数理解析研究所講究録 1425 巻 2005 年 21-39

21

Rotating Navier-Stokes Equations with Initial Data
Nondecreasing at Infinity
北海道大学大学院 理学研究科数学専攻 博士課程 3 年
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan乾 勝也 (Katsuya Inui)

Abstract
This is a supplementary note of the paper [12] by Yoshikazu Giga, Alex Mahalov, Shin’ya
Matsui and me. In [12] locai-in-time unique existence of strong solutions was obtained forfor a class of initial data that contains somethe rotating Navier-Stokes equations in
nondecreasing functions at space infinity. The rotating Navier-Stokes equations has thedenotes the vertical unit vector. The Coriolis, whereCoriolis term of the form
, using its skewsolution operator is estimated uniformly in the Coriolis parameterrotating
Navier-Stokes
theforsynnmetry, Then it is shown that local existence time estimate.
equations is uniform in$\mathbb{R}^{3}$$e_{3}\mathrm{x}u$$\mathrm{e}_{3}$$\Omega\in \mathbb{R}$$\Omega\in \mathbb{R}$1Introduction

We consider the rotating Navier-Stokes equations in$u_{t}-$

(RNS)

$\{$

bu

$\mathbb{R}^{3}$$+(u, \nabla)u+\nabla p=-\Omega e_{3}\mathrm{x}$

:

for
for
for

$u$

$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$u|_{t=0}=u_{0}$$x\in \mathrm{R}^{3}$$x\in \mathbb{R}^{3}$$x\in \mathbb{R}^{3}$, $0<t<T$ ,, $0<t<T$ ,,field and $p=$
where $u=u(x, t)=(u^{1}(x, t),$ $u^{2}(x, t),u^{3}(x, t))$ is the unknown velocity vector$u_{41}=u_{0}(=(u_{0}^{1}(x), u_{0}^{2}(x),$is the given$p(x, t)$ is the unknown scalar pressure field, while
is a scalar fixed constant, $e_{3}=(0, 0, 1)$ , andinitial velocity vector field. Besides, $T>0$ ,$u=(\Omega u^{2}, -\Omega u^{1},0)$ .
represents the outer product, hence,. The constant
The equations (RNS) are the Navier-Stokes equations with the term
is called the Coliolis term, whichis called the Coliolis parameter and the term.
aroundvelocity
angularrepresents the Coliolis force when the fluid is rotating with
The Coliolis term has an another expression:$u_{0}^{3}(x))$$\Omega\in \mathbb{R}$$-\Omega e_{3}\mathrm{x}$$\mathrm{x}$$-\Omega e_{3}\mathrm{x}u$$u$$-\Omega e_{3}\mathrm{x}$$\Omega$$\Omega/2$$-\Omega e_{3}\mathrm{x}$with the skew-symmetric matrix$\mathrm{J}$$u=-\Omega \mathrm{J}u$$x_{3^{-}}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}$,defined by$\mathrm{J}=(\begin{array}{ll}0-1 010 000 0\end{array})$.

[3] and
For (RNS) in the case of periodic and cylindrical domains, Babin-Mahalov-Nicolaenko
in theuniformly
Mahalov-Nicolaenko [19] proved local existence and uniqueness of solutions

22

Coriolis parameter . Moreover, they proved global in time regularity of solutions when issufficiently large. The method of proving global regularity for large fixed 0 is based on the), nonlinear averaging andanalysis of fast singular oscillating limits (singular limit
cancellation of oscillations in the nonlinear interactions for the vorticity field. It uses harm onic
analysis tools of lemmas on restricted convolutions and Littlewood-Paley dyadic decomposition
equations whichto prove global regularity of the limit resonant three-dimensional
holds without any restriction on the size of initial data and strong convergence theorems forlarge 0.
for
Our aim is to prove local existence with its existence time estimate uniformly intermat space inifinity. For this purpose we transpose the Coriolis
nondecreasing initial data
(RNS) in the form
rewriteto$\Omega$$\Omega$$\Omegaarrow+\infty$$\mathrm{N}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{e}\mathrm{r}rightarrow \mathrm{S}\mathrm{t}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{s}$$\Omega\in \mathbb{R}$$u_{0}$$-\Omega e_{3}\mathrm{x}u=-\Omega \mathrm{J}u$$u_{\mathrm{f}}-$

An

for
for
for

$+\Omega \mathrm{J}u+(ll\mathit{4},, \nabla)u+\nabla p=0$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\{$$u|_{t=0}=\mathrm{n}_{0}$$x\in \mathbb{R}^{3}$$x\in \mathbb{R}^{3}$$x\in \mathbb{R}^{3}$, $0<t<T$,, $0<t<T$,,so that the Coliolis term is dealt with the diffusion term Au as a linear problem. Then we,, $j\leq 3$ formally to get the abstract
multiply the Helmholtz operator
ordinary differential equation$\mathrm{P}=$(A)$u_{t}$–$\Delta u+$$R_{j}$$\delta_{i,j}$

Pu $=u$$1\leq i$

Ju $+\mathrm{P}(u\nabla)\}u=0$$\mathrm{Q}$Here,is the Kronecker delta andget (A) we used the fact that$(\delta_{i,j}+RR_{j})_{i,j}$

for $t>0$ .

is the scalar Riesz operator whose symbol isfor divergence free vector field$\mathrm{i}\xi_{j}/|\xi|$

. To

(1.1)

$u$

.
and that PA
However, instead of (A), we consider the following equation:$=\Delta \mathrm{P}$(ABS)$v_{t},-$

bu$+\mathrm{Q}\mathrm{P}\mathrm{J}\mathrm{P}\mathrm{u}+\mathrm{P}(\mathrm{u}, \nabla)u=0$

for $t>0$ ,which is equivalent to (A) because PJu PJPu if divn $=0$ by (1.1).
The corresponding integral equation to (ABS) is written as:
$=$

(I)

where

$u(t)=\exp(-\mathrm{A}(\Omega)t)u_{0}-f_{0}^{t}\exp(-\mathrm{A}(\Omega)(t-s))\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}(u\otimes \mathrm{u}(\mathrm{t})ds$$\mathrm{A}(\Omega)=-\Delta$ $+$for $t>0$ ,QPJP. Hence, $\exp(-\mathrm{A}(\Omega)t)$ , the exponential of the operator$-\mathrm{A}(\Omega)t$, isrepresented by(1.2)$\exp(-\mathrm{A}(\Omega)t)=\exp(t\Delta)\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)$and can be called the ’$\mathrm{H}\mathrm{e}\mathrm{a}\mathrm{t}+\mathrm{C}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}$’solution operator.

In the case $\Omega=0$ , that is, on the Navier-Stokes equations (NS) without the Coriolis term,, the space of
belongs to
unique local existence of mild solution was proved if initial databounded solenoidal functions, in Cannon-Knightly [6], Cannone [7] and Giga-Inui-Matsui [11]$\prime u_{\{;}$$L_{\sigma}^{\infty}$

23

contains nondecreasing functions. There are several related works for
Of cource, the space
initial data $[7])[18]$ . The method in [11] is to use estimate for the derivative of the heat kernel, Giga-Matsuiobtained by Carpio [8]. For (NS) with initial datain the Hardy spacein the 2-dimensional
Sawada [13] obtained unique global existence of strong solution
and $p\in BMO$
when
solutionweakcase and J. Kato [17] proved uniqueness of$BMO$of functions of
space
the
is
Here,
in the -dimensional case with $n\geq 2$ (see also [14]).
$\mathrm{e}L_{\sigma}^{\infty}$$L^{\infty}$$L_{\sigma}^{\infty}$$\mathcal{H}^{1}$$u\in L_{\sigma}^{\infty}$$u\in L^{\infty}$$(u, \nabla p)$$n$bounded mean oscillations.
, that is, rotating case, the crucial step is to estimate the Coriolis solution oPIn the case$-R_{2}R_{1}u^{2}+$$\Omega\neq 0$that comes from the Coriolis term
$Rj$$-R_{3}R_{1}u^{2}+R_{3}R_{2}u^{1})$ . The difficulty is that the term contains the Riesz operator. Moreover, Carpio’s estimate does not aPPly to the term since itwhich is not bounded in
has no derivatives.
Hieber-Sawada [15] and Sawada [20] constructed a local solution for (RNS) with generalized
is zero for the solenoidal initial datawhosematrixCorilolis term Mu with 3
is a homogeneous Besov space including various periodic and almost. Here,, which is a subspace of
periodic functions, that do not decay at space infinity. The space$[21]$ (see Taniuchi [22] for, was first used to solve Boussinesq equations by
operator
recent improvement). The advantage of the Besov space is boundedness of the Riesz
space.
in it. They are successful in estimating the Coriolis term in the Besov(RNS) were transequations
the
,
since
on
depends
However, their existence time estimate
formed to the integral equation

erator$\mathrm{P}\mathrm{J}u=(-R_{1}R_{1}u^{2}+R_{1}R_{2}u^{1},$$\exp(-\Omega \mathrm{P}\mathrm{J}t)$$R_{2}R_{2}u^{1},$$L^{\infty}$$\mathrm{x}3$$u_{0}\in\dot{B}_{\infty,1}^{0}$$\mathrm{M}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\dot{B}_{\infty,1}^{0}$$\dot{B}_{\infty,1}^{0}$$\mathrm{S}\mathrm{a}\mathrm{w}\mathrm{a}\mathrm{d}\mathrm{a}arrow \mathrm{T}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{i}$$L^{\infty}$$\Omega$$u(t)= \exp(t\Delta)u_{0}-\int_{0}^{l}\exp((t-s)\Delta)\mathrm{P}\{\mathrm{d}\mathrm{i}\mathrm{v}(u\otimes u)(s)+\Omega e_{3}\mathrm{x}u(s)\}$$ds$

for $t>0$(RNS) into (I) toto regard the Coriolis term as a perturbation. In this paPer, we transformed”
0 by using skewparameter
term uniformly in the Coriolis
estimate the linear
the equation
rathersymmetric structure of the operator PJP. That is the reason that we dealasform(ABS) instead of (A). We estimate the Coiiolis solution operator in the. Smallness of the Coriol is term is not assumed. Thisin (1.2) instead of the form
is a major difference between our and their approach.
arises again in the linear term.
In the integral equation (I), the unboundedness problem incontains the Riesz transforms, one cannot
Since the Coriolis solution operator”
operator. There was still a possibility that the
expect its boundedness in. Unforis unbounded in, even if
is bounded inisoperatorsolution
tunately, our exact calculation of the symbol arrived at conclusion that the(see $[[12]$ ; Appendix ).
not bounded in. To introduce ourof
In this situation we are forced to restrict initial data to a subspace(2 dimensional 3 components) vector field part andnew subspace we split initial data into
-dependent part by taking vertical average$” \mathrm{H}\mathrm{e}\mathrm{a}\mathrm{t}+\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}$$\mathrm{e}3\iota \mathrm{p}(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}l)$$\exp(-\Omega \mathrm{P}\mathrm{J}t)$$L^{\infty}$$\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)$$‘ \mathrm{E}\mathrm{e}\mathrm{a}\mathrm{t}+\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}$$L^{\infty}$$L^{\infty}$$\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)$$L^{\infty}$$\exp(t\Delta)\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)$$L^{\infty}$$\mathrm{A}]$$L_{\sigma}^{\infty}$$2\mathrm{D}3\mathrm{C}$$x_{3}$

24

Definition 1.1. (Vertical averaging). We say that admits vertical averaging ifLet$u$$u\in L_{\sigma}^{\infty}(\mathbb{R}^{3})$$\lim_{Larrow+\infty}\frac{1}{2L}\oint_{-L}^{L}u(x_{1},x_{2},x\mathrm{s})dx_{3}\equiv\overline{u}(x_{1}, x_{2})$exists almost everywhere. The vector field$\overline{u}(x_{1}, x_{2})$is called vertical average ofDefinition 1.2. (Space for initial data) We define a subspace of{$L_{\sigma,a}^{\infty}=L_{\sigma,a}^{\infty}(\mathbb{R}^{3})=$$u\in L_{\sigma}^{\infty}(\mathbb{R}^{3});u$$L_{\sigma}^{\infty}$$u,(x_{1}, x_{2}, x_{3})$.

of the form

admits vertical averaging and$u^{[perp]}\in\dot{B}_{\infty,1}^{0}$}.

is a homogeneous Besov space (see subsection 3.2 on details of its definition andis a Banach space with the norm
properties). The spaceHere$\dot{B}_{\infty,1}^{0}$$L_{\sigma,a}^{\infty}(\mathbb{R}^{3})$$||u||_{L_{\sigma_{1}a}}\infty=||\overline{u}||_{L^{\infty}(\mathbb{R}^{2};\mathbb{R}^{3})}+||u^{[perp]}||_{\dot{B}_{\infty 1\prime}^{0}(\mathbb{R}^{3};1\mathrm{R}^{3})}$.

Now we introduce theorems obtained in [12].
Theorem 1.1. (Existence and uniqueness of mild solution u). ThenSuppose that V.5$T_{0}>0$
independent(1) There existof and a unique solution $u=u(t)$ of (I) such that$\in L_{\sigma,a}^{\infty}(\mathbb{R}^{3})$$\Omega$$u\in C([\delta,T_{0}]_{\mathrm{i}}L_{\sigma}^{\infty})$(2) The solution$u$$\cap C_{w}([0,T_{0}];L_{\sigma}^{\infty})$

for any$\delta$(1.3)$>0$ .

satisfies$\sup_{t\in(0,T\mathrm{o})}||t^{1/2}\nabla u||L_{\sigma}\infty<$

oo

and

for any$\nabla u\in C([\delta,T_{0}|;L_{\sigma}^{\infty})$$\delta>0$

.

(1.4)

Remark 1.1. (i) For a lower estimate for $T_{0}>0$ we get$T_{0}\geq C/||u_{0}||_{L_{\sigma,a}^{\varpi}}^{2}$

with independent of .
, then the solution $u\in C([0,T_{0}];BUC)$ . Here,(ii) If in addition we assume that.
$BUC$ denotes the space of all bounded uniformly continuous functions inin Theorem(I)obtained
of
Then
the
solution(iii) Letbe uniformly continuous.
1.1 satisfies$C$$\Omega$$\overline{u_{0}}\in BUC$$\mathbb{R}^{3}$$u$$u_{0}\in L_{\sigma,a}^{\infty}(\mathbb{R}^{3})$$\lim_{t\downarrow 0}t^{1/2}||\nabla u(t)||_{L^{\infty}(\mathrm{J}\mathrm{R}^{3})}=0$.

Theorem 1.2. (Existence of classical solution u). Let $u=u(t)$ be a solution of (1) satisfying (L3) cvnd (1.4)
Suppose that$u_{0}\in L_{\sigma_{1}a}^{\infty},(\mathbb{R}^{3})$set

Vp(t) $= \nabla\sum_{j,k=1}^{3}RjRku^{j}u^{k}(t)-\Omega$$(\begin{array}{l}R_{1}(R_{2}u^{1}-R_{1}u^{2})R_{2}(R_{2}u^{\mathrm{l}}-R_{1}u^{2})R_{3}(R_{2}u^{1}-R_{1}u^{2})\end{array})$

then the pair ( , Vp) is a classical solution of (RNS),$u$

for $t>0$ ,

if we
(1.5)

25

Such a solution (satisfying (1.3)-(1.5)) is unique. In fa$\mathrm{c}\mathrm{t}$a stronger version is available.

Theorem 1.3. (Uniqueness of classical solution u). Let
Suppose that$u_{0}\in L_{\sigma,a}^{\infty}(\mathbb{R}^{3})$$u\in L^{\infty}((0, T)\mathrm{x}$$\mathbb{R}^{3})$,$p\in L_{loc}^{1}([0,T);BMO)$

be a solution of (RNS) in a distributional sense for some $T>0$ . Then the pair
Furthermore, the relation (1 . 5) holds.

Remark 1.2. (i) The space, where$L_{\sigma,a}^{\infty}$$(u, \nabla p)$

is unique.

has a topological direct sum decomposition af the form

$L_{\sigma,a}^{\infty}=$$\mathcal{W}\oplus B^{0}$$\mathcal{W}=${.

$f\in L_{\sigma 1}^{\infty}$$\partial f^{i}/\partial x_{3}\equiv 0$

in distributional sense$\mathcal{B}^{0}=\{f\in\dot{B}_{\infty,1}^{0}\cap L_{\sigma}^{\infty}; \overline{f}(x_{1}, x_{2})\equiv 0 \mathrm{a}.\mathrm{e}.

$\mathbb{R}^{3}$for$\mathrm{i}=1$, 2, 3},.

(x_{1}, x_{2})\in \mathbb{R}^{2}\}$(ii) Existence of vertical average of initial data is not needed for the thorem , but the following$\mathrm{s}$representation is needed:with$\phi\in \mathcal{W}$$\psi$$\in B^{0}$(1.6)$=\phi(x_{1}, x_{2})+\psi(x_{1}, x_{2}, x_{3})$, that is,
3.4).
Remark(seeand$L_{\sigma,a}^{\infty}=\mathcal{W}\oplus B^{0}$no,$\prime u_{0}$belongs to the space$\mathcal{W}+B^{0}$, which is larger thanof theThis manuscript is organized as follows. In section 2, 3 and 4, we give a brief sketch
3,
and
2
In
section[12].
inproof of the theorems for readers’ convenience although it is given
respectively.
(I),
we estimate the nonlinear term and the linear term of the integral equation
homogeneous Besov
In section 4, we introduce Mikhlin-type theorem in the Hardy space and aoperator.
solutionspace, which is crucial for uniform boundedness of the Coriolis
l.l(ii) is not
In section 5, we show Remark 1.1(ii) and (iii). In [12], detailed proof of Remark
written and the assertion (iii) is not mentioned,

2

Estimate for nonlinear term(I) using an estimate
In this section we prepare estimate for the nonlinear term of the equation
obtained by Carpio.
for derivative of the heat kernel in the Hardy space$\mathcal{H}^{1}$for t $>0$ . Then there
Lemma 2.1 ([8]). Let $G_{t}=G_{t}(x)$ be the heat kernel
exists a constant C $>0$ (depending only on space dimension n) that satisfies$(4 \pi t)^{-n/2}.\exp(\frac{-|x|^{2}}{4t})$$||\nabla G_{t}||_{H^{1}(\mathbb{R}^{\eta}\rangle}\leq Ct^{-1/2}$

for $t>0$ .

is $BMO$ , the space of functions
Since it is well known that the dual space of the Hardy spaceof bounded mean oscillations, we immediately have
only on space dimension ) that satisfies
Lemma 2.2. There exists a constant C $>0$ ($H^{1}$$depen\Lambda \mathrm{i}_{t}ng$$||\nabla e^{t\Delta}f||_{L\infty}\leq Ct^{-1/2}||f||_{BMO}$

for $t>0$ ,$f$$\in BMO$ .

28

By the above two lemmas and Corollary 3.1, which will be given later, we get the follow ing
estim tes for the nonlinear term.

Proposition 2.1. (Estimates for the nonlinear term)
There eists a constant (independent of , and ) that satisfies$C$$\Omega$$t$$f$$||\exp(-\mathrm{A}(\Omega)t)\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}(f\otimes f)||L^{\varpi}\leq Ct^{-1/2}||f||_{L^{\varpi}}^{2}$$||\nabla\exp(-\mathrm{A}(\Omega)t)\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}(f \otimes ff)||_{L^{\infty}}$,$t>0$ ,$\leq Ct^{-1/2}||\nabla f||_{L^{\varpi}}||f||_{L^{\infty}}$

and
,

$t>0$

.
will
for all
Proof. The proof is given in [12] (Lemma 4.3) using symbol calculation of the operators, however,$f\in L^{\infty}$$\nabla f\in L^{\infty}$

here we give proof again without symbol expression. For the first statement we have$||\exp(-\mathrm{A}(\Omega)t)\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}F||L\infty$$\leq||\nabla\exp(t\Delta)||_{BMOarrow L^{\infty}}||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})||_{BMOarrow BMO}||\mathrm{P}||_{BMO-BMO}||f\otimes f||_{BMO}$$\leq Ct^{-1/2}||f$(&f $||_{BMO}$$\leq Ct^{-1/2}||f\otimes$ $f||_{L}\infty\leq Ct^{-1/2}||f||_{L^{\infty}}^{2}$.

Here, in the second inequality we used Lemma 2.2, Proposition 3.2 and the boundedness of theoperator in $BMO$ since the Riesz transform is bounded in $BMO$ . In the third inequality we$BMO$ . For the second assertion one sees similarly
also used the embedding$\mathrm{P}$$L^{\infty}\mathrm{L}.+$$||\nabla\exp(-\mathrm{A}(\Omega)t)\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}F||L\infty$$\leq||\nabla\exp(t\Delta)||_{BMOarrow L}\infty||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})||_{BMOarrow BMO}||\mathrm{P}||_{BMOarrow BMO}||\mathrm{d}\mathrm{i}\mathrm{v}(f\otimes f)||_{BMO}$$\leq Ct^{-1/2}||\mathrm{d}1\mathrm{v}(f\otimes f)||_{BMO}$$\leq Ct^{-1/2}||\mathrm{d}\mathrm{i}\mathrm{v}(f\otimes f)||_{L}\infty\leq Ct^{-1/2}||\nabla f||L^{\infty}||f||_{L^{\infty}}$.
$\square$3

Estimate for linear term

In this section we show boundedness of the solution operator for the linearlized equation for
nondecreasing initial data. By virtue of skew-symmetry of the operator PJP, that we use
is reduced to boundedness of, boundedness problem of
instead of
R. By
for some
we denote the symbol of a operator .
$\omega$

3.1

$\in$

$\exp(\omega R_{3})$$\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)$$\mathrm{P}\mathrm{J}$$T$$\sigma(T)$Poincare-Sobolev equations

The linealized equations of the Rotating Navier-Stokes equations is called the Poincare-Sobolev
equations and has the form:$u_{t}-\Delta u+\Omega \mathrm{J}u+\nabla p=0$

(PS)

$\{$

$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$u|_{t=0}=u_{0}$

for
for
for

$x\in \mathbb{R}^{3}$,$x\in \mathbb{R}^{3}$, $0<t<T$ ,$x\in \mathbb{R}^{3}$.

$0<t$ $<T$ ,27Multiplying the Helmholtz operator$u_{t}-$

Au$+$$\mathrm{P}$, the equations (PS) are transformed into
for $t>0$ ,QPJu $=0$(3.1)$u|t=0=u0$ .

Instead of (3.1), as mentioned in introduction we deal rather$u_{t}-$

Au

for $t>0$ ,$+\Omega \mathrm{P}\mathrm{J}\mathrm{P}u=0$

whose solution operator is expressed by (1.2).
Before calculating the symbol of the solution operator
by$u|t=0=l\mathit{1},0$$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})$

(3.2)

)

, we define the operator$\mathrm{R}$$- \frac{\frac{\xi_{2}}{1_{\xi_{1}^{\xi 1}}}}{|\xi|,0},$$\sigma(\mathrm{R})\equiv \mathrm{R}(\xi)=(-\frac{\epsilon_{3}^{0}}{\frac{\xi_{2}|\xi|}{|\xi|}}$We note that the symbol
property$\mathrm{R}(\xi)$is a3$\mathrm{R}^{2}=-\mathrm{I}$$\mathrm{x}$$3$$- \frac{\xi_{3}}{\frac{\xi_{1}|\xi|0}{|\xi|}}$

$)$

(3.3)

.

skew-symmetric matrix. Since the operator$\mathrm{R}$has the(3.4)

in divergence free vector fields,

the vector Riesz operator. Here, I denotes the identity operator.
Simple matrix multiplication and (3.4) give the following expression of the operator PJP.
Lemma 3.1 ([4]). (Symbol of the operator PJP)
0) We have

we call$\mathrm{R}$$\sigma(\mathrm{P}\mathrm{J}\mathrm{P})=\frac{\xi_{3}}{|\xi|}(-\frac{\xi}{\xi_{\frac{}{\xi 1},2\mathrm{s}0_{1}}|\xi,|},$$- \frac{\xi \mathrm{s}}{\frac{|\xi|\xi_{1}0}{|\xi|}}$$- \frac{\xi_{2}}{\frac{\xi_{1}|\xi|}{|\xi|0}})(=\frac{\xi_{3}}{|\xi|}\mathrm{R}(\xi))$

(3.5)

.

(2) In particular, in divergence free vector fields$\sigma((\mathrm{P}\mathrm{J}\mathrm{P})^{2})=-\frac{\xi_{3}^{2}}{|\xi|^{2}}\mathrm{I}(=(\mathrm{i}\frac{\xi_{3}}{|\xi|})(i\frac{\xi_{3}}{|\xi|})\mathrm{I})$

Remark 3.1. The matrix$\sigma(\mathrm{P}\mathrm{J}\mathrm{P})$is a3$\mathrm{x}3$,$\mathrm{i}.e.$,$(\mathrm{P}\mathrm{J}\mathrm{P})^{2}=R_{3}^{2}\mathrm{I}$

.

(3.6)

skew-symmetric matrix. This fact is key in theargument of the subsection 3.3.
By (3.5) and (3.6) we can calculate the symbol of the exponential of theby$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})$operator PJP defined$= \sum_{j=0}^{\infty}\frac{(-\Omega t)^{j}}{j!}(\mathrm{P}\mathrm{J}\mathrm{P})^{j}$to getProposition 3.1 ([4]). (Symbol of the operator$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})$$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})=\cos(\frac{\xi_{3}}{|\xi|}\Omega t)\mathrm{I}-\sin(\frac{\xi_{3}}{|\xi|}\Omega t)\mathrm{R}(\xi)$) There holds

for $t>0$ .

(3.7)

28

3.2

Homogeneous Besov spaces, in, the difficulty is that the
In order to estimate the linear term.
contains the Riesz operator that is not bounded in
Coriolis solution operator
Moreover, Carpio’s estimate does not apply the linear term since it has no derivatives.
even if the Coriolis
It is possible that the “Heat-hCoriolis” solution operator is bounded in. However, the calculation of the kernel $K(x)$ (seesolution operator is not bounded inAppendix A in [12] , that is, the function $K(x)$ , defined by the identity$L^{\infty}$$\exp(t\Delta)\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)v_{0}$$L^{\infty}$$\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)$$L_{\sigma}^{\infty}$$L_{\sigma}^{\infty}$$)$$\exp(t\Delta)\exp(-\Omega \mathrm{P}\mathrm{J}\mathrm{P}t)f=F^{-1}(e^{-t|\xi|^{2}}\cos(\frac{\xi_{3}}{|\xi|}\Omega t)\mathrm{I}-e^{-t|\xi|^{2}}\sin(\frac{\xi_{3}}{|\xi|}\Omega t)\mathrm{R}(\xi))*f=:K*f$,turned out to have the asymptotic behavior
for large$K(x) \sim C\frac{1}{|x|^{3}}$

$|x|$

.

since
The corresponding integral operator cannot be viewed as a bounded operator ina characteristic function of the outside of a large ball is always mapped to oo by this operator.
, in which, the Coriolis
In this situation we are forced to restrict initial data to a subspace ofsolution operator (in particular, the Riesz transform) is bounded. We follow the idea to use a, that was first used to solve Boussinesq equations by Sawadahomogeneous Besov space[21].
Taniuchi
Before introducing the homogeneous Besov spaces, we prepare some notations. By we denotethe class of rapidly decreasing functions. The dual of , the space of tempered distributions is
be the Littlewood-Paley dyadic decomposition satisfying
denoted by . Let$L^{\infty}(\mathbb{R}^{3})$$L_{\sigma}^{\infty}$$\dot{B}_{\infty,1}^{0}$$\mathrm{S}$$\mathrm{S}$$\mathrm{S}’$$\{\phi_{j}\}_{j=-\infty}^{\infty}$$\hat{\phi_{j}}(\xi)=\overline{\phi_{0}}(2^{-j}\xi)\in C_{c}^{\infty}(\mathbb{R}^{n})$,$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\overline{\phi_{0}}\subset\{1/2<|\xi|<2\}$

Definition 3.1. (See, e.g. [5] page 146)
for
The homogeneous Besov space$B_{p,q}^{s}(\mathbb{R}^{n})$,$(\xi\neq 0)$$\sum_{j=-\infty}^{\infty}\hat{\phi_{j}}(\xi)=1$

.

(3.8)

is defined by$n\in \mathrm{N}$$\tilde{B}_{p,q}^{s}(\mathbb{R}^{n}):=\{f\in \mathcal{Z}’;||f_{\mathrm{i}}\dot{B}_{p,q}^{s}||<\infty\}$for$s\in \mathbb{R}$and $1\leq p$ ,$q\leq\infty$, where$[ \sum_{j=-\infty}^{\infty}2^{jsq}||\phi_{j}*f_{)}.L^{\mathrm{p}}(\mathbb{R}^{n})||^{q}]1/q$

if$q<\infty$,$||f||_{\dot{B}_{\mathrm{p},q}^{\mathit{8}}(\mathit{1}\mathrm{R}^{n})}:=\{$$\sup_{-\infty\leq j\leq\infty}2^{js}||\phi_{j}*f;L^{p}(\mathbb{R}^{n})||$

Hereis the topological dual space of the space.
for all multi-indices$\mathcal{Z}’$$0$$\alpha=$$\mathcal{Z}$$q=\infty$, which is defined by.

$\mathcal{Z}\equiv\{f\in \mathrm{S}$;$D^{\alpha}\hat{f}(0)=$$(\alpha_{1}, \ldots, \alpha_{n})\}$

The above definition yields that all polynomials vanish inthat$\dot{B}_{p,q}^{s}(\mathbb{R}^{n})\cong$

if{$f\in \mathrm{S}’;||f||_{\dot{B}_{p,q}^{s}(\mathbb{R}^{n})}<$

oo and$\dot{B}_{p,q}^{s}(\mathbb{R}^{n})$, however, it is well known$f= \sum_{j=-\infty}^{\infty}\phi_{j}*$

$f$

in

$\mathrm{S}’$

}

(3.9)

29

(3.10)( $s=n/P$ and $q=1$ ).
can be regarded assatisfy (3.10), the space
Since indices of our target space
and the embedding(3.9). It is known that the inclusion. [20],[21],[22].
hold. For the details and examples one can consult

if $s<n/p$ or$\dot{B}_{\mathrm{p},q}^{s}$$\dot{B}_{\infty,1}^{0}(\mathbb{R}^{n})$$\dot{B}_{\infty,1}^{0}(\mathbb{R}^{n})\mathrm{L}arrow$$\dot{B}_{\infty,1}^{0}(\mathbb{R}^{n})\subset BUC(\mathbb{R}^{n})$$\mathrm{e}.\mathrm{g}$$L^{\infty}(\mathbb{R}^{n})\mathrm{c}arrow\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n})$Uniform estimate of the Coriolis solution operator

3.3

In this subsection we show boundedness of the Coriolis solution operator
defined in the previous subsection uniformly in$BMO$ and the Besov space$t>0$ . For the purpose it is sufficient to show boundedness of the operator of the form
. In fact, noting that $\cos x=(\exp(ix)+\exp(-ix))/2$ , we see
uniformly ininand$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})$$\Omega\in \mathbb{R}$$\dot{B}_{\infty,1}^{0}$$\omega$$\exp(\omega R_{3})$$\in \mathbb{R}$$\sigma(\cos(\frac{\xi_{3}}{|\xi|}\Omega t))=\sigma((\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{s}\mathrm{s}((-ii\frac{\xi_{3}}{|\xi|}\Omega t))=\cos(-\mathrm{i}R_{3}\Omega t)=\frac{1}{2}\{\exp(\Omega tR_{3})+\exp(-\Omega tR_{3})\}$and similarlya.

$( \sin(\frac{\xi_{3}}{|\xi|}\Omega t))=\frac{1}{2\acute{\iota}}\{\exp(\Omega tR_{3})-\exp(-\Omega tR_{3})\}$

Besides, the vector Riesz operator$\mathrm{R}$appeared in the symbol (3.7) of$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})$is bounded

and $BMO$ .
inis sufficient to get unique local existence, however, we
The boundedness in the one space
for all $1\leq q\leq\infty$ as follows;
could obtain boundedness in
Proposition 3.2. (Uniform boundedness of the operator $\exp(\omega Rj)$ )Let
for $1\leq q\leq\infty$ and $BMO$ . Then there holds$\dot{B}_{\infty,1}^{0}$$\dot{B}_{\infty,1}^{0}$$\dot{B}_{\infty,q}^{0}$$X=\dot{B}_{\infty,q}^{0}$

1$\exp(\omega R_{j})f||x\leq||f||x$and $j=1,2,3$ .
Remark 3.2. (i) The uniform boundedness in $BMO$ is used in Proposition 2.1 to get uniform
is used to estimate the linear
estimate of the nonlinear term. The uniform boundedness in
term.
(i.e., $q=\infty$ ) is used in the proof of the regularity result, Remark(ii) The boundedness in
1.1 (ii) (see section 5).
Proof By spectrum mapping theorem we have for $j=1,2_{7}3$

for $f\in X$ ,$\omega$$\in \mathbb{R}$$\dot{B}_{\infty,1}^{0}$$\dot{B}_{\infty,\infty}^{0}$$||\exp(\omega R_{j})||xarrow x$$=$$\sup\{|z|;z\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\exp(\omega R_{j}))\}$$=$$\sup\{|z|;z\in\exp(-\mathrm{i}\omega \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{i}R_{\acute{J}}))\}$$=$$\sup\{|\exp(-\mathrm{i}\omega z)|;z\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{i}R_{j})\}$.

denotes the spectrum set of an operator . Now consider the resolvent operator
satisfies the assumption. Since its symbolfor, that is,of, which givesof Mikhlin-type theorem (4.1) if A is not real, it follows that

Here,

$T$

$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(T)$$(\lambda-\mathrm{i}R_{j})^{-1}$$\lambda\in \mathrm{C}$$m(\xi)=1/(\lambda+_{1\xi 1}^{\xi}[perp].)$$\mathrm{i}R_{j}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{i}R_{j})\subset \mathbb{R}$$|| \exp(\omega R_{j})||xarrow x\leq\sup\{|\exp(-\mathrm{i}\omega z)|;z\in \mathbb{R}\}=1$

since $|\exp(-\mathrm{i}\omega z)|=1$ when$\square$$z\in \mathbb{R}$.

30Corollary 3.1. Let
independent of and
(1) 1
(2) I
$\Omega$$X=\dot{B}_{\infty,q}^{0}$$t$such thatfor$1\leq q\leq$$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f||x\leq C||f||x$$\exp(-\mathrm{A}(\Omega)t)f||_{L^{\infty}}\leq C||f||_{\dot{B}_{\infty,1}^{0}}$oo and $BMO$ . There exists a constant $C>0$

for $t>0$ , $f\in X$,
for $t>0$ ,$f\in\dot{B}_{\infty,1}^{0}$.

Proof The statement (1) is obvious from Proposition 3.2 and and the argument in the beginningof this subsection. For (2) one sees from$||\exp(-\mathrm{A}(\Omega)t)f||_{L^{\infty}}$$||G_{t}||_{1}=1,\dot{B}_{\infty,1}^{0}\llcorner_{arrow L^{\infty}}$$=$$\leq$$\leq$$\leq$and (1) that$||\exp(t\Delta)\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f||_{L}\infty$$||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f||_{L^{\infty}}$$||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f||_{\dot{B}_{\infty,1}^{0}}$$C||f||_{\dot{B}_{\infty,1}^{0}}$.
$\square$Vertical average

3.4

By combining Corollary 3.1(2) and the nonlinear estimate Proposition 2.1 at least for initial
local-in-time existence of (RNS) is guaranteed with its existence
with
data
Q.
time estimate is uniform in
However, we can see the following property of the Corilois solution operator:$\mathrm{d}\mathrm{i}\mathrm{v}u_{0}=0$$u_{0}\in\dot{B}_{\infty,1}^{0}$Remark 3.3. Let

$f$

be a

$2\mathrm{D}3\mathrm{C}$(2-dimensional 3-components) vector field, that is,$f=$ $(f^{1}(x_{1},x_{2})$ , $f^{2}(x_{1},x_{2})$ , $f^{3}(x_{1},x_{2}))$ ,Then,for $t>0$ .

$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f=f$

in all elements, hence, PJP
In fact, the symbol matrix of the operator PJP, (3.5), has a$=0$vector field . Hence its
for ain all components. Then there holds
has
vectorfor a,exponential operator becomes the identity operator,
field .
, which was defined in
If we care about the structure of the operator PJP, the classintroduction; Is allowed for local-in-time existence for initial data.
$\xi_{3}$$\mathrm{P}\mathrm{J}\mathrm{P}/$$\partial/\partial_{x3}$$\mathrm{i}\mathrm{e}.$$f$$2\mathrm{D}3\mathrm{C}$$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f=f$$f$$L_{\sigma,a}^{\infty}$

Proposition 3.3. There exists a constant C $>0$ independent of 0 such that$||\exp(-\mathrm{A}(\Omega)t)f||_{L^{\mathrm{r}}}\leq C||f||_{L_{\sigma_{1}a}^{\infty}}$

Proof Since by Remark 3.3 we see for$f\in L_{\sigma,a}^{\infty}$,

for $t>0$ ,$f\in L_{\sigma,a}^{\infty}$

.

that

,

$\exp(-\mathrm{A}(\Omega)t)f=\exp(t\Delta)\overline{u_{0}}+\exp(t\Delta)\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})u_{0}^{[perp]}$$2\mathrm{D}3\mathrm{C}$31

One has$||e^{\mathrm{t}\Delta}\overline{f}+e^{t\Delta}\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f^{[perp]}||_{L^{\varpi}}$$=$$||\exp(-\mathrm{A}(\Omega)t)f||_{L^{\infty}}$$||e^{t\Delta}\overline{f}||_{L^{\mathrm{R}}}+||e^{t\Delta}\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f^{[perp]}||_{L^{\mathrm{m}}}$$\leq$$||\overline{f}||_{L^{\varpi}}+||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f^{[perp]}||_{L^{\infty}}$$\leq$

Here, we used Corollary 3.1 and$\leq$$||\overline{f}||_{L^{\infty}}+||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})f^{[perp]}||_{\dot{B}_{\infty,1}^{0}}$$\leq$$||\overline{f}||_{L^{\infty}}+C||f^{[perp]}||_{\dot{B}_{\infty_{1}1}^{0}}$$\leq$$C||f||_{L_{\sigma,a}^{\varpi}}$$||||_{L^{\infty}}\leq||$.

.

$||_{\dot{B}_{\mathrm{o}\mathrm{e},1}^{0}}$.

$\square$Remark 3.4. (i) The above proof does not require the existence of vertical averege ofrequire only the representation of as in (1.6).
(ii) Similarly, we can get the derivative estimate of the linear term

$f$

but

$f$

$||\nabla\exp(-\mathrm{A}(\Omega)t)f||_{L^{\infty}}\leq Ct^{-1/2}||f||_{L_{\sigma,a}^{\infty}}$for$f\in L_{\sigma,a}^{\infty}$,$t>0$ ,

(see Lemma 4.2 in [12]).

The estimates Proposition 3.3 and Proposition 2.1 yield Theorem 1.1 by the following iteration;,$u_{1}(t)$$=\exp(-\mathrm{A}(\Omega)t)u_{0}$$v_{j+1}(t)$$= \exp(-\mathrm{A}(\Omega)t)u_{0}-\int_{0}^{t}\exp(-\mathrm{A}(\Omega)(t-s))\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}(u_{g-1}\otimes u_{j-1})(s)ds$(3.11)$\{$

for $j\geq 1$ . Lower estimate of existence time

and$K_{j}=K_{j}(T)= \sup_{0<\mathrm{s}<T}||u_{j}(s)||_{L^{\infty}}$$T_{0}$(Remark 1.1(i)) comes from uniform estimate forfor $T>0$ .

$K_{j}’=K_{j}’(T)= \sup_{0<s<T}s^{1/2}||\nabla u_{j}(s)||_{L}\infty$$\Omega=0$
We note that Theorem 1.2 follow from Theorem 1.1 as observed in [11], where the casecase
is discussed. We also note that Theorem 1.3 can be proved along the line of [17], where the$\Omega=0$ is discussed. We won’t repeat the proofs.
$\mathrm{s}$4Mikhlin-type theorems, the space of
We introduce Mikhlin-type theorems in 3 kinds of spaces- the Hardy space$1\leqq\leq\infty$ . The
forfunctions of bounded mean oscillations $BMO$ , and the Besov spaces
space
Hardy space version theorem is applied to estimate of nonlinear term, and the Besov
spacedimension
version is for linear term. All statements in this section are valid for generalN.
$?\{^{1}$$B_{\infty,q}^{0}$$n\in$

Lemma 4.1.
Let

7.30 in [10], [16]for so me integer $k>n/2$ satisfy$)$$((1),(2):\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}$$m(\xi)\in C^{k}(\mathbb{R}^{n}\backslash \{0\})$$|D^{\alpha}m(\xi)|\leq C_{\alpha}|\xi|^{-|\alpha|}$$(\xi\neq 0)$

for all$|\alpha|=\alpha_{1}+\cdots+\alpha_{n}\leq k$

.

(4.1)

32

Then the operator defined by $T_{m}=F^{-1}mF$ is bounded
to itself,(1) from
to itself, and(2) from.
(3) from
to itself for all$H^{1}(\mathbb{R}^{n})$$BMO(\mathbb{R}^{n})$$1\leq q\leq\infty$$\dot{B}_{\infty,q}^{0}(\mathbb{R}^{n})$In [12], the statement (3) is proved by a Lemma on boundedness of convolution-type operator
B.I and Remark B. ), however, here we will give another proof of (3) in the(seecase $k=n+1$ when $n>2$ , using the following lemma by Amann [1].
$[[12];\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a}$Lemma 4.2$\mathrm{I}]$$([1];\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a}4.2(\mathrm{i}))$. Assume$s\in \mathbb{R}$, $1\leq p$ ,satisfy$\mu_{j}:=\max_{\alpha||\leq n+1_{2^{\mathrm{j}-1}}}\sup_{\leq|\xi|\leq 2^{g+1}}|\xi|^{|\alpha|}|D^{\alpha}m(\xi)|<$

Then$F^{-1}(m\hat{\phi}_{j})\in L^{1}(\mathbb{R}^{n})$$q\leq\infty$. Let

for someoo$m\in C^{n+1}(\mathbb{R}^{n}\backslash \{0\})$

(4.2)

Z.

$j\in$

and$||F^{-1}(m\hat{\phi}_{j})||_{L^{1}(\mathbb{R}^{n})}\leq C\mu_{J}$,here $C=C(n)>0$ is independent of $m$ and .
$j$

Although we deal with only the scalar-valued Besov spaces with specific indices $p=\infty$ , $q\in[1, \infty]$, Amann [I] proved Mikhlin-type theorem in the vector-valued Besov
and $s=0$ , that is,. Here, $E$ is a Banach space without any restriction such as UMD nor HT
spaces, $1\leq p_{1}q\leq\infty$ . Though he mentions only the inhomogeneous
spaces (see also [2], [9]), and.
Besov spaces, his proof can be adapted to the homogeneous Besov spaces$\dot{B}_{\infty,q}^{0}$$B_{p,q}^{s}(\mathbb{R}^{n}, E)$$s\in \mathbb{R}$$\dot{B}_{p,q}^{s}(\mathbb{R}^{n}, E)$

Proof of Lemma 4.1(3): For$f\in\dot{B}_{\infty,q}^{0}$$\phi_{j}*(F^{-1}m)$ $*f=(F^{-1}(m\hat{\phi}_{j}))*f$with $1\leq q\leq\infty$ it follows from $\phi_{j}*(F^{-1}mFf)$

$=$

that

$||F^{-1}mFf||_{\dot{B}_{\infty q}^{0}}’=$$( \sum_{j\in \mathbb{Z}} \mathrm{I}(F^{-1}(m\hat{\phi}_{j}))*f||_{L^{\infty}}^{q})^{1/q}$.

By (3.9) and Young’s inequality we get$||F^{-1}mFf||_{\dot{B}_{\infty q}^{0}}‘=( \sum_{j,k\in \mathbb{Z},|j-k|\leq 2}||(F^{-1}(m\hat{\phi}_{j}))*f*\phi_{k}||_{L^{\infty}}^{q})^{1/q}$$\leq(\sum_{j,k\in \mathbb{Z},|j-k|\leq 2}||F^{-1}(m\hat{\phi}_{j})||_{L^{1}}^{q}||f*\phi_{k}||_{L^{\infty}}^{q})^{1/q}\leq C(\sum_{j,k\in \mathbb{Z},|j-k|\leq 2}\mu_{j}^{q}||f*\phi_{k}||_{L^{\infty}}^{q})^{1/q}$.

Since the assumption (4.1) (the case $k=n+1$ ) yields that$\sup_{j\in \mathbb{Z}}\mu_{j}\leq\max_{\alpha||\leq n+1}\sup_{\xi\in \mathrm{R}^{n}\backslash \{0\}}|\xi|^{|\alpha|}|D^{\alpha}m(\xi)|\leq C$

for some $C>0$ independent of , one sees$j$$||F^{-1}mFf||_{\dot{B}_{\varpi,q}^{0}} \leq C(\sup_{j\in \mathbb{Z}}\mu_{j})(\sum_{k\in \mathbb{Z}}||f*\phi_{k}||_{L^{\infty}}^{q})^{1/q}\leq C||f||_{\dot{B}_{\infty,q}^{0}}$.
$\square$33

Regularity of mild solution

5

In this section we prove Remark l.l(ii) and (iii). All lemmas in this section hold for general
although the Remark 1.1 is valid only for $n=3$ .
space dimension
Lemma 5.1. There exists a constant C $>0$ independent of f and g such that$n\in \mathrm{N}$$||f*g||_{\dot{B}_{\infty,1}^{0}\{\mathbb{R}^{n})}\leq C||f||_{\dot{B}_{1.1}^{0}(\mathbb{R}^{n})}||g||_{\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n})}$

.
and
for
Proof. By Young’s inequality we have$g\in\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n})$$f\in\dot{B}_{1,1}^{0}(\mathbb{R}^{n})$$||f*g||_{\dot{B}_{\infty,1}^{0}}$$\leq$$\sum_{\acute{J}\in \mathbb{Z}}||\phi_{j}*(f*g)||_{L^{\mathrm{r}}}\leq\sum_{j,k\in \mathbb{Z}}||\phi_{j}*(f*g)*\phi_{k}||_{L^{\varpi}}$$\leq$$\sum_{j,k\in \mathbb{Z},|j-k|\leq 2}||\phi_{j}*f||_{L^{1}}||g*\phi_{k}||_{L}\infty$3$\leq$$\sup_{k\in \mathbb{Z}}||g*\phi_{k}||_{L\infty}\sum_{j\in \mathbb{Z}}||\phi_{j}*f||_{L^{1}}\leq 3||g||_{\dot{B}_{\infty,\infty}^{0}}||f||_{\dot{B}_{1,1}^{0}}$$\square$Lemma 5.2. Let(1)$G_{t}$

be the heat kernel $(4\pi f)^{-n/2}$$||\nabla G_{t}(x)||_{\dot{B}_{1,1}^{0}(\mathbb{R}^{n})}\leq Ct^{-1/2}$(2).

$||\nabla e^{t\Delta}f||_{\dot{B}_{\infty\rangle 1}^{0}(\mathbb{R}^{n}\rangle}\leq Ct^{-1/2}||f||_{\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n}j}$

Proof. (1) Since$\phi_{j}(x)=2^{jn}\phi_{0}(^{\underline{q}j}x)$for$f\in\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n})$.

, we see(5.1).

$2^{j}||2^{jn}(\nabla\phi_{0})(2^{j}\cdot)||_{1}||G_{t}||_{1}\leq 2^{j}||\nabla\phi_{0}||_{1}||G_{t}||_{1}$$\leq$On the other hand, we get by the mean value theorem and$(\phi_{j}*\nabla G_{t})(x)$for t $>0$ . Then$|| \nabla(\phi_{j})*G_{t}||_{1}=2^{j}||\int_{\mathbb{R}^{n}}|2^{jn}(\nabla\phi_{0})(2^{j}y)G_{t}(x-y)|dy||_{1}$$=$$||\phi_{j}*\nabla G_{t}||_{1}$$\exp(\frac{-|x|^{2}}{4t})$$\int\phi_{0}(z)dz=0$$=$$\int_{\mathbb{R}^{n}}\phi_{j}(y)(\nabla G_{t})(x-y)dy$$=$$\int_{1\mathrm{R}^{n}}2^{jn}\phi_{0}(2^{j}y)(\nabla G_{t})(x-y)dy=\int_{\mathbb{R}^{n}}\phi_{0}(z)(\nabla G_{t})(x-2^{-\mathrm{j}}z)dz$$=$$\oint_{\mathbb{R}^{n}}\phi \mathrm{o}(z)\{(\nabla G_{t})(x-2^{-j}z)-(\nabla G_{t})(x)\}dz$$=$.

$\oint_{\mathbb{R}^{n}}\phi_{0}(z)2^{-j}z(\int_{0}^{1}(\nabla^{2}G_{t})(x-\theta 2^{-j}z)d\theta)dz$

Hence,$||\phi_{j}*\nabla G_{t}||_{1}$$\leq$$\leq$$2^{-j} \int_{\mathbb{R}^{n}}|\phi_{0}(z)z\int_{0}^{1}(\nabla^{2}G_{t})(x-\theta 2^{-j}z)d\theta|dz$$2^{-j} \int_{\mathbb{R}^{n}}|\phi_{0}(z)||z|||\nabla^{2}G_{t}||_{1}dz\leq 2^{-j}||\phi_{0}(z)|z|||_{1}||\nabla^{2}G_{t}||_{1}$

.

(5.2)

34

Putting$C_{0}=||\nabla\phi_{0}||_{1}$, $C_{1}=||\phi_{0}(z)|z|||_{1}$ , the inequalities (5.1), (5.2) and $||G_{t}||_{1}=1$ yield$||\phi_{j}*\nabla G_{t}||_{1}\leq\{\begin{array}{l}C_{0}\underline{.]}jC_{2}2^{-j}t^{-1}\end{array}$Here,$C_{2}=C_{1}||\nabla^{2}G_{t}||_{1}t$

is indepencent of . Thus we get for any$N\in \mathbb{Z}$$t$$||\nabla G_{t}(x)||_{\dot{B}_{1,1}^{0}(\mathbb{R}^{n})}$$=$$\sum_{j=-\infty}^{\infty}||\phi_{j}*\nabla G_{t}(x)||_{1}=(\sum_{j=-\infty}^{N}+\sum_{j=N}^{\infty})||\phi_{j}*\nabla G_{t}(x)||_{1}$$\leq$$C_{0} \sum_{j=-\infty}^{N}2^{j}+C_{2}t^{-1}\sum_{j=N}^{\infty}2^{-j}=C_{0}2^{N+1}+C_{2}^{\underline{\eta}-N}t^{-1}$.

such that $(C_{2}/C)t^{-1/2}\leq 2^{N}\leq(1/2C_{0})t^{-1/2}$ , we derive the result.
Taking(2) This is a direct consequence of (1) and Lemma 5.1.
$N\in \mathbb{Z}$$>0$ whereis the heat kernel
for small
since
and
2,2.
by Lemma
It is easy from the second inequality of Proposition

Proof of Remark 1.1(iii): Let$(4 \pi\eta)^{-3/2}\exp(\frac{-|x|^{2}}{4\eta})$ ,Then$v_{0}^{\eta},=G_{\eta}*u_{0}$$u_{0}^{\eta}\in L^{\infty}$and2.1 to see the nonlinear term$||\nabla u_{0}^{\eta}||_{\infty}\leq C\eta^{-1/2}||u_{0}||_{\infty}$$\square$$G_{\eta}$$\eta$$||u_{0}^{\eta}||_{\infty}\leq||G_{\eta}||_{1}||u_{0}||_{\infty}\leq||8\mathit{1}_{l}0||_{\infty}$$\nabla u_{0}^{\eta}\in\dot{B}_{\infty,1}^{0}$$t^{1/2} \int_{0}^{t}||\nabla\exp(-\mathrm{A}(\Omega)(t-s))\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}(u\otimes u)(s)||_{\infty}ds$

tends to 0 as$t\downarrow \mathrm{O}$. On the linear term we get by Lemma 2.2 and Corollary 3.1 that$t^{1/2}||\nabla\exp(-\mathrm{A}(\Omega)(t))u_{0}||_{\infty}$$\leq$

By$t^{1/2}$($||\nabla\exp(t\Delta)\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})(u_{0}-u_{0}^{\eta})||_{\infty}+||\nabla\exp(t\Delta)$

exp($-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})u_{0}^{\eta}||_{\infty}$)$\leq$$Ct^{1/2}t^{-1/2}||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})(u_{0}-u_{0}^{\eta})||_{BMO}+t^{1/2}||\exp(t\Delta)\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})\nabla u_{0}^{\eta}||_{\infty}$$\leq$$C||u_{0}-u_{0}^{\eta}||BMO+t^{1/2}||\exp(-\Omega 8\mathrm{P}\mathrm{J}\mathrm{P})\nabla u_{0}^{\eta}||_{\infty}$$||\cdot||_{BMO}\leq||\cdot||L\infty\leq||\cdot||_{B_{\infty 1}^{0}}\rangle$.

and uniform boundedness of$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})$(Corollary 3.1)

we estimate$t^{1/2}||\nabla\exp(-\mathrm{A}(\Omega)(t))_{\mathfrak{U}}||_{\infty}$$\leq$$C||u_{\{\}}-u_{0}^{\eta}||_{\infty}+t^{1/2}||\exp$($-\Omega t$

PJP )$\nabla u_{0}^{\eta}||_{\dot{B}_{\varpi,1}^{0}}$$\leq$$C||v_{10}-u_{0}^{\eta}||_{\infty}+Ct^{1/2}||\nabla u_{0}^{\eta}||_{\dot{B}_{\mathrm{m},1}^{0}}$$\leq$$C||u_{0}-u_{0}^{\eta}||_{\infty}+Ct^{1/2}\eta^{-1/2}||u_{0}||_{\dot{B}_{\varpi_{\gamma}\infty}^{0}}$,. Then the first term in the RHS
where we used Lemma 5.2(2). After taking $77=t^{1/2}$ , sendsince
is assumed to be uniformly continuous (see Lemma 5 in [11]). The
second term also tends to 0 since
is finite.
$\eta\downarrow 0$$C||u_{0}-u_{0}^{\eta}||_{\infty}arrow 0$$u_{0}$$\square$$||u_{0}||_{B_{\infty,\infty}^{0}}\leq||u_{0}||L\infty\leq||u_{0}||L_{\sigma,a}\infty$

35

Lemma 5.3. Let $0<\alpha\leq 1$ . Then there exists a constant $C_{\alpha}=C(\alpha)>0$ such that(1)(2)$||(-\Delta)^{\alpha}G_{t}||_{\dot{B}_{1,1}^{0}(\mathbb{R}^{n})}\leq C_{\alpha}t^{-\alpha}$

for $t>0$ ,for $t>0$ ,for $s>t>0$ ,$f\in\dot{B}_{\infty\prime\infty}^{0}(\mathbb{R}^{n})_{t}$$||(-\Delta)^{\alpha}\exp(t\Delta)f||_{\dot{B}_{\infty,1}^{0}(\mathbb{R}^{n})}\leq C_{\alpha}t^{-\alpha}||f||_{\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n})}$

(3) I
(4)

far $t>0$ ,$f\in D((-\Delta)^{\alpha})$$(\exp(t\Delta)-\mathrm{I})f||_{\dot{B}_{\infty,1}^{0}(\mathbb{R}^{n})}\leq C_{\alpha}t^{-\alpha}||(-\Delta)^{\alpha}f||_{\dot{B}_{\infty\prime\infty}^{0}(\mathbb{R}^{n})}$,$f\in\dot{B}_{\infty}^{0}\prime 1,(\mathbb{R}^{n})$$||(\exp(s\Delta)-\exp(t\Delta))f||_{\dot{B}_{\infty,1}^{0}(\mathbb{R}^{n})}\leq C_{\alpha}(s-t)^{\alpha}t^{-\alpha}||f||_{\dot{B}_{\infty\infty\prime}^{0}(\mathbb{R}^{n}\rangle}$Fere,$D((-\Delta)^{\alpha})=\{f\in\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n})\mathrm{i}(-\Delta)^{\alpha}f\in\dot{B}_{\infty,\infty}^{0}(\mathbb{R}^{n})\}$

Remark 5.1. By$\dot{B}_{\infty,1}^{0}\mathrm{c}arrow L^{\infty}\mathrm{c}arrow\dot{B}_{\infty,\infty}^{0}$

we$\mathrm{i}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\downarrow \mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}$$||(-\Delta)^{\alpha}\exp(t\Delta)f||_{\dot{B}_{\infty,\infty}^{0}}\leq C_{\alpha}t^{-\alpha}||f||_{\dot{B}_{\infty,\infty}^{0}}$.

.

see by (2)for $t>0$ ,$f\in\dot{B}_{\infty,\infty}^{0}$

(5.3)

.

Proof. The inequality (1) shall be proved in Appendix. The assertion (2) immediately follows

from (1) and Lemma 5.1. For (3) we see for $f\in D((-\Delta)^{\alpha})$ that$( \exp(t\Delta)-\mathrm{I})f=-\int_{0}^{t}(-\Delta)\exp(s\Delta)fds=-\int_{0}^{t}(-\Delta)^{1-\alpha}\exp(s\Delta)(-\Delta)^{\alpha}fds$.

Then by (2)$||(\exp(t\Delta)-\mathrm{I})f||_{\dot{B}_{\infty,1}^{0}}$$\leq$$\leq$For (4) let$I_{0}^{t}||(-\Delta)^{1-\alpha}\exp(s\Delta)||_{B_{\infty\infty}^{0}arrow\dot{B}_{\infty,1}^{0}}’||(-\Delta)^{\alpha}f||_{\dot{B}_{\infty.\infty}^{0}}ds$$C_{1-\alpha} \int_{0}^{t}s^{\alpha-1}d,s||(-\Delta)^{\alpha}f||_{\dot{B}_{\infty,\infty}^{0}}\leq C_{1-\alpha}\frac{1}{\alpha}t^{\alpha}||(-\Delta)^{\alpha}f||_{\dot{B}_{\infty,\infty}^{0}}$.

, hence. Then $\exp(t\Delta)f\in D((-\Delta)^{\alpha})$ for $t>0$ . In fact,
from.
follows
$t>0$
It
for
. So, (2) implies$f\in\dot{B}_{\infty,1}^{0}\subset L^{\infty}$$f\in\dot{B}_{\infty,1}^{0}$$(-\Delta)^{\alpha}\exp(t\Delta)f\in\dot{B}_{\infty,1}^{0}$$\exp(t\Delta)f\in L^{\infty}\subseteq\dot{B}_{\infty,\infty}^{0}$(5.3) and (3) that$||(\exp(s\Delta)-\exp(t\Delta))f||_{\dot{B}_{\infty,1}^{0}}$$=||(\exp((s-t)\Delta)-\mathrm{I})(-\Delta)^{-\alpha}||_{\dot{B}_{\infty,\infty}^{0}arrow\dot{B}_{\infty,1}^{0}}||(-\Delta)^{\alpha}\exp(t\Delta)f||_{\dot{B}_{\infty,\infty}^{0}}$$\leq C_{\alpha}(s-t)^{\alpha}t^{-\alpha}||f||_{\dot{B}_{\infty,\infty}^{0}}$.

Cl
be sequence of the succesive iteration (3.11). By the asis an analytic semigroup in $BUC$
since
we seesurnptionyields, hence(see e.g. Proposition A.l.l of [11]). On the other hand,by Corollary 3.1(1). Then,in $BUC$ a.gain. Thus
thanks to the semigroupbelongs to $BUC$ .
is uniformly continuous$BUC$ for all $j\geq 2$ . Since it is known that
Next we show), it is sufficient to show(see e.g. Lemma 5 in
if and only if $||\exp(\delta\Delta)f-f||_{L\infty}arrow 0$ as

Proof of Remark 1,1 (ii): Let$\overline{u_{0}}\in BUC$$\{u_{j}\}$$\exp(t\Delta)\overline{u_{0}}\in BUC$$\exp(t\Delta)$$u_{0}\in L_{\sigma,a}^{\infty}$$u_{0}^{[perp]}\in\dot{B}_{\infty,1}^{0}$$\exp(t\Delta)\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})u_{0}^{[perp]}\in BUC$$\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})u_{0}^{[perp]}\in\dot{B}_{\infty,1}^{0}\subset BUC$

$u_{1}=\exp(t\Delta)\overline{u\prime 0}+\exp(t\Delta)\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})v_{0}^{[perp]}$$\exp(t\Delta)$$f\in L^{\infty}$$u_{j}\in$$\delta$$\downarrow 0$$\lfloor 11]$

36

that

$vlj$

$\in L^{\infty}$

satisfies

as$||\exp(\delta\Delta)u_{j}-u_{j}||_{L^{\infty}}arrow 0$$6\downarrow 0$. We have for fixed $t\in(0,T_{0}]$ and any$\delta>0$$||\exp(\delta\Delta)u_{j}-u_{j}||_{L}\infty$$\leq||\exp(\delta\Delta)\exp(-\mathrm{A}(\Omega)t)\prime u_{0}.-\exp(-\mathrm{A}(\Omega)t)u_{0}||_{\dot{B}_{\infty.1}^{0}}$$+ \oint_{0}^{t}||(\exp(\delta\Delta)\exp(-\mathrm{A}(\Omega)(t-s))-\exp(-\mathrm{A}(\Omega)(t-s)))\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}(u_{j-1}t\otimes u_{j-1})(s)||_{L^{\infty}}$(is$\leq||\exp((t+\delta)\Delta)-\exp(t\Delta)||_{\dot{B}_{\infty,\infty}^{0}arrow\dot{B}_{\infty,1}^{0}}||\exp(-\Omega t\mathrm{P}\mathrm{J}\mathrm{P})||_{\dot{B}_{\infty,\infty}^{0}arrow\dot{B}_{\infty_{1}\infty}^{0}}||u_{\{)}||_{\dot{B}_{\infty,\infty}^{0}}$$+ \int_{0}^{t}||\nabla\cdot\exp(\frac{t-s}{2}\Delta)||_{\dot{B}_{\infty,\infty}^{0}arrow L^{\infty}}||\exp(-\Omega(t-s)\mathrm{P}\mathrm{J}\mathrm{P})||_{\dot{B}_{\infty,\infty}^{0}arrow\dot{B}_{\infty_{1}\infty}^{0}}$

1

$t-s$

$t-s$

$\exp((+\delta)\Delta)-\exp(\Delta)||_{\dot{B}_{\infty\infty}^{0}arrow B_{\infty,1}^{0}}.||u_{j-1}\otimes u_{j-1}||_{\dot{B}_{\infty,\infty}^{0}}(s)ds\overline{2}\overline{2}$’$\leq C_{\alpha}\delta^{\alpha}t^{-\alpha}||u_{0}||_{\dot{B}_{\infty\infty}^{0}}+C_{\alpha}\delta^{\alpha}’\int_{0}^{t}(t-s)^{-\alpha-\frac{1}{2}}||u_{j},\otimes u_{j}||_{L^{\infty}}(s)ds$. Choosewith all $0<ce$ $<1$ . Here, we used Lemma 5.3(4), Proposition 3.2 andfor$0<\alpha<1/2$ and send
to see RHS tends to 0, noting that$j\geq1$, which implies that its uniform
a1J $0\leq s\leq t$ . Thus we have proved that $u_{j}\in BUC$ for all
$u\in
BUC$
.
limit
$\dot{B}_{\infty,1}^{0}<arrow L^{\infty}$$\delta\downarrow 0$$||u_{j}\otimes u_{j}||L\infty(s)\leq||u_{j}||_{L^{\infty}}^{2}(s)$$\square$A

Appendix: Estimate for fractional power of Laplacian of theheat kernel

In this appendix we shall prove Lemma 5.3(1).

Proof of Lemma 5.3(1): Setting $x=t^{1/2}z$ , it is easy to see that$((-\Delta)^{\alpha}G_{t})(x)=t^{-\frac{n}{2}-\alpha}((-\Delta)^{\alpha}G_{1})(z)$

for

(A.I)

$t>0$ .

Hence, it is sufficient to show only the case $t=1$ . In fact, by scaling invariancefor
we get$\lambda^{-n}||f||_{\dot{B}_{1,1}^{0}(\mathbb{R}^{n})}$$||f(\lambda\cdot)||_{\dot{B}_{1_{1}1}^{0}(\mathbb{R}^{n})}\approx$$\lambda>0$.

$||((-\Delta)^{\alpha}G_{t})(x)||_{\dot{B}_{1,1}^{0}}=t^{-\frac{n}{2}-\alpha}||((-\Delta)^{\alpha}G_{1})(t^{-1/2}x)||_{\dot{B}_{1_{\}}1}^{0}}\leq Ct^{-\frac{n}{2}-\alpha}t^{\frac{n}{2}}||(-\Delta)^{\alpha}G_{1}||_{\dot{B}_{1,1}^{0}}\leq C_{\alpha}t^{-\alpha}$

For any fixed$j\in \mathbb{Z}$one sees that$\phi_{j}*(-\Delta)^{\alpha}G_{1}=\phi_{j}*(F^{-1}(|\xi|^{2\alpha}\overline{G_{1}}))=F^{-1}(\hat{\phi_{j}}|\xi|^{2\alpha}\overline{G_{1}})$

.

37

Since

$\hat{\phi_{j}}(\xi)=\overline{\phi_{0}}(2^{-j}\xi)$

we continue$\phi_{j}*(-\Delta)^{\alpha}G_{1}$$=$$\int e^{\dot{\tau}x\cdot\xi}\overline{\phi_{0}}(2^{-j}\xi)|\xi|^{2\alpha}\overline{G_{1}}(\xi)d\xi$$=$$\int e^{ix\cdot 2^{\mathrm{J}}\xi}\overline{\phi_{0}}(\xi)|2^{j}\xi|^{2\alpha}\overline{G_{1}}(2^{j}\xi)2^{jn}d\xi$$=$$2^{jn+j2\alpha} \int e^{i2^{j}x\cdot\xi}|\xi|^{2\alpha}\overline{\phi_{0}}(\xi)\overline{G_{1}}(2^{j}\xi)d\xi$$2^{jn+j2\alpha}[F^{-1}(|\xi|^{2\alpha}\overline{\phi_{0}}(\xi)\overline{G_{1}}(2^{j}\xi))](2^{j}x)$$=$$2^{jn+j2\alpha}[F^{-1}(|\xi|^{2\alpha}\overline{\phi_{0}}(\xi))*F^{-1}(\overline{G_{1}}(2^{j}\xi))](_{A}^{q^{j}}x)$$=$.

It follows from$F^{-1}( \overline{G_{1}}(2^{\acute{J}}\xi))=F^{-1}(\frac{1}{2^{jn}}[F(G_{1}(\frac{x}{\eta_{\mathrm{r}}j}))](\xi))=\frac{1}{2^{jn}}G_{1}(\frac{x}{2^{j}})$and$F^{-1}(|\xi|^{2\alpha}\overline{\phi_{0}}(\xi))=(-\Delta)^{\alpha}\phi_{0}$that$\phi_{j}*(-\Delta)^{\alpha}G_{1}=2^{j2\alpha}[(-\Delta)^{\alpha}\phi_{0}*G_{1}(.)](^{\underline{9}^{j}}x)\overline{2^{j}}$

(A.2)

.

Hence Young’s inequality yields$||\phi_{j}*(-\Delta)^{\alpha}G_{1}||_{1}$$=$$2^{j2\alpha}||[(-\Delta)^{\alpha}\phi_{0}*G_{1}(.)\}(2^{j}x)||_{1}\overline{2^{j}}$$=$$2^{j2\alpha-jn}||[(-\Delta)^{\alpha}\phi_{0}*G_{1}(.)](x)||_{1}\overline{2^{j}}$$\leq$$2^{j2\alpha-jn}||(-\Delta)^{\alpha}\phi_{0}||_{1}||G_{1}(.)||_{1}\overline{2^{j}}$$=$$\leq$

Here we used $||G_{1}||_{1}=1$ andtothe other hand we shift$2^{j2\alpha}||(-\Delta)^{\alpha}\phi 0||_{1}||G_{1}||_{1}$$C_{\alpha}2^{j2\alpha}$$||(-\Delta)^{\alpha}\phi_{0}||_{1}=||F^{-1}(|\xi|^{2\alpha}\overline{\phi 0})||_{1}\leq C_{\alpha}$$(-\Delta)^{\alpha}$$\phi_{j}*(-\Delta)^{\alpha}G_{1}$$G_{1}(_{\overline{2J}}.)$

$=$

$=$

Here we put
that$(-\Delta)^{\beta-\beta}=\mathrm{I}$

(A.3)

.

because$|\xi|^{2\alpha}\overline{\phi 0}\in \mathrm{S}$. On

in RHS of (A.2) to get$2^{j2\alpha}[\phi_{0}*(-\Delta)^{a}G_{1}(_{\overline{\eta j\sim}})](2^{j}x)$.

$2^{j2\alpha}[((-\Delta)^{-\beta}\phi_{0})*(-\Delta)^{\alpha+\beta}G_{1}()](2^{j}x)\overline{2^{j}}$with some $\beta>0$ . Then we get by Young’s inequality for any fixed$j\in \mathrm{N}$$||\phi_{j}*(-\Delta)^{\alpha}G_{1}||_{1}$

$=$

$=$

$\leq$$2^{\acute{J}^{2\alpha}}||[((-\Delta)^{-\beta}\phi_{0})*(-\Delta)^{\alpha+\beta}G_{1}\mathrm{t}_{\overline{2^{j}}}.)](2^{j}x)||_{1}$$2^{j2\alpha-jn}||[((-\Delta)^{-\beta}\phi_{0})*(-\Delta)^{\alpha+\beta}G_{1}(.)](x)||_{1}\overline{2^{j}}$$2^{j2\alpha-jn}||(-\Delta)^{-\beta}\phi_{0}||_{1}||(-\Delta)^{\alpha+\beta}(G_{1}\mathrm{t}_{\overline{2^{j}}}.))||_{1}$.

38

Noting that$(-\Delta)^{\gamma}(G_{1}$$||F^{-1}(|\xi|^{-2\beta}\overline{\phi_{0}})||_{1}\leq C_{\beta}$(:)
for some$)$$=a^{-2\gamma}((-\Delta)^{\gamma}G_{1})(_{\overline{a}}.)$$||\phi_{j}*(-\Delta)^{\alpha}G_{1}||_{1}$$C_{\beta}>0$

becausefor $a>0$ , $\gamma>0$ , andwe continue$||(-\Delta)$ $-\beta\phi_{0}||_{1}=$$|\xi|^{-2\beta}\overline{\phi_{0}}\in \mathrm{S}$$\leq$$C_{\beta}2^{j2\alpha-jn}|| \underline{9}^{-2(\alpha+\beta)j}((-\Delta)^{\alpha+\beta}G_{1})(\frac{x}{\eta_{\sim}j})||_{1}$$\leq$$C_{\beta}2^{j2\alpha-jn}2_{\sim}^{-2(\alpha+\beta)j_{9}jn}||((- \Delta)^{\alpha+\beta}G_{1})(\frac{x}{2^{j}})||_{1}$$=$$C_{\beta}2^{-2\beta j}||((-\Delta)^{\alpha+\beta}G_{1})(x)||_{1}$.

Because$||(-\Delta)^{\gamma}G_{1}||_{1}\leq C_{\gamma}$

for$\gamma>0$

we get$||\phi_{j}*(-\Delta)^{\alpha}G_{1}||_{1}\leq C_{\alpha,\beta}2^{-2\beta j}$

(A.4)

.

Fix $\beta>0$ to get from (A.3) and (A.4) that.

$||(- \Delta)^{\alpha}G_{t}(x)||_{\dot{B}_{1,1}^{0}(\mathrm{J}\mathrm{R}^{n})}\leq\sum_{j\leq 0}C_{\alpha}2^{j2\alpha}+\sum_{j>0}C_{\alpha}2^{-j2\beta}\leq C_{\alpha}$

口Acknowledgements
The author would like to thank Professor Yoshikazu Giga, Professor Alex Mahalov and Professor
Shin’ya Matsui for their encouragement. The author also would like to thank Dr. Juergen Saal,
Dr. Kazuyuki Yamauchi, Dr. Jun Kato, Dr. Okihiro Sawada and Mr. Yutaka Terasawa for theuseful discussions.

References[1] H. Amann (1997), Operator-Valued Fourier Multipliers, Vector-Valued Besov Spaces, andApplications, Math. Nachr., 186, p. 5-56.
[2] H. Amann [2003], Vector-Valued Distributions and Fourier Multipliers,
http://www.math.unizh.ch/amann/books.htm1.
[3] A. Babin, A. Mahalov and B. Nicolaenko (2001), 3D Navier-Stokes and Euler Equations
with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J,, 50, p.

1-35.
[4] A. Babin, A. Mahalov and B. Nicolaenko (1997), Regularity and integrability of the 3D
Euler and Navier-Stokes Equations for uniformly rotating fluids, Asympt. Anal, 15, No. 2,

p. 103-150.

[5] Joran Bergh and Jorgen Lofstrom [1976], Interpolation Spaces An Introduction, SpringerVerlag.
[6] J.R. Cannon and G.H.Knightly (1970), A note on the Cauchy problem for the Navier-Stokes
equations, SIAM J. A ppl. Math., 18, p. 641-644

39[7] M. Cannone (1995), Ondelettes, Praproduits et Navier-Stokes, Diderot Editeur, Arts etSciences Paris-New York-Amsterdam.

[8] A. Carpio (1996), Large-time behaviour in incompressible Navier-Stokes equations, SIAM
J. Math Anal, 27, P. 449-475.
[9] R. Denk, M. Hieber and J. Priiss [2003] , -Boundedness, Fourier Multipliers and Problemsof Elliptic and Parabolic Type, Memoirs of the American Mathematical Society, Vol. 166,
Num. 788.
[10] J. Garcia-Cuerva and J.L. Rubio De Francia (1985), Weighted Norm Inequalities and Re$\mathcal{R}$lated Topics, North-Holland.

[11] Y. Giga, K. Inui and S. Matsui (1999), On the Cauchy problem for the Navier-Stokes
equations with nondecaying initial data, Quaderni di Matematika, 4, P. 28-68.
[12] Y. Giga, K. Inui, A. Mahalov and S. Matsui, Navier-Stokes Equations in a Rotating Frame
with Initial Data Nondecreasing at Infinity, submitted for publication in Hokkaido
inMath. J.
[13] Y. Giga, S. Matsui and 0. Sawada (2001), Global existence of two-dimensional NavierStokes flow with nondecaying initial velocity, J. Math. Fluid Mech., 3, p. 302-315.
$\mathbb{R}^{3}$[14] Y. Giga, K. Inui, J. Kato and S. Matsui (2001), Remarks on the uniqueness ofsolutions of the Navier-Stokes equations, Nonlinear Analysis, 47, p. 4151-4156.
[15] M. Hieber and 0. Sawada, The Navier-Stokes equations in
data, Arch. Rational Meek Anal, to appear.

$\mathbb{R}^{n}$boundedwith linearly growing initial[16] J. Hounie (2003), A note on the Mihlin-Hormander multiplier theorem, preprint.
equations,[17| J. Kato (2003), The uniqueness of nondecaying solutions for the Navier-Stokes
Arch. Rational Meek Anal, 169, p. 159-175.

[18] H. Koch and D. Tataru (2001), Well-posedness for the Navier-Stokes
in Mathematics, 157, p. 22-35.

equations, Advances

Equations[19] A. Mahalov and B. Nicolaenko (2003), Global Regularity of the 3D Navier-Stokes
2 (350), p.
with Weakly Aligned Large Initial Vorticity, Russian Math. Surveys, 58, No.

287-318.
whole space,[20] 0. Sawada, The Navier-Stokes flow with linearly growing initial velocity in the
Bol Soc. Paran. Mat, to appear.
data, Funkcial.
[21] 0. Sawada and Y. Taniuchi, On Boussinesq flow with non-decaying initial
Ekvac., to appear.
with non-decaying[22] Y. Taniuchi, Remarks on global solvability of 2-D Boussinsq equations
initial data, preprint.