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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1998/15\hfil Stability estimate for strong
solutions
\hfil\folio}
\def\leftheadline{\folio\hfil Tadashi Kawanago
 \hfil EJDE--1998/15}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1998}(1998), No.~15, pp.~1--23.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or
http://ejde.math.unt.edu
\hfill\break 
ftp (login: ftp) 147.26.103.110 or 129.120.3.113\bigskip} }

\topmatter
\title
Stability estimate for strong solutions of the Navier-Stokes system
and its applications
\endtitle

\thanks 
{\it 1991 Mathematics Subject Classifications:} 35Q30, 76D05.\hfil\break\indent
{\it Key words and phrases:} Navier-Stokes system, strong solutions, 
stability, uniqueness, \hfil\break\indent non-blowup condition.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and
University of North Texas.\hfil\break\indent
Submitted February 17, 1998. Published June 3, 1998.
\endthanks
\author Tadashi Kawanago   \endauthor
\address Tadashi Kawanago  \hfill\break
Department of Applied Mathematics \hfill\break
Faculty of Engineering, Shizuoka University \hfill\break
Hamamatsu 432, Japan\endaddress 
\email tstkawa\@eng.shizuoka.ac.jp
\endemail

\abstract
We obtain a `stability estimate' for strong solutions
of the Navier--Stokes system, which is an $L^\alpha$-version, 
$1< \alpha <\infty$, of the  estimate that Serrin [Se] used in obtaining 
uniqueness of weak solutions to the Navier-Stokes system.
By applying this estimate, we obtain new results in stability and
uniqueness
of solutions, and non-blowup conditions for strong solutions. 
\endabstract
\endtopmatter

\def\e{\varepsilon}
\def\; {\,;\,}
\def\del {\partial}
\def\ddt {\frac {d}{dt}}
\def\tmax {t_{\text{max}}}
\def\glons {C_0([0,\infty)  \, ; \, PL^N)}

\document

\head 1. Introduction \endhead
We consider the  Navier-Stokes system in ${\Bbb R}^N$ ($N\ge 2$),
$$
\gathered
u_t-\Delta u + (u\cdot \nabla)u +\nabla \pi =0 \quad
\text{in }{\Bbb R}^N\times{\Bbb R}^+, \\
 \nabla\cdot u =0 \quad\text{in }{\Bbb R}^N\times {\Bbb R}^+,\\
u(x,0)=u_0 (x) \quad\text{in }{\Bbb R}^N\,,
\endgathered \tag NS
$$
where $u(x,t) =(u_1,\cdots,u_N)$  is the velocity field and
$\pi(x,t)$ is the scalar pressure.
Let $P$ be the Helmholtz projection, and  let $\|\cdot\|_p$ denote
the $L^p({\Bbb R}^N)$ norm. Kato [K]  showed that
for any $u_0\in PL^{N}$ the problem (NS)  has a unique local mild
 solution  
$$u(t \; u_0)\in C([0,T) \; PL^{N}) \cap L^r((0,T) \; PL^{q})\,,$$
 where $q, r > N$ and $N/q + 2/r=1$.
He also proved in [K, the end note] that 
if $\|u_0\|_N$ is sufficiently small then  
$$u(t;u_0) \in\glons:=
\{u\in C([0,\infty) \; PL^N) \ \; \lim_{t\to \infty} \, \|u(t)\|_N=0 
\}\,,$$ 
(see Theorem 2.1 below). See Section 2 for the definition of mild
solution.
This unique (local) mild solution also has the smoothing effect and is
regular (see Remark 2.3). Therefore, we  call
it {\it strong solution} of (NS) for the remaining of this paper.

Our main result  is Theorem 3.1 which establishes the 
`stability estimate'  (which we call) for strong
solutions just mentioned. This estimate leads us
to  corollaries on uniqueness and stability, and to a non-blowup
condition for strong solutions. 
First, we state a new uniqueness result.  

\proclaim {Corollary 1.1. (Uniqueness)} 
Mild solutions of (NS) are unique in the space 
$C([0,T) \; PL^N) \cap L^r_{\text{loc}}((0,T) \; PL^q)$ with 
a pair of numbers $(q,r)$ that satisfies 
$$
N<q<\infty \quad\text{and}\quad  \frac Nq + \frac 2 r = 1.   \tag 1.1
$$
\endproclaim
See Theorem 2.1 below for previous uniqueness results.
Our improvement consists of imposing fewer restriction on the behavior
of solution near $t=0$.
This uniqueness result and its proof were suggested in [B] and [K2,
Introduction]. 
Our proof in Section 3 is, however, 
different from the one suggested in [B] and [K2].
Next, we give a stability result.

\proclaim{Corollary 1.2. (Global Stability)} 
Let  $u(t;u_0)\in \glons$ be a global strong solution of  (NS).
We have the following.

 \item{(i)} There exist  constants
$\delta_0=\delta_0(N,u_0)\in{\Bbb R}^+$
and $C_0=C_0(N)\in{\Bbb R}^+$ such that if
$$
v_0\in PL^N \quad\text{and}\quad \|v_0-u_0\|_{N}\le \delta_0\,,  
$$
then  we have $u(t \; v_0)\in\glons$ and
$$
\|u(t \,; v_0)-u(t;u_0) \|_{N} \le \|v_0-u_0\|_{N}
\exp\,(C_0 \int_0^t \|u(s \,; u_0)\|_{N+2}^{N+2}\,ds)  \tag 1.2
$$
 for all $t\in {\Bbb R}^+$.

\item{(ii)}   In addition, we assume that $u_0\in PL^N \cap PL^{\alpha}$
for a constant $\alpha \in (1,\infty) - \{ N \}$.
Then there exist  constants $\delta_1=\delta_1(N,\alpha,u_0)\in (0,
\delta_0]$, $q=q(N,\alpha) \in (N,\infty)$
and $C_1=C_1(N,\alpha)\in{\Bbb R}^+$  such that if
$$
v_0\in PL^N \cap PL^{\alpha} \quad\text{and}\quad \|v_0-u_0\|_{N}\le
\delta_1  
$$
then  for $\alpha\in [2,\infty)$ we have 
$$
\|u(t \,; v_0)-u(t;u_0) \|_{\alpha} \le \|v_0-u_0\|_{\alpha}
\exp\,(C_1 \int_0^t \|u(s \,; u_0)\|_{q}^{r}\,ds)  \tag 1.3
$$
 for all $t\in {\Bbb R}^+$ and for $\alpha\in (1,2)$  we have (1.3) with 
 the norm$\|\cdot\|_{\alpha}$ replaced by $|\cdot|_{\alpha}$ 
for all $t\in {\Bbb R}^+$.
Here, $r$ is the constant that satisfies (1.1).
\endproclaim

See Notation (just after this section) for the difference between
$\|\cdot\|_{\alpha}$ and $|\cdot|_{\alpha}$. 
Note that $u(t;u_0) \in \glons$ implies $u(t;u_0) \in L^r ({\Bbb R}^+ \;
PL^q)$
for any $(q,r)$ satisfying (1.1) (see Proposition 2.2).
Global strong solutions in the class $\glons$ are  important since 
all global strong  solutions belong to this class provided that 
$2\le N\le 4$ and $u_0\in PL^2\cap PL^N$ (see Propositions 4.1 and 4.2).
Corollary 1.2 gives the first global $L^N$-stability result.
Related global $L^2 \cap L^q$-stability results with 
$q>N$ were given in [VS, Theorem A], [Wi, Theorem 2]. 
Global $H^1$-stability results for $N=3$ can be found in [PRST].  
We remark  that our  estimates (1.2) and (1.3) 
are simpler than those in the previous works, and we clarify that the
$L^{\alpha}$-estimate holds if  the $L^N$-norm of $v_0 - u_0$ is small.
In Section 4 we give another version of Corollary 1.2 for non-global
solutions (see Corollary 4.1). 

        We also have the following result. Let $[0, \tmax (u_0))$
be  the maximal interval in which the strong solution $u(t;u_0)$ exists.

\proclaim{Corollary 1.3 (Non-blowup condition) }
Let the strong solution $u(t;u_0)$ exist on $[0,T)$ with $T< \infty$. 
Then,  we have $\tmax (u_0) > T$  if and only if
$ u(t;u_0) \in L^{r} ( (0,T) \;  PL^{q})$. Here, $(q,r)$ is  a pair of
numbers that satisfies (1.1).
\endproclaim

This result has a `global'-version (see Proposition 2.2). For example,
we can apply Corollary 1.3 to obtain that $u(t;u_0) \in C_0([0,\infty)
\;
PL^2)$
for any $u_0\in PL^2$ when $N=2$ (see Proposition 4.1).

      The contents of this paper is presented as follows. In Section 2, 
preliminary results;  In Section 3, statement and proof of main
result; In Section 4, applications of our stability estimate
and proofs of Corollaries 1.1-1.3.  Notice that part of the contents of 
this paper was announced in [Ka3] and [Ka4].

\subhead Notation \endsubhead
\item{1.} ${\Bbb R}^+:=(0,\infty)$.
\item{2.} $L^q:=L^q({\Bbb R}^N\, \; {\Bbb R})$ or $L^q({\Bbb R}^N  \;
 {\Bbb R}^N)$.
\item{3.}  $f\in L_{\text{loc} }^r ((0,T) \, \; L^q)$ means
            $f\in L^r ((\e,T) \, \; L^q)$ for any $\e\in (0,T)$.
\item{4.}  We often write $C=C(\alpha,\beta,\gamma,\cdots)$ to indicate
that
 $C$ depends only on $\alpha,\beta,\gamma,\cdots$.
\item{5.}  For a Banach space $V$ with the norm $\|\cdot \|$ we set
$$
C_0([0,\infty) \; V) := \{u\in C([0,\infty) \; V) \;  \lim_{t\to \infty}
\, \|u(t)\|
=0  \}.
$$
\item{6.}  $P$ is the Helmholtz projection, i.e. the continuous
projection
from $L^p$ onto $\{ u=(u_1,\cdots,u_N)\in L^p \; \nabla\cdot u =0\}$. 
\item{7.} We denote by $u(t;u_0)$ the strong solution, i.e. the unique
mild
solution of 
(NS)  whose existence is ensured by Theorem 2.1. See Definitions 2.2.

\item{8.} For $u=(u_1,\cdots,u_N)\in L^q({\Bbb R}^N \; {\Bbb R}^N)$ we
write
$$ \gather
|u|:=\sqrt{|u_1|^2 + \cdots + |u_N|^2}, 
\quad   | \nabla u | := ( \sum_{i,j=1}^N | \del u_i / \del x_j |^2
)^{1/2}\\
|u|_q := ( \int |u|^q dx)^{1/q}
\quad\text{and}\quad \|u(t)\|_q:= (\sum_{j=1}^N \int |u_j|^q dx)^{1/q}.
\endgather$$
Note that $|\cdot |_q$ and $\| \cdot \|_q$ are equivalent $L^q$-norms.
\item{9.}  We often write $|u|^{q-1}u = u^q$ $(0<q<\infty)$ for vector
(or
scalar) $u$.
\item{10.} $\del_j := \del / \del x_j$.


\head 2. Preliminaries \endhead

\subhead Definition 2.1\endsubhead A mild solution
$u$ of (NS) on $[0,T)$ is a function $u\in C([0,T) \; PL^N)$
satisfying the  integral equation
$$\align
u(t) &=e^{t\Delta}u_0 - \int_0^t e^{(t-s)\Delta} 
P(u\cdot\nabla)u(s)ds                    \tag 2.1 \\
     &=e^{t\Delta}u_0 - \int_0^t P\nabla\cdot e^{(t-s)\Delta}
         (u\cdot u(s))ds
\endalign$$
for $t\in (0,T)$. Here, we assume that there exists a constant
$\alpha\in (1,\infty)$ such that 
$$
P\nabla\cdot e^{(t-\cdot)\Delta}(u\cdot u(\cdot)) 
\in L^1 ((0,t) \; PL^{\alpha})  \quad\text{for}\quad t\in (0,T).    \tag
2.2
$$

\subhead  Remark 2.1 \endsubhead
\item{(i)} The second equality in (2.1)
holds since $(u\cdot \nabla) u_j = \nabla\cdot (u_ju)$. 
\item{(ii)} The value of integral in (2.1) is 
independent of the choice of $\alpha$,  and is unique. 
We understand that for each $t\in [0,T)$ the equalities (2.1) hold 
for a.e. $x\in{\Bbb R}^N$.
The assumption $u\in C([0,T) \; PL^N)$ guarantees 
(2.2) for e.g. $\alpha =2N/3$
 in view of the next $L^p$-$L^q$ estimate
$$\aligned
\| P\nabla\cdot e^{(t-s)\Delta} (u\cdot u(s)) \|_{\alpha}   
\le&  C(t-s)^{-3/2+N/2\alpha}\|u\cdot u(s)\|_{N/2} \\           
\le&  C(t-s)^{-3/2+N/2\alpha}\|u(s)\|_N^2.  \endaligned\tag 2.3
$$
\item{(iii)}  The next Theorem 2.1 ensures that 
for any $u_0\in PL^N$
Problem (NS) has a local mild solution. 

\noindent The following result is a small extension of the results in
[K] and [G].

\proclaim{Theorem 2.1}
\item{(i)} (Existence) Let $1<\alpha <\infty$.
For any $u_0\in PL^{N} \cap PL^{\alpha}$ there exists $T\in{\Bbb R}^+$
 such that  (NS)  has a unique local mild solution 
$$
u(t)\in C([0,T] \; PL^{N} \cap PL^{\alpha}) \cap L^r((0,T) \; PL^q) 
\tag
2.4
$$ 
for any pair of numbers $(q,r)$  satisfying (1.1), and
$$
t^{(1-N/q)/2}u \in C([0,T] \; PL^q) 
\quad\text{with \, the \, value \, zero \, at}\quad t=0       \tag 2.5
$$
for any $q\in (N,\infty]$.

 \item{(ii)} (Uniqueness) \hfill\break
  (ii-a) Mild solutions of (NS) are unique in 
$C([0,T] \; PL^N) \cap L^r((0,T) \; PL^q)$ with 
a pair of numbers $(q,r)$ satisfying (1.1). \hfill\break
  (ii-b)  Mild solutions $u(t)\in C([0,T] \; PL^{N})$ of (NS) 
satisfying (2.5) for a number $q\in (N,\infty)$ are unique.

\item{(iii)} (Existence of global solutions) There exists a 
constant $\e_*=\e_*(N) \in{\Bbb R}^+$ such that 
if $\|u_0\|_N \le \e_*$ then 
(NS)  has a unique global mild solution 
$$
u(t)\in \glons \cap L^r({\Bbb R}^+ \; PL^q). 
$$ 
Here,  $(q,r)$ is any pair of numbers  which  satisfies (1.1).
\endproclaim

\subhead Remark 2.2\endsubhead
\item{(i)} More precisely, Giga [G] obtained the  
uniqueness of  solutions of (NS) in the class
$L^r((0,T) \; PL^q)$ which satisfy the integral equation (2.1).

\item{(ii)} Kato [K] and Giga [G] obtained Theorem 2.1 with (1.1)
replaced by
$$N< q < N^2/(N-2) \quad and \quad    N /q +  2 /r = 1\,, \tag 1.1'$$
which is more restrictive than (1.1).
We obtain Theorem 2.1 from these previous results and Lemma 2.1 below.

\subhead Definition 2.2\endsubhead
For $u_0\in PL^N$ we denote by $u(t;u_0)$ the unique mild solution
of (NS) satisfying (2.4).  We call $u(t;u_0)$ the {\it strong solution}
of
(NS).
We set 
$$
\tmax (u_0) :=\sup \{  T\in{\Bbb R}^+ \; u(t;u_0) 
\text{ exists on the time interval } [0,T]  \},
$$
i.e., $[0,\tmax(u_0))$ be the maximal time-interval
where the strong solution $u(t;u_0)$ exists.

\subhead Remark 2.3\endsubhead
\item{(i)} The strong solution $u(t;u_0)$ is regular for
$t\in (0,\tmax(u_0))$ (see e.g. [GM]). 
\item{(ii)} The strong solution has the semigroup property, i.e. 
$u(t \; u(s \; u_0)\,) =$ \linebreak
$u(s+t \; u_0)$ for $s,t >0$ and $s+t < \tmax(u_0)$.

\proclaim{Lemma 2.1} Let $u(t)$ be a mild solution on $[0,T)$. Assume
that
$u\in L^r((0,T) \; PL^q)$ with {\it a} pair of numbers $(q,r)$satisfying
(1.1). Then we have $u\in L^r((0,T) \; PL^q)$ for {\it any} pair of 
numbers $(q,r)$ satisfying  (1.1).
\endproclaim

\noindent{\bf Proof.} By (2.1), $u(t) = e^{t\Delta}u_0 - I(t)$,
where
$$
I(t):=  \int_0^t P\nabla\cdot e^{(t-s)\Delta} (u\cdot u(s)) \,ds.    
$$
In [G, Lemma (p.196)] we find that 
$$
e^{t\Delta}u_0\in L^r((0,T) \; PL^q)
$$
for any pair of numbers $(q,r)$ which satisfy (1.1).
We will estimate $I(t)$. Let $q' \ge q/2$. We apply the 
$L^p$-$L^q$ estimate  and have
$$
\| I(t) \|_{q'} \le C\int_0^t (t-s)^{-1/2- (2/q -1/q' )N/2} \| u(s)
\|_q^2 \, ds\,.
$$
Here, $C=C(N,q,q')\in{\Bbb R}^+$ is a constant.
By the generalized Young inequality (see [RS, p.31]) 
we have $(P) \Rightarrow (Q)$ and 
$(R) \Rightarrow (S)$, where $(P), (Q), (R), (S)$ are defined as
follows:

\item{(P)} $u\in L^r((0,T) \; PL^q)$ with  a  $(q,r)$ satisfying
(1.1) and $2N\le q$

\item{(Q)} $u\in L^{r'}((0,T) \; PL^{q'})$ for  any  $(q',r')$
satisfying
 $q/2 \le  q'$ and
$$
N <q' < \infty  \quad\text{and}\quad  \frac{N}{q' } + \frac{2}{r'} =1 
\tag 2.6
$$
\item{(R)} $u\in L^r((0,T) \; PL^q)$ with  a  $(q,r)$ satisfying
(1.1) and $N<q\le 2N$

\item{(S)} $u\in L^{r'}((0,T) \; PL^{q'})$ for  any  $(q',r')$
satisfying
 (2.6) and $q' < Nq/(2N-q)$.

It suffices to show that $u\in L^4((0,T) \; PL^{2N})$ is necessary and
sufficient condition
for $u\in L^r((0,T) \; PL^q)$ with a  pair of numbers $(q,r)$ satisfying
(1.1).  The sufficiency is obvious by $(P) \Rightarrow (Q)$  with
setting $q=2N$. 
We obtain the necessity in the case $q>2N$ 
(resp. $N<q<2N$) by applying $(P) \Rightarrow (Q)$ 
(resp.  $(R) \Rightarrow (S)\,$)  finitely many times. \qed

 The next result shows how $\tmax (u_0)$ depend on $u_0$.


\proclaim{Proposition 2.1}
Let $u_0\in PL^N$ and $q\in (N,\infty)$. Then there exists a
number $S_q=S_q(N,q)\in{\Bbb R}^+$ such that for any $\tau \in{\Bbb
R}^+$ if 
$$
t^{(1-N/q)/2} \|e^{t\Delta} u_0\|_q \le S_q  \quad\text{for}\quad t\in
[0,\tau]   \tag 2.7
$$ 
then the strong solution $u(t;u_0)$ exists on the time interval
$[0,\tau]$.
\endproclaim

\noindent{\bf Sketch of Proof.} 
We can verify this result by observing carefully the method for
the construction of strong solutions in the proof of [K, Theorem 1]
and [G, Section 2]. For the convenience of the reader,
we will sketch the proof.

     By induction we define the sequence of functions 
$\{U_n(t)\}_{n=0}^{\infty}$:
$$\align
U_0(t)&:=e^{t\Delta}u_0, \quad U_{n+1}(t):= U_0(t)
- \int_0^t P\nabla\cdot e^{(t-s)\Delta}
         (U_n \cdot U_n(s))ds.
\endalign$$
Set $\nu :=(1-N/q)/2$ and 
$K_n:=\sup_{t\in[0,\tau]} t^{\nu} \|U_n(t)\|_q$.
We denote by $C_j\in{\Bbb R}^+$ $(j=1,2,\cdots)$ constants depending
only
on $N$ and $q$. It follows from the usual $L^p$-$L^q$ estimates
(see e.g. [K, (2.3) and (2.3)']) that
$$\align
\|U_{n+1}(t)\|_q 
                  &\le \|U_0(t)\|_q 
         +  C_1 \int_0^t (t-s)^{-(1-\nu)} \|U_n(s)\|_q^2 ds \tag 2.8 \\
    &\le \|U_0(t)\|_q 
         + C_1 K_n^2 \int_0^t (t-s)^{-(1-\nu)} s^{-2\nu} ds 
\endalign$$
for $t \ge 0$.
It follows that
$$
K_{n+1} \le K_0 + C_2 K_n^2.
$$
If the algebraic equation $x= K_0 + C_2 x^2$ has real solutions,
i.e. $1-4C_2 K_0 \ge 0$ then we see by induction that 
$\{K_n\}_{n=1}^{\infty}$ is bounded and 
$$
K_n\le K_* := \frac{1-\sqrt{1-4C_2 K_0}}{2C_2} \quad\text{for}\quad n\ge
0\,.
$$
Here, $K_*$ is the smaller solution of the algebraic equation.
By an estimation similar to (2.8),
$$\align
&\|U_{n+1}(t) - U_{n}(t)\|_q        \\
    \le \, &  C_1 \int_0^t (t-s)^{-(1-\nu)} 
        (\|U_{n+1}(s)\|_q +\|U_{n}(s)\|_q) \|U_{n+1}(s) - U_{n}(s)\|_q\,
ds \\
    \le \, & 2C_1 K_* (\int_0^t (t-s)^{-(1-\nu)} s^{-2\nu} ds )
                \sup_{s\in [0,\tau]} \|U_{n+1}(s) - U_{n}(s)\|_q
\endalign$$
for $t\in [0,\tau]$, which leads to 
$$
\sup_{t\in [0,\tau]} t^{\nu} \|U_{n+1}(t) - U_{n}(t)\|_q
\le 2C_2 K_* \sup_{t\in [0,\tau]} t^{\nu} \|U_{n+1}(t) - U_{n}(t)\|_q\,.
$$
Now, set $S_q := K_0 < 1/4C_2$. Then $2C_2  K_*  <1$.
Therefore,  $\{ t^{\nu} U_n(t) \}_{n=1}^{\infty}$ is a Cauchy sequence
in
$C([0,\tau] \; PL^q)$ and $u(t) = \lim_{\,n\to\infty} U_n(t)$ exists on
$[0,\tau]$.  We verify that $u(t)$ is a strong solution on $[0,\tau]$,
i.e. $u(t) = u(t;u_0)$ (see [G]).
\qed

\subhead Remark 2.4\endsubhead  Under the same assumption of Proposition
2.1
we have
$$
\lim_{t\to +0} t^{(1-N/q)/2} \|e^{t\Delta}u_0\|_q =0.      \tag 2.9 
$$
This well-known result follows from the density of $L^q \cap L^N$ in
$L^N$
and the  estimate (2.11) below.

From Proposition 2.1, we  can immediately obtain 

\proclaim{Corollary 2.1}Let  $u(t;u_0)$  be a strong
solution on $[0,T)$ with $T < \infty$. Then statements (a) - (f) 
are equivalent.

 \item{(a)}  $\lim_{\,t \to +0} \sup_{s\in (T-\e,\,T)} 
t^{(1-N/q)/2}
\| e^{t\Delta} u(s\,; u_0)\|_q =0$ \quad for some $q\in (N,\infty)$ and 
$\quad \e\in (0,T)$.

\item{(b)} $\lim_{\,t \to +0} \sup_{s\in (T-\e,\,T)} 
t^{(1-N/q)/2} \| e^{t\Delta} u(s\,; u_0)\|_q  < S_q$
\quad for some $q\in (N,\infty)$ and $\quad \e\in (0,T)$,
where $S_q$ is the same constant in the statement of 
Proposition 2.1.

 \item{(c)} $\tmax(u_0) > T$, i.e. the strong solution $u(t;u_0)$ 
exists on $[0,\,T+\delta)$ with  a small constant $\delta\in{\Bbb R}^+$.

\item{(d)} $\liminf_{\, t\to T-0} \| u(t;u_0) \|_q < \infty $
 for some   \quad    $q\in (N,\infty)$.

\item{(e)} $\limsup_{t\to T-0} \| u(t;u_0) \|_{\infty} < \infty$.

\item{(f)} $\lim_{t\to T-0} u(t;u_0)$ exists in $L^N$.
\endproclaim

\noindent{\bf Proof}. First we will prove the equivalence of (a), (b) and
(c).

        $(a)\Rightarrow (b)$: This is obvious.
     
       $(b)\Rightarrow (c)$: We can choose $\delta >0$ satisfying 
$$
\sup_{s\in (T-\e, \,T)} t^{(1-N/q)/2} 
\| e^{t\Delta} u(s\,; u_0)\|_r \le S_q  \quad\text{for}\quad t\in
[0,\delta).
$$
 Thus, it follows from Proposition 2.1 and the semigroup property of the
mild solution (see Remark 2.3 (ii) ) that $u(t;u_0)$
exists on $[0, \,T+\delta)$.
\par
        $(c)\Rightarrow (a)$: By (c) and the definition of the strong
solution
(see Definition 2.2)  we have
$$
\sup_{s\in (T-\e, T)} \| u(s\,;u_0) \|_q < \infty.       \tag 2.10
$$
We obtain (a) from (2.10) and the basic inequality:
$$
\| e^{t\Delta} f \|_q  \le \|f\|_q.     \tag 2.11
$$
Thus, (a), (b) and (c) are equivalent.
\par
       $(c)\Rightarrow (d)$ and $(e)$ and $(f)$:  
This is obvious from Theorem 2.1.
\par
        $(d) \Rightarrow (c)$: 
We fix $\tau\in{\Bbb R}^+$ so small that 
$$
\liminf_{t\to T-0} \| u(t;u_0) \|_q < \tau^{-(1-N/q)/2} S_q .
$$
Then, we choose a constant $s$ such that
$$
T - \frac{\tau}{2} < s< T \quad\text{and}\quad  \| u(s\,;u_0) \|_q <
\tau^{-(1-N/q)/2} S_q .
$$
It follows that 
$$
t^{(1-N/q)/2} \| e^{t\Delta} u(s\,;u_0) \|_q \le t^{(1-N/q)/2} \|
u(s\,;u_0) \|_q  
< S_q \quad\text{for}\quad t\in [0,\tau].
$$
Thus, by Proposition 2.1 $u(t;u_0)$ exists on $[\,0,\, T + \tau /2 \,]$.
\par
       $(e) \Rightarrow (d)$:
Let $\e >0$ be a small number which will be determined later.
We set $u(t) := u(t;u_0)$. We have
$$
u(t+T-\e) = e^{t\Delta}u(T-\e)
  - \int_0^t P\nabla\cdot e^{(t-s)\Delta} 
(u\cdot u(s+T-\e))ds  \tag 2.12 
$$
for $t\in (0,\e)$. We fix a number $q\in (N,\infty)$.
We have
$$
\|u(t+T-\e)\|_q \le \|u(T-\e)\|_q + 
  C_0\int_0^t(t-s)^{-1/2} 
    \|u(s+T-\e)\|_{\infty} \|u(s+T-\e)\|_{q} ds,  \tag 2.13
$$
where $C_0=C_0(N,q)\in{\Bbb R}^+$.
We set
$$\gather
C_{\e}=2C_0\e^{1/2}
   \sup\,\{ \|u(\tau)\|_{\infty} \; \tau\in [T-\e, T) \},
\\
M(t)=\sup\,\{ \|u(\tau)\|_{q} \; \tau\in [T-\e, T-\e+t) \}.
\endgather $$
It follows from (2.13) that 
$$
M(t) \le \|u(T-\e)\|_{q} + C_{\e}M(t) \quad\text{for}\quad t\in
[0,\e).     \tag 2.14
$$
We fix $\e>0$ so small that $C_{\e} <1$. Then (2.14)
implies $M(\e) < \infty$. Thus, we obtain (d).
\par
         $(f) \Rightarrow (b)$: 
We write $u(t):=u(t;u_0)$ and
$$\gather
u(T):= \lim_{t\to T-0} u(t) \in PL^N.
\\
t^{(1-N/q)/2} \| e^{t\Delta} u(s\,;u_0) \|_q  
\le t^{(1-N/q)/2} \| e^{t\Delta} (u(s)-u(T)) \|_q + t^{(1-N/q)/2} \|
e^{t\Delta} u(T) \|_q. \endgather
$$
It follows from the $L^p$-$L^q$ estimate and  (2.9) that
$$ \gather
t^{(1-N/q)/2} \| e^{t\Delta} (u(s) -u(T)) \|_q \le C_1 \|  u(s)
-u(T)\|_N,
\\
\lim_{t\to +0} t^{(1-N/q)/2} \| e^{t\Delta} u(T) \|_q =0\,.\endgather
$$
Thus, we have (b). The proof is complete. \qed

\subhead Remark 2.5\endsubhead  It seems to be an open problem
whether (c) is equivalent to (d) with $q=n$.

          We will characterize the strong solutions belonging to
$\glons$.
The next result is a `global'-version of Corollary 1.3.

\proclaim{Proposition 2.2} Let $u(t)=u(t;u_0)$ be a global strong
solution of  (NS).
Then, we have $u\in \glons$ if and only if $u\in L^r({\Bbb R}^+ \;
PL^q)$
for a pair of (or equivalently for any pair of) numbers $(q,r)$  
satisfying (1.1). 
\endproclaim

\subhead Remark 2.6\endsubhead
      When $N=3$, Ponce et al [PRST] obtained a similar result
under an assumption: $u_0\in PL^2\cap H^1$.

\noindent{\bf Proof of Proposition 2.2.} 
Let $u\in \glons$.  Fix a constant $T\in{\Bbb R}^+$ such that
$\|u(T)\|_N \le \e_*$, where $\e_*$ is the constant 
appeared in the 
statement of Theorem 2.1.  Set $u(T)$ as the initial value
and apply Theorem 2.1 (iii). Then we have \linebreak
$u\in L^r([T,\infty) \; PL^q)$.
Combining this with $u\in L^r([0,T) \; PL^q)$, we conclude that 
$u\in L^r({\Bbb R}^+ \; PL^q)$.
\par
        Next, we will prove the inverse. 
Let $u \in L^r({\Bbb R}^+ \; PL^q)$. In view of Lemma 2.1,
we can assume without loss of generality that $N<q \le 2N$.
 Although the essence of the proof below is 
given in the proof of [K, Theorem 2'], we will describe 
it for the convenience of the reader. 
Applying the $L^p$-$L^q$ estimate to (2.1), we have
$$
\|u(t)\|_{N} \le \|e^{t\Delta}u_0\|_{N} 
      + C_1\int_0^t (t-s)^{-(2/q-1/N)N/2-1/2}
\|u(s)^2\|_{q/2}ds :=I_1(t)+I_2(t).
$$
The function $I_1(t)$ is a decreasing function. Moreover, we  can easily
verify
  from  the density of $L^1 \cap L^N$ in $L^N$ 
that 
$$
I_1(t)\to 0 \quad\text{as}\quad t\to \infty.        \tag 2.15
$$
Now we estimate $I_2(t)$.
By the H\"older's inequality, we obtain
$$\align
\frac 1 T \int_0^T I_2(t)dt 
       &= \frac{C_1}{(1-N/q)T} \int_0^T (T-s)^{1-N/q}\|u(s)\|_q^2 ds 
                                                  \tag 2.16 
\\
       &\le \frac{C_1}{1-N/q} T^{-N/q} \int_0^T \|u(s)\|_q^2 ds  \\
       &\le \frac{C_1}{1-N/q} \, 
           \biggl (\int_0^T \|u(s)\|_q^r \,ds\, \biggr)^{2/r}.
\endalign$$
It follows from (2.15) and (2.16) that
$$
\liminf_{t\to\infty} \|u(t)\|_N \le 
      \frac{C_1}{1-N/r} \, \biggl(\int_0^{\infty} \|u(s)\|_q^r ds
\biggr)^{2/r}.  
                      \tag 2.17
$$
If  we choose $u(T)$ as the initial value then we obtain from the same 
argument as above that
$$
 \liminf_{t\to\infty} \|u(t)\|_N \le 
      \frac{C_1}{1-N/q}\, \biggl(\int_T^{\infty} \|u(s)\|_q^r ds
\biggr)^{2/r}.  
                     \tag 2.18
$$
Here, $T\in{\Bbb R}^+$ is any constant. Therefore, we have 
$$
\liminf_{t\to\infty} \|u(t)\|_N =0.   
$$
It follows from Theorem 2.1 (iii) that 
$u(t;u_0)\in\glons$. \qed

        \heading 3. Main result and its proof\endheading

        Our main result is the following:

\proclaim{Theorem 3.1. (Stability estimate)}
Let $\alpha\in (1,\infty)$ be a constant, and let $u_0$,  $v_0$ be in 
$PL^N \cap PL^{\alpha}$. 
Let $q \in (N,q_*]$ be a constant, 
where $q_*=q_*(N,\alpha)$ is the number defined by
$$q_* =
\left\{
\aligned
&\frac{N\alpha}{(N- \alpha)(\alpha-1)}  
   \quad\text{if}\quad  1 <\alpha < 2 \text{ and } \alpha > N-2,
\\
&\frac{N\alpha}{2(\alpha-1)}  
   \quad\text{if}\quad  1 <\alpha < 2 \text{ and } \alpha \le N-2,
\\
& \infty  \quad\text{if}\quad \alpha=2 \text{ or } [N\le 4 \text{ and }
\alpha \ge N],
\\
& \frac{N\alpha}{\alpha-2}  
   \quad\text{if}\quad  2<\alpha <N-2,
\\
& \frac{N\alpha}{N-4}  
   \quad\text{if}\quad  N-2 \le \alpha  \text{ and } \alpha > 4  \text{
and } N\ge 5,
\\
& \frac{2N\alpha}{(\alpha-2)(N-\alpha)}  \quad\text{if}\quad N-2 \le
\alpha <N \text{ and } 2<\alpha \le
4.
\endaligned\right.
              \tag 3.1
$$
Then  there exist constants
$d_{\alpha q}$, $A_{\alpha q}\in {\Bbb R}^+$ depending only on 
$N$, $\alpha$ and $q$ such that  if
$$
0<T< \min (\,\tmax(u_0), \tmax(v_0) \,)  \tag 3.2
$$
and 
$$
\|u(t \,; v_0)-u(t;u_0) \|_{N} \le d_{\alpha q} \quad\text{for}\quad
t\in
[0,T]  \tag 3.3
$$
then we have the estimate
$$
\|u(t \,; v_0)-u(t;u_0) \|_{\alpha} \le \|v_0-u_0\|_{\alpha}
\exp\,(A_{\alpha q}\int_0^t \|u(s \,; u_0)\|_{q}^{r}\,ds)  
\quad\text{for}\quad t\in [0,T]                                         
\tag 3.4
$$
for the case $\alpha\in [2,\infty)$, and 
$$
|u(t \,; v_0)-u(t;u_0) |_{\alpha} \le |v_0-u_0|_{\alpha}
\exp\,(A_{\alpha q}\int_0^t \|u(s \,; u_0)\|_{q}^{r}\,ds)  
\quad\text{for}\quad t\in
[0,T]                                 \tag 3.5
$$
for the case $\alpha\in (1,2]$.
 Here, $r\in {\Bbb R}^+$ is the constant satisfying (1.1).
\par
        Moreover, for the special case $\alpha =2$ we can take
$d_{\alpha
q}=\infty$.
\endproclaim

\subhead Remark 3.1\endsubhead
\item{(i)} Theorem 3.1 applies to the
 $N$-$\alpha$ region:
$$
\{ \, (N,\alpha) \,;\, N \in {\bold N},\,N \ge 2 \text{ and } 
1 <\alpha < \infty \}.
$$

\item{(ii)} For the special case $\alpha=2$,
the estimate (3.4) was obtained by   [Se, Theorem 6] for more general 
(what we call) weak solutions. 

\item{(iii)} It seems difficult to state the contents of 
Theorem 3.1 by using only one norm $\| \cdot \|_q$ or $| \cdot |_q$.
The main reason is that the estimate (3.9) [resp. (3.31)] below 
does {\it not} hold in the case $1 < \alpha < 2$ [ resp. in the case 
$\alpha >2$].

\item{(iv)} It seems that $\alpha =2$ is the exceptional 
case in which we can take $d_{\alpha q} =\infty$. Indeed, if
$d_{\alpha q} = \infty$
then by setting $u(t;u_0) \equiv 0$ in (3.4) we have
$$
\| u(t;v_0) \|_{\alpha} \le \| v_0 \|_{\alpha}, \quad t \ge 0     \tag
3.6
$$
for any $v_0 \in PL^N \cap PL^{\alpha}$. 
This monotonicity is valid for the special case $\alpha=2$. However,
it does not seem to hold for any $\alpha\not =2$.
To confirm it for each case, we have only to find a initial value $v_0$
which does not satisfy (3.6). Combining the analytical method and
the numerical method, we will show in [Ka5] that
for the two cases $(N,\alpha)=(3,4), (3,3)$ there exist $v_0$ which do
not
satisfy (3.6).  We remark that it is possible to apply the same
arguments 
as in [Ka5] to the other cases $(N,\alpha)$ with $\alpha \not =2$.

\noindent We obtain immediately the following.

\proclaim{Corollary 3.1} Let $\alpha\in (1,\infty)$ and $q\in (N,q_*)$
be  constants and 
$u_0,v_0\in PL^N \cap PL^{\alpha}$. 
We set $d_q:=d_{N q}$ and  $A_q:=A_{N q}$.
Here, we use the notations in the statement of Theorem 3.1.
Assume (3.2)  and
$$
\|v_0-u_0\|_{N}
\exp\,(A_{N+2} \int_0^T \|u(s \,; u_0)\|_{N+2}^{N+2}\,ds)
< \min\, (d_{N+2},d_{\alpha q})
.                                          
\tag 3.7
$$
Then we have (3.4)  [resp. (3.5)] for the case $\alpha \in [2,\infty)$ 
[resp. for the case $\alpha \in (1,2)$].
\endproclaim

\noindent{\bf Proof of Corollary 3.1.}  The proof is complete if we
derive
(3.3). By Theorem 3.1 and (3.7) it suffices to 
obtain (3.4) for $(\alpha, q)=(N, N+2)$.
 To this end we will prove $t_*=T$, where
$$
t_* := \max \{ \tau\in [0,T] \; (3.4) \text{ with } (\alpha, q)=(N, N+2)
  \text{ holds for } t\in [0,\tau] \}.
$$
We proceed by contradiction. Suppose $t_*<T$. Then, it follows from the 
continuity of $\| u(t;v_0) -u(t;u_0) \|_N$ that there exists a small 
constant $\e >0 $ such that \linebreak
$\| u(t;v_0) -u(t;u_0) \|_N < d_{N+2}$ for $t\in [\,0, \,t_* + \e\,]$. 
Therefore,   by Theorem 3.1
we have (3.4) with $(\alpha, q)=(N, N+2)$ for $t\in [\,0, \,t_* +
\e\,]$. 
This contradicts the 
definition of $t_*$. Thus we conclude $t_*=T$. The proof is complete.
\qed \smallskip

 \noindent{\bf Proof of Theorem 3.1.}
Our method of proof is close to the argument in [N] and 
[Ka] for the porous media equations.
Set $w(t):=u(t;v_0) - u(t;u_0)$ and $u(t) :=u(t;u_0)$ for simplicity.
The solutions $u(t;u_0)$ and 
$u(t;v_0)$ are regular for $t\in (0,T]$. In particular,
$w,w_t \in C((0,T] \; W^{2,p})$ for $p \ge \min\, (N,\alpha)$ 
(see e.g. [K] and [GM, Section 3]).
 We verify that $w=(w_1, \cdots ,w_N)$ satisfies 
$$
\gathered
 w_t  -\Delta w + (w\cdot\nabla)w + (u\cdot\nabla)w 
    + (w\cdot\nabla)u 
+\nabla \pi = 0 \quad\text{in } {\Bbb R}^N\times {\Bbb R}^+,  
\\
 \nabla\cdot w =0 \quad\text{in}\quad {\Bbb R}^N\times {\Bbb R}^+.
\endgathered           \tag 3.8
$$
For short notation, we  use $\del_j :=\del/ \del x_j$. 

\subhead Case $\alpha >2$ \endsubhead
We denote  $w_j^p := |w_j|^{p-1}w_j$ $(0< p < \infty)$.
By the integration by parts, we have
$$\align
\frac 1 {\alpha} \ddt \|w(t)\|_{\alpha}^{\alpha}
&=\frac 1 {\alpha} \ddt \sum_{j=1}^N \int_{{\Bbb R}^N} |w_j(t)|^{\alpha} 
= \sum_{j=1}^N \int w_j^{\alpha-1} (w_j)_t \, dx        \tag 3.9 
\\
&= - \,\frac{4(\alpha-1)}{\alpha^2} J(w)^2  - I_1 - I_2 - I_3 - I_4, 
\endalign$$
where we set
$$ \gather
J(w) = (\sum_{j=1}^N \| \nabla w_j^{\alpha/2} \|_2^2)^{1/2}, \qquad
I_1 = \sum_{j=1}^N\int w_j^{\alpha-1}  (w\cdot\nabla)w_j, 
\\
I_2 = \sum_{j=1}^N\int w_j^{\alpha-1} (u\cdot\nabla)w_j,  \quad
I_3 = \sum_{j=1}^N\int w_j^{\alpha-1} (w\cdot\nabla)u_j,  \quad
I_4 = \sum_{j=1}^N\int w_j^{\alpha-1} \del_j\pi\,. \endgather
$$
 We will estimate $I_j$ ($j=1,2,3,4$).
$$
I_1 = \frac{1}{\alpha} \sum_{j=1}^N\int \nabla (|w_j|^{\alpha}) \cdot w
    = - \frac{1}{\alpha} \sum_{j=1}^N\int |w_j|^{\alpha} \, \nabla\cdot
w =0.
$$
It follows from the  integration by parts and the H\"older's inequality
that
$$\align
|I_2|+|I_3| &\le C \,\sum_{j=1}^N\int |u| |w|^{\alpha/2}|\nabla
w_j^{\alpha/2}|
\tag 3.10\\   
    &\le C (\int |u|^{2} |w|^{\alpha})^{1/2} J(w)   
    \le C |u|_q \, |w|_{\alpha q/(q-2)}^{\alpha/2}  J(w).    
\endalign$$
By the H\"older's inequality and the Sobolev inequality,
$$\gather
|w|_{\alpha q/(q-2)} \le |w|_{\alpha }^{1-N/q}  |w|_{N
\alpha/(N-2)}^{N/q},   \tag
3.11
\\
|w|_{N \alpha/(N-2)} = |\,|w|^{\alpha/2} |^{2/\alpha}_{2N/(N-2)} 
        \le C J(w)^{2/\alpha}  \quad\text{for}\quad  N\ge 3.   \tag 3.12
\endgather
$$
Combining (3.11) and (3.12), we have
$$
|w|_{ \alpha q/(q-2)} \le C \|w\|_{\alpha}^{1-N/q} J(w)^{2N/\alpha
q}.      \tag
3.13
$$
This estimate is what we call (a version of) 
 the Gagliardo-Nirenberg inequality (see e.g. [N]). Note that 
(3.12) holds for $N \ge 3$, but {\it not} for $N=2$. However, (3.13)
does hold 
for $N \ge 2$.
In what follows , we often use the H\"older's inequality 
and the Sobolev inequality in the same way  as above in order to make
clear the 
essence of the argument below. Although the case $N=2$
is exceptional in such situation, we will not mention it. 
But  it is  easy to rewrite it rigorously by the same argument as above.
It follows from (3.10) and (3.13) that 
$$
|I_2|+|I_3| \le C \|u\|_q \|w\|_{\alpha}^{\alpha (q-N)/2q} J(w)^{1+N/q}
\le \e J(w)^2 + C_{\e} \|u\|_q^r \|w\|_{\alpha}^{\alpha}.          \tag
3.14
$$
Next we will estimate $I_4$, which is the most difficult part.
It follows from the integration by parts
and the H\"older's inequality  that 
$$\align
| I_4 | &\le \frac{2(\alpha-1)}{\alpha} 
   \sum_{j=1}^N\int |\,\pi \, w_j^{\alpha/2 -1}\del_j w_j^{\alpha/2}|   
\tag
3.15  \\
      &\le \frac{2(\alpha-1)}{\alpha} 
(\,\sum_{j=1}^N\int  \pi^2 |w_j|^{\alpha -2} )^{1/2} J(w)  \\
      &\le C \| \pi^2 \|_a^{1/2} \, \| \,|w|^{\alpha-2}\, \|_{b}^{1/2}
J(w) 
        = C \| \pi \|_{2a} |w|_{b(\alpha-2)}^{\alpha/2-1} J(w). 
\endalign$$
Here, $a$ and $b$ are positive constants which satisfy
$$
\frac{1}{a} + \frac{1}{b} =1 \quad\text{and}\quad 1< b < \infty .   
\tag 3.16
$$
We will later determine  $a$ and $b$.
By (3.8) we have 
$$
-\Delta \pi = \sum_{i,j=1}^{N}\del_j w_i\cdot \del_i (2u_j +w_j) = 
\sum_{i,j=1}^{N} \del_i\del_j [\,w_i(2u_j+w_j)\,].    \tag
3.17
$$
With the aid of the Calder\'on - Zygmund inequality and the 
 H\"older's inequality,
$$
\|\pi\|_{2a} \le C \sum_{i,j} \|w_i(2u_j+w_j)\|_{2a} 
\le C  \|w\|_{4a}^2 + C \|u\|_{q} \|w\|_{2aq/(q-2a)}.    \tag 3.18
$$
Combining (3.15) and (3.18), we have
$$
| I_4 | \le C \|w\|_{4a}^2 \|w\|_{b(\alpha-2)}^{\alpha/2-1} J(w)
       +  C \|u\|_{q} \|w\|_{2aq/(q-2a)}
\|w\|_{b(\alpha-2)}^{\alpha/2-1} J(w). 
                                      \tag 3.19
$$
We choose $a$ and $b$ such that the following two conditions hold. 
\item{(P)} Both $4a$ and $b(\alpha -2)$ are between $N$ and $N\alpha
/(N-2)$

\item{(Q)} Both $2aq/(q-2a)$ and $b(\alpha -2)$ are between $\alpha$ and
$N\alpha
/(N-2)$

\noindent Assuming that (P) and (Q) hold,  from
the H\"older's inequality and (3.12) we obtain 
$$ \gather
\|w\|_{4a} \le 
 \|w\|_N^{1-\theta_1} \|w\|_{N\alpha /(N-2)}^{\theta_1}
\le C \|w\|_N^{1-\theta_1} J(w)^{2 \theta_1/ \alpha},     \tag 3.20
\\
\|w\|_{b(\alpha -2)} \le 
 \|w\|_N^{1-\theta_2} \|w\|_{N\alpha /(N-2)}^{\theta_2}
\le C \|w\|_N^{1-\theta_2} J(w)^{2 \theta_2/ \alpha},     \tag 3.21
\\
\|w\|_{2aq/(q-2a)} \le 
 \|w\|_{\alpha}^{1-\theta_3} \|w\|_{N\alpha /(N-2)}^{\theta_3}
\le C \|w\|_{\alpha}^{1-\theta_3} J(w)^{2 \theta_3/ \alpha},   \tag 3.22 
\\
\|w\|_{b(\alpha -2)} \le 
 \|w\|_{\alpha}^{1-\theta_4} \|w\|_{N\alpha /(N-2)}^{\theta_4}
\le C \|w\|_{\alpha}^{1-\theta_4} J(w)^{2 \theta_4/ \alpha}.  \tag 3.23
\endgather
$$
Here, $\theta_j \in [0,1]$ ($1\le j \le 4$) are constants, and exactly
$(\theta_1, \theta_2,\theta_3,\theta_4) = $
$$
( {{ \alpha  \left(  N -4 a \right) }\over 
     {4 a \left(  N- 2 -  \alpha   \right) }},\,\,
   {{ \alpha  \left( N+2 b -  \alpha  b   \right) }\over 
      {\alpha  b N - 2 b N + 4 b - {{ \alpha }^2} b      }},\,\,
   {N\over 2} - {{ \alpha  N }\over {4 a}} + 
    {{ \alpha  N }\over {2 q}}, \,\,
   {{  N \left( \alpha  b- \alpha  - 2 b    \right) }\over 
      {2 b (\alpha- 2)}} \,). \tag 3.24
$$
It follows from (3.16), (3.19) and (3.20)-(3.23) that
$$\align
|I_4| &\le C \|w\|_N^{2(1-\theta_1) + (\alpha /2 -1)(1- \theta_2)} 
J(w)^{1 \,+ \,4 \theta_1/ \alpha \,+\, (1- 2/ \alpha) \theta_2}\tag
3.25   \\
&\quad + C\|u\|_q \|w\|_{\alpha}^{(1-\theta_3) + (\alpha /2 -1)(1-
\theta_4)} 
         J(w)^{1 + 2 \theta_3/ \alpha + (1- 2/ \alpha) \theta_4}  \\
&=C \|w\|_N  J(w)^2 
+ C\|u\|_q \|w\|_{\alpha}^{\alpha (q-N)/2q}  J(w)^{1 + N/q}.
\endalign$$
Therefore, when $q > N$ we have
$$
|I_4|  \le C \|w\|_N  J(w)^2 
        + \e J(w)^2 + C_{\e} \|u\|_q^r \|w\|_{\alpha}^{\alpha}.\tag
3.26
$$
Thus we obtain 
$$
\frac{1}{\alpha} \ddt \|w(t)\|_{\alpha}^{\alpha}
     \le -(\frac{\alpha -1}{\alpha^2} - C_0 \|w\|_N) J(w)^2 
      + A_{\alpha q} \|u\|_{q}^{r}  \|w\|_{\alpha}^{\alpha}.      
$$
Set $d_{\alpha q} := (\alpha-1)/\alpha^2 C_0$. Then, by (3.3)  we have
$$
\frac{1}{\alpha} \ddt \|w(t)\|_{\alpha}^{\alpha}
     \le  A_{\alpha q} \|u\|_{q}^{r}  \|w\|_{\alpha}^{\alpha}
\quad\text{for}\quad
t\in [0,T]. 
$$
Therefore, we obtain (3.4) for $t\in [0,T]$. 
\par
        Finally, we observe how the conditions 
(P) and (Q)  ($\Longleftrightarrow 0\le \theta_j\le 1$ for $1\le j\le
4$)
 restrict the range of $q$. 
First, we study the case $2 < \alpha < N-2$. We have
$\alpha < {N\alpha}/(N-2) < N$. In order to satisfy (P) and (Q), we need
to
choose
 $b(\alpha -2) = N\alpha /(N-2)$.
Then we have
$$
a=\frac{N\alpha}{2(N+\alpha -2)},\qquad  
(\theta_1, \theta_2,\theta_3,\theta_4) 
=(\,\frac 1 2, \, 1, \, 1-\frac{\alpha (q-N)}{2q}, \,1 \,).
$$
The condition $0 \le \theta_3 \le 1$ is equivalent to 
$N \le q \le q_*:= N\alpha /(\alpha-2)$.
Next, we study the case $N-2 \le \alpha$.
The condition $0 \le \theta_1 \le 1$ is equivalent to 
$$
(N-4)b \le N \quad\text{and}\quad (N\alpha -4N +8)b \ge N\alpha\,. \tag
3.27
$$
By (3.24) we have
$$
\frac{1}{q} = \frac{1}{2} - \frac{1}{\alpha} 
      + \frac{2\theta_3}{N\alpha} - \frac{1}{2b}.   \tag 3.28
$$
Let $N-2 \le \alpha < N$ and $ 2< \alpha \le 4$. Since $\alpha < N \le
N\alpha /(N-2)$,
the conditions (P) and (Q) implies $N \le b(\alpha -2) \le N\alpha
/(N-2)$.
Thus we have
$$
\frac{N}{\alpha-2} \le b \le \frac{N\alpha}{(\alpha-2)(N-2)}.       \tag
3.29
$$
Remark that (3.29) leads to $\theta_2, \theta_4 \in [0,1]$ and (3.27).
We see that $q$ achieves the maximum at $(\theta_3, b)
=(0,N/(\alpha-2))$
and the minimum at $(\theta_3, b) = (1, \frac{N\alpha}{(\alpha-2)(N-2)}
)$.
Thus we have
$$
N < q \le q_* := \frac{2N\alpha}{(N-\alpha)(\alpha-2)}.
$$
Let $N-2 \le \alpha$ and $\alpha > 4$ and $N \ge 5$.
It follows from (3.27) and the condition
$\theta_2, \theta_4 \in [0,1]$ that
$$ \gather
\frac{N\alpha}{N(\alpha -4)+8} \le b \le \frac{N}{N-4},
\\
\frac{\max\,(N,\alpha)}{\alpha -2} \le b \le
\frac{N\alpha}{(N-2)(\alpha-2)}.
\endgather $$
Here, we see that
$$
\frac{\max\,(N,\alpha)}{\alpha -2} \le  \frac{N\alpha}{N(\alpha -4)+8}  
\quad\text{and}\quad 
\frac{N\alpha}{(N-2)(\alpha-2)} \le  \frac{N}{N-4}.
$$
It follows that
$$
 \frac{N\alpha}{N(\alpha -4)+8}  \le b \le
\frac{N\alpha}{(N-2)(\alpha-2)}.
$$
In view of this inequality and (3.28), 
$q$ achieves the maximum at 
$(\theta_3, b) =$ \linebreak
$(0, N\alpha /(N(\alpha -4)+8)\,)$
and the minimum at $(\theta_3, b) =(1, {N\alpha}/{(\alpha-2)(N-2)} )$.
Therefore, we have
$$
N < q \le q_* := \frac{N\alpha}{N-4}.
$$
Finally, let $N\le 4$ and $\alpha \ge N$.
Since $N \le \alpha$, the conditions (P) and (Q) lead to 
$\alpha \le b(\alpha-2) \le N\alpha/(N-2)$.
Thus we have
$$
\frac{\alpha}{\alpha -2} \le b \le \frac{N\alpha}{(N-2)(\alpha-2)},
$$
which implies that $\theta_2,\theta_4 \in [0,1]$
and also that $\theta_1 \in [0,1]$.
By this inequality and (3.28) we see that
$q$ achieves the maximum at $(\theta_3, b) =(0,\alpha/(\alpha-2))$
and the minimum at $(\theta_3, b) =(1, {N\alpha}/{(\alpha-2)(N-2)} )$.
We conclude that
$N < q \le q_* := \infty$.

\subhead Case $ \alpha =2$ \endsubhead
The above estimates of $I_1$, $I_2$, $I_3$ holds in this case.
Moreover, by the integration by parts we have
$$
I_4 = \sum_{j=1}^N \int_{{\Bbb R}^N} w_j \del_j \pi = -\int \pi \,
\nabla\cdot w
=0.
$$
Therefore, we have
$$
\frac 1 2  \ddt \|w(t)\|_2^2 \le A_{2,q} \|u\|_q^r \|w\|_2^2.      \tag
3.30
$$
Here, $q \in (N,\infty]$ is any number. The estimate (3.30) implies
(3.4)
with $\alpha =2$.

 \subhead Case: $1 < \alpha <2$\endsubhead
 We write $w^p := |w|^{p-1} w$ $(1<p<\infty)$ for simplicity.
By Lemma~3.2 (at the end of this section) and integration by parts, we
have
$$\align
\frac{1}{\alpha} \ddt |w(t)|_{\alpha}^{\alpha} &= 
          \frac{1}{\alpha} \ddt \int_{{\Bbb R}^N} |w(t)|^{\alpha}
                     =\int \frac{w \cdot w_t}{|w|^{2-\alpha}}  \tag
3.31   \\
&= (\,\int w^{\alpha -1} \cdot \Delta w \,) - I_1- I_2- I_3- I_4 \\
&\le -(\alpha -1)K(w)^2 - I_1- I_2- I_3- I_4,
\endalign$$
where we set
$$\gather
K(w)= ( \int_{{\Bbb R}^N} \frac{ |\nabla w|^2}{|w|^{2-\alpha}} )^{1/2},
\qquad
I_1 = \int w^{\alpha -1} \cdot (w\cdot \nabla) w = \int w^{\alpha} \cdot
\nabla
|w|,
\\
I_2 = \int w^{\alpha -1} \cdot (u\cdot \nabla) w,
\qquad
I_3 = \int w^{\alpha -1} \cdot (w\cdot \nabla) u,
\qquad
I_4 = \int w^{\alpha -1} \cdot \nabla \pi\,. \endgather
$$
It follows from (3.12) and Lemma 3.2 that
$$
|w|_{N \alpha/(N-2)}  \le C K(w)^{2/\alpha} \quad\text{for}\quad N \ge
3\,.\tag 3.32
$$
We have
$$
|I_1| \le \int |w|^{\alpha} | \nabla w | \le
|w|_{\alpha+2}^{(\alpha+2)/2} K(w)
        \le |w|_N |w|_{N\alpha /(N-2)}^{\alpha/2} K(w) \le C\|w\|_N
K(w)^2. 
\tag 3.33
$$
We note that (3.33) holds also for $N=2$ (see the argument just after
(3.13)\,).
By the integration by parts,
$$
|I_2|+|I_3| \le C\int |u| |w|^{\alpha -1} |\nabla w| 
        \le C (\int |u|^2 |w|^{\alpha})^{1/2} K(w).
$$
This is the same estimate as (3.10).
Therefore, the same argumentation as in the case $\alpha\in (2,\infty)$
leads to
$$
|I_2|+|I_3| \le \e K(w)^2 + C\|u\|_q^r |w|_{\alpha}^{\alpha}.    \tag
3.34
$$
\par
             Next, we will estimate $I_4$ in the similar way as in the 
case: $\alpha \in (2,\infty)$.
We denote by $R_k$ the Riesz operator, i.e.
$$
R_k := \Cal F^{-1} \frac{\xi_k}{|\xi|} \Cal F.
$$
Here, $\Cal F$ is the Fourier transform operator.
It follows from (3.17) that 
$$
- \del_k \pi = \sum_{i,j} R_k R_i \del_j [w_i (2u_j + w_j)]
                 = \sum_{i,j} R_k R_i  [(\del_j w_i) (2u_j + w_j)].   
\tag 3.35
$$
Let $a \in (1,2)$ be a  constant which we will determine later.
By $L^p$-boundedness of the Riesz operator (see [St, Chapter 3]), we
have
$$
|\nabla \pi|_a \le C \sum_{i,j} \| \del_j w_i  (2u_j + w_j) \| _a.   
$$
Therefore, we obtain
$$\aligned
|I_4|  \le& |\nabla \pi|_a |w|_{a(\alpha-1)/(a-1)}^{\alpha-1}       \\
      \le& C \sum_{i,j} \| \del_j w_i  (2u_j + w_j) \| _a \, 
           |w|_{a(\alpha-1)/(a-1)}^{\alpha-1} \\
   \le& C |w|_{a(4-\alpha)/(2-a)}^{(4-\alpha)/2}
|w|_{a(\alpha-1)/(a-1)}^{\alpha-1}
K(w) \\
  &     +  C |u|_q |w|_{aq(2-\alpha)/(2q-aq-2a)}^{(2-\alpha)/2}  
                |w|_{a(\alpha-1)/(a-1)}^{\alpha-1} K(w).
\endaligned \tag 3.36 $$
We estimate this in a similar way as in (3.25).
Choose $a$  such that the following two conditions hold.

\item{(R)}  Both $a(4-\alpha)/(2-a)$ and $a(\alpha-1)/(a-1)$ are between
$N$ 
and $N\alpha/(N-2)$

\item{(S)} Both $aq(2-\alpha)/(2q-aq-2a)$ and $a(\alpha-1)/(a-1)$ are 
between $\alpha$ and $N\alpha /(N-2)$

Assuming that (R) and (S) hold, it follows  from
the H\"older's inequality and (3.32) that
$$\gather
|w|_{a(4-\alpha)/(2-a)} \le 
 |w|_N^{1-\theta_1} |w|_{N\alpha /(N-2)}^{\theta_1}
\le C \|w\|_N^{1-\theta_1} K(w)^{2 \theta_1/ \alpha},     \tag3.37
\\
|w|_{a(\alpha-1)/(a-1)} \le 
 |w|_N^{1-\theta_2} |w|_{N\alpha /(N-2)}^{\theta_2}
\le C \|w\|_N^{1-\theta_2} K(w)^{2 \theta_2/ \alpha},     \tag3.38
\\
|w|_{aq(2-\alpha)/(2q-aq-2a)} \le 
 |w|_{\alpha}^{1-\theta_3} |w|_{N\alpha /(N-2)}^{\theta_3}
\le C |w|_{\alpha}^{1-\theta_3} K(w)^{2 \theta_3/ \alpha},    \tag3.39
\\
|w|_{a(\alpha-1)/(a-1)} \le 
 |w|_{\alpha}^{1-\theta_4} |w|_{N\alpha /(N-2)}^{\theta_4}
\le C |w|_{\alpha}^{1-\theta_4} K(w)^{2 \theta_4/ \alpha},   
\tag3.40\endgather
$$
where$(\theta_1, \theta_2,\theta_3,\theta_4) = $
$$
(\, \frac{\alpha (2N - a N + a \alpha -4a )}{a(4-\alpha)(N-\alpha-2)}
,\,\,
   \frac{\alpha (a N + a - a \alpha -N )}{a(\alpha -1)(N-\alpha-2)}
,\,\,
    \frac{N (a \alpha  + a q -  \alpha q )}{a q(2-\alpha)}, \,\,
   \frac{N (\alpha - a )}{2a(\alpha -1)} \,). \tag 3.41
$$
It follows from (3.36) and (3.37)-(3.41)  that
$$
|I_4| \le C \|w\|_N  K(w)^2 
+ C\|u\|_q |w|_{\alpha}^{\alpha (q-N)/2q}  K(w)^{1 + N/q}.  
$$
Therefore, when $q>N$ we have
$$
|I_4| \le C \|w\|_N  K(w)^2 + \e K(w)^2 + C_{\e} \|u\|_q^r \,
|w|_{\alpha}^{\alpha}.  
$$
Thus, we obtain
$$
\frac{1}{\alpha} \ddt |w(t)|_{\alpha}^{\alpha}
     \le - (\frac{\alpha -1}{2} - C_0 \|w\|_N) K(w)^2 
      + A_{\alpha q} \|u\|_{q}^{r} \, |w|_{\alpha}^{\alpha}.      \tag
3.42
$$
Set $d_{\alpha q} := (\alpha -1)/2C_0$. Then, by (3.3) and (3.42) we
have
(3.5) for $t\in [0,T]$.
\par
        Finally, we  determine the available range of $q$.
First we consider the case: $1 < \alpha \le N-2$.
We have $\alpha < N\alpha /(N-2) \le N$. By (R) and (S), we need to
choose
$a(\alpha-1)/(a-1):={N\alpha}/(N-2)$, i.e.
$$
a:= \frac{N\alpha}{N+2(\alpha-1)} \quad (<2)\,.
$$
It leads to 
$$
\theta_1= \frac{2-\alpha}{4-\alpha} \in (0,1), 
\qquad    
\theta_3 = 1- \frac{\alpha (q-N)}{q(2-\alpha)}.
$$
By the conditions $\theta_3\in [0,1]$ and $q > N$ we have
$$
N < q \le \frac{N\alpha}{2(\alpha-1)} := q_*.
$$
Next, let $N-2 < \alpha < 2$. Since $\alpha < N < N\alpha /(N-2)$, (R)
and (S) 
implies
\linebreak
 $N \le a(\alpha -1) /(a-1) \le N\alpha /(N-2)$, i.e.
$$
\frac{N\alpha}{N + 2\alpha -2} \le a \le  \frac{N}{N-\alpha +1}.    \tag
3.43
$$
We remark that (3.43) is equivalent to $\theta_2$, $\theta_4 \in [0,1]$.
We verify that (3.43) also implies $\theta_1 \in [0,1]$.
By (3.41) we have
$$
\frac{1}{q} = \frac{2-\alpha}{N\alpha}\,\theta_3 + \frac{1}{a} -
\frac{1}{\alpha}.
$$
Thus, $q$ achieves the minimum at 
$(\theta_3,a) =(\,1,\,{N\alpha}/(N+ 2\alpha -2)\,)$
and the maximum at $(\theta_3,a) =(\,0,\, {N}/(N-\alpha +1)\,)$.
Combining this and the condition $q > N$, we have
$$
N < q \le \frac{N\alpha}{(N-\alpha)(\alpha-1)} := q_* .
$$
The proof is complete.
\qed

\proclaim{Lemma 3.2}. Let $\alpha \in (1,2)$ be a number and 
$w=(w_1,\cdots,w_N) \in W^{2,\alpha} ({\Bbb R}^N \; {\Bbb R}^N)$.
Then we have $|w|^{\alpha/2} \in H^1 (:= W^{1,2})$ and 
$$
\frac{4(\alpha -1)}{\alpha^2}  \| \nabla |w|^{\alpha/2} \|_2^2 
       \le (\alpha-1)\int_{{\Bbb R}^N} \frac{|\nabla
w|^2}{|w|^{2-\alpha}}
   \le - \int_{{\Bbb R}^N} w^{\alpha-1} \cdot \Delta w \, dx \,\,(<
\infty).       \tag
3.44
$$
Here, we define ${|\nabla w|^2} / {|w|^{2-\alpha}} = 0$ when $|w(x)|=0$.
\endproclaim

\noindent{\bf Proof.}  Since $| \nabla |w|\, | \le |\nabla w|$, we have
$$
\int_{{\Bbb R}^N} \frac{|\nabla w|^2}{|w|^{2-\alpha}} 
        \ge \int_{{\Bbb R}^N} |w|^{\alpha-2} | \nabla |w|\, |^2
        =\frac{4}{\alpha^2} \int | \nabla |w|^{\alpha/2} \,|^2.
$$
Thus we obtained the first inequality in (3.44).

To show the second inequality in (3.44), it suffices to derive
$$
-\int_{{\Bbb R}^N} |w|^{\alpha-2}w \cdot \del_j^2 w  
        \ge (\alpha-1) \int_{{\Bbb R}^N} \frac{|\del_j
w|^2}{|w|^{2-\alpha}}.      \tag
3.45
$$
Let $j=1$. (We omit the other cases: $j \not =1$ since they are  same.)
With the aid of the Fubini Theorem we have
$$
-\int_{{\Bbb R}^N} |w|^{\alpha-2}w \cdot \del_1^2 w 
     = -\int_{{\Bbb R}^{N-1}} dx' \int_{{\Bbb R}} |w|^{\alpha-2}w \cdot
\del_1^2 w \,
dx_1.
                                              \tag 3.46
$$
Here, $x':= (x_2,\cdots, x_N)$.
By the assumption: $w \in W^{2,\alpha}$ we have
$$
w(\cdot, x') \in C^1({\Bbb R} \; {\Bbb R}^N)   \tag 3.47
$$
for a.e.  $x' \in {\Bbb R}^{N-1}$.
We fix any $x'$ such that (3.47) holds and set
$$
W(\cdot) := w(\cdot, x') \in C^1({\Bbb R} \; {\Bbb R}^N).
$$
Then, there exist countable
numbers of open intervals $\{ I_k \}_{k=1}^{\infty}$ such that
$W(x)\not = 0$ on  each $I_k$. Actually, $W(x)$ is positive
definite or negative definite on each interval $I_k$. Each $\del I_k$
(:= the boundary of $I_k$ in ${\Bbb R}$) consists of at most two points.
Therefore, $\bigcup_{k=1}^{\infty} \del I_k$ is a set whose measure is
zero.
It follows from the  integration by parts that
$$\align
-\int_{{\Bbb R}} |W|^{\alpha-2}W \cdot  W_{xx} \,dx 
        &= - \sum_{k=1}^{\infty} \int_{I_k}  |W|^{\alpha-2}W \cdot 
W_{xx}   
\tag 3.48  \\
        &= \sum_{k=1}^{\infty} \int_{I_k} [ \,-(2-\alpha) 
|W|^{\alpha-2}  |W|_{x}
W_{x}
                + |W|^{\alpha-2} (W_{x})^2 \,]  \\
        &\ge \sum_{k=1}^{\infty} \int_{I_k} [\, -(2-\alpha) 
|W|^{\alpha-2} 
(W_{x})^2 
                + |W|^{\alpha-2} (W_{x})^2 \,]        \\
        &= (\alpha-1) \int_{{\Bbb R}} |W|^{\alpha-2} (W_{x})^2 .
\endalign$$
Combining (3.46) and (3.48), we obtain (3.45).  \qed

\head 4. Applications \endhead

              In this section we give the proofs of Corollaries 1.1-1.3 
and some  other applications of our Theorem 3.1.\smallskip

\noindent{\bf Proof of Corollary 1.1.}
Let $v$ be any mild solution of (NS) which satisfies
$v\in C([0,T)\, \; PL^N) \cap L_{\text{loc}}^r ((0,T) \, \; PL^q)$.
It suffices to show that
$$
v(t)=u(t\,\;  v(0))  \qquad\text{on} \qquad [0,T).    
$$
Here, $u(t\,\;  v(0))$ is a strong solution, which satisfies
$$
u(t\,\;  v(0)) \in C([0,T)\, \; PL^N) \cap L^{N+2} ((0,T) \, \;
PL^{N+2})
$$
(see Definition 2.2).
We set $u(t):=u(t \,\; v(0)\,)$ and  $w(t):=v(t)- u(t)$.  
Let $d_{N+2}$ and $A_{N+2}$ be
the  constants defined in Corollary 3.1.
We choose a small constant $\tau\in (0,T)$ such that
$$
\| w(t) \|_N \le d_{N+2} \quad\text{for}\quad t\in [0,\tau].
$$
Let $\e\in (0,\tau)$.
By the semigroup property of the mild solution, $v(t)$ is a 
strong solution on $[\e,T)$ with the initial value $v(\e)$.
We apply Theorem 3.1 and obtain
$$
\| w(t ) \|_{N} \le \| w(\e) \|_{N}
\exp\,( A_{N+2} \int_{\e}^{\tau} \|u(s)\|_{N+2}^{N+2}\,ds) 
\quad\text{for}\quad t\in [\e, \tau].
$$
Let $\e \to +0$. Then we have $w(t)=0$ for $t\in [0,\tau]$.
Now, we easily verify 
from the continuity of $w(t)$ that 
$T=\sup \,\{ \tau \; w(t)=0 \quad\text{for}\quad t\in [0,\tau] \}$.
\qed
\smallskip
        The next result is a `local'-version of Corollary 1.2.

\proclaim{Corollary 4.1. (Local Stability)}  Let $u_0\in PL^N \cap
PL^{\alpha}$
with a constant $\alpha \in (1,\infty)$ and $u(t;u_0)$ be a strong
solution.
Then, for any $T < \tmax (u_0)$ there exist
  constants $\delta_0=\delta_0(N,u_0,T)\in {\Bbb R}^+$
and $C_0=C_0(N)\in{\Bbb R}^+$  such that if
$$
v_0\in PL^N  \quad\text{and}\quad \|v_0-u_0\|_{N}\le \delta_0  
$$
then  we have $\tmax (v_0) > T$ and
$$
\|u(t \,; v_0)-u(t;u_0) \|_{N} \le \|v_0-u_0\|_{N}
\exp\,(C_0 \int_0^t \|u(s \,; u_0)\|_{N+2}^{N+2}\,ds)  \tag 4.1
$$
 for $t\in [0,T]$. 
\par
          Moreover, when $\alpha \not = N$ there exist constants 
$\delta_1=\delta_1(N,\alpha,u_0,T)\in (0,\delta_0]$, 
\linebreak
$q=q(N,\alpha)\in (N,\infty)$ and $C_1=C_1(N,\alpha)\in{\Bbb R}^+$ such
that 
if
$v_0\in PL^N \cap PL^{\alpha}$  and $\|v_0-u_0\|_{N}\le \delta_1$
then we have for $\alpha \in [2,\infty)$
$$
\|u(t \,; v_0)-u(t;u_0) \|_{\alpha} \le \|v_0-u_0\|_{\alpha}
\exp\,(C_1 \int_0^t \|u(s \,; u_0)\|_{q}^{r}\,ds)  \tag 4.2
$$
 for $t\in [0,T]$, and for $\alpha\in (1,2)$ the estimate (4.2) with the
norm 
$\|\cdot\|_{\alpha}$ replaced by $|\cdot|_{\alpha}$. 
Here, $r$ is the constant which satisfies (1.1).
\endproclaim

\noindent{\bf Proof.} We use the notation in the statement of
Theorem 3.1 and 
Corollary 3.1.  We denote $u(t):=u(t;u_0)$ and
$v(t):=u(t;v_0)$. We fix constants $\alpha_0\in (N,\infty)$ and $q_0\in
(N,q_*(N,\alpha_0))$.
For example, set $(\alpha_0,q_0)=(2N,2N)$.  We choose $\delta_0\in {\Bbb
R}^+$ 
such that
$$
\delta_0 < \min\,(d_{N+2},d_{\alpha_0,q_0}) 
                \exp\,( - A_{N+2} \int_0^T \|u(s)\|_{N+2}^{N+2}\,ds)
$$
and assume $\| v_0-u_0 \|_N \le \delta_0$.  
Let  $C_0 := A_{N+2}$.
Then, by Corollary 3.1 we have (4.1) for $t\in [0,T_0)$ and 
$$
\|v(t) -u(t) \|_{N} < \min (d_{N+2},d_{\alpha_0,q_0})
\quad\text{for}\quad
t\in [0,T_0).    \tag 4.3
$$
Here, $T_0 := \min (T, \tmax (v_0))$. We complete the proof if we
prove
$\tmax(v_0) > T$. We fix a constant $t_0 \in (0,T_0)$. 
Then, by (4.3) and Theorem 3.1 we have
$$
\|v(t)-u(t) \|_{\alpha_0} \le \| v(t_0) - u(t_0) \|_{\alpha_0}
\exp\,(A_{\alpha_0,q_0} \int_0^T \|u(s)\|_{q_0}^{r_0}\,ds)  < \infty
$$
 for $t\in (0,T_0)$, where $r_0$ is the constant which satisfies
$N/q_0 +2/r_0 = 1$.  Thus, $\| v(t) \|_{\alpha_0}$ is bounded on
$[0,T_0)$.
In view of Corollary 2.1 we have $T_0 = T < \tmax (v_0)$.
\par
         When $\alpha \not = N$, we choose $\delta_1 \in{\Bbb R}^+$ such
that 
$$
\delta_1 < \min\,(d_{N+2},d_{\alpha_0,q_0}, d_{\alpha q}) 
                \exp\,( - A_{N+2} \int_0^T \|u(s)\|_{N+2}^{N+2}\,ds).
$$
Here, $q\in (N,q_*(N,\alpha))$ is a constant.  Then we can verify (4.2)
 in the same argument as above. \qed
\smallskip
         
 The following lemma will be used in the proof of Corollary 1.2.

\proclaim{Lemma 4.2} Let $v(t)=u(t;v_0)\in \glons$ be a global 
strong solution.
And let $\alpha \in [N,\infty)$ be a constant. Then there exists a
constant
$T_* = T_*(N,\alpha,v_0)\in {\Bbb R}^+$ such that $\| u(t;v_0)
\|_{\alpha}$ is 
non-increasing on $[T_*,\infty)$.
\endproclaim

\noindent{\bf Proof.} We use the notation in the statement of Theorem3.1 
and Corollary 3.1. We choose $T_*$ such that 
$\| v(T_*) \|_{N} < \min (d_{N+2},d_{\alpha \alpha})$.
Let $u_0 := 0$. Then $u(t;u_0)$ is a trivial solution. 
It follows from Corollary 3.1 that
$\| v(t) \|_{\alpha} \le  \| v(T_*) \|_{\alpha} < \min\,(d_{N+2},
d_{\alpha\alpha})$ 
for $t \ge T_*$. 
Applying Corollary 3.1 again, we see that $\| u(t;v_0) \|_{\alpha}$ is
non-increasing
for $t \ge T_*$.
\qed

\subhead Remark 4.1\endsubhead Actually, we can verify the decay
estimate
$$
\|v(t)\|_{\alpha} \le Ct^{-(1-N/ \alpha)/2} \quad\text{for}\quad t\ge
1.  \tag
4.4
$$
See [K, Theorem 4].  We can prove (4.4) using the same
argument as in the proof of [Ka, Theorem 4.1]. We omit it since we do
not use (4.4) in this paper.\smallskip

\noindent{Proof of Corollary 1.2.} We use the notation in
the statement of Theorem 3.1 and Corollary 3.1.
We fix constants $\alpha_0\in (N,\infty)$ and $q_0\in
(N,q_*(N,\alpha_0))$.
For example, set $(\alpha_0,q_0)=(2N,2N)$. 
\par
        (i) We write $u(t):= u(t;u_0)$ and $v(t):=u(t;v_0)$. 
We choose $\delta_0 \in {\Bbb R}^+$ such that
$$
\delta_0 \exp\,(A_{N+2} \int_0^{\infty} \|u(s)\|_{N+2}^{N+2}\,ds)
        < \min\,( d_{N+2},d_{\alpha_0,q_0}, \frac{1}{2} \e_* ).
$$
Here, $\e_*$ is the constant in the statement
of Theorem 2.1 (iii).
We set $C_0 := A_{N+2}$. Let $\| v_0 -u_0 \|_N \le \delta_0$.
Then, by Corollary 3.1 we have (1.2) and
$$
\|v(t) -u(t) \|_{N} < \min (d_{N+2},d_{\alpha_0,q_0}, \frac{1}{2} \e_*
)      \tag 4.5
$$
for $t\in (t_0,\tmax(v_0))$. We will show $\tmax (v_0) =\infty$.
We fix a constant $t_0 \in (0, \tmax(v_0))$. By (4.5) and Theorem 3.1
we have 
$$
\|v(t)-u(t) \|_{\alpha_0} \le \| v(t_0) - u(t_0) \|_{\alpha_0}
\exp\,(A_{\alpha_0,q_0} \int_{t_0}^{\infty} \|u(s)\|_{q_0}^{r_0}\,ds)  <
\infty  
\tag 4.6
$$
for $t\in (t_0, \tmax(v_0))$, where $r_0$ is the constant which
satisfies
$N/q_0 +2/r_0 = 1$.  Since $\|u(t)\|_{\alpha_0}$ is bounded on
$[t_0,\infty)$ 
(see Lemma 4.2), $\|v(t)\|_{\alpha_0}$ is bounded on $[t_0,\tmax(v_0))$.
With the aid of  Corollary 2.1, we have $\tmax (v_0)=\infty$. Thus,
$v(t)$
is a
global solution. We choose a constant $T_1$ such that 
$\| u(T_1) \|_N < \e_* /2$.  Then, by (4.5) we have  $\| v(T_1) \|_N <
\e_* $.
Therefore, we obtain from Theorem 2.1 that $v(t) \in \glons$.
\par
        (ii) We choose a constant $q$ such that $N < q < q_*(N,\alpha)$.
We choose $\delta_1\in {\Bbb R}^+$ such that
$$
\delta_1 <  \min\,( d_{N+2},d_{\alpha_0,q_0}, \frac{1}{2} \e_* ,
d_{\alpha q})
\exp\,( - A_{N+2} \int_0^{\infty} \|u(s)\|_{N+2}^{N+2}\,ds).
$$ 
 Then we can easily 
verify (1.3) for $t\in {\Bbb R}^+$ by the same argument as in (i).  
\qed\smallskip

\noindent{\bf Proof of Corollary 1.3.} We denote $u(t) := u(t;u_0)$. 
It follows from Theorem 2.1 that $\tmax (u_0) >T$ implies 
$u(t)\in L^r( (0,T) \; PL^q )$.  Next, we will prove the inverse.
Let $u(t) \in L^r((0,T) \; PL^q)$.
In view of Corollary 2.1, it suffices 
to prove  that $\lim_{\,t\to T-0} \,u(t)$ exists in $PL^N$.
By Lemma 2.1 we have $u(t;u_0)\in L^{N+2}((0,T) \; PL^{N+2})$.
Let $\{ t_j \}_{j=1}^{\infty}$ be any monotone increasing 
sequence such that
$t_1 \ge T/2$ and  $t_j \to T$ $(j\to\infty)$. 
Setting $u_0 := u(t_k- T/2)$ and  $v_0 := u(t_l- T/2)$ $(k < l)$,
we apply Corollary 3.1.  We have
$$\align
&\|u(t_l)-u(t_k)\|_N \\
   \le\, &\|u(t_l-T/2)-u(t_k-T/2)\|_N 
      \exp ( A_{N+2} \int_0^{T/2} \| u(s+ t_k-T/2) \|_{N+2}^{N+2}\,ds)
\\
\to \, & 0 \quad (k\to\infty).
\endalign$$
 Thus,  $\lim_{\,j\to \infty} u(t_j)$ exists in $PL^N$. 
\qed
\medbreak
        Finally we mention the topological structure of the space
of strong solutions.  We set 
$$
A:=\{\, u_0\in PL^{N} \; u(t \; u_0) \in \glons \,\}. \tag 4.7
$$
Then $A$ is  open in $PL^{N}$ by Corollary 1.2.
When $N=3$ the open set $A$ is unbounded in $PL^3$ since 
$u_0\in A$ if $u_0 (x)$ is axially symmetric and 
$u_0\in PL^2\cap PL^3$ (which was shown in [UI]).
We also set 
$$\align
B:=& \{ u_0\in PL^N \; \tmax(u_0)<\infty  \}  \tag 4.8 \\
    =&\{ u_0\in PL^N \; \| u(t;u_0) \|_{\infty}  \,
\text{ blows up in finite time}  \, \}.
\endalign$$
The second equality holds by the equivalence of (e) and (c) in 
Corollary 2.1.

\proclaim{Proposition 4.1}  When $N=2$, we  have $A=PL^2$. 
\endproclaim

This result was proved in [KM], [M] and [Wi]. However, we give
a simple and different proof by using Corollary 1.3  and 
 Proposition 2.2.\smallskip
 
\noindent{\bf Proof.} Let fix any $u_0 \in PL^2$. We write $u(t) :=
u(t;u_0)$.
 We have the energy equality
$$
\|u(t)\|_2^2 +2\int_0^t \|\nabla u(s)\|_2^2ds= \|u_0\|_2^2
  \quad\text{for}\quad t\in [0,\tmax(u_0)).  \tag 4.9
$$
 By the Gagliardo-Nirenberg  inequality
$$
\|u(t)\|_4 \le C\|u(t)\|_2^{1/2}\|\nabla u(t)\|_2^{1/2}.   \tag 4.10
$$
In view of (4.9) and (4.10) we have $u(t;u_0)\in L^4((0,T) \; PL^4)$.
Here $T\in {\Bbb R}^+$ is a constant such that $T\le \tmax(u_0)$.
By Corollary 1.3 we have $T< \tmax(u_0)$. Thus, we obtain 
$\tmax(u_0)=\infty$, i.e. $u(t;u_0)$ is a global solution. 
Now we have $u(t;u_0) \in L^4({\Bbb R}^+ \; PL^4)$.
Therefore, we apply Proposition 2.2 
to conclude $u(t;u_0)\in\glons$. \qed

\proclaim{Proposition 4.2} Let $N=3,4$. Then we have the following 
results:
\item{(i)} For $u_0\in PL^2 \cap PL^N$ the solution $u(t;u_0)$
blows up in finite time or $u(t;u_0)\in\glons$, i.e. we have
$$
(A\cup B)\cap PL^2=PL^2\cap PL^N.  \tag 4.11
$$
\item{(ii)} the set $B\cap PL^2$ is closed in $PL^2 \cap PL^N$.
\item{(iii)} the set $B$ is empty or $B$ is not open in $PL^N$.
\endproclaim


\noindent{\bf Proof.}  (i) Let $u_0\in PL^2 \cap PL^N$ and
$\tmax(u_0)=\infty$.
In view of the energy equality (4.9),  $\|u(t)\|_2$ is bounded for $t\ge
0$
and $\liminf_{t\to\infty} \|\nabla u(t)\|_2 = 0$. Thus, by the Gagliardo
-
Nirenberg 
inequality, we have $\liminf_{t\to\infty} \|u(t)\|_N = 0$. 
It follows from Theorem 2.1 that  $u\in \glons$.
Hence, we have (4.11).

     (ii)  By Corollary 1.2, the set $A\cap PL^2$ is open 
in $PL^2 \cap PL^N$.
 Therefore, $B\cap PL^2 = (PL^2\cap PL^N)-(A \cap PL^2)$ is closed in 
$PL^2\cap PL^N$.  

     (iii) We proceed by contradiction. Suppose that $B$ 
is non-empty open
set in $PL^N$. Since $PL^2\cap PL^N$ is dense in $PL^N$, 
there exists
$u_0\in (PL^2 \cap B)- \{ 0 \}$. We set 
$$
C:=\{ \tau\in {\Bbb R} \; \tau u_0 \in A \}.
$$
In view of Corollary 1.2 and Theorem 2.1 (iii), the set $C$ is 
non-empty open set in ${\Bbb R}$. Since $C\not ={\Bbb R}$, we have 
$\del C\not =\phi$. Set $B_1 := \{ \tau u_0 \; \tau\in \del C \}$.
Then, we obtain from (4.11) that $B_1 \subset \del A\cap B$.
This implies $A \cap B \not = \emptyset$,
which is a contradiction. Thus, $B$ is empty or $B$ is not open in 
$PL^N$. \qed

 \subhead Acknowledgment\endsubhead
The author would like to express his sincere
gratitude to  Professors  Tosio Kato, Yoshikazu Giga,  Hideo Kozono, 
 Toshiaki Hishida,  and Takayoshi Ogawa for their  useful comments and 
information on literature. In particular, Professor Tosio Kato suggested 
a uniqueness problem (Corollary 1.1).

\head References \endhead

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