数理解析研究所講究録
第 1905 巻 2014 年 148-163

148

maximal regularity and its application$L_{p}-L_{q}$Yoshihiro SHIBATA*

Department of Mathematics and Research Institute of Science and Engineering,

Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan.
e-mail address: yshibata@waseda.jp

Miho MURATA
Department of Pure and Applied Mathematics, Graduate School of

Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan.
-mail address: m-murata@ruri.waseda.jp$e$1

Introduction

We consider a local in time unique existence theorem for the motion of a compressibleof the $N$ dimensional Euclidean spaceviscous fluid occupying a domain
with the boundary slip condition. Let $\rho=\rho(x, t)$ be the density of thefluid, $v=v(x, t)=(v_{1}(x, t), \ldots, v_{N}(x, t))$ the velocity vector field, andthe pressurefunction with $x=(x_{1}, \ldots, x_{N})\in\Omega$ and being the time variable. The motion is described$\Omega$$\mathbb{R}^{N}(N\geq 2)$$P(\rho)$$t$by the following equations:(1.1)$\{\begin{array}{ll}\partial_{t}\rho+div (pu) =0 in \Omega\cross(0, T) ,\rho(\partial_{t}u+u\cdot\nabla u)-DivS(u)+\nabla P(\rho)=0 in\Omega\cross(O, T) ,D(u)n- < D(u)n, n>n=0, u\cdot n=0 on\Gamma\cross(O, T) ,(\rho, u)|_{t=0}=(\rho_{*}+\theta_{0}, u_{0}) in \Omega\end{array}$is a positive constant describing the mass density of thebodyboundary
theof and the unit outward normal to . Moreover,reference
is afunction defined on $\rho>0$ satisfying the condition: $P’(\rho)>0$ for $\rho>0$ and$S(u)$ the stress tensor of the form:$(cf.[4,6$where$\partial_{t}=\partial/\partial t,$$\Omega,$$\Gamma$$\rho_{*}$$\Omega$$n$$\Gamma$$C^{\infty}$$P(\rho)$$S(u)=\alpha D(u)+(\beta-\alpha)divuI,$

where and are positive constants describing the first and second viscosity coeﬃcients,
respectively, $D(v)$ denotes the deformation tensor whose $(j, k)$ components are $D_{jk}(v)=$$\alpha$$\beta$*Partially supported by JST CREST and JSPS Grant-in-aid for Scientific Research (S)$\#$24224004

149, and I the $N\cross N$ identity matrix. Finally, for any matrix
with$k=1$ , . . . , $N$ , the quantity $DivK$ is an $N$-vector with
field
with components
j-th component, and also for any vector of functions $u=(u_{1}, \ldots, u_{N})$ we set
divu
and
with
.
A local in time unique existence theorem was proved by Burnat and Zaj4czkowski [2]in a bounded domain of 3-dimensional Euclidean space , where their velocity field anddensity of the fluid belong to Sobolev-Slobodetskii spaces
and$\alpha\in(1/2,1)$, respectively. Moreover, a global in time unique existence theorem waswithproved by Kobayashi and Zaj4czkowski [5] in the same class as in the local in time uniqueexistence theorem by the energy method. The purpose of this paper is to prove a local intime unique existence theorem in a umiform
and our velocity field and densityof the fluid belong toandwith $2<p<\infty$ and$N<q<\infty$ , where we have set$\partial_{j}v_{k}+\partial_{k}v_{j}$$\partial_{j}=\partial/\partial x_{j}$$K$$K_{jk},$$j,$$\sum_{k=1}^{N}\partial_{k}K_{jk}$$= \sum_{j=1}^{N}$$u\cdot\nabla=\sum_{j=1}^{N}u_{j}\partial_{j}$$\nabla=(\partial_{1}, \ldots, \partial_{N})$$\mathbb{R}^{3}$$u$$W_{2}^{2+\alpha,1+\alpha/2}$$W_{2}^{1+\alpha,1/2+\alpha/2}$$\rho$$W_{q}^{3-1/q_{*}}$$W_{q,p}^{2_{)}1}(\Omega\cross(0, T))$$\rho$$u$$W_{q,p}^{1,1}(\Omega\cross(0, T))$

$W_{q,p}^{l,m}(\Omega\cross(0, T))=L_{p}((0, T), W_{q}^{l}(\Omega))\cap W_{p}^{m}((0, T), L_{q}(\Omega))$

.

(1.2)

One of the merits of our approach is less compatibility condition than [2],
To obtain such local in time unique existence theorem, it is key to prove themaximal regularity for the linearized problem of the following form:$L_{p}-L_{q}$$\{\begin{array}{l}\partial_{t}\rho+\gamma_{2}divu=f in\Omega\cross(0, \infty) ,\gamma_{0}\partial_{t}u-DivS(u)+\nabla(\gamma_{1}\rho)=g in\Omega\cross(O, \infty) ,(1.3)\alpha[D(u)u-<D(u)n, n>n]=h-<h, n>n, u\cdot n=\tilde{h} on\Gamma\cross(O, \infty) ,(\rho, u)|_{t=0}=(\rho_{0}, u_{0}) in \Omega.\end{array}$are uniformly continuous functions defined on$\gamma_{i}=\gamma_{i}(x)(i=0,1,2)$$\overline{\Omega}$satisfying theassumptions:$\rho_{*}/2\leq\gamma_{0}(x)\leq 2\rho_{*},$$0\leq\gamma_{k}(x)\leq\rho_{I}(x\in\Omega, k=1,2)$,$\Vert\nabla\gamma_{\ell}\Vert_{L_{r}(\Omega)}\leq\rho_{1}(\ell=0,1,2)$(1.4)
with some positive constantIn order to show
maximal regularity, we prove the existence of -bounded
solution operator to the following generalized resolvent problem corresponding to timedependent problem (1.3):$\rho_{1}.$$L_{p}-L_{q}$$\mathcal{R}$(1.5)$\{\begin{array}{ll}\lambda\theta+\gamma_{2}divv=f in\Omega,\gamma_{0}\lambda v-DivS(v)+\nabla(\gamma_{1}\theta)=g in\Omega,\alpha[D(v)n-<D(v)n, n>n]=h-<h, n>n, v\cdot n=\tilde{h} on\Gamma.\end{array}$

In fact, we prove that for any $\epsilon\in(0, \pi/2)$ , there exist a constantand anoperator family$f\in
W_{q}^{1}(\Omega,$Holsuch that for any$g\in L_{q}(\Omega)^{N},$$h\in W_{q}^{1}(\Omega)^{N}$, problem (1.5) admits a unique soluand
tion
and
is
bounded forwith value in. Here,is the$\lambda_{0}\geq 1$$R(\lambda)\in$$(\Sigma_{\epsilon,\lambda_{0}}, \mathcal{L}(\mathcal{X}_{q}(\Omega), W_{q}^{2}(\Omega)^{N}))$$\tilde{h}\in W_{q}^{2}(\Omega)$$(\rho, v)=R(\lambda)(f, g, \lambda^{1/2}h, \nabla h, \lambda\tilde{h}, \lambda^{1/2}\nabla\tilde{h}, \nabla^{2}\tilde{h})$$\lambda\in\Sigma_{\epsilon,\lambda_{0}}\cap K_{\epsilon}$*The definition of$W_{q}^{3-1/q}$$(\lambda, \lambda^{1/2}\nabla P_{v}, \nabla^{2}P_{v})R(\lambda)$$\mathcal{L}(\mathcal{X}_{q}(\Omega), W_{q}^{1}(\Omega)\cross L_{q}(\Omega)^{\tilde{N}})$

domain is given in Definition 1.1, below.

$P_{v}$$\mathcal{R}-$150
projection such that $P_{v}(\rho, u)=u,$ $\tilde{N}=N+N^{2}+N^{3},$$\Sigma_{\epsilon,\lambda_{0}}=\{\lambda\in \mathbb{C}||\lambda|\geq\lambda_{0},|\arg\lambda|\leq\pi-\epsilon\},$$K_{\epsilon}= \{\lambda\in \mathbb{C}|(\lambda+\frac{\gamma}{\alpha+\beta}+\epsilon)^{2}+({\rm Im}\lambda)^{2}\geq(\frac{\gamma}{\alpha+\beta}+\epsilon)^{2}\}$

(1.6)

,

$\mathcal{X}_{q}(\Omega)=\{F=(F_{1}, \ldots, F_{7})|F_{1}\in W_{q}^{1}(\Omega) ,F_{5}\in L_{q}(\Omega) , F_{2}, F_{3}, F_{6}\in L_{q}(\Omega)^{N},$$F_{4}, F_{7}\in L_{q}(\Omega)^{N^{2}}\},$are independent variand
, and
with
$f,$
, respectively. Moreover,andables corresponding toholomorphic
functions defined onvalued
of
all
the
set
denotesHol
the set of all bounded linear operators from a Banacha complex domain and$X$to problem (1.3) is rep. Since the solution
space
space
into another Banachto problem (1.5), so that
resented by the inverse Laplace .transform of the solution
help
(1.3)
of Weis’ operator valued
is obtained withresult for problem
the maximal
Fourier multiplier theorem [11].
Before ending the introduction, we summarize several symbols and functional spaces
used throughout the paper. For the diﬀerentiations of scalar and $N$ -vector , we use
the following symbols:$F_{1},$$\gamma=\sup_{x\in\overline{\Omega}}\gamma_{1}(x)\gamma_{2}(x)$$g,$$F_{2},$$\lambda^{1/2}h\nabla h,$$(U, \mathcal{L}(X, Y))$

$F_{4},$$F_{3},$$\lambda\tilde{h},$$F_{5},$$\lambda^{1/2}\nabla\tilde{h}$$F_{7}$$F_{6}$$\nabla^{2}\tilde{h}$$\mathcal{L}(X, Y)$$U$$\mathcal{L}(X, Y)$$Y$$(\rho, u)$$(\theta, v)$$L_{p}-L_{q}$$f$$\nabla f=(\partial_{1}f, \ldots, \partial_{N}f) , \nabla^{2}f=(\partial_{i}\partial_{j}f|i,j=1, \ldots, N)$$g$,$\nabla g=(\partial_{i}g_{j}|i,j=1, \ldots, N) , \nabla^{2}g=(\partial_{i}\partial_{j}g_{k}|i,j, k=1, \ldots, N)$

denotes the -product. For any Banach space $X$ with norm
withfor the sake of simplicity.
instead ofspace of $X$ , while its norm is denoted by$D,$$L_{q}(D)$space and Sobolev
Lebesgue
usual
denotetheandFor any domain
denote their norms, respectively. We set
space, while
and$\Vert\cdot\Vert_{X},$$\partial_{j}=\partial/\partial x_{j}$$X^{d}$$d$$\Vert\cdot\Vert_{X^{d}}$$\Vert\cdot\Vert_{X}$$W_{q}^{m}(D)$$\Vert\cdot\Vert_{L_{q}(D)}$$\Vert\cdot\Vert_{W_{q}^{m}(D)}$$W_{q}^{m,l}(D)=\{(f, g)|f\in W_{q}^{m}(D), g\in W_{q}^{\ell}(D)^{N}\}$

with $W_{q}^{0}(D)=L_{q}(D)$ . For $1\leq p\leq\infty,$ $L_{p}((a, b), X)$ and $W_{p}^{m}((a, b), X)$ denote the usual
Lebesgue space and Sobolev space of $X$ -valued functions defined on the interval $(a, b)$ ,denote their norms, respectively. Set
and
while
$\Vert\cdot\Vert_{L_{p}((a,b)_{)}X)}$$\Vert\cdot\Vert_{W_{p}^{m}((a,b),X)}$$L_{p,\gamma_{1}}(\mathbb{R}, X)=\{f(t)\in L_{p,1oc}(\mathbb{R}, X)|e^{-\gamma_{1}}tf(t)\in L_{p}(\mathbb{R}, X$$L_{p,\gamma_{1},0}(\mathbb{R}, X)=\{f(t)\in L_{p,\gamma_{1}}(\mathbb{R}, X)|f(t)=0(t<0$$W_{p,\gamma_{1}}^{rn}(\mathbb{R}, X)=\{f(t)\in L_{p_{)}\gamma_{1}}(\mathbb{R}, X)|e^{-\gamma_{1}t}\partial_{t}^{J}f(t)\in L_{p}(\mathbb{R}, X)(j=1$$W_{p,\gamma_{1},0}^{m}(\mathbb{R}, X)=W_{p,\gamma_{1}}^{m}(\mathbb{R}, X)\cap L_{p,\gamma_{1},0}(\mathbb{R}, X)$, . . . , $m$,$\Vert e^{-\gamma t}f\Vert_{W_{p}^{m}(I,X)}=\sum_{j=0}^{m}(\int_{I}(e^{-\gamma t}\Vert\partial_{t}^{j}f(t)\Vert_{X})^{p}dt)^{1/p}$

denote the Fourier transform and the Fourier inverse
transform, respectively, which are defined byLet$\mathcal{F}_{x}=\mathcal{F}$and$\mathcal{F}_{\xi}^{-1}=\mathcal{F}^{-1}$$\mathcal{F}_{x}[f](\xi)=\int_{\mathbb{R}^{N}}e^{-ix\cdot\xi}f(x)dx, \mathcal{F}_{\xi}^{-1}[g](x)=\frac{1}{(2\pi)^{N}}\int_{1k^{N}}e^{ix\cdot\xi}g(\xi)d\xi.$151
We also write. Let anddenote the Laplace transform and theLaplace inverse transform, respectively, which are defined by$\hat{f}(\xi)=\overline{J^{-}}_{x}[f](\xi)$$\mathcal{L}^{-1}$$\mathcal{L}$$\mathcal{L}[f](\lambda)=\int_{-\infty}^{\infty}e^{-\lambda t}f(t)dt, \mathcal{L}^{-1}[9](\lambda)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{\lambda t}g(\tau)d\tau$with$\lambda=\gamma+i\tau\in \mathbb{C}$. Given$s\in \mathbb{R}$and $X$ -valued function $f(t)$ , we set.

$\Lambda_{\gamma}^{s}f(t)=\mathcal{L}_{\lambda}^{-1}[\lambda^{s}\mathcal{L}[f](\lambda)](t)$Moreover, the Bessel potential space of $X$ -valued functions of order
following:$s$

is defined by thefor any$H_{p,\gamma_{1}}^{s}(\mathbb{R}, X)=\{f\in L_{p}(\mathbb{R}, X)|e^{-\gamma t}\Lambda_{\gamma}^{S}[f](t)\in L_{p}(\mathbb{R}, X)$$\gamma\geq\gamma_{1}\},$$H_{p,\gamma_{1},0}^{s}(\mathbb{R}, X)=\{f\in H_{p,\gamma_{1}}^{s}(\mathbb{R}, X)|f(t)=0(t<0$depends on. The
The letter denotes generic constants and the constant
may change from line to line. and denote the set ofvalues of constants andall natural numbers and complex numbers, respectively, and we set
domain,Finally, we introduce the definition of a uniform
boundedness ofoperator families and the Weis operator valued Fourier multiplier theorem.
$C$$C_{a,b},\cdots$

$C$

$a,$

$b,$

$\cdots$$\mathbb{C}$$\mathbb{N}$$C_{a,b},\cdots$$\mathbb{N}_{0}=\mathbb{N}\cup\{0\}.$$W_{r}^{3-1/r}$$\mathcal{R}$Definition 1.1. Let $1<r<\infty$ and let be a domain inwith boundary . We saythat is a uniform
domain if there exists positive constants
and $K$ such thatfor any $x_{0}=(x_{01}, \ldots, x_{0N})\in\Gamma$ there exist a coordinate number and afunction$h(x’)(x’=(x_{1}, \ldots,\hat{x}_{j}, \ldots, x_{N}))$ defined on
andwithsuch that$\mathbb{R}^{N}$$\Omega$$\Gamma$$W_{r}^{3-1/r}$$\Omega$$\alpha,$$\beta$$W_{r}^{3-1/r}$$j$$B_{\alpha}’(x\’{o})$$x_{0}’=(x_{01}, \ldots,\hat{x}_{0j}, \ldots x_{0N})$$\Vert h\Vert_{W_{r}^{3-1/r}(B_{\alpha}’(x_{0}))}\leq K$$\Omega\cap B_{\beta}(x_{0})=\{x\in \mathbb{R}^{N}|x_{j}>h(x’)(x’\in B_{\alpha}’(x_{0}’))\}\cap B_{\beta}(x_{0})$,$\Gamma\cap B_{\beta}(x_{0})=\{x\in \mathbb{R}^{N}|x_{j}=h(x’)(x’\in B_{\alpha}’(x_{0}’))\}\cap B_{\beta}(x_{0})$.

Here, $B_{\alpha}’(x_{0}’)=\{x’\in \mathbb{R}^{N-1}||x’-x_{0}’|<\alpha\},$ $B_{\beta}(x_{0})=\{x\in \mathbb{R}^{N}||x-x_{0}|<\beta\}$ and( (x\’o)) denotes the set of all function( (x\’o)) such that
$W_{r}^{3-1/r}$$h\in W_{r}^{2}$$B_{\alpha}’$$B_{\alpha}’$$\{\int\int_{B_{\alpha}’(x_{O})\cross B_{\alpha}’(x_{0})}\frac{|\partial_{k}\partial_{l}h(x’)-\partial_{k}\partial_{l}h(y’)|^{r}}{|x-y’|^{N-2+r}}dx’dy’\}^{1/r}<\infty$

for$k,$ $l\neq j$

with$\partial_{k}\partial_{l}h=\partial^{2}h/\partial x_{k}\partial x_{l}.$,Definition 1.2. A family of operators
is called -bounded on
if there exist constants $C>0$ and $p\in[1, \infty$ ) such that for any
and sequencesof independent, symmetric, $\{-1, 1\}$ -valued randomvariables on , 1 there holds the inequality:$\mathcal{T}\subset \mathcal{L}(X, Y)$$\mathcal{L}(X, Y)$$\mathcal{R}$$n\in \mathbb{N},$$\{f_{j}\}_{j=1}^{n}\subset X$$\{T_{j}\}_{j=1}^{n}\subset \mathcal{T},$$\{r_{j}(u)\}_{j=1}^{n}$

$[0$

$]$

$\{\int_{0}^{1}\Vert\sum_{j=1}^{n}r_{j}(u)T_{j}x_{j}\Vert_{Y}^{p}du\}^{\frac{1}{r}}\leq C\{\Vert\sum_{j=1}^{n}r_{j}(u)x_{j}\Vert_{X}^{p}du\}^{\frac{1}{p}}$The smallest such $C$ is called$\mathcal{R}$-bound of$\mathcal{T}$, which is denoted by$\mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T})$.

152functions having compact
be the set of all $X$ valued
and$X$ valued functions, respectively,
decreasing
rapidly
spaceofsupports and the Schwartz, we define the operator. Givenwhile

Let$C^{\infty}$$S(\mathbb{R}, X)$$\mathcal{D}(\mathbb{R}, X)$$M\in L_{1,1oc}(\mathbb{R}\backslash \{0\}, X)$$\mathcal{S}’(\mathbb{R}, X)=\mathcal{L}(S(\mathbb{R}, \mathbb{C}), X)$

by$T_{M}:\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}, X)arrow S’(\mathbb{R}, Y)$(1.7)$T_{M}\phi=\mathcal{F}^{-1}[M^{\overline{r-}}[\phi]], (\mathcal{F}[\phi]\in \mathcal{D}(\mathbb{R}, X$

The following theorem is obtained by Weis [11].
Theorem 1.3. Let $X$ andfunction in

be two $UMD$ Banach spaces and $1<p<\infty$ . Let $M$ be asuch that$Y$$C^{1}(\mathbb{R}\backslash \{0\}, \mathcal{L}(X, Y))$$\mathcal{R}_{\mathcal{L}(X,Y)}(\{(\tau\frac{d}{d\tau})^{\ell}M(\tau)|\tau\in \mathbb{R}\backslash \{0\})\leq\kappa<\infty (\ell=0,1)$

with some constant . Then, the operator
defined in (1.7) is extended to a bounded. Moreover, denoting this extension byintolinear operator fromwe have$T_{M}$$\kappa$$T_{M},$$L_{p}(\mathbb{R}, Y)$$L_{p}(\mathbb{R}, X)$$\Vert T_{M}\Vert_{\mathcal{L}(L_{p}(\mathbb{R},X),L_{p}(\mathbb{R},Y))}\leq C\kappa$

for some positive constant$C$depending on

$p,$

$X$

and $Y.$

Remark 1.4. For the definition of UMD space, we refer to a book due to Amann [1].
and Sobolev space
For $1<q<\infty$ , Lebesgue space
are both UMD spaces.
$L_{q}(\Omega)$2$W_{q}^{m}(\Omega)$

Main Results

The following local in time unique existence theorem for (1.1) is our main result.

heorem 2.1. Let $2<p<\infty,$ $N<q<\infty,$ $R>0$ and assume that be a uniform. Let be a positive constant and let
be adomain infunction$\rho_{1}<P’(\rho)<\rho_{2}$ with some positive constants$\rho>0$
on
such
that
and
for
definedany. Let
be the space defined in (2.2). Then, there exists atimedepending onsuch that for any initial datawithsatisfying the range condition:$T$$\Omega$$W_{q}^{3-1/q}$$\mathbb{R}^{N}$$P(\rho)$$\rho_{*}$$C^{\infty}$$\rho_{1}$$\rho\in(\rho_{*}/4,4\rho_{*})$$T$$\rho_{2}$$B_{q,p}^{2(1-1/p)}(\Omega)$$(\theta_{0}, u_{0})\in W_{q}^{1}(\Omega)\cross B_{q,p}^{2(1-1/p)}(\Omega)$$R$$\Vert\theta_{0}\Vert_{W_{q}^{1}(\Omega)}+\Vert u_{0}\Vert_{B_{q,p}^{2(1-1/p)}(\Omega)}\leq R$(2.1)$\rho_{*}/2<\rho_{*}+\theta_{0}(x)<2\rho_{*} (x\in\Omega)$

problem (1.1) admits a unique solution$(\rho, u)$with$\rho\in W_{q,p}^{1,1}(\Omega\cross(0, T u\in W_{q_{)}p}^{2_{)}1}(\Omega\cross(0, T$

To obtain Theorem 2.1, first, we show the following.result for the equation (1.3) with
$f=0,$ $g=0,$ $h=0$ and$\tilde{h}=0.$

Theorem 2.2. Let $1<p,$ $q<\infty,$ $N<r<\infty,$ $\max(q, q’)\leq r(q’=q/(q-1$$0<\epsilon<\pi/2,$. Assume that is a uniform
anddomain and that
. Set$\delta_{0}>0$$\lambda_{0}>0$$\Omega$$W_{r}^{3-1/r}$$|\delta|\leq\delta_{0}$$B_{q,p}^{2(1-1/p)}(\Omega)=(W_{q}^{1,0}(\Omega), \mathcal{D}_{q}(\Omega))_{1-1/p,p}$

(2.2)

153
with real interpolation functor. Then, for any
problem (1.3) with $f=0,$ $g=0,$ $h=0$ andadmits a unique solution$(\rho_{0}, u_{0})\in W_{q}^{1}(\Omega)\cross B_{q,p}^{2(1-1/p)}(\Omega)$$)_{\theta,p}$$\tilde{h}=0$$(\rho, u)$

,

with

$\rho\in W_{p,\gamma_{1}}^{1}((0, \infty), W_{q}^{1}(\Omega)) , u\in L_{p,\gamma_{1}}((0, \infty), W_{q}^{2}(\Omega))\cap W_{p,\gamma_{1}}^{1}((0, \infty), L_{q}(\Omega))$possessing the estimate:$\Vert e^{-\gamma t}\rho\Vert_{W_{p}^{1}(0,\infty),W_{q}^{1}(\Omega))}+\Vert e^{-\gamma t}\partial_{t}u\Vert_{L_{p}((0,\infty),L_{q}(\Omega))}+\Vert e^{-\gamma t}u\Vert_{L_{p}((0,\infty)_{)}W_{q}^{2}(\Omega))}$$\leq C(\Vert\rho_{0}\Vert_{W_{q}^{1}(\Omega)}+\Vert u_{0}\Vert_{B_{q,p}^{2(1-1/p)}(\Omega)})$for any$\gamma\geq\gamma_{1}$with some constant$C$

Secondly, we show the maximal
and $u_{0}=0.$depending on $p,$

$L_{p}-L_{q}$$q,$$\gamma_{1}$and $N.$

regularity result for the equation (1.3) with$\rho_{0}=0$

Theorem 2.3. Let $1<p,$ $q<\infty,$ $N<r<\infty,$ $\max(q, q’)\leq r(q’=q/(q-1$ $0<\epsilon<$and. Assume that is a uniform
domain and that
, $h\in$
Then, there exists a positive constant such that for any$\pi/2,$$\delta_{0}>0$$W_{r}^{3-1/r}$$\Omega$$\lambda_{0}>0$$|\delta|\leq\delta_{0}.$$(f, g)\in L_{p,\gamma_{2},0}(\mathbb{R}, W_{q}^{1,0}(\Omega))$$\gamma_{2}$, andproblem (1.3) with $\rho=0$ and $u=0$ admits a unique solution$L_{p,\gamma_{2},0}(\mathbb{R}, W_{q}^{1}(\Omega)^{N})\cap H_{p,\gamma_{2},0}^{1/2}(\mathbb{R}, L_{q}(\Omega)^{N})$$\tilde{h}\in L_{p_{)}\gamma_{2)}0}(\mathbb{R}, W_{q}^{2}(\Omega))\cap W_{p,\gamma_{2},0}^{1}(\mathbb{R}, L_{q}(\Omega))$$(\rho, u)$

,

with

$\rho\in W_{p,\gamma_{2},0}^{1}(\mathbb{R}, W_{q}^{1}(\Omega)) , u\in L_{p,\gamma_{2},0}(\mathbb{R}, W_{q}^{2}(\Omega)^{N})\cap W_{p,\gamma_{2},0}^{1}(\mathbb{R}, L_{q}(\Omega)^{N})$,possessing the estimate$\Vert e^{-\gamma t}(\gamma\rho, \partial_{t}\rho)\Vert_{L_{p}(\pi,W_{q}^{1}(\Omega))}+\Vert e^{-\gamma t}(\gamma u, \partial_{t}u)\Vert_{L_{p}(\pi,L_{q}(\Omega))}+\Vert e^{-\gamma t}u\Vert_{L_{p}(\mathbb{R},W_{q}^{2}(\Omega))}$$\leq C(\Vert e^{-\gamma t}(f, g)\Vert_{L_{p}(\mathbb{R},W_{q}^{1,0}(\Omega))}+\Vert e^{-\gamma t}(\nabla h, \Lambda_{\gamma}^{1/2}h)\Vert_{L_{p}(\mathbb{R},L_{q}(\Omega))}$$+\Vert e^{-\gamma t}\tilde{h}\Vert_{W_{p}^{1}(\mathbb{R},L_{q}(\Omega))}+\Vert e^{-\gamma t}\tilde{h}\Vert_{L_{p}(\mathbb{R},W_{q}^{2}(\Omega))})$

for any$\gamma\geq\gamma_{2}$with some constant$C$depending on $N,$

$p$

and

(2.3)

$q.$

Remark 2.4. As was seen in Shibata and Shimizu [8], we know that
$H_{p,\gamma_{2},0}^{1/2}(\mathbb{R}, W_{q}^{1}(\Omega))\subset L_{p_{)}\gamma_{2},0}(\mathbb{R}, W_{q}^{2}(\Omega)^{N})\cap W_{p,\gamma_{2},0}^{1}(\mathbb{R}, L_{q}(\Omega)^{N})$,$\Vert e^{-\gamma t}\Lambda_{\gamma}^{1/2}\nabla f\Vert_{L_{p}(\pi,L_{q}(\Omega))}\leq C\{\Vert e^{-\gamma t}f\Vert_{W_{p}^{1}(\pi,L_{q}(\Omega))}+\Vert e^{-\gamma t}f\Vert_{L_{p}(\pi,W_{q}^{2}(\Omega))}\}.$

3$\mathcal{R}$-boundedness

Employing the similar argumentation to that in Enomoto and Shibata [3], Shibata andShimizu [8], we can show Theorem 2.2 and Theorem 2.3 by -boundedness of solution
operators to the generalized resolvent problem. Thus, the main part of this section isdevoted to the proof of the existence of -bounded solution operators to problem (1.5).
First, we are concerned with generalized resolvent problem (1.5).
$\mathcal{R}$$\mathcal{R}$154
Theorem 3.1. Let $1<q<\infty,$ $N<r<\infty,$ $\max(q, q’)\leq r(q’=q/(q-1$$0<\epsilon<\pi/2$ . Assume thatandis a uniform
domain. Let. Moreover, we define the space
the sets defined in (1.6) and set$W_{r}^{3-1/r}$$\Omega$$\Sigma_{\epsilon,\lambda_{0}},$$K_{\epsilon}$andbe$\mathcal{X}_{q}(\Omega)$$\Lambda_{\epsilon,\lambda_{0}}=\Sigma_{\epsilon,\lambda_{0}}\cap K_{\epsilon}$$X_{q}(\Omega)$$by$$X_{q}(\Omega)=\{(f, g, h,\tilde{h})|(f, g)\in W_{q}^{1,0}(\Omega), h\in W_{q}^{1}(\Omega)^{N}, \tilde{h}\in W_{q}^{2}(\Omega)\}.$

Then, there exist a positive constant$\lambda_{0}$and an operator family$R(\lambda)\in Ho1(\Lambda_{\epsilon,\lambda_{0}}, \mathcal{L}(\mathcal{X}_{q}(\Omega), W_{q}^{1,2}(\Omega)))$such that for any $(f, g, h,\tilde{h})\in X_{q}(\Omega)$ and$\lambda\in\Lambda_{\epsilon,\lambda_{0}},$is a unique solution to problem (1.5).
depending on
and $N$ such that$(\theta, v)=R(\lambda)(f, g, \lambda^{1/2}h, \nabla h, \lambda\tilde{h}, \lambda^{1/2}\nabla\tilde{h}, \nabla^{2}\tilde{h})$Moreover, there exists a constant$C$$\lambda_{0},$$\epsilon,$$q$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),W_{q}^{1,0}(\Omega))}(\{(\tau\partial_{\tau})^{\ell}(\lambda R(\lambda))|\lambda\in\Lambda_{\epsilon,\lambda_{0}}\})\leq C,$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),W_{q}^{1,0}(\Omega))}(\{(\tau\partial_{\tau})^{\ell}(\gamma R(\lambda))|\lambda\in\Lambda_{\epsilon,\lambda_{0}}\})\leq C,$(3.1)$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega)^{N^{2}})}(\{(\tau\partial_{\tau})^{p}(\lambda^{1/2}\nabla P_{v}R(\lambda))|\lambda\in\Lambda_{\epsilon,\lambda_{0}}\})\leq C,$$\mathcal{R}_{\mathcal{L}(\mathcal{X}_{q}(\Omega),L_{q}(\Omega)^{N^{3}})}(\{(\tau\partial_{\tau})^{\ell}(\nabla^{2}P_{v}R(\lambda))|\lambda\in\Lambda_{\epsilon,\lambda_{0}}\})\leq C$with$\lambda=\gamma+i\tau$

and $\ell=0$ , 1.

, inserting the formula: $\theta=\lambda^{-1}(f-\gamma_{1}(x)divv)$
To prove Theorem 3.1 in the caseinto the second equation in (1.5), we have$\lambda\neq 0$$\lambda v-\gamma DivS(v)-\lambda^{-1}\gamma_{0}^{-1}\nabla(\gamma_{1}\gamma_{2}divv)=g-\lambda^{-1}\gamma_{0}^{-1}\nabla(\gamma_{1}f)$.

Thus, instead of (1.5), we mainly consider the equations:$\{\begin{array}{ll}\gamma_{0}\lambda v-DivS(v)-\delta\nabla(\gamma_{3}divv)=f in\Omega\alpha[D(v)n-<D(v)n, n>n]=h-<h, n>n, v\cdot n=\tilde{h} on\Gamma,\end{array}$(3.2).

with$\gamma_{3}=\gamma_{1}\gamma_{2}$(C1). Let $0<\epsilon<\pi/2$ and$\delta=\lambda^{-1}$(C2)$\delta\in\Sigma_{\epsilon}$(C3)${\rm Re}\delta\geq 0$

and

with

$\lambda\in\Lambda_{\epsilon,\lambda_{0}}$${\rm Re}\delta<0,$, and$\lambda\in \mathbb{C}$$\lambda_{0}>0$. As , we consider the following three cases:$\delta$;$\lambda\in \mathbb{C}$

with

with

${\rm Re} \lambda\geq|\frac{Re\delta}{{\rm Im}\delta}||{\rm Im}\lambda|$${\rm Re}\lambda\geq\lambda_{0}|{\rm Im}\lambda|$

and

and

$|\lambda|\geq\lambda_{0}$;$|\lambda|\geq\lambda_{0}.$In the following, for
we assume thatin any cases of (C1), (C2) and (C3).
In particular, in (C1),
In (C2), we note that, becauseand
O. We may include the case where $\delta=0$ in (C1), which is corresponding to theLam\’e system. The case (C1) is used to prove the existence of an -bounded solution
operator pertaining to (1.5) and the cases (C2) and (C3) enable us the application ofa homotopic argument in proving the exponential stability of the analytic semigroup$\delta_{0}>0$$\delta_{0}=\lambda_{0}^{-1}$$|\delta|\leq\delta_{0}$${\rm Im}\delta\neq 0$$\delta\in\Sigma_{\epsilon}$${\rm Re}\delta<$$\mathcal{R}$155associated with (1.3) in a bounded domain. Such homotopic argument already appearedin [9] and [3] in the non-slip condition case. For the sake of simplicity, we introduce thesetdefined by$\Gamma_{\epsilon,\lambda_{0},\delta_{0}}$(3.3)$\Gamma_{\epsilon,\lambda_{0},\delta_{0}}=\{\begin{array}{ll}\Lambda_{\epsilon,\lambda_{0}} for (C1)\{\lambda\in \mathbb{C}|{\rm Re}\lambda\geq|\frac{{\rm Re}\delta}{{\rm Im}\delta}||{\rm Im}\lambda|, |\lambda|\geq\lambda_{0}\} for (C2)\{\lambda\in \mathbb{C}|{\rm Re}\lambda\geq\lambda_{0}|{\rm Im}\lambda|, |\lambda|\geq\lambda_{0}\} for (C3)\end{array}$withWe have Theorem 3.1 immediately with help of the following Theorem 3.2 in the case
(C1).
$|\delta|\leq\delta_{0}.$Theorem 3.2. Let $1<q<\infty,$ $N<r<\infty,$ $\max(q, q’)\leq r(q’=q/(q-1$ $0<\epsilon<\pi/2,$
. Let
and
O. Assume thatis a uniform
domain and thatbe the set defined in (3.3). Set$\delta_{0}>0$$W_{r}^{3-1/r}$$\Omega$$\lambda_{0}>$$|\delta|\leq\delta_{0}$$\Gamma_{\epsilon,\lambda_{0},\delta_{0}}$$Y_{q}(\Omega)=\{(f, h,\tilde{h})|f\in L_{q}(\Omega)^{N}, h\in W_{q}^{1}(\Omega)^{N}, \tilde{h}\in W_{q}^{2}(\Omega)\},$$\mathcal{Y}_{q}(\Omega)=\{F=(F_{2}, \ldots, F_{7})|F_{2}, F_{3}, F_{6}\in L_{q}(\Omega)^{N}, F_{4}, F_{7}\in L_{q}(\Omega)^{N^{2}} F_{5}\in L_{q}(\Omega)\}.$(3.4)
Then, there exist a positive constantand an operator familysuch that for any
is a unique solution to problem (3.2), and$\lambda_{0}$$v=\mathcal{A}(\lambda)F_{\lambda}(f, h,\tilde{h})$andsatisfies the following$(f, h,\tilde{h})\in Y_{q}(\Omega)$$\mathcal{A}(\lambda)\in Ho1(\Gamma_{\epsilon,\lambda_{0},\delta_{0}}, \mathcal{L}(\mathcal{Y}_{q}(\Omega), W_{q}^{2}(\Omega)^{N}))$$\mathcal{A}(\lambda)$$\lambda\in\Lambda_{\epsilon,\lambda_{0}},$estimates:$\mathcal{R}_{\mathcal{L}(\mathcal{Y}_{q}(\Omega),L_{q}(\Omega)^{N^{-}})}(\{(\tau\frac{d}{d\tau})^{\ell}(G_{\lambda}\mathcal{A}(\lambda))|\lambda\in\Lambda_{\epsilon,\lambda_{0}}\})\leq C (\ell=0,1)$

with some constantdepending on
lowing, we set $\tilde{N}=N^{3}+N^{2}+2N,$$C$$(f, \lambda^{1/2}h, \nabla h, \lambda\tilde{h}, \lambda^{1/2}\nabla\tilde{h}, \nabla^{2}\tilde{h})$$\lambda_{0},$$\epsilon,$$\delta_{0},$

$a,$

$b,$

$q$

and N. Here and in the fol, and$G_{\lambda}v=(\lambda v, \gamma v, \lambda^{1/2}\nabla v, \nabla^{2}v)$$F_{\lambda}(f, h,\tilde{h})=$.

, we are concerned with time dependent problem (1.3). Let
operator defined by$s_{eCOI1}dly$$\mathcal{B}(\theta, v)=(-\gamma_{2}divv, \gamma DivS(v)-\gamma_{0}^{-1}\nabla(\gamma_{1}\theta))$for$\mathcal{B}$be a linear$(\theta, v)\in W_{q}^{1}(\Omega)\cross \mathcal{D}_{q}(\Omega)$with $\mathcal{D}_{q}(\Omega)=\{v\in W_{q}^{2}(\Omega)| [D(v)n-< D(v)n, n>n]|_{\Gamma}=0, v\cdot n=0\}$ . Since
Definition 1.2 with $n=1$ implies the boundedness of the operator family , it follows
from Theorem 3.1 thatis contained in the resolvent set of and for any$(f,g,h,\tilde{h})\in
X_{q}(\Omega)$,and$\mathcal{T}$$\mathcal{B}$$\lambda\in\Lambda_{\epsilon,\lambda_{0}}$$\Lambda_{\epsilon,\lambda_{0}}$$|\lambda|\Vert\theta\Vert_{W_{q}^{1}(\Omega)}+\Vert(\lambda v, \lambda^{1/2}\nabla v, \nabla^{2}v)\Vert_{L_{q}(\Omega)}$$\leq C(\Vert(f, g)\Vert_{W_{q}^{1,0}(\Omega)}+\Vert(\lambda^{1/2}h, \nabla h)\Vert_{L_{q}(\Omega)}+\Vert(\lambda\tilde{h}, \lambda^{1/2}\nabla\tilde{h}, \nabla^{2}\tilde{h})\Vert_{L_{q}(\Omega)})$By (3.5), we have the following theorem.

.

(3.5)

156
heorem 3.3. Let $1<q<\infty,$ $N<r<\infty,$ $\max(q, q’)\leq r(q’=q/(q-1$ $0<$
is a uniform
domain and that
O. Assume thatand.
. Then, the operator generates an analytic semigroupon$(f, g)\in W_{q}^{1,0}(\Omega)$,
Moreover, there exists constants
and $M>0$ such that for any$(\rho(t), u(t))=T(t)(f, g)$ solves (1.3) with $f=0,$ $g=0,$ $h=0$ andand satisfies thefollowing estimate:$T$$\epsilon<\pi/2,$$\delta_{0}>0$$W_{r}^{3-1/r}$$\Omega$$\lambda_{0}>$$|\delta|\leq\delta_{0}$$W^{1,0}(\Omega)$$\{T(t)\}_{t\geq 0}$$\mathcal{B}$$\gamma_{0}>0$$\tilde{h}=0$$\Vert T(t)(f, g)\Vert_{W_{q}^{1,0}(\Omega)}+t^{1/2}\Vert\nabla P_{v}T(t)(f, g)\Vert_{L_{q}(\Omega)}+t\Vert\nabla^{2}P_{v}T(t)(f, g)\Vert_{L_{q}(\Omega)}$$\leq Me^{\gamma 0t}\Vert(f, g)\Vert_{W_{q}^{1,0}(\Omega)}$

.

(3.6)

Combining Theorem 3.3 with a real interpolation method (cf. Shibata and Shimizu[8, Proof of Theorem 3.9]), we have Theorem 2.2.
Employing the similar argumentation to that in Shibata and Shimizu [8], we can show
the existence part of Theorem 2.3 by using Theorem 3.1 and Theorem 1.3. Moreover, theuniqueness of solutions to (1.3) can be proved by using the existence of solutions to thedual problem as was seen also in Shibata and Shimizu [8]. Thus, we may omit the proofof Theorem 2.2 and Theorem 2.3.

4

Local in time unique existence theorem for nonlinear problem (1.1)

As was done in Burnat and Zajaczkowski [2] and Kobayashi and Zajaczkowski [5], inorder to prove Theorem 2.1, we formulate (1.1) in Lagrangian coordinates. Let velocity
fields $v(\xi, t)$ and $u(x, t)$ be known as vectors of functions of the Lagrange coordinates
and the Euler coordinates of the same fluid particle, respectively. In this case, theconnection between the Lagrange coordinate and the Euler coordinate is written in the
form:
$x$

$\xi$

$x= \xi+\int_{0}^{t}v(\xi, s)ds\equiv X_{v}(\xi, t)$

(4.1)

,

and $v(\xi, t)=u(x, t)$ . Let
be the Jacobi matrix of the transformation $x=X_{v}(\xi, t)$with element. There exists a small number such that$\det
A_{v}\neq0$, provided thatis invertible, that is$A_{v}$$a_{ij}= \delta_{ij}+\int_{0}^{t}(\partial v_{i}/\partial\xi_{j})(\xi, s)ds$$\sigma$$A_{v}$$i,j1 \ldots,N\max_{=}\Vert\int_{0}^{t}(\partial v_{i}/\partial\xi_{j})(\cdot, s)ds\Vert_{L_{\infty}(\Omega)}<\sigma (0<t<T)$

.

(4.2)

In this case, we have $\nabla_{x}=A_{v}^{-1}\nabla_{\xi}=(I+V_{0}(\int_{0}^{t}\nabla v(\xi, s)ds))\nabla\xi$ , where $V_{0}(K)$ is an$N\cross N$ matrix offunctions with respect to $K=(k_{ij})$ defined on $|K|<2\sigma$ and$V_{0}(0)=0$ , where
are corresponding variables to. Assume thatand $u(x, t)$ are solutions of (1.1) in the Euler coordinate. Setting $\rho(X_{v}(\xi, t), t)=$, we write (1.1) in the Lagrangian coordinate introduced by (4.1) as$C^{\infty}$$k_{ij}$$\rho(x, t)$$\rho_{*}+\theta_{0}(\xi)+\theta(\xi, t)$$\int_{0}^{t}(\partial v_{i}/\partial\xi_{j})(\cdot, s)ds$

157
follows:$\{\begin{array}{ll}\partial_{t}\theta+(\rho_{*}+\theta_{0})divv=F(\theta, v) in \Omega\cross(0, T) ,(\rho_{*}+\theta_{0})\partial_{t}v-DivS(v)+\nabla(P’(\rho_{*}+\theta_{0})\theta)=g+G(\theta, v) in \Omega\cross(0, T) ,\alpha[D(v)n-<D(v)n, n>n]=H(v) on \Gamma\cross(0, T) , (4.3)v\cdot n=-v\cdot(\hat{n}_{v}-n) on \Gamma\cross(0, T) ,(\theta, v)|_{t=0}=(0, u_{0}) in \Omega,\end{array}$where $\hat{n}_{v}=n(X_{v}(\xi, t g=-P’(\rho_{*}+\theta_{0})\nabla\theta_{0}$ andfunctions of the following forms:$F(\theta, v),$ $G(\theta, v)$

and $H(v)$ are nonlinear$F( \theta, v)=-\theta divv-(\rho_{*}+\theta_{0}+\theta)V_{1}(\int_{0}^{t}\nablavds)\nabla v,$$G( \theta, v)=-\theta\partial_{t}v+Div[\alpha V_{2}(\int_{0}^{t}\nabla vds)\nabla v+(\beta-\alpha)V_{1}(\int_{0}^{t}\nabla vds)\nabla vI]$$+ V_{3}(\int_{0}^{t}\nabla vds)\nabla[\alpha D(v)+\alpha V_{2}(\int_{0}^{t}\nabla vds)\nabla v$$+( \beta-\alpha)divvI+(\beta-\alpha)V_{1}(\int_{0}^{t}\nabla vds)\nabla v]$$- \nabla(\int_{0}^{1}P"(\rho_{*}+\theta_{0}+\tau\theta)(1-\tau)d\tau\theta^{2})-P’(\rho_{*}+\theta_{0}+\theta)V_{0}(\int_{0}^{t}\nabla vds)\nabla(\theta_{0}+\theta)$,$H(v)=-\alpha D(v)(\hat{n}_{v}-n)-\alpha\{<D(v)\hat{n}_{v}, \hat{n}_{v}>\hat{n}_{v}-<D(v)n, n>n\}$$- \alpha\{(V_{2}(\int_{0}^{t}\nabla vds)\nabla v)\hat{n}_{v}-<(V_{2}(\int_{0}^{t}\nabla vds)\nabla v)\hat{n}_{v}, \hat{n}_{v}>\hat{n}_{v}\}$

Here, $V_{0}(K)$ , $V_{1}(K)$ , $V_{2}(K)$ and $V_{3}(K)$ are some matrices of
to $K$ defined on $|K|\leq\sigma$ , which satisfy conditions:$C^{\infty}$

.

(4.4)

functions with respect(4.5)$V_{0}(0)=0, V_{1}(0)=0, V_{2}(0)=0, V_{3}(0)=0$

and relations: $A_{v}^{-1}=I+ V_{0}(\int_{0}^{t}\nabla vds)$ , $div_{x}w=div_{\xi}\hat{w}+V_{1}(\int_{0}^{t}\nabla\hat{w}ds)\nabla w,$ $D_{x}(w)=$and $Div_{x}K(w)=Div_{\xi}\hat{K}+V_{3}(\int_{0}^{t}\nabla\hat{w}ds)\nabla\hat{K}$ with$w(X_{v}(\xi, t), t)$ and $\hat{K}=K(X_{v}(\xi, t), t)$ .
Since we can show eventually that the correspondence $x=X_{v}(\xi, t)$ is invertible byusing the argument due to Str\"ohmer [10], our main task is to prove the following theorem.
$\hat{w}=$$D_{\xi}(\hat{w})+V_{2}(\int_{0}^{t}\nabla\hat{w}ds)\nabla\hat{w}$Theorem 4.1. Let $N<q<\infty,$ $2<p<\infty$ and $R>$ O. If the initial datafor(1.1) satisfy the condition (2.1) and, then there exists atime $T>0$ depending on such that the problem $(4.3)q’ p$ admits a unique solution
with$(\theta_{0}, u_{0})$$\Vert\theta_{0}\Vert_{W_{q}^{1}(\Omega)}+\Vert u_{0}\Vert_{B^{2(1-1/p)}}(\Omega)\leq R$$R$$\theta\in W_{p}^{1}((0, T), W_{q}^{1}(\Omega)) , v\in W_{p}^{1}(0, T) , L_{q}(\Omega))\cap L_{p}((0, T), W_{q}^{2}(\Omega))$$(\theta, v)$.

158
To prove Theorem 4.1, first we show the maximal
time local linear problem:$L_{p}-L_{q}$regularity for the following$\{\begin{array}{ll}\partial_{t}\rho+\gamma_{2}divu=f, \gamma_{0}\partial_{t}u-DivS(u)+\nabla(\gamma_{1}\rho)=g in\Omega\cross(O, T) ,\alpha[D(u)11-<D(u)n, n>n]=h-<h, n>n, u\cdot n=\tilde{h} on \Gamma\cross(0, T),(\rho, u)|_{t=0}=(\rho_{0}, u_{0}) in \Omega.\end{array}$(4.6)
For this purpose, we have to replace the nonlocal operator
bywith value in. For this purpose, according to Shibata [7], wethe local operator with value inhaving the following properties:
introduce the extension map :$\Lambda_{\gamma}^{1/2}$$L_{q}(\Omega)$$W_{q}^{-1}(\Omega)$$\partial_{t}$$L_{1,1oc}(\Omega)arrow L_{1,1oc}(\mathbb{R}^{N})$$\iota$,(e-1) For any $1<q<\infty$ and$\ell=0$, 1 with some constantfor$f\in W_{q}^{1}(\Omega)$$f\in W_{q}^{1}(\Omega)$$C_{q}$$q,$$r$, $\iota f=f$ in
depending on$\iota f\in W_{q}^{1}(\mathbb{R}^{N})$$C_{q}\Vert f\Vert_{W_{q}^{\ell}(\Omega)}$(e-2) For any $1<q<\infty$ anddepending onsome constant),$C_{q}$$\Omega$

$q,$

and
and

$r$

$\Vert\iota f\Vert_{W_{q}^{\ell}(\mathbb{R}^{N})}\leq$$\Omega.$$\Vert(1-\triangle)^{-1/2}\iota(\nabla f)\Vert_{L_{q}(\mathbb{R}^{N})}\leq C_{q}\Vert f\Vert_{L_{q}(\Omega)}$

and

with

$\Omega.$

Here, $(1-\triangle)^{-1/2}$ is the operator defined by
the following, such extension map is fixed. We define$(1-\triangle)^{-1/2}f=\mathcal{F}_{\xi}^{-1}[(1+|\xi|^{2})^{1/4}\mathcal{F}[f](\xi)]$$\iota$$W_{q}^{-1}(\Omega)$

. In

by

$W_{q}^{-1}(\Omega)=\{f\in L_{1,1oc}(\Omega)|(1-\triangle)^{-1/2}\iota f\in L_{q}(\Omega)\}.$

Employing the similar argumentation to the proof of Proposition 2.8 in Shibata andShimizu [8] (cf. also the appendix in Shibata [7]), we have
$W_{p_{)}\gamma_{0)}0}^{1}(\mathbb{R}, W_{q}^{-1}(\Omega))\cap L_{p,\gamma 0,0}(\mathbb{R}, W_{q}^{1}(\Omega))\subset H_{p,\gamma 0,0}^{1/2}(\mathbb{R}, L_{q}(\Omega))$

,

(4.7)

$\Vert e^{-\gamma t}\Lambda_{\gamma}^{1/2}f\Vert_{L_{p}(\mathbb{R},L_{q}(\Omega))}\leq C\{\Vert e^{-\gamma t}\partial_{t}[(1-\triangle)^{-1/2}(\iota f)]\Vert_{L_{p}(\mathbb{R},L_{q}(\Omega))}+\Vert e^{-\gamma t}f\Vert_{L_{p}((\mathbb{R},W_{q}^{1}(\Omega))}\}.$(4.8)

Combining Theorem 2.2, Theorem 2.3 and (4.7), we have the following theorem.
Theorem 4.2. Let $1<p,$ $q<\infty,$ $N<r<\infty$ and $\max(q, q’)\leq r(q’=q/(q-1$Letandbe any positive numbers andbe the same set as in (2.2). Assume
that
Then, there
satisfies the range condition (2.1) andexists a positive number
depending on and such that for any initial data
and right members $f,$ $g,$ and with$T$

$B_{q,p}^{2(1-1/p)}(\Omega)$$R$$\theta_{0}\in W_{q}^{1}(\Omega)$$\Vert\nabla\theta_{0}\Vert_{L_{q}(\Omega)}\leq R.$$R$$\gamma_{0}=\gamma_{0}(R)$$(\rho_{0}, u_{0})\in W_{q}^{1}(\Omega)\cross B_{q,p}^{2(1-1/p)}(\Omega)$$\rho_{*}$$\tilde{h}$$h$$f\in W_{p}^{1}((0, T), W_{q}^{1}(\Omega)^{N})) , g\in L_{p}((0, T), L_{q}(\Omega))$$h\in L_{p}((0, T), W_{q}^{1}(\Omega))\cap W_{p}^{1}((0, T), W_{q}^{-1}(\Omega))$,,$\tilde{h}\in L_{p}((0, T), W_{q}^{2}(\Omega)\cap W_{p}^{1}((0, T), L_{q}(\Omega))$satisfying the conditions: $h|_{t=0}=0$ andand with$\rho$$\tilde{h}|_{t=0}=0$, problem (4.6) admit unique solutions$u$$\rho\in W_{p}^{1}((0, T), W_{q}^{1}(\Omega)) , u\in L_{p}((0, T), W_{q}^{2}(\Omega))\cap W_{p}^{1}((0, T), L_{q}(\Omega))$159
possessing the estimate:$\Vert\rho\Vert_{W_{p}^{1}((0,t),W_{q}^{1}(\Omega))}+\Vert u\Vert_{L_{p}((0,t),W_{q}^{2}(\Omega))}+\Vert u\Vert_{W_{p}^{1}((0,t),L_{q}(\Omega))}$$\leq C(R)e^{\gamma t}\{\Vert(\rho_{0}, u_{0})\Vert_{E_{q,p}(\Omega)}+\Vert(f, g)\Vert_{L_{p}((0,t),W_{q}^{1,0}(\Omega))}+\Vert h\Vert_{L_{p}((0,t),W_{q}^{1}(\Omega))}$$+\Vert\partial_{s}[(1-\triangle)^{-1/2}\iota h]\Vert_{L_{p}((0,t),L_{q}(\Omega))}+\Vert\tilde{h}\Vert_{L_{p}((0,t),W_{q}^{2}(\Omega))}+\Vert\tilde{h}\Vert_{W_{p}^{1}((0,t),L_{q}(\Omega))}\}$

for any $t\in(0, T] and \gamma\geq\gamma_{0} with some$ constant $C(R)$ depending on
and $t\in(O, T$].
of$R$

but independent$\gamma\geq\gamma_{0}$Proof of Theorem 4.1. In the following, we assume that $2<p<\infty$ and $N<q<$is a uniform. By Sobolev’s embedding theorem
domain in
we have$\infty$, that$W_{q}^{3-1/q}$$\Omega$$\mathbb{R}^{N}(N\geq 2)$$W_{q}^{1}( \Omega)\subset L_{\infty}(\Omega) , \Vert\prod_{j=1}^{m}f_{j}\Vert_{W_{q}^{1}(\Omega)}\leq C\prod_{j=1}^{m}\Vert f_{j}\Vert_{W_{q}^{1}(\Omega)}$

Let

$T$

and

$L$

be any positive numbers and we define a space$\mathcal{I}_{L,T}=\{(\theta, v)|\theta\in W_{p}^{1}((0, T), W_{q}^{1}(\Omega))$$(\theta, v)|_{t=0}=(0, u_{0})$in$\Omega,$,$\mathcal{I}_{L,T}$

.

(4.9)

by

$v\in W_{p}^{1}((0, T), L_{q}(\Omega))\cap L_{p}((0, T), W_{q}^{2}(\Omega))$$\Vert\theta\Vert_{W_{p}^{1}((0,T),W_{q}^{1}(\Omega))}+\mathbb{I}_{v}(0, T)\leq L\})$(4.10)

where we have set. Since we choose$T>0$ small enough eventually, we may assume that $0<T\leq 1$ in the following. Given, let and be solutions to problem:$\mathbb{I}_{v}(0, T)=\Vert v\Vert_{L_{p}((0,T),W_{q}^{2}(\Omega))}+\Vert\partial_{t}v\Vert_{L_{p}((0,T),L_{q}(\Omega))}$$(\kappa, w)\in \mathcal{I}_{L,T}$$\theta$$v$$\{\begin{array}{ll}\partial_{t}\theta+(\rho_{*}+\theta_{0})divv=F(\kappa, w) in \Omega\cross(0, T) ,(\rho_{*}+\theta_{0})\partial_{t}v-DivS(v)+\nabla(P’(\rho_{*}+\theta_{0})\theta)=g+G(\kappa, w) in \Omega\cross(0, T) ,\alpha[D(v)n-\leq D(v)n, n>n]=H(w) on \Gamma\cross(0, T) ,v\cdot n=-w\cdot(\hat{n}_{w}-n) on \Gamma\cross(0, T) ,(\theta, v)|_{t=0}=(0, u_{0}) in \Omega,\end{array}$(4.11)First, we estimate the right-hand sides of (4.11). By (4.9), H\"older’s inequality and theidentity:)s) $ds$ , we have$t$$\kappa$$= \int_{0}^{t}\partial_{s}\kappa$$\sup_{t\in(0,T)}\Vert\int_{0}^{t}\nabla w$$s)ds\Vert_{L_{\infty}(\Omega)}\leq M_{1}T^{1/p’}L,$$\sup_{t\in(0,T)}\Vert\int_{0}^{t}\nabla w$$s)ds\Vert_{W_{q}^{1}(\Omega)}\leq CT^{1/p’}L,$$\sup_{t\in(0,T)}\Vert\kappa(\cdot, t)\Vert_{L_{\infty}(\Omega)}\leq M_{1}T^{1/p’}L, \sup_{t\in(0,T)}\Vert\kappa(\cdot, t)\Vert_{W_{q}^{1}(\Omega)}\leq CT^{1/p’}L,$(4.12)

with $p’=p/(p-1)$ . Here and in the following, denotes a generic constant independentof andand we use the lettersto denote a special constants independent ofand . The value of may change from line to line. To treat polynomials of functions
$C$

$T$

$R$

$L$

$M_{1}$

$C$

$T$

160
with respect to$\kappa$$M_{1}\tau^{1/p’}L\leq\rho_{*}/4$

and
and

$\int_{0}^{t}\nabla w$

so small that$T$

$M_{1}\tau^{1/p’}L\leq\sigma$$\rho_{*}/4\leq\rho_{*}+\theta_{0}+\tau\kappa\leq 4\rho_{*}$

Recall thats) $ds$ , in view of (2.1) and (4.2) we choose, so that$( \tau\in[0,1 \sup_{t\in(0,T)}\Vert\int_{0}^{t}\nabla w s)ds\Vert_{L_{\infty}(\Omega)}<\sigma$$\Vert\theta_{0}\Vert_{W_{q}^{1}(\Omega)}+\Vert u_{0}\Vert_{B_{q,p}^{2(1-1/p)}}(\Omega)\leq R$.

(4.13). By (4.12), (4.13), and (4.5), we have$\sup_{t\in(0,T)}\Vert V_{i}(\int_{0}^{t}\nabla w t)ds)\Vert_{W_{q}^{1}(\Omega)}\leq CT^{1/p’}L,$$\sup_{t\in(0,T)}\Vert\nabla W(\int_{0}^{t}\nabla w t)ds)\Vert_{L_{q}(\Omega)}\leq CT^{1/p’}L$$\sup_{t\in(0,T)}\Vert\nabla\int_{0}^{1}P"(\rho_{*}+\theta_{0}+\tau\kappa)(1-\tau)d\tau\Vert_{L_{q}(\Omega)}\leq C(R+T^{1/p’}L)$,(4.14)

where $i=1$ , 2, 4, 5 and 6, and $W=W(K)$ is any matrix of polynomials with respect toK. By (4.9), (4.4), (4.12), (4.13) and (4.14), we have$\Vert F(\kappa, w)\Vert_{W_{p}^{1}((0,T),W_{q}^{1}(\Omega))}\leq C(L^{2}T^{1/p’}+L^{3}(T^{1/p’})^{2})$,$\Vert g\Vert_{L_{p}((0,T),L_{q}(\Omega))}\leq CRT^{1/p},$$\Vert G(\kappa, w)\Vert)\leq C\{L^{2}T^{1/p’}+L^{3}(T^{1/p’})^{2}$$+(1+R+LT^{1/p’})(LT^{1/p’})^{2}T^{1/p}+(R+LT^{1/p’})(LT^{1/p’})T^{1/p}\}.$(4.15)

To obtain$\sup_{t\in(0,T)}\Vert w(\cdot, t)\Vert_{B_{q,p}^{2(1-1/p)}}(\Omega)\leq C(\mathbb{I}_{w}(0, T)+e^{\gamma T}\Vert u_{0}\Vert_{B_{q,p}^{2(1-1/p)}}(\Omega))$,(4.16)

we use the embedding relation:(4.17)$L_{p}((0, \infty), X_{1})\cap W_{p}^{1}((0, \infty), X_{0})\subset BUC(J, [X_{0}, X_{1}]_{1-1/p,p})$for any two Banach spaces[1]). Sinceand (4.13)$X_{0}$

such thatis dense in
and $1<p<\infty$ (cf.
as follows from the assumption: $2<p<\infty$ , by (4.16)

and$B_{q,p}^{2(1-1/p)}(\Omega)\subset W_{q}^{1}(\Omega)$$X_{1}$$X_{1}$$X_{0}$$\sup_{t\in(0,T)}\Vert w(\cdot, t)\Vert_{W_{q}^{1}(\Omega)}\leq C(L+e^{\gamma T}\Vert u_{0}\Vert_{B_{q,p}^{2(1-1/p)}(\Omega)})$,(4.18)$\sup_{t\in(0,T)}\Vert\partial_{t}W(\int_{0}^{t}\nabla w s)ds)\Vert_{L_{q}(\Omega)}\leq C(L+e^{\gamma T}1u_{0}\Vert_{B_{q,p}^{2(1-1/p)}}(\Omega))$.

In the following, we assume that $LT^{1/p’}\leq 1$ and $0<\sigma\leq 1$ to obtain$\sup_{t\in(0,T)}\Vert\hat{n}_{w}-n\Vert_{W_{q}^{1}(\Omega)}\leq C(R+L)T$

we writeestimates.

$\hat{n}_{w}-n=\int_{0}^{1}(\nabla n)(\xi+\tau\int_{0}^{t}w(\xi, \mathcal{S})ds)d\tau(\int_{0}^{t}w(\xi, s)ds)$$\Vert n\Vert_{W_{\infty}^{1}(\pi)}N\leq C,$$\Vert\hat{n}_{w}\Vert_{W_{\infty}^{1}(\Omega)}\leq C,$(4.19). Using the following$\Vert(\nabla n)(\cdot+\tau\int_{0}^{t}w s)$ $ds)\Vert_{L_{\infty}(\Omega)}\leq C(\tau\in[0$,1(4.20)

161(4.9) and (4.18) we have $\Vert\hat{n}_{w}-n\Vert_{L_{q}(\Omega)}\leq C(R+L)T$ . To estimate
with$\nabla(\hat{n}_{w}-n)$, we write$\nabla(\hat{n}_{w}-n)=A_{1}+A_{2}$$A_{1}= \int_{0}^{1}(\nabla n)(\xi+\tau\int_{0}^{t}w(\xi, s)ds)d\tau(\int_{0}^{t}\nabla w(\xi, s)ds)$,$A_{2}= \int_{0}^{1}(\nabla^{2}n)(\xi+\tau\int_{0}^{t}w(\xi, s)ds)(I+\tau\int_{0}^{t}\nabla w(\xi, s)ds)d\tau(\int_{0}^{t}w(\xi, s)ds)$.

By (4.18) and (4.20), we have
. On the other hand, by propertiesof uniform
domain and (4.12) we have
which combined with (4.18) furnishes (4.19). We also have$\Vert A_{1}\Vert_{L_{q}(\Omega)}\leq C(R+L)T$$W_{r}^{3-1/r}$$\Vert A_{2}\Vert_{L_{q}(\Omega)}\leq\sup_{t\in(0},{}_{T)}CT\Vert w(\cdot, t)\Vert_{W_{q}^{1}(\Omega)},$$\Vert f(\nabla^{2}n)\Vert_{L_{q}(\Omega)}\leq C\Vert f\Vert_{L_{q}(\Omega)}, \Vert f(\nabla^{2}\hat{n}_{w})\Vert_{L_{q}(\Omega)}\leq C\Vert f\Vert_{W_{q}^{1}(\Omega)}$.

(4.21)By (4.9), (4.14), (4.18), (4.19), (4.20), and (4.21) we have$\Vert H(w)\Vert_{L_{p}((0,T),W_{q}^{1}(\Omega))}\leq C((R+L)LT+L^{2}T^{1/p’})$,$\Vert w\cdot(\hat{n}_{w}-n)\Vert_{L_{p}((0,T),W_{q}^{2}(\Omega))}\leq C((R+L)LT+(R+L)T^{1/p})$Since$\partial_{t}\hat{n}_{w}=(\nabla n)(X_{w})w$(4.22).

, by (4.20) we have$\Vert\partial_{t}\hat{n}_{w}\Vert_{L_{q}(\Omega)}\leq C\Vert w(\cdot, t)\Vert_{L_{q}(\Omega)}$,(4.23)

so that by (4.9), (4.18), (4.19) and (4.23), we have$\Vert\partial_{t}(w\cdot(\hat{n}_{w}-n))\Vert_{L_{p}((0,T),L_{q}(\Omega))}\leq C((R+L)LT+(R+L)^{2}T^{1/p})$To estimate$\Vert\partial_{t}[(1-\triangle)^{-1/2}\iota H(w)]\Vert_{L_{p}((0,T),L_{q}(\pi^{N})}$.

(4.24), we prepare the following lemma.

Lemma 4.3. Let $1<p<\infty,$ $N<q,$ $r<\infty$ and let be a uniformbe the extension map satisfying the properties (e-1) and (e-2). Then,$\Omega$$W_{r}^{3-1/r}$

domain. Let$\iota$$\Vert\partial_{t}[(1-\triangle)^{-1/2}\iota((\nabla f)g)]\Vert_{L_{p}((0,T),L_{q}(\pi))}N$$\leq C\{(\int_{0}^{T}(\Vert\partial_{t}f(\cdot, t)\Vert_{L_{q}(\Omega)}\Vert g(\cdot, t)\Vert_{W_{q}^{1}(\Omega)})^{p}dt)^{1/p}$$+( \int_{0}^{T}(\Vert\nabla f(\cdot, t)\Vert_{L_{q}(\Omega)}\Vert\partial_{t}g(\cdot, t)\Vert_{L_{q}(\Omega)})^{p}dt)^{1/p}\}.$

Applying Lemma 4.3 and using (4.20) and (4.23), we have$\Vert\partial_{t}[(1-\triangle)^{-1/2}H(w)]\Vert_{L_{p}((0,T),L_{q}(R^{N}))}^{p}\leq C\int_{0}^{t}\Vert\partial_{s}w(\cdot, s)\Vert_{L_{q}(\Omega)}^{p}\Vert\hat{n}_{w}-n\Vert_{W_{q}^{1}(\Omega)}^{p}ds$$+ \int_{0}^{t}\Vert\nabla w(\cdot, s)\Vert_{L_{q}(\Omega)}^{p}\Vert w(\cdot, s)\Vert_{L_{q}(\Omega)}^{p}ds+\int_{0}^{t}\Vert\partial_{s}w(\cdot, s)\Vert_{L_{q}(\Omega)}^{p}\Vert V_{2}(\int_{0}^{s}\nabla w r)dr)\Vert_{W_{q}^{1}(\Omega)}^{p}ds$$+ \int_{0}^{t}(\Vert\nabla w(\cdot, s)\Vert_{L_{q}(\Omega)}^{2}+\Vert V_{2}(\int_{0}^{s}\nabla w r)dr)\Vert_{L_{\infty}(\Omega)}\Vert\nabla w(\cdot, s)\Vert_{L_{q}(\Omega)}\Vert w(\cdot, s)\Vert_{L_{q}(\Omega)})^{p}d_{\mathcal{S}}.$(4.25)

162
Applying (4.13), (4.14), (4.18) and (4.19) to the right-hand side of (4.25), we have$\Vert\partial_{t}[(1-\triangle)^{-1/2}H(w)]\Vert_{L_{p}((0,T),L_{q}(\mathbb{R}^{N}))}\leq C((R+L)LT+(R+L)^{2}T+L^{2}T^{1/p’})$Noting that satisfies the condition (2.1) and(4.22), (4.24) and (4.26), we have$\theta_{0}$$\Vert\nabla\theta_{0}\Vert_{L_{q}(\Omega)}\leq R$. (4.26), by Theorem 4.2, (4.15),$\Vert\theta\Vert_{W_{p}^{1}((0,T)_{)}W_{q}^{i}(\Omega))}+\mathbb{I}_{v}(0, T)\leq C(R)\{\Vert u_{0}\Vert_{B_{q,p}^{2(1-1/p)}}(\Omega)+C(R, L, T$(4.27)

where we have set$C(R, L, T)=M_{2}\{L^{2}T^{1/p’}+L^{3}(T^{1/p’})^{2}+(1+R+LT^{1/p’})(LT^{1/p’})^{2}T^{1/p}$$+(R+LT^{1/p’})(LT^{1/p’})T^{1/p}+(R+L)LT+(R+L)^{2}T+(R+L)T^{1/p}\}$independent of and . Recalling thatwith some special constant, setting $L=2C(R)R$ and choosing $T\in(0,1)$ so small that $C(R, L, T)\leq R$ , by (4.27)
we have(4.28).
$R$$M_{2}$$T$$\Vert u_{0}\Vert_{B_{q,p}^{2(1-1/p)}(\Omega)}\leq$$R$$\Vert\theta\Vert_{W_{p}^{1}((0,T)_{)}W_{q}^{1}(\Omega))}+\mathbb{I}_{v}(0, T)\leq L$

If we define a map
by $\Phi(\kappa, w)=(\theta, v)$ , then by (4.28)
is the map from,
into itself. Considering the diﬀerencefor
employing the same argument and choosing $T\in(0,1)$ smaller if necessary, we see that, so that by the Banach fixed point theorem there exists
is a contraction map on
a unique fixed pointsuch that $(\theta, v)=\Phi(\theta, v)$ which solves the nonlinear
equation (4.3). Since the existence of solutions to (4.3) is proved by the contraction
mapping principle, the uniqueness of solutions automatically follows, which completes
the proof of Theorem 4.1.
$\Phi$$\Phi$$\Phi(\kappa_{1}, w_{1})-\Phi(\kappa_{2}, w_{2})$$\Phi$$\mathcal{I}_{L,T}$$(\kappa_{i_{\rangle}}w_{i})\in \mathcal{I}_{L,T}(i=1,2)$$\mathcal{I}_{L,T}$$(\theta, v)\in \mathcal{I}_{L,T}$

Proof of Theorem 2.1. Following Str\"ohmer [10], we see that the correspondence $x=$
is injective for any $t\in[0, T]$ provided that (4.2) holds with some small constant$\sigma>0$ . Let
be defined by $F(x)=0$ locally. Sinceis parallel toso, which furnishes thatthat the fact thaton implies that
$F(X_{v}(\xi, t))=F(X_{v}(\xi, 0))=F(\xi)=0$ for. Therefore, we havefor each $t\in(0, T)$ . Note that
. Let us fix$t\in(O, T)$ . Employing the same argument as in proving the inverse mapping theorem for
map, we see thata non-degenerate
gives us a local diﬀeomorphism. We see that$\{X_{v}(\xi, t)|\xi\in\Gamma\}$ is non-empty, open and closed subset of, so that the connectednessof implies that $\Gamma=\{X_{v}(\xi, t)|\xi\in\Gamma\}$ . Thus, we also see that
because of the injectivity of. Since is a connected open set, we also see that$\Omega=\{X_{v}(\xi, t)|\xi\in\Omega\}$ . Moreover, the inverse map: $\xi=X_{v}^{-1}(x, t)\in W_{p}^{1}((0, T), W_{q}^{2}(\Omega))$ ,and therefore setting $\rho=\theta(X_{v}^{-1}(x, t), t)$ and $u=v(X_{v}^{-1}(x, t), t)$ , we see thatsatisfies the non-linear equations (1.1) and$X_{v}(\xi, t)$$\Gamma$$(\nabla F)(X_{v}(\xi,$ $t$$\hat{n}_{v}$$v\cdot\hat{n}_{v}=0$$\partial F(X_{v}(\xi, t))/\partial t=0$$\Gamma$$\{X_{v}(\xi, t)|\xi\in\Gamma\}\subset$$\xi\in\Gamma$$\Gamma$$X_{v}\in C^{0}([0, T], W_{q}^{2}(\Omega)\cap W_{p}^{1}((0, T), W_{q}^{2}(\Omega))$$C^{1}$$X_{v}$$\Gamma$$\Gamma$$\{X_{v}(\xi, t)|\xi\in\Omega\}\subset\Omega,$$X_{v}$$\Omega$$(\rho, u)$$\rho\in W_{q}^{1}((0, T), L_{q}(\Omega))\cap L_{p}((0, T), W_{q}^{1}(\Omega))$,$u\in W_{p}^{1}((0, T), L_{q}(\Omega))\cap L_{p}((0, T), W_{q}^{2}(\Omega))$.

The uniqueness for (1.1) follows from the uniqueness of solutions to problem (4.3), which
completes the proof of Theorem 2.1.

163

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