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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 299, pp. 1--14.\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.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/299\hfil 
 Burgers-complex-Ginzburg-Landau equations]
{Global solutions for 2D coupled Burgers-complex-Ginzburg-Landau
equations}

\author[H. Gao, L. Lin, Y. Chu \hfil EJDE-2015/299\hfilneg]
{Hongjun Gao, Lin Lin, Yajun Chu}

\address{Hongjun Gao \newline
Institute of Mathematics, School of Mathematical Sciences,
Nanjing Normal University, Nanjing 210023, China}
\email{gaohj@njnu.edu.cn}

\address{Lin Lin (corresponding author) \newline
Institute of Mathematics, School of Mathematical Sciences,
Nanjing Normal University, Nanjing 210023, China}
\email{linlin@njnu.edu.cn}

\address{Yajun Chu \newline
Institute of Mathematics, School of Mathematical Sciences,
Nanjing Normal University, Nanjing 210023, China}
\email{chuyjnjnu@163.com}

\thanks{Submitted June 2, 2015. Published 7, 2015.}
\subjclass[2010]{35M13, 35D35}
\keywords{Burgers-CGL equation; prior estimate; global solution;
\hfill\break\indent Gagliardo-Nirenberg inequality; H\"older inequality}

\begin{abstract}
 In this article, we study the periodic initial-value problem of the 2D 
 coupled Burgers-complex-Ginzburg-Landau (Burgers-CGL) equations.
 Applying the Brezis-Gallout inequality which is available in 2D case
 and establishing some prior estimates, we obtain the existence and 
 uniqueness of a global solution under certain conditions.
\end{abstract}

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

\section{Introduction}

 In this article, we are concerned with the periodic initial-value problem for
the following 2D coupled Burgers-complex-Ginzburg-Landau (Burgers-CGL) equations:
\begin{gather}
\begin{gathered}
P_t=\xi{P}+(1+iu)\Delta P-(1+iv)|P|^2P-\nabla P\cdot\nabla{Q}-r{P}\Delta Q
+f(x),\\
(x,t)\in\Omega\times{\mathbb{R}^+},
\end{gathered} \label{e1.1}\\
Q_t=m\Delta Q-\frac{1}{2}|\nabla{Q}|^2-\omega|P|^2+g(x),\quad
(x,t) \in\Omega\times{\mathbb{R}^+},
\label{e1.2}\\
P(x_i+2L,t)=P(x_i,t),Q(x_i+2L,t)=Q(x_i,t),\quad
(x,t)\in\Omega\times{\mathbb{R}^+},
\label{e1.3}\\
 P(x,0)=P_0(x),\quad Q(x,0)=Q_0(x),\quad x\in\Omega, \label{e1.4}
\end{gather}
where $\Omega=[-L,L]\times{[-L,L]}$, $2L$ is the period,
$f(x)$ and $g(x)$ are given real functions. The complex function
$P(x, t)$ is the rescaled amplitude of the flame oscillations,
the real function $Q(x, t)$ is the deformation of the first front.
The Landau coefficients $u$, $v$ are real and the coefficients $\xi$, $m$
are positive, while the coupling coefficients $\omega>0$ and $r=r_1+i r_2$.

 The coupled Burgers-CGL equations was derived from the nonlinear evolution
of the coupled long-scale oscillatory and monotonic instabilities of a
uniformly propagating combustion wave governed by a sequential chemical reaction,
having two flame fronts corresponding to two reaction zones with a finite
separation distance between them (see \cite{b2,g4,k1,01,p1,s1}).
It describes the interaction of the excited oscillatory mode and the
damped monotonic mode, that is  $\xi>0, m>0$.

 Let us recall some previous works which are related to our results.
If there are no terms involved Q in \eqref{e1.1}, then \eqref{e1.1}
reduces to the well-known
complex Ginzburg-Landau equation (CGL) that describes the weakly nonlinear
evolution of a long-scale instability (see \cite{n1}). Many results on the
well-posedness and global attractors for the CGL equation have been established,
one can find the details in \cite{g1,g2,g3,l1,l2,y1}.
And if taking the coupling coefficient $\omega=0$, \eqref{e1.1} would
be the well-known Burgers equation (see \cite{b3}). There are also many
works concerning the Burgers equation, one can see \cite{c1,z1} and so on.
So far, the mathematical analysis and physical study about the coupled
Burgers-CGL have been done by a few researchers. For the periodic initial-value
 problem of the 1-D Burgers-Ginzburg-Landau equation, well-posedness
and global attractors are obtained in \cite{g5}.
By some prior estimates and so-called continuity method,
the authors in \cite{g6} firstly showed the global existence of solutions
and attractors to 1-D coupled Burgers-CGL equations for flames
governed by sequential reaction, and then obtained the existence of
the global attractor $\mathcal{A}\subset H^2(\Omega)\times H^3(\Omega)$
for the problem. In addition, they established the estimates of the upper
bounds of Hausdorff and fractal dimensions for the attractors.
 However, to our knowledge, there is few work on the 2D coupled Burgers-CGL
equations. We will prove global well-posedness of
problem \eqref{e1.1}-\eqref{e1.4} as conjectured in \cite{g6}.
The method used here is well established but need to be applied
skillfully to get the desired result.

  The purpose of this article is to study the global solutions of
\eqref{e1.1}-\eqref{e1.4}. The existence and uniqueness of the global
solutions to \eqref{e1.1}-\eqref{e1.4} is obtained by virtue of
the Brezis-Gallout inequality, which is available in the 2D case.
The main result in our paper is as follows.

\begin{theorem} \label{thm1.1}
Suppose that $P_0(x)\in H^2(\Omega)$, $Q_0(x)\in H^3(\Omega)$,
$f(x)\in H^1(\Omega)$ and $g(x)\in H^2(\Omega)$, for any given $T>0$,
then \eqref{e1.1}-\eqref{e1.4}
has the unique solution $P(x,t)$ and $Q(x,t)$ satisfying
\begin{gather*}
P(x,t)\in L^\infty\big(0,T;H^2(\Omega)\big),\quad
P_t(x,t)\in L^\infty\big(0,T;L^2(\Omega)\big),\\
Q(x,t)\in L^\infty\big(0,T;H^3(\Omega)\big),\quad
Q_t(x,t)\in L^\infty\big(0,T;H^1(\Omega)\big).
\end{gather*}
\end{theorem}

 The rest of this article is organized as follows.
In section 2, we briefly give some preliminaries.
Some prior estimates for the solutions to  \eqref{e1.1}-\eqref{e1.2}
are established  in section 3.
Then, by the standard method, we can extend the local solutions to global
 solutions and subsequently the proof of Theorem \ref{thm1.1} is presented.


\section{Preliminaries}
In this section, We firstly recall some concepts and conventional notation.
Let $W^{k, p}(\Omega)$ denote the usual $k$th order Sobolev space with its norm
$$
\|u\|_{W^{k,p}(\Omega)}:=
\begin{cases}
  \Big(\sum_{|\alpha|\leq k}\int_\Omega|D^\alpha u|^p\,{\rm d}x\Big)^{1/p},& 1\leq p<\infty,\\
\sum_{|\alpha|\leq k}\operatorname{ess\,sup}_\Omega|D^\alpha u|, & p=\infty.
\end{cases}
$$
When $p=2$, we  denote $\|u\|_{H^k(\Omega)}=\|u\|_{W^{k,2}(\Omega)}$.
The inner product in $L^2(\Omega)$ is defined by
$$
(u,v)= \operatorname{Re}\int_{\Omega}u(x)\bar{v}(x)dx.
$$
For simplicity, we write $\|u\|=\|u\|_{L^2}$. Without any ambiguity,
we denote a generic positive constant by C which may vary from line to line.
The following inequalities play an important role in the proof of
the main results in $\mathbb{R}^2$.

\begin{lemma}[Gagliardo-Nirenberg inequality \cite{f1}] \label{lem2.1}
 Let $\Omega$ be a bounded domain with $\partial\Omega$ in $C^m$, and let
$u$ be any function in $W^{m,r}(\Omega)\cap L^q(\Omega)$,
$1\leq q, r\leq\infty$. For any integer $j$, $0\leq j\leq m,$ and for any number
$a$ satisfying $j/m\leq a\leq 1$, set
$$
\frac{1}{p}=\frac{j}{n}+a\Big(\frac{1}{r}-\frac{m}{n}\Big)+(1-a)\frac{1}{q}.
$$
If $m-j-n/r$ is not a non-negative integer, then
\begin{equation} \label{e2.1}
 \|D^ju\|_{L^p}\leq C\|u\|^a_{W^{m,r}}\|u\|^{1-a}_{L^q}.
\end{equation}
If $m-j-n/r$ is a non-negative integer, then \eqref{e2.1} holds
for $a=j/m$. The constant $C$ depends only on $\Omega, r, q, j$ and $a$.
\end{lemma}

\begin{lemma}[Brezis-Gallout inequality \cite{b1}] \label{lem2.2}
  Let $\Omega$ be a bounded domain in $\mathbb{R}^2$, and let $u $ be
any function in $H^2(\Omega)$. If $\|u\|_{H^1}\leq 1$, then there holds
\begin{equation} \label{e2.2}
 \|u\|_{L^\infty}\leq C\big(1+\sqrt{\log(1+\|u\|_{H^2})}\big).
\end{equation}
\end{lemma}


\section{Existence and uniqueness of solutions}

In this section, we firstly establish some  prior estimates for the solutions
to \eqref{e1.1}-\eqref{e1.4} which guarantee the existence of the global solutions.

\begin{lemma} \label{lem3.1}
Assume that $g(x)\in{L}^2(\Omega)$, then for the problem \eqref{e1.1}-\eqref{e1.4},
it holds that
 $$
\frac{d}{{\rm d}t}(\|Q\|^2)\leq E_0
\big(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3})\big)
\big(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\big)+E_1,
$$
 where $E_0,E_1$ are constants.
\end{lemma}

\begin{proof}
Multiplying \eqref{e1.2} by $Q$ and integrating with respect to $x$
over $\Omega$ gives
\begin{equation} \label{e3.1}
 \frac{d}{{\rm d}t}\|Q\|^2=-2m\|\nabla
Q\|^2-\int_\Omega|\nabla Q|^2Q\,{\rm d}x-2\omega\int_\Omega|P|^2Q\,{\rm d}x+
2\int_\Omega gQ\,{\rm d}x.
\end{equation}
Applying the Brezis-Gallout inequality (Lemma \ref{lem2.2})
and H\"{o}lder inequality, we deduce that
\begin{equation} \label{e3.2}
\begin{aligned}
\big|-\int_\Omega |\nabla Q|^2Q\,{\rm d}x\big|
&\leq \| Q\|_{L^\infty}\|\nabla Q\|^2 \\
&\leq C\Big(1+\sqrt{\log(1+\|Q\|_{H^2})}\Big)\|\nabla Q\|^2\\
&\leq C\left(1+\log(1+\|Q\|_{H^2})\right)\|Q\|^2_{H^1}
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.3}
\begin{aligned}
\big|-2\omega\int_\Omega|P|^2Q\,{\rm d}x\big|
&\leq 2\omega\|Q\|_{L^\infty}\|P\|^2\\
&\leq C\left(1+\sqrt{\log(1+\|Q\|_{H^2})}\right)\|P\|^2\\
&\leq C\left(1+\log(1+\|Q\|_{H^2})\right)\|P\|^2.
\end{aligned}
\end{equation}
Since $g(x)\in{L}^2(\Omega)$, obviously
\begin{equation} \label{e3.4}
\big|2\int_\Omega gQ\,{\rm d}x\big|
\leq \|Q\|^2+\|g\|^2\leq\|Q\|^2+C_1.
\end{equation}
Substituting \eqref{e3.2}-\eqref{e3.4} back into \eqref{e3.1}, we infer that
\begin{equation} \label{e3.5}
\begin{aligned}
\frac{d}{{\rm d}t}\|Q\|^2
&\leq -2m\|\nabla Q\|^2+C\left(1+\log(1+\|Q\|_{H^2})\right)\|Q\|^2_{H^1}\\
&\quad +C\left(1+\log(1+\|Q\|_{H^2})\right)\|P\|^2+\|Q\|^2+C_1\\
& \leq C(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3}))\left(\|P\|^2_{H^2}
  +\|Q\|^2_{H^3}\right)+C_1\\
&=E_0\left(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3})\right)
  \left(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\right)+E_1.
\end{aligned}
\end{equation}
Thus, the proof is complete.
\end{proof}

\begin{lemma} \label{lem3.2}
Assume that $f(x)\in{L}^2(\Omega)$ and $g(x)\in{L}^2(\Omega)$,
then for  \eqref{e1.1}-\eqref{e1.4}, we have
$$
\frac{d}{{\rm d}t}\left(\|P\|^2+\|\nabla Q\|^2\right)
\leq E_2\left(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3})\right)
\left(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\right)+E_3,
$$
where $E_2,E_3$ are constants.
\end{lemma}

\begin{proof}
By differentiating \eqref{e1.2} with respect to $x$ and setting
\begin{equation} \label{e3.6}
 W=\nabla{Q},
\end{equation}
Equations \eqref{e1.1}-\eqref{e1.2} become
\begin{gather} 
P_t=\xi{P}+(1+iu)\Delta{P}-(1+iv)\mid{P}\mid^2{P}-\nabla{P}
\cdot W-rP\operatorname{div}W+f, \label{e3.7}
\\
W_t=m\nabla(\operatorname{div}W)-\frac{1}{2}\nabla(W^2)
-\omega\nabla(|P|^2)+\nabla{g}. \label{e3.8}
\end{gather}
Multiplying \eqref{e3.7} by $\overline{P}$, integrating with respect
to $x$ over $\Omega$ and taking the real part gives
\begin{equation} \label{e3.9}
\begin{aligned}
\frac{1}{2}\frac{d}{{\rm d}t}\|P\|^2
&=\xi \|P\|^2-\|\nabla P\|^2-\|P\|_{L^4}^4
- \operatorname{Re}\int_\Omega\nabla{P}\cdot
W\overline{P}\,{\rm d}x \\
&\quad - \operatorname{Re}\int_\Omega r|P|^2\operatorname{div}W\,{\rm d}x
+\operatorname{Re}\int_\Omega{f}\overline{P}\,{\rm d}x,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.10}
-\operatorname{Re}\int_\Omega\nabla{P}\cdot W\overline{P}\,{\rm d}x
=-\frac{1}{2}\int_\Omega W\cdot\nabla(|P|^2)\,{\rm d}x
=\frac{1}{2}\int_\Omega |P|^2\operatorname{div}W\,{\rm d}x.
\end{equation}
While multiplying \eqref{e3.8} by $W$ and integrating over $\Omega$ yields
\begin{equation} \label{e3.11}
\begin{aligned}
\frac{1}{2}\frac{d}{{\rm d}t}\|W\|^2
&=-m\|\operatorname{div}W\|^2-\frac{1}{2}\int_\Omega
\nabla(W^2)\cdot W\,{\rm d}x+\int_\Omega\nabla{g}\cdot W\,{\rm d}x\\
&\quad -\omega\int_\Omega\nabla(|P|^2)\cdot W\,{\rm d}x,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.12}
 -\omega\int_\Omega\nabla(|P|^2)\cdot W\,{\rm d}x
=\omega\int_\Omega|P|^2\operatorname{div}W\,{\rm d}x.
\end{equation}
Combining \eqref{e3.9} with \eqref{e3.12}, we arrive at
\begin{equation} \label{e3.13}
\begin{aligned}
&\frac{d}{{\rm d}t}(\|P\|^2+\|W\|^2)\\
&= 2\xi \|P\|^2-2\|\nabla P\|^2-2m
\|\operatorname{div}W\|^2-2\|P\|_{L^4}^4\,{\rm d}x\\
&\quad +(2\omega+1-2r_1)\int_\Omega |P|^2\operatorname{div}W\,{\rm d}x
-\int_\Omega \nabla(W^2)\cdot W\,{\rm d}x \\
&\quad +2\operatorname{Re}\int_\Omega{f}\overline{P}\,{\rm d}x
 +2\int_\Omega\nabla{g}\cdot W\,{\rm d}x
=:\sum_{i=1}^5 I_i,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.14}
\begin{gathered}
I_1=2\xi\,\|P\|^2-2\|\nabla P\|^2
 -2m\|\operatorname{div}W\|^2-2\|P\|_{L^4}^4\,, \\
I_2=(2\omega+1-2r_1)\int_\Omega |P|^2\operatorname{div}W\,{\rm d}x ,\quad
I_3=-\int_\Omega \nabla(W^2)\cdot W\,{\rm d}x,\\
I_4=2 \operatorname{Re}\int_\Omega{f}\overline{P}\,{\rm d}x, \quad
I_5=2\int_\Omega\nabla{g}\cdot W\,{\rm d}x.
\end{gathered}
\end{equation}
We now estimate each term on the right-hand side of \eqref{e3.13}.
By the Brezis-Gallout inequality and H\"{o}lder inequality,
$I_2$ and $I_3$ can be estimated as follows
\begin{equation} \label{e3.15}
\begin{aligned}
|I_2|&\leq |2\omega+1-2r_1|\|P\|^2_{L^\infty}|\Omega|^{1/2}
 \|\operatorname{div}W\|\\
&\leq \frac{m}{2}\|\operatorname{div}W\|^2+C\|P\|^4_{L^\infty}\\
&\leq \frac{m}{2}\|\operatorname{div}W\|^2+C(1+\sqrt{\log(1+\|P\|_{H^2})})^4\\
&\leq \frac{m}{2}\|\operatorname{div}W\|^2+C(1+\log^2(1+\|P\|_{H^2}))\\
&\leq \frac{m}{2}\|\operatorname{div}W\|^2+C\|P\|_{H^2}^2+C\\
&\leq C(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2}))(\|P\|^2_{H^2}
 +\|W\|^2_{H^2})\\
&\quad +\frac{m}{2}\|\operatorname{div}W\|^2+C
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.16}
\begin{aligned}
|I_3|
&\leq \|W\|_{L^\infty}\|W\|\|\operatorname{div}W\|
 \leq \frac{m}{2}\|\operatorname{div}W\|^2
  +C\|W\|^2_{L^\infty}\|W\|^2\\
&\leq\frac{m}{2}\|\operatorname{div}W\|^2+C(1+\log(1+\|W\|_{H^2}))\|W\|^2\\
&\leq C(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2}))(\|P\|^2_{H^2}+\|W\|^2_{H^2})
+\frac{m}{2}\|\operatorname{div}W\|^2.
\end{aligned}
\end{equation}
For $I_4$ and $I_5$, one can deduce that
\begin{equation} \label{e3.17}
\begin{aligned}
|I_4|
&\leq\|P\|^2+\|f\|^2 \leq\|P\|_{H^2}+C_2\\
&\leq \left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
\left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)+C_2
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.18}
|I_5|\leq 2\|g\|\|\operatorname{div}W\|
\leq m\|\operatorname{div}W\|^2+C_3,
\end{equation}
provided $f(x)\in{L}^2(\Omega)$ and $g(x)\in{L}^2(\Omega)$.
Substituting the above estimates \eqref{e3.15}--\eqref{e3.18}
back into \eqref{e3.13} leads to
\begin{equation} \label{e3.19}
\begin{aligned}
&\frac{d}{{\rm d}t}\left(\|P\|^2+\|W\|^2\right)\\
&\leq2\xi\|P\|^2-2\|\nabla P\|^2-2\int_\Omega |P|^4\,{\rm d}x
+C\big(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\big)\\
&\quad\times \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)+C+C_2+C_3\\
&=E_2\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
\left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)+E_3.
\end{aligned}
\end{equation}
Noting \eqref{e3.6}, we finally obtain
\begin{equation} \label{e3.20}
\begin{aligned}
&\frac{d}{{\rm d}t}\left(\|P\|^2+\|\nabla Q\|^2\right) \\
&\leq E_2\left(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3})\right)
 \left(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\right)+E_3,
\end{aligned}
\end{equation}
which is the desired result.
\end{proof}

To obtain the higher order estimates with the complex amplitude of
the flame oscillations $P(x,t)$ and the deformation of the first
front $Q(x,t)$, we can use the transformed equations
\eqref{e3.7}-\eqref{e3.8} and have the following lemmas.

\begin{lemma} \label{lem3.3}
 Assume that $f(x)\in{L}^2(\Omega$ and $g(x)\in{H}^1(\Omega)$,
then for the problem \eqref{e1.1}-\eqref{e1.4}, we have
\begin{align*}
&\frac{d}{{\rm d}t}\left(\|\nabla P\|^2+\|\Delta Q\|^2\right)\\
&\leq E_4\left(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3})\right)
 \left(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\right)+E_5,
\end{align*}
where $E_4,E_5$ are constants.
\end{lemma}

\begin{proof}
Multiplying \eqref{e3.7} by $(-\overline{\Delta P})$, integrating with
respect to $x$ over $\Omega$ and taking the real part, we obtain
\begin{equation} \label{e3.21}
\begin{aligned}
\frac{1}{2}\frac{d}{{\rm d}t}\|\nabla P\|^2
&= \xi\|\nabla P\|^2-\|\Delta P\|^2+\operatorname{Re}\int_\Omega
(1+iv)|P|^2P\,\overline{\Delta P}\,{\rm d}x \\
&\quad  +\operatorname{Re}\int_\Omega \nabla P\cdot W
 \overline{\Delta P}\,{\rm d}x+\operatorname{Re}
 \int_\Omega rP\operatorname{div}W\overline{\Delta P}\,{\rm d}x\\
&\quad -\operatorname{Re}\int_\Omega f\overline{\Delta P}\,{\rm d}x.
\end{aligned}
\end{equation}
On the other hand, multiplying \eqref{e3.8} by $(-\nabla(\operatorname{div}W))$
and integrating with respect to $x$ over $\Omega$ yields
\begin{equation} \label{e3.22}
\begin{aligned}
\frac{1}{2}\frac{d}{{\rm d}t}\|\operatorname{div}W\|^2
&= -m\|\nabla (\operatorname{div}W)\|^2+\frac{1}{2}\int_\Omega \nabla(W^2)\cdot
\nabla(\operatorname{div}W)\,{\rm d}x\\
& \quad +\omega\int_\Omega\nabla(|P|^2)\cdot\nabla (\operatorname{div}W)\,{\rm d}x-\int_\Omega\nabla g\cdot\nabla
(\operatorname{div}W)\,{\rm d}x.
\end{aligned}
\end{equation}
Adding \eqref{e3.22} and \eqref{e3.21}, one has
\begin{equation} \label{e3.23}
\begin{aligned}
&\frac{d}{{\rm d}t}\left(\|\nabla P\|^2+\|\operatorname{div}W\|^2\right)\\
&= 2\xi \|\nabla P\|^2-2\|\Delta P\|^2
 -2m\|\nabla(\operatorname{div}W)\|^2\\
&\quad +2\operatorname{Re}\int_\Omega (1+iv)|P|^2P\overline{\Delta P}\,{\rm d}x
 +2\operatorname{Re}\int_\Omega \nabla P\cdot W\overline{\Delta P}\,{\rm d}x\\
&\quad +2\operatorname{Re}\int_\Omega r
P\operatorname{div}W\overline{\Delta P}\,{\rm d}x
+\int_\Omega \nabla(W^2)\cdot\nabla(\operatorname{div}W)\,{\rm d}x\\
&\quad +2\omega\int_\Omega\nabla(|P|^2)\cdot\nabla( \operatorname{div}W)\,{\rm d}x-2\operatorname{Re}\int_\Omega f\overline{\Delta P}\,{\rm d}x\\
&\quad -2\int_\Omega\nabla g\cdot\nabla(\operatorname{div}W)\,{\rm d}x
=\sum_{i=1}^8 J_i,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.24}
\begin{gathered}
J_1=2\xi \|\nabla P\|^2-2\|\Delta P\|^2
 -2m\|\nabla(\operatorname{div}W)\|^2,\\
J_2= 2\operatorname{Re}\int_\Omega (1+iv)|P|^2P\overline{\Delta P}\,{\rm d}x,\quad
J_3=\operatorname{Re}\int_\Omega \nabla P\cdot W\overline{\Delta P}\,{\rm d}x,\\
J_4 =2\operatorname{Re}\int_\Omega r P\operatorname{div}
 W\overline{\Delta P}\,{\rm d}x,\quad
J_5=\int_\Omega \nabla(W^2)\cdot\nabla(\operatorname{div}W)\,{\rm d}x,\\
J_6=\omega\int_\Omega\nabla(|P|^2)\cdot\nabla( \operatorname{div}W)\,{\rm d}x,\quad
J_7=-2\operatorname{Re}\int_\Omega f\overline{\Delta P}\,{\rm d}x,\\
J_8=-2\int_\Omega\nabla g\cdot\nabla(\operatorname{div}W)\,{\rm d}x.
\end{gathered}
\end{equation}
We now estimate $J_i$ ($i=2,\dots,8$) in \eqref{e3.23}.
From Lemma \ref{lem2.2}, it follows that
\begin{equation} \label{e3.25}
\begin{aligned}
|J_2|
&\leq 2|1+iv|\|P\|^2_{L^\infty}\|P\|\|\Delta P\|
 \leq C\|P\|^2_{L^\infty}\|P\|^2_{H^2}\\
&\leq C\left(1+\log(1+\|P\|_{H^2})\right)\|P\|^2_{H^2}\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right),
\end{aligned}
\end{equation}
\begin{equation} \label{e3.26}
\begin{aligned}
|J_3|&\leq 2\|W\|_{L^\infty}\|\nabla P\|\|\Delta P\|
 \leq 2\|W\|_{L^\infty}\|P\|^2_{H^2}\\
&\leq C(1+\log(1+\|W\|_{H^2}))\|P\|^2_{H^2}\\
&\leq C(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2}))(\|P\|^2_{H^2}
 +\|W\|^2_{H^2})
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.27}
\begin{aligned}
|J_4|
&\leq 2|r|\|P\|_{L^\infty}\|\operatorname{div}W\|\|\Delta P\|
 \leq C\|P\|^2_{L^\infty}\|\Delta P\|^2+\|\operatorname{div}W\|^2\\
&\leq C\left(1+\sqrt{\log(1+\|P\|_{H^2})}\right)^2\|\Delta P\|^2
+\|\operatorname{div}W\|^2\\
&\leq C\left(1+\log(1+\|P\|_{H^2})\right)\|P\|_{H^2}+C\|W\|_{H^2}\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right),
\end{aligned}
\end{equation}
where we used the Brezis-Gallout and H\"{o}lder inequalities.
 By  direct computations, we have
\begin{equation} \label{e3.28}
\begin{aligned}
&|J_5|\\
&\leq 2\|W\|_{L^\infty}\|\nabla W\|\|\nabla(\operatorname{div}W)\|
 \leq \frac{1}{m}\|W\|^2_{L^\infty}\|\nabla
  W\|^2+m\|\nabla(\operatorname{div}W)\|^2\\
&\leq C\left(1+\log(1+\|W\|_{H^2})\right)\|W\|^2_{H^2}
+m\|\nabla(\operatorname{div}W)\|^2\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)+m\|\nabla(\operatorname{div}W)\|^2
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.29}
\begin{aligned}
|J_6|
&\leq 4\omega\|P\|_{L^\infty}\|\nabla P\|\|\nabla(\operatorname{div}W)\|
 \leq \frac{8\omega^2}{m}\|P\|^2_{L^\infty}\|\nabla P\|^2\\
&\quad  +\frac{m}{2}\|\nabla(\operatorname{div}W)\|^2\\
&\leq C\left(1+\log(1+\|P\|_{H^2})\right)\|P\|^2_{H^2}
 +\frac{m}{2}\|\nabla(\operatorname{div}W)\|^2\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)\\
&\quad +\frac{m}{2}\|\nabla(\operatorname{div}W)\|^2.
\end{aligned}
\end{equation}
In addition, $J_7$ and $J_8$ can be estimated as follows
\begin{equation} \label{e3.30}
\begin{aligned}
|J_7|
&\leq \|\Delta P\|^2+\|f\|^2
 \leq\|P\|^2_{H^2}+C_4\\
&\leq \left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)+C_4
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.31}
\begin{aligned}
|J_8|
&\leq 2\|\nabla g\|\|\nabla(\operatorname{div}W)\|
 \leq \frac{m}{2}\|\nabla(\operatorname{div}W)\|^2+\frac{2}{m}\|\nabla g\|^2\\
&\leq \frac{m}{2}\|\nabla(\operatorname{div}W)\|^2+C_5.
\end{aligned}
\end{equation}
 Substituting the above estimate \eqref{e3.25}-\eqref{e3.31}
into \eqref{e3.23} leads to
\begin{equation} \label{e3.32}
\begin{aligned}
&\frac{d}{{\rm d}t}(\|\nabla P\|^2+\|\operatorname{div}W\|^2)\\
&\leq 2\xi\|\nabla P\|^2-2\|\Delta P\|^2 +(C+1)
 \big(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\big) \\
&\quad\times (\|P\|^2_{H^2}+\|W\|^2_{H^2})+C_4+C_5\\
&\leq E_4(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2}))(\|P\|^2_{H^2}+\|W\|^2_{H^2})+E_5.
\end{aligned}
\end{equation}
From the transformation \eqref{e3.6}, we finally obtain
\begin{equation} \label{e3.33}
\begin{aligned}
&\frac{d}{{\rm d}t}(\|\nabla P\|^2+\|\Delta Q\|^2)\\
&\leq E_4(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3}))(\|P\|^2_{H^2}+\|Q\|^2_{H^3})+E_5,
\end{aligned}
\end{equation}
which completes the proof of this lemma.
\end{proof}

\begin{lemma} \label{lem3.4}
 Assume that $f(x)\in{H}^1(\Omega)$ and $g(x)\in{H}^2(\Omega)$,
then for the problem \eqref{e1.1}-\eqref{e1.4}, we have
\begin{align*}
&\frac{d}{{\rm d}t}\left(\|\Delta P\|^2+\|\nabla\Delta Q\|^2\right)\\
&\leq E_6\left(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3})\right)
 \left(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\right)+E_7,
\end{align*}
where $E_6$ and $E_7$ are constants.
\end{lemma}

\begin{proof}
Multiplying \eqref{e3.7} by $\overline{\Delta^2P}$, integrating with respect
to $x$ over $\Omega$ and taking the real part gives
\begin{equation} \label{e3.34}
\begin{aligned}
 \frac{1}{2}\frac{d}{{\rm d}t}\|\Delta P\|^2
&=\xi\|\Delta P\|^2-\|\nabla\Delta P\|^2-\operatorname{Re}
 \int_\Omega(1+iv)|P|^2P\overline{\Delta^2P}\,{\rm d}x\\
&\quad -\operatorname{Re}\int_\Omega \nabla P\cdot W\overline{\Delta^2P}\,{\rm d}x
-\operatorname{Re}\int_\Omega rP\operatorname{div}W\overline{\Delta^2P}\,{\rm d}x\\
&\quad +\operatorname{Re}\int_\Omega f\overline{\Delta^2P}\,{\rm d}x.
\end{aligned}
\end{equation}
Taking the inner product of \eqref{e3.8} with $\nabla\Delta (\operatorname{div}W)$
over $\Omega$, one has
\begin{equation} \label{e3.35}
\begin{aligned}
&\frac{1}{2}\frac{d}{{\rm d}t}\|\nabla
(\operatorname{div}W)\|^2\\
&=-m\|\Delta (\operatorname{div}W)\|^2-\frac{1}{2}\int_\Omega
\nabla(W^2)\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x\\
&\quad -\omega\int_\Omega\nabla(|P|^2)\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x+\int_\Omega\nabla
g\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x.\\
\end{aligned}
\end{equation}
Adding \eqref{e3.35} and \eqref{e3.34}, we have
\begin{equation} \label{e3.36}
\begin{aligned}
&\frac{d}{{\rm d}t}(\|\Delta P\|^2+\|\nabla
 (\operatorname{div}W)\|^2)\\
&=2\xi\|\Delta P\|^2-2\|\nabla\Delta
 P\|^2-2m\|\Delta (\operatorname{div}W)\|^2 \\
&\quad -2\operatorname{Re}\int_\Omega(1+iv)|P|^2P\overline{\Delta^2P}\,{\rm d}x
 -2\operatorname{Re}\int_\Omega \nabla P\cdot W\overline{\Delta^2P}\,{\rm d}\\
&\quad -2\operatorname{Re}\int_\Omega rP\operatorname{div}W\overline{\Delta^2P}
 \,{\rm d}x-\int_\Omega \nabla(W^2)\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x\\
&\quad -2\omega\int_\Omega\nabla(|P|^2)\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x
 +2\operatorname{Re}\int_\Omega f\overline{\Delta^2P}\,{\rm d}x\\
&\quad +2\int_\Omega\nabla g\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x
=:\sum_{i=1}^8 K_i,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.37}
\begin{gathered}
K_1=2\xi\|\Delta P\|^2-2\|\nabla\Delta
P\|^2-2m\|\Delta (\operatorname{div}W)\|^2\\
K_2= -2\operatorname{Re}\int_\Omega(1+iv)|P|^2P\overline{\Delta^2P}\,{\rm d}x,\quad
K_3=-2\operatorname{Re}\int_\Omega \nabla P\cdot W\overline{\Delta^2P}\,{\rm d},\\
K_4=-2\operatorname{Re}\int_\Omega rP\operatorname{div}
 W\overline{\Delta^2P}\,{\rm d}x,\quad
K_5=-\int_\Omega \nabla(W^2)\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x\\
K_6=-2\omega\int_\Omega\nabla(|P|^2)\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x,\quad
K_7=2\operatorname{Re}\int_\Omega f\overline{\Delta^2P}\,{\rm d}x,\\
K_8=2\int_\Omega\nabla g\cdot\nabla\Delta (\operatorname{div}W)\,{\rm d}x.
\end{gathered}
\end{equation}
Next we analyze the each integrand in \eqref{e3.36}.
It follows from Lemmas \ref{lem2.1} \ref{lem2.2} that
 \begin{align}
|K_2|&\leq  6|1+iv|\|P\|^2_{L^\infty}\|\nabla P\|\|\nabla\Delta P\|
\leq \frac{1}{2}\|\nabla\Delta P\|^2+C\|P\|^4_{L^\infty}\|\nabla P\|^2
\nonumber \\
&\leq  \frac{1}{2}\|\nabla\Delta P\|^2+C\left(1+\log^2(1+\|P\|_{H^2})\right)
 \|P\|\|\Delta P\| \nonumber \\
&\leq \frac{1}{2}\|\nabla\Delta P\|^2+C(1+\|P\|_{H^2})\|P\|\| P\|_{H^2} \nonumber \\
&\leq \frac{1}{2}\|\nabla\Delta P\|^2+C\|P\|^2_{H^2}
 +C|\Omega|^\frac{1}{2}\|P\|_{L^\infty}\|P\|^2_{H^2}\nonumber  \\
&\leq \frac{1}{2}\|\nabla\Delta P\|^2
 +C\left(1+\log(1+\|P\|_{H^2})\right)\|P\|^2_{H^2} \nonumber \\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right) \nonumber \\
&\quad +\frac{1}{2}\|\nabla\Delta P\|^2. \label{e3.38}
\end{align}
With respect to the term $K_3$, by direct computation, one has
\begin{equation} \label{e3.39}
\begin{aligned}
|K_3|
&=\Big|2\operatorname{Re}\int_\Omega
(D^2P\cdot W\cdot\overline{\nabla\Delta P}+\nabla P\cdot\nabla
W\cdot\overline{\nabla\Delta P})\,{\rm d}x\Big|\\
&\leq 2\|W\|_{L^\infty}\|D^2P\|\|\nabla\Delta P\|+\|\nabla
P\|_{L^4}\|\nabla W\|_{L^4}\|\nabla\Delta P\|\\
&\leq \frac{1}{2}\|\nabla\Delta
P\|^2+4\|W\|^2_{L^\infty}\|D^2P\|^2+\|\nabla P\|^2_{L^4}\|\nabla
W\|^2_{L^4}\\
&\leq \frac{1}{2}\|\nabla\Delta
P\|^2+4\|W\|^2_{L^\infty}\|D^2P\|^2+\frac{1}{2}\|\nabla
P\|^4_{L^4}+\frac{1}{2}\|\nabla W\|^4_{L^4}.
\end{aligned}
\end{equation}
Applying the Gagliardo-Nirenberg inequality, it is sufficient to see that
\begin{equation} \label{e3.40}
\|\nabla P\|_{L^4}^4\leq\|P\|^2\|\Delta P\|^2,
\end{equation}
which enable us to estimate $K_3$ as
\begin{equation} \label{e3.41}
\begin{aligned}
&|K_3|\\
&\leq \frac{1}{2}\|\nabla\Delta P\|^2+4\|W\|^2_{L^\infty}\|D^2P\|^2
 +\frac{1}{2}\|P\|^2_{H^2}\|P\|^2_{L^\infty}
 +\frac{1}{2}\|W\|^2_{H^2}\|W\|^2_{L^\infty}\\
& \leq \frac{1}{2}\|\nabla\Delta P\|^2
 +C\left(1+\log(1+\|W\|_{H^2})\right)\|P\|^2_{H^2}\\
&\quad +C\left(1+\log(1+\|P\|_{H^2})\right)\|P\|^2_{H^2}
 +C\left(1+\log(1+\|W\|_{H^2})\right)\|W\|^2_{H^2}\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)+\frac{1}{2}\|\nabla\Delta P\|^2.
\end{aligned}
\end{equation}
Similarly to \eqref{e3.41}, we have
\begin{equation} \label{e3.42}
\begin{aligned}
&|K_4|\\
&=\big|2\operatorname{Re}\int_\Omega r(\operatorname{div}W\nabla
P\cdot\overline{\nabla\Delta
P}+P\nabla(\operatorname{div}W)\cdot\overline{\nabla\Delta P})\,{\rm d}x\big|\\
&\leq 2|r|\|\operatorname{div}W\|_{L^4}\|\nabla
P\|_{L^4}\|\nabla\Delta
P\|+2|r|\|P\|_{L^\infty}\|\nabla(\operatorname{div}W)\|\|\nabla\Delta P\|\\
&\leq \frac{1}{2}\|\nabla\Delta P\|^2+ C\|\nabla W\|^2_{L^4}\|\nabla
P\|^2_{L^4}+C\|P\|^2_{L^\infty}\|\nabla(\operatorname{div}W)\|^2 \\
&\leq\frac{1}{2}\|\nabla\Delta P\|^2
 +C\|\nabla W\|^4_{L^4}+C\|\nabla P\|^4_{L^4}
 +C\|P\|^2_{L^\infty}\|W\|^2_{H^2}\\
&\leq \frac{1}{2}\|\nabla\Delta
P\|^2+C\|W\|^2_{H^2}\|W\|^2_{L^\infty}
+C\|P\|^2_{H^2}\|P\|^2_{L^\infty}+C\|P\|^2_{L^\infty}\|W\|^2_{H^2}\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right) +\frac{1}{2}\|\nabla\Delta P\|^2.
\end{aligned}
\end{equation}
To estimate  $K_5$, we denote $W=(W^1,W^2)$ for simplicity. Note that
\begin{equation} \label{e3.43}
\begin{aligned}
&\big|\int_\Omega \Delta (W^2)\Delta (\operatorname{div}W)\,{\rm d}x\big|\\
&=\big|\int_\Omega(2W^1_{x_1}W^1_{x_1}+2W^1W^1_{x_1x_1}
 +2W^2_{x_1}W^2_{x_1}+2W^2W^2_{x_1x_1}\\
&\quad +2W^1_{x_2}W^1_{x_2} +2W^1W^1_{x_2x_2}+2W^2_{x_2}W^2_{x_2}
 +2W^2W^2_{x_2x_2})\Delta (\operatorname{div}W)\,{\rm d}x\big|\\
&\leq \big|\int_\Omega(2W^1_{x_1}W^1_{x_1}+2W^2_{x_1}W^2_{x_1}
 +2W^1_{x_2}W^1_{x_2}+2W^2_{x_2}W^2_{x_2})\Delta (\operatorname{div}W)\,{\rm d}x|\\
&\quad +|\int_\Omega(2W^1W^1_{x_1x_1}+2W^2W^2_{x_1x_1}
+2W^1W^1_{x_2x_2}\\
&\quad +2W^2W^2_{x_2x_2})\Delta (\operatorname{div}W)\,{\rm d}x|\\
&\leq 8\|\nabla W\|^2_{L^4}\|\Delta (\operatorname{div}W)\|
 +8\|W\|_{L^\infty}\|D^2W\|\|\Delta (\operatorname{div}W)\|.
\end{aligned}
\end{equation}
By \eqref{e3.43}, we find that
\begin{equation} \label{e3.44}
\begin{aligned}
|K_5|
&=\big|\int_\Omega \Delta (W^2)\Delta (\operatorname{div}W)\,{\rm d}x\big|\\
&\leq 8\|\nabla W\|^2_{L^4}\|\Delta (\operatorname{div}W)\|
 +8\|W\|_{L^\infty}\|D^2W\|\|\Delta (\operatorname{div}W)\|\\
&\leq \frac{m}{3}\|\Delta (\operatorname{div}W)\|^2
 +\frac{96}{m}\|W\|^2_{H^2}\|W\|^2_{L^\infty}
 +\frac{96}{m}\|W\|^2_{L^\infty}\|W\|^2_{H^2}\\
&\leq  C\left(1+\log(1+\|W\|_{H^2})\right)\|W\|^2_{H^2}
 +\frac{m}{3}\|\Delta (\operatorname{div}W)\|^2\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)\\
&\quad +\frac{m}{3}\|\Delta (\operatorname{div}W)\|^2.
\end{aligned}
\end{equation}
For the last three terms of \eqref{e3.36}, we have
\begin{equation} \label{e3.45}
\begin{aligned}
|K_6|
&=\big|2\omega\int_\Omega\left(\Delta P\overline{P}+2\nabla
P\cdot\nabla\overline{P}+P\Delta\overline{P}\right)\Delta (\operatorname{div}W)\,{\rm d}x\big|\\
&\leq 4\omega\|P\|_{L^\infty}\|\Delta P\|\|\Delta (\operatorname{div}W)\|
 +4\omega\|\nabla P\|^2_{L^4}\|\Delta (\operatorname{div}W)\|\\
&\leq \frac{m}{3}\|\Delta (\operatorname{div}W)\|^2
 +C\|P\|^2_{L^\infty}\|P\|^2_{H^2}+C\|P\|^2_{H^2}\|P\|^2_{L^\infty}\\
& \leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)\\
&\quad +\frac{m}{3}\|\Delta (\operatorname{div}W)\|^2,
\end{aligned}
\end{equation}
\begin{equation} \label{e3.46}
|K_7|\leq 2\|\nabla f\|\|\nabla\Delta P\|\leq
\frac{1}{2}\|\nabla\Delta P\|^2+C_6
\end{equation}
and
\begin{equation} \label{e3.47}
\begin{aligned}
|K_8|\leq 2\|\Delta g\|\|\Delta (\operatorname{div}W)\|
\leq \frac{m}{3}\|\Delta (\operatorname{div}W)\|^2+C_7
\end{aligned}
\end{equation}
because  $f(x)\in{H}^1(\Omega)$ and $g(x)\in{H}^2(\Omega)$.
Substituting \eqref{e3.38}-\eqref{e3.47} into \eqref{e3.36},
we obtain the following estimates
\begin{equation} \label{e3.48}
\begin{aligned}
&\frac{d}{{\rm d}t}\left(\|\Delta P\|^2
 +\|\nabla (\operatorname{div}W)\|^2\right)\\
&\leq C\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2})\right)
 \left(\|P\|^2_{H^2}+\|W\|^2_{H^2}\right)+2\xi\|\Delta P\|^2\\
&\quad +C_6+C_7\\
&\leq E_6\left(1+\log(1+\|P\|_{H^2}+\|W\|_{H^2}))(\|P\|^2_{H^2}
 +\|W\|^2_{H^2}\right)+E_7,
\end{aligned}
\end{equation}
where $\|\Delta P\|\leq \|P\|_{H^2}$ and $E_6=2\xi+C,E_7=C_6+C_7$.
\end{proof}

Under the conditions of Lemmas \ref{lem3.1}--\ref{lem3.4}, we have the following result.

\begin{lemma} \label{lem3.5}
Assume that $f(x)\in{H}^1(\Omega)$ and $g(x)\in{H}^2(\Omega)$, then for the
problem \eqref{e1.1}-\eqref{e1.4}, there exist positive constants
$E,E',E'',E'''$ such that for any $T>0$,
\begin{gather*}
\|P\|_{H^2}\leq E''\exp\big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}+\|Q_0\|_{H^3})}\big),\\
\|Q\|_{H^3}\leq E'''\exp\big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}+\|Q_0\|_{H^3})}\big).
\end{gather*}
\end{lemma}

\begin{proof}
From  Lemmas \ref{lem3.1}--\ref{lem3.4}, it is easy to show that
\begin{equation} \label{e3.49}
\begin{aligned}
&\frac{d}{{\rm d}t}\left(\|Q\|^2+\|P\|^2+\|\nabla Q\|^2+\|\nabla
P\|^2+\|\Delta Q\|^2+\|\Delta P\|^2+\|\nabla\Delta
Q\|^2\right) \\
&\leq\left(E_0+E_2+E_4+E_6\right)\left(1+\log(1+\|P\|_{H^2}
 +\|Q\|_{H^3})\right)\left(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\right)\\
&\quad +E_3+E_5+E_7\\
&= E_8\left(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3})\right)
\left(\|P\|^2_{H^2}+\|Q\|^2_{H^3}\right)+E_9.
\end{aligned}
\end{equation}
Denote
\begin{gather*}
\|Q\|^2+\|\nabla Q\|^2+\|\Delta Q\|^2+\|\nabla\Delta Q\|^2=[Q]^2_{H^3},\\
\|P\|^2+\|\nabla P\|^2+\|\Delta P\|^2=[P]^2_{H^2}.
\end{gather*}
Inequality \eqref{e3.49} becomes
\begin{equation} \label{e3.50}
\begin{aligned}
&\frac{d}{{\rm d}t}([P]^2_{H^2}+[Q]^2_{H^3})\\
&\leq E_8(1+\log(1+\|P\|_{H^2}+\|Q\|_{H^3}))(\|P\|^2_{H^2}+\|Q\|^2_{H^3})+E_9.
\end{aligned}
\end{equation}
We always use the notation
$\|\cdot\|_{H^s}=\big(\sum_{|\alpha|\leq s}|\partial^\alpha|_{L^2}\big)^{1/2}$,
thus $[\cdot]_{H^2}$ and $[\cdot]_{H^3}$  are equivalent norms to
$\|\cdot\|_{H^2}$ and $\|\cdot\|_{H^3}$ respectively.
 Furthermore, there exists a positive constant $C_8$ such that
$\|\cdot\|_{H^s}\leq C_8[\cdot]_{H^s}$, $s=2,3$. Applying Cauchy inequality,
from \eqref{e3.50} it follows that
\begin{equation} \label{e3.51}
\begin{aligned}
&\frac{d}{{\rm d}t}\left([P]^2_{H^2}+[Q]^2_{H^3}\right)\\
&\leq E_8\left(1+\log(1+C_8[P]_{H^2}+C_8[Q]_{H^3})\right)
\left(C^2_8[P]^2_{H^2}+C^2_8[Q]^2_{H^3}\right)+E_9\\
&\leq C^2_8E_8\left(1+\log(1+C^2_8+[P]^2_{H^2}+C^2_8+[Q]^2_{H^3})\right)
\left([P]^2_{H^2}+[Q]^2_{H^3}\right)+E_9\\
&\leq C^2_8E_8\left(1+\log(1+2C^2_8)\right)
\left(1+\log(1+[P]^2_{H^2}+[Q]^2_{H^3})\right)
\left([P]^2_{H^2}+[Q]^2_{H^3}\right) \\
&\quad +E_9\\
&=E_{10}\left(1+\log(1+[P]^2_{H^2}+[Q]^2_{H^3})\right)
\left([P]^2_{H^2}+[Q]^2_{H^3}\right)+E_9.
\end{aligned}
\end{equation}
If there exists a positive constant $N$ such that $[P]^2_{H^2}+[Q]^2_{H^3}\leq N$
for all $T>0$, then Lemma \ref{lem3.5} holds. Otherwise, there exists a positive
constant $E$ such that
\begin{equation} \label{e3.52}
\frac{d}{{\rm d}t}([P]^2_{H^2}+[Q]^2_{H^3})
\leq E([P]^2_{H^2}+[Q]^2_{H^3})\log([P]^2_{H^2}+[Q]^2_{H^3}),
\end{equation}
which implies that
\begin{equation} \label{e3.53}
\begin{aligned}
\log\log([P]^2_{H^2}+[Q]^2_{H^3})
&\leq ET+\log\log([P]^2_{H^2}(0)+[Q]^2_{H^3}(0))\\
& \leq ET+E'(\|P_0\|_{H^2}+\|Q_0\|_{H^3}).
\end{aligned}
\end{equation}
Therefore,
\begin{gather*}
[P]_{H^2}\leq \exp\Big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}+\|Q_0\|_{H^3})}\Big),\\
[Q]_{H^3}\leq \exp\Big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}+\|Q_0\|_{H^3})}\Big).
\end{gather*}
Namely,
\begin{gather*}
\|P\|_{H^2}\leq E''\exp\Big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}
+\|Q_0\|_{H^3})}\big),\\
\|Q\|_{H^3}\leq E'''\exp\Big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}+\|Q_0\|_{H^3})}\Big).
\end{gather*}
This proof is complete.
\end{proof}

Based on the previous lemmas, we are ready to prove the main result.

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 From Lemmas \ref{lem3.1}--\ref{lem3.4}, when
$P_0(x)\in H^2(\Omega)$, $Q_0(x)\in H^3(\Omega)$,
$f(x)\in H^1(\Omega)$, $g(x)\in H^2(\Omega)$ and for any $T>0$, we obtain that
\begin{gather*}
\|P\|_{H^2}\leq E''\exp\Big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}
+\|Q_0\|_{H^3})}\Big),\\
\|Q\|_{H^3}\leq E'''\exp\Big(\frac{1}{2}e^{ET+E'(\|P_0\|_{H^2}
+\|Q_0\|_{H^3})}\Big),
\end{gather*}
where the positive constant $C$ depends  only  on $\|P_0\|_{H^2},\ \|Q_0\|_{H^3}$
and $T$. Furthermore, by the standard method, we can extend the local solutions
$P(x,t)$ and $Q(x,t)$ of the periodic initial value problem
\eqref{e1.1}-\eqref{e1.4} to global solutions. Thus, the proof
is complete.
\end{proof}

\subsection*{Acknowledgments}
The work wass supported  in part by China NSF Grant
Nos. (11171158, 11271192), PAPD of Jiangsu Higher Education Institutions,
the Jiangsu Collaborative Innovation Center for Climate Change and the NSF
of the Jiangsu Higher Education Committee of China  No. 14KJB110016.

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\end{document}