\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 223, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/223\hfil Osgood type regularity criterion]
{Osgood type regularity criterion for the 3D Newton-Boussinesq equation}

\author[Z. Zhang, S. Gala \hfil EJDE-2013/223\hfilneg]
{Zujin Zhang, Sadek Gala} 

\address{Zujin Zhang \newline
School of Mathematics and Computer Science,
Gannan Normal University, Ganzhou 341000, China}
\email{zhangzujin361@163.com}

\address{Sadek Gala  \newline
Department of Mathematics,
University of Mostaganem, Box 227, Mostaganem, 27007, Algeria}
\email{sadek.gala@gmail.com}

\thanks{Submitted June 26, 2013. Published October 11, 2013.}
\subjclass[2000]{35B65, 76B03, 76D03}
\keywords{Newton-Boussinesq equations; regularity criterion; Osgood type}

\begin{abstract}
 In this article, we show an Osgood type regularity criterion for
 the three-dimensional Newton-Boussinesq equations, which improves
 the recent results in \cite{g3}.
\end{abstract}

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

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

 In this article, we consider the  three-dimensional Newton-Boussinesq equation
\begin{equation} \label{NB}
 \begin{gathered}
 \boldsymbol{\omega}_t+(\mathbf{u}\cdot\nabla)\boldsymbol{\omega}
 -\Delta \boldsymbol{\omega}=\nabla\times (\theta \mathbf{e}_3),\\
 \theta_t+(\mathbf{u}\cdot\nabla)\theta-\Delta\theta=0,\\
 \nabla\cdot \mathbf{u}=0,\\
 \mathbf{u}(0)=\mathbf{u}_0,\quad \theta(0)=\theta_0,
\end{gathered}
\end{equation}
 where $\boldsymbol{\omega}=\nabla\times \mathbf{u}$, and $\mathbf{u}$ is the velocity field,
$\theta$ is the scalar temperature, while $\mathbf{u}_0$, $\theta_0$ are
the prescribed initial data with $\nabla\cdot\mathbf{u}_0=0$ in distributional sense.

System \eqref{NB} arises from the study of B\'enard flow \cite{c1}.
Guo \cite{g1,g2} investigated the two-dimensional (2D) periodic case
by using spectral methods and nonlinear Galerkin methods.
 Meanwhile, the existence and regularity of a global attractor for the 2D
 Newton-Boussinesq equations were obtained in \cite{f2}.
Consequently, it is desirable to consider the regularity criteria for
\eqref{NB}. Noticing that the convective term
$(\mathbf{u}\cdot\nabla)\mathbf{u}$, $(\mathbf{u}\cdot\nabla)\theta$ are the
same as that in the 3D Boussinesq equations
\begin{gather*}
 \mathbf{u}_t+(\mathbf{u}\cdot\nabla)\mathbf{u}
-\Delta\mathbf{u}+\nabla \pi=\theta \mathbf{e}_3,\\
 \theta_t+(\mathbf{u}\cdot\nabla)\theta-\Delta \theta=0,\\
 \nabla\cdot\mathbf{u}=0,\\
 \mathbf{u}(0)=\mathbf{u}_0,\quad \theta(0)=\theta_0,
\end{gather*}
we could prove many regularity conditions as that for the Boussinesq equations.

 For the 3D Boussinesq equations, Ishimura nad Morimoto \cite{i1}
showed that if
 \begin{equation}
 \nabla\mathbf{u}\in L^1(0,T;L^\infty(\mathbb{R}^3)),
 \end{equation}
then the solution is smooth on $(0,T)$. Fan and Zhou \cite{f1} established
the regularity of the solution provided that
\begin{equation}
 \boldsymbol{\omega}=\nabla\times\mathbf{u}\in L^1
\big(0,T;\dot B^0_{\infty,\infty}(\mathbb{R}^3)\big),
\end{equation}
where $B^0_{\infty,\infty}(\mathbb{R}^3)$ is the homogeneous Besov spaces which
will be introduced in Section \ref{sect:Pre}. The interested readers can
find more result in \cite{q1,q2} and references cited therein.

For the 3D Newton-Boussinesq equations \eqref{NB}, Guo and Gala \cite{g3}
obtained some regularity criteria in terms of Morrey spaces and Besov spaces.
One of them reads
 \begin{equation} \label{GS_12_CPAA}
 \boldsymbol{\omega}=\nabla\times\mathbf{u}\in L^1(0,T;\dot B^0_{\infty,\infty}(\mathbb{R}^3)).
 \end{equation}
A blow-up criterion for the 2D Newton-Boussinesq equations was established
in \cite{q3}.

As we know, Osgood type conditions play an important role in solving
uniqueness of solutions to the incompressible fluid equations.
Motivated by the recent result \cite{z1} for the 3D MHD equations
\begin{gather*}
 \mathbf{u}_t-\Delta \mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u}-(\mathbb{b}\cdot\nabla)\mathbb{b}
 +\nabla \pi=\mathbb{0},\\
 \mathbb{b}_t-\Delta\mathbb{b}+(\mathbf{u}\cdot\nabla)\mathbb{b}
 -(\mathbb{b}\cdot\nabla)\mathbf{u}=\mathbb{0},\\
 \nabla\cdot\mathbf{u}=\nabla\cdot\mathbb{b}=0,\\
 \mathbf{u}(0)=\mathbf{u}_0,\quad \mathbb{b}(0)=\mathbb{b}_0,
\end{gather*}
 we would like to improve \eqref{GS_12_CPAA}. Precisely, we will prove the
following theorem.

 \begin{theorem}\label{thm:main}
 Let $(\mathbf{u}_0,\theta_0)\in H^1(\mathbb{R}^3)$ with $\nabla\cdot\mathbf{u}_0=0$ in
distributional sense. Assume that
 \begin{equation} \label{thm:main:eq}
 \sup_{q\geq 2}\int_0^T \frac{\|\bar S_q\nabla\mathbf{u}\|_{L^\infty}}{q\ln q}\,\mathrm{d} \tau<\infty,
 \end{equation}
 with $\bar S_q=\sum_{l=-q}^q \dot \Delta_l$, $\dot \Delta_l$
being the Fourier localization operator. Then the solution pair
 $(\mathbf{u},\theta)$ to \eqref{NB} with initial data $(\mathbf{u}_0,\theta_0)$
is smooth on $[0,T]$.
 \end{theorem}

 \begin{remark} \label{rmk1} \rm
 Since
\[
 \frac{\|\bar S_q\nabla\mathbf{u}\|_{L^\infty}}{q\ln q}
 \leq\frac{1}{q\ln q}\sum_{l=-q}^q \|\dot\Delta_l\nabla\mathbf{u}\|_{L^\infty}
 \leq C \|\nabla\times\mathbf{u}\|_{\dot B^0_{\infty,\infty}},
\]
 we indeed improve the regularity condition \eqref{GS_12_CPAA}
established in \cite{g3}.
 \end{remark}

\begin{remark} \label{rmk2} \rm
 When $\theta=0$, \eqref{NB} reduces to the Navier-Stokes equations,
thus our result covers the case for the Navier-Stokes equations.
 \end{remark}

The rest of this article is organized as follows.
In Section \ref{sect:Pre}, we recall the definition of Besov spaces,
and some interpolation inequalities. Section \ref{sect:proof} is
 devoted to proving Theorem \ref{thm:main}.

\section{Preliminaries}\label{sect:Pre}

 Let $\mathscr{S}(\mathbb{R}^3)$ be the Schwartz class of rapidly decreasing functions.
For $f\in \mathscr{S}(\mathbb{R}^3)$, its Fourier transform $\mathscr{F} f=\hat f$ is defined by
\[
 \hat f(\xi)=\int_{\mathbb{R}^3} f(x)e^{-i x\cdot \xi}\,\mathrm{d} x.
\]
 Let us choose a nonnegative radial function $\varphi\in \mathscr{S}(\mathbb{R}^3)$ such that
\[
 0\leq \hat \varphi(\xi)\leq 1,\quad
\hat \varphi(\xi)=\begin{cases}
 1,&\text{if }|\xi|\leq 1,\\
 0,&\text{if }|\xi|\geq 2,
\end{cases}
\]
 and let
\[
 \psi(x)=\varphi(x)-2^{-3}\varphi(x/2),\quad
\varphi_j(x)=2^{3j}\varphi(2^jx),\quad
\psi_j(x)=2^{3j}\psi(2^jx),\quad j\in \mathbb{Z}.
\]
 For $j\in \mathbb{Z}$, the Littlewood-Paley projection operators $S_j$
and $\dot \Delta_j$ are, respectively, defined by
\[
 S_jf=\varphi_j*f,\quad \dot \Delta_jf=\psi_j*f.
\]
 Observe that $\dot \Delta_j =S_j-S_{j-1}$. Also, it is easy to
check that if $f\in L^2(\mathbb{R}^3)$, then
\[
 S_jf\to 0,\text{ as } j\to-\infty;\quad
 S_jf\to f,\text{ as } j\to +\infty,
\]
 in the $L^2$ sense. By telescoping the series,
we thus have the following Littlewood-Paley decomposition
 \begin{equation} \label{LPD}
 f=\sum_{j=-\infty}^{+\infty} \dot \Delta_jf,
 \end{equation}
for all $f\in L^2(\mathbb{R}^3)$, where the summation is the $L^2$ sense.
Note that
\[
 \dot\Delta_jf=\sum_{l=j-2}^{j+2}\dot\Delta_l\dot\Delta_jf
=\sum_{l=j-2}^{j+2}\psi_l*\psi_j*f,
\]
 then from Young's inequality, it readily follows that
\begin{equation} \label{Bern}
 \|\dot\Delta_jf\|_{L^q}\leq C2^{3j(1/p-1/q)}\|\dot \Delta_j f\|_{L^p},
\end{equation}
where $1\leq p\leq q\leq \infty$, and $C$ is an absolute constant independent
of $f$ and $j$.


Let $-\infty<s<\infty,\ 1\leq p,q\leq \infty$, the homogeneous Besov
space $\dot B^s_{p,q}$ is defined by the full-dyadic decomposition such as
\[
 \dot B^s_{p,q}=\{f\in \mathscr{Z}'(\mathbb{R}^3);\ \|f\|_{\dot B^s_{p,q}}<\infty\},
\]
where
\[
 \|f\|_{\dot B^s_{p,q}}
=\|\big\{2^{js}\|\dot\Delta_j f\|_{L^p}\big\}_{j=-\infty}^{+\infty}\|_{\ell^q},
\]
and $\mathscr{Z}'(\mathbb{R}^3)$ is the dual space of
\[
 \mathscr{Z}(\mathbb{R}^3)=\{f\in \mathscr{S}(\mathbb{R}^3);
 D^\alpha \hat f(0)=0,\ \forall\ \alpha\in \mathbb{N}^3\}.
\]
 Also, it is well-known that
 \begin{equation} \label{Besov_Sob}
 \dot H^s(\mathbb{R}^3)=\dot B^s_{2,2}(\mathbb{R}^3),\quad \forall s\in \mathbb{R}.
 \end{equation}
We refer the reader to \cite{t1} for more detailed properties.

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

 This section is devoted to proving Theorem \eqref{thm:main}.
Taking the inner products of \eqref{NB}$_1$, \eqref{NB}$_2$ with
$\boldsymbol{\omega}$, $-\Delta\theta$ in $L^2(\mathbb{R}^3)$ respectively, we have
\begin{gather*}
 \frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d} t}\|\boldsymbol{\omega}\|_{L^2}^2
 +\|\nabla\boldsymbol{\omega}\|_{L^2}^2
 =\int_{\mathbb{R}^3} \nabla\times (\theta \mathbf{e}_3)\cdot \boldsymbol{\omega}\,\mathrm{d} x,\\
 \frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d} t}\|\nabla\theta\|_{L^2}^2
 +\|\Delta \theta\|_{L^2}^2
 =\int_{\mathbb{R}^3}[(\mathbf{u}\cdot\nabla)\theta]\cdot\Delta \theta\,\mathrm{d} x.
\end{gather*}
 Adding together yields
 \begin{equation} \label{I}
\begin{aligned}
 &\frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d} t}
 [\|\boldsymbol{\omega}\|_{L^2}^2+\|\nabla\theta\|_{L^2}^2]
 +[\|\nabla\boldsymbol{\omega}\|_{L^2}^2+\|\Delta \theta\|_{L^2}^2 ] \\
 &=\int_{\mathbb{R}^3} \nabla\times (\theta \mathbf{e}_3)\cdot \boldsymbol{\omega}\,\mathrm{d} x
 +\int_{\mathbb{R}^3}[(\mathbf{u}\cdot\nabla)\theta]\cdot\Delta \theta\,\mathrm{d} x\\
 &\leq \|\nabla\theta\|_{L^2}\|\nabla\boldsymbol{\omega}\|_{L^2}
 -\int_{\mathbb{R}^3}[(\nabla\mathbf{u}\cdot\nabla)\theta]\cdot\nabla\theta\,\mathrm{d} x\\
 &\leq \frac{1}{2}\|\nabla\theta\|_{L^2}^2
 +\frac{1}{2}\|\nabla\boldsymbol{\omega}\|_{L^2}^2
 -\int_{\mathbb{R}^3}[(\nabla\mathbf{u}\cdot\nabla)\theta]\cdot\nabla\theta\,\mathrm{d} x.
\end{aligned}
 \end{equation}
We are now in a position to estimate
 \begin{equation} \label{I_notice}
 I=-\int_{\mathbb{R}^3}[(\nabla\mathbf{u}\cdot\nabla)\theta]\cdot\nabla\theta\,\mathrm{d} x.
 \end{equation}
 Applying the Littlewood-Paley decomposition as in \eqref{LPD},
 \begin{equation} \label{grad_LPD}
 \nabla\mathbf{u}=\sum_{l<-q}\dot \Delta\nabla\mathbf{u}
 +\sum_{l=-q}^q\dot \Delta\nabla\mathbf{u}
 +\sum_{l>q}\dot \Delta\nabla\mathbf{u},
 \end{equation}
where $q$ is a positive integer to be determined later on.
Substituting \eqref{grad_LPD} in $I$, we see that
 \begin{equation} \label{I123}
 \begin{aligned}
 I&\leq \sum_{l<-q}\int_{\mathbb{R}^3}\|\dot\Delta_l\nabla\mathbf{u}\|
 \cdot \|\nabla\theta\|^2\,\mathrm{d} x
  +\int_{\mathbb{R}^3}\big|\sum_{l=-q}^q \dot\Delta_l\nabla\mathbf{u}\big|
 \cdot \|\nabla\theta\|^2\,\mathrm{d} x\\
&\quad +\sum_{l>q} \int_{\mathbb{R}^3}
 \|\dot\Delta_l\nabla\mathbf{u}\|\cdot
 \|\nabla\theta\|^2\,\mathrm{d} x\\
 &\equiv I_1+I_2+I_3.
\end{aligned}
 \end{equation}
 For $I_1$, we have
 \begin{equation} \label{J1}
\begin{aligned}
 I_1&\leq \sum_{l<-q} \|\dot\Delta_l\nabla\mathbf{u}\|_{L^\infty}  \|\nabla\theta\|_{L^2}^2\\
 &\leq C\sum_{l<-q} 2^{3l/2}  \|\dot\Delta_l\nabla\mathbf{u}\|_{L^2}
 \|\nabla\theta\|_{L^2}^2 \quad\text{(by \eqref{Bern})}
\\
 &\leq C\Big(\sum_{l<-q}2^{\frac{3l}{2}\cdot 2}\Big)^{1/2}
 \cdot \Big(\sum_{l<-q}\|\dot \Delta_l\nabla\mathbf{u}\|_{L^2}^2\Big)^{1/2}
 \|\nabla\theta\|_{L^2}^2
\\
 &\leq C 2^{-3q/2}
 |\nabla\mathbf{u}|_{L^2}\|\nabla\theta\|_{L^2}^2\quad\text{(by \eqref{Besov_Sob})}\\
 &=[C2^{-q/2}\|\nabla\theta\|_{L^2}]^3.
\end{aligned}
 \end{equation}
 For $I_2$, we have
 \begin{equation} \label{J2}
 I_2=\int_{\mathbb{R}^3}|\bar S_q \nabla\mathbf{u}|\cdot |\nabla\theta|^2\,\mathrm{d} x
 \leq \|\bar S_q\nabla\mathbf{u}\|_{L^\infty}
 \|\nabla\theta\|_{L^2}^2.
 \end{equation}
 Finally, for $I_3$, we have
 \begin{equation} \label{J3}
\begin{aligned}
 I_3&\leq \sum_{l>q} \|\Delta_l\nabla\mathbf{u}\|_{L^3}
 \|\nabla\theta\|_{L^3}^2\\
 &\leq C\sum_{l>q}2^{1/2}
 \|\Delta_l\nabla\mathbf{u}\|_{L^2}
 \|\nabla\theta\|_{L^2}
 \|\Delta\theta\|_{L^2}\\
 &\quad\text{by \eqref{Bern} and Gagliardo-Nireberg inequality}\\
 &\leq C\Big(\sum_{l>q} 2^{-\frac{l}{2}\cdot 2}\Big)^{1/2}
 \cdot \Big(\sum_{l>q}2^{l\cdot 2}\|\dot \Delta_l\nabla\mathbf{u}\|_{L^2}^2\Big)^{1/2}
 \|\nabla\theta\|_{L^2}\|\Delta\theta}_{L^2\|\\
 &\leq [C2^{-q/2}\|\nabla\theta\|_{L^2}]
 [\|\nabla\boldsymbol{\omega}\|_{L^2}^2
 +\|\Delta\theta\|_{L^2}^2]\quad\text{(by \eqref{Besov_Sob})}.
\end{aligned}
 \end{equation}
 Gathering \eqref{J1}, \eqref{J2} and \eqref{J3} together, and plugging
them into \eqref{J}, we deduce
 \begin{equation} \label{J}
 I  \leq [C2^{-q/2}\|\nabla\theta\|_{L^2}]^3
 +\|\bar S_q\nabla\mathbf{u}\|_{L^\infty}\|\nabla\theta\|_{L^2}^2
 +[C2^{-q/2}\|\nabla\theta\|_{L^2}]\cdot
 [\|\nabla\boldsymbol{\omega}\|_{L^2}^2
 +\|\Delta\theta\|_{L^2}^2].
 \end{equation}
 Substituting \eqref{J} into \eqref{I}, we find
 \begin{equation} \label{Gronwall}
\begin{aligned}
 &\quad \frac{\,\mathrm{d}}{\,\mathrm{d} t}
 [\|\boldsymbol{\omega}\|_{L^2}^2+\|\nabla\theta\|_{L^2}^2]
 +[\|\nabla\boldsymbol{\omega}\|_{L^2}^2+\|\Delta \theta\|_{L^2}^2]\\
 &\leq \|\nabla\theta\|_{L^2}^2
 +[C2^{-q/2}\|\nabla\theta\|_{L^2}]^3\\
 &\quad
 +\frac{\|\bar S_q\nabla\mathbf{u}\|_{L^\infty}}{q\ln q}\cdot q\ln q\|\nabla\theta\|_{L^2}^2
 +[C2^{-q/2}\|\nabla\theta\|_{L^2}]
 [\|\nabla\boldsymbol{\omega}\|_{L^2}^2
 +\|\Delta\theta\|_{L^2}^2].
\end{aligned}
 \end{equation}
Taking
\[
 q=[\frac{2}{\ln 2}\ln^+(C\|\nabla\theta\|_{L^2})]+3,
\]
 where $[t]$ is the largest integer smaller that $t\in\mathbb{R}$, and
$\ln^+t=\ln(e+t)$, then \eqref{Gronwall} implies that
\begin{align*}
 &\frac{\,\mathrm{d}}{\,\mathrm{d} t}
 [\|\boldsymbol{\omega}\|_{L^2}^2+\|\nabla\theta\|_{L^2}^2]
 +\frac{1}{2}[\|\nabla\boldsymbol{\omega}\|_{L^2}^2
+\|\Delta \theta\|_{L^2}^2] \\
 &
 \leq \|\nabla\theta\|_{L^2}^2
 +C
 +\frac{\|\bar S_q\nabla\mathbf{u}\|_{L^\infty}}{q\ln q}
 \ln^+(\|\nabla\theta\|_{L^2})
 \ln^+\ln^+(\|\nabla\theta\|_{L^2})
 \|\nabla\theta\|_{L^2}^2.
 \end{align*}
 Applying Gronwall inequality three times, we deduce
\begin{align*}
&[\|\boldsymbol{\omega}\|_{L^2}^2+\|\nabla\theta\|_{L^2}^2]
 +\int_0^t[\|\nabla\boldsymbol{\omega}\|_{L^2}^2
+\|\Delta \theta\|_{L^2}^2]\,\mathrm{d} \tau\\
&\leq C\exp\exp\exp\Big(\int_0^t\frac{\|\bar S_q\nabla\mathbf{u}\|_{L^\infty}}{q\ln q}
\,\mathrm{d} \tau\Big).
\end{align*}
By \eqref{thm:main:eq},  the solutions $(\mathbf{u},\theta)$ are
uniformly bounded in $L^\infty(0,T;H^1(\mathbb{R}^3))$, and thus smooth.
 This completes the proof of Theorem \ref{thm:main}.

\subsection*{Acknowledgements}
Zujin Zhang was partially supported by the (Youth) Natural
Science Foundation of Jiangxi Province (20132BAB211007, 20122BAB201014),
the Science Foundation of Jiangxi Provincial Department of Education
(GJJ13658, GJJ13659), the National Natural Science Foundation of
 China (11361004).

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\end{document}