\documentclass[twoside]{article}
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\pagestyle{myheadings}
\markboth{\hfil Local Regularity of Solutions \hfil EJDE--1998/12}%
{EJDE--1998/12\hfil Mike O'Leary \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~12, pp. 1--9. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
Pressure conditions for the local regularity of solutions
of the Navier--Stokes equations
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35Q30, 76D05.
\hfil\break\indent
{\em Key words and phrases:} Navier-Stokes, regularity, pressure.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted January 26, 1998. Published May 13, 1998.} }
\date{}
\author{Mike O'Leary}
\maketitle

\begin{abstract}
We obtain a relationship between the integrability of the pressure
gradient and the the integrability of the velocity for local solutions
of the Navier--Stokes equations with finite energy. In particular, we show that
if the pressure gradient is sufficiently integrable, then the corresponding
velocity is locally bounded and smooth in the spatial variables. The result
is proven by using De Giorgi type estimates in $L^{\text{weak}}_p$ spaces.
\end{abstract}

\def\pl{\partial}
\def\diver{\operatorname{div}}
\def\meas{\operatorname{meas}}
\def\essup{\operatornamewithlimits{ess\,sup}}
\newtheorem{Theorem}{Theorem}
\renewcommand\u{\mathbf u}
\renewcommand\v{\mathbf v}
\newcommand\dint[1]{\iint_{\!\!\! #1}}
\newcommand\dna{\downarrow}

\section{Introduction and statement of results}

One of the most important unresolved questions in the theory of the
Navier--Stokes equations is the
local behavior of solutions in three spatial dimensions, in particular,
questions of their local regularity.
This problem was studied by Serrin \cite{Serr62}, who showed that if the
velocity is sufficiently
integrable, then it is locally bounded and smooth in the spatial
variables. Indeed, let $\Omega\subset\mathbb R^N$
be an open set, let $T>0$, and set $\Omega_T=\Omega\times(0,T)$. Suppose that
${\v}\in V^2_{{\rm loc}}(\Omega_T)=L_{2,{\rm loc}}(0,T;W^1_{2,{\rm
loc}}(\Omega))\cap
L_{\infty,{\rm loc}}(0,T;L_{2,{\rm loc}}(\Omega))$
is a weak solution of the Navier--Stokes system
\begin{equation}\begin{gathered}
 {\v}_t-\Delta{\v}+({\v}\cdot\nabla){\v}+\nabla p=0\,,\\
  \diver{\v}=0\,.
\end{gathered} \label{eq-2}
\end{equation}
Serrin showed that if $\v\in L_{q,r}(\Omega_T)=L_r(0,T;L_q(\Omega))$ for some
$q,r$ satisfying
\begin{equation}
\label{params}
 \frac{N}{q}+\frac {2}{r}<1
\end{equation}
then $\v$ is locally bounded and smooth in the spatial variables. Kahane
\cite{Kaha69} later showed
that this also implies that the velocity is locally analytic in the
spatial variables. The case
where the inequality in \eqref{params} is replaced by an equality was
studied by Takahashi, \cite{Taka90}.

 In a sequence of papers including \cite{Sche76,Sche77,Sche78,Sche80},
Scheffer adopted a different
approach. Rather than finding conditions which guaranteed regularity, he
constructed solutions to initial
and initial boundary value problems that were partially regular; that is
the Hausdorff dimension in
space and time of the set of possible singularities could be estimated from
above. Caffarelli, Kohn and
Nirenberg in \cite{CKN82} extended these techniques to show that if
$\Omega_T\subset{{\mathbb R}}^3\times{{\mathbb R}}$ and
${\v}$ is a local suitable weak solution of \eqref{eq-2}, then the
one--dimensional Hausdorff measure in
space and time of the set of possible singularities is zero. A suitable
weak solution is a pair of
functions ${\v}\in V^2_{{\rm loc}}(\Omega_T)$ and $p\in L_{5/4,{\rm
loc}}(\Omega_T)$ that
satisfy \eqref{eq-2} and a
generalized energy inequality
\cite[Equation 2.5]{CKN82}.

 Since this partial regularity result requires some regularity of the
pressure, a natural question is
the relationship between the regularity of the pressure and the regularity
of the velocity. The purpose
of this paper is to examine this relationship and to prove the following
results.

\begin{Theorem}
Let $({\v},p)$ be a local weak solution of the Navier--Stokes equations in
a domain
$\Omega_T\subset{{\mathbb R}}^3\times{{\mathbb R}}$, and suppose that ${\v}\in
V^2_{{\rm loc}}(\Omega_T)$ and
$\nabla p\in L_{\mu,{\rm loc}}(\Omega_T)$. Then $\v\in L_{q,{\rm
loc}}(\Omega_T)$ for any $q$ that
satisfies
\begin{equation}
 \label{eq-3}
 q<\frac{5}{5/\mu-2}\,,
\end{equation}
where $1<\mu\leq\frac 53$. Furthermore, if $\mu>\tfrac 53$, then ${\v}$ is
locally bounded and smooth in the spatial variables.
\end{Theorem}

This can be generalized to domains in an arbitrary number of spatial
dimensions in the following fashion.
\begin{Theorem}
Let $({\v},p)$ be a local weak solution of the Navier--Stokes equations in
a domain $\Omega_T\subset{{\mathbb
R}}^N\times{{\mathbb R}}$, and suppose that
${\v}\in V^2_{{\rm loc}}(\Omega_T)$ and $\nabla p\in L_{\mu,{\rm
loc}}(\Omega_T)$. Then $\v\in
L_{q,{\rm loc}}(\Omega_T)$ for any $q$ that satisfies
\begin{equation}
 \label{eq-4}
 q<\frac{N+2}{(N+2)/\mu-2}\,,
\end{equation}
where $1<\mu\leq \tfrac {N+2}3$. Furthermore, if $\mu>\tfrac {N+2}3$, then
${\v}$ is locally bounded and smooth in the spatial variables.
\end{Theorem}

A consequence of these results is the fact that singularities of the
velocity ${\v}$ are only possible at singularities of
the pressure gradient $\nabla p$.

We point out that the norm $\|\nabla p\|_{L_{(N+2)/3}(\Omega_T)}$
is dimensionless in the same sense that the norms $\|{\v}\|_{L_{q,r}(\Omega_T)}$
are dimensionless when $N/q+2/r=1$. Indeed, if ${\v}(x,t)$ and $p(x,t)$
satisfy the
Navier-Stokes equations \eqref{eq-2} in a domain $\Omega\times(0,T)$, then the
functions ${\v}_\lambda(x,t)=\lambda{\v}(\lambda x,\lambda^2t)$ and
$p_\lambda(x,t)=\lambda^2p(\lambda
x,\lambda^2t)$ also satisfy
\eqref{eq-2} in the dilated domains $\Omega/\lambda\times(0,T/\lambda^2)$
for each
$\lambda>0$;
however, the given norms of ${\v}_\lambda$ and $p_\lambda$ are independent of
$\lambda$. Indeed,
\begin{equation}
 \begin{split}
  \|\v_\lambda\|_{L_{q,r}(\Omega/\lambda\times(0,T/\lambda^2))}
     &=\left\{\int_0^{T/\lambda^2}\left(\int_{\Omega/\lambda}|\lambda\v
     (\lambda x,\lambda^2 t)|^q
       \>dx\right)^{r/q}dt\right\}^{1/r} \\
     &=\left\{\int_0^T\left(\int_\Omega\lambda^q|\v(x,t)|^q\frac{dx}{\lambda^N}
       \right)^{r/q}\frac{dt}{\lambda^2}\right\}^{1/r} \\
     &=\lambda^{1-\left(N/q+ 2/r\right)}\|\v\|_{L_{q,r}(\Omega_T)}
       =\|\v\|_{L_{q,r}(\Omega_T)}
 \end{split}
\end{equation}
and
\begin{equation}
 \begin{split}
  \| \nabla p_\lambda\|_{L_{(N+2)/3}(\Omega/\lambda\times(0,T/\lambda^2))}&\\
    &\hspace{-1.0 in}=\left\{\int_0^{T/\lambda^2}
        \int_{\Omega/\lambda} |\lambda^2\cdot\lambda
        \nabla p(\lambda x,\lambda^2 t)|^{(N+2)/3}dx\>dt\right\}^{3/(N+2)} \\
    &\hspace{-1.0 in}=\left\{\int_0^T\int_\Omega \lambda^{N+2} |\nabla
p|^{(N+2)/3}
        \frac{dx}{\lambda^N}\frac{dt}{\lambda^2}\right\}^{3/(N+2)} \\
    &\hspace{-1.0 in}=\|\nabla p\|_{L_{(N+2)/3}(\Omega_T)}.
 \end{split}
\end{equation}
In this sense, Theorems 1 and 2 are
analogues of the aforementioned result of Serrin \cite{Serr62}.

The basic idea of the proof is to use as a test function
a smoothed and cutoff variant of
$(v^i\mp k)_\pm^\epsilon$.
Provided $\epsilon$ is sufficiently small, the nonlinear term is
integrable and we can remove the smoothing to
obtain a local energy estimate. From this we can obtain an estimate of
$\meas[|v^i|>k]$ and
consequently of $v^i$ in $L_{p,{\rm loc}}^{\text{weak}}(\Omega_T)$.

Recall the definitions of the spaces $L_q^{\text{weak}}(\mathcal U)$; a
measurable
function $f$ is an element of $L_q^{\text{weak}}(\mathcal U)$ if and only if
\begin{equation}
 |f|_{L_q^{\text{weak}}(\mathcal U)}\equiv\sup_{k>0}
k\big(\meas\{x\in\mathcal U:|f(x)|>k\}\big)^{1/q}<\infty\,.
\end{equation}
The quantity $|f|_{L_q^{\text{weak}}(\mathcal U)}$ is not a norm, but it is a
quasi-norm.
The inequality
\begin{equation}
 |f|_{L_q^{\text{weak}}(\mathcal U)}\leq\| f\|_{L_q(\mathcal U)}
\end{equation}
follows immediately from
\begin{equation}
 k^q\meas[|f|>k]\leq\int_\mathcal U |f|^q\>\chi[|f|>k]\>dx\leq\int_\mathcal
U |f|^q\>dx
\end{equation}
so that $L_q(\mathcal U)\subset L_q^{\text{weak}}(\mathcal U)$. However
$L_q^{\text{weak}}(\mathcal U)\neq L_q(\mathcal U)$, as the function $f(x)=1/x$
satisfies $f\in L_1^{\text{weak}}(0,1)$, but $f\notin L_1(0,1)$.
Finally, if $q'<q$ and $\mathcal U$ is bounded, then
$L_q^{\text{weak}}(\mathcal U)\subset L_{q'}(\mathcal U)$;
indeed
\begin{equation}
 \begin{split}
  \| f\|^{q'}_{L_{q'}(\mathcal U)}&=q'\int_0^\infty k^{q'-1}\meas[|f|>k]\>dk \\
  &\leq q'\meas \mathcal U +q'|f|^q_{L_q^{\text{weak}}(\mathcal U)}
      \int_1^\infty k^{q'-q-1}\>dk<\infty.
 \end{split}
\end{equation}
For further details about the spaces $L_q^{\text{weak}}(\mathcal U)$
see \cite[Chp. 1]{BeLo76} or \cite[IX.4]{ReSi75}.

Once we have the the energy inequality, we can show that $\v\in
L_{\beta,{\rm loc}}(\Omega_T)$ implies
$v^i\in L_{\alpha(\beta),{\rm loc}}^{\text{weak}}$ for some $\alpha(\beta)$ and
each $i$;
consequently $\v\in L_{q,{\rm loc}}(\Omega_T)$ for
all $q<\alpha(\beta)$. Carefully iterating the process yields the result.
\medskip

\noindent{\bf Remark:} The techniques of this paper exploit the structure of the
nonlinear term
and the fact that $\v$ is solenoidal; in particular these techniques fail
for non-solenoidal solutions
of the more general system
\begin{equation}
 \u_t-\Delta\u+(\u\cdot\nabla)\u=0\,.
\end{equation}

\section{Proof of Theorem 1}
For simplicity, suppose that
$\Omega_T\subset{\mathbb R}^3\times{\mathbb R}$, and let
\begin{equation*}
 Q_R=Q_R(x_o,t_o)=B_R(x_o)\times(t_o-R^2,t_o)\subset\Omega_T.
\end{equation*}
Let $t_o-R^2<\tau<t_o$ and define $Q_R^\tau=B_R(x_o)\times(t_o-R^2,\tau)$.
Choose $0<\sigma<1$, and let $\zeta\in C^\infty(\Omega_T)$ be a cutoff
function so that $\zeta(x,t)=1$ if $(x,t)\in Q_{\sigma R}$, so that
$\zeta(x,t)=0$
if either $|x-x_o|=R$ or $t=t_o-R^2$, and so that
$|\zeta_t|+|\nabla\zeta|^2\leq C_{\sigma,R}$. Further, let
$\{J_\eta(x,t)\}_{\eta>0}$
be a family of symmetric mollifying kernels in space and time;
given a function $f(x,t)$, we shall denote the mollification $(J_\eta*f)$
by $f_\eta$

Let $k>0$, $\omega>0$, and choose $0<\epsilon<\tfrac 23$ so small that
$\tfrac{10}{10-3\epsilon}\leq\mu$. Suppose that
${\v}\in V^2_{{\rm loc}}(\Omega_T)$; recall the Sobolev embedding
$V^2_{{\rm loc}}(\Omega_T)\hookrightarrow L_{10/3,{\rm loc}}(\Omega_T)$
when $\Omega_T\subseteq{{\mathbb R}}^3\times{\mathbb R}$. Fix
$i\in\{1,2,3\}$, and
consider the function
\begin{equation}
 \phi^i(x,t)=\left\{[(v^i_\eta(x,t)-k)_+
    +\omega]^\epsilon\zeta^2(x,t)
   \right\}_\eta.
\end{equation}
If $\eta$ is sufficiently small, $\phi^i$ is a valid test function;
multiplying the $i^{\text{th}}$ component of the
first of \eqref{eq-2} and integrating over $\Omega_T$ we obtain
\begin{equation*}
 \begin{split}
  0&=\iint_{\!\!Q_R^\tau}\!\!\left(\frac{\pl}{\pl t}v^i_\eta\right)
       [(v^i_\eta-k)_++\omega]^\epsilon\zeta^2\>dx\>dt \\
   &\hspace{0.6 in}+\sum_{j=1}^3\iint_{\!\!Q_R^\tau}
       \!\!\left(\frac{\pl}{\pl x_j}
       v^i_\eta\right)\frac{\pl}{\pl x_j}
       \big\{[(v^i_\eta-k)_++\omega]^\epsilon\zeta^2\big\}dx\>dt \\
   &\hspace{0.6 in}+\sum_{j=1}^3\iint_{\!\!Q_R^\tau}\left[v^j\frac{\pl v^i}
       {\pl x_j}\right]_\eta[(v^i_\eta-k)_++\omega]^\epsilon\zeta^2\>dx\>dt \\
   &\hspace{0.6 in}+\iint_{\!\!Q_R^\tau}\frac{\pl p_\eta}{\pl x_i}
       [(v^i_\eta-k)_++\omega]^\epsilon\zeta^2\>dx\>dt \\
   &=I_1+I_2+I_3+I_4
 \end{split}
\end{equation*}
for each fixed $i$.

To estimate $I_1$, first let $\omega\dna 0$ so that
\begin{equation}
 \begin{split}
  \lim_{\omega\dna 0}I_1
     &=\dint{Q_R^\tau}\left\{\frac{\pl}{\pl t}(v^i_\eta-k)_+\right\}
          (v^i_\eta-k)_+^{\epsilon}\zeta^2\>dx\>dt \\
     &=\frac{1}{\epsilon+1}\dint{Q_R^\tau}\left\{\frac{\pl}{\pl
t}\left[(v^i_\eta-k)_+^{\epsilon+1}\right]\right\}\zeta^2\>dx\>dt \\
     &=\frac{1}{\epsilon+1}\int_{B_R}(v^i_\eta-k)_+\zeta^2\Big|_{t=\tau}dx
           -\frac
2{\epsilon+1}\dint{Q^\tau_R}(v^i_\eta-k)_+^{\epsilon+1}\zeta\zeta_t\>dx\>dt.
\end{split}
\end{equation}
Then, sending $\eta\downarrow 0$ we obtain for almost every $\tau$
\begin{multline}
 \lim_{\eta\dna 0}\lim_{\omega\dna 0}I_1=
        \frac{1}{\epsilon+1}\int_{B_R(x_o)}
        (v^i-k)_+^{\epsilon+1}\zeta^2\Big|_{t=\tau}\>dx \\
     -\frac{2}{\epsilon+1}\iint_{\!\!Q_R^\tau}
       \!\!(v^i-k)_+^{\epsilon+1}\zeta\zeta_t\>dx\>dt.
\end{multline}

To estimate the $I_2$ term, first rewrite the integral as
\begin{equation}
 \begin{split}
 I_2&=\epsilon\sum_{j=1}^3\dint{Q_R^\tau}
      \left\{\frac{\pl}{\pl x_j}
      [(v^i_\eta-k)_++\omega]\right\}^2[(v^i_\eta-k)_+
      +\omega]^{\epsilon-1}\zeta^2\>dx\>dt\\
   &\hspace{0.5 in}+2\sum_{j=1}^3\dint{Q_R^\tau}
      \frac{\pl v^i_\eta}
      {\pl x_j}[(v^i_\eta-k)_++\omega]^\epsilon
      \zeta\zeta_{x_j}\>dx\>dt\\
   &=\frac{4\epsilon}{(\epsilon+1)^2}\sum_{j=1}^3
      \dint{Q_R^\tau}\left\{\frac{\pl}{\pl x_i}[(v^i_\eta-k)_+
      +\omega]^{\frac{\epsilon+1}2}\right\}^2\zeta^2\>dx\>dt \\
    &\hspace{0.5 in}+2\sum_{j=1}^3\dint{Q_R^\tau}\frac{\pl v_i}
      {\pl x_j}[(v^i_\eta-k)_++\omega]^\epsilon\zeta\zeta_{x_j}\>dx\>dt.
\end{split}
\end{equation}
Using Fatou's lemma in the first term and the fact that $\epsilon\leq 1$ in the
second, we send $\omega\dna 0$
to obtain
\begin{multline}
   \liminf_{\omega\dna 0}I_2\geq C_\epsilon
     \dint{Q_R^\tau}\left|\nabla
     \left\{(v^i_\eta-k)_+^{\frac{\epsilon+1}{2}}\right\}
     \right|^2\zeta^2\>dx\>dt\\
  \hspace{0.5 in}-2\dint{Q_R^\tau}|\nabla(v^i_\eta-k)_+|
       \>(v^i_\eta-k)_+^\epsilon\zeta\>|\nabla\zeta|\>dx\>dt.
\end{multline}
Using Young's inequality and Fatou's lemma once more, we obtain
\begin{multline}
  \liminf_{\eta\dna 0}\liminf_{\omega\dna 0}I_2
     \geq C_\epsilon\dint{Q_R^\tau}\left|\nabla
     \left\{(v^i-k)_+^{\frac{\epsilon+1}{2}}
     \right\}\right|^2\zeta^2\>dx\>dt \\
  -C_\epsilon\dint{Q_R^\tau}(v^i-k)_+^{\epsilon+1}|
      \nabla\zeta|^2\>dx\>dt.
\end{multline}

To estimate $I_3$, note that because $\epsilon\leq 2/3$ the integral is
uniformly bounded
and we can pass to the limit as $\omega\dna 0$ then as $\eta\dna 0$ to obtain
\begin{equation}
 \begin{split}
   \lim_{\eta\dna 0}\lim_{\omega\dna 0} I_3
       &=\sum_{j=1}^3\dint{Q_R^\tau} v^j \frac{\pl v^i}{\pl x_j}
           (v^i-k)_+^\epsilon\zeta^2\>dx\>dt\\
       &=\frac 1{\epsilon+1}\sum_{j=1}^3\dint{Q_R^\tau}v^j\left\{\frac{\pl}
           {\pl x_j}(v^i-k)_+^{\epsilon+1}\right\}\zeta^2\>dx\>dt.
 \end{split}
\end{equation}
Then, because $\v$ is solenoidal we can integrate by parts to obtain
\begin{equation}
    \lim_{\eta\dna 0}\lim_{\omega\dna 0}I_3=\frac{-2}{\epsilon+1}\sum_{j=1}^3
    \dint{Q_R^\tau}v^j(v^i-k)_+^{\epsilon+1}\zeta\zeta_{x_j}\>dx\>dt.
\end{equation}

As for $I_4$, we have
\begin{equation}
 I_4=\dint{Q_R^\tau}\frac{\pl p_\eta}{\pl x_i}
     \left[(v^i_\eta-k)_++\omega\right]^\epsilon\zeta^2\>dx\>dt
\end{equation}
so since $\nabla p\in L_{\mu,{\rm loc}}(\Omega_T)$, $|\v|\in L_{10/3,{\rm
loc}}(\Omega_T)$ and
$\frac{10}{10-3\epsilon}\leq\mu$,
we can pass to the limit, obtaining
\begin{equation}
 \lim_{\eta\dna 0}\lim_{\omega\dna 0}I_4=\dint{Q_R^\tau}|\nabla
p|(v^i-k)_+^\epsilon\zeta^2\>dx\>dt.
\end{equation}

Combine these results and use the arbitrariness of $\tau$ to obtain
\begin{equation}
 \begin{split}
 \label{eq-9}
 \|(v^i-k)_+^{(\epsilon+1)/2}\zeta\|^2_{V^2(Q_R)}&
       =\essup_{t_o-R^2<\tau<t_o}\int_{B_R}\left|(v^i-k)_+^{(\epsilon+1)/2}
          \zeta\right|^2\Bigg|_{\tau}dx\\
       &\hspace{0.5in}+\dint{Q_R}\left|\nabla\left\{(v^i-k)_+^{(\epsilon+1)/2}
          \zeta\right\}\right|^2\>dx\>dt\\
       &\leq C_{\epsilon,\sigma,R}\dint{Q_R}(v^i-k)^{\epsilon+1}_+\>dx\>dt\\
       &\hspace{0.5 in}+C_{\epsilon,\sigma,R}\dint{Q_R}|{\v}|
         (v^i-k)^{\epsilon+1}_+\>dx\>dt\\
       &\hspace{0.5 in}+C_{\epsilon}\dint{Q_R}|\nabla
p|(v^i-k)_+^\epsilon\>dx\>dt
 \end{split}
\end{equation}
which is our energy estimate.

The restriction $\epsilon\leq\tfrac 23$, needed to pass to the limit in $I_3$
implies that $\tfrac{\epsilon+1}{2}<1$ so we can not use this inequality to
directly prove strong $L_q$ estimates. We can, however, use this technique
to prove a weak-$L_q$ estimate. Indeed,  estimate the left side as follows.
\begin{equation}
 \begin{split}
 \|(v^i-k)_+^{(\epsilon+1)/2}\zeta\|^2_{V^2(Q_R)}&\geq C
       \|(v^i-k)_+^{(\epsilon+1)/2}\|^2_{L_{10/3}(Q_{\sigma R})}\\
     &\geq C\left\{\dint{Q_{\sigma R}}(v^i-k)_+^{5(\epsilon+1)/3}\>dx\>dt
        \right\}^{3/5}\\
     &\geq Ck^{\epsilon+1}\left[\meas\{(x,t)\in Q_{\sigma R}:v^i(x,t)>2k\}
        \right]^{3/5}.
 \end{split}
\end{equation}
Suppose that $\v\in L_{\beta,{\rm loc}}(\Omega_T)$ for some $\beta\geq
10/3$. We can
then estimate the first term on the
right side of \eqref{eq-9} as
\begin{equation}
 \begin{split}
    \dint{Q_R}(v^i-k)_+^{\epsilon+1}\>dx\>dt&\leq\left(\dint{Q_R}(v^i-k)_+^\beta
         \>dx\>dt\right)^{(\epsilon+1)/\beta}\left(\meas[v^i>k]
         \right)^{1-(\epsilon+1)/\beta}\\
    &\leq\|\v\|_{L_\beta(Q_R)}^{\epsilon+1}\left\{|\v|_{L_\beta^{\text{weak}}
         (Q_R)}^\beta\frac 1{k^\beta}\right\}^{1-(\epsilon+1)/\beta}\\
    &\leq\|\v\|_{L_\beta(Q_R)}^\beta\left(\frac 1k\right)^{\beta-\epsilon-1}.
 \end{split}
\end{equation}
Similarly,
\begin{equation}
 \dint{Q_R}|\v|(v^i-k)_+^{\epsilon+1}\>dx\>dt\leq\|\v\|_{L_\beta(Q_R)}^\beta
       \left(\frac 1k\right)^{\beta-\epsilon-2}.
\end{equation}
Lastly
\begin{equation}
 \begin{split}
  \dint{Q_R}|\nabla p|(v^i-k)_+^\epsilon\>dx\>dt\leq &\left(\dint{Q_R}
        |\nabla p|^\mu\>dx\>dt\right)^{1/\mu}\times \\
     &\hspace{-0.5in}\left(\dint{Q_R}|\v|^\beta\>dx\>dt
         \right)^{\epsilon/\beta}\left(\meas[v^i>k]
         \right)^{1-1/\mu-\epsilon/\beta}\\
     &\hspace{-1.0 in}\leq\|\nabla p\|_{L_\mu(Q_R)}
         \|\v\|_{L_\beta(Q_R)}^\epsilon
         \left(|\v|^\beta_{L_\beta^{\text{weak}}(Q_R)}\frac 1{k^\beta}
         \right)^{1-1/\mu-\epsilon/\beta}\\
     &\hspace{-1.0 in}\leq\|\nabla p\|_{L_\mu(Q_R)}
         \|\v\|_{L_\beta(Q_R)}^{\beta\left(1-1/\mu\right)}
         \left(\frac 1k\right)^{\beta\left(1- 1/\mu\right)-\epsilon}.
 \end{split}
\end{equation}
Combining these estimates, we find
\begin{multline}
 \meas\{(x,t)\in Q_{\sigma R}:v^i(x,t)>2k\}
         \leq C\|\v\|_{L_\beta(Q_R)}^{5\beta/3}
         \left(\frac 1k\right)^{5\beta/3}\\
    +C\|\v\|_{L_\beta(Q_R)}^{5\beta/3}\left(\frac 1k\right)^{5(\beta-1)/3}
    +C\|\nabla p\|_{L_\mu(Q_R)}^{5/3}
         \|\v\|_{L_\beta(Q_R)}^{5\beta\left(1-1/\mu\right)/3}
         \left(\frac 1k\right)^{5\left[\beta\left(1-1/\mu\right)
         +1\right]/3}.
\end{multline}
Repeating this process with test functions $(v^i+k)_-$, we obtain an estimate
of the form
\begin{equation}
\label{eq-27}
 |v^i|_{L_{\alpha(\beta)}^{\text{weak}}(Q_{\sigma R})}
    \leq C(\epsilon,\beta,R,\sigma,\mu,\|\nabla
p\|_{L_\mu(Q_R)},\|\v\|_{L_\beta(Q_R)})
\end{equation}
for each $i$, where
\begin{equation}
 \alpha(\beta)=\min\left\{\frac 53(\beta-1),\frac 53
       \left[\beta\left(\frac{\mu-1}\mu\right)+1\right]\right\}.
\end{equation}
Thus $\v\in L_{\beta,{\rm loc}}(\Omega_T)$ implies $\v\in L_{q,{\rm
loc}}(\Omega_T)$ for every
$q<\alpha(\beta)$.

Our result is proven by iteration. Since $V^2_{{\rm
loc}}(\Omega_T)\hookrightarrow
L_{10/3,{\rm loc}}(\Omega_T)$, set $\beta_o=\tfrac {10}3$, and inductively
define
\begin{equation}
 \beta_{n+1}=\alpha(\beta_n)=\min\left\{\frac 53(\beta_n-1),
      \frac 53\left[\beta_n\left(\frac{\mu-1}{\mu}\right)+1\right]\right\}.
\end{equation}
Now \eqref{eq-27} implies that ${\v}\in L_{q,{\rm loc}}(\Omega_T)$ for every
$q<\beta_n$, for every $n$. Set
\begin{align*}
 \gamma(\beta)&=\tfrac 53(\beta-1), \\
 \delta(\beta)&=\tfrac 53\left[\beta\left(\frac{\mu-1}{\mu}\right)+1\right].
\end{align*}
If $\beta\geq\tfrac {10}3$, then $\gamma(\beta)\geq\tfrac 76\beta$. On the other
hand, $\delta(\beta)\geq\beta$ if and only if
\begin{equation}
 \beta\leq\frac{5}{5/\mu-2}
\end{equation}
so that the sequence $\beta, \delta(\beta), \delta(\delta(\beta)), \dots$
converges to
$\tfrac{5}{5/\mu-2}$ independently of the choice
of $\beta$. Thus $\v\in L_{q,{\rm loc}}(\Omega_T)$ for all
$q<\tfrac{5}{5/\mu-2}$. If
$\mu>\tfrac 53$, we can choose
$q>5$ and apply the regularity result of Serrin \cite{Serr62} which
guarantees the local boundedness
and smoothness of $\v$ in the spatial variables.
\medskip

\noindent{\bf Remark.} Theorem 2, the general result in $N$ spatial
dimensions, is proven in the same fashion as Theorem 1.

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{\sc Mike O'Leary}\\
Department of Mathematics, University of California Santa Cruz\\
Santa Cruz, CA 95064 USA.\\
E-mail address: oleary@cats.ucsc.edu

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