数理解析研究所講究録
1495 巻 2006 年 137-153

137$L_{p}- L_{q}$maximal regularity for the Stokes equation
with first order boundary condition
Yoshihiro SHIBATA*\dagger

Department of Mathcmatical Sciences, School of Sciencc and Engineering,
University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan.
-mail address: yshibata@,wa‘s$\mathrm{W}\mathrm{a}‘ \mathrm{s}e\mathrm{d}\mathrm{a}$$\mathrm{t}^{\backslash }.\mathrm{d}\mathrm{a}.\mathrm{j}\mathrm{p}$1ProblemThoughout this paper St denotes a bounded or exterior doinain in
and $0<T\leqq\infty$ . We consider the generalized Stokes cquation:$\mathbb{R}^{n}(n\geqq 2)$

with$C^{2,1}$boundary$\partial’\Omega$,$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=f$$\mathrm{d}\mathrm{i}\mathrm{v}u=g=\mathrm{d}\mathrm{i}\mathrm{v}\tilde{g}$

in

(1.1)

$\Omega \mathrm{x}(0, T)$

which is obtained as a linealization of the Navier-Stokes cquation describing the rnotion of aviscous incompressible Newtonian fluid in St. Here, is a givcn exterior force field, andarc given functions, and $u=(u_{1}, , . ., u_{71})$ and are the unknown velocity and pressure field,
respcctively. In (1.1) we dcnoted thc stress tensor of the fluid by
$f$

$g$

$\tilde{g}$$\pi$$S(u, \pi)=D(u)-\pi I,$ $D(u)=(D_{ij}(u))$$D_{ij}(u)=\partial_{i}\prime u_{j}+\partial_{j}u_{i},$$I:n\mathrm{x}n\mathrm{i}(\mathrm{l}\mathrm{c}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{y}$Inatrix

A large numbcr of papers have been devoted to the study of thcsc equations. However, theoverwhelming majority of thosc works is concerned with the Stokes cquation coupled with nonat the boundary. The, that is, the fluid is required to be atslip boundary condition:, but nonetheless physicallypurpose of this paper is to study the Stokes equation withrcasonable, boundary condition. Narnely, as the boundary condition, we consider$\mathrm{r}\mathrm{c}^{\backslash }.\mathrm{s}\mathrm{t}$$u|_{\partial\Omega}=0$$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{c}\mathrm{r}\mathrm{e}^{\backslash },\mathrm{I}\mathrm{l}\mathrm{t}_{l}$$\nu\cdot u|_{\partial\Omega}=0$(1.2)$B(u, \pi)|_{\partial\Omega}=T_{1}^{\alpha}(u, \pi)|_{\theta\Omega}=\{$$(S(u, \pi)\nu-(S(u, \pi)\nu,$ $\nu)\nu+\alpha u)|_{\theta\Omega}=h|_{\partial\Omega}$a.s well as$B(u, \pi)|_{\mathit{0}\Omega}=T_{2}(u, \pi)|_{\partial\Omega}=S(u, \pi)\nu|_{\partial\Omega}=h|_{\theta\Omega}$(1.3)

is called slip boundary condition when.
denotes the unit outer normal to$\alpha>01$),
rcspectively. The boundary condition
a $=0$ and Robin boundary condition when
admits motion of the fluid at the boundary in tangcntial dircctions. On the other hand,where$\partial’\Omega$$\nu$$T_{1}^{\alpha}(u, \pi)$$T_{1}^{\alpha}$“Partly support\’e by Grant-in-Aid for Scientific Research (B) 15340204, Minigtry of Education, Sciences,
Sports and Culture, Japan\dagger The results about the Ncumann boundary condition casc and Robin boundary condition case were obtaincd
by joint works with S. Shimizu (Shizuoka Univ.) and R. Shimada (Waseda Univ), respcctively.
1)
Throughout the paper, we assume that a 10.
-

138
is called Neumann boundary condition, which is derived as
the bcundary condition
boimdary
problcrn for the Navicr-Stokes cquations which will befree
a linearization of somediscussed in section 5.
Complementing equations (1.1), (1.2) and (1.3) with initial condition for the velocity field,
wc arrivc at the following initial boundary value problern to be studied$T_{2}(u, \pi)$$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=f$in$\Omega \mathrm{x}(0, T)$$\mathrm{d}\mathrm{i}\mathrm{v}u=g=\mathrm{d}\mathrm{i}\mathrm{v}\tilde{g}$in$\Omega \mathrm{x}(0, T)$

on OStin$B(u, \pi)=h$$u|_{t=0}=u_{0}$$\mathrm{x}(0, T)$(1.4)$\Omega$

Analytic semigroup approach to (1.4)

2

In order to treat (1.4) in the analytic semigroup framework, in this section we consider the
initial-boundary value problem:$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}S(\mathrm{u}, \pi)=0$$\{$$B(u, \pi)|_{\theta\Omega}=0$

,

,

in

$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\Omega \mathrm{x}(0, \infty)$(2.1)$u|_{t=0}=u_{0}$First we consider the slip or R,obin boundary condition case. Namely, $B(u, \pi)$ is given by thefollowing formula:$S(u, \pi)\nu-(\nu\cdot S(u, \pi)\nu)\nu+\alpha u=D(u)\nu-(\nu\cdot D(u)\nu)\nu+\alpha u$$B(u, \pi)=T_{1}^{\alpha}(u, \pi)=\{$$\nu\cdot u$

To eliminate the pressure term
sition:$\pi$

in the first equation of (2.1) we use the Helmholtz decompo-$L_{q}(\Omega)^{n}=J_{(l}(\mathrm{t}l)\oplus G_{q}(1l)$$J_{q}(\Omega)=\overline{\{u\in C_{0}^{\infty}(\Omega)^{n}|\mathrm{d}\mathrm{i}\mathrm{v}u=0\}}$$=${$u\in L_{q}(\Omega)^{n}|\mathrm{d}\mathrm{i}\mathrm{v}u=0$in$\Omega,$$\nu\cdot u|_{1},$

$=0$

}

$G_{q}(\Omega)=\{\nabla\pi|\pi\in X_{q}(\Omega)\}$The space$X_{q}(\Omega)$

is defined in the following way: When

$X_{q}(\Omega)=$

When$\Omega${$\pi|\pi\in W_{q}^{1}$(il),$\int_{\Omega}\pi dx=0$$\Omega$},

is a boundcd domain,$||\pi||_{\mathrm{x}_{q}\langle\Omega)}=||\pi||_{W_{q}^{1}(\Omega)}$

is an exterior domain and $1<q<n$ ,$X_{q}(\Omega)=\{\pi\in L_{nq/(n-q)}(\Omega)|\nabla\pi\in L_{q}(\Omega), ||\pi||_{\mathrm{x}_{q}(\Omega)}<\infty\}$$||\pi||_{\mathrm{x}_{q^{(\Omega)}}}=||\pi||_{\iota_{nq/(n-q)^{(\Omega)}}}+||\nabla\pi||_{r\prime q(\Omega)}+||\pi(d_{q})^{-1}||_{L_{\mathrm{Q}}(\Omega)}$

When$\Omega$

is an exterior domain and $n\leqq q<\infty$ ,$X_{q}(\Omega)=\{\pi\in L_{q,1\mathrm{o}\mathrm{c}}(\overline{\ddagger l})|\nabla\pi\in L_{q}(\Omega), ||\pi||_{\mathrm{x}_{q^{(\Omega)}}}<\infty\}$$||\pi||_{x_{q^{(\Omega)}}}=||\nabla\pi||_{L_{q}(\Omega)}+||\pi(d_{q})^{-1}||_{Lq(\Omega)}$139
Here, $d_{q}(x)$ is a weight function defined by thc relations:$d_{q}(x)=\{$

when $q\neq n,$ $1<q<\infty$
when $q=n$$1+|x|$$(1+|x|)\log(2+|x|)$

In fact, such decomposition was proved by Fujiwara and Morimoto [9], Farwig and Sohr [8],Galdi [11], Miyakawa [16], Simader andand references therein. In this case, given, wc choose
as a weak solution to the Neurnann problem:$\mathrm{S}\mathrm{o}\mathrm{I}\mathrm{l}\mathrm{r}[24]$$\pi\in X_{q}(\Omega)$$u\in L_{q}(\mathrm{f}l)^{n}$in$\Delta\pi=\nabla\cdot u$(2.2)$\Omega,,\frac{\partial\pi}{\partial\nu}|_{\partial\Omega}=\nu\cdot u|_{\partial\Omega}$

And then, we define the lincar continuous projection:
relation: $P_{q}u=u-\nabla\pi$ . Noticing that
when
Stokes operator
corresponding to (2.1) with $B(u, \pi)=T_{1}^{\alpha}(u, \pi)$ by$P_{q}$$L_{q}(\mathrm{J}l)^{n}arrow J_{q}(\Omega)$$\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=\Delta u-\nabla\pi$

alongby the, we define the$G_{q}(\Omega)$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$A_{q}^{1}$(2.3)$A_{q}^{1}=P(-\Delta)$

with domain:.

$D(A_{q}^{1})=\{u\in J_{q}(Il)\cap W_{q}^{2}(\Omega)^{n}|(D(u)\nu-(\nu\cdot D(u)\nu)\nu+\alpha u)|_{\theta\Omega}=0\}$

Concerning the generation of an analytic semigroup by the Stokes operator
Robin boundary condition, we have following theorem.

$A_{q}^{1}$$(\mathrm{S}\mathrm{D})$with slip or

Theorem 2.1. Let $1<q<\infty an,d$ let be a bounded domain or an exterior domain in$(n\geqq 2)$ withboundary.
on. Then, generates an analytic sernigroupMoreover, when is bounded,is exponentially stable. In partic
for $t>0$ wehave the following estimates:$\mathbb{R}^{n}$$\Omega$$C^{2,1}$$\partial’\Omega$$\Omega$$\{T_{q}^{1}(t)\}_{t\geqq 0}$$A_{q}^{1}$$ular_{f}$$\{T_{q}^{1}(t)\}_{t\geqq 0}$$||T_{q}^{1}(t)u_{0}||_{L_{q}(\Omega)}\leqq C_{q}e^{-c_{q}\iota}||u_{0}||_{\iota_{q^{(\Omega\rangle}}}$$||T_{q}^{1}(t)u_{0}||_{\iota \mathrm{v}_{q}^{\mathrm{z}_{\langle\Omega)}}}\leqq C_{\mathrm{J}}‘ t^{-1}e^{-\mathrm{c}_{q}t}||\mathrm{u}_{0}||_{L_{q}(\Omega)}$$||T_{q}^{1}(t)u_{0}||_{w_{q}^{2_{(\Omega)}}}\leqq C_{q}e^{-c_{q}\ell}’||u_{0}||_{W_{q}^{2}(\Omega)}$with some positive constants$c_{q}$

and$C_{q}$$J_{q}(\Omega)$$\forall u_{0}\in J_{q}(\Omega)$$\forall u_{0}\in J_{q}(\Omega)$$\forall u_{0}\in D(A_{q}^{1})$.

Remark 2.2. In the boundcd domain case, the generation of the analytic semigroup of the
Stokes operatoron
was proved by Giga [12]. Its exponential stability was proved by
Shibata-Shirnada [17]. In the exterior domain case\dagger the generation of tho analytic semigroup of
the Stokes operatoron
was proved by Shibata-Shimada [17]. The asyrnptotic behaviour
not been proved yet in the exterior domain casc, but employing thc similar argument tothat due to Iwashita [13] in the non-slip boundary condition case, we can show the local energy
decay anddecay estiInate. We will discuss this elesewhere.
$J_{q}(\Omega)$$A_{q}^{1}$$J_{q}(\Omega)$$A_{q}^{1}$$\mathrm{h}\mathrm{a}_{\iota}\mathrm{s}$$L_{q^{-}}L_{f}$Next, we consider the Neumann boundary condition case. Namely, let us consider the
initial-value problexn:$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=0$,$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\{$$S(u, \pi)\nu|_{\partial\Omega}=0_{!}$$u|_{t=0}=u_{0}$in$\Omega \mathrm{x}(0, \infty)$

(2.4)

140
To

$\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{r}\mathrm{r}\dot{\mathrm{u}}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}$thc pressure term$\pi$

we use the second Helmholtz decomposition:$L_{q}(\Omega)^{n}=\hat{J}_{q}(\Omega)\oplus\hat{G}_{q}(\zeta])${$\hat{J}_{q}(\Omega)=$in$u\in L_{q}(\Omega)^{n}|\mathrm{d}\mathrm{i}\mathrm{v}u=0$}$\Omega$$\hat{G}_{q}(\Omega)=\{\nabla\pi|\pi\in X_{q}(\Omega), \pi|_{1}, =0\}$

was proved by Grubb and Solonnikov [10] and Shibata and Shimizu $[21, 22]$ . In this case,as a weak solution to the Dirichet problem:, we choose
given

whic,

$\mathrm{h}$$\pi\in X_{q}(\Omega)$$u\in L_{q}(\ddagger l)^{n}$$\Delta\pi=\mathrm{d}\mathrm{i}\mathrm{v}u$in$\Omega,$$\pi|_{1},$And then, we define the continuous projeetion fromby the relation: $\hat{P}_{q}u=u-\nabla\pi$ .
In order to introduce the reduced Stokes operator
the resolvent problein:$\hat{P}_{q}$$\lambda u-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=f$,:$A_{q}^{2}$

(2.5)

$=0$

$L_{q}(\Omega)^{n}arrow J_{(l}(\Omega)$along$G_{q}(\Omega)$

definedcorresponding to (2.4), let us considcr$\mathrm{d}\mathrm{i}\mathrm{v}u=0$in$\Omega$

(2.6)

$\{$

$S(u, \pi)\nu|_{\theta\Omega}=0$

Let

$g$

and

$\theta$be the second Helmholtz decomposition of :$f$$f=g+\nabla\theta,$$\mathrm{d}\mathrm{i}\mathrm{v}g=0,$$\Delta\theta=\mathrm{d}\mathrm{i}\mathrm{v}f$in$\Omega,$$\theta|_{\Gamma}=()$

Then, we have,$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\lambda u-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi-\theta)=g$

in$\Omega$$\{$$S(u, \pi-\theta)\nu|_{\partial\Omega}=0$

Setting $\kappa=\pi-\theta$ , we consider$\lambda u-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \kappa)=g$,$\mathrm{d}\mathrm{i}\mathrm{v}u=0$in$\Omega$

(2.7)

$\{$

$S(u, \kappa)\nu|_{\partial\Omega}=0$

under the condition thatand using the facts that$\mathrm{d}\mathrm{i}\mathrm{v}g=0$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$

in . Applying the divergence to thc first equation in (2.7)
in , we have
and (
$Il$

$\Omega$$\mathrm{l}\mathrm{i}\mathrm{v}g=0$

in$\Delta\kappa=0$(2.8)$\Omega$andSince $(D(u)\nu-\kappa\nu)|_{\Gamma}=0$ , Inultiplying the boundary condition by , using
complementing the resultant boundary condition with (2.8), we arrivc at the equation for as$\nu$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\pi$follows:$\Delta\kappa=0$

If$u\in W_{q}^{2}(\Omega)$in$\Omega,$$\kappa|_{\Gamma}=(\nu\cdot D(u)\nu-\mathrm{d}\mathrm{i}\mathrm{v}u)|_{\Gamma}$, then we have a unique solution$\kappa\in X_{q}(\mathrm{t}?)$(2.9)

of (2.9) such that$||\pi||_{X_{q}(\Omega)}\leqq C||u||_{W_{q}^{2}(\Omega)}$

Let $K$ be the operator:
be written in the form:$W_{q}^{2}(\mathrm{t}l)arrow X_{q}(\Omega)$

defined by the formula: $K(u)=\kappa$ . Then, (2.7) can$\lambda u-\mathrm{D}\mathrm{i}\mathrm{v}S(u, K(\mathrm{u}))=g$

in$\Omega,$$S(u, K(u))\nu|_{1},$ $=0$

(2.10)

141
We see that (2.10) is equivalent to (2.7) when$g\in\hat{J}_{q}(\Omega)$. Therefore, we set(2.11)$A_{q}^{2}u=-\mathrm{D}\mathrm{i}\mathrm{v}S(u, K(u))$

with dornain$D(A_{q}^{2})=\{u\in W_{q}^{2}(\Omega)\cap\hat{J}_{q}(\Omega)|S(u, K(u))|_{\Gamma}=0\}$

Then, we have the following theorern.

Theorem 2.3 (Grubb-Solonnikov [10], Shibata-Shimizu [21, 22]). Let $1<q<\infty$ .
generates an analytic semigroupon.
$\{T_{\mathrm{r}}^{2},(t)\}_{t\geqq 0}$$A_{q}^{2}$$\hat{J}_{q}(\Omega)$

In order to get the exponential stability ofin the bounded dornain case, we haveto restriet ourself to the subspace which is orthogonal to the rigid space defined by$\{T_{(\mathit{1}}^{2}(t)\}_{t\geqq 0}$$\mathcal{R}$$\mathcal{R}=${ $Ax+b|A$ : anti-symmetric rnatrix,

Note that $D(u)=0$ if and only if
and that if$M=n(n-1)/2+n$ , be the orthogonal basis of, we set
closed subspace $X$ of$u\in \mathcal{R}$$u\in \mathcal{R}$$1,$$\mathcal{R}$$\ldots,$$b\in \mathbb{R}^{n}$

then

}

$\mathrm{d}\mathrm{i}\mathrm{v}u=0$such tIlat. Let$p_{m},$$(p_{rn},p_{\ell})_{\Omega}=\delta_{m\ell}$$m=$. Given$L_{q}(\Omega)$$X\backslash \mathcal{R}=\{u\in X|(u,p\ell)_{\Omega}=0, \ell=1, \ldots, M\}$

Theorem 2.4 (Shibata-Shimizu [22]). Let $1<q<\infty$ . $Assum,e$ thatwithgenerates an analytic semigroupboundary. Then,enjoys the estimates:$C^{2,1}$$\{T_{q}^{2}(t)\}_{t\geqq 0}$$A_{q}^{2}$$||T_{q}^{2}(t)u_{0}||W_{q}^{2}(\Omega)\leqq C_{q}t^{-1}e^{-c_{q}t}||u_{0}||_{\iota_{q^{(\Omega)}}}$$||T_{q}^{2}(t)u_{0}||_{W_{q}^{2}(\Omega)}\leqq C_{q}e^{-c_{q}t}||u_{0}||_{W_{q}^{2}\langle\Omega)}$$u_{0}\in\hat{J}_{q}(\Omega)\backslash R$$\mathbb{R}^{n}\backslash \Omega\subset B_{R_{0}}$,on$\hat{J}_{q}(\Omega)\backslash \mathcal{R}$which$\forall u_{0}\in\hat{J}_{q}(\Omega)$$\forall u_{0}\in D(A_{q}^{2})$with some positive constants

Now, let us consider tbc asymptotic behaviour ofstate the theorems, we set

is a bounded domain$\forall u_{0}\in\hat{J}_{q}(\mathrm{f}l)$$||T_{q}^{2}(t)u_{0}||_{L_{q}(\Omega)}\leqq C_{q}e^{-r_{d}t}q||u_{0}||_{Lq(\Omega)}$

for any $t>0$ and$\Omega$$C_{q,\mathrm{k}}>0$$\{T_{q}^{2}(t)\}_{t\geqq 0}$

and$c_{q}$.

in the exterior domain. To$B_{R_{0}}=\{x\in \mathbb{R}^{n}||x|<R_{0}\}$$L_{q,R}(\Omega)=\{f\in L_{q}(\mathrm{f}l)|f(x)=0, x\not\in B_{R}\}$.

Then, Shibata and Shimizu [21] provcd thc following two thcorems.

Theorem 2.5 (Local energy decay estimate). Assume that is an exterior domain in. Then, wewitha that $n\geqq 3$ . Let $1<q<\infty$ and$\Omega$$C^{2,1}bo\mathrm{u}ndar\uparrow \mathit{1}$$R\geqq R_{0}$$r|,d$$||e^{-\Lambda_{q}^{2}}{}^{t}\hat{P}_{q}\mathrm{u}_{0}||_{W_{q}^{2}(\Omega\cap B_{R})}\leqq C_{q,R}t^{-\frac{n}{2}}||u_{0}||_{L_{q}(\Omega)}$

for all$u_{0}\in L_{q,R}(\Omega)$

and $t\geqq 1$ .

$h,a\uparrow\prime e$$\mathbb{R}^{n}$

142
$e$

$u’(^{\mathrm{J}},$is an ederior dorr’ am infollowzng estimates:$Assum,e$ that

Theorem 2.6 ( $L_{q}-L_{r}$ estimat ).
$boundan/and$ that $n\geqq 3$ . Then.

$ft,a\uparrow fCt_{\text{ノ}}he$$\zeta$]‘$\frac{1}{r}$$||e^{-\Lambda_{q}t}u_{0}||_{\iota_{r}(\Omega)}\leqq C_{q,r}t^{-\frac{n}{2}(\frac{1}{q}}$)‘for all$u_{0}\in\hat{J}_{q}(\Omega),$$1\leqq q\leqq r\leqq\infty(q\neq\infty, r\neq 1)$

with$C^{2,1}$(2.12)$||u_{0}||_{L_{q}(\Omega)}$$||\nabla e^{-\Lambda_{q}t}u_{0}||_{L_{r}(\Omega)}\leqq C_{q,r}t^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{f})}$$\mathbb{R}^{n}$$\frac{1}{2}||u_{()}||_{\Gamma_{q},(\Omega)}$(2.13)

and $t>0$ .

, instead of the Neumann boundary
If we consider the non-slip boundary condition$1\leqq
q\leqqr\leqq n(r\neq 1),$ which was proved bycondition,
we know that (2.13) holds only for, were done by several authors:
Iwashita [13]. Later on, several refinements of Iwashita’sand Dan-Shibata $[4, 5]$ . Further referMaremonti-Solonnikov [15],
4,
6,
5].
In particular, Marernonti-Solonriikov [15]ences can be found in the references of [15,is unavoidable in the non-slip condition
proved that the.se. Therefore the cstimate (2.13) shows the eventual character of the NellIIla\iota lIl boundary
boundary implies the better decay rate for the gradientcondition. Namely the null force onof the solutions.
$u|_{\Gamma}=0$$\mathrm{t}\mathrm{I}\iota \mathrm{e}\mathrm{n}$$\mathrm{r}\mathrm{c}\mathrm{s}\iota 1\mathrm{l}\mathrm{t}$$\mathrm{D}\mathrm{a}\mathrm{n}- \mathrm{K}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{y}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{i}- \mathrm{S}\mathrm{h}\mathrm{i}\mathrm{b}_{\dot{r}}\iota \mathrm{t}\mathrm{a}[6]$$\mathrm{a}_{\iota}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:1\leqq q\leqq r\leqq n(r\neq 1)$$\mathrm{c}\mathrm{a}$$\mathrm{t}\}_{1}\mathrm{e}$3maximal regularity of solutions to (1.4)$L_{p^{-}}L_{q}$rnaximal regularity of solutions to (1.4). We consider only
In this section, we consider the
maximal regularitythat is a bounded domain in what follows. To state ourthc$X:$ Banach space,: Let
theorem, first of all we introduce some functional$L_{p^{-}}L_{q}$$\Omega$$L_{\mathrm{p}^{-}}L_{q}$$\mathrm{c}\mathrm{a}_{\iota}\mathrm{s}\mathrm{e}$$\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}^{\backslash },\mathrm{c}\mathrm{s}$$0<T\leqq\infty$$1\leqq p,$ $q\leqq\infty,$.

$W_{q,p}^{\ell,m}(D\mathrm{x}I)=L_{p}(I, W_{q}^{\ell}(D))\cap W_{p}^{m}(I, L_{q}(D))$$||u||_{W_{q,p}^{\ell,m_{(D\mathrm{x}I)}}}=||u||+||u||_{W_{\mathrm{p}}^{m}(I,L_{q}(D))}L_{\mathrm{p}}(I,|\nu_{q}^{\ell_{\langle D)\rangle}}$$W_{\mathrm{p},0}^{1}((0, T),$

Given$\alpha\in \mathrm{R}$

$X)=$

{

$u\in W_{p}^{1}((-\infty,$ $T),$$X)|u=0$ for $t<0$ }, we set$<D_{l}>^{\alpha}u(t)=F^{-1}[(1+s^{2})^{\alpha/2}Fu(s)](t)$$H_{p}^{\alpha}(\mathbb{R}, X)=\{u\in L_{p}(\mathbb{R}, X)|<D_{t}>^{\alpha}u\in L_{p}(\mathbb{R}, X)\}$$||u||_{H_{\mathrm{p}}^{\alpha}(\mathrm{R}X)}=||<D_{t}>^{\alpha}u||_{L_{P}(\mathrm{R}X)}+||u||_{L_{\mathrm{p}}(\mathrm{R}\mathrm{X})}$$\mathcal{F}$

and
Set

$F^{-1}$denote thc Fourier transform and its inverse formula, respectively.

$H_{q,p}^{1,1/2}(D\mathrm{x}\mathbb{R})=H_{p}^{1/2}(\mathbb{R}, L_{q}(D))\cap L_{p}(\mathbb{R}, \mathrm{T}V_{q}^{1}(D))$$H_{q,p,0}^{1,1/2}(D\mathrm{x}(0, \infty))=${$u\in H_{q,p}^{1,1/2}(D\mathrm{x}\mathbb{R})|u=0$$||u||H_{q,\mathrm{p}}^{1,1/2_{(D\cross \mathrm{R})}}=||u||_{\iota \mathrm{r}_{\mathrm{p}}^{1/2_{(\mathrm{R}^{\gamma_{d}}q^{(D))}}}}+||u||_{\iota_{t},(\mathrm{R}\mathrm{t}\iota_{q^{1}}^{J}(D))}$for $t<0$ }143Finally, given $0<T\leqq\infty$ wc set$H_{\mathrm{r}/,p}^{1,1/2}(D\mathrm{x}(0, T))=\{u|^{\exists}v\in H_{q,p}^{1,1/2}(D\mathrm{x}\mathbb{R}), u=v\mathrm{o}\mathrm{I}1D\mathrm{x}(\mathrm{O}, T)\}$$||u||H_{q,\mathrm{p}}^{1,1/z_{(D\cross(0^{r}\tau))}},= \inf\{||v||H_{q,\mathrm{p}}^{1,1/2}(D\mathrm{x}\mathrm{R})|$$\forall_{v}\in H_{q,p}^{1,1/2}(D\cross \mathbb{R}),$$H_{q,p,0}^{1,1/2}(D\cross(0, T))=${$v=u$ on$u|^{\exists}v\in H_{q,p,0}^{1,1/2}(D\mathrm{x}(0,$$\infty)),$$D\mathrm{x}(\mathrm{O}, T)\}$$u=v$ on $D\mathrm{x}(0,$ $T)$}$||u||H_{q,\mathrm{p},0}^{1,1/2}(D \mathrm{x}(0,T))=\inf\{||v||H_{q,p}^{1,1/2}(D\mathrm{x}\mathrm{R})|$$\forall_{v}\in H_{q,p,0}^{1,1/2}(D\mathrm{x}(0, \infty)),$$v=u$ on $D\mathrm{x}(0, T)\}$$B_{q,p}^{2(1-1/p)}(\Omega)=[L_{q}(ll), W_{q}^{2}(\Omega)]_{1-1/p,p}$$[J_{q}(\zeta)), D(A_{q}^{1})]_{1-1/p,p}$$D_{q,p}(\Omega)=\{$$[\hat{J}_{q}(\Omega), D(A_{q}^{2})]_{1-1/p,p}$

Theorem 3.1 (maximal regularity). Let $1<p,$ $q<\infty,$ $0<T<\infty,$
satisfy the condition:$L_{p^{-}}L_{q}$$u_{0},$$f,$$g,\tilde{g}$

and

$h$

$u_{0}\in D_{q,p}(\Omega),$$f\in L_{p}((0, T),$ $L_{q}(\Omega))^{n},$$\tilde{g}\in W_{p,0}^{1}((0, T),$ $L_{q}(l1))^{n},$$g\in H_{q,\mathrm{p},0}^{1,1/2}(\Omega \mathrm{x}(0, T))$

(3.1)

$h\in H_{q,p,0}^{1,1/2}(\Omega \mathrm{x}(0, T))^{n}$In the slip or Robin boundary condi tion case, in addition we assume the compatibility condition:. Then, (1.4) adrnits a unique solution$\nu\cdot h|_{\Gamma}=0$$(u, \pi)\in\nu V_{q,p}^{2,1}(\Omega\cross(0, T))^{n}\mathrm{x}L_{p}((0, T),$$|vh,ich$$W_{q}^{1}(\Omega))$(3.2)

enjoys the estimate:$||u||_{W_{q,\mathrm{p}}^{2,1}\langle\Omega\cross(0,T))}+||\pi||L_{p}((0,T\rangle,W_{q}^{1}(\Omega))\leqq C_{T}\{||u_{0}||_{v_{q,p}(\Omega)}$$+||f||_{L_{\mathrm{p}}((0,T),L_{q}(\Omega))}+||(g, h)||+||\overline{g}||_{W_{\mathrm{P}}^{1}(0,J),l_{q},(\Omega))},.\}H_{q,\mathrm{p},0}^{1,1/2}(\Omega \mathrm{x}(0,T))$

where $C_{T}>0$ is in,dependent of
the Neurnann condition case.

$T$

(3.3)

in the slip or Robin condition case and $C_{T}=C(1+T)$ in

Theorem 3.2 (Exponential stability). Let $T=\infty$ in (1.4). Assume that (3.1) holds with
$T=\infty$ . In the Neumann condition case,in addition we assume th, $at$$(u_{0},p_{\ell})_{\Omega}=0$,$(f(t),p\ell)_{\Omega}+(g(t),p\ell)_{\Gamma}=0$$\ell=1,$$\ldots,$$M$$\ell=1,$ $\ldots,$ $M$, a.e.t $>0$

Then, (1.4) admits a solution $(u, \pi)u\prime hich$ satisfies not only (3.2) and (3.3) with $T=\infty$ but
also the following estim,ate:$||e^{\gamma l}u||_{w_{q,\mathrm{p}}^{2,\iota_{(\Omega \mathrm{x}(0,\infty))}}}+||e^{\gamma t}\pi||Lp((0,\infty),W_{q}^{1}(\Omega))\leqq C\{||u_{0}||_{0_{q,p^{(\Omega)}}}$$+||e^{\gamma l}f||_{\iota_{\mathrm{p}^{((0,\infty),L_{q}\langle\Omega))}}}+||e^{\gamma\ell}(g, h)||_{;\iota_{q,p}^{1,1/2_{(\Omega \mathrm{x}(0,\infty))}}}+||e^{\gamma t}\tilde{g}||_{\nu_{p^{(0,\infty),L_{q}(\Omega))}}^{1}},\}$

where is any constant inwith some $\gamma 0>0$ .
Moreover, in the Neumann boundary condition case, the solution $u(t)$ satisfies the orthogonal, M.
condition: $(u(t),p_{\ell})_{\Omega}=0$ for all $t>0$ and $\ell=1,$$\gamma$$[0, \gamma_{0}]$$\ldots$

144
Remark 3.3. (1) hi the slip or Robin boundary condition case, Theoreos 3.1 and 3.2 wereproved by Shibata-Shimtlda [18]. In the Neumann bounda.ry condition case,wasstated by Solonnikov [25] under thc assumption: $3<p=q<\infty$ , which is combined witll
Bencdek-Caldcr\’on-Panzonc theorcm [2] (cf. also [1]) implies that Theorem 3.1 also holds under
the a.ssumption: $3<q<\infty$ and $1<p<\infty$ . But, Theorem 3.2 is a quite new result comparedwith [25].
, we introduce the set(2) In order to state the relationship between
andas follows:$\mathrm{T}\mathrm{h}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}\iota \mathrm{n}3.1$$B_{q,p}^{2(1-1/p)}(\Omega)$$D_{q,p}(\Omega)$$B_{q,p}^{2(1-1/p)}(\Omega)$$B_{(\mathit{1},p}^{2(1-1/p)}(\Omega)=${in$u\in B_{q,p}^{2(1-1/p)}(\Omega)^{n}|\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\Omega$}Then, it follows from Triebel [28] and Steiger [26] that therc hold thc following relations: Whenthe boundary condition is a slip or R,obin one and $2(1-1/p)>1+1/q$ ,$D_{q,p}(\Omega)=\{u\in B_{q,p}^{2(1-1/p)}(\Omega)|\nu\cdot u|_{\theta\Omega}=0, (D(u)\nu-(\nu\cdot(D(u)\nu))\nu)|_{\partial\Omega}=0\}$,when the boundary (ioIldition is a slip or Robin one and $1/q<2(1-1/p)<1+1/q$ ,$D_{q,\mathrm{p}}(\Omega)=\{u\in B_{q,p}^{2(1-1/p)}(\Omega)|\nu\cdot u|_{\partial\Omega}=0\}$,

and when the boundary condition is a slip or Robin one \‘and $2(1-1/p)<1/q$ ,$D_{q,p}(\Omega)=B_{q,p}^{2(1-1/p)}(\Omega)$

On the other hand, when the boundary condition is a Neumann onc and $2(1-1/p)>1+1/q$ ,$D_{q,\mathrm{p}}(\zeta))=\{u\in B_{q,\rho}^{2(1-1/p)}(\Omega)|S(u, K(u))|_{\partial\Omega}=0\}$,

and when the boundary condition is a Neumann one an $\mathrm{d}2(1-1/p)<1+1/q$ ,$D_{q,p}(\mathrm{f}l)=\mathcal{B}_{t/,\mathrm{P}}^{2(1-1/p)}(\Omega)$

4

A skech of our proof of the maximal regularity results in the
Neumann boundary condition case

4.1

First step

We consider the maximal rcgularity of the problem:,$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=f$$S(u, \pi)\nu|_{\Gamma}=0$

where

,

$f\in C_{0}^{\infty}(\mathbb{R}, L_{q}(\Omega))\subset L_{p}(\mathbb{R}, L_{q}(\Omega))$$(f(t),p_{m})_{\Omega}=0,$$t\in \mathbb{R},$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$in f)$\mathrm{x}(0, \infty)$$u|_{t=0}=0$$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\cdot,\mathrm{h}$satisfies the conditions:$m=1,$$\ldots,$$\Lambda f;f(t)=0$

By Duhamcl’s principlc we have$u(t)=| \int_{0}^{t}T_{q}^{2}(t-s)f(s)ds$

for $t<0$ .

(4.1)

145
and therefore by Theorem 2.4 we have$||u(t)||_{W_{q}^{1}(\Omega)}\leqq cJ_{0}^{t}.(t-s)^{-1/2}e^{-2\gamma(t-s)}||f(s)||_{L_{q}(\Omega)}ds$

with suitable $\gamma>0$ , which implis that
(4.2)$||e^{\gamma t}u||L_{p}((0,\infty),W_{q}^{1}(\Omega))\leqq C_{\gamma,q}||e^{\gamma t}f||_{L_{\mathrm{p}}((0,\infty),L_{q}(\Omega))}$The estimate (4.2) is used to cstimate the perturbation terms which are obtained by the localization procedure.

4.2

2nd step

We consider
maximal rcgularity in the wholc space and the half-space. We used theoperator valued Fourier rnultiplier theorem. To state this thcorern, first we start with theandboundcdncss of thc opcrator family. Lct $X$ and be two Banach spaces with norms$X$, respectively. Let
denote the set of all bounded lincar operators from into ,and we set $B(X)=B(X, X)$ .
$L_{p^{-}}L_{q}$$\mathcal{R}$$Y$$||\cdot||_{X}$$B(X, \mathrm{Y})$$||\cdot||_{Y}$$\mathrm{Y}$is called -bounded
Deflnition 4.1 ( -boundedness). A family of opcrators
being the set of
if there exist constants $C>0$ and $p\in(1, \infty)$ such that for each$x_{j}\inX$symrnetric,
independent,
scquenccs
ofnumbers,forall
all naturaland$\{-1,1\}$ -valued random variables on $[0,1]$ therc holds the inequality:$\mathcal{T}\subset B(X, \mathrm{Y})$$\mathcal{R}$$m\in \mathrm{N},$$\mathcal{R}$$\mathrm{N}$$\{r_{i}(u)\}$$T_{j}\in \mathcal{T},$$\int_{0}^{\perp}||\sum_{j=1}^{m}r_{\mathrm{j}}(u)T(x_{j})||_{Y}^{p}du\leqq C\int_{0}^{1}||\sum_{j=1}^{m}r_{j}(u)x_{j}||_{\mathrm{x}}^{p}du$The smallest such

Given

$C$

is called$\mathcal{R}$-bound of$M\in L_{1,1\mathrm{o}\mathrm{c}}(\mathbb{R}, B(X, \mathrm{Y}))$$\mathcal{T}$, which is denotcd by

we define an operator$T_{M}$$\mathcal{R}(\mathcal{T})$.

by$T_{M}\phi=\mathcal{F}^{-1}[MF\phi]$

Then the following theorexn wus proved by Weis ([29]).

Theorem 4.2 (Operator Valued Fourier Multiplier Theorem). Suppose that $X$ andbe such that theare $UMD$ Banach spaces and $1<p<\infty$ . Let
following conditions are satisfied:$\mathrm{Y}$$M\in C^{1}(\mathbb{R}\backslash \{0\}, \mathcal{B}(X, Y))$$\mathcal{R}(\{M(\rho)|\rho\in \mathbb{R}\backslash \{0\}\})=\kappa_{0}<\infty$$\mathcal{R}(\{\rho M(\rho)’|\rho\in \mathbb{R}\backslash \{0\}\})=\kappa_{1}<\infty$

Then, the operator$T_{M}$

is a bounded linear operator from

$L_{p}(\mathbb{R}, X)$

into$L_{p}(\mathbb{R}, \mathrm{Y})$with no $rm$ :$||T_{M}||_{\mathcal{B}(L_{\mathrm{p}}(\mathrm{R}X),L_{\mathrm{p}}(\mathrm{R}Y))}\leqq C(\kappa_{0}+\kappa_{1})$where $C>0$ depends only$or|,$$p,$$X$

and Y.

To show thc $R$ -boundedhess, the following two propositions are useful (cf. Denk-HieberPr\"uss [7]

$)$

.

146
Proposition 4.3. Let $1<q<\infty$ and
. Set

be a

$\{k_{s}(x)|s\in \mathbb{R}\backslash \{0\}\}$$fam,\iota \mathfrak{l},\tau/$of$L_{1,1\mathrm{o}\mathrm{c}}(\mathbb{R}^{r\iota})$functions$k_{s}(x)$$K_{s}f(x)= \int_{W}k_{s}(x-y)f(y)dy$$s\in \mathbb{R}\backslash \{()\}$

Suppose that there exists a constant $C>0$ independent of$|1K_{\mathit{8}}f||_{L_{2}(\mathrm{N}^{1})}\leqq C||f||_{L_{2}(\mathrm{N}^{1})}$$s\in \mathbb{R}\backslash \{()\}$$f\in L_{2}(\mathbb{R}^{n}),$such that$s\in \mathbb{R}\backslash \{0\}$$x\in \mathbb{R}^{n}\backslash \{0\},$$s\in \mathbb{R}\backslash \{0\}$$\sum_{|\alpha|=1}|\partial_{x}^{\alpha}k_{\theta}(x)|\leqq C|x|^{-(n+1)}$

Then,$\{K_{s}|s\in \mathbb{R}\backslash \{0\}\}$$C_{n},{}_{q}C$with some constantis$\mathcal{R}$$C_{n,q}$-bounded on
depending only on

and itsand .

$\mathcal{R}$$\mathcal{B}(L_{q}(\mathbb{R}^{n}))$

Proposition 4.4. Let $1<q<\infty$ . Let
be a $fam,ily$ of the kemel operators:

$G$

$n$

be a domain in$[T_{\mu}f](x):=.

-bound is less than or equal to$q$$\mathbb{R}^{n}$and$\mathcal{T}=\{T_{\mu}|\mu\in \mathcal{M}\}\subset B(L_{q}(G))$\int_{G}k_{\mu}(x, y)f(y)dy$

for $x\in G$ and $f\in L_{q}(G)$ . Suppose that there exists asuch that$k_{0}(x, y)$$|k_{l^{4}}(x, y)|\leqq k_{0}(x, y)$for almost all$x,$$y\in G$and for$a\mathit{7}|,y\mu\in \mathcal{M}$. Set$[T_{0}f](x)= \int_{G}k_{0}(x, y)f(y)dy$

If$T_{0}\in B(L_{q}(G))$$C_{q,G}||T_{0}||_{B(L_{q}(G))}$, tfien,, wl,ere$\mathcal{T}$is$C_{q,G}$$\mathcal{R}$-bounded on $B(L_{q}(G))$ and its

depen$l,ds$ only on

In order to explain how to show themodel problem in the half-space:$q$$L_{p^{-}}L_{q}$$u_{\ell}-\Delta u+u+\nabla\pi=0,$

and

$G$

$\mathcal{R}$.

-bound is less than or equal tomaximal regularity, we consider the following$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$(, \frac{\partial’u_{j}}{\partial x_{n}}+\frac{\partial’u_{n}}{\partial^{r}x_{j}})|_{x_{n}=0}=\ell_{j},$in$j=1,$$\mathbb{R}_{+}^{n}\mathrm{x}\mathbb{R}_{+}$$\ldots,$$n-1$$(2 \frac{\partial^{t}u_{n}}{\partial’x_{n}}-\pi)|_{x_{\hslash}=0}=\ell_{n}$The solution formula for (4.3) consists of the following forrnulas:$w_{1}(x, t)= \int_{0}^{\infty}\mathcal{L}^{-1}[B_{\lambda}(\xi’)^{-1}e^{-B_{\lambda}(\xi’)(x_{n}+y_{n})}\hat{h}(\xi’, y_{n}, \lambda)](x’, t)dy_{n}$$B_{\lambda}(\xi’)=\sqrt{|\lambda|+|\xi’|^{2}+1},$$\xi’=(\xi_{1}, \ldots, \xi_{n-1}),$ $\lambda=\gamma+i\tau$$w_{2}(x, t)= \int_{0}^{\infty}\prime \mathcal{L}^{-1}[A_{\lambda}(\xi’)|\xi’|e^{-B_{\lambda}(\xi’)(x_{n}+y_{n})}\hat{h}(\xi’, y_{n}, \lambda)](x ‘, t)dy_{n}$$w_{3}(x, t)= \int_{0}^{\infty}\mathcal{L}^{-1}[A_{\lambda}(\xi’)|\xi’|^{2}\mathcal{M}_{\lambda}(\xi’, x_{n}+y_{n})\hat{h}(\xi’, y_{n}, \lambda)](x’, t)dy_{n}$$\mathcal{M}_{\lambda}(\xi’, x_{n})=\frac{e^{-B_{\lambda}(\xi’)x_{\hslash}}-e^{-|\xi’|x_{n}}}{B_{\lambda}(\xi^{f})-|\xi’|}$

(4.3)

147
Hcre,

$A_{\lambda}(\xi’)$is a$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\cdot \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{x}\mathrm{l}$satisfying the multiplier condition:$|\partial_{\xi}^{\alpha’},A_{\lambda}(\xi’)|\leqq C_{\alpha’}(|\lambda|^{1/2}+|\xi’|+1)^{-2}|\xi|^{-|(x’|}$for anytheNamely,aiidtransform of , and$\lambda=\gamma+i\tau\in[0, \infty)\cross i\mathbb{R}$$\mathrm{F}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}- \mathrm{L}\mathrm{a}\mathrm{p}1_{\dot{\mathrm{f}}}\iota \mathrm{o}\mathrm{e}$whereis independent of anddenotesdenotes the inverse Fourier-Laplace transform.

$\xi’\in \mathbb{R}^{n-1}$$h$$\lambda$$C_{\alpha’}$$\mathcal{L}^{-1}$$\xi’,\hat{h}$$\hat{h}(\xi’, y_{n}, \lambda)=J_{W}e^{-i\xi’\cdot x’-(\gamma+i\tau)t}h(x’, x_{n}, t)dx’dt=F_{(x’,t)}[e^{-\gamma t}h](\xi’, x_{n}, \tau)$$\mathcal{L}^{-1}[g](x’, t)=(2\pi)^{-n}\int_{\mathrm{r}}\mathrm{e}^{i\xi’\cdot x’+(\gamma+i\tau)t}g(\xi’, \tau)dx’d\tau=e^{\gamma t}F_{(\xi,\tau)}^{-1},[g](x’, t)$

For examplc, let us explain how to estimate the following term by using Theorem 4.2:
$e- \gamma t’\partial_{t}w_{1}=\int_{0}^{\infty}F_{(\xi,\tau)}^{-1},[\frac{\lambda e^{-B_{\lambda}(\xi’)(x_{n}+y_{n})}}{B_{\lambda}(\xi)},\hat{h}(\xi’, y_{n}, \lambda)](x’, t)dy_{n}$Set$k_{\gamma}( \tau, x)=F_{\xi}^{-1},[\frac{\lambda e^{-B_{\lambda}(\xi’)x_{n}}}{B_{\lambda}(\xi^{f})}](x’)$$K_{\gamma}( \tau)g=\int_{\Psi_{1}}k_{\gamma}(x^{f}-y’, x_{n}+y_{n})g(y)dy$

Then, we have$e^{-\gamma t}\partial_{t}\prime w_{1}=F_{\tau}^{-1}[K_{\gamma}(\tau)F_{t}[e^{-\gamma t}h](\tau)](t)$If we show that$|k_{\gamma}(\tau, x)|\leqq A|x|^{-n},$(4.4)$|\tau\partial_{\tau}\prime k_{\gamma}(\tau, x)|\leqq A|x|^{-n}$then by Proposition 4.4 wc havc$\mathcal{R}\{K_{\gamma}(\tau)|\tau\in \mathbb{R}\backslash \{0\}\}\leqq CA$

with some constant $C>0$ independent of 7 and , because we know theof the operator:$\tau$$L_{q}(\mathbb{R}_{+}^{n})$boundcdncss$T_{0}[g](x)= \int_{\mathrm{R}_{+}^{n}}A[|x’-y’|^{2}+(x_{n}+y_{n})^{2}]^{-n/2}g(y’, y_{n})dy’dy_{n}$Therefore, applying Theorem 4.2 we have$||e-\gamma \mathrm{t}’\partial_{l}w_{1}||_{Lp^{(}\mathrm{R}r_{q}(\Omega))},\leqq C||e^{-\gamma t}h||_{L_{\mathrm{p}}(\mathrm{R}L_{q}(\Psi_{1})\rangle}$, where is indcpcndcnt offor any.
To show (4.4), wc use thc following lernrna.
$\gamma\geqq 0$$C$$\gamma\geqq 0$

Lemma 4.5 (Shibata-Shimizu [19]). Let $X$ be a Banach, space andits no $7m$ . Let a
be a number $>-n$ and set $\alpha=N+\sigma-n$ where $N\geqq 0$ is an integer and $0<\sigma\leqq 1$ . Let
be a function insuch that$||\cdot||_{\chi}$$f(\xi)$$C^{\infty}(\mathbb{R}^{n}\backslash \{0\}, X)$$\theta_{\xi}^{\gamma}f(\xi)\in L_{1}(\mathbb{R}^{n}, X)$

for$||’\partial_{\xi}^{\gamma}f(\xi)||_{X}\leqq C_{\gamma}|\xi|^{\alpha-|\gamma|}$$|\gamma|\leqq N$$\forall\xi\neq 0,$$\forall\gamma\in \mathrm{N}_{0}^{n}$

148
Set

$g(x)=\mathrm{J}_{\mathbb{R}^{n}}e^{-ix\cdot\xi}f(\xi)d\xi$. Then, we$ha^{J}|;e$$||g(x)||_{X}\leqq C_{n,\alpha}(\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{x}C_{\gamma})|x|^{-(n+\alpha)}|\gamma|\leqq N+2$

where

$C_{n,\alpha}$

is a constan, dependi$t$$7\iota g$

only on$n$$\forall x\neq 0$

and or.

and second derivatives with respect to variables of
estirnated in the same spirit, and therefore we arrive at the estiinate:$\partial_{t}w_{2},$$x$$\partial_{\ell}^{r}w_{3}$$w_{1},$$w_{2}$

and$w_{3}$

can be$||e^{-\gamma t}(u_{t}, \nabla^{2}u, \nabla\pi)||_{L\mathrm{p}(\mathrm{R}L_{q}(\mathbb{R}_{\{}))}$$\leqq C\{||(\nabla(e^{-\gamma t}\ell), <D_{t}>^{1/2}(e^{-\gamma l}\ell), \sqrt{\gamma}e^{-\gamma t}\ell)||_{L_{\mathrm{p}}(\mathrm{R}L_{q}(r_{+}))}\}\gamma\geqq 0$

is a solution to thc equation (4.3).

where

$u$

4.3

3rd Step

By using the localization technique, we reduce the problcm to the whole space problcm and thehalf-space problern and by using the result in 2nd step and by rcprescnting in terms of and, we can show thc a priori estirnate:
$\pi$

$f$

$u$

$||(u_{l}, \nabla^{2}u)||_{L_{p}(\mathrm{R}L_{q}(\Omega))}\leqq C\{||f||_{L_{p}(\mathrm{R}L_{q}(\Omega))}+||u||J_{p}\lrcorner(\mathrm{R}W_{q}^{1}(\Omega))\}$

where

$u$

(4.5)

is a solution to$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=f,$$S(u, \pi)\nu|_{\Gamma}=0,$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$in$\Omega \mathrm{x}(0, \infty)$$u|_{\iota-0}=()$

Finally, applying (4.2) in the 1st stop to (4.5), we have$||u_{t}||_{L_{p}(\mathrm{R}L_{q}(\Omega))}+||u||_{\iota_{p}(\mathrm{R}W_{q}^{2}(\Omega))}+||\pi||L_{\mathrm{p}}(\mathrm{R}w_{q^{(\Omega))}}^{1}\leqq C||f||_{L_{p}(\mathrm{R}L_{9}(\Omega))}$

4.4
Let

4th step$u$

be a solution to thc non-homogeneous boundary value problem:$u_{l}-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=0,$$S(u, \pi)\nu|_{\Gamma}=h,$To get a priori estimatc of$\mathrm{u}_{t}$$\Omega \mathrm{x}(0, \infty)$, we use the solution th to the adjoint problcm:$S(\psi, \theta)\nu|_{\Gamma}=0,$$\phi\in L_{p’}(\mathbb{R}, L_{q’}(\Omega))$$\mathrm{d}\mathrm{i}\mathrm{v}$

th $=0$ in$\Omega\cross(-\infty, T)$$\psi|_{t_{-}T}=0$

which vanishes for $t>T$ , we know the existence of solution
for $t>T$ and satisfy thc
such that Cb and$(\psi, \theta)\in W_{qp}^{2,1},,’(\Omega \mathrm{x}\mathbb{R})^{n}\mathrm{x}L_{\mathrm{p}’}(\mathbb{R}, W_{q}^{1},(\Omega))$estimate:in$u|_{t=0}=0$$\psi_{t}+\mathrm{D}\mathrm{i}\mathrm{v}S(\psi, \theta)=\phi,$

Given $T>0$ and$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\theta \mathrm{v}\mathrm{a}\iota \mathrm{l}\mathrm{i}\mathrm{s}\mathrm{I}\mathrm{l}$$||\psi_{t}||_{L_{\mathrm{p}’}(\mathrm{R}L_{q’}(\Omega))}+||\psi||L_{\mathrm{p}’}(\mathrm{n}\iota \mathrm{v}^{2},(\Omega))+||\theta||qL_{p’}(\mathrm{R}W^{1},(\Omega))q\leqq C||\phi||_{\iota_{\mathrm{p}’}\mathrm{t}\mathrm{R}L_{q},(\Omega))}$

(4.6)

149
where $1/p+1/p’=1/q+1/q’=1$ .

$()$bserving that$(u_{t}, \phi)_{\Omega \mathrm{x}\mathrm{R}}=(u_{t}, \psi_{t}+\mathrm{D}\mathrm{i}\mathrm{v}S(\psi, \theta))_{\Omega\cross \mathrm{R}}$$=-(u_{tt}-\mathrm{D}\mathrm{i}\mathrm{v}S(u_{t}, \pi_{t}),$ $\psi)_{\Omega\cross \mathrm{R}}+(S(u_{t}, \pi_{t})\nu,$$=(h_{l}, \psi)_{\Gamma}$$\psi)_{\Gamma \mathrm{x}\mathrm{N}}$xm$=(\nabla’(\nu h_{\ell}), \psi)_{\Omega \mathrm{x}\mathrm{R}}+(\nu h_{t}, \nabla\psi)_{\Omega \mathrm{x}\mathrm{N}}$$=-(\nabla\cdot(\nu h), \psi_{t})_{\Omega \mathrm{x}\mathrm{R}}+(\nu<D_{\ell}>^{-1/2}h_{\ell}, <D_{\ell}>^{1/2}\nabla\psi)_{\Omega \mathrm{x}\mathrm{R}}$,we have$|(u_{t}, \phi)_{\Omega \mathrm{x}\mathbb{R}}|\leqq C(||h||L_{\mathrm{p}}(\mathrm{R}W_{q}^{1}(\Omega))+||<D_{\ell}>^{1/2}h||_{L_{\mathrm{p}}(\mathrm{R}L_{q}\{\Omega)\rangle})\cross$($||\psi_{t}||_{L_{p’}(\mathrm{R}L_{q},(\Omega))}+||<D_{t}>^{1/2}\nabla\psi||_{J_{z}(\mathrm{R}J_{d}(\Omega))}$(4.7)

Since we know the intcrpolation irlequality:$||<D_{t}>^{1/2}\nabla\psi||_{L_{p’}(\mathrm{R}L_{q},(\Omega))}\leqq C(||\psi_{t}||_{\iota_{\mathrm{p}’}(\mathrm{R}L_{q’}(\Omega))}+||\psi||,)L_{\mathrm{p}’}(\mathrm{R}W^{2}(\Omega))q$by (4.6) and (4.7) we have$|(u_{t}, \phi)_{\Omega \mathrm{x}\mathrm{I}\mathrm{R}}|\leqq C(||h||_{\iota_{1},(\mathrm{R}W_{q}^{1}(\Omega))}+||<D_{t}>^{1/2}h||_{L_{\mathrm{p}}(\mathrm{R}L_{q}(\Omega))})||\phi||_{L_{p’}(\mathrm{n}\iota_{q},(\Omega))}$which implies that$||u_{t}||_{L_{\mathrm{p}}(\mathrm{N},L_{q}\{\Omega))}\leqq C(||h||+||<D_{t}>^{1/2}h||_{L_{\mathrm{p}}\{\mathrm{R}L_{q}(\Omega))})Lp(\mathrm{R}W_{q}^{1}(\Omega))$hi this way, we can show the5$L_{p^{-}}L_{q}$maximal rcgularity.

A free boundary problem for the Navier-Stokes equations

In this section, we consider a time dependcnt problem with free surfacc for the Navier-Stokes
equations which describes the motion of an isolated finite volume of viscous incompressible fluid$n\geqq 2$ , occupied by the fluid
without taking surface tension into account. The region
is given only on the initial time $t=0$ , whilc for $t>0$ it is to be determined. The velocity
satisfy the Navier-Stokes
forvector field $v(x, t)=\downarrow(v_{1}, \ldots, v_{n})$ and the pressurc
equations (cf. [25]):$\zeta]_{b}\subset \mathbb{R}^{n},$$\theta(x, t)$$v_{\ell}+(v\cdot\nabla)v-\mathrm{D}\mathrm{i}\mathrm{v}S(v, \theta)=f(x, t)$$\mathrm{d}\mathrm{i}\mathrm{v}v=0$$v|_{t=0}=v_{0}$denotes the transposed $M,$.
normal to at the point
defined by the formula:${}^{t}M$$\Gamma_{t}$$x\in\Gamma_{t}$$\Gamma_{t}$

in
in

in
on

$S(v, \theta)\nu_{t}+\theta_{0}(x, t)\nu_{t}=0$Here,$x\in\Omega_{l}$$\Omega_{t},$$t>0$$\Omega_{t},$$t>0$$\Gamma_{t},$$t>$$\Omega$denotes the boundary ofwith$\nabla=(’\partial_{1}, \ldots,’\partial_{n})$

.

(5.1)

$\Omega_{\iota}$$\partial_{i}’=\partial/\partial’x_{i}$$S(v, \theta)=D(v)-\theta I$

$()$

.

and

is the unit outeris the strcss tensor$\nu_{t}(x)$$S(v, \theta)$

150
where $D(v)$ is the deformation tensor of the velocities with elements
identity matrix. For thc
matrix of functions $S=(S_{ij})$
is theand$D_{i_{J}}(v)=\partial_{i}’v_{j}+\partial_{j}^{t}v_{i}$$I$$n\mathrm{x}n$$n\mathrm{x}n$$\mathrm{D}\mathrm{i}\mathrm{v}S={}^{t}(\sum_{j=1}^{n}’\partial_{j}S_{1j}, \ldots,\sum_{j=1}^{n}’\partial_{j}S_{nj})$

are functions defined on
The external force $f(x, t)$ and the pressure
whole space.
$\theta_{0}(x,t)=0$, since we can arrive at this case by rcplacing
Below, we shall always assume thatby.
Aside frorn the dynamical boundary condition, a further kinematic condition foris sat$x(\xi,t)$$x=x(\xi,t),$, wherea set of points
is the solution of theisfied which givcs
Cauchy problem:$\theta_{0}(x, t)$$\theta(x, t)$$\mathrm{t}\}_{1}\mathrm{e}$$\theta+\theta_{0}$$\Gamma_{t}$$\xi\in\Gamma_{0}$$\Gamma_{l}\mathrm{a}_{\hslash}\mathrm{s}$$\frac{dx}{dt}=v(x, t)$,$x|_{l=0}=\xi$

.

(5.2)

This expresses the fact that thc free surfacc consists for all $t>0$ of the same fluid particlcs,
which do not leave it and are not incident on it from inside . It is clear that $\Omega_{t}=\{x=$and $\Gamma_{t}=\{x=x(\xi, t)|\xi\in\Gamma_{0}\}$ .
The problem (5.1) can therefore be written as an initial boundary valuc problem in the givcn
regionif wc go over the Euler coordinates
to Lagrangc coordinatcs
conncctcd$u(\xi,i)=(u_{1},\ldots,u_{n})$
is known as a function of thewith by (5.2). If a velocity vector field
Lagrange coordinatcs , then this connection can be written in the form:$\Gamma_{l}$$\Omega_{t}$$x(\xi, t)|\xi\in \mathrm{f}l_{0}\}$$x\in\Omega_{\ell}$$\Omega_{0}$$\xi\in\Omega$

$x$

$\xi$

(5.3)

$x= \xi+\int_{0}^{t}u(\xi, \tau)d\tau\equiv X_{u}(\xi, t)$Passing to Lagrange coordinate in (5.1) and setting$\theta(X_{u}(\xi, t),$$t)=\pi(\xi, t)$, we obtain$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}[S(u, \pi)+U(u, \pi)]=f(X_{u}(\xi, t),$ $t)$in$\Omega\cross(0, T)$$\mathrm{d}\mathrm{i}\mathrm{v}u+E(u)=\mathrm{d}\mathrm{i}\mathrm{v}[u+\tilde{E}(u)]=0$

in$\Omega \mathrm{x}(0, T)$$[S(u, \pi)+U(u, \pi)]\nu=0$$\mathrm{o}\mathrm{I}\downarrow\Gamma \mathrm{x}(0, T)$$u|_{t=0}=v_{0}$in$\Omega$.

(5.4)$n\geqq 2$ , whose boundary
Here and hereafter, is a bounded domain in
is assumed to$U(u,\pi),$$E(u)$
compact hypersurface, is the unit outer normal to
be a
are
and
nonlinear terms of the following forms:$\Omega$$\Gamma$$\mathrm{R}^{n},$$C^{2,1}$$\nu$$\Gamma,$$\tilde{E}(u)$$U(u, \pi)=V_{1}(\int_{0}^{t}\nabla ud\tau)\nabla u+V_{2}(\int_{0}^{t}\nabla ud\tau)\pi$$E(u)=V_{3}.( \int_{()}^{t}\nabla ud\tau)\nabla u$$\tilde{E}(u)=V_{4}(\int_{0}^{l}\nabla ud\tau)u$ofwith some polynomials$j=1,2,3,4$ , such as $V_{j}(0)=0$ .
As a linearizcd problem of (5.4), we have the Stokes equation with Neumann boundary
condition:$V_{j}(\cdot)$$\int_{0}^{t}\nabla ud\tau,$$u_{t}-\mathrm{D}\mathrm{i}\mathrm{v}S(u, \pi)=f$$\mathrm{d}\mathrm{i}\mathrm{v}u=g=\mathrm{d}\mathrm{i}\mathrm{v}\tilde{g}$$S(u, \pi)\nu|_{\Gamma}=h$,in$\Omega\cross(0, T)$in$\Omega \mathrm{x}(0, T)$$u|_{t=0}=u_{0}$(5.5)

By using Theorems 3.1 and 3.2 and the contraction mapping principle, for (5.4) we have thefollowing two thcorems ([23]).

151
Theorem 5.1. Let 2 $<p<\infty an,dn<q<\infty$ . Thcn, given $v_{0}\in D_{q,p}(\Omega)$ and $f\in$which fiwas bounded deri vatives with respect to for each , there exists a $T=$that (5.4) adrnits a unique solu$L_{p}(\mathbb{R}, L_{q}(\mathbb{R}^{n}))$

$t$

$x$

$T(||v_{0}||_{D_{q,\mathrm{p}^{(\Omega\rangle}}}, ||f||_{L_{p}(\mathrm{R}L_{q}(\mathrm{N}^{l}))}, \sup_{t\geqq 0}||\nabla f(\cdot, t)||_{L(\mathrm{N}^{\hslash})}\infty)>0sucf\iota$tion$(u, \pi)\in W_{q,p}^{2,1}(\Omega \mathrm{x}(0, T))^{n}\mathrm{x}L_{p}((0, T),$

which,$l\psi_{(\mathit{1}}^{1}(\Omega))$satisfies the estimate:$||u||_{W_{q,\mathrm{p}}^{2,1}(\Omega \mathrm{x}(0,T))}+||\pi||L_{p}((0’.\Gamma),W_{q}^{1}(\Omega)\rangle\leqq C\{||v_{0}||_{\mathcal{D}_{q,\mathrm{p}(\Omega)}}+||f||_{\iota_{n^{((0,T\rangle,L_{q}(\mathrm{N}^{1}))}}}\}$

Theorem 5.2. Let $2<p<\infty$ and $n<q<\infty$ . Then, there exist positive numbers andsuch that if
and $(v_{0},p\ell)_{\Omega}=0for\ell=1,$ , A4, then (5.4) with
$T=\infty$ andadmitsasolutionunique$f=0$$\epsilon$$v_{0}\in D_{q,p}(\Omega),$$||v_{0}||_{0_{q,\mathrm{p}^{(\Omega)}}}\leqq\epsilon$$\gamma$$\ldots$$(u, \pi)\in W_{q,p}^{2,1}(\Omega\cross((), \infty))^{n}\cross L_{p}((0, \infty),$ $W_{q}^{1}(\Omega))$

which satisfies the estimate:$||e^{\gamma\ell}u||_{W_{q,p}^{2,1}(\Omega \mathrm{x}(0,\infty))}+||e^{\gamma t}\pi||L_{p}((0,\infty),\mathrm{t}V_{q}^{1}(\Omega)\rangle\leqq C||v_{0}||_{\mathcal{D}_{q,\mathrm{p}}(\Omega)}$for sorne$\gamma>0$

and the condition:$(u(\cdot, t),p\ell)_{\Omega}=0$

for$\ell=1,$ $\ldots,$ $M$

and$t\geqq 0$

Remark 5.3. Theorems 5.1 and 5.2 have been proved by Solonnikov [25] when $p=q$ , but ourapproach is cornpletely different from [25]. Moreover, sincc wc show the iaximal regularityfor (1.7) with $T=\infty$ (cf.
result with exponential stability on the whole tirne axis
with the corresponding proof
Theorcm 3.2), our proof of Thcorezn 5.2 is vcry simple
in \S 4 of Solonnikov [25].
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