\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{cite,amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 97, pp. 1--7.\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-2015/97\hfil Regularity criteria]
{Regularity criteria for 3D Boussinesq equations
  with zero thermal diffusion}

\author[Z. Ye \hfil EJDE-2015/97\hfilneg]
{Zhuan Ye}

\address{Zhuan Ye \newline
School of Mathematical Sciences,
Beijing Normal University.
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
Beijing 100875,China}
\email{yezhuan815@126.com, Phone +86 10 58807735, Fax +86 10 58808208}

\thanks{Submitted January 14, 2015. Published April 14, 2015.}
\makeatletter
\@namedef{subjclassname@2010}{\textup{2010} Mathematics Subject Classification}
\makeatother
\subjclass[2010]{35Q35, 35B65, 76W05}
\keywords{3D Boussinesq equations;  Besov spaces; regularity criterion}

\begin{abstract}
 In this article, we consider the three-dimensional (3D) incompressible Boussinesq
 equations with zero thermal diffusion. We establish a regularity criterion for
 the local smooth solution in the framework of Besov spaces in terms of the velocity
 only.
\end{abstract}

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

\section{Introduction}\label{intro}

In this article, we consider the 3D Boussinesq equations with zero 
thermal diffusion,
\begin{equation}\label{Bouss}
\begin{gathered}
\partial_{t}u+(u \cdot \nabla) u-\mu\Delta u+\nabla P=\theta e_{3},
 \quad x\in \mathbb{R}^{3},\; t>0, \\
\partial_{t}\theta+(u \cdot \nabla) \theta=0, \quad x\in \mathbb{R}^{3},\; t>0,\\
\nabla\cdot u=0, \quad x\in \mathbb{R}^{3},\; t>0,\\
u(x, 0)=u_{0}(x),  \quad \theta(x,0)=\theta_{0}(x), \quad x\in \mathbb{R}^{3},
\end{gathered}
\end{equation}
where $\mu\geq 0$ is the viscosity, 
$u=u(x,t)\in \mathbb{R}^{3}$ is the velocity, $P=P(x,t)\in \mathbb{R}$
is the scalar pressure, $\theta=\theta(x,t)\in \mathbb{R}^{3}$ 
is the temperature, and
$e_{3}=(0,0,1)^{\mathrm{T}}$. The Boussinesq equations are of relevance
 to study a number of models coming from atmospheric or oceanographic 
turbulence (see  \cite{MB,PG}).

It is easy to check that in the case $\theta=0$, the
system \eqref{Bouss} reduces to the 3D classical Navier-Stokes equations.
Although the local existence and uniqueness of smooth solutions for the
system \eqref{Bouss} with large initial data were easily obtained 
(see \cite{CKN,MB}), whether the unique local smooth solution
can exist globally is an outstanding challenging open problem. 
Therefore, it is important to study the mechanism of blowup and structure 
of possible singularities of smooth solutions to the system \eqref{Bouss}.
For this reason, many researchers were devoted to finding sufficient conditions 
to ensure the smoothness of the solutions; see
 \cite{FZ,FO,GGRM,QDY1,QYWL,Xiang,XZZ,YZ1,Zhang2014} and so forth. 
For many interesting results on the high dimensional Boussinesq equations 
with axisymmetric data, we refer the readers to \cite{AHS,HR1,HR2,MZ,MZ1}.  
We remark that the 2D Boussinesq equations also has recently attracted considerable
attention, just name a few (see  \cite{CW,CW1,Chae,DP,HKR1,HKR2,HL,LLT,JMWZ}).


The aim of this paper is to improve the previous regularity criterion 
results on the system \eqref{Bouss}. Since the concrete value of $\mu$ 
does not play a special role in our discussion, for simplicity, we set $\mu=1$.
 Now we state the main results as follows

\begin{theorem}\label{Th1} 
Assume that $(u_{0}, \theta_{0})\in H^{3}(\mathbb{R}^{3})\times H^{3}(\mathbb{R}^{3})$.
 Let $(u, \theta)$ be a local smooth solution of the system \eqref{Bouss}. 
If the following condition holds
\begin{equation}\label{R1}
\int_{0}^{T}{\|u(t)\|_{\dot{B}_{p, \infty}^{\frac{3}{p}+\frac{2}{q}-1}}^{q}
\,dt}<\infty,
\end{equation}
with $\frac{3}{p}+\frac{2}{q}\leq2$ and $(p, q)\neq (\infty, \infty)$ for 
$1<p, q\leq\infty$, then the solution pair $(u, \theta)$ can be extended
beyond time $T$.
 Here $\dot{B}_{p, q}^{s}$ stands for the homogeneous Besov space.
In other words, if $T<\infty$ is the maximal existence time, then
 $$
\int_{0}^{T}{\|u(t)\|_{\dot{B}_{p, \infty}^{\frac{3}{p}+\frac{2}{q}-1}}^{q}
\,dt}=+\infty.
$$
\end{theorem}

\begin{remark} \label{rmk1.2} \rm
When the thermal diffusion $\Delta\theta$ was added in the second equation of 
the system \eqref{Bouss}, Xiang \cite{Xiang} obtained the same regularity 
result \eqref{R1}. As a result, Theorem \ref{Th1} significantly improves
the result (Theorem 1.1) in \cite{Xiang}. Moreover, we chose not to apply 
the Littlewood-Paley decomposition on the
system itself, in contrast to the proof of \cite{Xiang}.
\end{remark}

\begin{remark} \label{rmk1.3} \rm
The system \eqref{Bouss} has scaling property that if
$(u, \theta, P)$ is a solution of the system \eqref{Bouss}, then for
any $\lambda>0$ the functions
$$
u_{\lambda}(x,t)=\lambda u(\lambda x, \lambda^{2}t),
\quad  \theta_{\lambda}(x,t)=\lambda^{3} \theta(\lambda x, \lambda^{2}t), \quad
P_{\lambda}(x,t)=\lambda^{2}P(\lambda x, \lambda^{2}t),
$$
 are also solutions of \eqref{Bouss} with the corresponding
initial data $u_{0, \lambda}(x)=\lambda u_{0}(\lambda x)$ and 
$ \theta_{0, \lambda}(x)=\lambda^{3} \theta_{0}(\lambda x)$.  It is an obvious fact
that the assumption \eqref{R1} does belong to the invariant spaces.
\end{remark}


The method may also be adapted with almost no change to the study of
the following B\'enard system:
\begin{equation}\label{Benard}
\begin{gathered}
\partial_{t}u+(u \cdot \nabla) u-\mu\Delta u+\nabla P=\theta e_{3}, 
\quad x\in \mathbb{R}^{3}, \; t>0, \\
\partial_{t}\theta+(u \cdot \nabla) \theta=u_{3}, \quad x\in \mathbb{R}^{3}, \; t>0,\\
\nabla\cdot u=0,  \quad x\in \mathbb{R}^{3}, \; t>0,\\
u(x, 0)=u_{0}(x),  \quad \theta(x,0)=\theta_{0}(x),  \quad x\in \mathbb{R}^{3},
\end{gathered} 
\end{equation}
which describes convective motions in a heated incompressible fluid
(see \cite[Chap. 6]{AP}). Because of the
similar structure to Boussinesq system \eqref{Bouss}, it is not
difficult to show that  B\'enard system \eqref{Benard}
admits the same conclusion as Theorem \ref{Th1}, namely, we have the following
result.

\begin{theorem}\label{Th2} 
Assume that $(u_{0}, \theta_{0})\in
H^{3}(\mathbb{R}^{3})\times H^{3}(\mathbb{R}^{3})$. Let $(u, \theta)$ 
be a local smooth solution of the system \eqref{Benard}. 
If the following condition holds
\begin{equation}\label{R3}
\int_{0}^{T}{\|u(t)\|_{\dot{B}_{p, \infty}^{\frac{3}{p}+\frac{2}{q}-1}}^{q}
\,dt}<\infty,
\end{equation}
with $\frac{3}{p}+\frac{2}{q}\leq2$ and $(p, q)\neq (\infty, \infty)$ 
for $1<p, q\leq\infty$, then the solution pair $(u, \theta)$ can be 
extended beyond time $T$.
In other words, if $T<\infty$ is the maximal existence time, then
 $$
\int_{0}^{T}{\|u(t)\|_{\dot{B}_{p, \infty}^{\frac{3}{p}+\frac{2}{q}-1}}^{q}
\,dt}=+\infty.
$$
\end{theorem}


\section{Proof of Theorem \ref{Th1}}
 
As stated above that the local smooth solution was obtained, we only need 
to establish {\it a priori} estimates.
 Throughout the paper, $C$ represents a real positive constant
which may be different in each occurrence.

\begin{proof}[Proof of  Theorem \ref{Th1}]
Multiplying the second equation of \eqref{Bouss} by $|\theta|^{p-2}\theta$ and and
integrating the resulting equation over $\mathbb{R}^{3}$ yield that
\begin{equation}\label{BE01}
\|\theta(t)\|_{L^{p}}\leq \|\theta_{0}\|_{L^{p}},\quad \forall p\in
[1, \infty].
\end{equation}
Testing $\eqref{Bouss}_{1}$ and $\eqref{Bouss}_{2}$ by $u$ and $\theta$, 
respectively, it gives
\begin{equation}\label{BE02}
\frac{1}{2}\frac{d}{dt}(\|u(t)\|_{L^{2}}^{2}+\|\theta(t)\|_{L^{2}}^{2})+ \|\nabla u\|_{L^{2}}^{2}\leq \|u\|_{L^{2}}\|\theta\|_{L^{2}},
\end{equation}
which together with \eqref{BE01} implies that
\begin{equation}\label{BE03}
\| u(t)\|_{L^{2}}^{2}+\|
\theta(t)\|_{L^{2}}^{2}+\int_{0}^{t}{\|\nabla u (\tau)\|_{L^{2}}^{2}\,d\tau}
\leq C<\infty.
\end{equation}
Multiplying  equation $\eqref{Bouss}_{1}$ by
$\Delta u$, integration by parts and taking the divergence free property into
account, one concludes that
\begin{equation}\label{BE04}
\frac{1}{2}\frac{d}{dt}\|\nabla u(t)\|_{L^{2}}^{2}+\|\Delta
u\|_{L^{2}}^{2}
=-\int_{\mathbb{R}^{3}}{\theta e_{3}\cdot\Delta
u\,dx}+\int_{\mathbb{R}^{3}}{(u\cdot\nabla u)\cdot\Delta
u\,dx}.
\end{equation}
Integrating by parts and using Young inequality, we obtain
\begin{equation}\label{BE05}
-\int_{\mathbb{R}^{3}}{\theta e_{3}\cdot\Delta
u\,dx}\leq \|\Delta u\|_{L^{2}}\| \theta\|_{L^{2}}
\leq \frac{1}{4}\|\Delta u\|_{L^{2}}^{2}+C\| \theta\|_{L^{2}}^{2}.
\end{equation}
To bound the remainder term, we split it into the following two cases:
\smallskip

\noindent\textbf{Case 1: $2<q\leq\infty$.}
The following bilinear estimate (see \cite{YZ})
$$
\|ff\|_{\dot{B}_{2,2}^{s}}\leq C\|f\|_{\dot{B}_{\infty,\infty}^{-\alpha}}
\|f\|_{\dot{B}_{2,2}^{s+\alpha}},\quad \text{for any } s>0, \; \alpha>0
$$
and Young inequality allow us to show that
\begin{equation}\label{BE06}
\begin{aligned}
&\int_{\mathbb{R}^{3}}{(u\cdot\nabla u)\cdot\Delta u\,dx}\\
&\leq  \int_{\mathbb{R}^{3}}{\nabla\cdot(u\otimes u)\cdot\Delta
u\,dx} \\
&\leq  C\|u\otimes u\|_{\dot{H}^{1}}\|\Delta u\|_{L^{2}} \\
&\leq  C
(\|u\|_{\dot{B}_{\infty,\infty}^{-\beta}}\|u\|_{\dot{B}_{2,2}^{1+\beta}})
\|\Delta u\|_{L^{2}}\quad (0<\beta\leq 1) \\
&\leq  C \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\beta}}
 (\|\nabla u\|_{L^{2}}^{1-\beta}\|\Delta u\|_{L^{2}}^{\beta})\|\Delta u\|_{L^{2}} \\
&\leq  \frac{1}{4}\|\Delta u\|_{L^{2}}^{2}+C\|u\|_{\dot{B}_{p,\infty}
 ^{\frac{3}{p}-\beta}}^{\frac{2}{1-\beta}} \|\nabla u\|_{L^{2}}^{2} \\
&= \frac{1}{4}\|\Delta u\|_{L^{2}}^{2}+C\|u\|_{\dot{B}_{p, \infty}
^{\frac{3}{p}+\frac{2}{q}-1}}^{q} \|\nabla u\|_{L^{2}}^{2} 
\quad \Big(q=\frac{2}{1-\beta}\in (2, \infty]\Big),
\end{aligned}
\end{equation}
where we have used
\begin{gather*}
\|u\|_{\dot{B}_{2,2}^{1+\beta}}\thickapprox \|u\|_{\dot{H}^{1+\beta}}
\leq C\|\nabla u\|_{L^{2}}^{1-\beta}\|\Delta u\|_{L^{2}}^{\beta},\quad 
\text{for  } 0\leq\beta\leq1; \\
\|u\|_{\dot{B}_{\infty,\infty}^{-\beta}}
\leq C\|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\beta}},\quad \text{for }
 1\leq p\leq \infty.
\end{gather*}

\noindent\textbf{Case 2: $1<q\leq2$.}
Now we recall the following interpolation inequality due to
Meyer-Gerard-Oru \cite{MGO} (see also \cite[Theorem 2.42]{BCD})
\begin{equation}\label{t205}
\|f\|_{L^{m}}\leq C\|\Lambda^{s}f\|_{L^{2}}^{\frac{2}{m}}
\| f\|_{\dot{B}_{\infty, \infty}^{-\alpha}}^{\frac{m-2}{m}},
\end{equation}
for any $ f\in \dot{H}^{s}\cap \dot{B}_{\infty, \infty}^{-\alpha}$,
 $s=\alpha(\frac{m}{2}-1)>0$ and $2<m<\infty$.
Hence,  by using  \eqref{t205} with $m=4$, we obtain
\begin{equation}\label{BE0600}
\begin{aligned}
&\int_{\mathbb{R}^{3}}{(u\cdot\nabla u)\cdot\Delta
u\,dx}\\
&= -\int_{\mathbb{R}^{3}}{(\partial_{k}u\cdot\nabla u)\cdot\partial_{k}
u\,dx}\quad (\nabla\cdot u=0) \\
&\leq  C\|\nabla u\|_{L^{2}}\|\nabla u\|_{L^{4}}^{2} \\
&\leq  C\|\nabla u\|_{L^{2}} \|\Lambda^{\alpha} \nabla u\|_{L^{2}}
 \|\nabla u\|_{\dot{B}_{\infty, \infty}^{-\alpha}}\quad (0<\alpha\leq1) \\
&\leq  C\|\nabla u\|_{L^{2}} (\|\nabla u\|_{L^{2}}^{1-\alpha}
 \|\Delta u\|_{L^{2}}^{\alpha})\|\nabla u\|_{\dot{B}_{\infty, \infty}^{-\alpha}} \\
&\leq  C\|\nabla u\|_{L^{2}}^{2-\alpha}\|\Delta u\|_{L^{2}}^{\alpha}
 \| u\|_{\dot{B}_{p, \infty}^{1-\alpha+\frac{3}{p}}} \\
&\leq  \frac{1}{4}\|\Delta u\|_{L^{2}}^{2}+C\| u\|_{\dot{B}_{p, \infty}
 ^{1-\alpha+\frac{3}{p}}}^{\frac{2}{2-\alpha}} \|\nabla u\|_{L^{2}}^{2} \\
&= \frac{1}{4}\|\Delta u\|_{L^{2}}^{2}+C\|u\|_{\dot{B}_{p, \infty}
 ^{\frac{3}{p}+\frac{2}{q}-1}}^{q} \|\nabla u\|_{L^{2}}^{2} 
\quad \Big(q=\frac{2}{2-\alpha}\in (1, 2]\Big),
\end{aligned}
\end{equation}
where the following fact has been applied
$$
\|\nabla u\|_{\dot{B}_{\infty, \infty}^{-\alpha}}
\leq C \| u\|_{\dot{B}_{p, \infty}^{1-\alpha+\frac{3}{p}}}.
$$
Substituting \eqref{BE05} and \eqref{BE06} (or \eqref{BE0600}) into \eqref{BE04},
 we arrive at
\begin{equation}\label{BE07}
\frac{d}{dt}\|\nabla u(t)\|_{L^{2}}^{2}+\|\Delta u\|_{L^{2}}^{2}
\leq C\| \theta\|_{L^{2}}^{2}+C\|u\|_{\dot{B}_{p, \infty}
^{\frac{3}{p}+\frac{2}{q}-1}}^{q} \|\nabla u\|_{L^{2}}^{2}.
\end{equation}
It thus follows from the Gronwall inequality
that
\begin{equation}\label{BE08}
\begin{aligned}
&\|\nabla u(t)\|_{L^{2}}^{2}+\int_{0}^{t}{\|\Delta
u(\tau)\|_{L^{2}}^{2}\,d\tau} \\
&\leq  (\|\nabla u_{0}\|_{L^{2}}^{2}+1){\rm
exp}\Big[C\int_{0}^{T}{\big(\|\theta(\tau)\|_{L^{2}}^{2}+\|u(\tau)\|_{\dot{B}_{p, \infty}^{\frac{3}{p}+\frac{2}{q}-1}}^{q} \big) \,d\tau}\Big]-1 \\
&\leq  C<\infty.
\end{aligned}
\end{equation}
The H\"older inequality and Gagliardo-Nirenberg inequality lead to
\begin{equation}\label{BE09}
\begin{aligned}
\|u\cdot \nabla u\|_{L^{3+\delta}}
&\leq C\|u\|_{L^{\frac{18+6\delta}{3-\delta}}}\|\nabla u\|_{L^{6}} \\
&\leq C\|u\|_{L^{2}}^{\frac{3}{6+2\delta}}\|\Delta u\|_{L^{2}}
 ^{\frac{3+2\delta}{6+2\delta}} \|\Delta u\|_{L^{2}} \\
&\leq C\|u\|_{L^{2}}^{\frac{3}{6+2\delta}}\|\Delta u\|_{L^{2}}
 ^{\frac{9+4\delta}{6+2\delta}},
\end{aligned}
\end{equation}
where $0<\delta<3$.
It follows from the bounds \eqref{BE01}, \eqref{BE03} and \eqref{BE08} that
\begin{gather}\label{BE10}
u\cdot \nabla u\in L^{\frac{2(6+2\delta)}{9+4\delta}}
\big(0,T; L^{3+\delta}(\mathbb{R}^{3})\big), \\
\label{BE11}
\theta\in L^{\frac{2(6+2\delta)}{9+4\delta}}
\big(0,T; L^{3+\delta}(\mathbb{R}^{3})\big).
\end{gather}
Recall the first equation of \eqref{Bouss}, namely
\begin{equation}\label{BE12}
\partial_{t}u-\Delta u+\nabla P=f:=-(u \cdot \nabla) u+\theta e_{3}.
\end{equation}
As a consequence of \eqref{BE10} and \eqref{BE11}, it leads to
\begin{equation}\label{BE13}
f\in L^{\frac{2(6+2\delta)}{9+4\delta}}\big(0,T; L^{3+\delta}(\mathbb{R}^{3})\big).
\end{equation}
According to the divergence-free condition, we can rewrite equation \eqref{BE12} as
\begin{equation}\label{BE14}
\partial_{t}u-\Delta u=(I+\mathcal{R}_{i
}\mathcal{R}_{j})f,
\end{equation}
where the singular operator $\mathcal{R}_{i}$ is the classical Riesz operator,
 more precisely 
$$ 
\mathcal{R}_{i}=\frac{{\partial_{x_{i}}}}{\sqrt{-\Delta}}.
$$
Now we recall the following Maximal $L_t^{q}L_x^{p}$
regularity for the heat kernel (see \cite{LPG})

\begin{proposition}\label{Pr1}
The operator $A$ defined by
$$
Af(x,t):=\int_{0}^{t}{e^{(t-s)\Delta}\Delta f(s,x)\,ds}
$$
is bounded from $L^{p}(0,T; L^{q}(\mathbb{R}^{n}))$ to $L^{p}(0,T;
L^{q}(\mathbb{R}^{n}))$ for very $(p,q)\in (1,\infty)\times
(1,\infty)$ and $T\in(0,\infty]$.
\end{proposition}

Applying operator $\Delta$ to  \eqref{BE14}, we have that
the velocity $\Delta u$ can be solved by the Duhamel's Principle,
\begin{equation}\label{BE15}
\Delta u(x,t)=e^{t\Delta}\Delta u_{0}(x)
+\int_{0}^{t}{e^{(t-s)\Delta}\Delta(I+\mathcal{R}_{i}\mathcal{R}_{j})f(x,s)\,ds}.
\end{equation}
By Proposition \ref{Pr1}, one  concludes from \eqref{BE15} that
\begin{equation}\label{BE16}
\begin{aligned}
&\|\Delta u\|_{L_{T}^{\frac{12+4\delta}{9+4\delta}}L_{x}^{3+\delta}} \\
&\leq \|e^{t\Delta}\Delta u_{0}\|_{L_{T}^{\frac{12+4\delta}{9+4\delta}}
L_{x}^{3+\delta}}+
\big\|\int_{0}^{t}{e^{(t-s)\Delta}\Delta(I+\mathcal{R}_{i
}\mathcal{R}_{j})f(x,s)\,ds}\big\|_{L_{T}^{\frac{12+4\delta}{9+4\delta}}
L_{x}^{3+\delta}} \\
&\leq  C\|H(t,x)\|_{L_{T}^{\frac{12+4\delta}{9+4\delta}}L_{x}^{1}}\|\Delta
u_{0}\|_{L_{x}^{3+\delta}}+C \big\|(I+\mathcal{R}_{i
}\mathcal{R}_{j})f(x,s)\big\|_{L_{T}^{\frac{12+4\delta}{9+4\delta}}
 L_{x}^{3+\delta}} \\
&\leq  C(T)\| u_{0}\|_{H^{3}}+C\|f\|_{L_{T}^{\frac{12+4\delta}{9+4\delta}}
L_{x}^{3+\delta}} \\
&\leq  C<\infty,
\end{aligned}
\end{equation}
where we have used the boundedness of the Calderon-Zygmund operator
between the $L^{p}$ ($1<p<\infty$) space and 
$H^{3}(\mathbb{R}^{3})\hookrightarrow L^{3+\delta}(\mathbb{R}^{3})$ for $0<\delta<3$.

Now we deduce that from the bounds \eqref{BE03}, \eqref{BE08} and \eqref{BE16} that
\begin{equation}\label{BE17}
 u\in L^{\frac{12+4\delta}{9+4\delta}}\big(0,T; W^{2, 3+\delta}(\mathbb{R}^{3})\big).
\end{equation}
Thus, we have
\begin{equation}\label{BE18}
\nabla u\in L^{1}\big(0,T; L^{\infty}(\mathbb{R}^{3})\big).
\end{equation}
The above key estimate \eqref{BE18} as well as the local well-posedness result 
ensures implies that the local smooth solution pair $(u, \theta)$ 
can be extended beyond time $T$.
This completes the proof.
\end{proof}


\section{Proof of Theorem \ref{Th2}}

\begin{proof}[Proof of  Theorem \ref{Th2}] 
The proof  is largely the same as  Theorem
\ref{Th1} with only some modifications, thus we only say some words.

Testing $\eqref{Benard}_{1}$ and $\eqref{Benard}_{2}$ by $u$ and $\theta$, 
respectively, adding them up, we obtain
\begin{equation}\label{BEN1}
\frac{1}{2}\frac{d}{dt}(\|u(t)\|_{L^{2}}^{2}+\|\theta(t)\|_{L^{2}}^{2})+ \|\nabla u\|_{L^{2}}^{2}\leq 2\|u\|_{L^{2}}\|\theta\|_{L^{2}},
\end{equation}
which together with Gronwall inequality yields 
\begin{equation}\label{BEN2}
\| u(t)\|_{L^{2}}^{2}+\| \theta(t)\|_{L^{2}}^{2}
+\int_{0}^{t}{\|\nabla u (\tau)\|_{L^{2}}^{2}\,d\tau}\leq C<\infty.
\end{equation}
The Sobolev interpolation together with \eqref{BEN2} gives
\begin{equation}\label{BEN3} 
u\in L^{\frac{4p}{3(p-2)}}\big(0,T; L^{p}(\mathbb{R}^{3})\big),\quad 
2\leq p\leq 6.
\end{equation}
Recalling the second equation of \eqref{Benard}
$$
\partial_{t}\theta+(u \cdot \nabla) \theta=u_{3},
$$
it is easy to see that
\begin{equation}\label{BEN4} 
\theta\in L^{\frac{4p}{3(p-2)}}\big(0,T; L^{p}(\mathbb{R}^{3})\big),\quad 
2\leq p\leq 6.
\end{equation}
Thus,
$$
\theta\in L^{\frac{2(6+2\delta)}{9+4\delta}}
\big(0,T; L^{3+\delta}(\mathbb{R}^{3})\big),
$$
where $\delta$ is the stated in previous section. 
Thus, we can obtain the desired result immediately with only some
modifications correspondingly.
\end{proof}

\subsection*{Acknowledgements}
The author would like to thank an anonymous referee for the careful reading 
and helpful suggestions.



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