_{t}(Q_{\tau,\tau\cdot\succ h})o_{2\alpha,1+a/z}(k)\sim if$$\varphi\in c_{\chi}^{1}:_{t^{a,}}\circ(1+a)J2(\Gamma\equiv(\Gamma\Uparrow\Omega_{\sim})\dot{i}_{-}\gamma,\tau+h_{k}(k)\sim$$\tau+h\rfloor$$R_{\kappa}’\sim\varphi C$)

and satisfies the estimates$|D_{f}^{r}D_{\chi}^{s}IR_{k}’,’\varphi_{i}\leqq C_{7}(t-\tau)^{(2-2r-|s\}+a)/z_{1}}|^{1}|\varphi\Vert_{ik}(;_{)\tau,\tau+h}x)(2r\vdash|s|\leq 2)$$|\triangle_{x}^{x}D_{t}^{r}D_{x}^{s}R_{k}’\varphi|\leqq C_{7}|x-x’|^{\alpha}||\varphi\Vert^{t1}:^{a)}jk)\tau\tau+h(2r+|s|=2)$’$|l_{-}\backslash _{t}^{t}$$D_{t}^{r}D_{\chi}^{s}R_{k}’\varphi|\leq C_{7}|t-t’|(2-2r-|s\{+\sigma)/2!|\varphi||^{t1}$\d ag er$(k)\tau,$

–13–

,

,

$\tau+h(0\langle 2r’+|s|\leqq 2)$

,

136

where

$R_{k}’’\cdot\varphi=\Pi_{\chi}^{z}]_{\overline{\iota}_{k}^{J’}\prime}^{-}\varphi$$\overline{R}_{k}’$means the norm of the space$\tau+h$$H_{1}(k^{*}$$B=\Pi_{y}^{\chi}B$$K:=F_{\chi}^{z}(\Omega_{k}\sim()I^{\urcorner})$,, $\varphi=^{b}t_{\tau\backslash K_{1}’}d\tau’\backslash H_{1}^{tk}(z-\overline{z}’ ’. t-\tau’)\overline{\overline{\zeta}}_{k}\sim(\overline{z}’)\overline{\varphi}(\overline{z}’, \tau’)d\overline{z}’$$||\cdot||(n\ddagger^{\sigma})(k)\tau,$and,is the Poisson kernel for$(\xi k .

\tau :

V_{x}))$

with$\nabla$,$C_{\gamma,}^{n}:_{t^{a,}}(n+a)/2(I^{-,tk)})\tau,\tau+h\sim$$A=\Pi_{y}^{\chi}A(\xi_{\sim ,k}, \tau, w’ ; \nabla_{\chi})$repiaced by$\nabla_{z}$in (10) .
$\tau$

and$\tau+h$

The similar assertions to those in Lemma 5 are true in the caseLemma 6.

Under the same assumptions as those in Lemma 4,$\dot{\mathfrak{k}}_{-}^{-}- c_{\chi}^{2};_{t^{a,1+a/2}}(Q_{\tau,\tau})\circ.tk)’\sim_{+h}$if$R_{k}’\sim\varphi\in$$\varphi\in c_{x}^{2};_{t^{a,}}\circ(2+\langle 1)/2(\Gamma_{r}(k:)+h_{k}^{---(I^{\neg}\cap\Omega)\cross[\tau}’\sim \tau+h])$and satisfies the estiimates$|D_{*}^{r}D_{x}^{s}R_{k}’\varphi|\leq_{-}C_{7}^{\backslash }(t-\tau)^{t2-2r-|s|+\alpha)/2}||\varphi_{I}^{1}|^{t2};_{)\tau,\tau+h}^{a)}(k(2r+|s|_{-<}-- 2)$$|p_{-} \bigwedge_{\chi}^{\chi}D_{t}^{r}D_{\chi}^{s}R_{k}’\sim\varphi i\leqq C_{7}|$x-)$x’|||\varphi a|1\langle
2a$ $\tau\cdot\tau+h(2r+|s|$||:\langle k$
)

,

2)$ ,

’

$|\Delta_{t}^{t}$$D_{\ell}^{r}D_{\chi}^{s}R_{k^{-}}’.\varphi|\leqq C_{7}|t-t’|(2-Ir-|S\}+\alpha)/2||\varphi\Vert^{(2}:^{a}tk)’,$$\tau+h(0\langle 2r+||s|\leqq 2)$

where$ft_{k^{\wedge\prime}}’\varphi^{--}\Pi_{\chi}^{z}\overline{R}_{k}’\sim\varphi$,$K_{2}’=\Pi_{\chi}^{z}(\Omega_{\sim^{i} ,k}^{;\backslash _{1}}I^{\urcorner})$,$\overline{R}_{k}’\varphi--c|d\tau^{\prime\backslash \nearrow\vee\sim}tf- I_{1}^{tk)}(.\sim-\overline{\tau}’ t-\tau’)\overline{\zeta}(\overline{z}’)\overline{\varphi}(\overline{z}_{:}’\tau’)d\tilde{\nearrow\vee}-’$

–14–

,

$k=k$ ”.

,

137

and

$H_{1}^{tk)}\sim$$B=\Pi_{y}^{\chi}B$ $(\xi)

means the norm of the space$\Vert\cdot||^{(n}:^{a)}(k)\tau\tau+h$

is the Po $is$ son kerne1 fork’ \tau :

These

with

V_{x} )$

$le\mathfrak{n}maS$$C^{n}x:t^{\phi,}t_{J}’\iota+a$

)

$+h$

$A=\Pi_{y}^{x}A$ $(\xi k^{\vee} \tau, w’ :repl aced by$V_{y}$$/2t’\backslash \Gamma_{\tau}^{tk}:$$\nabla_{z}$\nabla_{x})$in (10)and the same arguments as those in)$\tau,$

and$\tau+h$$[7,8]$

yield thefollowing theorem:

Theorem 7.
$\varphi=(\varphi_{1}, \varphi_{2}, \varphi_{3})$,Suppose that$\varphi_{1}\in C_{x}^{2};_{t^{a.1+a/2}}(\Gamma_{T})$$\varphi_{3}\in C_{x}^{2};_{t^{\alpha 1+\alpha/2}}(7_{T})$$\varphi_{2},$$I^{\urcorner}\in C^{2+a}$

,

,

,

$\Phi\triangleright C_{xi,\prime}^{a,a/2}(\overline{Q}_{T})$$\varphi_{2},$(10) ,$\varphi_{3}\in C_{xt}^{1+a,}(1+a)/2(\Gamma_{T})$$7_{\tau^{=\bigcup_{0\leqq t\leqq T}}}\{x\in\Gamma|K(x, t)=1\}$

Then there ex ists a un ique solut ion,,.

$uj_{\sim}^{\ulcorner C_{\chi}^{2}:_{t^{\alpha,1+a/2}}}(\overline{Q}_{T})$ofwhich sat isfies$-$$|D_{t}^{r}D_{x}^{s}w|\leq(C_{{}^{t}J}+C_{10}M_{2})^{N_{1}}t(2- 2r\vee|s\{+\alpha)/2\{\Vert_{\Phi^{1^{}}}||_{-+}Q^{(a)}\tau\Gamma_{T}^{+a_{J}}||i$$+|^{1^{}}|(\backslash \varphi_{2}, \varphi_{3})||^{t1+\alpha)}-\vdash||(\varphi_{2}I_{\Gamma}^{\urcorner}’\varphi_{3})!|^{(2+a)}\}\gamma TA$$(2 r-\}-|s|\leqq 2)$$|i\Delta_{x}^{x}\prime D_{t}^{r}D_{x}^{s}u|-\leq(C_{9}+C_{10}M_{2})^{N_{1}}|x-x’|^{\alpha}\{\cdots\}_{A}$,$(2 r+|s|=2)$ ,’$|[\Delta_{t}^{t}$$D_{t}^{r}D_{\chi}^{s}w$$|\leq(C_{\{}, +C_{10}M_{2})^{N_{1}}|$t-t$|^{t2- 2r- \mathfrak{l}s|+\alpha)/2}\{\cdots\}A$$(0\langle 2r+|s|\leqq 2)$

where$C=C$ $(T, M)(\geqq 1)$
9cally $inT$

9

and

1

and

$M_{1}$,$C_{10}-->0$

increases monotonically in$T$,$(T, M)$

$=C$

$C$

10

10

as

$Tarrow 0$

and

and

$M_{2}$$M_{1}$–15–

increase monotoni-$1$$N_{1}=N_{1}$.

(T. $M_{1},$$M_{2}$

)

,

138

Returning to the problem (10), it is clear that$\varphi=-B$ $(x, t ; \nabla)v_{0}$implies that$\Vert(\varphi_{2}.

$||(\varphi_{2}, \varphi_{3})_{1}|||^{t1+\alpha)}\Gamma_{T}$$\varphi_{1}-- 0$,\varphi_{3})\Vert\gamma^{(2+\alpha)}T\leq- C_{11}$.

From (6), (7) and (I1) it follows that$||\rho!|_{- ,Q_{T}^{\langle 1+a)}}\leqq C_{12}(T, M_{1})\perp c_{13}(T, M_{1} )$]$\nu f_{2}$

.

hence
$\Vert\Phi_{\dagger}||_{Q_{T}^{(\alpha)}}-\leqq C_{12}(T, M_{1})+C_{t3}(T, M_{1} )M_{2}$where$C_{12}(-\geq 1)$

and$C_{13}$,have the same properties as$C_{9}$

andrespectively.

Therefore we obtain$[^{||w\Vert_{Q_{T}^{(2)}}}-\leqq[C_{9}(T,M.)+C(T,M_{1})M_{2}\neg A_{3}tTM_{1}M_{2})(\Gamma_{\lrcorner}^{a1+a/z}\{/I^{\tau})xx_{L}\ulcorner C_{\iota\iota}^{1}+C_{12}^{10}(7^{\backslash },1M_{1})-\}C_{1}^{\rfloor^{1}}(T’,M_{1}’)M_{2’}^{-}|$,(17)$\uparrow_{(}’\Sigma|D_{\chi}^{2}w||S_{I}^{}=2x^{(a}Q_{T}^{)}\leqq\underline{!^{-}}C_{9}(T, M_{1})+C_{10}(T, M_{1})M_{2}\rfloor^{N_{1}}(T.M_{1\prime}At_{2})x$

XL$C_{11}^{\backslash }\vdash C_{12}(T, M_{1})+C_{13}(T, 11f_{1})M_{2}$].

Next let us consider the following linearized problem of (9):in$\ovalbox{\tt\small REJECT}\frac{\partial\sigma}{\partial t}\Lambda_{=0}’(x\sigma|_{t=0}^{--} t, w’, \sigma’)\Delta\sigma+\Psi_{on\Omega}(x,t, w’, \sigma’)$(18)$((1-\kappa_{e})\nabla\sigma\cdot n-\kappa_{e}\sigma=\psi(x, t)$on$I_{T}^{\tau}$.

$Q_{T}$,$C_{10}$

139

Here

$(w’ \sigma’)\in \mathscr{J}_{T}$.

The similar, but easier, arguments to those for (10) yield

Theorem 8.

Suppose that$\in C_{\chi}^{1}:_{t^{\alpha}}\cdot t\iota+a)/2(\Gamma_{T})$$\Gamma\subset- C^{2+a}$

and moreover,$\Psi\in C_{xt}^{a,a/2}(\overline{Q}_{T})$$\psi^{\underline{p}}C_{\chi}^{2}:_{t^{a}}1+a/2$,$\psi’\subset$(For 1‘$(\Gamma_{?}’.)$$\acute{T}$,see

Theorem in \S 1).
Then there exists a unique solution$\sigma\in C_{\chi}^{2}:_{t^{a}}1+a/2(\overline{Q}_{T})$of (18)which satisfies$|D_{t}^{\gamma}D_{\chi}^{s}\sigma|\leq(C_{14}-\vdash C_{15}M_{2})^{N_{2}}t(2-2r-|s|+a)/2\{_{1}^{1}||\Psi\Vert_{-,Q_{\Gamma}}^{(a)}\vdash$$+!|\psi||^{t1+a)}+||\psi\Vert^{(2+a)}\}_{B}\Gamma_{T}\Gamma_{T}’$$(2 \gamma+|s|\leq 2)$,’$(2 r\{-|s|=2)$ ,$D_{t}^{r}D_{x}^{s}\sigma|\leqq(C_{14}+C_{15}M_{2})^{N_{2}}|x-x’|^{\{I}\{\cdots\}_{B}$$|^{f}\Delta_{x}^{x}$’$|\triangle_{t}^{t}$$D_{\ell}^{r}D_{\chi}^{s}\sigma|\leqq(C_{1A}\vdash C_{15}M_{2})^{N_{2}}|t-t’|(2-2r-\{S|+a)/2\{\cdots\}_{B}$$(0\langle 2r+|s|\leqq 2)$

where$C_{14}=C_{1*}(T, M_{1})(\geqq\dagger)$have the same properties of,$C_{15}=C_{15}(T, M_{1})$$C_{9}$,$C_{10}$and$N_{1}$

and$N_{2}=N_{2}(T,M_{1}M_{2})$, respectiveiy.

Therefore we obtain$[^{|^{1}|\sigma||^{t_{\frac{2}{Q}}}}T^{)}\leqq {}_{\dot{L}}C_{14}(T,flf_{1})+C_{15}(T, \Lambda M_{1})]|/l]_{8}^{N_{2,}(TAf_{1}M_{2})}(T^{a}-\vdash T^{1+a/2})xx|_{-}^{-}C_{48}’-\vdash C_{17}(T,M_{1}),’-$

,

–17–

,

140

(19)
(T.

$M_{1\prime}M_{2}$$\backslash x\underline{|^{-}}C_{16}\dashv- C_{17}(7^{\tau}, M_{1})+C_{18}(7^{\tau}, M_{1} )M_{2-}|$

where

and

$t_{-\cdot 17}^{\backslash }$$C_{1}$,

have the same properties as)$x$$|_{1^{}}\Sigma_{s\}-\ell}^{i}\vee’|D_{\chi}^{2}\sigma|_{\chi\prime\tau}^{(a_{\frac{)}{Q}}}\leq-[C_{14}(T, M_{1})+C_{15}(7’, M_{1})M_{2}]^{N_{2}}$,and$C_{9}$$C_{10}$respectively.
From the estimates
$w$

and

$\sigma$

and

(17)

(19)

of (10) and of (18) belong to

we conclude that the solutions
for some$y_{T_{0}}$Indeed, it is sufficient to choose a constant$M_{2}$$\prime l_{t\}}^{\tau_{\underline{\Gamma}}}(0, T$

so as to belarger than$|^{-}C_{9}$$(T, M_{1} )$ $\perp M|^{v_{1}}(y^{\iota}, M_{1^{l}}M)-\vee|C_{11}$ } $C_{12}(\Gamma, M_{1})+M|\neq$$+|C_{14}(T, M_{1})+1\triangleright 1|^{N_{2}}(T, J1t_{1}r_{C_{\downarrow\epsilon},-\}-}tt)-(_{-}:_{1i}(T, M_{1})aM^{1_{i}}$for any positive number$M$, and then$T_{0}\mathfrak{t}_{-}^{-}(0, T$]

such that$\{|C_{\backslash }\llcorner^{-}((T_{0}, M_{1})\vdash M|N_{1}(’\Gamma_{0}=M_{1’}M)-!.C_{11}\neq C_{12}-(’1_{0}^{\backslash }, M_{1})+\Uparrow\gamma_{-}]+$} $C_{17}(T_{0}, M_{1})+M|$$\}_{1_{-}^{--}}C_{14}(T_{0}, M_{1})-\}M]^{N_{2}tT_{0’}M_{1}\cdot M)-}\dot{\llcorner}(^{\neg},$$1G$$\backslash _{-}\prime_{-}M_{1}$$C_{10}(’1_{0}^{\tau}, M_{1})M_{2}$,$(T_{0}^{a}\downarrow 7_{0}^{1+\sigma/}\underline’)\leqq$,$C_{A’3}(’F_{1}, , M_{t,\wedge})M_{2}$$C_{18}(T_{0},\dot{/}\psi j_{1})M_{2}\leq/\nu 1$

For simplicity, we take},

$C_{I5}(’\overline{r}_{0}, M_{1})M_{p}$,.

$T=\prime I_{0}^{\backslash }$from the beginning.

–18–

].

141
2.2 Nonlinear problem (8) and (9)We construct the sequence $\{(w_{n}, \sigma_{n})(x, t)\}$of the successive

approximate solutions as follows$\{()(w_{0},\sigma)-0\epsilon_{g}y_{(w’,\sigma}w_{n}and_{as^{--}}\sigma f(18)^{0}su^{n}\min^{are^{T}define,d_{)}a}sth^{e_{1}so1uti_{\ulcorner}}\sigma_{n- 1})_{-\iota}^{on}\Psi_{T}^{s},$$wand\sigma ofrespectively.(10)$Then the results in \S 2.1 imply thatexists and belongs to$C\Psi_{T}$$(n=0, \ddagger, 2, \cdots)$$(uJ_{n}, \sigma_{n})(x, t)$uniquely.

Applying the estimates in \S 2. 1 to the equations concerning$w_{n}-w_{n\vee 1}$

and$\sigma_{n}-\sigma_{n-1}$

we obtain$|||(w_{n}, \sigma_{n})-(w_{n-1}, \sigma_{r-1_{Q}})\Vert_{-:}^{(2\alpha)}\leq$$\leq C_{19}(T, A^{I}1_{1}, M_{2})\Vert(w_{n- 1}, \sigma_{n-1})-(w_{r\iota-2}, \sigma_{n-2})||_{\frac{2}{Q}:}^{(a)}$where$C_{19}arrow 0$

as

Therefore the sequence
uniformly if we take$Tarrow 0$,.

$\{(w_{n}, \sigma_{n})\}$$T’\overline{e}(0, T$

converges to some function

so as to satisfy]$(w, \sigma)$$C_{19}(T’ . M_{1}, M_{2})\langle t$.

The uniqueness of the solution to the problem (8) and (9) is provedby the fact that the difference of two solutions supposed to exist satisfy

the inequality analogous to (20).
The positivity and the boundedness of$\rho$

andfrom our method for constructing the solution.

Thererefore our main theorem has been proved.

–19–$\theta$are obvious142

References[1]

Ito, S. ,Fundamental solutions of parabolic equations and boundary

value problems, Japan J. Math. , 27 (1957), 55-102.
[2]$IIauwl\iota e\#CKafl,$

CJl

[3]

$0$

fl

XRI

$0$

$K0l^{\backslash }l\#$$0cuot0BCK\Pi i$Il a1$0_{tU0I0\S\cap.K1\dot{\#}}$

T $Be\beta Q0i$

, B. F. ,($:\#\Downarrow|JR0UI0\#e,$[4]. A. 1

Serrin, J. ,$\rho\vee\phi e\rho oi$, B.
,$1^{\urcorner}$$Be_{\vee}^{\rho}$.,T.

$0cso6oQB0\dot{\#}$$\eta?\int$,$0cso60\#roiN088\beta X\mathbb{I}0r\uparrow\#$$3a1\cdot\#ay\iota$$\underline{13}$nonepxnocrn(\d ag er 976),

xano

25-30.

$RtNulfe\uparrow\beta IQ\mathfrak{g}0u$. $cevt8\cdot\# 0MM$ , 52(1975), $160-I74$ .

Mathematical principles of classical fluid mechanics,

Handbuch der Physik, Bd.8, Springer-Verlag, 1959.
[5]$Co\# 0\#IitK0B$

CM

$C$

\d ag er $e|!$II, B. A. ,$\phi\phi e\mathfrak{g}Q\mathbb{I}$a$\mathfrak{g}b$(1

I $HX$$f.\rho aP.B\downarrow’;xona\iota$$Y\beta$aBIl $e\#$ I$i$

a $\eta axA181\#\#ei\#wx$$06\#$ ] $C^{\Gamma}1C$$B\#\#$a,$\uparrow py\mu$Il

Ia $\rho a6c\#s\iota$ ecxtx$M\mathfrak{l}l\Lambda H$,$\underline{83}$(I965) ,3-162.
[6]$r_{0uo\#\#\#K0B}^{\backslash },$$c\uparrow aQ\# 0\#$

[7]

$3$

. A. }$1Na\# l\# 0R$, B. B.

a fl $0iClC\uparrow euHH$ aa e-C$\beta$Tani, A. ,$b$$\uparrow 0\kappa ca,$06

$0\#H0iK\beta a\mathfrak{e}h0i3aua\eta$$\iota_{DY0H}^{\backslash }$ae$\Lambda\lambda A$

M14 AH, 125 (1973), 196-210.

On the first initial-boundary value problem of compress-

ible viscous fluid motion, Publ. RIMS, Kyoto Univ. , 13 (1977), 193-

253.
[8]

Tani, A. ,On the free boundary value problem for compressible

viscous fluid motion, J. Math. Kyoto Univ. , 21 (1981), 839-859.
[9]Tani, A. ,Two-phase free boundary problem for compressible viscous

fluid motion, J. Math. Kyoto Univ. , 24 (1984), 243-267.
[10] Tani, A. ,The initial value problem for the equations of the motionof compressible viscous fluid with some slip boundary condition,

in Patterns and Waves-Qualitative analysis of nonlinear differential)
equations-, Studies in Mathematics and ts applications 18, Ed. by$i$

H. Fuji , M. Mimura and T. Nishida, $North-Holland/Kinokuniya$ , I986,$i$675-684.