数理解析研究所講究録 1204 巻 2001 年 58-70

58

Uniqueness and Regularity of solutions
to the Navier-Stokes equations.
Hideo Kozono
Mathematical Institute, Tohoku University, JAPAN英雄 (東北大・理)小薗Introduction
The purpose of this article is to give asurvey on the recent development of well-posedness
on the Navier-Stokes equations. We are mainly concerned with the results given by the:author. Consider the Navier-Stokes equations in$\mathbb{R}^{n}(n\geq 2)$$. \frac{\partial’u}{\partial t}-\Delta\tau\iota+u\cdot$(N-S)$\nabla u+\nabla p=0$$\{$$x\in \mathbb{R}^{n}$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$,,$t\in((\mathrm{I}, T^{1})$$x\in \mathbb{R}^{n}$,$t\in(0, \prime I^{1})$,,$u|_{t=0}=a$ ,, $u^{n}(x, t))$ and $p=p(x, t)$ denote the unknown velocity
where $u=u(x, t)=(u^{1}(x, t)$ ,vector and the pressure of the fluid at the point $(x, t)\in \mathbb{R}^{n}\cross(0, T)$ , respectively, while$a=a(x)=(a^{1}(x), \cdots, a^{n}(x))$ is the given initial velocity vector field. For simplicity, we
assume that the external force has ascalar potential and is included into the pressure
gradient.
vectorthe set of all
Let us first introduce some function spaces. We denote by, such that.
is thewith compact support infunctions $\phi=$,)
pairing
betweenduality(.
therespect
to thenorm
denoteswithclosure ofspace over,(vector-valued)$1/r+1/r’=1$
.
for
the
usual
standsandwhere, where $1<r<\infty$ .
with respect to the norm
denotes the closure of$\cdots$$C^{\infty}$$C_{0.\sigma}^{\infty}$$(\phi^{1}, \cdots, \phi^{n})$$L^{r}$$C_{0.\sigma}^{\infty}$$L^{r’}$$L^{r}$$\mathbb{R}^{r\iota}$$||\cdot||_{r}$$\cdot$$\mathrm{d}\mathrm{i}\mathrm{v}\phi=0$$\cdot$$L^{r}$$L^{r}$$\mathbb{R}^{n}$$L_{\sigma}^{r}$$H_{0,\sigma}^{1}$$C_{0,\sigma}^{\infty}$$||\phi||_{H^{1}}=||\phi||_{2}+||\nabla\phi||_{2}$,, . For an interval I in
, $i,j=1$ ,and aBanach space $X$ ,$L^{p}(IjX)$ and $C^{m}(I;X)$ denote the usual Banach spaces of functions ofand-classon I with values in $X$ , respectively, where $1\leq p\leq\infty$ , $m=0.1$ , .
Our definition of aweak solution of (N-S) now reads

where$\nabla\phi=(’\partial\phi^{i}/\partial x_{j})$$\cdots$$\mathbb{R}^{1}$$n$$L^{p}$$C^{tn}$$\cdots$. A measurable function ti on
Definition 0.1 Let
solution of (N-S) on $(0, T)$ if$a\in L_{\sigma}^{2}$(i) u$\in L^{\infty}(0, T;L_{\sigma}^{2})\cap L^{2}(0, T;H_{0,\sigma}^{1})$;$\mathbb{R}^{n}\cross(0.T)$

is called a weak

59(ii) For every $\Phi\in H^{1}(0, T;H_{0,\sigma}^{1}\cap L^{n})$ with $\Phi(T)=0$ ,(0.1)$\int_{0}^{T}${ $-(u,$$\partial_{t}\Phi)+(\nabla u$, $\nabla\Phi)+(u$ . Vu,$\Phi)$} $dt=(a, \Phi(0))$ .
Hopf [7].

Concerning existence of the weak solutions, we have Leray [13] andTheorem 0.2 (Leray-Hopf) For every
such thatof (N-S) on$a\in L_{\sigma}^{2}$, there exists at least one weak solution$(0, \infty)$

$u$

(0.2)

$||u(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla u(\tau)||_{2}^{2}d\tau\leq||a||_{2’}^{2}$and$||u(t)-a||_{2}arrow 0$$0\leq t<\infty$,

as $tarrow+0$ .

We are interested in the following problems on well-posednessto (N-S);

Problems.

(I) Uniqueness and regularity of weak solutions(II) Global existence of regular solutions for large data
(III) Blow-up; dose there exist $T_{*}<\infty$ such thatfor $0<t<T_{*}$ , but$u(t)\in C^{\infty}(\mathbb{R}^{n})$$a$$u(T_{*})\not\in C^{\infty}(\mathbb{R}^{n})$?

1Uniqueness and regularity
Let us introduce the class $L^{s}(0, T;L^{r})$ with the norm$||\cdot$$||_{L^{s}(0,T;L^{r})}$;$||u||_{L^{s}(0,T;L^{r})}=( \int_{0}^{T}||u(t)||_{r}^{s}dt)^{1/s}=(\int_{0}^{T}(\int_{\mathbb{R}^{n}}|u(x, t)|^{r}dx)^{s/r}dt)^{1/s}$

The classical result on uniqueness and regularity of weak solutions
was given by Foias-Serrin-Masuda [3], [16], [17], [14]:in the class $L^{s}(0, T;L^{r})$.
Theorem 1.1 (Foias-Serrin-Masuda) Let athat u satisfies(i) Let u and v be trno weak solutions of (N-S) on (0, T). Suppose$\in L_{\sigma}^{2}$(1.1)

Assume that.

$u\in L^{s}(0, T;L^{r})$$v$

for $2/s+n/r=1$ with $n<r\leq\infty$ .

fulfills the energy inequality (0.1) for $0\leq t<T$ . Then we$[0_{\backslash }T)$(ii) Eery weak solution(1.2)

for all multi-indices$u$$\frac{\partial u}{\partial t}$of (N-S) in the class (1.1) satisfies,$\frac{\partial^{\alpha_{1}+\cdot\cdot+\alpha_{n}}u}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{n}}}\in C(\mathbb{R}^{n}\cross(0,T))$$\alpha=(\alpha_{1}, \cdots, \alpha_{n})$with$|\alpha|=\alpha_{1}+\cdots+\alpha_{n}\leq 2$

.

have

$u\equiv v$

on60
Remark 1.2 (i) In Theorem 1.1 (i), v need not belong to the class (1.1). On the other
hand, every weak solution u with (1.1) fulfills the energy identity(1.3)$||u(t)||_{2}^{2}+2 \int_{0}^{t}||\nabla u(\tau)||_{2}^{2}d\tau=||a||_{2}^{2}$,$0\leq t\leq T$.

It seems to be an interesting question whether every weak solution satisfies the energy
inequality (0.2).
(ii) If is merely in the Leray-Hopf class, then there exists so, with $2/s_{0}+n/r_{0}=n/2$
such that $u\in L^{s_{0}}(0, T;L^{r_{0}})$ . For example, we may takeand $r_{0}=2r\iota/(7l-2)$ . In
particular, by Therem 1.1 with the aid of interpolation inequality$u$$r_{0}$$s\mathrm{o}=2$$||u||_{L^{r_{0}}(\mathbb{R}^{2})}\leq C||u||_{L^{2}(\mathrm{R}^{2})}^{r_{0}/2}||\nabla u||_{L^{2}(\mathbb{R}^{2})}^{1-r_{0}/2}$.

$2<r\circ<\infty$

for all$u\in H^{1}(\mathbb{R}^{2})$,we see that every weak solution of (N-S) in the 2-dimensional case is unique and regular,
so Problems (I), (II) and (III) are completely solved i. Notice that if ti is regular,then and can be taken arbitrarily large, which makes the quantity $2/s+n/r$ smaller.
(iii) The class (1.1) is important from viewpoint of the scaling invariance. It can beeasily seen that if $\{u,p\}$ is apair of the solution to (N-S) on, then so is thefamily, where$\mathbb{R}^{2}$$\mathrm{n}$

$s$

$r$

$\mathbb{R}^{n}\cross(0, \infty)$$\{u_{\lambda},p_{\lambda}\}_{\lambda>0}$$u\lambda(x, t)\equiv\lambda \mathrm{t}\mathrm{t}(\lambda x, \lambda^{2}t)$,$\mathrm{u}\mathrm{x}(\mathrm{x}, t)\equiv\lambda^{2}p(\lambda x, \lambda^{2}t)$.

Scaling invariance means that there holds$||u_{\lambda}||_{L^{\epsilon}(0,\infty;L^{r})}(=\lambda^{1-(\frac{2}{e}+\frac{n}{\mathrm{r}})}||u||_{L^{s}(L^{r}))}0,\infty j=||u||_{L^{\delta}(0,\infty;L^{r})}$for all$\lambda>0$

if and only if$2/s+n/r=1$ .

The solution $\{u,p\}$ with the property that $u_{\lambda}(x, t)=u(x, t)$ .
for allis called aself-similar solution. For (N-S), the self-similar solution has the form
such as$p_{\lambda}(x, t)=p(\iota\cdot, t)$$\lambda>0$$u(x, t)= \frac{1}{\sqrt{t}}U(\frac{x}{\sqrt{t}})$,$p(x, t)= \frac{1}{t}P(\frac{x}{\sqrt{t}})$.

where $U=$ ( $U^{1}(y)$ , , Un(y)), $P=P(y)$ is the functions forpresisely, the above solution is called aforward self-similar solution.
$\cdots$$y=(y_{1}, \cdots, y_{n})\in \mathbb{R}^{n}$. MoreWe shall next deal with the critical case with $s=\infty$ and $r=n$ in (1.1).

Theorem 1.3 (Masuda [14],$[11]$ , [12]) Let.
(i) (uniqueness) Let and be weak solutions of (N-S). Suppose that $u\in L^{\infty}(0, T;L^{n})$satisfies the energy inequality (0.2) for $0\leq t<T$ . Then weon$\mathrm{K}\mathrm{o}\mathrm{z}\mathrm{o}\mathrm{n}\infty \mathrm{S}\mathrm{o}\mathrm{h}\mathrm{r}$

$u$

$v$

$1\mathrm{n}\Pi 1\backslash andthat$$a\in L_{\sigma}^{2}$$v$$l\iota ave$$u\equiv v$

61(ii) (regularity) There exists a positive constant(N-S) in $L^{\infty}(0.T;L^{n})$ with the property
(1.4)
then

$\epsilon_{0}$$\lim_{tarrow t_{*}}\sup_{-0}||u(t)||_{n}^{n}\leq||u(t_{*})||_{n}^{n}+\in 0$

$u$

for

is a weak solution of$u$$t_{*}\in(0, T)$,satisfies

(1.4)

where

such that if$\frac{\partial^{1}u}{\partial t}\backslash$$\alpha=$for some $\rho>0$ ,$\frac{\dot{(}p_{1}+\cdots+\alpha_{n}u}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n^{n}}^{\alpha}}\in C(\mathbb{R}^{n}\cross(t_{*}-\rho, t_{*}+\rho))$is an arbitrar$ry$ multi-index with $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}\leq 2$ . In
has the property (1.4) for every $t_{*}\in(0, T)$ , then is regular on$(\alpha_{1}, \cdots, \alpha_{n})$particular, if
as in (1.2).

$u$$\mathbb{R}^{n}\cross(0., T)$$u$Remark 1.4 (i) Masuda [14] proved that if $u\in L^{\infty}(0.T;L^{n})$ is continuous from the right’on0-Sohr [11][$0.T)$ . Later on,
on $[0, T)$ in the norm of , then there holds
$((],
T)$
becomes necessarilyof (N-S) on
showed that every weak solution incontinuous from the right in the norm of .
(ii) By the above theorem, every weak solution in $C([0, T);L^{n})$ is unique and regular.
proofThis was proved by Giga [5] and von Wahl [20]. In Section 2, we shall give anotherby adifferent method.
(iii) Recently, Hishida-Izumida [8] improved the condition (1.4). They proved regularity of under tlle weaker assumption that$\mathrm{K}\mathrm{o}\mathrm{z}$$u\equiv v\mathrm{o}11$$L^{n}$$u$$L^{\infty}(0, T;L^{n})$$L^{n}$$u$$\lim_{tarrow t_{*}}\inf_{-0}||u(t)||_{n}^{7\downarrow\leq}||u(t_{*})||_{n}^{n}+\epsilon \mathrm{i}_{0}$.

It seems to be an interesting question whether or not every weak solution
is regular.

$\tau\iota\in L^{\infty}(0, T;L^{n})$Finally in this section, we investigate the size of singular sets of weak solutions in the
we denote by $S(u)$ the singular
3-dimensi0nal case. For aweak solution iset defined by$u$$S(u)\equiv${$(x,$ $t)\in \mathbb{R}^{3}\cross(0$,$\mathrm{n}$$\mathbb{R}^{3}\cross(0, T)$$T);u\not\in L^{\infty}(B_{\rho}$

( , )
$x$

$t$

$)$

for

$\forall\rho>0$

where $B_{\rho}(x, t)=\{(y_{\backslash }s)\in \mathbb{R}^{3}\cross(0.T);|y-il\cdot| <\rho, |s -t|<\rho\}$. For each
$S_{t}(u)=\{x\in \mathbb{R}^{3}; (x, t)\in S(u)\}$},$t\in(0, T)$.

Theorem 1.5 (Neustupa [15]) Let $=3$ . There is an absolute constantweak solution in $L^{\infty}(0, T;L^{3})$ fulfills
that$r\iota$$e\uparrow\prime e\gamma\cdot y$$u$$\# S_{t}(u)\leq(\frac{1}{\epsilon_{0}}\cdot\sup_{0<\tau<\mathrm{T}^{1}}||u(\tau)||_{3})^{3}$for all $t\in(0.T)$ . Here$\# S$

we setdenotes the number of elements of the set$S$.

$\epsilon 0$$>0$ such

62
Remark 1.6 (i) Caffarelli-Kohn-Nirenberg [2] showed if the weak solution u satisfies thegeneralized energy inequality(1.6) 2$\int\int_{\mathrm{R}^{3}\mathrm{x}(0.T)}|\nabla u|^{2}\phi dxdt\leq\int\int_{\mathbb{R}^{3}\mathrm{x}(0,T)}[|u|^{2}(\partial_{t}\phi+\Delta\phi)+(|\tau\iota|^{2}+2p)u\cdot\nabla\phi]dxdt$

for allwith, then $H^{1}(6’)=0$ , where $H^{1}(S)$ denotes the
one-dimensional Hausdorff measure of the set in tlie space-time.
(ii) Taniuchi [19] found aclass of weak solutions satisfying (1.6). His class is larger
than that of Serrin’s (1.1).
$\phi\in C_{0}^{\infty}(\mathbb{R}^{3}\cross(0, T))$$\phi\geq 0$$S$$\mathbb{R}^{3}\cross(0, \infty)$

Finally in this section, we investigate local properties of weak solutions in. Let
be aweak solution of (N-S) on. We callaregular point if there
are $\delta>0$ and $\rho>0$ such that$\mathbb{R}^{3}$$u$$(0, \mathrm{I}^{1})$$\frac{\partial u}{\partial t}$,$(\mathrm{a}\mathrm{r}\mathrm{O}, t_{0})\in \mathbb{R}^{\mathrm{i}3}\cross(0, T)$$\frac{\partial^{\alpha_{1}+\cdots+\alpha_{\iota}}\cdot \mathrm{e}\iota}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{r\iota}^{\alpha_{\iota}}’}\in C(B_{\delta}(x_{0})\cross(t_{0}-\rho.t_{0}+\rho))$for all multi-indiceswith. Herc $B_{\delta}(x_{0})=\{y\in$. The pointis called singular unless it is regular, is alled
on aspace-time $Q=D\cross(a, b)$ if every point of is aregular one.
$\alpha=(\alpha_{1}, \cdots, \alpha_{n})$$\mathbb{R}^{3};|y-x\mathrm{o}|<\delta\}$$|\alpha|=\alpha_{1}+\cdots+\alpha_{r\}}\leq 2$$(x\circ, fo)$$u$$\mathrm{c}$$7^{\cdot}e,gular$$Q$

Theorem 1.7 (Kozono [10]) Let $n=3$ . There is an absolute constantthe following property. If is etteak solution of (N-S) on
and
$u$

$a$

$(0, \mathrm{I}’)$$(x_{0}, t_{0})\in \mathbb{R}^{3}\cross(0, T)$(1.7)$if\uparrow\iota$$\sup_{t_{0}-\rho<t<t_{0}+\rho}||u(t)||_{L_{\dot{\mathrm{W}}}^{3}(B_{\delta}(x\mathrm{o}))}\leq\epsilon_{0}$for some
weakwithsatisfies at$\epsilon_{0}>0$$L^{3}$$\delta>0$-norm

and $\rho>0$ , thenis a regular point. Here$(x_{0}, t_{0})$$||\cdot||_{L_{\mathrm{W}}^{3}(B_{\delta}(x_{0}))}denot,es$$||u||_{L_{\mathrm{W}}^{3}(B_{\delta}(x_{0}))}= \sup_{R>0}R\mu\{x\in B_{\delta}(x_{0});|u(x)|>R\}^{\frac{1}{3}}$the( ;Lebesgue measure).
$l^{l}$

Corollary 1.8 (Removable Singularities) Let $n=3$ . There is an absolute constantwith the following property. Suppose that is a weak solutionof (N-S) on $(0, T)$ . If
( $x0$ , to) is an isolated singular point of satisfying$\epsilon 0$

$u$

$u$

(1.8)

$\lim_{xarrow x_{0}},\sup_{tarrow t_{0}}|x-x_{0}||u(x, t)|<\epsilon_{0}$thenis a regular point.
In particular, if tz behaves at,$(x_{0}, t_{0})$(1.9)

unifomly with respect to$(x_{0}, t_{0})$like$u(x, t)=o(|x-x_{0}|^{-1})$$t$

as

in some neighbourhood of$xarrow x_{0}$$t_{0}$, then$(x_{0}, t_{0})$is a regular point.

63
Remark 1.9 (i) Serrin [16] and Takahashi [18] showed that every weak solution u of(N-S) satisfying$\int_{a}^{b}(\int_{D}|u(x, t)|^{r}dx)^{\frac{\theta}{r}}dt<\infty$on acylinder $D\cross(a, b)\subset\Omega\cross(0, T)$ ,in the space variables. Our theorem deals
for $2/s+3/r\leq 1$ with $r>3$ is of classisand $r=3$ . Furthermore, our weak spacewith the marginal case when$L^{3}(D)$regularityof. Under the condition (1.7), we obtain interior
larger than the usual
not only in the space but also in the space-time variables, while Serrin [16] imposed the
additional assumption that
$C^{\infty}$$L_{\mathrm{W}}^{3}(D)$$s=\circ \mathrm{p}$$u$for some $s\geq 1$ .

$\partial_{t}u\in Ls(a, b;L^{2}(D))$

with the following(ii) Caffarelli-Kohn-Nirenberg [2] gave an absolute constanterty. Let be aweak solution of (N-S) on $(0, T)$ with the generalized energy inequality(1.6). Suppose that and its the associated pressure satisfy$\epsilon_{1}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{I}\succ$

$u$

$u$

$p$

$R^{-2} \iint_{Q_{R}(x_{\mathrm{O}},t_{\mathrm{O}})}(|u|^{3}+|u||p|)dxdt+R^{-13/4}\int_{t_{0}-R^{2}}^{t_{0}}(\int_{|x-x_{0}|<R}|p|dx)^{5/4}dt$(1.10)$\leq$$\epsilon_{1}$.

where $Q_{R}(x_{0}, t_{0})=\{(x, t);|x-x_{0}|<R, t_{0}-R^{2}<t<t_{0}\}$ denotes the parabolic cylinder.
Then is regular in $Q_{R/2}(x_{0}, t_{0})$ . In Theorem 1.7 we do not need any energy inequality
and show that the condition on the pressure is redundant. Moreover, the advantage ofof such as
our theorem enables us to handle the singularity
$u$

$p$

$(x_{0}, t_{0})$$u(x, t)=\mathrm{o}(|x-x_{0}|^{-1})$

asuniformly with respect to in some neighbourhood of
we have in (1.10)
their paper because for such
$t$

$u$

$xarrow x_{0}$$t_{0}$, the case of which is excluded in$(x_{0}, t_{0})$$\int\int_{Q_{R}(x_{0},t_{0})}|u(x, t)|^{3}dxdt=\infty$.

2Local existence and uniqueness of strong solutions
In this section, we investigate the solution with (1.1). To this end, we define the strong
solutions.

. A measurable function
Definition 2.1 Let
strong solution of (N-S) on $(0, T)$ if$a\in L_{\sigma}^{n}$$u$

defined on$\mathbb{R}^{n}\cross(0, T)$

(i)
(2.1)

$u\in C([0, T);L_{\sigma}^{n})$,$\frac{\partial u}{\partial t}$, $Au\in C((0, T);L_{\sigma}^{n})$ ;is called

$a$

64
(ii)

$u$

satisfies$-\partial u\tau t+Au+P(u\cdot\nabla u)=0$

(2.2)

$\{$

,

in

$L_{\sigma}^{n}$$u(0)=a$ .

for $0<t<T$ ,In the above definition, $P$ denotes the Helm holtz-Weyl projection from$1<r<\infty$ . More precisely, $P=\{P_{jk}\}_{j,k=1,\cdots,n}$ can be represented as,, $j=1$ ,whereis the Kronecker symbol and$A=-P\Delta$transforms(F; Fourier transform).
is the Stokes operator.

$L^{r}$

for. $+RjRk,$ ,onto$L_{\sigma}^{r}$$P_{jk}=\delta_{jk}$$R_{j}=F^{-1}( \frac{\sqrt{-1}\xi_{j}}{|\xi|^{2}}F)$$\delta_{jk}$$\cdots$$n$

are thc Riesz

Remark 2.2 It is easy to see that every strong solution u of (N-S) on (0, T) is regular asin (1.2).
Concerning the existence and uniqueness of the strong solution, we have
Theorem 2.3 (Kato [9], Giga-Miyakawa [6], Brezis [1]) For $n<r<\infty$ , there isconstant $\gamma=\gamma(n, r)>0$ with the following property. If the initial data
and $T_{*}>0$
satisfy(2.3)$a$$a\in L_{\sigma}^{7l}$$\sup_{0<t\leq T_{*}}t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{})}.||e^{-tA}a||_{r}<\gamma$

then there exists a unique strong solution $u(t)$ of (N-S) on
solution has the property
with. Moreover, such$a$$t^{\frac{n}{2}(\frac{1}{n}-^{\underline{1}})}..u(\cdot)\in C([0, T_{*});L^{r})$

$u$

(2.4)

$\lim_{tarrow+0}t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||u(t)||_{r}=0$

If, in addition,$(0, T_{*})$$[0, T_{*})$$a\in L_{\sigma}^{n}\cap L_{\sigma}^{2}$.

satisfies (2.3), then$u$.

is also a weak solution of (N-S) on

Under the condition (2.3) we can construct astrong solution on the intervalby thesuccessive approximation. To verify (2.3), we make use of the following $L^{p}-L^{r}$ -estimates;
for the Stokes semigroup$(0, \prime l_{*}^{1})$$u$$\{e^{-tA}\}_{t\geq 0}$(2.5)$||e^{-tA}a||_{r}\leq Ct^{-\frac{n}{2}(1/p-1/r)}||a||_{p}$,$1\leq p\leq r\leq\infty$$\{$$||\nabla e^{-tA}a||_{r}\leq Ct^{-\frac{n}{2}(1/p-1/r)-1/2}||a||_{p}$,hold for all
and all $t>0$ , where $C=C(n,p, r)$ . Hence, if$n<r<\infty$ , then (2.3) can be achieved in such away that$a\in L_{\sigma}^{p}$(2.6)$a\in L_{\sigma}^{n}\cap L^{r}$for some$T_{*}=( \frac{\gamma}{C||a||_{r}})^{\frac{2r}{r-n}}$with the same constant $C$ as in (2.5). If
we can takeso that$\tilde{a}\in C_{0,\sigma}^{\infty}$$t^{\frac{n}{2}\mathrm{t}\frac{1}{n}-\frac{1}{r})}||e^{-tA}a||_{r}$

(2.3)

,

$1\leq p\leq r<\infty$$\leq$$\leq$$\leq$, by the density argument, for every. Hence by (2.5) with $p=n$ , we have$a\in L_{\sigma}^{n}$$||a-\tilde{a}||_{n}<\epsilon$$t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||e^{-tA}(a-\tilde{a})||_{r}+t^{\frac{n}{2}(\frac{1}{l}-^{\underline{1}})}...||e^{-tA}\tilde{a}||_{r}$$C||a-\tilde{a}||_{n}+Ct^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||\tilde{a}||_{r}$$C\epsilon$$+Ct^{\frac{\triangleright}{2}(\frac{1}{n}-\frac{1}{r})}.||\tilde{a}||_{r}$,$\epsilon$$>0$ ,

65
which yields$\lim\sup[] \mathrm{H}^{(\mathrm{n}}\mathrm{g}$)$||etAa||_{r}\ovalbox{\tt\small REJECT}$

Ce. Since

is arbitrary, we obtain$\epsilon$$t-+0$(2.8)$\lim_{tarrow+0}t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||e^{-tA}a||_{r}=0$. However, this convergence is not uniform
which ensures existence of in (2.3) forfor. So, it is not clear whether the interval
for in any fixed bounded subset ofcan be characterized in terms ofexistence of strong solution with the initial data
-norm of such as (2.6). To overcome this difficulty, Brezis [1] considered aclassthe.
of precompact subsets in$a\in L_{\sigma}^{n}$$T_{*}$$T_{*}$$L_{\sigma}^{n}$$a$$a\in L_{\sigma}^{n}$$L^{n}$$a$$L_{\sigma}^{l}’$there
Proposition 2.4 (Brezis) Let $n<r<\infty$ . For every precompact set $K$ in
$t>0$
with
of
exists a monotone non-decreasing and $unifo7mly$ bounded function
such that$L_{\acute{\sigma}}^{\prime\iota}$$\delta_{r}(t;K)$$\iotaarrow+01\mathrm{i}_{1}\mathrm{n}\delta_{r}(t;K)=0$(2.9)$t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||e^{-tA}a||_{r}\leq\delta_{r}$( ; If)$t$holds for all $a\in K$ and all $t>0$ . In particular, we can take $T_{*}=T_{*}(K)$ so that (2.3)
holds for all $a\in K$ .
can be given by the following definition
Proof.
$\delta_{r}(t;K)$$\delta_{r}(t;K)\equiv\sup_{a\in K}(\sup_{0<\tau\leq\dagger}\tau^{\frac{n}{2}(\frac{1}{n}-\frac{1}{?})}||e^{-\tau A}a||_{r})$, it is bounded. Hence there is aconstant $L>0$. By (2.5) we see that the right hand side of the above
is well-defined with

Indeed, since $K$ is precom pact infor allstich thatdefinition is finite and that$||a||_{n}\leq L$$a\in K$

$L_{\sigma}^{n}$$5\mathrm{r}(\mathrm{t};K)$$\delta_{r}(t;K)\leq CL$,$\forall t,$

$>0$

.

This implies uniform boudedness. Obviously by definition,
decreasing function of $t>0$ . Now, it suffices to show that$tarrow+01\mathrm{i}_{111}\delta_{r}(t; K)=0$$\delta_{r}(t;K)$

is amonotone non-.

$=\{b\in L_{\sigma}^{7l}; ||b-a||_{n}<\epsilon\}$ . For any. Since$>0$ , there holds
Letsuch that,,,is compact, we can select finitely many points
for all
that
me
,
may
assu
we
is dense in. Since$a\inK$
there is some
. For any
$1\leq j\leq m$ . Define
as in (2.7)
way
same
the
in
have
we
such
. For
$1\leq j_{0}\leq m$ such thatwith the aid of (2.5$\overline{K}\subset\bigcup_{a\in\overline{K}}U_{\epsilon}(a)$$\epsilon$$U_{\vee}\sim(0)$$a_{1}(\epsilon)$$\overline{K}$$\overline{K}\subset\bigcup_{j=1}^{m}U_{\epsilon}(a_{j}(\epsilon))$$a_{2}(\epsilon)$$a_{m}(\epsilon)\in\overline{h}’$$a_{j}(\epsilon)\in C_{0,\sigma}^{\infty}$$L_{\sigma}^{n}$$C_{0,\sigma}^{\infty}$$\cdots$$M_{\epsilon}\equiv{\rm Max}\{||a_{1}(\epsilon)||_{r}, \cdots, ||a_{m}(\epsilon)||_{r}\}$$a\in U_{\epsilon}(a_{j_{0}}(\epsilon))$$\tau^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||e^{-\tau A}a||_{r}$$\leq$$\leq$$\leq$$j_{0}$$C||a-a_{j\mathrm{o}}(\epsilon)||_{n}+\tau^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||e^{-\tau A}a_{j\mathrm{o}}(\epsilon)||_{r}$$C,\epsilon$$C,\epsilon$$+C\tau^{\frac{\prime l}{2}(\frac{1}{n}-^{\underline{1}})}’||a_{j\mathrm{o}}(\epsilon)||_{r}$$+C\Lambda I_{\wedge}t^{\frac{n}{2}(\frac{1}{n\iota}--\frac{1}{})}\vee$’66
for all $0<\tau\leq t$ . Taking the supremum of the above estimate for $\tau\in(0, t]$ and $a\in K$ , weobtain$\delta_{r}(t;K)\leq C\epsilon+CM_{\epsilon}t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}.$

Lettingin both sides of the above, we have$tarrow+\mathrm{O}$.

$1\mathrm{i}1\mathrm{n}\llcorner\backslash ^{\urcorner}\mathrm{u}\mathrm{p}\delta_{r}(t;K)\mathrm{t}arrow+0\leq Ce$. Since$\overline{\mathrm{c}}>0$

isarbitrary, this implies that$\lim_{tarrow+0}\delta_{r}(t;K)=0$.
$\square$Proposition 2.4 has two applications. One is refinement of the classical theorem on uniqueness of strong solutions, and another is simplification of the proof of regularity criterion on
weak solutions in $C([0,1^{\tau});L^{n})$ . Although both of them are relatively well known for theexperts of the Navier-Stokes equations, we give here asketch of proofs. In particular, weshould notice that our investigation is closely related to the question on regularity given
by Remark 1.4 (iii).
First, we consider uniqueness of strong solutions in Theorem 2.3. In the classical resultof Fujita-Kato [4] and Kato [9], they imposed the restriction (2.4) on the behaviour near$t=0$ offor $n<r<\infty$ . Later on, Brezis [1] showed that (2.4) is redundant by
proving that every strong solution of (N-S) necessarily fulfills (2.4).
By Duhamel’s principle, (2.2) can be reduced to the following integral equation.
$||u(t)||_{r}$$u$(2.10),$\mathrm{u}(\mathrm{t})=e^{-tA}a-\int_{0}^{t}e^{-(t-\tau)A}P(u\cdot\nabla u)(\tau)d\tau$$0<t<T$ .

The classical result on existence uniqueness reads as follows.

Theorem 2.5 (Fujita-Kato [4], Kato [9]) Let
and let $n<r<\infty$ .
(i) If andsatisfy (2.3), then we can construct a solution $u(t)$ of (2.10) on
in the classwith the pr.operty (2.4).
$((0, T);L^{r})$ .
(ii) Suppose that is a solution of (2.10) in
If satisfies(2.4), then is the only solution of (2.10).
$a\in L_{\sigma}^{n}$$T_{*}$$a$$[\mathrm{t}\mathrm{I}, \prime l_{*}\urcorner)$$C([0, T_{*});L_{\sigma}^{n})\cap \mathrm{C}((0, T_{*});L^{r})$$u$$C([0,\mathit{1}^{\tau});L_{\sigma}^{n})\cap C$
’

$u$

$u$

To show that (2.4) is redundant for uniqueness, we need
Proposition 2.6 Let $K$ be a precompact set inand let $n<r<\infty$ . Suppose thatis the same function of $t>0$ as in Proposition 2.4. Then there exists $T_{*}>0$
such that for every $a\in K$ we can construct a solution $u(t)$ of (2.10) on $[0, T_{*})$ in the class$C([0, T_{*});L_{\sigma}^{n})\cap C((0, T_{*});L^{r})$ . Moreover,stich a solution satisfies$L_{\sigma}^{n}$$\delta_{r}(t;K)$(2.11)

In particular,$t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||u(t)||_{r}\leq 2\delta_{r}(t;K)$$u$

for all $0<t<T_{*}$ .

fulfills (2.4).

Remark 2.7 This proposition asserts that the time-intervalof existence of solutions
to (2.10) can be taken unifomly on each precompact subset K of the initial data in.
$T_{*}$$L_{\sigma}^{n}$

67, we can choose $T_{*}>0$ so thatis amonotone, where is the same constant as in (2.3). Since
non-decreasing of , we have by (2.9) that

Proof of Proposition 2.6. Since$\delta_{r}(T_{*}; K)<\gamma$$\lim_{tarrow+0}\delta_{f}(t;K)=0$$\delta_{f}(t;K)$$\gamma$$t$

for all $a\in K$ .

$\sup_{0<t<T_{*}}t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||e^{-tA}a||_{r}<\gamma$

Then it follows from Theorem 2.5 (i) that for every $a\in K$ there is asolution $u(t)$ of (2.10)
on $[0, T_{*})$ in the class $C([0, T_{*});L_{\sigma}^{n})\cap C((0, T_{*});L^{r})$ with the property (2.4). Let us defineby$l|\prime I(t)$$M(t) \equiv\sup_{0<\tau\leq t}\tau^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||u(\tau)||_{r}$.

By (2.4), we see that $M\in C([0, T_{*}))$ . Then by (2.5) and (2.10) there holds$||u(t)||_{r}$$\leq$$||e^{-tA}a||_{r}+ \int_{0}^{t}||P\nabla\cdot e^{-(t-\tau)A}(u\otimes u)(\tau)||_{r}d\tau$$\leq$$||e^{-tA}a||_{r}+C \int_{0}^{t}(t-\tau)^{-\frac{n}{2r}-\frac{1}{2}}||u(\tau)||_{r}^{2}d\tau$$\leq$$||e^{-tA}a||_{r}+CM(t)^{2} \int_{0}^{t}(t-\tau)^{-\frac{n}{2r}-\frac{1}{2}}\tau^{\frac{n}{r}-1}d\tau$$\leq$$||e^{-tA}a||_{r}+C\beta M(t)^{2}t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}$,$0<t<T_{*}$ ,where $\beta=B(1/2-n/2r, n/r)$ , $C=C(n, r)$ . Applying Proposition 2.4 to the above
estimate, we have,$t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{7})}||u(t)||_{r}\leq\delta_{r}(t;K)+C\beta M(t)^{2}$

Since both$\delta_{r}(t;K)$

and(2.12)

Since$\Lambda I(t)$are non-decreasing functions of $t>0$ , this implies$M(t)\leq\delta_{r}(t;K)+C\beta M(t)^{2}$$tarrow+01\mathrm{i}\mathrm{n}1\delta_{r}(t;K)=0$$0<t<T_{*}$ .

, we may assume$T_{*}$$0<t<T_{*}$ .

,satisfies also$\delta_{r}(T_{*}; K)<\frac{1}{4C\beta}$.

Hence by (2. 12), there holds(2.13)$\Lambda I(t)$$\leq$$\frac{1-\sqrt{1-4C\beta\delta_{r}(t,K)}}{2C\beta}.(\leq 2\delta_{r}(t;K))$or(2.14)$M(t)$$\geq$$\frac{1+\sqrt{1-4C\beta\delta_{r}(t\cdot K)}}{2C\beta},(\geq\frac{1}{2C\beta})$

for all $0<t<T_{*}$ . Since $M(t)$ is continuous on $[0, T_{*})$ with $\lim_{tarrow+0}M(t)=0$ (see (2.4)),
the latter case (2.14) cannot occur. Hence we obtain from (2.13)$M(t)\leq 2\delta_{r}(t;K)$,$0<\forall t<T_{*}$

68
This proves Proposition 2.6.
Because of Theorem 2.5 (ii), to prove assertion on uniqueness in Theorem 2.3, we may
show the following lemma.

and let $n<r<\infty$ . Every solution
Lemma 2.8 (Brezis [1]) Let
the class $C([0, T);L_{\sigma}^{n})\cap C((0, T);L^{r})$ fulfills (24).
$a\in L_{\sigma}^{n}$

Proof. We first define

$K$

$u$

of (2.10) in

as$K\equiv \mathrm{u}(\mathrm{t})0<t<\prime I/2\}$.

Since $u\in C([0, T);L_{\sigma}^{n})$ , $K$ is aprecompact subset of . For this $K$ , we take the function
given by Proposition 2.4. Furthermore, by Proposition 2.6 we can take $T_{*}>0$. Let us denote thisfor every initial dataof (2.10) on
and asolution$L_{\sigma}^{n}$$\delta_{r}(t;K)$$\tilde{u}(t)$$\tilde{a}\in K$$(0, T_{*})$$\tilde{u}(t)$by$\tilde{u}(t)\equiv S(t)\tilde{a}$,$0<t<T_{*}$By (2.11), there holds(2.15)$t^{\frac{n}{2}\mathrm{t}\frac{1}{n}-\frac{1}{r})}||S(t)\tilde{a}||_{r}\leq 2\delta_{r}(t;K)$,$0<t<T*$

arbitrarily as $0<s<{\rm Min}.\{T/2, T_{*}\}$ . Then we have
. Hence it follows$u(s)\in K$ . Since $u\in C((0, T);L^{r})$ , we see

for all$\tilde{a}\in K$. Let us take$s$$\lim_{tarrow+0}t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||\tau\iota(t+s)||_{r}=0$

from Theorem 2.5 (ii) and definition of the map $S(t)$ that$u(t+s)=S(t)u(s)$ ,$0<t<T_{*}$.

From (2.15) we obtain$t^{\frac{\prime l}{2}\mathrm{t}\frac{1}{n}-\frac{1}{r})}||u(t+s)||_{r}=t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||S(t)u(s)||_{r}\leq 2\delta_{r}(t;K)$

Since $u\in C((0, T);L^{r})$ , by letting$sarrow \mathrm{O}$,$0<t\leq T_{*}$.

in the above estimate we have$t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||u(t)||_{r}\leq 2\delta_{r}(t;K)$,$0<\forall t<T_{*}$.

Since $\lim_{tarrow+0}\delta_{r}(t;K)=0$, this yields$\lim_{tarrow+0}t^{\frac{n}{2}\mathrm{t},}\frac{1}{n}-^{\underline{1}}.)||u(t)||_{r}=0$.
$\square$

We shall next apply Proposition 2.4 to the proof of regularity of weak solutions in
Theorem 2.9 (Giga [5], von Wahl [20])Let
in $C([0, T);L^{n})$ is regular as in (1.2)$a\in L_{\sigma}^{2}$. Ever weak solution$ry$$C([0, \mathrm{I}’);L^{n})$$u$of (N-S).

69
Proof. Let us define the set$K$by$K=\{u(t):0<t<T\}$ .

. We take some
, $K$ is aprecompact subset ofSince $u\in C([0, T);L^{n})$ with
$T_{*}=T_{*}(K,r)$ such$n<r<\infty$ . Then it follows from Proposition 2.4 that there exists
that,(2.16)$L_{\sigma}^{n}$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$$\sup_{0<t<T_{*}}t^{\frac{n}{2}(\frac{1}{n}-\frac{1}{r})}||e^{-tA}a||_{r}\leq\delta_{r}(T*;K)<\gamma$

for all $a\in K$ where is the same constant as in (2.3). Let
we have by (2.16) that

. For every$t_{*}\in(0, \mathrm{I}’)$of (N-S) with$v|_{t-t_{*}-\rho}--=$$\rho\equiv T_{*}/2$$\gamma$$\sup t^{\frac{1}{2}(\frac{1}{n}-\frac{1}{1^{\cdot}})}’||e^{-tA}u(t_{*}-\rho)||_{r}<\gamma$.

$0<t<T_{*}$

By Theorem 2.3 and Remark 2.2, there exists astrong solution$u(t_{*}-\rho)$ such that
(2.17)$v\in C([t_{*}-/J, t_{*}+\rho);L^{n})$,$\frac{\partial\iota)}{\partial t}$,$v$$\frac{\partial^{\alpha_{1}+\cdot+\alpha_{n_{?f}}}}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{1l}}}\in C(\mathbb{R}^{n}\cross(t_{*}-\rho.t_{*}+\rho))$is an arbitrary multi-index with $|\alpha|=\alpha_{1}4-\cdots+\alpha_{n}\leq 2$ . Notice
wherethat is also aweak solution. Then uniqueness result of Theorem 1.3 (i) yields$\alpha=(c\nu_{1}, \cdots, \alpha_{n})$$v$$\mathrm{u}(\mathrm{t})\equiv v(t)$Since$t_{*}\in$$(0, T)$for $t\in[t_{*}-\rho, t_{*}+\rho)$ .

can be taken arbitrarily, we can conclude that$u$is regular as in (1.2).
口

To deal with the problem on regularity of weak solutions in $L^{\infty}(0, T;L^{n})$ , the above proof
proposes us the following question.
Question. For every weak solution in $L^{\infty}(0, T:L^{7l})$ is the set$u$$K=\{u(t);0<t<T\}$

precompact in$L_{\sigma}^{n}$?

References[1] Brezis, H., Remarks on the preceding paper by M. $Ben$ -Artzi, “Global solutions of
trnO-dimension al Navier-Stokes and Euler equations”. Arch. Rational Mech. Anal.
128, 359 360 (1994).

[2] Caffarelli, L., Kohn, R., Nirenberg, L., Partial regularity of suitable weak solutions of
the Navier-Stokes equations. Conlnl. Pure Appl. Math. 35, 771-831 (1982).
[3] C. Foias, Une remarque sur Vunicite des solutions des equations de Navier-Stoke s en
dimension n, Bull. Soc. Math. France 89 (1961), 1-8

70[4] Fujita, H., Kato, T., On the Navier-Stokes initial value problem 1. Arch. Rational
Mech. Anal. 16, 269-315 (1964).
[5] Giga, Y., Solutions for semilinear parabolic equations inand regularity of weaksolutions of the Navier-Stokes system. J. Differential Eq. 62, 182-212 (1986).
$L^{p}$[6] Giga, Y., Miyakawa, T., Solution inof the Navier-Stokes initial value problem.
Arch. Rational Mech. Anal. 89, 267 -281 (1985).
$L_{r}$[7] Hopf, E., Uber die Anfangswertaufgabe fiir die hydrodyanamischen Grundgleichungen.
Math. Nach. 4, 213-231 (1950/51).
[8] Hishida, T., Izumida, K., Remarks on a regularity criterion for weak solutions to theNavier-Stokes equations inAnalysis 20, 191 200 (2000).
$\mathbb{R}^{n}$[9] Kato, T., Strong -solutions of the Navier-Stokes equation in
to weak solutions. Math. Z. 187, 471-480 (1984).
$L^{p}$$R^{m}$, with applications[10] Kozono, H., Removable singularities of weak solutions to the Navier-Stokes equations.
Communications in Partial Differential Equations 23 (1998), 949-966.

[11] Kozono, H., Sohr, H., Remark on uniqueness of weak solutions to the Navier-Stokes
equations. Analysis 16, 255 -271 (1996).
[12] Kozono, H., Sohr, H., Regularity criterion on weak solutions to the Navier-Stokes
equations. Advances in Diff. Eq. 2, 535-554(1997).

[13] Leray, J., Sur le mouvement d’un liquide visqeux emplissant Vespace. Acta. Math.
63. )3-248 (1934).
$1^{(}.$[14] Masuda, K., Weak solutions of Navier-Stokes equations. Tohoku Math. J. 36, 623 646(1984).
[15] Neustupa, J., Partial regularity of weak solutions to the Navier-Stokes equations in
the class. J. Math. Fluid Mech. 1, 1-17 (1999).
$L^{\infty}(0, T;L^{3}(\Omega))$[16] Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations.
Arch. Rational Mech. Anal. 9,187-195 (1962).
[17] Serrin, J., The initial value problem for the Navier-Stokes equations. Nonlinear
Problems, R. E. Langer ed., Madison: University of Wisconsin Press, 69-98 (1963).

[18] Takahashi, S., On interior regularity criteria for weak solutions of the Navier-Stokes
equations. Manuscripta Math.69, 237-254 (1990).
[19] Taniuchi, Y.,
On generalized energy equality
Manuscripta Math. 94, 365-384 (1997).

of the Navier-Stokes equations.

[20] Wahl, W. von, Regularity of weak solutions of the Navier-Stokes equations. Proceedings of the 1983 Summer Institute on Nonlinear Functional Analysis and Applications,
Proc. Symposia in Pure Mathematics 45, F. E. Brower ed., Providence Rhode Island:Amer. Math. Soc. 497-503 (1986).