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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 99, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/99\hfil Global regularity criteria]
{Global regularity criteria for the $n$-dimensional Boussinesq
equations with fractional dissipation}

\author[Z. Zhang \hfil EJDE-2016/99\hfilneg]
{Zujin Zhang}

\address{Zujin Zhang \newline
School of Mathematics and Computer Sciences,
Gannan Normal University,
Ganzhou 341000, Jiangxi, China}
\email{zhangzujin361@163.com, phone (86) 07978393663}


\thanks{Submitted February 23, 2016. Published April 19, 2016.}
\subjclass[2010]{35B65, 35Q30, 76D03}
\keywords{Regularity criteria; Generalized Boussinesq equations;
\hfill\break\indent fractional diffusion}

\begin{abstract}
 We consider the $n$-dimensional Boussinesq equations with fractional
 dissipation, and establish a regularity criterion in terms of the
 velocity gradient in Besov spaces with negative order.
\end{abstract}

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

\section{Introduction}\label{sect:intro}

In this article, we study the $n$-dimensional Boussinesq equations
with fractional dissipation,
\begin{equation}\label{GB}
\begin{gathered}
\partial_t\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u} 
+ \varLambda^{2\alpha}\mathbf{u}
 +\nabla \varPi=\vartheta \mathbf{e}_n,\\
\partial_t\vartheta+(\mathbf{u}\cdot\nabla)\vartheta=0,\\
\nabla\cdot\mathbf{u}=0,\\
\mathbf{u}(0)=\mathbf{u}_0,\quad \vartheta(0)=\vartheta_0,
\end{gathered}
\end{equation}
where $\mathbf{u}: \mathbb{R}^+\times \mathbb{R}^n\to \mathbb{R}^n$ 
is the velocity field;
$\vartheta: \mathbb{R}^+\times \mathbb{R}^n\to \mathbb{R}$ is a scalar 
function representing
the temperature in the content of thermal convection
(see \cite{m1}) and the density in the modeling of geophysical
fluids (see \cite{p1}); $\varPi$ is the the fluid pressure;
$\mathbf{e}_n$ is the unit vector in the $x_n$ direction; and
 $\varLambda:=(-\Delta)^\frac{1}{2}$, $\alpha\geq 0$ is a real number.

When $\alpha=1$, Equation \eqref{GB} reduces to the classical Boussinesq equations,
which are frequently used in the atmospheric sciences and oceanographic
turbulence where rotation and stratification are important
(see \cite{m1,p1}).
If $\vartheta=0$, then \eqref{GB} becomes the generalized Navier-Stokes equation,
which was first considered by Lions \cite{l1}, where he showed
the global regularity once $\alpha\geq \frac{1}{2}+\frac{n}{4}$.
One may refer the reader to \cite{k2,t1} for recent
advances.  Xiang-Yan \cite{x2},
Yamazaki \cite{y1} and Ye \cite{y2} were able
to extend Lions's result to system \eqref{GB}, where there is no diffusion
in the $\vartheta$ equation. And it remains an open problem for the global-in-time
smooth for \eqref{GB} with $0<\alpha<\frac{1}{2}+\frac{n}{4}$.
The purpose of the present paper is to establish a blow-up criterion as follows.

\begin{theorem}\label{thm1}
Let $0<\alpha<\frac{1}{2}+\frac{n}{4}$, $(\mathbf{u}_0,\vartheta_0)\in H^s(\mathbb{R}^n)$ with
$s>1+\frac{n}{2}$ and $\nabla\cdot\mathbf{u}_0=0$. Assume that $(\mathbf{u},\vartheta)$ be the
smooth local unique solution pair to \eqref{GB} with initial data
$(\mathbf{u}_0,\vartheta_0)$. If additionally,
\begin{equation} \label{thm:reg}
\nabla\mathbf{u}\in L^\frac{2\alpha}{2\alpha-\gamma}(0,T;\dot B^{-\gamma}_{\infty,\infty}(\mathbb{R}^n))
\end{equation}
for some $0<\gamma<2\alpha$, then the solution $(\mathbf{u},\vartheta)$ can be extended
smoothly beyond $T$.
\end{theorem}

Here, $\dot B^{-\gamma}_{\infty,\infty}(\mathbb{R}^n)$ is the homogeneous Besov
space with negative order, which contains classical Lebesgue space
$L^\frac{n}{\gamma}(\mathbb{R}^n)$, see
 \cite[Chapter 2]{b1}.
In the proof of Theorem \ref{thm1} in Section \ref{sect:proof}, we shall
frequently use the following refined Gagliardo-Nirenberg inequality.

\begin{lemma}[{\cite[Theorem 2.42]{b1}}]
Let $2<q<\infty$ and $\gamma$ be a positive real number.
Then a constant $C$ exists such that
\begin{equation} \label{lem:inter:ineq}
\|f\|_{L^q}\leq C\|f\|_{\dot B^{-\gamma}_{\infty,\infty}}^{1-\frac{2}{q}}
\|f\|_{\dot H^{\gamma(\frac{q}{2}-1)}}^{2/q}.
\end{equation}
\end{lemma}


\begin{remark} \rm
Our result extends that of Kozono-Shimada \cite{k3}.
Indeed, the Navier-Stokes equations corresponds to \eqref{GB} with
$\vartheta=0$ and $\alpha=1$.
\end{remark}

\begin{remark} \rm
In \cite{g2} (see also the end-point smallness
condition in \cite{g1}),
Geng-Fan proved a regularity criterion
\begin{equation} \label{Geng-Fan}
\mathbf{u}\in L^\frac{2}{1-r}(0,T;\dot B^{-r}_{\infty,\infty}(\mathbb{R}^3))
\quad (-1<r<1,\ r\neq 0)
\end{equation}
for system \eqref{GB} with $\alpha=1$ and $n=3$. Thus our result generalizes
\eqref{Geng-Fan} also, in view of the fact that
\[
C_1\|\nabla f\|_{\dot B^{-1-r}_{\infty,\infty}} \leq
\|f\|_{\dot B^{-r}_{\infty,\infty}}
\leq C_2 \|\nabla f\|_{\dot B^{-1-r}_{\infty,\infty}}.
\]
Moreover, our result \eqref{thm:reg} is valid for \eqref{GB}
with arbitrarily large $n$ and arbitrarily small $\alpha$.
\end{remark}

Interested readers are referred to \cite{x1}
for blow-up criterion for \eqref{GB} without diffusion in the $\mathbf{u}$ equation.


\section{Proof of Theorem \ref{thm1}}\label{sect:proof}

 It is not difficult to prove that there exists a $T_0>0$ and a
unique smooth solution $(\mathbf{u},\vartheta)$ to \eqref{GB} on $[0,T_0]$.
We only need to establish the a priori estimates. Therefore, in the
following calculations, we assume that the solution $(\mathbf{u},\vartheta)$
is sufficiently smooth on $[0,T]$.

First, taking the inner product of \eqref{GB}$_1$ and \eqref{GB}$_2$ with
$\mathbf{u},\vartheta$ in $L^2(\mathbb{R}^n)$ respectively, we obtain
\[
\frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d} t}\|(\mathbf{u},\vartheta)\|_{L^2}^2
+\|\varLambda^\alpha\mathbf{u}\|_{L^2}^2
=\int_{\mathbb{R}^n} \vartheta\mathbf{e}_n\cdot\mathbf{u}\,\mathrm{d} x
\leq \frac{1}{2}\|(\mathbf{u},\vartheta)\|_{L^2}^2.
\]
Applying Gronwall inequality, we deduce
\begin{equation} \label{zero-order}
\|(\mathbf{u},\vartheta)\|_{L^\infty(0,t;L^2(\mathbb{R}^n))}
+\|\varLambda^\alpha\mathbf{u}\|_{L^2(0,t;L^2(\mathbb{R}^n))}\leq C.
\end{equation}

For $k>0$, applying $\varLambda^k$ to $\eqref{GB}_1$, and testing
the resulting equations by $\varLambda^k \mathbf{u}$ respectively, we obtain
 \begin{equation} \label{I}
 \begin{aligned}
&\frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d} t}
 \|\varLambda^k\mathbf{u}\|_{L^2}^2
 +\|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^2\\
&= -\int_{\mathbb{R}^n} \varLambda^k[(\mathbf{u}\cdot\nabla)\mathbf{u}]\cdot \varLambda^k\mathbf{u}\,\mathrm{d} x
 +\int_{\mathbb{R}^n}\varLambda^k(\vartheta \mathbf{e}_n)\cdot \varLambda^k \mathbf{u}\,\mathrm{d} x\\
&= -\int_{\mathbb{R}^3} \big\{\varLambda^k[(\mathbf{u}\cdot\nabla)\mathbf{u}]-(\mathbf{u}\cdot\nabla)(\varLambda^k\mathbf{u})\big\}
 \cdot \varLambda^k\mathbf{u}\,\mathrm{d} x
 +\int_{\mathbb{R}^n}\varLambda^k(\vartheta \mathbf{e}_n)\cdot \varLambda^k \mathbf{u}\,\mathrm{d} x\\
&\equiv I_1^k+I_2^k.
 \end{aligned}
 \end{equation}
We may use the following commutator estimates of Kato-Ponce
 \cite{k1}:
 \begin{equation} \label{Kao-Ponce}
\|\varLambda^k(fg)-f\varLambda^kg\|_{L^p}
\leq C \big[
 \|\nabla f\|_{L^{p_1}}
 \|\varLambda^{k-1}g\|_{L^{p_2}}
 +\|\varLambda^kf\|_{L^{p_3}}
 \|g\|_{L^{p_4}}  \big]
\end{equation}
 with
\[
1<p,p_2,p_3<\infty,\quad
 1\leq p_1,p_4\leq\infty,\quad
 \frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}
 =\frac{1}{p_3}+\frac{1}{p_4}
\]
 to bound $I_1^k$ as
\begin{equation} \label{I1}
 \begin{aligned}
 I_1^k
&\leq C \|\varLambda^k[(\mathbf{u}\cdot\nabla)\mathbf{u}]
-(\mathbf{u}\cdot\nabla)(\varLambda^k\mathbf{u})\|_{L^\frac{4(k+\gamma+\alpha-1)}{2k+3\gamma+2\alpha-2}}
 \|\varLambda^k \mathbf{u}\|_{L^\frac{4(k+\gamma+\alpha-1)}{2k+\gamma+2\alpha-2}}\\
&\leq C\|\nabla\mathbf{u}\|_{L^\frac{2(k+\gamma+\alpha-1)}{\gamma}}
 \|\varLambda^k \mathbf{u}\|_{L^\frac{4(k+\gamma+\alpha-1)}{2k+\gamma+2\alpha-2}}\cdot
 \|\varLambda^k \mathbf{u}\|_{L^\frac{4(k+\gamma+\alpha-1)}{2k+\gamma+2\alpha-2}}\\
&\leq C\|\nabla\mathbf{u}\|_{\dot B^{-\gamma}_{\infty,\infty}}^\frac{k+\alpha-1}{k+\gamma+\alpha-1}
 \|\nabla\mathbf{u}\|_{\dot H^{k+\alpha-1}}^\frac{\gamma}{k+\gamma+\alpha-1}
 \Big( \|\varLambda^k\mathbf{u}\|_{\dot B^{-(k-1+\gamma)}_{\infty,\infty}}
^\frac{\gamma}{2(k+\gamma+\alpha-1)}
 \|\varLambda^k\mathbf{u}\|_{\dot H^\frac{\gamma(k+\gamma-1)}{2k+\gamma+2\alpha-2}}
^\frac{2k+\gamma+2\alpha-2}{2(k+\gamma+\alpha-1)} \Big)^2\\
&\leq C\|\nabla\mathbf{u}\|_{\dot B^{-\gamma}_{\infty,\infty}}^\frac{k+\alpha-1}{k+\gamma+\alpha-1}
 \|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^\frac{\gamma}{k+\gamma+\alpha-1}
 \|\nabla\mathbf{u}\|_{\dot B^{-\gamma}_{\infty,\infty}}^\frac{\gamma}{k+\gamma+\alpha-1}
 \|\varLambda^k \mathbf{u}\|_{\dot H^\frac{\gamma(k+\gamma-1)}{2k+\gamma+2\alpha-2}}
^\frac{2k+\gamma+2\alpha-2}{k+\gamma+\alpha-1}\\
&\leq C\|\nabla\mathbf{u}\|_{\dot B^{-\gamma}_{\infty,\infty}}
 \|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^\frac{\gamma}{k+\gamma+\alpha-1} \\
&\quad\times  \Big( \|\varLambda^k\mathbf{u}\|_{L^2}^{1-\frac{\gamma(k+\gamma-1)}{\alpha(2k+\gamma+2\alpha-2)}}
 \|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^\frac{\gamma(k+\gamma-1)}{\alpha(2k+\gamma+2\alpha-2)}
 \Big)^\frac{2k+\gamma+2\alpha-2}{k+\gamma+\alpha-1}\\
&\leq C\|\nabla\mathbf{u}\|_{\dot B^{-\gamma}_{\infty,\infty}}
 \|\varLambda^k\mathbf{u}\|_{L^2}^\frac{2\alpha-\gamma}{\alpha}
 \|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^\frac{\gamma}{\alpha}\\
&\leq C\|\nabla\mathbf{u}\|_{\dot B^{-\gamma}_{\infty,\infty}}^\frac{2\alpha}{2\alpha-\gamma}
 \|\varLambda^k\mathbf{u}\|_{L^2}^2
 +\frac{1}{2}  \|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^2.
\end{aligned}
 \end{equation}
 Substituting \eqref{I1} in \eqref{I}, we find
 \begin{equation} \label{I_second}
 \frac{\,\mathrm{d}}{\,\mathrm{d} t}
 \|\varLambda^k\mathbf{u}\|_{L^2}^2
 +\|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^2
 \leq C\|\nabla\mathbf{u}\|_{\dot B^{-\gamma}_{\infty,\infty}}^\frac{2\alpha}{2\alpha-\gamma}
 \|\varLambda^k\mathbf{u}\|_{L^2}^2
 +2I_2^k.
 \end{equation}
 Now, we treat $2I_2^k$ step by step. If $0<k\leq \alpha$, then
 \begin{equation}\label{I2:1}
 \begin{aligned}
 2I_2^k
&=2\int_{\mathbb{R}^n}\vartheta\mathbf{e}_n\cdot \varLambda^{2k}\mathbf{u}\,\mathrm{d} x\\
 &\leq 2\|\vartheta\|_{L^2}\|\varLambda^{2k}\mathbf{u}\|_{L^2}\\
 &\leq C\|\vartheta\|_{L^2}\big(\|\mathbf{u}\|_{L^2}
+\|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}\big)\quad\big(H^{k+\alpha}(\mathbb{R}^n)
\subset \dot H^{2k}(\mathbb{R}^n)\big)\\
&\leq C+\frac{1}{2}\|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2}^2\quad
(\text{by \eqref{zero-order}}).
 \end{aligned}
 \end{equation}
Substituting \eqref{I2:1} into \eqref{I_second}, we  apply Gronwall
inequality to deduce
 \begin{equation} \label{goal:zero}
 \|\varLambda^k(\mathbf{u},\vartheta)\|_{L^\infty(0,t;L^2(\mathbb{R}^n))}
 +\|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2(0,t;L^2(\mathbb{R}^n))}\leq C\quad (0<k\leq \alpha).
 \end{equation}
 Suppose we have already the statement for some $0\leq l\in\mathbb{N}$,
 \begin{equation} \label{ass}
 \|\varLambda^k(\mathbf{u},\vartheta)\|_{L^\infty(0,t;L^2(\mathbb{R}^n))}
 +\|\varLambda^{k+\alpha}\mathbf{u}\|_{L^2(0,t;L^2(\mathbb{R}^n))}
\leq C\quad (\forall\ l\alpha<k\leq (l+1)\alpha),
 \end{equation}
we wish to deduce higher-order estimate
 \begin{equation} \label{goal}
 \|\varLambda^{k+\alpha}(\mathbf{u},\vartheta)\|_{L^\infty(0,t;L^2(\mathbb{R}^n))}
 +\|\varLambda^{k+2\alpha}\mathbf{u}\|_{L^2(0,t;L^2(\mathbb{R}^n))}\leq C.
 \end{equation}
 Indeed, as long as \eqref{ass} holds, we may dominate $2I_2^{k+\alpha}$ as
 \begin{equation} \label{I2:general k}
 \begin{aligned}
 2I_2^{k+\alpha}
&=2\int_{\mathbb{R}^n} \varLambda^{k+\alpha}(\vartheta \mathbf{e}_n)\cdot \varLambda^{k+\alpha}\mathbf{u}\,\mathrm{d} x\\
 &=2\int_{\mathbb{R}^n}\varLambda^k(\vartheta \mathbf{e}_n)\cdot \varLambda^{k+2\alpha}\mathbf{u}\,\mathrm{d} x\\
 &\leq 2\|\varLambda^k\vartheta\|_{L^2}\|\varLambda^{k+2\alpha}\mathbf{u}\|_{L^2}\\
 &\leq 2\|\varLambda^k\vartheta\|_{L^2}^2+\frac{1}{2}\|\varLambda^{k+2\alpha}\mathbf{u}\|_{L^2}^2.
 \end{aligned}
 \end{equation}
 Putting \eqref{I2:general k} into \eqref{I_second} with $k$ replaced
by $k+\alpha$, and using \eqref{ass}, we deduce \eqref{goal} as desired.

 Now prove that \eqref{goal:zero} and \eqref{ass} imply  \eqref{goal}, 
we see readily that
 \begin{equation} \label{goal:final}
 \|\varLambda^s\mathbf{u}\|_{L^\infty(0,t;L^2(\mathbb{R}^n))}
 +\|\varLambda^{s+\alpha}\mathbf{u}\|_{L^2(0,t;L^2(\mathbb{R}^n))}
 \leq C.
 \end{equation}
 With this good estimate of the velocity field, we are now in a position
to treat that of $\vartheta$. Applying $\varLambda^s$ to $\eqref{GB}_2$, and
testing the resultant equation by $\varLambda^s\vartheta$, we obtain
 \begin{equation} \label{vtt}
\begin{aligned}
&\frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d} t}\|\varLambda^s\vartheta\|_{L^2}^2\\
&=-\int_{\mathbb{R}^n} \varLambda^s[(\mathbf{u}\cdot\nabla)\vartheta]\cdot \varLambda^s \vartheta\,\mathrm{d} x\\
&=-\int_{\mathbb{R}^n} \{\varLambda^s[(\mathbf{u}\cdot\nabla)\vartheta]-(\mathbf{u}\cdot\nabla)\varLambda^s \vartheta\}\cdot \varLambda^s \vartheta\,\mathrm{d} x\\
&\leq C\Big( \|\nabla\mathbf{u}\|_{L^\infty}
 \|\varLambda^s \vartheta\|_{L^2}
 +\|\nabla\vartheta\|_{L^\infty}\|\varLambda^s \mathbf{u}\|_{L^2}
 \Big)\|\varLambda^s \vartheta\|_{L^2}\quad (\text{by \eqref{Kao-Ponce}})\\
&\leq C\Big(\|\mathbf{u}\|_{L^2}+\|\varLambda^s\mathbf{u}\|_{L^2}\Big)\|\varLambda^s\vartheta\|_{L^2}^2
 +\Big(\|\vartheta\|_{L^2}+\|\varLambda^s\vartheta\|_{L^2}\Big)\|\varLambda^s\mathbf{u}\|_{L^2}
\|\varLambda^s\vartheta\|_{L^2}\\
 &\quad(\text{by }H^s(\mathbb{R}^n)\subset W^{1,\infty}(\mathbb{R}^n))\\
&\leq C+C\|\varLambda^s\vartheta\|_{L^2}^2\quad(\text{by \eqref{zero-order} and
\eqref{goal:final}}).
\end{aligned}
 \end{equation}
 Applying Gronwall inequality, we obtain
\[
 \|\varLambda^s\vartheta\|_{L^\infty(0,t;L^2(\mathbb{R}^n))}\leq C.
\]
 With this estimate and \eqref{goal:final}, we complete
the proof.

\subsection*{Acknowledgements}
 This work is supported by the Natural Science Foundation of Jiangxi 
(grant no. 20151BAB201010), the National Natural Science Foundation of 
China (grant nos. 11501125, 11361004) and the Supporting the Development
 for Local Colleges and Universities Foundation of China
 -- Applied Mathematics Innovative Team Building.

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\end{document}