# Global Strong Solutions to the Compressible Magnetohydrodynamic Equations
with Slip Boundary Conditions in 3D Bounded Domains

Yazhou CHEN, Bin HUANG, Xiaoding SHI
College of Mathematics and Physics,
Beijing University of Chemical Technology, Beijing 100029, P. R. China

###### Abstract

We deal with the barotropic compressible magnetohydrodynamic equations in
three-dimensional (3D) bounded domain with slip boundary condition and vacuum.
By a series of a priori estimates, especially the boundary estimates, we prove
the global well-posedness of classical solution and the exponential decay rate
to the initial-boundary-value problem of this system for the regular initial
data with small energy but possibly large oscillations. The initial density of
such a classical solution is allowed to contain vacuum states. Moreover, it is
also shown that the oscillation of the density will grow unboundedly with an
exponential rate when the initial state contains vacuum.

Keywords: compressible magnetohydrodynamic equations; global classical
existence; priori estimates; slip boundary condition; vacuum.

AMS subject classifications: 35Q55, 35K65, 76N10, 76W05

##  1 Introduction

In this paper, we consider the viscous compressible magnetohydrodynamic (MHD)
equations for barotropic flows in a domain $\Omega\subset\mathbb{R}^{3}$,
which can be written as

$\begin{cases}\rho_{t}+\mathop{\mathrm{div}}\nolimits(\rho u)=0,\\\ (\rho
u)_{t}+\mathop{\mathrm{div}}\nolimits(\rho u\otimes u)+\nabla P=\mu\Delta
u+(\mu+\lambda)\nabla\mathop{\mathrm{div}}\nolimits u+(\nabla\times H)\times
H,\\\ H_{t}-\nabla\times(u\times H)=-\nu\nabla\times(\nabla\times H),\\\
\mathop{\mathrm{div}}\nolimits H=0,\end{cases}$ (1.1)

where $(x,t)\in\Omega\times(0,T]$, $t\geq 0$ is time, and
$x=(x_{1},x_{2},x_{3})$ is the spatial coordinate. The unknown functions
$\rho,u=(u^{1},u^{2},u^{3}),P=P(\rho),$ and $H=(H^{1},H^{2},H^{3})$ denote the
fluid density, velocity, pressure and magnetic field, respectively. Here we
consider the barotropic flows with $\gamma$-law pressure
$P(\rho)=a\rho^{\gamma},$ where $a>0$ and $\gamma>1$ are some physical
parameters. The constants $\mu$ and $\lambda$ are the shear viscosity and bulk
coefficients respectively satisfying the following physical restrictions
$\mu>0$ and $2\mu+{3}\lambda\geq 0$. The constant $\nu>0$ is the resistivity
coefficient which is inversely proportional to the electrical conductivity
constant and acts as the magnetic diffusivity of magnetic fields. In addition,
the system is solved subject to the given initial data

$\displaystyle\rho(x,0)=\rho_{0}(x),\quad\rho u(x,0)=\rho_{0}u_{0}(x),\quad
H(x,0)=H_{0}(x),\quad x\in\Omega,$ (1.2)

and slip boundary conditions

$\displaystyle u\cdot n=0,\,\,\,{\rm curl}u\times n=0,$
$\displaystyle\text{on}\,\,\,\partial\Omega,$ (1.3) $\displaystyle H\cdot
n=0,\,\,\,{\rm curl}H\times n=0,$
$\displaystyle\text{on}\,\,\,\partial\Omega,$ (1.4)

where $n$ is the unit outward normal vector to $\partial\Omega$.

The choice of boundary conditions is very important for hydrodynamics. For the
velocity field, one of the well-accepted choices is the no-slip boundary
condition (i.e. Dirichlet boundary condition), which has been successfully
applied to many hydrodynamical problems on the macroscopic scale. However,
experimental studies [47] reveal that for flows on the micro-andnanoscale, the
empirical non-slip boundary condition may break down, depending on the
interfacial roughness and in-terfacial interactions between solids and fluids.
Another choice is the slip boundary condition, which has different behaviors
between the macro-scopic scale and microscale. The earliest slip boundary
condition is proposed by Navier in [33], which indicates that there is a
stagnant layer of fluid close to the wall allowing a fluid to slip and the
slip velocity is proportional to the shear stress, that is

$\displaystyle\displaystyle u\cdot n=0,\,\,(D(u)n)_{\tau}+\vartheta
u_{\tau}=0\,\,\,\text{on}\,\,\,\partial\Omega,$ (1.5)

where $D(u)=(\nabla u+(\nabla u)^{*})/2$ is the shear stress, $\vartheta$ is a
scalar friction function, subscript $\tau$ denotes the tangential component on
$\partial\Omega$. With the development of micro/nano test technologies and
molecular-dynamic simulation technology, the Navier-type slip boundary has
been more concerned and well studied in numerical studies and analysis for
various fluid mechanical problems, see, for instance [6, 9, 16, 4, 25, 43, 41,
42] and the references therein. Additional, as shown in [6, 41], the Navier-
slip condition (1.5) is written to the following generalized one

$\displaystyle u\cdot n=0,\,\,\,{\rm curl}u\times n=-Au\,\,\,$
$\displaystyle\text{on}\,\,\,\partial\Omega,$ (1.6)

where $A$ is a smooth symmetric matrix defined on $\partial\Omega$, especially
when $A=0$, it is strongly related to (1.5). Hence, the boundary condition
(1.3) presented in this paper can be regarded as a Navier-type slip boundary
condition. For the magnetic field, the boundary condition (1.4) describes that
the boundary $\partial\Omega$ is a perfect conductor (see [14, 42]), that
means the magnetic field is confined inside and separated from the exterior.
We also observed that (1.4) is adaptable to the system since it ensured the
boundary balance of the quantities on the boundary. Therefore, it is
appropriate to consider the compressible MHD equations with the boundary
conditions (1.3)-(1.4).

The compressible MHD system (1.1) has been attracted a lot of attention of
physicists and mathematicians due to its physical importance and mathematical
challenges, including the strong coupling and interplay interaction between
fluid motion and magnetic field, and significant progress has been made in the
analysis of the well-posedness and dynamic behavior to the solutions of the
system, see, for example, [7, 8, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 26,
27, 30, 31, 32, 36, 37, 39, 40, 44, 45, 46] and their references. Among them,
we briefly review the results related to well-posedness of solutions for the
multi-dimensional compressible MHD equation. The local existence of strong
solutions to the compressible MHD equations was obtained by Vol’pert and
Hudjaev [37] for the Cauchy problem with large initial data and the initial
density being strictly positive. Fan and Yu [14] extended the result to the
case that the initial density may contain vacuum for the whole space or a
bounded domain with non-slip boundary condition. Lv and Huang [31] obtained
the local existence of strong and classical solutions in $\mathbb{R}^{2}$ with
vacuum as far field density. The global existence of solutions to the
compressible MHD equations has been studied in many works. Kawashima [26]
obtained the global existence of smooth solutions to the general electro-
magneto-fluid equations in two dimensions when the initial data are small
perturbations of a given constant state. Hu and Wang [18, 21] and Fan and Yu
[13] proved the global existence of renormalized solutions to the compressible
MHD equations for general large initial data. Recently, Li, Xu and Zhang [27]
established the global existence and uniqueness of classical solutions with
constant state as far field in $\mathbb{R}^{3}$ with large oscillations and
vacuum. Hong, Hou, Peng and Zhu [17] generalized the result for large initial
data when $\gamma-1$ and $\nu^{-1}$ are suitably small. Lv, Shi and Xu [32]
got the global existence of unqiue classical solutions in two-dimensional
space and obtained some better a priori decay with rates.

However, all of the above results only concern with the whole space or with
non-slip boundary conditions. It is rather complicated to investigate the
well-posedness and dynamical behaviors of the compressible MHD system with
slip boundary condition due to the compatibility issues of the nonlinear terms
with the slip boundary conditions. Tang and Gao [36] consider the local strong
solutions to the compressible MHD equations with initial vacuum, in which the
velocity field satisfies the Navier-slip condition. Considering the full
compressible MHD system, Xi and Hao [40] proved the local existence of the
classical solutions to the initial-boundary value problem with slip boundary
condition for the full compressible MHD system without thermal conductivity,
where the initial data contains vacuum and satisfies some initial layer
compatibility condition. However, to our best knowledge, whether the strong
(classical) solution for general bounded smooth domains $\Omega$ with density
containing vacuum initially to the MHD system exists globally in time is still
open. Recently, for the barotropic compressible Navier-Stokes equations in a
bounded domain $\Omega$ with slip boundary condition, Cai and Li [6] proved
that the classical solution of the initial-boundary-value problem exists
globally with vacuum and small energy but possibly large oscillations and
adopt some new techniques to obtain necessary a priori estimates, especially
the boundary estimates, compared with the work in [24, 29] for the Cauchy
problem of the compressible Navier-Stokes equations. The main purpose of this
paper is to establish the global well-posedness of classical solutions of the
compressible MHD system (1.1)-(1.4) in a bounded domain
$\Omega\subset\mathbb{R}^{3}$. We would like to obtain the time-independent
upper bound of the density and the time-dependent higher-norm estimates of
$(\rho,u,H)$ and extend the classical solution globally in time, which is
motivated by the works of Cai and Li [6] and Li, Xu and Zhang [27].

Before formulating our main result, we first explain the notation and
conventions used throughout the paper. For integer $k\geq 1$ and $1\leq
q<+\infty$, We denote the standard Sobolev space by $W^{k,q}(\Omega)$ and
$H^{k}(\Omega)\triangleq W^{k,2}(\Omega)$. For some $s\in(0,1)$, the
fractional Sobolev space $H^{s}(\Omega)$ is defined by

$H^{s}(\Omega)\triangleq\left\\{u\in
L^{2}(\Omega)~{}\text{:}\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^{2}}{|x-y|^{3+2s}}dxdy<+\infty\right\\},\,\,\text{with
the norm:}$
$\|u\|_{H^{s}(\Omega)}\triangleq\|u\|_{L^{2}(\Omega)}+\left(\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^{2}}{|x-y|^{3+2s}}dxdy\right)^{\frac{1}{2}}.$

For simplicity, we denote $L^{q}(\Omega)$, $W^{k,q}(\Omega)$, $H^{k}(\Omega)$
and ${H^{s}(\Omega)}$ by $L^{q}$, $W^{k,q}$, $H^{k}$ and ${H^{s}}$
respectively, and set

$\int fdx\triangleq\int_{\Omega}fdx,\quad\int_{0}^{T}\int
fdxdt\triangleq\int_{0}^{T}\int_{\Omega}fdxdt.$

For two $3\times 3$ matrices $A=\\{a_{ij}\\},\,\,B=\\{b_{ij}\\}$, the symbol
$A\colon B$ represents the trace of $AB^{*}$, where $B^{*}$ is the transpose
of $B$, that is,

$A\colon B\triangleq\text{tr}(AB^{*})=\sum\limits_{i,j=1}^{3}a_{ij}b_{ij}.$

Finally, for $v=(v^{1},v^{2},v^{3})$, we denote
$\nabla_{i}v\triangleq(\partial_{i}v^{1},\partial_{i}v^{2},\partial_{i}v^{3})$
for $i=1,2,3,$ and the material derivative of $v$ by $\dot{v}\triangleq
v_{t}+u\cdot\nabla v$.

The initial total energy of (1.1) is defined as

$\displaystyle\displaystyle
C_{0}=\int_{\Omega}\left(\frac{1}{2}\rho_{0}|u_{0}|^{2}+G(\rho_{0})+\frac{1}{2}|H_{0}|^{2}\right)dx.$
(1.7)

where

$\displaystyle\displaystyle
G(\rho)\triangleq\rho\int_{\bar{\rho}}^{\rho}\frac{P(s)-\bar{P}}{s^{2}}ds,\quad\bar{\rho}\triangleq\frac{1}{|\Omega|}\int_{\Omega}\rho_{0}dx,\quad\bar{P}\triangleq
P(\bar{\rho}).$ (1.8)

Now we can state our main result, Theorem 1.1, concerning existence of global
classical solutions to the problem (1.1)-(1.4).

###### Theorem 1.1

Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^{3}$ and its
smooth boundary $\partial\Omega$ has a finite number of 2-dimensional
connected components. For $q\in(3,6)$ and some given constants
$M_{1},M_{2}>0$, $s\in(\frac{1}{2},1]$, and $\hat{\rho}\geq\bar{\rho}+1$ , and
the initial data $(\rho_{0},u_{0},H_{0})$ satisfy the boundary conditions
(1.3)-(1.4) and

$\displaystyle 0\leq\rho_{0}\leq\hat{\rho},\quad(\rho_{0},P(\rho_{0}))\in
H^{2}\cap W^{2,q},$ (1.9) $\displaystyle u_{0}\in H^{2},\quad H_{0}\in
H^{2},\quad{\rm div}H_{0}=0,$ (1.10) $\displaystyle\|u_{0}\|_{H^{s}}\leq
M_{1},\quad\|H_{0}\|_{H^{s}}\leq M_{2},$ (1.11)

and the compatibility condition

$\displaystyle\displaystyle-\mu\triangle
u_{0}-(\mu+\lambda)\nabla\mathop{\mathrm{div}}\nolimits u_{0}+\nabla
P(\rho_{0})-(\nabla\times H_{0})\times H_{0}=\rho_{0}^{1/2}g,$ (1.12)

for some $g\in L^{2}.$ Then there exists a positive constant $\varepsilon$
depending only on $\mu$, $\lambda$, $\nu$, $\gamma$, $a$, $\bar{\rho}$,
$\hat{\rho}$, $s$, $\Omega$, $M_{1}$ and $M_{2}$ such that the system
(1.1)-(1.4) has a unique global classical solution $(\rho,u,H)$ in
$\Omega\times(0,\infty)$ satisfying

$\displaystyle\displaystyle 0\leq\rho(x,t)\leq
2\hat{\rho},\quad(x,t)\in\Omega\times(0,\infty),$ (1.13)
$\begin{cases}(\rho-\bar{\rho},P-\bar{P})\in C([0,\infty);H^{2}\cap
W^{2,q}),\\\ \nabla u\in C([0,\infty);H^{1})\cap L^{\infty}_{\rm
loc}(0,\infty;H^{2}\cap W^{2,q}),\\\ u_{t}\in L^{\infty}_{\rm
loc}(0,\infty;H^{2})\cap H^{1}_{\rm loc}(0,\infty;H^{1}),\\\ H\in
C([0,\infty);H^{2})\cap L^{\infty}_{\rm loc}(0,\infty;H^{4}),\\\ H_{t}\in
C([0,\infty);L^{2})\cap H^{1}_{\rm loc}(0,\infty;H^{1})\cap L^{\infty}_{\rm
loc}(0,\infty;H^{2}),\end{cases}$ (1.14)

provided the initial total energy $C_{0}\leq\varepsilon$. Moreover, for any
$r\in[1,\infty)$ and $p\in[1,6],$ there exist positive constants $C$ and
$\eta_{0}$ depending only on $\mu,$ $\lambda,$ $\nu,$ $\gamma,$ $a$, $s$,
$\bar{\rho}$, $\hat{\rho}$, $\Omega$, $M_{1},M_{2},$ $r$ and $p$ such that for
$t>0,$

$\displaystyle\displaystyle\|\rho(\cdot,t)-\bar{\rho}\|_{L^{r}}+\|u(\cdot,t)\|_{W^{1,p}}+\|(\sqrt{\rho}\dot{u})(\cdot,t)\|^{2}_{L^{2}}+\|H(\cdot,t)\|_{H^{2}}\leq
Ce^{-\eta_{0}t}.$ (1.15)

Then, thanks to the exponential decay rate (1.15), taking the similar
procedure as in [6, 28], we can directly deduce the following large-time
behavior of the gradient of the density when the initial density contains
vacuum state.

###### Theorem 1.2

Under the conditions of Theorem 1.1, assume further that there exists some
point $x_{0}\in\Omega$ such that $\rho_{0}(x_{0})=0.$ Then the unique global
classical solution $(\rho,u,H)$ to the problem (1.1)-(1.4) obtained in Theorem
1.1 satisfies that for any $r_{1}>3,$ there exist positive constants
$\tilde{C}_{1}$ and $\tilde{C}_{2}$ depending only on $\mu$, $\lambda$, $\nu$,
$\gamma$, $a$, $s$, $\bar{\rho}$, $\hat{\rho}$, $\Omega$, $M_{1}$, $M_{2}$ and
$r_{1}$ such that for any $t>0$,

$\displaystyle\displaystyle\|\nabla\rho(\cdot,t)\|_{L^{r_{1}}}\geq\tilde{C}_{1}e^{\tilde{C}_{2}t}.$
(1.16)

###### Remark 1.1

From Sobolev’s inequality and (1.14)1 with $q>3$, it follows that

$\displaystyle\rho,\nabla\rho\in C(\bar{\Omega}\times[0,T]).$ (1.17)

Moreover, it also follows from (1.14)2-5 that

$\displaystyle u,H,\nabla u,\nabla H,\nabla^{2}u,\nabla^{2}H,u_{t},H_{t}\in
C(\bar{\Omega}\times[\tau,T]),$ (1.18)

due to the following simple fact that

$L^{2}(\tau,T;H^{1})\cap H^{1}(\tau,T;H^{-1})\hookrightarrow
C([\tau,T];L^{2}).$

Finally, by (1.1)${}_{1},$ we have

$\displaystyle\rho_{t}=-u\cdot\nabla\rho-\rho{\rm div}u\in
C(\bar{\Omega}\times[\tau,T]).$ (1.19)

Hence the solution obtained in Theorem 1.1 becomes a classical one away from
the initial time.

###### Remark 1.2

When we consider the general slip boundary (1.6) for the velocity field, and
assume that the matrix $A$ is smooth and positive semi-definite, and even if
the restriction on $A$ is relaxed to $A\in H^{3}$ and the negative eigenvalues
of $A$ (if exist) are small enough, Theorem 1.1 will still hold. This can be
achieved by a similar way as in [6]. When $H=0$, i.e., there is no
electromagnetic field effect, the compressible MHD system (1.1) turns to be
the compressible Navier-Stokes equations, and Theorem 1.1 is the same as the
result of Cai and Li [6]. Roughly speaking, we generalize the results of [6]
to the compressible MHD equations.

###### Remark 1.3

When the initial state contains vacuum, Theorem 1.2 implies that the
oscillation of the density will grow unboundedly with an exponential rate,
which is somewhat surprisingly compared with the Cauchy problem [27] where
there is no results concerning the growth rate of the gradient of the density.

###### Remark 1.4

In our case, $\Omega$ is a bounded domain in $\mathbb{R}^{3}$, for the small
initial energy, we need the boundedness assumptions on the $H^{s}$-norm, for
$s\in(1/2,1]$, of the initial velocity and magnetic field, which is analogous
to the $\dot{H}^{\beta}$-norm in the whole space case [27, 29]. Thus, compared
with the results in [26], the conditions on the initial velocity may be
optimal under the smallness conditions on the initial energy.

We now sketch the main idea used in the proof of Theorem 1.1. Similar to the
argument in [6, 27], the key issue in our proof is to derive the time-
independent upper bound of the density in Proposition 3.1 and the time-
dependent higher-norm estimates of $(\rho,u,H)$. It is worth pointing out that
the effective viscous flux $F$ and the vorticity $\omega$ (see (2.5) for the
definition) play an important role in the proof. However, unlike [27], it can
not get the $L^{p}$-norm ($2\leq p\leq 6$) of $\nabla u$ by the standard
elliptic estimate, due to the bounded domain with the slip boundary condition
(1.3). To deal with this difficulty, we consider the Petrovsky type Lamé’s
system (A.5) (see [1]) and obtain the estimates of $W^{k,q}$-norm to the slip
boundary condition (see Lemma A.3). Thanks to [2, 38], Lemma A.4 shows that
the inequality $\|\nabla u\|_{L^{q}}\leq C(\|{\rm div}u\|_{L^{q}}+\|{\rm
curl}u\|_{L^{q}})\,\,\,\text{for any}\,\,\,q>1$ holds for $u\in W^{1,q}$ with
$u\cdot n=0$ on $\partial\Omega$. These fact allows us to control $\nabla u$
by means of ${\rm div}u$ and ${\rm curl}u$. For the magnetic field, with the
help of the magnetic diffusivity structure, we obtain the estimates of ${\rm
curl}H$ and ${\rm curl}^{2}H$ to control $\nabla H$ and $\nabla^{2}H$, which
can be used to deal with the strong coupling and interplay interaction between
the fluid motion and the magnetic field, such as the magnetic force
$(\nabla\times H)\times H$ and the convection term $\nabla\times(u\times H)$.
In addition, the slip boundary also makes the time-independent estimates of
$A_{1}(T)$ and $A_{2}(T)$ more difficult. Our observation is that due to the
boundary condition $u\cdot n=0$, which yields $u\cdot\nabla u\cdot
n=-u\cdot\nabla n\cdot u$. This equality is the key to estimate the integrals
on the boundary $\partial\Omega$ and we obtain the estimate of $\dot{u}$ and
$\nabla\dot{u}$ (see (2.34) and (2.35)).

The rest of the paper is organized as follows. In Section 2, we derive the
elementary energy estimates for the system (1.1)-(1.4) and some key a priori
estimates. Section 3 and Section 4 are devoted to deriving the necessary time-
independent lower-order estimates and time-dependent higher-order estimates,
which can guarantee the local classical solution to be a global classical one.
In Section 5, the proof of Theorem 1.1 will be completed. In Appendix A, we
list some elementary inequalities and important lemmas that we use intensively
in the paper.

##  2 Preliminaries

In this section, we derive the elementary energy estimates for the system
(1.1)-(1.4) and some key a priori estimates. Let $T>0$ be a fixed time and
$(\rho,u,H)$ be a smooth solution to (1.1)-(1.4) on $\Omega\times(0,T]$. For
$H$ and $u$ sufficiently smooth, there are some formulas based on ${\rm
div}H=0$:

$\begin{cases}(\nabla\times H)\times H={\rm div}(H\otimes
H-\frac{1}{2}|H|^{2}I_{3})=H\cdot\nabla H-\frac{1}{2}\nabla|H|^{2},\\\
\nabla\times(u\times H)=(H\cdot\nabla)u-(u\cdot\nabla)H-H{\rm
div}u.\end{cases}$ (2.1)

Then we rewrite (1.1) in the following form:

$\begin{cases}\rho_{t}+\mathop{\mathrm{div}}\nolimits(\rho u)=0,\\\ \rho
u_{t}+\rho u\cdot\nabla u-(\lambda\\!+\\!2\mu)\nabla{\rm
div}u+\mu\nabla\\!\times\\!\omega+\nabla(P\\!-\\!\bar{P})=H\\!\cdot\\!\nabla
H+\nabla\frac{|H|^{2}}{2},\\\ H_{t}+u\cdot\nabla H-H\cdot\nabla u+H{\rm
div}u=-\nu\nabla\times{\rm curl}H,\\\ \mathop{\mathrm{div}}\nolimits
H=0,\end{cases}$ (2.2)

where we used the fact $-\Delta u=-\nabla{\rm div}u+\nabla\times\omega$ and
$\omega\triangleq\nabla\times u,{\rm curl}H\triangleq\nabla\times H$.
Multiplying $\eqref{CMHD1}_{1}$ by $G^{\prime}(\rho)$, $\eqref{CMHD1}_{2}$ by
$u$ and $\eqref{CMHD1}_{3}$ by $H$ respectively, integrating by parts over
$\Omega$, summing them up, by (1.3) and (1.4), we have

$\displaystyle\left(\int\Big{(}G(\rho)+\frac{1}{2}\rho|u|^{2}+\frac{1}{2}|H|^{2}\Big{)}dx\right)_{t}+(\lambda+2\mu)\int({\rm
div}u)^{2}dx$ $\displaystyle+\mu\int|\omega|^{2}dx+\nu\int|{\rm
curl}H|^{2}dx=0,$ (2.3)

which, integrated over $(0,T)$, leads to the following elementary energy
estimates.

###### Lemma 2.1

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) on ${\Omega}\times(0,T]$.
Then

$\displaystyle\sup_{0\leq t\leq
T}\left(\frac{1}{2}\|\rho^{\frac{1}{2}}u\|_{L^{2}}^{2}+\|G(\rho)\|_{L^{1}}+\frac{1}{2}\|H\|_{L^{2}}^{2}\right)$
$\displaystyle+\int_{0}^{T}(\lambda+2\mu)\|{\rm
div}u\|_{L^{2}}^{2}+\mu\|\omega\|_{L^{2}}^{2}+\nu\|{\rm
curl}H\|_{L^{2}}^{2})dt\leq C_{0}.$ (2.4)

Next, similarly to the compressible Navier-Stokes equations, let us set

$\displaystyle
F\triangleq(\lambda+2\mu)\text{div}u-(P-\bar{P})-\frac{1}{2}|H|^{2},$ (2.5)

where $F$ denotes the effective viscous flux, which plays an important role in
our following analysis. For $F$, $\omega$, $\nabla u$ and $\nabla H$, we give
the following conclusion, which is a key to a priori estimates.

###### Lemma 2.2

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) on ${\Omega}\times(0,T]$.
Then for any $p\in[2,6],\,\,1<q<+\infty,$ there exists a positive constant $C$
depending only on $p$, $q$, $\mu$, $\lambda$ and $\Omega$ such that

$\displaystyle\|\nabla u\|_{L^{q}}\leq C(\|{\rm
div}u\|_{L^{q}}+\|\omega\|_{L^{q}}),$ (2.6) $\displaystyle\|\nabla
H\|_{L^{q}}\leq C\|{\rm curl}H\|_{L^{q}},$ (2.7) $\displaystyle\|\nabla
F\|_{L^{p}}\leq C(\|\rho\dot{u}\|_{L^{p}}+\|H\\!\cdot\nabla H\|_{L^{p}}),$
(2.8) $\displaystyle\|\nabla\omega\|_{L^{p}}\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|H\cdot\nabla H\|_{L^{p}}+\|\nabla u\|_{L^{2}}),$
(2.9) $\displaystyle\|F\|_{L^{p}}\leq$ $\displaystyle
C(\|\rho\dot{u}\|_{L^{2}}+\|H\\!\cdot\\!\nabla
H\|_{L^{2}})^{(3p-6)/(2p)}(\|\nabla
u\|_{L^{2}}+\|P\\!-\\!\bar{P}\|_{L^{2}}+\|H\|^{2}_{L^{4}})^{(6-p)/(2p)}$
$\displaystyle+C(\|\nabla
u\|_{L^{2}}+\|P\\!-\\!\bar{P}\|_{L^{2}}+\|H\|^{2}_{L^{4}}),$ (2.10)
$\displaystyle\|\omega\|_{L^{p}}\leq$ $\displaystyle
C(\|\rho\dot{u}\|_{L^{2}}+\|H\cdot\nabla H\|_{L^{2}})^{(3p-6)/(2p)}\|\nabla
u\|_{L^{2}}^{(6-p)/(2p)}+C\|\nabla u\|_{L^{2}},$ (2.11)

Moreover,

$\displaystyle\|\nabla u\|_{L^{p}}\leq$ $\displaystyle
C(\|\rho\dot{u}\|_{L^{2}}+\|H\\!\cdot\\!\nabla
H\|_{L^{2}})^{(3p-6)/(2p)}(\|\nabla
u\|_{L^{2}}+\|P\\!-\\!\bar{P}\|_{L^{2}}+\|H\|^{2}_{L^{4}})^{(6-p)/(2p)}$
$\displaystyle+C(\|\nabla
u\|_{L^{2}}+\|P\\!-\\!\bar{P}\|_{L^{p}}+\||H|^{2}\|_{L^{p}}).$ (2.12)

Proof. The inequality (2.6)-(2.7) is a direct result of Lemma A.4, since
$u\cdot n=0,H\cdot n=0$ on $\partial\Omega$. Moreover, noticing that (1.1)3
and $H\cdot n=0,{\rm curl}H\times n=0$ on $\partial\Omega$, by Lemma A.4-A.5,
for any integer $k\geq 1$, we obtain

$\displaystyle\|H\|_{W^{k+1,q}}\leq C\|{\rm curl}H\|_{W^{k,q}}\leq C(\|{\rm
curl}^{2}H\|_{W^{k-1,q}}+\|{\rm curl}H\|_{L^{p}}),$ (2.13)

where ${\rm curl}^{2}H\triangleq{\rm curl}{\rm curl}H$ and we have used the
fact ${\rm div}{\rm curl}H=0$.

By (1.1)2, (2.1) and the slip boundary condition (1.3), one can find that the
viscous flux $F$ satisfies

$\begin{cases}\Delta F={\rm div}(\rho\dot{u}-H\cdot\nabla
H)~{}~{}&in\,\,\Omega,\\\ \frac{\partial F}{\partial
n}=(\rho\dot{u}-H\cdot\nabla H)\cdot n\,\,&on\,\,\partial\Omega.\end{cases}$
(2.14)

It follows from Lemma 4.27 in [35] that

$\displaystyle\displaystyle\|\nabla F\|_{L^{q}}\leq
C(\|\rho\dot{u}\|_{L^{q}}+\|H\cdot\nabla H\|_{L^{q}}),$ (2.15)

which gives (2.8). Moreover, for any integer $k\geq 0$,

$\|\nabla\\!F\|_{W^{k+1,q}}\\!\leq\\!C(\|\rho\dot{u}\|_{L^{q}}\\!\\!+\\!\|H\\!\\!\cdot\\!\nabla\\!H\|_{L^{q}}\\!\\!+\\!\|\nabla\\!(\rho\dot{u})\|_{W^{k,q}}\\!\\!+\\!\|\nabla\\!(H\\!\\!\cdot\\!\nabla\\!H)\|_{W^{k,q}}).$
(2.16)

On the other hand, one can rewrite $\eqref{CMHD}_{2}$ as

$\displaystyle\mu\nabla\times\omega=\nabla F-\rho\dot{u}+H\cdot\nabla H.$
(2.17)

Noticing that $\omega\times n=0$ on $\partial\Omega$ and
$\mathop{\mathrm{div}}\nolimits\omega=0$, by Lemma A.5, we get

$\displaystyle\displaystyle\|\nabla\omega\|_{L^{q}}\leq
C(\|\nabla\times\omega\|_{L^{q}}+\|\omega\|_{L^{q}})\leq
C(\|\rho\dot{u}\|_{L^{q}}+\|H\cdot\nabla H\|_{L^{q}}+\|\omega\|_{L^{q}}),$
(2.18)

and for any integer $k\geq 0$,

$\displaystyle\quad\|\nabla\omega\|_{W^{k+1,q}}\leq
C(\|\nabla\times\omega\|_{W^{k+1,q}}+\|\omega\|_{L^{q}})$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{q}}+\|H\cdot\nabla
H\|_{L^{q}}+\|\nabla(\rho\dot{u})\|_{W^{k,q}}+\|\nabla(H\cdot\nabla
H)\|_{W^{k,q}}+\|\omega\|_{L^{q}}),$ (2.19)

where we have taken advantage of (2.15) and (2.16). By Sobolev’s inequality
and (2.18), for $p\in[2,6]$,

$\displaystyle\|\nabla\omega\|_{L^{p}}$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|H\cdot\nabla H\|_{L^{p}}+\|\omega\|_{L^{p}})$
$\displaystyle\leq C(\|\rho\dot{u}\|_{L^{p}}+\|H\cdot\nabla
H\|_{L^{p}}+\|\rho\dot{u}\|_{L^{2}}+\|H\cdot\nabla
H\|_{L^{2}}+\|\omega\|_{L^{2}})$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|H\cdot\nabla H\|_{L^{p}}+\|\nabla u\|_{L^{2}}),$
(2.20)

which implies (2.9).

Furthermore, one can deduce from (A.1) and (2.8) that for $p\in[2,6]$,

$\displaystyle\|F\|_{L^{p}}\leq$ $\displaystyle
C\|F\|_{L^{2}}^{(6-p)/(2p)}\|\nabla F\|_{L^{2}}^{(3p-6)/(2p)}+C\|F\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C(\|\rho\dot{u}\|_{L^{2}}+\|H\\!\cdot\\!\nabla
H\|_{L^{2}})^{(3p-6)/(2p)}(\|\nabla
u\|_{L^{2}}+\|P\\!-\\!\bar{P}\|_{L^{2}}+\|H\|^{2}_{L^{4}})^{(6-p)/(2p)}$
$\displaystyle+C(\|\nabla
u\|_{L^{2}}+\|P\\!-\\!\bar{P}\|_{L^{2}}+\|H\|^{2}_{L^{4}}),$ (2.21)

similarly, by (A.1) and (2.18),

$\displaystyle\|\omega\|_{L^{p}}\leq$ $\displaystyle
C\|\omega\|_{L^{2}}^{(6-p)/(2p)}\|\nabla\omega\|_{L^{2}}^{(3p-6)/(2p)}+C\|\omega\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle C(\|\rho\dot{u}\|_{L^{2}}+\|H\cdot\nabla
H\|_{L^{2}}+\|\nabla u\|_{L^{2}})^{(3p-6)/(2p)}\|\nabla
u\|_{L^{2}}^{(6-p)/(2p)}+C\|\nabla u\|_{L^{2}},$ $\displaystyle\leq$
$\displaystyle C(\|\rho\dot{u}\|_{L^{2}}+\|H\cdot\nabla
H\|_{L^{2}})^{(3p-6)/(2p)}\|\nabla u\|_{L^{2}}^{(6-p)/(2p)}+C\|\nabla
u\|_{L^{2}},$ (2.22)

and so (2.10)-(2.11) are established. By virtue of (2.6), (2.10), (2.11),
(2.8) and (2.18), it implies that (2.12) holds. This completes the proof.

###### Remark 2.1

It is easy to check that there exists a positive constant $C$ depending only
on $a,\gamma,\hat{\rho},\bar{\rho}$ such that

$\displaystyle\displaystyle C^{-1}(\rho-\bar{\rho})^{2}\leq G(\rho)\leq
C(\rho-\bar{\rho})^{2},$ (2.23)

which together with (2.4), (2.6) and (2.7) gives

$\displaystyle\displaystyle\sup_{0\leq t\leq
T}\|\rho-\bar{\rho}\|_{L^{2}}^{2}+\int_{0}^{T}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})dt\leq CC_{0}.$ (2.24)

###### Remark 2.2

From (2.13), we can get the estimate of $\|\nabla^{2}H\|_{L^{p}}$ and
$\|\nabla^{3}H\|_{L^{p}}$ for $p\in[2,6]$,

$\displaystyle\|\nabla^{2}H\|_{L^{p}}\leq C\|{\rm curl}H\|_{W^{1,p}}\leq
C(\|{\rm curl}^{2}H\|_{L^{p}}+\|{\rm curl}H\|_{L^{p}}),$ (2.25)

and

$\displaystyle\|\nabla^{3}H\|_{L^{p}}\leq C\|{\rm curl}H\|_{W^{2,p}}\leq
C(\|{\rm curl}^{2}H\|_{W^{1,p}}+\|{\rm curl}H\|_{L^{p}}).$ (2.26)

On the other hand, we can get the estimates of $\|\nabla^{2}u\|_{L^{p}}$ and
$\|\nabla^{3}u\|_{L^{p}}$ for $p\in[2,6]$ by Lemma A.4, which will be devoted
to giving higher order estimates in Section 4. In fact, by Lemma A.4, (2.15)
and (2.16), for $p\in[2,6]$,

$\displaystyle\quad\|\nabla^{2}u\|_{L^{p}}\leq C(\|{\rm
div}u\|_{W^{1,p}}+\|\omega\|_{W^{1,p}})$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|H\cdot\nabla H\|_{L^{p}}+\|\nabla
P\|_{L^{p}}+\|P-\bar{P}\|_{L^{p}}$
$\displaystyle\quad+\||H|^{2}\|_{L^{p}}+\|\nabla H\cdot H\|_{L^{p}}+\|\nabla
u\|_{L^{2}}),$ (2.27)

and

$\displaystyle\quad\|\nabla^{3}u\|_{L^{p}}\leq C(\|{\rm
div}u\|_{W^{2,p}}+\|\omega\|_{W^{2,p}})$ $\displaystyle\leq
C(\|\nabla(\rho\dot{u})\|_{L^{p}}+\|\nabla(H\cdot\nabla
H)\|_{L^{p}}+\|\nabla^{2}P\|_{L^{p}}+\|\nabla(\nabla H\cdot
H)\|_{L^{p}}+\|H\cdot\nabla H\|_{L^{p}}$
$\displaystyle\quad+\|\rho\dot{u}\|_{L^{p}}+\|\nabla
P\|_{L^{p}}+\|P-\bar{P}\|_{L^{p}}+\||H|^{2}\|_{L^{p}}+\|\nabla H\cdot
H\|_{L^{p}}+\|\nabla u\|_{L^{2}}).$ (2.28)

The lemma below gives an a priori estimate on the
$L^{2}(\Omega\times(0,T))$-norm of $\rho-\bar{\rho}.$

###### Lemma 2.3

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) on ${\Omega}\times(0,T]$.
Then there exists a positive constant $C$ depending only on $p$, $q$, $\mu$,
$\lambda$, $\nu$ and $\Omega$ such that

$\displaystyle\displaystyle\int_{0}^{T}\int(\rho-\bar{\rho})^{2}dxdt\leq
CC_{0}.$ (2.29)

Proof. From Lemma A.7, multiplying $\eqref{CMHD}_{2}$ by
$\mathcal{B}[\rho-\bar{\rho}]$ and integrating over $\Omega,$ one has

$\displaystyle\int(P-\bar{P})(\rho-\bar{\rho})dx$ $\displaystyle=$
$\displaystyle\left(\int\rho
u\cdot\mathcal{B}[\rho-\bar{\rho}]dx\right)_{t}-\int\rho
u\cdot\nabla\mathcal{B}[\rho-\bar{\rho}]\cdot udx-\int\rho
u\cdot\mathcal{B}[\rho_{t}]dx$ $\displaystyle+\mu\int\nabla
u\cdot\nabla\mathcal{B}[\rho-\bar{\rho}]dx+(\lambda+\mu)\int(\rho-\bar{\rho}){\rm
div}udx$ $\displaystyle-\int H\cdot\nabla\mathcal{B}[\rho-\bar{\rho}]\cdot
Hdx-\int(\rho-\bar{\rho})|H|^{2}/2dx$ $\displaystyle\leq$
$\displaystyle\left(\int\rho
u\cdot\mathcal{B}[\rho-\bar{\rho}]dx\right)_{t}+C\|\rho^{\frac{1}{2}}u\|_{L^{4}}^{2}\|\rho-\bar{\rho}\|_{L^{2}}+C\|\rho
u\|_{L^{2}}^{2}$ $\displaystyle+C\|\rho-\bar{\rho}\|_{L^{2}}\|\nabla
u\|_{L^{2}}+C\|\rho-\bar{\rho}\|_{L^{2}}\|H\|_{L^{4}}^{2}$ $\displaystyle\leq$
$\displaystyle\left(\int\rho
u\cdot\mathcal{B}[\rho-\bar{\rho}]dx\right)_{t}+\delta\|\rho-\bar{\rho}\|_{L^{2}}^{2}+C(\delta)(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2}),$ (2.30)

integrating it over $(0,T]$ together with (2.24) gives

$\displaystyle\displaystyle\int_{0}^{T}\int(\rho-\bar{\rho})^{2}dxdt$
$\displaystyle\leq C\Big{(}\sup_{0\leq t\leq
T}(\|\rho^{\frac{1}{2}}u\|_{L^{2}}^{2}+\|\rho-\bar{\rho}\|_{L^{2}}^{2})+\int_{0}^{T}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})dt\Big{)}$ $\displaystyle\leq
CC_{0},$ (2.31)

and we finish the proof.

Next we need the estimates on the material derivative of $u$. Since $u\cdot
n=0$ on $\partial\Omega$, it follows that

$\displaystyle\displaystyle u\cdot\nabla u\cdot n=-u\cdot\nabla n\cdot u,$
(2.32)

which implies

$\displaystyle\displaystyle(\dot{u}+(u\cdot\nabla n)\times u^{\perp})\cdot
n=0\mbox{ on }\partial\Omega,$ (2.33)

where $u^{\perp}\triangleq-u\times n$ on $\partial\Omega$. We review the
following Poincare-type inequality of $\dot{u}$, which depends on this
observation (see [6], Lemma 3.2).

###### Lemma 2.4

If $(\rho,u,H)$ is a smooth solution of (1.1) with slip condition (1.3)-(1.4),
then there exists a positive constant $C$ depending only on $\Omega$ such that

$\displaystyle\|\dot{u}\|_{L^{6}}\leq C(\|\nabla\dot{u}\|_{L^{2}}+\|\nabla
u\|_{L^{2}}^{2}),$ (2.34) $\displaystyle\|\nabla\dot{u}\|_{L^{2}}\leq C(\|{\rm
div}\dot{u}\|_{L^{2}}+\|{\rm curl}\dot{u}\|_{L^{2}}+\|\nabla
u\|_{L^{4}}^{2}).$ (2.35)

To this end, we recall the following local existence theorem of classical
solution of (1.1)-(1.4), which can be proved in a similar manner as that in
[36, 40], base on the standard contraction mapping principle.

###### Lemma 2.5

Assume that the initial date $(\rho_{0},u_{0},H_{0})$ satisfy the conditions
(1.9), (1.10) and (1.12). Then there exist a positive time $T_{0}>0$ and a
unique classical solution $(\rho,u,H)$ of the system (1.1)-(1.4) in
$\Omega\times(0,T_{0}]$, satisfying that $\rho\geq 0$, and that for
$\tau\in(0,T_{0})$,

$\begin{cases}(\rho-\bar{\rho},P-\bar{P})\in C([0,T_{0});H^{2}\cap
W^{2,q}),\\\ \nabla u\in C([0,T_{0});H^{1})\cap
L^{\infty}(\tau,T_{0};H^{2}\cap W^{2,q}),\\\ u_{t}\in
L^{\infty}(\tau,T_{0};D^{1}\cap H^{2})\cap H^{1}(\tau,T_{0};H^{1}),\\\ H\in
C([0,T_{0});H^{2})\cap L^{\infty}(\tau,T_{0};H^{4}),\\\ H_{t}\in
C([0,T_{0});L^{2})\cap H^{1}(\tau,T_{0};H^{1})\cap
L^{\infty}(\tau,T_{0};H^{2}).\end{cases}$ (2.36)

##  3  A priori estimates(I): lower order estimates

In this section, we will establish the time-independent a priori bounds of the
solutions of the problem (1.1)-(1.4). Let $T>0$ be a fixed time and
$(\rho,u,H)$ be a smooth solution to (1.1)-(1.4) on $\Omega\times(0,T]$ with
smooth initial data $(\rho_{0},u_{0},H_{0})$ satisfying $u_{0}\in
H^{s},H_{0}\in H^{s}$ for some $s\in(\frac{1}{2},1]$ and $0\leq\rho_{0}\leq
2\hat{\rho}$.

Set $\sigma=\sigma(t)\triangleq\min\\{1,t\\},$ we define

$\displaystyle A_{1}(T)\triangleq\sup_{0\leq t\leq T}\sigma\left(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2}\right)$
$\displaystyle\qquad\qquad+\int_{0}^{T}\sigma\int\left(\rho|\dot{u}|^{2}+|{\rm
curl}^{2}H|^{2}+|H_{t}|^{2}\right)dxdt,$ (3.1) $\displaystyle
A_{2}(T)\triangleq\sup_{0\leq t\leq
T}\sigma^{2}\int\left(\rho|\dot{u}|^{2}+|{\rm
curl}^{2}H|^{2}+|H_{t}|^{2}\right)dx$
$\displaystyle\qquad\qquad+\int_{0}^{T}\int\sigma^{2}\left(|\nabla\dot{u}|^{2}+|\nabla
H_{t}|^{2}\right)dxdt,$ (3.2) $\displaystyle A_{3}(T)\triangleq\sup_{0\leq
t\leq T}\int|H|^{3}dx,$ (3.3) $\displaystyle A_{4}(T)\triangleq\sup_{0\leq
t\leq T}\sigma^{\frac{3-2s}{4}}\left(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}\right)$
$\displaystyle\qquad\qquad+\int_{0}^{T}\sigma^{\frac{3-2s}{4}}\int\left(\rho|\dot{u}|^{2}+|{\rm
curl}^{2}H|^{2}+|H_{t}|^{2}\right)dxdt,$ (3.4) $\displaystyle
A_{5}(T)\triangleq\sup_{0\leq t\leq T}\int\rho|u|^{3}dx,$ (3.5)

where $s\in(\frac{1}{2},1]$ and $\dot{v}=v_{t}+u\cdot\nabla v$ is the material
derivative.

Now we will give the following key a priori estimates in this section, which
guarantees the existence of a global classical solution of (1.1)–(1.4).

###### Proposition 3.1

Under the conditions of Theorem 1.1, for
$\delta_{0}\triangleq\frac{2s-1}{9s}\in(0,\frac{1}{9}]$, there exists a
positive constant $\varepsilon$ depending on $\mu$, $\lambda$, $\nu$, $a$,
$\gamma$, $\bar{\rho}$, $\hat{\rho}$, $s$, $\Omega$, $M_{1}$ and $M_{2}$ such
that if $(\rho,u,H)$ is a smooth solution of (1.1)-(1.4) on
$\Omega\times(0,T]$ satisfying

$\begin{cases}\sup\limits_{\Omega\times[0,T]}\rho\leq 2\hat{\rho},\quad
A_{1}(T)+A_{2}(T)\leq 2C_{0}^{1/2},\\\ A_{3}(T)\leq 2C_{0}^{\delta_{0}},\quad
A_{4}(\sigma(T))+A_{5}(\sigma(T))\leq 2C_{0}^{\delta_{0}},\end{cases}$ (3.6)

then the following estimates hold

$\begin{cases}\sup\limits_{\Omega\times[0,T]}\rho\leq 7\hat{\rho}/4,\quad
A_{1}(T)+A_{2}(T)\leq C_{0}^{1/2},\\\ A_{3}(T)\leq C_{0}^{\delta_{0}},\quad
A_{4}(\sigma(T))+A_{5}(\sigma(T))\leq C_{0}^{\delta_{0}},\end{cases}$ (3.7)

provided $C_{0}\leq\varepsilon.$

Proof. Proposition 3.1 is a consequence of the following Lemmas 3.3, 3.7-3.9
below.

In the following, we will use the convention that $C$ denotes a generic
positive constant depending on
$\mu,\lambda,\nu,\gamma,a,\bar{\rho},\hat{\rho},s,\Omega$, $M_{1}$ and $M_{2}$
and use $C(\alpha)$ to emphasize that $C$ depends on $\alpha$. We begin with
the following standard energy estimate for $(\nabla u,\nabla H)$.

###### Lemma 3.2

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) satisfying (3.6). Then
there is a positive constant $C$ such that

$\displaystyle\displaystyle\int_{0}^{T}(\|\nabla u\|_{L^{2}}^{4}+\|\nabla
H\|_{L^{2}}^{4})dt\leq CC_{0}^{2\delta_{0}},$ (3.8)

provided $C_{0}\leq 1$.

Proof. By (3.6), together with (2.4), (2.24), we have

$\displaystyle\int_{0}^{T}(\|\nabla u\|_{L^{2}}^{4}+\|\nabla
H\|_{L^{2}}^{4})dt$ $\displaystyle\leq$ $\displaystyle
C\int_{0}^{\sigma(T)}(\|\nabla u\|_{L^{2}}^{4}+\|\nabla
H\|_{L^{2}}^{4})dt+\int_{\sigma(T)}^{T}(\|\nabla u\|_{L^{2}}^{4}+\|\nabla
H\|_{L^{2}}^{4})dt$ $\displaystyle\leq$ $\displaystyle C\sup_{0\leq
t\leq\sigma(T)}\left(\sigma^{\frac{3-2s}{4}}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})\right)^{2}\int_{0}^{\sigma(T)}\sigma^{\frac{2s-3}{2}}dt$
$\displaystyle+C\sup_{\sigma(T)\leq t\leq T}\sigma\left(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2}\right)\int_{\sigma(T)}^{T}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})dt$ $\displaystyle\leq$
$\displaystyle CC_{0}^{2\delta_{0}}+CC_{0}^{3/2}~{}\leq CC_{0}^{2\delta_{0}},$
(3.9)

since $s\in(1/2,1]$ and $\delta_{0}\in(0,1/9]$. The proof of Lemma 2.1 is
completed.

Now, we give the estimate of $A_{3}(T)$.

###### Lemma 3.3

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) satisfying (3.6). Then
there is a positive constant $\varepsilon_{1}>0$, depending on $\mu,$
$\lambda,$ $\nu,$ $a$, $\gamma$, $\bar{\rho}$, $\hat{\rho}$ and $\Omega$ such
that

$\displaystyle\displaystyle A_{3}(T)\leq C_{0}^{\delta_{0}},$ (3.10)

provided $C_{0}\leq\varepsilon_{1}$.

Proof. Multiplying (2.2)3 by $3|H|H$ and integrating by parts over $\Omega$,
we have

$\displaystyle\frac{d}{dt}\|H\|^{3}_{L^{3}}=$ $\displaystyle-3\nu\int{\rm
curl}H\cdot{\rm curl}(|H|H)dx+3\int|H|H\cdot\nabla u\cdot Hdx-2\int|H|^{3}{\rm
div}udx$ $\displaystyle\leq$ $\displaystyle C\|H\|_{L^{\infty}}\|\nabla
H\|_{L^{2}}^{2}+C\|\nabla u\|_{L^{2}}\|H\|_{L^{6}}^{3}$ $\displaystyle\leq$
$\displaystyle C\|\nabla H\|^{5/2}_{L^{2}}\|{\rm
curl}^{2}H\|_{L^{2}}^{1/2}+C\|\nabla H\|^{2}_{L^{2}}+C\|\nabla
H\|^{4}_{L^{2}}+C\|\nabla u\|_{L^{2}}^{4},$ (3.11)

which together with (3.6) and (3.8) indicates that

$\displaystyle\displaystyle\sup_{0\leq t\leq T}\|H\|_{L^{3}}^{3}$
$\displaystyle\leq$ $\displaystyle\|H_{0}\|_{L^{3}}^{3}+C\int_{0}^{T}\|\nabla
H\|^{5/2}_{L^{2}}\|{\rm
curl}^{2}H\|_{L^{2}}^{1/2}dt+CC_{0}+CC_{0}^{2\delta_{0}}$ $\displaystyle\leq$
$\displaystyle\|H_{0}\|_{L^{3}}^{3}+C\int_{0}^{\sigma(T)}\left(\sigma^{\frac{3-2s}{4}}\|\nabla
H\|_{L^{2}}^{2}\right)^{5/4}\left(\sigma^{\frac{3-2s}{4}}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}\right)^{1/4}\sigma^{\frac{3(2s-3)}{8}}dt$
$\displaystyle+C\|\nabla H\|_{L^{2}}\int_{\sigma(T)}^{T}\left(\|\nabla
H\|_{L^{2}}^{2}\right)^{3/4}\left(\sigma\|{\rm
curl}^{2}H\|_{L^{2}}^{2}\right)^{1/4}dt+CC_{0}+CC_{0}^{2\delta_{0}}$
$\displaystyle\leq$ $\displaystyle C_{1}C_{0}^{3\delta_{0}/2}$ (3.12)

where in the last inequality we have used the simple fact

$\displaystyle\displaystyle\|H_{0}\|_{L^{3}}^{3}\leq
C\|H_{0}\|_{L^{2}}^{\frac{3(2s-1)}{2s}}\|H_{0}\|_{H^{s}}^{\frac{3}{2s}}\leq
C(M_{2})C_{0}^{27\delta_{0}/4}.$ (3.13)

Thus it follows from (3.13) that (3.10) holds provided
$C_{0}\leq\varepsilon_{1}\triangleq\min\\{1,C_{1}^{-\frac{2}{\delta_{0}}}\\}.$
The proof of 3.3 is completed.

The following lemma shows the preliminary $L^{2}$ bounds for $\nabla H$.

###### Lemma 3.4

Let $(\rho,u,H)$ be a smooth solution of (1.1)–(1.4) satisfying (3.6). Then
there is a positive constant $C$ such that

$\displaystyle\displaystyle\sup_{0\leq t\leq T}\left(\sigma\|\nabla
H\|_{L^{2}}^{2}\right)+\int_{0}^{T}\sigma\left(\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2}\right)dt\leq CC_{0}.$ (3.14)

Moreover, for any $\theta\in[0,1]$, one has

$\displaystyle\displaystyle\sup_{0\leq t\leq T}\left(\sigma^{1-\theta}\|\nabla
H\|_{L^{2}}^{2}\right)+\int_{0}^{T}\sigma^{1-\theta}\left(\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2}\right)dt\leq
C\|H_{0}\|_{H^{\theta}}^{2}.$ (3.15)

Proof. Multiplying (2.2)3 by $H$ and integrating by parts over $\Omega$, by
(1.4), (A.1) and (2.7), we have

$\displaystyle\displaystyle\left(\frac{1}{2}\|H\|_{L^{2}}^{2}\right)_{t}+\nu\|{\rm
curl}H\|_{L^{2}}^{2}\leq\|\nabla
u\|_{L^{2}}\|H\|^{2}_{L^{4}}\leq\frac{\nu}{2}\|{\rm
curl}H\|_{L^{2}}^{2}+C\|\nabla u\|^{4}_{L^{2}}\|H\|^{2}_{L^{2}},$ (3.16)

which together with (2.7), (3.8) and Gronwall inequality gives

$\displaystyle\displaystyle\sup_{0\leq t\leq
T}\|H\|_{L^{2}}^{2}+\int_{0}^{T}\|\nabla H\|_{L^{2}}^{2}dt\leq
C\|H_{0}\|^{2}_{L^{2}}.$ (3.17)

By Lemma 2.2, one easily deduces from (2.2)3 and (1.4) that

$\displaystyle\left(\frac{\nu}{2}\|{\rm
curl}H\|_{L^{2}}^{2}\right)_{t}+\nu^{2}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|^{2}_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\int|H\cdot\nabla u-u\cdot\nabla H-H{\rm div}u|^{2}dx$
$\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{2}_{L^{2}}\|H\|^{2}_{L^{\infty}}+C\|u\|^{2}_{L^{6}}\|\nabla
H\|^{2}_{L^{3}}$ $\displaystyle\leq$ $\displaystyle C\|\nabla
u\|^{2}_{L^{2}}\|\nabla H\|_{L^{2}}\|{\rm curl}^{2}H\|_{L^{2}}+C\|\nabla
u\|^{2}_{L^{2}}\|\nabla H\|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{\nu^{2}}{2}\|{\rm curl}^{2}H\|^{2}_{L^{2}}+C(\|\nabla
u\|^{2}_{L^{2}}+\|\nabla u\|^{4}_{L^{2}})\|\nabla H\|_{L^{2}}^{2},$ (3.18)

using (2.7), (3.8) and Gronwall inequality, we get

$\displaystyle\displaystyle\sup_{0\leq t\leq T}\|\nabla
H\|_{L^{2}}^{2}+\int_{0}^{T}\left(\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2}\right)dt\leq C\|\nabla
H_{0}\|^{2}_{L^{2}}.$ (3.19)

On the other hand, multiplying (3) by $\sigma$ and integrating it over
$(0,T)$, by (3.8) and (3.17), we obtain

$\displaystyle\displaystyle\sup_{0\leq t\leq T}\left(\sigma\|\nabla
H\|_{L^{2}}^{2}\right)+\int_{0}^{T}\sigma\left(\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2}\right)dt\leq
C\|H_{0}\|^{2}_{L^{2}}\leq CC_{0}.$ (3.20)

This finishes the proof of (3.14). Note that for fixed $u$ (smooth), the
solution operator $H_{0}\mapsto H(\cdot,t)$ is linear, by the standard Stein-
Weiss interpolation argument [5], one can deduce from (3.19) and (3.20) that
(3.15) holds for any $\theta\in[0,1]$. The proof of Lemma 3.4 is completed.

Then, we give the estimate of $A_{1}(T)$ and $A_{2}(T)$.

###### Lemma 3.5

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) satisfying (3.6). Then
there is a positive constant $\varepsilon_{2}$ depending only on $\mu$,
$\lambda$, $\nu$, $a$, $\gamma$, $\bar{\rho}$, $\hat{\rho}$ and $\Omega$ such
that

$\displaystyle\displaystyle A_{1}(T)\leq CC_{0}+C\int_{0}^{T}\int\sigma|\nabla
u|^{3}dxdt,$ (3.21) $\displaystyle\displaystyle A_{2}(T)\leq
CC_{0}^{1/2+2\delta_{0}/3}+CA_{1}(T)+C\int_{0}^{T}\int\sigma^{2}|\nabla
u|^{4}dxdt,$ (3.22)

provided $C_{0}\leq\varepsilon_{2}$.

Proof. Let $m\geq 0$ be a real number which will be determined later.
Multiplying $\eqref{CMHD}_{2}$ by $\sigma^{m}\dot{u}$ and then integrating the
reslting equality over $\Omega$ lead to

$\displaystyle\int\sigma^{m}\rho|\dot{u}|^{2}dx$
$\displaystyle=-\int\sigma^{m}\dot{u}\cdot\nabla
Pdx+(\lambda+2\mu)\int\sigma^{m}\nabla{\rm div}u\cdot\dot{u}dx$
$\displaystyle\quad-\mu\int\sigma^{m}\nabla\times\omega\cdot\dot{u}dx+\int\sigma^{m}(H\cdot\nabla
H-\nabla|H|^{2}/2)\cdot\dot{u}dx$ $\displaystyle\triangleq
I_{1}+I_{2}+I_{3}+I_{4}.$ (3.23)

We neglect the calculation details similar to that in [6] and only show the
part related to the magnetic field and the boundary terms. Firstly, By
$\eqref{CMHD}_{1}$ and Lemma 2.2, a direct calculation gives

$\displaystyle I_{1}=$
$\displaystyle-\int\sigma^{m}u_{t}\cdot\nabla(P-\bar{P})dx-\int\sigma^{m}u\cdot\nabla
u\cdot\nabla Pdx$ $\displaystyle=$
$\displaystyle\left(\int\sigma^{m}(P-\bar{P})\,{\rm
div}u\,dx\right)_{t}-m\sigma^{m-1}\sigma^{\prime}\int(P-\bar{P})\,{\rm
div}u\,dx$ $\displaystyle+\int\sigma^{m}P\nabla u:\nabla
udx+(\gamma-1)\int\sigma^{m}P({\rm
div}u)^{2}dx-\int_{\partial\Omega}\sigma^{m}Pu\cdot\nabla u\cdot nds$
$\displaystyle\leq$ $\displaystyle\left(\int\sigma^{m}(P-\bar{P})\,{\rm
div}u\,dx\right)_{t}+C\|\nabla
u\|_{L^{2}}^{2}+C\|\rho-\bar{\rho}\|_{L^{2}}^{2}.$ (3.24)

Similarly, by (2.32), it indicates that

$\displaystyle I_{2}$
$\displaystyle=(\lambda+2\mu)\int_{\partial\Omega}\sigma^{m}{\rm
div}u\,(\dot{u}\cdot n)ds-(\lambda+2\mu)\int\sigma^{m}{\rm div}u\,{\rm
div}\dot{u}dx$
$\displaystyle=(\lambda+2\mu)\int_{\partial\Omega}\sigma^{m}{\rm
div}u\,(u\cdot\nabla u\cdot
n)ds-\frac{\lambda+2\mu}{2}\left(\int\sigma^{m}({\rm div}u)^{2}dx\right)_{t}$
$\displaystyle\quad+\frac{\lambda+2\mu}{2}\int\sigma^{m}({\rm
div}u)^{3}dx-(\lambda+2\mu)\int\sigma^{m}{\rm div}u\,\nabla u:\nabla udx$
$\displaystyle\quad+\frac{m(\lambda+2\mu)}{2}\sigma^{m-1}\sigma^{\prime}\int({\rm
div}u)^{2}dx$ (3.25)

For the first term on the righthand side of (3), we obtain

$\displaystyle(\lambda+2\mu)\int_{\partial\Omega}\sigma^{m}{\rm
div}u\,(u\cdot\nabla u\cdot n)ds$ $\displaystyle=$
$\displaystyle-\int_{\partial\Omega}\sigma^{m}Fu\cdot\nabla n\cdot
uds-\int_{\partial\Omega}\sigma^{m}(P-\bar{P})u\cdot\nabla n\cdot
uds-\int_{\partial\Omega}\sigma^{m}\frac{|H|^{2}}{2}u\cdot\nabla n\cdot uds$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{\partial\Omega}\sigma^{m}|F||u|^{2}ds+\int_{\partial\Omega}\sigma^{m}|u|^{2}ds+\int_{\partial\Omega}\sigma^{m}|H|^{2}|u|^{2}ds\right)$
$\displaystyle\leq$ $\displaystyle C\sigma^{m}(\|\nabla
F\|_{L^{2}}\|u\|_{L^{4}}^{2}+\|F\|_{L^{6}}\|u\|_{L^{3}}\|\nabla
u\|_{L^{2}}+\|F\|_{L^{2}}\|u\|^{2}_{L^{4}})+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}$ $\displaystyle+C\sigma^{m}(\|\nabla
H\|_{L^{2}}\|H\|_{L^{6}}\|u\|_{L^{6}}^{2}+\|H\|_{L^{6}}^{2}\|u\|_{L^{6}}\|\nabla
u\|_{L^{2}}^{2}+\|H\|_{L^{4}}^{2}\|u\|_{L^{4}}^{2})$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\sigma^{m}\|\rho\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+C\sigma^{m}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})(\|\nabla u\|_{L^{2}}^{2}+1),$ (3.26)

where we have used

$\displaystyle\displaystyle\|H\cdot\nabla H\|_{L^{2}}\leq
C\|H\|_{L^{3}}\|\nabla H\|_{L^{6}}\leq CC_{0}^{\delta_{0}/3}(\|{\rm
curl}^{2}H\|_{L^{2}}+\|\nabla H\|_{L^{2}}).$ (3.27)

Therefore,

$\displaystyle\displaystyle I_{2}\leq$
$\displaystyle-\frac{\lambda+2\mu}{2}\left(\int\sigma^{m}({\rm
div}u)^{2}dx\right)_{t}+C\sigma^{m}\|\nabla
u\|_{L^{3}}^{3}+\frac{1}{2}\sigma^{m}\|\rho\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}$ $\displaystyle+C\sigma^{m}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})\|\nabla u\|_{L^{2}}^{2}+C(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2}).$ (3.28)

Next, by (1.3), a straightforward computation shows that

$\displaystyle\displaystyle I_{3}$
$\displaystyle=-\frac{\mu}{2}\left(\int\sigma^{m}|\omega|^{2}dx\right)_{t}+\frac{\mu
m}{2}\sigma^{m-1}\sigma^{\prime}\int|\omega|^{2}dx$
$\displaystyle\quad-\mu\int\sigma^{m}(\nabla
u^{i}\times\nabla_{i}u)\cdot\omega
dx+\frac{\mu}{2}\int\sigma^{m}|\omega|^{2}\,{\rm div}udx$
$\displaystyle\leq-\frac{\mu}{2}\left(\int\sigma^{m}|\omega|^{2}dx\right)_{t}+Cm\sigma^{m-1}\sigma^{\prime}\|\nabla
u\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla u\|_{L^{3}}^{3}.$ (3.29)

Finally, by (1.3), a direct calculation yields

$\displaystyle I_{4}=$ $\displaystyle\left(\int\sigma^{m}(H\cdot\nabla
H-\nabla|H|^{2}/2)\cdot
udx\right)_{t}-m\sigma^{m-1}\sigma^{\prime}\int(H\cdot\nabla
H-\nabla|H|^{2}/2)\cdot udx$ $\displaystyle+\int\sigma^{m}\big{(}(H\otimes
H)_{t}:\nabla u-(|H|^{2}/2)_{t}{\rm div}u\big{)}dx+\int\sigma^{m}(H\cdot\nabla
H-\nabla|H|^{2}/2)\cdot u\cdot\nabla udx$ $\displaystyle\leq$
$\displaystyle\left(\int\sigma^{m}(H\cdot\nabla H-\nabla|H|^{2}/2)\cdot
udx\right)_{t}+C(\|\nabla H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2})$
$\displaystyle+C\sigma^{m}(\|H_{t}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})+C\sigma^{m}\|\nabla
u\|_{L^{3}}^{3}+C\sigma^{m}\|\nabla H\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2}$
$\displaystyle+C\sigma^{m}\|\nabla H\|_{L^{2}}^{2}(\|\nabla
H\|_{L^{2}}^{4}+\|\nabla u\|_{L^{2}}^{4}),$ (3.30)

Making use of the results (3), (3),(3) and (3), it follows from $\eqref{I0}$
that

$\displaystyle\left((\lambda+2\mu)\int\sigma^{m}({\rm
div}u)^{2}dx+\mu\int\sigma^{m}|\omega|^{2}dx\right)_{t}+\int\sigma^{m}\rho|\dot{u}|^{2}dx$
$\displaystyle\leq$ $\displaystyle\left(\int\sigma^{m}(P-\bar{P})\,{\rm
div}udx\right)_{t}+\left(\int\sigma^{m}(H\cdot\nabla H-\nabla|H|^{2}/2)\cdot
udx\right)_{t}$ $\displaystyle+C(\|\rho-\bar{\rho}\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2})+C\sigma^{m}\|\nabla u\|_{L^{3}}^{3}$
$\displaystyle+C\sigma^{m}(\|H_{t}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})+C\sigma^{m}\|\nabla H\|_{L^{2}}^{2}(\|\nabla
H\|_{L^{2}}^{4}+\|\nabla u\|_{L^{2}}^{4})$ $\displaystyle+C\sigma^{m}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})(\|\nabla u\|_{L^{2}}^{2}+1),$ (3.31)

integrating over $(0,T]$, by (2.6), (2.29), Lemma 2.1 and Young’s inequality,
we conclude that for any $m>0$,

$\displaystyle\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}+\int_{0}^{T}\int\sigma^{m}\rho|\dot{u}|^{2}dxdt$
$\displaystyle\leq$ $\displaystyle
CC_{0}+C\int_{0}^{T}\sigma^{m}(\|H_{t}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})dt$ $\displaystyle+C\int_{0}^{T}\sigma^{m}(\|\nabla
H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2})(\|\nabla H\|_{L^{2}}^{4}+\|\nabla
u\|_{L^{2}}^{4})dt$ $\displaystyle+C\int_{0}^{T}\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})dt+C\int_{0}^{T}\sigma^{m}\|\nabla u\|_{L^{3}}^{3}dt.$ (3.32)

Choose $m=1,$ together with (3.6), (3.8) and (3.14), we obtain $\eqref{a01}$.

Now we will claim (3.22). Operating $\sigma^{m}\dot{u}^{j}[\partial/\partial
t+{\rm div}(u\cdot)]$ to $(\ref{2m2})^{j},$ summing with respect to $j$, and
integrating over $\Omega,$ together with $\eqref{CMHD}_{1}$, we get

$\displaystyle\left(\frac{\sigma^{m}}{2}\int\rho|\dot{u}|^{2}dx\right)_{t}-\frac{m}{2}\sigma^{m-1}\sigma^{\prime}\int\rho|\dot{u}|^{2}dx$
$\displaystyle=\int\sigma^{m}(\dot{u}\cdot\nabla F_{t}+\dot{u}^{j}\,{\rm
div}(u\partial_{j}F))dx$
$\displaystyle\quad+\mu\int\sigma^{m}(-\dot{u}\cdot\nabla\times\omega_{t}-\dot{u}^{j}{\rm
div}((\nabla\times\omega)^{j}\,u))dx$
$\displaystyle\quad+\int\sigma^{m}(\dot{u}\cdot({\rm div}(H\otimes
H))_{t}+\dot{u}^{j}{\rm div}(({\rm div}(H\otimes H^{j})\,u))dx$
$\displaystyle\triangleq J_{1}+J_{2}+J_{3}.$ (3.33)

Let us estimate $J_{1},J_{2}$ and $J_{3}$. By (1.3) and (2.2)1, a direct
computation yields

$\displaystyle J_{1}$
$\displaystyle=\int_{\partial\Omega}\sigma^{m}F_{t}\dot{u}\cdot
nds-\int\sigma^{m}F_{t}\,{\rm
div}\dot{u}dx-\int\sigma^{m}u\cdot\nabla\dot{u}^{j}\partial_{j}Fdx$
$\displaystyle=\int_{\partial\Omega}\sigma^{m}F_{t}\dot{u}\cdot
nds-(\lambda+2\mu)\int\sigma^{m}({\rm
div}\dot{u})^{2}dx+(\lambda+2\mu)\int\sigma^{m}{\rm div}\dot{u}\,\nabla
u:\nabla udx$ $\displaystyle\quad-\gamma\int\sigma^{m}P{\rm div}\dot{u}\,{\rm
div}udx+\int\sigma^{m}{\rm div}\dot{u}\,u\cdot\nabla
Fdx-\int\sigma^{m}u\cdot\nabla\dot{u}^{j}\partial_{j}Fdx$
$\displaystyle\quad+\int\sigma^{m}{\rm div}\dot{u}\,H\cdot
H_{t}dx+\int\sigma^{m}{\rm div}\dot{u}\,u\cdot\nabla H\cdot Hdx$
$\displaystyle\leq\int_{\partial\Omega}\sigma^{m}F_{t}\dot{u}\cdot
nds-(\lambda+2\mu)\int\sigma^{m}({\rm
div}\dot{u})^{2}dx+\frac{\delta}{12}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|^{4}_{L^{4}}$ $\displaystyle\quad+C\sigma^{m}\big{(}\|\nabla
u\|_{L^{2}}^{2}\|\nabla F\|_{L^{3}}^{2}+C_{0}^{2\delta_{0}/3}\|\nabla
H_{t}\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}\|\nabla H\|_{L^{2}}^{2}\|\nabla
H\|_{L^{6}}^{2}\big{)}$ (3.34)

where in the second equality we have used

$\displaystyle\displaystyle F_{t}$ $\displaystyle=(2\mu+\lambda){\rm
div}\dot{u}-(2\mu+\lambda)\nabla u:\nabla u-u\cdot\nabla F+u\cdot\nabla H\cdot
H+\gamma P{\rm div}u-H\cdot H_{t}.$

For the first term on the righthand side of (3), we have

$\displaystyle\int_{\partial\Omega}\sigma^{m}F_{t}\dot{u}\cdot
nds=-\int_{\partial\Omega}\sigma^{m}F_{t}\,(u\cdot\nabla n\cdot u)ds$
$\displaystyle=$
$\displaystyle-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}+m\sigma^{m-1}\sigma^{\prime}\int_{\partial\Omega}(u\cdot\nabla
n\cdot u)Fds$
$\displaystyle\quad+\int_{\partial\Omega}\sigma^{m}\big{(}F\dot{u}\cdot\nabla
n\cdot u+Fu\cdot\nabla n\cdot\dot{u}\big{)}ds$
$\displaystyle\quad-\int_{\partial\Omega}\sigma^{m}\big{(}F(u\cdot\nabla)u\cdot\nabla
n\cdot u+Fu\cdot\nabla n\cdot(u\cdot\nabla)u\big{)}ds$ $\displaystyle\leq$
$\displaystyle-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}+Cm\sigma^{m-1}\sigma^{\prime}\|\nabla
u\|_{L^{2}}^{2}\|F\|_{H^{1}}$
$\displaystyle\quad+\frac{\delta}{12}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{4}+C\sigma^{m}\|\nabla u\|_{L^{2}}^{2}\|F\|_{H^{1}}^{2}$
$\displaystyle\quad+C\|\nabla F\|_{L^{6}}\|\nabla
u\|^{3}_{L^{2}}+C\|F\|_{H^{1}}\|\nabla u\|_{L^{2}}\left(\|\nabla
u\|^{2}_{L^{4}}+\|\nabla u\|^{2}_{L^{2}}\right).$ (3.35)

Together with Lemma 2.2, (2.9),(2.34),(2.35),(3.27), (3) and (3), we have

$\displaystyle J_{1}\leq-(\lambda+2\mu)\int\sigma^{m}({\rm
div}\dot{u})^{2}dx-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}$
$\displaystyle\quad+\frac{\delta}{3}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}C_{0}^{2\delta_{0}/3}\|\nabla
H_{t}\|_{L^{2}}^{2}+C\sigma^{m}(\|\nabla H\|_{L^{2}}^{4}+\|\nabla
u\|_{L^{2}}^{4})\|{\rm curl}^{2}H\|_{L^{2}}^{2}$
$\displaystyle\quad+C\sigma^{m}\|\nabla
u\|^{4}_{L^{4}}+C\sigma^{m}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})\|\nabla u\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
H\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{6}$
$\displaystyle\quad+C\sigma^{m}\|\nabla u\|_{L^{2}}^{2}(1+\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4}+\|\nabla H\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{4}+\|\nabla H\|_{L^{2}}^{6})$
$\displaystyle\quad+Cm\sigma^{m-1}\sigma^{\prime}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})$
$\displaystyle\quad+Cm\sigma^{m-1}\sigma^{\prime}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})\|\nabla u\|_{L^{2}}^{2}.$ (3.36)

Next, by $\omega_{t}={\rm curl}\dot{u}-u\cdot\nabla\omega-\nabla
u^{i}\times\partial_{i}u$ and a straightforward calculation leads to

$\displaystyle J_{2}$ $\displaystyle=-\mu\int\sigma^{m}|{\rm
curl}\dot{u}|^{2}dx+\mu\int\sigma^{m}{\rm curl}\dot{u}\cdot(\nabla
u^{i}\times\nabla_{i}u)dx$
$\displaystyle\quad+\mu\int\sigma^{m}u\cdot\nabla\omega\cdot{\rm
curl}\dot{u}dx+\mu\int\sigma^{m}u\cdot\nabla\dot{u}\cdot(\nabla\times\omega)dx$
$\displaystyle\leq-\mu\int\sigma^{m}|{\rm
curl}\dot{u}|^{2}dx+\frac{\delta}{3}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|_{L^{4}}^{4}$ (3.37)

Finally, a directly computation shows that

$\displaystyle\displaystyle J_{3}$
$\displaystyle=-\int\sigma^{m}\nabla\dot{u}:(H\otimes
H)_{t}dx-\mu\int\sigma^{m}H\cdot\nabla H^{j}u\cdot\nabla\dot{u}^{j}dx$
$\displaystyle\leq
C\sigma^{m}(\|\nabla\dot{u}\|_{L^{2}}\|H\|_{L^{3}}\|H_{t}\|_{L^{6}}+\|\nabla\dot{u}\|_{L^{2}}\|H\|_{L^{6}}\|\nabla
H\|_{L^{6}}\|u\|_{L^{6}})$
$\displaystyle\leq\frac{\delta}{3}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}(\|\nabla
H\|_{L^{2}}^{4}+\|\nabla u\|_{L^{2}}^{4})\|{\rm curl}^{2}H\|_{L^{2}}^{2}$
$\displaystyle\quad+C\sigma^{m}C_{0}^{2\delta_{0}/3}\|\nabla
H_{t}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla H\|_{L^{2}}^{4}\|\nabla
u\|_{L^{2}}^{2}.$ (3.38)

Combining (3), (3) with (3), we deduce from (3) that

$\displaystyle\left(\frac{\sigma^{m}}{2}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}\right)_{t}+(\lambda+2\mu)\sigma^{m}\|{\rm
div}\dot{u}\|_{L^{2}}^{2}+\mu\sigma^{m}\|{\rm curl}\dot{u}\|_{L^{2}}^{2}$
$\displaystyle\leq-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}+\delta\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}C_{0}^{2\delta_{0}/3}\|\nabla
H_{t}\|_{L^{2}}^{2}$ $\displaystyle\quad+C\sigma^{m}\|\nabla
u\|^{4}_{L^{4}}+C\sigma^{m}(\|\nabla H\|_{L^{2}}^{4}+\|\nabla
u\|_{L^{2}}^{4})\|{\rm curl}^{2}H\|_{L^{2}}^{2}$
$\displaystyle\quad+C\sigma^{m}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})\|\nabla u\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
H\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{6}$
$\displaystyle\quad+C\sigma^{m}\|\nabla u\|_{L^{2}}^{2}(1+\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4}+\|\nabla H\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{4}+\|\nabla H\|_{L^{2}}^{6})$
$\displaystyle\quad+Cm\sigma^{m-1}\sigma^{\prime}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})$
$\displaystyle\quad+Cm\sigma^{m-1}\sigma^{\prime}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{2})\|\nabla u\|_{L^{2}}^{2}.$ (3.39)

By (2.35) and Lemma 2.1, choosing $\delta$ small enough, and integrating (3)
over $(0,T]$, for $m>0$, we get

$\displaystyle\quad\sigma^{m}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\int_{0}^{T}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}dt$
$\displaystyle\leq-\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds+CC_{0}^{2\delta_{0}/3}\int_{0}^{T}\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}dt$ $\displaystyle\quad+C\int_{0}^{T}\sigma^{m}\|\nabla
u\|^{4}_{L^{4}}dt+CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq T}\sigma^{m}(\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2})$
$\displaystyle\quad+CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq
T}\sigma^{m}(\|\nabla u\|_{L^{2}}^{2}\|\nabla H\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2})+CC_{0}\sup_{0\leq t\leq\sigma(T)}\sigma^{m-1}\|\nabla
u\|_{L^{2}}^{2}$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}m\sigma^{m-1}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})dt+CC_{0}.$ (3.40)

For the boundary term in the right-hand side of (3) , from Lemma 2.2, we have

$\displaystyle\quad\int_{\partial\Omega}(u\cdot\nabla n\cdot u)Fds\leq
C\|\nabla u\|_{L^{2}}^{2}\|F\|_{H^{1}}$
$\displaystyle\leq\frac{1}{2}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+CC_{0}^{2\delta_{0}/3}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+C(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4}).$ (3.41)

Therefore,

$\displaystyle\sigma^{m}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\int_{0}^{T}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}dt-
C_{2}C_{0}^{2\delta_{0}/3}\int_{0}^{T}\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}dt$ $\displaystyle\leq C\int_{0}^{T}\sigma^{m}\|\nabla
u\|^{4}_{L^{4}}dt+CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq T}\sigma^{m}(\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2})$
$\displaystyle\quad+CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq
T}\sigma^{m}(\|\nabla u\|_{L^{2}}^{2}\|\nabla H\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2})+CC_{0}\sup_{0\leq t\leq\sigma(T)}\sigma^{m-1}\|\nabla
u\|_{L^{2}}^{2}$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}m\sigma^{m-1}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})dt+CC_{0}$
$\displaystyle\quad+CC_{0}^{2\delta_{0}/3}\sigma^{m}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+C\sigma^{m}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4}).$ (3.42)

Next, we need to estimate the term $\|\nabla H_{t}\|_{L^{2}}$. Noticing that

$\begin{cases}H_{tt}-\nu\nabla\times({\rm curl}H_{t})=(H\cdot\nabla
u-u\cdot\nabla H-H{\rm div}u)_{t},&\text{in}\quad\Omega,\\\ H_{t}\cdot
n=0,\quad{\rm curl}H_{t}\times n=0,&\text{on}\quad\partial\Omega,\\\
\end{cases}$ (3.43)

and after directly computations we obtain

$\displaystyle\quad\left(\frac{\sigma^{m}}{2}\|H_{t}\|_{L^{2}}^{2}\right)_{t}+\sigma^{m}\|{\rm
curl}H_{t}\|_{L^{2}}^{2}-\frac{m}{2}\sigma^{m-1}\sigma^{\prime}\|H_{t}\|_{L^{2}}^{2}$
$\displaystyle=\int\sigma^{m}(H_{t}\cdot\nabla u-u\cdot\nabla H_{t}-H_{t}{\rm
div}u)\cdot H_{t}dx$
$\displaystyle\quad+\int\sigma^{m}(H\cdot\nabla\dot{u}-\dot{u}\cdot\nabla
H-H{\rm div}\dot{u})\cdot H_{t}dx$
$\displaystyle\quad-\int\sigma^{m}(H\cdot\nabla(u\cdot\nabla u)-(u\cdot\nabla
u)\cdot\nabla H-H{\rm div}(u\cdot\nabla u))\cdot H_{t}dx$
$\displaystyle\triangleq K_{1}+K_{2}+K_{3}.$ (3.44)

By Lemma A.1 and Lemma 2.2, a direct calculation leads to

$\displaystyle K_{1}$ $\displaystyle\leq
C\sigma^{m}(\|H_{t}\|_{L^{3}}\|H_{t}\|_{L^{6}}\|\nabla
u\|_{L^{2}}+\|u\|_{L^{6}}\|H_{t}\|_{L^{3}}\|\nabla H_{t}\|_{L^{2}})$
$\displaystyle\leq\frac{\delta}{4}\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla u\|_{L^{2}}^{4}\|H_{t}\|_{L^{2}}^{2}.$
(3.45)

Similarly,

$\displaystyle K_{2}$ $\displaystyle\leq
C\sigma^{m}\|H\|_{L^{3}}\|H_{t}\|_{L^{6}}\|\nabla\dot{u}\|_{L^{2}}-\int_{\partial\Omega}\sigma^{m}(\dot{u}\cdot
n)(H\cdot H_{t})ds$ $\displaystyle\quad+\int\sigma^{m}{\rm div}\dot{u}\,H\cdot
H_{t}dx+\int\sigma^{m}\dot{u}\cdot\nabla H_{t}\cdot Hdx$
$\displaystyle\leq\int_{\partial\Omega}\sigma^{m}(u\\!\cdot\\!\nabla n\cdot
u)(H\\!\cdot\\!H_{t})ds+CC^{\delta_{0}/3}\sigma^{m}(\|\nabla\dot{u}\|_{L^{2}}^{2}\\!+\\!\|\nabla
u\|_{L^{2}}^{4}\\!+\\!\|\nabla H_{t}\|_{L^{2}}^{2}),$ (3.46)

where in the lase inequality we have used the fact that $\dot{u}\cdot
n=-u\cdot\nabla n\cdot u$ on $\partial\Omega$. By Sobolev trace theorem and
Lemma 2.2, it indicates that

$\displaystyle\quad\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)(H\cdot H_{t})ds$ $\displaystyle\leq C\sigma^{m}(\|u\|_{L^{6}}\|\nabla
u\|_{L^{2}}\|H\|_{L^{6}}\|H_{t}\|_{L^{6}}+\|u\|_{L^{6}}^{2}\|\nabla
H\|_{L^{2}}\|H_{t}\|_{L^{6}}$ $\displaystyle\quad+\|u\|_{L^{6}}^{2}\|\nabla
H_{t}\|_{L^{2}}\|H\|_{L^{6}}+\|u\|_{L^{4}}^{2}\|H\|_{L^{3}}\|H_{t}\|_{L^{6}})$
$\displaystyle\leq\frac{\delta}{4}\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla u\|_{L^{2}}^{4}\|\nabla
H\|_{L^{2}}^{2}+CC^{\delta_{0}/3}\sigma^{m}\|\nabla u\|_{L^{2}}^{4}.$ (3.47)

Combining (3) and (3), we have

$\displaystyle K_{2}$ $\displaystyle\leq\frac{\delta}{4}\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla u\|_{L^{2}}^{4}\|\nabla
H\|_{L^{2}}^{2}$
$\displaystyle\quad+CC^{\delta_{0}/3}\sigma^{m}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{4}+\|\nabla H_{t}\|_{L^{2}}^{2}).$ (3.48)

By Sobolev trace theorem and Lemma 2.2 again, a direct computation yields

$\displaystyle K_{3}$ $\displaystyle=\int\sigma^{m}H\cdot\nabla
H_{t}\cdot(u\cdot\nabla u)dx+\int_{\partial\Omega}\sigma^{m}H\cdot
H_{t}\,(u\cdot\nabla u\cdot n)ds$
$\displaystyle\quad-\int\sigma^{m}u\cdot\nabla u\cdot\nabla H_{t}\cdot Hdx$
$\displaystyle\leq\frac{\delta}{2}\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}+C\sigma^{m}(\|\nabla u\|_{L^{2}}^{4}+\|\nabla
H\|_{L^{2}}^{4})(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})$ $\displaystyle\quad+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}\|\nabla H\|_{L^{2}}^{2}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}+1)$
$\displaystyle\quad+CC^{\delta_{0}/3}\sigma^{m}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{4}+\|\nabla H_{t}\|_{L^{2}}^{2}).$ (3.49)

Putting (3), (3) and (3) into (3), choosing $\delta$ small enough, we have

$\displaystyle\quad\left(\sigma^{m}\|H_{t}\|_{L^{2}}^{2}\right)_{t}+\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}-CC^{\delta_{0}/3}\sigma^{m}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})$ $\displaystyle\leq C\sigma^{m}(\|\nabla
u\|_{L^{2}}^{4}+\|\nabla
H\|_{L^{2}}^{4})(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})$
$\displaystyle\quad+C\sigma^{m}\|\nabla u\|_{L^{2}}^{2}\|\nabla
H\|_{L^{2}}^{2}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}+1)+Cm\sigma^{m-1}\sigma^{\prime}\|H_{t}\|_{L^{2}}^{2}.$ (3.50)

Integrating over $(0,T]$, then by Lemma 2.1, for $m>0$, we get

$\displaystyle\quad\sigma^{m}\|H_{t}\|_{L^{2}}^{2}+\int_{0}^{T}\sigma^{m}\|\nabla
H_{t}\|_{L^{2}}^{2}dt-
C_{3}C^{\delta_{0}/3}\int_{0}^{T}\sigma^{m}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\leq CC_{0}^{2\delta_{0}}\sup_{0\leq
t\leq T}\sigma^{m}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})+CC_{0}\sup_{0\leq t\leq
T}\sigma^{m}\|\nabla u\|_{L^{2}}^{2}$
$\displaystyle\quad+CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq
T}\sigma^{m}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})+C\int_{0}^{\sigma(T)}m\sigma^{m-1}\|H_{t}\|_{L^{2}}^{2}dt.$
(3.51)

Now take $m=2$ in (3) and (3), we deduce after adding them together that

$\displaystyle\sigma^{2}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})+\int_{0}^{T}\sigma^{2}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\quad-
C_{2}C_{0}^{2\delta_{0}/3}\int_{0}^{T}\sigma^{2}\|\nabla
H_{t}\|_{L^{2}}^{2}dt-
C_{3}C^{\delta_{0}/3}\int_{0}^{T}\sigma^{2}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\leq C\int_{0}^{T}\sigma^{2}\|\nabla
u\|^{4}_{L^{4}}dt+CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq
T}\sigma^{2}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})$
$\displaystyle\quad+CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq
T}\sigma^{2}(\|\nabla u\|_{L^{2}}^{2}\|\nabla H\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2})+CC_{0}\sup_{0\leq t\leq T}\sigma\|\nabla u\|_{L^{2}}^{2}$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}\sigma(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})dt+CC_{0}$
$\displaystyle\quad+CC_{0}^{2\delta_{0}/3}\sigma^{2}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+C\sigma^{2}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4})$ $\displaystyle\leq
C\int_{0}^{T}\sigma^{2}\|\nabla
u\|^{4}_{L^{4}}dt+CC_{0}^{2\delta_{0}+1/2}+CA_{1}(T)+CC_{0}+CC_{0}^{2\delta_{0}/3+1/2}.$
(3.52)

Thus we have

$\displaystyle\quad\sup_{0\leq t\leq
T}\sigma^{2}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})+\int_{0}^{T}\sigma^{2}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\leq C\int_{0}^{T}\sigma^{2}\|\nabla
u\|^{4}_{L^{4}}dt+CA_{1}(T)+CC_{0}^{2\delta_{0}/3+1/2}.$ (3.53)

provided that $C_{0}$ is chosen to satisfy

$\displaystyle
C_{0}\leq\varepsilon_{2}\triangleq\min\\{\varepsilon_{1},(4C_{2})^{-3/{2\delta_{0}}},(4C_{3})^{-3/{\delta_{0}}}\\}.$

Finally, by Lemma A.1 and (1.1)3, it holds

$\displaystyle\|{\rm curl}^{2}H\|_{L^{2}}$ $\displaystyle\leq
C(\|H_{t}\|+\|{\rm curl}^{2}H\|_{L^{2}}^{1/2}\|\nabla
H\|_{L^{2}}^{1/2}\|\nabla u\|_{L^{2}}+\|\nabla H\|_{L^{2}}\|\nabla
u\|_{L^{2}})$ $\displaystyle\leq\frac{1}{2}\|{\rm
curl}^{2}H\|_{L^{2}}+C(\|H_{t}\|+\|\nabla H\|_{L^{2}}\|\nabla
u\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}\|\nabla u\|_{L^{2}}).$ (3.54)

Thus, by (3.6) and (3), we have

$\displaystyle\quad\sup_{0\leq t\leq T}\sigma^{2}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}$ $\displaystyle\leq C\sup_{0\leq t\leq
T}\sigma^{2}(\|H_{t}\|^{2}+\|\nabla H\|_{L^{2}}^{2}\|\nabla
u\|_{L^{2}}^{4}+\|\nabla H\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{2})$
$\displaystyle\leq C\int_{0}^{T}\sigma^{2}\|\nabla
u\|^{4}_{L^{4}}dt+CA_{1}(T)+CC_{0}^{2\delta_{0}/3+1/2}.$ (3.55)

Combining (3) and (3), we give (3.22) and complete the proof of Lemma 3.5.

###### Lemma 3.6

Assume that $(\rho,u,H)$ is a smooth solution of (1.1)-(1.4) satisfying (3.6)
and the initial data condition (1.10), then there exist positive constants $C$
and $\varepsilon_{3}$ depending only on
$\mu,\,\,\lambda,\,\,\nu,\,\,\gamma,\,\,a,\,\,\bar{\rho},\,\,\hat{\rho},\,\,s,\,\,\Omega$,
$M_{1}$ and $M_{2}$ such that

$\displaystyle\sup_{0\leq t\leq\sigma(T)}t^{1-s}\|\nabla
u\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}t^{1-s}\int\rho|\dot{u}|^{2}dxdt\leq
C(\hat{\rho},M_{1},M_{2}),$ (3.56) $\displaystyle\sup_{0\leq
t\leq\sigma(T)}t^{2-s}\int\left(\rho|\dot{u}|^{2}+|\mathop{\rm
curl}^{2}\nolimits H|^{2}+|H_{t}|^{2}\right)dx$
$\displaystyle\qquad+\int_{0}^{\sigma(T)}\int
t^{2-s}\left(|\nabla\dot{u}|^{2}+|\nabla H_{t}|^{2}\right)dxdt\leq
C(\hat{\rho},M_{1},M_{2}),$ (3.57)

provide that $C_{0}<\varepsilon_{3}$.

Proof. Suppose $w_{1}(x,t)$, $w_{2}(x,t)$ and $w_{3}(x,t)$ solve problems

$\displaystyle\begin{cases}Lw_{1}=0\,\,&in\,\,\Omega,\\\
w_{1}(x,0)=w_{10}(x)\,\,&in\,\,\Omega,\\\ w_{1}\cdot n=0\,\,\text{and}\,\,{\rm
curl}w_{1}\times n=0\,\,&on\,\,\partial\Omega,\end{cases}$ (3.58)
$\displaystyle\begin{cases}Lw_{2}=-\nabla(P-\bar{P})\,\,&in\,\,\Omega,\\\
w_{2}(x,0)=0\,\,&in\,\,\Omega,\\\ w_{2}\cdot n=0\,\,\text{and}\,\,{\rm
curl}w_{2}\times n=0\,\,&on\,\,\partial\Omega,\end{cases}$ (3.59)

and

$\displaystyle\begin{cases}Lw_{3}=(\nabla\times B)\times
H\,\,&in\,\,\Omega,\\\ w_{3}(x,0)=0\,\,&in\,\,\Omega,\\\ w_{3}\cdot
n=0\,\,\text{and}\,\,{\rm curl}w_{3}\times
n=0\,\,&on\,\,\partial\Omega,\end{cases}$ (3.60)

where $Lf\triangleq\rho\dot{f}-\mu\Delta f-(\lambda+\mu)\nabla{\rm div}f$, $B$
is the solution of (1.1)3 with fixed smooth $u$ and initial data $B_{0}(x)$.
Note that $B$ satisfies (3.14) and (3.15) of Lemma 3.4.

Just as we have done in the proof of Lemma 2.2, by Lemma A.3 and Sobolev’s
inequality, for any $p\in[2,6],$ we have

$\displaystyle\|\nabla^{2}w_{1}\|_{L^{2}}\leq
C(\|\rho\dot{w}_{1}\|_{L^{2}}+\|w_{1}\|_{L^{2}})\leq
C(\|\rho\dot{w}_{1}\|_{L^{2}}+\|\nabla w_{1}\|_{L^{2}}),$ (3.61)
$\displaystyle\|\nabla w_{1}\|_{L^{p}}\leq C\|w_{1}\|_{W^{2,2}}\leq
C(\|\rho\dot{w}_{1}\|_{L^{2}}+\|\nabla w_{1}\|_{L^{2}}),$ (3.62)
$\displaystyle\|\nabla F_{w_{2}}\|_{L^{p}}\leq C\|\rho\dot{w}_{2}\|_{L^{p}},$
(3.63) $\displaystyle\|F_{w_{2}}\|_{L^{p}}$ $\displaystyle\leq C(\|\nabla
F_{w_{2}}\|_{L^{2}}+\|F_{w_{2}}\|_{L^{2}})\leq
C(\|\rho\dot{w}_{2}\|_{L^{2}}+\|\nabla w_{2}\|_{L^{2}}),$ (3.64)
$\displaystyle\|\nabla w_{2}\|_{L^{p}}$ $\displaystyle\leq
C\|\rho^{\frac{1}{2}}\dot{w}_{2}\|_{L^{2}}^{\frac{3p-6}{2p}}(\|\nabla
w_{2}\|_{L^{2}}+\|P-\bar{P}\|_{L^{2}})^{\frac{6-p}{2p}}$ (3.65)
$\displaystyle\quad+C(\|\nabla w_{2}\|_{L^{2}}+\|P-\bar{P}\|_{L^{p}}),$

where $F_{w_{2}}=(\lambda+2\mu){\rm div}w_{2}-(P-\bar{P}).$

A similar way as for the proof of (2.4) shows that

$\displaystyle\displaystyle\sup_{0\leq
t\leq\sigma(T)}\int\rho|w_{1}|^{2}dx+\int_{0}^{\sigma(T)}\int|\nabla
w_{1}|^{2}dxdt\leq C\int|w_{10}|^{2}dx,$ (3.66)
$\displaystyle\displaystyle\sup_{0\leq
t\leq\sigma(T)}\int\rho|w_{2}|^{2}dx+\int_{0}^{\sigma(T)}\int|\nabla
w_{2}|^{2}dxdt\leq CC_{0},$ (3.67)

and

$\displaystyle\sup_{0\leq
t\leq\sigma(T)}\int\\!\rho|w_{3}|^{2}dx\\!+\\!\int_{0}^{\sigma(T)}\\!\int|\nabla
w_{3}|^{2}dxdt\leq C\int_{0}^{\sigma(T)}\\!\int|\nabla B|^{2}dxdt\leq
C\|B_{0}\|_{L^{2}}^{2}.$ (3.68)

Multiplying (3.58) by $w_{1t}$ and integrating over $\Omega,$ by (3.62),
(3.6), Sobolev’s and Young’s inequalities, we obtain

$\displaystyle\quad\left(\frac{\lambda+2\mu}{2}\int({\rm
div}w_{1})^{2}dx+\frac{\mu}{2}\int|{\rm
curl}w_{1}|^{2}dx\right)_{t}+\int\rho|\dot{w}_{1}|^{2}dx$
$\displaystyle=\int\rho\dot{w}_{1}\cdot(u\cdot\nabla w_{1})dx\leq
C_{4}C_{0}^{\frac{\delta_{0}}{3}}(\|\rho^{\frac{1}{2}}\dot{w}_{1}\|_{L^{2}}^{2}+\|\nabla
w_{1}\|_{L^{2}}^{2}).$ (3.69)

Together with (3.66), and by Gronwall’s inequality and Lemma A.4, it yields

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{1}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\int\rho|\dot{w}_{1}|^{2}dxdt\leq
C\|\nabla w_{10}\|_{L^{2}}^{2},$ (3.70)

and

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}t\|\nabla
w_{1}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}t\int\rho|\dot{w}_{1}|^{2}dxdt\leq
C\|w_{10}\|_{L^{2}}^{2},$ (3.71)

provided
$C_{0}<\hat{\varepsilon}_{1}\triangleq(2C_{4})^{-\frac{3}{\delta_{0}}}$.

Since the solution operator $w_{10}\mapsto w_{1}(\cdot,t)$ is linear, by the
standard Stein-Weiss interpolation argument [5], one can deduce from (3.70)
and (3.71) that for any $\theta\in[s,1],$

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}t^{1-\theta}\|\nabla
w_{1}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}t^{1-\theta}\int\rho|\dot{w_{1}}|^{2}dxdt\leq
C\|w_{10}\|_{H^{\theta}}^{2},$ (3.72)

with a uniform constant $C$ independent of $\theta.$

Multiplying (3.59) by $w_{2t}$ and integrating over $\Omega$ give that

$\displaystyle\quad\left(\frac{\lambda+2\mu}{2}\int({\rm
div}w_{2})^{2}dx+\frac{\mu}{2}\int|{\rm curl}w_{2}|^{2}dx-\int(P-\bar{P}){\rm
div}w_{2}dx\right)_{t}+\int\rho|\dot{w}_{2}|^{2}dx$
$\displaystyle=\int\rho\dot{w}_{2}\cdot(u\cdot\nabla
w_{2})dx-\frac{1}{\lambda+2\mu}\int(P-\bar{P})(F_{w_{2}}{\rm div}u+\nabla
F_{w_{2}}\cdot u)dx$
$\displaystyle\quad-\frac{1}{2(\lambda+2\mu)}\int(P-\bar{P})^{2}{\rm
div}udx+\gamma\int P{\rm div}u\,{\rm div}w_{2}dx$ $\displaystyle\leq
C(\|\rho^{\frac{1}{2}}\dot{w}_{2}\|_{L^{2}}\|\rho^{\frac{1}{3}}u\|_{L^{3}}\|\nabla
w_{2}\|_{L^{6}}+\|\nabla u\|_{L^{2}}\|F_{w_{2}}\|_{L^{2}}+\|\nabla
F_{w_{2}}\|_{L^{2}}\|u\|_{L^{2}})$
$\displaystyle\quad+C(\|P-\bar{P}\|_{L^{2}}\|\nabla u\|_{L^{2}}+\|\nabla
u\|_{L^{2}}\|\nabla w_{2}\|_{L^{2}})$ $\displaystyle\leq
C_{5}C_{0}^{\frac{\delta_{0}}{3}}\|\rho^{\frac{1}{2}}\dot{w}_{2}\|_{L^{2}}^{2}+\frac{1}{4}\|\rho^{\frac{1}{2}}\dot{w}_{2}\|_{L^{2}}^{2}+C(\|\nabla
w_{2}\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}+\|P-\bar{P}\|_{L^{2}}^{2/3}),$
(3.73)

where we have utilized (3.6), (3.63)-(3.65), Hölder’s, Poincaré’s and Young’s
inequalities. As a result,

$\displaystyle\quad\left((\lambda+2\mu)\|{\rm
div}w_{2}\|_{L^{2}}^{2}+\mu\|{\rm
curl}w_{2}\|_{L^{2}}^{2}-2\int(P-\bar{P}){\rm
div}w_{2}dx\right)_{t}+\int\rho|\dot{w}_{2}|^{2}dx$ $\displaystyle\leq
C\left(\|\nabla w_{2}\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2}+\|P-\bar{P}\|_{L^{2}}^{\frac{2}{3}}\right),$ (3.74)

provide that
$C_{0}<\hat{\varepsilon}_{2}\triangleq(4C_{5})^{-\frac{3}{\delta_{0}}}$.

By Gronwall’s inequality, (3.67) and Lemmas A.4, 2.1, one has

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{2}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\int\rho|\dot{w}_{2}|^{2}dxdt\leq
CC_{0}^{1/3}.$ (3.75)

Similarly, multiplying (3.60) by $w_{3t}$ and integrating over $\Omega$ give
that

$\displaystyle\quad\left(\frac{\lambda+2\mu}{2}\int({\rm
div}w_{3})^{2}dx+\frac{\mu}{2}\int|{\rm
curl}w_{3}|^{2}dx\right)_{t}+\int\rho|\dot{w}_{3}|^{2}dx$
$\displaystyle=\left(\int(H\cdot\nabla B-\nabla B\cdot H)\cdot
w_{3}dx\right)_{t}-\int(H\cdot\nabla B-\nabla B\cdot H)_{t}\cdot w_{3}dx$
$\displaystyle\quad+\int\rho\dot{w}_{3}\cdot(u\cdot\nabla w_{3})dx$
$\displaystyle\triangleq\frac{d}{dt}L_{0}+L_{1}+L_{2}.$ (3.76)

By (A.1) and (3.10), we have

$\displaystyle L_{0}\leq C\|H\|_{L^{3}}\|\nabla
B\|_{L^{2}}\|w_{3}\|_{L^{6}}\leq\frac{\mu}{4}\|\nabla
w_{3}\|_{L^{2}}^{2}+CC_{0}^{2\delta_{0}/3}\|\nabla B\|_{L^{2}}^{2}.$ (3.77)

Using (3.14), a directly computation yields

$\displaystyle L_{1}$ $\displaystyle\leq C(\|H_{t}\|_{L^{2}}\|\nabla
B\|_{L^{3}}\|\nabla w_{3}\|_{L^{2}}+\|B_{t}\|_{L^{2}}\|H\|_{L^{3}}\|\nabla
w_{3}\|_{L^{6}}+\|B_{t}\|_{L^{2}}\|\nabla w_{3}\|_{L^{2}}\|\nabla
H\|_{L^{3}})$ $\displaystyle\leq
CC^{\delta_{0}/3}(\|\rho^{1/2}\dot{u}\|_{L^{2}}^{2}+\|\nabla
w_{3}\|_{L^{2}}^{2})+C(\|B_{t}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}B\|_{L^{2}}^{2}+\|\nabla B\|_{L^{2}}^{2})$
$\displaystyle\quad+C(\|{\rm curl}^{2}H\|_{L^{2}}\|\nabla H\|_{L^{2}}+\|\nabla
H\|_{L^{2}}^{2})\|\nabla w_{3}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}\|\nabla
B\|_{L^{2}}\|\nabla w_{3}\|_{L^{2}}$ $\displaystyle\quad+\|{\rm
curl}^{2}B\|_{L^{2}}^{1/2}\|\nabla B\|_{L^{2}}^{1/2}\|H_{t}\|_{L^{2}}\|\nabla
w_{3}\|_{L^{2}},$ (3.78)

where we have used the fact (3.65). Similarly, using Lemma A.1 yields

$\displaystyle L_{2}$ $\displaystyle\leq
CC^{\delta_{0}/3}(\|\rho^{1/2}\dot{w_{3}}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}B\|_{L^{2}}^{2}+\|\nabla B\|_{L^{2}}^{2}+\|\nabla
w_{3}\|_{L^{2}}^{2}).$ (3.79)

Putting (3.77), (3) and (3.79) into (3), we obtain

$\displaystyle\quad(\|\nabla
w_{3}\|_{L^{2}}^{2})_{t}+\int\rho|\dot{w}_{3}|^{2}dx-(CC_{0}^{2\delta_{0}/3}\|\nabla
B\|_{L^{2}}^{2})_{t}$ $\displaystyle\leq
C_{6}C^{\delta_{0}/3}(\|\rho^{1/2}\dot{w_{3}}\|_{L^{2}}^{2}+\|\nabla
w_{3}\|_{L^{2}}^{2})+C(\|B_{t}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}B\|_{L^{2}}^{2}+\|\nabla B\|_{L^{2}}^{2})$
$\displaystyle\quad+C(\|{\rm curl}^{2}H\|_{L^{2}}\|\nabla H\|_{L^{2}}+\|\nabla
H\|_{L^{2}}^{2})\|\nabla w_{3}\|_{L^{2}}^{2}+C\|H_{t}\|_{L^{2}}\|\nabla
B\|_{L^{2}}\|\nabla w_{3}\|_{L^{2}}$ $\displaystyle\quad+C\|{\rm
curl}^{2}B\|_{L^{2}}^{1/2}\|\nabla B\|_{L^{2}}^{1/2}\|H_{t}\|_{L^{2}}\|\nabla
w_{3}\|_{L^{2}}.$ (3.80)

Thus, if $C_{0}$ is chosen to be such that
$C_{0}\leq\hat{\varepsilon}_{3}\triangleq(2C_{6})^{-3/\delta_{0}}$, we obtain

$\displaystyle\quad\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{3}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\int\rho|\dot{w}_{3}|^{2}dxdt$
$\displaystyle\leq C\|\nabla
B_{0}\|_{L^{2}}^{2}+CC_{0}^{\delta_{0}}\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{3}\|_{L^{2}}^{2}+CC_{0}^{\delta_{0}/2}\sup_{0\leq t\leq\sigma(T)}(\|\nabla
B\|_{L^{2}}^{2}+\|\nabla w_{3}\|_{L^{2}}^{2})$
$\displaystyle\quad+CC_{0}^{\delta_{0}/2}\left(\int_{0}^{\sigma(T)}\|{\rm
curl}^{2}B\|_{L^{2}}^{2}dt\right)^{1/4}\sup_{0\leq t\leq\sigma(T)}\|\nabla
B\|_{L^{2}}^{1/2}\|\nabla w_{3}\|_{L^{2}}$ $\displaystyle\leq C\|\nabla
B_{0}\|_{L^{2}}^{2}+C_{7}C_{0}^{\delta_{0}/2}\sup_{0\leq
t\leq\sigma(T)}\|\nabla w_{3}\|_{L^{2}}^{2},$ (3.81)

where we have used (3.14) and (3.8) (with $H,H_{0}$ being replaced by
$B,B_{0}$, respectively). Thus, it follows from (3) that

$\displaystyle\quad\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{3}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\int\rho|\dot{w}_{3}|^{2}dxdt\leq
C\|\nabla B_{0}\|_{L^{2}}^{2},$ (3.82)

provided
$C_{0}\leq\hat{\varepsilon}_{4}\triangleq\min\\{\hat{\varepsilon}_{3},(2C_{7})^{-2/\delta_{0}}\\}$.
Similarly, multiplying (3) by $t$, integrating it over $(0,\delta(T))$, for
$C_{0}\leq\hat{\varepsilon}_{4}$, we obtain that

$\displaystyle\quad\sup_{0\leq t\leq\sigma(T)}(t\|\nabla
w_{3}\|_{L^{2}}^{2})+\int_{0}^{\sigma(T)}t\int\rho|\dot{w}_{3}|^{2}dxdt\leq
C\|B_{0}\|_{L^{2}}^{2},$ (3.83)

which we have used (3.6), (3.8) and (3.14). Since the solution operators
$B_{0}\mapsto B$ and $B\mapsto w_{3}$ are linear, by the standard Stein-Weiss
interpolation argument [5], one can deduce from (3) and (3.82) that for any
$\theta\in[s,1],$

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}t^{1-\theta}\|\nabla
w_{3}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}t^{1-\theta}\int\rho|\dot{w_{3}}|^{2}dxdt\leq
C\|B_{0}\|_{H^{\theta}}^{2},$ (3.84)

Now let $w_{10}=u_{0}$ and $B_{0}=H_{0}$, so that $w_{1}+w_{2}+w_{3}=u$ and
$B=H$, we derive (3.56) from (3.72), (3.75) and (3.84) directly under certain
condition
$C_{0}<\bar{\varepsilon}_{3}\triangleq\min\\{\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{\varepsilon}_{4}\\}$.

In order to prove (3.57), taking $m=2-s$ in (3), (3), and integrating over
$(0,\sigma(T)]$ instead of $(0,T]$, in a similar way as we have gotten (3), we
obtain

$\displaystyle\sup_{0\leq
t\leq\sigma(T)}\sigma^{2-s}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})+\int_{0}^{\sigma(T)}\sigma^{2-s}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\sigma^{2-s}\|\nabla
u\|^{4}_{L^{4}}dt+CC_{0}^{2\delta_{0}/3}\sigma^{2-s}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+C(\hat{\rho},M_{1},M_{2}).$ (3.85)

where we have taken advantage of (3.56). Next, from (3), we have

$\displaystyle\quad\sup_{0\leq t\leq\sigma(T)}t^{2-s}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}$ $\displaystyle\leq C\sup_{0\leq
t\leq\sigma(T)}t^{2-s}\|H_{t}\|_{L^{2}}^{2}+C\sup_{0\leq
t\leq\sigma(T)}(\sigma^{1-s}\|\nabla
H\|_{L^{2}}^{2})(\sigma^{(3-2s)/4}\|\nabla u\|_{L^{2}}^{2})^{2}+CC_{0}^{1/2}$
$\displaystyle\leq C\sup_{0\leq
t\leq\sigma(T)}t^{2-s}\|H_{t}\|_{L^{2}}^{2}+C(\hat{\rho},M_{1},M_{2}).$ (3.86)

Combining (3) and (3), we obtain

$\displaystyle\sup_{0\leq
t\leq\sigma(T)}\sigma^{2-s}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})+\int_{0}^{\sigma(T)}\sigma^{2-s}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\sigma^{2-s}\|\nabla
u\|^{4}_{L^{4}}dt+C(\hat{\rho},M_{1},M_{2}),$ (3.87)

provided $C_{0}\leq\varepsilon_{2}$. By (2.12) and (3.56), we have

$\displaystyle\quad\int_{0}^{\sigma(T)}t^{2-s}\|\nabla u\|_{L^{4}}^{4}dt$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}t^{2-s}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{3}+\|{\rm
curl}^{2}H\|_{L^{2}}^{3})(\|\nabla u\|_{L^{2}}+\|P-\bar{P}\|_{L^{2}}+\|\nabla
H\|_{L^{2}})dt$ $\displaystyle\quad+C\int_{0}^{\sigma(T)}t^{2-s}(\|\nabla
u\|_{L^{2}}^{4}+\|P-\bar{P}\|_{L^{2}}^{4}+\|P-\bar{P}\|_{L^{4}}^{4}+\|\nabla
H\|_{L^{2}}^{4}+\||H|^{2}\|_{L^{4}}^{4})dt$ $\displaystyle\leq
CC_{0}^{\delta_{0}}\sup_{0\leq
t\leq\sigma(T)}(t^{2-s}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}))+C,$ (3.88)

since it follows from (3.6) that for $s\in(1/2,1]$,

$\displaystyle\quad\int_{0}^{\sigma(T)}t^{2-s}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{3}(\|\nabla
u\|_{L^{2}}+\|\nabla H\|_{L^{2}})dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}t^{\frac{2s-3}{4}}(t^{\frac{3-2s}{4}}\big{(}\|\nabla
u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})\big{)}^{\frac{1}{2}}(t^{\frac{3-2s}{4}}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2})^{\frac{1}{2}}(t^{2-s}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2})dt$
$\displaystyle\leq CC_{0}^{\delta_{0}}\sup_{0\leq
t\leq\sigma(T)}(t^{2-s}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}),$ (3.89)

and

$\displaystyle\int_{0}^{\sigma(T)}t^{2-s}\||H|^{2}\|_{L^{4}}^{4}dt$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}t^{2-s}\|H\|_{L^{\infty}}^{4}\|H\|_{L^{4}}^{4}dt$
$\displaystyle\leq C\int_{0}^{\sigma(T)}t^{2-s}(\|\nabla H\|_{L^{2}}^{4}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{6})dt$ $\displaystyle\leq
CC_{0}^{2\delta_{0}}\sup_{0\leq t\leq\sigma(T)}(t^{2-s}\|{\rm
curl}^{2}H\|_{L^{2}}^{2})+C.$ (3.90)

Then combining (3) and (3), we have

$\displaystyle\sup_{0\leq
t\leq\sigma(T)}\sigma^{2-s}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})+\int_{0}^{\sigma(T)}\sigma^{2-s}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\leq C_{8}C_{0}^{\delta_{0}}\sup_{0\leq
t\leq\sigma(T)}\sigma^{2-s}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2})+C(\hat{\rho},M_{1},M_{2}),$
(3.91)

Therefore, if we choose $C_{0}$ to be such that
$C_{0}\leq\varepsilon_{3}\triangleq\min\\{\bar{\varepsilon}_{3},(2C_{8})^{-1/\delta_{0}}\\}$,
(3) and (3) implies (3.57). The proof of Lemma 3.6 is completed.

###### Lemma 3.7

If $(\rho,u,H)$ is a smooth solution of (1.1)-(1.4) satisfying (3.6) and the
initial data condition (1.10), then there exists a positive constant
$\varepsilon_{4}$ depending only on
$\mu,\lambda,\nu,\gamma,a,\bar{\rho},\hat{\rho},s,\Omega$, $M_{1}$ and $M_{2}$
such that

$\displaystyle\displaystyle A_{4}(\sigma(T))+A_{5}(\sigma(T))\leq
C_{0}^{\delta_{0}},$ (3.92)

provided $C_{0}<\varepsilon_{4}$.

Proof. Multiplying $\eqref{CMHD}_{2}$ by $3|u|u$, and integrating the
resulting equation over ${\Omega}$ lead to

$\displaystyle\quad\left(\int\rho|u|^{3}dx\right)_{t}$ $\displaystyle\leq
C\int|u||\nabla u|^{2}dx+C\int|P-\bar{P}||u||\nabla u|dx+C\int|H||\nabla
H||u|^{2}dx$ $\displaystyle\leq C\|\nabla
u\|_{L^{2}}^{\frac{5}{2}}(\|\rho\dot{u}\|_{L^{2}}^{\frac{1}{2}}+\|{\rm
curl}^{2}H\|_{L^{2}}^{\frac{1}{2}})+C\|\nabla
u\|_{L^{2}}^{\frac{5}{2}}\|\nabla
H\|_{L^{2}}^{\frac{1}{2}}+CC_{0}^{\frac{1}{12}}\|\nabla
u\|_{L^{2}}^{\frac{5}{2}}$ $\displaystyle\quad+C\|\nabla
u\|_{L^{2}}^{3}+CC_{0}^{\frac{1}{3}}\|\nabla u\|_{L^{2}}^{2}+C(\|\nabla
H\|_{L^{2}}^{4}+\|\nabla u\|_{L^{2}}^{4}).$ (3.93)

Hence, integrating (3) over $(0,\sigma(T))$ and using (3.6), (3.8), we get

$\displaystyle\quad\sup_{0\leq t\leq\sigma(T)}\int\rho|u|^{3}dx$
$\displaystyle\leq C\int_{0}^{\sigma(T)}(t^{\frac{3-2s}{4}}\|\nabla
u\|_{L^{2}}^{2})^{\frac{5}{4}}(t^{\frac{3-2s}{4}}(\|\rho\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2}))^{\frac{1}{4}}t^{\frac{3(2s-3)}{8}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}(t^{\frac{3-2s}{4}}\|\nabla
u\|_{L^{2}}^{2})^{\frac{5}{4}}(t^{\frac{3-2s}{4}}\|\nabla
H\|_{L^{2}}^{2})^{\frac{1}{4}}t^{\frac{3(2s-3)}{8}}dt$
$\displaystyle\quad+CC_{0}^{\frac{1}{12}}\int_{0}^{\sigma(T)}(t^{\frac{3-2s}{4}}\|\nabla
u\|_{L^{2}}^{2})^{\frac{5}{4}}t^{\frac{5(2s-3)}{16}}dt+CC_{0}^{2\delta_{0}}+\int\rho_{0}|u_{0}|^{3}dx$
$\displaystyle\leq CC_{0}^{3\delta_{0}/2}+\int\rho_{0}|u_{0}|^{3}dx\leq
C_{9}C_{0}^{3\delta_{0}/2},$ (3.94)

where we have used the fact $\delta_{0}\in(0,1/9]$, $s\in(1/2,1]$ and

$\displaystyle\displaystyle\int\rho_{0}|u_{0}|^{3}dx\leq
C\|\rho_{0}^{\frac{1}{2}}u_{0}\|_{L^{2}}^{{3(2s-1)}/{2s}}\|u_{0}\|_{H^{s}}^{{3}/{2s}}\leq
CC_{0}^{3\delta_{0}/2},$ (3.95)

Finally, set
$\varepsilon_{4}\triangleq\min\\{\varepsilon_{3},(C_{9})^{-\frac{2}{\delta_{0}}}\\}$,
we get $A_{5}({\sigma(T)})\leq C_{0}^{\delta_{0}}$. Next, it remains to
estimate $A_{4}({\sigma(T)})$. Using (3.56), we have

$\displaystyle A_{4}({\sigma(T)})\leq$ $\displaystyle\sup_{0\leq
t\leq\sigma(T)}(t^{1-s}\|\nabla u\|_{L^{2}}^{2})^{\frac{2s+1}{4s}}\sup_{0\leq
t\leq\sigma(T)}(t\|\nabla u\|_{L^{2}}^{2})^{\frac{2s-1}{4s}}$
$\displaystyle+\sup_{0\leq t\leq\sigma(T)}(t^{1-s}\|\nabla
H\|_{L^{2}}^{2})^{\frac{2s+1}{4s}}\sup_{0\leq t\leq\sigma(T)}(t\|\nabla
H\|_{L^{2}}^{2})^{\frac{2s-1}{4s}}$
$\displaystyle+\left(\int_{0}^{\sigma(T)}t^{1-s}\|\rho^{1/2}\dot{u}\|_{L^{2}}^{2}dt\right)^{\frac{2s+1}{4s}}\left(\int_{0}^{\sigma(T)}t\|\rho^{1/2}\dot{u}\|_{L^{2}}^{2}dt\right)^{\frac{2s-1}{4s}}$
$\displaystyle+\left(\int_{0}^{\sigma(T)}t^{1-s}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}dt\right)^{\frac{2s+1}{4s}}\left(\int_{0}^{\sigma(T)}t\|{\rm
curl}^{2}H\|_{L^{2}}^{2}dt\right)^{\frac{2s-1}{4s}}$ $\displaystyle\leq$
$\displaystyle CA_{1}(T)^{\frac{2s-1}{4s}}\leq CC_{0}^{\frac{2s-1}{8s}}\leq
C_{0}^{\delta_{0}},$ (3.96)

provided $C_{0}\leq\varepsilon_{4}$ and
$\delta_{0}\triangleq\frac{2s-1}{9s}<\frac{2s-1}{8s}$. The proof of Lemma 3.7
is completed.

###### Lemma 3.8

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) on ${\Omega}\times(0,T]$
satisfying (3.6). Then there exists a positive constant $\varepsilon_{5}$
depending only on $\mu,$ $\lambda,$ $\nu,$ $\gamma,$ $a$, $s$, $\bar{\rho}$,
$\hat{\rho}$, $\Omega$, $M_{1}$ and $M_{2}$ such that

$\displaystyle\displaystyle A_{1}(T)+A_{2}(T)\leq C_{0}^{\frac{1}{2}},$ (3.97)

provided $C_{0}\leq\varepsilon_{5}$.

Proof. By (A.1) and (2.24), one can check that

$\displaystyle\int_{0}^{T}\sigma\|\nabla u\|_{L^{3}}^{3}dt$ $\displaystyle\leq
C\int_{0}^{T}\sigma(\|\nabla u\|_{L^{2}}\|\nabla u\|_{L^{4}}^{2}+\|\nabla
u\|_{L^{2}}^{3})dt$ $\displaystyle\leq CC_{0}+C\int_{0}^{T}\sigma^{2}\|\nabla
u\|_{L^{4}}^{4}dt,$ (3.98)

which, along with (3.21) and (3.22) gives

$\displaystyle\displaystyle A_{1}(T)+A_{2}(T)\leq
C(C_{0}^{1/2+\delta_{0}/2}+\int_{0}^{T}\sigma^{2}\|\nabla u\|_{L^{4}}^{4}dt).$
(3.99)

So it reduces to estimate $\int_{0}^{T}\sigma^{2}\|\nabla u\|_{L^{4}}^{4}dt$.
On one hand, by (2.12), (3.6), (2.29) and Lemma 2.1 again, it indicates that

$\displaystyle\quad\int_{0}^{\sigma(T)}t^{2}\|\nabla u\|_{L^{4}}^{4}dt$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}t^{2}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{3}+\|{\rm
curl}^{2}H\|_{L^{2}}^{3})(\|\nabla u\|_{L^{2}}+\|P-\bar{P}\|_{L^{2}}+\|\nabla
H\|_{L^{2}})dt$ $\displaystyle\quad+C\int_{0}^{\sigma(T)}t^{2}(\|\nabla
u\|_{L^{2}}^{4}+\|P-\bar{P}\|_{L^{2}}^{4}+\|P-\bar{P}\|_{L^{4}}^{4}+\|\nabla
H\|_{L^{2}}^{4}+\||H|^{2}\|_{L^{4}}^{4})dt$ $\displaystyle\leq
CC_{0}^{1/2+\delta_{0}}+CC_{0}+CC_{0}^{1/2+2\delta_{0}}\leq
CC_{0}^{1/2+\delta_{0}},$ (3.100)

since it follows from (3.6) that for $s\in(1/2,1]$,

$\displaystyle\quad\int_{0}^{\sigma(T)}t^{2}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{3}(\|\nabla
u\|_{L^{2}}+\|\nabla H\|_{L^{2}})dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}t^{\frac{2s-3}{4}}(t^{\frac{3-2s}{4}}\big{(}\|\nabla
u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})\big{)}^{\frac{1}{2}}(t^{\frac{3-2s}{4}}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2})^{\frac{1}{2}}(t^{2}\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2})dt$
$\displaystyle\leq CC_{0}^{1/2+\delta_{0}},$ (3.101)

and

$\displaystyle\quad\int_{0}^{\sigma(T)}t^{2}\||H|^{2}\|_{L^{4}}^{4}dt\leq
C\int_{0}^{\sigma(T)}t^{2}\|H\|_{L^{3}}^{2}\|\nabla H\|_{L^{2}}^{4}\|\nabla
H\|_{L^{6}}^{2}dt$ $\displaystyle\leq C\int_{0}^{\sigma(T)}t^{2}(\|\nabla
H\|_{L^{2}}^{4}\|{\rm curl}^{2}H\|_{L^{2}}^{2}+\|\nabla H\|_{L^{2}}^{6})dt\leq
CC_{0}^{1/2+2\delta_{0}}.$ (3.102)

On the other hand, by (3.6), (2.24) and Lemma 3.6, we have

$\displaystyle\quad\int_{\sigma(T)}^{T}\sigma^{2}\|\nabla u\|_{L^{4}}^{4}dt$
$\displaystyle\leq
C\int_{\sigma(T)}^{T}\sigma^{2}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{3}+\|{\rm
curl}^{2}H\|_{L^{2}}^{3})(\|\nabla u\|_{L^{2}}+\|P-\bar{P}\|_{L^{2}}+\|\nabla
H\|_{L^{2}})dt$ $\displaystyle\quad+C\int_{\sigma(T)}^{T}\sigma^{2}(\|\nabla
u\|_{L^{2}}^{4}+\|P-\bar{P}\|_{L^{2}}^{4}+\|P-\bar{P}\|_{L^{4}}^{4}+\|\nabla
H\|_{L^{2}}^{4}+\||H|^{2}\|_{L^{4}}^{4})dt$ $\displaystyle\leq
CC_{0}+CC_{0}^{1/2+2\delta_{0}}\leq CC_{0}^{1/2+2\delta_{0}}.$ (3.103)

Combining (3) and (3), it follows from (3.99) that

$\displaystyle\displaystyle A_{1}(T)+A_{2}(T)\leq
C_{10}C_{0}^{1/2+\delta_{0}/2}.$ (3.104)

Set
$\varepsilon_{5}\triangleq\min\\{\varepsilon_{4},(C_{10}^{-2/\delta_{0}}\\}$,
from (3.104), (3.97) holds when $C_{0}<\varepsilon_{5}$. The proof of Lemma
3.8 is completed.

We now proceed to proof the uniform (in time) upper bound for the density.

###### Lemma 3.9

Let $(\rho,u,H)$ be a smooth solution of (1.1)-(1.4) on ${\Omega}\times(0,T]$
satisfying (3.6). Then there exists a positive constant $\varepsilon_{6}$
depending only on $\mu,$ $\lambda,$ $\nu,$ $\gamma,$ $a$, $s$, $\bar{\rho}$,
$\hat{\rho}$, $\Omega$, $M_{1}$ and $M_{2}$ such that

$\displaystyle\displaystyle\sup_{0\leq t\leq
T}\|\rho(t)\|_{L^{\infty}}\leq\frac{7\hat{\rho}}{4},$ (3.105)

provided $C_{0}\leq\varepsilon_{6}.$

Proof. First, the equation of mass conservation $\eqref{CMHD}_{1}$ can be
equivalently rewritten in the form

$\displaystyle\displaystyle D_{t}\rho=g(\rho)+b^{\prime}(t),$ (3.106)

where

$\displaystyle\displaystyle
D_{t}\rho\triangleq\rho_{t}+u\cdot\nabla\rho,\,g(\rho)\triangleq-\frac{\rho(P-\bar{P})}{2\mu+\lambda},\,b(t)\triangleq-\frac{1}{2\mu+\lambda}\int_{0}^{t}\rho(F+\frac{|H|^{2}}{2})dt.$
(3.107)

Naturally, we shall prove our conclusion by Lemma A.2. It is sufficient to
check that the function $b(t)$ must verify (A.3) with some suitable constants
$N_{0}$, $N_{1}$.

For $t\in[0,\sigma(T)],$ one deduces from (A.1), (A.2), (2.8), (2.9), (2.34),
(3.6) and Lemmas 2.1, 3.6 that for $\delta_{0}$ as in Proposition 3.1 and for
all $0\leq t_{1}\leq t_{2}\leq\sigma(T)$,

$\displaystyle\quad|b(t_{2})-b(t_{1})|=\frac{1}{\lambda+2\mu}\left|\int_{t_{1}}^{t_{2}}\rho(F+\frac{|H|^{2}}{2})dt\right|\leq
C\int_{0}^{\sigma(T)}(\|F\|_{L^{\infty}}+\|H\|^{2}_{L^{\infty}})dt$
$\displaystyle\leq C\int_{0}^{\sigma(T)}\|F\|_{L^{6}}^{\frac{1}{2}}\|\nabla
F\|_{L^{6}}^{\frac{1}{2}}dt+C\int_{0}^{\sigma(T)}\|F\|_{L^{2}}dt+C\int_{0}^{\sigma(T)}\|H\|_{L^{6}}\|\nabla
H\|_{L^{6}}dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{\frac{1}{2}}+\|{\rm
curl}^{2}H\|_{L^{2}}^{\frac{1}{2}})\|\nabla\dot{u}\|_{L^{2}}^{\frac{1}{2}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{\frac{1}{2}}+\|{\rm
curl}^{2}H\|_{L^{2}}^{\frac{1}{2}})\|\nabla H\|_{L^{2}}^{\frac{1}{4}}\|{\rm
curl}^{2}H\|_{L^{2}}^{\frac{3}{4}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{\frac{1}{2}}+\|{\rm
curl}^{2}H\|_{L^{2}}^{\frac{1}{2}})(\|\nabla u\|_{L^{2}}+\|\nabla
H\|_{L^{2}})dt$ $\displaystyle\quad+C\int_{0}^{\sigma(T)}(\|\nabla
u\|_{L^{2}}^{\frac{1}{2}}+\|\nabla
H\|_{L^{2}}^{\frac{1}{2}}+\|P-\bar{P}\|_{L^{2}}^{\frac{1}{2}})\|\nabla\dot{u}\|_{L^{2}}^{\frac{1}{2}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}(\|\nabla
u\|_{L^{2}}^{\frac{1}{2}}+\|\nabla
H\|_{L^{2}}^{\frac{1}{2}}+\|P-\bar{P}\|_{L^{2}}^{\frac{1}{2}})\|\nabla
H\|_{L^{2}}^{\frac{1}{4}}\|{\rm curl}^{2}H\|_{L^{2}}^{\frac{3}{4}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}(\|\nabla
u\|_{L^{2}}^{\frac{1}{2}}+\|\nabla
H\|_{L^{2}}^{\frac{1}{2}}+\|P-\bar{P}\|_{L^{2}}^{\frac{1}{2}})(\|\nabla
u\|_{L^{2}}+\|\nabla H\|_{L^{2}})dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}(\|\nabla u\|_{L^{2}}+\|\nabla
H\|_{L^{2}}+\|P-\bar{P}\|_{L^{2}})dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}\|\nabla H\|_{L^{2}}(\|\nabla
H\|_{L^{2}}+\|{\rm curl}^{2}H\|_{L^{2}})dt\triangleq\sum_{i=1}^{8}B_{i}.$
(3.108)

We have to estimate $B_{i},i=1,2,\cdots,8$ one by one. A directly computation
gives

$\displaystyle B_{1}$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\big{(}t^{\frac{3-2s}{4}}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})\big{)}^{\frac{1}{4}}\big{(}t^{2-s}\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}^{\frac{1}{4}}t^{\frac{6s-11}{16}}dt$
$\displaystyle\leq
C\left(\int_{0}^{\sigma(T)}t^{\frac{3-2s}{4}}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})dt\right)^{\frac{1}{4}}$
$\displaystyle\qquad\times\left(\int_{0}^{\sigma(T)}t^{2-s}\|\nabla\dot{u}\|_{L^{2}}^{2}dt\right)^{\frac{1}{4}}\left(\int_{0}^{\sigma(T)}t^{\frac{6s-11}{8}}dt\right)^{\frac{1}{2}}\leq
CC_{0}^{\delta_{0}/4},$ (3.109)

similarly,

$\displaystyle B_{2}$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\big{(}t^{2-s}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})\big{)}^{\frac{1}{4}}\big{(}t^{\frac{3-2s}{4}}\|\nabla
H\|_{L^{2}}^{2}\big{)}^{\frac{1}{8}}$
$\displaystyle\qquad\times\big{(}t^{\frac{3-2s}{4}}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}\big{)}^{\frac{3}{8}}t^{\frac{4s-7}{8}}dt\leq
CC_{0}^{\delta_{0}/2},$ (3.110) $\displaystyle B_{3}$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\big{(}t^{\frac{3-2s}{4}}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|{\rm
curl}^{2}H\|_{L^{2}}^{2})\big{)}^{\frac{1}{4}}$
$\displaystyle\qquad\times\big{(}t^{\frac{3-2s}{4}}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})\big{)}^{\frac{1}{2}}t^{\frac{3(2s-3)}{8}}dt\leq
CC_{0}^{3\delta_{0}/4},$ (3.111) $\displaystyle B_{4}$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\big{(}\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2}+\|P-\bar{P}\|_{L^{2}}^{2}\big{)}^{\frac{1}{4}}\big{(}t^{2-s}\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}^{\frac{1}{4}}t^{\frac{s-2}{4}}dt$
$\displaystyle\leq CC_{0}^{1/4},$ (3.112) $\displaystyle B_{5}$
$\displaystyle\leq C\int_{0}^{\sigma(T)}\big{(}t^{\frac{3-2s}{4}}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})\big{)}^{\frac{1}{4}}\big{(}t^{\frac{3-2s}{4}}\|\nabla
H\|_{L^{2}}^{2}\big{)}^{\frac{1}{8}}$
$\displaystyle\qquad\times\big{(}t^{\frac{3-2s}{4}}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}\big{)}^{\frac{3}{8}}t^{\frac{3(2s-3)}{16}}dt$
$\displaystyle\quad+CC_{0}^{\frac{1}{4}}\int_{0}^{\sigma(T)}\big{(}t^{\frac{3-2s}{4}}\|\nabla
H\|_{L^{2}}^{2}\big{)}^{\frac{1}{8}}\big{(}t^{\frac{3-2s}{4}}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}\big{)}^{\frac{3}{8}}t^{\frac{2s-3}{8}}dt$
$\displaystyle\leq CC_{0}^{3\delta_{0}/4}+CC_{0}^{1/4+\delta_{0}/2}\leq
CC_{0}^{3\delta_{0}/4},$ (3.113) $\displaystyle B_{6}$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\big{(}t^{\frac{3-2s}{4}}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})\big{)}^{\frac{3}{4}}t^{\frac{3(2s-3)}{16}}dt$
$\displaystyle+CC_{0}^{\frac{1}{4}}\int_{0}^{\sigma(T)}\big{(}t^{\frac{3-2s}{4}}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})\big{)}^{\frac{1}{2}}t^{\frac{2s-3}{8}}dt$ $\displaystyle\leq
CC_{0}^{3\delta_{0}/4}+CC_{0}^{1/4+\delta_{0}/2}\leq CC_{0}^{3\delta_{0}/4},$
(3.114) $\displaystyle B_{7}$ $\displaystyle=\int_{0}^{\sigma(T)}(\|\nabla
u\|_{L^{2}}+\|\nabla H\|_{L^{2}}+\|P-\bar{P}\|_{L^{2}})dt\leq CC_{0}^{1/2},$
(3.115) $\displaystyle B_{8}$
$\displaystyle\leq\int_{0}^{\sigma(T)}\big{(}t^{\frac{3-2s}{4}}\|\nabla
H\|_{L^{2}}^{2}\big{)}^{\frac{1}{2}}\big{(}t^{\frac{3-2s}{4}}\|{\rm
curl}^{2}H\|_{L^{2}}^{2}\big{)}^{\frac{1}{2}}t^{\frac{2s-3}{4}}dt+CC_{0}dt$
$\displaystyle\leq CC_{0}^{1/2+\delta_{0}/2}.$ (3.116)

Putting (3)-(3.116) into (3), we have

$\displaystyle|b(t_{2})-b(t_{1})|\leq C_{11}C_{0}^{\delta_{0}/4}.$ (3.117)

Combining (3.117) with (3.106) and choosing $N_{1}=0$,
$N_{0}=C_{11}C_{0}^{\delta_{0}/4}$, $\bar{\zeta}=\hat{\rho}$ in Lemma A.2 give

$\displaystyle\displaystyle\sup_{t\in[0,\sigma(T)]}\|\rho\|_{L^{\infty}}\leq\hat{\rho}+C_{11}C_{0}^{\delta_{0}/4}\leq\frac{3\hat{\rho}}{2},$
(3.118)

provided
$C_{0}\leq\hat{\varepsilon}_{6}\triangleq\min\\{\varepsilon_{5},\left(\frac{\hat{\rho}}{2C_{11}}\right)^{\frac{4}{\delta_{0}}}\\}.$

On the other hand, for $t\in[\sigma(T),T],\,\,\sigma(T)\leq t_{1}\leq
t_{2}\leq T,$ it follows from (2.8), (2.34), (3.6), (2.29) and Lemma 2.1 that

$\displaystyle\quad|b(t_{2})-b(t_{1})|\leq
C\int_{t_{1}}^{t_{2}}(\|F\|_{L^{\infty}}+\|H\|^{2}_{L^{\infty}})dt$
$\displaystyle\leq\frac{a}{\lambda+2\mu}(t_{2}-t_{1})+C\int_{t_{1}}^{t_{2}}\|F\|_{L^{\infty}}^{8/3}dt+C\int_{t_{1}}^{t_{2}}\|H\|_{L^{\infty}}^{2}dt$
$\displaystyle\leq\frac{a}{\lambda+2\mu}(t_{2}-t_{1})+CC_{0}^{\frac{1}{6}}\int_{\sigma(T)}^{T}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}\|{\rm curl}^{2}H\|_{L^{2}}^{3}+\|\nabla H\|_{L^{2}}^{4})dt$
$\displaystyle\quad+CC_{0}+C\int_{t_{1}}^{t_{2}}(\|\nabla H\|_{L^{2}}\|{\rm
curl}^{2}H\|_{L^{2}}+\|\nabla H\|_{L^{2}}^{2})dt$
$\displaystyle\leq\frac{a}{\lambda+2\mu}(t_{2}-t_{1})+C_{12}C_{0}^{2/3}.$
(3.119)

Now we choose $N_{0}=C_{12}C_{0}^{2/3}$, $N_{1}=\frac{a}{\lambda+2\mu}$ in
(A.3) and set $\bar{\zeta}=\frac{3\hat{\rho}}{2}$ in (A.4). Since for all
$\zeta\geq\bar{\zeta}=\frac{3\hat{\rho}}{2}>\bar{\rho}+1$,

$g(\zeta)=-\frac{a\zeta}{2\mu+\lambda}(\zeta^{\gamma}-\bar{\rho}^{\gamma})\leq-\frac{a}{\lambda+2\mu}=-N_{1}.$

Together with (3.106) and (3), by Lemma A.2, we have

$\displaystyle\displaystyle\sup_{t\in[\sigma(T),T]}\|\rho\|_{L^{\infty}}\leq\frac{3\hat{\rho}}{2}+C_{12}C_{0}^{2/3}\leq\frac{7\hat{\rho}}{4},$
(3.120)

provided
$C_{0}\leq\varepsilon_{6}\triangleq\min\\{\hat{\varepsilon}_{6},(\frac{\hat{\rho}}{4C_{12}})^{3/2}\\}$.
The combination of (3.118) with (3.120) completes the proof of Lemma 3.9.

##  4  A priori estimates (II): higher order estimates

In this section, we derive the time-dependent higher order estimates, which
are necessary for the global existence of classical solutions. Here we adopt
the method of the article [6, 27, 29], and follow their work with a few
modifications. We sketch it here for completeness. Let $(\rho,u,H)$ be a
smooth solution of (1.1)-(1.4) satisfying Proposition 3.1 and the initial
energy $C_{0}\leq\varepsilon_{6}$, and the positive constant $C$ may depend on
$T,$ $\mu$, $\lambda$, $\nu$, $a$, $\gamma$, $\bar{\rho},$ $\hat{\rho},$ $s,$
$\Omega$, $M_{1},M_{2}$, $\|\nabla u_{0}\|_{H^{1}},\|\nabla
H_{0}\|_{H^{1}},\|\rho_{0}-\bar{\rho}\|_{W^{2,q}},\|g\|_{L^{2}},\|P(\rho_{0})-\bar{P}\|_{W^{2,q}}$
for $q\in(3,6)$ where $g\in L^{2}(\Omega)$ is given by compatibility condition
(1.12).

###### Lemma 4.1

There exists a positive constant $C,$ such that

$\displaystyle\sup_{0\leq t\leq T}(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
H\|_{L^{2}}^{2})+\int_{0}^{T}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2}+\|\nabla^{2}H\|_{L^{2}}^{2})dt\leq
C,$ (4.1) $\displaystyle\sup_{0\leq t\leq
T}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2}+\|\nabla^{2}H\|_{L^{2}}^{2})+\int_{0}^{T}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt\leq C,$ (4.2) $\displaystyle\sup_{0\leq t\leq
T}(\|\nabla\rho\|_{L^{6}}+\|u\|_{H^{2}})+\int_{0}^{T}(\|\nabla
u\|_{L^{\infty}}+\|\nabla^{2}u\|_{L^{6}}^{2})dt\leq C.$ (4.3)

Proof. First, taking $\theta=1$ in (3.15) and taking $s=1$ in (3.56) along
with (2.25) gives (4.1). Then choosing $m=0$ in (3) and (3), integrating them
over $(0,T)$, by (3), (4.1) and the compatibility condition (1.12), we have

$\displaystyle\quad\sup_{0\leq t\leq
T}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|H_{t}\|_{L^{2}}^{2}+\|\nabla^{2}H\|_{L^{2}}^{2})+\int_{0}^{T}(\|\nabla\dot{u}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})dt$ $\displaystyle\leq
C+C\int_{0}^{T}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{3}+\|\nabla^{2}H\|_{L^{2}}^{3}+\|\nabla
H\|_{L^{2}}^{4}\|\nabla^{2}H\|_{L^{2}}^{2})dt$ $\displaystyle\leq
C+\frac{1}{2}\sup_{0\leq t\leq
T}(\|\rho^{\frac{1}{2}}\dot{u}\|_{L^{2}}^{2}+\|\nabla^{2}H\|_{L^{2}}^{2}),$
(4.4)

where we have also used Lemma A.1, Lemma 2.2, (3.27) and (3), then we deduce
(4.2) from (4). Based on the Beale-Kato-Majda type inequality (see Lemma A.6),
we can derive (4.3), the estimates on the gradient of density and velocity, in
arguments similar to [6]. This finishes the proof.

###### Lemma 4.2

There exists a positive constant $C$ such that

$\displaystyle\sup_{0\leq t\leq
T}\|\rho^{\frac{1}{2}}u_{t}\|_{L^{2}}^{2}+\int_{0}^{T}\int|\nabla
u_{t}|^{2}dxdt\leq C,$ (4.5) $\displaystyle\sup\limits_{0\leq t\leq
T}\left(\|{\rho\\!-\\!\\!\bar{\rho}}\|_{H^{2}}\\!+\\!\|{P\\!-\\!\\!\bar{P}}\|_{H^{2}}\\!+\\!\|\rho_{t}\|_{H^{1}}\\!+\\!\|P_{t}\|_{H^{1}}\right)\\!+\\!\\!\int_{0}^{T}\\!\\!\left(\|\rho_{tt}\|_{L^{2}}^{2}\\!+\\!\|P_{tt}\|_{L^{2}}^{2}\right)dt\leq
C,$ (4.6) $\displaystyle\sup\limits_{0\leq t\leq T}\sigma(\|\nabla
u_{t}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2})+\int_{0}^{T}\sigma(\|\rho^{\frac{1}{2}}u_{tt}\|_{L^{2}}^{2}+\|H_{tt}\|_{L^{2}}^{2})dt\leq
C.$ (4.7)

Proof. Based on Lemma 4.1, (4.5)-(4.6) can be obtained by the same method as
that in [6]. It remains to prove (4.7). Introducing the function

$K(t)=(\lambda+2\mu)\int({\rm
div}u_{t})^{2}dx+\mu\int|\omega_{t}|^{2}dx+\nu\int|{\rm curl}H_{t}|^{2}dx.$

Since $u_{t}\cdot n=0,H_{t}\cdot n=0$ on $\partial\Omega$, by Lemma A.4, we
have

$\displaystyle\displaystyle\|\nabla u_{t}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2}\leq C(\Omega)K(t).$ (4.8)

Differentiating $\eqref{CMHD}_{2,3}$ with respect to $t,$

$\displaystyle\rho u_{tt}\\!-\\!(\lambda\\!+\\!2\mu)\nabla{\rm
div}u_{t}\\!+\\!\mu\nabla\\!\times\\!\omega_{t}\\!=-\nabla\\!P_{t}\\!-\\!\rho_{t}u_{t}\\!-\\!(\rho
u\\!\cdot\\!\nabla\\!u)_{t}\\!+\\!(H\\!\cdot\\!\nabla\\!H\\!-\\!\nabla|H|^{2}/2)_{t},$
(4.9)

and

$\displaystyle\quad H_{tt}-\nu\nabla\times{\rm curl}H_{t}=(H\cdot\nabla
u-u\cdot\nabla H-H{\rm div}u)_{t},$ (4.10)

then multiplying (4.9) by $2u_{tt}$, multiplying (4.10) $2H_{tt}$
respectively, we obtain

$\displaystyle\quad\frac{d}{dt}K(t)+2\int(\rho|u_{tt}|^{2}+|H_{tt}|^{2})dx$
$\displaystyle=\frac{d}{dt}\Big{(}-\int\rho_{t}|u_{t}|^{2}dx-2\int\rho_{t}u\cdot\nabla
u\cdot u_{t}dx+2\int P_{t}{\rm div}u_{t}dx$
$\displaystyle\qquad\quad-\int(2(H\otimes H)_{t}:\nabla u_{t}-|H|^{2}_{t}{\rm
div}u_{t}dx)\Big{)}$
$\displaystyle\quad+\int\rho_{tt}|u_{t}|^{2}dx+2\int(\rho_{t}u\cdot\nabla
u)_{t}\cdot u_{t}dx-2\int\rho(u\cdot\nabla u)_{t}\cdot u_{tt}dx$
$\displaystyle\quad-2\int P_{tt}{\rm div}u_{t}dx+\int(2(H\otimes
H)_{tt}:\nabla u_{t}-|H|^{2}_{tt}{\rm div}u_{t})dx$
$\displaystyle\quad+2\int(H\cdot\nabla u-u\cdot\nabla H-H{\rm div}u)_{t}\cdot
H_{tt}dx$
$\displaystyle\triangleq\frac{d}{dt}K_{0}+\sum\limits_{i=1}^{6}K_{i}.$ (4.11)

Let us estimate $K_{i}$, $i=0,1,\cdots,6.$ We conclude from
$\eqref{CMHD}_{1}$, (2.34), (4.2), (4.3), (4.5), (4.6), (4.8) and Sobolev’s,
Poincaré’s inequalities that

$\displaystyle K_{0}$ $\displaystyle\leq\left|\int{\rm div}(\rho
u)\,|u_{t}|^{2}dx\right|+C\|\rho_{t}\|_{L^{3}}\|u\|_{L^{\infty}}\|\nabla
u\|_{L^{2}}\|u_{t}\|_{L^{6}}+C\|P_{t}\|_{L^{2}}\|\nabla u_{t}\|_{L^{2}}$
$\displaystyle\quad+C\|H\|_{L^{\infty}}\|H_{t}\|_{L^{2}}\|\nabla
u_{t}\|_{L^{2}}$ $\displaystyle\leq\frac{1}{2}K(t)+C,$ (4.12) $\displaystyle
K_{1}$
$\displaystyle\leq\left|\int\rho_{tt}\,|u_{t}|^{2}dx\right|=\left|\int{\rm
div}(\rho u)_{t}\,|u_{t}|^{2}dx\right|=2\left|\int(\rho_{t}u+\rho
u_{t})\cdot\nabla u_{t}\cdot u_{t}dx\right|$ $\displaystyle\leq C\|\nabla
u_{t}\|_{L^{2}}^{2}K(t)+C\|\nabla u_{t}\|_{L^{2}}^{2}+C,$ (4.13)
$\displaystyle K_{2}$ $\displaystyle+K_{3}+K_{4}\leq
C\|\rho_{tt}\|_{L^{2}}^{2}+C\|\nabla
u_{t}\|_{L^{2}}^{2}+\|\rho^{{1/2}}u_{tt}\|_{L^{2}}^{2}+C\|P_{tt}\|_{L^{2}}^{2}+C,$
(4.14) $\displaystyle K_{5}$
$\displaystyle\leq\frac{1}{2}\|H_{tt}\|_{L^{2}}^{2}+C\|H_{t}\|_{L^{2}}^{2}K(t)+C(\|\nabla
H_{t}\|_{L^{2}}^{2}+\|\nabla u_{t}\|_{L^{2}}^{2}),$ (4.15) $\displaystyle
K_{6}$ $\displaystyle\leq\frac{1}{2}\|H_{tt}\|_{L^{2}}^{2}+C(\|\nabla
H_{t}\|_{L^{2}}^{2}+\|\nabla u_{t}\|_{L^{2}}^{2}).$ (4.16)

Consequently, multiplying (4) by $\sigma$, together with (4)-(4.16), we get

$\displaystyle\quad\frac{d}{dt}(\sigma K(t)-\sigma
K_{0})+\sigma\int(\rho|u_{tt}|^{2}+|H_{tt}|^{2})dx$ $\displaystyle\leq
C(1+\|\nabla u_{t}\|_{L^{2}}^{2})\sigma K(t)+C(1+\|\nabla
u_{t}\|_{L^{2}}^{2}+\|\nabla
H_{t}\|_{L^{2}}^{2}+\|\rho_{tt}\|_{L^{2}}^{2}+\|P_{tt}\|_{L^{2}}^{2}),$ (4.17)

By Gronwall’s inequality, (4.2), (4.5), (4.6) and (4), we derive that

$\displaystyle\displaystyle\sup_{0\leq t\leq T}(\sigma
K(t))+\int_{0}^{T}\sigma(\|\rho^{\frac{1}{2}}u_{tt}\|_{L^{2}}^{2}+\|H_{tt}\|_{L^{2}}^{2})dt\leq
C.$ (4.18)

As a result, by (4.8), we get (4.7). This finishes the proof .

###### Lemma 4.3

There exists a positive constant $C$ so that for any $q\in(3,6),$

$\displaystyle\sup_{t\in[0,T]}\left(\|\rho-\bar{\rho}\|_{W^{2,q}}+\|P-\bar{P}\|_{W^{2,q}}\right)\leq
C,$ (4.19) $\displaystyle\sup_{t\in[0,T]}\sigma(\|\nabla
u\|_{H^{2}}^{2}+\|\nabla H\|_{H^{2}}^{2})$
$\displaystyle\qquad+\int_{0}^{T}\left(\|\nabla u\|_{H^{2}}^{2}+\|\nabla
H\|_{H^{2}}^{2}+\|\nabla^{2}u\|^{p_{0}}_{W^{1,q}}+\sigma\|\nabla
u_{t}\|_{H^{1}}^{2}\right)dt\leq C,$ (4.20)

where $p_{0}=\frac{9q-6}{10q-12}\in(1,\frac{7}{6}).$

Proof. Let’s start with (4.20). By Lemma 4.1 and Poincaré’s, Sobolev’s
inequalities, one can check that

$\displaystyle\|\nabla(\rho\dot{u})\|_{L^{2}}$
$\displaystyle\leq\||\nabla\rho||u_{t}|\|_{L^{2}}\\!+\\!\|\rho\nabla
u_{t}\|_{L^{2}}\\!+\\!\||\nabla\rho||u||\nabla
u|\|_{L^{2}}\\!+\\!\|\rho|\nabla
u|^{2}\|_{L^{2}}\\!+\\!\|\rho|u||\nabla^{2}u|\|_{L^{2}}$ $\displaystyle\leq
C+C\|\nabla u_{t}\|_{L^{2}}.$ (4.21)

Consequently, together with (4.6) and Lemma 4.1, it yields

$\displaystyle\|\nabla^{2}u\|_{H^{1}}$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{H^{1}}+\|H\cdot\nabla
H\|_{H^{1}}+\|P-\bar{P}\|_{H^{2}}+\||H|^{2}\|_{H^{2}}+\|u\|_{L^{2}})$
$\displaystyle\leq C+C\|\nabla u_{t}\|_{L^{2}}.$ (4.22)

It then follows from (4), (4.3), (4.5) and (4.7) that

$\displaystyle\displaystyle\sup\limits_{0\leq t\leq T}\sigma\|\nabla
u\|_{H^{2}}^{2}+\int_{0}^{T}\|\nabla u\|_{H^{2}}^{2}dt\leq C.$ (4.23)

Next, from (2.2)3, (2.26), it follows

$\displaystyle\|\nabla^{2}H\|_{H^{1}}$ $\displaystyle\leq
C(\|H_{t}\|_{H^{1}}+\|u\cdot\nabla H\|_{H^{1}}+\|H\cdot\nabla
u\|_{H^{1}}+\|H{\rm div}u\|_{H^{1}}+\|\nabla H\|_{L^{2}})$ $\displaystyle\leq
C+C\|\nabla H_{t}\|_{L^{2}}.$ (4.24)

Similarly, from (4), (4.1) and (4.3), we obtain

$\displaystyle\displaystyle\sup\limits_{0\leq t\leq T}\sigma\|\nabla
H\|_{H^{2}}^{2}+\int_{0}^{T}\|\nabla H\|_{H^{2}}^{2}dt\leq C.$ (4.25)

Next, we deduce from Lemma 4.1 and (4.6) that

$\displaystyle\displaystyle\|\nabla^{2}u_{t}\|_{L^{2}}$ $\displaystyle\leq
C(\|(\rho\dot{u})_{t}\|_{L^{2}}+\|\nabla P_{t}\|_{L^{2}}+\|((\nabla\times
H)\times H)_{t}\|_{L^{2}}+\|u_{t}\|_{L^{2}})$ $\displaystyle\leq
C\|\rho^{\frac{1}{2}}u_{tt}\|_{L^{2}}+C\|\nabla u_{t}\|_{L^{2}}+C\|\nabla
H_{t}\|_{L^{2}}+C,$ (4.26)

where in the first inequality, we have utilized the $L^{p}$-estimate for the
following elliptic system

$\begin{cases}\mu\Delta u_{t}+(\lambda+\mu)\nabla{\rm
div}u_{t}=(\rho\dot{u})_{t}+\nabla P_{t}+((\nabla\times H)\times
H)_{t}\,\,\,&\text{in}\,\,\Omega,\\\ u_{t}\cdot
n=0\,\,\,\text{and}\,\,\,\omega_{t}\times
n=0\,\,&\text{on}\,\,\partial\Omega.\end{cases}$ (4.27)

Together with (4) and (4.7) yields

$\displaystyle\displaystyle\int_{0}^{T}\sigma\|\nabla
u_{t}\|_{H^{1}}^{2}dt\leq C.$ (4.28)

By Sobolev’s inequality, (2.34), (4.3), (4.6) and (4.7), we get for any
$q\in(3,6)$,

$\displaystyle\displaystyle\|\nabla(\rho\dot{u})\|_{L^{q}}$ $\displaystyle\leq
C\|\nabla\rho\|_{L^{q}}(\|\nabla\dot{u}\|_{L^{q}}+\|\nabla\dot{u}\|_{L^{2}}+\|\nabla
u\|_{L^{2}}^{2})+C\|\nabla\dot{u}\|_{L^{q}}$ $\displaystyle\leq
C\sigma^{-\frac{1}{2}}+C\|\nabla
u\|_{H^{2}}+C\sigma^{-\frac{1}{2}}(\sigma\|\nabla
u_{t}\|_{H^{1}}^{2})^{\frac{3(q-2)}{4q}}+C.$ (4.29)

Integrating this inequality over $[0,T],$ by (4.2) and (4.28), we have

$\displaystyle\displaystyle\int_{0}^{T}\|\nabla(\rho\dot{u})\|_{L^{q}}^{p_{0}}dt\leq
C.$ (4.30)

On the other hand, (4.6) gives

$\displaystyle\displaystyle(\|\nabla^{2}P\|_{L^{q}})_{t}$ $\displaystyle\leq
C\|\nabla u\|_{L^{\infty}}\|\nabla^{2}P\|_{L^{q}}+C\|\nabla^{2}u\|_{W^{1,q}}$
$\displaystyle\leq C(1+\|\nabla
u\|_{L^{\infty}})\|\nabla^{2}P\|_{L^{q}}+C(1+\|\nabla
u_{t}\|_{L^{2}})+C\|\nabla(\rho\dot{u})\|_{L^{q}},$ (4.31)

where in the last inequality we have used the following simple fact that

$\displaystyle\displaystyle\|\nabla^{2}u\|_{W^{1,q}}$ $\displaystyle\leq
C(1+\|\nabla
u_{t}\|_{L^{2}}+\|\nabla(\rho\dot{u})\|_{L^{q}}+\|\nabla^{2}P\|_{L^{q}}),$
(4.32)

due to (2.2), (2.2), (4.2) and (4.6).

Hence, applying Gronwall’s inequality in (4), we deduce from (4.3), (4.5) and
(4.30) that

$\displaystyle\displaystyle\sup_{t\in[0,T]}\|\nabla^{2}P\|_{L^{q}}\leq C,$
(4.33)

which along with (4.5), (4.6), (4.32) and (4.30) also gives

$\displaystyle\displaystyle\sup_{t\in[0,T]}\|P-\bar{P}\|_{W^{2,q}}+\int_{0}^{T}\|\nabla^{2}u\|_{W^{1,q}}^{p_{0}}dt\leq
C.$ (4.34)

Similarly, one has

$\displaystyle\displaystyle\sup\limits_{0\leq t\leq
T}\|\rho-\bar{\rho}\|_{W^{2,q}}\leq C,$ (4.35)

which together with (4.34) gives (4.19). The proof of Lemma 4.3 is finished.

###### Lemma 4.4

There exists a positive constant $C$ such that

$\displaystyle\sup_{0\leq t\leq
T}\sigma\left(\|\rho^{1/2}u_{tt}\|_{L^{2}}+\|H_{tt}\|_{L^{2}}+\|\nabla
u_{t}\|_{H^{1}}+\|\nabla H_{t}\|_{H^{1}}+\|\nabla^{2}H\|_{H^{2}}+\|\nabla
u\|_{W^{2,q}}\right)$ $\displaystyle+\int_{0}^{T}\sigma^{2}(\|\nabla
u_{tt}\|_{2}^{2}+\|\nabla H_{tt}\|_{2}^{2})dt\leq C,$ (4.36)

for any $q\in(3,6)$.

Proof. Differentiating $\eqref{CMHD}_{2,3}$ with respect to $t$ twice,
multiplying them by $2u_{tt}$ and $2H_{tt}$ respectively, and integrating over
$\Omega$ lead to

$\displaystyle\quad\frac{d}{dt}\int(\rho|u_{tt}|^{2}+|H_{tt}|^{2})dx$
$\displaystyle\quad+2(\lambda+2\mu)\int({\rm
div}u_{tt})^{2}dx+2\mu\int|\omega_{tt}|^{2}dx+2\nu\int|{\rm
curl}H_{tt}|^{2}dx$ $\displaystyle=-8\int\rho u^{i}_{tt}u\cdot\nabla
u^{i}_{tt}dx-2\int(\rho u)_{t}\cdot\left[\nabla(u_{t}\cdot u_{tt})+2\nabla
u_{t}\cdot u_{tt}\right]dx$
$\displaystyle\quad-2\int(\rho_{tt}u+2\rho_{t}u_{t})\cdot\nabla u\cdot
u_{tt}dx-2\int(\rho u_{tt}\cdot\nabla u\cdot u_{tt}-P_{tt}{\rm div}u_{tt})dx$
$\displaystyle\quad-2\int(H\cdot\nabla
H-\nabla|H|^{2}/2)_{tt}u_{tt}dx+2\int(H\cdot\nabla u-u\cdot\nabla H-H{\rm
div}u)_{tt}H_{tt}dx$ $\displaystyle\triangleq\sum_{i=1}^{6}R_{i}.$ (4.37)

Let us estimate $R_{i}$ for $i=1,\cdots,6$. Hölder’s inequality and (4.3) give

$\displaystyle\displaystyle R_{1}$ $\displaystyle\leq
C\|\sqrt{\rho}u_{tt}\|_{L^{2}}\|\nabla
u_{tt}\|_{L^{2}}\|u\|_{L^{\infty}}\leq\delta\|\nabla
u_{tt}\|_{L^{2}}^{2}+C(\delta)\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}.$ (4.38)

By (4.2), (4.5), (4.6) and (4.7), we conclude that

$\displaystyle R_{2}$ $\displaystyle\leq\delta\|\nabla
u_{tt}\|_{L^{2}}^{2}+C(\delta)\|\nabla u_{t}\|_{L^{2}}^{3}+C(\delta)\|\nabla
u_{t}\|_{L^{2}}^{2},$ (4.39) $\displaystyle R_{3}$
$\displaystyle\leq\delta\|\nabla
u_{tt}\|_{L^{2}}^{2}+C(\delta)\|\rho_{tt}\|_{L^{2}}^{2}+C(\delta)\|\nabla
u_{t}\|_{L^{2}}^{2},$ (4.40) $\displaystyle R_{4}$
$\displaystyle\leq\delta\|\nabla
u_{tt}\|_{L^{2}}^{2}+C(\delta)\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+C(\delta)\|P_{tt}\|^{2}_{L^{2}},$
(4.41) $\displaystyle R_{5}$ $\displaystyle\leq\delta\|\nabla
u_{tt}\|_{L^{2}}^{2}+C(\delta)\|H_{tt}\|^{2}_{L^{2}}+C(\delta)\|\nabla
H_{t}\|^{3}_{L^{2}},$ (4.42) $\displaystyle R_{6}$
$\displaystyle\leq\delta(\|\nabla u_{tt}\|_{L^{2}}^{2}+\|\nabla
H_{tt}\|_{L^{2}}^{2})+C(\delta)\|H_{tt}\|^{2}_{L^{2}}$
$\displaystyle\quad+C(\delta)(\|\nabla u_{t}\|_{L^{2}}\|\nabla
H_{t}\|^{2}_{L^{2}}+\|\nabla u_{t}\|^{2}_{L^{2}}\|\nabla
H_{t}\|^{2}_{L^{2}}),$ (4.43)

Substituting these estimates of $R_{i}(i=1,\cdots,6)$ into (4), utilizing the
fact that

$\displaystyle\displaystyle\|\nabla u_{tt}\|_{L^{2}}\leq C(\|{\rm
div}u_{tt}\|_{L^{2}}+\|\omega_{tt}\|_{L^{2}}),\quad\|\nabla
H_{tt}\|_{L^{2}}\leq C\|{\rm curl}H_{tt}\|_{L^{2}},$ (4.44)

due to Lemma A.4 since $u_{tt}\cdot n=0,H_{tt}\cdot n=0,$ on $\partial\Omega,$
and then choosing $\delta$ small enough, we can get

$\displaystyle\frac{d}{dt}(\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+\|H_{tt}\|^{2}_{L^{2}})+\|\nabla
u_{tt}\|_{L^{2}}^{2}+\|\nabla H_{tt}\|_{L^{2}}^{2}$ $\displaystyle\leq
C(\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+\|H_{tt}\|^{2}_{L^{2}}+\|\rho_{tt}\|^{2}_{L^{2}}+\|P_{tt}\|^{2}_{L^{2}}+\|\nabla
u_{t}\|^{3}_{L^{2}}+\|\nabla H_{t}\|^{3}_{L^{2}})$
$\displaystyle\quad+C(\|\nabla u_{t}\|_{L^{2}}\|\nabla
H_{t}\|^{2}_{L^{2}}+\|\nabla u_{t}\|^{2}_{L^{2}}\|\nabla
H_{t}\|^{2}_{L^{2}}),$ (4.45)

which together with (4.6), (4.7), and by Gronwall’s inequality yields that

$\displaystyle\quad\sup_{0\leq t\leq
T}\sigma^{2}(\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+\|H_{tt}\|^{2}_{L^{2}})+\int_{0}^{T}\sigma^{2}(\|\nabla
u_{tt}\|_{L^{2}}^{2}+\|\nabla H_{tt}\|_{L^{2}}^{2})dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\sigma(\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+\|H_{tt}\|^{2}_{L^{2}})dt+\int_{0}^{T}\sigma^{2}(\|\rho_{tt}\|^{2}_{L^{2}}+\|P_{tt}\|^{2}_{L^{2}}+\|\nabla
u_{t}\|^{2}_{L^{2}})dt$ $\displaystyle\quad+\int_{0}^{T}\sigma^{2}(\|\nabla
u_{t}\|^{3}_{L^{2}}+\|\nabla H_{t}\|^{3}_{L^{2}}+\|\nabla
u_{t}\|_{L^{2}}\|\nabla H_{t}\|^{2}_{L^{2}}+\|\nabla
u_{t}\|^{2}_{L^{2}}\|\nabla H_{t}\|^{2}_{L^{2}})dt$ $\displaystyle\leq C.$
(4.46)

Furthermore, it follows from (2.26), (2.2), (4) and (4.7) that

$\displaystyle\quad\sup_{0\leq t\leq
T}(\sigma\|\nabla^{2}u_{t}\|_{L^{2}}+\sigma\|\nabla^{2}H_{t}\|_{L^{2}})$
$\displaystyle\leq
C\sigma(1+\|\rho^{1/2}u_{tt}\|_{L^{2}}+\|H_{tt}\|_{L^{2}}+\|\nabla
u_{t}\|_{L^{2}}+\|\nabla H_{t}\|_{L^{2}})\leq C.$ (4.47)

Finally, we deduce from (4.7), (4.19), (4.20), (4), (4), (4.32), (4) and (4)
that

$\displaystyle\displaystyle\quad\sigma\|\nabla^{2}u\|_{W^{1,q}}\leq
C\sigma(1+\|\nabla u_{t}\|_{L^{2}}+\|\nabla
H_{t}\|_{L^{2}}+\|\nabla(\rho\dot{u})\|_{L^{q}}+\|\nabla^{2}P\|_{L^{q}})$
$\displaystyle\leq C(1+\sigma\|\nabla
u\|_{H^{2}}+\sigma^{\frac{1}{2}}(\sigma\|\nabla
u_{t}\|_{H^{1}}^{2})^{\frac{3(q-2)}{4q}})\leq
C+C\sigma^{\frac{1}{2}}(\sigma^{-1})^{\frac{3(q-2)}{4q}}\leq C,$ (4.48)

and

$\displaystyle\displaystyle\sigma\|\nabla^{2}H\|_{H^{2}}\leq
C\sigma(1+\|\nabla H_{t}\|_{H^{1}}+\|\nabla u\|_{H^{2}}\|\nabla
H\|_{H^{2}})\leq C,$ (4.49)

together with (4) and (4) yields (4.4) and this completes the proof of Lemma
4.4.

##  5 Proof of Theorem 1.1

With all the a priori estimates in Section 3 and Section 4 at hand, we are
going to prove the main result of the paper in this section.

Proof of Theorem 1.1. By Lemma 2.5, there exists a $T_{*}>0$ such that the
system (1.1)-(1.4) has a unique classical solution $(\rho,u,H)$ on
$\Omega\times(0,T_{*}]$. One may use the a priori estimates, Proposition 3.1
and Lemmas 4.2-4.4 to extend the classical solution $(\rho,u,H)$ globally in
time.

First, by the definition of (3.1)-(3.5), the assumption of the initial data
(1.10) and (3.95), one immediately checks that

$\displaystyle\displaystyle
0\leq\rho_{0}\leq\hat{\rho},\,\,A_{1}(0)+A_{2}(0)=0,\,\,A_{3}(0)\leq
C_{0}^{\delta_{0}},\,\,A_{4}(0)+A_{5}(0)\leq C_{0}^{\delta_{0}}.$ (5.1)

Therefore, there exists a $T_{1}\in(0,T_{*}]$ such that

$\begin{cases}0\leq\rho_{0}\leq 2\hat{\rho},\,\,A_{1}(T)+A_{2}(T)\leq
2C_{0}^{\frac{1}{2}},\\\ A_{3}(T)\leq
2C_{0}^{\delta_{0}},\,\,A_{4}(\sigma(T))+A_{5}(\sigma(T))\leq
2C_{0}^{\delta_{0}},\end{cases}$ (5.2)

hold for $T=T_{1}.$ Next, we set

$\displaystyle\displaystyle T^{*}=\sup\\{T\,|\,{\rm\eqref{pf2}\ holds}\\}.$
(5.3)

Then $T^{*}\geq T_{1}>0$. Hence, for any $0<\tau<T\leq T^{*}$ with $T$ finite,
it follows from Lemmas 4.1-4.4 that

$\begin{cases}\rho-\bar{\rho}\in C([0,T];H^{2}\cap W^{2,q}),\\\ (u,H)\in
C([\tau,T];H^{2}),\quad(\nabla u_{t},\nabla H_{t})\in
C([\tau,T];L^{q});\end{cases}$ (5.4)

where one has taken advantage of the standard embedding

$L^{\infty}(\tau,T;H^{1})\cap H^{1}(\tau,T;H^{-1})\hookrightarrow
C\left([\tau,T];L^{q}\right),\quad\mbox{ for any }q\in[2,6).$

Due to (4.5), (4.7), (4.4) and $\eqref{CMHD}_{1}$, we obtain

$\displaystyle\quad\int_{\tau}^{T}\left|\left(\int\rho|u_{t}|^{2}dx\right)_{t}\right|dt\leq\int_{\tau}^{T}\left(\|\rho_{t}|u_{t}|^{2}\|_{L^{1}}+2\|\rho
u_{t}\cdot u_{tt}\|_{L^{1}}\right)dt$ $\displaystyle\leq
C\int_{\tau}^{T}\left(\|\rho^{\frac{1}{2}}|u_{t}|^{2}\|_{L^{2}}\|\nabla
u\|_{L^{\infty}}+\|u\|_{L^{6}}\|\nabla\rho\|_{L^{2}}\|u_{t}\|^{2}_{L^{6}}+\|\rho^{\frac{1}{2}}u_{tt}\|_{L^{2}}\right)dt\leq
C,$

which together with (5.4) yields

$\displaystyle\displaystyle\rho^{1/2}u_{t},\quad\rho^{1/2}\dot{u}\in
C([\tau,T];L^{2}).$ (5.5)

Finally, we claim that

$\displaystyle\displaystyle T^{*}=\infty.$ (5.6)

Otherwise, $T^{*}<\infty$. Then by Proposition 3.1, it holds that

$\begin{cases}0\leq\rho\leq\frac{7}{4}\hat{\rho},\,\,A_{1}(T^{*})+A_{2}(T^{*})\leq
C_{0}^{\frac{1}{2}},\\\ A_{3}(T^{*})\leq
C_{0}^{\delta_{0}},\,\,A_{4}(\sigma(T^{*}))+A_{5}(\sigma(T^{*}))\leq
C_{0}^{\delta_{0}},\end{cases}$ (5.7)

It follows from Lemmas 4.3, 4.4 and (5.5) that
$(\rho(x,T^{*}),u(x,T^{*}),H(x,T^{*}))$ satisfies the initial data condition
(1.9)-(1.10), (1.12), where
$g(x)\triangleq\sqrt{\rho}\dot{u}(x,T^{*}),\,\,x\in\Omega.$ Thus, Lemma 2.5
implies that there exists some $T^{**}>T^{*}$ such that (5.2) holds for
$T=T^{**}$, which contradicts the definition of $T^{*}.$ As a result, (5.6)
holds. By Lemmas 2.5 and 4.1-4.4, it indicates that $(\rho,u,H)$ is in fact
the unique classical solution defined on $\Omega\times(0,T]$ for any
$0<T<T^{*}=\infty.$

Finally, with (2), (2.6), (3.17), (2.23), (2), (3) and (3) at hand, (1.15) can
be obtained in similar arguments as used in [6], and we omit the details. The
proof of Theorem 1.1 is finished.

##  Appendix A Some basic theories and lemmas

In this appendix, we review some elementary inequalities and important lemmas
that are used extensively in this paper.

First, we recall the well-known Gagliardo-Nirenberg inequality (see [34]).

######  Lemma A.1 (Gagliardo-Nirenberg)

Assume that $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^{3}$. For
$p\in[2,6],\,q\in(1,\infty),$ and $r\in(3,\infty),$ there exist two generic
constants $C_{1},\,\,C_{2}>0$ which may depend on $p$, $q$ and $r$ such that
for any $f\in H^{1}({{\Omega}})$ and $g\in L^{q}({\Omega})\cap
D^{1,r}({\Omega}),$

$\|f\|_{L^{p}({\Omega})}\leq C_{1}\|f\|_{L^{2}}^{\frac{6-p}{2p}}\|\nabla
f\|_{L^{2}}^{\frac{3p-6}{2p}}+C_{2}\|f\|_{L^{2}},$ (A.1)
$\|g\|_{C\left(\overline{{\Omega}}\right)}\leq
C_{1}\|g\|_{L^{q}}^{q(r-3)/(3r+q(r-3))}\|\nabla
g\|_{L^{r}}^{3r/(3r+q(r-3))}+C_{2}\|g\|_{L^{2}}.$ (A.2)

Moreover, if $f\cdot n|_{\partial\Omega}=0$ ,$g\cdot n|_{\partial\Omega}=0$ ,
then the constant $C_{2}=0.$

In order to get the uniform (in time) upper bound of the density $\rho,$ we
need the following Zlotnik inequality in [48].

###### Lemma A.2

Suppose the function $y$ satisfy

$\displaystyle y^{\prime}(t)=g(y)+b^{\prime}(t)\mbox{ on }[0,T],\quad
y(0)=y^{0},$

with $g\in C(R)$ and $y,b\in W^{1,1}(0,T).$ If $g(\infty)=-\infty$ and

$b(t_{2})-b(t_{1})\leq N_{0}+N_{1}(t_{2}-t_{1})$ (A.3)

for all $0\leq t_{1}<t_{2}\leq T$ with some $N_{0}\geq 0$ and $N_{1}\geq 0,$
then

$\displaystyle
y(t)\leq\max\left\\{y^{0},\overline{\zeta}\right\\}+N_{0}<\infty\mbox{ on
}[0,T],$

where $\overline{\zeta}$ is a constant such that

$g(\zeta)\leq-N_{1}\quad\mbox{ for }\quad\zeta\geq\overline{\zeta}.$ (A.4)

Consider the Lamé’s system

$\begin{cases}-\mu\Delta u-(\lambda+\mu)\nabla{\rm div}u=f\,\,&in~{}\Omega,\\\
u\cdot n=0\,\,\text{and}\,\,{\rm curl}u\times
n=0\,\,&on\,\,\partial\Omega,\end{cases}$ (A.5)

Then, the following estimate is standard (see [1]).

###### Lemma A.3

For the Lamé’s equation (A.5), one has

(1) If $f\in W^{k,q}$ for some $q\in(1,\infty),\,\,k\geq 0,$ then there exists
a unique solution $u\in W^{k+2,q},$ such that

$\|u\|_{W^{k+2,q}}\leq C(\|f\|_{W^{k,q}}+\|u\|_{L^{q}});$

(2) If $f=\nabla g$ and $g\in W^{k,q}$ for some $q\geq 1,\,\,k\geq 0,$ then
there exists a unique weak solution $u\in W^{k+1,q},$ such that

$\|u\|_{W^{k+1,q}}\leq C(\|g\|_{W^{k,q}}+\|u\|_{L^{q}}).$

The following two lemmas are given in Theorem 3.2 in [38] and Propositions
2.6-2.9 in [2].

###### Lemma A.4

Let $k\geq 0$ be a integer, $1<q<+\infty$, and assume that $\Omega$ is a
simply connected bounded domain in $\mathbb{R}^{3}$ with $C^{k+1,1}$ boundary
$\partial\Omega$. Then for $v\in W^{k+1,q}$ with $v\cdot n=0$ on
$\partial\Omega$, it holds that

$\|v\|_{W^{k+1,q}}\leq C(\|{\rm div}v\|_{W^{k,q}}+\|{\rm curl}v\|_{W^{k,q}}).$

In particular, for $k=0$, we have

$\|\nabla v\|_{L^{q}}\leq C(\|{\rm div}v\|_{L^{q}}+\|{\rm curl}v\|_{L^{q}}).$

###### Lemma A.5

Let $k\geq 0$ be a integer, $1<q<+\infty$. Suppose that $\Omega$ is a bounded
domain in $\mathbb{R}^{3}$ and its $C^{k+1,1}$ boundary $\partial\Omega$ only
has a finite number of 2-dimensional connected components. Then for $v\in
W^{k+1,q}$ with $v\times n=0$ on $\partial\Omega$, we have

$\|v\|_{W^{k+1,q}}\leq C(\|{\rm div}v\|_{W^{k,q}}+\|{\rm
curl}v\|_{W^{k,q}}+\|v\|_{L^{q}}).$

In particular, if $\Omega$ has no holes, then

$\|v\|_{W^{k+1,q}}\leq C(\|{\rm div}v\|_{W^{k,q}}+\|{\rm curl}v\|_{W^{k,q}}).$

Next, similar to [3, 23, 22], we need a Beale-Kato-Majda type inequality with
respect to the slip boundary condition (1.3) which is given in [6].

###### Lemma A.6

For $3<q<\infty$, assume that $u\cdot n=0$ and ${\rm curl}u\times n=0$ on
$\partial\Omega$, $u\in W^{2,q}$, then there is a constant $C=C(q,\Omega)$
such that the following estimate holds

$\displaystyle\begin{aligned} \|\nabla u\|_{L^{\infty}}\leq C\left(\|{\rm
div}u\|_{L^{\infty}}+\|{\rm
curl}u\|_{L^{\infty}}\right)\ln(e+\|\nabla^{2}u\|_{L^{q}})+C\|\nabla
u\|_{L^{2}}+C.\end{aligned}$

Finally, we consider the problem

$\begin{cases}{\rm div}v=f\,\,\,\,in\,\,\Omega,\\\
v=0\,\,\,\text{on}\,\,\,{\partial\Omega}.\end{cases}$ (A.6)

One has the following conclusion (see [15], Theorem III.3.1).

###### Lemma A.7

There exists a linear operator operator
$\mathcal{B}=[\mathcal{B}_{1},\mathcal{B}_{2},\mathcal{B}_{3}]$ enjoying the
properties:

1)

$\mathcal{B}:\\{f\in
L^{p}({\Omega})|\int_{\Omega}fdx=0\\}\mapsto(W^{1,p}_{0}({\Omega}))^{3}$

is a bounded linear operator, that is,

$\|\mathcal{B}[f]\|_{W^{1,p}_{0}({\Omega})}\leq
C(p)\|f\|_{L^{p}({\Omega})},\mbox{ for any }p\in(1,\infty),$ (A.7)

2) The function $v=\mathcal{B}[f]$ solve the problem (A.6).

3) if $f$ can be written in the form $f={\rm div}g$ for a certain $g\in
L^{r}({\Omega}),g\cdot n|_{\partial{\Omega}}=0,$ then

$\|\mathcal{B}[f]\|_{L^{r}({\Omega})}\leq C(r)\|g\|_{L^{r}({\Omega})},\mbox{
for any }r\in(1,\infty).$ (A.8)

## Acknowledgements

This research was partially supported by National Natural Sciences Foundation
of China No. 11671027, 11901025, 11971020, 11971217.

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