数理解析研究所講究録
1146 巻 2000 年 39-52

39

Bilinear estimates and critical Sobolev inequality in $BMO$ ,with applications to the Navier-Stokes and the Euler equations
Hideo Kozono (小–英雄)
Mathematical InstituteTohoku University
Sendai 980-8578 JAPANYasushi Taniuchi (谷内靖)
Graduate School of Mathematics
Nagoya University
Nagoya 464-8602 JAPANIntroduction.
In this paper we prove that the $BMO$ norm of the velocity and the vorticity controls
the blow-up phenomena of smooth solutions to the Navier-Stokes and the Euler equations.
Our result is applied to the criterion on regularity of weak solutions to the Navier-Stokes
equations.
$n\geq 3$ :We consider the Navier-Stokes and the Euler equations in$R^{n},$(N-S)$\frac{\partial u}{\partial t}-\Delta u+u\cdot\nabla u+\nabla p=0$$\{$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$

in $x\in R^{n},$ $t>0$ ,$u|_{t=0}=a$ ,$\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p=0$

(E)

,

,

$\mathrm{d}\mathrm{i}\mathrm{v}u=0$

in $x\in R^{n},$ $t>0$ ,$\{$$u|_{t=0}=a$$u^{n}(x, t))$ and $p=p(X,t)$ denote the unknown velocity vector
where $u=(u^{1}(X, t),$ $u^{2}(x,t),$and the unknown pressure of the fluid at the point $(x, t)\in R^{n}\cross(0, \infty)$ , respectively, while$a=(a^{1}(X), a(2x),$$a^{n}(X))$ is the given initial velocity vector.
It is proved by Fujita-Kato [10] that for every
with$s>n/2-1$ , there exist $T>0$ and a unique solution $u(t)$ of (N-S) on $[0,T)$ in the class$\cdots,$$\cdots,$$a\in H_{\sigma}^{s}\equiv\{v\in H^{s}; \mathrm{d}\mathrm{i}_{\mathrm{V}v}=0\}$$(\mathrm{C}\mathrm{N})_{s}$$u\in C([0, T);H_{\sigma}^{S})\cap C^{1s}((0, T);H)\cap C((0, T);H^{S+2})$.

Concerning the Euler equations, Kato-Lai [15] and Kato-Ponce [16] proved that for every
for $s>n/p+1,1<p<\infty$ , there are $T>0$ and a unique solution of (E) on the$u$$a\in W_{\sigma}^{s,p}$interval $[0, T)$ in the class$(\mathrm{C}\mathrm{E})_{s,p}$$u\in C([0, T);W^{s}’ p)\sigma\cap C^{12,p}([\mathrm{o},\tau);W_{\sigma}^{S}-)$,

40

where subindex means the divergence free. It is an interesting question whether the solution$u(t)$ really blows up as. Giga [11] showed that if the strong solution in $(CN)_{s}$ satisfies$\sigma$$t\uparrow T$

(Se)

$u$

$\int_{0}^{T}||u(t)||\kappa dL^{r}t<\infty$

for $2/\kappa+n/r=1$ with $n<r\leq\infty$ ,then can be continued to the solution in the class $(oN)_{s}$ beyond $t=T$ . Concerning therot and proved that
Euler equations, Beale-Kato-Majda [1] dealt with the vorticity
under the condition$u$$\omega=$$u$$\int_{0}^{T}||\omega(t)||_{L}\infty dt<\infty$

can never break down its regularity at $t=T$ . To prove this assertion, in [1] they madeuse of the logarithmic inequality such as$u(t)$(0.1)$||\nabla u||L\infty\leq C(1+||\mathrm{r}\mathrm{o}\mathrm{t}u||L^{\infty}(1+\log^{+}||u||W^{S+1,\mathrm{p}})+||\mathrm{r}\mathrm{o}\mathrm{t}u||_{L^{2}})$,$sp>n$, where $\log^{+}a=\log$ if $a\geq 1,$ $=0$ if $0<a<1$ .
for all vector functions with
The purpose of this paper is to extend these results to the marginal space $BMO$ whichis larger than.
$\mathrm{d}\mathrm{i}\mathrm{v}u=0$

$u$

$a$

$L^{\infty}$

1

Results.

denote the set
Before stating our results, we introduce some function spaces. Let, such thatwith compact support inof allvector functions;
with respect to the-norm.
is the closure ofdenotes the, where $1/r+1/r’=1$ .
duality pairing between
andstands for the usual (vector$1\leq r\leq\infty$ .
valued)denotes the closure of-space overwith respect to the$s\geq0$,.
-norm$C_{0,\sigma}^{\infty}$$\phi=(\phi^{1}, \phi^{2}, \cdots, \phi^{n})$$C^{\infty}$$\mathrm{d}\mathrm{i}\mathrm{v}\phi=0$$L_{\sigma}^{r}$$R^{n}$$L^{r}$$C_{0,\sigma}^{\infty}$$L^{r}$$L^{r}$$L^{r’}$$R^{n},$$||\cdot||_{r}$$(\cdot, \cdot)$$L^{r}$$H_{\sigma}^{s}$$C_{0,\sigma}^{\infty}$$||\phi||H^{\epsilon}=||(1-\Delta)^{\frac{s}{2}\phi 1}|_{2}$$H^{s}$

Our result on continuation of strong solutions of (N-S) now reads:
Theorem 1 Let $s>n/2-1$ and letin the class $(CN)_{s}$ on $(0, T)$ . If
(1.1)

then

$a\in H_{\sigma}^{s}$. Suppose that$\int_{\epsilon_{0}}^{T}||u(t)||_{BMO}^{2}dt<\infty$$u$

for some$u$is the strong solution of (N-S)$0<\epsilon_{0}<T$,

can be continued to the strong solution in the class $(CN)_{s}$ on $(0, T’)$ for some $T’>T$ .

Corollary 1 Let be the strong solution of (N-S) in the class $(CN)_{s}$ on $(0, T)$ for $s>n/2-1$ .
Suppose that is maximal, $i.e.,$ cannot be continued in the class $(CN)_{s}$ on $(0, T/)$ for any$T’>T$ . Then
$u$

$T$

(1.2)

$u$

$\int_{\epsilon}^{T}||u(t)||_{BMO}^{2}dt=\infty$

for all $0<\epsilon<T$ .

41

For the space $BMO$ , we refer to Stein [24]. Since $s>n/2-1$ , there holds $H^{s+2}\subset BMO$ ,and hence for every in the classon $(0, T)$ , we have $u\in C((\mathrm{O}, T);BMo)$ .
$u$$(\mathrm{C}\mathrm{N})_{s}$We next consider a criterion on uniqueness and regularity of weak solutions to (N-S). Ourdefinition of a weak solution is as follows.
Definition 1. Let
solution of (N-S) on $(0, T)$ if$a\in L_{\sigma}^{2}$(i). A measureble function$u\in L^{\infty}(0,T;L_{\sigma}^{2})\cap L^{2}(0,T;H_{\sigma}^{1})$

$u$

on

$R^{n}\mathrm{x}(0, T)$is called a weak;(ii) $u(t)$ is continuous on $[0, T]$ in the weak topology of$L_{\sigma}^{2}$

;

(iii)

(1.3)

$\int_{s}^{t}\{-(u, \partial_{\mathcal{T}}\Phi)+(\nabla u, \nabla\Phi)+(u\cdot\nabla u, \Phi)\}d\tau=-(u(t), \Phi(t))+(u(s), \Phi(s))$

for every $0\leq s\leq t<T$ and every $\Phi\in H^{1}((s, t);H^{1}\sigma\cap L^{n})$ .

Our result on weak solutions of (N-S) now reads:
Theorem 2 (1) (uniqueness) Let$(0, T)$ . Suppose that
(1.4)

and that$v$$a\in L_{\sigma}^{2}$and let $u,$$v$

be two weak solutions of (N-S) on$u\in L^{2}(0,T;BMo)$satisfies the energy inequality
,

(1.5)

$||v(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla v||2d2\tau\leq||a||_{2}^{2}$$0<t<T$ .

on $[0, T]$ .
Then we haveand let be a weak solution with the additional property (1.4).
(2) (regularity) Let
is actually a strong solution of (N-S) on
in the class
Then for every $0<\epsilon<T,$$u\equiv v$$a\in L_{\sigma}^{2}$$u$$(\epsilon, T)$$u$$(CN)_{s}$for $s>n/2-1$ .

Remark. Theorem 2 may be regarded as an extension of Serrin’s criterion [22], [23] on
uniqueness and regularity of weak solutions in the class
$u$

(1.6)

$u\in L^{\kappa}(\mathrm{O}, T;Lr)$for $2/\kappa+n/r=1$ with $n<r\leq\infty$ .

in (1.6). Moreover, by virtue
Our class (1.4) is larger than the marginal case$||u||BMO\leq
C||\nablau||M^{n}$
[13],
of
we
see that the weak solution
John-Nirenberg
of the estimate
denotes the Morrey space which iswith $\nabla u\in L^{2}(0, T;Mn)$ becomes regular, where
da Veiga [2].
. Seelarger than$L^{2}(\mathrm{o}, \tau;L^{\infty})$$M^{n}$$u$$L^{n}$$\mathrm{B}\mathrm{e}\mathrm{i}\mathrm{r}\tilde{\mathrm{a}}0$

We shall next investigate continuation of the strong solution in terms of the vorticity
and the deformation tensor Def.

$\omega=\mathrm{r}\mathrm{o}\mathrm{t}u\equiv(\partial_{j}u^{kj}-\partial_{k}u)1\leq j,k\leq n$$u\equiv(\partial_{j}u^{k}+\partial_{k}u^{j})_{1}\leq j,k\leq n$

42Theorem 3 Let $s>n/2-1$ . Suppose that$(CN)_{s}$$u$is the strong solution of (N-S) in the classon $(0,T)$ . If either(1.7)$\int_{\epsilon_{0}}^{T}||\omega(t)||_{B}Modt<\infty$

$or$

(1.8)

$\int_{\epsilon_{0}}^{T}||\mathrm{D}\mathrm{e}\mathrm{f}u(t)||_{BMO}dt<\infty$

holds for some $0<\epsilon_{0}<T$ , thenon $(0, T’)$ for some $T’>T$ .
Corollary 2 Suppose that$s>n/2-1$ . Assume that

$u$

$T$

(1.9)

$u$

can be continued to the strong solution in the$clas\mathit{8}(CN)_{s}$is the strong solution of (N-S) in the class $(CN)_{s}$ on $(0, T)$ for
as in Corollary 1. Then both
is maximal in the same$\mathit{8}ense$$\int_{\epsilon}^{T}||\omega(t)||_{BM}odt=\infty$

and$\int_{\epsilon}^{T}||\mathrm{D}\mathrm{e}\mathrm{f}u(t)||_{BM}odt=\infty$

hold for all $0<\epsilon<T$ .

Theorem 3 yields the following regularity criterion on weak solutions of (N-S) by mean ofrot and Def .
$u$

$u$

Theorem 4 Let$a\in L_{\sigma}^{2}$(1.10). Suppose that$u$$\omega\in L^{1}(\mathrm{o}, \tau;BMo)$holds, then for every $0<\epsilon<T,$on
for $s>n/2-1$ .

$u$is a weak solution of (N-S) on $(0, T)$ . If either
or Def $u\in L^{1}(\mathrm{o}, \tau;BMo)$is actually a strong solution of (N-S) in the class $(CN)_{s}$$(\epsilon, T)$

da Veiga [2] proved the regularity criterion in the class$2/\kappa+n/r=2$
with $1<\kappa<\infty,$ $n/2<r<\infty$ . Theorem 4 covers the borderline casefor$\kappa=1$
and $r=\infty$ .

Remark.

$\nabla u\in L^{\kappa}(\mathrm{O}, T;Lr)$$\mathrm{B}\mathrm{e}\mathrm{i}\mathrm{r}\tilde{\mathrm{a}}0$

Our result on (E) reads as follows.
Theorem 5 Let $1<p<\infty,$ $s>n/p+1$ . Suppose that$(CE)_{s,p}$ on $(0, T)$ . If either(1.11)$u$is the solution of $(E)$ in the class$\int_{0}^{T}||\omega(t)||BModt(\equiv M_{0})<\infty$$or$(1.12)

holds, then$\int_{0}^{T}||\mathrm{D}\mathrm{e}\mathrm{f}u(t)||BModt(\equiv M_{1})<\infty$$u$

can be continued to the solution in the$clas\mathit{8}(CE)_{S},p$on $(0, T’)$ for some $T’>T$ .

be the solution of $(E)$ in the class $(CE)_{S},p$ on $(0, T)$ for $1<p<\infty$ ,Corollary 3 Let$s>n/p+1$ . Assume that is maximal, $i.e.,$ cannot be continued to the solution in theclass $(CE)_{s,p}$ on $(0, T’)$ for any $T’>T$ . Then both
$u$

$T$

$\int_{0}^{T}||\mathrm{r}\mathrm{o}\mathrm{t}u(t)||BMOdt=\infty$

hold.

$u$

and

$\int_{0}^{T}||\mathrm{D}\mathrm{e}\mathrm{f}u(t)||BModt=\infty$

43
Bilinear estimates and critical Sobolev inequality in $BMO$ .

2

In this section we shall prepare some lemmas. In what follows we shall denote by $C$
various constants. In particular, $C=C(*, \cdots, *)$ denotes constants depending only on thequantities appearing in the parenthesis.
We first prove the following key estimate.

Lemma 2.1 (Bilinear estimates) Let $1<r<\infty$ . Then we have
(i)
(2.1)

for all$||f\cdot\nabla g||_{r}\leq C(||f||r||(-\triangle)^{\frac{1}{2}}g||_{BM}\mathit{0}+||(-\triangle)^{\frac{1}{2}f1}|_{B}MO||g||_{r})$$f,$ $g\in W^{1,r}$(ii) Letwith$\nabla f,$$\nabla g\in BMO$$\alpha=(\alpha_{1}, \cdots, \alpha_{n}),$with $C=C(n,r)$ .
be multi-indices with $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}\geq 1$$\beta=(\beta_{1}, \cdots,\beta_{n})$and $|\beta|=\beta_{1}+\cdots+\beta_{n}\geq 1$ . Then
(2.2)

for all$||\partial^{\alpha}f\cdot\partial^{\beta}g||_{2}\leq C(||f||BMo||(-\triangle)^{\frac{|\alpha|+|\beta|}{2}}g||_{2}+||(-\triangle)^{\frac{|\alpha|+|\beta|}{2}f1}|_{2}||g||BMo)$$f,$ $g\in BMO\cap H^{|\alpha|+|\beta|}$with $C=C(n, \alpha,\beta)$ , where$\partial^{\alpha}=\frac{\partial^{|.\alpha|}}{\partial x_{1}^{\alpha_{1}}\cdot\cdot\partial Xn^{n}\alpha}$.

The proof of this lemma is based on the following proposition due to Coifman-Meyer [6,Chapter V. Proposition 2].

Proposition 2.1 (Coifman-Meyer) Let$\sigma=\sigma(\xi, \eta)\in C^{\infty}(R^{n}\cross R^{n}\backslash \{(0,0)\})$$|\partial_{\xi}^{\alpha}\partial_{\eta}^{\beta}\sigma(\xi, \eta)|\leq C(|\xi|+|\eta|)-|\alpha|-|\beta|$

for all mluti-indice8$\alpha,\beta$,$(\xi, \eta)\in R^{n}\cross R^{n}\backslash \mathrm{f}(\mathrm{o}, \mathrm{o})\}$

with $C=C(\alpha,\beta)$ . Suppose that$\sigma(\xi, 0)=0$

Then the bilinear operator(2.3)

satisfy$\sigma(D)(\cdot, \cdot)$.

defined by$\sigma(D)(f, g)(x)\equiv\iint_{R^{n}\cross R^{n}}e\sigma(\xi, \eta)f(\xi)g(\wedge\eta)dix\cdot(\zeta+\eta)\wedge\xi d\eta$,$x\in R^{n}$satisfies(2.4)$||\sigma(D)(f, g)||_{p}\leq C||f||p||g||_{B}Mo$$(1<p<\infty)$

with $C=C(n,p)$ .

Proof of Lemma 2.1. Here we prove only (2.2). The proof of (2.1) is similar to that of-function on $[0, \infty)$ such thatbe a(2.2). Let
for $0\leq t\leq 1/2$ , and let $\Phi_{2}=1-\Phi_{1}$ . Then we have$\Phi_{1}$$C^{\infty}$$\partial^{\alpha}f(X)\partial^{\beta}g(x)$$=$$C \int\int_{R^{n}\mathrm{x}R^{n}}e^{i(}\xi\alpha\beta\wedge\wedge(\xi)g(\eta)d\xi dx\cdot\xi+\eta)\eta f\eta$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\Phi_{1}\subset[0,1),$$0\leq\Phi_{1}\leq 1,$ $\Phi_{1}(t)\equiv 1$

44

$=$

$C \int\int_{R^{n}\cross R^{n}}e^{ix\cdot(\xi}+\eta)\cross$$( \frac{\xi^{\alpha}\eta^{\beta}}{|\eta|^{|\alpha}|+|\beta|}\Phi_{1}(|\xi|/|\eta|)f(\xi\wedge)|\eta|^{|\alpha}|+|\beta|\wedge(g\eta)+\frac{\xi^{\alpha}\eta^{\beta}}{|\xi|^{|\alpha|+|\beta|}}\Phi 2(|\xi|/|\eta|)|\xi|^{1}\alpha|+|\beta|^{\wedge}f(\xi)^{\wedge}g(\eta)\mathrm{I}d\xi d\eta$$=$,$C( \sigma_{1}(D)(f, (-\Delta)\frac{|\alpha|+|\rho \mathrm{I}}{2}g)(_{X})+\sigma 2(D)((-\triangle)\frac{|\alpha|+|\beta \mathrm{I}}{2}f, g)(X))$

where$\sigma_{1(\xi,\eta)=}\frac{\xi^{\alpha}\eta^{\beta}}{|\eta|^{|\alpha}|+|\beta|}\Phi_{1}(|\xi|/|\eta|)$Since$|\alpha|\geq 1$

ans$|\beta|\geq 1$,$\sigma_{2}(\xi, \eta)=\frac{\xi^{\alpha}\eta^{\beta}}{|\xi|^{|\alpha|+|\beta|}}\Phi_{2}(|\xi|/|\eta|)$,$\sigma_{2}(\xi, 0)=0$, we see that$\sigma_{1}(0, \eta)=0$

and that$\sigma_{1}$and$\sigma_{2}$.

satisfy the hypotheses of Proposition 2.1. Hence there holds$||\sigma_{1}(D)(f, (-\Delta)^{\frac{|\alpha|+|\beta \mathrm{I}}{2}}g)||_{2}$$|| \sigma_{2}(D)((-\Delta)\frac{|\alpha|+\mathrm{I}\rho \mathrm{I}}{2}f,g)||_{2}$$\leq$$\leq$$C||f||_{B}Mo||(-\Delta)^{\frac{|\alpha|+|\beta|}{2}}g||_{2}$$C||(- \Delta)\frac{|\alpha|+|\beta|}{2}f||2||g||BMO$,,

which yields (2.2). This proves Lemma 2.1.
The next lemma plays an important role to show the energy identity of weak solutions in
the class (1.4) and (1.10).

Lemma 2.2 (i) Let
we have(2.5)(ii) Let $w,$

$w\in L^{\infty}(\mathrm{O}, T;L_{\sigma}^{2})\cap L^{2}(\mathrm{o}, \tau;H_{\sigma}^{1})$and $u\in L^{2}(0, T;H^{1}\sigma\cap BMO)$ . Then$\int_{0}\tau\tau(w\cdot\nabla u,u)d=0$$u\in L^{\infty}(\mathrm{O}, T;L_{\sigma}^{2})\cap L^{2}(0, T;H_{\sigma}^{1})$.

. Suppose that either

rot

$w$

, rot $u\in L^{1}(0,T;BMo)$

Def

$w$

, Def $u\in L^{1}(\mathrm{o}, \tau;BMo)$$or$holds. Then we have
(2.6)$\int_{0}^{T}(w\cdot\nabla u,u)d\tau=0$

To prove (2.5), we use the estimate of(2.7)-Meyer-Semmes [5]:$\mathrm{C}\mathrm{o}\mathrm{i}\mathrm{f}\mathrm{m}\mathrm{a}\mathrm{n}-\mathrm{L}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{s}}$with$w\cdot\nabla u\in \mathcal{H}^{1}$.

$||w\cdot\nabla u||_{\mathcal{H}^{1}}\leq C||w||_{2}||\nabla u||_{2}$,. For detail, see [5].
wheredenotes the Hardy space on
To prove (2.6), we use (2.1) and the Biot-Savart law. Indeed, by the Biot-Savart law, wehave the representation$R^{n}$$\mathcal{H}^{1}$

(2.8)
(2.9)

$\frac{\partial u}{\partial x_{j}}$$\frac{\partial u^{l}}{\partial x_{j}}$$=$$R_{j}(R\cross\omega)$$=$$R_{j}$($\sum_{k=1}^{n}R_{k}$,Def$j=1,$ $\cdots,n$ ,$u_{kl}$),$j,$$l=1,$

where$\cdots,$ $n$$\omega=\mathrm{r}\mathrm{o}\mathrm{t}u$,;where Def$u_{kl}= \frac{\partial u^{k}}{\partial x_{l}}+\frac{\partial u^{l}}{\partial x_{k}}$.

45Here $R=(R_{1}, \cdots, R_{n})$ , and$R_{j}= \frac{\partial}{\partial x_{j}}(-\Delta)^{-\frac{1}{2}}$denote the Riesz transforms. Since $R$ is abounded operator in $BMO$ , we have by (2.8), (2.9) and assumption that
(2.10)$\nabla u,$$\nabla w\in L^{1}(0, T;BMo)$.

It follows from Lemma 2.1 (2.1) and (2.10) thatis well-defined. For detailsof the proof of Lemma 2.2 we refer to [17].
Using the usual mollifier argument, by Lemma 2.2, we have the following energy identity
for weak solutions with (1.4) or (1.10).
$\int_{0}^{T}(w\cdot\nabla u,u)d\mathcal{T}$Lemma 2.3 Let $n\geq 3$ and letsatisfying one of the additional(2.11)$a\in L_{\sigma}^{2}$.

that is a weak solution of (N-S) on $(0,T)$and (1.10). Then fulfills the energy identity$u$$s_{upp_{\mathit{0}\mathit{8}}e}$$condition\mathit{8}(\mathit{1}.\mathit{4})$$u$for all $0\leq s\leq t<T$ .

$||u(t)||_{2}^{2}+2 \int_{s}^{t}||\nabla u||2d2\tau=||u(s)||_{2}^{2}$

Now we prove the following lemma which is an extension of (0.1).

Lemma 2.4 (Critical Sobolev Inequality) Let $1<p<\infty$ and let $s>n/p$ . There is aconstant $C=C(n,p, s)$ such that the estimate(2.12)$||f||_{\infty}\leq C(1+||f||_{BM}o(1+\log^{+}||f||_{W^{S}},\mathrm{p}))$holds for all $f\in W^{s,p}$ .
to the right hand side ofRemark. Compared with (0.1), we do not need to add(2.12). This makes it easier to derive an apriori estimate of solutions to the Euler equations.
than
Proof of Lemma 2.4.
We shall make use of the Littlewood-Paley decomposition; there exists a non-negative( ; the Schwartz class) such thatand such thatfunction$||f||_{L^{2}}$$\mathrm{B}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{e}_{-}\mathrm{K}\mathrm{a}\mathrm{t}\mathrm{o}-\mathrm{M}\mathrm{a}\mathrm{j}\mathrm{d}\mathrm{a}[1]$$\varphi\in S$$S$$\sum_{k=-\infty}^{\infty}\varphi(2-k\xi)=1$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\varphi\subset\{2^{-1}\leq|\xi|\leq 2\}$

for$\xi\neq 0$. See$\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{h}-\mathrm{L}\ddot{\mathrm{O}}\mathrm{f}_{\mathrm{S}\mathrm{t}\ddot{\mathrm{o}}}\mathrm{r}\mathrm{m}$[ , Lemma 6.1.7]. Let us define$3$$\phi_{0}$and$\phi_{1}$as$\phi_{0}(\xi)=\sum_{k=1}^{\infty}\varphi(2^{k}\xi)$and$\phi_{1}(\xi)=\sum_{k=-\infty}^{-1}\varphi(2^{k}\xi)$,

respectively. Then we have that $\phi_{0}(\xi)=1$ for $|\xi|\leq 1/2,$ $\phi_{0}(\xi)=0$ for $|\xi|\geq 1$ and that$\phi_{1}(\xi)=0$ for $|\xi|\leq 1,$ $\phi_{1}(\xi)=1$ for. It is easy to see that for every positive integer $N$
there holds the identity$|\xi|\geq 2$(2.13)$\phi_{0}(2^{N}\xi)+\sum_{k=-N}^{N}\varphi(2-k\xi)+\phi_{1}(2^{-N}\xi)=1$

and sinceis dense in$s>n/p$ , it suffices to prove (2.12) for

Since$C_{0}^{\infty}$$W^{s,p}$$W^{s,p}$,$\xi\neq 0$.

is continuously embedded in $BMO$ , implied by. For such we have the representation$f$$f\in C_{0}^{\infty}$$f(x)= \int_{y\in R^{n}}K(x-y)\cdot\nabla f(y)dy$

with$K(y)= \frac{1}{n\omega_{n}}\frac{y}{|y|^{n}}$,46

for all $x\in R^{n}$ , where
into three parts:$\omega_{n}$denotes the volume of the unit ball in$R^{n}$. By (2.13) we decompose$f$$f(x)$$=$$\int_{y\in R^{n}}K(_{X}-y)\mathrm{x}$$\cross(\phi_{0}(2(NX-y))+\sum_{Nk=-}^{N}\varphi(2^{-}(k-yX))+\phi_{1}(2^{-N}(x-y))\mathrm{I}\cdot\nabla f(y)dy$(2.14)$\equiv$$f_{0}(x)+g(X)+f1(_{X)}$for all $x\in R^{n}$ .
We can show that
(2.15)$|f_{0}(X)|\leq C2^{-\beta N}||f||_{W^{s},\mathrm{p}}$is a positive constant. For detail, see [18].
for all $x\in R^{n}$ , where
By integration by parts we have$\beta=\beta(n,p, \mathit{8})$. $*f(x)$ ,$g(x)= \sum_{k=-N}^{N}(\mathrm{d}\mathrm{i}\mathrm{V}\Psi)_{2}\mathrm{t}$$x\in R^{n}$,where $\Psi(x)=K(x)\varphi(x)$ and $\psi_{t}(x)=t^{-n}\psi(x/t)$ for $t>0$ . Since $\Psi\in S$ with the property
that,$\int_{R^{n}}\mathrm{d}\mathrm{i}\mathrm{V}\Psi(X)dx=0$

it follows from Stein [24, Chap. IV, 4.3.3] that$||g||_{\infty}$(2.16)$\leq$$\sum_{k=-N}^{N}||(\mathrm{d}\mathrm{i}\mathrm{v}\Psi)_{2^{k^{*}}}f||_{\infty}$$\leq$$\sum_{k=-N}^{N}\sup_{>t0}||(\mathrm{d}\mathrm{i}\mathrm{V}\Psi)t*f||_{\infty}$$\leq$$CN||f||BMo$ ,where $C=C(n)$ is independent of $N$ .
Integrating by parts, we have by a direct calculation$|f_{1}(X)|$(2.17)$=$$\leq$$| \int_{y\in R}n\mathrm{d}\mathrm{i}\mathrm{v}_{y}(K(x-y)\phi 1(2^{-N}(x-y)))f(y)dy|$$C2^{-N\cdot\frac{n}{\mathrm{p}}}||f||_{p}$for all $x\in R^{n}$ , where $C=C(n,p)$ is independent of $N$ .
Now it follows from (2.14) and $(2.15)-(2.17)$ that
(2.18)$||f||_{\infty}\leq C(2^{-\gamma N}||f||_{W^{s},\mathrm{p}}+N||f||_{BMO})$,with $\gamma={\rm Min}.\{\beta, n/p\}$ , where $C=C(n, s,p)$ is independent of $N$ and . If$N$
so large that the first term of the right handthen we may take $N=1$ ; otherwise, we take$f$$||f||_{W^{s,\mathrm{p}}}\leq 1$

47side of (2.18) is dominated by 1, i.e.,( ; Gauss symbol) and (2.18)$N \equiv[\frac{\log||f||W^{S},\mathrm{p}}{\gamma\log 2}]+1$$[\cdot]$

becomes$||f||_{\infty} \leq C\{1+||f||BMo(\frac{\log||f||W^{s,\mathrm{p}}}{\gamma\log 2}+1)\}$.

In both cases, (2.12) holds. This proves Lemma 2.4.

3

Proof of Theorems 1-4

3.1

Proof of Theorem 1.

It is proved by Kato [14] and Giga [11] that, for the initial data $a\in H^{s}$ with $s>n/2-1$ ,of the strong solution of (N-S) in the classthe local existence time intervalcanbe estimated from below as
,
(3.1)
$T$

$u$

$(\mathrm{C}\mathrm{N})_{s}$$T \geq\frac{C}{||a||^{\frac{2}{H^{\delta}s-(n/2-1)}}}$

where $C=C(n, s)$ . Actually, for $a\in L^{r}$ with $r>n$ , Giga [11, Theorem 1
a way that
(3.2)

$T= \frac{C}{||a||^{2/(r-n}rr)}$$(\mathrm{i}\mathrm{i})$] gave$T$in such,for $1/r=1/2-s/n$ , we obtain (3.1). Henceso from the continuous embeddingby the standard argument of continuation of local solutions, it suffices to prove the following
apriori estimate$H^{s}\subset L^{r}$(3.3)$\sup_{\epsilon_{0}<t<T}||u(t)||_{H}[S]+1\leq||u(\epsilon 0)||H^{\mathrm{I}}S3+1\exp(C\int_{\epsilon}^{\tau}0||u||_{BMO}2dt)$,

where $C=C(n, s)$ is independent of .
be a multi-index with $|\alpha|\equiv\alpha_{1}+\cdots+\alpha_{n}\leq[s]+1,$
Let$T$$\alpha=(\alpha_{1}, \cdots, \alpha_{n})$$v= \partial^{\alpha}u=\frac{\partial^{|\alpha|}.u}{\partial x_{1}^{\alpha_{1}}\cdot\cdot\partial Xn^{n}\alpha}$. Applying(3.4)$\frac{\partial v}{\partial t}-\triangle v+u\cdot\nabla v+\nabla q=F$

where
(3.5)

$q=\partial^{\alpha}p$$\partial^{\alpha}$to (N-S), we have for,$v$

and.. letthe equation$\epsilon_{0}<t<T$,

and$F=- \sum_{|\beta|\leq|\alpha|-1}\alpha C\beta\partial\alpha-\beta\nabla(\partial^{\beta}u\cdot u)$.

between (3.4) and , and then integrating the result identity
Taking the inner product in, we obtainon the time interval$L^{2}$$v$$(\epsilon 0, t)$(3.6)$||v(t)||_{2}^{2}+2 \int_{\epsilon_{0}}^{t}||\nabla v||2d2\tau\leq||v(\Xi_{0})||^{2}2+2\int_{\epsilon_{0}}^{t}||F||_{2}||v||_{2}d\tau$On the other hand, by (2.2), we have$||F||_{2} \leq c||u||_{BMo}||(-\Delta)\frac{|\alpha|+1}{2}u||2$

,

.

48

from which and (3.6) it follows that$|| \partial^{\alpha}u(t)||_{2}2+2\int_{\epsilon_{0}}^{t}||\nabla(\partial\alpha u)||^{2}2d\tau$$\leq$$|| \partial^{\alpha}u(_{\mathcal{E}}0)||^{2}2+\int_{\epsilon_{0}}^{t}||(-\Delta)^{\frac{|\alpha|+1}{2}}u||^{2}2\mathcal{T}d+\mathit{0}\int_{\epsilon_{0}}^{t}||u||^{2}BMO||\partial\alpha u||2d2\mathcal{T}$with $C$ independent of . Summing over$t$$\alpha$with $0\leq|\alpha|\leq[s]+1$ , we have$||u(t)||_{H[]+}^{2} \theta 1\leq||u(\mathcal{E}0)||2H[s]+1+C\int_{\epsilon_{0}}^{t}||u||_{BM}2|\mathit{0}|u||^{2}H^{\mathfrak{c}}s\mathrm{J}+1dT$. Now the Gronwall inequality yields (3.3). This proves Theorem 1.

for all$\epsilon_{0}\leq t<T$

3.2

Proof of Theorem 2.

(1) Let us first prove uniqueness. We follow the argument of Masuda [20, Theorems 2,
3]. We can show that
(3.7)

.

$\int_{0}^{t}\{2(\nabla u, \nabla v)+(v\cdot\nabla v, u)-(u\cdot\nabla v,u)\}d\tau=-(u(t), v(t))+||a||_{2}^{2}$

See Masuda [20, p.640 (4.4)]. By Lemma 2.3,
(3.8)

$u$

satisfies the energy identity$||u(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla u||^{2}2\tau d=||a||_{2}^{2}$.

Addition of (3.7) (multiplied by $-2$ ), $(3.8)$ and (1.5) yields(3.9)$||w(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla w||2d2\tau\leq 2\int_{0}^{t}(w\cdot\nabla v, u)d\tau=2\int_{0}^{t}(w\cdot\nabla w, u)d\tau$where $w=v-u$ . In the last identity, we have used (2.5). By (2.7) we have

RHS of (3.9)$\leq$$C \int_{0}^{t}||w\cdot\nabla w||_{\mathcal{H}^{1}}||u||_{BM}od\tau$$\leq$$C \int_{0}^{t}||w||2||\nabla w||_{2}||u||BMod\tau$$\leq$$\int_{0}^{t}||\nabla w||^{2}2\mathcal{T}d+\mathit{0}\int_{0}^{t}||w||^{2}2||u||2d\mathcal{T}BMo$

Hence by (3.9)$||w(t)||_{2}^{2} \leq C\int_{0}^{t}||w||^{2}2||u||2d\mathcal{T}BMo$’

Since $u\in L^{2}(0, \tau;BMo)$ , the Gronwall inequality yields$||w(t)||^{2}2=0$

from which we get the desired uniqueness.

,$0\leq t<T$$0\leq t<T$

.

.

,

49

(2) We next prove regularity. Since $u\in L^{2}(0,T;H^{1}\sigma\cap BMO)$ , for every $0<\epsilon<T$ , there
is $0<\delta<\epsilon$ such thatfor $n<r<\infty$ . Hence it follows fromthe local existence theorem of Kato [14] and Giga [11] that there are
and a uniquesolution onwith, such that$u(\delta)\in H_{\sigma}^{1}\cap BMO\subset L_{\sigma}^{2}\cap L_{\sigma}^{r}$$T_{*}>\delta$$[\delta, T_{*})$$\tilde{u}$(3.10)

Since$\tilde{u}|_{t=\delta}=u(\delta)$

for $s>n/2-1$ .

$\tilde{u}\in C([\delta,T_{*});H^{1}\sigma\cap L_{\sigma}^{r})\cap C^{1+2}((\delta, T_{*});H^{s})$$u$satisfies the energy identity(3.11)$||u(t)||_{2}^{2}+2 \int_{\delta}^{t}||\nabla u||2d2\tau=||u(\delta)||_{2}^{2}$,$\delta\leq t<T$,

implied by Lemma 2.3, we have by the uniqueness criterion of Serrin-Masuda [23], [20](3.12)$u\equiv\tilde{u}$By (3.10) and (3.12), we may regard$\delta<\delta’<\epsilon$

$u$

on

$[\delta,T_{*})$.

as a strong solution in the class.

$(\mathrm{C}\mathrm{N})_{s}$.

on$(\delta’, \tau_{*})$

forIn fact, there holds $T_{*}=T$ . Suppose that $T_{*}<T$ . Then there exists $T_{0}<T$ such thatis a strong solution in the classon ( To), but
be continued in the classon
for any. By assumption, we have$(\mathrm{C}\mathrm{N})_{s}$$(\delta’,\tilde{T})$cann.ot$\delta’,$$\tilde{T}>T_{0}$$u$$(\mathrm{C}\mathrm{N})_{s}$.

(3.13)$\int_{\mathit{5}}^{T_{0}},||u||^{2}BMo^{d}\tau\leq\int_{0}^{T}||u||_{BMo^{d}}2\mathcal{T}<\infty$This contradicts Corollary 1, so we get $T_{*}=T$ . This proves Theorem 2.

3.3

Proof of Theorems 3-4.

Proof of Theorem 3:
On account of (3.2), it suffices to prove(3.14)$\sup_{\epsilon_{0}<t<\tau}||u(t)||_{r}\leq||u(\epsilon_{0})||r\exp(C\int_{\epsilon 0}T||\nabla u||_{B}Mod_{T}\mathrm{I},$$r>n$ .

In the same way as in (2.10), we see that the hypothesis (1.7) or (1.8) yields(3.15)

Since$\int_{\epsilon_{0}}^{T}||\nabla u||_{B}Mo^{d}t<\infty$$u\in C([\epsilon_{0},\tau);Hs+2)\subset C([\mathit{6}_{0}, \tau);W1,\infty),$$C^{1}((\epsilon 0, \tau);W^{1},r)\cap C((\epsilon_{0}, \tau);W^{3},r)$(3.16)

for all$u$.

is actually the solution in $C([\epsilon_{0},T);W1,r)\cap$and has the integral representation:$2\leq r<\infty$$u(t)=e^{(t-\epsilon 0)\Delta}u( \mathcal{E}0)-\int_{\epsilon_{0}}^{t}e^{(tS)\triangle}-P(u\cdot\nabla u)(s)d_{S}$,$\epsilon_{0}<t<T$.

is the well-known heat oparator and $P=\{P_{kl}\}_{k,l=}1,\ldots,n$ is theSee Kato [14]. Here
Helmholtz projection defined by $P_{kl}=\delta_{kl}+R_{k}R_{l}$ .
$e^{t\triangle}$

50

Since

$||e^{t\Delta}||_{B}(Lr,L^{r})\leq 1$$||u(t)||_{r}$for all $t>0$ , it follows from Lemma 2.1 (i) and (3.16) that$\leq$$||u( \epsilon 0)||_{r}+C\int_{\epsilon_{0}}^{t}||u\cdot\nabla u||_{r}d\mathcal{T}$$\leq$$||u( \epsilon 0)||_{r}+C\int_{\epsilon_{0}}^{t}||\nabla u||_{B}Mo||u||_{r}d\tau$,$\epsilon_{0}<t<T$.

From this and the Gronwall inequality, we obtain the desired apriori estimate (3.14), which
proves Theorem 3.
Proof of Theorem 4:
The proof of Theorem 4 is parallel to that of Theorem 2.

4Proof of Theorem 5.

We follow the argument of Beale-Kato-Majda [1]. It is proved by Kato-Lai [15] and Katoof
Ponce [16] that for the given initial data $a\in W^{s,p}$ for $\mathit{8}>1+n/p$ , the time interval
depends only on. Hencethe existence of the solution to (E) in the classby the standard argument of continuation of local solutions, it suffices to establish an apriori
or $T,$
in terms of $T,$
according to (1.11) or (1.12). Indeed,
estimate for in$u(t)$
on $(0, T)$ is subject to the following
in the class
we shall show that the solution
estimate:
$T$

$u$

$u$

$W^{s,\mathrm{p}}$$||a||_{W^{\mathit{8}},\mathrm{p}}$$(\mathrm{C}\mathrm{E})_{S},p$$a,$$M_{0}$$a,$$M_{1}$$(\mathrm{C}\mathrm{E})_{s,p}$(4.17)$\sup_{0<t<T}||u(t)||_{W}\epsilon,\mathrm{p}\leq(||a||_{W^{\mathit{8}},\mathrm{p}}+e)^{\alpha_{j}}\exp(c\tau\alpha_{j})$with$\alpha_{j}=e^{CM_{j}}$,$j=0,1$ ,where $C=C(n,p, s)$ is a constant independent of and .
We shall first prove (4.17) under (1.11). It follows from the commutator estimate ingiven by Kato-Ponce [16, Proposition 4.2] that$a$(4.18)$T$$||u(t)||W^{\theta}, \mathrm{p}\leq||a||_{W}\theta,p\exp(C\int_{0}^{t}||\nabla u(\tau)||_{\infty)}d\tau,$$0<t<T$ ,where $C=C(n,p, s)$ .
By the Biot-Savard law (2.8), we have
(4.19)$||\nabla u||_{BM}\mathit{0}\leq C||\omega||_{BM}\mathit{0}$with $C=C(n)$ . Hence it follows from (4.19) and Lemma 2.4 that
(4.20)$||\nabla u(t)||\infty\leq C(1+||\omega(t)||_{BMo(1}+\log^{+}||u(t)||Ws,\mathrm{p}))$for all $0<t<T$ with $C=C(n,p, s)$ . Substituting (4.20) to (4.18), we have$||u(t)||_{W^{\mathit{8}}},p+e$$\leq$$(||a||W^{\theta}, \mathrm{p}+e)\exp(C\int_{0}^{t}\{1+||\omega(\mathcal{T})||_{B}Mo\log(||u(\mathcal{T})||ws,\mathrm{p}+e)\}d\tau)$$L^{p}$

51

for all $0<t<T$ . Defining $z(t)\equiv\log(||u(t)||_{W^{\epsilon},p}+e)$ , we obtain from the above estimate$z(t) \leq z(\mathrm{O})+CT+C\int_{0}^{t}||\omega(\mathcal{T})||_{BM}o^{\mathcal{Z}}(\tau)d\tau$,$0<t<T$ .

Now (1.11) and the Gronwall inequality yield$z(t)$$\leq$$(z( \mathrm{O})+CT)\exp(C\int_{0}^{t}||\omega(\tau)||BMod\tau)$$\leq$$(z(0)+CT)\alpha 0$

for all $0<t<T$ with $C=C(n,p, s)$ , which implies (4.17) for $j=0$ .
Similarly we prove (4.17) for $j=1$ under (1.12). This proves Theorem 5.

Acknowledgment
The authors would like to express their thanks to Professor Takayoshi Ogawa for his valuablesuggestions.

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