\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 62, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/62\hfil Blow-up criterion]
{Blow-up criterion for the zero-diffusive Boussinesq equations 
 via the velocity components}

\author[W. Wang  \hfil EJDE-2015/62\hfilneg]
{Weihua Wang}

\address{Weihua Wang \newline
School of Mathematics and Statistics,
Hubei University, Wuhan 430062, China}
\email{wwh73@hubu.edu.cn}

\thanks{Submitted September 29, 2014. Published March 11, 2015.}
\subjclass[2000]{35Q35, 76D05}
\keywords{Zero-diffusive Boussinesq equations; blow up criterion; 
 Lorentz spaces}

\begin{abstract}
 This article  concerns the blow up for the smooth solutions of the
 three-dimensional Boussinesq equations with zero diffusivity.
 It is shown that if any two components of the velocity field $u$
 satisfy
 \begin{equation*}
 \int_0^T  \frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} }
  {1+\ln ( e+\|\nabla u\|^2_{L^2}) } ds<\infty,\quad
  \frac{2}{q}+\frac{3}{p}=1,\quad 3<p<\infty,
 \end{equation*}
 then the local smooth solution $(u,\theta)$ can be continuously
 extended to $(0,T_1)$ for some $T_1>T$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Since the famous laboratory experiments on turbulence derived by
Reynolds in 1883, the mathematical models which described the motion
of the viscous incompressible fluid flow have attracted more and
more attention. Those mathematical models are usually controlled by
the nonlinear partial differential equations. In this study, we
consider a dynamical model  of the ocean and atmosphere dynamics
\cite{Ma,Pe} which is so-called Boussinesq equations
\begin{equation}  \label{1.1}
\begin{gathered}
 \partial_t u  +u\cdot\nabla u +\nabla  p  =   \nu\Delta u+\theta e_3,\\
 \operatorname{div}u=0, \\
 \partial_t\theta +u\cdot\nabla\theta  =  \kappa\Delta \theta,
\end{gathered}
\end{equation}
where $u(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))$ and $\theta(x,t)$ are
 the unknown velocity vector field and the unknown scalar temperature,
$ p (x,t)$ is the unknown scalar pressure field.
  $\nu>0, \kappa\geq 0$ are the constants
kinematic viscosity and the thermal diffusivity,  $e_3=(0,0,1)^T$.

As an important mathematical model in the atmospheric sciences
\cite{Ma}, the Boussinesq equations have play an important role in
many geophysical applications \cite{Pe}. When $\theta=0$, the
Boussinesq equations \eqref{1.1} become  the classic  Navier-Stokes
equations
\begin{equation}  \label{1.2}
\begin{gathered} 
\partial_t u  +u\cdot\nabla u +\nabla  p  =   \nu\Delta u,\\
 \operatorname{div}u=0.
\end{gathered}
\end{equation}
From the viewpoint of mathematics, the Boussinesq system 
is the generalization of the Navier-Stokes equations. There is a
large body of  literature on the existence, uniqueness and regularity of
solutions for the Boussinesq equations. In the two-dimensional case,
when $\nu,\kappa>0$, the global existence and uniqueness of smooth
solution Boussinesq equations are obtained  by   Cannon and
DiBenedetto \cite{CD}. When $\nu=0,\kappa>0$ or $\nu>0,\kappa=0$,
the global regularity of local smooth solution of the Boussinesq
equations is also well studied in \cite{CW,Ch,HL,MZ,Xu}.

In the three-dimensional case,  corresponding three-dimensional
Navier-Stokes equations \cite{DZ14,Li}, the global regularity or
finite time singularity of weak solutions for the Boussinesq
equations \eqref{1.1} with positive dissipation is a big challenging
problem. Therefore, it is an important problem to consider the
blow-up issue  for the three-dimensional Boussinesq equations
\eqref{1.1} and related fluid dynamical
 models such as the Navier-Stokes equations and micropolar fluid flows
(refer to \cite{DC092,DZ10a,DJC}).  Ishimura and Morimoto \cite{IM}
(see also \cite{QDY})first proved the Beale-Kato-Majda blow-up
criteria of local smooth solution for the Boussinesq equations
\eqref{1.1}. That is to say, if   $T$ is the maximal existence time
of the local smooth solution for the Boussinesq equations
\eqref{1.1}, then
\begin{equation}\label{1.3}
T< \infty\Rightarrow\ \int_0^{T} \|\nabla u(s)\|_{L^{\infty}} ds
 =+\infty
\end{equation}

When $\kappa=0,$ the diffusive equation in Boussinesq
equations\eqref{1.1} is reduced to a transport equation
$$
\partial_t\theta +u\cdot\nabla\theta  = 0,
$$
and Boussinesq system \eqref{1.1} namely becomes the following
parabolic-hyperbolic system (for simplicity taking $\nu=1$)
\begin{equation}  \label{1.4}
\begin{gathered} \partial_t u
 +u\cdot\nabla u +\nabla  p  =  \Delta u+\theta e_3,\\
 \operatorname{div}u=0, \\
 \partial_t\theta +u\cdot\nabla\theta  = 0
\end{gathered}
\end{equation}
together with the initial data
\begin{equation}\label{1.5}
u(x,0)=u_0,\quad  \theta(x,0)=\theta_0.
\end{equation}

It should be mentioned that the temperature function $ \theta(x,t)$
in the transport equation does not  gain  smoothness whatsoever. The
blow-up issue of the zero-diffusive Boussinesq equations
\eqref{1.4}-\eqref{1.5} is more difficult compared with that of
Boussinesq system \eqref{1.1} with full viscosities. Fan and Zhou
\cite{FZ} recently studied the blow-up criterion of the local smooth
solution of the zero-diffusive Boussinesq equations
\eqref{1.4}-\eqref{1.5} and derived the following Beale-Kato-Majda
criterion
\begin{equation}\label{1.6}
\int_0^T \| \nabla \times u\|_{\dot{B}^{0}_{\infty,\infty}(\mathbb{R}^3)}ds
      < \infty
 \end{equation}
Jia, Zhang and Dong \cite{JZD} further refined the blow-up criterion
for local smooth solutions of zero-diffusive Boussinesq equations
\eqref{1.4}-\eqref{1.5} in the large critical Besov space
 \begin{equation}\label{1.9}
 \int_0^T \|u\|^p_{B^s_{q,\infty}(\mathbb{R}^3)}ds < \infty
\end{equation}
with
$\frac{2}{p}+\frac{3}{q}=1+s$ and
$$\
 \frac{3}{1+s}<p\leq\infty,\quad -1<s\leq 1,\quad
 (p,s)\neq(\infty,1). 
$$

To the author's knowledge, there are a few results on the blow-up
criterion for local smooth solution of zero-diffusive Boussinesq
equations \eqref{1.4}-\eqref{1.5} in terms of the components of the
velocity. The main purpose of this study is to investigate the
blow-up criterion for local smooth solution via horizontal velocity
$u_1,u_2 $ in the critical Lorentz spaces. More precisely, we 
show the following blow-up criterion for local smooth solution of
zero-diffusive Boussinesq equations \eqref{1.4}-\eqref{1.5}
\begin{equation*}
 \int_0^T\frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} }
      {1+\ln ( e+\|\nabla u\|^2_{L^2}) } ds<\infty,\quad
  \frac{2}{q}+\frac{3}{p}=1,\quad 3<p<\infty\,,
\end{equation*}
where $L^{p,\infty}$ is Lorentz space (see the definition in the
next section).


\section{Preliminaries and main results}

In this section, we first recall some basic notation. We denote by
$C$ the positive constant which may be different from line to line.
We denote by $L^q(\mathbb{R}^3)$ with $1\leq p\leq \infty $ the usual
vector or scalar Lebesgue space under the norm
\begin{equation*}
\|\varphi\|_{L^{p}} =\begin{cases}
\Big(\int_{\mathbb{R}^3}|\varphi(x)|^p\,dx\Big)^{1/p},
& 1\leq p<\infty,\\
 \operatorname{ess\,sup}_{x\in\mathbb{R}^3} |\varphi(x)|, & p=\infty.
\end{cases}
\end{equation*}
We also denote by   $H^k(\mathbb{R}^3)$    the usual Sobolev space 
$ \{ \varphi\in L^2(\mathbb{R}^3); \|\nabla^k \varphi\|_{L^2}<\infty\}$.

We denote by  $L^{p,q}(\mathbb{R}^3)$ with $1\leq p$, $q\leq \infty $ the
Lorenz space with the norm \cite{Tr}
\[
\|\varphi\|_{L^{p,q}}= \Big(\int_0^{\infty}t^q(m(\varphi,t))^{q/p}\
\frac{dt}{t} \Big)^{1/q}<\infty\quad \mbox{for }\ 1\leq q<\infty,
\]
where $m(\varphi,t) $ is the Lebesgue measure of the set $\{x\in
\mathbb{R}^3:|\varphi(x)|> t\}$, \emph{i.e.}
$$
m(\varphi,t):=m\{x\in \mathbb{R}^3:|\varphi(x)|> t\}.
$$
In particular, when $q=\infty $,
\[
\|\varphi\|_{L^{p,\infty}}
= \sup_{t\geq 0}\{t(m(\varphi,t))^{\frac1p}\}   <\infty\,.
\]
The  Lorents space $L^{p,\infty}$ is also called weak $L^p$ space.
The norm is equivalent to the norm
\begin{equation*}
 \|f\|_{L^{q,\infty}}
 = \sup_{0<|E|<\infty}|E|^{1/q-1}\int_{E}|f(x)|   dx.
\end{equation*}

As stated by Triebel \cite{Tr}, Lorentz space $L^{p,q}(\mathbb{R}^3)$ may
be  defined by real interpolation methods
\begin{equation}\label{2.1}
L^{p,q}(\mathbb{R}^3)
=(L^{p_{1}}(\mathbb{R}^3),\,L^{p_{2}}(\mathbb{R}^3))_{\alpha,q},
\end{equation}
with
$$
\frac{1}{p}=\frac{1-\alpha}{p_{1}}+\frac{\alpha}{p_{2}},\quad 
1\leq p_{1}<p<p_{2}\leq \infty.
$$
We now recall some basic inequality which will be used in the next
section.

\begin{lemma}[O'Neil \cite{ON}] \label{lem2.1}
Assume $1\leq  p_a$, $p_b\leq \infty$,
$1\leq  q_a$, $q_b\leq \infty $ and 
$u\in L^{p_a,q_a}(\mathbb{R}^3)$,  $v\in L^{p_b,q_b}(\mathbb{R}^3)$.
 Then $uv\in L^{p_c,q_c}(\mathbb{R}^3)$ with
\begin{equation*}
\frac1{p_c}=\frac1{p_a}+\frac1{p_b},\quad 
\frac1{q_c}\leq \frac1{q_a}+\frac1{q_b}
\end{equation*}
 and  the  inequality
\begin{equation}\label{2.3}
\|uv\|_{L^{p_c,q_c}}\leq C \|u\|_{L^{p_a,q_a}}\|v\|_{L^{p_b,q_b}}
\end{equation}
is valid.
\end{lemma}


Our main results are read as follows.

\begin{theorem} \label{thm2.1}
Assume $(u, \theta)$ is the local smooth solution of zero-diffusive
Boussinesq equations \eqref{1.4}-\eqref{1.5} satisfying that
\begin{eqnarray*}
 (u,\theta)\in C([0,T);H^m(\mathbb{R}^3)), \quad m>3.
\end{eqnarray*}
If $T$ is the maximal existence time of the  solution $(u,\theta) $,
then for
$$
 \frac{2}{q}+\frac{3}{p}=1,\quad\,\,
3<p<\infty,
$$
the following necessary blow-up condition
\begin{equation}\label{2.4}
T< \infty\Rightarrow\
  \int_0^T\frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} }
                {1+\ln ( e+\|\nabla u\|^2_{L^2}) } ds
                =+\infty
 \end{equation}
holds.
\end{theorem}

The above theorem obviously implies the following corollary.

\begin{corollary}
 Assume $(u, \theta)$ is the local smooth solution of zero-diffusive
Boussinesq equations \eqref{1.4}-\eqref{1.5} satisfying 
\[
 (u,\theta)\in C([0,T);H^m(\mathbb{R}^3)),\quad  m>3.
\]
If the velocity satisfies
\begin{equation*}
  \int_0^T\frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} }
  {1+\ln ( e+\|\nabla u\|^2_{L^2}) } ds<\infty,\quad
  \frac{2}{q}+\frac{3}{p}=1,\quad 3<p<\infty
\end{equation*}
  then the  solution  $(u,\theta)$   can be continually extended to
   the interval $(0,T_1)$ for some $T_1>T$.
\end{corollary}



\begin{remark} \label{rmk2.1} \rm
When $\nu=\kappa=0,$ the existence and uniqueness  of local smooth
solution $(u,\theta)$ for zero-dissipation   Boussinesq equations
\eqref{1.1} have been investigated by Chae and   Nam \cite{CN},
therefore, we only need to prove  the blow-up criterion of Theorem
\ref{thm2.1}. Moreover, once the proof of Theorem \ref{thm2.1} is obtained, 
the proof of Corollary 2.1  follows directly from Theorem \ref{thm2.1} and we omit
it here.
\end{remark}

\section{Proof of Theorem \ref{thm2.1}}

\subsection{$L^p$ estimate for $ \theta$}

Multiplying both sides of the transport equation of zero-diffusive
Boussinesq equations \eqref{1.4}-\eqref{1.5} by
$|\theta|^{p-2}\theta$ and integrating in $\mathbb{R}^3$, we have
\begin{equation}
\frac{d}{dt}\int_{\mathbb{R}^3} |\theta|^p\,dx=0, \ \ \ p\geq2
\end{equation}
where we have used
$$
\int_{\mathbb{R}^3} u\cdot\nabla\theta \theta dx=0.
$$
 Integrating in time becomes
\begin{equation}\label{3.1}
\operatorname{ess\,sup}_{0<t<T} \|\theta\|_{L^p}\leq \|\theta_0\|_{L^p}, \quad
p\geq2
\end{equation}

\subsection{Energy estimate for $(u, \theta)$}

Taking the inner product of the zero-diffusive Boussinesq equations
\eqref{1.4}-\eqref{1.5} with $u$, we obtain
\begin{equation}
\frac12\frac{d}{dt}\int_{\mathbb{R}^3} |u|^2\,dx+ \int_{\mathbb{R}^3} |\nabla u|^2\,dx=\int_{\mathbb{R}^3} \theta e_3 u dx
\end{equation}
where we have also used
$$
\int_{\mathbb{R}^3} u\cdot\nabla u  u\, dx=0, \quad  
\int_{\mathbb{R}^3} \nabla p u\, dx=0.
$$
Thanks to
\[
 \int_{\mathbb{R}^3} \theta e_3 u\,dx \leq
  \|\theta\|_{L^2}\|u\|_{L^2}\leq \|\theta_0\|_{L^2}\|u\|_{L^2},
\]
we have
\[
 \frac12\frac{d}{dt}\int_{\mathbb{R}^3} |u|^2\,dx+ \int_{\mathbb{R}^3} |\nabla u|^2\,
dx \leq \| \theta_0\|_{2}\| u\|_{2};
\]
 the Gronwall inequality  gives
\begin{equation}\label{3.2}
\sup_{0\leq t<T} \|  u(t)\|_{L^2}^2+ 2 \int_0^{T} \|  \nabla
u(\tau)\|_{L^2}^2d\tau~\leq~C (u_0,\theta_0).
\end{equation}


\subsection{Uniform estimate for $\|\nabla u \|_{L^2}$}

Multiplying both sides of the momentum equations  zero-diffusive
Boussinesq equations \eqref{1.4}-\eqref{1.5} with $\Delta u$ and
integrating in $\mathbb{R}^3$, it follows that
 \begin{equation}\label{3.3}
 \frac12\frac{d}{dt}\int_{\mathbb{R}^3} |\nabla u|^2\,dx
+ \int_{\mathbb{R}^3} |\Delta u|^2\,dx
= -\int_{\mathbb{R}^3} u\cdot \nabla u \Delta  u \,dx
\end{equation}
where we have used
$$
   \int_{\mathbb{R}^3} \nabla p \Delta u \,dx=0.
 $$
Integrating by parts and using the divergence free condition
$\sum_{k=1}^3 \partial_ku_k=0$, we have
 \begin{equation}\label{3.4}
\begin{aligned}
&-\int_{\mathbb{R}^3} u\cdot \nabla u \Delta  u dx \\
&=  -\sum_{i, j, k=1}^3\int_{\mathbb{R}^3}\partial_{kk}u_ju_i  \partial_i u_j \,dx \\
&= \sum_{i, j, k=1}^3 \int_{\mathbb{R}^3}\partial_{k}( u_i
                          \partial_i u_j)\  \partial_{ k}u_j\,dx \\
&=  \sum_{i, j, k=1}^3 \int_{\mathbb{R}^3}\partial_{k} u_i\partial_{k}u_j
 \partial_i u_j \,dx  +\frac{1}{2} \sum_{i, j, k=1}^3 \int_{\mathbb{R}^3} u_i
                          \partial_i (\partial_{k}u_j \partial_{ k}u_j)\, dx \\
&= I+J.
\end{aligned}
\end{equation}

We now estimate $I$ and $J$. When $i=1,2$ or $j=1,2$, by integrating by
parts,
 \begin{equation}\label{3.6}
\begin{aligned}
I &=   \sum_{i, j =1,2}   \sum_{k=1}^3 \int_{\mathbb{R}^3}\partial_{k}
               u_i    \partial_{ k}u_j\ \partial_i u_j  dx    \\
&=\sum_{i=1}^2  \sum_{j,  k=1}^3\int_{\mathbb{R}^3}\partial_{k} u_i
  \partial_{ k}u_j    \partial_i u_j   dx
  +\sum_{j=1}^2\sum_{k=1}^3\int_{\mathbb{R}^3}\partial_{k} u_3
               \partial_{ k}u_j  \partial_3 u_j  dx \\
&\leq  C\int_{\mathbb{R}^3}  (|u_1|+|u_2|) |\nabla u| |\Delta u| \,dx.
\end{aligned}
 \end{equation}

When $i=j=3$,  applying the fact
$$
-\partial_3u_3= \partial_1u_1+\partial_2u_2
$$
and integrating by parts, we have
 \begin{equation}\label{3.7}
\begin{aligned}
 -\sum_{i, j, k=1}^3\int_{\mathbb{R}^3}u_i
     \partial_i u_j \partial_{kk}u_j\,dx   
&=  -     \sum_{i, j=3}\sum_{ k=1}^3 \int_{\mathbb{R}^3}\partial_{k}( u_i
                          \partial_i u_j)\  \partial_{ k}u_j\,dx \\
&=  \sum_{k=1}^3\int_{\mathbb{R}^3}\partial_{k} u_3
                    \partial_{ k}u_3   \partial_3 u_3 \, dx  \\
&= \sum_{k=1}^3\int_{\mathbb{R}^3}\partial_{k} u_3
                   \partial_{ k}u_3  (\partial_1u_1+\partial_2u_2)\, dx \\
&\leq   C\int_{\mathbb{R}^3}  (|u_1|+|u_2|) |\nabla u| |\Delta u| \,dx\, dx.
\end{aligned}
 \end{equation}
Inserting the inequalities \eqref{3.6} and\eqref{3.7} in
\eqref{3.5}, we have
 \begin{equation}\label{3.8}
I  \leq   C\int_{\mathbb{R}^3}  (|u_1|+|u_2|) |\nabla u| |\Delta u| \,dx\,dx.
 \end{equation}

For $J$,
 \begin{equation}\label{3.5}
\frac{1}{2} \sum_{i, j, k=1}^3 \int_{\mathbb{R}^3} u_i
\partial_i (\partial_{k}u_j \partial_{ k}u_j)\,dx 
= - \frac{1}{2} \sum_{i, j, k=1}^3 \int_{\mathbb{R}^3}\partial_i  u_i
 (\partial_{k}u_j \partial_{ k}u_j)\,dx 
= 0.
 \end{equation}

Substituting the estimates $I,J$ in the right hand side of
\eqref{3.3}, we obtain
 \begin{equation}\label{3.9}
 \frac{d}{dt}\int_{\mathbb{R}^3} |\nabla u|^2\,dx+ 2\int_{\mathbb{R}^3} |\Delta u|^2\,dx
 \leq C\int_{\mathbb{R}^3}  (|u_1|+|u_2|) |\nabla u| |\Delta u|\, dx\,dx.
 \end{equation}

To control the right hand side of \eqref{3.9}, with the aid
of the H\"{o}lder inequality,the Young inequality and Lemma \ref{lem2.1}, it
follows that
\begin{align*}
&\int_{\mathbb{R}^3}  (|u_1|+|u_2|) |\nabla u| |\Delta u| dx \\
&\leq  C\| (|u_1|+|u_2|) |\nabla u|\|_{L_2}\| \Delta   u\|_{L^2}\\
&\leq C \| (|u_1|+|u_2|) |\nabla u|\|^2_{L^2}
                 +\frac12\| \Delta u  \|^2_{L^2}\\
&\leq C \| (|u_1|+|u_2|) |\nabla u|\|^2_{L^2}
                 +\frac12\| \Delta u  \|^2_{L^{2,2}}\\
& \leq  C  \||u_1|+|u_2|\|^2_{L^{p,\infty}}\|\nabla
              u\|^2_{L^{\frac{2p}{p-2},2}}
                 + \frac{1}{2}\| \Delta u  \|^2_{L^2},
 \end{align*}
thus we rewrite the inequality \eqref{3.9} as
 \begin{equation}\label{3.10}
 \frac{d}{dt}\int_{\mathbb{R}^3} |\nabla u|^2\,dx
           + \frac32\int_{\mathbb{R}^3} |\Delta u|^2\,dx
 \leq  C  \||u_1|+|u_2|\|^2_{L^{p,\infty}}\|\nabla
              u\|^2_{L^{\frac{2p}{p-2},2}}
 \end{equation}
Since
$$
L^{{\frac{2p}{p-2},2}}(\mathbb{R}^3)=\left(L^{{\frac{2p_1}{p_1-2} }}(\mathbb{R}^3),
L^{{\frac{2p_2}{p_2-2} }}(\mathbb{R}^3)\right)_{\frac{1}{2},2}
$$
with
$$
3<p_1<p<p_2<\infty,\quad \frac{2}{p}=\frac{1}{p_1}+\frac{1}{p_2}
$$
it follows that
  \begin{align*}
 \|g\|_{L^{\frac{2p}{p-2},2}}
&\leq  C\|g\|^{1/2}_{L^{\frac{2p_1}{p_1-2} }}
             \|g\|^{1/2}_{L^{\frac{2p_2}{p_2-2} }}\\
&\leq   C\Big(\|g\|^{\frac{p_1-3}{p_1}}_{L^2}
            \|\nabla g \|^{\frac{3}{p_2}}_{L^2}\Big)^{1/2}
                 \Big(\|g\|^{\frac{p_2-3}{p_2 }}_{L^2}
            \|\nabla  g\|^{\frac{3}{p_2}}_{L^2}\Big)^{1/2}\\
&\leq  C\|g\|^{\frac{p-3}{p }}_{L^2}   \|\nabla g\|^{\frac{3}{p}}_{L^2}
\end{align*}
which implies
$$
\|\nabla u\|^2_{L^{\frac{2p}{p-2},2}}
\leq C \|\nabla u\|_{L^2}^{\frac{2(p-3)}{p}}\|\Delta
             u\|^{6/p}_{L^2}
$$
Hence inserting the above inequality into the right hand side of
\eqref{3.10} and applying the Young inequality, one shows that
 \begin{equation}\label{3.10b}
\begin{aligned}
& \frac{d}{dt}\int_{\mathbb{R}^3} |\nabla u|^2\,dx
           + \frac32\int_{\mathbb{R}^3} |\Delta u|^2\,dx
 \\
&\leq C  \||u_1|+|u_2|\|^2_{L^{p,\infty}}
             \|\nabla u\|_{L^2}^{\frac{2(p-3)}{p}}\|\Delta
             u\|^{6/p}_{L^2}  \\
&\leq C   \||u_1|+|u_2|\|^q_{L^{p,\infty}}
             \|\nabla u\|_{L^2}^{2}
                 +  \frac12 \| \Delta u  \|^2_{L^2}
\end{aligned}
 \end{equation}
where we have used that
$q=2p/(p-3)$.
Thus we derive
 \begin{equation}\label{3.10c}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^3} |\nabla u|^2\,dx
           +  \int_{\mathbb{R}^3} |\Delta u|^2\,dx  \\
& \leq  C    \||u_1|+|u_2|\|^q_{L^{p,\infty}}
                         \|\nabla u\|_{L^2}^{2} \\
& \leq  C  \frac{  \||u_1|+|u_2|\|^q_{L^{p,\infty}}}{ 1+\ln (
          e+\|\nabla u\|^2_{L^2})} (1+\ln ( e+\|\nabla
                  u\|^2_{L^2}))    \|\nabla u\|_{L^2}^{2}.
\end{aligned}
 \end{equation}
Taking the Gronwall inequality into consideration, it follows that
 \begin{equation}
\|\nabla u\|_{L^2}^2
 \leq  \|\nabla u_0\|_{L^2}^2
           \exp\Big\{\int_0^T \Big(
           \frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} }
                {1+\ln ( e+\|\nabla u\|^2_{L^2}) }
      \{1+\ln ( e + \|\nabla u\|_{L^2}^2 )\}
       \Big) dt \Big\}.
\end{equation}
 Hence we have
 \begin{equation}
\begin{aligned}
\ln (e+  \|\nabla u\|_{L^2}^2)
&\leq \ln (e+  \|\nabla u_0\|_{L^2}^2) \\
&\quad +\int_0^T \Big(
           \frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} }
                {1+\ln ( e+\|\nabla u\|^2_{L^2}) }
     \{1+\ln ( e + \|\nabla u\|_{L^2}^2 )\} \Big) dt.
\end{aligned}
\end{equation}
Taking the Gronwall inequality into account again, we have
\begin{equation}\label{3.11}
   \ln\{e+  \|\nabla u\|_{L^2}^2 \}
 \leq   C(u_0)\exp\Big\{ \int_0^T
           \frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} }
                {1+\ln ( e+\|\nabla u\|^2_{L^2}) } ds \Big\}<
                \infty
\end{equation}
which implies the uniform estimates of $\nabla u$,
\begin{equation}\label{3.12}
\operatorname{ess\,sup}_{0<t<T} \|\nabla u\|_{L^2}^2  < \infty.
\end{equation}

\subsection{Uniform $H^m $estimate for $(u, \theta)$}

Since
$$
 \Delta u  =\partial_t u  +\nabla  p +u\cdot\nabla u  - \theta
 e_3,
$$
by the standard  elliptic regularity theory, we can derive 
\begin{equation}\label{3.13}
\operatorname{ess\,sup}_{0<t<T}\|u \|_{ H^2(\mathbb{R}^3) }\leq C,
\end{equation}
from which and together with the standard bootstrap technique, we
can obtain uniform $ H^m$ estimates
\begin{equation}\label{3.14}
\sup_{0\leq t<T_1} (\| u\|^2_{H^m} +\|\theta\|^2_{H^m}) \leq C.
\end{equation}
The detail argument can be found  in \cite{JZD}, 
we omit it here.
The proof of Theorem \ref{thm2.1} is complete.


\subsection*{Acknowledgments}
 The author  wants to express her
sincere thanks to the editor and the referee for their valuable
comments and suggestions which improve this article.

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\end{document}