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

Guoli Zhou  Chongqing University, P.R. China

###### Abstract

Under general boundary conditions we consider the finiteness of the Hausdorff
and fractal dimensions of the global attractor for the strong solution of the
3D moist primitive equations with viscosity. Firstly, we obtain time-uniform
estimates of the first-order time derivative of the strong solutions in
$L^{2}(\mho)$. Then, to prove the finiteness of the Hausdorff and fractal
dimensions of the global attractor, the common method is to obtain the uniform
boundedness of the strong solution in $H^{2}(\mho)$ to establish the squeezing
property of the solution operator. But it is difficult to achieve due to the
boundary conditions and complicated structure of the 3D moist primitive
equations. To overcome the difficulties, we try to use the uniform boundedness
of the derivative of the strong solutions with respect to time $t$ in
$L^{2}(\mho)$ to prove the uniform continuity of the global attractor.
Finally, using the uniform continuity of the global attractor we establish the
squeezing property of the solution operator which implies the finiteness of
the Hausdorff and fractal dimensions of the global attractor.

Keywords: Moist primitive equations, global attractor, fractal dimension,
Hausdorff dimension

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, 36, 38, 39] 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, 41,
50, 51, 52] 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 [32] 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. Recently, the authors in [54]
proved the existence of the global attractor of the strong solutions to the 3D
moist primitive equations.

In this paper, we investigate the finiteness of the the Hausdorff and fractal
dimensions of the global attractor for the strong solutions to the 3D moist
primitive equations with viscosity. The common method is to obtain uniformly
estimates for the strong solution in $H^{2}(\mho)$ space and then use the
squeezing property of the solution operator to show our main result. But it is
difficult to get the time-uniform estimates for the strong solution in the
space $H^{2}(\mho)$. Because, firstly, the boundary conditions in this work is
general and consists with the boundary conditions assuring the global well-
posedness of the moist primitive equations without any extra conditions which
help us to use integration by parts formula to obtain $a\ priori$ estimates in
space with higher regularity. Secondly, the structure of the moist primitive
equations is even more complicated than the oceanic primitive equations
studied in [12]. For example, in the horizontal momentum equation, there is a
challenging term of the gradient of temperature times the mixing ratio of
water vapor. The temperature equations and the mixing ratio of water vapor
equations also have the similar challenging terms which present essential
difficulties for $a$ $priori$ estimates in the $H^{2}(\mho)$ norm. To overcome
the difficulties, inspired by [28] we try to use the uniform continuity of the
global attractor to prove the squeezing property of the solution operator
which implies the finiteness of the the Hausdorff and fractal dimensions of
the global attractor for the strong solutions to the 3D moist primitive
equations. This method can be applied to other dissipative equations with
physical boundary conditions.

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 of the first-order time derivatives of the solution
in $L^{2}(\mho)$ is obtained in section $3$. Section $4$ is for the uniform
continuity and the dimensions of the 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

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

In our article, we will frequently use the following inequalities. So we state
them as the lemmas below. For their proof, one can refer to [20] and
$\cite[cite]{[\@@bibref{}{ZG}{}{}]}.$

###### Lemma 2.4

Let $v\in H^{2}(T\mho|TS^{2}),\mu\in H^{1}(T\mho|TS^{2})\Big{(}\mu\in
H^{1}(\mho)\Big{)}$ and $\nu\in L^{2}(T\mho|TS^{2})\Big{(}\nu\in
L^{2}(\mho)\Big{)}.$ Then, there exists a positive constant $c$ independent of
$v,\mu$ and $\nu$ such that

$\displaystyle\Big{|}\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v(t;\theta,\phi,\xi^{\prime})d\xi^{\prime}\Big{)}\mu,\nu\rangle\Big{|}$
$\displaystyle\leq$ $\displaystyle
c|\mathrm{div}v|_{2}^{\frac{1}{2}}(|\mathrm{div}v|_{2}^{\frac{1}{2}}+|\Delta
v|_{2}^{\frac{1}{2}})|\mu|_{2}^{\frac{1}{2}}(|\nabla_{e_{\theta}}\mu|_{2}^{\frac{1}{2}}+|\nabla_{e_{\varphi}}\mu|_{2}^{\frac{1}{2}}+|\Delta\mu|_{2}^{\frac{1}{2}})|\nu|_{2}$
$\displaystyle\Big{(}$ $\displaystyle\leq$ $\displaystyle
c|\mathrm{div}v|_{2}^{\frac{1}{2}}(|\mathrm{div}v|_{2}^{\frac{1}{2}}+|\Delta
v|_{2}^{\frac{1}{2}})|\mu|_{2}^{\frac{1}{2}}(|\nabla\mu|_{2}^{\frac{1}{2}}+|\Delta\mu|_{2}^{\frac{1}{2}})|\nu|_{2}\Big{)}.$

###### Lemma 2.5

Let $v\in H^{1}(T\mho|TS^{2}),\mu\in H^{1}(T\mho|TS^{2})\Big{(}\mu\in
H^{1}(\mho)\Big{)}$ and $\nu\in H^{1}(T\mho|TS^{2})\Big{(}\nu\in
H^{1}(\mho)\Big{)}.$ Then, there exists a positive constant $c$ independent of
$v,\mu$ and $\nu$ such that

$\displaystyle\Big{|}\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v(t;\theta,\phi,\xi^{\prime})d\xi^{\prime}\Big{)}\mu,\nu\rangle\Big{|}$
$\displaystyle\leq$ $\displaystyle
c|\mathrm{div}v|_{2}|\mu|_{2}^{\frac{1}{2}}(|\mu|_{2}^{\frac{1}{2}}+|\nabla_{e_{\theta}}\mu|_{2}^{\frac{1}{2}}+|\nabla_{e_{\varphi}}\mu|_{2}^{\frac{1}{2}})|\nu|_{2}^{\frac{1}{2}}(|\nu|_{2}^{\frac{1}{2}}+|\nabla_{e_{\theta}}\nu|_{2}^{\frac{1}{2}}+|\nabla_{e_{\varphi}}\nu|_{2}^{\frac{1}{2}})$
$\displaystyle\Big{(}$ $\displaystyle\leq$ $\displaystyle
c|\mathrm{div}v|_{2}|\mu|_{2}^{\frac{1}{2}}(|\mu|_{2}^{\frac{1}{2}}+|\nabla\mu|_{2}^{\frac{1}{2}})|\nu|_{2}^{\frac{1}{2}}(|\nu|_{2}^{\frac{1}{2}}+|\nabla\nu|_{2}^{\frac{1}{2}})\Big{)}.$

In the process of obtaining absorbing ball for the strong solution $U$ in
$H^{2}(T\mho|TS^{2})\times H^{2}(\mho)\times H^{2}(\mho),$ the uniform
Gronwall lemma is used extensively. Therefore, for sake of convenience, we
cite it here.

###### Lemma 2.6

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 dimensions of the global attractor for the strong
solutions to the moist primitive equations, we recall the definitions of
strong solution to $(1.11)-(1.17).$

###### Definition 2.1

Suppose $Q_{1},Q_{2}\in L^{2}(\mho)$ and $Q_{1}|_{\xi=1},Q_{2}|_{\xi=1}\in
L^{2}(S^{2}).$ Let $U_{0}:=(v_{0},T_{0},q_{0})\in V$ and $\tau>0.$
$U:=(v,T,q)$ 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
$\tau>0$ given, the global strong solution $U$ of the system $(1.11)-(1.17)$
is unique on the interval $[0,\tau]$. Moreover, the strong solution $U$ is
continuous with respect to initial data in $H.$

###### Remark 2.1

In fact, we proved in [54] that for any $Q_{1},Q_{2}\in L^{2}(\mho)$ and
$Q_{1}|_{\xi=1},Q_{2}|_{\xi=1}\in L^{2}(S^{2}),$ the global strong solution
$U$ of the system $(1.11)-(1.17)$ is unique on the interval $[0,\tau]$ with
any $\tau>0$. Moreover, the strong solution $U$ is continuous with respect to
initial data in $V.$

For the reader’s convenience, we introduce the definition of global attractor
in the following. For more details, we refer to [22, 48] 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

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

In $\cite[cite]{[\@@bibref{}{ZG}{}{}]},$ we prove the result below about the
existence of global attractor for the solutions to 3D moist primitive
equations.

###### Theorem 2.1

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 $V.$

Therefore, the main result of this work is the finite Hausdorff and fractal
dimensions of the global attractor in space $V.$ To prove the properties of
global attractor, we use a theorem in $\cite[cite]{[\@@bibref{}{L0}{}{}]}.$
For reader’s convenience, we cite it below.

###### Theorem 2.2

Let $X$ be a Hilbert space with norm $\|\cdot\|_{X}$. $S:X\mapsto X$ be a map
and $\mathcal{A}\subset X$ be a compact set such that
$S(\mathcal{A})=\mathcal{A}.$ Suppose that there exist a positive constant $c$
and $\delta\in(0,1),$ such that $\forall a_{1},a_{2}\in\mathcal{A}.$

$\displaystyle\mathrm{(i)}$ $\displaystyle\ \|S(a_{1})-S(a_{2})\|_{X}\leq
c\|a_{1}-a_{2}\|_{X},$ $\displaystyle\mathrm{(ii)}$ $\displaystyle\
\|Q_{N}[S(a_{1})-S(a_{2})]\|_{X}\leq\delta\|a_{1}-a_{2}\|_{X},$

where $Q_{N}$ is the projection in $X$ onto some subspace $(X_{N})^{\bot}$ of
co-dimension $N\in\mathbb{N}.$ Then

$\displaystyle d_{H}(\mathcal{A})\leq d_{F}(\mathcal{A})\leq
N\frac{\mathrm{ln}(\frac{8{G_{a}}^{2}c^{2}}{1-\delta^{2}})}{\mathrm{ln}(\frac{2}{1+\delta^{2}})},$

where $d_{H}(\mathcal{A})$ and $d_{F}(\mathcal{A})$ are the Hausdorff and
fractal dimensions of $\mathcal{A}$ respectively and $G_{a}$ is the Gauss
constant:

$\displaystyle
G_{a}:=\frac{1}{2\pi}\beta(\frac{1}{4},\frac{1}{2})=\frac{2}{\pi}\int_{0}^{1}\frac{dx}{\sqrt{1-x^{4}}}=0.8346268....$

##  3 Uniform estimates and absorbing balls for $\partial_{t}U$ in the space
$H$

###### Theorem 3.1

Assume $Q_{1},Q_{2}\in L^{2}(\mho)$ and $Q_{1}|_{\xi=1},Q_{2}|_{\xi=1}\in
L^{2}(S^{2}).$ Let $U_{0}:=(v(0),T(0),q(0))\in V$ and $\partial_{t}U(0)\in H.$
Then, there exists a unique strong solution $U:=(v,T,q)$ of $(1.11)-(1.17)$
such that

$\displaystyle\partial_{t}U\in L^{\infty}([0,\infty);H)\cap
L^{2}([0,\infty);V).$

Furthermore, there exists a bounded absorbing ball for $U_{t}:=\partial_{t}U$
in space $H.$

###### Proof.

The uniqueness of such solutions follows by Proposition 2.1. To obtain uniform
estimates of $\partial_{t}U$ in $H,$ we introduce some notations denoted by

$\displaystyle u:=v_{t}=\partial_{t}v,\ \theta:=T_{t}=\partial_{t}T,\
\eta:=q_{t}=\partial_{t}q.$

To prove $\partial_{t}U$ is uniformly bounded in $H$ with respect to $t,$ we
first prove that there exists a positive constant c independent of $t$ such
that

$\displaystyle\int_{t}^{t+1}|\partial_{s}U(s)|_{2}^{2}ds<c$ (3.24)

for arbitrary $t\geq 0.$ By Lemma 2.1, we have

$\displaystyle\int_{\mho}q_{t}\Delta qd\mho=-\int_{\mho}\nabla
q_{t}\cdot\nabla qd\mho=-\frac{1}{2}\frac{d}{dt}|\nabla q|_{2}^{2}.$ (3.25)

By the boundary conditions $(1.15)-(1.16)$, we arrive at

$\displaystyle\int_{\mho}q_{t}\partial_{\xi\xi}qd\mho$ $\displaystyle=$
$\displaystyle\int_{\mho}[\partial_{\xi}(q_{t}q_{\xi})-(\partial_{\xi}q_{t})q_{\xi}]d\mho$
(3.26) $\displaystyle=$
$\displaystyle\int_{S^{2}}-\beta_{s}q_{t}|_{\xi=1}q|_{\xi=1}dS^{2}-\frac{1}{2}\frac{d}{dt}|q_{\xi}|_{2}^{2}$
$\displaystyle=$
$\displaystyle-\frac{\beta_{s}}{2}\frac{d}{dt}|q|_{\xi=1}|_{2}^{2}-\frac{1}{2}\frac{d}{dt}|q_{\xi}|_{2}^{2}.$

Using $(3.25)-(3.26)$ and taking the inner product of equation $(1.13)$ with
$q_{t}$ yields

$\displaystyle\frac{1}{2}\frac{d}{dt}|\nabla q|_{2}^{2}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\frac{d}{dt}|q_{\xi}|_{2}^{2}+\frac{\beta_{s}}{2}\frac{d}{dt}|q|_{\xi=1}|_{2}^{2}+|\eta|_{2}^{2}$
(3.27) $\displaystyle=$
$\displaystyle-\langle\nabla_{v}q,\eta\rangle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)}\partial_{\xi}q,\eta\rangle+\langle
Q_{2},\eta\rangle$ $\displaystyle\leq$ $\displaystyle|\eta|_{2}|v|_{4}|\nabla
q|_{4}+c|\eta|_{2}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}\|q\|_{1}^{\frac{1}{2}}\|q\|_{2}^{\frac{1}{2}}+|Q_{2}|_{2}|\eta|_{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon|\eta|_{2}^{2}+c|Q_{2}|_{2}^{2}+c\|v\|_{1}^{2}\|q\|_{2}^{2}+c\|v\|_{1}\|v\|_{2}\|q\|_{1}\|q\|_{2},$

where the first inequality follows by Lemma $2.4.$ Then $(3.27)$ implies that
for $t>0$

$\displaystyle\|q(t+1)\|_{1}^{2}$ $\displaystyle+$
$\displaystyle\int_{t}^{t+1}|\eta(s)|_{2}^{2}ds\leq\|q(t)\|_{1}^{2}+|Q_{2}|_{2}^{2}$
(3.28)
$\displaystyle+c\int_{t}^{t+1}(\|v(s)\|_{1}^{2}\|q(s)\|_{2}^{2}+c\|v(s)\|_{1}\|v(s)\|_{2}\|q(s)\|_{1}\|q(s)\|_{2})ds.$

It is proved in [54] that there exists a positive constant $c$ independent of
$t$ such that

$\displaystyle\|U(t)\|_{1}^{2}+\int_{t}^{t+1}\|U(s)\|_{2}^{2}ds<c$ (3.29)

for arbitrary $t\geq 0.$ Therefore, by $(3.28)$ and $(3.29)$, we obtain the
uniform boundedness for $\int_{t}^{t+1}|\eta(s)|_{2}^{2}ds$ with respect $t.$
Taking an similar argument of $(3.27),$ we have

$\displaystyle\frac{1}{2}\frac{d}{dt}|\nabla T|_{2}^{2}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\frac{d}{dt}|T_{\xi}|_{2}^{2}+\frac{\alpha_{s}}{2}\frac{d}{dt}|T|_{\xi=1}|_{2}^{2}+|\theta|_{2}^{2}$
(3.30) $\displaystyle=$
$\displaystyle-\langle\nabla_{v}T,\theta\rangle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)}\partial_{\xi}T,\theta\rangle$
$\displaystyle+\langle
Q_{1},\theta\rangle+\langle\frac{bP}{p}(1+aq)\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)},\theta\rangle$
$\displaystyle\leq$ $\displaystyle|\theta|_{2}|\nabla
T|_{4}|v|_{4}+c|\theta|_{2}\|T\|_{1}^{\frac{1}{2}}\|T\|_{2}^{\frac{1}{2}}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}$
$\displaystyle+|\theta|_{2}|Q_{1}|_{2}+c|\theta|_{2}\|v\|_{1}+c|\theta|_{2}\|q\|_{1}^{\frac{1}{2}}\|q\|_{2}^{\frac{1}{2}}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon|\theta|_{2}^{2}+c\|T\|_{2}^{2}\|v\|_{1}^{2}+c\|T\|_{1}\|T\|_{2}\|v\|_{1}\|v\|_{2}$
$\displaystyle+c|Q_{1}|_{2}^{2}+c\|v\|_{1}^{2}+c\|q\|_{1}\|q\|_{2}\|v\|_{1}\|v\|_{2},$

where the first inequality follows by Lemma $2.4.$ For $t>0,$ integrating
$(3.30)$ with respect to $t$ yields

$\displaystyle\|T(t+1)\|_{1}^{2}+\int_{t}^{t+1}|\theta(s)|_{2}^{2}ds$ (3.31)
$\displaystyle\leq$
$\displaystyle\|T(t)\|_{1}^{2}+c\int_{t}^{t+1}\|T(s)\|_{2}^{2}\|v(s)\|_{1}^{2}ds$
$\displaystyle+c\int_{t}^{t+1}\|T(s)\|_{1}\|T(s)\|_{2}\|v(s)\|_{1}\|v(s)\|_{2}ds$
$\displaystyle+c|Q_{1}|_{2}^{2}+c\int_{t}^{t+1}\|v(s)\|_{1}^{2}ds$
$\displaystyle+c\int_{t}^{t+1}\|q(s)\|_{1}\|q(s)\|_{2}\|v(s)\|_{1}\|v(s)\|_{2}ds,$

which combined with $(3.29)$ implies the uniform boundednss of
$\int_{t}^{t+1}|\theta(s)|_{2}^{2}ds$ with respect to $t$. Since by Lemma 2.1
and boundary conditions $(1.15)-(1.16)$, we have

$\displaystyle\int_{\mho}\mathrm{grad}\Phi_{s}(\theta,\phi,t)\cdot
ud\mho=\int_{\mho}\mathrm{grad}\Phi_{s}(\theta,\phi,t)\cdot v_{t}d\mho=0,$
(3.32) $\displaystyle-\int_{\mho}v_{t}\cdot\Delta
vd\mho=\int_{\mho}(\nabla_{e_{\theta}}v_{t}\cdot\nabla_{e_{\theta}}v+\nabla_{e_{\phi}}v_{t}\cdot\nabla_{e_{\phi}}v+v_{t}\cdot
v)dt,$ (3.33)
$\displaystyle-\int_{\mho}v_{t}\cdot\partial_{\xi\xi}vd\mho=\frac{1}{2}\frac{d}{dt}|\partial_{\xi}v|_{2}^{2}.$
(3.34)

Then taking inner product of $(1.11)$ with $v_{t}$ and using equalities
$(3.32)-(3.34)$ yields

$\displaystyle\frac{1}{2}\frac{d}{dt}\|v\|_{1}^{2}+|u|_{2}^{2}$ (3.35)
$\displaystyle=$
$\displaystyle-\langle\nabla_{v}v,u\rangle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}\
v(t;\theta,\varphi,\xi^{\prime})d\xi^{\prime}\Big{)}\partial_{\xi}v,u\rangle$
$\displaystyle-\langle\frac{f}{R_{0}}v^{\bot},u\rangle-\langle\Big{(}\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)T]d\xi^{\prime}\Big{)},u\rangle$
$\displaystyle\leq$
$\displaystyle|v|_{\infty}\|v\|_{1}|u|_{2}+c\|v\|_{1}\|v\|_{2}|u|_{2}+|v|_{2}|u|_{2}$
$\displaystyle+c|\nabla T|_{2}|u|_{2}+c|T|_{4}|\nabla q|_{4}|u|_{2}+c|\nabla
T|_{4}|q|_{4}|u|_{2}$ $\displaystyle\leq$
$\displaystyle\varepsilon|u|_{2}^{2}+c|v|_{2}^{2}+c\|T\|_{1}^{2}+c\|v\|_{1}^{2}\|v\|_{2}^{2}+c\|T\|_{1}^{2}\|q\|_{2}^{2}+c\|q\|_{1}^{2}\|T\|_{2}^{2}.$

Using uniform estimates $(3.29)$ and integrating $(3.25)$ with respect to $t,$
we prove that $\int_{t}^{t+1}|u(s)|_{2}^{2}ds$ is uniformly bounded with
respect to $t(t\geq 0).$ With these time-uniform $a\ priori$ estimates of

$\displaystyle\int_{t}^{t+1}|u(s)|_{2}^{2}ds,\ \
\int_{t}^{t+1}|\theta(s)|_{2}^{2}ds\ \ \mathrm{and}\
\int_{t}^{t+1}|\eta(s)|_{2}^{2}ds,$

we prove $(3.24).$ In the following, we will show the existence of the
absorbing ball of $U_{t}$ in $H.$ Taking derivative with respect to $t$ in
$(1.13),$ we have

$\displaystyle\eta_{t}+L_{3}\eta+\nabla_{u}q+\nabla_{v}\eta+w_{t}q_{\xi}+w\eta_{\xi}=0.$
(3.36)

Taking inner product of $(3.36)$ with $\eta$ and using boundary conditions
$(1.15)-(1.16),$ we have

$\displaystyle\frac{1}{2}\frac{d}{dt}|\eta|_{2}^{2}+|\nabla\eta|_{2}^{2}+|\eta_{\xi}|_{2}^{2}+\beta_{s}|\eta|_{\xi=1}|_{2}^{2}$
$\displaystyle=$
$\displaystyle-\langle\nabla_{u}q,\eta\rangle-\langle\nabla_{v}\eta,\eta\rangle-\langle
w_{t}q_{\xi},\eta\rangle-\langle w\eta_{\xi},\eta\rangle$ $\displaystyle=$
$\displaystyle-\langle\nabla_{u}q,\eta\rangle-\langle
w_{t}q_{\xi},\eta\rangle,$

where the second equality follows by Lemma $2.3.$ Therefore, by
H$\mathrm{\ddot{o}}$lder inequality, Lemma $2.4,$ interpolation inequality and
Young’s inequality we obtain

$\displaystyle\frac{d}{dt}|\eta|_{2}^{2}+\|\eta\|_{1}^{2}$ (3.37)
$\displaystyle\leq$ $\displaystyle c|\nabla
q|_{2}|u|_{4}|\eta|_{4}+c|\mathrm{div}u|_{2}|q_{\xi}|_{2}^{\frac{1}{2}}|\nabla
q_{\xi}|_{2}^{\frac{1}{2}}|\eta|_{2}^{\frac{1}{2}}|\nabla\eta|_{2}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
c\|q\|_{1}|u|_{2}^{\frac{1}{4}}\|u\|_{1}^{\frac{3}{4}}|\eta|_{2}^{\frac{1}{4}}\|\eta\|_{1}^{\frac{3}{4}}++c|\mathrm{div}u|_{2}|q_{\xi}|_{2}^{\frac{1}{2}}|\nabla
q_{\xi}|_{2}^{\frac{1}{2}}|\eta|_{2}^{\frac{1}{2}}|\nabla\eta|_{2}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon\|\eta\|_{1}^{2}+\varepsilon\|u\|_{1}^{2}+c|\eta|_{2}^{2}+c\|q\|_{1}^{8}|u|_{2}^{2}+c\|q\|_{1}^{2}\|q\|_{2}^{2}|\eta|_{2}^{2}.$

Taking derivative with respect to $t$ in $(1.12),$ we have

$\displaystyle\partial_{t}\theta$ $\displaystyle+$ $\displaystyle
L_{2}\theta+\nabla_{u}T+\nabla_{v}\theta+w_{t}T_{\xi}+w\theta_{\xi}$ (3.38)
$\displaystyle-\frac{bP}{p}a\eta w-\frac{bP}{p}(1+aq)w_{t}=0.$

Since by Lemma $2.3,$

$\displaystyle\langle\nabla_{v}\theta+w\theta_{\xi},\theta\rangle=0.$

Consequently, multiplying $(3.38)$ by $\theta$ and integrating over $\mho$
yields

$\displaystyle\frac{1}{2}\frac{d}{dt}|\theta|_{2}^{2}+|\nabla\theta|_{2}^{2}+|\theta_{\xi}|_{2}^{2}+\alpha_{s}|\theta|_{\xi=1}|_{2}^{2}$
(3.39) $\displaystyle=$
$\displaystyle\langle\nabla_{u}T+w_{t}T_{\xi},\theta\rangle$
$\displaystyle-\langle\frac{bP}{p}a\eta
w,\theta\rangle-\langle\frac{bP}{p}(1+aq)w_{t},\theta\rangle$
$\displaystyle=:$ $\displaystyle I_{1}+I_{2}+I_{3}.$

By H$\mathrm{\ddot{o}}$lder inequality, Lemma $2.4$, interpolation inequality
and Young’s inequality we have

$\displaystyle I_{1}$ $\displaystyle\leq$ $\displaystyle|u|_{4}|\nabla
T|_{4}|\theta|_{2}+c\|u\|_{1}|T_{\xi}|_{2}^{\frac{1}{2}}\|T_{\xi}\|_{1}^{\frac{1}{2}}|\theta|_{2}^{\frac{1}{2}}\|\theta\|_{1}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
c|\theta|_{2}|u|_{2}^{\frac{1}{4}}\|u\|_{1}^{\frac{3}{4}}\|T\|_{1}^{\frac{1}{4}}\|T\|_{2}^{\frac{3}{4}}+c\|u\|_{1}|T_{\xi}|_{2}^{\frac{1}{2}}\|T_{\xi}\|_{1}^{\frac{1}{2}}|\theta|_{2}^{\frac{1}{2}}\|\theta\|_{1}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon\|u\|_{1}^{2}+\varepsilon\|\theta\|_{1}^{2}+c|u|_{2}^{2}+c|\theta|_{2}^{2}\|T\|_{2}^{2}+c|\theta|_{2}^{2}\|T\|_{1}^{2}\|T\|_{2}^{2}.$

To estimate $I_{2},$ using H$\mathrm{\ddot{o}}$lder inequality, interpolation
inequality and Young’s inequality we reach

$\displaystyle I_{2}$ $\displaystyle\leq$ $\displaystyle
c|\theta|_{2}|\eta|_{2}^{\frac{1}{2}}\|\eta\|_{1}^{\frac{1}{2}}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon\|\eta\|_{1}^{2}+c|\eta|_{2}^{2}+c|\theta|_{2}^{2}\|v\|_{1}\|v\|_{2}.$

Taking an analogous argument as $I_{2},$ we deduce

$\displaystyle I_{3}$ $\displaystyle\leq$ $\displaystyle
c\|u\|_{1}|\theta|_{2}+c|q|_{2}^{\frac{1}{2}}\|q\|_{1}^{\frac{1}{2}}\|u\|_{1}|\theta|_{2}^{\frac{1}{2}}\|\theta\|_{1}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon\|u\|_{1}^{2}+\varepsilon\|\theta\|_{1}^{2}+c|\theta|_{2}^{2}|q|_{2}^{2}\|q\|_{1}^{2}.$

Substituting the estimates of $I_{1}-I_{3}$ into $(3.39),$ we reach

$\displaystyle\frac{d}{dt}|\theta|_{2}^{2}+\|\theta\|_{1}^{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon(\|u\|_{1}^{2}+\|\theta\|_{1}^{2}+\|\eta\|_{1}^{2})+c(|u|_{2}^{2}+|\eta|_{2}^{2})$
(3.40)
$\displaystyle+c|\theta|_{2}^{2}(\|T\|_{2}^{2}+\|v\|_{2}^{2}+\|q\|_{1}^{4}+\|T\|_{1}^{2}\|T\|_{2}^{2}).$

Taking derivative with respect to $t$ in $(1.11),$ we have

$\displaystyle\frac{d}{dt}u$ $\displaystyle+$ $\displaystyle
L_{1}u+\nabla_{u}v+\nabla_{v}u+w_{t}v_{\xi}+wu_{\xi}+\frac{f}{R_{0}}u^{\bot}+\mathrm{grad}(\partial_{t}\Phi_{s})$
(3.41) $\displaystyle+$
$\displaystyle\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}(a\eta
T)d\xi^{\prime}+\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)\eta]d\xi^{\prime}=0.$

By the aid of Lemma $2.3$ and Lemma $2.1,$ it follows that

$\displaystyle\langle\nabla_{v}u+wu_{\xi},u\rangle=\langle\mathrm{grad}(\partial_{t}\Phi_{s}),u\rangle=\langle\frac{f}{R_{0}}u^{\bot},u\rangle=0.$

Therefore, multiplying $(3.41)$ by $u$ and integrating over $\mho$ yields

$\displaystyle\frac{1}{2}\frac{d}{dt}|u|_{2}^{2}+\|u\|_{1}^{2}$
$\displaystyle=$ $\displaystyle-\langle\nabla_{u}v,u\rangle-\langle
w_{t}v_{\xi},u\rangle$ (3.42)
$\displaystyle-\langle\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}(a\eta
T)d\xi^{\prime},u\rangle-\langle\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}[(1+aq)\eta]d\xi^{\prime},u\rangle$
$\displaystyle:=$ $\displaystyle J_{1}+J_{2}+J_{3}+J_{4}.$

In view of H$\mathrm{\ddot{o}}$lder inequality, Sobolev inequality and Young’s
inequality, we have

$\displaystyle J_{1}\leq|u|_{2}|\nabla
v|_{4}|u|_{4}\leq\varepsilon\|u\|_{1}^{2}+c|u|_{2}^{2}\|v\|_{2}^{2}.$

Thanks to Lemma $2.5$ and Young’s inequality, we reach

$\displaystyle J_{2}$ $\displaystyle\leq$ $\displaystyle
c\|u\|_{1}\|v\|_{1}^{\frac{1}{2}}\|v\|_{2}^{\frac{1}{2}}|u|_{2}^{\frac{1}{2}}\|u\|_{1}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon\|u\|_{1}^{2}+c|u|_{2}^{2}\|v\|_{1}^{2}\|v\|_{2}^{2}.$

Similarly, by the Lemma $2.4$, Lemma $2.5$ and Young’s inequality, we obtain
the estimates of $J_{3}$

$\displaystyle J_{3}$ $\displaystyle\leq$ $\displaystyle
c\|\eta\|_{1}|T|_{2}^{\frac{1}{2}}\|T\|_{1}^{\frac{1}{2}}|u|_{2}^{\frac{1}{2}}\|u\|_{1}^{\frac{1}{2}}+c\|T\|_{1}^{\frac{1}{2}}\|T\|_{2}^{\frac{1}{2}}|\eta|_{2}^{\frac{1}{2}}\|\eta\|_{1}^{\frac{1}{2}}|u|_{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon\|\eta\|_{1}^{2}+\varepsilon\|u\|_{1}^{2}+c\|T\|_{1}^{2}\|T\|_{2}^{2}(|u|_{2}^{2}+|\eta|_{2}^{2}).$

Finally, taking an analogously argument, we deduce

$\displaystyle J_{4}$ $\displaystyle\leq$ $\displaystyle
c\|\eta\|_{1}|q|_{2}^{\frac{1}{2}}\|q\|_{1}^{\frac{1}{2}}|u|_{2}^{\frac{1}{2}}\|u\|_{1}^{\frac{1}{2}}+c|u|_{2}\|q\|_{1}^{\frac{1}{2}}\|q\|_{2}^{\frac{1}{2}}|\eta|_{2}^{\frac{1}{2}}\|\eta\|_{1}^{\frac{1}{2}}+c\|\eta\|_{1}|u|_{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon\|\eta\|_{1}^{2}+\varepsilon\|u\|_{1}^{2}+c|u|_{2}^{2}+c\|q\|_{1}^{2}\|q\|_{2}^{2}(|u|_{2}^{2}+|\eta|_{2}^{2}).$

Substituting these estimates of $J_{1}-J_{4}$ into $(3.42),$ we have

$\displaystyle\frac{d}{dt}|u|_{2}^{2}+\|u\|_{1}^{2}\leq\varepsilon(\|u\|_{1}^{2}+\|\eta\|_{1}^{2})$
$\displaystyle+c(|u|_{2}^{2}+|\eta|_{2}^{2})(1+\|v\|_{2}^{2}+\|v\|_{1}^{2}\|v\|_{2}^{2}+\|T\|_{1}^{2}\|T\|_{2}^{2}+\|q\|_{1}^{2}\|q\|_{2}^{2}).$
(3.43)

For $t\geq 0,$ denote by

$\displaystyle
f(t):=1+\|q(t)\|_{1}^{8}+\|T(t)\|_{2}^{2}+\|v(t)\|_{2}^{2}+\|q(t)\|_{1}^{2}\|q(t)\|_{2}^{2}+\|T(t)\|_{1}^{2}\|T(t)\|_{2}^{2}+\|v(t)\|_{1}^{2}\|v(t)\|_{2}^{2}.$

Then, it follows from the time-uniform estimates $(3.29)$ that
$\int_{t}^{t+1}f(s)ds$ is uniformly bounded with respect to $t.$ Summing
$(3.37),(3.40)$ and $(3.43),$ we obtain

$\displaystyle\frac{d}{dt}|\partial_{t}U(t)|_{2}^{2}+\|\partial_{t}U(t)\|_{1}^{2}\leq
c|\partial_{t}U(t)|_{2}^{2}f(t).$ (3.44)

Applying the uniform boundedness of $\int_{t}^{t+1}f(s)ds$ and the uniform
Gronwall inequality to $(3.44),$ the conclusions of the theorem follows.
$\square$ ∎

##  4 Dimensions of the global attractor

###### Theorem 4.1

Assume $Q_{1},Q_{2}\in L^{2}(\mho)$ and $Q_{1}|_{\xi=1},Q_{2}|_{\xi=1}\in
L^{2}(S^{2}).$ Let $U_{0}\in\mathcal{A}.$ Then there is a positive constant
$c$ independent of initial condition $U_{0}$ and time $\tau(\tau>0)$ such that

$\displaystyle\int_{0}^{\tau}\|U(t)\|_{2}^{2}dt<c(\tau^{\frac{1}{2}}+\tau).$

###### Proof.

By direct calculation, we reach

$\displaystyle||v(\tau)|_{2}^{2}-|v(0)|_{2}^{2}|$ $\displaystyle=$
$\displaystyle 2|\int_{0}^{\tau}\langle v_{t}(t),v(t)\rangle dt|\leq c\tau.$

Similarly,

$\displaystyle||T(\tau)|_{2}^{2}-|T(0)|_{2}^{2}|$ $\displaystyle=$
$\displaystyle 2|\int_{0}^{\tau}\langle T_{t}(t),T(t)\rangle dt|\leq c\tau$

and

$\displaystyle||q(\tau)|_{2}^{2}-|q(0)|_{2}^{2}|$ $\displaystyle=$
$\displaystyle 2|\int_{0}^{\tau}\langle q_{t}(t),q(t)\rangle dt|\leq c\tau.$

Taking inner product of $(1.13)$ with $A_{3}q$ and integrating over $[0,\tau]$
with respect to $t$ yields

$\displaystyle 2\int_{0}^{\tau}\langle q_{t}(t),A_{3}q(t)\rangle
dt=\|q(\tau)\|_{1}^{2}-\|q(0)\|_{1}^{2}.$

Consequently, we have

$\displaystyle|\|q(\tau)\|_{1}^{2}-\|q(0)\|_{1}^{2}|$ $\displaystyle\leq$
$\displaystyle 2\int_{0}^{\tau}|q_{t}(t)|_{2}|A_{3}q(t)|_{2}dt$
$\displaystyle\leq$ $\displaystyle
2\sup\limits_{t\in[0,\tau]}|q_{t}(t)|_{2}\int_{0}^{\tau}|A_{3}q(t)|_{2}dt$
$\displaystyle\leq$ $\displaystyle c\tau^{\frac{1}{2}}.$

Similarly, we deduce that

$\displaystyle|\|T(\tau)\|_{1}^{2}-\|T(0)\|_{1}^{2}|$ $\displaystyle\leq$
$\displaystyle 2\int_{0}^{\tau}|T_{t}(t)|_{2}|A_{2}T(t)|_{2}dt$
$\displaystyle\leq$ $\displaystyle
2\sup\limits_{t\in[0,\tau]}|T_{t}(t)|_{2}\int_{0}^{\tau}|A_{2}T(t)|_{2}dt$
$\displaystyle\leq$ $\displaystyle c\tau^{\frac{1}{2}}$

and

$\displaystyle|\|v(\tau)\|_{1}^{2}-\|v(0)\|_{1}^{2}|$ $\displaystyle\leq$
$\displaystyle 2\int_{0}^{\tau}|v_{t}(t)|_{2}|A_{1}v(t)|_{2}dt$
$\displaystyle\leq$ $\displaystyle
2\sup\limits_{t\in[0,\tau]}|v_{t}(t)|_{2}\int_{0}^{\tau}|A_{1}v(t)|_{2}dt$
$\displaystyle\leq$ $\displaystyle c\tau^{\frac{1}{2}}.$

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

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

which combined with the uniform-time estimates proved in [54] implies

$\displaystyle\int_{0}^{\tau}\|v_{\xi}(t)\|_{1}^{2}dt$ $\displaystyle\leq$
$\displaystyle||v_{\xi}(\tau)|_{2}^{2}-|v_{\xi}(0)|_{2}^{2}|$ (4.45)
$\displaystyle+c\int_{0}^{\tau}[(\|\bar{v}(t)\|_{1}^{8}+|\tilde{v}(t)|_{4}^{8})|v_{\xi}(t)|_{2}^{2}+|T(t)|_{2}^{2}+|q(t)|_{4}^{4}+|T(t)|_{4}^{4}]dt$
$\displaystyle\leq$ $\displaystyle c\tau^{\frac{1}{2}}+c\tau.$

In view of the energy estimates proved in
$\cite[cite]{[\@@bibref{}{ZG}{}{}]},$ it follows that

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

which implies that

$\displaystyle\int_{0}^{\tau}|\Delta v(t)|_{2}^{2}dt$ $\displaystyle\leq$
$\displaystyle|\|v(\tau)\|_{1}^{2}-\|v(0)\|_{1}^{2}|+||v(\tau)|_{2}^{2}-|v(0)|_{2}^{2}|$
(4.46)
$\displaystyle+c\int_{0}^{\tau}(|\nabla_{e_{\theta}}v(t)|_{2}^{2}+|\nabla_{e_{\varphi}}v(t)|_{2}^{2})$
$\displaystyle\cdot(1+|v_{\xi}(t)|_{2}^{4}+|v_{\xi}(t)|_{2}^{2}|\nabla_{e_{\theta}}v_{\xi}(t)|_{2}^{2}++|v_{\xi}(t)|_{2}^{2}|\nabla_{e_{\varphi}}v_{\xi}(t)|_{2}^{2})dt$
$\displaystyle+c\int_{0}^{\tau}(1+|\nabla T(t)|_{2}^{2}+|\nabla
q(t)|_{2}^{2})dt$ $\displaystyle\leq$ $\displaystyle
c\tau^{\frac{1}{2}}+c\tau.$

According to $\cite[cite]{[\@@bibref{}{ZG}{}{}]},$ we know the following
estimates

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

and

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

Since it is proved in [54] that $|T|_{\xi=1}|_{4}$ and $|q|_{\xi=1}|_{4}$ are
uniformly bounded with respect to $t,$ summing the above estimates about $T$
and $q$ and integrating with respect to $t$ over $[0,\tau]$ yields

$\displaystyle\int_{0}^{\tau}(|\nabla
T_{\xi}(t)|_{2}^{2}+|T_{\xi\xi}(t)|_{2}^{2}+|\Delta
T(t)|_{2}^{2}+\alpha_{s}|\nabla T(t)|_{\xi=1}|_{2}^{2})dt$ (4.47)
$\displaystyle+\int_{0}^{\tau}(|\nabla
q_{\xi}(t)|_{2}^{2}+|q_{\xi\xi}(t)|_{2}^{2}+|\Delta
q(t)|_{2}^{2}+\beta_{s}|\nabla q(t)|_{\xi=1}|_{2}^{2})dt$ $\displaystyle\leq$
$\displaystyle
c\int_{0}^{\tau}(1+\|T(t)\|_{1}^{2}+\|q(t)\|_{1}^{2}+\|v(t)\|_{2}^{2}+|T|_{\xi=1}(t)|_{4}^{4}+|q|_{\xi=1}(t)|_{4}^{4})dt$
$\displaystyle+\tau(|Q_{1}|_{\xi=1}|_{2}^{2}+|Q_{2}|_{\xi=1}|_{2}^{2}+|Q_{1}|_{2}^{2}+|Q_{2}|_{2}^{2})$
$\displaystyle+|\|T(\tau)\|_{1}^{2}-\|T(0)\|_{2}^{2}|+|\|q(\tau)\|_{1}^{2}-\|q(0)\|_{2}^{2}|$
$\displaystyle\leq$ $\displaystyle c(\tau^{\frac{1}{2}}+\tau).$

Combining $(4.45)-(4.47),$ we prove the last inequality of the theorem.
$\square$ ∎

Next, we state our main result of this paper.

###### Theorem 4.2

Assume $Q_{1},Q_{2}\in L^{2}(\mho)$ and $Q_{1}|_{\xi=1},Q_{2}|_{\xi=1}\in
L^{2}(S^{2}).$ Then the global attractor $\mathcal{A}$ has finite Hausdorff
and fractal dimensions measured in the $V$ space.

###### Proof.

Recalling that in [54] we prove that the strong solution $U$ to
$(1.11)-(1.17)$ is Lipschitz continuous with respect to initial data in space
$V.$ Therefore, in order to prove our theorem, according to Theorem $2.2$, we
only need to show the condition $(ii)$ of Theorem $2.2.$

Let initial data $U_{i}=(v_{i},T_{i},q_{i}),i=1,2$ be two strong solutions to
$(1.11)-(1.17)$ with initial data
$U_{i}(0)=(v_{i,0},T_{i,0},q_{i,0})\in\mathcal{A},$ where $\mathcal{A}$ is the
global attractor given by Theorem $2.1.$ In view of the invariance property of
global attractor and Proposition 2.1, we can assume $U_{i}(0)\in
D(A_{1})\times D(A_{2})\times D(A_{3})$ with $i=1,2.$ By virtue of energy
estimates we can easily show that $|\partial_{t}U_{i}(0)|_{2}$ are finite for
$i=1,2.$ Consequently, according to Theorem $3.1,$ we know that
$|\partial_{t}U_{i}(t)|$ are uniformly bounded with respect to $t(t\geq 0)$.
Meanwhile, we also have $U(t)\in D(A_{1})\times D(A_{2})\times D(A_{3})$ for
any $t\geq 0$ in view of Theorem $2.1.$ Let’s define

$\displaystyle\mu=v^{1}-v^{2},\ \ \vartheta=T^{1}-T^{2},\ \ \rho=q^{1}-q^{2},\
\
\Phi(\theta,\varphi,t)=\Phi^{1}_{s}(\theta,\varphi,t)-\Phi^{2}_{s}(\theta,\varphi,t).$

Then we derive from $(1.11)-(1.17)$ that $\mu,\vartheta$ and $\rho$ satisfy

$\displaystyle\partial_{t}\mu$ $\displaystyle+$ $\displaystyle
L_{1}\mu+\nabla_{v_{1}}\mu+\nabla_{\mu}v_{2}+\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}\mu$
(4.48)
$\displaystyle+\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}v_{2}+\frac{f}{R_{0}}\mu^{\bot}+\mathrm{grad}\Phi_{s}$
$\displaystyle+\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}\vartheta
d\xi^{\prime}+\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(q_{1}\vartheta)d\xi^{\prime}+\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(\rho
T_{2})d\xi^{\prime}=0,$ $\displaystyle\partial_{t}\vartheta$ $\displaystyle+$
$\displaystyle
L_{2}\vartheta+\nabla_{v_{1}}\vartheta+\nabla_{\mu}T_{2}+\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}\vartheta$
(4.49)
$\displaystyle+\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}T_{2}-\frac{bP}{p}\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}$
$\displaystyle-\frac{abP}{p}q_{1}\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}-\frac{abP}{p}\rho\Big{(}\int_{\xi}^{1}\mathrm{div}v_{2}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}=0,$
$\displaystyle\partial_{t}\rho$ $\displaystyle+$ $\displaystyle
L_{3}\rho+\nabla_{v_{1}}\rho+\nabla_{\mu}q_{2}+\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}\rho$
(4.50)
$\displaystyle+\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}q_{2}=0.$

Since operators $A_{i},i=1,2,3$ are positive selfadjoint with compact
resolvent, we denote by $\\{\lambda_{i,k}\\}_{k\geq 1}$ the corresponding
eigenvalues of $A_{i}.$ Obviously
$0\leq\lambda_{i,1}<\lambda_{i,2}<\lambda_{i,3}\cdots$ and
$\lambda_{i,k}\rightarrow\infty$ as $k\rightarrow\infty$ for arbitrary
$i\in\\{1,2,3\\}.$ Let $P_{i,n},i=1,2,3$ be the orthogonal projector in
$H_{i}$ onto the subspace spanned by first $n$ eigenvectors of $A_{i}.$ Denote
$Q_{i,n}=I-P_{i,n},i=1,2,3$ and $n\in\mathbb{N}.$

Taking inner product of $(4.48)$ with $A_{1}Q_{1,n}\mu$ in $L^{2}(\mho)$, we
reach

$\displaystyle\frac{1}{2}\frac{d}{dt}\|Q_{1,n}\mu\|_{1}^{2}+|A_{1}Q_{1,n}\mu|_{2}^{2}$
(4.51) $\displaystyle=$
$\displaystyle-\langle\nabla_{v_{1}}\mu,A_{1}Q_{1,n}\mu\rangle-\langle\nabla_{\mu}v_{2},A_{1}Q_{1,n}\mu\rangle$
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}v_{2},A_{1}Q_{1,n}\mu\rangle$
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}\mu,A_{1}Q_{1,n}\mu\rangle$
$\displaystyle-\langle\int_{\xi}^{1}\frac{bP}{p}\mathrm{grad}\vartheta
d\xi^{\prime},A_{1}Q_{1,n}\mu\rangle-\langle(\frac{f}{R_{0}}\mu^{\bot}+\mathrm{grad}\Phi_{s}),A_{1}Q_{1,n}\mu\rangle$
$\displaystyle-\langle\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(q_{1}\vartheta)d\xi^{\prime},A_{1}Q_{1,n}\mu\rangle-\langle\int_{\xi}^{1}\frac{abP}{p}\mathrm{grad}(\rho
T_{2})d\xi^{\prime},A_{1}Q_{1,n}\mu\rangle$ $\displaystyle=:$
$\displaystyle\sum_{m=1}^{8}I_{m}.$

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

$\displaystyle I_{1}$ $\displaystyle\leq$
$\displaystyle|v_{1}|_{\infty}(|\nabla_{e_{\theta}}\mu|_{2}+|\nabla_{e_{\varphi}}\mu|_{2})|A_{1}Q_{1,n}\mu|_{2}$
$\displaystyle\leq$ $\displaystyle
c\|v_{1}\|_{1}^{\frac{1}{2}}|A_{1}v_{1}|_{2}^{\frac{1}{2}}\|\mu\|_{1}|A_{1}Q_{1,n}\mu|_{2}$
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}Q_{1,n}\mu|_{2}^{2}+c\|v_{1}\|_{1}|A_{1}v_{1}|_{2}\|\mu\|_{1}^{2}.$

Similarly, we obtain

$\displaystyle
I_{2}\leq|\mu|_{\infty}\|v_{2}\|_{1}|A_{1}Q_{1,n}\mu|_{2}\leq\varepsilon|A_{1}Q_{1,n}\mu|_{2}^{2}+c\|\mu\|_{1}\|\mu\|_{2}\|v_{2}\|_{1}^{2}.$

In view of Lemma $2.4$ and Young’s inequality, we reach

$\displaystyle I_{3}+I_{4}$ $\displaystyle\leq$ $\displaystyle
c|A_{1}Q_{1,n}\mu|_{2}\|\mu\|_{1}^{\frac{1}{2}}\|\mu\|_{2}^{\frac{1}{2}}\|v_{2}\|_{1}^{\frac{1}{2}}\|v_{2}\|_{2}^{\frac{1}{2}}$
$\displaystyle+c|A_{1}Q_{1,n}\mu|_{2}\|\mu\|_{1}^{\frac{1}{2}}\|\mu\|_{2}^{\frac{1}{2}}\|v_{1}\|_{1}^{\frac{1}{2}}\|v_{1}\|_{2}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}Q_{1,n}\mu|_{2}^{2}+c\|\mu\|_{1}\|\mu\|_{2}\|v_{1}\|_{1}\|v_{1}\|_{2}$
$\displaystyle+c\|\mu\|_{1}\|\mu\|_{2}\|v_{2}\|_{1}\|v_{2}\|_{2}.$

Similarly, we have the estimates of $I_{7}$ and $I_{8}$ that

$\displaystyle I_{7}+I_{8}$ $\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}Q_{1,n}\mu|_{2}^{2}+c|q_{1}|_{2}\|q_{1}\|_{1}\|\vartheta\|_{1}\|\vartheta\|_{2}$
$\displaystyle+c\|q_{1}\|_{1}\|q_{1}\|_{2}|\vartheta|_{2}\|\vartheta\|_{1}$
$\displaystyle+c|T_{2}|_{2}\|T_{2}\|_{1}\|\rho\|_{1}\|\rho\|_{2}$
$\displaystyle+c\|T_{2}\|_{1}\|T_{2}\|_{2}|\rho|_{2}\|\rho\|_{1}$
$\displaystyle\leq$
$\displaystyle\varepsilon|A_{1}Q_{1,n}\mu|_{2}^{2}+c\|q_{1}\|_{1}\|q_{1}\|_{2}\|\vartheta\|_{1}\|\vartheta\|_{2}$
$\displaystyle+c\|T_{2}\|_{1}\|T_{2}\|_{2}\|\rho\|_{1}\|\rho\|_{2},$

where the last inequality follows by Sobolev inequality. Estimates of $I_{5}$
and $I_{6}$ follows by H$\mathrm{\ddot{o}}$lder inequality and Young’s
inequality

$\displaystyle
I_{5}+I_{6}\leq\varepsilon|A_{1}Q_{1,n}\mu|_{2}^{2}+c\|\vartheta\|_{1}^{2}+c|\mu|_{2}^{2}.$

Combining $(4.51)$ and the estimates of $I_{1}-I_{8},$ we have

$\displaystyle\frac{d}{dt}\|Q_{1,n}\mu\|_{1}^{2}+|A_{1}Q_{1,n}\mu|_{2}^{2}$
$\displaystyle\leq$ $\displaystyle
c\|v_{1}\|_{1}\|v_{1}\|_{2}\|\mu\|_{1}^{2}+c\|\mu\|_{1}\|\mu\|_{2}\|v_{2}\|_{1}^{2}$
(4.52) $\displaystyle+c\|\mu\|_{1}\|\mu\|_{2}\|v_{1}\|_{1}\|v_{1}\|_{2}$
$\displaystyle+c\|\mu\|_{1}\|\mu\|_{2}\|v_{2}\|_{1}\|v_{2}\|_{2}$
$\displaystyle+c\|q_{1}\|_{1}\|q_{1}\|_{2}\|\vartheta\|_{1}\|\vartheta\|_{2}$
$\displaystyle+c\|T_{2}\|_{1}\|T_{2}\|_{2}\|\rho\|_{1}\|\rho\|_{2}$
$\displaystyle+c\|\vartheta\|_{1}^{2}+c\|\mu\|_{1}^{2}.$

Taking inner product of $(4.49)$ with $A_{2}Q_{2,n}\vartheta$ in
$L^{2}(\mho)$, we reach

$\displaystyle\frac{1}{2}\frac{d}{dt}\|Q_{2,n}\vartheta\|_{1}^{2}+|A_{2}Q_{2,n}\vartheta|_{2}^{2}$
(4.53) $\displaystyle=$
$\displaystyle-\langle\nabla_{v_{1}}\vartheta,A_{2}Q_{2,n}\vartheta\rangle-\langle\nabla_{\mu}T_{2},A_{2}Q_{2,n}\vartheta\rangle$
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}v_{1}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}\vartheta,A_{2}Q_{2,n}\vartheta\rangle$
$\displaystyle-\langle\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)}\partial_{\xi}T_{2},A_{2}Q_{2,n}\vartheta\rangle$
$\displaystyle+\langle\frac{bP}{p}\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)},A_{2}Q_{2,n}\vartheta\rangle$
$\displaystyle+\langle\frac{abP}{p}q_{1}\Big{(}\int_{\xi}^{1}\mathrm{div}\mu(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)},A_{2}Q_{2,n}\vartheta\rangle$
$\displaystyle+\langle\frac{abP}{p}\rho\Big{(}\int_{\xi}^{1}\mathrm{div}v_{2}(x,y,\xi^{\prime},t)d\xi^{\prime}\Big{)},A_{2}Q_{2,n}\vartheta\rangle.$

Obviously, $(4.53)$ is very similar to $(4.51),$ so taking an analogous
argument as $(4.52)$ we have

$\displaystyle\frac{d}{dt}\|Q_{2,n}\vartheta\|_{1}^{2}+|A_{2}Q_{2,n}\vartheta|_{2}^{2}$
$\displaystyle\leq$ $\displaystyle
c\|v_{1}\|_{1}\|v_{1}\|_{2}\|\vartheta\|_{1}^{2}+c\|\mu\|_{1}\|\mu\|_{2}\|T_{2}\|_{1}^{2}$
(4.54)
$\displaystyle+c\|v_{1}\|_{1}\|v_{1}\|_{2}\|\vartheta\|_{1}\|\vartheta\|_{2}$
$\displaystyle+c\|\mu\|_{1}\|\mu\|_{2}\|T_{2}\|_{1}\|T_{2}\|_{2}$
$\displaystyle+c\|\mu\|_{1}\|\mu\|_{2}|q_{1}|_{2}\|q_{1}\|_{1}$
$\displaystyle+c\|v_{2}\|_{1}\|v_{2}\|_{2}|\rho|_{2}\|\rho\|_{1}+c\|\mu\|_{1}^{2}.$

Similarly, we obtain the estimates of $Q_{3,n}\rho:$

$\displaystyle\frac{d}{dt}\|Q_{3,n}\rho\|_{1}^{2}+|A_{3}Q_{3,n}\rho|_{2}^{2}$
$\displaystyle\leq$ $\displaystyle
c\|v_{1}\|_{1}\|v_{1}\|_{2}\|\rho\|_{1}^{2}+c\|\mu\|_{1}\|\mu\|_{2}\|q_{2}\|_{1}^{2}$
(4.55)
$\displaystyle+c\|v_{1}\|_{1}\|v_{1}\|_{2}\|\rho\|_{1}\|\rho\|_{2}+c\|\mu\|_{1}\|\mu\|_{2}\|q_{2}\|_{1}\|q_{2}\|_{2}.$

For $t\geq 0,$ let

$\displaystyle\varphi(t):=\|Q_{1,n}\mu(t)\|_{1}^{2}+\|Q_{2,n}\vartheta(t)\|_{1}^{2}+\|Q_{3,n}\rho(t)\|_{1}^{2},$
$\displaystyle\psi(t):=\|\mu(t)\|_{1}^{2}+\|\vartheta(t)\|_{1}^{2}+\|\rho(t)\|_{1}^{2}.$

Summing $(4.52),(4.54)$ and $(4.55)$ and using the uniform-time estimates of
the strong solution to $(1.11)-(1.17)$ in space $V$ we get

$\displaystyle\frac{d}{dt}\varphi+\lambda_{n}\varphi$ $\displaystyle\leq$
$\displaystyle c\psi(1+\|v_{1}\|_{2}+\|v_{2}\|_{2})$ (4.56)
$\displaystyle+c\|\mu\|_{1}\|\mu\|_{2}(1+\|v_{1}\|_{2}+\|v_{2}\|_{2}+\|T_{2}\|_{2}+\|q_{2}\|_{2})$
$\displaystyle+c\|\vartheta\|_{1}\|\vartheta\|_{2}(\|q_{1}\|_{2}+\|v_{1}\|_{2})$
$\displaystyle+c\|\rho\|_{1}\|\rho\|_{2}(\|T_{2}\|_{2}+\|v_{1}\|_{2}+\|v_{2}\|_{2}),$

where $\lambda_{n}:=min\\{\lambda_{1,n},\lambda_{2,n},\lambda_{3,n}\\}.$
Obviously $0\leq\lambda_{n}\rightarrow\infty$ as $n\rightarrow\infty.$
Integration $(4.56)$ with respect to $t$ over $[0,T]$ yields

$\displaystyle\varphi(T)$ $\displaystyle\leq$ $\displaystyle
e^{-\lambda_{n}T}\varphi(0)+ce^{-\lambda_{n}T}\int_{0}^{T}e^{\lambda_{n}t}\psi(t)dt$
(4.57)
$\displaystyle+ce^{-\lambda_{n}T}\int_{0}^{T}e^{\lambda_{n}t}\psi(t)(\|v_{1}(t)\|_{2}+\|v_{2}(t)\|_{2})dt$
$\displaystyle+ce^{-\lambda_{n}T}\int_{0}^{T}e^{\lambda_{n}t}\psi(t)^{\frac{1}{2}}(\|\mu(t)\|_{2}+\|\vartheta(t)\|_{2}+\|\rho(t)\|_{2})$
$\displaystyle\cdot(\|v_{1}(t)\|_{2}+\|v_{2}(t)\|_{2}+\|T_{2}(t)\|_{2}+\|q_{1}(t)\|_{2}+\|q_{2}(t)\|_{2})dt$
$\displaystyle=:$ $\displaystyle\sum_{m=1}^{4}J_{m}.$

To verify that $\varphi(T)$ satisfies the condition $(ii)$ of Theorem $2.2,$
we need an estimate derived from [54] that

$\displaystyle\psi(t)+\int_{0}^{t}(\|\mu(t)\|_{2}^{2}+\|\vartheta(t)\|_{2}^{2}+\|\rho(t)\|_{2}^{2})dt\leq\gamma(t)\psi(0),$
(4.58)

where $\gamma(t),t\geq 0,$ is a positive continuous and non-decreasing
function independent of initial data. Therefore, the estimates of $J_{2}$
follows that

$\displaystyle J_{2}\leq
c\gamma(T)\psi(0)e^{-\lambda_{n}T}\int_{0}^{T}e^{\lambda_{n}t}dt\leq
c\gamma(T)\psi(0)\lambda_{n}^{-1}.$

Using $(4.58)$ again and Theorem $4.1$ we have

$\displaystyle J_{3}$ $\displaystyle\leq$ $\displaystyle
c\gamma(T)\psi(0)e^{-\lambda_{n}T}\int_{0}^{T}e^{\lambda_{n}t}(\|v_{1}\|_{2}+\|v_{2}\|_{2})dt$
$\displaystyle\leq$ $\displaystyle
c\gamma(T)\psi(0)\int_{0}^{T}(\|v_{1}\|_{2}+\|v_{2}\|_{2})dt$
$\displaystyle\leq$ $\displaystyle
c\gamma(T)\psi(0)T^{\frac{1}{2}}(\int_{0}^{T}(\|v_{1}\|_{2}^{2}+\|v_{2}\|_{2}^{2})dt)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle c\gamma(T)\psi(0)(T+T^{\frac{3}{4}}).$

Similarly, by $(4.58)$, Theorem $4.1$ and H$\mathrm{\ddot{o}}$lder inequality
we obtain the estimates of $J_{4}$

$\displaystyle J_{4}$ $\displaystyle\leq$ $\displaystyle
c\gamma(T)^{\frac{1}{2}}\psi(0)^{\frac{1}{2}}\Big{(}\int_{0}^{T}(\|\mu\|_{2}^{2}+\|\vartheta\|_{2}^{2}+\|\rho\|_{2}^{2})dt\Big{)}^{\frac{1}{2}}$
$\displaystyle\cdot\Big{(}\int_{0}^{T}(\|q_{1}\|_{2}^{2}+\|v_{1}\|_{2}^{2}+\|v_{2}\|_{2}^{2}+\|T_{2}\|_{2}^{2})dt\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
c\gamma(T)^{\frac{1}{2}}\psi(0)^{\frac{1}{2}}\gamma(T)^{\frac{1}{2}}\psi(0)^{\frac{1}{2}}(T^{\frac{1}{2}}+T)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
c\gamma(T)\psi(0)(T^{\frac{1}{2}}+T)^{\frac{1}{2}}.$

Summing $(4.57)$ and estimates of $J_{1}-J_{4}$, we get

$\displaystyle\varphi(T)\leq
c\psi(0)[e^{-\lambda_{n}T}+\gamma(T)\lambda_{n}^{-1}+\gamma(T)(T+T^{\frac{3}{4}})+\gamma(T)(T^{\frac{1}{4}}+T^{\frac{1}{2}})].$

Since $U_{i}(0)\in\mathcal{A}$ and $\mathcal{A}$ is bounded in $V$, $\psi(0)$
is bounded. We can choose $T$ which is small enough such that for
$\delta\in(0,1)$

$\displaystyle
c\gamma(T)(T+T^{\frac{3}{4}})+c\gamma(T)(T^{\frac{1}{4}}+T^{\frac{1}{2}})\leq\frac{\delta}{2}.$

Then let $\lambda_{n}$ be big enough such that

$\displaystyle
ce^{-\lambda_{n}T}+c\gamma(T)\lambda_{n}^{-1}\leq\frac{\delta}{2}.$

Consequently, for small $T$ and large $\lambda_{n},$ we have

$\displaystyle\varphi(T)\leq\delta\psi(0),$

which combined with Theorem 2.2 proves this theorem. $\square$ ∎

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