\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 45, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/45\hfil Regularity of weak solutions]
{Regularity of weak solutions of the Navier-Stokes
equations near the smooth boundary}
\author[Z. Skal\' ak\hfil EJDE-2005/45\hfilneg]
{Zden\v ek Skal\' ak }  

\address{Czech Technical University\\
Faculty of Civil Engineering\\
Thakurova 7\\ 
166 29 Prague 6\\ 
Czech Republic}
\email{skalak@mat.fsv.cvut.cz}

\date{}
\thanks{Submitted May 19, 2004. Published April 24, 2005.}
\thanks{Supported by project MSM 6840770003 from the
Ministry of Education of the Czech \hfill\break\indent Republic
and by grant A2060302 from the Grant Agency of the Academy of
Sciences \hfill\break\indent of the Czech Republic.}
\subjclass[2000]{35Q35, 35B65} 
\keywords{Navier-Stokes equations; weak solutions; boundary regularity}

\begin{abstract}
 Any weak solution $u$ of the Navier-Stokes equations in
 a bounded domain satisfying the Prodi-Serrin's conditions
 locally near the smooth boundary cannot have singular
 points there. This local-up-to-the-boundary boundedness
 of $u$ in space-time implies the H\"{o}lder continuity
 of $u$ up-to-the-boundary in the space variables.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

Let  $\Omega$ be a bounded domain in $\mathbb{R}^3$ with a
smooth boundary $\partial \Omega$, let $T>0$ and $Q_T = \Omega
\times (0,T)$. We consider the Navier-Stokes initial-boundary
value problem describing the evolution of the velocity
$u=(u_1,u_2,u_3)$ and the pressure $\phi$ in $Q_T$:
\begin{gather}
\label{eq:e1}  \frac {\partial u}{\partial t} - \nu \Delta u +
u \cdot \nabla u + \nabla \phi = 0 \quad \text{in} \ Q_T, \\
\label{eq:e2} \nabla\cdot u = 0 \quad \text{in} \ Q_T,  \\
\label{eq:e3} u = 0 \quad \text{on} \ \partial \Omega \times (0,T), \\
\label{eq:e4}  u \vert _{t=0} = u_0,
\end{gather}
where $\nu > 0$ is the viscosity coefficient. The initial data
$u_0$ satisfy the compatibility conditions $u_0 \vert _{\partial
\Omega} = 0$ and $\nabla \cdot u_0 = 0$ and for our purposes we
can suppose without loss of generality that $u_0$ is sufficiently
smooth. The existence of a weak solution $u \in
L^2(0,T;W_0^{1,2}(\Omega)^3) \cap
L^\infty(0,T;L^2_\sigma(\Omega))$ of (\ref{eq:e1})--(\ref{eq:e4})
is well known (see e.g. \cite{Ga} or \cite{Te}). The associated
pressure $\phi$ is a scalar function such that $u$ and $\phi$
satisfy the equation (\ref{eq:e1}) in $Q_T$ in the sense of
distributions.

Let $q>1$. $L_\sigma^q(\Omega)$ denotes the closure of $\{
\varphi \in (C_0^\infty (\Omega))^3; \nabla \cdot \varphi = 0
\text{ in } \Omega\}$ in ($L^q(\Omega))^3$. There exists a
continuous projection $P_\sigma^q$ from ($L^q(\Omega))^3$ onto
$L_\sigma^q(\Omega)$. If $\Delta$ denotes the Laplacian then the
famous Stokes operator is defined as $A_q = -P_\sigma^q \Delta$.
It is known that $-A_q$ generates a bounded analytic semigroup in
$L_\sigma^q(\Omega)$ (see e.g. \cite{GiSo}).

In the paper we use both scalar and vector functions and for the
sake of simplicity we denote by $S$ any space $S^3$ of vector
functions with the exception of the notation in Lemma~\ref{L:L4}.
We use the standard notation for the Lebesgue spaces
$L^p(\Omega)$ and their norms $\Vert \cdot \Vert _{p,\Omega}$.
The Sobolev spaces are denoted by $W^{k,p}(\Omega)$. Sometimes we
drop $\Omega$ and write only $L^p$, $\Vert \cdot \Vert _{p}$ and
$W^{k,p}$. Further, if $A = B \times (t_1,t_2)$ then $L^{p,q}$ or
$L^{p,q}(A)$ denote the space $L^q(t_1,t_2;L^p(B))$ with the norm
$\Vert \cdot \Vert_{p,q,A}$ or simply $\Vert \cdot \Vert_{p,q}$.
$L^{p,p}(A)$ is also denoted as $L^p(A)$ or $L^p$.
$C^\beta(\overline\Omega)$ is the space of H\"{o}lder continuous
functions on $\Omega$ with the norm $\Vert f \Vert
_{C^\beta(\overline\Omega)} = \sup_{x \in \Omega} \vert f(x)
\vert + \sup_{x,y \in \Omega, x \neq y} \vert f(x)-f(y) \vert /
\vert x-y \vert ^\beta$.

For $(x_0,t_0) \in \overline \Omega \times (0,T)$ and $r>0$ we
will denote $B_r = B_r(x_0)$ the ball centered at $x_0$ with
radius $r$, $D_r = D_r(x_0) = B_r(x_0) \cap \Omega$, $Q_r =
Q_r(x_0,t_0) = D_r(x_0) \times (t_0-r^2,t_0+r^2)$.

A point $(x_0,t_0) \in \overline\Omega \times (0,T)$ is called a
regular point of a weak solution $u$ if $u \in L^{\infty} (Q_r)$
for some $r>0$. Otherwise, $(x_0,t_0)$ is called a singular point
of $u$.

In his famous paper (see \cite{Ser}) J.Serrin proved the
following interior regularity result. If $Q_r \subset Q_T$ for
some $(x_0,t_0) \in Q_T$ and $r>0$ and a weak solution $u$ of
(\ref{eq:e1})--(\ref{eq:e4}) satisfies the Prodi-Serrin's
conditions in $Q_r$, that is
 \begin{equation}
\label{eq:e52} u \in L^{p,q}(Q_r), \quad \frac {3} {p} + \frac {2}
{q} < 1, \; p,q \in (1,\infty),
 \end{equation}
then $u$ is necessarily a $L^{\infty}$ function on compact
subsets of $Q_r$ and smooth in the space variables. This result
was extended by M.Struwe in \cite {St} for the case of $p,q \in
(1,\infty)$, $3/p + 2/q \le 1$. A local version up to the
boundary of the Serrin-Struwe's results was proved by
S.Takahashi. He showed in \cite{Ta1} and \cite{Ta2} that if $u
\in L^{p,q}(Q_r)$, where $(x_0,t_0) \in \partial \Omega \times
(0,T)$, $r>0$, $p,q \in (1,\infty)$ and $3/p + 2/q \le 1$ then $u
\in L^{\infty}(Q_{\tilde{r}})$ for any $\tilde{r} \in (0,r)$
provided that $B_r \cap \partial \Omega$ is a part of a plane.

In this paper we improve the Takahashi's result in two
directions. Firstly, we show, that $\partial \Omega$ can be an
arbitrary smooth boundary, that is $B_r \cap \partial \Omega$
needn't be a part of a plane. Secondly, we show that $u$ is
locally a H\"{o}lder continuous function in the space variables
up to the boundary in the neighborhood of the point $x_0$.
Precisely, we prove the following theorem.

 \begin{theorem} \label{T:T1}
Let $u$ be an arbitrary weak solution of
\eqref{eq:e1}--\eqref{eq:e4}, $(x_0,t_0) \in \partial \Omega \times (0,T)$,
$r>0$. We suppose that $u \in L^{p,q}(Q_r)$, where $2/q+3/p=1$
and $p,q \in (1,\infty)$. Then
 \begin{equation}
\label{eq:e53} u \in
L^\infty(t_0-\tilde{r}^2,t_0+\tilde{r}^2;C^\beta(\overline{D}_{\tilde{r}}
))
 \end{equation}
for every $\beta \in (0,1)$ and $\tilde{r} \in (0,r)$.
 \end{theorem}

In \cite{Ne} Neustupa proved a similar result. He supposed that
$u \in L^q(t_1,t_2;L^p(U_r^*))$ for some $r>0$, $0<t_1<t_2<T$,
$p,q \in (1,\infty)$ with $3/p+2/q=1$, where $U_r^* = \{ x \in
\Omega; \mathop{\rm dist}(x,\partial \Omega) < r \}$. He proved under
this assumption that if $u$ is a weak solution of
(\ref{eq:e1})--(\ref{eq:e4}) satisfying the strong energy
inequality then $u \in
L^\infty(t_1+\zeta,t_2-\zeta;W^{2+\delta,2}(U_\rho^*))$ and
$\partial u/\partial t, \nabla \phi \in
L^\infty(t_1+\zeta,t_2-\zeta;W^{\delta,2}(U_\rho^*))$ for each
$\delta \in [0,1/2)$, $\rho \in (0,r)$ and such $\zeta>0$ that
$t_1+\zeta < t_2-\zeta$.

The proof of the Neustupa's result was based on the fact (see
\cite{Ne}, Lemma 1) that $\partial u/\partial t, \nabla \phi$ and
their space derivatives of an arbitrary order belong to
$L^\alpha(t_1+\zeta,t_2-\zeta;L^\infty(\Omega_2))$ for each
$\alpha \in [1,2)$ and $\zeta \in (0,(t_2-t_1)/2)$ if $u \in
L^q(t_1,t_2;L^p(\Omega_1))$ for some $p,q \in (1,\infty)$ with
$3/p+2/q \le 1$, where $\Omega_1$ and $\Omega_2$ are such
sub-domains of $\Omega$ that $\overline \Omega_2 \subset \Omega_1
\subset \Omega$. Using this result together with the cut-off
function technique, it was then possible to show that the right
hand side $h$ of the localized equations and its space
derivatives of an arbitrary order belong to the space
$L^\alpha(t_1+\zeta,t_2-\zeta;L^\infty(\Omega))$ for each $\alpha
\in [1,2)$ (see \cite[(6)]{Ne}). This regularity of $h$ produced
better regularity of $u$ near the whole boundary $\partial
\Omega$ which further improved the regularity of $\phi$ and
consequently of $h$. Repeating this procedure several times one
of the main results of \cite{Ne} presented in the preceding
paragraph was obtained.

We assume in Theorem~\ref{T:T1} that $u$ satisfies the
Prodi-Serrin's conditions only in a space-time neighborhood of
$(x_0,t_0) \in \partial \Omega \times (0,T)$. Thus, the cut-off
function technique does not produce in this case the right hand
side $h$ which is from the space
$L^\alpha(t_1+\zeta,t_2-\zeta;L^\infty(\Omega_2))$, $\alpha \in
[1,2)$ and the procedure from \cite{Ne} mentioned in the
preceding paragraph cannot be used. Instead, we use at the
beginning the regularity results of Giga,Sohr (see \cite{GiSo}).
They lead, however, to worse regularity results for $u$ in
Theorem~\ref{T:T1} in comparison with the results from \cite{Ne}.

The local boundary regularity of $u$ was also studied in
\cite{Cho}, \cite{Se} and \cite{Ka}. It was proved in \cite{Cho}
that a suitable weak solution $u$ is bounded locally near the
boundary if $u \in L^{p,q}$, $3/p+2/q=1$, $p,q \in (1,\infty)$
and the pressure $\phi$ is bounded at the boundary. Moreover,
better regularity of $\phi$ gives better local regularity of $u$.
G.A.Seregin presented in \cite{Se} a condition for local
H\"{o}lder continuity for suitable weak solutions near the plane
boundary which has the form of the famous
Caffarelli-Kohn-Nirenberg condition for boundedness of suitable
weak solutions in a neighborhood of an interior point of $Q_T$.
Finally, in \cite{Ka} K.Kang studied boundary regularity of weak
solutions in the half-space. He proved that a weak solution $u$
which is locally in the class $L^{p,q}$ with $3/p+2/q=1$ and $p,q
\in (1,\infty)$ near the boundary is  H\"{o}lder continuous up to
the boundary. The main tool in the proof of this result is a
pointwise estimate for the fundamental solution of the Stokes
system.

 \section{Auxiliary Lemmas}

 In this section we present a few lemmas
which will be used in the proof of Theorem~\ref{T:T1}. We
consider the Stokes problem:
 \begin{gather}
\label{eq:e16}  \frac {\partial u}{\partial t} - \nu \Delta u
+ \nabla \phi = f \quad \text{in }  Q_T, \\
\label{eq:e17} \nabla\cdot u = 0 \quad \text{in }  Q_T,  \\
\label{eq:e18} u = 0 \quad \text{on }  \partial \Omega \times (0,T), \\
\label{eq:e19}  u \vert _{t=0} = 0.
 \end{gather}
It was proved in \cite[Theorem 2.8]{GiSo}, that if $f \in
L^{\beta,\beta'}$, where $\beta, \beta' \in (1,\infty)$, then
there exists a unique solution $(u,\phi)$ of (\ref{eq:e16}) -
(\ref{eq:e19}) such that
 \begin{equation}
\label{eq:e71} \big\Vert \frac {\partial u} {\partial t}
\big\Vert _{\beta,\beta'} + \Vert A_\beta u \Vert
_{\beta,\beta'} + \Vert \nabla \phi \Vert _{\beta,\beta'} \le c
\Vert f \Vert _{\beta,\beta'}, \quad c=c(\beta,\beta').
 \end{equation}

 \begin{lemma} \label{L:L1}
Let $\beta, \beta' \in (1,\infty)$, $\gamma \in [\beta,\infty)$,
$\gamma' \in [\beta',\infty)$ and
 \begin{equation}
\label{eq:e20} \frac {2} {\beta'} + \frac {3} {\beta} = \frac {2}
{\gamma'} + \frac {3} {\gamma} + 1.
 \end{equation}
Then for every $f \in L^{\beta,\beta'}$ there exists a unique
solution $u$ of \eqref{eq:e16}--\eqref{eq:e19} such that
$\nabla u \in L^{\gamma,\gamma'}$ and
 \begin{equation}
\label{eq:e21} \Vert \nabla u \Vert _{\gamma,\gamma'} \le c \Vert
f \Vert _{\beta,\beta'}, \quad c=c(\beta,\beta',\gamma,\gamma').
 \end{equation}
 \end{lemma}

\begin{proof} Equation (\ref{eq:e16}) can be written
as
 \begin{equation}
\nonumber \frac {\partial u}{\partial t} - \nu \Delta u = f -
\nabla \phi,
 \end{equation}
where $\Vert f - \nabla \phi \Vert _{\beta,\beta'} \le c \Vert f
\Vert _{\beta,\beta'}$. Lemma~\ref{L:L1} now follows immediately
from the following Lemma~\ref{L:L2}.
\end{proof}

 \begin{lemma} \label{L:L2}
Let the assumptions of Lemma~\ref{L:L1} be satisfied. Consider
the problem
 \begin{gather}
\label{eq:e23}  \frac {\partial u}{\partial t} - \nu \Delta u
= f \quad \text{in }  Q_T, \\
\label{eq:e24}  u = 0 \quad \text{on } \partial \Omega \times (0,T), \\
\label{eq:e25}  u \vert _{t=0} = 0.
 \end{gather}
Then for every $f \in L^{\beta,\beta'}$ there exists a unique
solution $u$ of \eqref{eq:e23}--\eqref{eq:e25} such that
 \begin{equation}
\label{eq:e73} \big\Vert \frac {\partial u} {\partial t}
\big\Vert _{\beta,\beta'} + \Vert \nabla ^2 u \Vert
_{\beta,\beta'} \le c \Vert f \Vert _{\beta,\beta'}, \quad
c=c(\beta,\beta').
 \end{equation}
Moreover, $\nabla u \in L^{\gamma,\gamma'}$ and
 \begin{equation}
\label{eq:e26} \Vert \nabla u \Vert _{\gamma,\gamma'} \le c \Vert
f \Vert _{\beta,\beta'}, \quad c=c(\beta,\beta',\gamma,\gamma').
 \end{equation}
 \end{lemma}

\begin{proof} The existence of a unique solution $u$ of
\eqref{eq:e23}--\eqref{eq:e25} satisfying (\ref{eq:e73}) follows
from \cite[Theorem 2.1]{GiSo}. We will prove that $u$ satisfies
also (\ref{eq:e26}).

Let us suppose at first that $\Omega$ is a half space, i.e.
$\Omega = \mathbb{R}^3_+$, where $\mathbb{R}^3_+ = \{x=(x_1,x_2,x_3) \in
\mathbb{R}^3; x_3 > 0 \}$. We extend $f$ to the whole space $\mathbb{R}^3$
in such a way that $f(x_1,x_2,x_3) = -f(x_1,x_2,-x_3)$ for any
$x=(x_1,x_2,x_3) \in \mathbb{R}^3$ and denote the extended function by
$f$. Then the unique solution $u$ of (\ref{eq:e23}) - (\ref{eq:e25})
can be written as
 \begin{equation}
\label{eq:e29} u(x,t) = \int_0^t \int_{\mathbb{R}^3} K(x-\xi,t-\tau)
f(\xi,\tau) d\xi d\tau,
 \end{equation}
where
 $$
K(x,t) = \frac {1}{2^3 \pi^{3/2} t^{3/2}} e^{-\frac {\vert x
\vert^2} {4t}}, \quad x \in \mathbb{R}^3, t>0.
 $$
It is possible to compute that
 \begin{equation}
\label{eq:e27} \Vert \nabla K(\cdot,t) \Vert _{s,\mathbb{R}^3} = c
t^{-2+\frac{3}{2s}}
 \end{equation}
for any $s \in [1,\infty)$, where $c$ depends only on $s$. Let
$u_0 \in L^\beta(\mathbb{R}^3)$. If we define
 $$
v(x,t) = \int_{\mathbb{R}^3} K(x-y,t) u_0(y) dy,
 $$
then
 $$
\nabla v(x,t) = \int_{\mathbb{R}^3} \nabla K(x-y,t) u_0(y) dy.
 $$
There exists $s \in [1,\infty)$ such that $1/\gamma = 1/s +
1/\beta - 1$. According to \cite[estimate (9.2), p. 85]{Ga}
and (\ref{eq:e27}), we have
 \begin{equation}
\label{eq:e28} \Vert \nabla v(\cdot,t) \Vert _{\gamma,\mathbb{R}^3}
\le c
t^{-\frac{1}{2}-\frac{3}{2}\left(\frac{1}{\beta}-\frac{1}{\gamma}\right)}
\Vert u_0 \Vert _{\beta,\mathbb{R}^3}, \quad c=c(\beta,\gamma).
 \end{equation}
It follows from (\ref{eq:e29}) that
 $$
\Vert \nabla u(\cdot,t) \Vert_{\gamma,\mathbb{R}^3} \le \int_0^t \Vert
\int_{\mathbb{R}^3} \nabla K(x-\xi,t-\tau) f(\xi,\tau) d\xi
\Vert_{\gamma ,\mathbb{R}^3} d\tau
 $$
and using (\ref{eq:e28}), we get
 \begin{equation}
\label{eq:e30} \Vert \nabla u(\cdot,t) \Vert_{\gamma,\mathbb{R}^3} \le
c \int_0^t
(t-\tau)^{-\frac{1}{2}-\frac{3}{2}\left(\frac{1}{\beta}-\frac{1}{\gamma}\right)}
\Vert f(\cdot,\tau) \Vert_{\beta,\mathbb{R}^3} d\tau.
 \end{equation}
Applying now the Hardy-Littlewood-Sobolev inequality to
(\ref{eq:e30}) we get
 $$
\Vert \nabla u \Vert _{\gamma,\gamma',\mathbb{R}^3\times(0,T)} \le c
\Vert f \Vert _{\beta,\beta',\mathbb{R}^3\times(0,T)}, \quad
c=c(\beta,\beta',\gamma,\gamma')
 $$
and the inequality (\ref{eq:e26}) for the case $\Omega =
\mathbb{R}^3_+$, that is
 $$
\Vert \nabla u \Vert _{\gamma,\gamma',\mathbb{R}^3_+\times(0,T)} \le c
\Vert f \Vert _{\beta,\beta',\mathbb{R}^3_+\times(0,T)},
 $$
follows immediately.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$. Let $x_0
\in \partial\Omega$ be chosen arbitrarily. Let us choose a local
system of coordinates with the origin at $x_0$ and with the axis
$x_3$ perpendicular to $\partial\Omega$ and pointing into
$\Omega$. Thus, the axes $x_1$ and $x_2$ form the tangent plane
to $\partial\Omega$ at the point $x_0$. Let us define for
$\varepsilon>0$
 \begin{equation}
\label{eq:e31} \Omega_{x_0}^{\varepsilon} = \{ x=(x_1,x_2,x_3)
\in \mathbb{R}^3; \sqrt{x_1^2+x_2^2}<\varepsilon \wedge
\varphi(x_1,x_2) < x_3 < \varphi(x_1,x_2) + \varepsilon \},
 \end{equation}
where the function $\varphi$ describes locally the boundary
$\partial\Omega$ near the point $x_0$. Let $\psi \in
C^\infty(\overline\Omega)$ be a cut-off function such that
$\psi(x) = 1$ if $x \in \Omega_{x_0}^{\varepsilon/2}$, $\psi(x) =
0$ if $x \in \Omega \setminus \Omega_{x_0}^{\varepsilon}$ and
$\psi(x) \in [0,1]$ for every $x\in \Omega$.

If we put $v=\psi u$ then $v$ solves the system
 \begin{gather}
\label{eq:e32}  \frac {\partial v}{\partial t} - \nu \Delta v
= h \quad \text{in } Q_T, \\
\label{eq:e33}  v = 0 \quad \text{on } \partial \Omega \times (0,T), \\
\label{eq:e34}  v \vert _{t=0} = 0,
 \end{gather}
where $h = \psi f - 2\nu \nabla \psi \cdot \nabla u - \nu \Delta
\psi u$ and it follows from (\ref{eq:e73}) that
 \begin{equation}
\label{eq:e35} \Vert h \Vert _{\beta,\beta'} \le c \Vert f \Vert
_{\beta,\beta'}.
 \end{equation}
Let
 $$
\Phi _{x_0}^{\varepsilon} = \{ x=(x_1,x_2,x_3) \in \mathbb{R}^3;
x_1^2+x_2^2<\varepsilon \wedge 0 < x_3 < \varepsilon \}
 $$
The following equations describe the transformation between $\Phi
_{x_0}^{\varepsilon}$ and $\Omega _{x_0}^{\varepsilon}$:
 \begin{equation}
\label{eq:e36} x_1' = x_1, \ x_2' = x_2, \ x_3' = x_3 -
\varphi(x_1,x_2).
 \end{equation}
If we define $v'$ on $\Phi _{x_0}^{\varepsilon}$ by the equation
 \begin{equation}
\label{eq:e37} v'(x_1',x_2',x_3') = v(x_1,x_2,x_3),
 \end{equation}
then $v'$  satisfies the equation
 \begin{equation} \label{eq:e38}
\begin{aligned}
\frac {\partial v'} {\partial t} - \nu \Delta' v'
& =  h - \frac {\partial^2 v} {\partial x_3^2}
\big[ \big( \frac{\partial \varphi} {\partial x_1} \big)^2
+ \big( \frac {\partial \varphi} {\partial x_2} \big)^2 \big]
- 2 \frac {\partial \varphi} {\partial x_1} \frac {\partial^2 v} {\partial
x_1 \partial x_3} \\
&\quad - 2 \frac {\partial
\varphi} {\partial x_2} \frac {\partial^2 v} {\partial x_2
\partial x_3} - \frac {\partial v} {\partial x_3}
\big( \frac {\partial^2 \varphi} {\partial x_1^2} + \frac {\partial^2
\varphi} {\partial x_2^2} \big) \quad \text{in } \mathbb{R}^3_+ \times
(0,T)
\end{aligned}
 \end{equation}
and the boundary and initial conditions
 \begin{gather}
\label{eq:e39}  v' = 0 \quad \text{on} \ \partial \mathbb{R}^3_+
\times (0,T), \\
\label{eq:e40}  v' \vert _{t=0} = 0.
 \end{gather}
We denote the right hand side of (\ref{eq:e38}) by $H$. $v'$ is a
solution of (\ref{eq:e23}) - (\ref{eq:e25}) for $\Omega =
\mathbb{R}_+^3$ with the right hand side $H$ instead of $f$ and
according to the first half of this proof  (\ref{eq:e26}) holds,
that is
 \begin{equation}
\label{eq:e42} \Vert \nabla' v' \Vert _{\gamma,\gamma'} \le c
\Vert H \Vert _{\beta,\beta'}.
 \end{equation}
Since $v \in L^{\beta'}(0,T,W^{2,\beta}(\Omega))$ and $\Vert v
\Vert_{L^{\beta'}(0,T,W^{2,\beta}(\Omega))} \le c \Vert f
\Vert_{\beta,\beta'}$ - see (\ref{eq:e73}), it follows from
(\ref{eq:e35}), the smoothness of $\varphi$ and the substitution
theorem that
 \begin{equation}
\label{eq:e41} \Vert H \Vert _{\beta,\beta'} \le c \Vert f \Vert
_{\beta,\beta'}.
 \end{equation}
Further, the smoothness of the transformation (\ref{eq:e36})
gives the inequality
 \begin{equation}
\label{eq:e43} \Vert \nabla v \Vert _{\gamma,\gamma'} \le c \Vert
\nabla' v' \Vert _{\gamma,\gamma'}.
 \end{equation}
Summing up (\ref{eq:e43}), (\ref{eq:e42}) and (\ref{eq:e41}) we
have
 \begin{equation}
\nonumber \Vert \nabla v \Vert _{\gamma,\gamma'} \le c \Vert f
\Vert _{\beta,\beta'}
 \end{equation}
and thus
 \begin{equation}
\label{eq:e44} \Vert \nabla u \Vert
_{\gamma,\gamma',\Omega_{x_0}^{\varepsilon/2}} \le c \Vert f
\Vert _{\beta,\beta'}.
 \end{equation}
The estimate (\ref{eq:e44}) can also be proved in the same way
for the sets $\Omega^{\varepsilon}_I = \{ x\in \Omega;
\mathop{\rm dist}(x,\partial\Omega)>\varepsilon \}$, where
$\varepsilon$ is an arbitrary positive number. To conclude the
proof, it is now sufficient to realize, that there exist $n \in
N$, points $x_0^i \in \partial \Omega, \ i=1,2,3, \cdots n$ and
positive numbers $\varepsilon, \varepsilon_i, \ i=1,2,3,\cdots,
n$ such that
 $$
\Omega \subset \cup_{i=1}^n \Omega_{x_0^i}^{\varepsilon_i/2} \cup
\Omega^{2\varepsilon}_I.
 $$
and use (\ref{eq:e44}).
\end{proof}

\begin{proof}[Another proof of Lemma \ref{L:L1}]
Let the assumptions of
Lemma~\ref{L:L1} be satisfied and $(u,\phi)$ be a unique solution
of (\ref{eq:e16}) - (\ref{eq:e19}) satisfying the inequality
(\ref{eq:e71}). We use the integral representation of
$u(\cdot,t)$ by means of the semigroup $e^{-A_\beta t}$:
 \begin{equation}
\label{eq:e74} u(\cdot,t) = \int _0^t e^{-A_\beta(t-\tau)}
(u'+A_\beta u) d\tau.
 \end{equation}
If $\alpha \in [0,1]$ then
 \begin{equation}
\label{eq:e75} \Vert A_\beta^\alpha u(\cdot,t) \Vert_\beta \le
\int _0^t \frac {1} {(t-\tau)^\alpha} (\Vert u' \Vert_\beta +
\Vert A_\beta u \Vert_\beta) d\tau.
 \end{equation}
Let us take $\alpha \in [1/2,1]$ such that $1 + 1/\gamma' =
\alpha + 1/\beta'$. The Hardy-Littlewood-Sobolev inequality gives
that
 \begin{equation}
\label{eq:e76} \Vert A_\beta^\alpha u \Vert_{\beta,\gamma'} \le c
(\Vert u' \Vert_{\beta,\beta'} + \Vert A_\beta u
\Vert_{\beta,\beta'}).
 \end{equation}
It further follows from \cite{SoWa} that
 \begin{equation}
\label{eq:e77} \Vert A_\beta^\alpha u \Vert_\beta \ge c \Vert
A^{1/2}_\gamma u \Vert_\gamma.
 \end{equation}
Let us show now that
 \begin{equation}
\label{eq:e78} \text{the space } D(A_m^\eta) \text { is
continuously embedded into the space } W^{2\eta,m},
 \end{equation}
if $m>1$ and $\eta \in (0,1)$. It is known that $D(A_m^\eta) =
D(B_m^\eta) \cap L^m_\sigma$, where $B_m = -\Delta$ is the
Laplace operator with zero boundary condition in $L^m$ (see
\cite{Gi}, Theorem 3). It follows from Theorem 1.15.3. in \cite
{Tr}, p.103, that $D(B_m^\eta)$ is the complex interpolation
space $[L^m,D(B_m)]_\eta$, thus $D(B_m^\eta) = [L^m,W^{2,m} \cap
W_0^{1,m}]_\eta$. Since $W^{2,m} \cap W_0^{1,m}$ is continuously
embedded into $W^{2,m}$, it follows from \cite{BeLo}, 2.4.(3),
that $D(B_m^\eta)$ is continuously embedded into
$[L^m,W^{2,m}]_\eta = W^{2\eta,m}$ and (\ref{eq:e78}) is proved.
The last equality follows from \cite[Theorem 4.3.1/2, p.315]{Tr}.

The following inequality is a special case of (\ref{eq:e78}):
 \begin{equation}
\label{eq:e79} \Vert A^{1/2}_\gamma u \Vert_\gamma \ge c \Vert
\nabla u \Vert _\gamma.
 \end{equation}
The inequality (\ref{eq:e21}) now follows from (\ref{eq:e76}),
(\ref{eq:e77}), (\ref{eq:e79}) and (\ref{eq:e71}) and the proof
of Lemma~\ref{L:L1} is completed.
\end{proof}

 \begin{lemma} \label{L:L3}
Let $2/q+3/p=1$, $p,q\in (1,\infty)$, $b \in L^{p,q}$,
$2/\theta'+3/\theta=3$, $p/(p-1)<\theta<3 $, $\theta'>2$,
$1/\alpha = 1/\theta - 1/3$ and $v \in L^{2,\infty}$, $\nabla v
\in L^{2,2}$, $v \in L^{\alpha,\theta'}$, $\nabla v \in
L^{\theta,\theta'}$. Let $r,r',l,l' \in (1,\infty)$, $1/r = 1/l -
1/p$, $1/r' = 1/l' - 1/q$, $r \ge \theta$, $r' \ge \theta'$ and
$h \in L^{l,l'}$. Suppose further that the function $v$ is a weak
solution of the linearized Navier-Stokes system, that is
 \begin{gather}
\label{eq:e45} \int_0^T \int_\Omega \big( -\frac {\partial
\varphi}{\partial t} - \Delta \varphi \big) \cdot v \,dx\,dt =
\int_0^T \int_\Omega (h - b \cdot \nabla v) \cdot \varphi \,dx\,dt,
\\ \label{eq:e46}  \nabla \cdot v = 0 \quad\text{in } Q_T, \\
\label{eq:e47}  v = 0 \quad \text{on} \ \partial \Omega \times
(0,T)
 \end{gather}
for every $\varphi \in C_0^\infty ([0,T) \times \Omega)$, $\nabla
\cdot \varphi = 0$. There exists a positive constant $\varepsilon
= \varepsilon(l,l',p)$ such that if $\Vert b \Vert_{p,q} <
\varepsilon$ then
 \begin{gather}
\label{eq:e48}  \nabla v \in L^{r,r'} \text{ and } \Vert \nabla
v \Vert _{r,r'} \le c \Vert h \Vert _{l,l'}, \\
\label{eq:e49}
\nabla v \in L^{m,l'} \text{ and } \Vert \nabla v \Vert _{m,l'}
\le c \Vert h \Vert _{l,l'}, \text {  if } l \in (1,3) \text{ and
} \frac {1}{m} = \frac {1}{l} - \frac {1}{3}, \\
\label{eq:e50}
 \nabla \phi, \frac {\partial v} {\partial t} \in L^{l,l'}
\text{ and } \Vert \nabla \phi \Vert _{l,l'},
\big\Vert \frac {\partial v} {\partial t} \big\Vert _{l,l'} \le
c \Vert h \Vert _{l,l'},
 \end{gather}
where $\phi$ is the pressure associated to $v$.
 \end{lemma}

\begin{proof}
This lemma was proved in \cite[Proposition 4.1, Theorem 4.1]{Ta1}
 for $\Omega = \mathbb{R}^3_+$. If
$\Omega$ is a bounded domain with a smooth boundary, the proof
proceeds in the same way and so we present only the main steps of
it.

We suppose without loss of generality that $h \in
C_0^\infty(Q_T)$. Let further $b_k \in C_0^\infty(Q_T)$ such that
$b_k \rightarrow b$ in $L^{p,q}$ if $k\rightarrow \infty$. By
\cite{Ta1}, Theorem 4.1 and the citation there, there exists a
smooth solution $(v_k,\phi_k)$ of the problem
 \begin{gather}
\label{eq:e63}  \frac {\partial v_k}{\partial t} - \nu \Delta
_k + b_k \cdot \nabla v_k + \nabla \phi_k = h \quad \text{in }
Q_T, \\
 \label{eq:e64}  \nabla\cdot v_k = 0 \quad \text{in } Q_T,  \\
\label{eq:e65}  v_k = 0 \quad \text{on } \partial \Omega \times (0,T), \\
 \label{eq:e66}  v_k \vert _{t=0} = 0.
 \end{gather}
If we choose $l_\theta$ and $l'_\theta$ so that $1/l_\theta =
1/\theta + 1/p$ and $1/l'_\theta = 1/\theta' + 1/q$ then $1 <
l_\theta \le l$, $1 < l'_\theta \le l'$ and $h \in
L^{l_\theta,l'_\theta}$. By the application of Lemma~\ref{L:L1}
to the system (\ref{eq:e63}) - (\ref{eq:e66}) we get
 \begin{equation}
\label{eq:e67} \Vert \nabla v_k \Vert_{\theta,\theta'} \le c
\Vert h - b_k \cdot \nabla v_k \Vert_{l_\theta,l'_\theta} \le c (
\Vert h \Vert_{l_\theta,l'_\theta} + \Vert b_k \Vert_{p,q} \Vert
\nabla v_k \Vert_{\theta,\theta'}),
 \end{equation}
where $c$ is independent of $k$. If $\Vert b \Vert_{p,q}$ is
sufficiently small, we get from (\ref{eq:e67}) that $\Vert \nabla
v_k \Vert_{\theta,\theta'} \le c \Vert h
\Vert_{l_\theta,l'_\theta}$, $\Vert v_k \Vert_{\alpha,\theta'}
\le c \Vert h \Vert_{l_\theta,l'_\theta}$ and consequently, from
the sequence $\{ v_k \}_{k\in N}$ we can select a subsequence
which we denote again $\{ v_k \}_{k\in N}$ such that
 \begin{gather}
\label{eq:e68} \nabla v_k \to \nabla \tilde v \quad
\text{weakly in } L^{\theta,\theta'}, \\
\label{eq:e69} v_k \to \tilde v \quad
\text{ weakly in } L^{\alpha,\theta'}.
 \end{gather}
Using (\ref{eq:e68}) and (\ref{eq:e69}) it is possible to show
that $\tilde v$ satisfies the equations (\ref{eq:e45}) -
(\ref{eq:e47}). It implies that $\tilde v=v$.

Applying again Lemma~\ref{L:L1} to the system (\ref{eq:e63}) -
(\ref{eq:e66}) we get
 \begin{equation}
\nonumber \Vert \nabla v_k \Vert_{r,r'} \le c \Vert h - b_k \cdot
\nabla v_k \Vert_{l,l'} \le c (\Vert h \Vert_{l,l'} + \Vert b_k
\Vert_{p,q} \Vert \nabla v_k \Vert_{r,r'})
 \end{equation}
and thus
 \begin{equation}
\label{eq:e70} \Vert \nabla v_k \Vert_{r,r'} \le c \Vert
h\Vert_{l,l'}.
 \end{equation}
It follows from (\ref{eq:e68}) and (\ref{eq:e70}) that (again
after selecting a subsequence) $\nabla v_k \to \nabla
v \text{ weakly in } L^{r,r'}$ which gives (\ref{eq:e48}).

$v$ and its associated pressure $\phi$ satisfy  the equations
(\ref{eq:e63}) - (\ref{eq:e66}) (with $b_k$ replaced by $b$) and
according to \cite{GiSo}, Theorem 2.8 and (\ref{eq:e48}) we have
 \begin{equation}
\label{eq:e72}  \big\Vert \frac {\partial v} {\partial t}
\big\Vert_{l,l'} + \Vert \nabla^2 v \Vert_{l,l'} + \Vert \nabla
\phi \Vert_{l,l'} \le c \Vert h - b \cdot \nabla v \Vert_{l,l'}
\le c \Vert h \Vert_{l,l'}.
 \end{equation}
Inequality (\ref{eq:e50}) is an immediate consequence of (\ref{eq:e72}) and
(\ref{eq:e49}) follows from (\ref{eq:e72}) and the fact that
$\Vert \nabla v \Vert _{m,l'} \le c \Vert \nabla^2 v
\Vert_{l,l'}$ if $l$ and $m$ are given by (\ref{eq:e49}). The
proof is complete.
\end{proof}

For the proof of the following lemma see e.g.
\cite[Theorem 3.2, Chap.III.3]{Ga}.

 \begin{lemma} \label{L:L4}
Let $D$ be a bounded Lipschitz domain in $\mathbb{R}^3$, $\Gamma$ be an
open subset of $\partial D$, $r \in (1,\infty)$, $j \in N \cup
\{0\}$. There exists a bounded linear operator $K=K_{j,r}:
W_0^{j,r}(D)\to W_0^{j+1,r}(D)^3$ such that
 \begin{itemize}
\item[(i)] $ \nabla \cdot Kg = g$ for all
$g \in W_0^{j,r}(D)$  such that $\int_D g dx = 0$

\item[(ii)] $\Vert \nabla ^{j+1} Kg \Vert _r \le c \Vert \nabla ^j g \Vert_r$
 for all $g \in W_0^{j,r}(D)$, $c=c(j,r,D)$

\item[(iii)] $\mathop{\rm supp} Kg \subset D \cup \Gamma$
 if $\mathop{\rm supp} g \subset D \cup \Gamma$.
 \end{itemize}
 \end{lemma}

In this lemma, $W_0^{j,r}(D)$ is the completion of
$C_0^\infty(D)$ with respect to the standard norm of the space
$W^{j,r}(D)$. It is possible to show that $K_{j,r}(g) =
K_{l,s}(g)$ if $g \in W_0^{j,r}(D) \cap W_0^{l,s}(D)$, where $r,s
\in (1,\infty)$ and $j,l \in N \cup \{0\}$ and so in the rest of
the paper the operator $K_{j,r}$ is denoted only by $K$.

\section{Proof of Theorem~\ref{T:T1}}

In this section, we assume that the hypotheses  of
Theorem~\ref{T:T1} are satisfied and $\phi$ is the associated
pressure to $u$. We can suppose without loss of generality that
$\Vert u \Vert_{p,q,Q_r}$ is sufficiently small - see
$\varepsilon$ from Lemma~\ref{L:L3}. Let $\tilde r \in (0,r)$.
Let us localize the problem (\ref{eq:e1}) - (\ref{eq:e4}) in a
standard way: Let $\psi \in C^\infty(\overline Q_T)$ be a cut-off
function such that $\psi(x,t)=0$ if $(x,t) \in Q_T \setminus
\overline Q_{2r/3+\tilde r/3}$, $\psi(x,t)=1$ if $(x,t) \in
Q_{r/3+2\tilde r/3}$ and $\psi(x,t) \in [0,1]$ for every $(x,t)
\in Q_T$. We put $w = K(\nabla \cdot (\psi u))$, $v = \psi u -
w$. Then $v$ satisfies the following system of equations:
\begin{gather} \label{eq:e5}
\begin{aligned}
 \frac {\partial v}{\partial t} - \nu \Delta v +
u \cdot \nabla v + \nabla (\psi \phi)
&= -\nu \Delta \psi u - 2\nu \nabla \psi \cdot \nabla u
 +  u \cdot \nabla \psi u - \phi \nabla \psi \\
&\quad - \frac {\partial w}{\partial t} + \nu \Delta w - u \cdot \nabla w
 - \frac {\partial \psi}{\partial t} u
\quad \text{in} \ Q_T,
\end{aligned}  \\
 \label{eq:e6}  \nabla\cdot v = 0 \quad \text{in } Q_T,
 \\
 \label{eq:e7}  v = 0 \quad \text{on }\partial \Omega \times (0,T), \\
  \label{eq:e8}  v \vert_{t=0} = 0.
\end{gather}
We denote the right hand side of (\ref{eq:e5}) by $h$ and show at
first that
 \begin{equation}
\label{eq:e14} h \in L^{l,l'},
\text{ for every $l'\in(1,2)$, $l\in (3/2,3)$ such that
$\frac {2} {l'} + \frac {3} {l} =3$.}
\end{equation}
We will use the global estimates for $u$ and $\phi$ derived in
\cite[Theorem 3.1]{GiSo}:
 \begin{gather}
 \label{eq:e9}  \big\Vert \frac {\partial u} {\partial t}
\big\Vert_{q,s} + \Vert \nabla^2 u \Vert_{q,s} + \Vert \nabla
\phi \Vert_{q,s}< \infty, \quad s \in (1,2), \;q \in (1,3/2), \frac
{2}{s} + \frac {3}{q} = 4,\\
 \label{eq:e11}  \Vert \nabla u \Vert_{h,\rho} < \infty, \quad  h \in (1,3),
 \;\rho \in (1,\infty), \frac {2}{\rho} + \frac {3}{h} = 3,\\
 \label{eq:e12}  \Vert u \Vert_{h^*,\rho} < \infty, \quad
h^* \in (3/2,\infty), \; \rho \in (1,\infty),\; \frac {2}{\rho} +
\frac{3}{h^*} = 2, \\
 \label{eq:e13}  \Vert \phi \Vert_{r,s} < \infty, \quad
r \in (3/2,3), \; s \in (1,2), \frac {2}{s} + \frac {3}{r} = 3.
 \end{gather}
It is supposed in  (\ref{eq:e13}) that $\int _{\Omega} \phi(x,t)
dx = 0$ for every $t \in (0,T)$. Thus, let $l,l'$ satisfy the
conditions from (\ref{eq:e14}). We have immediately from
(\ref{eq:e13}) that $\phi \nabla \psi \in L^{l,l'}$. It follows
further from Lemma~\ref{L:L4} that
 $$
\big\Vert \frac {\partial w} {\partial t} \big\Vert _{l,l'}
=\big\Vert \frac {\partial} {\partial t} (K (\nabla \psi \cdot
u)) \big\Vert _{l,l'}
= \big\Vert K \big(\frac {\partial}
{\partial t} (\nabla \psi \cdot u) \big) \big\Vert _{l,l'}
\le c \big\Vert \frac {\partial} {\partial t} (\nabla \psi \cdot
u) \big\Vert _{q,l'},
 $$
where $1/q=1/l+1/3$. Since $2/l'+3/q=4$, we have $\partial w /
\partial t \in L^{l,l'}$ by (\ref{eq:e9}). Similarly, $\nu \Delta
w \in L^{l,l'}$, as follows from Lemma~\ref{L:L4} and
(\ref{eq:e11}). Finally,
 $$
\Vert u \cdot \nabla w \Vert_{l,l'} \le \Vert u \Vert_{p,q} \Vert
\nabla w \Vert_{\frac {pl} {p-l},\frac {ql'} {q-l'}}
 $$
and since
 $$
\Vert \nabla w \Vert_{\frac {pl} {p-l}} = \Vert \nabla K (\nabla
\psi \cdot u) \Vert_{\frac {pl} {p-l}} \le c \Vert \nabla \psi
\cdot u \Vert_{\frac {pl} {p-l}} \le c \Vert u \Vert_{\frac {pl}
{p-l}},
 $$
we have
 $$
\Vert u \cdot \nabla w \Vert_{l,l'} \le \Vert u \Vert_{p,q} \Vert
u \Vert_{\frac {pl} {p-l},\frac {ql'} {q-l'}}.
 $$
Thus, $u \cdot \nabla w \in L^{l,l'}$ as a consequence of
$3(p-l)/pl + 2(q-l')/ql' = 3/l+2/l' - (3/p+2/q) = 2$ and
(\ref{eq:e12}). The remaining terms of $h$ belong obviously to
the space $L^{l,l'}$ and (\ref{eq:e14}) is proved.

 \begin{lemma} \label{L:L5}
Let us consider the equations \eqref{eq:e5}--\eqref{eq:e8}.
Let $l' \in (2q/(q+2),2)$ and $l \in (3/2, 3p/(p+3))$, that is $m<p$
for $m$ such that $1/m = 1/l - 1/3$. If $h \in L^{l,l'}$ and
$\psi\phi \in L^{l,l'}$, then $h \in L^{m,l'}$ and $\psi\phi \in L^{m,l'}$.
 \end{lemma}

\begin{proof}
Let us define $r,r'$ as in
Lemma~\ref{L:L3}. Then $r>3/2$ and $r'>2$. There exist
$\theta,\theta'$ such that $p/(p-1)<\theta<3/2$, $2<\theta'<r'$
and $2/\theta'+3/\theta=3$. It follows from (\ref{eq:e11}) that
$\nabla v \in L^{\theta,\theta'}$ and $v \in L^{\alpha,\theta}$,
where $1/\alpha=1/\theta-1/3$. Further, $v$ is a solution of
(\ref{eq:e45}) - (\ref{eq:e47}) with $u$ instead of $b$ and with
$h$ being the right hand side of (\ref{eq:e5}). Thus, the
assumptions of Lemma~\ref{L:L3} are obviously satisfied and we
get by (\ref{eq:e48})--(\ref{eq:e50}) that
 \begin{equation}
\label{eq:e51} \Vert \nabla v \Vert _{r,r'}, \Vert \nabla (\psi
\phi) \Vert _{l,l'}, \Vert \nabla v \Vert _{m,l'},
\big\Vert \frac {\partial v}{\partial t} \big\Vert _{l,l'} \le c \Vert h
\Vert _{l,l'}.
 \end{equation}
Now, one can show that $h \in L^{m,l'}$ using (\ref{eq:e51}), the
assumption $\psi\phi \in L^{l,l'}$ from Lemma~\ref{L:L5} and
Lemma~\ref{L:L4}. It is possible to proceed in the same way as
was done in the paragraph preceding Lemma~\ref{L:L5} and during
the process we can possibly diminish, if necessary, without loss
of generality the support of the cut-off function $\psi$.
\end{proof}

Now, we use twice Lemma~\ref{L:L5}. According to (\ref{eq:e14})
and (\ref{eq:e13}) we have that $h,\phi \in L^{l,l'}$, where
$l,l'$ satisfy the assumptions of Lemma~\ref{L:L5} and
$2/l'+3/l=3$. By the first application of Lemma~\ref{L:L5} we get
that $h,\psi\phi \in L^{m,l'}$, where $ 1/m = 1/l - 1/3$, $m \in
(3,p)$ and $2/l' + 3/m = 2$. Consequently, $h,\psi\phi \in
L^{l,l'}$, where $l,l'$ satisfy the assumptions from
Lemma~\ref{L:L5} and $2/l' + 3/l < 3$. By the second application
of Lemma~\ref{L:L5} we get that $h \in L^{m,l'}$ and $2/l' + 3/m
< 2$. Lemma~\ref{L:L3}, (\ref{eq:e48}) now produces that $\nabla
v \in L^{r,r'}$, where $1/r=1/m-1/p$ and $1/r'=1/l'-1/q$.
Consequently,
 $$
\Vert u \cdot \nabla v \Vert _{m,l'} \le \Vert u \Vert_{p,q}
\Vert \nabla v \Vert _{\frac {pm}{p-m} \frac {ql'}{q-l'}} \le c
\Vert \nabla v \Vert _{r,r'} < \infty
 $$
and $v$ satisfies the equation
 \begin{equation}
\label{eq:e54} \frac {\partial v}{\partial t} - \nu \Delta v +
\nabla (\psi \phi) = h - u \cdot \nabla v \quad \text{in} \ Q_T
 \end{equation}
and equations (\ref{eq:e6}) - (\ref{eq:e8}), where
 \begin{equation} \label{eq:e57}
 h - u \cdot \nabla v \in L^{m,l'}
\text{ for every $m,l'$  such that $\frac {2} {l'} + \frac {3} {m} < 2$,
$l'\in (1,2), m \in (3,p)$.}
 \end{equation}
Using the integral representation of $v(\cdot,t)$ by means of the
semigroup $e^{-A_mt}$, we have
 \begin{equation}
\label{eq:e55} v(\cdot,t) = \int _0^t e^{-A_m(t-\tau)}
P_\sigma^m(h - u \cdot \nabla v) d\tau.
 \end{equation}
Let $\alpha < 1/2$. We can choose $l'$ such that $\alpha
l'/(l'-1) < 1$ and obtain the estimate
 \begin{equation} \label{eq:e56}
\begin{aligned}
\Vert A_m^\alpha v(\cdot,t) \Vert_m
&\le \int _0^t \Vert A_m^\alpha e^{-A_m(t-\tau)} P_\sigma^m(h - u \cdot \nabla
v) \Vert_m d\tau \\
&\le \int _0^t \frac {\Vert h - u \cdot \nabla v
\Vert_m}{(t-\tau)^\alpha} d\tau \\
& \le \Big(\int _0^t \frac {d\tau}{(t-\tau)^{\alpha l'/(l'-1)}}
\Big)^{\frac{l'-1}{l'}} \Vert h - u \cdot \nabla v \Vert_{m,l'} \le c.
\end{aligned}
\end{equation}
The space $D(A_m^\alpha)$ is continuously embedded into the space
$W^{2\alpha,m}$ - see (\ref{eq:e78}). It further follows from
 in \cite[Theorem 4.6.1(e), p.327]{Tr} that
 \begin{equation}
\label{eq:e80} \text{the space $W^{2\alpha,m}$  is
continuously embedded into the H\"{o}lder space
$C^\beta(\overline\Omega)$,}
 \end{equation}
if $\beta = 2\alpha - 3/m > 0$. By the suitable choice of
$\alpha$ and $m$ we can have $\beta$ as close to $1-3/p$ as
possible and so it follows from (\ref{eq:e78}), (\ref{eq:e80}),
(\ref{eq:e56}) and (\ref{eq:e57}) that
 \begin{equation}
\label{eq:e58} v \in L^\infty(0,T,C^\beta(\overline\Omega)) \text
{ for every } \beta \in (0,1-3/p).
 \end{equation}
Thus, $v \in L^\infty(Q_T)$ and consequently,
 \begin{equation}
\label{eq:e59} u \in L^\infty(Q_{r/3+2\tilde r/3}).
 \end{equation}
We can now use this last information on local regularity of $u$,
go through this section once again and get that
 \begin{equation}
\label{eq:e60} h - u \cdot \nabla v \in L^{m,l'} \quad
\text{ for every }  l' \in (1,2), m \in (3,\infty).
 \end{equation}
Using (\ref{eq:e78}), (\ref{eq:e80}), (\ref{eq:e56}) and
(\ref{eq:e57}) we obtain that
 \begin{equation}
\label{eq:e61} v \in L^\infty(0,T,C^\beta(\overline\Omega))
\quad \text{for every } \beta \in (0,1)
 \end{equation}
and (\ref{eq:e53}) follows immediately. The proof of
Theorem~\ref{T:T1} is completed.


\begin{thebibliography}{99}

\bibitem{BeLo} J.~Bergh, J.~L\"{o}fstr\"{o}m; {\it Interpolation
spaces}, Springer-Verlag, Berlin, Heidelberg, New York, (1976).

\bibitem{Cho} H.J.~Choe; {\it Boundary regularity of weak
solutions of the Navier-Stokes equations}, J. of Differential
Equations 149 (1998), 211--247.

\bibitem{Ga} G.P.~Galdi; {\it An Introduction to the
Navier-Stokes initial-boundary value problem}, in Fundamental
Directions in Mathematical Fluid Mechanics, editors G.P.~Galdi,
J.~Heywood and R.~Rannacher, series "Advances in Mathematical
Fluid Mechanics", Birkhauser-Verlag, Basel (2000), 1--98.

\bibitem{Gi} Y.~Giga; {\it Domains of fractional powers of the
Stokes operator in $L_r$ spaces}, Arch. Rational Mech. Anal. 89
(1985), 251--265.

\bibitem{GiSo} Y.~Giga, H.~Sohr; {\it Abstract $L^p$ estimates
for the Cauchy problem with applications to the Navier-Stokes
equations in exterior domains}, J. of Functional Analysis 102
(1991), 72--94.

\bibitem{Ka} K.~Kang; {\it On boundary regularity of the
Navier-Stokes equations}, Preprint, (2004) 1--33.

\bibitem{Ne} J.~Neustupa; {\it The boundary regularity of a weak
solution of the Navier-Stokes equation and its connection with the
interior regularity of pressure}, Applications of Mathematics 48
(2003), 547--558.

\bibitem{Se} G.A.~Seregin; {\it Local regularity of suitable weak
solutions to the Navier-Stokes equations near the boundary}, J.
Math. Fluid Mech 4 (2002), 1--29.

\bibitem{Ser} J.~Serrin; {\it On the interior regularity of weak
solutions of the Navier-Stokes equations}, Arch. Rational Mech.
Anal. 9 (1962), 187--195.

\bibitem{SoWa} H.~Sohr, W. von Wahl; {\it On the regularity of
the pressure of weak solutions of Navier-Stokes equations}, Arch.
Math. 46 (1986), 428--439.

\bibitem{St} M.~Struwe; {\it On partial regularity results for the
Navier-Stokes equations}, Comm. Pure Appl. Math. 41 (1988),
437--458.

\bibitem{Ta1} S.~Takahashi; {\it On a regularity criterion up to
the boundary for weak solutions of the Navier-Stokes equations},
Comm. in Partial Differential Equations 17 (1992), 261--285.

\bibitem{Ta2} S.~Takahashi; {\it Erratum to "On a regularity
criterion up to the boundary for weak solutions of the
Navier-Stokes equations"}, Comm. in Partial Differential
Equations 19 (1994), 1015--1017.

\bibitem{Te} R.~Temam; {\it Navier-Stokes equations, theory and
numerical analysis}, North@-Holland Publishing Company,
Amsterodam, New York, Oxford, (1979).

\bibitem{Tr} H.~Triebel; {\it Interpolation theory, Function
spaces, differential operators}, VEB Deutscher Verlag der
Wissenschaften, Berlin, (1978).

\end{thebibliography}

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