数理解析研究所講究録
第 745 巻 1991 年 53-77$L^{2}$

53

Decay for Navier-Stokes Flows in Unbounded Domains, with
Application to Exterior Stationary FlowsWOLFGANG BORCHERS & TETSURO MIYAKAWA(パーダーボルン大学)(広島大学)
宮川鉄朗

1. Introduction

The motion of a viscous incompressible fluid filling a domain
the Navier-Stokes initial value problem:$D\subset R^{?t}$$\frac{\partial u}{\partial t}+u\cdot\nabla u=\triangle tt-\nabla p+f$$(x\in D, t>0)$ ,$\nabla\cdot u=0$$(x\in D, t\geq 0)$;$u|_{S}=0$

is governed by

(NS)

,

$u|_{t=0}=a$for unknown velocity $u=(u_{j})_{j=1}^{n}$ and pressure . Hereis the boundary of $D,$ $x=$$(x_{1}, \cdots, x.)$is a point ofdenote, respectively, given initial velocity andandexternal force ; and. The
fluid density and the kinematic viscosity are normalized to be one. It is known [16] thatproblem (NS) possesses at least one weak solution for an arbitrary initial velocity in .
Uniqueness of weak solutions is proved by now only when $?t=2$ .
This paper studies the existence problem of a weak solution, in an arbitrary unboundeddomain, which goes to zero in , as
, with explicit rates. The
decay problem forNavier-Stokes flows was first raised by LERAY [14] in case $D=R^{3}$ . The first (affirmative)
answer was given by KATO [13] in case $D=R^{n},$ $\uparrow?=3,4$ , through his study of strong
solutions in generalspaces. A different approach was then given by SCHONBEK [20]which is based on the Fourier decomposition for the fluid velocity ; see also [12,21,23].
The idea of SCHONBEK was then applied by the present authors $[2,3]$ to the case where $D$$n\geq 2$ , or an exterior domain of$n\geq 3$ . In this paper we firstis a halfspace ofshow, in Sections 2 and 3, that the method developed in [2,3,12] can be modffied so that$n\leq 4$ . The $a_{fguments}$
it applies to the case of an arbitrary unbounded domain indeveloped in Sections 2 and 3 are then applied in Section 4 to the stability problem forexterior stationary flows in three-dimensions.
To state our main results, we use the standard no.tation :denotes the set ofsmooth and compactly supported solenoidal vector fields on $D$ . We denote by $H$ (resp. )the(resp.
) closure of.
following orthogonal decomposition
$S$

$p$

$R^{n},$$f$$a$$u\cdot\nabla u=\Sigma_{\dot{J}}u_{j}\partial_{j}u_{\dot{J}},$$\nabla/\cdot u=\Sigma_{J}\cdot\partial_{J}\cdot u_{J}\cdot,$$\nabla p=(\partial_{j}p)_{j=1}^{n},$$\partial_{j}=\partial/\partial x,\cdot$$a$$tarrow\infty$$L^{2}$$L^{2}$$L^{2}$$L^{p}$$u$$R^{n},$$R^{n},$$R^{n},$$C_{0^{\infty_{\sigma}}},(D)$$V$$L^{2_{-}}$$H^{1_{-}}$$C_{0}^{\infty_{\sigma}}(D)$$(L^{2}(D))^{n}=H\oplus H^{\perp}$,$T1_{1}e$$H^{\perp}=\{\nabla p\in(L^{\Gamma})(D))^{n} :1p\in L_{loc}^{2}(D)\}$,

54
is well known [22, Chap.I]. We denote by $P$ the associated orthogonal projector onto
$H$ .
To the bilinear forn
defined on $V\cross V$ , we associate a (unique) positive
in $H$ such that $D(A^{1/2})=V$ and $||A^{1/2}u||_{2}=||\nabla u\Vert_{2}$ , whereand self-adjoint operator$(1 \leq r\leq\infty)$
we denote the completion ofis the usual -norm. By, and byits dual space. For simplicity in notation we assume thatin the norm
$f=Pf$ , using the above orthogonal decomposition. Then the function$(\nabla u, \nabla v)$$A$$\hat{V}$$L^{r}$$||\cdot||_{r}$$C_{0^{\infty_{\sigma}}},(D)$$\hat{V}^{*}$$||\nabla\cdot||_{2}$$v(t)=e^{-tA}a+ \int_{0}^{t}e^{-(t-s)A}f(s)ds$

with $a\in H$ and $f\in L_{loc}^{2}([0, \infty);H)$ solves the nonstationary Stokes system$\frac{\partial v}{\partial t}-\triangle v=f-\nabla q$$(x\in D, t>0)$$\nabla\cdot v=0$$(x\in D, t\geq 0)$$v|_{S}=0$;(S)$v|_{t=0}=a$with an appropriate scalar distribution
into the integral equation :$q$; so the problem (NS) is formally transformed.

$u(t)=e^{-tA}a+ \int_{0}^{t}e^{-(t-s)A}[f(s)-P(u\cdot\nabla)u(s)]ds$(I)

Given and as above, a weakly continuous function) $arrow H$ is called a weak:solution of (NS) (or eqtlivalently of (I)) if it belongs to $L^{\infty}(0, T;H)\cap L^{2}(0, T;V)$ for all$T>0$ , and satisfies $u(O)=a$ and
$a$

$f$

$[0, \infty$$u$$(u(t), \phi(t))+\int_{s}^{t}[(\nabla u, \nabla\phi)+(u\cdot\nabla u, \phi)]d\tau=(u(s), \phi(s))+\int_{s}^{t}[(u, \phi’)+(f, \phi)]d\tau$(W)

for all $t\geq s\geq 0$ , and all $\phi\in C([0, \infty);V)\cap C^{1}([0, \infty))H)$ , whereis the standard
inner product and. The existence of a weak solution corresponding to arbitrary
and is well known ; see, e.g., [16]. All the weak solutions obtained so far satisfy theenergy inequality$(\cdot, \cdot)$$L^{2_{-}}$$\phi’=\partial\phi/\partial t$

$a$

$f$

$||u(t) \Vert_{2}^{2}+2\int_{0}^{t}||\nabla u||_{2}^{2}ds\leq||a||_{2}^{2}+2\int_{0}^{t}(f, u)ds$

for all $t\geq 0$ , and the equality sign holds in case $n=2$ . In Section 4 we shall deal with amore stringent form of the above energy inequality. Our main results are the following
Theoren 1.1. Let $n=3,4$ , and let
be an arbitrary n-dimensional unbounded
does not hold. If $a\in H$ ,for which the Poincar\’e inequality for functions in
if $f\in L_{loc}^{2}([0, \infty);H)\cap L^{1}(0, \infty;H)\cap L^{1}(0, \infty;\hat{V}^{*})_{f}$ and if$D$$do\uparrow\gamma lai\uparrow x$$C_{0^{\infty}}(D)$$\int_{0}^{\infty}t||f(t)||_{2}dt<+\infty$

then there is a weak solution$u$of (NS) such that, as
2,$tarrow\infty$

,

55
(i)

$||u(t)||_{2}arrow 0$.

(ii) If $||e^{-tA}a||_{2}=O(t^{-\alpha})$ for some $\alpha>0$ , then,$\Vert u(t)||_{2}=\{\begin{array}{l}O(t^{-\alpha})if\alpha<1/2O(t^{\epsilon-1/2})if\alpha\geq 1/2\end{array}$where $0<\epsilon<1/2$ is arbitrary in case $n=3$ , and $c’=0$ in case $n=4$ .

Theorem 1.2. Let
be an $arb\iota trary$ unbounded domain for which the Poincar\’einequality does not hold. Given $a\in H$ and as in Theorem 1.1, there is a unique weak,solution such that, as$D\subset R^{2}$$f$$tarrow\infty$

$u$

(i)

$\Vert u(t)||_{2}arrow 0$.

(ii) If $||e^{-tA}a||_{2}=O(t^{-\alpha})$ for some $\alpha>0$ , then$||u(t)||_{2}=O((\log(t+e))^{-n/2})$.
for all integers(iii) If in addition$m\geq 1$

and$||a||_{2}$$\int_{0^{\infty}}||f||_{2}ds$

are small enough and$f\in L^{2}(0, \infty;\hat{V}^{*})$, then$||u(t)||_{2}=\{\begin{array}{l}O(t^{-\alpha})if\alpha<1/2O(t^{\epsilon-1/2})if\alpha\geq 1/2\end{array}$where$\epsilon>0$(iv)

size of

If$\Vert a\Vert_{2}$is arbitrary.
$a\in R(A^{\alpha})$andfor some $0<\alpha\leq 1/2$ , then assertion (iii) holds$\int_{0}^{\infty}\Vert f||_{2}ds$$irrespect\iota ve$of the.

or the halfspace $R_{+)}^{n}n\geq 2$ , and $f=0$ , it is known
When $D$ is the entire space[2, 12,21,23] that there exists a weak solution satisfying$R^{n}$$u$$|1u(t)\Vert_{2}=\{\begin{array}{l}O(t^{-\alpha})for\alpha<(n+2)/4O(t^{-(n+2)/4})for\alpha\geq(7\not\supset+\underline{9})/4\end{array}$provided thatfor some $\alpha>0$ . When $n\geq 3,$ $f=0$ , andis anexterior domain with smooth boundary, we have recently proved in [3] the existence of a
weak solution such that, under the assumption $||e^{-tA}a||_{2}=O(t^{-\alpha})$ ,$D$$\Vert e^{-tA}a\Vert_{2}=O(t^{-\alpha})$$u$$||u(t)||_{2}=\{\begin{array}{l}O(t^{-\alpha})if\alpha<n/4O(t^{-n/4})if\alpha\geq n/4\end{array}$

All these results were deduced by essential use of various properties of the operator inspaces. In our present case, however, the class of domains $D$ is so large thatgeneralwe cannot appeal to -theories. So we restrict ourselves to the case of space dimensions$n\leq 4$ and deduce our results by applying only
-theory of the operator . We note that$A$

$L^{r}$$L^{r}$$L^{2}$

3

$A$

56
Theorem 1.2 partially extends our previous result in [3] to the case of two-dimensional
exterior domains (with nonsmooth boundaries).
We prove Theorem 1.1 in Section 2, using a specific approximation scheme. Since theuniqueness of weak solutions remains open in case $n\geq 3$ , we first consider in Section 2
the decay problem for generaI weak solutions satisfying the energy inequality and showthat the time-averageof any such weak solution decays in the same way
as stated in Theorem 1.1 ; see Theorem 2.1. It turns out that Theorem 1.1 immediately
follows from Theorem 2.1.
Our proofs of Theorems 1.1 and 2.1 systematically use the weak version of the H\"older
and Young inequalities, thereby avoiding a direct application of the spectral measure associated to the self-adjoint operator . However, this approach does not work in twodimensional case. So, we give in Section 3 a detailed proof of Theorem 1.2 which uses the
spectral decomposition for . This approach was first suggested by SCHONBEK [20] andthen systematically studied by [2,3,12]. It is also possible to prove Theorem 1.1 by using
the spectral decomposition. But, we do not employ this approach, since our argument
in Section 2 provides not only Theorem 2.1, which is difficult to obtain by applying thespectral decomposition, but also a stability result given in Section 4, which directly deals
with a non-self-adjoint linearized operator instead of the self-adjoint operator .
In both of Theorems 1.1 and 1.2, it is in general difficult to completely characterize
the class of functions $a\in H$ satisfying the condition $||e^{-tA}a||_{2}=O(t^{-\alpha})$ . In Section 3 weshow that this condition holds for in somespaces. This result is deduced from the factthat the rangeof the fractional power
remains invariant under the Navier-Stokes
flow if $\alpha>0$ satisfies an appropriate condition depending on the space dimension ; andthis invariance property not only enables us to prove assertion (iv) of Theorem 1.2, butalso implies the following$t^{-1} \int_{0}^{t}||u||_{2}ds$

$u$

$A$

$A$

$A$

$a$

$R(A^{\alpha})$$L^{r}$$A^{\alpha}$

Corollary 1.3. (i) If $n=2$ and $a\in H\cap(L^{r}(D))^{2}$ for some $1<r<2$ , then thecorresponding weak solution satisfies$u$$||u(t)||_{2}=O(t^{-(1/r-1/2)})$

satisfies the assumption in Theorem 1.2 (iii).
(ii) If $n=3$ and $a\in H\cap(L^{r}(D))^{3}$ for some $6/5\leq r\leq 3/2$ , then there is a weakprovided thatsolution$u$$f$with $u(0)=a$ such that$||u(t)||_{2}=O(t^{-3(1/r-1/2)/2})$.

(iii) If $n=4$ and $a\in H\cap(L^{4/3}(D))^{4}$ , there is a weak solutionthat$||u(t)||_{2}=O(t^{-1/2})$

4

.

$u$

with $u(0)=a$ such

57
The problem of
decay for Navier-Stokes flows is closely connected with the notionof energy stability in viscous fluid motions (see [6]). Indeed, Theorems 1.1 and 1.2 assert inparticular that the trivial steady state: $u=0$ is globally asymptotically stable in this sense
in arbitrary unbounded domains. In Section 4 we apply the method of proof of Theorems
1.1 and 1.2 to the stability problem for exterior stationary flows in three-dimensions. We
prove that an exterior stationary flow is globally asymptotically stable in energy sense
provided that the associated Reynolds number is small enough. See Theorem 4.2. This
result improves and supplements the known results as given for instance in [6,8,9,10,15]. A
novel feature of our result is that we deal with global -norms of disturbances and deduce
their explicit decay rates. However, we believe that our result in this section is not the
optimal one.
We thank Professors J. G. HEYWOOD and A. MATSUMURA for their interest in
decay $pro$ blem. Parts of our results are announced in [24].
$L^{2}$$L^{2}$$L^{2}$

2. Proof of Theorem 1.1
First we deal with general weak solutions satisfying the energy inequality :(E)$||u(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla u||_{2}^{2}ds\leq||a||_{2}^{2}+2\int_{0}^{t}(f, u)ds$

for all $t\geq 0$ . The following result can be regarded as a refinement of the decay result of
MASUDA [16].

Theorem 2.1. Let the assumptions in Theorem 1.1 be satisfied.
solution satisfying the energy inequality (E), then, as

If

$u$

is any weak$tarrow\infty_{f}$(i)$\int_{t}^{t+1}||u||_{2}dsarrow 0$

(ii)

If

$||e^{-tA}a||_{2}=O(t^{-\alpha})$for some$\alpha>0$.

, then,$\frac{1}{t}\int_{0}^{t}||u||_{2}ds=\{o(t^{-\alpha})O(t_{\epsilon-1})_{/2}$where $0<\epsilon<1/2$ is arbitrary in case $n=3$ , and$ifif$$\epsilon=0$$\alpha\geq\alpha<11(_{2^{)}}^{\underline{9}}$.

in case $n=4$ .

For the proof we prepare three lenimas.

Lemma 2.2. Letfunctions on the real line

$L_{w}^{p}=L_{w}^{p}(R),$

$f$

$R$

$1<p<\infty$ ,denote the Banach space of measurablewith norm$||f||_{p,w} \equiv\sup_{E}|E|^{-1+1/p}\int_{E}|f|ds<\infty$

5

58
where $|E|$ is the Lebesgue measure of a measrable set $E$

If

(i)

$f\in L_{w}^{p},$and $1/p+1/q=1/r$ , then$g\in L_{w}^{q}$

,

, and

$fg\in L_{w}^{r}$$|1fg||_{r,w}\leq C||f||_{p,w}||g||_{q,w}$

with $C>0$ depending only on

$p$

and

$q$

,

and $1/p+1/q=1+1/r$ , then the convolution $f*g$ is in(ii) If
and there is a constant $C>0$ depending only on and so that$f\in L_{w}^{p},$$g\in L_{w}^{q}$

$p$

llf
(iii)

If

$f\in L_{w}^{p}$$L_{w}^{r}$$q$$*g||_{r,w}\leq C||f||_{p,w}||g||_{q,w}$.

and $g\in L^{1}$ , then $f*g\in L_{w}^{p}$ , and$||f*g||_{p,w}\leq||f||_{p,w}||g||_{1}$It is easy to see thatfunctions that we need is$L^{p}\subset L_{w}^{p}$.

with continuous injection. A typical example of$L_{w^{-}}^{p}$$f(t)=\{\begin{array}{l}t^{-1/p}(t>0)0(t\leq 0)\end{array}$

Lemma 2.2 (i) is the weak version of H\"older’s inequality, while (ii) and (iii) are the weakversions of Young’s inequality. Although these inequalities seem to be well known (see [19,p.32] for (ii)), we give here an elementary proof for the reader’s convenience.
Proof. (i) First observe that

$f$

is in

$L_{w}^{p}$

if and only if$||f||_{p,w}^{*} \equiv\sup_{t>0}t|E(|f|>t)|^{1/p}<+\infty$,

where $E(|f|>t)=\{s\in R;|f(s)|>t\}$ , and that$||f||_{p,w}^{*} \leq||f\Vert_{p,w}\leq\frac{p}{p-1}||f||_{p,w}^{*}$as shown for instance in [7, p. 585]. Applying the classical Young’s inequality :$|fg| \leq\frac{r}{p}\epsilon^{p/r}|f|^{p/r}+\frac{r}{q}\epsilon^{-q/r}|g|^{q/r}$

for any $e>0$ , we get$E(|fg|>t)\subset E(|f|>c_{1}\epsilon^{-1}t^{r/p})\cup E(|g|>C^{}\underline{)}\epsilon t^{r/q})$with$c_{1}$

and$c_{2}$

depending only on
$($

llf

$p$

and . Direct calculation thus gives$q$$g||_{r,w}^{*})^{r}\leq c_{16^{p}}(||f||_{p,w}^{*})p+c_{2^{\mathcal{E}^{-q}}}(||g||_{q,w}^{*})q$

6

59
and
depend only on, wherefor allthe minimum with respect to $e>0$ .
$\epsilon>0$$C_{1}$$C_{2}$$p$and . The result now follows by taking$q$(ii) Fix an arbitrary $\alpha>0$ and let$K_{1}(\tau)=\{0^{f(\tau)1}|$$((|\begin{array}{l}f(\tau)f(\tau)\end{array}|\leq\alpha)>\alpha)$

and $K_{2}(\tau)=|f(\tau)|-Ii_{1}^{r}(\tau)$ . Then$|f*g|(s)\leq A_{1}^{\nearrow}*|g|(s)+K_{2}*|g|(s)\equiv I_{1}(s)+I_{2}(s)$.

(2. 1)

By the definition of the Lebesgue integral,$I_{1}(s)= \int_{0}^{\infty}dt\int_{E_{1}}|g(\tau)|d\tau\leq\Vert g||_{q,w}\int_{0}^{\infty}|E_{t}|^{1-1/q}dt$,, and $|E_{t}|=0$ if $t>\alpha$ . Since
where $E_{t}=\{\tau,\cdot If_{1}(s-\tau)>t\}$ , so$p(1-1/q)<1$ by assuInption, we obtain, with $1/q’=1-1/q$ ,$|E_{t}|\leq(||f||_{p,w}^{*})^{p}t^{-p}$$I_{1}(s) \leq||g||_{q,w}(||f||_{p,w}^{*})^{p/q’}\int_{0}^{a}t^{-p/q’}dt=C\alpha^{1-p/q’}||g||_{q,w}(||f||_{p,w}^{*})^{p/q’}$

with

$C$

depending only on$p$,and . We thus have
$q$

(2.2)

$\int_{E}I_{1}(s)ds\leq C\alpha^{1-p/q’}|E|\cdot\Vert g\Vert_{q,w}(||f||_{p,w}^{*})^{p/q}$

On the other hand, denoting by$\int_{E}I_{2}(s)ds$$1_{E}$

the indicator function of the set$E$, we have$=$$\int\int 1_{E}(s)iK_{2}(\dot{s}-\tau)|g(\tau)|dsd\tau$$=$$\int|g(\tau)|d\tau\int 1_{E}(s+\tau)K_{2}(s)ds=\int K_{2}(s)ds\int 1_{E}(s+\tau)|g(\tau)|d\tau$$\leq$$||g||_{q,w}|E|^{1-1/q} \int I\iota_{2}^{\nearrow}(s)ds$.

Since $|E(|f|>t)|\leq(\Vert f\Vert_{p,w}^{*})^{p}t^{-p}$ , the definition of the Lebesgue integral shows$\int K_{2}(s)ds$

with

$C$

depending only on

$p$

$=$

$\alpha|E(|f|>\alpha)|+\int_{\alpha}^{\infty}|E(|f|>t)|dt$$\leq$$C(\Vert f\Vert_{p,w}^{*})^{p}\alpha^{1-p}$

and . Hence
$q$

(2.3)

$\int_{E}I_{2}(s)ds\leq C\alpha^{1-p}|E|^{1-1/q}\Vert g\Vert_{q,w}(||f||_{p,w}^{*})^{p}$

Combining (2.1), (2.2) and (2.3) gives$\int_{E}|f*g|ds\leq C||g||_{q,w}[\alpha^{1-p/q’}(||f||_{p,w}^{*})^{p/q’}|E|+\alpha^{1-p}(||f||_{p,w}^{*})^{p}|E|^{1-1/q}]$

7

.

60
Taking the minimum with respect to $\alpha>0$ yields$\int_{E}|f*g|ds\leq C||f||_{p,w}^{*}||g||_{q,w}|E|^{1-1/r}$

and this proves (ii).
(iii) Direct calculation gives$\int_{E}|f*g|ds$$\leq$$\int\int 1_{E}(s)|f(s-\tau)|\cdot|g(\tau)|dsd\tau$$=$$\int|g(\tau)|d\tau\int 1_{E}(s+\tau)|f(s)|ds\leq||g||_{1}||f\Vert_{p,w}|E|^{1-}$.

This proves (iii).
Lemma 2.3. Let $f\geq 0$ be a measurable function on R. Suppose there exist constants$M>0,$ $C>0$ and $p>q>1$ so that $0\leq f\leq M$ and$\int_{E}fds\leq C(|E|^{1-1/p}+|E|^{1-1/q})$

for all measurable subsets E. Then, there is another constant $C’>0$ such that$\int_{E}fds\leq C’|E|^{1-1/p}$

for all measurable$E$ ,Proof. Since $1-1/p>1-1/q$ , the result is obvious forassume $|E|<1$ . Then, since$\int_{E}fds\leq 2C|E|^{1-1/q}$$E$with $|E|\geq 1$ . So we may,

H\"older’s inequality yields, with $\theta=1-q/p$ ,$\int_{E}fds$$\leq$$M^{\theta} \int_{E}f^{1-\theta}ds$$\leq$$M^{\theta}|E|^{\theta}[ \int_{E}fds]^{1-\theta}\leq M^{\theta}(2C)^{1-\theta}|E|^{1-1/p}$,

which completes the proof.
Lemma 2.4. Let $n=3,4$ . Then for all $v\in V$ and $w\in H^{1}(R^{n})$ with $\nabla\cdot w=0$ ,$||e^{-tA}P(w\cdot\nabla)v||_{2}\leq Ct^{-1/2}(||w||_{2}||v||_{2})^{1-n/4}(||\nabla w||_{2}||\nabla v||_{2})^{n/4}$

Proof. Let$\phi\in C_{0^{\infty_{\sigma}}},(D)$. Since$||\nabla e^{- tA}\phi I12=||A^{1/2}e^{- tA}\phi||_{2}\leq t^{-1/2}||\phi||_{2}$

8

6

$i$

direct calculation gives$|(e^{-tA}P(w\cdot\nabla)v, \phi)|$$=$$|(v, w\cdot\nabla e^{-tA}\phi)|\leq||v||_{4}||w||_{4}||\nabla e^{-tA}\phi||_{2}$$\leq t^{-1/2}||v||_{4}||w||_{4}||\phi||_{2}$.

The result follows by applying the Sobolev inequality :$||f||_{4}\leq C||f\Vert_{2}^{1- n/4}\Vert\nabla f||_{2}^{n/4}$

Proof of Theorem 2.1. First observe that the energy inequality (E) gives$||u(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla u\Vert_{2}^{2}ds\leq||a||_{2}^{2}+.\int_{0}^{t}$lIf112$(1+||u||_{2}^{2})ds$.

Applying Gronwall’s lemma yields$\Vert u(t)||_{2}^{2}+2\int_{0}^{t}||\nabla u\Vert_{2}^{2}ds\leq(||a||_{2}^{2}+\int_{0}^{\infty}\Vert f\Vert_{2}ds)\exp(\int_{0}^{\infty}||f||_{2}ds)$

Thus,
and
into (W) we obtain$||u||2\in L^{\infty}$$||\nabla u||_{2}^{2}\in L^{1}$. Now, substituting $\phi(\tau)=e^{-(t-\tau)A}\psi$ , with.
$\psi\in C_{0^{\infty_{\sigma}}},(D)$,.

$(u(t), \psi)=(e^{-(t-s)A}u(s), \psi)-\int_{s}^{t}(u\cdot\nabla u(\tau), e^{-(t-\tau)A}\psi)d\tau+\int^{t}(f, e^{-(t-\tau)A}\psi)d\tau$

We apply Lemma 2.4 to estimate the nonlinear term, to get(2.4)$||u(t)||_{2} \leq||e^{-(t-s)A}u(s)||_{2}+C\int_{s}^{t}(t-\tau)^{-1/2}(||u||_{2}^{1/2}||\nabla u||_{2}^{3/2}+\Vert f||_{\dot{V}^{*}})d\tau$when $n=3$ ; and(2.5)$||u(t) \Vert_{2}\leq\Vert e^{-(t-s)A}u(s)||_{2}+c_{c}\int_{S}^{t}(t-\tau)^{-1/2}(||\nabla u||_{2}^{2}+||f||_{\dot{V}^{*}})d\tau$when $n=4$ , with independent of $s\geq 0$ . From now on we regard
andand
as defined to be zero for. Suppose first that $n=3$ . Since, it follows from Lemma 2.2 that the last term of right-hand side of (2.4) belong5
toand the norm is bounded by$C$$||u\Vert_{2}$$\Vert\nabla u\Vert_{2}^{3/2}||u||_{2}^{1/2}\in$$\tau\leq s$$\Vert f\Vert_{\hat{V}^{*}}$$\Vert\nabla u\Vert_{2}$$L^{4/3}\subset L_{w}^{4/3}$$L_{w}^{4}+L_{w}^{2}$$C[ \int_{s}^{\infty}||\nabla u||_{2}^{2}d\tau]^{3/4}+C\int_{s}^{\infty}||f||_{\dot{V}^{*}}d\tau$with independent of $s>0$ . For an arbitrary $e>0$ , we choose $s>0$ so that the above
quantity is less than . Then, by the definition of -norm we get for all $t>s$ ,$C$$L_{w}^{p}$$\epsilon$$\int_{t}^{t+1}||u||_{2}d\tau\leq\int_{t}^{t+1}||e^{-(\tau-s)A}u(s)||_{2}d\tau+e$

9

.

62
Since $||e^{-tA}a||_{2}arrow 0$ because of the injectivity oftheorem yields$A$, application of the bounded convergence$\lim_{Parrow}\sup_{\infty}\int_{t}^{t+1}||u||_{2}d\tau\leq\epsilon$

treated by using (2.5) insteadand this shows (i) in case $n=3$ . The case $n=4$ isof (2.4). To prove (ii), we consider (2.4) and (2.5) with $s=0$ and integrate both sides in, to get(2.6)$si_{1}\dot{m}1arly$$t$$\frac{1}{t}I_{0}^{t}||u||_{2}ds\leq\frac{1}{t}\int_{0}^{t}||e^{-sA}a||_{2}ds+C(t^{-\beta}+t^{-1/2})$

where $\beta=1/4$ when $n=3$ , and $\beta=1/2$ when $n=4$ . This proves (ii) with $n=4$ . To
prove (ii) with $n=3$ , we systematically apply Lemmas 2.2 and 2.3. First observe that$(2.4)$ shows that
(2.6) shows the result for $\alpha<1/4$ . Whenis bounded from, Lemma 2.3 implies
above by a function belonging to. Since, and so
. Thus, by Lemma 2.2 (i),
that
with
$1/p=$ 1/8+3/4. Since $1/2+1/p=1+1/4+1/8$ , Lemma 2.2 (ii) implies$\alpha\geq 1/4,$$L_{w}^{1/\alpha}+L_{w}^{4}+L_{w}^{2}$$||u||_{2}$$||u||_{2}\in L^{\infty}$$||u||_{2}^{1/2}||\nabla u||_{2}^{3/2}\in L_{w}^{p}$$||u||_{2}^{1/2}\in L_{w}^{8}$$||u||_{2}\in L_{w}^{4}$$\frac{1}{t}\int_{0}^{t}||u||_{2}ds\leq C(t^{-\alpha}+t^{-1/q}+t^{-1/2})$

with $1/q=$ 1/4+1/8, and this shows the result for $\alpha<1/q$ . Whenis bounded from above by a function in
that
; so
2.3. Thus, the same argument as above gives$L_{w}^{1/\alpha}+L_{w}^{q}+L_{w}^{2}$$||u||_{2}$$\alpha\geq 1/q,$

shows
by Lemma$(2.4)$$||u||_{2}\in L_{w}^{q}$$\int_{0}^{t}(t-s)^{-1/2}||u||_{2}^{1/2}||\nabla u||_{2}^{s/2}ds\in L_{w}^{r}$with $1/r=$ 1/4+1/8+1/16. Hence$\frac{1}{t}\int_{0}^{t}||u||_{2}ds\leq C(t^{-\alpha}+t^{-1/r}+t^{-1/2})$and this proves the result for $\alpha<1/r$ . Repeating these processes eventually proves thedesired result all the way. The proof is complete.
Proof of Theorem 1.1. We first construct approximate solutions of (NS), for $n=$
$3,4$ , by solving$u_{k}(t)=e^{-tA}a_{k}- \int_{0}^{t}e^{-(t-s)A}(P(\overline{u}_{k}\cdot\nabla)u_{k}-f_{k})(s)ds$,$k=1,2,$$\cdots$,(IE)

where $a_{k}=(I+k^{-1}A)^{-1}a,$ $f_{k}=(I+k^{-1}A)^{-1}f$ , andis the spatial mollificationof theof . The unique solvability of (IE) in the space $C([0, T];V)$ as
well as the fact thatsolves, in $L^{2}(0, T;D(A))\cap W^{1,2}(0, T;H)$ , the problem$\overline{u}_{k}=J_{1/k}\tilde{u}_{k}$$zerc\succ extension\tilde{u}_{k}$$u_{k}$$u_{k}$$\frac{du_{k}}{dt}+Au_{k}+P(\overline{u}_{k}\cdot\nabla)u_{k}=f_{k}$10, a.e. $t>0$ ; $u_{k}(0)=a_{k}$

(2.7)

63
can be shown as in $[2,18]$ . From (2.7) we get, for $t\geq s\geq 0$ ,
.

(2.8)

$||u_{k}(t)||_{2}^{2}+2 \int_{s}^{t}||\nabla u_{k}||_{2}^{2}d\tau=||u_{k}(s)\Vert_{2}^{2}+2\int_{s}^{t}(f_{k}, t4_{k})d\tau$Upon taking $s=0$ and using$u_{k}$$|a_{k}||\leq||a||_{2},$$(2.8)$

implies

is bounded in $L^{\infty}(0, T)H)\cap L^{2}(0, T;V)$ .

Hence we may assume that a subsequence of
converges wealdy-star in $L^{\infty}(0, T, H)$
and weakly in. Moreover, a standard argument ([22, Chap.III]) applies toshow that if we define $v_{k}(t)=u_{k}(t)$ for $t\in(0, T)$ and $v_{k}(t)=0$ otherwise, then thein , remain bounded
fractional derivatives, defined via the Fourier transform ofin $L^{2}(R;H)$ provided $0<\gamma<1/4$ . We thus conclude that a subsequence, denoted again, converges in $L_{loc}^{2}([0, T]\cross D)$ to a function , and it is readily seen (cf. [16]) that the$linlit$ functionasis a weak solution of (NS). In view of the convergence mode of
decays inmentioned above, we need only show thatas indicated in Theorem 1.1$unif_{olJ}dy$ in.
Now the energy equality (2.8) implies that$u_{k}$$L^{2}(0, T,\cdot V)$$v_{k}(t)$$D_{t}^{\gamma}v_{k}$

$t$

$u$

$u_{k}$$u$$u_{k}$$L^{2}$$u_{k}(t)$$k$$||u_{k}(t)||_{2}^{2}+2 \int_{s}^{t}||\nabla u_{k}||_{2}^{2}d\tau\leq||u_{k}(s)||_{2}^{2}+\int_{s}^{t}||f||_{2}(1+||u_{k}||_{2}^{2})d\tau$so that, by Gronwall’s lemma,$\Vert u_{k}(t)||_{2}^{2}+2l^{t}||\nabla u_{k}||_{2}^{2}d\tau\leq C(||u_{k}(s)|1_{2}^{2}+\int^{t}11f||_{2}d\tau)$

where $C= \exp(\int_{0^{\infty}}||f||2ds)$ . Hence$\int_{0^{\infty}}||\nabla v_{k}||\frac{Q}{2}ds$

is bounded uniformly in$\Vert u_{k}(t)||_{2}\leq C[||u_{k}(s)||_{2}+(\int_{s}^{t}||f||_{2}d\tau)^{1/2}]$

Integrating this in

$s$

$k$

and

.

gives

$\Vert v_{k}(t+1)||_{2}\leq C[\int^{t+1}||u_{k}||_{2}ds+(\int_{t}^{t+1}||f||_{2}ds)^{1/2}]$

(2.9)

and
$\Vert u_{k}(t)||_{2}$$\leq$$Ct^{-1} \iota\int_{0}^{t}||u_{k}||_{2}ds+Ct^{-1/2}[\int_{0}^{t}\tau||f(\tau)||_{2}d\tau]^{1/2}$(2.10)$\leq$$Ct^{-1} \int_{0}^{t}||u_{k}||_{2}ds+Ct^{-1/2}$.

On the other hand, applying Lemma 2.3 to (IE) gives$||u_{k}( \tau)||_{2}\leq||e^{-(\tau-s)A}u_{k}(s)||_{2}+C\int_{s}^{\tau}(\tau-\sigma)^{-1/2}(||\nabla u_{k}||_{2}^{2}+||f||_{\dot{V}^{*}})d\sigma$

11(2.11)

64
when $n=4$ ; and(2.12)$||u_{k}( \tau)||_{2}\leq||e^{-(\tau-s)A}u_{k}(s)||_{2}+C\int_{s}^{\tau}(\tau-\sigma)^{-1/2}(||u_{k}||_{2}^{1/2}||\nabla u_{k}||_{2}^{3/2}+||f||_{\dot{V}^{*}})d\sigma$. Sincewhen $n=3$ . Note that we have usedand$||e^{-tA}a_{k}||_{2}\leq||e^{-tA}a||_{2}$ , and sinceanduniformly in , we see from1
(2.11) and (2.12) that the assertions of Theorem 2.1 hold foruniformly in . Hence(2.10) implies thathave the decay properties listed in Theorem 1.1 uniformly in . This
completes the proof of Theorem 1.1.
$||\nabla\overline{u}_{k}||_{2}\leq||\nabla u_{k}||_{2}$$||\overline{u}_{k}||_{2}\leq||u_{k}||_{2}$$u_{k}||_{2}\in L^{\infty}$$k$$||\nabla u_{k}||_{2}^{2}\in L^{1}$$k$$u_{k}$$k$$u_{k}$

3. Proof of Theorem 1.2

In this section we first prove Theorem 1.2, using the spectral decomposition of thepositive self-adjoint operator . To prove assertion (iv) of Theorem 1.2, we establish theinvariance of some of the ranges. As a byproduct weof the fractional powers
obtain Corollary 1.3. The argument below originates from those of [12,21,23] and, as in
Section 2, relies on the following
Lemma 3.1. (i) Let
be the spectral decomposition of A. If $n=2$ , we$v\in
V_{f}$
have the following estimate : for$A$$R(A^{\alpha})$$A^{\alpha}$$A= \int_{0}^{\infty}\lambda dE_{\lambda}$$||E_{\lambda}P(v\cdot\nabla)v||_{2}\leq C\lambda^{1/2}||v||_{2}||\nabla v||_{2}$,$\lambda>0$.

(ii) Under the same assumption as in (i), we have$||e^{-tA}P(v\cdot\nabla)v||_{2}\leq Ct^{-1/2}||v||_{2}||\nabla v||_{2}$Proof. By the definition of

$P$

and

$|(E_{\lambda}P(v\cdot\nabla)v, \phi)|$$=$$E_{\lambda}$,$t>0$ .

we easily see that, for$\phi\in C_{0}^{\infty_{\sigma}}(D)$,$|(v, v\cdot\nabla E_{\lambda}\phi)|\leq\Vert v\Vert_{4}^{2}\Vert\nabla E_{\lambda}\phi\Vert_{2}$$=||v||_{4}^{2}||A^{1/2}E_{\lambda}\phi||_{2}\leq\lambda^{1/2}||v||_{4}^{2}||\phi||_{2}$Applying the Sobolev inequality :$||f||_{4}\leq C||f||_{2}^{1/2}||\nabla f||_{2}^{1/2}$,

to the last term yields the desired estimate. This proves (i). Assertion (ii) is proved in thesame way as in Lemma 2.4. The proof is complete.

Proof of Theorem 1.2. The standard theory of the two-dimensional Navier-Stokes
equations as given in $[16,22]$ asserts the existence of a unique weak solution satisfying
the energy equality$u$$||u(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla u||_{2}^{2}ds=||a||_{2}^{2}+2\int_{0}^{t}(f, u)ds$

12

65
for all $t\geq 0$ . Hence we have

Using the estimate$2\Vert f||_{2}||u||_{2}\leq||f||_{2}(1+||u||_{2}^{2})$(3. 1).

$\frac{d}{dt}||u\Vert_{2}^{2}+2||\nabla u||_{2}^{2}=2(f, u)\leq 2||f||_{2}||u||_{2}$and Gronwall’s lelnma, we easily see that$||u(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla u||_{2}^{2}ds\leq(||a||_{2}^{2}+\int_{0}^{\infty}||f\Vert_{2}ds)\exp(1_{0}^{\infty}||f||_{2}ds)$

and therefore
estimates$||u||_{2}\in L^{\infty}$

and$||\nabla u||_{2}^{2}\in L^{1}$(3.2). Now, since $||\nabla u||_{2}=||A^{1/2}u||_{2}$ , using the$2||A^{1/2}u||_{2}^{2}=2 \int_{0}^{\infty}\lambda d||E_{\lambda}u||_{2}^{2}\geq 2p\int_{\rho}^{\infty}d||E_{\lambda}u||_{2}^{2}=2\rho(||u||_{2}^{2}-||E_{\rho}u||_{2}^{2})$and$||E_{\rho}u\Vert_{2}\leq||u||_{2}$, we see from (3.1) that
(3.3)$\frac{d}{dt}||u||_{2}+\rho||u||_{2}\leq\rho||E_{\rho}u||_{2}+||f||_{2}$for all $\rho>0$ . To deal with the right-hand side we substitute $\phi(\tau)=e^{-(t-\tau)A}E_{\rho}\psi,$, into (W) and get$\psi\in$$C_{0^{\infty_{\sigma}}},(D)$$(E_{\rho}u(t), \psi)=(E_{\rho}e^{-tA}a, \psi)+\int_{0}^{t}(E_{\rho}e^{-(t-s)A}f, \psi)ds-\int_{0}^{t}(E_{\rho}e^{-(t-s)A}P(u\cdot\nabla u), \psi)ds$

so that, by duality and Lemma 3.1 (i),$||E_{\rho}u(t)||_{2} \leq||e^{-tA}a.||_{2}+C\rho^{1/2}\int_{0}^{t}(||u|\}_{2}||\nabla u||_{2}+||f||_{\dot{V}^{*}})ds$.

Combining this with (3.3) and applying H\"older’s inequality gives$\frac{d}{dt}||u||_{2}+\rho||u||_{2}\leq C\rho[||e^{-tA}a||_{2}+\rho^{1/2}(\int_{0}^{t}||u||_{2}^{2}ds)^{1/2}+\rho^{1/2}]+||f||_{2}$sinceby (3.2). We take in (3.4)
both sides by $(\log(t+e))^{2}$ to obtain$||\nabla u||_{2}^{2}\in L^{1}$(3.4)

and then multiply$\rho=2/(t+e)\log(t+e)$$\frac{d}{dt}((\log(t+e))^{2}||u||_{2})\leq 2C(t+e)^{-1}\log(t+e)[\Vert e^{-tA}a||_{2}+C(\log(t+e))^{-1/2}]$$+C(\log(t+e))^{2}(t+e)^{-1}||f||_{2}(t+e)$

Sincesince $||e^{-1A}a||_{2}arrow 0$ as $tarrow\infty$ because
is finite by assumption, we obtain$||e^{-tA}a\Vert_{2}\leq\Vert a|1_{2}$$\int_{0}^{\infty}||f||_{2}(t+e)dt$.
$A$

is injective, and since$||u(t)||_{2} \leq(\log(t+e))^{-2}[||a||_{2}+C\int_{0}^{t}(s+e)^{-1}\log(s+e)||e^{-sA}a||_{2}ds]$$+C(\log(t+e))^{-1/2}+C(\log(t+e))^{-2}arrow 0$

13

as$tarrow\infty$.

66
The proof of (i) is complete. We next prove (ii). Since $||e^{-tA}a||_{2}\leq C(t+e)^{-\alpha}$ by $assum_{1^{J}}tion$ ,the proof of (i) shows$||u(t)\Vert_{2}\leq C[(\log(t+e))^{-2}+(\log(t+e))^{-1/2}]$.

This shows (ii) for $m=1$ . From this we obtain (see [21,23])$\int_{0}^{t}||u||_{2}^{2}ds\leq C\int_{0}^{t}(\log(s+e))^{-1}ds\leq C(t+e)(\log(t+e))^{-1}$

so that, as in the proof of (i),$\frac{d}{dt}((\log(t+e))^{2}||u\Vert_{2})\leq C(t+e)^{-1}\log(t+e)[(t+e)^{-\alpha}+(\log(t+e))^{-1}]$$+(\log(t+e))^{2}(t+e)^{-1}\Vert f||_{2}(t+e)$.

Integrating this gives$||u(t)||_{2}\leq C(\log(t+e))^{-1}$.

This proves (ii) for $m=2$ . Now suppose (ii) is true for some $m\geq 2$ . Taking in (3.4)$\rho=(m+1)/(t+e)\log(t+e)$ and then multiplying by $(\log(t+e))^{m+1}$ we obtain, since$\int_{0}^{t}(\log(s+e))^{-m}ds\leq C_{m}(t+e)(\log(t+e))^{-m}$ ([21,23]),$\frac{d}{dt}((\log(t+e))^{m+1}||u||2)\leq C(t+e)^{-1}(\log(t+e))^{m}[(t+e)^{-\alpha}+(\log(t+e))^{-(m+1)/2}]$$+C(\log(t+e))^{m+1}(t+e)^{-1}||f||_{2}(t+e)$ .

Hence$||u(t)\Vert_{2}\leq C(\log(t+e))^{-(m+1)/2}$

and this completes the proof of (ii).
We next prove (iii), following [12, pp.142-143].
, into (W) and applying Lemma 3.1 (ii) gives

Inserting $\phi(\tau)=e^{-(t-\tau)A}\psi,$$\psi\in$$C_{0^{\infty_{\sigma}}},(D)$$\Vert u(t)\Vert_{2}\leq\Vert e^{-tA}a||_{2}+C\int_{0}^{t}(t-s)^{-1/2}(\Vert u\Vert_{2}\Vert\nabla u\Vert_{2}+||f||_{\dot{V}}.)$ $ds$

.

(3.5)

Assume first that $0<\alpha<1/2$ and choose $q>2$ so that $q\alpha<1$ . Since $1+1/q=$$1/2+(q+2)/2q$ , the Hardy-Littlewood-Sobolev inequality [19] applied to (3.5) gives$[ \int_{0}^{t}||u||_{2}^{q}ds]^{1/q}\leq C_{1}(t+1)^{1/q-\alpha}+C[\int_{0}^{t}(||u||_{2}||\nabla u||_{2})^{2q/(q+2)}ds]^{(q+2)/2q}$$+C[ \int_{0}^{t}||f||_{\dot{V}}^{2q_{*}/(q+2)}ds]^{(q+2)/2q}$

14

67
Notice that the last term is bounded in by the assumption
Since $(q+2)/2q=1/2+1/q$ , H\"older’s inequality and (3.2) together yield$f\in L^{p}(0, \infty;\hat{V}^{*})$$t$for $p=1,2$ .

$[ \int_{0}^{t}||u||_{2}^{q}ds]^{1/q}\leq C_{1}(t+1)^{1/q-\alpha}+C_{2}[_{\llcorner}\int_{0}^{t}||u||_{2}^{q}ds]^{1/q}+C_{3}$

where
and
are constants and $C_{2}=C_{2}(a, f)$ is the square root of the right-hand sideof (3.2). Here we assume
are so small as $C_{2}\leq 1/2$ , to obtain fromandthe above$C_{3}$$C_{1}$$||a||_{2}$$\int_{0}^{\infty}||f||_{2}dt$$[ \int_{0}^{t}||u||_{2}^{q}ds]^{1/q}\leq C(t+1)^{1/q-\alpha}+C\leq C(t+1)^{1/q-\alpha}$

because $1/q-\alpha>0$ . Inserting this into (3.4) gives$\frac{d}{dt}\Vert u||_{2}+\rho\Vert u||_{2}\leq C\rho[(t+1)^{-\alpha}+\rho^{1/2}(t+1)^{1/2- 1/q}(\int_{0}^{t}||u||_{2}^{q}ds)^{1/q}+p^{1/2}]+||f||_{2}$$\leq c_{\rho}((t+1)^{-\alpha}+\rho^{1/2}(t+1)^{1/2-\alpha})+|\}f||_{2}$.

Taking $\rho=m/(t+1)$ , nmltiplying by $(t+1)^{m}$ , and then proceeding as in the proof of (ii), thenwe obtain $||u(t)||_{2}=O((t+1)^{-}")$ , and this proves (iii) in case $\alpha<1/2$ . If
$q\alpha>1$ for all $q>2$ , and so. The foregoing argument thus gives$\alpha\geq 1/2$$\int_{0}^{\infty}||e^{-tA}a||_{2}^{q}dt<+\infty$$[ \int_{0}^{t}||u||_{2}^{q}ds]^{1/q}\leq C_{1}+C_{2}[\int_{0}^{t}||u||_{2}^{q}ds]^{1/q}+C_{3}$so that
if $C_{2}\leq 1/2$ . Inserting this into (3.4) and repeating the same
argument as in the case $\alpha<1/2$ , we obtain$[ \int_{0^{f}}||u||_{2}^{q}ds]^{1/q}\leq C$$||u(t)||_{2}\leq C((t+1)^{-a}+(t+1)^{-1/q})$.

Since $q>2$ was arbitrary, the proof of (iii) is now complete.
To prove (iv), we need nly show that the (unique) weak solution $u(t)$ belongs to; because it then follows thatfor all $t>0$ provided$0$$a\in R(A^{\alpha})$$R(A^{\alpha})$$||e^{-tA}u(s)||_{2}=O(t^{-\alpha})$for any fixed $s\geq 0$ and so the proof of (iii) applies if we choose $s\geq 0$ as the initial time soare small enough. To this end we usethat
and$||u(s)||_{2}$$\int_{s^{\infty}}||f||_{2}d\tau$

Proposition 3.2. Let $n=2$ and $0<\alpha\leq 1/2$,

If $v\in V$ , then for all$||(\lambda+A)^{-\alpha}P(v\cdot\nabla)v||_{2}\leq C||v||^{\frac{Q}{2}\alpha}||\nabla v||_{2}^{2-2\alpha}$

with

$C$

depending only on$\alpha$,Moreover, we have$||(\lambda+A)^{-\alpha}$f112$\leq||f||_{\dot{V}}^{2\alpha_{*}}||f||_{2}^{1-2\alpha}$

15.

$\lambda>0$,

68
We continue the proof of Theorem 1.2 (iv), admitting Proposition 3.2 for a moment.
Let;$u_{\lambda}(t)=(\lambda+A)^{-\alpha}u(t)$Since $a\in R(A^{\alpha})$ by assumption,$||a_{\lambda}||_{2}$inserting(i) we obtain$\phi(\tau)=(\lambda+A)^{-\alpha}e^{-(t-\tau)A}\psi,$$||u_{\lambda}(t)||_{2}$This shows thatthat.

$a_{\lambda}=(\lambda+A)^{-\alpha}a$. On the other hand,is uniformly bounded in, into (W) and applying Proposition 3.2$\lambda>0$$\psi\in C_{0^{\infty_{\sigma}}},(D)$$\leq$$||a_{\lambda}||_{2}+ \int_{0}^{t}||(\lambda+A)^{-\alpha}P(u\cdot\nabla)u(s)||_{2}ds+\int_{0}^{t}||(\lambda+A)^{-\alpha}f||_{2}ds$$\leq$$C_{1}+C_{2} \int_{0}^{t}||u||_{2}^{2\alpha}||\nabla u||_{2}^{2-2\alpha}ds+\int_{0}^{t}||f||_{\dot{V}}^{2\alpha_{*}}||f||_{2}^{1-2\alpha}ds$$\leq$$C_{1}+C[ \int_{0}^{t}||u||_{2}^{2}ds]^{\alpha}+C_{3}$.

remains bounded in $H$ for any fixed $t>0$ . Hence we may assume,
asweakly in $H$ , and therefore, for any$u_{\lambda}(t)$$u_{\lambda}(t)arrow w(t)$$\lambdaarrow 0$$\phi\in D(A^{\alpha})$.

$(u(t), \phi)=(u_{\lambda}(t), (\lambda+A)^{\alpha}\phi)arrow(w(t), A^{\alpha}\phi)$Hence $u(t)\in R(A^{\alpha})$ for all $t\geq 0$ and this proves (iv). The proof is complete.

Proof of Proposition 3.2. First we show the estimate(3.6)$|I\emptyset||_{p}\leq C||A^{\alpha}\phi||_{2}$for $1/p=1/2-\alpha$ , and $0\leq\alpha<1/2$ . This is deduced from the following : (i) The family
in the norm
forms a complexof the completionofinterpolation family (see, e.g. [17]) ; (ii) $D^{1/2}\subset BMO$ with continuous injection (see, e.g.
[7, Prop. 3.4]) ; and (iii) $[L^{2}, BMO]_{\theta}=L^{2/(1-\theta)},$ $0\leq\theta<1$ , where the bracket denotes thecomplex interpolation (see [11]). Now, estimate (3.6) implies that$\{D^{\alpha}; 0\leq\alpha\leq 1/2\}$$D^{\alpha}$$|((\lambda+A)^{-\alpha}P(v\cdot\nabla)v, \phi)|$$=$$D(A^{\alpha})$$||A^{\alpha}\cdot||_{2}$$|(v\cdot\nabla v, (\lambda+A)^{-\alpha}\phi)|$$\leq||\nabla v||_{2}||v||_{1/\alpha}||(\lambda+A)^{-\alpha}\phi||_{p}$$\leq C\Vert\nabla v||_{2}^{2-2a}\Vert v||_{2}^{2\alpha}||A^{\alpha}(\lambda+A)^{-\alpha}\phi||_{2}$$\leq C||\nabla v||_{2}^{2- 2\alpha}\Vert v||_{2}^{2\alpha}||\phi||_{2}$for $\alpha<1/2$ , and$|((\lambda+A)^{-1/2}P(v\cdot\nabla)v,\phi)|=|(v,v\cdot\nabla(\lambda+A)^{-1/2}\phi)|\leq||v||_{4}^{2}||\nabla(\lambda+A)^{-1/2}\phi||_{2}$,$=C||v||_{4}^{2}||A^{1/2}(\lambda+A)^{-1/2}\phi||_{2}\leq C||v||_{2}||\nabla v||_{2}||\phi\Vert_{2}$for $\alpha=1/2$ . This proves the first assertion. The second assertion is obtained as$||(\lambda+A)^{-\alpha}f||_{2}\leq||f||_{2}^{1-2\alpha}||(\lambda+A)^{-1/2}f||_{2}^{2\alpha}\leq||f||_{2}^{1-2\alpha}||f||_{\dot{V}}^{2\alpha_{*}}$

16

69
by using the estimate$||(\lambda+A)^{-1/2}f||_{2}\leq||f||_{\dot{V}*}$.

This completes the proof of Proposition 3.2.

Using the estimates :$\Vert\phi||_{p}\leq C||A^{a}\phi||_{2}$where $1/p=1/2-2\alpha/3,1/4\leq\alpha\leq 1/2$ , if $n=3$ ; and $p=4,$ $\alpha=1/2$ , if $n=4$ , we canalso prove

Proposition 3.3. Let $v\in V$ and $w\in H^{1}(R^{n})$ with $\nabla\cdot w=0$ .

If $n=3$ and(i)$1/4\leq\alpha\leq 1/2$

, then for all$\lambda>0$,$|1(\lambda+A)^{-\alpha}P(w\cdot\nabla)v||_{2}\leq C||u1||_{2}^{2\alpha-1/2}||\nabla w||_{\underline{9}}^{3/2-2\alpha}||\nabla v||_{2}$

with

$C$

depending only on$\alpha$.

If $n=4$ , then for all(ii)$\lambda>0$,$\Vert(\lambda+A)^{-1/2}P(w\cdot\nabla)v||_{2}\leq C\Vert\nabla w||_{2}||\nabla v||_{2}$

with

$C$

independent of(iii)$\alpha\leq 1/2$

$R(A^{\alpha})$

$v$

and

$w$

.

is invariant under the Navier-Stokes flow provided either $n=3,1/4\leq$; or $n=4,$ $\alpha=1/2$ .

Finally, Corollary 1.3 is immediately obtained from Theorems 1.1, 1.2 and the following

Corollary 3.4. There holds the inclusion :$H\cap(L^{r}(D))^{n}\subset R(A^{\alpha})$,$\alpha=n(1/r-1/2)/2$$=4$ More precisely,
where $1<r<2$ if $n=2$ ; $6/5\leq r\leq 3/2$ if $n=3$ ; and $r=4/3$
andto each $a\in H\cap(L^{r}(D))^{n}$ there corresponds a unique $b\in D(A^{\alpha})$ such that$||b||_{2}\leq C||a||_{r}u)ithC>0$ independent of .
$if\uparrow 1,$,$a=A^{\alpha}b$$a$Proof. We consider only the case $n=2$ ; the other cases are treated $si_{1}nilally$ . The
satisfies
function$a_{\lambda}=(\lambda+A)^{-\alpha}a$$|(a_{\lambda}, \phi)|$$=$$|(a, (\lambda+A)^{-\alpha}\phi)|\leq||a||_{r}||(\lambda+A)^{-\alpha}\phi||_{r’}$$\leq C||a||_{r}||A^{\alpha}(\lambda+A)^{-\alpha}\phi||_{2}\leq C||a||_{r}||\phi||_{2}$, is bounded infor all $\phi\in H$ , where $1/r’=1-1/r$ . This shows that$b\in
H$ with, to a function
subsequence converges weakdy in , as
But then,$a_{\lambda},$$H$$\lambda>0$$\lambdaarrow 0$$(a, \phi)=(a_{\lambda}, (\lambda+A)^{\alpha}\phi)arrow(b, A^{\alpha}\phi)$17; so a$||b||_{2}\leq C||a||_{r}$ .
$H$

70
for all$\phi\in D(A^{\alpha})$, so $a=A^{\alpha}b$ andis thus determined uniquely. The proof is complete.

$b$with smooth boundary, theark. When $n\geq 3$ and $D$ is an exterior domain of-theory of the operator as developed in [3] gives the estimate$Rem$$L^{r}$$R^{n}$$A$$\Vert\phi||_{p}\leq C||A^{\alpha}\phi||_{r}$is invariant underfor $1<r<n,$ $1<p<\infty$ and $1/p=1/r-2\alpha/n$ . This implies that
the Navier-Stokes flow (with $f=0$ ) provided $\alpha<n/4$ , and that the inclusion relation ofCorollary 3.4 holds for all $1<r<2$ . More generally, the -theory implies that
$R(A^{\alpha})$$L^{r}$$L_{\sigma}^{r}\cap L_{\sigma}^{q}\subset R(A_{q}^{\alpha})$whenever $n’<q<\infty,$ $1<r<\infty$ , and $1/r=1/q+2\alpha/n$ . Here $1/n’=1-1/\uparrow z;L_{\sigma}^{r}$ denotes; andis the operator regarded as a closed linear operator in
the -closure of. We can further show that the space $L^{r}\cap H$ is invariant under the Navier-Stokes flow
provided $1<r\leq n’$ . See [3, Sect.4-5] for the details.
$L^{r}$$A$$A_{q}$$C_{0^{\infty_{\sigma}}},(D)$$L_{\sigma}^{q}$

4. Energy Stability of Exterior Stationary Flows
This section deals with the problem$\frac{\partial v}{\partial t}+v\cdot\nabla v=\triangle v-\nabla p+f_{0}(x\in D, t>0)$$(x\in D, t>0)$ ,$\nabla\cdot v=0$$v|_{S}=v^{*};$$varrow v^{\infty}(|x|arrow\infty)$$v|_{t=0}=v_{0}$

,
(4.1)

,

,

in an exterior domain $D$ inwith smooth boundary . Here $v^{*}=v^{*}(x)$ is a given smooth
vector field on ;is a given constant vector; and $f_{0}=f_{0}(x)$ is a given external force.
Under some assumptions onand , FINN $[4,5]$ and BABENKO [1] proved theexistence of a stationary solution to problem (4.1) satisfying$S$$R^{3}$$S$$v^{\infty}$$v^{*},$$v^{\infty}$$f_{0}$$w$$w-v^{\infty}\in L^{3}(D)$; $\nabla w\in L^{3}(D)$ ; $C_{0} \equiv\sup_{D}|x|\cdot|w(x)-v^{\infty}|<+\infty$ .

(4.2)

In this section we study the stability of these stationary solutions with respect toand $a=v_{0}-w$ , thedisturbances. Given a stationary solution
and disturbances
time-evolution of $u=v-w$ is governed by$L^{2}$$w$$f$$\frac{\partial u}{\partial t}+u\cdot\nabla u=\triangle u-w\cdot\nabla u-u\cdot\nabla w-\nabla q+f$$(x\in D, t>0)$ ,$\nabla\cdot u=0$$(x\in D, t>0)$ ,$u|_{S}=0$; $uarrow 0(|x|arrow\infty)$ ,$u|_{t=0}=a$ .

18

(P)

71
The problem (P) is formally transformed into the integral equation :$u(t)=e^{-tL}a- \int_{0}^{t}e^{-(t-s)L}(P(U\cdot\nabla)u-f)(s)ds$

(4.3)

where
$L=A+B$ ; $Bu=P(w\cdot\nabla)u+P(u\cdot\nabla)w$ ,and $\{e^{-tL} ; t\geq 0\}$ is the analytic semigroup in $H$ generated by -L. (See [18].)
Given $a\in H$ , a weakly continuous function :) $arrow H$ is called a weak solutionof (P) if $u\in L^{\infty}(0, T;H)\cap L^{2}(0, T,\cdot V)$ for all $T>0,$ $u(0)=a$ , and satisfies$u$$[0, \infty$$(u(t), \phi(t))+\int_{s}^{t}[(\nabla u, \nabla\phi)+(w\cdot\nabla u+u\cdot\nabla w+u\cdot\nabla u, \phi)]d\tau$(4.4)$=(u(s), \phi(s))+\int_{s}^{t}(u_{\}\phi’)d\tau+J_{s}^{t}(f, \phi)d\tau$for all $t\geq s\geq 0$ and all $\phi\in C^{1}([0, \infty);H)\cap C^{0}([0, \infty);V)$ .
For the existence of weak solutions, we already have the following result:
Theorem 4.1([18]). Given E $H$ andproblem (P) possesses a
weak solution which, moreover, satisfies the energy $inequal\iota ty$ of the.following form :$f\in L_{oc}^{\frac{\prime)}{1}}([0, \infty);H)_{i}$

$a$

$u$

$\Vert u(t)||_{2}^{2}+2\int_{s}^{t}||\nabla u||_{2}^{2}d\tau+2\int_{s}^{t}(u\cdot\nabla_{1’}, u)d\tau\leq||u(s)||_{2}^{2}+2\int_{s}^{t}(f, u)d\tau$

for $s=0,$

$a,e$

(SE)

. $s>0$ , and all $t\geq s$ .

In this section we shall prove
Theorem 4.2. Let $C\prime 0<1/2,$ $a\in H$ , and suppose
satisfies the $assumpt\iota on$ of
Theorem 1.1. Then any weak solution of problem (P) satisfying the energy inequality(SE) has the following decay properties :
$f$

$u$

(i)

(ii)

$||u(t)||_{2}-0$

as$If||e^{-tL}a||_{2}=O(t^{-\alpha})$$tarrow\infty$ ,for some$\alpha>0$, thenfor an arbitrary $e>0$ ,$||u(t)||_{2}=O((\log t)^{\epsilon-1/2})$

(iii)

If

$v^{\infty}=0$.

and $||e^{-tL}a||_{2}=O(t^{-a})$ for some $\alpha>0$ , then$||u(t)||_{2}=\{\begin{array}{l}O(t^{-\alpha})if\alpha<1/9O(t^{e}\cdot-1/2)if\alpha\geq 1/2\end{array}$where$\epsilon>0$

is arbitrary.

Theorenls 4.1 and 4.2 mean that an exterior stationary flow is globally asymptoticallystable in energy sense provided $C_{0}<1/2$ . The enrgy inequality (SE) was first deduced by
$w$

19

72
LERAY [14] in the case $D=R^{3},$ $w=0$ and $f=0$ . Part (i) of Theorem 3.2 is alreadyproved in [18]. HEYWOOD $[8,9]$ and GALDI &RIONERO [6, p.41] discuss the decay of local-norms of strong solutions in case $C_{0}<1/2$ . Contrary to these works, our Theorem
4.2 deals with global -norms of weak solutions satisfying (SE). The decay properties ofstrong solutions in other function spaces are discussed in detail by HEYWOOD [10] andMASUDA [15]. For various problems $related$ to the stability of fluid motions, we refer to [6]and references therein.
$L^{2}$$L^{2}$We begin the proof of Theorem 4.2 by establishing the following

Lemma 4.3. Let $C_{0}<1/2$ , and let(i)denote the adjoint operator of L. Then:

and $\{e^{-tL^{*}} ; t\geq 0\}$ are contraction semigroups in $H$$\{e^{-tL} ; t\geq 0\}$(ii)

$L^{*}$$||e^{-tL}a||_{2}arrow 0$

as$tarrow\infty$,.

(iii) The estimate$||E_{\rho}e^{-tL}P(u\cdot\nabla)u||_{2}\leq C(\rho^{1/2}+\rho^{1/4})||v||_{2}^{1/2}||\nabla u||_{2}^{3/2}$

holds for all $u\in V$ and $\rho>0$ , where
(iv)

If

$v^{\infty}=0$$E_{p}$

is the spectral measure associated to$A$.

, then for all $u\in V$ and $t>0_{f}$$||e^{-tL}P(u\cdot\nabla)u\Vert_{2}\leq Ct^{-1/2}||u||_{2}^{1/2}||\nabla u||_{2}^{3/2}$

Proof. As shown in [15], we have the estimate$|(\phi\cdot\nabla w, \phi)|\leq 2C_{0}||\nabla\phi||_{2}^{2}$

.

(4.5)

Since $(w\cdot\nabla\phi, \phi)=0,$ $(4.5)$ implies(4.6)$(L\phi, \phi)=(\phi, L^{*}\phi)\geq(1-2C_{0})\Vert\nabla\phi\Vert_{2}^{2}$for all $\phi\in D(L)=D(L^{*})=D(A)$ . Hence, we get (i) if $C_{0}\leq 1/2$ . To show (ii), suppose
first that $a\in R(L)$ and hence $a=Lb$ for some $b\in D(L)$ . Then $e^{-tL}a=-v’(t)$ with$v(t)=e^{-tL}b$ . Since$v”(t)+Lv’(t)=0$ ,we get(4.7)$||v’(t)||_{2}^{2}+2(1-2C_{0}) \int_{s}^{t}||\nabla v’||_{2}^{2}d\tau\leq||v’(s)||_{2}^{2}$for all $t\geq s>0$ . This implies that
Sobolev inequality:$||\nabla v’||_{2}$is in$L^{2}$

on the interval$||f||_{6} \leq\frac{2}{\sqrt{3}}||\nabla f||_{2}$20$[1, \infty$). Next, from the
(4.8)

73
we see that $||Bv||_{2}\leq(||w||_{\infty}+C||\nabla w||_{3})||\nabla v||_{2}$. Direct calculation thus gives$(v,v’)=-(v’,Av)-(v’,Bv)\leq||\nabla v’||_{2}||\nabla v||_{2}+C||v’||_{2}||\nabla v||_{2}$$\leq||\nabla v’||_{2}^{2}+||\nabla v||_{2}^{2}+\frac{1}{2}||v’||_{2}^{2}+C||\nabla v||_{2}^{2}$

so that$||v’||_{2}^{2}\leq C(||\nabla v||_{2}^{2}+||\nabla v’||_{2}^{2})$and hence$||v’||_{2}\in L^{2}$

on$[1, \infty$). From (4.7) it follows that$(t-1)||v’(t)||_{2}^{2} \leq\int_{1}^{t}||v’||_{2}^{2}ds\leq\int_{1}^{\infty}||v’||_{2}^{2}ds<+\infty$

and we conclude that$||e^{-tL}a||_{2}=|Iv’(t)||_{2}arrow 0$

as $tarrow\infty$ . This shows (ii) in case E $R(L)$ . To complete the proof, it suffices to showthat $R(L)$ is dense in $H$ . But this follows from $N(L^{*})=0$ which is a consequence of theassumption $C_{0}<1/2$ . The proof of (ii) is complete.
$a$

To show (iii), observe that (4.6) gives$||\nabla e^{-tL^{*}}E_{\rho}\phi||_{2}\leq C||L^{*}e^{-tL^{*}}E_{\rho}\phi||_{2}^{1/2}||E_{p}\phi||_{2}^{1/2}\leq C||L^{*}E_{\rho}\phi||_{2}^{1/2}||\phi||_{2}^{1/2}$

By the definition of$L$

we easily see that$||L^{*}E_{\rho}\phi||_{2}\leq||AE_{\rho}\phi||_{2}+C(||w||_{\infty}+||\nabla w||_{3})||\nabla E_{\rho}\phi||_{2}$.

The H\"older and Sobolev inequalities then imply$|(e^{-tL}P(u\cdot\nabla)u, E_{\rho}\phi)|$$=$I$\leq$$C||u||_{4}^{2}||\nabla e^{-tL^{*}}E_{\rho}\phi||_{2}\leq C(\rho^{1/2}+\rho^{1/4})||u||_{2}^{1/2}||\nabla u||_{2}^{3/2}||\phi\Vert_{2}$$(u, u\cdot\nabla e^{-tL^{*}}E_{\rho}\phi)|$This proves (iii). To show (iv), consider the equation$\frac{dv}{dt}+e^{i\theta}L^{*}v=0$

in the complexification of the Hilbert space$H$; $v(0)=a$. The standard argument then gives$\Vert v(t)||_{2}^{2}+2{\rm Re}\int_{s}^{t}(e^{i\theta}L^{*}v, v)d\tau=||v(s)\{|_{2}^{2}$.

Estimating each term of${\rm Re}(e^{\theta}L^{*}v, v)={\rm Re}(e^{i\theta}Av, v)+{\rm Re}(e^{i\theta}v, w\cdot\nabla v)+{\rm Re}(e^{i\theta}v, v\cdot\nabla w)$

by (4.5) and the inequality$\int_{D}\frac{|v|^{2}}{|x|^{2}}dx\leq 4||\nabla v||_{2}^{2}$

21

,

.

74
we see that$||v(t)||_{2}^{2}+2[ \cos\theta-2C_{0}(1+2|\sin\theta|)]\int_{s}^{t}\Vert\nabla v||_{2}^{2}d\tau\leq||v(s)||_{2}^{2}$

.

Hence
$||v(t)||_{2}\leq||a||_{2}$is small enough. This means that $\{e^{-tL}" ; t\geq 0\}$ is a bounded analytic
providedselnigroup in $H[19]$ ; so we have$|\theta|$$||L^{*}e^{-tL^{*}}a||_{2}\leq Ct^{-1}||a||_{2}$.

Thus, (4.6) yields$||\nabla e^{-tL^{*}}\phi||_{2}^{2}\leq C\Vert L^{*}e^{-tL^{*}}\phi||_{2}||e^{-tL^{*}}\phi\Vert_{2}\leq Ct^{-1}||\phi\Vert_{2}^{2}$.

We therefore obtain$|(e^{-tL}P(u\cdot\nabla)u, \phi)|$$=$$\leq$$|(u, u\cdot\nabla e^{-tL^{*}}\phi)|$$Ct^{-1/2}||u||_{2}^{1/2}||\nabla u||_{2}^{3/2}||\phi||_{2}$,which proves (iv).

Proof of Theorem 4.2. Since $(Lu, u)=||\nabla u\Vert_{2}^{2}+(u\cdot\nabla w, u),$ $(4.6)$ and the energy
inequality (SE) yield(4.9)$||u(t)||_{2}^{2}+2(1-2C_{0}) \int_{s}^{t}||\nabla u||_{2}^{2}d\tau\leq||u(s)||_{2}^{2}+2\int^{t}(f, u)d\tau$for a.e. $s>0$ and all $t\geq s$ . Thus, if $C_{0}<1/2$ , then as in Section 2, we see that. As in Section 3 we get$||u\Vert_{2}\in L^{\infty}$,$||\nabla u||_{2}^{2}\in L^{1}$(4.10)$||u(t)||_{2}^{2}+g(t, s)\leq||u(s)||_{2}^{2}+h(t, s)$for a.e. $s>0$ and all $t\geq s$ , where$g(t, \tau)=C\int_{\tau}^{t}\rho||u\Vert_{2}^{2}d\sigma$:$h(t, \tau)=C(\int_{\tau}^{t}p||E_{\rho}u\Vert_{2}^{2}d\sigma+\int_{T}^{t}\Vert f\Vert_{2}d\sigma)$,and the function $\rho(\tau)>0$ of $\tau>0$ is to be fixed later. Since$\frac{\partial g}{\partial\tau}=-C\rho(\tau)||u(\tau)\Vert_{2}^{2}\leq-C\rho(\tau)[||u(t)||_{2}^{2}+g(t, \tau)-h(t, \tau)]$,assuming the existence of a function $F(\tau)>0$ with $F’=C\rho F$ , we obtain by Gronwall’s
lemma and (4.10),.

$F(t)||u(t)||_{2}^{2} \leq F(s)||u(s)||_{2}^{2}-\int_{s}^{t}F\frac{\partial h}{\partial\tau}d\tau$22

75
Letting $sarrow 0$ , we thus have$||?4(t)||_{2}^{2}$$\leq$$CF(t)^{-1} \Vert a||_{2}^{2}+CF(t)^{-1}\int_{0}^{t}F’(r)||E_{\rho}u||^{2}\underline{Q}d\tau$(4. 11)$+CF(t)^{-1} \int_{0}^{t}F(\tau)(1+\tau)^{-1}||f||_{2}(1+\tau)d\tau$

Next, we take in (4.4)$\phi(\tau)=e^{-(t-\tau)L^{*}}E_{\lambda}\psi,$$|(u(t), E_{\lambda}\psi)|$$\leq$$\psi\in C_{0^{\infty_{\sigma}}},(D)$.

, and set $s=0$ , to obtain$|| \psi||_{2}[||e^{-tL}a\Vert_{2}+\int_{0}^{t}\Vert E_{\lambda}e^{-(t-\tau)L}P(u\cdot\nabla)u||_{2}d\tau]$$+|| \psi\Vert_{2}\int_{0}^{t}||E_{\lambda}e^{-(t-\tau)L}f||_{2}d\tau$.

Applying Lemma 4.3 and H\"older’s inequality yields.

$\Vert E_{\lambda}u(s)||_{2}\leq||e^{-sL}o\Vert_{2}+C(\lambda^{1/2}+\lambda^{1/4})[(\int_{0}^{s}||u||_{2}^{2}d\tau)^{1/4}+\int_{0}^{s}\Vert f||_{\dot{V}}$$d\tau]$(4.12).

We here set $p(\tau)=2C/(\tau+e)1og(\tau+e)$ and $F=C(\log(\tau+e))^{2}$ . Assertions (i) and (ii)follow from (4.11) and (4.12), by repeating the argument in the proof of (i) and (ii) of
Theorem 1.2.

We finally prove (iii). As in Section 2, applying Gronwall’s lemma to (4.9) gives$\Vert u(t)||_{2}^{2}+2(1-2C_{0})\int_{s}^{t}||\nabla u||_{2}^{2}d\tau\leq C(||u(s)||_{2}^{2}+\int_{s}^{t}\Vert f||_{2}d\tau)$

for a.e. $s\in(0, t)$ . So we get$||u(t)||_{2}$$||u||_{2}\in L^{\infty},$$||\nabla u\Vert_{2}\in L^{2}$, and$\leq$$Ct^{-1} \int_{0}^{t}||u||_{2}ds+Ct^{-1}\int_{0}^{t}[\int_{s}^{t}||f||_{2}d\tau]^{1/2}ds$$\leq$$Ct^{-1} \int_{0}^{t}||u\Vert_{2}ds++Ct^{-1/2}[\int_{0}^{t}s||f(s)||_{2}ds]^{1/2}$$\leq$$Ct^{-1} \int_{0}^{t}||u||_{2}ds+Ct^{-I/2}$

On the other hand, taking
Lemma 4.3 (iv) yields$\phi(\tau)=e^{-(t-\tau)L^{*}}\psi,$(4.13).

$\psi\in C_{0^{\infty_{\sigma}}},(D)$, in (4.4) and then applying$||u( \tau)||_{2}\leq||e^{-(\tau-s)L}u(s)||_{2}+C\int_{s}^{\tau}(\tau-\sigma)^{-1/2}(||u||_{2}^{1/2}||\nabla u||_{2}^{3/2}+\Vert f||_{\hat{V}^{*}})d\sigma$.

(4.14)The result now follows from (4.13) and (4.14) in the same way as in the proof of Theorem
1.1 (ii). The proof is complete.

Acknowledgment. The first author is supported by Deutsche Forschungsgemeinschaft inFederal Republic of Germany, the second by a Grant-in-Aid for Scientific Research, theJapan Ministry of Education, Science and Culture.
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WOLFGANG BORCHERS
Department of Mathematics
University of Paderborn
D-4790 Paderborn, Germany

TETSURO MIYAKAWA
Department of Mathematics
Hiroshima University
Hiroshima 730, Japan

25