数理解析研究所講究録
第 1782 巻 2012 年 28-43

28

Mild solutions to the Navier-Stokes equationsin unbounded domains with unboundedboundary
Okihiro Sawada
Department of Mathematical and Design Engineering,
Gifu University, Yanagido 1-1, Gifu, 501-1193, Japanokihiro@gifu-u.ac.jp
Abstract

It is mathematically investigated the incompressible viscous flows
in domains
with nonslip boundary conditions in the framework-boundary, where has a possibly non-compact uniformofwith someand boundedness of the Helmholtz projection onto
$1<p<\infty$ . The key is to show that the Stokes operator generates an-regularity
admitting the maximal
analytic semigroup on
estimates. Moreover, the local-in-time existence and the uniqueness ofmild solutions to the Navier-Stokes equation in such and $p\in(n, \infty)$.
are proved, when the initial data belong to$\Omega\subset \mathbb{R}^{n}$$L_{\sigma}^{p}(\Omega)$$C^{3}$$\Omega$$L_{\sigma}^{p}(\Omega)$$\mathbb{P}_{p}$$L^{q}-L^{p}$$L_{\sigma}^{p}(\Omega)$$\Omega$$L_{\sigma}^{p}(\Omega)$

1

Introduction

This is a brief survey of the results related to [18], mainly.
, it is well-known that the Stokes operator
For any open set(with nonslip boundary conditions) is a self adjoint operator in
by Masuda [28]. Hence, $-A_{2}$ is the generator of an analytic contraction. Here,is defined by the solenoidal
semigroup
onto, wherepart of the Helmholtz decomposition of
intodenotes the direct sum, anddenotes the Helmholtz projection from. It seems to be natural to investigate whether this techniquetocan be applicable in general If-setting, that is,extends to an$\Omega\subset \mathbb{R}^{n}$$A_{2};=-\mathbb{P}_{2}\triangle$$L_{\sigma}^{2}(\Omega)$$\{e^{-tA_{2}}\}_{t\geq 0}$$L_{\sigma}^{2}(\Omega)$$L_{\sigma}^{2}(\Omega)$$L^{2}(\Omega)$$\oplus$$L^{2}(\Omega)$$L_{\sigma}^{2}(\Omega)\oplus G^{2}(\Omega)$$\mathbb{P}_{2}$$L_{\sigma}^{2}(\Omega)$$\{e^{-tA_{p}}\}_{t\geq 0}$

29analytic semigroup on an-space for some 1 $<p<\infty$ , and that there
are the maximal-regularity estimates for the solution of the associated$U_{\sigma}$$L^{q}-L^{p}$

Stokes equations. Once we obtain the above semigroup theory, we have achance to construct the local-in-time mild solutions to the Navier-Stokes
equations infor $n\leq p<\infty$ by the fixed point argument of Kato [26]or Giga-Miyakawa [22]. The notion of a mild solution was first introduced by
Fujita-Kato [14, 27] when the initial velocity belongs towith smoothbounded domains
via Duhamel principle at the almost same yearsof Browder [6] to study some equations of parabolic type.
It is clear to have the affirmative answer of the above question whenis the whole space or the half space (see Ukai [33] and Desch-Hieber-Pr\"uB[7] for any $p\in(1, \infty)$ . For bounded or exterior domains with smoothboundaries, the maximal-regularity estimates were firstly shown by
Solonnikov [30]. His proof makes use of potential theoretic arguments. Lateron, Giga [19, 20] also established the Stokes semigroup theory due to thebounded imaginary powers of the Stokes operator, Giga-Sohr [23] applied
the Dore-Venni theorem in two-dimension case, Grubb-Solonnikov [24] used
the pseudo-differential techniques, and Fr\"ohlich [13] made use of the conceptof weighted estimates with respect to Muckenhoupt weights. The reader canfind related results in the list of reference in Farwig-Sohr [11]. Furthermore,
the ca.se of a perturbed half space is treated by e.g. Noll-Saal [29]. For results
concerning infinite layers-like domains, we refer to the works of Abe-Shibata[1], Abels [2] and Abels-Wiegner [3]. Franzke [12] and Hishida [25] considered
the case of aperture domains. Farwig-Ri [10] derived the maximal
regularity estimates in infinite tube-like domains. In the domains listed-up
above the Helmholtz decomposition is valid.
The key of this approach is to show the boundedness of the Helmholtz
projectiononinto its solenoidal subspace. For example, ifisbounded, then the boundedness of ; this fact was first proved by FujiwaraMorimoto [15].
On the other hand, in the case of general domains , it is not clear
whether the Helmholtz decomposition makes sense, that is,or not, in general, unless $p=2$ . Indeed, $Bogovski_{\dot{1}}[4,5]$ gave examples$L_{\sigma}^{p}(\Omega)$$H_{\sigma}^{1/2}(\Omega)$$\Omega\subset \mathbb{R}^{3}$$s$$\Omega$$)$$L^{q}-L^{p}$$L^{q}-L^{p_{-}}$$\mathbb{P}_{p}$$L^{p}(\Omega)$$\Omega$$\mathbb{P}_{p}$$\Omega$$U(\Omega)=L_{\sigma}^{p}(\Omega)\oplus$$G^{p}(\Omega)$

30
of unbounded domains with smooth boundaries in which it is not enable
for certain . For details, seeto have the Helmholtz decomposition of If
also [16]. To overcome the difficulties, Farwig-Kozono-Sohr [9] introduced$\Omega$$(\Omega)$$p$$\tilde{L}^{p}(\Omega):=\{\begin{array}{ll}L^{2}(\Omega)\cap U(\Omega), 2\leq p<\infty,L^{2}(\Omega)+L^{p}(\Omega), 1<p<2.\end{array}$-boundaries, proved the existence ofwith uniform
for domains), and obtained the useful(assisted by
the Helmholtz projection in. Moreover, they proved that the Stokes operator
properties as usual in, andwith nonslip boundary conditions is well-defined in$C^{2}$$\Omega\subset \mathbb{R}^{3}$$L^{2}$$\tilde{I}f$$\tilde{\mathbb{P}}$$U(\Omega)$$A_{p}$$:=-\tilde{\mathbb{P}}\triangle$$\tilde{I}f_{\sigma}$

as well as the maximal
generates an analytic semigroup onto
regularity estimates in the class (Ll’) $:=L^{q}((0, T);\tilde{I}f(\Omega))$ for $T>0$$L^{q}-\tilde{I}f-$$\tilde{I}f_{\sigma}(\Omega)$$L^{q}$$\Vert u_{t}\Vert_{L^{q}(\tilde{L}^{p})}+\Vert u\Vert_{L^{q}(\tilde{L}^{p})}+\Vert\nabla^{2}u\Vert_{L^{q}(\tilde{L}^{p})}+\Vert\nabla\tilde{\pi}\Vert_{L^{q}(\tilde{L}^{p})}\leq C\Vert f\Vert_{L^{q}(\tilde{L}^{p})}$with some constant $C>0$ independent ofsolution to the Stokes equations in domains with

$f\in L^{q}(\tilde{I}f)$$\Omega$$f\in L^{q}(\tilde{L}^{p})$$u_{t}-\triangle u+\nabla\tilde{\pi}=f$in$\Omega\cross(0, T)$,$\nabla\cdot u=0$in$\Omega\cross(0, T)$

,

$u=0$

on

$\partial\Omega\cross(0, T)$$u|_{t=0}=0$

in

(1.1)

$\Omega$. Here$(u,\tilde{\pi})$

is a

:

,

.

In the paper [18] they however employed a different approach to [9]. For
having a uniformly -boundary with $p\in(1, \infty)$ , it is assumed that
. They actually showedexists bounded in
the Helmholtz projection, which comesthat $-A_{p}$ generates an analytic semigroup onto usualfrom the fact that solutions to the Stokes equation satisfies the maximal
If-regularity estimates in $L^{q}((0, T);L^{\rho}(\Omega))$ . They also obtained the localin-time existence of a unique mild solution to the Navier-Stokes equations inLPwith $p>n$ under the assumption of the existence of the Helmholtz
projection. Although it seems to be an interesting problem in the frameworkofwhich is excluded by [18], the author has no idea to overcome theif $p\neq q$ ) so far.
difficulties (for example, it is not clear whether
This paper is organized as follows. In Sections 2 we will state the main
results of [18]. In Section 3 the strategy of their approach is explained.
$C^{3}$$\Omega\subset \mathbb{R}^{n}$$\mathbb{P}_{p}$$L^{p}(\Omega)$$U_{\sigma}(\Omega)$$L^{q_{-}}$$(\Omega)$$L_{\sigma}^{n}(\Omega)$$\mathbb{P}_{p}=\mathbb{P}_{q}$312Main Results

In this section we mention the main results in [18]. Here and hereafter, let. The definition of uniform
next section. For any open set$n\geq 2$$C^{k}$-domain forwill be given in theand for $p\in(1, \infty)$ , we set$k\in \mathbb{N}$$\Omega\subset \mathbb{R}^{n}$$G^{p}(\Omega)$$:=\{u\in L^{p}(\Omega);u=\nabla\tilde{\pi}$for some$\tilde{\pi}\in W_{loc}^{1,p}(\Omega)\}$,$L_{\sigma}^{p}(\Omega):=\overline{\{u\in C_{c}^{\infty}(\Omega);\nabla\cdot u=0in\Omega\}}^{\Vert\cdot\Vert_{p}}$We say that the Helmholtz projection
can be decomposed into$\mathbb{P}$$:=\mathbb{P}_{p}$exists for $U(\Omega)$ , whenever$L^{p}(\Omega)$$L^{p}(\Omega)=If_{\sigma}(\Omega)\oplus G^{p}(\Omega)$.

In this case, there naturally exists a unique projection
having the properties
andas its null space. A well-known factby e.g. [16] is that the Helmholtz projection exists forfor $p\in(1, \infty)$ if
and only if for every $f\in L^{p}(\Omega)$ , there exists a unique function
satisfying$\mathbb{P}_{p}:If(\Omega)arrow L_{\sigma}^{p}(\Omega)$$\mathbb{P}_{p}^{2}=\mathbb{P}_{p}$$G^{p}(\Omega)$$L^{\rho}(\Omega)$$u\in\hat{W}^{1,p}(\Omega)$$\langle\nabla u,$$\nabla\varphi\}=\{f,$$\nabla\varphi\rangle$,$\varphi\in\hat{W}^{1,p’}(\Omega)$.

Thus the Helmholtz projection exists forif and only if for every $f\in$the above weak Neumann problem is uniquely solvable within the class. We now state the maximal-regularity estimate for solutions
to the Stokes equations, which is one of the main results of [18].
$\nu(\Omega)$$L^{p}(\Omega)$$\hat{W}^{1,p}(\Omega)$$L^{q}-L^{p}$

Theorem 2.1. Let $n\geq 2_{f}p,$ $q\in(1, \infty)$ and $T>0$ . Assume thatis a domain with uniform -boundary and that the Helmholtz projection
exists for. Let $f\in L^{q}((0, T);U(\Omega))$ . Then equation (1.1) admits aunique solution
in the class$\Omega\subset \mathbb{R}^{n}$$C^{3}$$\mathbb{P}_{p}$$L^{p}(\Omega)$$(u,\tilde{\pi})$$u\in W^{1,q}(L^{p})\cap L^{q}(W^{2,p}\cap W_{0}^{1,p}\cap L_{\sigma}^{p})$and$\tilde{\pi}\in L^{q}(\hat{W}^{1,p})$,and there exists a constant $C>0$ such that$\Vert u_{t}\Vert_{Lq(L)}p+\Vert u\Vert_{L(L)}qp+\Vert\nabla^{2}u\Vert_{L^{q}(L)}p+\Vert\nabla\tilde{\pi}\Vert_{Lq(Lp})\leq C\Vert f\Vert_{Lq(Lp})$.

32
Assuming as in the above theorem that the Helmholtz projectionas, we may define the Stokes operator $A=A_{p}$ infor$U(\Omega)$$\mathbb{P}_{p}$exists$L_{\sigma}^{p}(\Omega)$$D(A_{p}):=W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\cap If_{\sigma}(\Omega)$

(2.1)

for

$A_{p}u:=-\mathbb{P}_{p}\triangle u$$u\in D(A_{p})$,.

The definition of the function spaces are found in e.g. the book of Triebel, the following corollary holds[32]. Concerning the Cauchy problem intrue for the abstract equation with valued in the solenoidal subspace$U_{\sigma}(\Omega)$$t>0$ ,$u’(t)+A_{p}u(t)=f(t)$ ,(2.2)$u(0)=u_{0}$ .

Corollary 2.2. Let $n\geq 2,$

is a domain with uniform. Thenexists foronto
semigroup$p,$ $q\in(1, \infty)$$C^{3}$$U(\Omega)$

and $T>0$ . Assume that$\Omega\subset \mathbb{R}^{n}$-boundary and that the Helmholtz projection
defined as in (2.1) genemtes an analytic. Moreover, the solution to the problem$\mathbb{P}_{p}$$C_{0^{-}}$$-A_{p}$$\{e^{-tA_{p}}\}_{t\geq 0}$$u$$\nu_{\sigma}(\Omega)$(2.2) satisfies$\Vert u’\Vert_{L^{q}(L^{p})}+\Vert A_{p}u\Vert_{L^{q}(L^{p})}\leq C(\Vert f\Vert_{L^{q}(L^{p})}+\Vert u_{0}\Vert_{B_{p,q}^{2-2/q}})$with some constant $C>0$ independent of $f\in L^{q}((0, T);U_{\sigma}(\Omega))$ and$B_{p,q}^{2-2/q}(\Omega)\cap\nu_{\sigma}(\Omega)$$u_{0}\in$.

for $f\in If(\Omega)$ , where II denotes the
Setting$:=(\lambda+A)^{-1}$ , we can also obtain the following
identity matrix andresults for the Stokes resolvent problem$\nabla\tilde{\pi}=(II-\mathbb{P})\triangle R(\lambda, A)f$

$R(\lambda, A)$

(2.3)

for

$\lambda\in\Sigma_{\theta}$$\lambda u-\triangle u+\nabla\tilde{\pi}=f$in$\Omega$,$\nabla\cdot u=0$in$\Omega$

,

$u=0$

on

$:=\{\lambda\in \mathbb{C};\lambda\neq 0, |\arg\lambda|<\theta\}$$\partial\Omega$for some$\theta\in(0, \pi)$.

. Thenas above andCorollary 2.3. Let $1<p<\infty,$
and $f\in U(\Omega)$ there exists
there exists
such that for all
satisfying
a unique solution(2.3). Moreover, there exists $C>0$ such that$\theta\in(0, \pi)$$\Omega\subset \mathbb{R}^{n}$$\lambda_{0}\in \mathbb{R}$$\lambda\in\lambda_{0}+\Sigma_{\theta}$$(u,\tilde{\pi})\in(W^{2,p}(\Omega)\cap W_{0}^{1,p}(\Omega)\cap U_{\sigma}(\Omega))\cross\hat{W}^{1,p}(\Omega)$,$|\lambda|\Vert u\Vert_{L^{p}(\Omega)}+\Vert\nabla^{2}u\Vert_{L^{p}(\Omega)}+\Vert\nabla\tilde{\pi}\Vert_{L^{p}(\Omega)}\leq C\Vert f\Vert_{L^{p}(\Omega)}$$\lambda\in\lambda_{0}+\Sigma_{\theta},$$f\in L^{p}(\Omega)$.

33
The semigroupondescribed in Corollary 2.2 admits thefollowing U- smoothing properties, which are well known for the situationof bounded or exterior domains.
$\{e^{-tA_{p}}\}_{t\geq 0}$$U_{\sigma}(\Omega)$$L^{}$$s\in(1, \infty)$ such that $s\leq p\leq r,$
Proposition 2.4. Let,
and $T>0$ . Then there exists a $C>0$ such that for $t\in(0, T)$$f\in L^{s}(\Omega)^{n}$$p,$ $r,$$F\in L^{s}(\Omega)^{n\cross n}$for$\Vert e^{-tA_{p}}\mathbb{P}_{p}f\Vert_{r}\leq Ct^{-\frac{n}{2}(\frac{1}{s}-\frac{1}{r})}\Vert f\Vert_{s}$$\underline{1}_{-}\underline{2}\leq\underline{1}$

$n$

$p$

$\Vert\nabla e^{-tA_{p}}\mathbb{P}_{p}f\Vert_{r}\leq Ct^{-\frac{n}{2}(\frac{1}{s}-\frac{1}{r})-\frac{1}{2}}\Vert f\Vert_{s}$$\Vert e^{-tA_{p}}\mathbb{P}_{p}\nabla\cdot F\Vert_{r}\leq Ct^{-\frac{n}{2}(\frac{1}{s}-\frac{1}{r})-\frac{1}{2}}\Vert F\Vert_{s}$$\underline{1}\leq\underline{1}+\underline{2}$

’
$r$

$s$

$p$

.

$n$

for

$\frac{1}{p}-\frac{1}{n}\leq\frac{1}{r}$,$\frac{1}{s}\leq\frac{1}{p}+\frac{1}{n}$.

for$\frac{1}{p}-\frac{1}{n}\leq\frac{1}{r}$,$\frac{1}{s}\leq\frac{1}{p}+\frac{1}{n}$.

The proof of this proposition can be found in [18]. So, we omit it in here.
We finally consider the Navier-Stokes equations$u_{t}-\triangle u+(u\cdot\nabla)u+\nabla\tilde{\pi}=0$$\nabla\cdot u=0$

(2.4)
$u=0$
$u|_{t=0}=u_{0}$in$\Omega\cross(0, T)$,

in $\Omega\cross(0, T)$ ,on$\partial\Omega\cross(0, T)$in$\Omega$,.

We prove the following local well-posedness results for (2.4). To this end,
assume thatis a domain such that the Helmholtz projection
existsfor. Then, by the notion of a mild solution of (2.4), it is understood afunctionfor some $T>0$ satisfying the integral equation$\Omega\subset \mathbb{R}^{n}$$\mathbb{P}_{p}$$L^{p}(\Omega)$$u\in C([0, T);L_{\sigma}^{p}(\Omega))$$u(t)=e^{-tA_{p}}u_{0}- \int_{0}^{t}e^{-(t-s)A_{p}}\mathbb{P}_{p}\nabla\cdot(u(s)\otimes u(s))ds$,$0\leq t<T$ .

Theorem 2.5. Let $n\geq 2$ . Assume thatis a domain with uniform-boundary and that the Helmholtz projection
exists forfor some. Then there exist $T_{0}>0$ and a unique mild solution.
$p>n$ . Let$\Omega\subset \mathbb{R}^{n}$$C^{3}$$\mathbb{P}_{p}$$U(\Omega)$$u_{0}\in L_{\sigma}^{p}(\Omega)$

The proof follows the lines of the well-known iteration procedure describedin [21, 22, 26] with Proposition 2.4. We will not give a detailed proof here.

34

Outline of the proof

3

In this section we give the outline of the proof of Theorem 2.1. We referto the localization procedure and the divergence equation. Starting from, the main problem is that thethe corresponding results for the half space
usual localization procedure known from elliptic problem does not transfer tothe situation of the Stokes equation. Indeed, the usual localization proceduredoes not respect the condition on the divergence. In [17], a new localization
procedure for the Stokes resolvent problem (2.3) respecting the condition on$\mathbb{R}_{+}^{n}$the divergence was introduced.
Throughout this section, let,be an unbounded domain. For given-domain, if there exist constants
is called a uniform
a domain$K,$$\beta>0$ such that for each
there exists a Cartesian coordinate
system with origin at , coordinates $y=(y’, y_{n})$ and $h\in C^{k}((-\alpha, \alpha)^{n-1})$such that the neighborhood
with$k\in \mathbb{N}$$\Omega$$C^{k}$$\Omega\subset \mathbb{R}^{n}$$x_{0}\in\partial\Omega$$\alpha,$$x_{0}$$\Vert h\Vert_{C^{k}}\leq K$$U(x_{0}):=\{(y^{l}, y_{n})\in \mathbb{R}^{n};h(y’)-\beta<y_{n}<h(y’)+\beta, |y^{l}|<\alpha\}$of$x_{0}$

satisfying $\partial\Omega\cap U(x_{0})=\{(y’, h(y’));|y’|<\alpha\}$ and$U^{-}(x_{0}):=\{(y’, y_{n})\in \mathbb{R}^{n};h(y’)-\beta<y_{n}<h(y’), |y’|<\alpha\}=U(x_{0})\cap\Omega$.

Let us note that our assumption implies that one may choose for some
$K$ , balls$:=B_{r}(x_{j})$ with centers$r\in(O, \alpha)$ , depending only on
Stsuch thatiffor $j\in N$ and -functions$\alpha,$$C^{3}$$\overline{\Omega}\subset\bigcup_{j=1}^{\infty}B_{j}$$h_{j}$,$x_{j}\in$$B_{j}$$\beta,$$x_{j}\in\partial\Omega$$\overline{B_{j}}\subset U(x_{j})$if$x_{j}\in\partial\Omega$,$\overline{B_{j}}\subset\Omega$if$x_{j}\in\Omega$.

Moreover, we may construct this covering in such a way that not more thanof these balls can have a nonempty intersection.
a finite fixed number, their common intersection
Thus, choosing $N_{0}+1$ different balls, there exists a partition of unity
is empty. For given the covering.
andin , satisfying$N_{0}\in \mathbb{N}$$B_{1},$ $B_{2},$$\ldots$$\{B_{j}\}_{j=1}^{\infty}$$\varphi_{j}\in C_{c}^{\infty}(\mathbb{R}^{n}),$$\sum_{j}\varphi_{j}\equiv 1$$0\leq\varphi_{j}\leq 1$$supp\varphi_{j}\subset B_{j}$$\Omega$

We now consider the case when
In order to explain the main idea of [17], let us consider(i) Compact Boundary.

$\tilde{u}$$:= \sum_{j=1}^{\infty}\varphi_{j}u_{j}$

and

it:

$= \sum_{j=1}^{\infty}\varphi_{j}\pi_{j}$.

$\partial\Omega$is compact.

35
Here

is the solution to the Stokes resolvent equations (2.3) in thewhole space within the right hand side if, andis thepush-forward of the solution
to the Stokes resolvent equations in thehalf space$(u_{j}, \pi_{j})$$x_{j}\in\Omega$$\psi_{j}f$$(u_{j}, \pi_{j})$$(\hat{u}_{j},\hat{\pi}_{j})$in$\lambda\hat{u}_{j}-\triangle\hat{u}_{j}+\nabla\hat{\pi}_{j}=\hat{f}_{j}$(3.1)$\nabla\cdot\hat{u}_{j}=0$

in$\hat{u}=0$on$\mathbb{R}_{+}^{n}$,,$\mathbb{R}_{+}^{n}$$\partial \mathbb{R}_{+}^{n}$with the right hand side defined by a suitable affine transformation ofif, whereinandsatisfying. Define the solution operatorbyand$\hat{f}_{j}$$x_{j}\in\partial\Omega$$\psi_{j}\in C_{c}^{\infty}(\mathbb{R}^{n})$$\psi_{j}\equiv 1$$B_{2r}(x_{j})$$\hat{U}_{\lambda}$$\psi_{j}f$$supp\psi_{j}\subset D_{j}$$B_{j}$$:=$$\hat{\Pi}_{\lambda}$

へ.

$(U_{\lambda}\hat{f}_{j}, \Pi_{\lambda}^{}\hat{f}_{j}):=(\hat{u}_{j},\hat{\pi}_{j})$Since we assume that
, we may construct thehas boundary of classpull-back and push-forward mappings in such a way that they preserve the$C^{3}$$\Omega$condition on the divergence. Hence,

ever,$\tilde{u}$$u_{j}$

is solenoidal by construction. How-

is not solenoidal, in general, since$\nabla\cdot\tilde{u}=\sum_{j=1}^{\infty}(\nabla\varphi_{j})\cdot u_{j}\neq 0$.

Therefore, we use the modified ansatz
(3.2)

where

$\overline{u}$$B_{j}$

denotes the. Inserting$\overline{U_{j}^{-}}=B_{j}\cap\overline{\Omega}$$:= \sum_{j=1}^{\infty}(\varphi_{j}u_{j}+B_{j}(\nabla\cdot(\varphi_{j}u_{j})))$$Bogovski_{\dot{1}}$$(\overline{u},\overline{\pi})$,

operator onsuch thatin (2.3), we thus obtain$U_{j}^{-}:=B_{j}\cap\Omega$$\lambda\overline{u}-\triangle\overline{u}+\nabla\overline{\pi}=f+T_{\lambda}f$$\nabla\cdot\overline{u}=0$$\overline{u}=0$$in\Omega$,$in\Omega$,$on\partial\Omega$$supp\nabla\varphi_{j}\subset$

,

where
denotes the correction terms. In order to show thatis small forlarge, it is crucial to estimate the correction terms involving the pressure it$T_{\lambda}$$\lambda$$T_{\lambda}$

36
and theoperator. Note that, for domains with compact boundary
it is enough to consider the divergence problem on suitable bounded domains,since one can get the convergence of the right hand side of (3.2). If the domain
does not have a compact boundary it seems to be necessary to correct thedivergence term on an unbounded domain, because it is not clear how toprove the convergence of (3.2).
$Bogovski_{\dot{1}}$is not(ii) Non-compact Boundary. We now consider the case whencompact. In order to circumvent these difficulties, we present an approach to
the Stokes problem on domains which non-compact boundaries which relies$\partial\Omega$correction
on the above localization procedure where, however, theof the weak Neumann problem:
term (3.2) is replaced by the solution$Bogovski_{\dot{1}}$$v_{j}$$\triangle v=\nabla\cdot f$

in$\frac{\partial v}{\partial\nu}=f\cdot\nu$

on

(3.3)

$\Omega$,$\partial\Omega$.

To be more precise, we use the other ansatz$u:= \sum_{j=1}^{\infty}\varphi_{j}u_{j}+\nabla v_{j}$. Notewithwhich solves the weak Neumann problem (3.3) withis guaranteed since the Helmholtzthat the existence and uniqueness ofprojection exists by assumption. By construction we then obtain$f=\varphi_{j}u_{j}$$v_{j}$$v_{j}$$\nabla\cdot u=\sum_{j=1}^{\infty}\nabla\cdot(\varphi_{j}u_{j})+\triangle v_{j}=0$.

However, the tangential component of does not vanish at the boundary
anymore. This leads to additional correction terms. In our main linear
satisfying
result we show that (2.3) has a unique solution for any
the usual resolvent estimates. Replacing norm bounds by -bounds (see e.g.
-estimate in[8] in the arguments above, we even obtain the maximal
view of the vector-valued version of Mihklin theorem due to Weis [34].
, theTo explain more details, we prepare the notation. For each
local coordinate corresponding to is defined as a coordinate obtained from$u$$f\in L_{\sigma}^{q}(\Omega)$$\mathcal{R}$$\nu-L^{q}$

$)$

$s$

$x_{j}\in\partial\Omega$$x_{j}$

37
the original ones by some affine transform which moves
to the origin andafter which the positive -axis has the direction of the interior normal toat . Letand choose local coordinates corresponding to .
By definition of a uniform -boundary, there exists an open neighborhood$:=U_{j}$containing $x_{j}=0$ with
andopen, and a height function
satisfying $\partial\Omega\cap U=\{x=(x’, x_{n})\in$$U;x_{n}=h_{j}(x’)\}$ and $\Omega\cap U=\{x\in U;x_{n}>h_{j}(x’)\}$ . Note that choosing theradius of small, we may assume that(independent of j)
is as small as we like. Next we define$x_{j}$$x_{n}$$\partial\Omega$$x_{j}\in\partial\Omega$$x_{j}$$x_{j}$$C^{3}$$U$$:=V_{1}\cross V_{2}\subset \mathbb{R}^{n}$$V_{1}\subset \mathbb{R}^{n-1}$$V_{2}\subset \mathbb{R}$$h_{j}\in C^{3}(\overline{V_{1}})$$V_{1}$(3.4)$\Vert h_{j}\Vert_{\infty}+\Vert\nabla h_{j}\Vert_{\infty}$$g_{j}(x):=(g_{j}^{1}(x), \ldots, g_{j}^{n}(x)):=(x’, x_{n}-h_{j}(x’))$,$x\in U$.

We obtain an injection
satisfying $\Omega\cap U=\{x\in U;g_{j}^{n}(x)>0\}$
and $\partial\Omega\cap U=\{x\in U : g_{j}^{n}(x)=0\}$ . Since
is a uniform -boundary, all$:=g_{j}(U_{j})$ ) up to order 3 may bederivatives of and of(defined onassumed to be bounded by a constant independent of .
For a function :, we call the push-forwardon
defined by $v(y)$ $:=u(g_{j}^{-1}(y))$ , locally. Due to the regularity of the boundary,
this transformation is an isomorphism
for
all $p\in(1, \infty)$ and $s\in[-2,2]$ . Similarly, for a vector-valued function :we define the push-forward
for the solenoidal spaces$:=J_{g}(u(g^{-1}(y)))$ , wherebydenotes the Jacobian of . In fact, thelinear transformation
is an isomorphisms fromto.
Furthermore, it is an isomorphism fromfor
all $p\in(1, \infty)$ and $s\in[-2,2]$ . The corresponding pull-back mappings
andare defined in a similar way. Note, that we may choose $h=0$ if, that is,.
For any $\epsilon\in(0,1)$ , letbe a family of locally finite covers ofsuch thathas -regularity,$g_{j}\in C^{3}(\overline{U}, \mathbb{R}^{n})$$C^{3}$$\partial\Omega$$\hat{U}_{j}$$g_{j}^{-1}$$g_{j}$$x_{j}$$u$$v=\mathcal{G}u$$U_{j}\cap\Omegaarrow \mathbb{R}$$\hat{U}_{j}\cap \mathbb{R}_{+}^{n}$$W^{s,p}(U_{j}\cap\Omega)arrow W^{s,p}(\hat{U}_{j}\cap \mathbb{R}_{+}^{n})$$u$$U\cap\Omegaarrow \mathbb{R}^{n}$$v_{\sigma}=\mathcal{G}_{\sigma}u$$v_{\sigma}(y)$$J_{g}$$g$$\mathcal{G}_{\sigma}$$L_{\sigma}^{p}(U_{j}\cap\Omega)$$U_{\sigma}(\hat{U}_{j}\cap \mathbb{R}_{+}^{n})$$W^{s,p}(U_{j}\cap\Omega)arrow W^{s,p}(\hat{U}_{j}\cap \mathbb{R}_{+}^{n})$$\mathcal{G}^{-1}$$\mathcal{G}_{\sigma}^{-1}$$U_{j}\cap\partial\Omega=\emptyset$$x_{j}\in\Omega$$\Omega$$\{\Omega_{j}^{\epsilon}\}_{j\in \mathbb{N}}$$U_{j}\subset\Omega_{j}^{\epsilon},$$\partial\Omega_{j}^{\epsilon}$$C^{3}$(3.5)$\Vert\nabla h_{j}^{\epsilon}\Vert_{\infty}<\epsilon$

(3.6)

where

,

$\sum_{j\in \mathbb{N}}\chi_{\Omega_{j}^{\epsilon}}\leq C$$\chi_{\Omega_{j}^{\epsilon}}$is the characteristic function on

function corresponding to$\Omega_{j}^{\epsilon}$,

for each $j,$is the height, and $C>0$ is a constant independent of . For$\Omega_{j}^{\epsilon}$$h_{j}^{\epsilon}$$\epsilon$

38
subordi, we choose a partition of unity
each such covering
theandnate to this covering. Furthermore, denote bycorresponding push-forward mappings and pull-back mappings.
can beandoffor
The commutator
contains second order terms (highest) ofsplit into two parts:
contains all lower order terms. In particular, by (3.5)
only and
there exists a constant $C>0$ such that$\{\varphi_{j}^{\epsilon}\}_{j\in N}$$\{\Omega_{j}^{\epsilon}\}_{j\in N}$$\mathcal{G}_{\sigma,j}^{-1,\epsilon}$$\mathcal{G}_{j}^{-1,\epsilon}$$\mathcal{G}_{\sigma,j}^{\epsilon},$$\mathcal{G}_{j}^{\epsilon},$$\triangle$$\hat{u}_{j}\in W^{2,p}(\mathbb{R}_{+}^{n})$$[\triangle, \mathcal{G}_{j,\sigma}^{-1,\epsilon}]\hat{u}_{j}$$\mathcal{G}_{j,\sigma}^{-1,\epsilon}$$\hat{u}_{j}$$[\triangle, \mathcal{G}_{j,\sigma}^{-1,\epsilon}]_{h}\hat{u}_{j}$$[\triangle, \mathcal{G}_{j,\sigma}^{-1,\epsilon}]_{l}\hat{u}_{j}$$\Vert[\triangle, \mathcal{G}_{j,\sigma}^{-1,\epsilon}]_{h}\hat{u}_{j}$Il,$Lp(\Omega_{j})\leq C\epsilon\Vert\hat{u}_{j}\Vert_{\overline{W}^{2,p}(\hat{\Omega}_{j}^{\epsilon})}$$\Vert[\triangle, \mathcal{G}_{j,\sigma}^{-1,\epsilon}]_{l}\hat{u}_{j}\Vert_{L^{p}(\Omega_{j})}\leq C\Vert\hat{u}_{j}\Vert_{W^{1,p}(\hat{\Omega}_{j}^{\epsilon})}$,$\epsilon\in(0,1),$$j\in N,\hat{u}_{j}\in W^{2,p}(\hat{\Omega}_{j})$$j\in \mathbb{N},\hat{u}_{j}\in W^{2,p}(\hat{\Omega}_{j})$$\epsilon\in(0,1),$,.

denotes the transformation by the j-th push
Here and in the following,
denotes the function living on theforward map of . In the same way
half space
which is connected with through the j-th push forward map.
Similarly, there exists a constant $C>0$ such that$\hat{\Omega}_{j}^{\epsilon}$$\hat{u}_{j}^{\epsilon}$$\Omega_{j}^{\epsilon}$$\mathbb{R}_{+}^{n}$$u_{j}^{\epsilon}$,$\Vert[\nabla, \mathcal{G}_{j}^{-1,\epsilon}]\hat{\pi}_{j}\Vert_{L^{p}(\Omega_{j})}\leq C\epsilon\Vert\hat{\pi}_{j}\Vert_{\overline{W}^{1,p}(\mathbb{R}_{+}^{n})}$$\epsilon\in(0,1),$$j\in \mathbb{N},\hat{\pi}_{j}\in\hat{W}^{1,p}(\hat{\Omega}_{j}^{\epsilon})$.

’s operator to construct localized data forandour localization procedure. For a bounded Lipschitz domain$g\in If(\Omega’)$ with
$g=0Bogovski\dot{1}’ S$ operator
is a solution operator to
the divergence equation as follows

As in [17], we use$Bogovski_{\dot{1}}$$\Omega’\subset\Omega$$B_{\Omega’}$$\int_{\Omega},$(3.7)$\{\begin{array}{l}divu = g in \Omega’,u = 0 on\partial\Omega’.\end{array}$

By the definition of$\Omega_{j}^{\epsilon}$, there exists $C>0$ independent of$\Vert B_{\Omega_{j}^{\epsilon}}f\Vert_{Lp(\Omega_{j}^{\epsilon})}\leq C\Vert f\Vert_{L^{p}(\Omega_{j}^{\epsilon})}$We finally choose cut-off functions. For $f\in X_{f}$onterms by$supp\varphi_{j}^{\epsilon}$$\psi_{j}^{\epsilon}\equiv 1$,$\epsilon\in(0,1),$$j\in \mathbb{N},$$f_{j}^{\epsilon}:=\psi_{j}^{\epsilon}f-B_{\Omega_{j}^{\epsilon}}((\nabla\psi_{j}^{\epsilon})f)$

and letdenote the extension to
the uniform boundedness of$\hat{f}_{j}^{\epsilon}$$\mathbb{R}_{+}^{n}$$Bogovski_{\dot{1}}$(3.8)

such that$f\in L^{p}(\Omega_{j}^{\epsilon})$.

andsuch that, we define the local external force$0\leq\psi_{j}^{\epsilon}\leq 1$$\psi_{j}^{\epsilon}\in C_{c}^{\infty}(\Omega_{j}^{\epsilon})$$:=\nu_{\sigma}(\Omega)$$j\in \mathbb{N}$,by of the push-forward
operator, we obtain$0$$\Vert\hat{f}_{j}^{\epsilon}\Vert_{L^{p}(\mathbb{R}_{+}^{n})}\leq C\Vert f\Vert_{Lp(\Omega_{j}^{\epsilon})}$$\mathcal{G}_{\sigma,j}^{\epsilon}f_{j}^{\epsilon}$$\hat{f}_{j}^{\epsilon}\in U_{\sigma}(\mathbb{R}_{+}^{n})$

. By

and

39
with some $C>0$ independent of$\epsilon,$

$j$

and

$f$

. Hence, (3.6) yields that
へ

(3.9)

$((S_{j}^{1,\epsilon})_{j\in \mathbb{N}})_{\epsilon\in(0,1)}\subset \mathcal{L}(X_{f}, \ell^{p}(X_{f}))$is uniformly bounded, where$S_{j}^{1,\epsilon}f$$:=\hat{f}_{j}^{\epsilon}$. Similarly, for $(a, b)\in X_{a,b}:=\{a\in$, we define the local$W^{1-1/p,p}(\partial\Omega);a\cdot\nu=0\}\cross\{b\in W^{2-1/p,p}(\partial\Omega);b\cdot\nu=0\}$boundary dataHere,is the restriction of$a_{j}^{\epsilon}$$:=\psi_{j}^{\epsilon}a,$$b_{j}^{\epsilon}$andto the boundary of . Again, we see$:=\psi_{j}^{\epsilon}b,\hat{a}_{j}^{\epsilon}$$:=\mathcal{G}_{j,\sigma}^{\partial\Omega,\epsilon}\psi_{j}^{\epsilon}a$$\hat{b}_{j}^{\epsilon}$$:=\mathcal{G}_{j,\sigma}^{\partial\Omega,\epsilon}\psi_{j}^{\epsilon}b$.

$\Omega$$\mathcal{G}_{j,\sigma}^{\partial\Omega,\epsilon}$$\mathcal{G}_{j,\sigma}^{\epsilon}$へ(3.10)$((S_{j}^{2,\epsilon}(a, b))_{j\in N})_{\epsilon\in(0,1)}\subset \mathcal{L}(X_{a,b}, \ell^{p}(X_{a,b}))$is uniformly bounded, where$S_{j}^{2,\epsilon}$$:=S_{j}^{2,\epsilon}(a, b)$$:=(\hat{a}_{j}^{\epsilon},\hat{b}_{j}^{\epsilon})$. We now set$U_{\lambda}^{\epsilon}(f, a, b):= \sum_{j\in N}\varphi_{j}^{\epsilon}\mathcal{G}_{j,\sigma}^{-1,\epsilon}\hat{U}_{\lambda}S_{j}^{\epsilon}(f, a, b)-\nabla \mathcal{N}(\sum_{j\in N}\varphi_{j}^{\epsilon}\mathcal{G}_{j,\sigma}^{-1,\epsilon}\hat{U}_{\lambda}S_{j}^{\epsilon}(f, a, b))$whereis the solution operator of the weak Neumann problem and$:=$. Here, similarly to (3.2), we add a correction
term in order to have a solenoidal ansatz. However, in contrast to thecase (i), the correction term is based on the solution operator of the weak
Neumann problem instead ofoperator. Inserting,
we calculate$\mathcal{N}$$S_{j}^{\epsilon}(f, a, b)$$S_{j}^{\epsilon}$$:=(S_{j}^{1,\epsilon}f, S_{j}^{2,\epsilon}(a, b))$$U_{\lambda}^{\epsilon}$$Bogovski_{\check{1}S}$$\lambda u-\mathbb{P}\triangle u=f+\mathcal{T}_{\lambda}^{1,\epsilon}(f, a, a)$(3.11)$\nabla\cdot u=0$

$u=a+\mathcal{T}_{\lambda}^{2,\epsilon}(f, a, a)$

$u$

in

$\Omega$,in$\Omega$,on$\partial\Omega$$:=U_{\lambda}^{\epsilon}(f, a, a)$

,

where
$\mathcal{T}_{\lambda}^{\epsilon}(f, a, b):=(\mathcal{T}_{\lambda}^{1,\epsilon}(f, a, b), \mathcal{T}_{\lambda}^{2,\epsilon}(f, a, b)):=T_{1,\lambda}^{\epsilon}(f, a, b)+\cdots+T_{6,\lambda}^{\epsilon}(f, a, b)$with$T_{1,\lambda}^{\epsilon}(f, a, b):=( \mathbb{P}\sum_{j\in \mathbb{N}}\varphi_{j}^{\epsilon}[\nabla, \mathcal{G}_{j}^{-1,\epsilon}]\hat{\Pi}_{\lambda}S_{j}^{\epsilon}(f, a, b),$$T_{2,\lambda}^{\epsilon}(f, a, b);=( \mathbb{P}\sum_{j\in N}(\nabla\varphi_{j}^{\epsilon})\mathcal{G}_{j}^{-1,\epsilon}\Pi_{\lambda}^{へ}S_{j}^{\epsilon}(f, a,b),$$0,0)$ ,$0,0)$ ,40$T_{3,\lambda}^{\epsilon}(f, a, b):=(- \mathbb{P}\sum_{j\in N}[\varphi_{j}^{\epsilon}, \triangle]\mathcal{G}_{j,\sigma}^{-1,\epsilon^{へ}}U_{\lambda}S_{j}^{\epsilon}(f, a, b),$$0,0)$ ,$T_{4,\lambda}^{\epsilon}(f, a, b):=(- \mathbb{P}\sum_{j\in N}\varphi_{j}^{\epsilon}[\triangle, \mathcal{G}_{j}^{-1,\epsilon^{へ}}]_{h}U_{\lambda}S_{j}^{\epsilon}(f, a, b),$$T_{5,\lambda}^{\epsilon}(f, a, b):=(- \mathbb{P}\sum_{j\in N}\varphi_{j}^{\epsilon}[\triangle, \mathcal{G}_{j}^{-1,\epsilon}]_{l}\hat{U}_{\lambda}S_{j}^{\epsilon}(f, a, b),$$T_{6,\lambda}^{\epsilon}(f, a, b)$$:=(0,$$-\nabla \mathcal{N}V^{\epsilon},$$-\nabla \mathcal{N}V^{\epsilon})$$0,0)$ ,$0,0)$ ,.

. This means that we obtain a soluHeretion of the Stokes resolvent problem in which is given by$V^{\epsilon}$$:= \sum_{j\in N}\varphi_{j}^{\epsilon}\mathcal{G}_{j,\sigma}^{-1,\epsilon_{U_{\lambda}S_{j}^{\epsilon}(f,a,b)|_{\partial\Omega}}^{へ}}$$\Omega$(3.12),$R^{\epsilon}(\lambda)f$$:=U_{\lambda}^{\epsilon}(1+ \mathcal{T}_{\lambda}^{\epsilon})^{-1}(f, 0,0)=U_{\lambda}^{\epsilon}\sum_{k\in N_{0}}(\mathcal{T}_{\lambda}^{\epsilon})^{k}(f,0,0)$

provided if the above Neumann series converges.
In the following we show that the Neumann series exists for some$(0,1)$ , which hence yields the existence of a solution to (3.11). The uniquenessof the solution follows from a standard duality argument. Hence, we finally$\epsilon\in$

In order to estimate it, we set $X$ $:=X_{f}\cross X_{a,b}$ .
Then, the representation formula (3.12) can be written as

obtain$R^{\epsilon}(\lambda):=(\lambda+A_{p})^{-1}$$R^{\epsilon}( \lambda)f=U_{\lambda}^{\epsilon}\sum_{k\in N_{0}}(\mathcal{T}_{\lambda}^{\epsilon})^{k}(f, 0,0)=U_{\lambda}^{\epsilon}K_{\lambda}^{-1}\sum_{k\in N_{0}}(K_{\lambda}\mathcal{T}_{\lambda}^{\epsilon}K_{\lambda}^{-1})^{k}K_{\lambda}(f, 0,0)$$=U_{\lambda}^{\epsilon}K_{\lambda}^{-1} \sum_{k\in N_{0}}(K_{\lambda}\mathcal{T}_{\lambda}^{\epsilon}K_{\lambda}^{-1})^{k}(f, 0,0)$,

provided if the above series converges. Here

$K_{\lambda}:=(\begin{array}{lll}1 0 00 \lambda^{l-\frac{1}{3p}} 00 0 1\end{array})$In the following lemma we show that
(3.13)$\mathcal{R}_{X}\{K_{\lambda}\mathcal{T}_{\lambda}^{\epsilon}K_{\lambda}^{-1};\lambda\in\lambda_{0}+\Sigma_{\theta}\}<1$. Hence,is well defined for some $\epsilon\in(0,1)$
for sufficient large
with large . The following lemma is crucial.
and all$\lambda_{0}>0$$\lambda\in\lambda_{0}+\Sigma_{\theta}$$R^{\epsilon}(\lambda)$$\lambda_{0}$41Lemma 3.1. For $\alpha\in(0,1/2p’)$ there exist$\epsilon_{0}\in(0,1)$

and $C>0$ such that$\mathcal{R}_{X}\{K_{\lambda}\mathcal{T}_{1,\lambda}^{\epsilon}K_{\lambda}^{-1};\lambda\in 1+\Sigma_{\theta}\}\leq 1/4$,,$\mathcal{R}_{X}\{\lambda^{\alpha}K_{\lambda}\mathcal{T}_{2,\lambda}^{\epsilon}K_{\lambda}^{-1};\lambda\in 1+\Sigma_{\theta}\}\leq C$$\mathcal{R}_{X}\{\lambda^{1/2}K_{\lambda}T_{3,\lambda}^{\epsilon}K_{\lambda}^{-1};\lambda\in 1+\Sigma_{\theta}\}\leq C$,$\mathcal{R}_{X}\{K_{\lambda}\mathcal{T}_{4,\lambda}^{\epsilon}K_{\lambda}^{-1};\lambda\in 1+\Sigma_{\theta}\}\leq 1/4$,,$\mathcal{R}_{X}\{\lambda^{1/2}K_{\lambda}\mathcal{T}_{5,\lambda}^{\epsilon}K_{\lambda}^{-1};\lambda\in 1+\Sigma_{\theta}\}\leq C$$\mathcal{R}_{X}\{\lambda^{1/2p}K_{\lambda}\mathcal{T}_{6,\lambda}^{\epsilon}K_{\lambda}^{-1};\lambda\in 1+\Sigma_{\theta}\}\leq C$.

The reader can find the proof of the above lemma in [18]. This lemma
leads us to (3.13) if
is taken sufficient large. That is the outline of proofof Theorem 2.1.
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