# Global Well-posedness of Classical Solutions to the Compressible Navier-
Stokes-Poisson 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 consider the initial-boundary-value problem of the isentropic compressible
Navier-Stokes-Poisson equations subject to large and non-flat doping profile
in 3D bounded domain with slip boundary condition and vacuum. The global well-
posedness of classical solution is established with small initial energy but
possibly large oscillations and vacuum. The steady state (except velocity) and
the doping profile are allowed to be of large variation.

††E-mail addresses<EMAIL_ADDRESS>(Y. Chen<EMAIL_ADDRESS>(B.
Huang<EMAIL_ADDRESS>(X. Shi)

Keywords: Navier-Stokes-Poisson equations; global classical solutions; slip
boundary condition; large oscillations; non-flat doping profile.

AMS Subject Classifications: 35Q35, 35B40, 76N10

##  1 Introduction

In this paper, we consider the compressible Navier-Stokes-Poisson (NSP)
equations for the dynamics of charged particles of electrons (see [25]) in a
domain $\Omega\subset\mathbb{R}^{3}$, which can be written as

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

where $(x,t)\in\Omega\times(0,T]$, $t\geq 0$ is time, $x\in\mathbb{R}^{3}$ is
the spatial coordinate. The unknown functions
$\rho,u=(u^{1},u^{2},u^{3}),P=P(\rho)$ and $\Phi$ denote the electron density,
the particle velocity, pressure and the electrostatic potential, respectively.
Here we consider the isentropic flows with $\gamma$-law pressure
$P(\rho)=a\rho^{\gamma}$, where $a>0$ and $\gamma>1$ are some physical
parameters. The given function $\tilde{\rho}=\tilde{\rho}(x)>0$ is the doping
profile, which describes the density of fixed, positively charged background
ions. The constants $\mu$ and $\lambda$ are the shear viscosity and bulk
coefficients respectively satisfying $\mu>0,2\mu+3\lambda\geq 0.$ In addition,
the system is solved subject to the given initial data

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

and boundary condition

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

where $n$ is the unit outer normal to $\partial\Omega$. The boundary condition
(1.3) on the velocity is a special Navier-type slip boundary condition (see
[7, 37]), in which 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. This type of boundary condition was first introduced by Navier in
[26], which was followed by great many applications, numerical studies and
analysis for various fluid mechanical problems, see, for instance [8, 14, 38,
37, 6] and the references therein. For the electrostatic potential, we
consider the Neumann boundary condition (1.4), which describes that the
boundary of the domain is insulated (see[11]). It should be pointed out that

$\displaystyle\int(\rho-\tilde{\rho})dx=0.$ (1.5)

Furthermore, we observed that the solvability of the system (1.1)-(1.4)
implies that the initial density $\rho_{0}$ and the doping profile
$\tilde{\rho}$ must satisfy

$\displaystyle\int(\rho_{0}-\tilde{\rho})dx=0.$ (1.6)

Moreover, when we consider the Poisson equation with the Neumann boundary
condition in a bounded domain, we need the following uniqueness condition on
$\Phi$,

$\displaystyle\int\Phi dx=0.$ (1.7)

Next, we consider the steady state of the system (1.1) with the non-flat
doping profile $\tilde{\rho}(x)$. Assume that $(\rho_{s},u_{s},\Phi_{s})$ with
$u_{s}\equiv 0$ is the stationary solution of (1.1). Then it follows that

$\begin{cases}\nabla
P(\rho_{s})=\rho_{s}\nabla\Phi_{s}\,\,\,&\text{in}\,\,\,\Omega,\\\
\Delta\Phi_{s}=\rho_{s}-\tilde{\rho}\,\,\,&\text{in}\,\,\,\Omega,\\\
\nabla\Phi_{s}\cdot n=0\,\,\,&\text{on}\,\,\,\partial\Omega.\end{cases}$ (1.8)

Let $Q(\rho)$ be the enthalhy function defined by

$\displaystyle Q^{\prime}(\rho)=P^{\prime}(\rho)/\rho,$ (1.9)

which can change (1.8) to a quasilinear elliptic PDE and the classical fixed-
point therem or a variational method can be used to prove the existence of
stationary solution, without the smallness assumption on the oscillation of
$\tilde{\rho}(x)$ (see [11, 15]). We will record the existence and uniqueness
of the solution to (1.8) in Lemma 2.1.

The compressible Navier-Stokes-Poisson system (1.1) has been attracted a lot
of attention and significant progress has been made in the analysis of the
well-posedness and dynamic behavior to the solutions of the system. We briefly
review some results related to the global existence of strong (classical)
solutions to Cauchy problem of the multi-dimensional compressible Navier-
Stokes-Poisson system (1.1)-(1.2). When the doping profile is flat,
$i.e.,\tilde{\rho}(x)\equiv\bar{\rho}>0$, the steady state of (1.1) is the
trivial constant one $(\bar{\rho},0,0)$. Global existence and the decay rates
of the classical solution to its steady state were studied in [12, 16, 19, 33,
34, 41, 35, 20, 4] in the Sobolev spaces or Besove spaces framework. This is
extended to non-isentropic Navier-Stokes-Poisson systems, see [31, 40, 39, 30,
36]. While the doping profile is non-flat, global well-posedness of classical
solutions and stability of the steady state were obtained in [29, 10], under
analogous smallness conditions as the ones in [19, 34]. All of the above
results need the smallness assumption on the oscillations between the initial
data and the steady state. In particular, the initial density is near the non-
vacuum steady density, which indicates that the density is uniformly away from
the vacuum. Recently, for Cauchy problem of the compressible Navier-Stokes-
Poisson system (1.1)-(1.2) subject to large and non-flat doping profile,
global existence and uniqueness of strong solutions with large oscillations
and vacuum was established in [24], provided the initial data are of small
energy and the steady state is strictly away from vacuum.

For the initial-boundary-value problem of the compressible Navier-Stokes-
Poisson system (1.1)-(1.2) with non-slip boundary condition for the velocity
field, in the case that the doping profile is flat, the local and global
existence of weak solution were obtained in [9, 18] and the local existence of
unique strong solutions with vacuum was proved in [32] with the initial data
$\rho_{0}$ and $u_{0}$ satisfy a nature compatibility condition. Recently, for
the case that the doping profile is non-flat, global existence of smooth
solutions near the steady state for compressible Navier-Stokes-Poisson
equations was established with the exponential stability in [23], with the
smallness assumption on the oscillations between the initial data and the
steady state but lager doping profile. We also mentioned that the global
existence of solutions to compressible Navier-Stokes-Poisson equations with
the large initial data on a domain exterior to a ball was proved in [22] with
the radial symmetry assumption.

However, there are no works about the global existence of the strong
(classical) solution to the initial-boundary-value problem (1.1)-(1.4) for
general bounded smooth domains $\Omega\in\mathbb{R}^{3}$ with initial density
containing vacuum, at least to the best of our knowledge. Recently, for the
barotropic compressible Navier-Stokes equations in $\Omega$ with slip boundary
condition, the global classical solutions with large oscillations and vacuum
to the initial-boundary-value problem was established in [7], with some new
estimates on boundary integrals related to the slip boundary condition.

Based on the above research works, we study the global existence of the
classical solutions with large oscillations and vacuum to the initial-boundary
value problem (1.1)-(1.4) subject to large and non-flat doping profile in a
bounded domain $\Omega\subset\mathbb{R}^{3}$. Motivated by the works in [7,
24], we would like to obtain the time-independent upper bound of the density
and the time-dependent higher-norm estimates of $(\rho,u,\Phi)$, and extend
the classical solution globally in time. In our system, we need to conquer the
difficulties arising from the coupling of the density $\rho$ with the electric
field $\nabla\Phi$ and the slip boundary conditions (1.3). Firstly, compared
with the previous results (see [24, 17]) where they treated the Cauchy
problem, 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. To deal with this difficulty, we utilize $L^{p}$-theory
for the div-curl system to control $\nabla u$ by means of ${\rm div}u$ and
${\rm curl}u$ (Lemma A.3). For the electric field $\nabla\Phi$, with the help
of the the classical regularity theory for the Neumann problem of elliptic
equation, we obtain the estimates (2.19) and (2.23), which can be used to deal
with the coupling between the density and the electric field. Secondly, owing
to the coupling term $(\rho-\rho_{s})\nabla\Phi_{s}$ and
$\rho\nabla(\Phi-\Phi_{s})$ in (3.8), we can not obtain the time-independent
estimates of $\|\rho-\rho_{s}\|_{L^{2}(0,T;L^{2})}$ or
$\|P-P_{s}\|_{L^{2}(0,T;L^{2})}$. It is also difficult to get time-independent
estimates of $\|\nabla(\Phi-\Phi_{s})\|_{L^{2}(0,T;L^{2})}$ due to the
ellipticity of the Poisson equation (2.21). As a result, we can not close the
time-independent estimates $A_{1}(T)$ (see (3.1)) in a similar manner as that
in [17, 7]. In order to overcome this difficult point, we divide the time-
independent estimates $A_{1}(T)$ into two time intervals $(0,\sigma(T))$ and
$(\sigma(T),T)$. On the one hand, we can derive the short-time estimates
(3.70) with the help of time-weighted estimates (see Lemma 3.5). On the other
hand, we obtain the long-time estimates (3.79) by the time-piecewise iterative
argument, which was used in [24]. Furthermore, we also obtain the time-
dependent estimate $B[t_{1},t_{2}]$, which can be bounded by the initial
energy and the factor $t_{2}-t_{1}$ for any $1\leq t_{1}\leq t_{2}\leq T$ (see
(3.72)). These estimates help us to derive the uniform (in time) upper bound
for the density. Additionally, we also need to pay more attention to control
the boundary integrals during we derive the time-independent estimates of
$A_{1}(T)$. It should be pointed out that the boundary condition $u\cdot n=0$
yields

$\displaystyle\displaystyle u\cdot\nabla u\cdot n=-u\cdot\nabla n\cdot u,$
(1.10)

which 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 Lemma 3.3).
In order to estimate the high order derivatives of the solutions, we recall
the similar Beale-Kato-Majda-type inequality (see Lemma A.5) with the respect
to the slip boundary condition to prove the important estimates on the
gradients of the density and velocity.

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 $\beta\in(0,1)$, the
fractional Sobolev space $H^{\beta}(\Omega)$ is defined by

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

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

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

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 $\dot{v}\triangleq
v_{t}+u\cdot\nabla v$.

Assume $\Omega$ is a simply connected bounded domain in $\mathbb{R}^{3}$ and
its smooth boundary $\partial\Omega$ has a finite number of 2-dimensional
connected components. The initial total energy of (1.1) is defined as

$\displaystyle\displaystyle
C_{0}\triangleq\int_{\Omega}\left(\frac{1}{2}\rho_{0}|u_{0}|^{2}+G(\rho_{0})+\frac{1}{2}|\nabla(\Phi_{0}-\Phi_{s})|^{2}\right)dx.$
(1.11)

where $\Phi_{0}=\Phi(x,0)$ satisfies $\Delta\Phi_{0}=\rho_{0}-\tilde{\rho}$ in
$\Omega$ and $\nabla\Phi_{0}\cdot n=0$ on $\partial\Omega$, and $G(\rho)$ is
the potential energy density given by

$\displaystyle\displaystyle
G(\rho)\triangleq\int_{\rho_{s}}^{\rho}\int_{\rho_{s}}^{\xi}\frac{P^{\prime}(\zeta)}{\zeta}d\zeta
d\xi=\rho\int_{\rho_{s}}^{\rho}\frac{P(\xi)-P_{s}}{\xi^{2}}d\xi.$ (1.12)

where $P_{s}=P(\rho_{s})$.

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 $(\rho_{s},\Phi_{s})$ be the stationary solutions of (1.8). Assume that
the smooth function $\tilde{\rho}(x)$ satisfy
$0<\underline{\rho}\leq\tilde{\rho}(x)\leq\bar{\rho}$. For $q\in(3,6)$ and
some given constants $M>0$, $\beta\in(\frac{1}{2},1]$, and
$\hat{\rho}\geq\bar{\rho}+1$ , suppose that the initial data
$(\rho_{0},u_{0})$ satisfy (1.3), (1.6) and

$\displaystyle(\rho_{0},P(\rho_{0}))\in H^{2}\cap W^{2,q},\quad u_{0}\in
H^{2},$ (1.13) $\displaystyle
0\leq\rho_{0}\leq\hat{\rho},\quad\|u_{0}\|_{H^{\beta}}\leq M,$ (1.14)

and the compatibility condition

$\displaystyle\displaystyle-\mu\triangle
u_{0}-(\mu+\lambda)\nabla\mathop{\mathrm{div}}\nolimits u_{0}+\nabla
P(\rho_{0})=\rho_{0}^{1/2}g,$ (1.15)

for some $g\in L^{2}.$ Then there exists a positive constant $\varepsilon$
depending only on $\mu$, $\lambda$, $\gamma$, $a$, $\bar{\rho}$, $\hat{\rho}$,
$\beta$, $\Omega$ and $M$ such that for the initial energy $C_{0}$ as in
(1.11) if

$\displaystyle C_{0}\leq\varepsilon,$ (1.16)

the initial-boundary-value problem (1.1)-(1.4), (1.7) has a unique global
classical solution $(\rho,u,\Phi)$ in $\Omega\times(0,\infty)$ satisfying

$\displaystyle\displaystyle 0\leq\rho(x,t)\leq
2\hat{\rho},\quad(x,t)\in\Omega\times(0,\infty),$ (1.17)
$\begin{cases}(\rho,P)\in C([0,\infty);H^{2}\cap W^{2,q}),\\\ \nabla u\in
C([0,\infty);H^{1})\cap L^{\infty}_{\rm loc}(0,\infty;H^{2}\cap W^{2,q}),\\\
u_{t}\in L^{\infty}_{\rm loc}(0,\infty;H^{2})\cap H^{1}_{\rm
loc}(0,\infty;H^{1}),\\\ \nabla\Phi\in C([0,\infty);H^{2}\cap W^{2,q})\cap
L^{\infty}(0,\infty;H^{3}\cap W^{3,q}),\\\ \nabla\Phi_{t}\in
C([0,\infty);L^{2})\cap L^{\infty}(0,\infty;H^{2}).\end{cases}$ (1.18)

###### Remark 1.1

Compared with the results about global existence of the classical solutions
mentioned above (see, for instance, [19, 29, 23]), our conclusion does not
need the smallness assumption on the oscillations between the initial data and
the steady state. Although it has small energy, its oscillations could be
arbitrarily large. In particular, the initial vacuum states are allowed.

###### Remark 1.2

From (1.18), Sobolev’s inequality and the embedding

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

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

###### Remark 1.3

When we consider the following general slip boundary for the velocity field

$\displaystyle u\cdot n=0,\,\,\,{\rm curl}u\times
n=-Au\,\,\,\text{on}\,\,\,\partial\Omega,$ (1.19)

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 [7]. Roughly speaking, we
generalize the results of [7] to the compressible Navier-Stokes-Poisson
equations.

The rest of the paper is organized as follows. In Section 2, the existence
theorem of the steady-state solution and the local strong solution, and some
key a priori estimates needed in later analysis are collected. Section 3 is
devoted to deriving the necessary a priori estimates on classical solutions
which can guarantee the local classical solution to be a global classical one.
In Section 4, 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 Auxiliary lemma

In this section, we recall the steady-state solution and the local strong
(classical) solution of the system (1.1)-(1.4). We also derive some key a
priori estimates, which will be used frequently later.

First, similar to the proof of [15, 11], we have the following existence and
uniqueness of the solution to (1.8).

###### Lemma 2.1

Assume that the smooth function $\tilde{\rho}(x)$ satisfy
$0<\underline{\rho}\leq\tilde{\rho}(x)\leq\bar{\rho}$. Then the problem (1.8)
has a uniqueness classical solution $(\rho_{s},\Phi_{s})$. Moreover

$\displaystyle\underline{\rho}\leq\rho_{s}(x)\leq\bar{\rho},$ (2.1)
$\displaystyle\|\nabla\rho_{s}\|_{H^{3}}+\|\nabla\Phi_{s}\|_{H^{4}}\leq C,$
(2.2)

where $C$ depends only on $a,\gamma,\tilde{\rho}(x)$ and $\Omega$.

The following local existence theorem of classical solution of (1.1)-(1.4) can
be proved in a similar manner as that in [32], base on the standard
contraction mapping principle.

###### Lemma 2.2

Assume that the initial date $(\rho_{0},u_{0})$ satisfy the conditions (1.13)
and (1.15). Then there exist a positive time $T_{0}>0$ and a unique classical
solution $(\rho,u,\Phi)$ of the system (1.1)-(1.4) in
$\mathbb{R}^{3}\times(0,T_{0}]$, satisfying that $\rho\geq 0$, and that for
$\tau\in(0,T_{0})$,

$\begin{cases}(\rho,P)\in C([0,T_{0});H^{2}\cap W^{2,q}),\\\ \nabla u\in
C([0,T_{0});H^{1})\cap L^{\infty}(\tau,T_{0};H^{2}\cap W^{2,q}),\\\ u_{t}\in
L^{\infty}(\tau,T_{0};H^{2})\cap H^{1}(\tau,T_{0};H^{1}),\\\ \nabla\Phi\in
C([0,T_{0});H^{2}\cap W^{2,q})\cap L^{\infty}(\tau,T_{0};H^{3}\cap
W^{3,q}),\\\ \nabla\Phi_{t}\in C([0,T_{0});L^{2})\cap
L^{\infty}(\tau,T_{0};H^{2}).\end{cases}$ (2.3)

Next, we denote the effective viscous flux $F$ and the vorticity $\omega$ by

$\displaystyle
F\triangleq(\lambda+2\mu)\text{div}u-(P-P_{s}),\quad\omega\triangleq\nabla\times
u,$ (2.4)

which plays an important role in our following analysis, similarly to that for
the compressible Navier-Stokes equations (see [13, 21, 17]). For $F$, $\omega$
and $\nabla u$, we give the following conclusion, which is a key to a priori
estimates.

###### Lemma 2.3

Let $(\rho,u,\Phi)$ be a smooth solution of (1.1)-(1.4). Then for any
$p\in[2,6],$ there exists a positive constant $C$ depending only on $p$,
$\mu$, $\lambda$ and $\Omega$ such that

$\displaystyle\|\nabla F\|_{L^{p}}\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{p}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{p}}),$
(2.5) $\displaystyle\|\nabla\omega\|_{L^{p}}\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{p}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{p}}+\|\nabla
u\|_{L^{2}}),$ (2.6) $\displaystyle\|F\|_{L^{p}}\leq
C(\|\rho\dot{u}\|_{L^{2}}\\!+\\!\|\rho\nabla(\Phi\\!-\\!\Phi_{s})\|_{L^{2}}\\!+\\!\|(\rho\\!-\\!\rho_{s})\nabla\Phi_{s}\|_{L^{2}})^{\frac{3p-6}{2p}}(\|\nabla
u\|_{L^{2}}\\!+\\!\|P\\!-\\!P_{s}\|_{L^{2}})^{\frac{6-p}{2p}}$
$\displaystyle+C(\|\nabla u\|_{L^{2}}+\|P-P_{s}\|_{L^{2}}),$ (2.7)
$\displaystyle\|\omega\|_{L^{p}}\leq
C(\|\rho\dot{u}\|_{L^{2}}\\!+\\!\|\rho\nabla\\!(\Phi\\!-\\!\Phi_{s})\|_{L^{2}}\\!+\\!\|(\rho\\!-\\!\rho_{s})\nabla\\!\Phi_{s}\|_{L^{2}})^{\frac{3p-6}{2p}}\|\nabla\\!u\|_{L^{2}}^{\frac{6-p}{2p}}\\!+\\!C\|\nabla\\!u\|_{L^{2}},$
(2.8)

Moreover,

$\displaystyle\|\nabla u\|_{L^{p}}\leq$ $\displaystyle
C(\|\rho\dot{u}\|_{L^{2}}\\!+\\!\|\rho\nabla(\Phi\\!-\\!\Phi_{s})\|_{L^{2}}\\!+\\!\|(\rho\\!-\\!\rho_{s})\nabla\Phi_{s}\|_{L^{2}})^{\frac{3p-6}{2p}}(\|\nabla
u\|_{L^{2}}\\!+\\!\|P\\!-\\!P_{s}\|_{L^{2}})^{\frac{6-p}{2p}}$
$\displaystyle+C(\|\nabla u\|_{L^{2}}+\|P\\!-\\!P_{s}\|_{L^{p}}).$ (2.9)

Proof. By (1.1)2, one can find that the viscous flux $F$ satisfies

$\begin{cases}\Delta F={\rm
div}(\rho\dot{u}-\rho\nabla(\Phi-\Phi_{s})-(\rho-\rho_{s})\nabla\Phi_{s})~{}~{}&in\,\,\Omega,\\\
\frac{\partial F}{\partial n}=\rho\dot{u}\cdot
n\,\,&on\,\,\partial\Omega.\end{cases}$ (2.10)

It follows from Lemma 4.27 in [28], for $1<q<+\infty$, that

$\displaystyle\displaystyle\|\nabla F\|_{L^{q}}\leq
C(\|\rho\dot{u}\|_{L^{q}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{q}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{q}}),$
(2.11)

so that (2.5) holds. On the other hand, one can rewrite $\eqref{NSP}_{2}$ as

$\displaystyle\mu\nabla\times\omega=\nabla
F-\rho\dot{u}+\rho\nabla(\Phi-\Phi_{s})+(\rho-\rho_{s})\nabla\Phi_{s}.$ (2.12)

Notice that $\omega\times n=0$ on $\partial\Omega$ and
$\mathop{\mathrm{div}}\nolimits\omega=0$, by Lemma A.4, we get

$\displaystyle\displaystyle\|\nabla\omega\|_{L^{q}}$ $\displaystyle\leq
C(\|\nabla\times\omega\|_{L^{q}}+\|\omega\|_{L^{q}})$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{q}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{q}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{q}}+\|\omega\|_{L^{q}}),$
(2.13)

where we have taken advantage of (2.11). By Sobolev’s inequality and (2.13),
for $p\in[2,6]$,

$\displaystyle\|\nabla\omega\|_{L^{p}}$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{p}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{p}}+\|\nabla\omega\|_{L^{2}}+\|\omega\|_{L^{2}})$
$\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{p}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{p}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{p}}+\|\nabla
u\|_{L^{2}}),$ (2.14)

which implies (2.6).

Furthermore, one can deduce from (A.1) and (2.5) that for $p\in[2,6]$,

$\displaystyle\|F\|_{L^{p}}\leq C\|F\|_{L^{2}}^{\frac{6-p}{2p}}\|\nabla
F\|_{L^{2}}^{\frac{3p-6}{2p}}+C\|F\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle
C(\|\rho\dot{u}\|_{L^{2}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{2}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{2}})^{\frac{3p-6}{2p}}(\|\nabla
u\|_{L^{2}}+\|P-P_{s}\|_{L^{2}})^{\frac{6-p}{2p}}$ $\displaystyle+C(\|\nabla
u\|_{L^{2}}+\|P-P_{s}\|_{L^{2}}),$ (2.15)

similarly, by (A.1) and (2.6),

$\displaystyle\|\omega\|_{L^{p}}\leq
C\|\omega\|_{L^{2}}^{\frac{6-p}{2p}}\|\nabla\omega\|_{L^{2}}^{\frac{3p-6}{2p}}+C\|\omega\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
C(\|\rho\dot{u}\|_{L^{2}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{2}}+\|(\rho-\rho_{s})\nabla\Phi_{s}\|_{L^{2}})^{\frac{3p-6}{2p}}\|\nabla
u\|_{L^{2}}^{\frac{6-p}{2p}}+C\|\nabla u\|_{L^{2}},$ (2.16)

and so (2.7)-(2.8) are established. By virtue of (A.6), (2.7) and (2.8), it
implies that (2.9) holds. This completes the proof.

###### Remark 2.1

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.3, which will be devoted
to give higher order estimates in Section 3. In fact, by Lemma A.3 and (2.11)
, for $p\in[2,6]$,

$\displaystyle\quad\|\nabla^{2}u\|_{L^{p}}\leq C(\|{\rm
div}u\|_{W^{1,p}}+\|\omega\|_{W^{1,p}})$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{L^{p}}\\!\\!+\\!\|\rho\nabla\\!(\Phi\\!\\!-\\!\Phi_{s})\|_{L^{p}}\\!\\!+\\!\|(\rho\\!-\\!\\!\rho_{s})\nabla\\!\Phi_{s}\|_{L^{p}}\\!\\!+\\!\|\nabla\\!(P\\!\\!-\\!\\!P_{s})\|_{L^{p}}\\!\\!+\\!\|P\\!\\!-\\!P_{s}\|_{L^{p}}\\!\\!+\\!\\!\|\nabla\\!u\|_{L^{2}}),$
(2.17)

and

$\displaystyle\quad\|\nabla^{3}u\|_{L^{p}}\leq C(\|{\rm
div}u\|_{W^{2,p}}+\|\omega\|_{W^{2,p}})$ $\displaystyle\leq
C(\|\nabla(\rho\dot{u})\|_{L^{p}}\\!+\\!\|\nabla\\!(\rho\nabla\\!(\Phi\\!-\\!\Phi_{s}))\|_{L^{p}}\\!+\\!\|\nabla\\!((\rho\\!-\\!\rho_{s})\nabla\\!\Phi_{s})\|_{L^{p}}\\!+\\!\|\nabla^{2}(P\\!-\\!P_{s})\|_{L^{p}}\\!+\\!\|\rho\dot{u}\|_{L^{p}}$
$\displaystyle\quad+\\!\|\rho\nabla\\!(\Phi\\!-\\!\Phi_{s})\|_{L^{p}}\\!+\\!\|(\rho\\!-\\!\rho_{s})\nabla\\!\Phi_{s}\|_{L^{p}}\\!+\\!\|\nabla\\!(P\\!\\!-\\!P_{s})\|_{L^{p}}\\!+\\!\|P\\!\\!-\\!P_{s}\|_{L^{p}}\\!+\\!\|\nabla\\!u\|_{L^{2}}).$
(2.18)

To this end, we derive the estimates of $\nabla(\Phi-\Phi_{s})$ and
$\nabla(\Phi-\Phi_{s})_{t}$.

###### Lemma 2.4

Let $(\rho,u,\Phi)$ be a smooth solution of (1.1)-(1.4). Then for any integer
any integer $k\geq 0$, $q>1$, there exists a positive constant $C$ depending
only on $k$, $q$ and $\Omega$ such that

$\displaystyle\|\nabla(\Phi-\Phi_{s})\|_{W^{k+1,q}}\leq
C\|\rho-\rho_{s}\|_{W^{k,q}},$ (2.19)
$\displaystyle\|\nabla(\Phi-\Phi_{s})_{t}\|_{W^{k+1,q}}\leq C(\|\rho
u\|_{L^{q}}+\|\rho u\|_{W^{k,q}}).$ (2.20)

Proof. From (1.1)3, (1.4) and (1.8), we have

$\begin{cases}\Delta(\Phi\\!-\\!\Phi_{s})=\rho-\rho_{s}\,\,\,&\,in\,\,\,\Omega,\\\
\frac{\partial(\Phi-\Phi_{s})}{\partial
n}=0\,\,\,&\,on\,\,\,\partial\Omega,\end{cases}\displaystyle$ (2.21)

which, in view of the classical regularity theory for the Neumann problem of
elliptic equation [5], leads to (2.19). From (1.1)1 and (2.21), we obtain

$\begin{cases}\displaystyle\Delta(\Phi-\Phi_{s})_{t}=-{\rm div}(\rho
u)\,\,\,&in\,\,\Omega,\\\ \frac{\partial(\Phi-\Phi_{s})_{t}}{\partial
n}=0\,\,\,&on\,\,\partial\Omega.\end{cases}$ (2.22)

Using the same method as Lemma 2.3, we obtain, for $1<q<+\infty$, that

$\displaystyle\displaystyle\|\nabla(\Phi-\Phi_{s})_{t}\|_{L^{q}}\leq C\|\rho
u\|_{L^{q}},$ (2.23)

Moreover, for any integer $k\geq 0$,

$\displaystyle\|\nabla(\Phi-\Phi_{s})_{t}\|_{W^{k+1,q}}\leq C(\|\rho
u\|_{L^{q}}+\|\rho u\|_{W^{k,q}}).$ (2.24)

The proof of Lemma (2.4) is completed.

###### Remark 2.2

It follows from (2.23), if $\rho\leq 2\hat{\rho}$, that

$\displaystyle\displaystyle\|\nabla(\Phi-\Phi_{s})_{t}\|_{L^{2}}\leq C\|\rho
u\|_{L^{2}}\leq C\|\nabla u\|_{L^{2}},$ (2.25)

where $C$ depends on $\hat{\rho}$ and $\Omega$.

##  3  A priori estimates

In this section, we will establish some necessary a priori bounds for smooth
solutions to the problem (1.1)-(1.4) to extend the local classical solutions
guaranteed by Lemma 2.2.

Let $T>0$ be a fixed time and $(\rho,u,\Phi)$ be a smooth solution to
(1.1)-(1.4) on $\Omega\times(0,T]$ with smooth initial data $(\rho_{0},u_{0})$
satisfying $u_{0}\in H^{\beta}$ for some $\beta\in(\frac{1}{2},1]$ and
$0\leq\rho_{0}\leq 2\hat{\rho}$. Set
$\sigma=\sigma(t)\triangleq\min\\{1,t\\},$ we define

$\displaystyle A_{1}(T)\triangleq\sup_{0\leq t\leq T}\big{(}\sigma\|\nabla
u\|_{L^{2}}^{2}+\sigma^{3}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\big{)},$ (3.1)
$\displaystyle A_{2}(T)\triangleq\sup_{0\leq t\leq T}\int\rho|u|^{3}dx,$ (3.2)
$\displaystyle
B[t_{1},t_{2}]\triangleq\int_{t_{1}}^{t_{2}}\big{(}\sigma\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\sigma^{3}\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}dt,$
(3.3)

where $0\leq t_{1}<t_{2}\leq T$ 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
$\delta_{0}\triangleq\frac{2\beta-1}{4\beta}\in(0,\frac{1}{4}]$, there exists
a positive constant $\varepsilon$ depending on $\mu$, $\lambda$, $a$,
$\gamma$, $\bar{\rho}$, $\hat{\rho}$, $\beta$, $\Omega$ and $M$ such that if
$(\rho,u,\Phi)$ is a smooth solution of (1.1)-(1.4) on $\Omega\times(0,T]$
satisfying

$\sup\limits_{\Omega\times[0,T]}\rho\leq 2\hat{\rho},\quad A_{1}(T)\leq
2C_{0}^{1/2},\quad A_{2}(\sigma(T))\leq 2C_{0}^{\delta_{0}},$ (3.4)

then the following estimates hold

$\sup\limits_{\Omega\times[0,T]}\rho\leq 7\hat{\rho}/4,\quad A_{1}(T)\leq
C_{0}^{1/2},\quad A_{2}(\sigma(T))\leq C_{0}^{\delta_{0}},$ (3.5)

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

Proof. Proposition 3.1 is a consequence of the following Lemmas 3.6-3.8.

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

###### Lemma 3.2

Let $(\rho,u,\Phi)$ be a smooth solution of (1.1)–(1.4) on
${\Omega}\times(0,T]$ satisfying $\rho\leq 2\hat{\rho}$. Then there is a
positive constant $C$ depending only on $\mu,$ $\lambda,$ $a$, $\gamma$,
$\hat{\rho}$ and $\Omega$ such that

$\displaystyle\sup_{0\leq t\leq
T}\left(\frac{1}{2}\|\sqrt{\rho}u\|_{L^{2}}^{2}+\|G(\rho)\|_{L^{1}}+\frac{1}{2}\|\nabla(\Phi-\Phi_{s})\|_{L^{2}}^{2}\right)$
$\displaystyle+\int_{0}^{T}(\lambda+2\mu)\|{\rm
div}u\|_{L^{2}}^{2}+\mu\|\omega\|_{L^{2}}^{2})dt\leq C_{0},$ (3.6)

and

$\displaystyle\displaystyle\sup_{0\leq t\leq
T}\|\rho-\rho_{s}\|_{L^{2}}^{2}+\int_{0}^{T}\|\nabla u\|_{L^{2}}^{2}dt\leq
CC_{0}.$ (3.7)

Proof. Note that $-\Delta u=-\nabla{\rm div}u+\nabla\times\omega$ and (1.8),
we rewrite $\eqref{NSP}_{2}$ as

$\rho u+\rho u\cdot\nabla u-(\lambda+2\mu)\nabla{\rm
div}u+\mu\nabla\times\omega\\!+\nabla(P\\!-\\!P_{s})=\rho\nabla(\Phi\\!-\\!\Phi_{s})+(\rho\\!-\\!\rho_{s})\nabla\Phi_{s}.$
(3.8)

Multiplying (3.8)2 by $u$ and (1.1)1 by $G^{\prime}(\rho)$ respectively,
integrating over $\Omega$, summing them up, by (1.3)-(1.4) and (2.21), we have

$\displaystyle\left(\int\Big{(}G(\rho)+\frac{1}{2}\rho|u|^{2}\Big{)}dx\right)_{t}+(\lambda+2\mu)\int({\rm
div}u)^{2}dx+\mu\int|\omega|^{2}dx$
$\displaystyle=\int\mathop{\mathrm{div}}\nolimits(\rho
u)G^{\prime}(\rho)dx\\!+\\!\int
u\cdot\nabla(P\\!-\\!P_{s})dx\\!+\\!\int(\rho\\!-\\!\rho_{s})u\cdot\nabla\Phi_{s}dx\\!+\\!\int\rho
u\cdot\nabla(\Phi\\!-\\!\Phi_{s})dx$ $\displaystyle=\int\rho u\cdot\nabla
Q(\rho)+\int u\cdot\nabla
Pdx-\left(\int\frac{1}{2}|\nabla(\Phi-\Phi_{s})|^{2}dx\right)_{t}$
$\displaystyle=-\left(\int\frac{1}{2}|\nabla(\Phi-\Phi_{s})|^{2}dx\right)_{t},$
(3.9)

where we have used the fact $G^{\prime}(\rho)=Q(\rho)-Q(\rho_{s})$ and
$Q^{\prime}(\rho)=P^{\prime}(\rho)/\rho$. Then integrating (3) over $(0,T)$
yields (3.6). Finally, it is easy to check that there exists a positive
constant $C,$ depending only on $a,\gamma$ and $\hat{\rho}$, such that

$C^{-1}(\rho-\rho_{s})^{2}\leq G(\rho)\leq C(\rho-\rho_{s})^{2},$

which together with (A.6) and (3.6) gives (3.7). The proof of Lemma 3.2 is
completed.

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

$\displaystyle\displaystyle(\dot{u}+(u\cdot\nabla n)\times u^{\perp})\cdot
n=0\mbox{ on }\partial\Omega,$ (3.10)

where $u^{\perp}\triangleq-u\times n$ on $\partial\Omega$. In view of this
observation, we review the following Poincare-type inequality of $\dot{u}$
(see [7], Lemma 3.2).

###### Lemma 3.3

If $(\rho,u,\Phi)$ 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

$\displaystyle\|\dot{u}\|_{L^{6}}\leq C(\|\nabla\dot{u}\|_{L^{2}}+\|\nabla
u\|_{L^{2}}^{2}),$ (3.11) $\displaystyle\|\nabla\dot{u}\|_{L^{2}}\leq C(\|{\rm
div}\dot{u}\|_{L^{2}}+\|{\rm curl}\dot{u}\|_{L^{2}}+\|\nabla
u\|_{L^{4}}^{2}).$ (3.12)

Then, we give the estimate of $A_{1}(T)$ and $B[0,T]$.

###### Lemma 3.4

Let $(\rho,u,\Phi)$ be a smooth solution of (1.1)-(1.4) satisfying (3.4). Then
there is a positive constant $C$ depending only on $\mu$, $\lambda$, $a$,
$\gamma$, $\rho_{s}$, $\hat{\rho}$ and $\Omega$ such that

$\displaystyle\displaystyle A_{1}(T)+B[0,T]\leq
CC_{0}+C\int_{0}^{T}\big{(}\sigma\|\nabla u\|_{L^{3}}^{3}+\sigma^{3}\|\nabla
u\|_{L^{4}}^{4}\big{)}dt.$ (3.13)

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

$\displaystyle\int\sigma^{m}\rho|\dot{u}|^{2}dx$
$\displaystyle=-\int\sigma^{m}\dot{u}\cdot\nabla(P-P_{s})dx+(\lambda+2\mu)\int\sigma^{m}\nabla{\rm
div}u\cdot\dot{u}dx$
$\displaystyle\quad-\mu\int\sigma^{m}\nabla\times\omega\cdot\dot{u}dx+\int\sigma^{m}\rho\dot{u}\nabla(\Phi-\Phi_{s})dx$
$\displaystyle\quad+\int\sigma^{m}(\rho-\rho_{s})\dot{u}\nabla\Phi_{s}dx\triangleq\sum_{i=1}^{5}I_{i}.$
(3.14)

We have to estimate $I_{i}(i=1,\cdots,5)$ one by one. Firstly, By
$\eqref{NSP}_{1}$, one can check that

$\displaystyle\displaystyle(P-P_{s})_{t}+u\cdot\nabla(P-P_{s})+\gamma P{\rm
div}u+u\cdot\nabla P_{s}=0,$ (3.15)

A direct calculation by applying $\eqref{Pu1}$ gives

$\displaystyle I_{1}=$ $\displaystyle\left(\int\sigma^{m}(P-P_{s})\,{\rm
div}u\,dx\right)_{t}-m\sigma^{m-1}\sigma^{\prime}\int(P-P_{s})\,{\rm
div}u\,dx$ $\displaystyle-\int\sigma^{m}u\cdot\nabla P_{s}{\rm
div}u\,dx+\int\sigma^{m}(P-P_{s})\nabla u:\nabla udx$
$\displaystyle+\int\sigma^{m}(\gamma P-P+P_{s})({\rm
div}u)^{2}dx-\int_{\partial\Omega}\sigma^{m}(P-P_{s})u\cdot\nabla u\cdot nds$
$\displaystyle\leq$ $\displaystyle\left(\int\sigma^{m}(P-P_{s})\,{\rm
div}u\,dx\right)_{t}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}+Cm\sigma^{m-1}\sigma^{\prime}\|\rho-\rho_{s}\|_{L^{2}}^{2}$
$\displaystyle+\int_{\partial\Omega}\sigma^{m}(P-P_{s})u\cdot\nabla n\cdot
uds.$ (3.16)

where in last term on the right-hand side of (3) we have used the fact (1.10)
on $\partial\Omega$ and we conclude that

$\displaystyle\displaystyle\int_{\partial\Omega}\sigma^{m}(P-P_{s})u\cdot\nabla
n\cdot uds$ $\displaystyle\leq C\int_{\partial\Omega}\sigma^{m}|u|^{2}ds\leq
C\sigma^{m}\|\nabla u\|_{L^{2}}^{2}.$ (3.17)

Hence,

$\displaystyle\displaystyle I_{1}\leq\left(\int\sigma^{m}(P-P_{s})\,{\rm
div}u\,dx\right)_{t}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}+Cm\sigma^{m-1}\sigma^{\prime}\|\rho-\rho_{s}\|_{L^{2}}^{2}.$
(3.18)

Similarly,

$\displaystyle I_{2}$
$\displaystyle\leq(\lambda+2\mu)\int_{\partial\Omega}\sigma^{m}{\rm
div}u\,(u\cdot\nabla u\cdot
n)ds-\frac{\lambda+2\mu}{2}\left(\int\sigma^{m}({\rm div}u)^{2}dx\right)_{t}$
$\displaystyle\quad C\sigma^{m}\|\nabla
u\|_{L^{3}}^{3}+Cm\sigma^{m-1}\sigma^{\prime}\|\nabla u\|_{L^{2}}^{2}$ (3.19)

For the first term on the righthand side of (3), by (2.4) and Lemma 2.3, we
obtain

$\displaystyle(\lambda+2\mu)\int_{\partial\Omega}\sigma^{m}{\rm
div}u\,(u\cdot\nabla u\cdot n)ds$ $\displaystyle=$
$\displaystyle-\int_{\partial\Omega}\sigma^{m}Fu\cdot\nabla n\cdot
uds-\int_{\partial\Omega}\sigma^{m}(P-P_{s})u\cdot\nabla n\cdot uds$
$\displaystyle\leq$ $\displaystyle C\sigma^{m}(\|\nabla
F\|_{L^{2}}\|u\|_{L^{4}}^{2}+\|F\|_{L^{6}}\|u\|_{L^{3}}\|\nabla
u\|_{L^{2}}+\|F\|_{L^{2}}\|u\|^{2}_{L^{4}})+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\sigma^{m}\|\rho\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}(\|\nabla u\|_{L^{2}}^{2}+1),$ (3.20)

Therefore,

$\displaystyle\displaystyle I_{2}\leq$
$\displaystyle-\frac{\lambda+2\mu}{2}\left(\int\sigma^{m}({\rm
div}u)^{2}dx\right)_{t}+C\sigma^{m}\|\nabla
u\|_{L^{3}}^{3}+\frac{1}{4}\sigma^{m}\|\rho\dot{u}\|_{L^{2}}^{2}$
$\displaystyle+C\sigma^{m}(\|\nabla u\|_{L^{2}}^{2}+1)\|\nabla
u\|_{L^{2}}^{2}.$ (3.21)

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

$\displaystyle\displaystyle I_{3}$
$\displaystyle=-\frac{\mu}{2}\left(\int\sigma^{m}|\omega|^{2}dx\right)_{t}+\frac{\mu
m}{2}\sigma^{m-1}\sigma^{\prime}\int|\omega|^{2}dx-\mu\int\sigma^{m}\omega\cdot{\rm
curl}(u\cdot\nabla u)dx$
$\displaystyle\leq-\frac{\mu}{2}\left(\int\sigma^{m}|\omega|^{2}dx\right)_{t}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla u\|_{L^{3}}^{3}.$ (3.22)

In view of (2.19) and (2.25), it follows that

$\displaystyle I_{4}=$ $\displaystyle\left(\int\sigma^{m}\rho
u\cdot\nabla(\Phi-\Phi_{s})dx\right)_{t}-m\sigma^{m-1}\sigma^{\prime}\int\rho
u\cdot\nabla(\Phi-\Phi_{s})dx$ $\displaystyle-\int\sigma^{m}\rho
u\cdot\nabla(\Phi-\Phi_{s})_{t}dx-\int\sigma^{m}\rho
u\cdot\nabla^{2}(\Phi-\Phi_{s})\cdot udx$ $\displaystyle\leq$
$\displaystyle\left(\int\sigma^{m}\rho
u\cdot\nabla(\Phi-\Phi_{s})dx\right)_{t}+Cm\sigma^{m-1}\sigma^{\prime}\|\nabla(\Phi-\Phi_{s})\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2},$ (3.23)

and similarly,

$\displaystyle I_{5}\leq$
$\displaystyle\left(\int\sigma^{m}(\rho-\rho_{s})u\cdot\nabla\Phi_{s}dx\right)_{t}+Cm\sigma^{m-1}\sigma^{\prime}\|\rho-\rho_{s}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2},$ (3.24)

Making use of the results (3.18), (3),(3) and (3), it follows from
$\eqref{I0}$ that

$\displaystyle\left((\lambda+2\mu)\int\sigma^{m}({\rm
div}u)^{2}dx+\mu\int\sigma^{m}|\omega|^{2}dx\right)_{t}+\int\sigma^{m}\rho|\dot{u}|^{2}dx$
$\displaystyle\leq$ $\displaystyle\left(\int\sigma^{m}\Big{(}(P-P_{s})\,{\rm
div}u+(\rho-\rho_{s})u\cdot\nabla\Phi_{s}+\rho
u\cdot\nabla(\Phi-\Phi_{s})\Big{)}dx\right)_{t}+C\sigma^{m}\|\nabla
u\|_{L^{3}}^{3}$ $\displaystyle+C\sigma^{m}\|\nabla u\|_{L^{2}}^{2}(1+\|\nabla
u\|_{L^{2}}^{2})+Cm\sigma^{m-1}\sigma^{\prime}(\|\nabla(\Phi-\Phi_{s})\|_{L^{2}}^{2}+\|\rho-\rho_{s}\|_{L^{2}}^{2}),$
(3.25)

integrating over $(0,T]$, choosing $m=1$, by (A.6), (3.4), Lemma 3.2 and
Young’s inequality, we conclude that

$\displaystyle\displaystyle\sup_{0\leq t<T}\sigma\|\nabla
u\|_{L^{2}}^{2}+\int_{0}^{T}\sigma\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}dt\leq
CC_{0}+C\int_{0}^{T}\sigma\|\nabla u\|_{L^{3}}^{3}dt.$ (3.26)

Next, operating $\sigma^{m}\dot{u}^{j}[\partial/\partial t+{\rm div}(u\cdot)]$
to $\eqref{NSP-2}^{j},$ summing with respect to $j$, and integrating over
$\Omega,$ together with $\eqref{NSP}_{1}$, we get

$\displaystyle\left(\frac{\sigma^{m}}{2}\int\rho|\dot{u}|^{2}dx\right)_{t}-\frac{m}{2}\sigma^{m-1}\sigma^{\prime}\int\rho|\dot{u}|^{2}dx$
$\displaystyle=\int\sigma^{m}(\dot{u}\cdot\nabla F_{t}+\dot{u}^{j}\,{\rm
div}(u\partial_{j}F))dx$
$\displaystyle\quad+\mu\int\sigma^{m}(-\dot{u}\cdot\nabla\times\omega_{t}-\dot{u}^{j}{\rm
div}((\nabla\times\omega)^{j}\,u))dx$
$\displaystyle\quad+\int\sigma^{m}(\dot{u}\cdot\nabla\Phi_{s}(\rho-\rho_{s})_{t}+\dot{u}^{j}{\rm
div}((\rho-\rho_{s})\partial_{j}\Phi_{s}\,u))dx$
$\displaystyle\quad+\int\sigma^{m}(\dot{u}\cdot(\rho\nabla(\Phi-\Phi_{s}))_{t}+\dot{u}^{j}{\rm
div}(\rho\partial_{j}(\Phi-\Phi_{s})\,u))dx\triangleq\sum_{i=1}^{4}J_{i}.$
(3.27)

Let us estimate $J_{1},J_{2},J_{3}$ and $J_{4}$. By (1.3) and (3.15), a direct
computation yields

$\displaystyle J_{1}$
$\displaystyle=\int_{\partial\Omega}\sigma^{m}F_{t}\dot{u}\cdot
nds-(\lambda+2\mu)\int\sigma^{m}({\rm
div}\dot{u})^{2}dx+(\lambda+2\mu)\int\sigma^{m}{\rm div}\dot{u}\,\nabla
u:\nabla udx$ $\displaystyle\quad-\gamma\int\sigma^{m}P{\rm div}\dot{u}\,{\rm
div}udx+\int\sigma^{m}{\rm div}\dot{u}\,u\cdot\nabla
Fdx-\int\sigma^{m}u\cdot\nabla\dot{u}\cdot\nabla Fdx$
$\displaystyle\quad-\int\sigma^{m}{\rm div}\dot{u}\,u\cdot\nabla P_{s}dx$
$\displaystyle\leq\int_{\partial\Omega}\sigma^{m}F_{t}\dot{u}\cdot
nds-(\lambda+2\mu)\int\sigma^{m}({\rm
div}\dot{u})^{2}dx+\frac{\delta}{12}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|^{4}_{L^{4}}$ $\displaystyle\quad+C\sigma^{m}\big{(}\|\nabla
u\|_{L^{2}}^{2}\|\nabla F\|_{L^{3}}^{2}+\|\nabla u\|_{L^{2}}^{2}\big{)}$
(3.28)

where in the first equality we have used

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

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

$\displaystyle\int_{\partial\Omega}\sigma^{m}F_{t}\dot{u}\cdot
nds=-\int_{\partial\Omega}\sigma^{m}F_{t}\,(u\cdot\nabla n\cdot u)ds$
$\displaystyle=$
$\displaystyle-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}+m\sigma^{m-1}\sigma^{\prime}\int_{\partial\Omega}(u\cdot\nabla
n\cdot u)Fds$
$\displaystyle\quad+\int_{\partial\Omega}\sigma^{m}\big{(}F\dot{u}\cdot\nabla
n\cdot u+Fu\cdot\nabla n\cdot\dot{u}\big{)}ds$
$\displaystyle\quad-\int_{\partial\Omega}\sigma^{m}\big{(}F(u\cdot\nabla)u\cdot\nabla
n\cdot u+Fu\cdot\nabla n\cdot(u\cdot\nabla)u\big{)}ds$ $\displaystyle\leq$
$\displaystyle-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}+Cm\sigma^{m-1}\sigma^{\prime}\|\nabla
u\|_{L^{2}}^{2}\|F\|_{H^{1}}+\frac{\delta}{12}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}$
$\displaystyle+C\sigma^{m}\|\nabla u\|_{L^{2}}^{4}+\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}\|F\|_{H^{1}}^{2}+C\sigma^{m}\|\nabla F\|_{L^{6}}\|\nabla
u\|^{3}_{L^{2}}+C\sigma^{m}\|\nabla u\|^{2}_{L^{4}}.$ (3.29)

Combining these estimates of the boundary terms with (3), using (2.5),(3.11)
and (3.12) gives

$\displaystyle J_{1}\leq-(\lambda+2\mu)\int\sigma^{m}({\rm
div}\dot{u})^{2}dx-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}$
$\displaystyle\quad+\frac{\delta}{4}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|^{4}_{L^{4}}+C\sigma^{m}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4})$
$\displaystyle\quad+C\sigma^{m}\|\nabla u\|_{L^{2}}^{2}(1+\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4})$
$\displaystyle\quad+Cm\sigma^{m-1}\sigma^{\prime}(\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4}).$ (3.30)

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

$\displaystyle J_{2}$ $\displaystyle=-\mu\int\sigma^{m}|{\rm
curl}\dot{u}|^{2}dx+\mu\int\sigma^{m}(\omega\times\nabla
u^{i})\cdot\nabla_{i}\dot{u}dx$ $\displaystyle\quad-\mu\int\sigma^{m}{\rm
curl}\dot{u}\cdot(\nabla u^{i}\times\nabla_{i}u)dx-\mu\int\sigma^{m}{\rm
div}u\,\omega\cdot{\rm curl}\dot{u}dx$
$\displaystyle\leq-\mu\int\sigma^{m}|{\rm
curl}\dot{u}|^{2}dx+\frac{\delta}{4}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|_{L^{4}}^{4}.$ (3.31)

In view of (2.2) and (3.11), we obtain that

$\displaystyle\displaystyle J_{3}$
$\displaystyle=\int\sigma^{m}((\rho-\rho_{s})u\cdot\nabla^{2}\Phi_{s}\cdot\dot{u}dx-\int\sigma^{m}\dot{u}\cdot\nabla\Phi_{s}{\rm
div}(\rho_{s}u)dx$ $\displaystyle\leq
C\sigma^{m}(\|\dot{u}\|_{L^{6}}\|\rho-\rho_{s}\|_{L^{2}}\|\nabla
u\|_{L^{2}}+\|\nabla\dot{u}\|_{L^{2}}\|\nabla
u\|_{L^{2}})\|\nabla\Phi_{s}\|_{H^{2}}$
$\displaystyle\leq\frac{\delta}{4}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}+C\sigma^{m}\|\nabla u\|_{L^{2}}^{4},$ (3.32)

and similarly, by (3.11), (2.19), (2.25) and (2.23), we have

$\displaystyle\displaystyle J_{4}$
$\displaystyle=\int\sigma^{m}\rho\dot{u}\cdot\nabla(\Phi-\Phi_{s})_{t}dx+\int\sigma^{m}\rho
u\cdot\nabla^{2}(\Phi-\Phi_{s})\cdot\dot{u}dx$ $\displaystyle\leq
C\sigma^{m}(\|\sqrt{\rho}\dot{u}\|_{L^{2}}\|\nabla(\Phi-\Phi_{s})_{t}\|_{L^{2}}+\|\nabla^{2}(\Phi-\Phi_{s})\|_{L^{2}}\|\sqrt{\rho}u\|_{L^{3}}\|\dot{u}\|_{L^{6}})$
$\displaystyle\leq\frac{\delta}{4}\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}+C\sigma^{m}(\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2}).$ (3.33)

Combining (3), (3), (3) and (3) with (3), by (3.12) and choosing $\delta$
small enough, we have

$\displaystyle\quad\left(\sigma^{m}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\right)_{t}+\sigma^{m}\|\nabla\dot{u}\|_{L^{2}}^{2}$
$\displaystyle\leq-\left(\int_{\partial\Omega}\sigma^{m}(u\cdot\nabla n\cdot
u)Fds\right)_{t}+C\sigma^{m}\|\nabla
u\|_{L^{4}}^{4}+C\sigma^{m}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}$
$\displaystyle\quad+C\sigma^{m}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4})+C\sigma^{m}\|\nabla
u\|_{L^{2}}^{2}(1+\|\nabla u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4})$
$\displaystyle\quad+Cm\sigma^{m-1}\sigma^{\prime}(\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4}).$ (3.34)

For the boundary term in the right-hand side of (3), from Lemma 2.3, we have

$\displaystyle\int_{\partial\Omega}(u\cdot\nabla n\cdot u)Fds\leq C\|\nabla
u\|_{L^{2}}^{2}\|F\|_{H^{1}}\leq\frac{1}{2}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+C(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{4}).$ (3.35)

Now integrating (3) with $m=3$ over $(0,T)$, using (3.4), (3.26) and (3.35),
yields

$\displaystyle\sup_{0\leq t\leq
T}\sigma^{3}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\\!+\\!\int_{0}^{T}\\!\\!\sigma^{3}\|\nabla\dot{u}\|_{L^{2}}^{2}dt\leq
CC_{0}\\!+\\!C\int_{0}^{T}\\!\\!\big{(}\sigma\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\\!+\\!\sigma^{3}\|\nabla
u\|_{L^{4}}^{4}\big{)}dt.$ (3.36)

Combining (3.26) with (3.36), we immediately obtain (3.13) and complete the
proof of Lemma 3.4.

###### Lemma 3.5

Assume that $(\rho,u,\Phi)$ is a smooth solution of (1.1)-(1.4) satisfying
(3.4) and the initial data condition (1.14), then there exist positive
constants $C$ and $\varepsilon_{1}$ depending only on
$\mu,\,\,\lambda,\,\,\gamma,\,\,a,\,\,\rho_{s},\,\,\hat{\rho},\,\,\beta,\,\,\Omega$
and $M$ such that

$\displaystyle\sup_{0\leq t\leq\sigma(T)}t^{1-s}\|\nabla
u\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}t^{1-s}\int\rho|\dot{u}|^{2}dxdt\leq
C(\hat{\rho},M),$ (3.37) $\displaystyle\sup_{0\leq
t\leq\sigma(T)}t^{2-s}\int\rho|\dot{u}|^{2}dx+\int_{0}^{\sigma(T)}\int
t^{2-s}|\nabla\dot{u}|^{2}dxdt\leq C(\hat{\rho},M),$ (3.38)

provide that $C_{0}<\varepsilon_{1}$.

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

$\displaystyle\begin{cases}Lw_{1}=0\,\,&in\,\,\Omega,\\\
w_{1}(x,0)=w_{10}(x)\,\,&in\,\,\Omega,\\\ w_{1}\cdot n=0\,\,\text{and}\,\,{\rm
curl}w_{1}\times n=0\,\,&on\,\,\partial\Omega,\end{cases}$ (3.39)
$\displaystyle\begin{cases}Lw_{2}=-\nabla(P-P_{s})\,\,&in\,\,\Omega,\\\
w_{2}(x,0)=0\,\,&in\,\,\Omega,\\\ w_{2}\cdot n=0\,\,\text{and}\,\,{\rm
curl}w_{2}\times n=0\,\,&on\,\,\partial\Omega,\end{cases}$ (3.40)

and

$\displaystyle\begin{cases}Lw_{3}=\rho\nabla(\Phi-\Phi_{s})+(\rho-\rho_{s})\nabla\Phi_{s}\,\,&in\,\,\Omega,\\\
w_{3}(x,0)=0\,\,&in\,\,\Omega,\\\ w_{3}\cdot n=0\,\,\text{and}\,\,{\rm
curl}w_{3}\times n=0\,\,&on\,\,\partial\Omega,\end{cases}$ (3.41)

where $Lf\triangleq\rho\dot{f}-\mu\Delta f-(\lambda+\mu)\nabla{\rm div}f$.

A similar way as for the proof of (3.6) shows that

$\displaystyle\displaystyle\sup_{0\leq
t\leq\sigma(T)}\int\rho|w_{1}|^{2}dx+\int_{0}^{\sigma(T)}\int|\nabla
w_{1}|^{2}dxdt\leq C\int|w_{10}|^{2}dx,$ (3.42)
$\displaystyle\displaystyle\sup_{0\leq
t\leq\sigma(T)}\int\rho|w_{2}|^{2}dx+\int_{0}^{\sigma(T)}\int|\nabla
w_{2}|^{2}dxdt\leq CC_{0},$ (3.43)

and

$\displaystyle\sup_{0\leq
t\leq\sigma(T)}\int\rho|w_{3}|^{2}dx+\int_{0}^{\sigma(T)}\int|\nabla
w_{3}|^{2}dxdt\leq CC_{0}.$ (3.44)

Multiplying (3.39) by $w_{1t}$ and integrating over $\Omega,$ by (3.4),
Sobolev’s and Young’s inequalities, we obtain

$\displaystyle\quad\left(\frac{\lambda+2\mu}{2}\int({\rm
div}w_{1})^{2}dx+\frac{\mu}{2}\int|{\rm
curl}w_{1}|^{2}dx\right)_{t}+\int\rho|\dot{w}_{1}|^{2}dx$ $\displaystyle\leq
C\|\sqrt{\rho}\dot{w}_{1}\|_{L^{2}}\|\rho^{\frac{1}{3}}u\|_{L^{3}}\|\nabla
w_{1}\|_{L^{6}}\leq
C_{1}C_{0}^{\frac{\delta_{0}}{3}}(\|\sqrt{\rho}\dot{w}_{1}\|_{L^{2}}^{2}+\|\nabla
w_{1}\|_{L^{2}}^{2}),$ (3.45)

since it follows from (3.39), with the similar methods to obtain (2.11) as in
Lemma (2.3) that

$\displaystyle\|\nabla w_{1}\|_{L^{6}}\leq C\|w_{1}\|_{W^{2,2}}\leq
C(\|\rho\dot{w}_{1}\|_{L^{2}}+\|\nabla w_{1}\|_{L^{2}}).$ (3.46)

Together (3) with (3.42), and by Gronwall’s inequality and Lemma A.3, it
yields

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{1}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\int\rho|\dot{w}_{1}|^{2}dxdt\leq
C\|\nabla w_{10}\|_{L^{2}}^{2},$ (3.47)

and

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}t\|\nabla
w_{1}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}t\int\rho|\dot{w}_{1}|^{2}dxdt\leq
C\|w_{10}\|_{L^{2}}^{2},$ (3.48)

provided
$C_{0}<\hat{\varepsilon}_{1}\triangleq(2C_{1})^{-\frac{3}{\delta_{0}}}$. Since
the solution operator $w_{10}\mapsto w_{1}(\cdot,t)$ is linear, by the
standard Stein-Weiss interpolation argument [3], one can deduce from (3.47)
and (3.48) that for any $\theta\in[s,1],$

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}t^{1-\theta}\|\nabla
w_{1}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}t^{1-\theta}\int\rho|\dot{w_{1}}|^{2}dxdt\leq
C\|w_{10}\|_{H^{\theta}}^{2},$ (3.49)

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

Multiplying (3.40) by $w_{2t}$ and integrating over $\Omega$ give that

$\displaystyle\quad\left(\frac{\lambda+2\mu}{2}\int({\rm
div}w_{2})^{2}dx+\frac{\mu}{2}\int|{\rm curl}w_{2}|^{2}dx-\int(P-P_{s}){\rm
div}w_{2}dx\right)_{t}+\int\rho|\dot{w}_{2}|^{2}dx$
$\displaystyle=\int\rho\dot{w}_{2}\cdot(u\cdot\nabla
w_{2})dx-\frac{1}{\lambda+2\mu}\int(P-P_{s})(F_{w_{2}}{\rm div}u+\nabla
F_{w_{2}}\cdot u)dx$ (3.50)
$\displaystyle\quad-\frac{1}{2(\lambda+2\mu)}\int(P-P_{s})^{2}{\rm
div}udx+\gamma\int P{\rm div}u\,{\rm div}w_{2}dx+\int u\cdot\nabla P_{s}\,{\rm
div}w_{2}dx$ $\displaystyle\leq
C(\|\sqrt{\rho}\dot{w}_{2}\|_{L^{2}}\|\rho^{\frac{1}{3}}u\|_{L^{3}}\|\nabla
w_{2}\|_{L^{6}}+\|\nabla u\|_{L^{2}}\|F_{w_{2}}\|_{L^{2}}+\|\nabla
F_{w_{2}}\|_{L^{2}}\|u\|_{L^{2}})$
$\displaystyle\quad+C(\|P-P_{s}\|_{L^{2}}\|\nabla u\|_{L^{2}}+\|\nabla
u\|_{L^{2}}\|\nabla w_{2}\|_{L^{2}})$ $\displaystyle\leq
C_{2}C_{0}^{\frac{\delta_{0}}{3}}\|\sqrt{\rho}\dot{w}_{2}\|_{L^{2}}^{2}+\frac{1}{4}\|\sqrt{\rho}\dot{w}_{2}\|_{L^{2}}^{2}+C(\|\nabla
w_{2}\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}+\|P-P_{s}\|_{L^{6}}^{2}),$ (3.51)

where we have utilized (3.4), (3.15) and the following estimates

$\|\nabla F_{w_{2}}\|_{L^{2}}\leq
C\|\rho\dot{w}_{2}\|_{L^{2}},\quad\|F_{w_{2}}\|_{L^{2}}\leq
C(\|\rho\dot{w}_{2}\|_{L^{2}}+\|\nabla w_{2}\|_{L^{2}}),$ (3.52)
$\displaystyle\|\nabla w_{2}\|_{L^{6}}\leq
C(\|\rho\dot{w}_{2}\|_{L^{2}}+\|\nabla
w_{2}\|_{L^{2}}+\|P-\bar{P}\|_{L^{6}}),$ (3.53)

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

$\displaystyle\quad\left((\lambda+2\mu)\|{\rm
div}w_{2}\|_{L^{2}}^{2}+\mu\|{\rm curl}w_{2}\|_{L^{2}}^{2}-2\int(P-P_{s}){\rm
div}w_{2}dx\right)_{t}+\int\rho|\dot{w}_{2}|^{2}dx$ $\displaystyle\leq
C\left(\|\nabla w_{2}\|_{L^{2}}^{2}+\|\nabla
u\|_{L^{2}}^{2}+\|P-P_{s}\|_{L^{2}}^{\frac{2}{3}}\right),$ (3.54)

provide that
$C_{0}<\hat{\varepsilon}_{2}\triangleq(4C_{2})^{-\frac{3}{\delta_{0}}}$.
Integrating (3) over $(0,\sigma(T))$, with (3.43) and Lemmas A.3, 3.2, one has

$\displaystyle\displaystyle\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{2}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\int\rho|\dot{w}_{2}|^{2}dxdt\leq
CC_{0}^{1/3}.$ (3.55)

Similarly, multiplying (3.41) by $w_{3t}$ and integrating over $\Omega$ give
that

$\displaystyle\quad\left(\frac{\lambda+2\mu}{2}\int({\rm
div}w_{3})^{2}dx+\frac{\mu}{2}\int|{\rm
curl}w_{3}|^{2}dx\right)_{t}+\int\rho|\dot{w}_{3}|^{2}dx$
$\displaystyle=\left(\int(\rho-\rho_{s})w_{3}\cdot\nabla\Phi_{s}dx\right)_{t}+\int{\rm
div}((\rho-\rho_{s})u)w_{3}\cdot\nabla\Phi_{s}dx$ $\displaystyle\quad+\int{\rm
div}(\rho_{s}u)w_{3}\cdot\nabla\Phi_{s}dx+\int\rho_{s}\dot{w_{3}}\cdot\nabla(\Phi-\Phi_{s})dx$
$\displaystyle\quad+\int\rho_{s}u\cdot\nabla
w_{3}\cdot\nabla(\Phi-\Phi_{s})dx+\int\rho u\cdot\nabla w_{3})\cdot
dot{w}_{3}dx$ $\displaystyle\triangleq\frac{d}{dt}L_{0}+\sum_{i=1}^{5}L_{i}.$
(3.56)

By (2.2) and (A.1), we have

$\displaystyle L_{0}\leq
C\|\nabla\Phi_{s}\|_{L^{3}}\|\rho-\rho_{s}\|_{L^{2}}\|w_{3}\|_{L^{6}}\leq\frac{\mu}{4}\|\nabla
w_{3}\|_{L^{2}}^{2}+CC_{0}.$ (3.57)

Using the similar methods to obtain (2.11) as in Lemma (2.3), it follows from
(3.41) that

$\displaystyle\displaystyle\|\nabla w_{3}\|_{L^{6}}$ $\displaystyle\leq
C(\|\rho\dot{w}_{3}\|_{L^{2}}+\|(\rho-\bar{\rho})\nabla\Phi_{s}\|_{L^{2}}+\|\rho\nabla(\Phi-\Phi_{s})\|_{L^{2}}+\|\rho
w_{3}\|_{L^{2}})$ $\displaystyle\leq C(\|\rho\dot{w}_{3}\|_{L^{2}}+\|\rho
w_{3}\|_{L^{2}}+CC_{0}^{\frac{1}{2}}).$ (3.58)

Then, by (2.2), (3.6) and (3), a directly computation yields

$\displaystyle L_{1}$ $\displaystyle=-\int(\rho-\rho_{s})(u\cdot\nabla
w_{3}\cdot\nabla\Phi_{s}+u\cdot\nabla^{2}\Phi_{s}\cdot w_{3})dx$
$\displaystyle\leq
C\|\rho-\rho_{s}\|_{L^{2}}\|u\|_{L^{6}}(\|w_{3}\|_{L^{6}}+\|\nabla
w_{3}\|_{L^{6}})\|\nabla\Phi_{s}\|_{H^{2}}$
$\displaystyle\leq\frac{1}{4}\|\sqrt{\rho}\dot{w}_{3}\|_{L^{2}}^{2}+C(\|\nabla
u\|_{L^{2}}^{2}+\|\nabla w_{3}\|_{L^{2}}^{2}+C_{0}).$ (3.59)

similarly, using (2.2), (2.19), (2.23), (3.6) and (3.4), we obtain

$\displaystyle\sum_{i=2}^{5}L_{i}\leq$ $\displaystyle C(|\nabla
u\|_{L^{2}}\|\nabla\Phi_{s}\|_{L^{3}}+\|\nabla\rho_{s}\|_{L^{2}}\|\nabla\Phi_{s}\|_{L^{6}}\|u\|_{L^{6}})\|w_{3}\|_{L^{6}}$
$\displaystyle+C\|\sqrt{\rho}\dot{w_{3}}\|_{L^{2}}\|\nabla(\Phi-\Phi_{s})\|_{L^{2}}$
$\displaystyle+C\|\rho^{1/3}u\|_{L^{3}}\|\nabla
w_{3}\|_{L^{6}}(\|\nabla(\Phi-\Phi_{s})\|_{L^{2}}+\|\sqrt{\rho}\dot{w_{3}}\|_{L^{2}})$
$\displaystyle\leq$ $\displaystyle C(\|\nabla u\|_{L^{2}}^{2}+\|\nabla
w_{3}\|_{L^{2}}^{2}+C_{0})+(\frac{1}{4}+CC^{\delta_{0}/3})\|\sqrt{\rho}\dot{w_{3}}\|_{L^{2}}^{2}.$
(3.60)

Putting (3.57), (3) and (3) into (3) yields

$\displaystyle\quad\left(\|\nabla
w_{3}\|_{L^{2}}^{2}\right)_{t}+\int\rho|\dot{w}_{3}|^{2}dx$ $\displaystyle\leq
C_{3}C^{\delta_{0}/3}\|\sqrt{\rho}\dot{w_{3}}\|_{L^{2}}^{2}+C(\|\nabla
w_{3}\|_{L^{2}}^{2}+\|\nabla w_{3}\|_{L^{2}}^{2}+C_{0}).$ (3.61)

Thus, integrating it over $(0,\sigma(T)]$, choosing
$C_{0}\leq\hat{\varepsilon}_{3}\triangleq(2C_{3})^{-3/\delta_{0}}$, we obtain

$\displaystyle\quad\sup_{0\leq t\leq\sigma(T)}\|\nabla
w_{3}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\int\rho|\dot{w}_{3}|^{2}dxdt\leq
CC_{0}.$ (3.62)

Now let $w_{10}=u_{0}$, so that $w_{1}+w_{2}+w_{3}=u$, we derive (3.37) from
(3.49), (3.55) and (3.62) directly under certain condition
$C_{0}<\varepsilon_{1}\triangleq\min\\{\hat{\varepsilon}_{1},\hat{\varepsilon}_{2},\hat{\varepsilon}_{3}\\}$.

In order to prove (3.38), taking $m=2-s$ in (3), and integrating over
$(0,\sigma(T)]$ instead of $(0,T]$, in a similar way as we have gotten (3.36),
we obtain

$\displaystyle\sup_{0\leq
t\leq\sigma(T)}\sigma^{2-s}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\int_{0}^{\sigma(T)}\sigma^{2-s}\|\nabla\dot{u}\|_{L^{2}}^{2}dt$
$\displaystyle\leq C\int_{0}^{\sigma(T)}\sigma^{2-s}\|\nabla
u\|^{4}_{L^{4}}dt+C(\hat{\rho},M).$ (3.63)

where we have taken advantage of (3.37). By (3.4) and (3.37), we have

$\displaystyle\quad\int_{0}^{\sigma(T)}\sigma^{2-s}\|\nabla u\|^{4}_{L^{4}}dt$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}t^{2-s}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{3}\|\nabla
u\|_{L^{2}}dt+C\int_{0}^{\sigma(T)}t^{2-s}\|\nabla u\|_{L^{2}}^{4}dt+C$
$\displaystyle\leq C\int_{0}^{\sigma(T)}t^{\frac{2s-1}{2}}(t^{1-s}\|\nabla
u\|_{L^{2}}^{2})^{\frac{1}{2}}(t^{2-s}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})^{\frac{1}{2}}(t^{1-s}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})dt+C$
$\displaystyle\leq C(\hat{\rho},M)\left(\sup_{0\leq
t\leq\sigma(T)}t^{2-s}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\right)^{\frac{1}{2}}+C,$
(3.64)

which together with (3) yields (3.38). The proof of Lemma 3.5 is completed.

###### Lemma 3.6

If $(\rho,u,\Phi)$ is a smooth solution of (1.1)-(1.4) satisfying (3.4) and
the initial data condition (1.14), then there exists a positive constant
$\varepsilon_{2}$ depending only on
$\mu,\lambda,\gamma,a,\bar{\rho},\hat{\rho},\beta,\Omega$, and $M$ such that

$\displaystyle\displaystyle A_{3}(\sigma(T))\leq C_{0}^{\delta_{0}},$ (3.65)

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

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

$\displaystyle\left(\int\rho|u|^{3}dx\right)_{t}+3(\lambda+2\mu)\int{\rm
div}u\,{\rm div}(|u|u)dx+3\mu\int\omega\cdot{\rm curl}(|u|u)dx$
$\displaystyle-3\int(P-P_{s}){\rm
div}(|u|u)dx-3\int(\rho\nabla(\Phi-\Phi_{s})+(\rho-\rho_{s})\nabla\Phi_{s})\cdot|u|udx=0.$
(3.66)

By (2.2), (2.9), (2.19), (3.4) and (3.6), it follows that

$\displaystyle\quad\left(\int\rho|u|^{3}dx\right)_{t}$ $\displaystyle\leq
C\|u\|_{L^{6}}\|\nabla u\|_{L^{2}}\|\nabla
u\|_{L^{3}}+C\|P-P_{s}\|_{L^{3}}\|u\|_{L^{6}}\|\nabla u\|_{L^{2}}$
$\displaystyle\quad+C\|\rho^{\frac{1}{3}}u\|_{L^{3}}^{2}\|\nabla(\Phi-\Phi_{s})\|_{L^{3}}+\|\rho-\rho_{s}\|_{L^{2}}\|\nabla
u\|_{L^{2}}^{2}\|\nabla\Phi_{s}\|_{L^{6}}$ $\displaystyle\leq C\|\nabla
u\|_{L^{2}}^{\frac{5}{2}}\|\rho\dot{u}\|_{L^{2}}^{\frac{1}{2}}+C\|\nabla
u\|_{L^{2}}^{3}+CC_{0}^{\frac{1}{12}}\|\nabla u\|_{L^{2}}^{\frac{5}{2}}$
$\displaystyle\quad+CC_{0}^{\frac{1}{3}}\|\nabla
u\|_{L^{2}}^{2}+CC_{0}^{\frac{2\delta_{0}}{3}+\frac{1}{2}}.$ (3.67)

Hence, integrating (3) over $(0,\sigma(T))$ and using (3.4), we get

$\displaystyle\quad\sup_{0\leq t\leq\sigma(T)}\int\rho|u|^{3}dx$
$\displaystyle\leq C\int_{0}^{\sigma(T)}\|\nabla
u\|_{L^{2}}^{\frac{5}{2}}\|\rho\dot{u}\|_{L^{2}}^{\frac{1}{2}}dt+C\int_{0}^{\sigma(T)}\|\nabla
u\|_{L^{2}}^{3}dt$
$\displaystyle\quad+CC_{0}^{\frac{1}{12}}\int_{0}^{\sigma(T)}\|\nabla
u\|_{L^{2}}^{\frac{5}{2}}dt+CC_{0}^{2\delta_{0}}+\int\rho_{0}|u_{0}|^{3}dx$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}(t^{1-\beta}\|\rho\dot{u}\|_{L^{2}}^{2})^{\frac{1}{4}}(t^{1-\beta}\|\nabla
u\|_{L^{2}}^{2})^{\frac{5-8\delta_{0}}{4}}\|\nabla
u\|_{L^{2}}^{4\delta_{0}}t^{\frac{(\beta-1)(4\delta_{0}-3)}{2}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}(t\|\rho\dot{u}\|_{L^{2}}^{2})^{\frac{1}{2}}t^{1-\beta}\|\nabla
u\|_{L^{2}}^{2}t^{\frac{\beta-2}{2}}dt$
$\displaystyle\quad+CC_{0}^{\frac{1}{12}}\int_{0}^{\sigma(T)}(t^{1-\beta}\|\nabla
u\|_{L^{2}}^{2})^{\frac{5}{4}}t^{\frac{5(\beta-1)}{4}}dt+CC_{0}^{2\delta_{0}}+\int\rho_{0}|u_{0}|^{3}dx$
$\displaystyle\leq CC_{0}^{2\delta_{0}}+\int\rho_{0}|u_{0}|^{3}dx\leq
C_{4}C_{0}^{2\delta_{0}},$ (3.68)

where we have used the fact
$\delta_{0}=\frac{2\beta-1}{4\beta}\in(0,\frac{1}{4}]$, $\beta\in(1/2,1]$ and

$\displaystyle\displaystyle\int\rho_{0}|u_{0}|^{3}dx\leq
C\|\rho_{0}^{\frac{1}{2}}u_{0}\|_{L^{2}}^{{3(2\beta-1)}/{2\beta}}\|u_{0}\|_{H^{\beta}}^{{3}/{2\beta}}\leq
CC_{0}^{3\delta_{0}}.$ (3.69)

Finally, set
$\varepsilon_{2}\triangleq\min\\{\varepsilon_{1},(C_{4})^{-\frac{1}{\delta_{0}}}\\}$,
we get (3.65) and the proof of Lemma 3.6 is completed.

###### Lemma 3.7

Let $(\rho,u,\Phi)$ be a smooth solution of (1.1)-(1.4) on
${\Omega}\times(0,T]$ satisfying (3.4) and the initial data condition
$\|u_{0}\|_{H^{\beta}}\leq M$ in (1.14). Then there exists a positive constant
$C$ and $\varepsilon_{3}$ depending only on $\mu,$ $\lambda,$ $\gamma,$ $a$,
$\beta$, $\bar{\rho}$, $\hat{\rho}$, $M$ and $\Omega$ such that

$\displaystyle A_{1}(\sigma(T))+B[0,\sigma(T)]\leq$ $\displaystyle
CC_{0}^{\frac{3}{4}},$ (3.70) $\displaystyle A_{1}(T)\leq$ $\displaystyle
C_{0}^{\frac{1}{2}},$ (3.71)

provided $C_{0}\leq\varepsilon_{3}$. Furthermore, if $T>1$, then for any
$1\leq t_{1}<t_{2}\leq T$,

$\displaystyle B[t_{1},t_{2}]\leq CC_{0}^{3/4}+CC_{0}(t_{2}-t_{1}).$ (3.72)

Proof. The proof proceeds in two steps. First, for $t\in(0,\sigma(T)),T\geq
1$, by (2.9), (3.4) and Lemma 3.5, one can check that

$\displaystyle\quad\int_{0}^{\sigma(T)}\sigma\|\nabla
u\|^{3}_{L^{3}}dt+\int_{0}^{\sigma(T)}\sigma^{3}\|\nabla u\|^{4}_{L^{4}}dt$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}t\left(\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{\frac{3}{2}}\|\nabla
u\|_{L^{2}}^{\frac{3}{2}}+\|\nabla
u\|_{L^{2}}^{3}+C_{0}^{\frac{3}{4}}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{\frac{3}{2}}+C_{0}^{\frac{1}{2}}\|\nabla
u\|_{L^{2}}^{\frac{3}{2}}+C_{0}\right)dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}t^{3}\left(\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{3}\|\nabla
u\|_{L^{2}}+\|\nabla
u\|_{L^{2}}^{4}+C_{0}^{\frac{1}{2}}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{3}+C_{0}\|\nabla
u\|_{L^{2}}+C_{0}\right)dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}t^{\frac{3\beta-2}{4}}(\|\nabla
u\|_{L^{2}}^{2})^{\frac{3}{4}}(t^{2-\beta}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})^{\frac{3}{4}}+CC_{0}^{\frac{3}{4}}\int_{0}^{\sigma(T)}t^{\frac{3\beta-2}{4}}(t^{2-\beta}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})^{\frac{3}{4}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}t^{\frac{2\beta-1}{2}}(\|\nabla
u\|_{L^{2}}^{2})^{\frac{1}{2}}(t^{3}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})^{\frac{1}{2}}(t^{2-\beta}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})dt+CC_{0}$
$\displaystyle\quad+CC_{0}^{\frac{1}{2}}\int_{0}^{\sigma(T)}t^{\frac{2\beta-1}{2}}(t^{3}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})^{\frac{1}{2}}(t^{2-\beta}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})dt+CC_{0}^{\frac{3}{4}}$
$\displaystyle\leq CC_{0}^{\frac{3}{4}},$ (3.73)

which, along with (3.13), gives (3.70).

Second, for $t\in(\sigma(T),T),T\geq 1$, we shall show that (3.71) holds on
each small time-interval. It should be pointed out that (3.70) implies that on
$t=1$,

$\|\nabla u(1)\|_{L^{2}}^{2}+\|\sqrt{\rho}\dot{u}(1)\|_{L^{2}}^{2}\leq
CC_{0}^{3/4}.$ (3.74)

Integrating (3) and (3) over $[1,3]$, summing them up, from (3.4), (2.9) and
(3.70), we can obtain that there exists a positive $C$, independent of $T$,
such that

$\displaystyle\sup_{1\leq t\leq 3}(\|\nabla
u\|_{L^{2}}^{2}+\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})+B[1,3]$
$\displaystyle\leq$ $\displaystyle\|\nabla
u(1)\|_{L^{2}}^{2}+\|\sqrt{\rho}\dot{u}(1)\|_{L^{2}}^{2}+CC_{0}+C\int_{1}^{3}\|\nabla
u\|_{L^{3}}^{3}dt+C\int_{1}^{3}\|\nabla u\|_{L^{4}}^{4}dt$ $\displaystyle\leq$
$\displaystyle
CC_{0}^{3/4}+C\int_{1}^{3}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{\frac{3}{2}}\|\nabla
u\|_{L^{2}}^{\frac{3}{2}}dt+C\int_{1}^{3}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{3}\|\nabla
u\|_{L^{2}}dt$ $\displaystyle\leq$ $\displaystyle CC_{0}^{3/4},$ (3.75)

For $T\geq 3$, Let $[T]$ be the largest integer less or equal to $T$. For each
integer $k=2,3,\cdots,[T]-1$, we introduce the function
$\sigma_{k}(t)\triangleq\sigma(t+1-k)=\min\\{1,t+1-k\\}$. Then, for
$t\in[k-1,k+1]$, by replacing $\sigma(t)$ with $\sigma_{k}(t)$ and repeating
the process of Lemma 3.4, we obtain that (3) and (3) still holds with
$\sigma_{k}(t)$ instead of $\sigma(t)$. Therefore, integrating them over
$[k-1,k+1]$ respectively, summing them up, from (3.4), (2.9) and (3.70), we
can obtain that there exists a positive $C$, independent of $k$ and $T$, such
that

$\displaystyle\sup_{k-1\leq t\leq k+1}(\sigma_{k}(t)\|\nabla
u\|_{L^{2}}^{2}+\sigma^{3}_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})+\int_{k-1}^{k+1}\big{(}\sigma_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\sigma^{3}_{k}(t)\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}dt$
$\displaystyle\leq$ $\displaystyle
CC_{0}^{3/4}+C\int_{k-1}^{k+1}\sigma_{k}(t)\|\nabla
u\|_{L^{3}}^{3}dt+C\int_{k-1}^{k+1}\sigma^{3}_{k}(t)\|\nabla
u\|_{L^{4}}^{4}dt$ $\displaystyle\leq$ $\displaystyle
CC_{0}^{3/4}+C\int_{k-1}^{k+1}\sigma_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{\frac{3}{2}}\|\nabla
u\|_{L^{2}}^{\frac{3}{2}}dt+C\int_{k-1}^{k+1}\sigma^{3}_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{3}\|\nabla
u\|_{L^{2}}dt$ $\displaystyle\leq$ $\displaystyle CC_{0}^{3/4}+C\sup_{1\leq
t\leq T}(\|\nabla
u\|_{L^{2}}^{4}+\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{4})\int_{0}^{T}\|\nabla
u\|_{L^{2}}^{2}dt+\frac{1}{2}\int_{k-1}^{k+1}\sigma_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}dt$
$\displaystyle\leq$ $\displaystyle
CC_{0}^{3/4}+\frac{1}{2}\int_{k-1}^{k+1}\sigma_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}dt,$
(3.76)

which implies that

$\displaystyle\sup_{k-1\leq t\leq k+1}(\sigma_{k}(t)\|\nabla
u\|_{L^{2}}^{2}+\sigma^{3}_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})$
$\displaystyle+\int_{k-1}^{k+1}\big{(}\sigma_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\sigma^{3}_{k}(t)\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}dt\leq
CC_{0}^{3/4},\quad\text{for}\,k=2,\cdots,[T]-1.$ (3.77)

Similarly, choosing
$\sigma_{[T]}(t)\triangleq\sigma(t+1-[T])=\min\\{1,t+1-[T]\\}$, one has

$\displaystyle\sup_{[T]-1\leq t\leq T}(\sigma_{k}(t)\|\nabla
u\|_{L^{2}}^{2}+\sigma^{3}_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})$
$\displaystyle+\int_{[T]-1}^{T}\big{(}\sigma_{k}(t)\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\sigma^{3}_{k}(t)\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}dt\leq
CC_{0}^{3/4}.$ (3.78)

Since $\sigma_{k}(t)=1$ for $t\in[k,k+1]$ and $\sigma_{[T]}(t)=1$ for
$t\in[[T],T]$, (3) and (3) yields that

$\displaystyle\displaystyle\sup_{1\leq t\leq T}(\|\nabla
u\|_{L^{2}}^{2}+\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})\leq CC_{0}^{3/4}.$ (3.79)

Combining (3.70) and (3.79) yields that

$\sup_{0\leq t\leq T}(\sigma\|\nabla
u\|_{L^{2}}^{2}+\sigma^{3}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})\leq
C_{5}C_{0}^{3/4}.$ (3.80)

Set $\varepsilon_{3}\triangleq\min\\{\varepsilon_{2},(C_{5}^{-4}\\}$, (3.71)
holds when $C_{0}<\varepsilon_{3}$. Noting that
$[0,\infty]=\cup_{k=1}^{\infty}[k-1,k+1]$ and the estimate (3) holds for each
$k=2,3,\cdots$, we thus conclude that (3.71) is valid in the case when
$T=\infty$.

Finally, we proceed to prove (3.72). Integrating (3) and (3) over
$(t_{1},t_{2})$ with $1\leq t_{1}<t_{2}\leq T$, summing them up, using (2.9),
(3.71) and (3.79), yields

$\displaystyle\quad B[t_{1},t_{2}]\leq
CC_{0}^{3/4}+C\int_{t_{1}}^{t_{2}}(\|\nabla u\|_{L^{3}}^{3}+\|\nabla
u\|_{L^{4}}^{4})dt$ $\displaystyle\leq
CC_{0}^{3/4}+C\int_{t_{1}}^{t_{2}}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{\frac{3}{2}}\|\nabla
u\|_{L^{2}}^{\frac{3}{2}}dt+C\int_{t_{1}}^{t_{2}}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{3}\|\nabla
u\|_{L^{2}}dt$ $\displaystyle\leq CC_{0}^{3/4}+CC_{0}(t_{2}-t_{1}),$ (3.81)

and finishes the proof of Lemma 3.7.

We now proceed to derive a uniform (in time) upper bound for the density,
which turns out to be the key to obtain all the higher order estimates and
thus to extend the classical solution globally.

###### Lemma 3.8

There exists a positive constant $\varepsilon_{4}$ depending on $\mu$,
$\lambda$, $\gamma$, $a$, $\bar{\rho}$, $\hat{\rho}$, $\beta$, $\Omega,$ and
$M$ such that, if $(\rho,u,\Phi)$ is a smooth solution of (1.1)-(1.4) on
${\Omega}\times(0,T]$ satisfying (3.4) and the initial data condition (1.14),
then

$\displaystyle\displaystyle\sup_{0\leq t\leq
T}\|\rho(t)\|_{L^{\infty}}\leq\frac{7\hat{\rho}}{4},$ (3.82)

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

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

$\displaystyle\displaystyle D_{t}\rho=g(\rho)+b^{\prime}(t),$ (3.83)

where

$\displaystyle\displaystyle D_{t}\rho\triangleq\rho_{t}+u\cdot\nabla\rho,\quad
g(\rho)\triangleq-\frac{\rho(P-P_{s})}{2\mu+\lambda},\quad
b(t)\triangleq-\frac{1}{2\mu+\lambda}\int_{0}^{t}\rho Fdt.$ (3.84)

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), (2.5), (2.6), (3.11),
(3.4) and Lemmas 3.2, 3.5 that for $\delta_{0}$ as in Proposition 3.1 and for
all $0\leq t_{1}\leq t_{2}\leq\sigma(T)$,

$\displaystyle\quad|b(t_{2})-b(t_{1})|=\frac{1}{\lambda+2\mu}\left|\int_{t_{1}}^{t_{2}}\rho
Fdt\right|\leq C\int_{0}^{\sigma(T)}\|F\|_{L^{\infty}}dt$ $\displaystyle\leq
C\int_{0}^{\sigma(T)}\|F\|_{L^{6}}^{\frac{1}{2}}\|\nabla
F\|_{L^{6}}^{\frac{1}{2}}dt+C\int_{0}^{\sigma(T)}\|F\|_{L^{2}}dt$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}(\|\sqrt{\rho}\dot{u}\|_{L^{2}}+C_{0}^{\frac{1}{6}}+\|\nabla
u\|_{L^{2}})^{\frac{1}{2}}(\|\nabla\dot{u}\|_{L^{2}}+\|\nabla
u\|_{L^{2}}^{2}+C_{0}^{\frac{1}{6}})^{\frac{1}{2}}dt+CC_{0}^{\frac{1}{2}}$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{\frac{1}{2}}\|\nabla\dot{u}\|_{L^{2}}^{\frac{1}{2}}dt+C\int_{0}^{\sigma(T)}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{\frac{1}{2}}\|\nabla
u\|_{L^{2}}dt$ $\displaystyle\quad+C\int_{0}^{\sigma(T)}\|\nabla
u\|_{L^{2}}^{\frac{1}{2}}\|\nabla\dot{u}\|_{L^{2}}^{\frac{1}{2}}dt+CC_{0}^{\frac{1}{12}}$
$\displaystyle\leq
C\int_{0}^{\sigma(T)}\big{(}t^{2-\beta}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\big{)}^{\frac{1-3\delta_{0}}{4}}\big{(}t\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\big{)}^{\frac{3\delta_{0}}{4}}\big{(}t^{2-\beta}\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}^{\frac{1}{4}}t^{\frac{(\beta-2)(2-3\delta_{0})-3\delta_{0}}{4}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}\big{(}t^{1-\beta}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}\big{)}^{\frac{1}{4}}\big{(}t\|\nabla
u\|_{L^{2}}^{2}\big{)}^{\frac{1}{2}}t^{\frac{2\beta-3}{4}}dt$
$\displaystyle\quad+C\int_{0}^{\sigma(T)}\big{(}t^{2-\beta}\|\nabla\dot{u}\|_{L^{2}}^{2}\big{)}^{\frac{1}{4}}\big{(}t\|\nabla
u\|_{L^{2}}^{2}\big{)}^{\frac{1}{4}}t^{\frac{\beta-3}{4}}dt+CC_{0}^{\frac{1}{12}}$
$\displaystyle\leq C_{6}C_{0}^{\delta_{0}/4}.$ (3.85)

Combining (3) with (3.83) and choosing $N_{1}=0$,
$N_{0}=C_{6}C_{0}^{\delta_{0}/4}$, $\bar{\zeta}=\hat{\rho}$ in Lemma A.2 give

$\displaystyle\displaystyle\sup_{t\in[0,\sigma(T)]}\|\rho\|_{L^{\infty}}\leq\hat{\rho}+C_{1}C_{0}^{\delta_{0}/4}\leq\frac{3\hat{\rho}}{2},$
(3.86)

provided
$C_{0}\leq\hat{\varepsilon}_{4}\triangleq\min\\{\varepsilon_{3},\left(\frac{\hat{\rho}}{2C_{6}}\right)^{\frac{4}{\delta_{0}}}\\}.$

On the other hand, for any $1\leq t_{1}\leq t_{2}\leq T,$ it follows from
(2.5), (3.11), (3.4) and Lemma 3.2 that

$\displaystyle\quad|b(t_{2})-b(t_{1})|\leq
C\int_{t_{1}}^{t_{2}}\|F\|_{L^{\infty}}dt$ $\displaystyle\leq
C\int_{t_{1}}^{t_{2}}\|F\|_{L^{6}}^{\frac{1}{2}}\|\nabla
F\|_{L^{6}}^{\frac{1}{2}}dt+C\int_{t_{1}}^{t_{2}}\|F\|_{L^{2}}dt$
$\displaystyle\leq
CC_{0}^{\frac{1}{12}}\int_{t_{1}}^{t_{2}}(\|\nabla\dot{u}\|_{L^{2}}^{\frac{1}{2}}+1)dt$
$\displaystyle\leq
C_{7}C_{0}^{\frac{3}{4}}+C_{8}C_{0}^{\frac{1}{12}}(t_{2}-t_{1}).$ (3.87)

Now we choose $N_{0}=C_{7}C_{0}^{3/4}$, $N_{1}=C_{8}C_{0}^{\frac{1}{12}}$ in
(A.3) and set $\bar{\zeta}=\frac{3\hat{\rho}}{2}$ in (A.4). Since for all
$\zeta\geq\bar{\zeta}=\frac{3\hat{\rho}}{2}>\rho_{s}+1$,

$g(\zeta)\leq-\frac{a}{\lambda+2\mu}\leq-C_{8}C_{0}^{\frac{1}{12}}=-N_{1}.$

Together with (3.83) and (3), by Lemma A.2, we have

$\displaystyle\displaystyle\sup_{t\in[\sigma(T),T]}\|\rho\|_{L^{\infty}}\leq\frac{3\hat{\rho}}{2}+C_{7}C_{0}^{3/4}\leq\frac{7\hat{\rho}}{4},$
(3.88)

provided
$C_{0}\leq\varepsilon_{4}\triangleq\min\\{\hat{\varepsilon}_{4},(\frac{\hat{\rho}}{4C_{7}})^{4/3},(\frac{a}{(2\mu+\lambda)C_{8}})^{12}\\}$.
The combination of (3.86) with (3.88) completes the proof of Lemma 3.8.

The following Lemmas deal with some necessary higher order estimates, which
make sure that one can extend the strong solution globally in time. The proofs
are similar to the ones in [13, 17, 21, 7], and are sketched here for
completeness. From now on, we always assume that the initial energy
$C_{0}\leq\varepsilon_{6}$, and the positive constant $C$ may depend on $T$,
$\mu$, $\lambda$, $a$, $\gamma$, $\bar{\rho},$ $\hat{\rho},$ $\Omega$, $M$,
$\rho_{s},\Phi_{s}$ and $g$, where $g\in L^{2}(\Omega)$ is given as in (1.15).

###### Lemma 3.9

There exists a positive constant $C,$ such that

$\displaystyle\sup_{0\leq t\leq T}(\|\nabla
u\|_{L^{2}}^{2}+\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2})+\int_{0}^{T}(\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\|\nabla\dot{u}\|_{L^{2}}^{2})dt\leq
C,$ (3.89) $\displaystyle\sup_{0\leq t\leq
T}(\|\nabla\rho\|_{L^{6}}+\|u\|_{H^{2}})+\int_{0}^{T}(\|\nabla
u\|_{L^{\infty}}+\|\nabla^{2}u\|_{L^{6}}^{2})dt\leq C,$ (3.90)
$\displaystyle\sup_{0\leq t\leq
T}\|\sqrt{\rho}u_{t}\|_{L^{2}}^{2}+\int_{0}^{T}\int|\nabla u_{t}|^{2}dxdt\leq
C.$ (3.91)

Proof. First, taking $s=1$ in (3.37) along with (3.71) and (3.72) gives

$\displaystyle\sup_{t\in[0,T]}\|\nabla
u\|_{L^{2}}^{2}+\int_{0}^{T}\int\rho|\dot{u}|^{2}dxdt\leq C.$ (3.92)

Choosing $m=0$ in (3), integrating it over $(0,T)$, by (2.19), (3.6), (3.92)
and the compatibility condition (1.15), we have

$\displaystyle\sup_{0\leq t\leq
T}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2}+\int_{0}^{T}\|\nabla\dot{u}\|_{L^{2}}^{2}dt\leq
C+C\int_{0}^{T}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{3}dt\leq
C+\frac{1}{2}\sup_{0\leq t\leq T}\|\sqrt{\rho}\dot{u}\|_{L^{2}}^{2},$

which along with (3.92) gives (3.89). Based on the Beale-Kato-Majda type
inequality (see Lemma A.5), we can derive (3.90), in arguments similar to [7]
and we omit the details. (3.89) and (3.90) directly yields (3.91). This
finishes the proof.

###### Lemma 3.10

There exists a positive constant $C$ such that

$\displaystyle\sup\limits_{0\leq t\leq
T}\left(\|\rho\|_{H^{2}}\\!+\\!\|P\|_{H^{2}}\\!+\\!\|\rho_{t}\|_{H^{1}}\\!+\\!\|P_{t}\|_{H^{1}}\right)\\!+\\!\\!\int_{0}^{T}\\!\\!\left(\|\rho_{tt}\|_{L^{2}}^{2}\\!+\\!\|P_{tt}\|_{L^{2}}^{2}\right)dt\leq
C,$ (3.93) $\displaystyle\sup\limits_{0\leq t\leq T}\sigma\|\nabla
u_{t}\|_{L^{2}}^{2}+\int_{0}^{T}\sigma\|\sqrt{\rho}u_{tt}\|_{L^{2}}^{2}dt\leq
C,$ (3.94) $\displaystyle\sup\limits_{0\leq t\leq
T}\left(\|\nabla(\Phi-\\!\Phi_{s})\|_{H^{3}}+\|\nabla\Phi_{t}\|_{H^{2}}+\|\nabla\Phi_{tt}\|_{L^{2}}\right)\leq
C.$ (3.95)

Proof. From (2.1), (2.1), (2.19), (3.11) and Lemma 3.9, we have

$\|\nabla^{3}u\|_{L^{2}}\leq
C\|\nabla\dot{u}\|_{L^{2}}+C\|\nabla^{2}P\|_{L^{2}}+C,$ (3.96)

which can help us to get (3.93) by the same method as that in [7]. So is
(3.94) and we omit the details. From (1.1), we have

$\Delta\Phi_{t}=\rho_{t},\qquad\Delta\Phi_{tt}=-{\rm div}(\rho_{t}u+\rho
u_{t}).$ (3.97)

With the similar arguments used in Lemma 2.4, combining (3.93) with (2.19) and
(3.97) leads to (3.95). The proof is completed.

###### Lemma 3.11

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

$\displaystyle\sup_{t\in[0,T]}\sigma\|\nabla
u\|_{H^{2}}^{2}+\int_{0}^{T}\left(\|\nabla
u\|_{H^{2}}^{2}+\|\nabla^{2}u\|^{p_{0}}_{W^{1,q}}+\sigma\|\nabla
u_{t}\|_{H^{1}}^{2}\right)dt\leq C,$ (3.98)
$\displaystyle\sup_{t\in[0,T]}\left(\|\rho-\rho_{s}\|_{W^{2,q}}+\|P-P_{s}\|_{W^{2,q}}+\|\nabla(\Phi-\Phi_{s})\|_{W^{3,q}}\right)\leq
C,$ (3.99)

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

Proof. Let’s start with (3.98). By Lemma 3.9, (3.94), (3.95) and Poincaré’s,
Sobolev’s inequalities, one can check that

$\displaystyle\|\nabla^{2}u\|_{H^{1}}$ $\displaystyle\leq
C(\|\rho\dot{u}\|_{H^{1}}\\!+\\!\|\rho\nabla(\Phi\\!-\\!\Phi_{s})\|_{H^{1}}\\!+\\!\|(\rho\\!-\\!\rho_{s})\nabla\Phi_{s}\|_{H^{1}}\\!+\\!\|P\\!-\\!P_{s}\|_{H^{2}}\\!+\\!\|u\|_{L^{2}})$
$\displaystyle\leq C+C\|\nabla u_{t}\|_{L^{2}}.$ (3.100)

It then follows from (3), (3.91) and (3.94) that

$\displaystyle\displaystyle\sup\limits_{0\leq t\leq T}\sigma\|\nabla
u\|_{H^{2}}^{2}+\int_{0}^{T}\|\nabla u\|_{H^{2}}^{2}dt\leq C.$ (3.101)

Next, we deduce from Lemmas 3.9-3.10 that

$\displaystyle\displaystyle\|\nabla^{2}u_{t}\|_{L^{2}}$ $\displaystyle\leq
C(\|(\rho\dot{u})_{t}\|_{L^{2}}+\|\nabla
P_{t}\|_{L^{2}}+\|(\rho\nabla\Phi)_{t}\|_{L^{2}}+\|u_{t}\|_{L^{2}})$
$\displaystyle\leq C\|\sqrt{\rho}u_{tt}\|_{L^{2}}+C\|\nabla
u_{t}\|_{L^{2}}+C,$ (3.102)

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

$\begin{cases}\mu\Delta u_{t}+(\lambda+\mu)\nabla{\rm
div}u_{t}=(\rho\dot{u})_{t}+\nabla
P_{t}+(\rho\nabla\Phi)_{t}\,\,\,&\text{in}\,\,\Omega,\\\ u_{t}\cdot
n=0\,\,\,\text{and}\,\,\,\omega_{t}\times
n=0\,\,&\text{on}\,\,\partial\Omega.\end{cases}$ (3.103)

Together with (3) and (3.94) yields

$\displaystyle\displaystyle\int_{0}^{T}\sigma\|\nabla
u_{t}\|_{H^{1}}^{2}dt\leq C.$ (3.104)

By Sobolev’s inequality, (3.11), (3.90) and (3.94), we get for any
$q\in(3,6)$,

$\displaystyle\displaystyle\|\nabla(\rho\dot{u})\|_{L^{q}}$ $\displaystyle\leq
C\|\nabla\rho\|_{L^{q}}(\|\nabla\dot{u}\|_{L^{q}}+\|\nabla\dot{u}\|_{L^{2}}+\|\nabla
u\|_{L^{2}}^{2})+C\|\nabla\dot{u}\|_{L^{q}}$ $\displaystyle\leq C(\|\nabla
u_{t}\|_{L^{2}}+1)+C\|\nabla u_{t}\|_{L^{2}}^{\frac{6-q}{2q}}\|\nabla
u_{t}\|_{L^{6}}^{\frac{3(q-2)}{2q}}$
$\displaystyle\quad+C(\|u\|_{L^{\infty}}\|\nabla^{2}u\|_{L^{q}}+\|\nabla
u\|_{L^{\infty}}\|\nabla u\|_{L^{q}})$ $\displaystyle\leq
C\sigma^{-\frac{1}{2}}+C\|\nabla
u\|_{H^{2}}+C\sigma^{-\frac{1}{2}}(\sigma\|\nabla
u_{t}\|_{H^{1}}^{2})^{\frac{3(q-2)}{4q}}+C.$ (3.105)

Integrating this inequality over $[0,T],$ by (3.89) and (3.104), we have

$\displaystyle\displaystyle\int_{0}^{T}\|\nabla(\rho\dot{u})\|_{L^{q}}^{p_{0}}dt\leq
C.$ (3.106)

On the other hand, the combination of (3.15) with (2.1), (2.1), (3.89) and
(3.94) gives

$\displaystyle\displaystyle(\|\nabla^{2}P\|_{L^{q}})_{t}$ $\displaystyle\leq
C\|\nabla u\|_{L^{\infty}}\|\nabla^{2}P\|_{L^{q}}+C\|\nabla^{2}u\|_{W^{1,q}}$
$\displaystyle\leq C(1+\|\nabla
u\|_{L^{\infty}})\|\nabla^{2}P\|_{L^{q}}+C(1+\|\nabla
u_{t}\|_{L^{2}})+C\|\nabla(\rho\dot{u})\|_{L^{q}},$ (3.107)

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

$\displaystyle\displaystyle\|\nabla^{2}u\|_{W^{1,q}}$ $\displaystyle\leq
C(1+\|\nabla
u_{t}\|_{L^{2}}+\|\nabla(\rho\dot{u})\|_{L^{q}}+\|\nabla^{2}P\|_{L^{q}}),$
(3.108)

due to (2.1), (2.1), (3.89) and (3.94). Hence, applying Gronwall’s inequality
in (3), we deduce from (3.90), (3.91) and (3.106) that

$\displaystyle\displaystyle\sup_{t\in[0,T]}\|\nabla^{2}P\|_{L^{q}}\leq C,$
(3.109)

which along with (3.91), (3.94) and (3.108) also gives

$\displaystyle\displaystyle\sup_{t\in[0,T]}\|P-P_{s}\|_{W^{2,q}}+\int_{0}^{T}\|\nabla^{2}u\|_{W^{1,q}}^{p_{0}}dt\leq
C.$ (3.110)

Similarly, one has

$\displaystyle\displaystyle\sup\limits_{0\leq t\leq
T}\|\rho-\rho_{s}\|_{W^{2,q}}\leq C,$ (3.111)

which together with (3.110) and (2.19) gives (3.99). The proof of Lemma 3.11
is finished.

###### Lemma 3.12

There exists a positive constant $C$ such that

$\displaystyle\displaystyle\sup_{0\leq t\leq
T}\sigma\left(\|\sqrt{\rho}u_{tt}\|_{L^{2}}+\|\nabla u_{t}\|_{H^{1}}+\|\nabla
u\|_{W^{2,q}}\right)+\int_{0}^{T}\sigma^{2}\|\nabla u_{tt}\|_{2}^{2}dt\leq C,$
(3.112)

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

Proof. Differentiating $\eqref{NSP-2}$ with respect to $t$ twice, multiplying
it by $u_{tt}$, and integrating over $\Omega$ lead to

$\displaystyle\quad\frac{d}{dt}\int\frac{\rho}{2}|u_{tt}|^{2}dx+(\lambda+2\mu)\int({\rm
div}u_{tt})^{2}dx+\mu\int|\omega_{tt}|^{2}dx$ $\displaystyle=-4\\!\int\rho
u^{i}_{tt}u\\!\cdot\\!\nabla\\!u^{i}_{tt}dx\\!-\\!\int(\rho
u)_{t}\\!\cdot\left[\nabla(u_{t}\\!\cdot\\!u_{tt})\\!+\\!\nabla
u_{t}\\!\cdot\\!u_{tt}\right]dx\\!-\\!\int(\rho_{tt}u\\!+\\!2\rho_{t}u_{t})\\!\cdot\\!\nabla
u\\!\cdot\\!u_{tt}dx$ $\displaystyle\quad-\\!\int(\rho u_{tt}\cdot\nabla
u\cdot u_{tt}\\!-\\!P_{tt}{\rm
div}u_{tt})dx\\!-\\!\int(\rho_{tt}\nabla\Phi\\!+\\!2\rho_{t}\nabla\Phi_{t}\\!+\\!\rho\nabla\Phi_{tt})u_{tt}dx$
$\displaystyle\triangleq\sum_{i=1}^{5}R_{i}.$ (3.113)

Let us estimate $R_{i}$ for $i=1,\cdots,5$. Hölder’s inequality, (3.89),
(3.90), (3.91), (3.93) and (3.94), give

$\displaystyle\displaystyle\sum_{i=1}^{4}R_{i}\leq$
$\displaystyle\delta\|\nabla
u_{tt}\|_{L^{2}}^{2}+C(\delta)\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+C(\delta)\|\nabla
u_{t}\|_{L^{2}}^{3}+C(\delta)\|\nabla u_{t}\|_{L^{2}}^{2}$
$\displaystyle+C(\delta)\|\rho_{tt}\|_{L^{2}}^{2}+C(\delta)\|P_{tt}\|^{2}_{L^{2}},$
(3.114)

By (3.89), (3.91), (3.93) and (3.94), we conclude that

$\displaystyle R_{5}$ $\displaystyle\leq
C\left(\|\rho_{tt}\|_{L^{2}}\|\nabla\Phi\|_{L^{3}}+\|\rho_{t}\|_{L^{3}}\|\nabla\Phi_{t}\|_{L^{6}}\right)\|u_{tt}\|_{L^{6}}+\|\nabla\Phi_{tt}\|_{L^{2}}\|u_{tt}\|_{L^{2}}$
$\displaystyle\leq\delta\|\nabla
u_{tt}\|_{L^{2}}^{2}+C(\delta)(\|\rho_{tt}\|^{2}_{L^{2}}+\|\nabla^{2}\Phi_{t}\|_{L^{2}}^{2}+\|\nabla\Phi_{tt}\|^{2}_{L^{2}}),$
(3.115)

Substituting these estimates into (3), utilizing the fact that

$\displaystyle\displaystyle\|\nabla u_{tt}\|_{L^{2}}\leq C(\|{\rm
div}u_{tt}\|_{L^{2}}+\|\omega_{tt}\|_{L^{2}}),$ (3.116)

due to Lemma A.3 since $u_{tt}\cdot n=0,$ on $\partial\Omega,$ and then
choosing $\delta$ small enough, we can get

$\displaystyle\frac{d}{dt}\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+\|\nabla
u_{tt}\|_{L^{2}}^{2}$ $\displaystyle\leq
C(\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+\|\rho_{tt}\|^{2}_{L^{2}}+\|P_{tt}\|^{2}_{L^{2}}+\|\nabla
u_{t}\|^{3}_{L^{2}}+\|\nabla u_{t}\|^{2}_{L^{2}}+C),$ (3.117)

which together with (3.93), (3.94), and by Gronwall’s inequality yields that

$\displaystyle\sup_{0\leq t\leq
T}\sigma^{2}\|\sqrt{\rho}u_{tt}\|^{2}_{L^{2}}+\int_{0}^{T}\sigma^{2}\|\nabla
u_{tt}\|_{L^{2}}^{2}dt\leq C.$ (3.118)

Furthermore, it follows from (2.1), (3) and (3.94) that

$\displaystyle\quad\sup_{0\leq t\leq T}\sigma\|\nabla^{2}u_{t}\|_{L^{2}}\leq
C\sigma(1+\|\sqrt{\rho}u_{tt}\|_{L^{2}}+\|\nabla u_{t}\|_{L^{2}})\leq C.$
(3.119)

Finally, we deduce from (3.94), (3.99), (3.98), (3), (3.108), (3.118) and
(3.119) that

$\displaystyle\displaystyle\sigma\|\nabla^{2}u\|_{W^{1,q}}$ $\displaystyle\leq
C\sigma(1+\|\nabla
u_{t}\|_{L^{2}}+\|\nabla(\rho\dot{u})\|_{L^{q}}+\|\nabla^{2}P\|_{L^{q}})$
$\displaystyle\leq C(1+\sigma\|\nabla
u\|_{H^{2}}+\sigma^{\frac{1}{2}}(\sigma\|\nabla
u_{t}\|_{H^{1}}^{2})^{\frac{3(q-2)}{4q}})$ $\displaystyle\leq
C+C\sigma^{\frac{1}{2}}(\sigma^{-1})^{\frac{3(q-2)}{4q}}\leq C,$ (3.120)

together with (3.118) and (3.119) yields (3.112) and this completes the proof
of Lemma 3.12.

##  4 Proof of Theorem 1.1

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

Proof of Theorem 1.1. By Lemma 2.2, there exists a $T_{*}>0$ such that the
system (1.1)-(1.4) has a unique classical solution $(\rho,u,\Phi)$ on
$\Omega\times(0,T_{*}]$. One may use the a priori estimates, Proposition 3.1
and Lemmas 3.10-3.12 to extend the classical solution $(\rho,u,\Phi)$ globally
in time.

First, by the definition of (3.1), the assumption of the initial data (1.14)
and (3.69), one immediately checks that

$\displaystyle\displaystyle
0\leq\rho_{0}\leq\hat{\rho},\,\,A_{1}(0)=0,\,\,A_{2}(0)\leq
C_{0}^{\delta_{0}}.$ (4.1)

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

$0\leq\rho_{0}\leq 2\hat{\rho},\,\,A_{1}(T)\leq
2C_{0}^{\frac{1}{2}},\,\,A_{2}(\sigma(T))\leq 2C_{0}^{\delta_{0}},$ (4.2)

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

$\displaystyle\displaystyle T^{*}=\sup\\{T\,|\,{\rm\eqref{pf2}\ holds}\\}.$
(4.3)

Then $T^{*}\geq T_{1}>0$. Hence, for any $0<\tau<T\leq T^{*}$ with $T$ finite,
it follows from Lemmas 3.9-3.12 that

$\begin{cases}(\rho-\rho_{s},\nabla(\Phi-\Phi_{s})\in C([0,T];H^{2}\cap
W^{2,q}),\\\ \nabla u\in C([\tau,T];H^{1}),\quad\nabla u_{t}\in
C([\tau,T];L^{q});\end{cases}$ (4.4)

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 (3.91), (3.94), (3.112) and $\eqref{NSP}_{1}$, we obtain

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

which together with (4.4) yields

$\displaystyle\displaystyle\sqrt{\rho}u_{t},\quad\sqrt{\rho}\dot{u}\in
C([\tau,T];L^{2}).$ (4.5)

Finally, we claim that

$\displaystyle\displaystyle T^{*}=\infty.$ (4.6)

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

$0\leq\rho\leq\frac{7}{4}\hat{\rho},\,\,A_{1}(T^{*})\leq
C_{0}^{\frac{1}{2}},\,\,A_{2}(\sigma(T^{*}))\leq C_{0}^{\delta_{0}},$ (4.7)

It follows from Lemmas 3.11, 3.12 and (4.5) that
$(\rho(x,T^{*}),u(x,T^{*}),H(x,T^{*}))$ satisfies the initial data condition
(1.13) and (1.15), where
$g(x)\triangleq\sqrt{\rho}\dot{u}(x,T^{*}),\,\,x\in\Omega.$ Thus, Lemma 2.2
implies that there exists some $T^{**}>T^{*}$ such that (4.2) holds for
$T=T^{**}$, which contradicts the definition of $T^{*}.$ As a result, (4.6)
holds. By Lemmas 2.2 and 3.9-3.12, it indicates that $(\rho,u,\Phi)$ is in
fact the unique classical solution defined on $\Omega\times(0,T]$ for any
$0<T<T^{*}=\infty.$ 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 [27]).

######  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 [42].

###### 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)

The following two lemmas are given in Propositions 2.6-2.9 in [1].

###### Lemma A.3

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

$\displaystyle\|v\|_{W^{k+1,q}}\leq C(\|{\rm div}v\|_{W^{k,q}}+\|{\rm
curl}v\|_{W^{k,q}}).$ (A.5)

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

$\displaystyle\|\nabla v\|_{L^{q}}\leq C(\|{\rm div}v\|_{L^{q}}+\|{\rm
curl}v\|_{L^{q}}).$ (A.6)

###### Lemma A.4

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

$\displaystyle\|v\|_{W^{k+1,q}}\leq C(\|{\rm div}v\|_{W^{k,q}}+\|{\rm
curl}v\|_{W^{k,q}}+\|v\|_{L^{q}}).$ (A.7)

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

$\displaystyle\|v\|_{W^{k+1,q}}\leq C(\|{\rm div}v\|_{W^{k,q}}+\|{\rm
curl}v\|_{W^{k,q}}).$ (A.8)

Finally, similar to [2, 17], we need a Beale-Kato-Majda type inequality with
respect to the slip boundary condition (1.3) which is given in [7].

###### Lemma A.5

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

$\displaystyle\|\nabla u\|_{L^{\infty}}\leq C\left(\|{\rm
div}u\|_{L^{\infty}}+\|{\rm
curl}u\|_{L^{\infty}}\right)\ln(e+\|\nabla^{2}u\|_{L^{q}})+C\|\nabla
u\|_{L^{2}}+C.$ (A.9)

## Acknowledgements

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

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