# The global attractor for the 3-D viscous primitive equations of large-scale
moist atmosphere ††thanks: This work was partially supported by NNSF of
China(Grant No. 11401057, No. 11301097), Natural Science Foundation Project of
CQ (Grant No. cstc2016jcyjA0326), Fundamental Research Funds for the Central
Universities(Grant No. 106112015CDJXY100005) , China Scholarship Council
(Grant No.201506055003) and GXNSF (Grant No. 2014GXNSFAA118016) .

Guoli Zhou Yanfeng Guo  Chongqing University, P.R. China Guangxi University of
Science and Technology, P.R. China

###### Abstract

Absorbing ball in $H^{1}(\mho)$ is obtained for the strong solution to the
three dimensional viscous moist primitive equations under the natural
assumption $Q_{1},Q_{2}\in L^{2}(\mho)$ which is weaker than the assumption
$Q_{1},Q_{2}\in H^{1}(\mho)$ in [20]. In view of the structure of the manifold
and the special geometry involved with vertical velocity, the continuity of
the strong solution in $H^{1}(\mho)$ is established with respect to time and
initial data. To obtain the existence of the global attractor for the moist
primitive equations, the common method is to obtain the absorbing ball in
$H^{2}(\mho)$ for the strong solution to the equations. But it is difficult
due to the complex structure of the moist primitive equations. To overcome the
difficulty, we try to use Aubin-Lions lemma and the continuous property of the
strong solutions to the moist primitive equations to prove the the existence
of the global attractor which improves the result, the existence of weak
attractor, in [20].

Keywords: Primitive equations, global attractor

Mathematics Subject Classification (2000): 35Q35, 86A10.

##  1 Introduction

The paper is concerned with the 3-dimensional viscous primitive equations in
the pressure coordinate system (see e.g. [19, 34, 36, 37] and the references
therein).

$\displaystyle\partial_{t}v+\nabla_{v}v+w\partial_{\xi}v+\frac{f}{R_{0}}v^{\bot}+\mathrm{grad}\Phi+L_{1}v=0,$
(1.1) $\displaystyle\partial_{\xi}\Phi+\frac{bP}{p}(1+aq)T=0,$ (1.2)
$\displaystyle\mathrm{div}v+\partial_{\xi}w=0,$ (1.3)
$\displaystyle\partial_{t}T+\nabla_{v}T+w\partial_{\xi}T-\frac{bP}{p}(1+aq)w-L_{2}T=Q_{1},$
(1.4) $\displaystyle\partial_{t}q+\nabla_{v}q+w\partial_{\xi}q+L_{3}q=Q_{2}.$
(1.5)

The unknowns for the primitive equations are the fluid velocity field
$(v,w)=(v_{\theta},v_{\varphi},w)\in\mathbb{R}^{3}$ with
$v=(v_{\theta},v_{\varphi})$ and $v^{\perp}=(-v_{\varphi},v_{\theta})$ being
horizontal, the temperature $T$, $q$ the mixing ratio of water vapor in the
air and the geopotential $\Phi.$ $f=2\mathrm{cos\theta}$ is the given Coriolis
parameter, $Q_{1}$ corresponds to the sum of the heating of the sun and the
heat added or removed by condensation or evaporation, $Q_{2}$ represents the
amount of water added or removed by condensation or evaporation, $a$ and $b$
are positive constants with $a\approx 0.618$, $R_{0}$ is the Rossby number,
$P$ stands for an approximate value of pressure at the surface of the earth,
$p_{0}$ is the pressure of the upper atmosphere with $p_{0}>0$ and the
variable $\xi$ satisfies $p=(P-p_{0})\xi+p_{0}$ where $0<p_{0}\leq p\leq P.$
The viscosity, the heat and the water vapor diffusion operators $L_{1},\
L_{2}$ and $L_{3}$ are given respectively as the following:

$L_{i}=-\nu_{i}\Delta-\mu_{i}\partial_{zz},i=1,2,3.$

Here the positive constants $\nu_{1},\mu_{1}$ are the horizontal and vertical
viscosity coefficients; the positive constant $\nu_{2},\mu_{2}$ are the
horizontal and vertical heat diffusivity coefficients; while the positive
constant $\nu_{3},\mu_{3}$ are the horizontal and vertical water vapor
diffusivity coefficients. The definitions of $\nabla_{v}v,\Delta v,\Delta
T,\Delta q,\nabla_{v}q,\nabla_{v}T,\mathrm{div}v,\mathrm{grad}\Phi$ will be
given in section 2.

The space domain of equations: $(1.1)-(1.5)$ is

$\mho=S^{2}\times(0,1),$

where $S^{2}$ is two-dimensional unit sphere. The boundary value conditions
are given by

$\displaystyle\xi=1(p=P):\partial_{\xi}v=0,\ \ w=0,\ \
\partial_{\xi}T=\alpha_{s}(T_{s}-T),\ \ \partial_{\xi}q=\beta_{s}(q_{s}-q),$
(1.6) $\displaystyle\xi=0(p=p_{0}):\partial_{\xi}v=0,\ \ w=0,\ \
\partial_{\xi}T=0,\ \ \partial_{\xi}q=0,$ (1.7)

where $\alpha_{s},\beta_{s}$ are positive constants, $T_{s}$ is the given
temperature on the surface of the earth, $q_{s}$ is the given mixing ratio of
water vapor on the surface of the earth. To simplify the notations, we set
$T_{s}=0$ and $q_{s}=0$ without losing any generality. For the case $T_{s}\neq
0$ and $q_{s}\neq 0,$ we can homogenize the boundary value conditions for
$T,q;$ see [19] for detailed discussion on this issue. Moreover, using
$(1.2),(1.3)$ and the boundary conditions $(1.6)-(1.7)$, we have

$\displaystyle w(t;\theta,\varphi,\xi)=\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime},$ (1.8)
$\displaystyle\int_{0}^{1}\mathrm{div}\ vd\xi=0,$ (1.9)
$\displaystyle\Phi(t;\theta,\varphi,\xi)=\Phi_{s}(t;\theta,\varphi)+\int_{\xi}^{1}\frac{bP}{p}(1+aq)Td\xi^{\prime},$
(1.10)

where $\Phi_{s}(t;\theta,\varphi)$ is a certain unknown function at the
isobaric surface $\xi=1.$ In this article, we assume that the constants
$v_{i}=\mu_{i}=1,i=1,2,3.$ For the general case, the results will still be
valid. Then using $(1.8)-(1.10)$, we obtain the following equivalent
formulation for system $(1.1)-(1.7)$ with initial condition

$\displaystyle\partial_{t}v$ $\displaystyle+$
$\displaystyle\nabla_{v}v+\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)}\partial_{\xi}v+\frac{f}{R_{0}}v^{\bot}+\mathrm{grad}\Phi_{s}$
(1.11)
$\displaystyle+\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}-\Delta
v-\partial_{\xi\xi}v=0,$ $\displaystyle\partial_{t}T$ $\displaystyle+$
$\displaystyle\nabla_{v}T+\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)}\partial_{\xi}T$ (1.12)
$\displaystyle-\frac{bP}{p}(1+aq)\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)}-\Delta
T-\partial_{\xi\xi}T=Q_{1},$
$\displaystyle\partial_{t}q+\nabla_{v}q+\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)}\partial_{\xi}q-\Delta
q-\partial_{\xi\xi}q=Q_{2},$ (1.13) $\displaystyle\int_{0}^{1}\mathrm{div}\
vd\xi=0,$ (1.14) $\displaystyle\xi=1:\partial_{\xi}v=0,\ \ w=0,\ \
\partial_{\xi}T=-\alpha_{s}T,\ \ \partial_{\xi}q=-\beta_{s}q,$ (1.15)
$\displaystyle\xi=0:\partial_{\xi}v=0,\ \ w=0,\ \ \partial_{\xi}T=0,\ \
\partial_{\xi}q=0,$ (1.16) $\displaystyle
v(0;\theta,\phi,\xi)=v_{0}(\theta,\phi,\xi),T(0;\theta,\phi,\xi)=T_{0}(\theta,\phi,\xi),q(0;\theta,\phi,\xi)=q_{0}(\theta,\phi,\xi).$
(1.17)

In order to understand the mechanism of long-term weather prediction, one can
take advantage of the historical records and numerical computations to detect
the future weather. Alternatively, one should also study the long time
behavior mathematically for the equations and models governing the motion. The
primitive equations represent the classic model for the study of climate and
weather prediction, describing the motion of the atmosphere when the
hydrostatic assumption is enforced $\cite[cite]{[\@@bibref{}{ G, Ha, HW, MT,
Ri}{}{}]}.$ But the resulting flow or the atmosphere is rich in its
organization and complexity (see [18, 23, 24]), the full governing equations
are too complicated to be treatable both from the theoretical and the
computational side. To overcome this difficulty, some simple numerical models
were introduced. The 2-D and 3-D quasi-geostrophic models have been the
subject of analytical mathematical study (see e.g., [3, 5, 10, 11, 15, 16, 39,
48, 49, 50] and references therein). To the best of our knowledge, the
mathematical framework of primitive equations was formulated in
$\cite[cite]{[\@@bibref{}{LTW1, LTW2, LTW3}{}{}]},$ where the definitions of
weak and strong solutions were given and the existence of weak solution was
proven, leaving the uniqueness of weak solution as an open problem for now.
Local well-posedness of strong solutions was obtained in
$\cite[cite]{[\@@bibref{}{GMR, TZ}{}{}]}.$ If the domains was thin, the global
well-posedness of 3D primitive equations was shown in
$\cite[cite]{[\@@bibref{}{HTZ}{}{}]}.$ Taking advantage of the fact that the
pressure is essentially two-dimensional in the primitive equations, global
well-posedness of the full three-dimensional case was established in [12] and
independently in $\cite[cite]{[\@@bibref{}{Kob1, Kob2}{}{}]}.$ In the
subsequent work [31] a different proof was developed which allows one to treat
non-rectangular domains. Recently, the results were improved in [7, 8, 9, 13]
by considering the system with partial dissipation, i.e. , with only partial
viscosities or only partial diffusion. For the inviscid primitive equations,
finite-time blowup was established in $\cite[cite]{[\@@bibref{}{CINT}{}{}]}.$
To study the long term behavior of primitive equations, the existence of
global attractor was established in [27] and dimensions were proven to be
finite in $\cite[cite]{[\@@bibref{}{JT}{}{}]}.$ When moisture is included, an
equation for the conservation of water must be added, which is the case in
e.g. $\cite[cite]{[\@@bibref{}{GH1,GH2, LTW1, PTZ}{}{}]}.$ In
$\cite[cite]{[\@@bibref{}{ZHKTZ}{}{}]},$ global well-posedness of quasi-strong
and strong solutions was obtained for the primitive equations of atmosphere in
presence of vapour saturation.

The understanding of asymptotic behavior of dynamical system is one of the
most important topics of modern mathematical physics. One way to solve the
problem for dissipative deterministic dynamical system is to consider its
global attractor (see its definition in section 2). Thus, in order to capture
the dynamical features of moist primitive equations, Guo and Huang in [20]
proved the existence of universal attractor which is weakly closed in $V$ (see
the notations in section $2$) and attracts any orbit in $V$-weak topology,
when time goes to $\infty.$ Then, one natural question arising from it is the
existence of global attractor which is compact in $V$ and attracts any orbit
in $V$-strong topology, as time goes to $\infty.$

As it is stated in [20] that laking information for the time derivative of the
vertical velocity in the moist primitive equations leads to the failure of
establishing the existence of the global attractor. How to overcome the
difficulty? Inspired by [27], we try to use Aubin-Lions lemma combined with
continuity of the strong solution in $V$ with respect to time and initial data
to prove the solution operation is compact in $V$. Then the compact property
of the solution operator implies that the existence of global attractor, i.e.,
the universal attractor is indeed compact in $V$ and attracts all bounded sets
in $V$ with respect to $V$-strong topology, when $t\rightarrow\infty$.

In this article, we first try to obtain time-uniform $a\ priori$ estimates in
various function spaces under the natural assumption $Q_{1},Q_{2}\in
L^{2}(\mho),$ reducing the stronger assumption $Q_{1},Q_{2}\in H^{1}(\mho)$ in
[20]. Then, by making delicate and careful estimates of $L^{4}$ norm for the
strong solution, estimates about $L^{3}$ norm which is required in [20] is
omitted. To obtain uniform boundedness with respect to time $t$ in $V,$
estimates about $\partial_{\xi}T,\partial_{\xi}q$ are carefully considered,
which is more complex than oceanic primitive equations. We recall that the
continuity of the strong solution with respect to initial data was shown to be
true in weak solution space $H$ in previous work due to no boundedness is
available for the derivatives of the vertical velocity. This is not sufficient
for our purpose. To overcome the difficulty, the structure of the manifold and
the special geometry involved with the vertical velocity are used to obtain
the continuity of the strong solution with respect to initial data in $V$,
improving the results before. Finally, in order to prove the absorbing ball is
compact in $V,$ the common method is to show that the ball is uniformly
bounded with respect to time $t$ in $H^{2}(\mho).$ But it is difficult to
achieve because of the high nonlinearity of the moist primitive equations.
Inspired by [27], we try to use an Aubin-Lions compactness lemma combined with
a continuity argument to show that the solution operator is compact in $V$ for
every time $t>0$, which further implies the existence of the global attractor
for the dynamical system generated by the primitive equations of large-scale
moist atmosphere.

The remaining of the paper is organized as follows. In section $2$, we present
the notations and recall some important facts which are crucial to later
analysis. Absorbing ball is obtained in section $3$, where we correct the
mistake $fv^{\bot}\times\Delta v=0$ in $\cite[cite]{[\@@bibref{}{GH2}{}{}]}.$
Section $4$ and section $5$ are for continuity of the strong solution with
respect to time $t$ and initial data in $V$ respectively. Finally, in section
$6$, using Aubin-Lions lemma and the continuity properties of the strong
solution we prove the existence of global attractor. As usual, the positive
constants $c$ may change from one line to the next, unless, we give a special
declaration.

##  2 Preliminaries

In this section we collect some preliminary results that will be used in the
rest of this paper, and we start with the following notations which will be
used throughout this work. Denote

$\displaystyle\bar{v}=\int_{0}^{1}vd\xi,\ \ \tilde{v}=v-\bar{v}.$

Then we have

$\displaystyle\nabla\cdot\bar{v}=0,\ \ \bar{\tilde{v}}=0.$ (2.18)

Now we give the definitions of some differential operators. Firstly, the
natural generalization of the directional derivative on the Euclidean space to
the covariant derivative on $S^{2}$ is given as follows. Let $T,q,\in
C^{\infty}(\mho),\Phi_{s}\in C^{\infty}(S^{2})$ and

$\displaystyle v=v_{\theta}e_{\theta}+v_{\varphi}e_{\varphi},\ \ \ \ \
u=u_{\theta}e_{\theta}+u_{\varphi}e_{\varphi}\ \ \ \ \in
C^{\infty}(T\mho|TS^{2}),$

where $C^{\infty}(T\mho|TS^{2})$ is the first two components of smooth vector
fields on $\mho.$ We define the covariant derivative of $u,T$ and $q$ with
respect to $v$ as follows

$\displaystyle\nabla_{v}u=(v_{\theta}\partial_{\theta}u_{\theta}+\frac{v_{\varphi}}{\mathrm{sin}\theta}\partial_{\varphi}u_{\theta}-v_{\varphi}u_{\varphi}\mathrm{cot}\theta)e_{\theta}+(v_{\theta}\partial_{\theta}u_{\varphi}+\frac{v_{\varphi}}{\mathrm{sin}\theta}\partial_{\varphi}u_{\varphi}+v_{\varphi}u_{\theta}\mathrm{cot}\theta)e_{\varphi},$
$\displaystyle\nabla_{v}T=v_{\theta}\partial_{\theta}T+\frac{v_{\varphi}}{\mathrm{sin}\theta}\partial_{\varphi}T,$
$\displaystyle\nabla_{v}q=v_{\theta}\partial_{\theta}q+\frac{v_{\varphi}}{\mathrm{sin}\theta}\partial_{\varphi}q.$

We give the definition of the horizontal gradient $\nabla=\mathrm{grad}$ for
$T$ and $\Phi_{s}$ on $S^{2}$ by

$\displaystyle\nabla
T=\mathrm{grad}T=(\partial_{\theta}T)e_{\theta}+\frac{1}{\mathrm{sin}\theta}(\partial_{\varphi}T)e_{\varphi},$
$\displaystyle\nabla\Phi_{s}=\mathrm{grad}\Phi_{s}=(\partial_{\theta}\Phi_{s})e_{\theta}+\frac{1}{\mathrm{sin}\theta}(\partial_{\varphi}\Phi_{s})e_{\varphi}.$

We define the divergence of $v$ by

$\displaystyle\mathrm{div}v=\mathrm{div}(v_{\theta}e_{\theta}+v_{\varphi}e_{\varphi})=\frac{1}{\mathrm{sin}\theta}(\partial_{\theta}(v_{\theta}\mathrm{sin}\theta)+\partial_{\varphi}v_{\varphi}).$

The horizontal Laplace-Beltrami operator of scalar functions $T$ and $q$ are

$\displaystyle\Delta
T=\mathrm{div}(\mathrm{grad}T)=\frac{1}{\mathrm{sin}\theta}[\partial_{\theta}(\mathrm{sin}\theta\partial_{\theta}T)+\frac{1}{\mathrm{sin}\theta}\partial_{\varphi\varphi}T],$
$\displaystyle\Delta
q=\mathrm{div}(\mathrm{grad}q)=\frac{1}{\mathrm{sin}\theta}[\partial_{\theta}(\mathrm{sin}\theta\partial_{\theta}q)+\frac{1}{\mathrm{sin}\theta}\partial_{\varphi\varphi}q].$

We define the horizontal Laplace-Beltrami operator $\Delta$ for vector
functions on $S^{2}$ as

$\displaystyle\Delta v=(\Delta
v_{\theta}-\frac{2\mathrm{cos}\theta}{\mathrm{sin}^{2}\theta}\partial_{\varphi}v_{\varphi}-\frac{v_{\theta}}{\mathrm{sin}^{2}\theta})e_{\theta}+(\Delta
v_{\varphi}+\frac{2\mathrm{cos}\theta}{\mathrm{sin}^{2}\theta}\partial_{\varphi}v_{\theta}-\frac{v_{\varphi}}{\mathrm{sin}^{2}\theta})e_{\varphi}.$

Consequently, by integration by parts, we have

$\displaystyle\int_{0}^{1}w\partial_{\xi}vd\xi=\int_{0}^{1}v\mathrm{div}vd\xi=\int_{0}^{1}\tilde{v}\mathrm{div}\tilde{v}d\xi,$
(2.19)
$\displaystyle\int_{0}^{1}\nabla_{v}vd\xi=\int_{0}^{1}\nabla_{\tilde{v}}\tilde{v}d\xi+\nabla_{\bar{v}}\bar{v}.$
(2.20)

Taking the average of equations $(1.11)$ in the $z$ direction, over the
interval $(0,1)$ and using $(2.18)-(2.20)$ and the boundary conditions
$(1.15)-(1.16),$ we arrive at

$\displaystyle\partial_{t}\bar{v}$ $\displaystyle+$
$\displaystyle\nabla_{\bar{v}}\bar{v}+\overline{\tilde{v}\mathrm{div}\tilde{v}+\nabla_{\tilde{v}}\tilde{v}}+\frac{f}{R_{0}}\bar{v}^{\bot}+\mathrm{grad}\Phi_{s}+\int_{0}^{1}\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}d\xi$
(2.21) $\displaystyle-\Delta\bar{v}=0\ \ \mathrm{in}\ S^{2}.$

By subtracting $(2.21)$ from $(1.11),$ we obtain the following equation

$\displaystyle\partial_{t}\tilde{v}$ $\displaystyle+$
$\displaystyle\nabla_{\tilde{v}}\tilde{v}+\Big{(}\int_{\xi}^{1}\mathrm{div}\tilde{v}d\xi^{\prime}\Big{)}\partial_{\xi}\tilde{v}+\nabla_{\tilde{v}}\bar{v}+\nabla_{\bar{v}}\tilde{v}-\overline{(\tilde{v}\mathrm{div}\tilde{v}+\nabla_{\tilde{v}}\tilde{v})}+\frac{f}{R_{0}}\tilde{v}^{\bot}$
(2.22)
$\displaystyle+\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}-\int_{0}^{1}\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}d\xi$
$\displaystyle-\Delta\tilde{v}-\partial_{\xi\xi}\tilde{v}=0\ \ \mathrm{in}\
\mho,$

with the following boundary value conditions

$\displaystyle\partial_{\xi}\tilde{v}=0\ \mathrm{on}\ \xi=1\ \mathrm{and}\
\xi=0.$ (2.23)

Let $e_{\theta},e_{\varphi},e_{\xi}$ be the unite vectors in $\theta,\varphi$
and $\xi$ directions of the space domain $\mho$ respectively,

$\displaystyle e_{\theta}=\partial_{\theta},\ \ \ \
e_{\varphi}=\frac{1}{\mathrm{sin\theta}}\partial_{\varphi},\ \ \ \
e_{\xi}=\partial_{\xi}.$

The inner product and norm on $T_{(\theta,\varphi,\xi)}\mho$ (the tangent
space of $\mho$ at the point $(\theta,\varphi,\xi)$ ) are defined by

$\displaystyle(u,v)=u\cdot v=\sum_{i=1}^{3}u_{i}v_{i},\ \ \ \ |u|=(u,\
u)^{\frac{1}{2}},$

where $u=u_{1}e_{\theta}+u_{2}e_{\varphi}+u_{3}e_{\xi}\in
T_{(\theta,\varphi,\xi)}\mho$ and
$v=v_{1}e_{\theta}+v_{2}e_{\varphi}+v_{3}e_{\xi}\in
T_{(\theta,\varphi,\xi)}\mho.$ For $1\leq p\leq\infty,$ let
$L^{p}(\mho),L^{p}(S^{2})$ be the usual Lebesgue spaces with the norm
$|\cdot|_{p}$ and $|\cdot|_{L^{p}(S^{2})}$ respectively. If there is no
confusion, we will write $|\cdot|_{p}$ instead of $|\cdot|_{L^{p}(S^{2})}$.
$L^{2}(T\Omega|TS^{2})$ is the first two components of $L^{2}$ vector fields
on $\mho$ with the norm
$|v|_{2}=(\int_{\mho}(|v_{\theta}|^{2}+|v_{\varphi}|^{2})d\mho)^{\frac{1}{2}},$
where $v=(v_{\theta},v_{\varphi}):\mho\rightarrow TS^{2}$. Denoted by
$C^{\infty}(S^{2})$ the functions of all smooth functions from $S^{2}$ to
$\mathbb{R}.$ Similarly, we can define $C^{\infty}(\mho)$. $H^{m}(\mho)$ is
the Sobolev space of functions which are in $L^{2},$ together with all their
covariant derivatives with respect to $e_{\theta},e_{\varphi},e_{\xi}$ of
order $\leq m,$ with the norm

$\displaystyle\|h\|_{m}=[\int_{\mho}(\sum_{1\leq k\leq
m}\sum_{i_{j}=1,2,3;j=1,...,k}|\nabla_{i_{1}}\cdots\nabla_{i_{k}}h|^{2}+|h|^{2})]^{\frac{1}{2}},$

where $\nabla_{1}=\nabla_{e_{\theta}},\nabla_{2}=\nabla_{e_{\varphi}}$ and
$\nabla_{3}=\partial_{\xi}$ which are defined above. Denote
$H^{m}(T\mho|TS^{2})=\\{v;v=(v_{\theta},v_{\varphi}):\mho\rightarrow
TS^{2},\|v\|_{m}<\infty\\},$ where the norm is similar to that of
$H^{m}(\mho)$ (i.e., let
$h=(v_{\theta},v_{\varphi})=v_{\theta}e_{\theta}+v_{\varphi}e_{\varphi}.$)

We will conduct our work in the following functional spaces. Let

$\displaystyle\mathcal{V}_{1}:=\\{v;v\in C^{\infty}(T\mho|TS^{2}),\
\partial_{\xi}v|_{\xi=0}=0,\ \partial_{\xi}v|_{\xi=1}=0,\
\int_{0}^{1}\mathrm{div}\ vd\xi=0\\},$
$\displaystyle\mathcal{V}_{2}:=\\{T;T\in C^{\infty}(\mho),\
\partial_{\xi}T|_{\xi=0}=0,\ \partial_{\xi}T|_{\xi=1}=-\alpha_{s}T\\},$
$\displaystyle\mathcal{V}_{3}:=\\{q;q\in C^{\infty}(\mho),\
\partial_{\xi}q|_{\xi=0}=0,\ \partial_{\xi}q|_{\xi=1}=-\beta_{s}q\\}.$

We denote by $V_{1},V_{2}$ and $V_{3}$ the closure spaces of
$\mathcal{V}_{1},\mathcal{V}_{2}$ and $\mathcal{V}_{3}$ in $H^{1}(\mho)$ under
$H^{1}-$ topology, respectively. In addition, we denote by $H_{1},H_{2}$ and
$H_{3}$ the closure of $\mathcal{V}_{1},\mathcal{V}_{2}$ and $\mathcal{V}_{3}$
in $L^{2}(\mho)$ under $L^{2}-$ topology. Let $H:=H_{1}\times H_{2}\times
H_{3}$ and $V=V_{1}\times V_{2}\times V_{3}$ with $V^{\prime}$ being dual
space of $V$. By definition, the inner products and norms on $V_{1},V_{2}$ and
$V_{3}$ are given by

$\displaystyle\langle
v,v_{1}\rangle_{V_{1}}=\int_{\mho}(\nabla_{e_{\theta}}v\cdot\nabla_{e_{\theta}}v_{1}+\nabla_{e_{\varphi}}v\cdot\nabla_{e_{\varphi}}v_{1}+\partial_{\xi}v\partial_{\xi}v_{1}+v\cdot
v_{1})d\mho,$ $\displaystyle\|v\|_{1}=\langle
v,v\rangle_{V_{1}}^{\frac{1}{2}},\ \ \ \ \forall\ v,\ v_{1}\in V_{1},$
$\displaystyle\langle
T,T_{1}\rangle_{V_{2}}=\int_{\mho}(\mathrm{grad}T\cdot\mathrm{grad}T_{1}+\partial_{\xi}T\partial_{\xi}T_{1})d\mho+\alpha_{s}\int_{S^{2}}TT_{1}dS^{2},$
$\displaystyle\|T\|_{1}=\langle T,T\rangle_{V_{2}}^{\frac{1}{2}},\ \ \
\forall\ T,T_{1}\in V_{2},$ $\displaystyle\langle
q,q_{1}\rangle_{V_{3}}=\int_{\mho}(\mathrm{grad}q\cdot\mathrm{grad}q_{1}+\partial_{\xi}q\partial_{\xi}q_{1})d\mho+\beta_{s}\int_{S^{2}}qq_{1}dS^{2},$
$\displaystyle\|q\|_{1}=\langle q,q\rangle_{V_{3}}^{\frac{1}{2}},\ \ \
\forall\ q,q_{1}\in V_{3}.$

Let $V_{i}^{\prime}(i=1,2,3)$ be the dual space of $V_{i}$ with
$\langle,\rangle$ being the inner products between $V_{i}^{\prime}$ and
$V_{i}.$ Without confusion, we also denote by $\langle,\rangle$ the inner
product in $L^{2}(\mho)$ and $L^{2}(S^{2}).$ Define the linear operator
$A_{i}:V_{i}\mapsto V_{i}^{\prime},i=1,2,3$ :

$\displaystyle\langle A_{1}u_{1},u_{2}\rangle=\langle u,v\rangle_{V_{1}},\ \ \
\forall\ u_{1},\ u_{2}\in V_{1};$ $\displaystyle\langle
A_{2}\theta_{1},\theta_{2}\rangle=\langle\theta_{1},\theta_{2}\rangle_{V_{2}},\
\ \ \forall\ \theta_{1},\theta_{2}\in V_{2};$ $\displaystyle\langle
A_{3}q_{1},q_{2}\rangle=\langle q_{1},q_{2}\rangle_{V_{3}},\ \ \ \forall\
q_{1},q_{2}\in V_{3}.$

Denote $D(A_{i})=\\{\eta\in V_{i},A_{i}\eta\in H_{i}\\}.$ Since $A_{i}$ is
positive self-adjoint with compact resolvent, according to the classic
spectral theory we can define the power $A_{i}^{s}$ for any $s\in\mathbb{R}.$
Then we have $D(A_{i}^{\frac{1}{2}})=V_{i}$ and
$D(A_{i}^{-\frac{1}{2}})=V_{i}^{\prime}.$ Moreover,

$\displaystyle D(A_{i})\subset V_{i}\subset H_{i}\subset V_{i}^{\prime}\subset
D(A_{i})^{\prime},$

where $D(A_{i})^{\prime}$ is the dual space of $D(A_{i})$ and the embeddings
above are all compact. In the following, we state some lemmas including
integrations by parts and uniform Gronwall lemma, which are frequently used in
our paper. For the proof of Lemma $2.1$-Lemma $2.3$, we can see
$\cite[cite]{[\@@bibref{}{GH1}{}{}]}.$ The proof of uniform Gronwall lemma was
given in $\cite[cite]{[\@@bibref{}{FP, T}{}{}]}.$

###### Lemma 2.1

Let $u=(u_{\theta},u_{\varphi}),v=(v_{\theta},v_{\varphi})\in
C^{\infty}(T\mho|TS^{2})$ and $p\in C^{\infty}(S^{2}).$ Then

$\displaystyle\int_{S^{2}}p\ \mathrm{div}\ udS^{2}=-\int_{S^{2}}\nabla p\cdot
udS^{2},$ $\displaystyle\int_{\mho}\nabla p\cdot vd\mho=0\ \ \ for\ any\ v\in
V_{1},$

and

$\displaystyle\int_{\mho}(-\Delta u)\cdot
vd\mho=\int_{\mho}(\nabla_{e_{\theta}}u\cdot\nabla_{e_{\theta}}v+\nabla_{e_{\varphi}}u\cdot\nabla_{e_{\varphi}}v+u\cdot
v)d\mho.$

###### Lemma 2.2

For any $h\in C^{\infty}(S^{2}),v\in C^{\infty}(T\mho|TS^{2}),$ we have

$\displaystyle\int_{S^{2}}\nabla_{v}hdS^{2}+\int_{S^{2}}h\ \mathrm{div}\
vdS^{2}=\int_{S^{2}}\mathrm{div}(hv)dS^{2}=0.$

###### Lemma 2.3

Let $u,v\in V_{1},T\in V_{2},q\in V_{3}.$ Then we have

$\displaystyle\int_{\mho}[\nabla_{u}v+(\int_{\xi}^{1}\mathrm{div}\
ud\xi^{\prime})\partial_{\xi}v]vd\mho=0,$
$\displaystyle\int_{\mho}[\nabla_{u}g+(\int_{\xi}^{1}\mathrm{div}\
ud\xi^{\prime})\partial_{\xi}g]gd\mho=0,\ \ for\ g=T\ or\ g=q,$
$\displaystyle\int_{\mho}\Big{(}\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}\cdot
u-\frac{bP}{p}(1+aq)T(\int_{\xi}^{1}\mathrm{div}\ ud\xi^{\prime})\Big{)}=0.$

###### Lemma 2.4

Let $f,g$ and $h$ be three non-negative locally integrable functions on
$(t_{0},\infty)$ such that

$\displaystyle\frac{df}{dt}\leq gf+h,\ \ \ \forall\ t\geq t_{0},$

and

$\displaystyle\int_{t}^{t+r}f(s)ds\leq a_{1},\ \ \int_{t}^{t+r}g(s)ds\leq
a_{2},\ \ \int_{t}^{t+r}h(s)ds\leq a_{3},\ \ \forall\ t\geq t_{0},$

where $r,a_{1},a_{2},a_{3}$ are positive constants. Then

$\displaystyle f(t+r)\leq(\frac{a_{1}}{r}+a_{3})e^{a_{2}},\ \ \ \forall\ t\geq
t_{0}.$

Before considering the long time behavior of the dynamics, we recall the
definitions of strong solution to $(1.11)-(1.17).$

###### Definition 2.1

Suppose $Q_{1},Q_{2}\in L^{2}(\mho),(v_{0},T_{0},q_{0})\in V$ and $\tau>0.$
$(v,T,q)$ is called a strong solution of $(1.11)-(1.17)$ on the time interval
$[0,\tau]$ if it satisfies $(1.11)-(1.13)$ in a week sense, and also

$\displaystyle v\in C([0,\tau];V_{1})\cap L^{2}([0,\tau];H^{2}(\mho)),$
$\displaystyle T\in C([0,\tau];V_{2})\cap L^{2}([0,\tau];H^{2}(\mho)),$
$\displaystyle q\in C([0,\tau];V_{3})\cap L^{2}([0,\tau];H^{2}(\mho)),$
$\displaystyle\partial_{t}v,\partial_{t}T,\partial_{t}q\in
L^{1}([0,\tau];L^{2}(\mho)).$

Now we state the global well-posedness theorem for the strong solution as
follows. For the proof of the therem, one can refer to
$\cite[cite]{[\@@bibref{}{GH2}{}{}]}.$

###### Proposition 2.1

Let $Q_{1},Q_{2}\in H^{1}(\mho),U_{0}=(v_{0},T_{0},q_{0})\in V.$ Then for any
$T>0$ given, the global strong solution $U$ of the system $(1.11)-(1.17)$ is
unique on the interval $[0,T]$. Moreover, the strong solution $U$ is
continuous with respect to initial data in $H.$

###### Remark 2.1

Notice that there are some gap between the Proposition $2.1$ and Definition
2.1. In fact, in section 3 of this work we will find that if the condition
$Q_{1},Q_{2}\in H^{1}(\mho)$ is relaxed as $Q_{1},Q_{2}\in L^{2}(\mho)$, the
result of Proposition $2.1$ still holds.

###### Remark 2.2

In the Proposition $2.1$, the result that the strong solution is continuous
with respect to initial data in $H$ is not sufficient for our purpose. We will
improve the result in section 5 by establishing the strong solution is
continuous with respect to initial data in $V.$

To see the difference between global attractor and universal attractor, we
introduce the two definitions in the following. For more details, we refer to
[20, 22, 46] and other references. Let $(X,d)$ be a separable metric space and
$S(t):X\rightarrow X,0\leq t<\infty,$ be a semigroup satisfying:

$(i)\ S(t)S(s)x=S(t+s)x,$ for all $t,s\in\mathbb{R}_{+}$ and $x\in X;$

$(ii)\ S(0)=I$ (Identity in $X$);

$(iii)\ S(t)$ is continuous in $X$ for all $t\geq 0.$

Typically, $S(t)$ is associated with a autonomous differential equation;
$S(t)x$ is the state at time $t$ of the solution whose initial data is $x.$

###### Definition 2.2

A subset $\mathcal{A}$ in $X$ is said to be a global attractor if it satisfies
the following properties:
$(i)$ $\mathcal{A}$ is compact in $X;$
$(ii)$ for every $t\geq 0,S(t)\mathcal{A}=\mathcal{A};$
$(iii)$ for every bounded set $B$ in $X,$ the set $S(t)B$ converges to
$\mathcal{A}$ in $X$, when $t\rightarrow\infty,i.e.,$

$\lim\limits_{t\rightarrow\infty}d(S(t)B,\mathcal{A})=0.$

Here, and in the following, for $A$and $B$ subsets of $X,d(A,B)$ is the semi-
distance given by

$d(A,B)=\sup\limits_{x\in A}\inf\limits_{y\in B}d(x,y).$

###### Definition 2.3

A subset $\mathcal{A}$ in $X$ is said to be a universal attractor or weak
attractor if it satisfies the following properties:
$(i)$ $\mathcal{A}$ is bounded and weakly closed in $X;$
$(ii)$ for every $t\geq 0,S(t)\mathcal{A}=\mathcal{A};$
$(iii)$ for every bounded set $B$ in $X,$ the set $S(t)B$ converges to
$\mathcal{A}$ with respect to $X$-weak topology, when
$t\rightarrow\infty,i.e.,$

$\lim\limits_{t\rightarrow\infty}d_{X}^{w}(S(t)B,\mathcal{A})=0$

where the distance $d_{X}^{w}$ is induced by the $X$-weak topology.

##  3 Uniform estimates and absorbing balls

In this section, we will obtain some useful uniform $a$ $priori$ estimates
about the solution to $(1.11)-(1.17)$ under the natural assumption
$Q_{1},Q_{2}\in L^{2}(\mho)$ and give a proof of the existence of the
absorbing ball in $V$ for the solution to the moist primitive equation. The
estimates of this section are rigorous without justification using Galerkin
approximation due to Proposition $2.1.$

###  3.1 $L^{2}$ estimates of $v,T,q$

Taking inner product of $(1.13)$ with $q$ in $L^{2}(\mho),$ by Lemma $2.3$ we
have

$\displaystyle\frac{1}{2}\frac{d|q|_{2}^{2}}{dt}+|\nabla
q|_{2}^{2}+|\partial_{\xi}q|_{2}^{2}+\beta_{s}|q|_{\xi=1}|_{2}^{2}=\int_{\mho}qQ_{2}d\mho.$
(3.24)

Since
$q(\theta,\varphi,\xi)=-\int^{1}_{\xi}\partial_{\xi}qd\xi^{\prime}+q|_{\xi=1},$
by H$\mathrm{\ddot{o}}$lder inequality and Cauchy-Schwarz inequality we have

$\displaystyle|q|_{2}^{2}\leq
2|\partial_{\xi}q|_{2}^{2}+2|q_{\xi=1}|_{2}^{2},$ (3.25)

which together with $(3.24)$ implies that there exists a positive constant $c$
such that

$\displaystyle\frac{d}{dt}|q|_{2}^{2}+c|q|_{2}^{2}\leq|Q_{2}|_{2}^{2}.$

Therefore, we have

$\displaystyle|q(t)|_{2}^{2}\leq e^{-ct}|q_{0}|_{2}^{2}+c|Q_{2}|_{2}^{2}.$
(3.26)

By $(3.24)$ and $(3.26),$ for arbitrary $t_{0}\geq 0$ we have

$\displaystyle\int_{t_{0}}^{t_{0}+1}(|\nabla
q(t)|_{2}^{2}+|\partial_{\xi}q(t)|_{2}^{2}+\beta_{s}|q|_{\xi=1}(t)|_{2}^{2})dt$
$\displaystyle\leq|q(t_{0})|_{2}^{2}+|Q|_{2}^{2}\leq
e^{-ct_{0}}|q_{0}|_{2}^{2}+c|Q_{2}|_{2}^{2}.$ (3.27)

Taking an analogous argument as $(3.24)$ , we have

$\displaystyle\frac{1}{2}\frac{d|T|^{2}_{2}}{dt}+|\nabla
T|_{2}^{2}+|\partial_{\xi}T|_{2}^{2}+\alpha_{s}|T|_{\xi=1}|_{2}^{2}=\int_{\mho}\frac{bP}{p}(1+aq)Twd\mho+\int_{\mho}Q_{1}Td\mho.$
(3.28)

Multiplying $v$ with respect to $(1.11)$ and integrating on $\mho,$ by Lemma
$2.1,$ Lemma $2.3$ and $(\frac{f}{R_{0}}\times v)\cdot v=0$ we have

$\displaystyle\frac{1}{2}\frac{d|v|_{2}^{2}}{dt}+|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2}+|v|_{2}^{2}+|\partial_{\xi}v|_{2}^{2}=-\int_{\mho}\Big{(}\int^{1}_{\xi}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}\Big{)}\cdot
v.$ (3.29)

Combining $(3.28)-(3.29)$ and Lemma $2.3$ yields

$\displaystyle\frac{1}{2}\frac{d(|v|_{2}^{2}+|T|_{2}^{2})}{dt}$
$\displaystyle+$
$\displaystyle|\nabla_{e_{\theta}}v|_{2}^{2}+\nabla_{e_{\varphi}}v|_{2}^{2}+|\partial_{\xi}v|_{2}^{2}+|v|_{2}^{2}$
(3.30) $\displaystyle+|\nabla
T|_{2}^{2}+|\partial_{\xi}T|_{2}^{2}+\alpha_{s}|T|_{\xi=1}|_{2}^{2}=\int_{\mho}Q_{1}Td\mho.$

Similarly to the deduction of $(3.25),$ we have

$\displaystyle|T|_{2}^{2}\leq
2|\partial_{\xi}T|_{2}^{2}+2|T_{\xi=1}|_{2}^{2},$

which together with $(3.29)$ and H$\mathrm{\ddot{o}}$lder inequality implies

$\displaystyle\frac{d(|v|_{2}^{2}+|T|_{2}^{2})}{dt}$ $\displaystyle+$
$\displaystyle|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2}+|\partial_{\xi}v|_{2}^{2}+|v|_{2}^{2}$
(3.31) $\displaystyle+|\nabla
T|_{2}^{2}+|\partial_{\xi}T|_{2}^{2}+\alpha_{s}|T|_{\xi=1}|_{2}^{2}\leq
c|Q_{1}|_{2}^{2}.$

Therefore, we conclude that

$\displaystyle|v(t)|_{2}^{2}+|T(t)|_{2}^{2}\leq
e^{-ct}(|v_{0}|_{2}^{2}+|T_{0}|_{2}^{2})+c|Q_{1}|_{2}^{2}.$ (3.32)

Combining $(3.30)$ and $(3.31)$ we arrive at

$\displaystyle\int_{t_{0}}^{t_{0}+1}|\nabla_{e_{\theta}}v(t)|_{2}^{2}+\nabla_{e_{\varphi}}v(t)|_{2}^{2}+|\partial_{\xi}v(t)|_{2}^{2}+|v(t)|_{2}^{2}dt$
$\displaystyle+\int_{t_{0}}^{t_{0}+1}|\nabla
T(t)|_{2}^{2}+|\partial_{\xi}T(t)|_{2}^{2}+\alpha_{s}|T|_{\xi=1}(t)|_{2}^{2}dt\leq
e^{-ct}(|v_{0}|_{2}^{2}+|T_{0}|_{2}^{2})+c|Q_{1}|_{2}^{2}.$ (3.33)

###  3.2 $L^{4}$ estimates of $q$

Multiplying $q^{3}$ on both sides of $(1.13)$ and integrating on $\mho$ yields

$\displaystyle\frac{1}{4}\frac{d|q|_{4}^{4}}{dt}+3||\nabla
q|q|_{2}^{2}+3||\partial_{\xi}q|q|_{2}^{2}+\beta_{s}|q|_{\xi=1}|_{4}^{4}$
(3.34) $\displaystyle=$
$\displaystyle\int_{\mho}Q_{2}q^{3}d\mho-\int_{\mho}[\nabla_{v}q+(\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime})\partial_{\xi}q]q^{3}.$

Since by Lemma 2.2 or Lemma 2.3 we have

$\displaystyle\int_{\mho}[\nabla_{v}q+(\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime})\partial_{\xi}q]q^{3}=0.$
(3.35)

Then $(3.34)$ and $(3.35)$ imply

$\displaystyle\frac{1}{4}\frac{d|q|_{4}^{4}}{dt}+3||\nabla
q|q|_{2}^{2}+3||\partial_{\xi}q|q|_{2}^{2}+\beta_{s}|q|_{\xi=1}|_{4}^{4}$
(3.36) $\displaystyle=$
$\displaystyle\int_{\mho}Q_{2}q^{3}d\mho\leq|Q_{2}|_{2}|q^{2}|_{3}^{\frac{3}{2}}\leq
c|Q_{2}|_{2}|q^{2}|_{2}^{\frac{3}{4}}\|q^{2}\|_{1}^{\frac{3}{4}}$
$\displaystyle=$ $\displaystyle
c|Q_{2}|_{2}|q|_{4}^{\frac{3}{2}}(|q|_{4}^{\frac{3}{2}}+||\nabla
q|q|_{2}^{\frac{3}{4}}+|q\partial_{\xi}q|_{2}^{\frac{3}{4}})$
$\displaystyle\leq$ $\displaystyle\varepsilon(||\nabla
q|q|_{2}^{2}+|q\partial_{\xi}q|_{2}^{2})+c|Q_{2}|_{2}^{\frac{8}{5}}|q|_{4}^{\frac{12}{5}}+c|Q_{2}|_{2}|q|_{4}^{3}.$

Since
$q^{4}(\theta,\varphi,\xi)=-\int_{\xi}^{1}\partial_{\xi^{\prime}}q^{4}d\xi^{\prime}+q^{4}|_{\xi=1},$
by H$\mathrm{\ddot{o}}$lder inequality we get

$\displaystyle|q|_{4}^{4}\leq
c|q|\partial_{\xi}q||_{2}^{2}+\frac{1}{2}|q|_{4}^{4}+|q|_{\xi=1}|_{4}^{4},$

which implies that there exists a positive constant $c_{1}$ such that

$\displaystyle\frac{d}{dt}|q|_{4}^{4}+c_{1}|q|_{4}^{4}\leq
c|Q_{2}|_{2}^{\frac{8}{5}}|q|_{4}^{\frac{12}{5}}+c|Q_{2}|_{2}|q|_{4}^{3}.$

That is

$\displaystyle\frac{d}{dt}|q|_{4}^{2}+c|q|_{4}^{2}\leq c|Q_{2}|_{2}^{2}.$

Therefore, we have

$\displaystyle|q(t)|_{4}^{2}\leq e^{-ct}|q_{0}|_{4}^{2}+c|Q_{2}|_{2}^{2}.$
(3.37)

Combining $(3.36)$ and $(3.37),$ we arrive at

$\displaystyle\int_{t_{0}}^{t_{0}+1}(||\nabla
q(t)|q(t)|_{2}^{2}+||\partial_{\xi}q(t)|q(t)|_{2}^{2}+\beta_{s}|q|_{\xi=1}(t)|_{4}^{4})dt\leq
ce^{-c_{1}t_{0}}|q_{0}|_{4}^{2}+c|Q_{2}|_{2}^{2}.$ (3.38)

###  3.3 $L^{4}$ estimate for $T$

Taking inner product of $(1.12)$ with $T^{3}$ in $L^{2}(\mho)$ yields,

$\displaystyle\frac{1}{4}\frac{|T|_{4}^{4}}{dt}+3||\nabla
T|T|_{2}^{2}+3||\partial_{\xi}T|T|_{2}^{2}+\alpha_{s}|T|_{\xi=1}|_{4}^{4}$
$\displaystyle=\int_{\mho}\frac{bP}{p}\Big{(}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}\Big{)}|T|^{2}Td\mho+\int_{\mho}\frac{abP}{p}\Big{(}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}\Big{)}q|T|^{2}Td\mho$
$\displaystyle-\int_{\mho}\Big{[}\nabla_{v}T+\Big{(}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}\Big{)}\partial_{\xi}T\Big{]}|T|^{2}Td\mho+\int_{\mho}Q_{1}|T|^{2}Td\mho.$
(3.39)

Using the estimate in [20], we know that

$\displaystyle|\int_{\mho}\frac{bP}{p}\Big{(}\int_{\xi}^{1}\mathrm{div}d\xi^{\prime}\Big{)}|T|^{2}T|$
$\displaystyle\leq
c(|\nabla_{e_{\theta}}v|_{2}+|\nabla_{e_{\varphi}}v|_{2})|T|_{4}^{2}(|\nabla
T|_{2}+|T|_{2}).$ (3.40)

Taking an similar argument as in [20], we get

$\displaystyle|\int_{\mho}\frac{abP}{p}q\Big{(}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}\Big{)}T^{3}|$
$\displaystyle\leq
c\int_{0}^{1}\Big{[}\Big{(}\int_{S^{2}}q^{4}dS^{2}\Big{)}^{\frac{1}{4}}\Big{(}\int_{S^{2}}T^{12}dS^{2}\Big{)}^{\frac{1}{4}}\Big{]}d\xi\sup\limits_{\xi\in[0,1]}|\frac{bP}{p}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}|_{L^{2}(S^{2})}.$

Therefore,

$\displaystyle|\int_{\mho}\frac{abP}{p}q\Big{(}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}\Big{)}T^{3}|$
(3.41) $\displaystyle\leq$ $\displaystyle
c\int_{0}^{1}|q|_{L^{4}(S^{2})}|T^{2}|_{L^{6}(S^{2})}^{\frac{3}{2}}d\xi|\mathrm{div}v|_{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{1}|q|_{L^{4}(S^{2})}|T^{2}|_{L^{2}(S^{2})}^{\frac{1}{2}}|\nabla
T^{2}|_{L^{2}(S^{2})}d\xi|\mathrm{div}v|_{2}$ $\displaystyle\leq$
$\displaystyle\varepsilon|T|\nabla
T||_{2}^{2}+c(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2})|T|_{4}^{2}|q|_{4}^{2}.$

To estimate the last term on the right of $(3.39),$ we use interpolation
inequality and H$\mathrm{\ddot{o}}$lder inequality to have

$\displaystyle\int_{\mho}Q_{1}T^{3}d\mho$ $\displaystyle\leq$
$\displaystyle|Q_{1}|_{2}|T|_{6}^{3}=|Q_{1}|_{2}|T^{2}|_{3}^{\frac{3}{2}}$
(3.42) $\displaystyle\leq$ $\displaystyle
c|Q_{1}|_{2}|T^{2}|_{2}^{\frac{3}{4}}(|T|\nabla
T||_{2}^{\frac{3}{4}}+|T|\partial_{\xi}T||_{2}^{\frac{3}{4}}+|T^{2}|_{2}^{\frac{3}{4}})$
$\displaystyle\leq$ $\displaystyle\varepsilon(|T|\nabla
T||_{2}^{2}+|T|\partial_{\xi}T||_{2}^{2})+c|Q_{1}|_{2}|T|_{4}^{3}+c|Q_{1}|_{2}^{\frac{8}{5}}|T|_{4}^{\frac{12}{5}}.$

By virtue of $(3.39)-(3.42),$ we have

$\displaystyle\frac{|T|_{4}^{2}}{2}\frac{d|T|_{4}^{2}}{dt}$
$\displaystyle\leq$ $\displaystyle
c(|\nabla_{e_{\theta}}v|_{2}+|\nabla_{e_{\varphi}}v|_{2})|T|_{4}^{2}(|\nabla
T|_{2}+|T|_{2})$
$\displaystyle+c(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2})|T|_{4}^{2}|q|_{4}^{2}$
$\displaystyle+c|Q_{1}|_{2}|T|_{4}^{3}+c|Q_{1}|_{2}^{\frac{8}{5}}|T|_{4}^{\frac{12}{5}},$

which implies

$\displaystyle\frac{d|T|_{4}^{2}}{dt}$ $\displaystyle\leq$ $\displaystyle
c(|\nabla_{e_{\theta}}v|_{2}+|\nabla_{e_{\varphi}}v|_{2})(|\nabla
T|_{2}+|T|_{2})$ (3.43)
$\displaystyle+c(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2})|q|_{4}^{2}$
$\displaystyle+c|Q_{1}|_{2}|T|_{4}+c|Q_{1}|_{2}^{\frac{8}{5}}|T|_{4}^{\frac{2}{5}}.$

Since $|T|_{4}\leq c\|T\|_{1}$ , by $(3.32),(3.33),(3.37)$ and the uniform
Gronwall lemma, we obtain the desired uniform boundedness of $|T|_{4}$ and it
gives the absorbing ball of $T$ in $L^{4}(\mho).$ That is to say there exists
a constant $c$ independent of $t$ such that

$\displaystyle|T|_{4}(t)\leq c$ (3.44)

for all $t\geq 0.$ Moreover, by $(3.39)$ we have the following uniform bound
on the time average

$\displaystyle\int_{t_{0}}^{t_{0}+1}(||\nabla
T(t)|T(t)|_{2}^{2}+||\partial_{\xi}T(t)|T(t)|_{2}^{2}+|T(t)|_{\xi=1}|_{4}^{4})dt$
(3.45) $\displaystyle\leq$ $\displaystyle
c|T(t_{0})|_{4}^{4}+c\int_{t_{0}}^{t_{0}+1}(|\nabla_{e_{\theta}}v(t)|_{2}+|\nabla_{e_{\varphi}}v(t)|_{2})|T(t)|_{4}^{2}(|\nabla
T(t)|_{2}+|T(t)|_{2})dt$
$\displaystyle+c\int_{t_{0}}^{t_{0}+1}(|\nabla_{e_{\theta}}v(t)|_{2}^{2}+|\nabla_{e_{\varphi}}v(t)|_{2}^{2})|T(t)|_{4}^{2}|q(t)|_{4}^{2}dt$
$\displaystyle+c\int_{t_{0}}^{t_{0}+1}|Q_{1}|_{2}|T(t)|_{4}^{3}dt+c\int_{t_{0}}^{t_{0}+1}|Q_{1}|_{2}^{\frac{8}{5}}|T(t)|_{4}^{\frac{12}{5}}dt\leq
c,$

where the constant $c$ is independent of $t_{0}.$

###### Remark 3.1

In this part, we try to use uniform Gronwall lemma to get the uniform
boundedness of $|T|_{4}^{2}$ instead of $|T|_{4}^{4}$. Therefore, from
$(3.43)-(3.45)$ we can see that the estimate of $|T|_{3}^{3}$ is not necessary
, which simplify the proof of the existence of the absorbing ball in space
$L^{4}(\mho)$ in $\cite[cite]{[\@@bibref{}{GH2}{}{}]}.$ This technique will be
used again in the following to get the uniform estimates for $v$ in
$L^{4}(\mho)$ with respect to time.

###  3.4 $L^{4}$ estimate for $v$

Taking inner product of equation $(2.22)$ with $|\tilde{v}|^{2}\tilde{v}$ in
$L^{2}(\mho)$, we get

$\displaystyle\frac{1}{4}\frac{d|\tilde{v}|_{4}^{4}}{dt}$ $\displaystyle+$
$\displaystyle\int_{\mho}\Big{(}|\nabla_{e_{\theta}}\tilde{v}|^{2}|\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2}|\tilde{v}|^{2}+\frac{1}{2}|\nabla_{e_{\theta}}|\tilde{v}|^{2}|^{2}+\frac{1}{2}|\nabla_{e_{\varphi}}|\tilde{v}|^{2}|^{2}+|\tilde{v}|^{4}\Big{)}d\mho$
(3.46)
$\displaystyle+\int_{\mho}\Big{(}|\tilde{v}_{\xi}|^{2}|\tilde{v}|^{2}+\frac{1}{2}|\partial_{\xi}|\tilde{v}|^{2}|^{2}\Big{)}d\mho=-\int_{\mho}\Big{[}\nabla_{\tilde{v}}\tilde{v}+\Big{(}\int_{\xi}^{1}\mathrm{div}\tilde{v}d\xi^{\prime}\Big{)}\partial_{\xi}\tilde{v}\Big{]}\cdot|\tilde{v}|^{2}\tilde{v}d\mho$
$\displaystyle-\int_{\mho}(\nabla_{\bar{v}}\tilde{v})\cdot|\tilde{v}|^{2}\tilde{v}d\mho-\int_{\mho}(\nabla_{\tilde{v}}\bar{v})\cdot|\tilde{v}|^{2}\tilde{v}d\mho$
$\displaystyle-\int_{\mho}\Big{(}\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}\cdot|\tilde{v}|^{2}\tilde{v}\Big{)}d\mho$
$\displaystyle+\int_{\mho}\Big{(}\int_{0}^{1}\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}d\xi\Big{)}\cdot|\tilde{v}|^{2}\tilde{v}d\mho$
$\displaystyle+\int_{\mho}\overline{(\tilde{v}\mathrm{div}\tilde{v}+\nabla_{\tilde{v}}\tilde{v})}\cdot|\tilde{v}|^{2}\tilde{v}d\mho-\int_{\mho}(\frac{f}{R_{0}}k\times\tilde{v})\cdot|\tilde{v}|^{2}\tilde{v}d\mho,$

where $\tilde{v}_{\xi}=\partial_{\xi}\tilde{v}.$ By Lemma $2.3,$ we have

$\displaystyle\int_{\mho}\Big{[}\nabla_{\tilde{v}}\tilde{v}+\Big{(}\int_{\xi}^{1}\mathrm{div}\tilde{v}d\xi^{\prime}\Big{)}\partial_{\xi}\tilde{v}\Big{]}|\tilde{v}|^{2}\tilde{v}d\mho=0.$
(3.47)

Using Lemma $2.3$ again and $(2.18)$, we obtain

$\displaystyle\int_{\mho}(\nabla_{\bar{v}}\tilde{v})\cdot|\tilde{v}|^{2}\tilde{v}d\mho=\frac{1}{4}\int_{\mho}\nabla_{\bar{v}}|\tilde{v}|^{4}d\mho=-\frac{1}{4}\int_{\mho}|\tilde{v}|^{4}\mathrm{div}\bar{v}d\mho=0.$
(3.48)

By virtue of Lemma $2.2$, we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mho}\mathrm{div}[(|\tilde{v}|^{2}\tilde{v}\cdot\bar{v})\tilde{v}]d\mho=\int_{\mho}\nabla_{\tilde{v}}(|\tilde{v}|^{2}\tilde{v}\cdot\bar{v})d\mho+\int_{\mho}|\tilde{v}|^{2}\tilde{v}\cdot\bar{v}\mathrm{div}\tilde{v}d\mho$
$\displaystyle=$
$\displaystyle\int_{\mho}[|\tilde{v}|^{2}\tilde{v}\cdot\nabla_{\tilde{v}}\bar{v}+\bar{v}\cdot\nabla_{\tilde{v}}(|\tilde{v}|^{2}\tilde{v})]d\mho+\int_{\mho}|\tilde{v}|^{2}\tilde{v}\cdot\bar{v}\mathrm{div}\tilde{v}d\mho.$

Therefore,

$\displaystyle\int_{\mho}[|\tilde{v}|^{2}\tilde{v}\cdot\bar{v}\mathrm{div}\tilde{v}+\bar{v}\cdot\nabla_{\tilde{v}}(|\tilde{v}|^{2}\tilde{v})]d\mho=-\int_{\mho}|\tilde{v}|^{2}\tilde{v}\cdot\nabla_{\tilde{v}}\bar{v}d\mho.$
(3.49)

Using integration by parts, we obtain

$\displaystyle\int_{\mho}\Big{[}\int_{0}^{1}(\tilde{v}\mathrm{div}\tilde{v}+\nabla_{\tilde{v}}\tilde{v})d\xi\Big{]}\cdot|\tilde{v}|^{2}\tilde{v}d\mho$
$\displaystyle=$
$\displaystyle-\int_{\mho}\Big{(}\int_{0}^{1}\tilde{v}_{\theta}\tilde{v}d\xi\Big{)}\cdot\nabla_{e_{\theta}}(|\tilde{v}|^{2}\tilde{v})d\mho$
(3.50)
$\displaystyle-\int_{\mho}\Big{(}\int_{0}^{1}\tilde{v}_{\varphi}\tilde{v}d\xi\Big{)}\cdot\nabla_{e_{\varphi}}(|\tilde{v}|^{2}\tilde{v})d\mho.$

Note that the minus on the right hand side of $(3.50)$ was missed in
$\cite[cite]{[\@@bibref{}{GH2}{}{}]}.$ In view of $(3.46)-(3.50)$ combined
with $(\frac{f}{R_{0}}\times\tilde{v})\cdot|\tilde{v}|^{2}\tilde{v}=0$, we
have

$\displaystyle\frac{1}{4}\frac{d|\tilde{v}|_{4}^{4}}{dt}$ $\displaystyle+$
$\displaystyle\int_{\mho}\Big{(}|\nabla_{e_{\theta}}\tilde{v}|^{2}|\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2}|\tilde{v}|^{2}+\frac{1}{2}|\nabla_{e_{\theta}}|\tilde{v}|^{2}|^{2}+\frac{1}{2}|\nabla_{e_{\varphi}}|\tilde{v}|^{2}|^{2}+|\tilde{v}|^{4}\Big{)}d\mho$
$\displaystyle+\int_{\mho}\Big{(}|\tilde{v}_{\xi}|^{2}|\tilde{v}|^{2}+\frac{1}{2}|\partial_{\xi}|\tilde{v}|^{2}|^{2}\Big{)}d\mho$
$\displaystyle=\int_{\mho}[|\tilde{v}|^{2}\tilde{v}\cdot\bar{v}\mathrm{div}\tilde{v}+\bar{v}\cdot\nabla_{\tilde{v}}(|\tilde{v}|^{2}\tilde{v})]d\mho$
$\displaystyle+\int_{\mho}\Big{(}\int_{0}^{1}\tilde{v}_{\theta}\tilde{v}d\xi\Big{)}\cdot\nabla_{e_{\theta}}(|\tilde{v}|^{2}\tilde{v})d\mho+\int_{\mho}\Big{(}\int_{0}^{1}\tilde{v}_{\varphi}\tilde{v}d\xi\Big{)}\cdot\nabla_{e_{\varphi}}(|\tilde{v}|^{2}\tilde{v})d\mho$
$\displaystyle+\int_{\mho}\Big{(}\int_{\xi}^{1}\frac{bP}{p}[(1+aq)T]d\xi^{\prime}\cdot\mathrm{div}(|\tilde{v}|^{2}\tilde{v})\Big{)}d\mho$
$\displaystyle-\int_{\mho}\Big{(}\int_{0}^{1}\int_{\xi}^{1}\frac{bP}{p}[(1+aq)T]d\xi^{\prime}d\xi\cdot\mathrm{div}(|\tilde{v}|^{2}\tilde{v})\Big{)}d\mho.$

Then by H$\mathrm{\ddot{o}}$lder inequality and the argument above, we obtain

$\displaystyle\frac{1}{4}\frac{d|\tilde{v}|_{4}^{4}}{dt}$ $\displaystyle+$
$\displaystyle\int_{\mho}\Big{(}|\nabla_{e_{\theta}}\tilde{v}|^{2}|\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2}|\tilde{v}|^{2}+\frac{1}{2}|\nabla_{e_{\theta}}|\tilde{v}|^{2}|^{2}+\frac{1}{2}|\nabla_{e_{\varphi}}|\tilde{v}|^{2}|^{2}+|\tilde{v}|^{4}\Big{)}d\mho$
(3.51)
$\displaystyle+\int_{\mho}\Big{(}|\tilde{v}_{\xi}|^{2}|\tilde{v}|^{2}+\frac{1}{2}|\partial_{\xi}|\tilde{v}|^{2}|^{2}\Big{)}d\mho$
$\displaystyle\leq$ $\displaystyle
c\int_{S^{2}}|\bar{v}|\Big{(}\int_{0}^{1}|\tilde{v}|^{3}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})^{\frac{1}{2}}d\xi\Big{)}dS^{2}$
$\displaystyle+c\int_{S^{2}}\Big{(}\int_{0}^{1}|\tilde{v}|^{2}d\xi\Big{)}\Big{(}\int_{0}^{1}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})^{\frac{1}{2}}d\xi\Big{)}dS^{2}$
$\displaystyle+c\int_{S^{2}}\Big{(}(|\bar{T}|+|\overline{qT}|)\int_{0}^{1}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})^{\frac{1}{2}}d\xi\Big{)}dS^{2}$
$\displaystyle=:$ $\displaystyle I_{1}+I_{2}+I_{3}.$

In the following, we estimate $I_{i},i=1,2,3,$ separately. By interpolation
inequality , H$\mathrm{\ddot{o}}$lder inequality and Minkowski inequality, we
have

$\displaystyle I_{1}$ $\displaystyle\leq$
$\displaystyle\int_{S^{2}}\Big{[}|\bar{v}|\Big{(}\int_{0}^{1}|\tilde{v}|^{4}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\xi\Big{)}^{\frac{1}{2}}\Big{]}dS^{2}$
$\displaystyle\leq$
$\displaystyle|\bar{v}|_{4}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\tilde{v}|^{4}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}\Big{(}\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle|\bar{v}|_{4}\Big{(}\int_{0}^{1}(\int_{S^{2}}|\tilde{v}|^{2\times
4}dS^{2})^{\frac{1}{2}}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c|\bar{v}|_{4}^{2}\int_{0}^{1}||\tilde{v}|^{2}|_{2}(||\tilde{v}|^{2}|_{2}+|\nabla|\tilde{v}|^{2}|_{2})d\xi$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c|\bar{v}|_{4}^{2}||\tilde{v}|^{2}|_{2}(||\tilde{v}|^{2}|_{2}+|\nabla|\tilde{v}|^{2}|_{2})$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+\varepsilon|\nabla|\tilde{v}|^{2}|_{2}^{2}+c(\|\bar{v}\|_{1}^{2}+|\bar{v}|_{2}^{2}\|\bar{v}\|_{1}^{2})|\tilde{v}|_{4}^{4}.$

Similarly,

$\displaystyle I_{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{S^{2}}(\int_{0}^{1}|\tilde{v}|^{2}d\xi)(\int_{0}^{1}|\tilde{v}|^{2}d\xi)^{\frac{1}{2}}\Big{(}\int_{0}^{1}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\xi\Big{)}^{\frac{1}{2}}dS^{2}$
$\displaystyle\leq$ $\displaystyle
c\Big{(}\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho\Big{)}^{\frac{1}{2}}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\tilde{v}|^{2}d\xi)^{3}dS^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c\Big{(}\int_{0}^{1}(\int_{S^{2}}|\tilde{v}|^{6}dS^{2})^{\frac{1}{3}}d\xi\Big{)}^{3}$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c\Big{(}\int_{0}^{1}|\tilde{v}|_{4}^{\frac{4}{3}}(|\tilde{v}|_{2}^{\frac{2}{3}}+|\nabla_{e_{\theta}}\tilde{v}|_{2}^{\frac{2}{3}}+|\nabla_{e_{\varphi}}\tilde{v}|_{2}^{\frac{2}{3}})d\xi\Big{)}^{3}$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c|\tilde{v}|_{4}^{4}|\tilde{v}|_{2}^{2}+c|\tilde{v}|_{4}^{4}(|\nabla_{e_{\theta}}\tilde{v}|_{2}^{2}+|\nabla_{e_{\varphi}}\tilde{v}|_{2}^{2}).$

Analogously, we have

$\displaystyle I_{3}$ $\displaystyle\leq$
$\displaystyle\int_{S^{2}}(|\overline{T}|+|\overline{qT}|)\Big{(}\int_{0}^{1}|\tilde{v}|^{2}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\xi\Big{)}^{\frac{1}{2}}dS^{2}$
$\displaystyle\leq$
$\displaystyle\Big{(}\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho\Big{)}^{\frac{1}{2}}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\tilde{v}|^{2}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}(|\overline{T}|_{4}+|\overline{qT}|_{4})$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c\int_{0}^{1}\Big{(}\int_{S^{2}}|\tilde{v}|^{4}dS^{2}\Big{)}^{\frac{1}{2}}d\xi(|\overline{T}|_{4}^{2}+|\overline{qT}|_{4}^{2})$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+c|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+|\tilde{v}|_{4}^{2}\Big{[}|\overline{T}|_{4}^{2}+\Big{(}\int_{S^{2}}(\int_{0}^{1}qTd\xi)^{4}dS^{2}\Big{)}^{\frac{1}{2}}\Big{]}$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+c|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c|\tilde{v}|_{4}^{2}|T|_{4}^{2}+c|\tilde{v}|_{4}^{2}\Big{(}\int_{0}^{1}|q|_{8}|T|_{8}d\xi\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+c|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c|\tilde{v}|_{4}^{2}|T|_{4}^{2}$
$\displaystyle+c|\tilde{v}|_{4}^{2}\Big{(}\int_{0}^{1}|q|_{4}^{\frac{1}{2}}(|q|_{2}^{\frac{1}{2}}+|\nabla
q|_{2}^{\frac{1}{2}})|T|_{4}^{\frac{1}{2}}(|T|_{2}^{\frac{1}{2}}+|\nabla
T|_{2}^{\frac{1}{2}})d\xi\Big{)}^{2}$ $\displaystyle\leq$
$\displaystyle\varepsilon\int_{\mho}|\tilde{v}|^{2}(|\nabla_{e_{\theta}}\tilde{v}|^{2}+c|\nabla_{e_{\varphi}}\tilde{v}|^{2})d\mho+c|\tilde{v}|_{4}^{2}|T|_{4}^{2}$
$\displaystyle+c|\tilde{v}|_{4}^{2}(|q|_{4}^{2}+|q|_{4}|\nabla
q|_{2})(|T|_{4}^{2}+|T|_{4}|\nabla T|_{2}).$

By the estimates of $I_{1}-I_{3},$ we have

$\displaystyle\frac{d|\tilde{v}|_{4}^{2}}{dt}\leq
c(\|v\|_{1}^{2}+|v|_{2}^{2}|\|v\|_{1}^{2})|\tilde{v}|_{4}^{2}+c|T|_{4}^{2}+c(|q|_{4}^{2}+|q|_{4}|\nabla
q|_{2})(|T|_{4}^{2}+|T|_{4}|\nabla T|_{2}).$

Therefore, by $(3.27),(3.32),(3.33),(3.37),(3.44)$ and uniformly Gronwall
Lemma, we have

$\displaystyle|\tilde{v}(t)|_{4}\leq c,$ (3.52)

where $t\geq 0$ and $c$ is independent of $t$. Furthermore, by $(3.51)$ and
estimates of $I_{1}-I_{3},$ we get

$\displaystyle\int_{t_{0}}^{t_{0}+1}||\nabla_{e_{\theta}}\tilde{v}(t)||\tilde{v}(t)||_{2}^{2}+||\nabla_{e_{\varphi}}\tilde{v}(t)||\tilde{v}(t)||_{2}^{2}dt$
(3.53) $\displaystyle\leq$
$\displaystyle|\tilde{v}(t_{0})|_{4}^{4}+c\int_{t_{0}}^{t_{0}+1}[(\|v(t)\|_{1}^{2}+|v(t)|_{2}^{2}|\|v(t)\|_{1}^{2})|\tilde{v}(t)|_{4}^{2}+|T(t)|_{4}^{2}]dt$
$\displaystyle+c\int_{t_{0}}^{t_{0}+1}(|q(t)|_{4}^{2}+|q(t)|_{4}|\nabla
q(t)|_{2})(|T(t)|_{4}^{2}+|T(t)|_{4}|\nabla T(t)|_{2})dt\leq c,$

where $c$ is independent of $t_{0}.$

###  3.5 $H^{1}$ estimates of $v,T,q$.

From $\cite[cite]{[\@@bibref{}{GH2}{}{}]},$ we have

$\displaystyle\frac{d\|\bar{v}\|_{1}}{dt}+|\Delta\bar{v}|_{2}^{2}\leq
c(\|\bar{v}\|_{1}^{2}+|\bar{v}|_{2}^{2}\|\bar{v}\|_{1}^{2})\|\bar{v}\|_{1}^{2}+c||\tilde{v}||\nabla_{e_{\theta}}\tilde{v}||_{2}^{2}+c||\tilde{v}||\nabla_{e_{\varphi}}\tilde{v}||_{2}^{2}.$
(3.54)

By the uniform Gronwall lemma and $(3.32)-(3.33)$, we obtain

$\displaystyle\|\bar{v}(t)\|_{1}\leq c,$ (3.55)

where $t\geq 0$ and $c$ is independent of $t.$ By Sobolev inequality, we have
that for all $t\geq 0$

$\displaystyle|\bar{v}(t)|_{L^{4}(S^{2})}\leq
c|\bar{v}(t)|_{L^{2}(S^{2})}+c|\nabla\bar{v}(t)|_{L^{2}(S^{2})}\leq c,$

which implies

$\displaystyle|v(t)|_{4}\leq|\tilde{v}(t)|_{4}+|\bar{v}(t)|_{4}\leq c.$ (3.56)

From $\cite[cite]{[\@@bibref{}{GH2}{}{}]},$ we have

$\displaystyle\frac{d|v_{\xi}|_{2}^{2}}{dt}+|\nabla_{e_{\theta}}v_{\xi}|_{2}^{2}+|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{2}+|v_{\xi}|_{2}^{2}+|v_{\xi\xi}|_{2}^{2}$
(3.57) $\displaystyle\leq$ $\displaystyle
c(\|\bar{v}\|_{1}^{8}+|\tilde{v}|_{4}^{8})|v_{\xi}|_{2}^{2}+c|T|_{2}^{2}+c|q|_{4}^{4}+c|T|_{4}^{4},$

where $\partial_{\xi\xi}v=v_{\xi\xi}.$ By the uniform Gronwall lemma, we
obtain

$\displaystyle|v_{\xi}(t)|_{2}\leq c,$ (3.58)

where $t\geq 0$ and $c$ is independent of $t.$ Therefore, by $(3.56)-(3.58)$
and uniformly Gronwall inequality we have

$\displaystyle\int_{t_{0}}^{t_{0}+1}(|\nabla_{e_{\theta}}v_{\xi}|_{2}^{2}+|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{2}+|v_{\xi\xi}|_{2}^{2})dt\leq
c,$ (3.59)

where $t_{0}\geq 0$ and $c$ is independent of $t_{0}.$ Taking inner product
with $-\Delta v$ in $L^{2}(\mho),$ we obtain

$\displaystyle\frac{1}{2}\frac{d(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2}+|v|_{2}^{2})}{dt}+|\Delta
v|_{2}^{2}+|\nabla_{e_{\theta}}v_{\xi}|_{2}^{2}+|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{2}+|v_{\xi}|_{2}^{2}$
(3.60) $\displaystyle=$ $\displaystyle\int_{\mho}\nabla_{v}v\cdot\Delta
vd\mho+\int_{\mho}\Big{(}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}\Big{)}\partial_{\xi}v\cdot\Delta
vd\mho$
$\displaystyle+\int_{\mho}\Big{(}\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}(1+aq)Td\xi^{\prime}\Big{)}\Delta
vd\mho$ $\displaystyle+\int_{\mho}\frac{f}{R_{0}}v^{\bot}\cdot\Delta
vd\mho+\int_{\mho}\mathrm{grad}\Phi_{s}\cdot\Delta vd\mho$ $\displaystyle=$
$\displaystyle J_{1}+J_{2}+J_{3}+J_{4}+J_{5}.$

By H$\mathrm{\ddot{o}}$lder inequality, interpolation inequality and Young’s
inequality, we have

$\displaystyle J_{1}$ $\displaystyle\leq$ $\displaystyle|\Delta
v|_{2}|v|_{4}(|\nabla_{e_{\theta}}v|_{4}+|\nabla_{e_{\varphi}}v|_{4})$
$\displaystyle\leq$ $\displaystyle c|\Delta
v|_{2}|v|_{4}(|\nabla_{e_{\theta}}v|_{2}^{\frac{1}{4}}+|\nabla_{e_{\varphi}}v|_{2}^{\frac{1}{4}})(|\Delta
v|_{2}^{\frac{3}{4}}+|\nabla_{e_{\theta}}v_{\xi}|_{2}^{\frac{3}{4}}+|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{\frac{3}{4}}+|\nabla_{e_{\varphi}}v|_{2}^{\frac{3}{4}}+|\nabla_{e_{\theta}}v|_{2}^{\frac{3}{4}})$
$\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
v|_{2}^{2}+\varepsilon|\nabla_{e_{\theta}}v_{\xi}|_{2}^{2}+\varepsilon|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{2}+c(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2})(|v|_{4}^{8}+|v|_{4}^{2})$
$\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
v|_{2}^{2}+\varepsilon|\nabla_{e_{\theta}}v_{\xi}|_{2}^{2}+\varepsilon|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{2}+c(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2}).$

To estimate $J_{2},$ we have

$\displaystyle J_{2}$ $\displaystyle\leq$
$\displaystyle\int_{S^{2}}\Big{(}\int_{0}^{1}|\mathrm{div}v|d\xi\Big{)}\Big{(}\int_{0}^{1}|\partial_{\xi}v||\Delta
v|d\xi\Big{)}dS^{2}$ (3.61) $\displaystyle\leq$
$\displaystyle\int_{S^{2}}\Big{(}\int_{0}^{1}|\mathrm{div}v|d\xi\Big{)}\Big{(}\int_{0}^{1}|\partial_{\xi}v|^{2}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}|\Delta
v|^{2}d\xi\Big{)}^{\frac{1}{2}}dS^{2}$ $\displaystyle\leq$ $\displaystyle
c|\Delta
v|_{2}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\partial_{\xi}v|^{2}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\mathrm{div}v|d\xi)^{4}dS^{2}\Big{)}^{\frac{1}{4}}$
$\displaystyle\leq$ $\displaystyle c|\Delta
v|_{2}\Big{(}\int_{0}^{1}(\int_{S^{2}}|\partial_{\xi}v|^{4}dS^{2})^{\frac{1}{2}}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}(\int_{S^{2}}|\mathrm{div}v|^{4}dS^{2})^{\frac{1}{4}}d\xi\Big{)}$
$\displaystyle\leq$ $\displaystyle c|\Delta
v|_{2}\Big{(}\int_{0}^{1}|v_{\xi}|_{2}(|v_{\xi}|_{2}+|\nabla_{e_{\theta}}v_{\xi}|_{2}+|\nabla_{e_{\varphi}}v_{\xi}|_{2})d\xi\Big{)}^{\frac{1}{2}}\cdot$
$\displaystyle\Big{(}\int_{0}^{1}|\mathrm{div}v|_{2}^{\frac{1}{2}}(|\mathrm{div}v|_{2}^{\frac{1}{2}}+|\Delta
v|_{2}^{\frac{1}{2}})d\xi\Big{)}$ $\displaystyle\leq$ $\displaystyle c|\Delta
v|_{2}|v_{\xi}|_{2}^{\frac{1}{2}}(|v_{\xi}|_{2}^{\frac{1}{2}}+|\nabla_{e_{\theta}}v_{\xi}|_{2}^{\frac{1}{2}}+|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{\frac{1}{2}})$
$\displaystyle\cdot|\mathrm{div}v|_{2}^{\frac{1}{2}}(|\mathrm{div}v|_{2}^{\frac{1}{2}}+|\Delta
v|_{2}^{\frac{1}{2}})$ $\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
v|_{2}^{2}+c(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2})$
$\displaystyle\cdot(1+|v_{\xi}|_{2}^{4}+|v_{\xi}|_{2}^{2}|\nabla_{e_{\theta}}v_{\xi}|_{2}^{2}+|v_{\xi}|_{2}^{2}|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{2}).$

Concerning estimate of $J_{3},$ there are some typos in $(5.80)$ of [20]. In
the first inequality of $(5.80)$, the integral region should be $S^{2}$
instead of $\Omega.$ For reader’s convenience, we estimate $J_{3}$ again.
Using H$\mathrm{\ddot{o}}$lder inequality, Minkowski inequality and
interpolation inequality, we have

$\displaystyle J_{3}$ $\displaystyle\leq$ $\displaystyle c|\nabla
T|_{2}|\Delta
v|_{2}+c\int_{S^{2}}(\int_{0}^{1}|\mathrm{grad}q|^{2}d\xi)^{\frac{1}{2}}(\int_{0}^{1}T^{2}d\xi)^{\frac{1}{2}}\int_{0}^{1}|\Delta
v|d\xi dS^{2}$ (3.62)
$\displaystyle+c\int_{S^{2}}(\int_{0}^{1}|\mathrm{grad}T|^{2}d\xi)^{\frac{1}{2}}(\int_{0}^{1}q^{2}d\xi)^{\frac{1}{2}}(\int_{0}^{1}|\Delta
v|d\xi)dS^{2}$ $\displaystyle\leq$ $\displaystyle c|\nabla T|_{2}|\Delta
v|_{2}+c|\Delta
v|_{2}|T|_{4}\Big{(}\int_{0}^{1}(\int_{S^{2}}|\mathrm{grad}q|^{4}dS^{2})^{\frac{1}{2}}d\xi\Big{)}^{\frac{1}{2}}$
$\displaystyle+c|\Delta
v|_{2}|q|_{4}\Big{(}\int_{0}^{1}(\int_{S^{2}}|\mathrm{grad}T|^{4}dS^{2})^{\frac{1}{2}}d\xi\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle c|\nabla T|_{2}|\Delta v|_{2}+c|\Delta
v|_{2}|T|_{4}|\nabla q|_{2}^{\frac{1}{2}}(|\nabla q|_{2}^{\frac{1}{2}}+|\Delta
q|_{2}^{\frac{1}{2}})$ $\displaystyle+c|\Delta v|_{2}|q|_{4}|\nabla
T|_{2}^{\frac{1}{2}}(|\nabla T|_{2}^{\frac{1}{2}}+|\Delta
T|_{2}^{\frac{1}{2}})$ $\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
v|_{2}^{2}+\varepsilon|\Delta T|_{2}^{2}+\varepsilon|\Delta
q|_{2}^{2}+c|\nabla T|_{2}^{2}+c|q|_{4}^{2}|\nabla T|_{2}^{2}$
$\displaystyle+c|T|_{4}^{2}|\nabla q|_{2}^{2}+c|q|_{4}^{4}|\nabla
T|_{2}^{2}+c|T|_{4}^{4}|\nabla q|_{2}^{2}.$

By H$\mathrm{\ddot{o}}$lder inequality, we have

$\displaystyle J_{4}\leq\varepsilon|\Delta v|_{2}^{2}+c|v|_{2}^{2}.$

By Lemma $2.1,$ we infer that $J_{5}=0.$ From $(3.60)$ and estimates of
$J_{1}-J_{5},$ we obtain

$\displaystyle\frac{d(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2}+|v|_{2}^{2})}{dt}$
(3.63) $\displaystyle\leq$ $\displaystyle
c(|\nabla_{e_{\theta}}v|_{2}^{2}+|\nabla_{e_{\varphi}}v|_{2}^{2})(1+|v_{\xi}|_{2}^{4}+|v_{\xi}|_{2}^{2}|\nabla_{e_{\theta}}v_{\xi}|_{2}^{2}+|v_{\xi}|_{2}^{2}|\nabla_{e_{\varphi}}v_{\xi}|_{2}^{2})$
$\displaystyle+c+c|\nabla T|_{2}^{2}+c|\nabla q|_{2}^{2}.$

In view of $(3.27),(3.33),(3.58),(3.59)$ and uniform Gronwall lemma, we get
that for all $t\geq 0$

$\displaystyle|\nabla_{e_{\theta}}v(t)|_{2}^{2}+|\nabla_{e_{\varphi}}v(t)|_{2}^{2}\leq
c,$ (3.64)

where $c$ is a positive constant which is independent of $t.$ Furthermore, by
virtue of $(3.60)$ we infer that for all $t_{0}\geq 0$

$\displaystyle\int_{t_{0}}^{t_{0}+1}|\Delta
v(t)|_{2}^{2}+|\nabla_{e_{\theta}}v_{\xi}(t)|_{2}^{2}+|\nabla_{e_{\varphi}}v_{\xi}(t)|_{2}^{2}dt\leq
c,$ (3.65)

where $c$ is independent of $t_{0}$.

###### Remark 3.2

To get the estimates of $T$ in $V_{2}$ space, we should first take derivative
with respect to $\xi$ in the temperature equation and then estimate $T_{\xi}$
in $L^{2}(\mho).$ If we take inner product of equation $(2.12)$ with
$-\partial_{\xi\xi}T$ in $L^{2}(\mho),$ it is difficult to obtain the energy
estimates of $T_{\xi}$ because of its higher nonlinear structure than the
oceanic primitive equation. Therefore, one can not even prove the global
existence of the strong solution in this way.

By integration by parts,

$\displaystyle\int_{\mho}Q_{1\xi}T_{\xi}d\mho$ $\displaystyle=$
$\displaystyle\int_{\mho}[\partial_{\xi}(Q_{1}T_{\xi})-Q_{1}T_{\xi\xi}]d\mho$
$\displaystyle=$
$\displaystyle-\alpha_{s}\int_{S^{2}}Q_{1}|_{\xi=1}T|_{\xi=1}dS^{2}-\int_{\mho}Q_{1}T_{\xi\xi}d\mho$
$\displaystyle\leq$
$\displaystyle\varepsilon|T_{\xi\xi}|_{2}^{2}+c|Q_{1}|_{\xi=1}|_{2}^{2}+c|T|_{\xi=1}|_{2}^{2}+|Q_{1}|_{2}^{2}.$

In view of estimates in [20] and the above argument, we have

$\displaystyle\frac{1}{2}\frac{d(|T_{\xi}|_{2}^{2}+\alpha_{s}|T|_{\xi=1}|_{2}^{2})}{dt}+|\nabla
T_{\xi}|_{2}^{2}+|T_{\xi\xi}|_{2}^{2}+\alpha_{s}|\nabla T|_{\xi=1}|_{2}^{2}$
(3.66) $\displaystyle\leq$
$\displaystyle\varepsilon(|T_{\xi\xi}|_{2}^{2}+|\nabla
T_{\xi}|_{2}^{2}+|q_{\xi\xi}|_{2}^{2}+|\nabla
q_{\xi}|_{2}^{2})+c|T_{\xi}|_{2}^{2}+c|q_{\xi}|_{2}^{2}+c\|v_{\xi}\|_{1}^{2}+c\|v\|_{1}^{2}$
$\displaystyle+c|\nabla
q|_{2}^{2}+|T|_{\xi=1}|_{4}^{4}+c|T|_{\xi=1}|_{2}^{2}+c|Q_{1}|_{\xi=1}||_{2}^{2}+c|Q_{1}|_{2}^{2}+c.$

Similarly, by the estimates in [20], we also have

$\displaystyle\frac{1}{2}\frac{d(|q_{\xi}|_{2}^{2}+\beta_{s}|q|_{\xi=1}|_{2}^{2})}{dt}+|\nabla
q_{\xi}|_{2}^{2}+|q_{\xi\xi}|_{2}^{2}+\beta_{s}|\nabla q|_{\xi=1}|_{2}^{2}$
(3.67) $\displaystyle\leq$
$\displaystyle\varepsilon(|q_{\xi\xi}|_{2}^{2}+|\nabla
q_{\xi}|_{2}^{2})+c|q_{\xi}|_{2}^{2}+c\|v\|_{1}^{2}+c\|v_{\xi}\|_{1}^{2}$
$\displaystyle+c|q|_{\xi=1}|_{4}^{4}+c|q|_{\xi=1}|_{2}^{2}++c|Q_{2}|_{\xi=1}||_{2}^{2}+c|Q_{2}|_{2}^{2}.$

Combining $(3.66)$ and $(3.67)$ yields,

$\displaystyle\frac{d(|T_{\xi}|_{2}^{2}+|q_{\xi}|_{2}^{2}+\alpha_{s}|T|_{\xi=1}|_{2}^{2}+\beta_{s}|q|_{\xi=1}|_{2}^{2})}{dt}+|\nabla
T_{\xi}|_{2}^{2}+|T_{\xi\xi}|_{2}^{2}$ $\displaystyle+|\nabla
q_{\xi}|_{2}^{2}+|q_{\xi\xi}|_{2}^{2}+\alpha_{s}|\nabla
T|_{\xi=1}|_{2}^{2}+\beta_{s}|\nabla q|_{\xi=1}|_{2}^{2}$ $\displaystyle\leq
c+c(|T_{\xi}|_{2}^{2}+|q_{\xi}|_{2}^{2})+c\|v_{\xi}\|_{1}^{2}+c\|v\|_{1}^{2}+c\|q\|_{1}^{2}+c|T|_{\xi=1}|_{4}^{4}+c|T|_{\xi=1}|_{2}^{2}$
$\displaystyle+c|q|_{\xi=1}|_{4}^{4}+c|q|_{\xi=1}|_{2}^{2}+c(|Q_{1}|_{\xi=1}|_{2}^{2}+|Q_{2}|_{\xi=1}|_{2}^{2})+c(|Q_{1}|_{2}^{2}+|Q_{2}|_{2}^{2}).$
(3.68)

Then in view of uniform Gronwall lemma, $(3.27),(3.33),(3.38),(3.45)$ and
$(3.59),$ we obtain for arbitrary $t_{0}\geq 0$

$\displaystyle|T_{\xi}(t_{0})|_{2}^{2}+|q_{\xi}(t_{0})|_{2}^{2}+\int_{t_{0}}^{t_{0}+1}(\|T_{\xi}(t)\|_{1}^{2}+\|q_{\xi}(t)\|_{1}^{2}+|\nabla
T|_{\xi=1}(t)|_{2}^{2}+|\nabla q|_{\xi}(t)|_{2}^{2})dt\leq c,$ (3.69)

where $c$ is independent of $t_{0}.$ By taking inner product of equation
$(1.12)$ with $-\Delta T,$ in $L^{2}(\mho),$ we reach

$\displaystyle\frac{1}{2}\frac{d|\nabla T|_{2}^{2}}{dt}+|\Delta
T|_{2}^{2}+|\nabla T_{z}|_{2}^{2}+\alpha_{s}|\nabla
T|_{\xi=1}|_{L^{2}(S^{2})}^{2}$ (3.70) $\displaystyle=$
$\displaystyle\int_{\mho}\nabla_{v}T\Delta
Td\mho+\int_{\mho}\int_{\xi}^{1}\mathrm{div}vd\xi^{\prime}T_{\xi}\Delta
Td\mho$ $\displaystyle-\int_{\mho}\frac{bP}{p}(1+aq)w\Delta
Td\mho-\int_{\mho}Q_{1}\Delta Td\mho$ $\displaystyle=$ $\displaystyle
l_{1}+l_{2}+l_{3}+l_{4}.$

To estimates $l_{1},$ using H$\mathrm{\ddot{o}}$lder inequality, interpolation
inequality and Young’s inequality we have

$\displaystyle l_{1}$ $\displaystyle\leq$ $\displaystyle|\Delta T|_{2}|\nabla
T|_{4}|v|_{4}$ $\displaystyle\leq$ $\displaystyle c|\Delta T|_{2}[|\nabla
T|_{2}^{\frac{1}{4}}(|\nabla T|_{2}^{\frac{3}{4}}+|\Delta
T|_{2}^{\frac{3}{4}}+|\nabla T_{\xi}|_{2}^{\frac{3}{4}})]|v|_{4}$
$\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
T|_{2}^{2}+\varepsilon|\nabla T_{\xi}|_{2}^{2}+c|\nabla
T|_{2}^{2}|v|_{4}^{2}+c|\nabla T|_{2}^{2}|v|_{4}^{8}$ $\displaystyle\leq$
$\displaystyle\varepsilon|\Delta T|_{2}^{2}+\varepsilon|\nabla
T_{\xi}|_{2}^{2}+c|\nabla T|_{2}^{2}.$

To estimates $l_{2},$ using H$\mathrm{\ddot{o}}$lder inequality, Minkowski
inequality, interpolation inequality and Young’s inequality we obtain

$\displaystyle l_{2}$ $\displaystyle\leq$
$\displaystyle\int_{S^{2}}\Big{(}\int_{0}^{1}|\mathrm{div}v|d\xi\int_{0}^{1}|T_{\xi}||\Delta
T|d\xi\Big{)}dS^{2}$ $\displaystyle\leq$
$\displaystyle\int_{S^{2}}\Big{(}\int_{0}^{1}|\mathrm{div}v|d\xi(\int_{0}^{1}T_{\xi}^{2}d\xi)^{\frac{1}{2}}(\int_{0}^{1}|\Delta
T|^{2}d\xi)^{\frac{1}{2}}\Big{)}dS^{2}$ $\displaystyle\leq$
$\displaystyle|\Delta
T|_{2}\Big{(}\int_{S^{2}}(\int_{0}^{1}T_{\xi}^{2}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\mathrm{div}v|d\xi)^{4}dS^{2}\Big{)}^{\frac{1}{4}}$
$\displaystyle\leq$ $\displaystyle|\Delta
T|_{2}\Big{(}\int_{0}^{1}(\int_{S^{2}}T_{\xi}^{4}dS^{2})^{\frac{1}{2}}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}(\int_{S^{2}}|\mathrm{div}v|^{4}dS^{2})^{\frac{1}{4}}d\xi\Big{)}$
$\displaystyle\leq$ $\displaystyle c|\Delta
T|_{2}\Big{(}\int_{0}^{1}|T_{\xi}|_{2}(|T_{\xi}|_{2}+|\nabla
T_{\xi}|_{2})d\xi\Big{)}^{\frac{1}{2}}$
$\displaystyle\cdot\Big{(}\int_{0}^{1}|\mathrm{div}v|_{2}^{\frac{1}{2}}(|\mathrm{div}v|_{2}^{\frac{1}{2}}+|\Delta
v|_{2}^{\frac{1}{2}})d\xi\Big{)}$ $\displaystyle\leq$ $\displaystyle c|\Delta
T|_{2}|T_{\xi}|_{2}^{\frac{1}{2}}(|T_{\xi}|_{2}^{\frac{1}{2}}+|\nabla
T_{\xi}|_{2}^{\frac{1}{2}})$
$\displaystyle\cdot|\mathrm{div}v|_{2}^{\frac{1}{2}}(|\mathrm{div}v|_{2}^{\frac{1}{2}}+|\Delta
v|_{2}^{\frac{1}{2}})$ $\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
T|_{2}^{2}+c(|T_{\xi}|_{2}^{2}+|T_{\xi}|_{2}|\nabla
T_{\xi}|_{2})(|\mathrm{div}v|_{2}^{2}+|\mathrm{div}v|_{2}|\Delta v|_{2})$
$\displaystyle\leq$ $\displaystyle\varepsilon|\Delta T|_{2}^{2}+c(1+|\nabla
T_{\xi}|_{2})(1+|\Delta v|_{2}).$

Taking an analogous argument as above we get

$\displaystyle l_{3}$ $\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
T|_{2}^{2}+c(1+|\nabla q|_{2})(1+|\Delta v|_{2}).$

Similarly, we have

$\displaystyle l_{4}\leq\varepsilon|\Delta T|_{2}^{2}+c|Q_{1}|_{2}^{2}.$

Combining $(3.70)$ and estimates of $l_{1}-l_{4}$ we get

$\displaystyle\frac{1}{2}\frac{d|\nabla T|_{2}^{2}}{dt}+|\Delta
T|_{2}^{2}+|\nabla T_{\xi}|_{2}^{2}+\alpha_{s}|\nabla T(\xi=1)|_{2}^{2}$
(3.71) $\displaystyle\leq$ $\displaystyle c|\nabla
T|_{2}^{2}+c|Q_{1}|_{2}^{2}+c(1+|\nabla T_{\xi}|_{2})(1+|\Delta v|_{2})$
$\displaystyle+c(1+|\nabla q|_{2})(1+|\Delta v|_{2}).$

According to $(3.27),(3.33),(3.65),(3.69)$ and uniform Gronwall inequality, we
have for all $t\geq 0$

$\displaystyle|\nabla T(t)|_{2}^{2}+\int_{t}^{t+1}(|\Delta
T(t)|_{2}^{2}+|\nabla T_{\xi}(t)|_{2}^{2})dt\leq c,$ (3.72)

where $c$ is a positive constant independent of $t.$ Analogously to the
deduction of $(3.71)$, we have

$\displaystyle\frac{1}{2}\frac{d|\nabla q|_{2}^{2}}{dt}+|\Delta
q|_{2}^{2}+|\nabla q_{\xi}|_{2}^{2}+\beta_{s}|\nabla q(\xi=1)|_{2}^{2}$ (3.73)
$\displaystyle\leq$ $\displaystyle\varepsilon|\Delta
q|_{2}^{2}+c|Q_{2}|_{2}^{2}+c|\nabla q|_{2}^{2}$ $\displaystyle+c(1+|\nabla
T_{\xi}|_{2})(1+|\Delta v|_{2}).$

According to $(3.27),(3.65),(3.72)$ and uniform Gronwall inequality, we have
for all $t\geq 0$

$\displaystyle|\nabla q(t)|_{2}^{2}\leq c,$ (3.74)

where $c$ is a positive constant independent of $t.$

Combining the above results, we have proved the existence of an absorbing ball
for the strong solution $(v,T,q)$ in the solution space $V$ for the 3D viscous
PEs of large-scale moist atmosphere.

##  4 Continuity of strong solution with respect to $t$.

In this section, we will consider the continuity of strong solution with
respect to time $t$ in $V$, which will be helpful for proving the existence
and connectedness of the global attractor for moist primitive equation. To
establish our result of this section, we need the following lemma which is a
consequence of a general result of $\cite[cite]{[\@@bibref{}{LM}{}{}]}.$ For
the proof, one can refer to $\cite[cite]{[\@@bibref{}{T}{}{}]}.$

###### Lemma 4.1

Let $V,H,V^{\prime}$ be three Hilbert spaces such that $V\subset
H=H^{\prime}\subset V^{\prime},$ where $H^{\prime}$ and $V^{\prime}$ are the
dual spaces of $H$ and $V$ respectively. Suppose $u\in L^{2}([0,T];V)$ and
$u^{\prime}\in L^{2}([0,T];V^{\prime}).$ Then $u$ is almost everywhere equal
to a function continuous from $[0,T]$ into $H.$

From $(1.11)$ and Lemma $2.1,$ we obtain for $\eta\in
V_{1}=D(A_{1}^{\frac{1}{2}}),$

$\displaystyle\langle\partial_{t}A_{1}^{\frac{1}{2}}v,\eta\rangle$
$\displaystyle=$
$\displaystyle\langle\partial_{t}v,A_{1}^{\frac{1}{2}}\eta\rangle$
$\displaystyle=$ $\displaystyle-\langle
A_{1}v,A_{1}^{\frac{1}{2}}\eta\rangle-\langle\nabla_{v}v,A_{1}^{\frac{1}{2}}\eta\rangle$
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi,t)d\xi\Big{)}v_{\xi},A_{1}^{\frac{1}{2}}\eta\rangle$
$\displaystyle-\langle\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime},A_{1}^{\frac{1}{2}}\eta\rangle$
$\displaystyle-\langle\frac{f}{R_{0}}v^{\bot},A_{1}^{\frac{1}{2}}\eta\rangle.$

Using H$\mathrm{\ddot{o}}$lder inequality, Agmon inequality and interpolation
inequality yields

$\displaystyle|\langle A_{1}v,A_{1}^{\frac{1}{2}}\eta\rangle|$
$\displaystyle\leq$ $\displaystyle|A_{1}v|_{2}\|\eta\|_{1},$
$\displaystyle|\langle(v\cdot\nabla)v,A_{1}^{\frac{1}{2}}\eta\rangle|$
$\displaystyle\leq$ $\displaystyle
c|v|_{\infty}\|v\|_{1}|A_{1}^{\frac{1}{2}}\eta|_{2}\leq
c\|v\|_{1}^{\frac{3}{2}}\|v\|_{2}^{\frac{1}{2}}|A_{1}^{\frac{1}{2}}\eta|_{2}.$

By H$\mathrm{\ddot{o}}$lder inequality, Minkowski inequality and interpolation
inequality, we have

$\displaystyle-\langle\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime},A_{1}^{\frac{1}{2}}\eta\rangle$
$\displaystyle\leq$ $\displaystyle
c\int_{S^{2}}\Big{(}\int_{0}^{1}(|\mathrm{grad}T|+|\mathrm{grad}(Tq)|)d\xi\int_{0}^{1}A_{1}^{\frac{1}{2}}\eta
d\xi\Big{)}dS^{2}$ $\displaystyle\leq$ $\displaystyle
c\|T\|_{1}|A_{1}^{\frac{1}{2}}\eta|_{2}+c|A_{1}^{\frac{1}{2}}\eta|_{2}\int_{0}^{1}|\mathrm{grad}T|_{4}|q|_{4}d\xi$
$\displaystyle+c|A_{1}^{\frac{1}{2}}\eta|_{2}\int_{0}^{1}|\mathrm{grad}q|_{4}|T|_{4}d\xi$
$\displaystyle\leq$ $\displaystyle
c\|T\|_{1}|A_{1}^{\frac{1}{2}}\eta|_{2}+c|A_{1}^{\frac{1}{2}}\eta|_{2}\int_{0}^{1}|\mathrm{grad}T|_{2}^{\frac{1}{2}}$
$\displaystyle\cdot(|\Delta
T|_{2}^{\frac{1}{2}}+|\mathrm{grad}T|_{2}^{\frac{1}{2}})|q|_{4}d\xi$
$\displaystyle+c|A_{1}^{\frac{1}{2}}\eta|_{2}\int_{0}^{1}|\mathrm{grad}q|_{2}^{\frac{1}{2}}$
$\displaystyle\cdot(|\Delta
q|_{2}^{\frac{1}{2}}+|\mathrm{grad}q|_{2}^{\frac{1}{2}})|T|_{4}d\xi$
$\displaystyle\leq$ $\displaystyle
c\|T\|_{1}|A_{1}^{\frac{1}{2}}\eta|_{2}+c|A_{1}^{\frac{1}{2}}\eta|_{2}\|T\|_{1}^{\frac{1}{2}}\|T\|_{2}^{\frac{1}{2}}|q|_{4}$
$\displaystyle+c|A_{1}^{\frac{1}{2}}\eta|_{2}\|q\|_{1}^{\frac{1}{2}}\|q\|_{2}^{\frac{1}{2}}|T|_{4}.$

Similarly, we have

$\displaystyle|\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}v_{\xi},A_{1}^{\frac{1}{2}}\eta\rangle|\leq
c\|v\|_{1}\|v\|_{2}\|\eta\|_{1}.$

In view of the above arguments, we conclude that

$\displaystyle\|\partial_{t}(A_{1}^{\frac{1}{2}}v)\|_{V_{1}^{\prime}}$
$\displaystyle\leq$ $\displaystyle c\|v\|_{2}+c\|v\|_{1}\|v\|_{2}+c\|T\|_{1}$
$\displaystyle+c(\|T\|_{1}^{\frac{1}{2}}\|T\|_{2}^{\frac{1}{2}}|q|_{4}+\|q\|_{1}^{\frac{1}{2}}\|q\|_{2}^{\frac{1}{2}}|T|_{4}).$

Since

$\displaystyle v\in L^{\infty}([0,\tau];V_{1})\cap
L^{2}([0,\tau];H^{2}(\mho)),\ \ \ \forall\ \tau>0,$

we have

$\displaystyle A_{1}^{\frac{1}{2}}v\in L^{2}([0,\tau];V_{1}),\ \ \ \ \
\partial_{t}A_{1}^{\frac{1}{2}}v\in L^{2}([0,\tau];V_{1}^{\prime}),\ \ \
\forall\ \tau>0.$

Therefore, by Lemma $4.1$, we infer that $v\in C([0,\tau];V_{1}).$ For any
$\phi\in V_{2}=D(A_{2}^{\frac{1}{2}}),$ we have

$\displaystyle\langle\partial_{t}A_{2}^{\frac{1}{2}}T,\phi\rangle$
$\displaystyle=$
$\displaystyle\langle\partial_{t}T,A_{2}^{\frac{1}{2}}\phi\rangle$
$\displaystyle=$ $\displaystyle-\langle
A_{2}T,A_{2}^{\frac{1}{2}}\phi\rangle-\langle\nabla_{v}T,A_{2}^{\frac{1}{2}}\phi\rangle$
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}T,A_{2}^{\frac{1}{2}}\phi\rangle$
$\displaystyle+\langle\frac{bP}{p}(1+aq)\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)},A_{2}^{\frac{1}{2}}\phi\rangle+\langle
Q_{1},A_{2}^{\frac{1}{2}}\phi\rangle.$

By H$\mathrm{\ddot{o}}$lder inequality, Minkowski inequality and interpolation
inequality, we have

$\displaystyle\langle\frac{bP}{p}(1+aq)\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)},A_{2}^{\frac{1}{2}}\phi\rangle$
$\displaystyle\leq$ $\displaystyle
c\|v\|_{1}\|\phi\|_{1}+c\int_{S^{2}}\Big{(}\int_{0}^{1}|\mathrm{div}v|d\xi\Big{)}\Big{(}\int_{0}^{1}|q||A_{1}^{\frac{1}{2}}\phi|d\xi\Big{)}dS^{2}$
$\displaystyle\leq$ $\displaystyle
c\|v\|_{1}\|\phi\|_{1}+c\int_{S^{2}}\Big{(}\int_{0}^{1}|\mathrm{div}v|d\xi\Big{)}\Big{(}\int_{0}^{1}q^{2}d\xi\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}|A_{2}^{\frac{1}{2}}\phi|^{2}d\xi\Big{)}^{\frac{1}{2}}dS^{2}$
$\displaystyle\leq$ $\displaystyle
c\|v\|_{1}\|\phi\|_{1}+c\|\phi\|_{1}\Big{(}\int_{0}^{1}|\mathrm{div}v|_{4}d\xi\Big{)}\Big{(}\int_{0}^{1}|q|_{4}^{2}d\xi\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
c\|v\|_{1}\|\phi\|_{1}+c\|\phi\|_{1}|q|_{4}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}.$

Consequently, taking an analogous argument about
$\|\partial_{t}A_{1}^{\frac{1}{2}}v\|_{V_{1}^{\prime}}$, we have

$\displaystyle\|\partial_{t}A_{2}^{\frac{1}{2}}T\|_{V_{2}^{\prime}}$
$\displaystyle\leq$ $\displaystyle
c(\|T\|_{2}+\|v\|_{1}+\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}\|T\|_{1}+|q|_{4}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}$
$\displaystyle+\|T\|_{1}^{\frac{1}{2}}\|T\|_{2}^{\frac{1}{2}}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}+|Q_{1}|_{2}).$

Similarly, we can infer that

$\displaystyle\|\partial_{t}A_{3}^{\frac{1}{2}}q\|_{V_{2}^{\prime}}\leq
c(\|q\|_{2}+\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}\|q\|_{1}+\|q\|_{1}^{\frac{1}{2}}\|q\|_{2}^{\frac{1}{2}}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}+|Q_{2}|_{2}).$

Since

$\displaystyle(v,T,q)\in L^{\infty}([0,\tau];V)\cap
L^{2}([0,\tau];H^{2}(\mho)),\ \ \ \forall\ \tau>0,$

we have

$\displaystyle A_{2}^{\frac{1}{2}}T\in L^{2}([0,\tau];V_{2}),\ \ \ \ \
\partial_{t}A_{2}^{\frac{1}{2}}T\in L^{2}([0,\tau];V_{2}^{\prime})$

and

$\displaystyle A_{3}^{\frac{1}{2}}q\in L^{2}([0,\tau];V_{3}),\ \ \ \ \
\partial_{t}A_{3}^{\frac{1}{2}}q\in L^{2}([0,\tau];V_{3}^{\prime}).$

By Lemma $4.1,$ we infer that $T,q\in C([0,+\infty);V_{2})$ and
$C([0,+\infty);V_{3}),$ respectively. So far, we obtain that

$\displaystyle(v,T,q)\in C([0,+\infty);V).$

##  5 Continuity in $V$ with respect to initial data.

It is shown in [20] that the strong solution to the 3D viscous primitive
equations of large-scale moist atmosphere is unique and Lipschitz continuous
with respect to the initial condition in $H.$ But what we need to do here is
to show the continuity property in $V.$

In the following, will prove that for any fixed $t>0,$ the mapping
$(v_{0},T_{0},q_{0})\mapsto(v(t),T(t),q(t))$ is Lipschitz continuous from $V$
into itself for all the strong solutions.

Assume $(v_{i},T_{i},q_{i}),i=1,2,$ are two strong solutions to the equation
$(1.11)-(1.17)$ with initial data $(v_{0,i},T_{0,i},q_{0,i})\in V.$ Let

$\displaystyle u=v_{1}-v_{2},\ \ \tau=T_{1}-T_{2},\ \ q=q_{1}-q_{2},\ \
\Phi_{s}(t;\theta,\varphi)=\Phi_{1,s}(t;\theta,\varphi)-\Phi_{2,s}(t;\theta,\varphi).$

Then we derive from $(1.11)-(1.17)$ that

$\displaystyle\partial_{t}u$ $\displaystyle+$ $\displaystyle
L_{1}u+\nabla_{v_{1}}u+\nabla_{u}v_{2}+\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}v$
(5.75)
$\displaystyle+\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}v_{2}+\frac{f}{R_{0}}v^{\bot}+\mathrm{grad}\Phi_{s}$
$\displaystyle+\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}Td\xi^{\prime}+\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(q_{1}T)d\xi^{\prime}+\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(qT_{2})d\xi^{\prime}=0,$
$\displaystyle\partial_{t}T$ $\displaystyle+$ $\displaystyle
L_{2}T+\nabla_{v_{1}}T+\nabla_{v}T_{2}+\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}T$
(5.76)
$\displaystyle+\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}T_{2}-\frac{bP}{p}\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}$
$\displaystyle-\frac{abP}{p}q_{1}\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}-\frac{abP}{p}q\Big{(}\int_{\xi}^{1}\mathrm{div}v_{2}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}=0,$
$\displaystyle\partial_{t}q$ $\displaystyle+$ $\displaystyle
L_{3}q+\nabla_{v_{1}}q+\nabla_{v}q_{2}+\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}q$
(5.77)
$\displaystyle+\Big{(}\int_{\xi}^{1}\mathrm{div}v(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}q_{2}=0,$
$\displaystyle u|_{t=0}=v_{0}^{1}-v_{0}^{2},\ \ T|_{t=0}=T_{0}^{1}-T_{0}^{2},\
\ q|_{t=0}=q_{0}^{1}-q_{0}^{2},$ (5.78) $\displaystyle\xi=1:\ \
\partial_{\xi}v=0,\ \ \partial_{\xi}T=-\alpha_{s}T,\ \
\partial_{\xi}q=-\beta_{s}q,$ (5.79) $\displaystyle\xi=0:\ \
\partial_{\xi}v=0,\ \ \partial_{\xi}T=0,\ \ \partial_{\xi}q=0.$ (5.80)

Taking inner product of $(5.75)$ with $A_{1}u$ in $L^{2}(\mho)$ we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u\|_{1}^{2}+|A_{1}u|_{2}^{2}$
$\displaystyle=$
$\displaystyle-\langle\nabla_{v_{1}}u,A_{1}u\rangle-\langle\nabla_{u}v_{2},A_{1}u\rangle$
(5.81)
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}u(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}v_{2},A_{1}u\rangle$
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}u,A_{1}u\rangle$
$\displaystyle-\langle\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}Td\xi^{\prime},A_{1}u\rangle-\langle(\frac{f}{R_{0}}u^{\bot}+\mathrm{grad}\Phi_{s}),A_{1}u\rangle$
$\displaystyle-\langle\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(q_{1}T)d\xi^{\prime},A_{1}u\rangle-\langle\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(qT_{2})d\xi^{\prime},A_{1}u\rangle$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{8}k_{i}.$

By H$\mathrm{\ddot{o}}$lder inequality, Agmon inequality and Young’s
inequality , we have

$\displaystyle k_{1}$ $\displaystyle\leq$
$\displaystyle|v_{1}|_{\infty}(|\nabla_{e_{\theta}}u|_{2}+|\nabla_{e_{\varphi}}u|_{2})|A_{1}u|_{2}$
$\displaystyle\leq$ $\displaystyle
c\|v_{1}\|_{1}^{\frac{1}{2}}|A_{1}v_{1}|_{2}^{\frac{1}{2}}\|u\|_{1}|A_{1}u|_{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}u|_{2}^{2}+c\|v_{1}\|_{1}|A_{1}v_{1}|_{2}\|u\|_{1}^{2}.$

Similarly, we obtain

$\displaystyle
k_{2}\leq|u|_{\infty}\|v_{2}\|_{1}|A_{1}u|_{2}\leq\varepsilon|A_{1}u|_{2}^{2}+c\|u\|_{1}^{2}\|v_{2}\|_{1}^{4}.$

Taking an analogous argument as $(3.61),$ we have

$\displaystyle
k_{3}+k_{4}\leq\varepsilon|A_{1}u|_{2}^{2}+c\|u\|_{1}^{2}\|v_{2}\|_{1}^{2}\|v_{2}\|_{2}^{2}+c\|u\|_{1}^{2}\|v_{1}\|_{1}^{2}\|v_{1}\|_{2}^{2}.$

In view of Lemma $2.1$ and H$\mathrm{\ddot{o}}$ler inequality, we obtain

$\displaystyle
k_{5}+k_{6}\leq\varepsilon|A_{1}u|_{2}^{2}+c\|T\|_{1}^{2}+c|u|_{2}^{2}.$

To estimate $k_{7}$, by H$\mathrm{\ddot{o}}$lder inequality, Minkowski
inequality and interpolation inequality we have

$\displaystyle k_{7}$ $\displaystyle\leq$ $\displaystyle
c\int_{S^{2}}(\int_{0}^{1}|\mathrm{grad}q_{1}||T|d\xi\int_{0}^{1}|A_{1}u|d\xi)dS^{2}$
$\displaystyle+c\int_{S^{2}}(\int_{0}^{1}|q_{1}||\mathrm{grad}T|d\xi\int_{0}^{1}|A_{1}u|d\xi)dS^{2}$
$\displaystyle\leq$ $\displaystyle
c|A_{1}u|_{2}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\mathrm{grad}q_{1}|^{2}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}\Big{(}\int_{S^{2}}(\int_{0}^{1}|T|^{2}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}$
$\displaystyle+c|A_{1}u|_{2}\Big{(}\int_{S^{2}}(\int_{0}^{1}|q_{1}|^{2}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}\Big{(}\int_{S^{2}}(\int_{0}^{1}|\mathrm{grad}T|^{2}d\xi)^{2}dS^{2}\Big{)}^{\frac{1}{4}}$
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}u|_{2}^{2}+c|\mathrm{grad}q_{1}|_{2}(|\mathrm{grad}q_{1}|_{2}+|\Delta
q_{1}|_{2})(|T|_{2}^{2}+|T|_{2}|\nabla T|_{2})$
$\displaystyle+c|\mathrm{grad}T|_{2}(|\mathrm{grad}T|_{2}+|\Delta
T|_{2})|q_{1}|_{4}^{2}$ $\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}u|_{2}^{2}+\varepsilon|\Delta
T|_{2}^{2}+c\|T\|_{1}^{2}(\|q_{1}\|_{1}^{4}+\|q_{1}\|_{2}^{2}).$

Similarly, we have

$\displaystyle k_{8}$ $\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}u|_{2}^{2}+c|\mathrm{grad}q|_{2}(|\mathrm{grad}q|_{2}+|\Delta
q|_{2})|T_{2}|_{4}^{2}$
$\displaystyle+c|\mathrm{grad}T_{2}|_{2}(|\mathrm{grad}T_{2}|_{2}+|\Delta
T_{2}|_{2})(|q|_{2}^{2}+|q|_{2}|\nabla q|_{2})$ $\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}u|_{2}^{2}+\varepsilon|\Delta
q|_{2}^{2}+c\|q\|_{1}^{2}(\|T_{2}\|_{1}^{4}+\|T_{2}\|_{2}^{2}).$

By $(5.81)$ and estimates of $k_{1}-k_{8},$ we get

$\displaystyle\frac{1}{2}\frac{d\|u\|_{1}^{2}}{dt}+|A_{1}u|_{2}^{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}u|_{2}^{2}+\varepsilon|A_{2}T|_{2}^{2}+\varepsilon|A_{3}q|_{2}^{2}$
(5.82)
$\displaystyle+c\|u\|_{1}^{2}(\|v_{1}\|_{1}^{2}\|v_{1}\|_{2}^{2}+\|v_{2}\|_{1}^{2}\|v_{2}\|_{2}^{2}+\|v_{2}\|_{1}^{4}+1)$
$\displaystyle+c\|T\|_{1}^{2}(\|q_{1}\|_{1}^{4}+\|q_{1}\|_{2}^{2})$
$\displaystyle+c\|q\|_{1}^{2}(\|T_{2}\|_{1}^{4}+\|T_{2}\|_{2}^{2}).$

Taking an analogous argument as above, from $(5.76)$ and $(5.77)$ we have

$\displaystyle\frac{1}{2}\frac{d\|T\|_{1}^{2}}{dt}+|A_{2}T|_{2}^{2}$ (5.83)
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{2}T|_{2}^{2}+\varepsilon|A_{1}u|_{2}^{2}+c\|q\|_{1}^{2}\|v_{2}\|_{1}\|v_{2}\|_{2}$
$\displaystyle+c\|T\|_{1}^{2}(1+\|v_{1}\|_{1}^{2}\|v_{1}\|_{2}^{2})$
$\displaystyle+c\|u\|_{1}^{2}(1+\|q_{1}\|_{1}^{4}+\|T_{2}\|_{2}^{2}+\|T_{2}\|_{1}^{2}\|T_{2}\|_{2}^{2})$

and

$\displaystyle\frac{1}{2}\frac{d\|q\|_{1}^{2}}{dt}+|A_{3}q|_{2}^{2}$ (5.84)
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{3}q|_{2}^{2}+c\|q\|_{1}^{2}(1+\|v_{1}\|_{1}^{2}\|v_{1}\|_{2}^{2})$
$\displaystyle+c\|u\|_{1}^{2}(\|q_{2}\|_{2}^{2}+\|q_{2}\|_{1}^{2}\|q_{2}\|_{2}^{2}).$

Let

$\displaystyle g_{1}$ $\displaystyle:=$ $\displaystyle
1+\|v_{1}\|_{1}^{2}\|v_{1}\|_{2}^{2}+\|v_{2}\|_{1}^{4}+\|v_{2}\|_{1}^{2}\|v_{2}\|_{2}^{2}+\|T_{2}\|_{2}^{2}$
$\displaystyle+\|T_{2}\|_{1}^{2}\|T_{2}\|_{2}^{2}+\|q_{1}\|_{1}^{4}+\|q_{2}\|_{2}^{2}+\|q_{2}\|_{1}^{2}\|q_{2}\|_{2}^{2},$
$\displaystyle
g_{2}:=1+\|q_{1}\|_{2}^{2}+\|q_{1}\|_{1}^{4}+\|v_{1}\|_{1}^{2}\|v_{1}\|_{2}^{2}$

and

$\displaystyle
g_{3}:=1+\|T_{2}\|_{2}^{2}+\|T_{2}\|_{1}^{4}+\|v_{2}\|_{1}\|v_{2}\|_{2}+\|v_{1}\|_{1}^{2}\|v_{1}\|_{2}^{2}.$

Obviously, for arbitrary $0\leq a<b<\infty$, we have

$\displaystyle\int_{a}^{b}(g_{1}(t)+g_{2}(t)+g_{3}(t))dt<\infty.$

Thereby, in view of $(5.82)-(5.84)$ we get

$\displaystyle\frac{d(\|u\|_{1}^{2}+\|T\|_{1}^{2}+\|q\|_{1}^{2})}{dt}\leq
c(g_{1}(t)+g_{2}(t)+g_{3}(t))(\|u\|_{1}^{2}+\|T\|_{1}^{2}+\|q\|_{1}^{2}),$

which combined with Gronwall lemma implies

$\displaystyle\|u(t)\|_{1}^{2}+\|T(t)\|_{1}^{2}+\|q(t)\|_{1}^{2}$
$\displaystyle\leq$ $\displaystyle
c(\|v_{0,1}-v_{0,2}\|_{1}^{2}+\|T_{0,1}-T_{0,2}\|_{1}^{2}+\|q_{0,1}-q_{0,2}\|_{1}^{2})e^{\int_{0}^{t}(g_{1}(s)+g_{2}(s)+g_{3}(s))ds}.$

So far, we have shown that for $t>0,\ (v(t),T(t),q(t))$ is Lipschitz
continuous in $V$ with respect to the initial data $(v(0),T(0),q(0)).$

##  6 The global attractor.

In this section, we present our main result, the existence of global
attractor, of this paper. To show our main result, we make use of the
following theorem from Teman $\cite[cite]{[\@@bibref{}{T}{}{}]}.$ For more
details about the theorem, we can see, e.g., [4, 14, 22, 32, 44, 45, 46] and
the references therein.

###### Theorem 6.1

Suppose that $X$ is a metric space and semigroup $\\{S(t)\\}_{t\geq 0}$ is a
family of operators from $X$ into itself such that

$(i)$ for any fixed $t>0,S(t)$ is continuous from $X$ into itself;

$(ii)$ for some $t_{0},S(t_{0})$ is compact from $X$ into itself;

$(iii)$ there exists a subset $B_{0}$ of $X$ which is bounded, a subset $U$ of
$X$ is open, such that

$B_{0}\subseteq U\subseteq X,$ and $B_{0}$ is the absorbing set of $U,$ i.e.
for any bounded subset $B\subset U,$ there

is a $t_{0}=t_{0}(B),$ such that

$\displaystyle S(t)B\subset B_{0},\ \ \ \ \ \forall\ t>t_{0}(B).$

Then $\mathcal{A}:=\omega(B_{0}),$ the $\omega-$limit set of $B_{0}$, is a
compact attractor which attracts all the bounded sets of $U,$ i.e. for any
$x\in U,$

$\displaystyle\lim\limits_{t\rightarrow\infty}dist(S(t)x,\mathcal{A})=0.$

The set $\mathcal{A}$ is the maximal bounded attractor in $U$ for the
inclusion relation.

Suppose in addition that $X$ is a Banach space, $U$ is a convex and

$(iv)\ \forall x\in X,S(t)x:\mathbb{R}_{+}\mapsto X$ is continuous.
Then $\mathcal{A}:=\omega(B_{0})$ is also connected.

If $U=X,\mathcal{A}$ is the global attractor of the semigroup
$\\{S(t)\\}_{t\geq 0}$ in $X.$

Next, we give our main result of the paper and complete the proof of the
theorem.

###### Theorem 6.2

Assume $Q_{1},Q_{2}\in L^{2}(\mho)$ and $Q_{1}|_{\xi=1},Q_{2}|_{\xi=1}\in
L^{2}(S^{2}).$ Then, for $t\geq 0,$ the solution operator $\\{S(t)\\}_{t\geq
0}$ of the $3D$ viscous PEs of large-scale moist atmosphere
$(1.11)-(1.17):S(t)(v_{0},T_{0},q_{0})=(v(t),T(t),q(t))$ defines a semigroup
in the space $V.$ Furthermore, the results below hold:

$(1)$ For any $(v_{0},T_{0},q_{0})\in V,t\rightarrow S(t)(v_{0},T_{0},q_{0})$
is a continuous map from $\mathbb{R}_{+}$ into $V.$

$(2)$ For any $t>0,S(t)$ is a continuous map in $V.$

$(3)$ For any $t>0,S(t)$ is a compact map in $V.$

$(4)$ $\\{S(t)\\}_{t\geq 0}$ possesses a global attractor $\mathcal{A}$ in
$V.$ The global attractor in $\mathcal{A}$ is compact and

connected in $V$ and is the maximal bounded attractor in $V$ in the sense of
set inclusion

relation; $\mathcal{A}$ attracts all bounded subset in $V$ in the norm of $V.$

To prove Theorem 6.2, we need to check the conditions $(i)-(iv)$ in Theorem
6.1. First, condition $(i)$, the continuous dependence on initial data of the
solution, is verified in section $5.$ Second, condition $(iii)$, the existence
of an absorbing ball in $V,$ is proved in section $3.$ Third, condition
$(iv)$, the regularity of the solution, is shown in section $4.$ Finally, only
the condition $(ii),$ compactness of the solution operator $\\{S(t)\\}_{t\geq
0},$ is left to be checked. We will use Aubin-Lions lemma stated below and
continuity argument to verify condition $(ii).$ For more details of the lemma,
we can see
$\cite[cite]{[\@@bibref{}{A}{}{}]},\cite[cite]{[\@@bibref{}{Li}{}{}]}$ and
references therein.

###### Lemma 6.1

Let $\mathcal{H}_{0},\mathcal{H},\mathcal{H}_{1}$ be Banach spaces such that
$\mathcal{H}_{0},\mathcal{H}_{1}$ are reflexive and
$\mathcal{H}_{0}\overset{c}{\subset}\mathcal{H}\subset\mathcal{H}_{1}.$
Define, for $0<\tau<\infty,$

$\displaystyle X:=\Big{\\{}u\Big{|}u\in L^{2}([0,\tau];\mathcal{H}_{0}),\ \
\frac{du}{dt}\in L^{2}([0,\tau];\mathcal{H}_{1})\Big{\\}}.$

Then $X$ is a Banach space equipped with the norm
$\|u\|_{L^{2}([0,\tau];\mathcal{H}_{0})}+\|u^{\prime}\|_{L^{2}([0,\tau];\mathcal{H}_{1})}.$
Moreover, $X\overset{c}{\subset}L^{2}([0,\tau];\mathcal{H}).$

Proof of Theorem $6.1.$ By the argument above, we only have to check condition
$(ii),$ the compactness of the solution operator. For any fixed $\tau>0,$ let
$\mathcal{B}$ be a bounded subset of $V$ and $A_{\tau}$ denote the subset of
the space $L^{2}([0,T];H):$

$\displaystyle
A_{\tau}:=\Big{\\{}\Big{(}A_{1}^{\frac{1}{2}}v,A_{2}^{\frac{1}{2}}T,A_{3}^{\frac{1}{2}}q\Big{)}\Big{|}(v_{0},T_{0},q_{0})\in\mathcal{B},(v(t),T(t),q(t))=S(t)(v_{0},T_{0},q(0)),t\in[0,\tau]\Big{\\}}.$

For $(v_{0},T_{0},q(0))\in\mathcal{B}$, it has been shown previously that the
strong solution $(v,T,q)$ satisfies

$\displaystyle(A_{1}^{\frac{1}{2}}v,A_{2}^{\frac{1}{2}}T,A_{3}^{\frac{1}{2}}q)\in
L^{2}([0,\tau];V),\ \
(\partial_{t}A_{1}^{\frac{1}{2}}v,\partial_{t}A_{2}^{\frac{1}{2}}T,\partial_{t}A_{3}^{\frac{1}{2}}q)\in
L^{2}([0,\tau];V^{\prime}).$

If we denote

$\displaystyle\mathcal{H}_{0}=V,\mathcal{H}=H,\mathcal{H}_{1}=V^{\prime},$

from Lemma $6.1$ we infer that $A_{\tau}$ is compact in $L^{2}([0,\tau];H).$

To prove solution operator $\\{S(t)\\}_{t\geq 0}$ is compact in $V,$ for any
bounded sequence $\\{(v_{0,n},T_{0,n},q_{0,n})\\}_{0}^{n}\subset\mathcal{B},$
we should show there exists a convergent subsequence of
$\\{S(t)(v_{0,n},T_{0,n},q_{0,n})\\}_{0}^{\infty}$ in $V.$

Since $A_{\tau}$ is compact in $L^{2}([0,\tau];H),$ there exists a function
$(v_{*},T_{*},q_{*})\in L^{2}([0,\tau];V)$ such that there is a subsequence of
$\\{S(\cdot)(v_{0,n},T_{0,n},q_{0,n})\\}_{0}^{\infty},$ still denoted as
$\\{S(\cdot)(v_{0,n},T_{0,n},q_{0,n})\\}_{0}^{\infty},$ satisfying

$\displaystyle\lim\limits_{n\rightarrow\infty}\int_{0}^{T}\|S(t)(v_{0,n},T_{0,n},q_{0,n})-(v_{*}(t),T_{*}(t),q_{*}(t))\|_{1}^{2}dt=0,$

which implies that there is a subsequence of
$\\{S(\cdot)(v_{0,n},T_{0,n},q_{0,n})\\}_{0}^{\infty},$ still denoted as
$\\{S(\cdot)(v_{0,n},T_{0,n},q_{0,n})\\}_{0}^{\infty}$ for simplicity of
notation, converging to $(v_{*},T_{*},q_{*})$ in $V$ a.e. in $(0,\tau):$

$\displaystyle\lim\limits_{n\rightarrow\infty}\|S(t)(v_{0,n},T_{0,n},q_{0,n})-(v_{*}(t),T_{*}(t),q_{*}(t))\|_{1}=0,\
\ a.e.\ t\ in\ (0,\tau).$

For any $t\in(0,\tau),$ we can choose a $t_{0}\in(0,t)$ such that

$\displaystyle\lim\limits_{n\rightarrow\infty}\|S(t_{0})(v_{0,n},T_{0,n},q_{0,n})-(v_{*}(t_{0}),T_{*}(t_{0}),q_{*}(t_{0}))\|_{1}=0.$

Therefore, for any $t>0,$ by the continuity of $S(t)$ in $V$, we have

$\displaystyle\lim\limits_{n\rightarrow\infty}\|S(t)(v_{0,n},T_{0,n},q_{0,n})-S(t-t_{0})(v_{*}(t_{0}),T_{*}(t_{0}),q_{*}(t_{0}))\|_{1}$
$\displaystyle=$
$\displaystyle\lim\limits_{n\rightarrow\infty}\|S(t-t_{0})S(t_{0})(v_{0,n},T_{0,n},q_{0,n})-S(t-t_{0})(v_{*}(t_{0}),T_{*}(t_{0}),q_{*}(t_{0}))\|_{1}=0,$

which implies $S(t)$ is a compact map in $V$ for any $t>0.$ Then by the above
argument of this section and Theorem 6.1, we obtain the existence of the
global attractor for the moist primitive equations $(1.11)-(1.17).$ $\square$

##  Bibliography

  * [1] J. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042–5044.
  * [2] V. Bjerknes, Das Problem der Wettervorhersage, betrachtet vom Standpunkte der Mechanik und der Physik, Meteorol. Z. 21( 1904), 1–7.
  * [3] A.J. Bourgeois, J.T. Beale, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J. Math. Anal. 25 (1994), 1023–1068.
  * [4] A. Babin and M. Vishik, Attractor of Evolution Equations, North-Holland, Amsterdam 1992\.
  * [5] D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math. 148 (1998), 1135–1152.
  * [6] C. Cao, S. Ibrahim, K. Nakanishi, E. S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys. 337(2015), 473–482.
  * [7] C. Cao, J. Li, E.S. Titi, Local and global well-posedness of strong solutions to the 3D primitive equations with vertical eddy diffusivity, Arch. Anal. Ration. Mech. 214(2014), 35–76.
  * [8] C. Cao, J. Li, E.S. Titi, Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity, J. Differ. Equ. 257(2014), 4108–4132.
  * [9] C. Cao, J. Li, E.S. Titi, Global well-posedness for the 3D primitive equations with only horizontal viscosity and diffusion, Commun. Pure Appl. Math. Vol.LXIX(2016), 1492–1531.
  * [10] P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (1994), 1495–1533.
  * [11] P. Constantin, A. Majda, E. Tabak, Singular front formation in a model for quasigeostrophic flow, Phys. Fluids 6 (1994), 9–11.
  * [12] C. Cao and E. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math. 166(2007), 245–267.
  * [13] C. Cao, E.S. Titi, Global well-posedness of the three-dimensional primitive equations with partial vertical turbulence mixing heat diffusion, Commun. Math. Phys. 310(2012), 537–568.
  * [14] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002.
  * [15] P. Constantin, J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 (1999), 937–948.
  * [16] P.F. Embid, A.J. Majda, Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity, Comm. Partial Differential Equations, 21 (1996), 619–658.
  * [17] C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967), 1–34.
  * [18] A. E. Gill, Atmosphere–Ocean Dynamics (International Geophysics Series vol 30) (San Diego, CA: Academic), 1982.
  * [19] B. Guo, D. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys. 47(2006), 083508.
  * [20] B. Guo, D. Huang, Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere, J.Differential Equations, 251(2011), 457–491.
  * [21] F. Guill$\acute{e}$n–Gonz$\acute{a}$ez, N. Masmoudi and M.A. Rodríguez–Bellido, Anisotropic estimates and strong solutions of the Primitive Equations, Diff. Integral Eq. 14(2001), 1381–1408.
  * [22] J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.
  * [23] G. J. Haltiner, Numerical Weather Prediction (New York: Wiley), 1971.
  * [24] G. J. Haltiner and R. T. Williams, Numerical Prediction and Dynamic Meteorology (New York: Wiley), 1980.
  * [25] J.R. Holton, An Introduction to Dynamic Meteorology, third edition, Academic Press, 1992.
  * [26] C. Hu, R. Temam and M. Ziane, The primitive equations on the large scale ocean under the small depth hypothesis, Discrete Contin. Dyn. Syst. 9(2003), 97–131.
  * [27] N.Ju, The global attractor for the solutions to the 3d viscous primitive equations, Discrete and Continuous Dynamical Systems, 17(2007), 159–179.
  * [28] N.Ju and R.Teman, Finite dimensions of the global attractor for 3d viscous primitive equations with viscosity, J Nonlinear Sci, 25(2015), 131–155.
  * [29] G.M. Kobelkov, Existence of a solution in the large for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris 343(2006), 283–286.
  * [30] G.M. Kobelkov, Existence of a solution in the large for ocean dynamics equations, J. Math. Fluid Mech. 9(2007), 588–610.
  * [31] I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity 20(2007), 2739–2753.
  * [32] O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991.
  * [33] J. Lions, Quelques M ethode de R esolution des Probl‘ emes aux Limites Non Lin eaires, Dunod, Paris, 1969.
  * [34] J. Li, J. Chou, Asymptotic behavior of solutions of the moist atmospheric equations, Acta Meteor. Sinica, 56(1998), 61–72 (in Chinese).
  * [35] J. Lions and B. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Springer–Verlag, New York, 1972.
  * [36] J.L. Lions, R. Temam, S. Wang, New formulations of the primitive equations of atmosphere and applications, Nonlinearity 5(1992), 237–288.
  * [37] J.L. Lions, R. Temam, S. Wang, On the equations of the large-scale ocean, Nonlinearity 5(1992), 1007–1053.
  * [38] J.L. Lions, R. Temam, S. Wang, Models of the coupled atmosphere and ocean (CAO I), Comput. Mech. Adv. 1(1993), 1–54.
  * [39] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lect. Notes Math., vol. 9, 2003.
  * [40] A. Miranville and R. Temam, Mathematical Modeling in Continuum Mechanics (Cambridge: Cambridge University Press), 2005.
  * [41] J. Pedlosky, Geophysical Fluid Dynamics, second edition, Springer-Verlag, Berlin/New York, 1987.
  * [42] M. Petcu, R. M. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of Numerical Analysis vol 14 Special vol Computational Methods for the Atmosphere and the Oceans (Amsterdam: Elsevier/North-Holland) pp 577–750, 2009.
  * [43] L.F. Richardson, Weather Prediction by Numerical Process(Cambridge Mathematical Library)(Cambridge: Cambridge University), 2007.
  * [44] J. Robinson, Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge University Press, Cambridge, 2001.
  * [45] G. Sell and Y. You, Dynamics of evolutionary equations, Springer-Verlag, New York, 2002.
  * [46] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, SpringVerlag, 1988, 2nd Edition, 1997.
  * [47] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, Handbook of mathematical fluid dynamics, vol 3, 2004.
  * [48] S. Wang, Attractors for the 3-D baroclinic quasi-geostrophic equations of large-scale atmosphere, J. Math. Anal. Appl. 165 (1) (1992), 266–283.
  * [49] J. Wang, Global solutions of the 2D dissipative quasi-geostrophic equations in Besov spaces, SIAM J. Math. Anal. 36 (2004), 1014–1030.
  * [50] J. Wang, The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation, Nonlinearity, 18 (2005), 139–154.
  * [51] M.C. Zelati, A.Huang, L. Kukavica, R.Teman and M. Ziane, The primitive equations of the atmosphere in presence of vapour saturation, Nonlinearlity, 28(2015), 625–668.