1706.10173.tex
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{\Large \bf %
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Some remarks on the regularity time of Leray} \\
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\mbox{} \vspace{-0.250cm} \\
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\mbox{\Large \bf %
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solutions to the Navier-Stokes equations} \\
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\nl
\mbox{} \vspace{-0.200cm} \\
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{\large \sc P\!\:\!.\;Braz e Silva,}$\mbox{}^{\!\!\;\!1}$
{\large \sc J.\;P\!\:\!.\;Zingano}$\mbox{}^{\:\!2}$
{\large \sc and P\!\:\!.\;R.\;Zingano}$\mbox{}^{\:\!2}$ \\
%
\mbox{} \vspace{-0.150cm} \\
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{\small $\mbox{}^{1}\:\!$Departmento de Matem\'atica} \\
\mbox{} \vspace{-0.670cm} \\
{\small Universidade Federal de Pernambuco} \\
\mbox{} \vspace{-0.670cm} \\
{\small Recife, PE 50740, Brazil} \\
%
\mbox{} \vspace{-0.300cm} \\
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$\mbox{}^{2}\:\!${\small Departamento de Matem\'atica Pura e Aplicada} \\
\mbox{} \vspace{-0.670cm} \\
{\small Universidade Federal do Rio Grande do Sul} \\
\mbox{} \vspace{-0.670cm} \\
{\small Porto Alegre, RS 91509, Brazil} \\
%
\nl
\mbox{} \vspace{-0.300cm} \\
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{\bf Abstract} \\
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\mbox{} \vspace{-0.525cm} \\
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{\small
%
\mbox{} \hspace{+0.200cm}
%
We strengthen the classic result
about the regularity time $t_{\ast} $
of arbitrary Leray solutions
of the Navier-Stokes equations
in $ \mathbb{R}^{n} $ ($n = 3, 4$),
which have the form
$ \;\!t_{\ast} \!\:\!\leq\!\;\! K_{\mbox{}_{\!\;\!3}} \,\nu^{-\,5}\:\!
\|\, \mbox{\boldmath $u$}_0 \;\!
\|_{L^{2}(\mathbb{R}^{3})}^{\:\!4} $
if $ \:\!n = 3 $,
and
$ \!\;\!\:\!t_{\ast} \!\:\!\leq\!\;\! K_{\mbox{}_{\!\;\!4}}
\,\nu^{-\,3}\:\!
\|\, \mbox{\boldmath $u$}_0 \;\!
\|_{L^{2}(\mathbb{R}^{4})}^{\:\!2} \!\;\!$
if $ \:\!n = 4 $
(in particular, by
reducing \linebreak
the current known values
for the constants
$ \:\!K_{\mbox{}_{\!\:\!3}} \!\;\! $,
$ \!\;\!K_{\mbox{}_{\!\:\!4}} $).
Some related results
are also included
in our discussion. \\
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% -----------------------------------------------------
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% *************************************************************
% *                                                           *
% *                 Section 1: Introduction                   *
% *                                                           *
% *************************************************************
%

{\bf 1. Introduction} \\
%
\mbox{} \vspace{-0.750cm} \\
%

In this note
we rederive and slightly improve
a fundamental result
originally obtained by J.\;Leray \cite{Leray1934}
in dimension $ n = 3 $
(see (1.3) below)
for the global weak solutions
introduced in \cite{Leray1934}
to solve the incompressible
Navier-Stokes equations \\
%
\mbox{} \vspace{-0.650cm} \\
%
\begin{equation}
\tag{1.1$a$}
\mbox{\boldmath $u$}_{t}
\;\!+\,
\mbox{\boldmath $u$} \!\cdot\! \nabla \mbox{\boldmath $u$}
\,+\,
\nabla p
\;=\;
\nu \,
\Delta\:\! \mbox{\boldmath $u$},
%
% \qquad
% \nabla \!\cdot  \mbox{\boldmath $u$}(\cdot,t) \,=\, 0,
%
\end{equation}
%
\mbox{} \vspace{-0.975cm} \\
%
\begin{equation}
\tag{1.1$b$}
\nabla \!\cdot  \mbox{\boldmath $u$}(\cdot,t) \,=\, 0,
\end{equation}
%
\mbox{} \vspace{-0.975cm} \\
%
\begin{equation}
\tag{1.1$c$}
\mbox{\boldmath $u$}(\cdot,0)
\,=\,
\mbox{\boldmath $u$}_0 \in
\mbox{\boldmath $L$}^{2}_{\sigma}(\mathbb{R}^{n}),
\end{equation}
%
\mbox{} \vspace{-0.250cm} \\
%
and which is valid more generally
for arbitrary solutions
$ {\displaystyle
\;\!
\mbox{\boldmath $ u $}(\cdot,t)
\in
L^{\infty}((0,\infty), \:\!\mbox{\boldmath $L$}^{2}_{\sigma}(\mathbb{R}^{n}))
} $
%
$ {\displaystyle
\cap \,
L^{2}((0,\infty), \:\!\dot{\mbox{\boldmath $H$}}
\mbox{}^{\!\;\!1}\!\;\!(\mathbb{R}^{n}))
\cap \;\!
C_{\mbox{\scriptsize w}}([\;\!0, \infty),
\mbox{\boldmath $L$}^{2}(\mathbb{R}^{n}))
} $
of (1.1),
where $ n = 3 $ or $4$,
satisfying
the so-called
{\em strong energy inequality} \\
%
\mbox{} \vspace{-0.725cm} \\
%
\begin{equation}
\tag{1.2}
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!2}
\!\:\!+\,
2 \, \nu \!\!\;\!
\int_{\mbox{}_{\mbox{\footnotesize \!\,\!$s$}}}
    ^{\mbox{\footnotesize \;\!$t$}}
\!\!\:\!
\|\, D \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!2}
d\tau
\,\leq\:
\|\, \mbox{\boldmath $u$}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!2}
\!\:\!,
\quad
\forall \;\;\! t \geq s
\end{equation}
%

%
\mbox{} \vspace{-0.750cm} \\
%
for a.e.$\;s \geq 0 $,
including $ s = 0 $.\footnote{%
%
% ----------------------------------------------
%
%              Footnote 1
%
% ----------------------------------------------
%
For the definition of
$ {\displaystyle
\:\!
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{n})}
\!\;\!
} $,
$ {\displaystyle
\|\, D \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{n})}
\!\:\!
} $
and other similar norms,
% of $ \mbox{\boldmath $u$}(\cdot,t) $,
see (1.5) next.
}
%
% ------------------------------- End of Footnote 1
%
Such solutions are now
called {\em Leray}
(or {\em Leray-Hopf\/})
solutions,
after \cite{Hopf1950, Leray1933, Leray1934}.
They have also been constructed
in higher dimensions
and other methods,
see e.g.\;\cite{% ConstantinFoias1988,
CaffarelliKohnNirenberg1982,
DoeringGibbon1995, FujitaKato1964,
Giga1986, Hopf1950, Kato1984, Ladyzhenskaya1969,
Lions1969, RobinsonRodrigoSadowski2016, Serrin1963,
ShinbrotKaniel1966, Sohr2001, Temam1984, Wahl1985}.
% and references therein.
% CaffarelliKohnNirenberg1982, %
% FujitaKato1964, % Galdi2000, Giga1986,
% GigaMiyakawa19xx, Heywood1980,
% Hopf1950, % KajikiyaMiyakawa1986,
% Kato1984, KiselevLadyzhenskaya1957, %
% Ladyzhenskaya1969,
% Lions1969, Masuda1984,
% Serrin1963,
% ShinbrotKaniel1966, Sohr2001, Temam1984,
% SohrWahl1987, Weissler1980}.
%
In (1.1),
$ \nu > 0 $ is a given constant,
\mbox{$ \mbox{\boldmath $u$} = \mbox{\boldmath $u$}(x,t) $}
and
$ p = p(x,t) $
are the unknowns
(the flow velocity and pressure, respectively),
and condition (1.1$c$)
is meant in $ \mbox{\boldmath $L$}^{2}(\mathbb{R}^{n}) $,
i.e.,
$ {\displaystyle
\,\!\,\!
\|\, \mbox{\boldmath $u$}(\cdot,t) - \mbox{\boldmath $u$}_0 \;\!
\|_{L^{2}(\mathbb{R}^{n})}
\!\;\!\rightarrow 0
\,\!\,\!\,\!
} $
as $ \,\!\,\! t \,\mbox{\footnotesize $\searrow$}\,0 $.
%
$\!\;\!$As usual,
$ \!\;\!\mbox{\boldmath $L$}^{2}_{\sigma}(\mathbb{R}^{n}) $
is the
space
of solenoidal fields
$ \:\!\mbox{\bf v} = (v_{1}, ..., v_{n}) \!\:\!\in
\mbox{\boldmath $L$}^{2}(\mathbb{R}^{n}) \equiv L^{2}(\mathbb{R}^{n})^{n} \!\:\!$
with
$ \nabla \!\cdot \mbox{\bf v} \!\;\!= 0 $
% \mbox{in $ {\cal D}^{\prime}(\mathbb{R}^{n}) $}
% (i.e., as distributions)
in the distributional sense, \linebreak
%
$ \dot{\mbox{\boldmath $H$}}\mbox{}^{1}(\mathbb{R}^{n}) =
\dot{H}^{1}(\mathbb{R}^{n})^{n} \!$,
where
$ \dot{H}^{1}(\mathbb{R}^{n}) $
denotes the homogeneous Sobolev space of order~1
(see e.g.\;\cite{BahouriCheminDanchin2011}, p.\;25),
and
$ \mbox{\boldmath $H$}\mbox{}^{m}(\mathbb{R}^{n}) =
H^{m}(\mathbb{R}^{n})^{n}\!$,
where
$ H^{m}(\mathbb{R}^{n}) $
is the Sobolev space of order $m$
(\cite{BahouriCheminDanchin2011}, p.\;38).
Also,
$ C_{\mbox{\scriptsize w}}(I, \:\!\mbox{\boldmath $L$}^{2}(\mathbb{R}^{n})) $
denotes the set of mappings from a
given interval~\mbox{$ I \subseteq \mathbb{R} $}
to
$ \mbox{\boldmath $L$}^{2}(\mathbb{R}^{n}) $
that are $L^{2}$-\;\!weakly continuous
at each $\:\! t \in I$. \\
%
\mbox{} \vspace{-0.900cm} \\
%

The result about Leray solutions
of (1.1)
that concerns us here
is the following.
%
In dimension $ n = 3 $ or $ 4 $,
it is known that
$ {\displaystyle
\mbox{\boldmath $u$}
\in
C^{\infty}(\mathbb{R}^{n} \!\times \!\:\![\, t_{\ast}, \infty))
} $
and \\
%
\mbox{} \vspace{-0.575cm} \\
%
\begin{equation}
\tag{1.3$a$}
\mbox{\boldmath $u$}(\cdot,t)
\in
C^{0}([\,t_{\ast}, \infty), \:\!
\mbox{\boldmath $H$}\mbox{}^{\!\:\!m}\!\;\!(\mathbb{R}^{n})),
\quad \;\,
\forall \;\;\! m \geq 0
\end{equation}
%
\mbox{} \vspace{-0.225cm} \\
%
for some
regularity time
$ \:\!t_{\ast} \!\geq 0 \:\!$
that satisfies \\
%
\mbox{} \vspace{-0.625cm} \\
%
\begin{equation}
\tag{1.3$b$}
t_{\ast}
% \,\leq\;
% \mbox{\small $ {\displaystyle \frac{1}{\;\!128\;\!\pi^{2}} }$}
\;\!\leq\, K_{\mbox{}_{\!\:\!3}}
\,
\nu^{-\,5} \;\!
\|\, \mbox{\boldmath $u$}_0 \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!4}
\quad \;\;
(\:\!\mbox{if }\;\! n = 3\:\!),
\end{equation}
%
\mbox{} \vspace{-0.750cm} \\
%
\begin{equation}
\tag{1.3$c$}
t_{\ast}
% \,\leq\;
% \mbox{\small $ {\displaystyle \frac{1}{\;\!128\;\!\pi^{2}} }$}
\;\!\leq\, K_{\mbox{}_{\!\:\!4}}
\,
\nu^{-\,3} \;\!
\|\, \mbox{\boldmath $u$}_0 \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\quad \;\;
(\:\!\mbox{if }\;\! n = 4\:\!),
\end{equation}
%
\mbox{} \vspace{-0.250cm} \\
%
with constants
$ K_{\mbox{}_{\!\:\!3}} \!\;\!$,
$ K_{\mbox{}_{\!\:\!4}} $
that are independent of $ \;\!\nu $,
$ \mbox{\boldmath $u$}_0 $
or the solution $ \mbox{\boldmath $u$} $
considered.
%
For example,
it follows from
(\cite{Heywood1994}, p.\:14;
\cite{Leray1934}, p.\:246;
\cite{LorenzZingano2003}, p.\:18)
that \\
%
\mbox{} \vspace{-0.600cm} \\
%
\begin{equation}
\notag
K_{\mbox{}_{\!\:\!3}}
\:\!\leq\:
\mbox{\small $ {\displaystyle \frac{1}{\,128\;\!\pi^{2}} }$}
\;\!<\,
\mbox{\small $0.000\,791\;\!572$},
\end{equation}
%
\mbox{} \vspace{-0.225cm} \\
%
which appears to be the best
(smallest)
estimate presently known
for the constant
$ \!\;\!K_{\mbox{}_{\!\:\!3}} \!\;\! $ \linebreak
in the literature.
In Section 2,
it is shown that \\
%
\mbox{} \vspace{-0.600cm} \\
%
\begin{equation}
\tag{1.4}
K_{\mbox{}_{\!\:\!3}}
\:\!<\:
\mbox{\small $0.000\,464\,504\,284$},
\quad \;\;\;\,
K_{\mbox{}_{\!\:\!4}}
<\:
\mbox{\small $0.002\,727\,993\,110$},
\end{equation}
%
\mbox{} \vspace{-0.225cm} \\
%
along with other improvements
regarding the monotonic behavior
of $ \;\!\|\;\! D \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}} $
for large $ \!\;\!\:\!t $.
%
\!Moreover,
by a similar argument,
we note in {\small \sc Remark 2.3}
that,
for each $m$,
$ \|\, D^{m} \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}} \!\:\!$
also becomes monotonically decreasing for $\;\!t \gg 1$.
%
% with
% $ \;\!t^{\:\!m/2} \;\!\|\, D^{m} \mbox{\boldmath $u$}(\cdot,t) \,
% \|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
% \!\rightarrow \:\!0 $
% as $ \;\!t \rightarrow \infty $.
%
\!Our analysis is
inspired by the interesting approach to these questions
developed in \cite{KreissHagstromLorenzZingano2002, %
KreissHagstromLorenzZingano2003, Zhou2007, Zingano2015}.

%
% ---------------------------------------------------
%
%                     Notation
%
% ---------------------------------------------------
%
\mbox{} \vspace{-0.750cm} \\
%
{\bf Notation.}
%
As shown above,
boldface letters
are used for
vector quantities,
as in
$ {\displaystyle
\;\!
\mbox{\boldmath $u$}(x,t)
=
} $
$ {\displaystyle
(\:\! u_{\mbox{}_{\!\:\!1}}\!\;\!(x,t), ...\,\!\,\!,
 \:\! u_{\mbox{}_{\scriptstyle \!\;\! n}}\!\;\!(x,t) \:\!)
} $.
$\!$Also,
$ \nabla p \;\!\equiv \nabla p(\cdot,t) $
denotes the spatial gradient of $ \;\!p(\cdot,t) $,
$ D_{\!\;\!j} \!\;\!=\:\! \partial / \partial x_{\!\;\!j} $,
%
$ {\displaystyle
\,\!
\nabla \!\cdot \mbox{\boldmath $u$}
\:\!=
  D_{\mbox{}_{\!\:\!1}} u_{\mbox{}_{\!\:\!1}} \!\;\!+
  ... \!\;\!+
  D_{\mbox{}_{\scriptstyle \!\;\!n}} \,\!
  u_{\mbox{}_{\scriptstyle \!\;\!n}}
} $
is the (spatial) divergence of
$ \:\!\mbox{\boldmath $u$}(\cdot,t) $.
%
$ |\,\!\cdot\,\!|_{\mbox{}_{2}} \!\,\!\,\!$
denotes the Euclidean norm
in $ \mathbb{R}^{n} \!$,
and
$ {\displaystyle
\,\!\,\!
\| \:\!\cdot\:\!
\|_{\scriptstyle L^{q}(\mathbb{R}^{n})}
\!\;\!
} $,
$ 1 \leq q \leq \infty $,
are the standard norms
of the Lebesgue spaces
$ L^{q}(\mathbb{R}^{n}) $,
with the vector counterparts \\
%
\mbox{} \vspace{-0.625cm} \\
%
\begin{equation}
\tag{1.5$a$}
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
\;\!=\;
\Bigl\{\,
\sum_{i\,=\,1}^{n} \int_{\mathbb{R}^{n}} \!
|\:u_{i}(x,t)\,|^{q} \,dx
\,\Bigr\}^{\!\!\:\!1/q}
\end{equation}
%
\mbox{} \vspace{-0.700cm} \\
%
\begin{equation}
\tag{1.5$b$}
\|\, D \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
\;\!=\;
\Bigl\{\,
\sum_{i, \,j \,=\,1}^{n} \int_{\mathbb{R}^{n}} \!
|\, D_{\!\;\!j} \;\!u_{i}(x,t)\,|^{q} \,dx
\,\Bigr\}^{\!\!\:\!1/q}
\end{equation}
%
\mbox{} \vspace{-0.350cm} \\
%
and, in general, \\
%
\mbox{} \vspace{-0.750cm} \\
%
\begin{equation}
\tag{1.5$c$}
\|\, D^{m} \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}
\;\!=\;
\Bigl\{\!\!
\sum_{\mbox{} \;\;i, \,j_{\mbox{}_{1}} \!,..., \,j_{\mbox{}_{m}} =\,1}^{n}
\!\;\! \int_{\mathbb{R}^{n}} \!
|\, D_{\!\;\!j_{\mbox{}_{1}}}
\!\!\!\;\!\cdot\!\,\!\cdot\!\,\!\cdot \!\:\!
D_{\!\;\!j_{\mbox{}_{m}}}
\!\:\! u_{i}(x,t)\,|^{q} \,dx
\,\Bigr\}^{\!\!\:\!1/q}
\end{equation}
%
\mbox{} \vspace{-0.175cm} \\
%
if $ 1 \leq q < \infty $\/;
%
if $\, q = \infty $,
then
$ {\displaystyle
\;\!
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!=\;\!
\max \, \bigl\{\,
\|\,u_{i}(\cdot,t)\,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!\!: \, 1 \leq i \leq n
\,\bigr\}
} $, \linebreak
%
\mbox{} \vspace{-0.530cm} \\
%
$ {\displaystyle
\|\, D \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!=\;\!
\max \, \bigl\{\,
\|\, D_{\!\;\!j} \;\! u_{i}(\cdot,t)\,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!\!: \:\! 1 \leq i, \:\!j \leq n
\,\bigr\}
} $
and,
for general \mbox{$m \!\;\!\geq\!\:\! 1$\/:} \\
%
\mbox{} \vspace{-0.500cm} \\
%
\begin{equation}
\tag{1.5$d$}
\|\, D^{m} \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!\;\!=\;
\max\,\Bigl\{\;\!
\|\;\! D_{\!\;\!j_{\mbox{}_{1}}}
\!\!\!\;\!\;\!\,\!\cdot \!\;\!\cdot \!\;\!\cdot \!\;\!\;\!
D_{\!\;\!j_{\mbox{}_{m}}}
\!\!\;\!\;\! u_{i}(\cdot,t)\,
\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}
\!\!\!\;\!: \;\!
1 \leq \!\;\!\;\!
i, \!\;\!\;\!j_{\mbox{}_{1}}\!\!\;\!\;\!, \!...\!\;\!\;\!,j_{\mbox{}_{m}}
\!\leq n
\!\;\!\;\!\Bigr\}.
\end{equation}
%
\mbox{} \vspace{-0.150cm} \\
%
The definitions chosen in (1.5) are very convenient
for the discussion that follows. \linebreak
%
\nl
\mbox{} \vspace{-0.450cm} \\
%

%
% *************************************************************
% *                                                           *
% *              Section 2: Improving K_3, K_4                *
% *                                                           *
% *************************************************************
%

{\bf 2. Derivation of (1.4) and related improvements} \\
% \mbox{\boldmath $K_{\mbox{}_{\!\:\!3}}$}} \\
%
\mbox{} \vspace{-0.750cm} \\
%

Here we
elaborate
on the method used to obtain (1.3$b$)
in
(\cite{KreissHagstromLorenzZingano2003}, p.\;235)
in order to improve the current estimates
on the regularity time $ \:\!t_{\ast} \!\:\!$
(in dimension $ n = 3, 4$)
as defined in (1.3) above.
We first recall the
elementary Sobolev inequalities \\
%
\mbox{} \vspace{-0.650cm} \\
%
\begin{equation}
\tag{2.1$a$}
\|\,\mbox{u}\,
\|_{\mbox{}_{\scriptstyle L^{3}(\mathbb{R}^{3})}}
\leq\:
\Gamma_{\mbox{}_{\!\,\!3}}
\,
\|\,\mbox{u}\,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\,\!
\|\, D \:\!\mbox{u}\,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\end{equation}
%
\mbox{} \vspace{-0.650cm} \\
%
and \\
%
\mbox{} \vspace{-1.200cm} \\
%
\begin{equation}
\tag{2.1$b$}
\|\,\mbox{u}\,
\|_{\mbox{}_{\scriptstyle L^{3}(\mathbb{R}^{4})}}
\leq\:
\Gamma_{\mbox{}_{\!\,\!4}}
\,
\|\,\mbox{u}\,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!1/3}
\,\!
\|\, D \:\!\mbox{u}\,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2/3}
\end{equation}
%
\mbox{} \vspace{-0.150cm} \\
%
for functions in $ H^{1}(\mathbb{R}^{3}) $
and $ H^{1}(\mathbb{R}^{4}) $, respectively.
In both cases,
extremals are given by
$ \;\!\mbox{u}(x) =\;\! C \!\cdot
\mbox{sech}^{2} \:\!(\:\!
\lambda \,|\,x - x_0 \;\!|_{\mbox{}_{2}}) $
for arbitrary $ C \!\in \mathbb{R} $,
$ \lambda \neq 0 $, $ x_0 \!\in \mathbb{R}^{n} \!$
(see \cite{Agueh2008}, p.\;761),
so that the optimal constants in (2.1)
satisfy \\
%
\mbox{} \vspace{-0.600cm} \\
%
\begin{equation}
\tag{2.2}
\Gamma_{\mbox{}_{\!\:\!3}} \!\;\!<\,
\mbox{\small $0.558\,901\,115\,737 $},
\qquad
\Gamma_{\mbox{}_{\!\:\!4}} \!\;\!<\,
\mbox{\small $0.419\,577\,519\,172$}.
\end{equation}

%
\mbox{} \vspace{-0.850cm} \\
%
We begin with the following lemma. \\
%
\mbox{} \vspace{-0.050cm} \\
%
% -----------------------------------------------
%
%                 lemma 2.1
%
% -----------------------------------------------
%
{\bf Lemma 2.1.}
\textit{%
%
Let $\;\! n = 3, 4 $,
and
$ \;\!{\bf u} = (\;\!u_{\mbox{}_{1}} \!\;\!,\!\;\!..., u_{\mbox{}_{\scriptstyle n}})
\in H^{2}(\mathbb{R}^{n})^{\:\!n} \!$.
%
Then \\
}
%
\mbox{} \vspace{-0.800cm} \\
%
\begin{equation}
\tag{2.3$a$}
\int_{\mbox{}_{\scriptstyle \mathbb{R}^{3}}}
\!\!\;\!\Bigl\{\!\!\!\!\:\!
\sum_{\mbox{}\;\;\,i, \,j, \,\ell \,=\,1}^{3}
\!\!\!\!
|\, D_{\ell} \;\!u_{i} \;\!| \:
|\, D_{\ell} \;\!u_{j} \;\!| \:
|\, D_{j} \;\!u_{i} \;\!|
\,\Bigr\}
\;\!dx
%
\:\leq\:
%
\Gamma_{\scriptscriptstyle \!\:\!3}^{\;\!3}
\,
\|\, D \:\!{\bf u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!3/2}
%
\|\, D^{2} \,\!{\bf u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!3/2}
\end{equation}
%
\mbox{} \vspace{-0.100cm} \\
%
\textit{%
if $\;\!n = 3 $,
and \\
}
%
\mbox{} \vspace{-0.750cm} \\
%
\begin{equation}
\tag{2.3$b$}
\int_{\mbox{}_{\scriptstyle \mathbb{R}^{4}}}
\!\!\;\!\Bigl\{\!\!\!\!\:\!
\sum_{\mbox{}\;\;\,i, \,j, \,\ell \,=\,1}^{4}
\!\!\!\!
|\, D_{\ell} \;\!u_{i} \;\!| \:
|\, D_{\ell} \;\!u_{j} \;\!| \:
|\, D_{j} \;\!u_{i} \;\!|
\,\Bigr\}
\;\!dx
%
\:\leq\:
%
\Gamma_{\scriptscriptstyle \!\:\!4}^{\;\!3}
\,
\|\, D \:\!{\bf u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
%
\|\, D^{2} \,\!{\bf u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}^{\:\!2}
\end{equation}
%
\mbox{} \vspace{-0.050cm} \\
%
\textit{%
if $\;\!n = 4 $,
where
$\:\!\Gamma_{\mbox{}_{\!\:\!3}}\!\;\!$,
$\Gamma_{\mbox{}_{\!\:\!4}}\!$
are the constants in
\:\!\mbox{\em (2.1)}, \mbox{\em (2.2)}\!
above.
}

%
% ------------------------------ End of Lemma 2.1
%

%
\mbox{} \vspace{+0.050cm} \\
%
% ----------------------------- Proof of lemma 2.1:
%
{\small
%
{\bf Proof:}
%
By repeated application
of the Cauchy-Schwarz inequality,
we have \\
%
\mbox{} \vspace{-0.100cm} \\
%
\mbox{} \hfill
$ {\displaystyle
\sum_{\mbox{}\;\;\,i, \,j, \,\ell \,=\,1}^{n}
\!\!\!\!\!\;\!
|\, D_{\ell} \;\!u_{i} \;\!| \:
|\, D_{\ell} \;\!u_{j} \;\!| \:
|\, D_{j} \;\!u_{i} \;\!|
\;\leq\!
\sum_{\mbox{}\;\, i, \,\ell \,=\,1}^{n}
\!\!\:\!
|\, D_{\ell} \;\!u_{i} \;\!|
\:
\Bigl\{\;\!
\sum_{j\,=\,1}^{n}
|\, D_{\ell} \;\!u_{j} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
\!\;\!
\Bigl\{\,
\sum_{j\,=\,1}^{n}
|\, D_{j} \:\!u_{i} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
} $ \\
%
\mbox{} \vspace{+0.050cm} \\
%
\mbox{} \hfill
$ {\displaystyle
\leq\:
\sum_{i\,=\,1}^{n}
\;
\Bigl\{\,
\sum_{j\,=\,1}^{n}
|\, D_{j} \:\!u_{i} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
\,\!
\Bigl\{\,
\sum_{\ell\,=\,1}^{n}
|\, D_{\ell} \;\!u_{i} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
\,\!
\Bigl\{\!
\sum_{\mbox{}\;\,j,\,\ell\,=\,1}^{n}
\!\!\!\;\!
|\, D_{\ell} \;\!u_{j} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
} $ \\
%
\mbox{} \vspace{+0.050cm} \\
%
\mbox{} \hfill
$ {\displaystyle
\leq\;
\Bigl\{\!
\sum_{\mbox{}\;j,\,\ell\,=\,1}^{n}
\!\!\!\;\!
|\, D_{\ell} \;\!u_{j} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
\:\!
\Bigl\{\!
\sum_{\mbox{}\;i,\,j\,=\,1}^{n}
\!\!\!\;\!
|\, D_{j} \:\!u_{i} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
\:\!
\Bigl\{\!
\sum_{\mbox{}\;i,\,\ell\,=\,1}^{n}
\!\!\!\;\!
|\, D_{\ell} \;\!u_{i} \;\!|^{2}
\;\!\Bigr\}^{\!\!\:\!1/2}
\!\!\!\!
} $, \\
%
\mbox{} \vspace{-0.350cm} \\
%
so that \\
%
\mbox{} \vspace{-0.700cm} \\
%
\begin{equation}
\notag
\int_{\mbox{}_{\scriptstyle \mathbb{R}^{n}}}
\!\!\;\!\Bigl\{\!\!\!\!\:\!
\sum_{\mbox{}\;\;\,i, \,j, \,\ell \,=\,1}^{n}
\!\!\!\!
|\, D_{\ell} \;\!u_{i} \;\!| \:
|\, D_{\ell} \;\!u_{j} \;\!| \:
|\, D_{j} \;\!u_{i} \;\!|
\,\Bigr\}
\;\!dx
\:\leq\;
\|\: \mbox{w} \:
\|_{\mbox{}_{\scriptstyle L^{3}(\mathbb{R}^{n})}}^{3}
\!,
\quad \;\,
\mbox{w}(x) :=
\Bigl\{\!\!\;\!
\sum_{\mbox{} \;i, \,j \,=\,1}^{n}
\!\!
|\, D_{j} \;\!u_{i} \;\!|^{\:\!2}
\,
\Bigr\}^{\!\!\;\!1/2}
\!\!\!\!.
\end{equation}
%
\mbox{} \vspace{+0.025cm} \\
%
Applying (2.1) to the function
$ \,\mbox{w} \,$
then gives the result,
%
since,
by (1.5),
we have \\
%
\mbox{} \vspace{-0.100cm} \\
%
\mbox{} \hspace{+1.500cm}
$ {\displaystyle
\|\,\mbox{w}\,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
=\;
\|\,D\:\!{\bf u}\,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\!\;\!,
\quad
\;\;
\|\,D\:\!\mbox{w}\,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\leq\;\!
\|\,D^{2} \,\!{\bf u}\,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\!\;\!
} $.
%
\mbox{} \hfill $\Box$ \\
%
%
% -------------------------------- End of Proof of lemma 2.1
%
%
}
%
\mbox{} \vspace{-0.500cm} \\
%

We are now in good standing
to reexamine (1.3)
of Section~1.
%
\!Starting with $ n = 3 $
and
recalling the
Leray's regularity time $ \:\!t_{\ast} $
given in (1.3$a$), (1.3$b$),
we have\:\!: \\
%
\mbox{} \vspace{-0.010cm} \\
%
% -----------------------------------------------
%
%                 Theorem 2.1
%
% -----------------------------------------------
%
{\bf Theorem 2.1.}
%
\textit{%
%
Let $\;\! n = 3 $,
and let $\;\!\mbox{\boldmath $u$}(\cdot,t) $
be any given Leray solution
to the Navier-Stokes system $\,(1.1)$.
Then
there exists $ \;\!t_{\ast\ast} \!\;\!$
satisfying
} \\
%
\mbox{} \vspace{-0.700cm} \\
%
\begin{equation}
\tag{2.4}
t_{\ast}
\;\!\leq\,
t_{\ast\ast}
%
% \,\!<\:
%
% % \mbox{\small $ {\displaystyle
% % \frac{1}{\,128\,\pi^{2}} }$}
%
% \mbox{\small $ 0.000\,464\,504\,284 $}
%
\,\!\leq\:
%
\mbox{\small $ {\displaystyle \frac{1}{2} }$}
\:
\Gamma_{\scriptscriptstyle \!\:\!3}^{\;\!12}
\cdot \;\!
\nu^{-\,5} \,
\|\, \mbox{\boldmath $u$}_0 \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!4}
\end{equation}
%
\mbox{} \vspace{-0.250cm} \\
%
\textit{%
such that
$ {\displaystyle
\;\!
\|\, D\:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\!\!\:\!
} $
is monotonically decreasing
everywhere in
$ [\,t_{\ast\ast}\!\;\!, \infty) $,
where
$ \Gamma_{\mbox{}_{\!\:\!3}} \!\:\! $
is given in \:\!\mbox{\em (2.1$a$)},
\mbox{\em (2.2)}\!\;\! above.
%
}
%
% ----------------------------------- End of Theorem 2.1
%

%
\mbox{} \vspace{-0.850cm} \\
%
% ---------------------------------- Proof of Theorem 2.1:
%
{\small
%
{\bf Proof:}
%
Consider
$ \:\!\hat{t} > 0 $
satisfying \\
%
\mbox{} \vspace{-0.625cm} \\
%
\begin{equation}
\tag{2.5}
\hat{t} \;>\:
\frac{\;\!1\;\!}{2} \:
\Gamma_{\scriptscriptstyle \!\:\!3}^{\;\!12}
\;\!\;\!
\nu^{-\,5}
\,
\|\, \mbox{\boldmath $u$}_{0} \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{4}
\!\!\!\;\! ~.
\end{equation}
%
\mbox{} \vspace{-0.350cm} \\
%
Since
(by (1.2))
$ {\displaystyle
\!\!\;\!
\int_{0}^{\:\!\mbox{\footnotesize $\hat{t}$}}
\!\!\:\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{2}
d\tau
\,\!\leq\;\!
\mbox{\small $ {\displaystyle \frac{1}{2\;\!\nu} }$}
\:
\|\, \mbox{\boldmath $u$}_0 \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}^{2}
\!\!\;\!
} $,
%
there exists
some set $ E \!\;\!\subseteq\!\;\! (\;\!0,\!\;\!\;\!\hat{t}\;\!) $
of positive measure
such that \\
%
\mbox{} \vspace{-0.600cm} \\
%
\begin{equation}
\tag{2.6}
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\,\leq\:
\frac{1}{\sqrt{\:\!2\;\!\nu\,}\,}
\:
\|\, \mbox{\boldmath $u$}_{0} \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\;\!
\hat{t}^{\:-\,1/2}
\quad \;\;\,
\forall \;\,
t^{\:\!\prime} \!\:\!\in E.
\end{equation}
%
\mbox{} \vspace{-0.150cm} \\
%
By the {\em epochs of regularity property\/}
\cite{Leray1934} (see also \cite{Heywood1988}),
we can then choose
$ \:\!t^{\:\!\prime} \!\in\!\;\! E $
such that $ \:\!\mbox{\boldmath $u$}(\cdot,\tau) $
is smooth for $\:\!\tau\!\;\!$ close to $ \:\!t^{\:\!\prime} \!$.
%
Hence,
by (2.5),
we have \\
%
\mbox{} \vspace{-0.600cm} \\
%
\begin{equation}
\tag{2.7}
\Gamma_{\mbox{}_{\!\:\!3}}^{\;\!3}
\,
\|\, \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\:\!
\|\, D\:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\!\,\!<\; \nu
\end{equation}
%
\mbox{} \vspace{-0.175cm} \\
%
for all $ \tau \geq t^{\:\!\prime} $
close to
the point $ t^{\prime} \!\:\!$.
%
This gives

%
% differentiating (1.1$a$)
% with respect to $ \:\!x_{\mbox{}_{\scriptstyle \!\;\!\ell}} $,
% multiplying by
% $ D_{\mbox{}_{\scriptstyle \!\:\!\ell}} \mbox{\boldmath $u$}(\cdot,t) $,
% integrating on $\,[\,t^{\prime} \!\:\!, \:\!t\;\!] $,
% and summing over $ \:\!\ell = 1, 2, 3$, \\
%
\mbox{} \vspace{-0.100cm} \\
%
\mbox{} \hspace{+3.500cm}
$ {\displaystyle
\|\, D \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\:
2 \,\nu \!\!\;\!
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
d\tau
} $ \\
%
\mbox{} \vspace{-0.625cm} \\
%
\mbox{} \hfill (2.8) \\
%
\mbox{} \vspace{-0.425cm} \\
%
\mbox{} \hfill
$ {\displaystyle
\leq\;\:\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\,
2 \hspace{-0.200cm}
\sum_{i, \, j, \, \ell\,=\,1}^{3}
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\int_{\mathbb{R}^{3}}
\!\!\;\!
|\, D_{\mbox{}_{\scriptstyle \!\ell}} \:\! u_{i}(x,\tau) \,|
\;
|\, D_{\mbox{}_{\scriptstyle \!\ell}} \:\! u_{j}(x,\tau) \,|
\;
|\, D_{\scriptstyle \!j} \;\!u_{i}(x,\tau) \,|
\;
dx \, d\tau
} $ \\
%
\mbox{} \vspace{+0.050cm} \\
%
\mbox{} \hspace{+0.850cm}
$ {\displaystyle
\leq\;
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\,
2 \!\!\;\!
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\Gamma_{\mbox{}_{\!\:\!3}}^{\;\!3}
\,
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!3/2}
%
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!3/2}
\,\!
d\tau
} $
\mbox{} \hfill
\mbox{[}$\,$by (2.3$a$)$\,$\mbox{]} \\
%
\mbox{} \vspace{+0.025cm} \\
%
\mbox{} \hfill
$ {\displaystyle
\leq\;
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\,
2 \!\!\;\!
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\bigl[\;\!\;\!
\Gamma_{\mbox{}_{\!\:\!3}}^{\;\!3}
\,
\|\, \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\:\!
\bigr]
\:
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}}}
  ^{\:\!2}
%
d\tau
} $ \\
%
\mbox{} \vspace{-0.025cm} \\
%
\mbox{} \hspace{+2.950cm}
$ {\displaystyle
\leq\:
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\,
2 \, \nu \!\!\;\!
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
d\tau
} $
\mbox{} \hfill
\mbox{[}$\,$by (2.7)$\,$\mbox{]} \\
%
\mbox{} \vspace{+0.100cm} \\
%
for all $ \;\!t \geq\,\! t^{\:\!\prime} \!\:\!$
close to $ t^{\:\!\prime} \!\,\!$,
where
in the fourth line above
we used the elementary estimate \\
%
\mbox{} \vspace{-0.575cm} \\
%
\begin{equation}
\notag
\|\, D \:\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\:\!\leq\;
\|\, \mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!1/2}
\,\!
\|\, D^{2} \mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!1/2}
\end{equation}
%
\mbox{} \vspace{-0.175cm} \\
%
(valid for any $\:\!n$),
which is easily obtained
using the Fourier transform.
%
This shows that \linebreak
%
$ {\displaystyle
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{3})}
\!\;\!
} $
stays bounded by
$ {\displaystyle
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{3})}
\!\;\!
} $,
and because
$ {\displaystyle
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{3})}
\!\;\!
} $
cannot surpass
$ {\displaystyle
\;\!
\|\, \mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{3})}
\!\;\!
} $
(in view of (1.2)),
it follows
that
we must then have \\
%
\mbox{} \vspace{-0.550cm} \\
%
\begin{equation}
\tag{2.9}
\Gamma_{\mbox{}_{\!\:\!3}}^{\;\!3}
\,
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\:\!
\|\, D\:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\!\;\!<\: \nu
%
% \qquad
% \forall \;\,
% t \geq t^{\:\!\prime}
% \!\:\!.
%
\end{equation}
%
\mbox{} \vspace{-0.200cm} \\
%
for {\em all} $ \, t \geq t^{\:\!\prime} \!$,
and in particular
$ \:\!\mbox{\boldmath $u$}(\cdot,t) $
%
% is smooth for $ \:\![\, t^{\:\!\prime} \!, \:\! \infty) $.
%
is smooth for all
$ \;\! t \in [\;\!t^{\:\!\prime} \!\,\!, \infty) $.
%
So,
all
the estimates
in (2.8) above
can be done
on {\em any\/} interval
$ [\,t_0, \:\!t\, ]  \subseteq
[\,t^{\:\!\prime}\!\:\!, \infty) $,
giving,
for any $ \;\! t \geq t_0 \!\;\!\geq t^{\:\!\prime}$: \\
%

%
\mbox{} \vspace{-0.900cm} \\
%
\mbox{} \hspace{+3.500cm}
$ {\displaystyle
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\:
2 \, \nu \!\!\;\!
\int_{\mbox{\footnotesize $ t_0 $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
d\tau
} $ \\
%
\mbox{} \vspace{-0.075cm} \\
%
\mbox{} \hfill
$ {\displaystyle
\leq\:
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\,
2 \!\!\;\!
\int_{\mbox{\footnotesize $ t_0 $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!\!\;\!
\bigl[\:
\Gamma_{\mbox{}_{\!\:\!3}}^{\;\!3}
\,
\|\, \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!1/2}
\:\!
\bigr]
\:
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\,\!
d\tau
} $ \\
%
\mbox{} \vspace{-0.075cm} \\
%
\mbox{} \hspace{+2.950cm}
$ {\displaystyle
\leq\:
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
\!\:\!+\:
2 \, \nu \!\!\;\!
\int_{\mbox{\footnotesize $ t_0 $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!\!\;\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
  ^{\:\!2}
d\tau
} $.
\mbox{} \hfill
\mbox{[}$\,$by (2.9)$\,$\mbox{]} \\
%
\mbox{} \vspace{+0.050cm} \\
%
This shows that
$ {\displaystyle
\;\!
\|\, D\:\! \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{3})}}
\!
} $
stays finite and is monotonically decreasing
% for $ \:\!t \geq t^{\:\!\prime} $
everywhere in
$ {\displaystyle
[\,t^{\:\!\prime}\!, \:\!\infty)
\supseteq\:\!
[\,\hat{t}, \:\!\infty)
} $.
%
$\!$Therefore,
by Leray's theory,
(1.3$a$) of Section~1
will be satisfied for
any
$ \;\!t_{\ast} \!> t^{\:\!\prime} \!\;\!$,
with
$ \;\!t^{\:\!\prime} \!\:\!<\:\! \hat{t\:\!} \!\;\!$.
%
Recalling (2.5),
this completes the proof of
{\sc Theorem 2.1}.
%
% since
% $ {\displaystyle
% \:\!
% 1/2
% \;\:\!
% \Gamma_{\mbox{}_{\!\:\!3}}^{\;\!12}
% \!\;\!<\:\!
% 0.000\,464\,504\,284
% } $.
%
%
% ------------------------- End of Proof of Theorem 2.1
%
}
%
\hfill \mbox{\small $\Box$} \\
%
%
% --------------------------------------------- Theorem 2.2:
%
%
\nl
%
%
In much the same way,
using (2.3$b$)
and the 4\mbox{\small D}
version of (2.8),
we can show$\:\!$: \\
%
% we obtain
% the following result. \\
%
%
\mbox{} \vspace{-0.050cm} \\
%
% -----------------------------------------------
%
%                 Theorem 2.2
%
% -----------------------------------------------
%
{\bf Theorem 2.2.}
%
\textit{%
%
Let $\;\! n = 4 $,
and let $\;\!\mbox{\boldmath $u$}(\cdot,t) $
be any given Leray solution
to the Navier-Stokes system $\,(1.1)$.
%
Let $\;\!t_{\ast} \!\;\!\geq 0 $
be defined in $\;\!(1.3$a$)$.
%
Then
there exists $ \;\!t_{\ast\ast} \!\;\!$
satisfying
} \\
%
\mbox{} \vspace{-0.650cm} \\
%
\begin{equation}
\tag{2.10}
t_{\ast}
\;\!\leq\,
t_{\ast\ast}
%
% \,\!<\:
%
% \mbox{\small $ 0.002\,727\,993\,110 $}
%
\:\!\leq\:
%
\mbox{\small $ {\displaystyle \frac{1}{2} }$}
\:
\Gamma_{\scriptscriptstyle \!\:\!4}^{\;\!6}
\cdot \;\!
\nu^{-\,3}
\,
\|\, \mbox{\boldmath $u$}_0 \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\end{equation}
%
\mbox{} \vspace{-0.275cm} \\
%
\textit{%
such that
$ {\displaystyle
\;\!
\|\, D\:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\!\!\:\!
} $
is monotonically decreasing
everywhere in
$ [\,t_{\ast\ast}\!\;\!, \infty) $,
where
$ \Gamma_{\mbox{}_{\!\:\!4}} \!\,\! $
is given in \,\!\mbox{\em (2.1$b$)},
\!\:\!\mbox{\em (2.2)}\!\:\! above.
%
}
%
% ----------------------------------- End of Theorem 2.2
%

%
\mbox{} \vspace{-0.050cm} \\
%
% --------------------------------- Proof of Theorem 2.2:
%
{\small
%
{\bf Proof:}
%
Consider
$ \:\!\hat{t} > 0 $
satisfying \\
%
\mbox{} \vspace{-0.625cm} \\
%
\begin{equation}
\tag{2.11}
\hat{t} \;>\:
\frac{\;\!1\;\!}{2} \:
\Gamma_{\scriptscriptstyle \!\:\!4}^{\;\!6}
\;\!\;\!
\nu^{-\,3}
\,
\|\, \mbox{\boldmath $u$}_{0} \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}^{2}
\!\!\!\;\! ~.
\end{equation}
%
\mbox{} \vspace{-0.350cm} \\
%
As before,
from (1.2)
it follows the existence
of some set
$ E \!\;\!\subseteq\!\;\! (\;\!0,\!\;\!\;\!\hat{t}\;\!) $
with positive measure
such that \\
%
\mbox{} \vspace{-0.800cm} \\
%
\begin{equation}
\tag{2.12}
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\,\leq\:
\frac{1}{\sqrt{\:\!2\;\!\nu\,}\,}
\:
\|\, \mbox{\boldmath $u$}_{0} \;\!
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\;\!
\hat{t}^{\:-\,1/2}
\quad \;\;\,
\forall \;\,
t^{\:\!\prime} \!\:\!\in E
\end{equation}
%
\mbox{} \vspace{-0.150cm} \\
%
and so,
by the epochs of regularity property,
we can again choose
$ \;\!t^{\:\!\prime} \!\in\!\;\! E \;\!$
such that $ \:\!\mbox{\boldmath $u$}(\cdot,\tau) $
is smooth for $\:\!\tau\!\;\!$ close to $ \:\!t^{\:\!\prime} \!$.
%
Hence,
by (2.11) and (2.12),
we have \\
%
\mbox{} \vspace{-0.600cm} \\
%
\begin{equation}
\tag{2.13}
\Gamma_{\mbox{}_{\!\:\!4}}^{\;\!3}
\;\!\;\!
\|\, D\:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\!\;\!<\; \nu
\end{equation}
%
\mbox{} \vspace{-0.175cm} \\
%
for all $ \tau \geq t^{\:\!\prime} $
close to
the point $ t^{\prime} \!\:\!$.
%
This then gives

%
\mbox{} \vspace{-0.100cm} \\
%
\mbox{} \hspace{+3.500cm}
$ {\displaystyle
\|\, D \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\:
2 \,\nu \!\!\;\!
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
d\tau
} $ \\
%
\mbox{} \vspace{-0.025cm} \\
%
\mbox{} \hfill
$ {\displaystyle
\leq\;\:\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\,
2 \hspace{-0.200cm}
\sum_{i, \, j, \, \ell\,=\,1}^{4}
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\int_{\mathbb{R}^{4}}
\!\!\;\!
|\, D_{\mbox{}_{\scriptstyle \!\ell}} \:\! u_{i}(x,\tau) \,|
\;
|\, D_{\mbox{}_{\scriptstyle \!\ell}} \:\! u_{j}(x,\tau) \,|
\;
|\, D_{\scriptstyle \!j} \;\!u_{i}(x,\tau) \,|
\;
dx \, d\tau
} $

\newpage
%
\mbox{} \vspace{-0.850cm} \\
%
\mbox{} \hspace{+0.850cm}
$ {\displaystyle
\leq\;
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\,
2 \!\!\;\!
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\Gamma_{\mbox{}_{\!\:\!4}}^{\;\!3}
\,
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
%
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\,\!
d\tau
} $
\mbox{} \hfill
\mbox{[}$\,$by (2.3$b$)$\,$\mbox{]} \\
%
\mbox{} \vspace{-0.025cm} \\
%
\mbox{} \hspace{+2.950cm}
$ {\displaystyle
\leq\:
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\,
2 \, \nu \!\!\;\!
\int_{\mbox{\footnotesize $ t^{\:\!\prime} $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
d\tau
} $
\mbox{} \hfill
\mbox{[}$\,$by (2.13)$\,$\mbox{]} \\
%
\mbox{} \vspace{-0.350cm} \\
%
\mbox{} \hfill (2.14) \\
%
\mbox{} \vspace{-0.375cm} \\
%
for all $ \;\!t \geq\,\! t^{\:\!\prime} \!\;\!$
close to $ t^{\:\!\prime} \!\,\!$.
%
This actually implies
that
%
$ {\displaystyle
\;\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{4})}
\!\;\!
} $
will stay bounded by
$ {\displaystyle
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t^{\:\!\prime}) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{4})}
\!\;\!
} $
for {\em all}
$ \,t \geq\,\! t^{\:\!\prime} \!\,\!$,
\;\!so that
we actually have \\
%
\mbox{} \vspace{-0.550cm} \\
%
\begin{equation}
\tag{2.15}
\Gamma_{\mbox{}_{\!\:\!4}}^{\;\!3}
\;\!\;\!
\|\, D\:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\!\;\!<\: \nu
%
\qquad
\forall \;\,
\tau \geq t^{\:\!\prime}
\!\:\!.
%
\end{equation}
%
\mbox{} \vspace{-0.200cm} \\
%
In particular,
it follows that
$ \;\!\mbox{\boldmath $u$}(\cdot,t) \:\!$
%
is smooth for every
$ \;\! t \in \:\![\, t^{\:\!\prime} \!, \:\! \infty) $.
%
This fact
and (2.15)
allow us to repeat
the derivation of
(2.14) above
on {\em any\/} interval
$ [\,t_0, \:\!t\, ]  \subseteq
[\,t^{\:\!\prime}\!\:\!, \infty) $,
obtaining \\
%
\mbox{} \vspace{-0.100cm} \\
%
\mbox{} \hspace{+3.500cm}
$ {\displaystyle
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\:
2 \, \nu \!\!\;\!
\int_{\mbox{\footnotesize $ t_0 $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
d\tau
} $ \\
%
\mbox{} \vspace{-0.075cm} \\
%
\mbox{} \hspace{+1.500cm}
$ {\displaystyle
\leq\:
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\,
2 \!\!\;\!
\int_{\mbox{\footnotesize $ t_0 $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!\!\;\!
\Gamma_{\mbox{}_{\!\:\!4}}^{\;\!3}
\;\!\;\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\;\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\,\!
d\tau
} $ \\
%
\mbox{} \vspace{-0.075cm} \\
%
\mbox{} \hspace{+2.975cm}
$ {\displaystyle
\leq\:
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\:
2 \, \nu \!\!\;\!
\int_{\mbox{\footnotesize $ t_0 $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!\!\;\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
d\tau
} $.
\mbox{} \hfill
\mbox{[}$\,$by (2.15)$\,$\mbox{]} \\
%
\mbox{} \vspace{+0.050cm} \\
%
This shows that
$ {\displaystyle
\;\!
\|\, D\:\! \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\!
} $
stays bounded and monotonically decreasing
% for $ \:\!t \geq t^{\:\!\prime} $
everywhere in
$ {\displaystyle
[\,t^{\:\!\prime}\!, \:\!\infty)
\supseteq\:\!
[\,\hat{t}, \:\!\infty)
} $.
%
$\!$As in the previous proof,
(1.3$a$)
must then be satisfied for
any
$ t_{\ast} \!> t^{\:\!\prime} \!\;\!$,
where
$ \;\!t^{\:\!\prime} \!\:\!<\:\! \hat{t\:\!} \!\;\!$.
%
Recalling (2.11),
this completes the proof of
{\sc Theorem 2.2}.
%
%
% ------------------------- End of Proof of Theorem 2.2
%
}
%
\hfill \mbox{\small $\Box$} \\
%
%
\nl
%
% -------------------------------------------------------------
%
%                          Remark 2.1
%
% -------------------------------------------------------------
%
{\bf Remark 2.1.}
\!\:\!In dimension $ \:\! n = 2 $,
it is well known that
$ \:\!\mbox{\boldmath $u$} \in C^{\infty}(\,\!\mathbb{R}^{2}
\!\times\!\:\! (\:\!0, \infty)) \,\!$
and \linebreak
%
\mbox{} \vspace{-0.625cm} \\
%
\begin{equation}
\tag{2.16}
\mbox{\boldmath $u$}(\cdot,t)
\in
C^{0}(\,\!(\:\!0, \infty), \:\!
\mbox{\boldmath $H$}\mbox{}^{\!\:\!m}\!\;\!(\mathbb{R}^{2}))
\quad \;\,
\forall \;\;\! m \geq 0,
\end{equation}
%
\mbox{} \vspace{-0.600cm} \\
%
with \\
%
\mbox{} \vspace{-0.900cm} \\
%
\begin{equation}
\tag{2.17}
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{2})}}
  ^{\:\!2}
\!\:\!+\:
2 \, \nu \!\!\;\!
\int_{\mbox{\footnotesize $ t_0 $}}
    ^{\mbox{\footnotesize $\:\!t$}}
\!
\|\, D^{2} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{2})}}
  ^{\:\!2}
d\tau
\;=\;
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{2})}}
  ^{\:\!2}
\end{equation}
%
\mbox{} \vspace{-0.125cm} \\
%
for all
$ \;\! t \geq t_0 \!\:\!> 0 $,
so that,
in particular,
$ {\displaystyle
\;\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{2})}}
\!\;\!
} $
is monotonically decreasing
in $(\:\!0, \infty) $.
%
Since
monotonic functions
$ f \!\;\!\in L^{1}(\:\!t_0, \infty) $
satisfy:
$ f(t) = o(1/t) $
as $ t \rightarrow \infty $
(because, in this case:
$ \;\!t \,| \:\!f(t)\;\!| \:\!\leq\,
\mbox{\small 2} \!\:\!\int_{\:\!t/2}^{\;\!t}
| \:\!f(\tau)\;\! | \, d\tau \,\!$
for all $ \:\!t \geq \mbox{\small 2} \;\!t_{0} $),
%
it follows
that \\
%
% $ {\displaystyle
% \lim_{t\,\rightarrow\,\infty}
% t^{1/2} \,
% \|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
% \|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
% \!=\;\! 0
% } $
% ($ 2 \leq n \leq 4 $),
%
\mbox{} \vspace{-0.600cm} \\
%
\begin{equation}
\tag{2.18}
\lim_{t\,\rightarrow\,\infty}
\,
t^{1/2} \,
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\;\!=\; 0
\end{equation}
%
\mbox{} \vspace{-0.150cm} \\
%
if $ \;\!2 \leq n \leq 4 $,
as partially observed in
\cite{KreissHagstromLorenzZingano2002, %
KreissHagstromLorenzZingano2003}.
%
\!(\:\!For a derivation of (2.18)
in the case of more general $n$,
see \cite{OliverTiti2000, SchonbekWiegner1996}.)
%
\newpage
%
\mbox{} \vspace{-0.750cm} \\
%
% -------------------------------------------------------------
%
%                          Remark 2.2
%
% -------------------------------------------------------------
%
{\bf Remark 2.2.}
%
From (2.18),
one can easily obtain
that
all Leray solutions to (1.1)
satisfy the asymptotic property \\
%
\mbox{} \vspace{-0.650cm} \\
%
\begin{equation}
\tag{2.19}
\lim_{t\,\rightarrow\,\infty}
\,
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\;\!=\; 0
\end{equation}
%
\mbox{} \vspace{-0.125cm} \\
%
if $ \;\!2 \leq n \leq 4 $,
as shown in
\cite{RigeloSchutzZinganoZingano2016, Zingano2015}.\footnote{%
%
% -------------------------------------- Footnote 2:
%
The validity or not of (2.19)
was left open by Leray in \cite{Leray1934} (p.\;248),
being first shown in \cite{Kato1984, Masuda1984}.
}
%
% ----------------------------------- End of Footnote 2
%
In fact,
in view that
$ \mbox{\boldmath $u$}(\cdot,t) $
is smooth
for large~\mbox{$ \:\!t $},
it can be written as \\
%
\mbox{} \vspace{-0.700cm} \\
%
\begin{equation}
\tag{2.20}
\mbox{} \hspace{+0.500cm}
\mbox{\boldmath $u$}(\cdot,t)
\;=\;
e^{\:\!\nu\:\!\Delta (t - t_0)}
\:\!
\mbox{\boldmath $u$}(\cdot,t_0)
\,-
\int_{\!\:\!t_0}^{\;\!t}
\!\!\:\!
e^{\:\!\nu\:\!\Delta (t - \tau)}
\:\!
\mbox{\boldmath $Q$}(\cdot,\tau)
\,
d\tau,
\quad \;\;\,
t > t_0
\end{equation}
%
\mbox{} \vspace{-0.150cm} \\
%
for $ \:\!t_0 \!\;\!$
large enough,
where
$ {\displaystyle
\:\!
\mbox{\boldmath $Q$} \!\;\!=\,\!
\mbox{\boldmath $u$} \!\:\!\cdot\!\:\!\nabla \mbox{\boldmath $u$}
\:\!+\!\;\! \nabla p
\;\!
} $
is the Helmholtz projection
of
$ \,\!\,\!\mbox{\boldmath $u$} \!\;\!\cdot\!\;\!\nabla \mbox{\boldmath $u$} $
in
$ \mbox{\boldmath $L$}^{2}_{\sigma}(\mathbb{R}^{n}) $,
and
$ \:\!e^{\:\!\nu\:\!\Delta t} \!\;\!$
denotes the heat semigroup.
%
%
% -------------------------- Proof of (2.19) for n = 4:
%
%
\!\:\!In the case $ n = 4 $,
we can then
get~(2.19) as follows:
from (2.20),
we obtain,
recalling
that
$ {\displaystyle
\,
\|\, \mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\!\;\!\leq\;\!
\|\, D \:\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
} $ \linebreak
%
\mbox{} \vspace{-0.600cm} \\
%
for arbitrary
$ \;\!\mbox{u} \in H^{1}(\mathbb{R}^{4}) $, \\
%
\mbox{} \vspace{-0.100cm} \\
%
\mbox{} \hspace{+0.250cm}
$ {\displaystyle
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\leq\;
\|\, \mbox{\boldmath $v$}_{0}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\:\!+
\int_{\!\:\!t_0}^{\;\!t}
\!\!\:\!
\|\, \mbox{\boldmath $u$}(\cdot,\tau) \!\;\!\cdot\!\;\!
\nabla \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\;\!
d\tau
} $ \\
%
\mbox{} \vspace{-0.000cm} \\
%
\mbox{} \hspace{+3.025cm}
$ {\displaystyle
\leq\;
\|\, \mbox{\boldmath $v$}_{0}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\:\!+\,
\sqrt{\:\!2\;\!\:\!} \!\!
\int_{\!\:\!t_0}^{\;\!t}
\!\!\:\!
\|\, \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\,\!
\|\, D \:\! \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\;\!
d\tau
} $ \\
%
\mbox{} \vspace{-0.000cm} \\
%
\mbox{} \hspace{+3.025cm}
$ {\displaystyle
\leq\;
\|\, \mbox{\boldmath $v$}_{0}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\:\!+\,
\sqrt{\:\!2\;\!\:\!} \!\!
\int_{\!\:\!t_0}^{\;\!t}
\!\!\:\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\,\!
\|\, D^{2} \,\! \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\;\!
d\tau
} $ \\
%
\mbox{} \vspace{+0.150cm} \\
%
since
$ {\displaystyle
\;\!
\|\, \mbox{\boldmath $Q$}(\cdot,\tau) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{n})}
\leq\;\!
\|\, \mbox{\boldmath $u$}(\cdot,\tau) \!\:\!\cdot\!\:\!
\nabla \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{n})}
} $
by the orthogonality of the
Helm\-holtz projector in
$ \mbox{\boldmath $L$}^{2}(\mathbb{R}^{n}) $
(see e.g.\;\cite{RigeloSchutzZinganoZingano2016}),
or directly using Fourier transform
\cite{KreissHagstromLorenzZingano2002, %
KreissHagstromLorenzZingano2003},
%
and where
$ {\displaystyle
\:\!
\mbox{\boldmath $v$}_{0}(\cdot,t)
\!:=
e^{\:\!\nu\:\!\Delta (t - t_0)}
\:\!
\mbox{\boldmath $u$}(\cdot,t_0)
} $.
%
\!This shows that,
given $ \epsilon > 0 $,
taking $ \:\!t_0 $
large enough
we get
$ {\displaystyle
\;\!
\|\, \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\scriptstyle L^{2}(\mathbb{R}^{4})}
\!\;\!< \epsilon
\;\!
} $
for all $ t > t_{0}$,
since
the integrand on the
righthand side above is in
$ L^{1}(\:\!t_{\ast}, \infty) $.
%
A similar argument for
$ n = 2, \,3 $
can be found in
\cite{RigeloSchutzZinganoZingano2016, %
Zingano2015}.
%
The proof of (2.19)
for arbitrary $ \;\!n \;\!$
is significantly harder
and given in \cite{Wiegner1987}.
%
\mbox{} \hfill $ \Box $ \\
%
\nl
\mbox{} \vspace{-0.450cm} \\
%
% -------------------------------------------------------------
%
%                          Remark 2.3
%
% -------------------------------------------------------------
%
{\bf Remark 2.3.}
%
\!\;\!Now that (2.18) and (2.19) are known,
it is possible to extend~the
monotonicity property
of
$ {\displaystyle
\;\!
\|\, D\,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{L^{2}(\mathbb{R}^{n})}
} $,
$ n \leq 4 $
(cf.\;(2.17) and
\mbox{\small \sc Theorems 2.1},~\mbox{\small \sc 2.2})
to higher order derivatives.
Let $ \;\!t_{\ast} $ be the regularity time
given in (1.3) for $ \;\!n = 3, \;\!4 $,
and $ \;\! t_{\ast} \!:= 0 \:\!$ if $ \:\!n = 2 $
(cf.\;(2.16) above).
We then have\:\!: \\
%
\mbox{} \vspace{+0.025cm} \\
%
% -----------------------------------------------
%
%                 Theorem 2.3
%
% -----------------------------------------------
%
{\bf Theorem 2.3.}
%
\textit{%
%
Let $\;\! 2 \leq n \leq 4 $,
and let $\;\!\mbox{\boldmath $u$}(\cdot,t) $
be any particular Leray solution
to the Navier-Stokes system $\,(1.1)$.
Then,
for each $ \;\! m \geq 1 \!\!\:\!:$
there exists
$ \;\!\:\!t_{\ast\ast}^{\:\!(m)} \!\geq\:\! t_{\ast} \!\;\!$
such that
$ {\displaystyle
\,
\|\, D^{m} \mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\!\!\:\!
} $
is monotonically decreasing
everywhere in
$ \;\![\,t_{\ast\ast}^{\:\!(m)}\!, \:\!\infty) $.
%
} \\
%
% ---------------------------------------- Proof of Theorem 2.3:
%
\nl
%
{\small
%
{\bf Proof\/:}
%
Let $\;\! t_0 \!\;\!> t_{\ast} $,
$ m \geq 1 $.
Given $ \;\! t > t_0 $,
we have from (1.1$a$), (1.1$b$)
the energy estimate \\
%
\mbox{} \vspace{-0.050cm} \\
%
\mbox{} \hspace{+0.750cm}
$ {\displaystyle
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!2}
\!\:\!+\;
2 \, \nu \!\!\;\!
\int_{\mbox{}_{\scriptstyle \!\:\!t_{\mbox{}_{0}}}}^{\;\!t}
\!\!\:\!
\|\, D^{m + 1} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!2}
d\tau
\;\leq\;\:\!
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!2}
} $ \\
%
\mbox{} \vspace{-0.550cm} \\
%
\mbox{} \hfill (2.21) \\
%
\mbox{} \vspace{-0.550cm} \\
%
\mbox{} \hspace{+1.250cm}
$ {\displaystyle
+\;\,
2 \,
% \sum_{\mbox{} \;\;i, \,j, \,j_{\mbox{}_{1}} \!,..., \,j_{\mbox{}_{m}} =\,1}^{n}
\sum
\int_{\mbox{}_{\scriptstyle \!\:\!t_{\mbox{}_{0}}}}^{\;\!t}
\!\!\;\!
\int_{\mbox{}_{\scriptstyle \:\!\mathbb{R}^{n}}}
\!\!\!\:\!
|\, D_{\!\;\!j} \;\!
D_{\!\;\!j_{\mbox{}_{1}}}
\!\!\!\;\!\cdot\!\,\!\cdot\!\,\!\cdot \!\:\!
D_{\!\;\!j_{\mbox{}_{m}}}
\!\:\! u_{i}(x,\tau)\,|
\cdot
|\, D_{\!\;\!j_{\mbox{}_{1}}}
\!\!\!\;\!\cdot\!\,\!\cdot\!\,\!\cdot \!\:\!
D_{\!\;\!j_{\mbox{}_{m}}}
\!\:\! (\:\! u_{i}(x,\tau)\, u_{j}(x,\tau) \,\!)\,|
\;dx \,d\tau
} $ \\
%
\mbox{} \vspace{+0.075cm} \\
%
where the sum
is over
all indices
$ \;\!1 \leq\:\! i, \;\!j, \;\!j_{\mbox{}_{1}}\!\:\!,
 ...\;\!, \;\!j_{\mbox{}_{m}} \!\:\!\leq\:\! n $.
%
Another important inequality is \\
%
\mbox{} \vspace{-0.550cm} \\
%
\begin{equation}
\tag{2.22}
\|\, D^{\ell} \,\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
\:\!\leq\;
\|\, \mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!1 - \theta}
\,\!
\|\, D^{m} \,\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}
  ^{\:\!\theta}
\!\:\!,
\quad \;\;
\theta \,=\:
\mbox{\small $ {\displaystyle \frac{\ell}{m} }$}
\end{equation}
%
\mbox{} \vspace{-0.100cm} \\
%
for every $\;\!0 \leq \ell \leq m $,
which can be obtained by Fourier transform.
%
From this point further, \linebreak
the argument becomes dependent on the dimension $\:\!n$,
and we illustrate the typical steps
by considering, say,
$ \:\! n = 4 $.
In this case,
we observe that,
from (2.21) and H\"older's inequality, \\
%
\mbox{} \vspace{-0.350cm} \\
%
\mbox{} \hspace{+0.750cm}
$ {\displaystyle
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\;
2 \, \nu \!\!\;\!
\int_{\mbox{}_{\scriptstyle \!\:\!t_{\mbox{}_{0}}}}^{\;\!t}
\!\!\:\!
\|\, D^{m + 1} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
d\tau
\;\leq\;\:\!
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
} $ \\
%
\mbox{} \vspace{-0.550cm} \\
%
\mbox{} \hfill (2.23) \\
%
\mbox{} \vspace{-0.550cm} \\
%
\mbox{} \hspace{+0.500cm}
$ {\displaystyle
+\;\,
K_{\!\;\!m}
\:\!
\sum_{\ell\,=\,0}^{m}
\;\!
\int_{\mbox{}_{\scriptstyle \!\:\!t_{\mbox{}_{0}}}}^{\;\!t}
\!\!\;\!
\|\, D^{\:\!\ell} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\;\!
\|\, D^{\:\!m - \ell} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\;\!
\|\, D^{\:\!m + 1} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\;\!
d\tau
} $ \\
%
\mbox{} \vspace{+0.175cm} \\
%
for some constant $ K_{\!\;\!m} $
that depends on $m$ only.
%
Recalling that
$ {\displaystyle
\;\!
\|\, \mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\!\;\!\leq\,
\|\, D \:\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
} $ \linebreak
%
\mbox{} \vspace{-0.575cm} \\
%
(for arbitrary
$ \mbox{u} \in H^{1}(\mathbb{R}^{4}) $),
we get,
using (2.22) above, \\
%
\mbox{} \vspace{-0.550cm} \\
%
\begin{equation}
\tag{2.24}
\|\, D^{\:\!\ell} \,\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\;\!
\|\, D^{\:\!m - \ell} \,\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{4}(\mathbb{R}^{4})}}
\:\!\leq\;
\|\, D \:\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\;\!
\|\, D^{\:\!m + 1} \,\!\mbox{u} \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\end{equation}
%
\mbox{} \vspace{-0.100cm} \\
%
for every
$ \;\! 0 \leq \ell \leq m $,
so that
(2.23) gives \\
%
\mbox{} \vspace{+0.000cm} \\
%
\mbox{} \hspace{+0.750cm}
$ {\displaystyle
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\;
2 \, \nu \!\!\;\!
\int_{\mbox{}_{\scriptstyle \!\:\!t_{\mbox{}_{0}}}}^{\;\!t}
\!\!\:\!
\|\, D^{m + 1} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
d\tau
\;\leq\;\:\!
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t_0) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
} $ \\
%
\mbox{} \vspace{-0.525cm} \\
%
\mbox{} \hfill (2.25) \\
%
\mbox{} \vspace{-0.500cm} \\
%
\mbox{} \hspace{+1.750cm}
$ {\displaystyle
+\;\;\!\;\!
(\:\!m + 1)
\,
K_{\!\;\!m}
\!\;\!
\int_{\mbox{}_{\scriptstyle \!\:\!t_{\mbox{}_{0}}}}^{\;\!t}
\!\!\;\!
\|\, D \:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\;\!
\|\, D^{\:\!m + 1} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\,
d\tau
} $. \\
%
\mbox{} \vspace{+0.175cm} \\
%
In particular,
choosing
$ \;\! t_0 >\:\! t_{\ast} $
so that
$ {\displaystyle
\;\!
(\:\!m + 1) \;\!\;\! K_{\!\;\!m} \,
\|\, D\:\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\!\;\!<\, \nu
\;\!
} $
for all
$ \:\!\tau > t_ 0 $
\mbox{[}\,which is possible
because
$ {\displaystyle
\;\!
\lim_{\tau \,\rightarrow\, \infty}
\|\, D \:\! \mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\!\;\!=\;\! 0
} $
(cf.\;(2.18)\,\!)\;\!\mbox{]},
it follows that \\
%
\mbox{} \vspace{-0.500cm} \\
%
\begin{equation}
\tag{2.26}
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\!\:\!+\;\;\!
\nu \!\!\;\!
\int_{\mbox{}_{\mbox{\footnotesize \!\:\!$s$}}}
    ^{\mbox{\footnotesize \;\!$t$}}
\!\!\;\!
\|\, D^{m + 1} \,\!\mbox{\boldmath $u$}(\cdot,\tau) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
d\tau
\;\leq\;\:\!
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,s) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
  ^{\:\!2}
\end{equation}
%
\mbox{} \vspace{-0.125cm} \\
%
for all
$ \;\! t > s > t_{0} $,
%
showing that
$ {\displaystyle
\;\!
\|\, D^{m} \,\!\mbox{\boldmath $u$}(\cdot,t) \,
\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{4})}}
\!\;\!
} $
is monotonically decreasing in
$ \;\![\;\!t_0, \:\!\infty ) $.
This concludes the proof
when $ \:\! n = 4 $.
%
The cases $ \:\!n = 2, \:\!3 $
are handled in a similar way,
using (2.19) instead of (2.18)
and appropriate
replacements for (2.23) and (2.24).
%
\mbox{} \hfill $ \Box $
%

%
}
%
% ------------------------------------------- End of Proof
%                                            for Theorem 2.3


%
% *************************************************************
% *                                                           *
% *                     Bibliography                          *
% *                                                           *
% *************************************************************
%
% *************************************************************
% *                                                           *
% *                      References                           *
% *                                                           *
% *************************************************************
%

\begin{thebibliography}{999}

%
% ------------------------------------------------------
%

\bibitem{Agueh2008}
\textsc{M. Agueh},
{\it Gagliardo-Nirenberg inequalities involving
the gradient $L^{2}\!$-norm},
C. R. Acad. Sci. Paris, Ser. I, {\bf 346} (2008),
757-762.

\bibitem{BahouriCheminDanchin2011}
{\sc H. Bahouri, J.-Y. Chemin and R. Danchin},
{\sf Fourier Analysis and Nonlinear Partial Differential Equations},
Springer, New York, 2011.

\bibitem{CaffarelliKohnNirenberg1982}
{\sc L. Caffarelli, R. Kohn and L. Nirenberg},
{\it Partial regularity of suitable weak solutions
of the Navier-Stokes equations},
Comm. Pure Appl. Math. {\bf 35} (1982),
771-831.

% \bibitem{ConstantinFoias1988}
% {\sc P. Constantin and C. Foias},
% {\sf Navier-Stokes equations},
% Univ. of Chicago Press, Chicago, 1988.

\bibitem{DoeringGibbon1995}
{\sc C. R. Doering and J. D. Gibbon},
{\sf Applied analysis of the Navier-Stokes equations},
Cambridge Univ. Press, Cambridge, 1995.

\bibitem{FujitaKato1964}
{\sc H. Fujita and T. Kato},
{\it On the Navier-Stokes initial value problem},
Arch. Rat. Mech. Anal. {\bf 16} (1964),
269-315.

\bibitem{Giga1986}
{\sc Y. Giga},
{\it Solutions for semilinear parabolic equations
in $L^{p}\!\;\!$ and regularity of weak solutions
of the Navier-Stokes system},
J. Diff. Eqs. {\bf 62} (1986),
186-212.

\bibitem{Heywood1988}
{\sc J. G. Heywood},
{\it The epochs of regularity for weak solutions
of the Navier-Stokes equations in unbounded domains},
T\^ohoku Math. J. {\bf 40} (1988),
293-313.

\bibitem{Heywood1994}
{\sc J. G. Heywood},
{\it Remarks on the possible global regularity of solutions
of the three-dimensional Navier-Stokes equations},
in: G. P. Galdi, J. M\'alek and J. Ne\u{c}as (Eds.),
{\sf Progress in Theoretical and Computational Fluid Mechanics},
Longman, New York, 1994, pp.\;1-32.

\bibitem{Hopf1950}
{\sc E. Hopf},
{\it \"Uber die Anfangswertaufgabe f\"ur die
hydrodynamischen Grundgleichungen},
Math. Nachr. {\bf 4} (1950),
213-231.

% \bibitem{KajikiyaMiyakawa1986}
% {\sc R. Kajikiya and T. Miyakawa},
% {\it On the $L^2$ decay of weak solutions of the
% Navier-Stokes equations in $\mathbb{R}^{n}\!\;\!$},
% Math. Z. {\bf 192} (1986), 135-148.

\bibitem{Kato1984}
{\sc T. Kato},
{\it Strong $L^p$-$\:\!$solutions of the Navier\;\!-\:\!Stokes equations
in $\mathbb{R}^{m}\!\,\!$, with applications to weak solutions},
Math. Z. {\bf 187} (1984), 471-480.

% \bibitem{KiselevLadyzhenskaya1957}
% {\sc A. A. Kiselev and O. A. Ladyzhenskaya},
% {\it On the existence and uniqueness of the solution
% of the nonstationary problem for a viscous incompressible fluid},
% Izv. Akad. Nauk SSSR, Ser. Mat., {\bf 21} (1957),
% 655-680.

% \bibitem{KiselevLadyzhenskaya1963}
% {\sc A. A. Kiselev and O. A. Ladyzhenskaya},
% {\it On the existence and uniqueness of solutions
% of the nonstationary problems for flows of noncompressible fluids},
% AMS Transl. {\bf 24} (1963),
% 79-106.

\bibitem{KreissHagstromLorenzZingano2002}
{\sc H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. R. Zingano},
{\it Decay in time of the solutions of the Navier-Stokes equations
for incompressible flows},
unpublished note,
University of New Mexico, Albuquerque, NM, 2002.

\bibitem{KreissHagstromLorenzZingano2003}
{\sc H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. R. Zingano},
{\it Decay in time of incompressible flows},
J. Math. Fluid Mech. {\bf 5} (2003),
231-244.

\bibitem{Ladyzhenskaya1969}
{\sc O. A. Ladyzhenskaya},
{\sf The Mathematical Theory of Viscous Incompressible Flow} (2nd ed.),
Gordon and Breach, New York, 1969.

\bibitem{Leray1933}
{\sc J. Leray},
{\it \'Etude de diverses \'equations int\'egrales non lin\'eaires
et de quelques probl\`emes que pose l'hydrodynamique},
J. Math. Pures Appl. {\bf 12} (1933),
1-82.

\bibitem{Leray1934}
{\sc J. Leray},
{\it Essai sur le mouvement d'un fluide visqueux emplissant l'espace},
Acta Math. {\bf 63} (1934), 193-248.

% \bibitem{Lions1959}
% {\sc J. L. Lions},
% {\it Sur l'existence de solutions des \'equations de Navier-Stokes},
% C. R. Acad. Sci. Paris {\bf 248} (1959),
% 2847-2849.

\bibitem{Lions1969}
{\sc J. L. Lions},
{\sf Quelques M\'ethodes de R\'esolution des Probl\`emes Nonlin\'eaires},
Dunod, Paris, 1969.

\bibitem{LorenzZingano2003}
{\sc J. Lorenz and P. R. Zingano},
{\it The Navier-Stokes equations for incompressible flows:
solution properties at potential blow-up times},
%
% Universidade Federal do Rio Grande do Sul,
% Porto Alegre, RS, 2003
% (available at
% http://arxiv.org/pdf/1503.01767.pdf)
% % http://www.arXiv.org/abs/1503.01767).
% % (freely available at http://www.arXiv.org).
%
Bol. Soc. Paran. Mat. {\bf 35} (2017),
127-158.

\bibitem{Masuda1984}
{\sc K.$\;$Masuda},
{\it Weak solutions of the Navier-Stokes equations},
T\^ohoku Math. Journal {\bf 36} (1984),
623-646.

\bibitem{OliverTiti2000}
\textsc{M. Oliver and E. S. Titi},
{\it Remark on the rate of decay of higher order derivatives
for solutions to the Navier-Stokes equations in $ \mathbb{R}^{n} \!$},
J. Funct. Anal. {\bf 172} (2000),
1-18.

\bibitem{RigeloSchutzZinganoZingano2016}
{\sc J. C. Rigelo, L. Sch\"utz, J. P. Zingano and P. R. Zingano},
{\it Leray's problem  for the Navier-Stokes equations revisited},
% Comptes rendus Math\'e\-matique (to appear).
C. R. Acad. Sci. Paris, Ser. I,
{\bf 354} (2016), 503-509.

\bibitem{RobinsonRodrigoSadowski2016}
{\sc J. C. Robinson, Jos\'e L. Rodrigo and W. Sadowski},
{\sf The three-dimensional Navier-Stokes equations},
Cambridge Univ. Press, Cambridge, 2016.

% \bibitem{Schonbek1986}
% {\sc M. E. Schonbek},
% {\it Large time behaviour of solutions of the Navier-Stokes equations},
% Comm. Partial Diff. Eqs. {\bf 11} (1986),
% 733-763.

% \bibitem{Schonbek1995}
% {\sc M. E. Schonbek},
% {\it The Fourier splitting method},
% in:
% P. Concus and K. Lancaster (Eds.),
% {\sf Advances in Geometric Analysis and Continuum Mechanics},
% International Press, Cambridge, 1995, pp.$\;$269-274.

% \bibitem{Schonbek1995}   % n = 2 only
% {\sc M. E. Schonbek},
% {\it Large time behaviour of solutions to the
% Navier-Stokes equations in $H^{m}\!\:\!$ spaces},
% Comm. Partial Diff. Eqs. {\bf 20} (1995),
% 103-117.

\bibitem{SchonbekWiegner1996}   % n <= 5
{\sc M. E. Schonbek and M. Wiegner},
{\it On the decay of higher-order norms of the
solutions of Navier-Stokes equations},
Proc. Roy. Soc. Edinburgh {\bf 126A} (1996),
677-685.

% \bibitem{SchutzZiebellZinganoZingano2015}
% {\sc L. Sch\"utz, J. S. Ziebell, J. P. Zingano and P. R. Zingano},
% {\it Sharp pointwise estimates for functions
% in the Sobolev spaces $ H^{s}(\mathbb{R}^{n}) $},
% Universidade Federal do Rio Grande do Sul,
% Porto Alegre, RS, Brazil,
% 2015
% (available at www.arXiv.org).
% (available at
% http://arxiv.org/pdf/1602.01902.pdf).

\bibitem{Serrin1963}
{\sc J. Serrin},
{\it The initial value problem
for the Navier-Stokes equations},
in: R. E. Langer (Ed.),
{\sf Nonlinear Problems},
Univ. of Wisconsin Press, Madison,
1963, pp.\;69-98.

\bibitem{ShinbrotKaniel1966}
{\sc M. Shinbrot and S. Kaniel},
{\it The initial value problem
for the Navier-Stokes equations},
Arch. Rat. Mech. Anal. {\bf 21} (1966),
270-285.

\bibitem{Sohr2001}
{\sc H. Sohr},
{\sf The Navier-Stokes Equations},
Birkh\"auser, Basel, 2001.

\bibitem{Temam1984}
{\sc R. Temam},
{\sf Navier-Stokes Equations:
theory and numerical analysis} (2nd ed.),
AMS/Chelsea,
Providence, 1984.

\bibitem{Wahl1985}
{\sc W. von Wahl},
{\sf The equations of Navier-Stokes
and abstract parabolic equations},
Friedr. Vieweg \mbox{\footnotesize \&} Sons,
Braunschweig, 1985.

% \bibitem{Weissler1980}
% {\sc F. B. Weissler},
% {\it The Navier-Stokes initial value problem in $L^p\!\;\!$},
% Arch. Rat. Mech. Anal. {\bf 74} (1980),
% 219-230.

\bibitem{Wiegner1987}
{\sc M. Wiegner},
{\it Decay results for weak solutions of the Navier-Stokes
equations on $ \mathbb{R}^{n}\!$},
J. London Math. Soc. {\bf 35} (1987), 303-313.

% \bibitem{Wiegner1990}
% {\sc M. Wiegner},
% {\it Decay and stability in $L^{p}\!$ for strong solutions
% of the Cauchy problem for the Navier-Stokes equations},
% in: J. G. Heywood, K. Masuda, R. Rautmann \mbox{\small \&} V.$\;$A.$\;$Solonnikov (Eds.),
% {\sf The Navier-Stokes Equations:$\:$Theory and} \linebreak
% {\sf Numerical Methods},
% Lecture Notes in Mathematics, {\bf 1431} (1990), pp.$\;$95-99.

% \bibitem{Zingano1999}
% {\sc P. R. Zingano},
% {\it Nonlinear $L^{2}\!$ stability under large disturbances},
% J. Comp. Appl. Math. {\bf 103} (1999),
% 207-219.

\bibitem{Zhou2007}
{\sc Y. Zhou},
{\it A remark on the decay of solutions
to the {\em 3-D} Navier-Stokes equations},
Math. Meth. Appl. Sci. {\bf 30} (2007),
1223-1229.

\bibitem{Zingano2015}
{\sc P. R. Zingano},
{\it Two problems in partial differential equations}\/ (Portuguese),
Universidade Federal do Rio Grande do Sul,
Porto Alegre, RS, August 2015
(available at http://lume.ufrgs.br).

\end{thebibliography}

%
% -------------------------------------------------------
%

%
\nl
\mbox{} \vspace{-0.250cm} \\
\nl
%
{\small
%
\begin{minipage}[t]{10.00cm}
\mbox{\normalsize \textsc{Pablo Gustavo Albuquerque Braz e Silva}} \\
Departmento de Matem\'atica \\
Universidade Federal de Pernambuco \\
Recife, PE 50740, Brazil \\
E-mail: {\sf pablo@dmat.ufpe.br} \\
\mbox{} \vspace{-0.620cm} \\
\mbox{} \hspace{+1.150cm}
        {\sf braz.pablo@gmail.com} \\
\end{minipage}
%
\nl
\mbox{} \vspace{-0.450cm} \\
\nl
%
\begin{minipage}[t]{10.00cm}
\mbox{\normalsize \textsc{Jana\'\i na Pires Zingano}} \\
Departamento de Matem\'atica Pura e Aplicada \\
Universidade Federal do Rio Grande do Sul \\
Porto Alegre, RS 91509, Brazil \\
E-mail: {\sf jzingano@mat.ufrgs.br} \\
\mbox{} \hspace{+1.150cm}
        {\sf janaina.zingano@ufrgs.br} \\
\end{minipage}
%
\nl
\mbox{} \vspace{-0.450cm} \\
\nl
%
\begin{minipage}[t]{10.00cm}
\mbox{\normalsize \textsc{Paulo Ricardo de Avila Zingano}} \\
Departamento de Matem\'atica Pura e Aplicada \\
Universidade Federal do Rio Grande do Sul \\
Porto Alegre, RS 91509, Brazil \\
E-mail: {\sf paulo.zingano@ufrgs.br} \\
\mbox{} \hspace{+1.150cm}
        {\sf zingano@gmail.com} \\
\end{minipage}
%
%
}
%
% -------------------------------------------------------------
%

\end{document} 
1706.10173.extracted_bib
{Agueh2008 \textsc{M. Agueh, {Gagliardo-Nirenberg inequalities involving the gradient $L^{2\!$-norm, C. R. Acad. Sci. Paris, Ser. I, {346 (2008), 757-762.
1706.10173.extracted_bib
{BahouriCheminDanchin2011 {\sc H. Bahouri, J.-Y. Chemin and R. Danchin, {\sf Fourier Analysis and Nonlinear Partial Differential Equations, Springer, New York, 2011.
1706.10173.extracted_bib
{CaffarelliKohnNirenberg1982 {\sc L. Caffarelli, R. Kohn and L. Nirenberg, {Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. {35 (1982), 771-831. %
1706.10173.extracted_bib
{ConstantinFoias1988 % {\sc P. Constantin and C. Foias, % {\sf Navier-Stokes equations, % Univ. of Chicago Press, Chicago, 1988.
1706.10173.extracted_bib
{DoeringGibbon1995 {\sc C. R. Doering and J. D. Gibbon, {\sf Applied analysis of the Navier-Stokes equations, Cambridge Univ. Press, Cambridge, 1995.
1706.10173.extracted_bib
{FujitaKato1964 {\sc H. Fujita and T. Kato, {On the Navier-Stokes initial value problem, Arch. Rat. Mech. Anal. {16 (1964), 269-315.
1706.10173.extracted_bib
{Giga1986 {\sc Y. Giga, {Solutions for semilinear parabolic equations in $L^{p\!\;\!$ and regularity of weak solutions of the Navier-Stokes system, J. Diff. Eqs. {62 (1986), 186-212.
1706.10173.extracted_bib
{Heywood1988 {\sc J. G. Heywood, {The epochs of regularity for weak solutions of the Navier-Stokes equations in unbounded domains, T\^ohoku Math. J. {40 (1988), 293-313.
1706.10173.extracted_bib
{Heywood1994 {\sc J. G. Heywood, {Remarks on the possible global regularity of solutions of the three-dimensional Navier-Stokes equations, in: G. P. Galdi, J. M\'alek and J. Ne\u{cas (Eds.), {\sf Progress in Theoretical and Computational Fluid Mechanics, Longman, New York, 1994, pp.\;1-32.
1706.10173.extracted_bib
{Hopf1950 {\sc E. Hopf, {\"Uber die Anfangswertaufgabe f\"ur die hydrodynamischen Grundgleichungen, Math. Nachr. {4 (1950), 213-231. %
1706.10173.extracted_bib
{KajikiyaMiyakawa1986 % {\sc R. Kajikiya and T. Miyakawa, % {On the $L^2$ decay of weak solutions of the % Navier-Stokes equations in $\mathbb{R^{n\!\;\!$, % Math. Z. {192 (1986), 135-148.
1706.10173.extracted_bib
{Kato1984 {\sc T. Kato, {Strong $L^p$-$\:\!$solutions of the Navier\;\!-\:\!Stokes equations in $\mathbb{R^{m\!\,\!$, with applications to weak solutions, Math. Z. {187 (1984), 471-480. %
1706.10173.extracted_bib
{KiselevLadyzhenskaya1957 % {\sc A. A. Kiselev and O. A. Ladyzhenskaya, % {On the existence and uniqueness of the solution % of the nonstationary problem for a viscous incompressible fluid, % Izv. Akad. Nauk SSSR, Ser. Mat., {21 (1957), % 655-680. %
1706.10173.extracted_bib
{KiselevLadyzhenskaya1963 % {\sc A. A. Kiselev and O. A. Ladyzhenskaya, % {On the existence and uniqueness of solutions % of the nonstationary problems for flows of noncompressible fluids, % AMS Transl. {24 (1963), % 79-106.
1706.10173.extracted_bib
{KreissHagstromLorenzZingano2002 {\sc H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. R. Zingano, {Decay in time of the solutions of the Navier-Stokes equations for incompressible flows, unpublished note, University of New Mexico, Albuquerque, NM, 2002.
1706.10173.extracted_bib
{KreissHagstromLorenzZingano2003 {\sc H.-O. Kreiss, T. Hagstrom, J. Lorenz and P. R. Zingano, {Decay in time of incompressible flows, J. Math. Fluid Mech. {5 (2003), 231-244.
1706.10173.extracted_bib
{Ladyzhenskaya1969 {\sc O. A. Ladyzhenskaya, {\sf The Mathematical Theory of Viscous Incompressible Flow (2nd ed.), Gordon and Breach, New York, 1969.
1706.10173.extracted_bib
{Leray1933 {\sc J. Leray, {\'Etude de diverses \'equations int\'egrales non lin\'eaires et de quelques probl\`emes que pose l'hydrodynamique, J. Math. Pures Appl. {12 (1933), 1-82.
1706.10173.extracted_bib
{Leray1934 {\sc J. Leray, {Essai sur le mouvement d'un fluide visqueux emplissant l'espace, Acta Math. {63 (1934), 193-248. %
1706.10173.extracted_bib
{Lions1959 % {\sc J. L. Lions, % {Sur l'existence de solutions des \'equations de Navier-Stokes, % C. R. Acad. Sci. Paris {248 (1959), % 2847-2849.
1706.10173.extracted_bib
{Lions1969 {\sc J. L. Lions, {\sf Quelques M\'ethodes de R\'esolution des Probl\`emes Nonlin\'eaires, Dunod, Paris, 1969.
1706.10173.extracted_bib
{LorenzZingano2003 {\sc J. Lorenz and P. R. Zingano, {The Navier-Stokes equations for incompressible flows: solution properties at potential blow-up times, % % Universidade Federal do Rio Grande do Sul, % Porto Alegre, RS, 2003 % (available at % http://arxiv.org/pdf/1503.01767.pdf) % % http://www.arXiv.org/abs/1503.01767). % % (freely available at http://www.arXiv.org). % Bol. Soc. Paran. Mat. {35 (2017), 127-158.
1706.10173.extracted_bib
{Masuda1984 {\sc K.$\;$Masuda, {Weak solutions of the Navier-Stokes equations, T\^ohoku Math. Journal {36 (1984), 623-646.
1706.10173.extracted_bib
{OliverTiti2000 \textsc{M. Oliver and E. S. Titi, {Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in $ \mathbb{R^{n \!$, J. Funct. Anal. {172 (2000), 1-18.
1706.10173.extracted_bib
{RigeloSchutzZinganoZingano2016 {\sc J. C. Rigelo, L. Sch\"utz, J. P. Zingano and P. R. Zingano, {Leray's problem for the Navier-Stokes equations revisited, % Comptes rendus Math\'e\-matique (to appear). C. R. Acad. Sci. Paris, Ser. I, {354 (2016), 503-509.
1706.10173.extracted_bib
{RobinsonRodrigoSadowski2016 {\sc J. C. Robinson, Jos\'e L. Rodrigo and W. Sadowski, {\sf The three-dimensional Navier-Stokes equations, Cambridge Univ. Press, Cambridge, 2016. %
1706.10173.extracted_bib
{Schonbek1986 % {\sc M. E. Schonbek, % {Large time behaviour of solutions of the Navier-Stokes equations, % Comm. Partial Diff. Eqs. {11 (1986), % 733-763. %
1706.10173.extracted_bib
{Schonbek1995 % {\sc M. E. Schonbek, % {The Fourier splitting method, % in: % P. Concus and K. Lancaster (Eds.), % {\sf Advances in Geometric Analysis and Continuum Mechanics, % International Press, Cambridge, 1995, pp.$\;$269-274. %
1706.10173.extracted_bib
{Schonbek1995 % n = 2 only % {\sc M. E. Schonbek, % {Large time behaviour of solutions to the % Navier-Stokes equations in $H^{m\!\:\!$ spaces, % Comm. Partial Diff. Eqs. {20 (1995), % 103-117.
1706.10173.extracted_bib
{SchonbekWiegner1996 % n <= 5 {\sc M. E. Schonbek and M. Wiegner, {On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. Roy. Soc. Edinburgh {126A (1996), 677-685. %
1706.10173.extracted_bib
{SchutzZiebellZinganoZingano2015 % {\sc L. Sch\"utz, J. S. Ziebell, J. P. Zingano and P. R. Zingano, % {Sharp pointwise estimates for functions % in the Sobolev spaces $ H^{s(\mathbb{R^{n) $, % Universidade Federal do Rio Grande do Sul, % Porto Alegre, RS, Brazil, % 2015 % (available at www.arXiv.org). % (available at % http://arxiv.org/pdf/1602.01902.pdf).
1706.10173.extracted_bib
{Serrin1963 {\sc J. Serrin, {The initial value problem for the Navier-Stokes equations, in: R. E. Langer (Ed.), {\sf Nonlinear Problems, Univ. of Wisconsin Press, Madison, 1963, pp.\;69-98.
1706.10173.extracted_bib
{ShinbrotKaniel1966 {\sc M. Shinbrot and S. Kaniel, {The initial value problem for the Navier-Stokes equations, Arch. Rat. Mech. Anal. {21 (1966), 270-285.
1706.10173.extracted_bib
{Sohr2001 {\sc H. Sohr, {\sf The Navier-Stokes Equations, Birkh\"auser, Basel, 2001.
1706.10173.extracted_bib
{Temam1984 {\sc R. Temam, {\sf Navier-Stokes Equations: theory and numerical analysis (2nd ed.), AMS/Chelsea, Providence, 1984.
1706.10173.extracted_bib
{Wahl1985 {\sc W. von Wahl, {\sf The equations of Navier-Stokes and abstract parabolic equations, Friedr. Vieweg \mbox{\footnotesize \& Sons, Braunschweig, 1985. %
1706.10173.extracted_bib
{Weissler1980 % {\sc F. B. Weissler, % {The Navier-Stokes initial value problem in $L^p\!\;\!$, % Arch. Rat. Mech. Anal. {74 (1980), % 219-230.
1706.10173.extracted_bib
{Wiegner1987 {\sc M. Wiegner, {Decay results for weak solutions of the Navier-Stokes equations on $ \mathbb{R^{n\!$, J. London Math. Soc. {35 (1987), 303-313. %
1706.10173.extracted_bib
{Wiegner1990 % {\sc M. Wiegner, % {Decay and stability in $L^{p\!$ for strong solutions % of the Cauchy problem for the Navier-Stokes equations, % in: J. G. Heywood, K. Masuda, R. Rautmann \mbox{\small \& V.$\;$A.$\;$Solonnikov (Eds.), % {\sf The Navier-Stokes Equations:$\:$Theory and \linebreak % {\sf Numerical Methods, % Lecture Notes in Mathematics, {1431 (1990), pp.$\;$95-99. %
1706.10173.extracted_bib
{Zingano1999 % {\sc P. R. Zingano, % {Nonlinear $L^{2\!$ stability under large disturbances, % J. Comp. Appl. Math. {103 (1999), % 207-219.
1706.10173.extracted_bib
{Zhou2007 {\sc Y. Zhou, {A remark on the decay of solutions to the {\em 3-D Navier-Stokes equations, Math. Meth. Appl. Sci. {30 (2007), 1223-1229.
1706.10173.extracted_bib
{Zingano2015 {\sc P. R. Zingano, {Two problems in partial differential equations\/ (Portuguese), Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, August 2015 (available at http://lume.ufrgs.br).