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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 217, pp. 1--12.\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-2013/217\hfil Regularity criteria of supercritical SQG]
{Regularity criteria of supercritical beta-generalized quasi-geostrophic
 equations \\ in terms of partial derivatives}

\author[K. Yamazaki \hfil EJDE-2013/217\hfilneg]
{Kazuo Yamazaki}  % in alphabetical order

\address{Kazuo Yamazaki \newline
Department of Mathematics, Oklahoma State University,
401 Mathematical Sciences, Stillwater, OK 74078, USA}
\email{kyamazaki@math.okstate.edu}

\thanks{Submitted June 25, 2013. Published September 30, 2013.}
\subjclass[2000]{35B65, 35Q35, 35Q86}
\keywords{Quasi-geostrophic equation; porous media equation; 
\hfill\break\indent regularity criteria}

\begin{abstract}
 We study the two-dimensional beta-generalized supercritical quasi-geostrophic
 equation, and in particular show that to obtain global
 regularity results, one needs to bound only its partial derivative.
 Results may be generalized to similar active scalars, e.g. the porous media
 equation.
\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 and statement of results}

We study the two-dimensional $\beta$-generalized surface quasi-geostrophic
equation ($\beta$-SQG)
\begin{equation} \label{e1}
\begin{gathered}
\frac{\partial\theta}{\partial t} + (u \cdot \nabla) \theta
 + \nu\Lambda^{2\alpha}\theta = 0\\
\theta(x,0) = \theta_{0}(x), \quad u = \Lambda^{1-2\beta}(-\mathcal{R}_2\theta,
\mathcal{R}_1\theta)
\end{gathered}
\end{equation}
and the three-dimensional porous media equation
\begin{equation} \label{e2}
\begin{gathered}
\frac{\partial\theta}{\partial t} + (u\cdot\nabla)\theta
 + \nu \Lambda^{2\alpha}\theta = 0\\
\theta(x, 0) = \theta_{0}(x), \quad u = - \kappa(\nabla p + g\gamma\theta),
\quad \nabla \cdot u = 0
\end{gathered}
\end{equation}
In both equations, $\theta$ is a scalar function and
$\Lambda = (-\Delta)^{1/2}$ with $\alpha \in (0, 1/2), \beta \in [1/2, 1)$
 as fixed parameters. In   \eqref{e1}, $\theta$ represents temperature
and we denoted Riesz transform by $\mathcal{R}$.
In \eqref{e2}, $\theta$ represents density, $\kappa$ the matrix medium
permeability divided by viscosity in different directions respectively,
$g$ the acceleration due to gravity, vector $\gamma$ the last canonical
vector e$_{3}$. Moreover, $p$ is the pressure and $u$ the liquid discharge
 by Darcy's law (flux per unit area). Hereafter, without loss
of generality we set $\nu = 1$ and denote
$\partial_{t} = \frac{\partial}{\partial t}$ and
$\partial_{i} = \frac{\partial}{\partial x_{i}}, i = 1, 2, 3$.
We denote by $L^{p}$ the Lebesgue space equipped with the norm
$\| \cdot\|_{L^{p}}$ while those of Sobolev space $H^{s}$ with
$\| \cdot\|_{H^{s}}$ and a horizontal gradient operator by
 $\nabla_{h} = (\partial_1, \partial_2)$.

The $\beta$-SQG was formally introduced in \cite{k2}. At $\beta = 1/2$,
the equation reduces to an important model in geophysical fluid dynamics
used in meteorology and oceanography (cf. \cite{p1}).
The physical and mathematical
similarity between the $\beta$-SQG at $\nu = 0, \beta = \frac{1}{2}$ and the
three-dimensional Euler equation as well as the $\beta$-SQG at
 $\nu > 0$, $\alpha = \beta = \frac{1}{2}$ and the Navier-Stokes equation
(NSE) are discussed in \cite{c7}. The latter case is also physically relevant,
modeling the Eckmann pumping effect in the boundary layer near the surface.
We also note that while the Navier-Stokes and the Stokes systems are both
microscopic equations, Darcy's law yields a macroscopic description of a
flow in the porous medium (cf. \cite{b1}).

Due to the rescaling of the solutions to the $\beta$-SQG and the fact that
the $L^{p}$-norms of the solutions for $1 \leq p \leq \infty$ are
non-increasing (cf. \cite{c12}), we consider the case
$\alpha + \beta \in (1, 3/2)$ the subcritical case, the case
$\alpha + \beta = 1$ the critical case, and the case
$\alpha + \beta \in (1/2, 1)$ the supercritical case.
We do so similarly for the system \eqref{e2} with $\beta = 1/2$.

The existence of global weak solution to \eqref{e1} in case
$\alpha > 0, \beta = 1/2$ was established in \cite{r1},
followed by the existence of the global unique smooth solution in case
 $\beta = 1/2$, $\alpha > 1/2$ in \cite{c9}.
The same result in the case $\beta = 1/2$, $\alpha = 1/2$
was completed in \cite{k5} while the authors in \cite{c1} showed that solutions with
initial data in $L^{2}$ are locally smooth for any space dimension.
These works were subsequently generalized to allow $\beta \neq 1/2$ in
\cite{c6,m1,m2,y3}; in particular, in \cite{k2}
the author proved the global regularity of the unique solution
in the critical case for $\alpha$ and $\beta$ in the range of our consideration.
More recently, the third and fourth proofs in the case
$\alpha = \beta = 1/2$ appeared in \cite{c8,k3}.

Concerning the system \eqref{e2}, the authors in \cite{c2} showed the existence
of the unique global solution in the critical and subcritical cases.
In the supercritical case, they obtained local results in
 $H^{s}, s > (5-2\alpha)/2$ and extended to be global under the condition
$\|\theta_{0}\|_{H^{s}} \leq c\nu$ for some fixed small constant $c > 0$.
In \cite{x3}, the author considered the supercritical case and obtained local
results using iterative process while in \cite{y6},
the authors considered  the critical case and obtained the global existence
and uniqueness  of the solution in critical Besov spaces
$\dot{B}_{p,1}^{3/p}(\mathbb{R}^3)$ with $1 \leq p \leq \infty$
by the method introduced in \cite{k5}.
Finally, in \cite{y2} the author considered a modified porous media equation,
analogously to the modified QG in \cite{c6} and obtained global regularity
results in the critical case with initial data
$\theta_{0} \in H^{s}, s > 5/2$.

Regularity criteria results for $\beta$-SQG started in \cite{c7}:
at $\beta = 1/2$, $\nu = 0$,
\begin{equation*}
\limsup_{t\nearrow T}\| \theta(t)\|_{H^{m}} < \infty
  \text{ if and only if }
 \int_{0}^{T}\|\nabla^{\bot}\theta(t)\|_{L^{\infty}}dt < \infty
\end{equation*}

Extension and improvements of this type of results were followed by many
(e.g. \cite{c3,c4,c10,c11,d1,f1,w1,x1,x2,y5,y7}).
 In particular, the author in \cite{c3} obtained a Serrin-type regularity
criteria which showed that at $\beta = 1/2, \alpha \in (0, \frac{1}{2}]$,
if the solution $\theta(x,t)$ satisfied
\begin{equation*}
\int_{0}^{T}\| \nabla^{\bot}\theta\|_{L^{p}}^{r}dt < \infty \quad
 \text{for some $p, r$  such that } \frac{2}{p} + \frac{2\alpha}{r}
\leq 2\alpha, \quad \frac{1}{\alpha} < p < \infty,
\end{equation*}
then there is no singularity up to time $T$. These results are in harmony
with the finding of \cite{c7} that $\nabla^{\bot}\theta$ plays an
analogous role for the system \eqref{e1} as the vorticity
of the three-dimensional Euler equation does. Recently, the author
in \cite{y1} provided a first regularity criteria that relies only
on a partial derivative of $\theta$ in the critical case.
In the proof, the fact that $\alpha + \beta = 1$ was crucial
and hence in the supercritical case, it was not clear whether
the global regularity relies only on a bound of a partial
derivative of any order. In this paper we provide an affirmative
answer to this question. Let us present our results:

\begin{theorem} \label{thm1.1}
Let $\alpha \in (0, \frac{1}{2}), \beta \in [1/2, 1)$
so that $\alpha + \beta \in (\frac{1}{2}, 1)$. If the solution
$\theta(x,t)$ to \eqref{e1} in $\mathbb{R}^{2}$ or $\mathbb{T}^{2}$ satisfied
\begin{equation} \label{e3}
\int_{0}^{T}\| \partial_1\theta\|_{L^{p}}^{r} dt < \infty
\end{equation}
for some $p, r$ such that
\begin{gather*}
\frac{2}{p} + \frac{2\alpha}{r} \leq 2\alpha, \quad
 \frac{2(1-\alpha)}{\alpha^{2}} < p < \infty \quad \text{if } \beta = 1/2\\
\frac{2}{p} + \frac{2\alpha}{r} \leq 2\alpha + 2\beta - 1,  \quad
\frac{2(1-\alpha)}{\alpha^{2} + 2\beta - 1} < p < \frac{2(1-\alpha)}{2\beta - 1}
\quad \text{if } \beta > 1/2
\end{gather*}
then there is no singularity up to time $T$. Also, if
\begin{equation*}
\sup_{t\in [0,T]}\|\partial_1\theta\|_{L^{\frac{2}{2\alpha + 2\beta - 1}}}
\end{equation*}
is sufficiently small, then there is no singularity up to time $T$.
\end{theorem}

\begin{theorem} \label{thm1.2}
Let $\alpha \in (0, 1/2)$. If the solution $\theta(x,t)$ to
\eqref{e2} in $\mathbb{R}^3$ or $\mathbb{T}^3$ satisfies
\begin{equation} \label{e4}
\int_{0}^{T}\|\nabla_{h}\theta\|_{L^{p}}^{r}dt < \infty
\end{equation}
for some $p,r$ such that
\begin{equation*}
\frac{3}{p} + \frac{2\alpha}{r} \leq 2\alpha, \quad
\frac{9-6\alpha}{2\alpha^{2}} < p < \infty,
\end{equation*}
then there is no singularity up to time $T$. Also, if
\begin{equation*}
\sup_{t\in [0,T]}\|\nabla_{h}\theta(t)\|_{L^{\frac{3}{2\alpha}}}
\end{equation*}
is sufficiently small, then there is no singularity up to time $T$.
\end{theorem}

Next theorem was partially inspired by the work in \cite{v1} where the author
obtained a regularity criteria for the NSE in terms of
$\operatorname{div}(u/| u|)$ (see also \cite{c5}). We make use of the
fact that that due to the incompressibility of $u$,
\[
-\sum_{i=1}^{2}\partial_1u_{i}\partial_{1i}\theta = (\partial_1\theta)^{2}\sum_{i=1}^{2}\partial_{i}\left(\frac{\partial_1u_{i}}{\partial_1\theta}\right) = (\partial_1\theta)^{2}\operatorname{div}\left(\frac{\partial_1u}{\partial_1\theta}\right)
\]

\begin{theorem} \label{thm1.3}
Let $\alpha \in (0, 1/2), \beta \in [1/2, 1)$ so that
 $\alpha + \beta \in (1/2, 1)$. If the solution $\theta(x,t)$
to \eqref{e1} in $\mathbb{R}^{2}$ or $\mathbb{T}^{2}$ satisfies
\begin{equation} \label{e5}
\int_{0}^{T}\| \operatorname{div}(\frac{\partial_1u}{\partial_1\theta})
\|_{L^{p}}^{r}dt < \infty
\end{equation}
for some $p, r$ such that
\begin{gather*}
\frac{2}{p} + \frac{2\alpha}{r} \leq 2\alpha, \quad
 \frac{2(1-\alpha)^{2}}{\alpha^3} < p < \infty \quad \text{if } \beta = 1/2\\
\frac{2}{p} + \frac{2\alpha}{r} \leq 2\alpha, \quad
\frac{2(1-\alpha)^{2}}{\alpha(\alpha^{2} + 2\beta - 1)} < p
< \frac{2(1-\alpha)^{2}}{\alpha(2\beta - 1)} \quad \text{if } \beta > 1/2
\end{gather*}
then there is no singularity up to time $T$. Also, if
\begin{equation*}
\sup_{t\in [0,T]}\| \operatorname{div}
(\frac{\partial_1u}{\partial_1\theta})\|_{L^{1/\alpha}} < \infty,
\end{equation*}
then there exists $C = C(\alpha,\beta)$ such that
\begin{equation*}
\|\theta_{0}\|_{L^{\infty}} < C
\end{equation*}
which implies that there is no singularity up to time $T$.
\end{theorem}

\begin{remark} \label{rmk1.1} \rm
(1) The Roles of $x_1, x_2, x_{3}$ in the hypothesis may be switched.

(2)  Theorem \ref{thm1.1} may be seen as a component reduction type results in
comparison to the \cite[Theorem 1.1]{c3}.
In the case of $\mathbb{T}^3$, one may also compare our Theorem \ref{thm1.2}
with \cite[Theorem 1.3]{y6} (wee also \cite[Remark 2.4]{c2}).
Upon succeeding the reduction component type result, it is common that
the range of $p,r$ become restrictive (cf. e.g. \cite{k6}).
From the proof, it becomes clear that one can also extend
Theorems \ref{thm1.1} and \ref{thm1.3}
 for dispersive SQG introduced in \cite{k4}.

(3) In fact, in the case of $\beta$-SQG with $\beta = \frac{1}{2}$,
because $\widehat{\Lambda u_2}(\xi) = \widehat{\partial_1\theta}(\xi)$
we have shown that
\begin{equation*}
\int_{0}^{T}\|\Lambda u_2\|_{L^{p}}^{r}dt < \infty
\end{equation*}
under the same condition of Theorem \ref{thm1.1} implies global regularity.
It is not difficult to show that there exists a constant $c> 0$ such that
\begin{equation*}
\| \Lambda u_2\|_{L^{p}} \leq c\| \nabla u_2\|_{L^{p}}
\end{equation*}
holds for $p \in (1,\infty)$. Hence, we have also shown that
\begin{equation*}
\int_{0}^{T}\|\nabla u_2\|_{L^{p}}^{r}dt < \infty
\end{equation*}
implies global regularity of the solution. In relevance,
we refer readers to \cite{w1} in which the author obtained a regularity
criteria that involves $ \nabla u$.

(4) Results of this type exist for NSE (e.g. \cite{k6}).
However, to the best of our knowledge, there does not exist any
regularity criteria in terms of partial derivatives for the generalized
NSE with fractional Laplacian with power arbitrary close to zero.

(5) In general for systems like NSE or MHD, one relies on a decomposition
 of nonlinear term such as the \cite[Lemma 2.2]{k6} that makes use of the
incompressibility of the solution itself. The key observation in this
paper is that if we start with the $L^{p}$ estimate of a partial
derivative for $p > 2$, then because the vector $u$ dotted with
a gradient within the nonlinear term is incompressible, the partial
derivatives within the nonlinear term may be shifted to miss one direction.

(6) It is not clear if we can obtain analogous results for different
range of $\alpha$ and $\beta$ as considered in \cite{m2}.
\end{remark}

In the next section, we prove Theorems \ref{thm1.1} and \ref{thm1.3}.
The proof of Theorem \ref{thm1.2} is similar to that of
Theorem \ref{thm1.1} and will  be addressed in \cite{y4} with different results;
hence we just sketch it in the Appendix.

\section{Proofs}

To prove Theorem \ref{thm1.1}, we need the following proposition.

\begin{proposition} \label{prop2.1}
Let $\alpha \in (0, \frac{1}{2}), \beta \in [1/2, 1)$ so that
 $\alpha + \beta \in (\frac{1}{2}, 1)$. If the solution $\theta(x,t)$
to \eqref{e1} in $[0,T]$ satisfies \eqref{e3}, then
\begin{align*}
&\sup_{t\in [0,T]}\| \partial_1\theta(t)\|
_{L^{\frac{2\alpha p}{2+p(1-2\beta)}}}^{\frac{2\alpha p}{2+p(1-2\beta)}}
+ \| \partial_2\theta(t)\|_{L^{\frac{2\alpha p}{2+p(1-2\beta)}}}
^{\frac{2\alpha p}{2+p(1-2\beta)}}\\
&+ \int_{0}^{T}\| \partial_1\theta\|_{L^{(\frac{\alpha}{1-\alpha})
[\frac{2p}{2+p(1-2\beta)}]}}^{\frac{2\alpha p}{2+p(1-2\beta)}}
+ \| \partial_2\theta\|_{L^{(\frac{\alpha}{1-\alpha})
[\frac{2p}{2+p(1-2\beta)}]}}^{\frac{2\alpha p}{2+p(1-2\beta)}} dt
 < \infty
\end{align*}
\end{proposition}

\begin{proof}
We fix $p$ and $r$ that satisfy \eqref{e3} and then define
\begin{equation*}
q = \frac{2\alpha p}{2+p(1-2\beta)}
\end{equation*}
One can check that $q \in (2, \infty)$. We apply $\partial_1$
on \eqref{e1}, multiply by $q| \partial_1\theta|^{q-2}\partial_1\theta$
and integrate in space to obtain
\[
\partial_{t} \| \partial_1\theta\|_{L^{q}}^{q}
+ q\int(\Lambda^{2\alpha}\partial_1\theta)|
\partial_1\theta|^{q-2}\partial_1\theta
= -q\int\partial_1((u\cdot\nabla)\theta)|
\partial_1\theta|^{q-2}\partial_1\theta
\]
\end{proof}
We use the following lemma from \cite{j1}.

\begin{lemma} \label{lem2.1}
For $\alpha \in [0,1]$, $x \in \mathbb{R}^{2}$,
$\mathbb{T}^{2}, \Lambda^{2\alpha}\theta\in L^{s}$, $s \geq 2$, we have
\begin{equation*}
2\int | \Lambda^{\alpha}\theta^{\frac{s}{2}}|^{2}
\leq s\int\Lambda^{2\alpha}\theta| \theta|^{s-2}\theta\,.
\end{equation*}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
By this lemma and the homogeneous Sobolev embedding
$\dot{H}^{\alpha}\hookrightarrow L^{\frac{2}{1-\alpha}}$,
we have
\begin{equation*}
c(q, \alpha)\| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
\leq 2\int| \Lambda^{\alpha}(\partial_1\theta)^{q/2}|^{2}
\leq q\int(\Lambda^{2\alpha}\partial_1\theta)|\partial_1\theta|^{q-2}
 \partial_1\theta
\end{equation*}
Expanding the right hand side now, we have
\[
q\int\partial_1((u\cdot\nabla)\theta)| \partial_1\theta|^{q-2}
\partial_1\theta = q\int \partial_1u\cdot\nabla\theta|
\partial_1\theta|^{q-2}\partial_1\theta + u\cdot\nabla\partial_1
\theta| \partial_1\theta|^{q-2}\partial_1\theta
\]
The second term vanishes by the incompressibility of $u$.
We further bound by
\begin{align*}
&-q\int\partial_1u_1\partial_1\theta| \partial_1
 \theta|^{q-2}\partial_1\theta
 + \partial_1u_2\partial_2\theta| \partial_1\theta|^{q-2}
 \partial_1\theta\\
&\leq  c\Big(\| \partial_1u_1\|_{L^{\frac{2q}{2(1-\alpha) - q(2\beta - 1)}}}\|
 \partial_1\theta\|_{L^{q}}^{q-1}\| \partial_1\theta\|
 _{L^{\frac{2q}{2\alpha + q(2\beta - 1)}}}\\
&\quad + \| \partial_1u_2\|_{L^{\frac{2q}{2(1-\alpha) - q(2\beta - 1)}}}\|
 \partial_2\theta\|_{L^{q}}\| \partial_1\theta\|_{L^{q}}^{q-2}\|
 \partial_1\theta\|_{L^{\frac{2q}{2\alpha + q(2\beta - 1)}}}\Big)
\\
&\leq  c\Big(\| \partial_1\mathcal{R}_2\theta\|_{L^{\frac{q}{1-\alpha}}}\|
  \partial_1\theta\|_{L^{q}}^{q-1}\| \partial_1\theta\|_{L^{\frac{2q}{2\alpha + q(2\beta - 1)}}}\\
&\quad + \| \partial_1\mathcal{R}_1\theta\|_{L^{\frac{q}{1-\alpha}}}\|
 \partial_2\theta\|_{L^{q}}\| \partial_1\theta\|_{L^{q}}^{q-2}\|
 \partial_1\theta\|_{L^{\frac{2q}{2\alpha + q(2\beta -1)}}}\Big)
\end{align*}
by H\"older's inequality and the homogeneous Sobolev embedding
\begin{equation*}
\dot{W}^{2\beta -1, \frac{q}{1-\alpha}}
\hookrightarrow L^{\frac{2q}{2(1-\alpha) - q(2\beta -1)}}.
\end{equation*}
Now we use continuity of the Riesz transform in $L^{s}, s \in (1, \infty)$
for a bound from above,
\begin{align*}
&c\Big(\| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}\| 
 \partial_1\theta\|_{L^{q}}^{q-1}\| \partial_1\theta\|_{L^{\frac{2q}{2\alpha 
 + q(2\beta - 1)}}}\\
&+ \| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}\| 
 \partial_2\theta\|_{L^{q}}\| \partial_1\theta\|_{L^{q}}^{q-2}\| 
\partial_1\theta\|_{L^{\frac{2q}{2\alpha + q(2\beta - 1)}}}\Big)\\
&\leq  \epsilon \| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q} 
+ c\| \partial_1\theta\|_{L^{\frac{2q}{2\alpha + q(2\beta -1)}}}
 ^{\frac{q}{q-1}}(\| \partial_1\theta\|_{L^{q}}^{q} 
 + \| \partial_2\theta\|_{L^{q}}^{q})
\end{align*}
for any $\epsilon > 0$ where we used Young's inequality twice.
Next, applying $\partial_2$ on \eqref{e1} and following similar
 procedure as above leads to
\begin{align*}
&\partial_{t} \| \partial_2\theta\|_{L^{q}}^{q}
 + c(q,\alpha)\| \partial_2\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}\\
&\leq  -q\int \partial_2u_1\partial_1\theta | \partial_2
 \theta|^{q-2}\partial_2\theta + \partial_2u_2\partial_2\theta
 | \partial_2\theta|^{q-2}\partial_2\theta\\
&= -q\int \partial_2u_1\partial_1\theta| \partial_2\theta|^{q-2}\partial_2\theta
 - \partial_1u_1\partial_2\theta | \partial_2\theta|^{q-2}\partial_2\theta\\
&\leq  \epsilon\| \partial_2\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
 + c\| \partial_1\theta\|_{L^{\frac{2q}{2\alpha + q(2\beta - 1)}}}
 ^{\frac{q}{q-1}}\| \partial_2\theta\|_{L^{q}}^{q}
\end{align*}
where the only difference from the previous estimate was the equality
which made use of the incompressibility of $u$ so that
$\partial_1u_1 = -\partial_2u_2$.

In summary, for $\epsilon > 0$ sufficiently small we have
\begin{align*}
&\partial_{t}(\|\partial_1\theta\|_{L^{q}}^{q}
 + \|\partial_2\theta\|_{L^{q}}^{q})
 + \|\partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
 + \|\partial_2\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}\\
&\leq  c\|\partial_1\theta\Vert_{L^{\frac{2q}{2\alpha
 + q(2\beta - 1)}}}^{\frac{q}{q-1}}(\|\partial_1\theta\|_{L^{q}}^{q}
 + \|\partial_2\theta\|_{L^{q}}^{q})
\end{align*}
Thus, by Gronwall's inequality and \eqref{e3},
\begin{align*}
&\sup_{t\in [0,T]}\| \partial_1\theta(t)\|_{L^{q}}^{q}
 + \| \partial_2\theta(t)\|_{L^{q}}^{q}
 + \int_{0}^{T} \| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
 + \| \partial_2\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}dt\\
&\leq  c(\| \partial_1\theta_{0}\|_{L^{q}}^{q}
 + \| \partial_2\theta_{0}\|_{L^{q}}^{q})e^{\int_{0}^{T}
 \| \partial_1\theta\|_{L^{p}}^{r}dt} < \infty
\end{align*}
This completes the proof.
\end{proof}

We will use the following well-known
commutator estimate due to \cite{k1}.

\begin{lemma} \label{lem2.2}
Let $p \in (1,\infty), s > 0, f, g \in W^{s, p}$.
 Then there exists some constant $c > 0$ independent of $f,g$
such that for $\frac{1}{p} = \frac{1}{p_1}+\frac{1}{p_2}
= \frac{1}{p_{3}} + \frac{1}{p_{4}}, p_2, p_{3} \in (1, \infty)$
\begin{equation*}
\| \Lambda^{s}(fg) - f\Lambda^{s}g\|_{L^{p}}
\leq c(\| \nabla f\|_{L^{p_1}}\| \Lambda^{s-1}g\|_{L^{p_2}}
+ \| \Lambda^{s}f\|_{L^{p_{3}}}\| g\|_{L^{p_{4}}})
\end{equation*}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
For any $s \in \mathbb{R}^{+}$, applying $\Lambda^{s}$ on \eqref{e1},
taking an $L^{2}$-inner product with $\Lambda^{s}\theta$ we have
\begin{align*}
&\frac{1}{2}\partial_{t}\|\Lambda^{s}\theta\|_{L^{2}}^{2}
 + \|\Lambda^{\alpha+s}\theta\|_{L^{2}}^{2}\\
&\leq  c\Big(\|\nabla u\|_{L^{\frac{2}{\alpha}}}\|\Lambda^{s-1}\nabla\theta\|_{L^{2}}
 + \|\Lambda^{s}u\|_{L^{\frac{1}{1-\beta}}}\|\nabla\theta\|_{L^{\frac{2}{2\beta
 + \alpha - 1}}}\Big)\|\Lambda^{s}\theta\|_{L^{\frac{2}{1-\alpha}}}\\
&\leq  c\|\nabla\theta\|_{L^{\frac{2}{\alpha + 2\beta - 1}}}
 \|\Lambda^{s}\theta\|_{L^{2}}\|\Lambda^{s+\alpha}\theta\|_{L^{2}}
\end{align*}
by the homogeneous Sobolev embeddings
\begin{equation*}
\dot{W}^{2\beta -1, \frac{2}{\alpha + 2\beta - 1}}
\hookrightarrow L^{\frac{2}{\alpha}}, \quad
 \dot{H}^{2\beta -1} \hookrightarrow L^{\frac{1}{1-\beta}}, \quad
\dot{H}^{\alpha} \hookrightarrow L^{\frac{2}{1-\alpha}}
\end{equation*}
By Young's and Gagliardo-Nirenberg inequalities we have the bound
\[
\frac{1}{2}\|\Lambda^{s+\alpha}\theta\|_{L^{2}}^{2}
+ c\|\theta\|_{L^{\infty}}^{2(\frac{2\alpha
+ 2\beta - 1}{1+\alpha})}\|\Lambda^{2+\alpha}\theta\|_{L^{2}}
^{2(\frac{2-\alpha-2\beta}{1+\alpha})}\|\Lambda^{s}\theta\|_{L^{2}}^{2}\,.
\]
Let us use the fact that (cf. \cite{c12})
\begin{equation*}
\| \theta(t)\|_{L^{\infty}} \leq \| \theta_{0}\|_{L^{\infty}}
\end{equation*}
Gronwall's inequality implies that to complete the proof it suffices
to show that
\begin{equation*}
\int_{0}^{T}\| \Lambda^{2+\alpha}\theta\|_{L^{2}}^{2}dt < \infty
\end{equation*}
We apply $\Delta$ to \eqref{e1} and take an $L^{2}$-inner product
with $\Delta\theta$ to estimate
\begin{align*}
&\frac{1}{2}\partial_{t}\| \Delta \theta\|_{L^{2}}^{2}
 + \| \Lambda^{\alpha}\Delta\theta\|_{L^{2}}^{2}\\
&\leq  c\Big(\| \nabla u\|_{L^{\frac{2}{\epsilon\alpha}}}
 \|\Lambda\nabla\theta\|_{L^{2}} + \| \Lambda^{2}u\|_{L^{\frac{1}{1-\beta}}}
\| \nabla \theta\|_{L^{\frac{2}{\epsilon\alpha + 2\beta -1}}}\Big)
\|\Delta\theta\|_{L^{\frac{2}{1-\epsilon\alpha}}}
\end{align*}
by Lemma \ref{lem2.2} where we choose
\begin{equation*}
\epsilon = \frac{2+p(1-2\beta) - 2\alpha}{p\alpha^{2}}
\end{equation*}
One can readily check using \eqref{e3} and the range of $\alpha$
and $\beta$ that $\epsilon \in (0,1)$.
Now by the homogeneous Sobolev embeddings of
\begin{equation*}
 \dot{W}^{2\beta - 1, \frac{2}{\epsilon\alpha + 2\beta -1}}
\hookrightarrow L^{\frac{2}{\epsilon\alpha}}, \quad
\dot{H}^{2\beta -1} \hookrightarrow  L^{\frac{1}{1-\beta}}
\end{equation*}
and Gagliardo-Nirenberg inequality, followed by Young's inequality
we bound the right-hand side as
\[
c\| \nabla \theta\|_{L^{\frac{2}{2\beta
 + \epsilon\alpha - 1}}}\|\Delta\theta\|_{L^{2}}^{2-\epsilon}\|
\Lambda^{2+\alpha}\theta\|_{L^{2}}^{\epsilon}
\leq \frac{1}{2}\|\Lambda^{\alpha}\Delta\theta\|_{L^{2}}^{2}
+ c\|\nabla\theta\|_{L^{\frac{2}{2\beta
+ \epsilon\alpha - 1}}}^{\frac{2}{2-\epsilon}}\|\Delta\theta\|_{L^{2}}^{2}
\]
Hence, absorbing the dissipative term, Gronwall's inequality implies
\[
\| \Delta\theta(t)\|_{L^{2}}^{2}
+ \int_{0}^{t}\|\Lambda^{\alpha}\Delta\theta \|_{L^{2}}^{2}dt
\leq  c\|\Delta\theta_{0}\|_{L^{2}}^{2}
e^{\int_{0}^{T}\|\nabla\theta\|_{L^{\frac{2}{2\beta
+ \epsilon\alpha - 1}}}^{\frac{2}{2-\epsilon}}dt}
\]
for any $t \in [0,T]$. Next, we will use the following elementary
inequality several times: for $a, b \geq 0$,
\begin{equation} \label{e6}
(a+b)^{s} \leq 2^{s}(a^{s} + b^{s}), \quad \forall s \geq 0
\end{equation}
Using this inequality, we observe that
\begin{align*}
&\int_{0}^{T}\| \nabla \theta\|_{L^{\frac{2}{2\beta + \epsilon\alpha - 1}}}
^{\frac{2}{2-\epsilon}}dt\\
&\leq  \int_{0}^{T}
 \Big(\int 2^{\frac{1}{2\beta + \epsilon\alpha -1}}
 [| \partial_1\theta|^{\frac{2}{2\beta+\epsilon\alpha - 1}}
 + | \partial_2\theta|^{\frac{2}{2\beta + \epsilon\alpha - 1}}]\Big)
 ^{\frac{2\beta + \epsilon\alpha - 1}{2-\epsilon}}dt\\
&\leq  2^{\frac{2\beta + \epsilon\alpha}{2-\epsilon}}
 \int_{0}^{T}\| \partial_1\theta\|_{L^{\frac{2}{2\beta
 + \epsilon\alpha - 1}}}^{^{\frac{2}{2-\epsilon}}}
 + \| \partial_2\theta\|_{L^{\frac{2}{2\beta
 + \epsilon\alpha - 1}}}^{\frac{2}{2-\epsilon}}dt\\
&\leq  c(\alpha, \beta, \epsilon, T)
 \int_{0}^{T}\| \partial_1\theta\|_{L^{(\frac{\alpha}{1-\alpha})
 \frac{2p}{2+p(1-2\beta)}}}^{\frac{2\alpha p}{2+p(1-2\beta)}}
 + \|\partial_2\theta\|_{L^{(\frac{\alpha}{1-\alpha})
 \frac{2p}{2+p(1-2\beta)}}}^{\frac{2\alpha p}{2+p(1-2\beta)}}dt < \infty\,,
\end{align*}
due to Proposition \ref{prop2.1}. This completes the proof of the first claim.

Now we prove the second claim. By the first claim, it suffices to show that
\begin{equation*}
\int_{0}^{T}\|\partial_2\theta\|_{L^{p}}^{r}dt < \infty
\end{equation*}
for some $p,r$ that satisfy\eqref{e3}.
We fix $p$ in such a range, apply $\partial_2$ on \eqref{e1},
 multiply by $p|\partial_2\theta|^{p-2}\partial_1\theta$ and
integrate in space to estimate
\begin{equation*}
\partial_{t}\|\partial_2\theta\|_{L^{p}}^{p}
+ p\int(\Lambda^{2\alpha}\partial_2\theta)|\partial_2\theta|^{p-2}
\partial_2\theta
= -p\int\partial_2((u\cdot\nabla)\theta)|\partial_2\theta|^{p-2}
\partial_2\theta
\end{equation*}
Since $p\geq 2$, by Lemma \ref{lem2.1} and the same homogeneous Sobolev
 embedding, as before, we see that due to the incompressibility
of $u$,
\begin{align*}
&\partial_{t}\|\partial_2\theta\|_{L^{p}}^{p}
 + c(p,\alpha)\|\partial_2\theta\|_{L^{\frac{p}{1-\alpha}}}^{p}\\
&\leq  -p\int\partial_2u_1\partial_1\theta|\partial_2\theta|^{p-2}
 \partial_2\theta + p\int\partial_1u_1\partial_2\theta|\partial_2\theta|
 ^{p-2}\partial_2\theta\\
&\leq  p\|\partial_2u_1\|_{L^{\frac{2p}{2(1-\alpha) - p(2\beta - 1)}}}
 \| \partial_1\theta\|_{L^{\frac{2}{2\alpha + 2\beta - 1}}}
 \|\partial_2\theta\|_{L^{\frac{p}{1-\alpha}}}^{p-1}
 + p\|\partial_1u_1\|_{L^{\frac{1}{\alpha}}}\|\partial_2\theta\|
 _{L^{\frac{p}{1-\alpha}}}^{p}
\end{align*}
by H\"older's inequality. Now Sobolev's embeddings
\begin{equation*}
\dot{W}^{2\beta -1, \frac{2}{2\alpha + 2\beta - 1}}
\hookrightarrow L^{\frac{1}{\alpha}}, \quad
 \dot{W}^{2\beta - 1, \frac{p}{1-\alpha}}\hookrightarrow L^{\frac{2p}{2(1-\alpha)
- p(2\beta - 1)}}
\end{equation*}
we obtain
\begin{equation*}
\partial_{t}\|\partial_2\theta\|_{L^{p}}^{p}
\leq \Big(c\sup_{t\in [0,T]}\|\partial_1\theta\|
_{L^{\frac{2}{2\alpha + 2\beta - 1}}} - c(p,\alpha)\Big)
\|\partial_2\theta\|_{L^{\frac{p}{1-\alpha}}}^{p}
\end{equation*}
This completes the proof of Theorem \ref{thm1.1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
We fix $p$ and $r$ that satisfy \eqref{e5}.
It is important for the proof below to note that
\begin{equation*}
\frac{2(1-\alpha)^{2}}{\alpha(\alpha^{2} + 2\beta - 1)}
> \frac{2}{\alpha(2\alpha + 2\beta - 1)}
\end{equation*}
Now define $q = p\alpha$ for which one can verify that $q \in (2, \infty)$.
 We apply $\partial_1$ on \eqref{e1}, multiply by
$q| \partial_1\theta|^{q-2}\partial_1\theta$ and integrate in space to obtain
\begin{equation*}
\partial_{t} \| \partial_1\theta\|_{L^{q}}^{q}
+ q\int(\Lambda^{2\alpha}\partial_1\theta)| \partial_1\theta|^{q-2}
\partial_1\theta = -q\int\partial_1((u\cdot\nabla)\theta)|
\partial_1\theta|^{q-2}\partial_1 \theta
\end{equation*}
Lemma \ref{lem2.1}, and the same homogeneous Sobolev embedding, as before, lead to
\begin{align*}
\partial_{t}\| \partial_1\theta\|_{L^{q}}^{q}
 + c(q,\alpha)\| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
&\leq  -q\int\partial_1((u\cdot\nabla)\theta)| \partial_1\theta|^{q-2}\partial_1\theta\\
&= -q\sum_{i=1}^{2}\int \partial_1u_{i}\partial_{i}\theta|
\partial_1\theta|^{q-2}\partial_1\theta
\end{align*}
due to the incompressibility of $u$. We integrate by parts once more
and obtain
\[
\partial_{t}\| \partial_1\theta\|_{L^{q}}^{q} + c(q, \alpha)
\| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
\leq c\| \theta\|_{L^{\infty}} \| \partial_1\theta\|
_{L^{\frac{q}{1-\alpha}}}\| \partial_1\theta\|_{L^{q}}^{q-1}
\| \operatorname{div}\big(\frac{\partial_1u}{\partial_1\theta}\big)
\|_{L^{\frac{q}{\alpha}}}
\]
by H\"older's inequality. Now Young's inequality gives
\begin{equation*}
\partial_{t}\| \partial_1\theta\|_{L^{q}}^{q} + c(q, \alpha)
\| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
\leq \epsilon\| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}
 + c\| \theta\|_{L^{\infty}}^{\frac{q}{q-1}}\| \partial_1\theta\|_{L^{q}}^{q}
\| \operatorname{div}\big(\frac{\partial_1u}{\partial_1\theta}\big)
\|_{L^{\frac{q}{\alpha}}}^{\frac{q}{q-1}}
\end{equation*}
As shown in \cite{c12}, for $\epsilon > 0$ sufficiently small Gronwall's
inequality implies
\begin{equation*}
\| \partial_1\theta(T)\|_{L^{q}}^{q} + \int_{0}^{T}\|
\partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}dt
\leq c\| \partial_1\theta_{0}\|_{L^{q}}^{q} e^{\int_{0}^{T}
\| \operatorname{div}\big(\frac{\partial_1u}{\partial_1\theta}\big)
\|_{L^{\frac{q}{\alpha}}}^{\frac{q}{q-1}}dt}
\end{equation*}
By the definition of $q$ and \eqref{e5}, we have
\begin{equation*}
\int_{0}^{T}\| \partial_1\theta\|_{L^{\frac{q}{1-\alpha}}}^{q}dt < \infty
\end{equation*}
That is, we have proved the hypothesis of Theorem \ref{thm1.1}.
 This completes the proof of the first claim.

Next, we prove the second claim. We take $p$ that satisfies \eqref{e3}.
An identical process as above gives
\[
\partial_{t}\|\partial_1\theta\|_{L^{p}}^{p}
 + c(p,\alpha)\|\partial_1\theta\|_{L^{\frac{p}{1-\alpha}}}^{p}
\leq  p(p-1)\|\theta_{0}\|_{L^{\infty}}\|\partial_1\theta
\|_{L^{\frac{p}{1-\alpha}}}^{p}
\|\operatorname{div}(\frac{\partial_1u}{\partial_1\theta})
\|_{L^{\frac{1}{\alpha}}}
\]
Thus,
\begin{equation*}
\partial_{t}\|\partial_1\theta\|_{L^{p}}^{p} \leq
\Big(p(p-1)\|\theta_{0}\|_{L^{\infty}}\sup_{t\in[0,T]}
\|\operatorname{div}(\frac{\partial_1u}{\partial_1\theta})(t)
\|_{L^{\frac{1}{\alpha}}} - c(p,\alpha)\Big)
\|\partial_1\theta\|_{L^{\frac{p}{1-\alpha}}}^{p}
\end{equation*}
This completes the proof of Theorem \ref{thm1.3}.
\end{proof}

\section{Appendix}

In this section we prove Theorem \ref{thm1.2}.
It was shown in \cite{c2} that $u$ of \eqref{e2} may be decomposed as
$u = c(\theta + \mathcal{P}\theta)$ for some constant $c > 0$
 where $\mathcal{P}$ is a type of singular integral operator bounded
in $L^{p}, p \in (1, \infty)$. Immediately, we have
\begin{equation} \label{e7}
\| u\|_{L^{p}} \leq c\| \theta\|_{L^{p}}, \quad p \in (1, \infty).
\end{equation}

\begin{proposition} \label{prop3.1}
Let $\alpha \in (0, 1/2)$. If the solution $\theta(x,t)$ to \eqref{e2}
in $[0,T]$ satisfies \eqref{e4}, then
\begin{equation*}
\sup_{t\in [0,T]}\| \partial_{3}\theta(t)\|_{L^{\frac{2p\alpha}{3}}}
^{\frac{2p\alpha}{3}} + \int_{0}^{T}\| \partial_{3}
\theta\|_{L^{\frac{2p\alpha}{3-2\alpha}}}^{\frac{2p\alpha}{3}} dt < \infty
\end{equation*}
\end{proposition}

\begin{proof}
We fix $p$ and $r$ that satisfy \eqref{e4} and define $q = \frac{2p\alpha}{3}$
for which it is easy to check that $q \in (2, \infty)$.
 We apply $\partial_{3}$ to \eqref{e2}, multiply by
$q| \partial_{3}\theta|^{q-2}\partial_{3}\theta$, integrate in space to obtain
\begin{equation*}
\partial_{t}\| \partial_{3}\theta\|_{L^{q}}^{q}
+ q\int(\Lambda^{2\alpha}\partial_{3}\theta)| \partial_{3}\theta|^{q-2}
\partial_{3}\theta = -q\int \partial_{3}((u\cdot\nabla)\theta)|
 \partial_{3}\theta|^{q-2}\partial_{3}\theta
\end{equation*}
The proof of Lemma \ref{lem2.1} is generalizable to higher dimension
(See \cite{c2,w2} for details); hence, by the homogeneous Sobolev embedding
\begin{equation*}
c(q, \alpha)\| \partial_{3}\theta\|_{L^{\frac{3q}{3-2\alpha}}}^{q}
\leq c\| (\partial_{3}\theta)^{q/2}\|_{\dot{H}^{\alpha}}^{2}
\leq q\int (\Lambda^{2\alpha}\partial_{3}\theta)| \partial_{3}\theta|^{q-2}
\partial_{3}\theta
\end{equation*}
On the right hand side, we have due to the incompressibility of $u$,
\[
\int\partial_{3}u_1\partial_1\theta| \partial_{3}\theta|^{q-2}\partial_{3}\theta
 + \partial_{3}u_2\partial_2\theta| \partial_{3}\theta|^{q-2}\partial_{3}\theta
+ (-\partial_1u_1 - \partial_2u_2)\partial_{3}\theta| \partial_{3}\theta|^{q-2}
\partial_{3}\theta
\]
which by H\"older's and Young's inequalities and \eqref{e7}, we can bound by
\begin{equation*}
c\| \partial_{3}\theta\|_{L^{\frac{3q}{3-2\alpha}}}\|\nabla_{h}
\theta\|_{L^{\frac{3q}{2\alpha}}}\|\partial_{3}\theta\|_{L^{q}}^{q-1}
\leq \epsilon\| \partial_{3}\theta\|_{L^{\frac{3q}{3-2\alpha}}}^{q}
+ c\|\nabla_{h}\theta\|_{L^{\frac{3q}{2\alpha}}}^{\frac{q}{q-1}}\|\partial_{3}
\theta\|_{L^{q}}^{q}
\end{equation*}
Thus, for $\epsilon > 0$ sufficiently small,
\begin{equation*}
\partial_{t}\| \partial_{3}\theta\|_{L^{q}}^{q}
+ \| \partial_{3}\theta\|_{L^{\frac{3q}{3-2\alpha}}}^{q}
\leq c\| \nabla_{h}\theta\|_{L^{\frac{3q}{2\alpha}}}
^{\frac{q}{q-1}}\|\partial_{3}\theta\|_{L^{q}}^{q}
\end{equation*}
Gronwall's inequality and \eqref{e4} completes the proof.
\end{proof}

The following proposition can be obtained by a similar process.

\begin{proposition} \label{prop3.2}
Let $\alpha \in (0, \frac{1}{2})$. If the solution $\theta(x,t)$ to \eqref{e2}
in $[0,T]$ satisfies \eqref{e4}, then
\begin{equation*}
\sup_{t\in [0,T]}\| \nabla_{h}\theta(t)\|_{L^{\frac{2p\alpha}{3}}}
^{\frac{2p\alpha}{3}}
+  \int_{0}^{T}\|\nabla_{h}\theta\|_{L^{\frac{2p\alpha}{3-2\alpha}}}
^{\frac{2p\alpha}{3}}dt < \infty
\end{equation*}
\end{proposition}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
Using \eqref{e4} and the range of $\alpha$, one can readily check that
\[
\frac{4p\alpha^{2}}{4p\alpha^{2} - 9 + 6\alpha} \leq \frac{2p\alpha}{3}
\]
With this in mind, by \eqref{e6}, Propositions \ref{prop3.1} and
\ref{prop3.2},  as before, we obtain
\[
\int_{0}^{T}\|\nabla\theta\|_{L^{\frac{2p\alpha}{3-2\alpha}}}
^{\frac{4p\alpha^{2}}{4p\alpha^{2} - 9 + 6\alpha}}dt
\leq c\int_{0}^{T}\|\nabla_{h}\theta\|_{L^{\frac{2p\alpha}{3-2\alpha}}}
^{\frac{2p\alpha}{3}} + \|\partial_{3}\theta\|
_{L^{\frac{2p\alpha}{3-2\alpha}}}^{\frac{2p\alpha}{3}}dt < \infty
\]
Now we use Lemma \ref{lem2.2}  to estimate \eqref{e2}, after applying $\Lambda^{2}$
and taking an $L^{2}$-inner product with $\Lambda^{2}\theta$,
\begin{align*}
&\partial_{t}\|\Lambda^{2}\theta\|_{L^{2}}^{2}
+ \|\Lambda^{2+\alpha}\theta\|_{L^{2}}^{2}\\
&\leq  c(\|\nabla u\|_{L^{\frac{3}{\epsilon\alpha}}}
\|\Lambda\nabla\theta\|_{L^{2}}
+ \|\Delta u\|_{L^{2}}\|\nabla\theta\|_{L^{\frac{3}{\epsilon\alpha}}})
\|\Delta\theta\|_{L^{\frac{6}{3-2\epsilon\alpha}}}\\
&\leq  c\|\nabla \theta\|_{L^{\frac{3}{\epsilon\alpha}}}
\|\Lambda^{2}\theta\|_{L^{2}}^{2-\epsilon}\|
\Lambda^{2+\alpha}\theta\|_{L^{2}}^{\epsilon}\\
&\leq  \frac{1}{2}\|\Lambda^{2+\alpha}\theta\|_{L^{2}}^{2}
+ c\|\nabla\theta\|_{L^{\frac{3}{\epsilon\alpha}}}
^{\frac{2}{2-\epsilon}}\|\Lambda^{2}\theta\|_{L^{2}}^{2}
\end{align*}
by \eqref{e7}, Gagliardo-Nirenberg and Young's inequalities.
We choose $\epsilon = \frac{3}{\alpha}(\frac{3-2\alpha}{2p\alpha})$
for which it is easy to check that $\epsilon \in (0,1)$.
 By Gronwall's inequality,
\begin{equation*}
\sup_{t\in [0,T]}\|\Lambda^{2}\theta(t)\|_{L^{2}}^{2}
+ \int_{0}^{T}\|\Lambda^{2+\alpha}\theta\|_{L^{2}}^{2}dt
\leq \|\Lambda^{2}\theta_{0}\|_{L^{2}}^{2}e^{\int_{0}^{T}
\|\nabla\theta\|_{L^{\frac{2p\alpha}{3-2\alpha}}}
^{\frac{4p\alpha^{2}}{4p\alpha^{2}-9+6\alpha}}dt}
\end{equation*}
Similarly, by Lemma \ref{lem2.2} choosing the same $\epsilon$, we see that
\begin{equation*}
\sup_{t\in [0,T]}\|\Lambda^3\theta(t)\|_{L^{2}}^{2}
+ \int_{0}^{T}\|\Lambda^{3+\alpha}\theta\|_{L^{2}}^{2}dt < \infty
\end{equation*}
In summary, by the inhomogeneous Sobolev embedding we have
\begin{equation*}
\int_{0}^{T}\|\nabla\theta\|_{L^{\infty}}dt
\leq c\int_{0}^{T}\|\theta\|_{H^3}dt
\leq c\sup_{t\in [0,T]}\| \theta(t)\|_{H^3}T < \infty
\end{equation*}
This completes the proof of the first claim.

For the second claim, let us denote by
$\nabla_{2,3} = (0, \partial_2,\partial_{3})$.
By the previous result, it suffices to show that
\begin{equation*}
\int_{0}^{T}\|\nabla_{2,3}\theta\|_{L^{p}}^{r}dt < \infty
\end{equation*}
for some $p,r$  that \eqref{e4} is satisfied.
We fix $p$ in such a range, apply $\nabla_{2,3}$ on \eqref{e2},
 multiply by $p|\nabla_{2,3}\theta|^{p-2}\nabla_{2,3}\theta$
and integrate in space to estimate as before
\begin{align*}
&\partial_{t}\|\nabla_{2,3}\theta\|_{L^{p}}^{p}
 + c(p,\alpha)\|\nabla_{2,3}\theta\|_{L^{\frac{3p}{3-2\alpha}}}^{p}\\
&\leq  -p\int\partial_2u_1\partial_1\theta|\nabla_{2,3}\theta|^{p-2}
 \partial_2\theta
 + \partial_{3}u_1\partial_1\theta|\nabla_{2,3}\theta|^{p-2}\partial_{3}\theta\\
&\quad +\partial_2u_2\partial_2\theta|\nabla_{2,3}\theta|^{p-2}\partial_2\theta
 + \partial_{3}u_2\partial_2\theta|\nabla_{2,3}\theta|^{p-2}\partial_{3}\theta\\
&\quad +\partial_2u_{3}\partial_{3}\theta|\nabla_{2,3}\theta|^{p-2}
 \partial_2\theta
 + (-\partial_1u_1 - \partial_2u_2)\partial_{3}\theta|\nabla_{2,3}\theta|^{p-2}
 \partial_{3}\theta\\
&\leq c\|\nabla_{h}\theta\|_{L^{\frac{3}{2\alpha}}}\|\nabla_{2,3}
\theta\|_{L^{\frac{3p}{3-2\alpha}}}^{p}
\end{align*}
Hence,
\begin{equation*}
\partial_{t}\|\nabla_{2,3}\theta\|_{L^{p}}^{p}
\leq c\Big(\sup_{t\in [0,T]}\|\nabla_{h}\theta(t)\|_{L^{\frac{3}{2\alpha}}}
- c(p,\alpha)\Big)\|\nabla_{2,3}\theta\|_{L^{\frac{3p}{3-2\alpha}}}^{p}
\end{equation*}
This completes the proof of Theorem \ref{thm1.2}.
\end{proof}

\subsection*{Acknowledgments}
The author expresses gratitude to Professors Jiahong Wu and
 David Ullrich for their teaching.


\begin{thebibliography}{00}

\bibitem{b1} J. Bear;
\emph{Dynamics of fluids in porous media}, American Elsevier, New York, 1972.

\bibitem{c1} L. Caffarelli, A. Vasseur;
\emph{Drift diffusion  equations with fractional diffusion and  the
quasi-geostrophic equation}, Ann. of Math. \textbf{171}, 3 (2010), 1903-1930.

\bibitem{c2} A. Castro, D. C\'ordoba, F. Gancedo, R. Orive;
 \emph{Incompressible flow in porous media with fractional diffusion},
 Nonlinearity, \textbf{22} (2009), 1791-1815.

\bibitem{c3} D. Chae;
\emph{On the regularity conditions for the dissipative quasi-geostrophic
equations}, SIAM J. Math. Anal., \textbf{37}, 5 (2006), 1649-1656.

\bibitem{c4} D. Chae;
 \emph{The quasi-geostrophic equation in the Triebel-Lizorkin spaces},
Nonlinearity, \textbf{16}, 2 (2003), 479-495.

\bibitem{c5} P. Constantin, C. Fefferman;
\emph{Direction of vorticity and the problem of global regularity for
the Navier-Stokes equations}, Indiana Univ. Math. J., \textbf{42} (1993), 775-789.

\bibitem{c6} P. Constantin, G. Iyer, J. Wu;
\emph{Global regularity for a modified critical dissipative quasi-geostrophic
equation}, Indiana Univ. Math. J., \textbf{57}, (2001), 97-107 Special Issue.

\bibitem{c7} P. Constantin, A. J. Majda, E. Tabak;
\emph{Formation of strong fronts in the 2-D quasi-geostrophic thermal
active scalar}, Nonlinearity \textbf{7} (1994), 1495-1533.

\bibitem{c8} P. Constantin, V. Vicol;
\emph{Nonlinear maximum principles for dissipative linear nonlocal operators
 and applications}, Geom. Funct. Anal., \textbf{22} (2012), 1289-1321.

\bibitem{c9} P. Constantin, J. Wu;
\emph{Behavior of solutions of 2D quasi-geostrophic equations},
SIAM J. Math. Anal., \textbf{30} (1999), 937-948.

\bibitem{c10} P. Constantin, J. Wu;
\emph{H\"older continuity of solutions of supercritical dissipative
hydrodynamic transport equations}, Ann. I. H. Poincar\'e Anal.
Non Lin\'eaire, \textbf{26} (2009), 159-180.

\bibitem{c11} P. Constantin, J. Wu;
\emph{Regularity of H\"older continuous solutions of the supercritical
quasi-geostrophic equation}, Ann. I. H. Poincar\'e
 Anal. Non Lin\'eaire \textbf{25}, (2008), 1103-1110.

\bibitem{c12} A. C\'ordoba, D. C\'ordoba;
\emph{A maximum principle applied to quasi-geostrophic equations},
 Comm. Math. Phys. \textbf{249} (2004), 511-528.

\bibitem{d1} H. Dong and N. Pavlovic;
\emph{A regularity criterion for the dissipative quasi-geostrophic equations},
 Ann. I. H. Poincar\'e Anal. Non Lin\'eaire, \textbf{26} (2009), 1607-1619.

\bibitem{d2} J. Z. Duoandikoetxea;
\emph{Fourier analysis}, American Mathematical Society, 2000.

\bibitem{f1} J. Fan, H. Gao and G. Nakamura;
\emph{Regularity criteria for the generalized magnetohydrodynamic equations
and the quasi-geostrophic equations},
Taiwanese J. Math., \textbf{15}, 3 (2011), 1059-1073.

\bibitem{j1} N. Ju; \emph{The maximum principle and the global attractor
for the dissipative 2D quasi-geostrophic equations}, Comm. Math. Phys.,
\textbf{255} (2005), 161-181.

\bibitem{k1} T. Kato and G. Ponce;
\emph{Commutator estimates and the Euler and Navier-Stokes equations},
Comm. Pure Appl. Math., \textbf{41} (1988), 891-907.

\bibitem{k2} A. Kiselev; \emph{Regularity and blow-up for active scalars},
 Math. Model. Nat. Phenom. \textbf{5}, 4 (2010), 225-255.

\bibitem{k3} A. Kiselev, F. Nazarov;
\emph{A variation on a theme of Caffarelli and Vasseur},
Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)
\textbf{370} (2009), 58-72.

\bibitem{k4} A. Kiselev, F. Nazarov;
\emph{Global regularity for the critical dispersive dissipative surface
 quasi-geostrophic equation}, Nonlinearity \textbf{23} (2010), 549-554.

\bibitem{k5} A. Kiselev, F. Nazarov, A. Volberg;
\emph{Global well-posedness for the critical 2D dissipative quasi-geostrophic
equation.} Invent. Math. \textbf{167}, 3 (2007), 445-453.

\bibitem{k6} I. Kukavica, M. Ziane;
\emph{Navier-Stokes equations with regularity in one direction},
J. Math. Phys. \textbf{48}, 065203 (2007).

\bibitem{m1} R. May;
\emph{Global well-posedness for a modified dissipative surface
quasi-geostrophic equation in the critical Sobolev space $H^{1}$},
J. Differential Equations, \textbf{250}, 1 (2011), 320-339.

\bibitem{m2} C. Miao, L. Xue;
\emph{Global wellposedness for a modified critical dissipative
quasi-geostrophic equation}, J. Differential Equations,
\textbf{252}, 1 (2012), 792-818.

\bibitem{p1} J. Pedlosky;
 \emph{Geophysical fluid dynamics}, Springer, New York, 1987.

\bibitem{r1} S. Resnick;
\emph{Dynamical problems in nonlinear advective partial differential equations},
 Ph.D. thesis, University of Chicago, 1995.

\bibitem{v1} A. Vasseur;
\emph{Regularity criterion for 3D Navier-Stokes equations in terms
of the direction of the vorticity}, Appl. Math. \textbf{54}, 1 (2009), 47-52.

\bibitem{w1} J. Wu;
\emph{Dissipative quasi-geostrophic equations with $L^{p}$ data},
 Electron. J. Differential Equations,  \textbf{2001},  (2001), no. 56.  1-13.

\bibitem{w2} J. Wu;
 \emph{Lower bounds for an integral involving fractional Laplacians and
the generalized Navier-Stokes equations in Besov spaces},
Comm. Math. Phys., \textbf{263} (2005), 803-831.

\bibitem{x1} Z. Xiang;
\emph{A regularity criterion for the critical and supercritical dissipative
quasi-geostrophic equations}, Appl. Math. Lett., \textbf{23} (2010), 1286-1290.

\bibitem{x2}  Y. Xinwei;
\emph{Remarks on the global regularity for the super-critical
2D dissipative quasi-geostrophic equation}, J. Math. Anal. Appl.,
\textbf{339} (2008), 359-371.

\bibitem{x3} L. Xue;
\emph{On the well-posedness of incompressible flow in porous media
with supercritical diffusion}, Appl. Anal., \textbf{88}, 4, (2009), 547-561.

\bibitem{y1} K. Yamazaki;
\emph{On the regularity criteria of a surface quasi-geostrophic equation},
Nonlinear Anal., \textbf{75} (2012), 4950-4956.

\bibitem{y2} K. Yamazaki;
\emph{Remarks on the method of modulus of continuity and the modified
dissipative porous media equation}, J. Differential Equations,
\textbf{250}, 4 (2011), 1909-1923.

\bibitem{y3} K. Yamazaki;
\emph{A remark on the global well-posedness of a modified critical
quasi-geostrophic equation}, arXiv:1006.0253v2 [math.AP], Aug. 21. 2011.

\bibitem{y4} K. Yamazaki;
\emph{Regularity criteria of N-dimensional porous media equation in
terms of one partial derivative or pressure scalar field}, submitted.

\bibitem{y5} B. Yuan;
\emph{Regularity condition of solutions to the quasi-geostrophic
equations in Besov spaces with negative indices}, \textbf{26},
Acta Math. Appl. Sin., \textbf{26}, 3 (2010), 381-386.

\bibitem{y6} B. Yuan, J. Yuan
\emph{Global well-posedness of incompressible flow in porous media
with critical diffusion in Besov spaces},
J. Differential Equations \textbf{246} (2009), 4405-4422.

\bibitem{y7} J. Yuan;
\emph{On regularity criterion for the dissipative quasi-geostrophic equations},
J. Math. Anal. Appl. \textbf{340} (2008), 334-339.

\end{thebibliography}

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