数理解析研究所講究録
第 1905 巻 2014 年 90-111

90

THE STABILITY AND THE RATE OF CONVERGENCE TO
STATIONARY SOLUTIONS OF THE TWO-DIMENSIONAL
NAVIER-STOKES EXTERIOR PROBLEMMASAO YAMAZAKI (山崎昌男)

FACULTY OF SCIENCE AND ENGINEERING, WASEDA UNIVERSITY(早稲田大学

理工学術院)

ABSTRACT. This paper is concerned with the stability of stationary solutions of the two-dimensional Navier-Stokes exterior problem. The stationary solutions are assumed to be small and enjoy certain pointwise
decay conditions. If the decay condition is critical, the domains and solutions are assumed to satisfy some symmetry condition as well. Under an
initial perturbation in the solenoidal -space, with the same symmetry
if the decay order of the stationary solution is critical, the solution of thenonstationary equation tends to the stationary solution in the solenoidal-class. Also given are the decay orders of the perturbation in otherfunction spaces.
$L^{2}$$L^{2}$

1. INTRODUCTION.
Let be an exterior domain with-boundary with some $\gamma>0.$
We consider the nonstationary Navier-Stokes equation onwith timeindependent external force $f(x)$ :$\Omega$$C^{3+\gamma}$$\Gamma$$\Omega$(1.1)$\partial u$$\overline{\partial t}^{u(x,t)}-\Delta u(x,t)+(u(x,t)\cdot\nabla)u(x,t)+\nabla p(x,t)=f(x)$

(1.2)

in

$\Omega,$$\nabla\cdot u(x,t)=0, in\Omega,$

on
as

(1.3)

$u(x,t)=a(x)$(1.4)$u(x,t)arrow 0$(1.5)$u(x,t)=u_{0}(x)$ in$\Gamma,$$|x|arrow\infty,$$\Omega,$2010 Mathematics Subject Classification. $35Q30.$
Key words and phrases. Navier-Stokes equations, Exterior problem, Uniqueness, Weaksolutions.
Partly supported partly by the Intemational Research Training Group (IGK 1529) on
Mathematical Fluid Dynamics funded by DFG and JSPS and associated with TU Darmstadt, Waseda University in Tokyo and the University of Tokyo, and by Grant-in-Aid for
Scientific Research (C) 25400185, Ministry of Education, Culture, Sports, Science andTechnology, Japan.

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where $a(x)$ satisfies the outflow condition(1.6)$\int_{\Gamma}a(x)\cdot n(x)ds(x)=0.$

We are concerned with the asymptotic stability of the stationary solutionsof this equation. Suppose that $(w(x), \pi(x))$ is a stationary solution of the: Namely,system$(1.1)-(1\backslash 5)$(1.7)$-\Delta w(x)+(w(x)\cdot\nabla)w(x)+\nabla\pi(x)=f(x)$in$\Omega,$(1.8)$\nabla\cdot w(x)=0, in\Omega,$(1.9)$w(x)=a(x)$ on(1.10)$w(x)arrow 0$

as$\Gamma,$$|x|arrow\infty.$

Putting $v(x,t)=u(x,t)-w(x)$ , $p(x,t)=\tilde{p}(x,t)-\pi(x)$ and $v_{0}(x)=u_{0}(x)-$$w(x)$ , we can rewrite the system $(1.1)-(1.5)$ into the system(1.11)$\frac{\partial v}{\partial\iota}(x,t)-\Delta v(x,t)+(w(x)\cdot\nabla)v(x,t)$$+(v(x,t)\cdot\nabla)w(x)+(v(x,t)\cdot\nabla)v(x,t)+\nabla p(x,t)=0$$in\Omega,$$\nabla\cdot v(x,t)=0$$in\Omega,$$(1_{:}12)$(1.13)$v(x,t)=0 on\Gamma,$(1.14)$v(x,t)arrow 0$(1.15)$v(x,0)=v_{0}(x)$ in

as$|x|arrow\infty,$$\Omega.$

We next introduce the Helmholtz decomposition. For $q\in(1,\infty)$ , there exists
a projection operator inonto the space$P_{q}$$(L^{q}(\Omega))^{2}$${\rm Im} P_{q}=L_{\sigma}^{q}(\Omega)=\{u\in(L^{q}(\Omega))^{2}|\nabla\cdot u=0$in$\Omega,$$n\cdot u=0$on$\Gamma\}$such that$KerP_{q}=G^{q}(\Omega)=\{\nabla f\in(L^{q}(\Omega))^{2}|f\in L_{1oc}^{q}(\Omega)\}.$

Since we haveon

, we abbreviate

in thesequel.
Applying the projection to the system $(1.11)-(1.14)$ and putting $A=$
, we obtain the abstract diﬀerential equation$P_{q}\equiv P_{r}$$(L^{q}(\Omega)\cap L^{r}(\Omega))^{2}$$P_{q}$

by

$P$

$P$

$-P\Delta$(1.16)$\frac{\partial v}{\partial t}(t)+Av(t)+P[(w\cdot\nabla)v(t)+(v(t)\cdot\nabla)w+(v(t)\cdot\nabla)v(t)]=0.$

This equation together with the initial condition (1.15) is formally equivalent to the following integral equation:(1.17)$v(t)=\exp(-tA)v_{0}$$- \int_{0}^{t}\exp(-(t-\tau)A)+P[(w\cdot\nabla)v(\tau)+(v(\tau)\cdot\nabla)w+(v(\tau)\cdot\nabla)v(\tau)]d\tau.$

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In order to consider the time-local unique solvability of (1.17), we introduceclasses of functions. For $s\in(2,\infty)$ and $T\in(O,\infty$], put$\mathscr{Y}(s,T)=\{u(t)|t^{1/2-1/s}u(t)\in BC((0, T),L_{\sigma}^{s}(\Omega))$,$t^{1/2}u(t)\in BC((0, T), (H_{0}^{1}(\Omega))^{2})\}$

equipped with the norm$\Vert u\Vert_{\mathscr{Y}(s,T)}=\sup_{0<t<T}\{t^{1/2-1/s}\Vert u(t)\Vert_{s}+t^{1/2}\Vert\nabla u(t)\Vert_{2}\}.$

Then the classbecomes a Banach space, and$\mathscr{Y}(s, T)$$\mathscr{Y}_{0}(s, T)=\{u(t)\in \mathscr{Y}(s, T)|\lim_{tarrow+0}t^{1/2-1/s}u(t)=0$in$L_{\sigma}^{s}(\Omega)$$\lim_{tarrow+0}t^{1/2}u(t)=0$in,$(H_{0}^{1}(\Omega))^{2}\}.$

is a closed subspace of. Then we have the following theorem on
the existence of time-local solutions:$\mathscr{Y}(s, T)$

Theorem 1.1. Suppose that $s>4$ and that is an exterior domain. Suppose moreover that $(w(x), \pi(x))$ is a solution of the system $(1.7)-(1.10)$$\Omega$such that$w(x)\in(L^{s}(\Omega)\cap\dot{H}^{1}(\Omega))^{2}$

Then, for every initial perturbation, there exists a positive number

such that the integral equation (1.17) admits a solutiononin the classsuch that$v(t)$ converges to$tarrow+0$inas. This solution belongs to the class$v_{0}(x)\in L_{\sigma}^{2}(\Omega)$$v(t)$$T_{0}$$(0, T_{0})$$\mathscr{Y}_{0}(s, T_{0})$$L_{\sigma}^{2}(\Omega)$$v0$$C([O, T_{0}),L_{\sigma}^{2}(\Omega))\cap C^{1}((0, T_{0}),L_{\sigma}^{2}(\Omega))\cap C((0, T_{0})$,$(H^{2}(\Omega))^{2})$,and is a solution of the abstract diﬀerential equation (1.16). Furthermore, if
belongs to the space
as well, then the number is estimatedandfrom below by$(H^{1}(\Omega))^{2}$$v_{0}$$s,$$\Vert\nabla w\Vert_{2},$$\Vert w\Vert_{s},$$T_{0}$$\Vert v_{0}\Vert_{2}$$\Vert\nabla v_{0}\Vert_{2}.$We also have the theorem for the uniqueness as follows:
Theorem 1.2. Suppose thatin Theorem 1.1. Suppose that$s,$$\Omega,$$T_{1},$$(w(x), \pi(x))$

and$v_{0}(x)$are the same as
unctions
and satisfies), where $T_{3}=$$T_{2}\in(0,\infty] and that thef$

are solutions of (1.17) on$v_{j}(0)=v_{0}$onfor $j=1,2$. Then we have$\mathscr{Y}(s, T_{j})\cap C([0, T_{j}),L_{\sigma}^{2}(\Omega))$$v_{1}(t)\equiv v_{2}(t)$$v_{j}(t)\in$$(0, T_{j})$$[0, T_{3}$$\min\{T_{1},T_{2}\}.$In order to state the main result on the asymptotic stability of theaforementioned stationary solution $(w(x), \pi(x))$ under initial perturbation$v_{0}(x)\in L_{\sigma}^{2}(\Omega)$. we put$\mathscr{X}(b)=\{w(x)\in C(\Omega)|\Vert w\Vert_{\mathscr{X}(b)}=\sup_{x\in\Omega}(1+|x|)^{b}|w(x)|<\infty\}$

for a positive number , and assume that one of the following conditions
holds:$b$93(C) The exterior domain$\Omega$

is invariant under the mappings$(x_{1},x_{2})\mapsto(-x_{1},x_{2}) , (x_{1},x_{2})\mapsto(x_{1}, -x_{2})$,and $(w(x), \pi(x))$ satisfies the symmetry conditions(U4)$\{\begin{array}{ll}fi(-x_{1},x_{2})=-fi(x_{1},x_{2}) , fi(x_{1}, -x_{2})=f](x_{1},x_{2}) ,f_{2}(-x_{1},x_{2})=f_{2}(x_{1},x_{2}) , f_{2}(x_{1}, -x_{2})=-f_{2}(x_{1},x_{2}) ,\end{array}$and $\pi(-x_{1},x_{2})=\pi(x_{1}, -x_{2})=\pi(x_{1},x_{2})$ .
andsuch that
small, andsatisfy (U4).
$(\mathscr{X}(1)\cap\dot{H}^{1}(\Omega))^{2}$$\Vert w\Vert_{\mathscr{X}(1)}$Furthermore,$1\nabla w\Vert_{2}$$w\in$

are suﬃciently$v_{0}(x)$(S) $w\in(\mathscr{X}(b)\cap\dot{H}^{1}(\Omega))^{2}$ with some $b>1$ such that$\Vert w\Vert_{\mathscr{X}(b)}$andare suﬃciently small.

$\Vert\nabla w\Vert_{2}$Remark 1.1. If $(w(x), \pi(x))$ satisfies$w(x)\in(L^{s}(\Omega))^{2}$
holds for every $s\in(2,\infty$].

$w\in(\mathscr{X}(b))^{2}$

with some $b\geq 1$ , thenRemark 1.2. If the condition (C) holds, then Theorem 1.2 implies thatsatisfies the condition (U4) for every$v(\cdot,t)$$t.$Remark 1.3. For the existence of the stationary solution satisfying the condition (S), the boundary value $a(x)$ must satisfy the condition (1.6).
Remark 1.4. The existence of stationary solutions satisfying the above con-ditions are already proved in [9] under more restrictive symmetry conditions
on the domain and the external forces.
Then our main result is the following:
Theorem 1.3. Under the assumption (C) or (S), there uniquely exists a solution $v(t)\in BC([O,\infty),L_{\sigma}^{2}(\Omega))$ of the integral equation (1.17) such thatand that $t^{1/2}v(t)\in BC((0,\infty),$. Furthermore, theis monotone-decreasing with respect to , and $v(t)$ enjoys
function
the decay properties$v(O)=v_{0}$$(H_{0}^{1}(\Omega))^{2})$$\Vert v(t)\Vert_{2}$(1.18)(1.19)(1.20)$\Vert v(t)\Vert_{q}=o(t^{1/q-1/2})$$\Vert\nabla v(t)\Vert_{2}=o(t^{-1/2})$$\Vert v(t)\Vert_{\infty}=o(t^{-1/2}\sqrt{\log t})$

$t$

as

$tarrow\infty$as$tarrow\infty,$as$tarrow\infty.$for$q\in[2,\infty$),

Remark 1.5. It follows from the assumption that the solution $v(t)$ enjoys theassumption of Theorem 1.2, from which the uniqueness follows.

Remark 1.6. This theorem asserts that the stationary solution $w(x)$ satisfying the condition (S) is the global attractor in.
$L_{\sigma}^{2}(\Omega)$

As will be seen later, this note is an abridged version of Galdi and Yamazaki [6] and Yamazaki [10]. However, I believe that it will be worthwhile
to provide a unified note of the separate papers on the same problem withthe same essential tools.

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2.

$0$

UTLINE OF THE PROOF OF THEOREMS 1.1 AND 1.2.

In this section we give a sketch of the proof of the theorems above. Detailestimates given by
is given in [6]. To this end we first review the
Borchers and Varnhom [2] and Dan and Shibata [3, 4].
$L^{q}-L^{r}$

Theorem 2.1. For the semigroup $exp(-tA)$ we have the following assertions:O. Then there(i) Assume that $1<q<\infty,$ $q\leq r\leq\infty$ andthat,
we havesuch
exists a constantfor every$\alpha\geq$$C$$u\in L_{\sigma}^{q}(\Omega)$$\Vert A^{\alpha}\exp(-tA)u\Vert_{r}\leq Ct^{-\alpha-1/q+1/r}\Vert u\Vert_{q}.$(ii) Assume that $1<q\leq r\leq 2$ .
$C$such that, for every

Then there exists a constant
we have$\Vert\nabla\exp(-tA)u\Vert_{r}\leq$$u\in L_{\sigma}^{q}(\Omega)$$Ct^{-1/2-1/q+1/r}\Vert u\Vert_{q}.$From this theorems we can prove the following lemmata.

Lemma 2.2. Suppose that $2<s<\infty$ . Then there exists a positive conand
stant such that the function $u(t)=\exp(-tA)u_{0}$ belongs toholds for every. Morethe estimate
over, we have $u(t)\in BC([O, 1),L_{\sigma}^{2}(\Omega))$ with $u(O)=u_{0}$ . Furthermore, if$C$$\mathscr{Y}_{0}(s, 1)$$u0\in L_{\sigma}^{2}(\Omega)$$\Vert u\Vert_{\mathscr{Y}(s,1)}\leq C\Vert u_{0}\Vert_{2}$$u0\in L_{\sigma}^{2}(\Omega)\cap(H^{1-2/s}(\Omega))^{2}$(2.1), the inequality$\Vert u\Vert_{\mathscr{Y}(s,T)}\leq C\Vert u_{0}\Vert_{H^{1-2/s}}T^{1/2-1/s}$

holds for every $T\in(O, 1$ ].
Lemma 2.3. Let
positive constant

$q$

$C$

and satisfy $1<q<2<s<\infty$ . Then there exists asuch that the following assertions hold.
$s$(i) Suppose that $u(t)\in C((0,T),L_{\sigma}^{q}(\Omega))$ with some $T\in(O, 1$ ], satisfies

the estimate $B= \sup_{0\leq t<T}t^{3/2-1/q}\Vert u(t)\Vert_{q}<\infty$ . Then the function $v(t)$define$d^{}$

by the formula $v(t)= \int_{0}^{t}\exp(-(t-\tau)A)u(\tau)d\tau$ belongs

holds. Moreover, $we$
have $v(t)\in BC((0, T),L_{\sigma}^{2}(\Omega))$ . Furthermore, for every $\alpha<1-$$1/q$ and every $\delta\in(0, T)$ ,thefunction $v(t)$ is H\"older continuous oforder with values inon.
(ii) If we assume in addition that, then weto$\mathscr{Y}(s, T)$, and the estimate$\Vert v\Vert_{\mathscr{Y}(s,T)}\leq CB$$(H_{0}^{1}(\Omega))^{2}$$a$$(\delta, T)$$\lim_{tarrow+0}t^{3/2-1/q}\Vert u(t)|\}_{q}=0$

have$v\in \mathscr{Y}_{0}(s, T)$, and $v(t)$ converges to

$0$

in

$L_{\sigma}^{2}(\Omega)$

as $tarrow+0.$

Lemma 2.4. Suppose that $2<s<\infty$. Then we have the following assertions:(i) There exists a positive constant such that the following assertion
holds. Let be a positive number such that $T\leq 1$ . Suppose that
$C$

$T$

95
$w(x)\in(L^{s}(\Omega)\cap\dot{H}^{1}(\Omega))^{2}$and that $u(t)$ , $v(t)\in \mathscr{Y}(s, T)$ . Put$S_{w}[v,u](t)=P[(w\cdot\nabla)v(t)+(v(t)\cdot\nabla)w+(u(t)\cdot\nabla)v(t)].$

Then we have $t^{1-1/s}S_{w}[v, u](t)\in BC([0,T),L_{\sigma}^{2s/(2+s)}(\Omega))$ with
(2.2)$\sup_{0<t<T}t^{1-1/s}\Vert S_{w}[v, u](t)\Vert_{2s/(2+s)}$$\leq C(T^{1/2-1/s}(\Vert w\Vert_{s}+\Vert\nabla w\Vert_{2})+\Vert u\Vert_{\mathscr{Y}(s,T)})\Vert v\Vert_{\mathscr{Y}(s,T)}.$(ii) Suppose that $u(t)$ and $v(t)$ are $H$ lder continuous with values inonin addition to the assumpfor some
$S_{w}[v,u](t)$
tion in Assertion (i). Thenis H\"older continuous withvalues inon.
(iii) Suppose thatorholds in addition to the assumption in Assertion (i). Then we have$(H_{0}^{1}(\Omega))^{2}$$(\delta, T)$$\delta\in(0, T)$$L_{\sigma}^{2s/(2+s)}(\Omega)$$(\delta, T)$$\lim_{tarrow+0}t^{1/2-1/s}\Vert u(t)\Vert_{s}=0$$\lim_{tarrow+0}t^{1/2}\Vert\nabla v(t)\Vert_{2}=0$$\lim_{tarrow+0}t^{1-1/s}\Vert S_{w}[v, u](t)\Vert_{2s/(2+s)}=0.$

The following corollary follows immediately from the lemmata above.

Corollary 2.5. Suppose that $s>2$ , there exists a constant such that thefollowing assertion holds. Suppose that $0<T\leq 1$ , and let $w(x)$ , $u(t)$ and$v(t)$ be the same as in Lemma 2.4. Put$C$$T_{w}[v, u](t)=- \int_{0}^{t}\exp(-(t-\tau)A)S_{w}[v, u](\tau)d\tau.$

Then we have $T_{w}[v, u](t)\in \mathscr{Y}(s, T)$ , and we have the estimate$\Vert T_{w}[v, u]\Vert_{\mathscr{Y}(s,T)}\leq C(T^{1/2-1/s}(\Vert w\Vert_{s}+\Vert\nabla w\Vert_{2})+\Vert u\Vert_{\mathscr{Y}(s,T)})\Vert v\Vert_{\mathscr{Y}(s,T)}.$

Furthermore, if $\lim_{tarrow+0}t^{1/2-1/s}\Vert u(t)\Vert_{s}=0$ or$\lim_{tarrow+0}t^{1/2}\Vert\nabla v(t)\Vert_{2}=0$

then we have $T_{w}[v,u](t)\in Y_{0}(s, T)$ and $T_{w}[v,u](t)arrow 0$ in
In particular, if, thenor$u\in \mathscr{Y}_{0}(s, T)$

Proofof Theorem 1.1. Put$L_{\sigma}^{2}(\Omega)$

as $tarrow+0.$$T_{w}[u, v]\in \mathscr{Y}_{0}(s, T)$$v\in \mathscr{Y}_{0}(s, T)$$\tilde{v}_{0}(t)=exp(-tA)v_{0}$forholds,.

.

Then], consider the$v_{0}\in L_{\sigma}^{2}(\Omega)$

Lemma 2.2 implies. Next, for every
mapping from
into itself defined by $U[v](t)=\tilde{v}_{0}(t)+T_{w}[v,v](t)$ .
Then Lemma 2.2 and Corollary 2.5 imply that the estimate$\tilde{v}_{0}\in \mathscr{Y}_{0}(s,\infty)$$U$$T_{0}’\in(0_{\}}1$$\mathscr{Y}_{0}(s, T_{0}’)$$\Vert U[v]\Vert_{\mathscr{Y}(s,T_{\acute{0}})}$$\leq\Vert\tilde{v}_{0}\Vert_{\mathscr{Y}(s,T_{\acute{0}})}+CT_{0}^{1/2-1/s}(\Vert w\Vert_{s}+\Vert\nabla w\Vert_{2})\Vert v\Vert_{\mathscr{Y}(s,T_{\acute{0}})}+C\Vert v\Vert_{\mathscr{Y}(s,T_{\acute{0}})^{2}}$holds with a constant $C\geq 1$ independent of $w,$

inequality(2.3)$\Vert\tilde{v}_{0}\Vert_{\mathscr{Y}(s,T_{\acute{0}})}<\frac{1}{16C}$$\tilde{v}_{0},$$v$

and $T_{0}’\in(0,1$ ]. If the96
holds with some $T_{0}’\in(0,1$ ], put$T_{0}= \min\{T_{0}’, (\frac{1}{2C(\Vert w\Vert_{s}+\Vert\nabla w\Vert_{2})})^{2s/(s-2)}\}.$

Then the quadratic equation
be the smaller one. Then, if
real roots. Let, it follows that$x=\{|\tilde{v}_{0}\Vert_{\mathscr{Y}(s,T_{0})}+x/2+Cx^{2}$$\alpha$has two distinct
satisfies$v\in \mathscr{Y}_{0}(s, T_{0})$$\Vert v\Vert_{\mathscr{Y}(s,T_{0})}\leq\alpha$$\Vert U[v]\Vert_{\mathscr{Y}(s,T_{0})}\leq\Vert\tilde{v}_{0}\Vert_{\mathscr{Y}(s,T_{0})}+CT_{0}^{\iota/2-1/s}(\Vert w\Vert_{s}+\Vert\nabla w\Vert_{2})\alpha+C\alpha^{2}\leq\alpha.$Hence, if the inequality (2.3) holds with some $T_{0}’\in(0,1$ ], the mapping $U$of center and radius into itself.
which satisfies (2.3) exists for every
We next show that the constant. There exists a constant such that, for every $T>0,$, we have the estimate
andmaps the closed ball in$0$$\mathscr{Y}_{0}(s,T_{0})$$\alpha$$T_{0}’$$C’$$v_{0}\in L_{\sigma}^{2}(\Omega)$$v_{0}\in$$v_{1}\in L_{\sigma}^{2}(\Omega)\cap(H^{1}(\Omega))^{2}$$L_{\sigma}^{2}(\Omega)$$\Vert\tilde{v}_{0}\Vert_{\mathscr{Y}(s,T)}\leq\Vert\exp(-tA)v_{1}\Vert_{\mathscr{Y}(s,T)}+\Vert\exp(-tA)(v_{0}-v_{1})\Vert_{\mathscr{Y}(s,T)}$$\leq C’T^{1/2-1/s}\Vert v_{1}\Vert_{H^{1}(\Omega)}+C’\Vert v_{0}-v_{1}\Vert_{2}.$Choose so that $\Vert v_{0}-v_{1}\Vert_{2}<1/32CC’$ , and then choose $T_{0}’\in(0,1$ ] forabove by $T_{0}’= \min\{1,$$v_{1}$$v_{1}$$(1/32CC’\Vert v_{1}\Vert_{H^{1}(\Omega)})^{2s/(s-2)}\}.$

If$v_{0}\in L_{\sigma}^{2}(\Omega)\cap(H^{1}(\Omega))^{2}$, we have$\Vert_{\tilde{V}_{0}}\Vert_{\mathscr{Y}(s,T)}\leq C’T^{1/2-1/s}\Vert v_{0}\Vert_{H^{1}(\Omega)}.$. Then we
In this case we put $T_{0}’= \min\{1,$by thehave (2.3) in both cases, and in the latter case we can chooseofthevaluesby. Hence we can choose
andvalues ofand. ThenNext, let $v(t)$ ,such that
we have$(1/64CC’\Vert v_{0}\Vert_{H^{1}(\Omega)})^{2s/(s-2)}\}$$T_{0}’$$s,$$\Vert v_{0}\Vert_{2},$$\Vert v_{0}\Vert_{2}$$\Vert\nabla v_{0}\Vert_{2},$$\Vert w\Vert_{s}$$T_{0}$$\Vert\nabla v_{0}\Vert_{2}$$1\nabla w\Vert_{2}.$$\tilde{v}(t)\in \mathscr{Y}_{0}(s,T_{0})$$\Vert v\Vert_{\mathscr{Y}(s,T_{0})},$$\Vert\tilde{v}\Vert_{\mathscr{Y}(s,T_{0})}\leq\alpha$$U[\tilde{v}](t)-U[v](t)=T_{w}[\tilde{v},\tilde{v}](t)-T_{w}[v,v](t)$$= \int_{0}^{t}\exp(-(t-\tau)A)(S_{w}(v,v)(\tau)-S_{w}(\tilde{v},\tilde{v})(\tau))d\tau$$= \int_{0}^{t}\exp(-(t-\tau)A)P[(w\cdot\nabla)v(\tau)+(v(\tau)\cdot\nabla)w+(v(\tau)\cdot\nabla)v(\tau)$$-(w\cdot\nabla)\tilde{v}(\tau)-(\tilde{v}(\tau)\cdot\nabla)w-(\tilde{v}(\tau)\cdot\nabla)\tilde{v}(\tau)]d\tau$$= \int_{0}^{t}\exp(-(t-\tau)A)P[(w\cdot\nabla)(v(\tau)-\tilde{v}(\tau))+((v(\tau)-\tilde{v}(\tau))\cdot\nabla)w$$+(\tilde{v}(\tau)\cdot\nabla)(v(\tau)-\tilde{v}(\tau))+((v(\tau)-\tilde{v}(\tau))\cdot\nabla)v(\tau)]d\tau$$=T_{w}[\tilde{v}-v,\tilde{v}](t)+T_{0}[v,\tilde{v}-v]$

$s,$

97
for every $t\in(O, T_{0})$ . Hence Corollary 2.5 implies that
(2.4)$\Vert U[\tilde{v}]-U[v]\Vert_{\mathscr{Y}(s,T_{0})}$$\leq C(T_{0}^{1/2-1/s}(\Vert w\Vert_{s}+\Vert\nabla w\Vert_{2})+\Vert\tilde{v}\Vert_{\mathscr{Y}(s,T_{0})}+\Vert v\Vert_{\mathscr{Y}(s,T_{0})})\Vert_{\tilde{\mathcal{V}}}-v\Vert_{\mathscr{Y}(s,T_{0})}$$\leq(\frac{1}{2}+2C\alpha)\Vert\tilde{v}-v\Vert_{\mathscr{Y}(s,T_{0})}.$

In view of the definition of , we haveHence (2.4) implies that the mapping is a contraction mapping from theclosed ball inof center and radius into itself, and therefore
it has a unique fixed point $v(t)$ in this ball. If, thenumber is determined byand$\frac{1}{2}+2C\alpha=1-\frac{\Vert\tilde{v}_{0}\Vert_{\mathscr{Y}(s,T_{0})}}{\alpha}<1.$$\alpha$$U$$0$$\mathscr{Y}_{0}(s, T_{0})$$\alpha$$v_{0}\in L_{\sigma}^{2}(\Omega)\cap(H^{1}(\Omega))^{2}$$T_{0}’$$s,$$\Vert v_{0}\Vert_{2},$$\Vert\nabla v_{0}\Vert_{2},$$\Vert w\Vert_{s}$$\square$$\Vert\nabla w\Vert_{2}.$Proofof Theorem 1.2. We first remark that we may assume that. Indeed, letandthe functions satisfying the as-.
$v_{1}(t)\in$$y_{1}(t)$$\mathscr{Y}_{0}(s, T_{1})$$y_{2}(t)$sumption of this theorem defined on $[0, T’]$ andrespectively. Let
be the solution constructed in Theorem 1.1. Applying
this theorem to $v_{1}(t)=v(t)$ and $v_{2}(t)=y_{1}(t)$ , we haveon$(0, \min\{T_{0},$. Hence, putting$[0, T_{2}]$$v(t)\in \mathscr{Y}_{0}(s, T_{0})$$v_{1}(t)\equiv y_{1}(t)$$T$$v_{1}(t)=\{\begin{array}{ll}v(t) if T’\leq T_{0},y_{1}(t) if \tau_{0}\leq\tau’\end{array}$we see that$v_{1}(t)\in \mathscr{Y}_{0}(s, T_{1})$, where $T_{1}= \max\{T_{0},$ $T$

Then it suﬃces toshow the identity
on the interval
for every $T_{4}\in(0, T_{3})$ .
From the assumption we see. Put $\tilde{v}(t)=v_{2}(t)-v_{1}(t)$ .
Then we have $\tilde{v}(t)=T_{w}[v_{2},v_{2}](t)-T_{w}[v_{1},v_{1}](t)$ , and hence$v_{1}(t)\equiv v_{2}(t)$$[0, T_{4}]$$v1(t)\in \mathscr{Y}_{0}(s, T_{1})$(2.5)$\tilde{v}(t)=-\int_{0}^{t}\exp(-(t-\tau)A)$$P[((w+v_{2}(t))\cdot\nabla)\tilde{v}(t)+(\tilde{v}(t)\cdot\nabla)(w+v_{1}(t))]d\tau$for every $t\in(O, T_{4}$ ]. Hence Lemmata 2.3 and 2.4 imply that there exists a
constant such that the estimate
$C$

(2.6)

$\Vert\tilde{v}\Vert_{\mathscr{Y}(s,T)}\leq C(T^{1/2-1/s}\Vert w\Vert_{s}+T^{1/2}\Vert\nabla w\Vert_{2}$$+ \sup_{0<\tau\leq T}\tau^{1/2-1/s}\Vert v_{2}(\tau)\Vert_{s}+\sup_{0<\tau\leq T}\tau^{1/2}\Vert\nabla v_{1}(\tau)\Vert_{2})\Vert\tilde{v}\Vert_{\mathscr{Y}(s,T)}$

98
holds for every $T\in(O, T_{4}$ ]. Then, in the same calculation as in the proof of
Theorem 1.1, we can find a positive constant such that
$T_{5}$$T_{5}^{1/2-1/s} \Vert w\Vert_{s}+T_{5}^{1/2}\Vert\nabla w\Vert_{2}+\sup\tau^{1/2-1/s}\Vert v_{2}(\tau)||_{s}$$0<\tau\leq T_{5}$$+ \sup\tau^{1/2}\Vert\nabla v_{1}(\tau)\Vert_{2}\leq\frac{1}{2C},$$0<\tau\leq T_{5}$with the same constant$C$

which implies that
For a positive number
consider the condition

as in (2.6). Then (2.6) implies that$\tilde{v}(t)\equiv 0$$\delta$on$\Vert\tilde{v}\Vert_{\mathscr{Y}(s,T_{5})}=0,$$[0,T_{5}].$determined later and a nonnegative integer$n,$holds on $[0, T_{5}+n\delta].$
Suppose that (2.7) holds with some , which we have already seen that wehave already verified for $n=0$ . Then the identity (2.5) can be rewritten as(2.7)$\tilde{v}(t)\equiv 0$$n$$\tilde{v}(t)=-\int_{T_{5}+n\delta}^{t}\exp(-(t-\tau)A)$$P[((w+v_{2}(\tau))\cdot\nabla)\tilde{v}(\tau)+(\tilde{v}(\tau)\cdot\nabla)(w+v_{1}(\tau))]d\tau$

for $t\in(T_{5}+n\delta,T_{4}]$ . Then Lemmata 2.3 and 2.4 imply that there exists aand such that the estimate
constant independent of$C$$v,$ $w$$n$$\Vert\tilde{v}(t)\Vert_{s}+\Vert\nabla\tilde{v}(t)\Vert_{2}$$\leq C\frac{2s}{s-2}(t-T_{5}-n\delta)^{(s-2)/2s} \sup (\Vert\tilde{v}(\tau)\Vert_{s}+\Vert\nabla\tilde{v}(\tau)\Vert_{2})$$T_{5}+n\delta\leq\tau\leq t$$(\Vert w\Vert_{s}+T_{5}^{1/s-1/2}\Vert v_{2}\Vert_{\mathscr{Y}(s,T_{3})}+\Vert\nabla w\Vert_{2}+T_{5}^{-1/2}\Vert\nabla v_{1}\Vert_{\mathscr{Y}(s,T_{3})})$

holds for $t\in[T_{5}+n\delta,T_{5}+n\delta+1]$ . Suppose that $T_{6}\in(T_{5}+n\delta, T_{5}+n\delta+1$ ].
Taking the supremum with respect to $t\in[T_{5}+n\delta, T_{6}]$ , we have$\sup_{T_{5}+n\delta\leq t\leq\tau_{6}}(\Vert\tilde{v}(t)\Vert_{s}+\Vert\nabla\tilde{v}(t)\Vert_{2})(1-C\frac{2s}{s-2}(T_{6}-T_{5}-n\delta)^{(s-2)/2s}\cross$$(\Vert w\Vert_{s}+T_{5}^{1/s-1/2}\Vert v_{2}\Vert_{\mathscr{Y}(s,T_{3})}+\Vert\nabla w\Vert_{2}+T_{5}^{-1/2}\Vert\nabla v_{1}\Vert_{\mathscr{Y}(s,T_{3})}))\leq 0.$

Now choose $\delta\in(0,1$ ] so small that it satisfies$C \frac{2s}{s-2}\delta^{(s-2)/2s}$$( \Vert w\Vert_{s}+T_{5}^{1/s-1/2}\Vert v_{2}\Vert_{\ovalbox{\tt\small REJECT}(s,T_{3})}+\Vert\nabla w\Vert_{2}+T_{5}^{-1/2}\Vert\nabla v_{1}\Vert_{\mathscr{Y}(s,T_{3})})\leq\frac{1}{2},$forand put $T_{6}= \min\{T_{5}+(n+1)\delta, T_{4}\}$ . Then we havefor $0\leq t\leq T_{4}$ . Otherwise we have
If $T_{6}=T_{4}$ , we conclude that
(2.7) with replaced by $n+1$ . Repeating the argument above, we can arrive$T_{6}=T_{4}$ in finite steps. This completes the proof.
$\tilde{v}(t)\equiv 0$$0\leq t\leq T_{6}.$$\tilde{v}(t)\equiv 0$$n$$\square$

99
3.

$0$

UTLINE OF THE PROOF OF THEOREM 1.3.

In order to obtain the decay rate of, we follow theand[7].
However,
method by Kato
this calculation requires the smallness of
the initial value. Hence, to prove the result for large initial value, anothermethod is needed to prove the global solvability and weak decay property.
For this purpose we employ the energy inequality.
$\Vert v(t)\Vert_{q}$$1\nabla v(t)\Vert_{2}$We first recall Hardy’s inequality as follows:
Lemma 3.1. Suppose that is an exterior domain. Then there exists a
constant such that, for every $u(x)\in H_{0}^{1}(U)$ ,
$U$

$C$

$\int_{U}\frac{|u(x)|^{2}}{|x|^{2}(1+|\log|x||)^{2}}dx\leq C\Vert\nabla u\Vert_{2^{2}}$

If

$U$

enjoys some symmetry property, we have the following improved

version, whose proof is found in Galdi [5].

Lemma 3.2. Suppose that is an exterior domain satisfying (D4). Then
there exists a constant such that, for every
satisfying (U4),
we have$U$$C$$u(x)\in\dot{H}_{0}^{1}(U)$$\int_{U}\frac{|u(x)|^{2}}{|x|^{2}}dx\leq C\Vert\nabla u\Vert_{2^{2}}$We now start the proof of Theorem 1.3. The proof consists of four steps

as follows:(i) Global solvability together with the boundedness (a priori estimate)(ii) Decay of(cannot grow so rapidly)(iii) Decay of(Slowness of energy dispersion)(iv) Decay rate ofandestimate for the per$\Vert\nabla v(t)\Vert_{2}$$\Vert\nabla v(t)\Vert_{2}$$\Vert v(t)\Vert_{2}$$\Vert v(t)\Vert_{q}$$||\nabla v(t)\Vert_{2}(L^{q}-L^{r}$turbed s\‘emigroup)
Detailed proof of Step (i)-Step (iii) is given in [6], and that of Step (iv) is

given in [10].
Step (i): Under the assumption of Theorem 1.3 we have the following
lemma, which implies the boundedness of$\Vert v(t)\Vert_{2}.$Lemma 3.3. We have the inequality$\frac{d}{dt}\Vert v(t)\Vert_{2^{2}}\leq(C\Vert w\Vert_{\mathscr{X}(b)}-1)\Vert\nabla v(t)\Vert_{2^{2}}$Proof Taking the inner product with $v(t)$ with the equality (1.16) and integrating by parts, we obtain the equality(3.1)$\frac{d}{dt}\Vert v(t)\Vert_{2^{2}}+\Vert\nabla v(t)\Vert_{2^{2}}-(v(t)\otimesw)\nabla v(t)=0.$

Employing Lemma 3.1 under Assumption (S) and Lemma 3.2 under Assumption (C), we can estimate(3.2)$\Vert v(t)\otimes w\Vert_{2}\leq C\Vert w\Vert_{\mathscr{X}(b)}\Vert\nabla v(t)\Vert_{2}.$Substituting this estimate into (3.1) we obtain the conclusion.

$\square$100
Lemma 3.3 implies the required estimates$0\leq s<t<\infty$ and(3.3)$\Vert v(t)\Vert_{2}\leq\Vert v(s)\Vert_{2}$

for

$s,$

$t$

with

$\int_{0}^{\infty}\Vert\nabla v(t)\Vert_{2^{2}}dt<\infty.$

In the same way we have the an estimate for a higher order derivative, which
we admit for the moment.

Lemma 3.4. We have the inequality$\frac{d}{dt}\Vert\nabla v(t)\Vert_{2^{2}}\leq C’(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla u\Vert_{2})^{4}\Vert v(t)\Vert_{2^{2}}$If
3.3 and 3.4 imply$\Vert w\Vert_{\mathscr{X}(b)}<1/2C’$, put $R=2C’(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2})^{4}$ . Then Lemmata$\frac{d}{dt}(R\Vert v(t)\Vert_{2^{4}}+\Vert\nabla v(t)\Vert_{2^{4}})\leq-R\Vert\nabla v(t)\Vert_{2^{2}}\Vert v(t)\Vert_{2^{2}}\leq 0.$, and hence Theorem 1. 1This estimate ensures the boundedness of
implies that the solution become a time-global one.
$\Vert\nabla v(t)\Vert_{2}$Proofof Lemma 3.4: We have the equality$\frac{1}{2}\frac{d}{dt}\Vert\nabla v(t)\Vert_{2^{2}}=(\frac{dv}{dt}(t),Av(t))$(3.4)$=(Av(t)-P[(v(t)\cdot\nabla)w+(w\cdot\nabla)v(t)+(v(t)\cdot\nabla)v(t)],Av(t))$$=-\Vert-\Delta v(t)\Vert_{2^{2}}+I_{1}+I_{2}+I_{3},$where$I_{1}=((v(t)\cdot\nabla)w,Av(t))$,$I_{2}=((w\cdot\nabla)v(t),Av(t))$,$I_{3}=((v(t)\cdot\nabla)v(t),Av(t))$.

By direct calculation we have $I_{3}=0$ . Next, in view of the interpolation, we can estimate
relation$(L^{2},H^{2})_{1/2,1}=B_{2,1}^{1}\subset L^{\infty}$$|I_{1}|\leq C\Vert v(t)\Vert_{2}^{1/2}\Vert\Delta v(t)\Vert_{2}^{3/2}\Vert\nabla w\Vert_{2},$$|I_{2}|\leq C\Vert v(t)\Vert_{2}^{1/2}\Vert\Delta v(t)\Vert_{2}^{3/2}\Vert w\Vert_{\mathscr{X}(1)}.$Substituting these estimates into (3.4) we obtain the conclusion.

$\square$

Step (ii): We can prove the following lemma, which implies thatcannot grow so rapidly.

$\Vert\nabla v(t)\Vert_{2}$Lemma 3.5. For and such that $1\leq t-1\leq s\leq t$ , we have the estimate
$s$

$t$

$\Vert\nabla v(s)\Vert_{2}\geq\Vert\nabla v(t)\Vert_{2}$$-C(t-s)^{1/3}( \Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+\sup_{t\geql}\Vert v(t)\Vert_{2}+\sup_{t\geq 1}\Vert\nabla v(t)\Vert_{2})^{2}$

101
Admitting this lemma for the moment, we can derivefrom (3.3). In view of this fact and the boundedness of
Gagliardo-Nirenberg inequality implies thatas$\Vert\nabla v(t)\Vert_{2}arrow 0$$tarrow\infty$$\Vert v(t)\Vert_{q}arrow 0$$q\in(2,\infty)$$\Vert v(t)\Vert_{2}$$tarrow\infty$

as

, the

for every.

Proof of Lemma 3.5: We have $v(t)=\exp(-(t-s)A)v(s)+\tilde{v}$, where(3.5)$\tilde{v}=-\int_{s}^{t}\exp(-(t-\tau)A)P[(v(\tau)\cdot\nabla)w+(w\cdot\nabla)v(\tau)+(v(\tau)\cdot\nabla)v(\tau)]d\tau.$Put$g_{1}(\tau)=P[(v(\tau)\cdot\nabla)w+(v(\tau)\cdot\nabla)v(\tau)]$and $g_{2}(\tau)=P(w\cdot\nabla)v(\tau)$ .

Then we have the estimates$\Vert g_{1}(\tau)\Vert_{3/2}\leq C(\Vert\nabla w\Vert_{2}+\sup_{t\geq 1}\Vert\nabla v(t)\Vert_{2})\sup_{t\geq 1}\Vert v(t)\Vert_{2^{1/3}}\sup_{t\geq 1}\Vert\nabla v(t)\Vert_{2^{2/3}}$

and$\Vert g_{2}(\tau)\Vert_{2}\leq C\Vert w\Vert_{\mathscr{X}(1)}\sup_{t\geq 1}\Vert\nabla v(t)\Vert_{2}.$Substituting these estimates into (3.5) we have$\Vert\nabla\tilde{v}\Vert_{2}\leq\int_{s}^{t}C(t-\tau)^{-2/3}d\tau(\Vert\nablaw\Vert_{2}+\sup_{t\geq 1}\Vert\nabla v(t)\Vert_{2})$$\sup\Vert V(f)\Vert_{2^{1/3}}\sup\Vert\nabla v(t)\Vert_{2^{2/3}}$$t\geq 1 t\geq 1$$+ \int_{S}^{t}C(t-T)^{-1/2}d\tau\Vert w\Vert_{\mathscr{X}(1)\sup_{t\geq 1}}\Vert\nabla v(t)\Vert_{2}$$\leq C(t-s)^{1/3}(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+\sup_{t\geq 1}\Vert v(t)\Vert_{2}+\sup_{t\geq 1}\Vert\nabla v(t)\Vert_{2})^{2}$Integrating this inequality on the interval $[s,t]$ we obtain the conclusion.
This completes the proof of Lemma 3.5.

$\square$

Step (iii): We show an estimate which dominates the increase of the energy far from the origin. Let
be a smooth function on such that$0\leq\chi(x)\leq 1,$
on , 1 andon [2, ). Then we have thefollowing lemma.
$\chi(x)$$\chi(x)\equiv 0$

$[0$

$]$

$\mathbb{R}$$\chi(x)\equiv 1$$\infty$

Lemma 3.6. We have the estimate(3.6)$\frac{d}{dt}\Vert\chi(\frac{|x|}{R})v(t)\Vert_{2}^{2}\leq C(\Vert w\Vert_{\mathscr{X}(b)}+\Vert v_{0}\Vert_{2})\Vert\nabla v(\cdot,t)\Vert_{2^{2}}$with a constant$C$

independent of $R>0.$102
Admitting this lemma for the moment, we complete the proof of Step(iii). Suppose that $s<t$ . Integrating (3.6) on the interval $[s,t]$ , we obtain$\int_{|x|\geq 2R}|v(x,t)|^{2}dx$$\leq\int_{|x|\geq R}|v(x,s)|^{2}dx+C(\Vert w\Vert_{\mathscr{X}(b)}+\Vert v_{0}\Vert_{2})\int_{s}^{t}\Vert\nabla v(\tau)\Vert_{2^{2}}d\tau.$

For every fixed$\epsilon>0$, choose$s$so large that$\int_{s}^{\infty}\Vert\nabla v\{\tau,s)\Vert_{2^{2}}d\tau<\frac{\epsilon}{4C(\Vert w\Vert_{\mathscr{X}(b)}+\Vert v_{0}\Vert_{2})}.$For this , choose $R>0$ so large that
$s$

(3.7)

$\int_{|x|\geq R}|v(x,s)|^{2}dx<\frac{\epsilon}{4}$$\int_{|x|\geq 2R}|v(x,t)|^{2}dx<\frac{\epsilon}{2}$On the other hand, it follows from the factthat there exists a constant $T\geq ssuc|1$ that
(3.8)$\int_{|x|\leq 2R}|v(x,t)|^{2}dx<\frac{\epsilon}{2}$. Then we havefor every $t\geq s.$$\Vert v(t)\Vert_{q}arrow 0$

as$tarrow\infty$

for $q>2$

for every $t\geq T.$

Then the required asymptotic stability follows from (3.7) and (3.8).

$\square$

Proofof Lemma 3.6: In the same way as in the proof of Lemma 3.3, we
obtain$\frac{1}{2}\frac{d}{dt}\Vert\chi(\frac{|x|}{R})v(x,t)\Vert_{2}^{2}$$=( \frac{d}{dt}(\chi(\frac{|x|}{R})v(x,t)),\chi(\frac{|x|}{R}v(t,x)))$(39)$=( \chi(\frac{|x|}{R})(-\Delta v(x,t)+P[(w(x)\cdot\nabla)v(x,t)$$+(v(x,t) \cdot\nabla)w(x)+(v(x,t)\cdot\nabla)v(x,t)]),\chi(\frac{|x|}{R})v(x,t))$$=I_{1}+I_{2}+I_{3}+I_{4},$where$I_{1}=(- \Delta v(x,t),\chi(\frac{|x|}{R})^{2}v(x,t))$,$I_{2}=((v(x,t) \cdot\nabla)v(x,t),P\chi(\frac{|x|}{R})^{2}v(x,t))$$I_{3}=((w(x) \cdot\nabla)v(x,t),P\chi(\frac{|x|}{R})^{2}v(x,t))$

,

,

103

$I_{4}=((v(x,t) \cdot\nabla)w(x),P\chi(\frac{|x|}{R})^{2}v(x,t))$We first estimate . Sincetion by parts yields$I_{1}$$\nabla v(t,x)\in L^{2}(\Omega)$.

and $v(t,x)=0$ on$\partial\Omega$, integra-$I_{1}=-\Vert\nabla v(\cdot,t)\Vert_{2^{2}}+(\nabla v(x,t),$ $( \nabla(\chi(\frac{|x|}{R})^{2}))v(x,t))$$=- \Vert\nabla v(\cdot,t)\Vert_{2^{2}}+\frac{1}{R}(\nabla v(x,t),2(\nabla\chi)(\frac{|x|}{R})\chi(\frac{|x|}{R})v(x,t))$.

It follows that
(3.10)$I_{1} \leq\frac{C}{R}\Vert\nabla v(\cdot,t)\Vert_{2}\Vert(\nabla\chi)(\frac{|x|}{R})(\frac{|x|}{R})v(x,t)\Vert_{2}$Since $v(x,t)=0$ on
estimate(3.11)$\partial\Omega$, we can apply the Poincar\’e inequality to obtain the$\Vert(\nabla\chi)(\frac{|x|}{R})\chi(\frac{|x|}{R})v(x,t)\Vert_{2}$$\leq C(\int_{\{x\in\Omega||x|\leq 2R\}}|\nabla v(x,t)|^{2}dx)^{1/2}\leq CR\Vert\nabla v\Vert_{2}.$Substituting this estimate into (3.10) we conclude(3.12)$I_{1}\leq C\Vert\nabla v(\cdot,t)\Vert_{2^{2}}$We next estimate the term
(3.13)$I_{2}$

as follows:$I_{2}\leq\Vert\nabla v(\cdot,t)\Vert_{2}\Vert v(\cdot,t)\Vert_{4^{2}}$$\leq C\Vert\nabla v(\cdot,t)\Vert_{2^{2}}\Vert v(\cdot,t)\Vert_{2}\leq C\Vert v(\cdot, T)\Vert_{2}\Vert\nabla v(\cdot,t)\Vert_{2^{2}}$for $t\geq T$ in view of the Gagliardo-Nirenberg inequality.
In view of (3.2), the term can be estimated as$I_{3}$(3.14)$I_{3}\leq C\Vert w\Vert_{\mathscr{X}(b)}\Vert\nabla v(\cdot,t)\Vert_{2^{2}}$Finally, in order to estimate we recall the construction of the Helmholtz
decomposition in exterior domains by Miyakawa. We have$I_{4}$$P \chi(\frac{|x|}{R})^{2}v(x,t)=\chi(\frac{|x|}{R})^{2}v(x,t)+\nabla q_{1}(x,t)+\nabla q_{2}(x,t)$

where $q_{1}(x,t)$ is the solution in(3.15)$\mathbb{R}^{2}$,of the equation$- \Delta q_{1}(x,t)=div(\chi(\frac{|x|}{R})^{2}v(x,t))=\frac{1}{R}(\nabla\chi^{2})(\frac{|x|}{R})\cdot v(x,t)$

104
and $q_{2}(x,t)$ is the solution of the Neumann problem$\{\begin{array}{l}-\Delta q_{2}(x,t)=0 in\Omega,(n\cdot\nabla)q_{2}(x,t)=-(n\cdot\nabla)(\chi(\frac{|x|}{R})v(x,t)+q_{1}(x,t))=-(n\cdot\nabla)q_{1}(x,t)on\partial\Omega.\end{array}$Then, integrating by parts, we have$I_{4}=(v(x,t)\otimes w(x),$ $- \nabla(\chi(\frac{|x|}{R})^{2}v(x,t))-\nabla^{2}q_{1}(x,t)-\nabla^{2}q_{2}(x,t))$.

It follows that
(3.16)$I_{4} \leq C\Vert w\Vert_{\mathscr{X}(b)}\Vert\nabla v\Vert_{2}(\Vert\chi(\frac{|x|}{R})^{2}\nabla v(x,t)\Vert_{2}+$$+ \frac{2}{R}\Vert(\nabla\chi)(\frac{|x|}{R})\chi(\frac{|x|}{R})v(x,t)\Vert_{2}+\Vert\nabla^{2}q_{1}(\cdot,t)\Vert_{2}+\Vert\nabla^{2}q_{2}(\cdot,t)\Vert_{2})$Then the$L^{2}$.

-boundedness of the Riesz transforms implies(3.17)$\Vert\nabla^{2}q_{1}(\cdot,\iota)\Vert_{2}\leq\frac{C}{R}\Vert(\nabla\chi)(\frac{|x|}{R})\chi(\frac{|x|}{R})\cdot v(x,t)\Vert_{2}\leq C\Vert\nabla v(\cdot,t)\Vert_{2}.$We next have$\Vert\nabla^{2}q_{2}(\cdot,t)\Vert_{2}\leq C\Vert(n\cdot\nabla)q_{2}(\cdot,t)\Vert_{H^{1/2}(\partial\Omega)}=C\Vert(n\cdot\nabla)q_{1}(\cdot,t)\Vert_{H^{1/2}(\partial\Omega)}$$\leq C\Vert\nabla^{2}q_{1}(\cdot,t)\Vert_{2}$It follows from (3.17) that
(3.18)$\Vert\nabla^{2}q_{2}(\cdot,t)\Vert_{2}\leq C\Vert\nabla v(\cdot,t)\Vert_{2}.$Substituting (3. 11), (3. 17) and (3. 18) into (3.16) we obtain(3.19)$I_{4}\leq C\Vert w\Vert_{\mathscr{X}(b)}\Vert\nabla v\Vert_{2^{2}}$Substituting (3.12), (3. 13), (3. 14) and (3.19) into (3.9) we conclude that$\frac{d}{dt}\Vert\chi(\frac{|x|}{R})v(x,t)\Vert_{2}^{2}\leq C(\Vert w\Vert_{\mathscr{X}(b)}+\Vert v(T)\Vert_{2})\Vert\nabla v(\cdot,t)\Vert_{2^{2}}$Now (3.6) follows from the monotonicity of$\square$$\Vert v(t)\Vert_{2}.$We now recall the estimate of coerciveness of the Stokes operator.

Lemma 3.7. We have the following assertions:(i) For $v\in D(A^{1/2})=L_{\sigma}^{2}(\Omega)\cap(H_{0}^{1}(\Omega))^{2}$ , we have(ii) For$\Vert\nabla v\Vert_{2}=\Vert A^{1/2_{\mathcal{V}\Vert_{2}}}.$$v\in D(A)=L_{\sigma}^{2}(\Omega)\cap(H_{0}^{1}(\Omega)\cap H^{2}(\Omega))^{2}$

a constant$C$,such that we have the estimate$C(\Vert Av\Vert_{2}+\Vert A^{1}/2_{\mathcal{V}}\Vert_{2})$.

there exists$\Vert\nabla^{2}v\Vert_{2}\leq$105
We next recall the resolvent estimates of the Stokes operator by Borchers
and Vamhorn [2] and Dan and Shibata [3, 4], from which estimates Theorem 2.1 follows.

Proposition 3.8. Put $D=\{\zeta\in \mathbb{C}|\zeta\neq 0, |\arg\zeta|\leq 3\pi/4\}$ . Then we havethe following assertions:(i) For every and such that $1<q\leq r\leq\infty$, there exists a positive, the operator $(\zeta+A)^{-1}$ is aconstantsuch that, for every
satisfying the estimate
tobounded operator from. In
for every.
particular, if $q\leq r<\infty$, we have(ii) For every and such that $1<q\leq r\leq 2$, there exists a positive, the operator
constantsuch that, for every
satisfying the esto
is a bounded operator fromtimatefor every $u\in$
$q$

$r$

$\zeta\in D$$C_{q,r}$$L_{\sigma}^{q}(\Omega)$$(L^{r}(\Omega))^{2}$$\Vert(\zeta+A)^{-1}u\Vert_{r}\leq C_{q,r}|\zeta|^{-1+1/q-1/r}\Vert u\Vert_{q}$$u\in L_{\sigma}^{q}(\Omega)$$(\zeta+A)^{-1}u\in L_{\sigma}^{r}(\Omega)$

$q$

$r$

$\zeta\in D$$C_{q,r}$$L_{\sigma}^{q}(\Omega)$$\nabla(\zeta+A)^{-1}$$(L^{r}(\Omega))^{4}$$\Vert\nabla(\zeta+A)^{-1}u\Vert_{r}\leq C_{q,r}|\zeta|^{-1/2+1/q-1/r}\Vert u\Vert_{q}$$L_{\sigma}^{q}(\Omega)$.

This proposition and Lemma 3.7 yield the following proposition.
Proposition 3.9. We have thefollowing assertions:(i) Suppose that $1<q\leq 2$ . Then there exists a constantsuch that,and every $t>0$, thefunction $\exp(-tA)u$ befor every, and satisfies the estimate
longs to the space$C_{q}’$$u\in L_{\sigma}^{q}(\Omega)$$(H_{0}^{1}(\Omega)\cap H^{2}(\Omega))^{2}$$\Vert\nabla^{2}\exp(-tA)u\Vert_{2}\leq C_{q,s}’t^{-1/q}(1+t^{-1/2})\Vert u\Vert_{q}.$such that, for every, thefunction $\exp(-tA)u$ satisfies the estimate(ii) There exists a constant$(H_{0}^{1}(\Omega))^{2}$$C_{s}"$$u\in L_{\sigma}^{2}(\Omega)\cap$$\Vert\nabla^{2}\exp(-tA)u\Vert_{2}\leq C_{s}"(1+t^{-1/2})\Vert\nabla u\Vert_{2}.$This proposition immediately implies the following corollary.

Corollary 3.10. Suppose that $1\leq s<3/2$ . Then we have the following
assertions:(i) Suppose that $1<q\leq 2$ . Then there exists a constantsuchthat, for every
and every $t>0$, the function $exp(-tA)u$, and satisfies the estimate
belongs to the space$C_{q,s}’$$u\in L_{\sigma}^{q}(\Omega)$$(H_{0}^{s}(\Omega))^{2}$$\Vert exp(-tA)u\Vert_{\dot{H}^{s}}\leq C_{q,s}’t^{-1/q}(1+l^{(s-1)/2})\Vert u\Vert_{q}.$(ii) There exists a constant $C”$ such that, for every$(H_{0}^{1}(\Omega))^{2}$,$u\in L_{\sigma}^{2}(\Omega)\cap$thefiznction $exp(-tA)u$ satisfies the estimate$\Vert\exp(-tA)u\Vert_{\dot{H}^{s}}\leq C_{s}"(1+t^{-(s-1)/2})\Vert\nabla u\Vert_{2}.$106
We now introduce a perturbation of the operator , and show somewith some $b\geq 1$ andproperties. Suppose that satisfies, and put $B[u]=P\{(w\cdot\nabla)u+(u\cdot\nabla)w\}$ . Then, for every$A$$w\in(\mathscr{X}(b))^{2}$$w$$\nabla w\in(L^{2}(\Omega))^{4}$$u\in D(A)=L_{\sigma}^{q}(\Omega)\cap(H_{q,0}^{1}(\Omega)\cap H_{q}^{2}(\Omega))^{2}$, which implies
with $1<q\leq 2$ , we have
We moreover have
if $1<q<2$ and $u\in$
if $q=2$ , which imply
in both cases. Hence$u\in
D(A)$ .
the operator $L_{w}[u]=Au+B[u]$ is well-defined on
In the sequel we obtain the resolvent estimate of this operator. For thispurpose Borchers and Miyakawa [1] expanded the resolvent into Neumann
series. Kozono and Yamazaki [8] extended the range of boundedness by
estimating the Neumann series by using fractional powers of the resolvent. However, we cannot employ this method straightforward due to thestrong limitation of the range of coerciveness. We get around this diﬃculty by obtaining the estimate for the fractional power $(\zeta+A)^{-1/2}$ definedby the spectral decomposition of onand estimate the operator$(\zeta+A)^{-1/2}B(\zeta+A)^{-1/2}$ by duality argument.
Letdenote the spectral measure associated with the operator on, we can write. Then, for$\nabla u\in(L^{q}(\Omega))^{4}$$(w\cdot\nabla)u\in$$(L^{q}(\Omega))^{2}$$u\in L_{\sigma}^{2q/(2-q)}(\Omega)$$(L^{\infty}(\Omega))^{2}$$(u\cdot\nabla)w\in(L^{q}(\Omega))^{2}$$A$$L_{\sigma}^{2}(\Omega)$$A$$\mu(\lambda)$$L_{\sigma}^{2}(\Omega)$$\zeta\in D$$( \zeta+A)^{-1}=\int_{0}^{\infty}\frac{1}{\zeta+\lambda}d\mu(\lambda)$,$( \zeta+A)^{-1/2}=\int_{0}^{\infty}\frac{1}{\sqrt{\zeta+\lambda}}d\mu(\lambda)$

Then the operator $(\zeta+A)^{-1/2}$ is holomorphic in the interior ofues in bounded linear operators on. Here we note that, and hence the branch ofplies
for every$L_{\sigma}^{2}(\Omega)$$\lambda\geq 0$$\zeta+\lambda\in D$$D$.

with valimis well-$\zeta\in D$$\sqrt{\zeta+\lambda}$

defined. It is easy to see that $\{(\zeta+A)^{-1/2}\}^{2}=(\zeta+A)^{-1}$ . For the operator$(\zeta+A)^{-1/2}$

we can prove the following lemmas by spectral decomposition.

Lemma 3.11. For every
constants$C_{q}$

and$C_{r}$$q$and satisfying $1<q\leq 2\leq r<\infty$, there exist$r$such that, for every $\zeta\in D$ we have the estimates$\Vert(\zeta+A)^{-1/2}u\Vert_{2}\leq C_{q}|\zeta|^{-1+1/q}\Vert u\Vert_{q}$for every$\Vert(\zeta+A)^{-\iota/2}u\Vert_{r}\leq C_{r}|\zeta|^{-1/r}\Vert u\Vert_{2}$for every

Lemma 3.12. There exists a constant, we have the estimate
every$u\in L_{\sigma}^{2}(\Omega)$$C_{2}$$u\in L_{\sigma}^{2}(\Omega)\cap L_{\sigma}^{q}(\Omega)$$u\in L_{\sigma}^{2}(\Omega)$,.

such that, for every$\zeta\in D$

and$\Vert\nabla(\zeta+A)^{-1/2}u\Vert_{2}\leq C_{2}\Vert u\Vert_{2}.$From these lemmas we can prove the following estimate.
Lemma 3.13. Suppose that
Suppose also that$(L^{2}(\Omega))^{4}$$w\in(\mathscr{X}(b))^{2}$$\zeta\in \mathbb{C}\backslash \{O\}$with some $b\geq 1$ andsatisfies$|\arg\zeta|\leq 3\pi/4$$\nabla w\in$. Then107
the operator $(\zeta+A)^{-1/2}B(\zeta+A)^{-1/2}$ is bounded in
the estimate$L_{\sigma}^{2}(\Omega)$, and itsatisfies$\Vert(\zeta+A)^{-1/2}B[(\zeta+A)^{-1/2}u]\Vert_{2}\leq C\Vert w\Vert_{\mathscr{X}(b)}\Vert u\Vert_{2},$

where

$C$

is a constant depending only on

Proof Suppose that$\varphi\in C_{0,\sigma}^{\infty}(\Omega)$$\nabla\cdot(\zeta+A)^{-1/2}u=0$, we have$\Omega.$. In view of the equalities$\nabla\cdot w=0$

and

(3.20)$|(\varphi, (\zeta+A)^{-1/2}P\{(w\cdot\nabla)(\zeta+A)^{-1/2}u+((\zeta+A)^{-1/2}u\cdot\nabla)w\})|$$=|-(\nabla(\zeta+A)^{-1/2}\varphi,w(\zeta+A)^{-1/2}u)|$$\leq\Vert\nabla(\zeta+A)^{-1/2}\varphi\Vert_{2}\Vert w(\zeta+A)^{-1/2}u\Vert_{2}.$In view of the fact $(\zeta+A)^{-1/2}u\in D(A^{1/2})$ , Lemma 3.12 and (3.2) imply(3.21)$\Vert w(\zeta+A)^{-1/2}u\Vert_{2}\leq C\Vert w\Vert_{\mathscr{X}(b)}\Vert\nabla(\zeta+A)^{-1/2}u\Vert_{2}\leq C\Vert w\Vert_{\mathscr{X}(b)}\Vert u\Vert_{2},$,is dense in
where the constant depends only on . Since
we obtain the conclusion by substituting Lemma 3.12 and the inequality(3.21) into (3.20).
$C$$\Omega$$C_{0,\sigma}^{\infty}(\Omega)$$L_{\sigma}^{2}(\Omega)$$\square$

For the operator$L_{w}$

we have the following proposition.

Proposition 3.14. For every
such that $1<q\leq 2\leq r<\infty$, there exist
satisfying
positive numbers andsuch that, for every, we have the estimates
and
$q,$

$A$

$\nabla w\in(L^{2}(\Omega))^{4}$$r$$w\in(\mathscr{X}(b))^{2}$$A_{q,r}$$\Vert w\Vert_{\mathscr{X}(b)}\leq A$$\Vert(\zeta+I_{\ovalbox{\tt\small REJECT}})^{-1}u\Vert_{r}\leq A_{q_{)}r}|\zeta|^{-1+1/q-1/r}\Vert u\Vert_{q},$$\Vert\nabla(\zeta+I_{\ovalbox{\tt\small REJECT}})^{-1}u\Vert_{2}\leq A_{q,2}|\zeta|^{-1+1/q}\Vert u\Vert_{q}$and every
for every
Proof Suppose that$u\in L_{\sigma}^{q}(\Omega)$$\zeta\in D.$$\Vert w\Vert_{\mathscr{X}(b)}\leq 1/2C$

operator$T$. Then Lemma 3.13 implies that thedefined by$T= \sum_{j=0}^{\infty}\{-(\zeta+A)^{-1/2}B(\zeta+A)^{-1/2}\}^{j}$is bounded on(3.22)

For
ply

$q$

$L_{\sigma}^{2}(\Omega)$

uniformly in$\zeta\in D$$(\zeta+A)^{-1/2}T(\zeta+A)^{-1/2}=(\zeta+A+B)^{-1}=(\zeta+L_{w})^{-1}$

and

$r$

as in the assumption, Lemmata 3.11 and 3.12 im-$\Vert(\zeta+A)^{-1}u\Vert_{2}\leq C_{q}|\zeta|^{-1+1/q}\Vert u\Vert_{q}$$\Vert(\zeta+A)^{-1}u\Vert_{r}\leq C_{r}|\zeta|^{-1/r}\Vert u\Vert_{2},$$L_{\sigma}^{2}(\Omega)$and satisfiesfor$u\in L_{\sigma}^{q}(\Omega)\cap L_{\sigma}^{2}(\Omega)$$\Vert\nabla(\zeta+A)^{-1}u\Vert_{2}\leq C_{2}\Vert u\Vert_{2}$. Hence the required estimates follow from these estimates.

,

andfor $u\in$$\square$108
Since we can obtain a semigroup by integrating the resolvent of the generator on an appropriate contour in the complex plane, we can deduce thenext theorem from the proposition above.
Theorem 3.15. Let be the same as in Proposition 3.14. Then the opera,$tor-L_{w}$ generates a bounded analytic-semigroup $\exp(-tL_{w})$ on
and for every and such that $1<q\leq 2\leq r<\infty$, there exists a constantand $t>0$, we have the estimatessuch that, for every$w$$C^{0}$$L_{\sigma}^{2}(\Omega)$

$r$

$q$

$u\in L_{\sigma}^{q}(\Omega)$$B_{q,r}$$\Vert\exp(-tL_{w})u\Vert_{r}\leq B_{q_{)}r}t^{-1/q+1/r}\Vert u\Vert_{q},$$\Vert\nabla\exp(-tL_{w})u\Vert_{2}\leq B_{q,2}t^{-1/q}\Vert u\Vert_{q}.$We now proceed to Step (iv). The conclusions of Step (ii) and Step (iii), there exists a positive number
imply that, for every
such that, forevery
we haveandNext, forsuch that $T_{0}<T_{1}<\infty$ , we put$\epsilon>0$$t\geq T_{0}$$\Vert v(t)\Vert_{2}<\epsilon,$$T_{0}$$\Vert v(t)\Vert_{4}<\epsilon$$\Vert\nabla v(t)\Vert_{2}<\epsilon.$$T_{1}$$\alpha(T_{1})=\sup_{\tau_{0}\leq t\leq T_{1}}\max\{(t-T_{0})^{1/4}\Vert v(t)\Vert_{4},$ $(\iota-T_{0})^{1/2}\Vert\nabla v(t)\Vert_{2}\}.$

Then the function, we can write$\alpha(T_{1})$is continuous and monotone-increasing. For$t\in$$[T_{0}, T_{1}]$$v(t)= \exp(-(t-T_{0})L_{w})v(T_{0})+\int_{T_{0}}^{t}\exp(-(t-\tau)L_{w})P[(v(\tau)\cdot\nabla)v(\tau)]d\tau.$

From this we can estimate$\Vert v(t)\Vert_{4}\leq B_{2,4}(t-T_{0})^{-1/4}\Vert v(T_{0})\Vert_{2}$$+C_{4/3} \int_{T_{0}}^{t}B_{4/3,4}(t-\tau)^{-1/2}\Vert v(\tau)\Vert_{4}\Vert\nabla v(\tau)\Vert_{2}d\tau$$\leq B_{2,4}(f-T_{0})^{-1/4}\epsilon+C_{4/3}\alpha(t)^{2}\int_{T_{0}}^{t}B_{4/3,4}(t-\tau)^{-1/2_{T}-3/4}d\tau$

where
to

$L_{\sigma}^{4/3}(\Omega)$(3.23)

denotes the operator norm of the projection. This implies$C_{4/3}$

$P$

from

$(L^{4/3}(\Omega))^{2}$$(t-T_{0})^{1/4}\Vert v(t)\Vert_{4}\leq B_{2,4}\epsilon+C_{4/3}B_{4/3.4}B(\begin{array}{l}11\overline{2}’\overline{4}\end{array})\alpha(T_{1})^{2}$

In the same way, from the estimate$\Vert\nabla v(t)\Vert_{2}\leq B_{2,2}(t-T_{0})^{-1/2}\epsilon+C_{4/3}\alpha(t)^{2}\int_{T_{0}}^{t}B_{4/3,2}(t-\tau)^{-3/4_{T}-3/4}d\tau,$

it follows that
(3.24)$(t-T_{0})^{1/2}\Vert\nabla v(t)\Vert_{2}\leq B_{2,2}\epsilon+C_{4/3}B_{4/3.2}B(\begin{array}{l}11\overline{4}’\overline{4}\end{array})\alpha(T_{1})^{2}.$Hence, putting$C_{1}= \max\{C_{4/3}B_{4/3.4}B(\begin{array}{l}1l\overline{2}’\overline{4}\end{array}),C_{4/3}B_{4/3.2}B(\begin{array}{l}11\overline{4}’\overline{4}\end{array})\},$$C_{2}= \max\{B_{2,4},B_{2,2}, 1\}$

109
and taking the maximum of (3.23) and (3.24), we see that$\max\{(t-T_{0})^{1/4}\Vert v(t)\Vert_{4},$ $(t-T_{0})^{1/2}\Vert\nabla v(t)\Vert_{2}\}\leq C_{1}\alpha(T_{1})^{2}+C_{2}\epsilon.$

Taking the supremum for(3.25)$t\in[T_{0}, T_{1}]$, we see that$\alpha(T_{1})$satisfies$\alpha(T_{1})\leq C_{1}\alpha(T_{1})^{2}+C_{2}\epsilon.$

We suppose that $\epsilon<1/4C_{1}C_{2}$ . Then there exists two distinct roots of theequation $C_{1}X^{2}-X+C_{2}\epsilon=0$ . Let
denote the smaller one. Then we, and the intermediate theorem implies that we havehave
$T_{1}>T_{0}$ifis suﬃciently close to . It follows that$f(\epsilon)$$\alpha(T_{1})<$$\epsilon<f(\epsilon)$$f(\epsilon)$$T_{0}$$\Vert\nabla v(T_{1})\Vert_{2}\leq f(\epsilon)(T_{1}-T_{0})^{-1/2}\leq\sqrt{2}f(\epsilon)T_{1}^{-1/2}$

for every. On the other hand, we have
the Gagliardo-Nirenberg inequality implies that the estimate$T_{1}\geq 2T_{0}$$\Vert v(T_{1})\Vert_{2}\leq\epsilon\leq f(\epsilon)$. Hence$\Vert v(T_{1})\Vert_{q}\leq$holds for every $T_{1}>2T_{0}$ and $q\in[2,\infty$ ). Since we have$f(\epsilon)arrow+0$ as, we conclude (1.18) and (1.19).
It remains only to show (1.20). First, sinceand since, we havecan be regarded as a closed subset of$C_{q}f(\epsilon)T_{1}^{-1/2+1/q}$$\epsilonarrow+0$$v(t)\in H_{0}^{1}(\Omega)$$H_{0}^{1}(\Omega)$$H_{0}^{1}(\mathbb{R}^{2})$$\Vert v(t)\Vert_{\dot{B}_{\infty,2}^{0}(\Omega)}\leq\Vert v(t)\Vert_{\dot{B}_{\infty,2}^{0}(\mathbb{R}^{2})}$$\leq C\Vert\nabla v(t)\Vert_{L^{2}(\mathbb{R}^{2})}=C\Vert\nabla v(t)\Vert_{L^{2}(\Omega)}=o(t^{-1/2})$

where$\dot{B}_{\infty,2}^{0}(\mathbb{R}^{2})$every fixed

denotes the homogeneous Besov space on], we can choose $T\geq 2$ so large that$\mathbb{R}^{2}$,. Then, for$\epsilon\in(0,1$$\sup_{t\geq T-1}\max\{t^{3/8}\Vert v(t)\Vert_{8},t^{1/2}\Vert\nabla v(t)\Vert_{2},t^{1/2}\Vert v(t)\Vert_{B_{\infty,2}^{0}}\}\leq\epsilon.$

Suppose that $t\geq T$ . Then, for every $\tau\in[t-1,t]$ , we have $\tau\geq t-1\geq t/2.$
We next recall the Littlewood-Paley decomposition. Let
be amonotone-decreasing -function on $(-1,\infty)$ such thaton $(-1,1$ ]and $\chi(s)\equiv 0$ on, put $\Phi(\xi)=\chi(|\xi|)$ and. Next, for$\varphi_{j}(\xi)=\chi(2^{-j}|\xi|)-\chi(2^{1-j}|\xi|)$ for. Then, for every
we have$\chi(s)$$C^{\infty}$$\chi(s)\equiv 1$$[2,\infty)$$\xi\in \mathbb{R}^{n}$$k\in \mathbb{Z}$$j\in \mathbb{Z}$$\Phi(2^{-k}\xi)+\sum_{j=k+1}^{\infty}\varphi_{j}(\xi)=1$

positive integer such that

. For a fixed $t\geq T$ , choose as the smallest$t\leq 2^{2k}$$k$; namely, $k\geq(\log_{2}t)/2$ . We then put$v^{(1)}(t)=\mathscr{F}^{-1}[\Phi(2^{k}\xi)\mathscr{F}[v(t)]],$$v^{(2)}(t)= \sum_{j=-k+1}^{k-1}[\varphi_{j}(\xi)\mathscr{F}[v(t)]],$$v^{(3)}(t)= \sum_{j=k}^{\infty}[\varphi_{j}(\xi)\mathscr{F}[v(t)]].$

Then we have $v(t)=v^{(1)}(t)+v^{(2)}(t)+v^{(3)}(t)$ .
We first have

110
(3.26)$\Vert v^{(1)}(t)\Vert_{\infty}$$\leq\Vert v(t)\Vert_{8}\Vert 2^{-2k}(\mathscr{F}^{-1}[\Phi])(2^{-k}\cdot)\Vert_{8/7}=C2^{-k/4}\Vert v(t)\Vert_{8}\leq Ct^{-1/2}\epsilon.$

We next observe that$\Vert v^{(2)}(t)\Vert_{\infty}\leq\sum_{j=-k+1}^{k-1}\Vert \mathscr{F}^{-1}[\varphi_{j}\mathscr{F}[v(t)]]\Vert_{\infty}$(3.27)$\leq\sqrt{2k-1}(\sum_{j=-k+1}^{k-1}\Vert \mathscr{F}^{-1}[\varphi_{j}\mathscr{F}[v(t)]]\Vert_{\infty}^{2})^{1/2}$$\leq C\sqrt{\log t}\Vert v(t)\Vert_{\dot{B}_{\infty,2}^{0}}\leq C\epsilon t^{-1/2}\sqrt{1\circ gt}.$

Finally, in order to estimate$\Vert v^{(3)}(t)\Vert_{\infty}$, we employ another representation(3.28) $v(t)=\exp(-A)v(t-1)$$+ \int_{t-1}^{t}\exp(-(t-\tau)A)P[(w\cdot\nabla)v(\tau)+(v(\tau)\cdot\nabla)w+(v(\tau)\cdot\nabla)v(\tau)]d\tau.$

Then the Sobolev embedding theorem and Corollary 3.10, (2) imply(3.29)$\Vert\exp(-A)v(t-1)\Vert_{\dot{C}^{1/3}}\leq C\Vert\exp(-A)v(t-1)\Vert_{\dot{H}^{4/3}}$$\leq C\Vert\nabla v(t-1)\Vert_{2}\leq C\epsilon(t-1)^{-1/2}\leq 2C\epsilon t^{-1/2}$

Next, for $\tau\in[t-1,t]$ , we have$\Vert(w\cdot\nabla)v(\tau)\Vert_{8/5}\leq\Vert w\Vert_{8}\Vert\nabla v(\tau)\Vert_{2}\leq C\Vert w\Vert_{\mathscr{X}(b)}\epsilon t^{-1/2},$$\Vert(v(\tau)\cdot\nabla)w\Vert_{8/5}\leq\Vert v(\tau)\Vert_{8}\Vert\nabla w\Vert_{2}\leq C\Vert\nabla w\Vert_{2}\epsilon t^{-3/8},$

and$\Vert(v(\tau)\cdot\nabla)v(\tau)\Vert_{8/5}\leq C\epsilon^{2_{f}-7./8}$

Summing up these estimates we conclude that$\Vert P[(w\cdot\nabla)v(\tau)+(v(\tau)\cdot\nabla)w+(v(\tau)\cdot\nabla)v(\tau)]\Vert_{8/5}$$\leq C\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-3/8}.$

Hence the Sobolev embedding theorem and Corollary 3.10, (2) imply(3.30)$\Vert g(t)\Vert_{\dot{C}^{1/3}}\leq C\int_{t-1}^{t}(t-\tau)^{-2/3-5/4+1}$$\Vert P[(w\cdot\nabla)v(\tau)+(v(\tau)\cdot\nabla)w+(v(\tau)\cdot\nabla)v(\tau)]\Vert_{8/5}d\tau$$\leq C\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-3/8}l_{-1}^{t}(t-\tau)^{-11/12}d\tau$$\leq C\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-3/8}$

111
It follows from (3.28), (3.29) and (3.30) that$\Vert v(t)\Vert_{\dot{C}^{1/3}}\leq C\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-3/8}$

Since$\dot{C}^{1/3}$

coincides with

$B_{\infty,\infty}^{1/3}$, we have$\Vert \mathscr{F}^{-1}[\varphi_{j}\mathscr{F}[v(t)]]\Vert_{\infty}\leq C2^{-j/3}\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-3/8}.$

Summing up we obtain$\Vert v^{(3)}(t)\Vert_{\infty}\leq\sum_{j=k}^{\infty}\Vert \mathscr{F}^{-1}[\varphi_{j}\mathscr{F}[v(t)]]\Vert_{\infty}$(3.31)$\leq C2^{-k/3}\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-3/8}$$\leq C\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-3/8-1/6}$$\leq C\epsilon(\Vert w\Vert_{\mathscr{X}(b)}+\Vert\nabla w\Vert_{2}+1)t^{-1/2}$

Summing up (3.26), (3.31) and (3.27) we conclude (1.20).
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