# Global Classical Large Solutions to Navier-Stokes Equations for Viscous
Compressible and Heat Conducting Fluids with Vacuum

Huanyao Wen1,2, Changjiang Zhu2 1 School of Mathematical Sciences South China
Normal University, Guangzhou 510631, P.R. China 2 The Hubei Key Laboratory of
Mathematical Physics School of Mathematics and Statistics Huazhong Normal
University, Wuhan 430079, P.R. China Corresponding author. Email:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider the 1D Navier-Stokes equations for viscous
compressible and heat conducting fluids (i.e., the full Navier-Stokes
equations). We get a unique global classical solution to the equations with
large initial data and vacuum. Because of the strong nonlinearity and
degeneration of the equations brought by the temperature equation and by
vanishing of density (i.e., appearance of vacuum) respectively, to our best
knowledge, there are only two results until now about global existence of
solutions to the full Navier-Stokes equations with special pressure, viscosity
and heat conductivity when vacuum appears (see [13] where the viscosity
$\mu=$const and the so-called variational solutions were obtained, and see [1]
where the viscosity $\mu=\mu(\rho)$ degenerated when the density vanishes and
the global weak solutions were got). It is open whether the global strong or
classical solutions exist. By applying our ideas which were used in our former
paper [8] to get $H^{3}-$estimates of $u$ and $\theta$ (see Lemma 3.10, Lemma
3.11, Lemma 3.12 and the corresponding corollaries), we get the existence and
uniqueness of the global classical solutions (see Theorem 1.1). In fact, the
existence of strong solutions would be done obviously by our estimates if the
regularity of the initial data is assumed to be weaker. Like [8], we get
$H^{4}-$regularity of $\rho$ and $u$ (see Theorem 1.2). We do not get further
regularity of $\theta$ such as $H^{4}-$regularity, because of the degeneration
and strong nonlinearity brought by vacuum and the term $(\mu uu_{x})_{x}$ in
the temperature equation. This can be viewed the first result on global
classical solutions to the 1D Navier-Stokes equations for viscous compressible
and heat conducting fluids which may be large initial data and contain vacuum.

Key Words: Compressible Navier-Stokes equations, heat conducting fluids,
vacuum, global classical solutions.
2000 Mathematics Subject Classification. 35Q30, 35K65, 76N10.

## Contents

1\. Introduction 2

2\. Preliminaries 7

3\. Proof of Theorem 1.1 9

4\. Proof of Theorem 1.2 26

References 36

##  1 Introduction

In this paper, we consider the Navier-Stokes equations for viscous
compressible and heat conducting fluids (i.e. the full Navier-Stokes
equations). The model, describing for instance the motion of gas, plays an
important role in applied physics. Mathematically, the model in one dimension
can be written as follows in sense of Eulerian coordinates:

$\displaystyle\begin{cases}\rho_{t}+(\rho u)_{x}=0,\ \rho\geq 0,\\\ (\rho
u)_{t}+(\rho u^{2})_{x}+P_{x}=(\mu u_{x})_{x},\\\ (\rho E)_{t}+(\rho
uE)_{x}+(Pu)_{x}=(\mu uu_{x})_{x}+(\kappa\theta_{x})_{x},\end{cases}$ (1.1)

for $(x,t)\in(0,1)\times(0,+\infty)$. Here $\rho=\rho(x,t)$, $u=u(x,t)$,
$P=P(\rho,\theta)$, $E$, $\theta$ and $\kappa=\kappa(\rho,\theta)$ denote the
density, velocity, pressure, total energy, absolute temperature and
coefficient of heat conduction, respectively. The total energy
$E=e+\frac{1}{2}u^{2}$, where $e$ is the internal energy. $\mu>0$ is the
coefficient of viscosity. $P$ and $e$ satisfy the second principle of
thermodynamics:

$P=\rho^{2}\frac{\partial e}{\partial\rho}+\theta\frac{\partial
P}{\partial\theta}.$ (1.2)

In the present paper, we consider the initial and boundary conditions:

$(\rho,\ u,\ \theta)\big{|}_{t=0}=(\rho_{0},\ u_{0},\ \theta_{0})(x)\ \
{\rm{in}}\ \ [0,1],$ (1.3)

and

$(u,\ \theta_{x})\big{|}_{x=0,1}=0,\ t\geq 0.$ (1.4)

Since the model is important, lots of works on the existence, uniqueness,
regularity and asymptotic behavior of the solutions were done during the last
five decades. While, because of the stronger nonlinearity in (1.1) compared
with the Navier-Stokes equations for isentropic fluids (no temperature
equation), many known mathematical results mainly focused on the absence of
vacuum (vacuum means $\rho=0$), refer for instance to [17, 18, 24, 25, 29, 30,
34] for classical solutions. More precisely, the local classical solutions to
the Navier-Stokes equations with heat-conducting fluid in Hölder spaces was
obtained respectively by Itaya in [17] for Cauchy problem and by Tani in [34]
for IBVP with $\inf\rho_{0}>0$, where the spatial dimension $N=3$. Using
delicate energy methods in Sobolev spaces, Matsumura and Nishida showed in
[29, 30] that the global classical solutions exist provided that the initial
data is small in some sense and away from vacuum and the spatial dimension
$N=3$. For large initial data and dimension $N=1$, Kazhikhov, Shelukhi in [25]
(for polytropic perfect gas with $\mu,\kappa=$const.) and Kawohl in [24] (for
real gas with $\kappa=\kappa(\rho,\theta),\ \mu=\mathrm{const.}$) respectively
got global classical solutions to (1.1) in Lagrangian coordinates with
boundary condition (1.4) and $\inf\rho_{0}>0$. The internal energy $e$ and the
coefficient of heat conduction $\kappa$ in [24] satisfy the following
assumptions for $\rho\leq\overline{\varrho}$ and $\theta\geq 0$ (we translate
these conditions in Eulerian coordinates)

$\begin{cases}e(\rho,0)\geq 0,\ \
\nu(1+\theta^{r})\leq\partial_{\theta}e(\rho,\theta)\leq
N(\overline{\varrho})(1+\theta^{r}),\\\
\kappa_{0}(1+\theta^{q})\leq\kappa(\rho,\theta)\leq\kappa_{1}(1+\theta^{q}),\\\
|\partial_{\rho}\kappa(\rho,\theta)|+|\partial_{\rho\rho}\kappa(\rho,\theta)|\leq\kappa_{1}(1+\theta^{q}),\end{cases}$
(1.5)

where $r\in[0,1]$, $q\geq 2+2r$, and $\nu$, $N(\overline{\varrho})$,
$\kappa_{0}$ and $\kappa_{1}$ are positive constants. For the perfect gas in
the domain exterior to a ball in $\mathbb{R}^{N}$ ($N=2$ or $3$) with
$\mu,\kappa=$const., Jiang in [18] got the existence of global classical
spherically symmetric large solutions in Hölder spaces.

In fact, Kawohl in [24] also considered the case of $\mu=\mu(\rho)$ for
another boundary condition with $\inf\rho_{0}>0$, where
$0<\underline{\mu}_{0}\leq\mu(\rho)\leq\overline{\mu}_{0}$ for any $\rho\geq
0$ and $\underline{\mu}_{0}$ and $\overline{\mu}_{0}$ are positive constant.
This result was generalized to the case $\mu(\rho)=\rho^{\alpha}$ by Jiang in
[19] for $\alpha\in(0,\frac{1}{4})$, and by Qin, Yao in [31] for
$\alpha\in(0,\frac{1}{2})$, respectively.

On the existence, asymptotic behavior of the weak solutions for full Navier-
Stokes equations (including the temperature equation) with $\inf\rho_{0}>0$,
please refer for instance to [20, 21, 23] for weak solutions in 1D and for
spherically symmetric weak solutions in bounded annular domains in
$\mathbb{R}^{N}$ ($N=2$, $3$), and refer to [12] for variational solutions in
a bounded domain in $\mathbb{R}^{N}$ ($N=2$, $3$).

In the presence of vacuum (i.e. $\rho$ may vanish), to our best knowledge, the
mathematical results about global well-posedness of the full Navier-Stokes
equations are usually limited to the existence of weak solutions with special
pressure, viscosity and heat conductivity (see [1, 13]). More precisely,
Feireisl in [13] got the existence of so-called variational solutions in
dimension $N\geq 2$. The temperature equation in [13] is satisfied only as an
inequality. Anyway, this work in [13] is the very first attempt towards the
existence of weak solutions to the full compressible Navier-Stokes equations
in higher dimensions, where the viscosity $\mu$ is constant and

$\begin{cases}\kappa=\kappa(\theta)\in C^{2}[0,\infty),\
\underline{\kappa}(1+\theta^{a})\leq\kappa(\theta)\leq\overline{\kappa}(1+\theta^{a})\
\ \mathrm{for}\ \mathrm{all}\ \theta\geq 0,\\\
P=P(\rho,\theta)=\mathcal{P}_{e}(\rho)+\theta\mathcal{P}_{\theta}(\rho)\ \
\mathrm{for}\ \mathrm{all}\ \rho\geq 0\ \mathrm{and}\ \theta\geq 0,\\\
\mathcal{P}_{e},\mathcal{P}_{\theta}\in C[0,\infty)\cap C^{1}(0,\infty);\
\mathcal{P}_{e}(0)=0,\ \mathcal{P}_{\theta}(0)=0,\\\
\mathcal{P}_{e}^{\prime}(\rho)\geq a_{1}\rho^{\overline{\gamma}-1}-b_{1}\ \
\mathrm{for}\ \mathrm{all}\ \rho>0;\ \mathcal{P}_{e}(\rho)\leq
a_{2}\rho^{\overline{\gamma}}+b_{1}\ \ \mathrm{for}\ \mathrm{all}\ \rho\geq
0,\\\ \mathcal{P}_{\theta}\ \ \mathrm{is}\ \ \mbox{non-decreasing}\
\mathrm{in}\ [0,\infty);\ \mathcal{P}_{\theta}(\rho)\leq
a_{3}(1+\rho^{\Gamma})\ \ \mathrm{for}\ \mathrm{all}\ \rho\geq 0,\end{cases}$
(1.6)

where $\Gamma<\frac{\overline{\gamma}}{2}$ if $N=2$ and
$\Gamma=\frac{\overline{\gamma}}{N}$ for $N\geq 3$; $a\geq 2$,
$\overline{\gamma}>1$, and $a_{1}$, $a_{2}$, $a_{3}$, $b_{1}$,
$\underline{\kappa}$ and $\overline{\kappa}$ are positive constants. Note that
the perfect gas equation of state (i.e. $P=R\rho\theta$ for some constant
$R>0$) is not involved in (1.6). In order that the equations are satisfied as
equalities in the sense of distribution, Bresch and Desjardins in [1] proposed
some different assumptions from [13], and obtained the existence of global
weak solutions to the full Navier-Stokes equations with large initial data in
$\mathbb{T}^{3}$ or $\mathbb{R}^{3}$. In [1], the viscosity $\mu=\mu(\rho)$
may vanish when vacuum appears, and $\kappa$, $P$ and $e$ are assumed to
satisfy

$\begin{cases}\kappa(\rho,\theta)=\kappa_{0}(\rho,\theta)(\rho+1)(\theta^{a}+1),\\\
P=r\rho\theta+p_{c}(\rho),\\\ e=C_{\upsilon}\theta+e_{c}(\rho),\end{cases}$
(1.7)

where $a\geq 2$, $r$ and $C_{\upsilon}$ are two positive constants,
$p_{c}(\rho)=O(\rho^{-\ell})$ and $e_{c}(\rho)=O(\rho^{-\ell-1})$ (for some
$\ell>1$) when $\rho$ is small enough, and $\kappa_{0}(\rho,\theta)$ is
assumed to satisfy

$\underline{c}_{0}\leq\kappa_{0}(\rho,\theta)\leq\frac{1}{\underline{c}_{0}},$

for $\underline{c}_{0}>0$. We have to mention that the smooth solutions in
$C^{1}\left([0,\infty);H^{d}(\mathbb{R}^{N})\right)$ ($d>2+[\frac{N}{2}]$)
would blow up when the initial density is of nontrivial compact support (see
[36]). On the local existence and uniqueness of strong solutions in
$\mathbb{R}^{3}$, please refer to [4] for the perfect gas with
$\mu,\kappa=$const.

It is still open whether global strong (or classical) solutions exist when
vacuum appears (i.e., the density may vanish). Our main concern here is to
show the existence and uniqueness of global classical solutions to (1.1)-(1.4)
with vacuum and large initial data. In fact, the existence of the strong
solutions to this problem is obvious if the regularity of initial data is
assumed to be weaker.

For compressible isentropic Navier-Stokes equations (i.e. no temperature
equation), there are so many results about the well-posedness and asymptotic
behaviors of the solutions when vacuum appears. Refer to [14, 22, 26, 28] and
[15, 27, 35, 38, 39, 40] for global weak solutions with constant viscosity and
with density-dependent viscosity, respectively. Refer to [6, 10] and [2, 3, 5,
33] for global strong solutions and for local strong (classical) solutions
with constant viscosity, respectively. Recently, Huang, Li, Xin in [16] and
Ding, Wen, Yao, Zhu in [8, 9] independently got existence and uniqueness of
global classical solutions, where the initial energy in [16] is assumed to be
small in $\mathbb{R}^{3}$ and $\rho-\widetilde{\rho}\in
C\left([0,T];H^{3}(\mathbb{R}^{3})\right)$, $u\in
C\left([0,T];D^{1}(\mathbb{R}^{3})\cap D^{3}(\mathbb{R}^{3})\right)\cap
L^{\infty}\left([\tau,T];D^{4}(\mathbb{R}^{3})\right)$ (for $\tau>0$) which
generalized the results in [3], and the initial data in [8, 9] could be large
for dimension $N=1$ and could be large but spherically symmetric for $N\geq
2$, and $(\rho,u)\in C([0,T];H^{4}(I))$ ($I$ is bounded in [8], and is bounded
or an exterior domain in [9]).

We would like to give some notations which will be used throughout the paper.

Notations:

(1) $I=[0,1]$, $\partial I=\\{0,1\\}$, $Q_{T}=I\times[0,T]$ for $T>0$.

(2) For $p\in[1,\infty]$, $L^{p}=L^{p}(I)$ denotes the $L^{p}$ space with the
norm $\|\cdot\|_{L^{p}}$. For $k\geq 1$ and $p\in[1,\infty]$,
$W^{k,p}=W^{k,p}(I)$ denotes the Sobolev space, whose norm is denoted as
$\|\cdot\|_{W^{k,p}}$, $H^{k}=W^{k,2}(I)$.

(3) For an integer $k\geq 0$ and $0<\alpha<1$, let $C^{k+\alpha}(I)$ denote
the Schauder space of functions on $I$, whose $k$th order derivative is Hölder
continuous with exponents $\alpha$, with the norm $\|\cdot\|_{C^{k+\alpha}}$.

In this paper, our assumptions are the following:

($A_{1}$): $\int_{I}\rho_{0}>0$.

($A_{2}$): $\mu=\mathrm{const.}>0$, $e=C_{0}Q(\theta)+e_{c}(\rho)$, $P=\rho
Q(\theta)+P_{c}(\rho)$, $\kappa=\kappa(\theta)$, for some constant $C_{0}>0$.

($A_{3}$): $P_{c}(\rho)\geq 0$, $e_{c}(\rho)\geq 0$, for $\rho\geq 0$;
$P_{c}\in C^{2}[0,\infty)$; $\rho|\frac{\partial e_{c}}{\partial\rho}|\leq
C_{1}e_{c}(\rho)$, for some constant $C_{1}>0$.

$(A_{4}):$ $Q(\cdot)\in C^{2}[0,\infty)$ satisfies

$\begin{cases}C_{2}\left(\beta+(1-\beta)\theta+\theta^{1+r}\right)\leq
Q(\theta)\leq C_{3}\left(\beta+(1-\beta)\theta+\theta^{1+r}\right),\\\
C_{4}(1+\theta^{r})\leq Q^{\prime}(\theta)\leq
C_{5}(1+\theta^{r}),\end{cases}\ \ \ \ \ \ \ \ \ \ \ \ \ $

for some constants $C_{i}>0$ ($i=2$, $3$, $4$, $5$) and $r\geq 0$, $\beta=0$
or $1$.

($A_{5}$): $\kappa\in C^{2}[0,\infty)$ satisfies

$C_{6}(1+\theta^{q})\leq\kappa(\theta)\leq C_{7}(1+\theta^{q}),$

for $q\geq 2+2r$, and some constants $C_{i}>0$ ($i=6$, $7$).

($A_{6}$): $Q,P_{c}\in C^{4}[0,\infty)$, and $\kappa$ satisfies

$|\partial_{\theta}^{3}\kappa(\theta)|\leq C_{8}(1+\theta^{q-3}),$

for $\theta>0$ and some constant $C_{8}>0$.

###### Remark 1.1

$(i)$ $(A_{1})$ is needed to get the upper bounds of $\theta$ and $\theta_{t}$
in terms of some norms by using mass conservation, Lemma 2.1 and Corollary
2.1.

$(ii)$ The case for the perfect gas (i.e. $P=R\rho\theta$, $e=C_{\nu}\theta$
for constants $R>0$ and $C_{\nu}>0$) is involved in the above assumptions.

$(iii)$ As it mentioned in [24], the restriction on $\mu$ ($\mu$=const., see
other restrictions on $\kappa$ and e in (1.5)) is not physically motivated.
Physically, it seems more importantly that the state functions e, $\mu$ and
$\kappa$ usually depend on both $\rho$ and $\theta$. Particularly, the
internal energy e grows as $\theta^{1+r}$ with $r\approx 0.5$, the
conductivity $\kappa$ grows as $\theta^{q}$ with $4.5\leq q\leq 5.5$ and
viscosity $\mu$ increases like $\theta^{p}$ with $0.5\leq p\leq 0.8$ (see [24,
31] and references therein). Because of mathematical technique, in the present
paper, we assume $\mu=$const. and $\kappa=\kappa(\theta)$ as in [13] (see
(1.6)). From ($A_{2}$)–($A_{5}$), we know that e and $\kappa$ grow
respectively as $\theta^{1+r}$ and $\theta^{q}$, where $q$ can be taken as
$q\in[4.5,5.5]$, and r can be taken as $r=0.5$ if we consider $\theta>0$.

$(iv)$ The restriction on $q$ in ($A_{5}$) (i.e. $q\geq 2+2r$) is same as
(1.5), and is the same as (1.6) and (1.7) when we take $r=0$. This assumption
plays an important role in the analysis.

Main results:

###### Theorem 1.1

In addition to $(A_{1})$-$(A_{5})$, we assume $\rho_{0}\geq 0$, $\rho_{0}\in
H^{2}$, $(\sqrt{\rho_{0}})_{x}\in L^{\infty}$, $u_{0}\in H^{3}\cap H_{0}^{1}$,
$\theta_{0}\in H^{3}$, $\partial_{x}\theta_{0}|_{x=0,1}=0$, and that the
following compatible conditions are valid:

$\displaystyle\begin{cases}\mu u_{0xx}-[P(\rho_{0},\
\theta_{0})]_{x}=\sqrt{\rho}_{0}g_{1},\\\
\left[\kappa(\theta_{0})\theta_{0x}\right]_{x}+\mu|u_{0x}|^{2}=\sqrt{\rho_{0}}\
g_{2},\ x\in I,\end{cases}$ (1.8)

for some $g_{1},g_{2}\in L^{2}$, and
$\left(\sqrt{\rho_{0}}g_{1}\right)_{x},\left(\sqrt{\rho_{0}}g_{2}\right)_{x}\in
L^{2}$. Then for any $T>0$ there exists a unique global solution $(\rho,\ u,\
\theta)$ to (1.1)-(1.4) such that

$\displaystyle\rho\in C([0,T];H^{2}),\ \rho_{t}\in C([0,T];H^{1}),\
\sqrt{\rho}\in W^{1,\infty}(Q_{T}),$ $\displaystyle u\in
L^{\infty}([0,T];H^{3}),\ \sqrt{\rho}u_{t}\in L^{\infty}([0,T];L^{2}),$
$\displaystyle\rho u_{t}\in L^{\infty}([0,T];H^{1}),\ \ u_{t}\in
L^{2}([0,T];H_{0}^{1}),\ \ \sqrt{\rho}e_{t}\in L^{\infty}([0,T];L^{2}),$
$\displaystyle\rho e_{t}\in L^{\infty}([0,T];H^{1}),\ \ \theta\in
L^{\infty}([0,T];H^{3}),\ \ \theta_{t}\in L^{2}([0,T];H^{1}).$

###### Remark 1.2

(i) (1.8) was proposed by Cho and Kim in [4] to get $\mathrm{local}$
$H^{2}$-regularity of $u$ and $\theta$ for $\mathrm{the\ polytropic\ perfect\
gas}$. The detailed reasons why such conditions were needed can be found in
[4]. Roughly speaking, $g_{1}$ and $g_{2}$ are equivalent to
$\sqrt{\rho}u_{t}$ and $\sqrt{\rho}e_{t}$ at $t=0$, respectively.

(ii) By the Sobolev embedding theorems (cf. [7]) and Lemma 2.3, we know from
Theorem 1.1

$\displaystyle\rho\in C\left([0,T];C^{1+\frac{1}{2}}(I)\right)\cap
C^{1}\left([0,T];C^{\frac{1}{2}}(I)\right),$ $\displaystyle u\in
C\left([0,T];C^{2+\sigma}(I)\right),\ (\rho u)_{t}\in
C\left([0,T];C^{\sigma}(I)\right),$ $\displaystyle\theta\in
C\left([0,T];C^{2+\sigma}(I)\right),\ (\rho e)_{t}\in
C\left([0,T];C^{\sigma}(I)\right),$

for any $T>0$ and $\sigma\in(0,\frac{1}{2})$. This implies $(\rho,u,\theta)$
is the classical solution to (1.1)-(1.4).

###### Theorem 1.2

In addition to $(A_{1})$-$(A_{6})$, we assume $\rho_{0}\geq 0$, $\rho_{0}\in
H^{4}$, $(\sqrt{\rho_{0}})_{x}\in L^{\infty}$, $u_{0}\in H^{4}\cap H_{0}^{1}$,
$\theta_{0}\in H^{3}$, $\partial_{x}\theta_{0}|_{x=0,1}=0$, $q>2+2r$, and that
the following compatible conditions are valid:

$\displaystyle\begin{cases}\mu u_{0xx}-[P(\rho_{0},\
\theta_{0})]_{x}=\rho_{0}g_{3},\\\
\left[\kappa(\theta_{0})\theta_{0x}\right]_{x}+\mu|u_{0x}|^{2}=\sqrt{\rho_{0}}\
g_{2},\ x\in I,\end{cases}$ (1.9)

for some $g_{3}\in H_{0}^{1}$,
$\left(\sqrt{\rho_{0}}\partial_{x}g_{3}\right)_{x}\in L^{2}$, and
$g_{2},\left(\sqrt{\rho_{0}}g_{2}\right)_{x}\in L^{2}$. Then for any $T>0$
there exists a unique global solution $(\rho,\ u,\ \theta)$ to (1.1)-(1.4)
satisfying:

$\displaystyle\rho\in C([0,T];H^{4}),\ \rho_{t}\in C([0,T];H^{3}),\
\sqrt{\rho}\in W^{1,\infty}(Q_{T}),$ $\displaystyle u\in C([0,T];H^{4})\cap
L^{2}([0,T];H^{5}),\ u_{t}\in L^{\infty}([0,T];H_{0}^{1})\cap
L^{2}([0,T];H^{3}),$ $\displaystyle(\rho u)_{t}\in C([0,T];H^{2}),\ \
\sqrt{\rho}u_{xxt}\in L^{\infty}([0,T];L^{2}),\ \ \sqrt{\rho}e_{t}\in
L^{\infty}([0,T];L^{2}),$ $\displaystyle(\rho e)_{t}\in
L^{\infty}([0,T];H^{1}),\ \ \theta\in L^{\infty}([0,T];H^{3})\cap
L^{2}([0,T];H^{4}),\ \ \theta_{t}\in L^{2}([0,T];H^{1}).$

###### Remark 1.3

(i) (1.9)1 was proposed by Cho and Kim in [3] where they consider the local
existence of classical solutions for $\mathrm{isentropic\ fluids}$ (no
temperature equation). Roughly speaking, $g_{3}$ is equivalent to $u_{t}$ at
$t=0$.

(ii) We could not get $\theta\in C([0,T];H^{4})$ (or
$L^{\infty}([0,T];H^{4})$) even if (1.9)2 is changed similarly to (1.9)1,
because of the strong nonlinearity and degeneration brought by $(\mu
uu_{x})_{x}$ in the temperature equation and the appearance of vacuum,
respectively.

(iii) Using ideas of Cho and Kim in [3], we can also get

$u\in L^{\infty}\left([\tau,T];H^{4}\right),\ \theta\in
L^{\infty}\left([\tau,T];H^{3}\right),$

for $\tau>0$. If we can obtain our estimates in higher dimensions, it will be
useful to investigate the local (global) existence of classical solutions to
the full Navier-Stokes equations (including the temperature equation) in
$\mathbb{R}^{N}$ ($N\geq 2$). For example, to guarantee (1.4) in
$\mathbb{R}^{N}$ ($N=2$ or $3$) is valid for all $t\geq 0$, it is necessary to
get $\theta\in L^{\infty}\left([0,T];H^{3}\right)$. We will consider these
problems in the near future.

The constants $C_{0}$ in $(A_{2})$ and the viscosity $\mu$ don’t play any role
in the analysis, we assume henceforth that $C_{0}=1$ and $\mu=1$ for
simplicity.

The rest of the paper is organized as follows. In Section 2, we present some
useful lemmas which will be used in the next sections. In Section 3, we prove
Theorem 1.1 by giving the initial density and the initial temperature a lower
bound $\delta>0$, getting a sequence of approximate solutions to (1.1)-(1.4),
and taking $\delta\rightarrow 0^{+}$ after making some estimates uniformly for
$\delta$. More precisely, based on Lemma 2.1 and the one-dimensional
properties of the equations, we get $H^{2}-$estimates of the solutions. Using
our ideas in [8, 9], we obtain $H^{3}-$estimates of $u$ and $\theta$. In
Section 4, using the similar arguments as in Section 3, we prove Theorem 1.2.

##  2 Preliminaries

###### Lemma 2.1

Let $\Omega=[\overline{a},\overline{b}]$ be a bounded domain in $\mathbb{R}$,
and $\rho$ be a non-negative function such that

$0<M\leq\int_{\Omega}\rho\leq K,$

for constants $M>0$ and $K>0$. Then

$\|v\|_{L^{\infty}(\Omega)}\leq\frac{K}{M}\|v_{x}\|_{L^{1}(\Omega)}+\frac{1}{M}\left|\int_{\Omega}\rho
v\right|,$

for any $v\in H^{1}(\Omega)$.

Proof. For any $x\in\Omega$, we have

$\displaystyle|v(x)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{M}\left|v(x)\int_{\Omega}\rho(y)dy\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{M}\left|\int_{\Omega}v(x)\rho(y)dy-\int_{\Omega}\rho(y)v(y)dy\right|+\frac{1}{M}\left|\int_{\Omega}\rho(y)v(y)dy\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{M}\left|\int_{\Omega}\int_{y}^{x}v_{\xi}(\xi)d\xi\rho(y)dy\right|+\frac{1}{M}\left|\int_{\Omega}\rho(y)v(y)dy\right|$
$\displaystyle\leq$
$\displaystyle\frac{K}{M}\|v_{x}\|_{L^{1}(\Omega)}+\frac{1}{M}\left|\int_{\Omega}\rho(y)v(y)dy\right|.$

$\Box$

###### Remark 2.1

The version of higher dimensions for Lemma 2.1 can be found in [12] or [13].

###### Corollary 2.1

Consider the same conditions in Lemma 2.1, and in addition assume $\Omega=I$,
and

$\|\rho v\|_{L^{1}(I)}\leq\overline{c}.$

Then for any $l>0$, there exists a positive constant $C=C(M,K,l,\overline{c})$
such that

$\|v^{l}\|_{L^{\infty}(I)}\leq C\|(v^{l})_{x}\|_{L^{2}(I)}+C,$

for any $v^{l}\in H^{1}(I)$.

Proof. By Lemma 2.1, we have

$\displaystyle\|v^{l}\|_{L^{\infty}(I)}\leq
C\|(v^{l})_{x}\|_{L^{2}(I)}+C\int_{I}\rho|v^{l}|.$

Case 1: $l\in(0,1]$.

In this case, we use the Young inequality to get

$\displaystyle\|v^{l}\|_{L^{\infty}(I)}$ $\displaystyle\leq$ $\displaystyle
C\|(v^{l})_{x}\|_{L^{2}(I)}+C\int_{I}\rho|v|+C\int_{I}\rho+C$
$\displaystyle\leq$ $\displaystyle C\|(v^{l})_{x}\|_{L^{2}(I)}+C.$

Case 2: $l\in(1,\infty)$.

In the case, we use the Young inequality again to get

$\displaystyle\|v^{l}\|_{L^{\infty}(I)}$ $\displaystyle\leq$ $\displaystyle
C\|(v^{l})_{x}\|_{L^{2}(I)}+C\|v^{l-1}\|_{L^{\infty}(I)}\int_{I}\rho|v|$
$\displaystyle\leq$ $\displaystyle
C\|(v^{l})_{x}\|_{L^{2}(I)}+\frac{1}{2}\|v^{l}\|_{L^{\infty}(I)}+C.$

This gives

$\displaystyle\|v^{l}\|_{L^{\infty}(I)}\leq C\|(v^{l})_{x}\|_{L^{2}(I)}+C.$

$\Box$

###### Lemma 2.2

For any $v\in H^{1}_{0}(I)$, we have

$\|v\|_{L^{\infty}(I)}\leq\|v_{x}\|_{L^{1}}.$

Proof. Since $v(0)=0$, we have for any $x\in I$

$\displaystyle|v(x)|=|v(x)-v(0)|=\left|\int_{0}^{x}v_{x}\right|\leq\|v_{x}\|_{L^{1}(I)}.$

This completes the proof. $\Box$

###### Lemma 2.3

([32]). Assume $X\subset E\subset Y$ are Banach spaces and
$X\hookrightarrow\hookrightarrow E$. Then the following imbedding are compact:

$\displaystyle(i)\ \ \left\\{\varphi:\varphi\in
L^{q}(0,T;X),\frac{\partial\varphi}{\partial t}\in
L^{1}(0,T;Y)\right\\}\hookrightarrow\hookrightarrow L^{q}(0,T;E),\ \ {\rm if}\
\ 1\leq q\leq\infty;$ $(ii)\ \ \left\\{\varphi:\varphi\in
L^{\infty}(0,T;X),\frac{\partial\varphi}{\partial t}\in
L^{r}(0,T;Y)\right\\}\hookrightarrow\hookrightarrow C([0,T];E),\ \ {\rm if}\ \
1<r\leq\infty.$

##  3  Proof of Theorem 1.1

In this section, we get a global solution to (1.1)-(1.4) with initial density
and initial temperature having a respectively lower bound $\delta>0$ by using
some a priori estimates of the solutions based on the local existence. Theorem
1.1 would be got after we make some a priori estimates uniformly for $\delta$
and take $\delta\rightarrow 0^{+}$.

Denote $\rho_{0}^{\delta}=\rho_{0}+\delta$ and
$\theta_{0}^{\delta}=\theta_{0}+\delta$ for $\delta\in(0,1)$. Throughout this
section, we denote $c$ to be a generic constant depending on $\rho_{0}$,
$u_{0}$, $\theta_{0}$, $T$ and some other known constants but independent of
$\delta$ for any $\delta\in(0,1)$.

Before proving Theorem 1.1, we need the following auxiliary theorem.

###### Theorem 3.1

Consider the same assumptions as in Theorem 1.1. Then for any $T>0$ and
$\delta\in(0,1)$ there exists a unique global solution $(\rho,u,\theta)$ to
(1.1)-(1.4) with initial data replaced by
($\rho_{0}^{\delta},u_{0},\theta_{0}^{\delta}$), such that

$\displaystyle\rho\in C([0,T];H^{2}),\ \ \ \rho_{t}\in C([0,T];H^{1}),\ \ \
\rho_{tt}\in L^{2}([0,T];L^{2}),\ \rho\geq\frac{\displaystyle\delta}{c}>0,$
$\displaystyle u\in C([0,T];H^{3}\cap H^{1}_{0}),\ u_{t}\in C([0,T];H^{1})\cap
L^{2}([0,T];H^{2}),\ \ \ u_{tt}\in L^{2}([0,T];L^{2}),\ $
$\displaystyle\theta\geq c_{\delta}>0,\ \theta\in C([0,T];H^{3}),\ \
\theta_{t}\in C([0,T];H^{1})\cap L^{2}([0,T];H^{2}),\ \ \theta_{tt}\in
L^{2}([0,T];L^{2}),$

where $c_{\delta}$ is a constant depending on $\delta$, but independent of
$u$.

Proof of Theorem 3.1:

The local solutions as in Theorem 3.1 can be obtained by the successive
approximations like in [4]. We omit it here for brevity. The regularities
guarantee the uniqueness (refer for instance to [4]). Based on it, Theorem 3.1
can be proved by some a priori estimates globally in time.

For any given $T\in(0,+\infty)$, let $(\rho,u,\theta)$ be the solution to
(1.1)-(1.4) as in Theorem 3.1. Then we have the following basic energy
estimate.

###### Lemma 3.1

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\displaystyle\int_{I}\rho\left(1+e_{c}(\rho)+\theta^{1+r}+u^{2}\right)(t)\leq
c.$

Proof. Integrating $(\ref{non-1.2})_{1}$ and $(\ref{non-1.2})_{3}$ over
$I\times[0,t]$, and using (1.4) , $(A_{2})$ and $(A_{4})$, we complete the
proof of Lemma 3.1. $\Box$

###### Lemma 3.2

Under the conditions of Theorem 3.1, it holds for any $(x,t)\in Q_{T}$

$\displaystyle\begin{cases}0<\rho(x,t)\leq c,\\\ \theta(x,t)>0.\end{cases}$

Proof. The proof of the upper bound of $\rho$ relies on constant viscosity
(i.e. $\mu=const.$). It is similar to [37].

Denote

$w(x,t)=\int_{0}^{t}(u_{x}-P-\rho u^{2})+\int_{0}^{x}\rho_{0}u_{0}.$ (3.1)

Differentiating (3.1) with respect to $x$, and using $(\ref{non-1.2})_{2}$, we
have

$w_{x}=\rho u.$

This together with Lemma 3.1 and the Cauchy inequality gives

$\int_{I}|w_{x}|\leq c.$

It follows from (3.1), (1.2), ($A_{2}$), ($A_{3}$), ($A_{4}$), (1.4), and
Lemma 3.1 that

$\left|\int_{I}w\right|\leq c.$

This gives for any $(x,t)\in Q_{T}$

$\displaystyle|w(x,t)|$ $\displaystyle\leq$
$\displaystyle\left|w(x,t)-\int_{I}w\right|+\left|\int_{I}w\right|$
$\displaystyle\leq$ $\displaystyle\left|\int_{I}\int_{y}^{x}w_{\xi}(\xi,t)d\xi
dy\right|+c$ $\displaystyle\leq$ $\displaystyle\int_{I}|w_{x}|+c\leq c,$

which implies

$\|w\|_{L^{\infty}(Q_{T})}\leq c.$ (3.2)

For any $(x,t)\in Q_{T}$, let $X(s;x,t)$ satisfy

$\displaystyle\begin{cases}\frac{dX(s;x,t)}{ds}=u\left(X(s;x,t),s\right),\
0\leq s<t,\\\ X(t;x,t)=x.\end{cases}$ (3.3)

Denote

$F(x,t)=\exp\left\\{w(x,t)\right\\}.$

It is easy to verify

$\displaystyle\frac{d(\rho F)\left(X(s;x,t),s\right)}{ds}$ $\displaystyle=$
$\displaystyle F\left(\rho_{s}+\frac{\partial\rho}{\partial
X}u+\rho\frac{\partial w}{\partial X}u+\rho w_{s}\right)$ (3.4)
$\displaystyle=$ $\displaystyle-\rho PF.$

Multiplying (3.4) by $\exp\left(\int_{0}^{s}P\right)$, we have

$\frac{d}{ds}\left\\{\rho F\exp\left(\int_{0}^{s}P\right)\right\\}=0.$

Integrating it over $(0,t)$, we have

$\rho(x,t)=\frac{F(X(0;x,t),0)}{F(x,t)}\rho_{0}^{\delta}\exp\left(-\int_{0}^{t}P\right),$
(3.5)

which implies

$\rho(x,t)>0,$

for any $(x,t)\in Q_{T}$.

By (3.2), (3.5) and $P\geq 0$, we get the upper bound of $\rho$. The lower
bound of $\theta$ can be got by (3.7) and the maximum principle for parabolic
equations. $\Box$

###### Lemma 3.3

Under the conditions of Theorem 3.1, it holds for any given $\alpha\in(0,1)$

$\int_{Q_{T}}\left(\frac{u_{x}^{2}}{\theta^{\alpha}}+\frac{(1+\theta^{q})\theta_{x}^{2}}{\theta^{1+\alpha}}\right)\leq
c,$

where $c$ may depend on $\alpha$.

###### Remark 3.1

$\alpha$ was usually taken as $1$ when the basic energy inequality was done
(see [1] and references therein). This depends on $\rho_{0}\log\theta_{0}\in
L^{1}$ which can not be got under the assumptions of Theorem 1.1 and Theorem
1.2, since $\theta_{0}$ may vanish.

Proof. From (1.2) and (1.1), we get

$\rho e_{\theta}\theta_{t}+\rho ue_{\theta}\theta_{x}+\theta
P_{\theta}u_{x}=u_{x}^{2}+\left(\kappa(\theta)\theta_{x}\right)_{x}.$ (3.6)

Substituting $e=Q(\theta)+e_{c}(\rho)$ and $P=\rho Q(\theta)+P_{c}(\rho)$ into
(3.6), we get

$\rho Q^{\prime}(\theta)\theta_{t}+\rho
uQ^{\prime}(\theta)\theta_{x}+\rho\theta
Q^{\prime}(\theta)u_{x}=u_{x}^{2}+\left(\kappa(\theta)\theta_{x}\right)_{x},$
(3.7)

or

$(\rho Q)_{t}+(\rho uQ)_{x}+\rho\theta
Q^{\prime}(\theta)u_{x}=u_{x}^{2}+\left(\kappa(\theta)\theta_{x}\right)_{x}.$
(3.8)

Multiplying (3.7) by $\theta^{-\alpha}$, integrating the resulting equation
over $Q_{T}$, and using integration by parts, we have

$\displaystyle\int_{Q_{T}}\left(\frac{u_{x}^{2}}{\theta^{\alpha}}+\frac{\alpha\kappa(\theta)\theta_{x}^{2}}{\theta^{1+\alpha}}\right)$
(3.9) $\displaystyle=$
$\displaystyle\int_{I}\rho\int_{0}^{\theta}\frac{Q^{\prime}(\xi)}{\xi^{\alpha}}-\int_{I}\rho_{0}\int_{0}^{\theta_{0}}\frac{Q^{\prime}(\xi)}{\xi^{\alpha}}+\int_{Q_{T}}\rho\theta^{1-\alpha}Q^{\prime}(\theta)u_{x}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\int_{0}^{\theta}\frac{1+\xi^{r}}{\xi^{\alpha}}+c\int_{I}\rho_{0}\int_{0}^{\theta_{0}}\xi^{r-\alpha}+c\int_{Q_{T}}\rho\theta^{1-\alpha}(1+\theta^{r})|u_{x}|$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho(1+\theta^{1+r})+c+\frac{1}{2}\int_{Q_{T}}\frac{u_{x}^{2}}{\theta^{\alpha}}+c\int_{Q_{T}}\rho^{2}\theta^{2-\alpha+2r}$
$\displaystyle\leq$ $\displaystyle
c+\frac{1}{2}\int_{Q_{T}}\frac{u_{x}^{2}}{\theta^{\alpha}}+c\int_{0}^{T}\max\limits_{x\in
I}\theta^{1+r-\alpha},$

where we have used ($A_{4}$), the Cauchy inequality, Lemma 3.1 and Lemma 3.2.
Now we estimate the last term of (3.9) as follows:

$\displaystyle c\int_{0}^{T}\max\limits_{x\in I}\theta^{1+r-\alpha}$
$\displaystyle\leq$ $\displaystyle
c+\int_{0}^{T}\|\theta^{r-\alpha}\theta_{x}\|_{L^{2}}$ (3.10)
$\displaystyle\leq$ $\displaystyle
c+c\int_{0}^{T}\left(\int_{I}\frac{\theta_{x}^{2}\theta^{2r-\alpha+1}}{\theta^{1+\alpha}}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
c+\frac{1}{2}\int_{Q_{T}}\frac{\alpha\kappa(\theta)\theta_{x}^{2}}{\theta^{1+\alpha}},$

where we have used Corollary 2.1, Lemma 3.1, ($A_{5}$) and the Cauchy
inequality. By (3.9), (3.10) and ($A_{5}$), we complete the proof. $\Box$

###### Corollary 3.1

Under the conditions of Theorem 3.1, it holds

$\displaystyle\int_{0}^{T}\|\theta\|_{L^{\infty}}^{q-\alpha+1}\leq c.$

Proof. By Corollary 2.1 and Lemma 3.1, we have

$\displaystyle\int_{0}^{T}\|\theta\|_{L^{\infty}}^{q-\alpha+1}$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\|\theta^{\frac{q-\alpha+1}{2}}\|_{L^{\infty}}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{T}\int_{I}\left(\theta^{\frac{q-\alpha-1}{2}}\theta_{x}\right)^{2}+c$
$\displaystyle=$ $\displaystyle
c\int_{0}^{T}\int_{I}\theta^{q-\alpha-1}\theta_{x}^{2}+c$ $\displaystyle\leq$
$\displaystyle c.$

$\Box$

###### Lemma 3.4

Under the conditions of Theorem 3.1, it holds

$\int_{Q_{T}}u_{x}^{2}\leq c.$

Proof. From (1.1)1 and (1.1)2, we get

$\rho u_{t}+\rho uu_{x}+P_{x}=u_{xx}.$ (3.11)

Multiplying (3.11) by $u$, integrating it over $I$, and using integration by
parts, we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho u^{2}+\int_{I}u_{x}^{2}$
$\displaystyle=$ $\displaystyle\int_{I}Pu_{x}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{x}^{2}+c\int_{I}\rho^{2}Q^{2}+c\int_{I}P_{c}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{x}^{2}+c\int_{I}\theta^{2+2r}+c,$

where we have used the Cauchy inequality, ($A_{2}$), ($A_{3}$), ($A_{4}$) and
Lemma 3.2. This implies

$\displaystyle\frac{d}{dt}\int_{I}\rho u^{2}+\int_{I}u_{x}^{2}\leq
c\sup\limits_{x\in I}\theta^{q-\alpha+1}+c.$

Integrating it over $(0,t)$, and using Corollary 3.1, we complete the proof of
Lemma 3.4. $\Box$

###### Lemma 3.5

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}(u_{x}^{2}+\rho\theta^{q+2+r})+\int_{Q_{T}}\left(\rho
u_{t}^{2}+(1+\theta^{q})^{2}\theta_{x}^{2}\right)\leq c.$

Proof. Multiplying (3.11) by $u_{t}$, integrating it over $I$, and using
integration by parts, Lemma 2.2, Lemma 3.2 and the Cauchy inequality, we have

$\displaystyle\int_{I}\rho u_{t}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}u_{x}^{2}$
$\displaystyle=$ $\displaystyle\frac{d}{dt}\int_{I}Pu_{x}-\int_{I}\rho
uu_{x}u_{t}-\int_{I}P_{t}u_{x}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\rho u_{t}^{2}+\frac{1}{2}\int_{I}\rho
u^{2}u_{x}^{2}+\frac{d}{dt}\int_{I}Pu_{x}-\int_{I}P_{t}u_{x}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}\rho
u_{t}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+\frac{d}{dt}\int_{I}Pu_{x}-\int_{I}P_{t}(u_{x}-P)-\frac{1}{2}\frac{d}{dt}\int_{I}P^{2},$

which implies

$\displaystyle\int_{I}\rho u_{t}^{2}+\frac{d}{dt}\int_{I}u_{x}^{2}\leq
c\left(\int_{I}u_{x}^{2}\right)^{2}+2\frac{d}{dt}\int_{I}Pu_{x}-\frac{d}{dt}\int_{I}P^{2}-2\int_{I}P_{t}(u_{x}-P).$
(3.12)

We are going to estimate the last term of the right side of (3.12). Using
($A_{2}$), (3.8), (1.1)1 and integration by parts, we have

$\displaystyle-2\int_{I}P_{t}(u_{x}-P)$ $\displaystyle=$
$\displaystyle-2\int_{I}(\rho Q)_{t}(u_{x}-P)-2\int_{I}(P_{c})_{t}(u_{x}-P)$
$\displaystyle=$
$\displaystyle-2\int_{I}\left[(\kappa\theta_{x})_{x}+u_{x}^{2}-(\rho
uQ)_{x}-\rho\theta Q^{\prime}(\theta)u_{x}\right](u_{x}-P)$
$\displaystyle+2\int_{I}P_{c}^{\prime}(\rho)(\rho_{x}u+\rho u_{x})(u_{x}-P)$
$\displaystyle=$ $\displaystyle
2\int_{I}\kappa\theta_{x}(u_{xx}-P_{x})-2\int_{I}u_{x}^{2}(u_{x}-P)-2\int_{I}\rho
uQ(u_{xx}-P_{x})$ $\displaystyle+2\int_{I}\rho\theta
Q^{\prime}(\theta)u_{x}(u_{x}-P)-2\int_{I}P_{c}u(u_{xx}-P_{x})-2\int_{I}P_{c}u_{x}(u_{x}-P)$
$\displaystyle+2\int_{I}\rho P_{c}^{\prime}(\rho)u_{x}(u_{x}-P).$

This, together with (3.11), ($A_{2}$), ($A_{4}$), Lemma 2.2, Lemma 3.2, the
Cauchy inequality, and $W^{1,1}(I)\hookrightarrow L^{\infty}(I)$, gives

$\displaystyle-2\int_{I}P_{t}(u_{x}-P)$ $\displaystyle\leq$ $\displaystyle
2\int_{I}\kappa\theta_{x}(\rho u_{t}+\rho
uu_{x})+2\|u_{x}-P\|_{L^{\infty}}\int_{I}u_{x}^{2}-2\int_{I}\rho uQ(\rho
u_{t}+\rho uu_{x})$ (3.13) $\displaystyle+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}u_{x}^{2}+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho Q^{2}+c\sup\limits_{x\in I}\theta^{1+r}$
$\displaystyle-2\int_{I}P_{c}u(\rho u_{t}+\rho
uu_{x})+c\int_{I}u_{x}^{2}+c\int_{I}\rho Q^{2}+c$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\int_{I}\rho
u_{t}^{2}+c\int_{I}\kappa^{2}\theta_{x}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+c\left(\|u_{x}-P\|_{L^{1}}+\|\rho
u_{t}+\rho uu_{x}\|_{L^{1}}\right)\int_{I}u_{x}^{2}$
$\displaystyle+c\int_{I}u_{x}^{2}\int_{I}\rho Q^{2}+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}(u_{x}^{2}+\rho Q^{2})+c\sup\limits_{x\in
I}\theta^{1+r}+c$ $\displaystyle\leq$ $\displaystyle\frac{1}{4}\int_{I}\rho
u_{t}^{2}+c\int_{I}\kappa^{2}\theta_{x}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+\frac{1}{4}\int_{I}\rho
u_{t}^{2}+c\int_{I}u_{x}^{2}\int_{I}\rho Q^{2}$
$\displaystyle+c\sup\limits_{x\in I}(1+\theta^{1+r})\int_{I}(u_{x}^{2}+\rho
Q^{2})+c\sup\limits_{x\in I}\theta^{1+r}+c.$

Substituting (3.13) into (3.12), we have

$\displaystyle\frac{1}{2}\int_{I}\rho u_{t}^{2}+\frac{d}{dt}\int_{I}u_{x}^{2}$
(3.14) $\displaystyle\leq$ $\displaystyle
c\left(\int_{I}u_{x}^{2}\right)^{2}+2\frac{d}{dt}\int_{I}Pu_{x}-\frac{d}{dt}\int_{I}P^{2}+c\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{I}u_{x}^{2}\int_{I}\rho
Q^{2}$ $\displaystyle+c\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}(u_{x}^{2}+\rho Q^{2})+c\sup\limits_{x\in
I}\theta^{1+r}+c.$

Integrating (3.14) over $(0,t)$, and using ($A_{2}$)-($A_{4}$), Lemma 3.2,
Corollary 3.1, Lemma 3.4 and the Cauchy inequality, we have

$\displaystyle\frac{1}{2}\int_{0}^{t}\int_{I}\rho u_{t}^{2}+\int_{I}u_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+2\int_{I}(\rho
Q+P_{c})u_{x}+c\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{0}^{t}\int_{I}u_{x}^{2}\int_{I}\rho\theta^{2+2r}$
$\displaystyle+c\int_{0}^{t}\sup\limits_{x\in
I}\theta^{1+r}\int_{I}u_{x}^{2}+c\int_{0}^{t}\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho\theta^{2+2r}+c$ $\displaystyle\leq$
$\displaystyle
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+\frac{1}{2}\int_{I}u_{x}^{2}+c\int_{I}\rho\theta^{2+2r}+c\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{0}^{t}\int_{I}u_{x}^{2}\int_{I}\rho\theta^{2+2r}$
$\displaystyle+c\int_{0}^{t}\sup\limits_{x\in
I}\theta^{1+r}\int_{I}u_{x}^{2}+c\int_{0}^{t}\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho\theta^{2+2r}+c.$

The second term of the right side can be absorbed by the left. After that, we
have

$\displaystyle\int_{0}^{t}\int_{I}\rho u_{t}^{2}+\int_{I}u_{x}^{2}$ (3.15)
$\displaystyle\leq$ $\displaystyle
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho\theta^{q+2+r}+c\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}+c\int_{0}^{t}\int_{I}u_{x}^{2}\int_{I}\rho\theta^{2+2r}$
$\displaystyle+c\int_{0}^{t}\sup\limits_{x\in
I}\theta^{1+r}\int_{I}u_{x}^{2}+c\int_{0}^{t}\sup\limits_{x\in
I}(1+\theta^{1+r})\int_{I}\rho\theta^{2+2r}+c.$

Here, we have used Lemma 3.2 and the Young inequality on the second term of
the right side. Note that the terms about $\theta$ in (3.15) need to be
handled. To do this, we make use of (3.7).

Multiplying (3.7) by $\int_{0}^{\theta}\kappa(\xi)d\xi$, integrating it over
$I$, and using integration by parts, ($A_{4}$) and ($A_{5}$), we have

$\displaystyle\frac{d}{dt}\int_{I}\rho\left[\int_{0}^{\theta}Q^{\prime}(\eta)\int_{0}^{\eta}\kappa(\xi)d\xi
d\eta\right]+\int_{I}\kappa^{2}\theta_{x}^{2}$
$\displaystyle=\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi-\int_{I}\rho\theta
Q^{\prime}(\theta)u_{x}\int_{0}^{\theta}\kappa(\xi)d\xi$ $\displaystyle\leq
c\|(1+\theta^{q})\theta\|_{L^{\infty}}\int_{I}u_{x}^{2}+c\|(1+\theta^{q})\theta\|_{L^{\infty}}\int_{I}\rho(1+\theta^{1+r})|u_{x}|.$
(3.16)

By Corollary 2.1 and ($A_{5}$), we get

$\|(1+\theta^{q})\theta\|_{L^{\infty}}\leq c\|\kappa\theta_{x}\|_{L^{2}}+c.$
(3.17)

Substituting (3.17) into (3), and using the Hölder inequality, the Cauchy
inequality and Lemma 3.2, we get

$\displaystyle\frac{d}{dt}\int_{I}\rho\left[\int_{0}^{\theta}Q^{\prime}(\eta)\int_{0}^{\eta}\kappa(\xi)d\xi
d\eta\right]+\int_{I}\kappa^{2}\theta_{x}^{2}$ $\displaystyle\leq$
$\displaystyle
c\|\kappa\theta_{x}\|_{L^{2}}\int_{I}u_{x}^{2}+c\int_{I}u_{x}^{2}+c\|\kappa\theta_{x}\|_{L^{2}}\int_{I}\rho(1+\theta^{1+r})|u_{x}|+c\int_{I}\rho(1+\theta^{1+r})|u_{x}|$
$\displaystyle\leq$ $\displaystyle
c\|\kappa\theta_{x}\|_{L^{2}}\left(\int_{I}u_{x}^{2}+\|\rho(1+\theta^{1+r})\|_{L^{2}}\|u_{x}\|_{L^{2}}\right)+c\int_{I}u_{x}^{2}+c\int_{I}\rho(1+\theta^{2+2r})$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\kappa^{2}\theta_{x}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho\theta^{2+2r}\int_{I}u_{x}^{2}+c\int_{I}\rho\theta^{2+2r}+c,$

which implies

$\displaystyle\frac{d}{dt}\int_{I}\rho\left[\int_{0}^{\theta}Q^{\prime}(\eta)\int_{0}^{\eta}\kappa(\xi)d\xi
d\eta\right]+\frac{1}{2}\int_{I}\kappa^{2}\theta_{x}^{2}$ $\displaystyle\leq$
$\displaystyle
c\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho\theta^{2+2r}\int_{I}u_{x}^{2}+c\int_{I}\rho\theta^{2+2r}+c.$

Integrating it over $(0,t)$, and using ($A_{4}$), ($A_{5}$), Lemma 3.1 and
Corollary 3.1, we get

$\int_{I}\rho\theta^{q+2+r}+\int_{0}^{t}\int_{I}\kappa^{2}\theta_{x}^{2}\leq
c\int_{0}^{t}\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{0}^{t}\left(\int_{I}\rho\theta^{2+2r}\int_{I}u_{x}^{2}\right)+c.$
(3.18)

By (3.15), (3.18), Corollary 3.1, Lemma 3.4, and the Gronwall inequality, we
complete the proof. $\Box$

###### Lemma 3.6

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}(\rho_{x}^{2}+\rho_{t}^{2})+\int_{Q_{T}}u_{xx}^{2}\leq c.$

Proof. Differentiating $(\ref{non-1.2})_{1}$ with respect to $x$, we have

$\rho_{xt}+\rho_{xx}u+2\rho_{x}u_{x}+\rho u_{xx}=0.$ (3.19)

Multiplying (3.19) by $2\rho_{x}$, integrating it over $I$ and using
integration by parts, we have

$\displaystyle\frac{d}{dt}\int_{I}\rho_{x}^{2}$ $\displaystyle=$
$\displaystyle-3\int_{I}\rho_{x}^{2}u_{x}-2\int_{I}\rho\rho_{x}u_{xx}$ (3.20)
$\displaystyle=$
$\displaystyle-3\int_{I}\rho_{x}^{2}(u_{x}-P)-3\int_{I}\rho_{x}^{2}P-2\int_{I}\rho\rho_{x}u_{xx}$
$\displaystyle\leq$ $\displaystyle
c\left(\|u_{x}-P\|_{L^{2}}+\|u_{xx}-P_{x}\|_{L^{2}}\right)\int_{I}\rho_{x}^{2}+c\int_{I}u_{xx}^{2}+c\int_{I}\rho_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\left(1+\|\sqrt{\rho}u_{t}\|_{L^{2}}\right)\int_{I}\rho_{x}^{2}+c\int_{I}u_{xx}^{2},$

where we have used (3.11), the Sobolev inequality, ($A_{2}$)-($A_{4}$), the
Cauchy inequality, Lemma 2.2, Lemma 3.2 and Lemma 3.5.

It follows from (3.11), Lemma 2.2, Lemma 3.2, ($A_{2}$)-($A_{4}$), Lemma 3.5
and the Cauchy inequality that

$\displaystyle\int_{I}u_{xx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho
u_{t}^{2}+c\left(\int_{I}u_{x}^{2}\right)^{2}+c\int_{I}\rho_{x}^{2}Q^{2}+c\int_{I}\rho^{2}\left[Q^{\prime}(\theta)\right]^{2}\theta_{x}^{2}+c\int_{I}\rho_{x}^{2}+c$
(3.21) $\displaystyle\leq$ $\displaystyle c\int_{I}\rho
u_{t}^{2}+c\sup\limits_{x\in
I}(1+\theta^{2+2r})\int_{I}\rho_{x}^{2}+c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c$
$\displaystyle\leq$ $\displaystyle c\int_{I}\rho u_{t}^{2}+c\sup\limits_{x\in
I}(1+\theta^{q-\alpha+1})\int_{I}\rho_{x}^{2}+c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c.$

Substituting (3.21) into (3.20), and using the Gronwall inequality, Corollary
3.1 and Lemma 3.5, we get

$\int_{I}\rho_{x}^{2}\leq c.$ (3.22)

By (3.21), (3.22), Corollary 3.1 and Lemma 3.5, we have

$\displaystyle\int_{Q_{T}}u_{xx}^{2}\leq c.$

It follows from (1.1)1, (3.22), Lemma 2.2, Lemma 3.2 and Lemma 3.5 that

$\int_{I}\rho_{t}^{2}\leq c.$

The proof of the lemma is complete. $\Box$

###### Lemma 3.7

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}\left(\rho
u_{t}^{2}+\theta_{x}^{2}\right)+\int_{Q_{T}}\left(u_{xt}^{2}+\rho\theta_{t}^{2}\right)\leq
c.$

Proof. Differentiating (3.11) with respect to $t$, we have

$\rho u_{tt}+\rho_{t}u_{t}+\rho_{t}uu_{x}+\rho u_{t}u_{x}+\rho
uu_{xt}+P_{xt}=u_{xxt}.$ (3.23)

Multiplying (3.23) by $u_{t}$, integrating the resulting equation over $I$, we
have

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
u_{t}^{2}+\int_{I}u_{xt}^{2}$ $\displaystyle=$ $\displaystyle-2\int_{I}\rho
uu_{t}u_{xt}-\int_{I}\rho_{t}uu_{x}u_{t}-\int_{I}\rho
u_{t}^{2}u_{x}+\int_{I}P_{t}u_{tx}$ $\displaystyle\leq$ $\displaystyle
2\|\sqrt{\rho}u_{t}\|_{L^{2}}\|\sqrt{\rho}u\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}+\|u_{t}\|_{L^{\infty}}\|u\|_{L^{\infty}}\|\rho_{t}\|_{L^{2}}\|u_{x}\|_{L^{2}}+\|u_{x}\|_{L^{\infty}}\int_{I}\rho
u_{t}^{2}$
$\displaystyle+\|P_{c}^{\prime}(\rho)\|_{L^{\infty}}\|\rho_{t}\|_{L^{2}}\|u_{xt}\|_{L^{2}}+\|Q(\theta)\|_{L^{\infty}}\|\rho_{t}\|_{L^{2}}\|u_{xt}\|_{L^{2}}+\|\rho
Q^{\prime}(\theta)\theta_{t}\|_{L^{2}}\|u_{xt}\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{xt}^{2}+c\int_{I}\rho
u_{t}^{2}+c+c\int_{I}u_{xx}^{2}\int_{I}\rho u_{t}^{2}+c\sup\limits_{x\in
I}\theta^{2+2r}+c\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$

Here, we have used $(\ref{non-1.2})_{1}$, integration by parts, the Hölder
inequality, the Cauchy inequality, the Sobolev inequality,
($A_{2}$)-($A_{4}$), Lemma 2.2, Lemma 3.2, Lemma 3.5 and Lemma 3.6.

The first term of the right side can be absorbed by the left. This implies

$\displaystyle\frac{d}{dt}\int_{I}\rho u_{t}^{2}+\int_{I}u_{xt}^{2}$ (3.24)
$\displaystyle\leq$ $\displaystyle c\int_{I}\rho
u_{t}^{2}+c+c\int_{I}u_{xx}^{2}\int_{I}\rho u_{t}^{2}+c\sup\limits_{x\in
I}\theta^{2+2r}+c\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$

Integrating (3.24) over $(0,t)$, and using Corollary 3.1 and Lemma 3.5, we
have

$\displaystyle\int_{I}\rho
u_{t}^{2}+\int_{0}^{t}\int_{I}u_{xt}^{2}\leq\int_{I}\rho
u_{t}^{2}(0)+c+c\int_{0}^{t}\int_{I}u_{xx}^{2}\int_{I}\rho
u_{t}^{2}+c\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$
(3.25)

Multiplying (3.11) by $\frac{1}{\sqrt{\rho}}$, taking $t\rightarrow 0^{+}$ and
using (1.8)1, we have

$\displaystyle|\sqrt{\rho}u_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|u_{0xx}-P(\rho_{0}^{\delta},\theta_{0}^{\delta})_{x}\right|}{\sqrt{\rho_{0}^{\delta}}}+\sqrt{\rho_{0}^{\delta}}|u_{0}u_{0x}|$
$\displaystyle\leq$
$\displaystyle\frac{\left|u_{0xx}-P(\rho_{0},\theta_{0})_{x}\right|}{\sqrt{\rho_{0}^{\delta}}}+\frac{|P(\rho_{0},\theta_{0})_{x}-P(\rho_{0}^{\delta},\theta_{0}^{\delta})_{x}|}{\sqrt{\rho_{0}^{\delta}}}+\sqrt{\rho_{0}^{\delta}}|u_{0}u_{0x}|$
$\displaystyle\leq$
$\displaystyle|g_{1}|+c\frac{\delta}{\sqrt{\rho_{0}^{\delta}}}(|\rho_{0x}|+|\theta_{0x}|)+c,$

which implies

$\int_{I}\rho u_{t}^{2}(0)\leq c.$ (3.26)

Substituting (3.26) into (3.25), we have

$\displaystyle\int_{I}\rho u_{t}^{2}+\int_{0}^{t}\int_{I}u_{xt}^{2}\leq
c+c\int_{0}^{t}\int_{I}u_{xx}^{2}\int_{I}\rho
u_{t}^{2}+c\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}.$
(3.27)

Multiplying (3.7) by $\left(\int_{0}^{\theta}\kappa(\xi)d\xi\right)_{t}$(i.e.
$\kappa(\theta)\theta_{t}$), integrating the resulting equation over $I$, and
using integration by parts, ($A_{4}$), ($A_{5}$), Lemma 2.2, Lemma 3.2, Lemma
3.5 and the Cauchy inequality, we have for any $\varepsilon>0$

$\displaystyle\int_{I}\rho
Q^{\prime}(\theta)\kappa(\theta)\theta_{t}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\kappa^{2}\theta_{x}^{2}$
$\displaystyle=$ $\displaystyle-\int_{I}\rho
uQ^{\prime}\kappa\theta_{x}\theta_{t}-\int_{I}\rho\theta Q^{\prime}\kappa
u_{x}\theta_{t}+\int_{I}u_{x}^{2}\left(\int_{0}^{\theta}\kappa(\xi)d\xi\right)_{t}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+c\int_{I}\rho
u^{2}Q^{\prime}\kappa\theta_{x}^{2}+c\int_{I}\rho Q^{2}Q^{\prime}\kappa
u_{x}^{2}$
$\displaystyle+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)-2\int_{I}u_{x}u_{xt}\int_{0}^{\theta}\kappa(\xi)d\xi$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c\left(1+\int_{I}u_{xx}^{2}\right)\int_{I}\rho(1+\theta^{q+r+2})$
$\displaystyle+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)+\varepsilon\int_{I}u_{xt}^{2}+c_{\varepsilon}\sup\limits_{x\in
I}(1+\theta^{q})^{2}\theta^{2},$

which combining Lemma 2.2, Lemma 3.2, Lemma 3.5 implies

$\displaystyle\int_{I}\rho
Q^{\prime}(\theta)\kappa(\theta)\theta_{t}^{2}+\frac{d}{dt}\int_{I}\kappa^{2}\theta_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c\int_{I}u_{xx}^{2}+c+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)+\varepsilon\int_{I}u_{xt}^{2}+c_{\varepsilon}\sup\limits_{x\in
I}(1+\theta^{q})^{2}\theta^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}u_{xx}^{2}+\frac{d}{dt}\left(\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi\right)+\varepsilon\int_{I}u_{xt}^{2}+c_{\varepsilon}\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c_{\varepsilon}.$

Integrating it over $(0,t)$, and using ($A_{4}$) and ($A_{5}$), Lemma 2.2,
Lemma 3.5, Lemma 3.6 and the Cauchy inequality, we obtain

$\displaystyle\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}+\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}u_{x}^{2}\int_{0}^{\theta}\kappa(\xi)d\xi+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}$
$\displaystyle\leq$ $\displaystyle c\sup\limits_{x\in
I}(1+\theta^{q})\theta+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}$
$\displaystyle\leq$ $\displaystyle
c\|(1+\theta^{q})\theta_{x}\|_{L^{2}}+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}+c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}.$

After the first term of the right side is absorbed by the left, we get

$\int_{0}^{t}\int_{I}\rho\left(1+\theta^{q+r}\right)\theta_{t}^{2}+\int_{I}(1+\theta^{q})^{2}\theta_{x}^{2}\leq
c\varepsilon\int_{0}^{t}\int_{I}u_{xt}^{2}+c_{\varepsilon}.$ (3.28)

Multiplying (3.28) by $2c$, adding the resulting inequality to (3.25), taking
$\varepsilon=\frac{1}{4c^{2}}$, and using the Gronwall inequality and Lemma
3.6, we complete the proof of Lemma 3.7. $\Box$

From Corollary 2.1, Lemma 3.1 and Lemma 3.7, we get the following corollary
immediately.

###### Corollary 3.2

Under the conditions of Theorem 3.1, it holds

$\|\theta\|_{L^{\infty}(Q_{T})}\leq c.$

###### Corollary 3.3

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\|u\|_{W^{1,\infty}(Q_{T})}+\int_{I}u_{xx}^{2}+\int_{Q_{T}}\theta_{xx}^{2}\leq
c.$

Proof. It follows from (3.21), Lemma 3.6, Lemma 3.7 and Corollary 3.2 that

$\int_{I}u_{xx}^{2}\leq c,$

which, combining Lemma 2.2, Lemma 3.5 and the Sobolev inequality, gives

$\|u\|_{W^{1,\infty}(Q_{T})}\leq c.$ (3.29)

By (3.7), Corollary 3.2, ($A_{4}$), ($A_{5}$), Lemma 2.2, Lemma 3.2, (3.29),
Lemma 3.7, the Hölder inequality, Sobolev inequality and Cauchy inequality, we
have

$\displaystyle\int_{I}\theta_{xx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\theta_{x}^{4}+c\int_{I}u_{x}^{4}+\int_{I}\rho\theta_{t}^{2}+c\int_{I}u^{2}\theta_{x}^{2}+c\int_{I}\theta^{2}u_{x}^{2}$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{x}\theta_{xx}\|_{L^{1}}\int_{I}\theta_{x}^{2}+\int_{I}\rho\theta_{t}^{2}+c$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{xx}\|_{L^{2}}+\int_{I}\rho\theta_{t}^{2}+c$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\theta_{xx}^{2}+\int_{I}\rho\theta_{t}^{2}+c.$

After the first term of the right side is absorbed by the left, we get

$\int_{I}\theta_{xx}^{2}\leq\int_{I}\rho\theta_{t}^{2}+c.$ (3.30)

Integrating (3.30) over $[0,T]$, and using Lemma 3.7, we get

$\int_{Q_{T}}\theta_{xx}^{2}\leq c.$

This proves Corollary 3.3. $\Box$

###### Lemma 3.8

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\|\rho\|_{W^{1,\infty}(Q_{T})}+\|\rho_{t}\|_{L^{\infty}(Q_{T})}+\int_{I}(\rho_{xx}^{2}+\rho_{xt}^{2})+\int_{Q_{T}}(\rho_{tt}^{2}+u_{xxx}^{2})\leq
c.$

Proof. Differentiating (3.19) w.r.t. $x$, we have

$\rho_{xxt}=-\rho_{xxx}u-3\rho_{xx}u_{x}-3\rho_{x}u_{xx}-\rho u_{xxx}.$ (3.31)

Multiplying (3.31) by $2\rho_{xx}$, integrating it over $I$, and using
integration by parts and the Hölder inequality, we have

$\displaystyle\frac{d}{dt}\int_{I}\rho_{xx}^{2}$ $\displaystyle=$
$\displaystyle-5\int_{I}\rho_{xx}^{2}u_{x}-6\int_{I}\rho_{x}\rho_{xx}u_{xx}-2\int_{I}\rho\rho_{xx}u_{xxx}$
$\displaystyle\leq$ $\displaystyle
5\|u_{x}\|_{L^{\infty}}\int_{I}\rho_{xx}^{2}+6\|\rho_{x}\|_{L^{\infty}}\|\rho_{xx}\|_{L^{2}}\|u_{xx}\|_{L^{2}}+2\|\rho\|_{L^{\infty}}\|\rho_{xx}\|_{L^{2}}\|u_{xxx}\|_{L^{2}}.$

By the Sobolev inequality, Cauchy inequality, Lemma 3.2, Lemma 3.6, and
Corollary 3.3, we have

$\displaystyle\frac{d}{dt}\int_{I}\rho_{xx}^{2}\leq
c\int_{I}\rho_{xx}^{2}+c\int_{I}u_{xxx}^{2}+c.$ (3.32)

The next step is to estimate the term $\int_{I}u_{xxx}^{2}$. Differentiating
(3.11) with respect to $x$, we have

$\displaystyle u_{xxx}=\rho_{x}u_{t}+\rho u_{xt}+\rho_{x}uu_{x}+\rho
u_{x}^{2}+\rho uu_{xx}+(P_{c})_{xx}+(\rho Q)_{xx}.$ (3.33)

By ($A_{3}$), ($A_{4}$), Lemma 2.2, Lemma 3.2, Lemma 3.6, Corollary 3.2,
Corollary 3.3 and the Sobolev inequality, we get

$\displaystyle\int_{I}u_{xxx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}u_{xt}^{2}+c\int_{I}\rho_{x}^{2}u_{t}^{2}+c\int_{I}\rho_{xx}^{2}+c\int_{I}\theta_{xx}^{2}+c$
(3.34) $\displaystyle\leq$ $\displaystyle
c\int_{I}u_{xt}^{2}+c\int_{I}\rho_{xx}^{2}+c\int_{I}\theta_{xx}^{2}+c.$

Substituting (3.34) into (3.32), and using the Gronwall inequality, Lemma 3.7
and Corollary 3.3, we get

$\int_{I}\rho_{xx}^{2}\leq c.$ (3.35)

By (3.35), Lemma 3.2, Lemma 3.6 and the Sobolev inequality, we have

$\|\rho\|_{W^{1,\infty}(Q_{T})}\leq c.$ (3.36)

By (3.34), (3.35), Lemma 3.7 and Corollary 3.3, we get

$\int_{Q_{T}}u_{xxx}^{2}\leq c.$

The estimates of $\rho_{xt}$ and $\rho_{tt}$ can be obtained directly by
(3.19), (1.1)1, (3.35), (3.36), Lemma 2.2, Lemma 3.2, Lemma 3.6, Lemma 3.7,
and Corollary 3.3. The proof of Lemma 3.8 is complete. $\Box$

###### Lemma 3.9

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}\rho\theta_{t}^{2}+\int_{Q_{T}}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c.$

Differentiating (3.7) w.r.t. $t$, we have

$\displaystyle\rho Q^{\prime}\theta_{tt}+\rho
Q^{\prime\prime}\theta_{t}^{2}+\rho_{t}Q^{\prime}\theta_{t}+(\rho
uQ^{\prime}\theta_{x})_{t}+(\rho\theta
Q^{\prime}u_{x})_{t}=2u_{x}u_{xt}+(\kappa\theta_{x})_{xt}.$ (3.37)

Multiplying (3.37) by $(\int_{0}^{\theta}\kappa(\xi)d\xi)_{t}$ (i.e.
$\kappa(\theta)\theta_{t}$), integrating it over $I$, and using integration by
parts, (1.1)1, ($A_{4}$), ($A_{5}$), Corollary 3.2, Lemma 3.2, Corollary 3.3
and the Hölder inequality, we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{I}\rho_{t}Q^{\prime}\kappa\theta_{t}^{2}-\frac{1}{2}\int_{I}\rho
Q^{\prime\prime}\theta_{t}^{3}\kappa+\frac{1}{2}\int_{I}\rho
Q^{\prime}\kappa^{\prime}\theta_{t}^{3}-\int_{I}(\rho
uQ^{\prime}\theta_{x})_{t}\kappa\theta_{t}$ $\displaystyle-\int_{I}(\rho\theta
Q^{\prime}u_{x})_{t}\kappa\theta_{t}+2\int_{I}u_{x}u_{xt}\kappa\theta_{t}$
$\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{I}(\rho
u)_{x}Q^{\prime}\kappa\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}-\int_{I}\rho
uQ^{\prime}(\kappa\theta_{x})_{t}\theta_{t}-\int_{I}\rho
u(Q^{\prime\prime}\kappa-Q^{\prime}\kappa^{\prime})\theta_{t}^{2}\theta_{x}$
$\displaystyle-\int_{I}(\rho
u)_{t}Q^{\prime}\theta_{x}\kappa\theta_{t}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}-\int_{I}\rho_{t}\theta
Q^{\prime}u_{x}\kappa\theta_{t}+c\|\kappa\theta_{t}\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}\|u_{x}\|_{L^{2}}.$

This, combining integration by parts, ($A_{4}$), ($A_{5}$), Lemma 2.1, Lemma
2.2, Corollary 3.2, Corollary 3.3, Lemma 3.7, Lemma 3.8 and the Cauchy
inequality, gives

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}$
$\displaystyle\leq$ $\displaystyle-\frac{1}{2}\int_{I}\rho
uQ^{\prime\prime}\theta_{x}\kappa\theta_{t}^{2}-\frac{1}{2}\int_{I}\rho
uQ^{\prime}\kappa^{\prime}\theta_{x}\theta_{t}^{2}-\int_{I}\rho
uQ^{\prime}\kappa\theta_{t}\theta_{xt}+c\|\kappa\theta_{t}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}$
$\displaystyle+c\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\|(\kappa\theta_{x})_{t}\|_{L^{2}}+c\|\theta_{x}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}$
$\displaystyle+c\|\kappa\theta_{t}\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{xx}\|_{L^{2}}\int_{I}\rho\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}\int_{I}\rho\theta_{t}^{2}+c\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\|(\kappa\theta_{x})_{t}\|_{L^{2}}+c\|\kappa\theta_{t}\|_{L^{\infty}}$
$\displaystyle+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\|\kappa\theta_{t}\|_{L^{\infty}}\|u_{xt}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
c\|\theta_{xx}\|_{L^{2}}\int_{I}\rho\theta_{t}^{2}+c\left(\|(\kappa\theta_{t})_{x}\|_{L^{2}}+\int_{I}\rho\kappa|\theta_{t}|\right)\int_{I}\rho\theta_{t}^{2}+c\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\|(\kappa\theta_{x})_{t}\|_{L^{2}}$
$\displaystyle+c\|(\kappa\theta_{t})_{x}\|_{L^{2}}+c\int_{I}\rho\kappa|\theta_{t}|+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\left(\|(\kappa\theta_{t})_{x}\|_{L^{2}}+\int_{I}\rho\kappa|\theta_{t}|\right)\|u_{xt}\|_{L^{2}},$

which together with the Cauchy inequality, Lemma 3.2, ($A_{5}$) and Corollary
3.2 gives

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq\frac{1}{2}\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\int_{I}\theta_{xx}^{2}+c\left(\int_{I}\rho\theta_{t}^{2}\right)^{2}+c\int_{I}u_{xt}^{2}+c,$

where we have used $(\kappa\theta_{t})_{x}=(\kappa\theta_{x})_{t}$. This gives

$\displaystyle\frac{d}{dt}\int_{I}\rho
Q^{\prime}\kappa\theta_{t}^{2}+\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c\int_{I}\theta_{xx}^{2}+c\left(\int_{I}\rho\theta_{t}^{2}\right)^{2}+c\int_{I}u_{xt}^{2}+c.$

Integrating it over $(0,t)$, and using ($A_{4}$), ($A_{5}$), Lemma 3.7 and
Corollary 3.3, we obtain

$\int_{I}\rho\theta_{t}^{2}+\int_{0}^{t}\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c\int_{I}\rho\theta_{t}^{2}(0)+c\int_{0}^{t}\left(\int_{I}\rho\theta_{t}^{2}\right)^{2}+c.$
(3.38)

Multiplying (3.7) by $\displaystyle\frac{1}{Q^{\prime}(\theta)\sqrt{\rho}}$,
taking $t\rightarrow 0^{+}$, and using (1.8)2, we have

$\displaystyle|\sqrt{\rho}\theta_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|u_{0x}^{2}+\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+|\sqrt{\rho_{0}^{\delta}}u_{0}\theta_{0x}|+|\sqrt{\rho_{0}^{\delta}}\theta_{0}^{\delta}u_{0x}|$
$\displaystyle\leq$
$\displaystyle\frac{\left|u_{0x}^{2}+\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+\frac{\left|\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}-\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+c$
$\displaystyle\leq$ $\displaystyle
c|g_{2}|+\frac{c\delta}{\sqrt{\rho_{0}^{\delta}}}(1+|\theta_{0xx}|)+c,$

which implies

$\displaystyle\int_{I}\rho\theta_{t}^{2}(0)$ $\displaystyle\leq$
$\displaystyle c\int_{I}g_{2}^{2}+c\int_{I}\theta_{0xx}^{2}+c$ (3.39)
$\displaystyle\leq$ $\displaystyle c.$

Substituting (3.39) into (3.38), using the Gronwall inequality and Lemma 3.7,
we complete the proof. $\Box$

###### Corollary 3.4

Under the conditions of Theorem 3.1, it holds

$\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}\leq c.$

Proof. By Lemma 3.2, ($A_{5}$), Corollary 2.1, Corollary 3.2, Lemma 3.9, and
$(\kappa\theta_{t})_{x}=(\kappa\theta_{x})_{t}$, we get

$\displaystyle\int_{0}^{T}\|\kappa\theta_{t}\|_{L^{\infty}}^{2}\leq
c\int_{0}^{T}\|(\kappa\theta_{t})_{x}\|_{L^{2}}^{2}+c\leq c.$

This combining ($A_{5}$) completes the proof. $\Box$

###### Corollary 3.5

Under the conditions of Theorem 3.1, it holds

$\int_{Q_{T}}\theta_{xt}^{2}\leq c.$

Proof. Since

$\kappa\theta_{xt}=(\kappa\theta_{x})_{t}-\kappa^{\prime}\theta_{t}\theta_{x},$

we obtain

$\displaystyle\int_{Q_{T}}\theta_{xt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{Q_{T}}\kappa^{2}\theta_{xt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{Q_{T}}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\int_{Q_{T}}(\kappa^{\prime})^{2}\theta_{t}^{2}\theta_{x}^{2}$
$\displaystyle\leq$ $\displaystyle c+c\int_{0}^{T}\sup\limits_{x\in
I}\theta_{t}^{2}\int_{I}\theta_{x}^{2}$ $\displaystyle\leq$ $\displaystyle c,$

where we have used ($A_{5}$), Lemma 3.7, Lemma 3.9, Corollary 3.2 and
Corollary 3.4. $\Box$

###### Corollary 3.6

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\|\theta\|_{W^{1,\infty}(Q_{T})}+\int_{I}\theta_{xx}^{2}+\int_{Q_{T}}\theta_{xxx}^{2}\leq
c.$

Proof. From (3.30) and Lemma 3.9, we have

$\displaystyle\int_{I}\theta_{xx}^{2}\leq c,$ (3.40)

which, combining Corollary 3.2, Lemma 3.7 and the Sobolev inequality, gives

$\|\theta\|_{W^{1,\infty}(Q_{T})}\leq c.$ (3.41)

Differentiating (3.7) w.r.t. $x$, we have

$\displaystyle\kappa\theta_{xxx}=-3\kappa^{\prime}\theta_{x}\theta_{xx}-\kappa^{\prime\prime}\theta_{x}^{3}-2u_{x}u_{xx}+\rho
Q^{\prime}\theta_{xt}+\rho_{x}Q^{\prime}\theta_{t}+\rho
Q^{\prime\prime}\theta_{x}\theta_{t}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+(\rho uQ^{\prime}\theta_{x})_{x}+(\rho\theta Q^{\prime}u_{x})_{x}.$ (3.42)

By (3.40), (3.41), (3), ($A_{4}$), ($A_{5}$), Lemma 3.8, Lemma 3.9 and
Corollary 3.3, we have

$\displaystyle\int_{I}\theta_{xxx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\theta_{x}^{2}\theta_{xx}^{2}+c\int_{I}\theta_{x}^{6}+c\int_{I}u_{x}^{2}u_{xx}^{2}+c\int_{I}\rho^{2}\theta_{xt}^{2}+c\int_{I}\rho_{x}^{2}\theta_{t}^{2}+c\int_{I}\rho^{2}\theta_{x}^{2}\theta_{t}^{2}$
(3.43) $\displaystyle+c\int_{I}\left|(\rho
uQ^{\prime}\theta_{x})_{x}\right|^{2}+c\int_{I}\left|(\rho\theta
Q^{\prime}u_{x})_{x}\right|^{2}+c$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}\theta_{xt}^{2}+c\int_{I}\rho_{x}^{2}\theta_{t}^{2}+c$
$\displaystyle\leq$ $\displaystyle c\int_{I}\theta_{xt}^{2}+c\sup\limits_{x\in
I}\theta_{t}^{2}+c.$

By (3.43), Corollary 3.4 and Corollary 3.5, we obtain

$\int_{Q_{T}}\theta_{xxx}^{2}\leq c.$

$\Box$

The next lemma, which we used in [8] to get $H^{4}-$estimates of velocity,
plays an important role in getting $H^{3}-$estimates of $\theta$ in the
following.

###### Lemma 3.10

Under the conditions of Theorem 3.1, it holds

$\|(\sqrt{\rho})_{x}\|_{L^{\infty}(Q_{T})}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}(Q_{T})}\leq
c.$

Proof. Multiplying $(\ref{non-1.2})_{1}$ by
$\displaystyle\frac{1}{2\sqrt{\rho}}$, we have

$(\sqrt{\rho})_{t}+(\sqrt{\rho})_{x}u+\frac{1}{2}\sqrt{\rho}u_{x}=0.$ (3.44)

Differentiating (3.44) with respect to $x$, we get

$(\sqrt{\rho})_{xt}+(\sqrt{\rho})_{xx}u+\frac{3}{2}(\sqrt{\rho})_{x}u_{x}+\frac{1}{2}\sqrt{\rho}u_{xx}=0.$

Denote $h=(\sqrt{\rho})_{x}$, we have

$h_{t}+h_{x}u+\frac{3}{2}hu_{x}+\frac{1}{2}\sqrt{\rho}u_{xx}=0,$

which implies

$\frac{d}{ds}\left\\{h\exp\left(\frac{3}{2}\int_{0}^{s}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)\right\\}=-\frac{1}{2}\sqrt{\rho}(\partial_{X}^{2}u)\exp\left(\frac{3}{2}\int_{0}^{s}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right),$
(3.45)

where $X(s;x,t)$ is the solution to (3.3).

Integrating (3.45) over $(0,t)$, we get

$\begin{array}[]{rl}h(x,t)=&\displaystyle\exp\left(-\frac{3}{2}\int_{0}^{t}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)h\left(X(0;x,t),0\right)\\\\[14.22636pt]
&\displaystyle-\frac{1}{2}\exp\left(-\frac{3}{2}\int_{0}^{t}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)\int_{0}^{t}\sqrt{\rho}(\partial_{X}^{2}u)\exp\left(\frac{3}{2}\int_{0}^{s}\partial_{X}u\left(X(\tau;x,t),\tau\right)d\tau\right)ds.\end{array}$

This together with Corollary 3.3, Lemma 3.8 and the Sobolev inequality,
implies

$\|(\sqrt{\rho})_{x}\|_{L^{\infty}(Q_{T})}\leq c.$ (3.46)

From (3.44), (3.46), Lemma 3.8 and Corollary 3.3, we get

$\|(\sqrt{\rho})_{t}\|_{L^{\infty}(Q_{T})}\leq c.$

This proves Lemma 3.10. $\Box$

The next lemma will be used to get $H^{3}-$estimates of $\theta$.

###### Lemma 3.11

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}+\int_{Q_{T}}\rho^{3}\theta_{tt}^{2}\leq
c.$

Proof. Multiply $(\ref{non-3.36})$ by
$\rho^{\gamma_{1}}(\kappa\theta_{t})_{t}$ (i.e.
$\rho^{\gamma_{1}}\kappa\theta_{tt}+\rho^{\gamma_{1}}\kappa^{\prime}\theta_{t}^{2}$,
where $\gamma_{1}$ is to be decided later), and using integration by parts, we
have

$\displaystyle\int_{I}\rho^{\gamma_{1}+1}\kappa
Q^{\prime}\theta_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{\gamma_{1}}\left|(\kappa\theta_{x})_{t}\right|^{2}$
(3.47) $\displaystyle=$
$\displaystyle\frac{\gamma_{1}}{2}\int_{I}\rho^{\gamma_{1}-1}\rho_{t}\left|(\kappa\theta_{x})_{t}\right|^{2}-\gamma_{1}\int_{I}\rho^{\gamma_{1}-1}\rho_{x}\kappa\theta_{tt}(\kappa\theta_{x})_{t}-\gamma_{1}\int_{I}\rho^{\gamma_{1}-1}\rho_{x}\kappa^{\prime}\theta_{t}^{2}(\kappa\theta_{x})_{t}$
$\displaystyle+2\int_{I}u_{x}u_{xt}(\rho^{\gamma_{1}}\kappa\theta_{tt}+\rho^{\gamma_{1}}\kappa^{\prime}\theta_{t}^{2})-\int_{I}\rho^{\gamma_{1}+1}Q^{\prime}\kappa^{\prime}\theta_{t}^{2}\theta_{tt}$
$\displaystyle-\int_{I}\left(\rho
Q^{\prime\prime}\theta_{t}^{2}+\rho_{t}Q^{\prime}\theta_{t}+(\rho
uQ^{\prime}\theta_{x})_{t}+(\rho\theta
Q^{\prime}u_{x})_{t}\right)\left(\rho^{\gamma_{1}}\kappa\theta_{tt}+\rho^{\gamma_{1}}\kappa^{\prime}\theta_{t}^{2}\right).$

We are going to look for the minimal of $\gamma_{1}$. It seems that the second
term of the right side plays an important role.

$\displaystyle-\gamma_{1}\int_{I}\rho^{\gamma_{1}-1}\rho_{x}\kappa\theta_{tt}(\kappa\theta_{x})_{t}$
$\displaystyle=$
$\displaystyle-2\gamma_{1}\int_{I}\rho^{\gamma_{1}-\frac{1}{2}}(\sqrt{\rho})_{x}\kappa\theta_{tt}(\kappa\theta_{x})_{t}$
(3.48) $\displaystyle\leq$
$\displaystyle\frac{1}{4}\int_{I}\rho^{\gamma_{1}+1}\kappa
Q^{\prime}\theta_{tt}^{2}+c\int_{I}\rho^{\gamma_{1}-2}|(\kappa\theta_{x})_{t}|^{2},$

where we have used Lemma 3.10, ($A_{4}$), ($A_{5}$), Corollary 3.2 and the
Cauchy inequality.

From Lemma 3.9, we know that $\int_{Q_{T}}|(\kappa\theta_{x})_{t}|^{2}\leq c$.
This implies that the minimal of $\gamma_{1}$ should be 2. Substituting
$\gamma_{1}=2$ into (3.47) and (3.48), and then substituting (3.48) into
(3.47), we have

$\displaystyle\frac{3}{4}\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\int_{I}\rho\theta_{t}^{4}+c\int_{I}u_{xt}^{2}+c\int_{I}\theta_{xt}^{2}+c$
$\displaystyle+c\left(\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}\right)^{\frac{1}{2}}\left\\{1+\left(\int_{I}\rho\theta_{t}^{4}\right)^{\frac{1}{2}}+\|\theta_{xt}\|_{L^{2}}+\|\sqrt{\rho}u_{t}\|_{L^{2}}+\|u_{xt}\|_{L^{2}}+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}\right\\}$
$\displaystyle\leq$ $\displaystyle\frac{1}{4}\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}+c\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c\int_{I}u_{xt}^{2}+c,$

where we have used ($A_{4}$), ($A_{5}$), Lemma 3.7, Lemma 3.8, Lemma 3.9,
Corollary 3.3, Corollary 3.6 and the Cauchy inequality. This implies

$\displaystyle\int_{I}\rho^{3}\kappa
Q^{\prime}\theta_{tt}^{2}+\frac{d}{dt}\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}\leq
c\int_{I}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c\int_{I}u_{xt}^{2}+c.$

Integrating it over $(0,t)$, and using ($A_{4}$), ($A_{5}$), Lemma 3.7, Lemma
3.9, Corollary 3.4 and Corollary 3.5, we have

$\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}+\int_{0}^{t}\int_{I}\rho^{3}\theta_{tt}^{2}\leq
c\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}(0)+c.$ (3.49)

By ($A_{5}$), (3.39), (3) and Corollary 3.2, we get

$\displaystyle\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}(0)$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}\theta_{xt}^{2}(0)+c\int_{I}\rho\theta_{t}^{2}(0)$ (3.50)
$\displaystyle\leq$ $\displaystyle
c\|\theta_{0}\|_{H^{3}}^{2}+c\|u_{0}\|_{H^{2}}^{2}+c\int_{I}\rho_{x}^{2}u_{t}^{2}(0)+c$
$\displaystyle\leq$ $\displaystyle c+c\int_{I}|(\sqrt{\rho})_{x}|^{2}\rho
u_{t}^{2}(0)$ $\displaystyle\leq$ $\displaystyle c.$

Substituting (3.50) into (3.49), we complete the proof. $\Box$

###### Corollary 3.7

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}\left(\theta_{xxx}^{2}+\rho^{2}\theta_{xt}^{2}\right)\leq c.$

Proof. A direct calculation gives

$\rho\kappa\theta_{xt}=\rho(\kappa\theta_{x})_{t}-\rho\kappa^{\prime}\theta_{t}\theta_{x},$

which implies

$\displaystyle\int_{I}\rho^{2}\theta_{xt}^{2}$ $\displaystyle\leq$
$\displaystyle
c\int_{I}\rho^{2}\left|(\kappa\theta_{x})_{t}\right|^{2}+c\|\theta_{x}\|_{L^{\infty}}^{2}\int_{I}\rho\theta_{t}^{2}$
(3.51) $\displaystyle\leq$ $\displaystyle c.$

Here we have used $(A_{5})$, Lemma 3.8, Lemma 3.9, Lemma 3.11 and Corollary
3.6.

From the second inequality of (3.43), we obtain

$\displaystyle\int_{I}\theta_{xxx}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{2}\theta_{xt}^{2}+c\int_{I}\rho_{x}^{2}\theta_{t}^{2}+c$
$\displaystyle\leq$ $\displaystyle
c\int_{I}|(\sqrt{\rho})_{x}|^{2}\rho\theta_{t}^{2}+c$ $\displaystyle\leq$
$\displaystyle c,$

where we have used Lemma 3.9, (3.51) and Lemma 3.10. $\Box$

The next lemma will be used to get $H^{3}-$estimates of $u$.

###### Lemma 3.12

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}\rho^{2}u_{xt}^{2}+\int_{Q_{T}}\rho^{3}u_{tt}^{2}\leq c.$

Proof. Similarly to Lemma 3.11, multiplying (3.23) by $\rho^{2}u_{tt}$, and
integrating it over $I$, we have

$\displaystyle\int_{I}\rho^{3}u_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{2}u_{xt}^{2}$
$\displaystyle=$
$\displaystyle\int_{I}\rho\rho_{t}u_{xt}^{2}-2\int_{I}\rho\rho_{x}u_{xt}u_{tt}-\int_{I}\rho^{2}u_{tt}(\rho_{t}u_{t}+\rho_{t}uu_{x}+\rho
u_{t}u_{x}+\rho uu_{xt}+P_{xt})$ $\displaystyle\leq$ $\displaystyle
c\int_{I}u_{xt}^{2}-4\int_{I}\rho^{\frac{3}{2}}(\sqrt{\rho})_{x}u_{xt}u_{tt}+\frac{1}{4}\int_{I}\rho^{3}u_{tt}^{2}+c\int_{I}\rho
u_{t}^{2}+c\int_{I}|(\rho Q)_{xt}|^{2}+c\int_{I}|(P_{c})_{xt}|^{2}+c$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}\rho^{3}u_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c.$

Here, we have used integration by parts, Lemma 3.7, Lemma 3.8, Lemma 3.10,
Corollary 3.3, Corollary 3.6 and the Cauchy inequality.

The first term of the right side can be absorbed by the left. After that, we
have

$\displaystyle\int_{I}\rho^{3}u_{tt}^{2}+\frac{d}{dt}\int_{I}\rho^{2}u_{xt}^{2}\leq
c\int_{I}u_{xt}^{2}+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\theta_{xt}^{2}+c.$

Integrating this inequality on both side over $(0,t)$, and using Lemma 3.7,
Corollary 3.4 and Corollary 3.5, we have

$\int_{0}^{t}\int_{I}\rho^{3}u_{tt}^{2}+\int_{I}\rho^{2}u_{xt}^{2}\leq\int_{I}\rho^{2}u_{xt}^{2}(0)+c.$
(3.52)

Similarly to (3.50), we use (3.26), (3.33), ($A_{3}$) and ($A_{4}$) to get

$\displaystyle\int_{I}\rho^{2}u_{xt}^{2}(0)$ $\displaystyle\leq$
$\displaystyle
c\|u_{0}\|_{H^{3}}^{2}+c\|\theta_{0}\|_{H^{2}}^{2}+c\|\rho_{0}\|_{H^{2}}^{2}+c\int_{I}\rho
u_{t}^{2}(0)+c$ (3.53) $\displaystyle\leq$ $\displaystyle c.$

Substituting (3.53) into (3.52), we complete the proof. $\Box$

By (3.34), Lemma 3.7, Lemma 3.8, Lemma 3.10, Lemma 3.12 and Corollary 3.6, we
get the following corollary.

###### Corollary 3.8

Under the conditions of Theorem 3.1, it holds for any $0\leq t\leq T$

$\int_{I}u_{xxx}^{3}\leq c.$

From the above estimates, we get

$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{2}}+\|\rho_{t}\|_{H^{1}}+\displaystyle\|u\|_{H^{3}}+\|\rho
u_{t}\|_{H^{1}}+\|\sqrt{\rho}u_{t}\|_{L^{2}}+\|\theta\|_{H^{3}}$
$\displaystyle+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{t}\|_{H^{1}}+\int_{Q_{T}}\left(u_{xt}^{2}+\rho_{tt}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\rho^{3}u_{tt}^{2}+\rho^{3}\theta_{tt}^{2}\right)\leq
c.$ (3.54)

###### Corollary 3.9

Under the conditions of Theorem 3.1, there exists a positive constant
$c_{\delta}$ depending on $\delta$ such that for any $(x,t)\in Q_{T}$, it
holds

$\begin{cases}\rho(x,t)\geq\displaystyle\frac{\delta}{c}>0,\\\ \theta(x,t)\geq
c_{\delta}>0.\end{cases}$ (3.55)

Proof. By (3.5), ($A_{3}$), ($A_{4}$), Lemma 3.8 and Corollary 3.6, we have
for any $(x,t)\in Q_{T}$

$\displaystyle\rho(x,t)\geq\frac{\delta}{c}.$

This gets (3.55)1. (3.55)2 can be got by (3.55)1, (3), (3.7) and the maximum
principle for parabolic equation. $\Box$

From (3), (3.55), (3.23) and (3.37), we obtain

$\displaystyle\|\rho\|_{H^{2}}+\|\rho_{t}\|_{H^{1}}+\|u\|_{H^{3}}+\|u_{t}\|_{H^{1}}+\|\theta\|_{H^{3}}+\|\theta_{t}\|_{H^{1}}$
$\displaystyle+\int_{Q_{T}}\left(u_{xt}^{2}+u_{xxt}^{2}+\rho_{tt}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\theta_{xxt}^{2}+u_{tt}^{2}+\theta_{tt}^{2}\right)\leq
c.$

This proves Theorem 3.1. $\Box$

Proof of Theorem 1.1:

Consider (1.1)-(1.4) with initial data replaced by ($\rho_{0}^{\delta}$,
$u_{0}$, $\theta_{0}^{\delta}$), we obtain from Theorem 3.1 that there exists
a unique solution ($\rho^{\delta}$, $u^{\delta}$, $\theta^{\delta}$), such
that (3) and (3.55) are valid when we replace ($\rho,$ $u$, $\theta$) by
($\rho^{\delta}$, $u^{\delta}$, $\theta^{\delta}$). With the estimates uniform
for $\delta$, we take $\delta\rightarrow 0^{+}$ (take subsequence if
necessary) to get a solution to (1.1)-(1.4) still denoted by ($\rho$, $u$,
$\theta$) which satisfies (3) by the lower semi-continuity of the norms. This
proves the existence of the solutions as in Theorem 1.1. The uniqueness of the
solutions can be proved by the standard method like in [4], we omit it for
brevity. The proof of Theorem 1.1 is complete. $\Box$

##  4  Proof of Theorem 1.2

In this section, we use the similar arguments as in Section 3 to prove Theorem
1.2. Throughout this section, we denote $c$ to be a generic constant depending
on $\rho_{0}$, $u_{0}$, $\theta_{0}$, $T$ and some other known constants but
independent of $\delta$ for any $\delta\in(0,1)$.

Denote $\rho_{0}^{\delta}=\rho_{0}+\delta$,
$\theta_{0}^{\delta}=\theta_{0}+\delta$ and
$P^{\delta}_{0}=P(\rho_{0}^{\delta},\ \theta_{0}^{\delta})$, where $\rho_{0}$
and $\theta_{0}$ satisfy the same conditions as those in Theorem 1.2. Note
that $\rho_{0}^{\delta}\in H^{4}(I)$, $\rho_{0}^{\delta}\geq\delta>0$,
$\theta_{0}^{\delta}\in H^{3}(I)$,
$\partial_{x}\theta_{0}^{\delta}|_{x=0,1}=\partial_{x}\theta_{0}|_{x=0,1}=0$,
and

$\begin{cases}\|\rho_{0}^{\delta}\|_{H^{4}}\leq c,\\\
\|\big{(}\sqrt{\rho_{0}^{\delta}}\big{)}_{x}\|_{L^{\infty}}\leq c,\\\
\|\theta_{0}^{\delta}\|_{H^{3}}\leq c.\end{cases}$ (4.1)

Different from Section 3, we need to mollify $g_{3}$. Denote
$g_{3}^{\delta}=J_{\delta}*\overline{g}_{3}$, then $g_{3}^{\delta}\in
C^{\infty}(I)$, where

$\overline{g}_{3}(x)=\begin{cases}-g_{3}(-x),\ x\in[-1,0),\\\ g_{3}(x),\ \ \ \
\ x\in I,\\\ -g_{3}(2-x),\ x\in(1,2],\end{cases}$

and $J_{\delta}(\cdot)=\frac{1}{\sqrt{\delta}}J(\frac{\cdot}{\sqrt{\delta}})$,
and $J$ is the usual mollifier such that $J\in C_{0}^{\infty}(\mathbb{R})$,
supp$J\in(-1,1)$, and $\int_{\mathbb{R}}J(x)dx=1$. Since $g_{3}\in
H_{0}^{1}(I)$, we have $\overline{g}_{3}\in H_{0}^{1}([-1,2])$ and

$\partial_{x}\overline{g}_{3}(x)=\begin{cases}g_{3}^{\prime}(-x),\
x\in[-1,0),\\\ g_{3}^{\prime}(x),\ \ \ \ x\in I,\\\ g_{3}^{\prime}(2-x),\
x\in(1,2].\end{cases}$

Claim:

$\displaystyle\begin{cases}g_{3}^{\delta}\rightarrow g_{3}\ \ \mathrm{in}\
H^{1}(I),\ \mathrm{as}\ \delta\rightarrow 0,\\\
\|g_{3}^{\delta}\|_{H^{1}(I)}\leq c\|\overline{g}_{3}\|_{H^{1}([-1,2])}\leq
c\|g_{3}\|_{H^{1}(I)},\ \mathrm{for}\ \mathrm{any}\ \delta\in(0,1),\\\
\|\sqrt{\rho^{\delta}_{0}}(g_{3}^{\delta})_{xx}\|_{L^{2}(I)}\leq c,\
\mathrm{for}\ \mathrm{any}\ \delta\in(0,1).\end{cases}$ (4.2)

In fact, the proof of (4.2)1 and (4.2)2 can be found in [7]. We are going to
prove (4.2)3.

$\displaystyle\sqrt{\rho_{0}^{\delta}}(g_{3}^{\delta})_{xx}$ $\displaystyle=$
$\displaystyle(\sqrt{\rho_{0}^{\delta}}-\sqrt{\rho_{0}})(g_{3}^{\delta})_{xx}+\sqrt{\rho_{0}}(g_{3}^{\delta})_{xx}$
(4.3) $\displaystyle=$
$\displaystyle\frac{\delta(g_{3}^{\delta})_{xx}}{\sqrt{\rho_{0}^{\delta}}+\sqrt{\rho}}+\sqrt{\rho_{0}}(g_{3}^{\delta})_{xx}$
$\displaystyle=$ $\displaystyle A_{1}+A_{2}.$

Recall
$J_{\delta}(\cdot)=\frac{1}{\sqrt{\delta}}J(\frac{\cdot}{\sqrt{\delta}})$, we
conclude

$\displaystyle\|A_{1}\|_{L^{2}(I)}$ $\displaystyle\leq$
$\displaystyle\sqrt{\delta}\|(g_{3}^{\delta})_{xx}\|_{L^{2}(I)}$ (4.4)
$\displaystyle\leq$ $\displaystyle
c\|(\overline{g}_{3})_{x}\|_{L^{2}([-1,2])}$ $\displaystyle\leq$
$\displaystyle c\|(g_{3})_{x}\|_{L^{2}(I)}.$

A direct calculation combining $\left(\sqrt{\rho_{0}}(g_{3})_{x}\right)_{x}\in
L^{2}(I)$ gives

$\displaystyle\int_{I}|A_{2}|^{2}\leq c.$ (4.5)

By (4.3), (4.4) and (4.5), we get (4.2)3.

Let $u_{0}^{\delta}$ be the solution to the following elliptic problem for
each $\delta\in(0,1)$:

$\displaystyle\begin{cases}u^{\delta}_{0xx}-(P_{0}^{\delta})_{x}=\rho^{\delta}_{0}g^{\delta}_{3},\\\
u_{0}^{\delta}|_{x=0,1}=0.\end{cases}$ (4.6)

Since $\rho_{0}^{\delta}=\rho_{0}+\delta\in H^{4}(I)$,
$\theta_{0}^{\delta}=\theta_{0}+\delta\in H^{3}(I)$, and $g_{3}^{\delta}\in
C^{\infty}(I)$, we obtain from the elliptic theory (see [7]), (4.1), (4.2) and
(4.6) that $u_{0}^{\delta}\in H^{4}(I)\cap H^{1}_{0}(I)$ with the following
properties:

$\displaystyle\begin{cases}u_{0}^{\delta}\rightarrow u_{0}\ \mathrm{in}\
H^{3}(I),\ \mathrm{as}\ \delta\rightarrow 0,\\\
\|u_{0}^{\delta}\|_{H^{4}(I)}\leq c\ \ \mathrm{for}\ \mathrm{any}\
\delta\in(0,1).\end{cases}$ (4.7)

###### Theorem 4.1

Consider the same assumptions as in Theorem 1.2. Then for any $T>0$ and
$\delta\in(0,1)$ there exists a unique global solution $(\rho,u,\theta)$ to
(1.1)-(1.4) with initial data replaced by
($\rho_{0}^{\delta},u_{0}^{\delta},\theta_{0}^{\delta}$), such that

$\displaystyle\rho\in C([0,T];H^{4}),\ \ \ \rho_{t}\in C([0,T];H^{3}),\ \ \
\rho_{tt}\in C([0,T];H^{1})\cap L^{2}([0,T];H^{2}),$
$\displaystyle\rho_{ttt}\in L^{2}(Q_{T}),\
\rho\geq\frac{\displaystyle\delta}{c}>0,\ u\in C([0,T];H^{4}\cap
H^{1}_{0})\cap L^{2}([0,T];H^{5}),$ $\displaystyle u_{t}\in C([0,T];H^{2})\cap
L^{2}([0,T];H^{3}),\ \ \ u_{tt}\in C([0,T];L^{2})\cap L^{2}([0,T];H_{0}^{1}),\
$ $\displaystyle\theta\in C([0,T];H^{3})\cap L^{2}([0,T];H^{4}),\
\theta_{t}\in C([0,T];H^{1})\cap L^{2}([0,T];H^{2}),\ $
$\displaystyle\theta_{tt}\in L^{2}([0,T];L^{2}),\ \theta\geq c_{\delta}>0,\ $

where $c_{\delta}$ is a constant depending on $\delta$, but independent of
$u$.

Proof of Theorem 4.1:

Similarly to the proof of Theorem 3.1, Theorem 4.1 can be proved by some a
priori estimates globally in time.

For any given $T\in(0,+\infty)$, let $(\rho,u,\theta)$ be the solution to
(1.1)-(1.4) as in Theorem 4.1. Then we have the following estimates.

###### Lemma 4.1

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{2}}+\|\rho_{t}\|_{H^{1}}+\displaystyle\|u\|_{H^{3}}+\|\rho
u_{t}\|_{H^{1}}+\|\sqrt{\rho}u_{t}\|_{L^{2}}+\|\theta\|_{H^{3}}$
$\displaystyle+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{t}\|_{H^{1}}+\int_{Q_{T}}\left(u_{xt}^{2}+\rho_{tt}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\rho^{3}u_{tt}^{2}+\rho^{3}\theta_{tt}^{2}\right)\leq
c.$

Proof. Though the initial velocity in Theorem 4.1 (i.e. $u_{0}^{\delta}$) is
different from that in Theorem 3.1 (i.e. $u_{0}$), both of them are bounded in
$H^{3}$. It suffices to check if (3.26) and (3.39) work here. If do, Lemma 4.1
will be obtained from (3).

By (3.11) and (4.6)

$\displaystyle|\sqrt{\rho}u_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|u_{0xx}^{\delta}-P(\rho_{0}^{\delta},\theta_{0}^{\delta})_{x}\right|}{\sqrt{\rho_{0}^{\delta}}}+\sqrt{\rho_{0}^{\delta}}|u_{0}^{\delta}u_{0x}^{\delta}|$
$\displaystyle=$
$\displaystyle\sqrt{\rho_{0}^{\delta}}|g_{3}^{\delta}|+\sqrt{\rho_{0}^{\delta}}|u_{0}^{\delta}u_{0x}^{\delta}|.$

This gives

$\int_{I}\rho u_{t}^{2}(0)\leq c.$

Therefore, (3.26) is valid here.

Multiplying (3.7) by $\displaystyle\frac{1}{Q^{\prime}(\theta)\sqrt{\rho}}$,
taking $t\rightarrow 0^{+}$, and using (1.9)2, we have

$\displaystyle|\sqrt{\rho}\theta_{t}(x,0)|$ $\displaystyle\leq$
$\displaystyle\frac{\left|(u^{\delta}_{0x})^{2}+\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+|\sqrt{\rho_{0}^{\delta}}u_{0}^{\delta}\theta_{0x}|+|\sqrt{\rho_{0}^{\delta}}\theta_{0}^{\delta}u_{0x}^{\delta}|$
$\displaystyle\leq$
$\displaystyle\frac{\left|u_{0x}^{2}+\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+\frac{c\left|u_{0x}^{\delta}-u_{0x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+\frac{\left|\left(\kappa(\theta_{0}^{\delta})\theta_{0x}\right)_{x}-\left(\kappa(\theta_{0})\theta_{0x}\right)_{x}\right|}{Q^{\prime}(\theta_{0}^{\delta})\sqrt{\rho_{0}^{\delta}}}+c$
$\displaystyle\leq$ $\displaystyle
c|g_{2}|+\frac{c\delta}{\sqrt{\rho_{0}^{\delta}}}(1+|\theta_{0xx}|)+\frac{c\left|u_{0x}^{\delta}-u_{0x}\right|}{\sqrt{\delta}}.$

Note that $\|u_{0x}^{\delta}-u_{0x}\|_{L^{2}(I)}\leq c\sqrt{\delta}$ by (1.9)1
and (4.6). This gives

$\displaystyle\int_{I}\rho\theta_{t}^{2}(0)\leq
c\int_{I}g_{2}^{2}+c\int_{I}\theta_{0xx}^{2}+c\leq c.$

Therefore, (3.39) is valid here. $\Box$

###### Lemma 4.2

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\int_{I}u_{xt}^{2}+\int_{Q_{T}}\rho u_{tt}^{2}\leq c.$

Proof. Multiplying (3.23) by $u_{tt}$, integrating it over $I$, and using
integration by parts, Lemma 2.2, Lemma 4.1 and the Cauchy inequality, we have

$\displaystyle\int_{I}\rho
u_{tt}^{2}+\frac{1}{2}\frac{d}{dt}\int_{I}u_{xt}^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{d}{dt}\int_{I}\rho_{t}u_{t}^{2}+\frac{1}{2}\int_{I}\rho_{tt}u_{t}^{2}-\frac{d}{dt}\int_{I}\rho_{t}uu_{x}u_{t}+\int_{I}\rho_{tt}uu_{x}u_{t}+\int_{I}\rho_{t}u_{t}^{2}u_{x}$
$\displaystyle+\int_{I}\rho_{t}uu_{xt}u_{t}-\int_{I}\rho
u_{t}u_{x}u_{tt}-\int_{I}\rho
uu_{xt}u_{tt}+\frac{d}{dt}\int_{I}P_{t}u_{xt}-\int_{I}P_{tt}u_{xt}$
$\displaystyle\leq$
$\displaystyle\frac{d}{dt}\int_{I}\left(P_{t}u_{xt}-\frac{1}{2}\rho_{t}u_{t}^{2}-\rho_{t}uu_{x}u_{t}\right)+c\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}$
$\displaystyle+c\int_{I}u_{xt}^{2}+c\int\rho_{tt}^{2}+\frac{1}{2}\int_{I}\rho
u_{tt}^{2}-\int_{I}P_{tt}u_{xt}+c.$

This gives

$\displaystyle\int_{I}\rho u_{tt}^{2}+\frac{d}{dt}\int_{I}u_{xt}^{2}$
$\displaystyle\leq$
$\displaystyle\frac{d}{dt}\int_{I}\left(2P_{t}u_{xt}-\rho_{t}u_{t}^{2}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}$
(4.8)
$\displaystyle+c\int\rho_{tt}^{2}-\frac{d}{dt}\int_{I}P_{t}^{2}-2\int_{I}P_{tt}(u_{xt}-P_{t})+c.$

We are going to estimate the last term of the right side of (4.8). By
($A_{2}$)-($A_{4}$), integration by parts, Lemma 4.1 and the Cauchy
inequality, we have

$\displaystyle-2\int_{I}P_{tt}(u_{xt}-P_{t})$ $\displaystyle=$
$\displaystyle-2\int_{I}(\rho
Q)_{tt}(u_{xt}-P_{t})-2\int_{I}(P_{c})_{tt}\left[u_{xt}-(\rho
Q)_{t}-(P_{c})_{t}\right]$ $\displaystyle\leq$
$\displaystyle-2\int_{I}\left[(\kappa\theta_{x})_{x}+u_{x}^{2}-(\rho
uQ)_{x}-\rho\theta Q^{\prime}u_{x}\right]_{t}(u_{xt}-P_{t})$
$\displaystyle+c\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c$
$\displaystyle=$ $\displaystyle
2\int_{I}(\kappa\theta_{x})_{t}(u_{xx}-P_{x})_{t}-4\int_{I}u_{x}u_{xt}(u_{xt}-P_{t})-2\int_{I}(\rho
uQ)_{t}(u_{xx}-P_{x})_{t}$ $\displaystyle+2\int_{I}\left[\rho\theta
Q^{\prime}u_{x}\right]_{t}(u_{xt}-P_{t})+c\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c.$

This, combining (3.11), ($A_{2}$)–($A_{4}$), Lemma 4.1 and the Cauchy
inequality, concludes

$\displaystyle-2\int_{I}P_{tt}(u_{xt}-P_{t})$ (4.9) $\displaystyle\leq$
$\displaystyle c+2\int_{I}(\kappa\theta_{x})_{t}(\rho u_{t}+\rho
uu_{x})_{t}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}$
$\displaystyle-2\int_{I}(\rho uQ)_{t}(\rho u_{t}+\rho
uu_{x})_{t}+2\int_{I}(\rho\theta
Q^{\prime}u_{x})_{t}(u_{xt}-P_{t})+c\int_{I}\rho_{tt}^{2}$ $\displaystyle\leq$
$\displaystyle c\int_{I}|(\kappa\theta_{x})_{t}|^{2}+\frac{1}{2}\int_{I}\rho
u_{tt}^{2}+c\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\int_{I}\rho_{tt}^{2}+c.$

Substituting (4.9) into (4.8), we get

$\displaystyle\frac{1}{2}\int_{I}\rho
u_{tt}^{2}+\frac{d}{dt}\int_{I}u_{xt}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{d}{dt}\int_{I}\left(2P_{t}u_{xt}-\rho_{t}u_{t}^{2}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{I}u_{xt}^{2}$
(4.10)
$\displaystyle+c\int\rho_{tt}^{2}-\frac{d}{dt}\int_{I}P_{t}^{2}+c\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c\int_{I}\rho\theta_{t}^{2}+c.$

Integrating (4.10) over $(0,t)$, and using (1.1)1, integration by parts,
(3.8), (3.11), (4.2)2, (4.6), and Lemma 4.1, we have

$\displaystyle\frac{1}{2}\int_{0}^{t}\int_{I}\rho
u_{tt}^{2}+\int_{I}u_{xt}^{2}$ $\displaystyle\leq$
$\displaystyle\int_{I}\left(2P_{t}u_{xt}+(\rho
u)_{x}u_{t}^{2}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c$
$\displaystyle=$ $\displaystyle\int_{I}\left(2P_{t}u_{xt}-2\rho
uu_{t}u_{xt}-2\rho_{t}uu_{x}u_{t}\right)+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}+c\int_{I}\rho^{2}u^{2}u_{t}^{2}+c\int_{I}\rho_{t}^{2}u^{2}u_{x}^{2}+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{I}u_{xt}^{2}+c\int_{I}\rho\theta_{t}^{2}++c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c,$

which implies

$\displaystyle\int_{0}^{t}\int_{I}\rho u_{tt}^{2}+\int_{I}u_{xt}^{2}\leq
c\int_{I}\rho\theta_{t}^{2}+c\int_{0}^{t}\int_{I}u_{xt}^{2}\int_{I}\rho_{tt}^{2}+c\int_{0}^{t}\int_{I}|(\kappa\theta_{x})_{t}|^{2}+c.$
(4.11)

Using the Gronwall inequality and Lemma 4.1, we complete the proof of the
lemma. $\Box$

###### Corollary 4.1

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\int_{Q_{T}}u_{xxt}^{2}\leq c.$

Proof. It follows from (3.23), Lemma 4.1 and ($A_{2}$)–($A_{4}$) that

$\displaystyle\int_{Q_{T}}u_{xxt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{Q_{T}}\rho
u_{tt}^{2}+c\int_{Q_{T}}\rho_{t}^{2}u_{t}^{2}+c\int_{Q_{T}}\rho_{t}^{2}u^{2}u_{x}^{2}+c\int_{Q_{T}}\rho^{2}u_{t}^{2}u_{x}^{2}$
$\displaystyle+c\int_{Q_{T}}\rho^{2}u^{2}u_{xt}^{2}+c\int_{Q_{T}}\left|(\rho
Q)_{xt}\right|^{2}+c\int_{Q_{T}}\left|(P_{c})_{xt}\right|^{2}$
$\displaystyle\leq$ $\displaystyle
c+c\int_{0}^{T}\|u_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}u_{xt}^{2}+c\int_{Q_{T}}\rho_{xt}^{2}+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}\theta_{xt}^{2}+c$
$\displaystyle\leq$ $\displaystyle c.$

This proves Corollary 4.1. $\Box$

###### Lemma 4.3

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\int_{I}\left(\rho_{xxx}^{2}+\rho_{xxt}^{2}+\rho_{tt}^{2}\right)+\int_{Q_{T}}\left(\rho_{xtt}^{2}+u_{xxxx}^{2}\right)\leq
c.$

Proof. Differentiating (3.31) with respect to $x$, we have

$\rho_{xxxt}=-\rho_{xxxx}u-4\rho_{xxx}u_{x}-6\rho_{xx}u_{xx}-4\rho_{x}u_{xxx}-\rho
u_{xxxx}.$ (4.12)

Multiplying (4.12) by $2\rho_{xxx}$, integrating the resulting equation over
$I$, and using integration by parts and the Hölder inequality, we have

$\displaystyle\frac{d}{dt}\int_{I}\rho_{xxx}^{2}$ $\displaystyle=$
$\displaystyle-7\int_{I}\rho_{xxx}^{2}u_{x}-12\int_{I}\rho_{xx}\rho_{xxx}u_{xx}-8\int_{I}\rho_{x}\rho_{xxx}u_{xxx}-2\int_{I}\rho\rho_{xxx}u_{xxxx}$
$\displaystyle\leq$ $\displaystyle
7\|u_{x}\|_{L^{\infty}}\int_{I}\rho_{xxx}^{2}+12\|u_{xx}\|_{L^{\infty}}\|\rho_{xx}\|_{L^{2}}\|\rho_{xxx}\|_{L^{2}}$
$\displaystyle+8\|\rho_{x}\|_{L^{\infty}}\|\rho_{xxx}\|_{L^{2}}\|u_{xxx}\|_{L^{2}}+2\|\rho\|_{L^{\infty}}\|\rho_{xxx}\|_{L^{2}}\|u_{xxxx}\|_{L^{2}}.$

By Lemma 4.1 and the Cauchy inequality, we get

$\frac{d}{dt}\int_{I}\rho_{xxx}^{2}\leq
c\int_{I}\rho_{xxx}^{2}+c\int_{I}u_{xxxx}^{2}+c.$ (4.13)

Differentiating (3.33) with respect to $x$, we have

$\displaystyle u_{xxxx}=\rho_{xx}u_{t}+2\rho_{x}u_{xt}+\rho
u_{xxt}+(\rho_{x}uu_{x})_{x}+(\rho u_{x}^{2})_{x}+(\rho uu_{xx})_{x}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ +(P_{c})_{xxx}+(\rho Q)_{xxx}.$ (4.14)

By (4), ($A_{6}$) and Lemma 4.1, we have

$\int_{I}u_{xxxx}^{2}\leq c\int_{I}\rho
u_{xxt}^{2}+c\int_{I}\rho_{xxx}^{2}+c.$ (4.15)

By (4.13), (4.15), Corollary 4.1 and the Gronwall inequality, we get

$\int_{I}\rho_{xxx}^{2}\leq c.$ (4.16)

It follows from (4.15), (4.16) and Corollary 4.1 that

$\int_{Q_{T}}u_{xxxx}^{2}\leq c.$

A direct calculation, combining (1.1)1, (3.31), (4.16), Lemma 4.1, Corollary
4.1 and Lemma 4.2, implies

$\int_{I}\left(\rho_{xxt}^{2}++\rho_{tt}^{2}\right)+\int_{Q_{T}}\rho_{xtt}^{2}\leq
c.$

The proof of Lemma 4.3 is complete. $\Box$

The next lemma play the most important role in getting $H^{4}$ estimates of
$u$.

###### Lemma 4.4

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\int_{I}\rho^{3}u_{tt}^{2}+\int_{Q_{T}}\rho^{2}u_{xtt}^{2}\leq c.$

Proof. Differentiating (3.23) with respect to $t$, we have

$(\rho u_{tt})_{t}+\rho_{tt}u_{t}+\rho_{t}u_{tt}+(\rho_{t}uu_{x}+\rho
u_{t}u_{x}+\rho uu_{xt})_{t}+P_{xtt}=u_{xxtt}.$ (4.17)

Multiplying (4.17) by $\rho^{\gamma_{2}}u_{tt}$ ($\gamma_{2}$ is to be decided
later), and integrating the resulting equation over $I$, we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}+\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}$
$\displaystyle=$
$\displaystyle\frac{\gamma_{2}-3}{2}\int_{I}\rho^{\gamma_{2}}\rho_{t}u_{tt}^{2}-\int_{I}[\rho_{tt}u_{t}+\rho_{tt}uu_{x}+2\rho_{t}u_{t}u_{x}+2\rho_{t}uu_{xt}+2\rho
u_{t}u_{xt}+P_{xtt}](\rho^{\gamma_{2}}u_{tt})$
$\displaystyle-\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}u_{x}-\int_{I}\rho^{\gamma_{2}+1}uu_{tt}u_{xtt}-\gamma_{2}\int_{I}\rho^{\gamma_{2}-1}\rho_{x}u_{tt}u_{xtt}$
$\displaystyle\leq$ $\displaystyle
c\|(\sqrt{\rho})_{t}\|_{L^{\infty}}\int_{I}\rho^{\gamma_{2}+\frac{1}{2}}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}-\int_{I}\rho^{\gamma_{2}}u_{tt}(\rho
Q)_{xtt}-\int_{I}\rho^{\gamma_{2}}u_{tt}(P_{c})_{xtt}$
$\displaystyle+c\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}-\int_{I}\rho^{\gamma_{2}+1}uu_{tt}u_{xtt}-2\gamma_{2}\int_{I}\rho^{\gamma_{2}-\frac{1}{2}}u_{xtt}u_{tt}(\sqrt{\rho})_{x}+c$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{\gamma_{2}+\frac{1}{2}}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}+\int_{I}\rho^{\gamma_{2}}u_{xtt}(\rho
Q)_{tt}+\gamma_{2}\int_{I}\rho^{\gamma_{2}-1}\rho_{x}u_{tt}(\rho Q)_{tt}$
$\displaystyle+c\|\rho_{xtt}\|_{L^{2}}^{2}+\frac{1}{4}\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}+c\int_{I}\rho^{\gamma_{2}+2}u_{tt}^{2}+c\int_{I}\rho^{\gamma_{2}-1}u_{tt}^{2}+c$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{\gamma_{2}-1}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}+\frac{1}{2}\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}+c\int_{I}\rho^{\gamma_{2}}\left|(\rho
Q)_{tt}\right|^{2}$
$\displaystyle+c\int_{I}\rho^{2\gamma_{2}-2}u_{tt}^{2}+c\int_{I}\rho\left|(\rho
Q)_{tt}\right|^{2}+c\|\rho_{xtt}\|_{L^{2}}^{2}+c.$

Here, we have used integration by parts, the Cauchy inequality, ($A_{2}$),
($A_{3}$), Lemma 2.2, Lemma 4.1, Lemma 4.2 and Lemma 4.3.

After the third term of the right side is absorbed by the left, we have

$\displaystyle\frac{d}{dt}\int_{I}\rho^{\gamma_{2}+1}u_{tt}^{2}+\int_{I}\rho^{\gamma_{2}}u_{xtt}^{2}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{\gamma_{2}-1}u_{tt}^{2}+c\int_{I}\rho^{2\gamma_{2}}u_{tt}^{2}+c\int_{I}\rho^{\gamma_{2}}\left|(\rho
Q)_{tt}\right|^{2}$ (4.18)
$\displaystyle+c\int_{I}\rho^{2\gamma_{2}-2}u_{tt}^{2}+c\int_{I}\rho\left|(\rho
Q)_{tt}\right|^{2}+c\|\rho_{xtt}\|_{L^{2}}^{2}+c.$

By Lemma 4.2, we know $\int_{Q_{T}}\rho u_{tt}^{2}\leq c$. This implies that
the minimum of $\gamma_{2}$ we should take in (4.18) is $2$. Substituting
$\gamma_{2}=2$ into (4.18), we have

$\frac{d}{dt}\int_{I}\rho^{3}u_{tt}^{2}+\int_{I}\rho^{2}u_{xtt}^{2}\leq
c\int_{I}\rho u_{tt}^{2}+c\int_{I}\rho\left|(\rho
Q)_{tt}\right|^{2}+c\int_{I}\rho_{xtt}^{2}+c.$ (4.19)

We are going to estimate $\int_{I}\rho\left|(\rho Q)_{tt}\right|^{2}$. Using
Lemma 4.1, ($A_{4}$) and Lemma 4.3, we have

$\displaystyle\int_{I}\rho\left|(\rho Q)_{tt}\right|^{2}$ $\displaystyle=$
$\displaystyle\int_{I}\rho\left|\rho_{tt}Q+2\rho_{t}Q^{\prime}\theta_{t}+\rho
Q^{\prime\prime}\theta_{t}^{2}+\rho Q^{\prime}\theta_{tt}\right|^{2}$ (4.20)
$\displaystyle\leq$ $\displaystyle
c+c\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{I}\rho^{3}\theta_{tt}^{2}.$

Substituting (4.20) into (4.19), integrating the resulting inequality over
$(0,t)$, and using Lemma 4.1, Lemma 4.2 and Lemma 4.3, we get

$\int_{I}\rho^{3}u_{tt}^{2}+\int_{0}^{t}\int_{I}\rho^{2}u_{xtt}^{2}\leq\int_{I}\rho^{3}u_{tt}^{2}(x,0)+c.$
(4.21)

Using (4.1), (4.2), (4.6), (4.7), (3.8), (3.23) and (4.2)3, we have

$\int_{I}\rho^{3}u_{tt}^{2}(x,0)\leq c,$

which combining (4.21) completes the proof. $\Box$

###### Lemma 4.5

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\displaystyle\int_{I}\rho
u_{xxt}^{2}+\int_{Q_{T}}\left(u_{xxxt}^{2}+\rho\theta_{xxt}^{2}\right)\leq c.$

Proof. By (3.23), we have

$\displaystyle\int_{I}\rho u_{xxt}^{2}$ $\displaystyle\leq$ $\displaystyle
c\int_{I}\rho^{3}u_{tt}^{2}+c\int_{I}\rho\rho_{t}^{2}u_{t}^{2}+c\int_{I}\rho\rho_{t}^{2}u^{2}u_{x}^{2}+\int_{I}\rho^{3}u_{t}^{2}u_{x}^{2}$
$\displaystyle+c\int_{I}\rho^{3}u^{2}u_{xt}^{2}+c\int_{I}\rho|(\rho
Q)_{xt}|^{2}+c\int_{I}\rho|(P_{c})_{xt}|^{2}$ $\displaystyle\leq$
$\displaystyle c,$

where we have used ($A_{2}$)-($A_{4}$), Lemma 2.2, Lemma 4.1, Lemma 4.2 and
Lemma 4.4.

It follows from (3.37) and ($A_{5}$) that

$\displaystyle\int_{Q_{T}}\rho\theta_{xxt}^{2}$ $\displaystyle\leq$
$\displaystyle
c\int_{Q_{T}}\rho|\kappa^{\prime}|^{2}\theta_{t}^{2}\theta_{xx}^{2}+c\int_{Q_{T}}\rho|\kappa^{\prime\prime}|^{2}\theta_{t}^{2}\theta_{x}^{4}+c\int_{Q_{T}}\rho|\kappa^{\prime}|^{2}\theta_{x}^{2}\theta_{xt}^{2}$
$\displaystyle+c\int_{Q_{T}}\rho^{3}|Q^{\prime}|^{2}\theta_{tt}^{2}+c\int_{Q_{T}}\rho^{3}|Q^{\prime\prime}|^{2}\theta_{t}^{4}+c\int_{Q_{T}}\rho\rho_{t}^{2}|Q^{\prime}|^{2}\theta_{t}^{2}$
$\displaystyle+c\int_{Q_{T}}\rho\left|(\rho
uQ^{\prime}\theta_{x})_{t}\right|^{2}+c\int_{Q_{T}}\rho\left|(\rho\theta
Q^{\prime}u_{x})_{t}\right|^{2}+c\int_{Q_{T}}\rho u_{x}^{2}u_{xt}^{2},$

which, combining ($A_{4}$), ($A_{5}$) and Lemma 4.1, gives

$\displaystyle\int_{Q_{T}}\rho\theta_{xxt}^{2}$ $\displaystyle\leq$
$\displaystyle
c\int_{Q_{T}}\theta_{xt}^{2}+c\int_{Q_{T}}\rho^{3}\theta_{tt}^{2}+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+\int_{Q_{T}}u_{xt}^{2}+c$
(4.22) $\displaystyle\leq$ $\displaystyle c.$

Differentiating (3.33) with respect to $t$, we get

$\displaystyle u_{xxxt}$ $\displaystyle=$ $\displaystyle
2(\sqrt{\rho})_{x}\sqrt{\rho}u_{tt}+\rho
u_{xtt}+\rho_{xt}u_{t}+\rho_{t}u_{xt}+\rho_{xt}uu_{x}+\rho_{t}u_{x}^{2}+\rho_{t}uu_{xx}+\rho_{x}u_{t}u_{x}$
$\displaystyle+2\rho u_{xt}u_{x}+\rho u_{t}u_{xx}+\rho_{x}uu_{xt}+\rho
uu_{xxt}+(\rho Q)_{xxt}+(P_{c})_{xxt}.$

This, together with (4.22), ($A_{6}$), Lemma 2.2, Lemma 4.1, Lemma 4.2, Lemma
4.3, Lemma 4.4 and Corollary 4.1, implies

$\displaystyle\int_{Q_{T}}u_{xxxt}^{2}\leq
c+c\int_{Q_{T}}\left(\rho\theta_{xxt}^{2}+\theta_{xt}^{2}\right)+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}\rho_{xxt}^{2}\leq
c.$

This completes the proof. $\Box$

From (4.15), Lemma 4.3 and Lemma 4.5, we get the following corollary
immediately.

###### Corollary 4.2

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\int_{I}u_{xxxx}^{2}\leq c.$

###### Corollary 4.3

Under the conditions of Theorem 4.1, it holds

$\int_{Q_{T}}\theta_{xxxx}^{2}\leq c.$

Proof. Differentiating (3) with respect to $x$, we have

$\displaystyle\kappa\theta_{xxxx}=-4\kappa^{\prime}\theta_{x}\theta_{xxx}-3\kappa^{\prime}\theta_{xx}^{2}-3\kappa^{\prime\prime}\theta_{x}^{2}\theta_{xx}-\left(\kappa^{\prime\prime}\theta_{x}^{3}\right)_{x}-2\left(u_{x}u_{xx}\right)_{x}+\left(\rho
Q^{\prime}\theta_{xt}\right)_{x}$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\left(\rho_{x}Q^{\prime}\theta_{t}\right)_{x}+\left(\rho
Q^{\prime\prime}\theta_{x}\theta_{t}\right)_{x}+(\rho
uQ^{\prime}\theta_{x})_{xx}+(\rho\theta Q^{\prime}u_{x})_{xx}.$

This, combining ($A_{5}$), ($A_{6}$), Lemma 3.3, Lemma 4.1 and Lemma 4.5,
implies

$\displaystyle\int_{Q_{T}}\theta_{xxxx}^{2}\leq
c+c\int_{Q_{T}}\left(\rho\theta_{xxt}^{2}+\theta_{xt}^{2}\right)+c\int_{0}^{T}\|\theta_{t}\|_{L^{\infty}}^{2}+c\int_{Q_{T}}|\kappa^{\prime\prime\prime}|^{2}\theta_{x}^{8}\leq
c.$

This proves Corollary 4.3. $\Box$

###### Lemma 4.6

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\int_{I}\rho_{xxxx}^{2}+\int_{Q_{T}}u_{xxxxx}^{2}\leq c.$

Proof. Differentiating (4.12) with respect to $x$, multiplying the resulting
equation by $2\rho_{xxxx}$, integrating over $I$, and using integration by
parts, Lemma 4.1, Lemma 4.3, Corollary 4.2 and the Cauchy inequality, we get

$\displaystyle\frac{d}{dt}\int_{I}\rho_{xxxx}^{2}$ $\displaystyle=$
$\displaystyle-9\int_{I}\rho_{xxxx}^{2}u_{x}-20\int_{I}\rho_{xxx}\rho_{xxxx}u_{xx}-20\int_{I}\rho_{xx}\rho_{xxxx}u_{xxx}$
(4.23)
$\displaystyle-10\int_{I}\rho_{x}\rho_{xxxx}u_{xxxx}-2\int_{I}\rho\rho_{xxxx}u_{xxxxx}$
$\displaystyle\leq$ $\displaystyle
c\int_{I}\rho_{xxxx}^{2}+c\int_{I}u_{xxxxx}^{2}+c.$

Now we estimate the second term of the right-hand side of (4.23).

Differentiating (4) with respect to $x$, we have

$\displaystyle u_{xxxxx}$ $\displaystyle=$
$\displaystyle\rho_{xxx}u_{t}+3\rho_{xx}u_{xt}+3\rho_{x}u_{xxt}+\rho
u_{xxxt}+(\rho_{x}uu_{x})_{xx}+(\rho u_{x}^{2})_{xx}$ $\displaystyle+(\rho
uu_{xx})_{xx}+(\rho Q)_{xxxx}+(P_{c})_{xxxx}.$

This, combining $(A_{6})$, Lemma 2.2, Lemma 4.1, Lemma 4.3 and Corollary 4.2,
concludes

$\displaystyle\int_{I}u_{xxxxx}^{2}\leq
c\int_{I}u_{xxt}^{2}+c\int_{I}u_{xxxt}^{2}+c\int_{I}\rho_{xxxx}^{2}+c\int_{I}\theta_{xxxx}^{2}+c.$
(4.24)

Substituting (4.24) into (4.23), and using Corollary 4.1, Corollary 4.3, Lemma
4.5 and the Gronwall inequality, we obtain

$\int_{I}\rho_{xxxx}^{2}\leq c.$ (4.25)

It follows from (4.24), (4.25), Corollary 4.1, Corollary 4.3 and Lemma 4.5
that

$\int_{Q_{T}}u_{xxxxx}^{2}\leq c.$

This completes the proof of Lemma 4.6. $\Box$

###### Corollary 4.4

Under the conditions of Theorem 4.1, it holds for any $0\leq t\leq T$

$\int_{I}\left(\rho_{xtt}^{2}+\rho_{xxxt}^{2}\right)+\int_{Q_{T}}\left(\rho_{ttt}^{2}+\rho_{xxtt}^{2}\right)\leq
c.$

Here we have used the following inequality when we get the upper bound of
$\rho_{ttt}$:

$\rho_{x}^{2}u_{tt}^{2}=2\left[(\sqrt{\rho})_{x}\sqrt{\rho}\right]^{2}u_{tt}^{2}\leq
c\rho u_{tt}^{2}.$

From the above estimates, we get

$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{4}}+\|\rho_{t}\|_{H^{3}}\displaystyle+\|\rho_{tt}\|_{H^{1}}+\|u\|_{H^{4}}$
$\displaystyle+\|u_{t}\|_{H^{1}}+\|\rho^{\frac{3}{2}}u_{tt}\|_{L^{2}}+\|\sqrt{\rho}u_{xxt}\|_{L^{2}}+\|\theta\|_{H^{3}}+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{xt}\|_{L^{2}}$
$\displaystyle+\int_{Q_{T}}\left(\rho^{2}u_{xtt}^{2}+\rho
u_{tt}^{2}+u_{xxt}^{2}+u_{xxxt}^{2}+u_{xxxxx}^{2}\right)$
$\displaystyle+\int_{Q_{T}}\left(\rho_{ttt}^{2}+\rho_{xxtt}^{2}+\theta_{xxxx}^{2}+\theta_{t}^{2}+\theta_{xt}^{2}+\rho\theta_{xxt}^{2}+\rho^{3}\theta_{tt}^{2}\right)\leq
c.$ (4.26)

From (4) and (3.55), we get

$\displaystyle\|\rho\|_{H^{4}}+\|\rho_{t}\|_{H^{3}}+\|\rho_{tt}\|_{H^{1}}+\|u\|_{H^{4}}+\|u_{t}\|_{H^{2}}+\|u_{tt}\|_{L^{2}}+\|\theta\|_{H^{3}}+\|\theta_{t}\|_{H^{1}}$
$\displaystyle+\int_{Q_{T}}\left(u_{xtt}^{2}+u_{xxxt}^{2}+u_{xxxxx}^{2}\right)+\int_{Q_{T}}\left(\rho_{ttt}^{2}+\rho_{xxtt}^{2}+\theta_{xxxx}^{2}+\theta_{xxt}^{2}+\theta_{tt}^{2}\right)\leq
c(\delta),$

where $c(\delta)$ is a positive constant, and may depend on $\delta$.

The proof of Theorem 4.1 is complete. $\Box$

Proof of Theorem 1.2:

Consider (1.1)-(1.4) with initial data replaced by ($\rho_{0}^{\delta}$,
$u_{0}^{\delta}$, $\theta_{0}^{\delta}$), we obtain from Theorem 4.1 that
there exists a unique solution ($\rho^{\delta}$, $u^{\delta}$,
$\theta^{\delta}$) such that (4) and (3.55) are valid when we replace ($\rho,$
$u$, $\theta$) by ($\rho^{\delta}$, $u^{\delta}$, $\theta^{\delta}$). With
this estimates uniform for $\delta$, we take $\delta\rightarrow 0^{+}$ ( take
subsequence if necessary) to get a solution to (1.1)-(1.4) still denoted by
($\rho$, $u$, $\theta$). By the lower semi-continuity of the norms, we have

$\displaystyle\|(\sqrt{\rho})_{x}\|_{L^{\infty}}+\|(\sqrt{\rho})_{t}\|_{L^{\infty}}+\|\rho\|_{H^{4}}+\|\rho_{t}\|_{H^{3}}\displaystyle+\|\rho_{tt}\|_{H^{1}}+\|u\|_{H^{4}}+\|u_{t}\|_{H^{1}}$
$\displaystyle+\|\sqrt{\rho}u_{xxt}\|_{L^{2}}+\|\theta\|_{H^{3}}+\|\sqrt{\rho}\theta_{t}\|_{L^{2}}+\|\rho\theta_{xt}\|_{L^{2}}+\int_{Q_{T}}\left({u}_{xxt}^{2}+{u}_{xxxt}^{2}+{u}_{xxxxx}^{2}\right)$
$\displaystyle+\int_{Q_{T}}\left({\rho}_{ttt}^{2}+{\rho}_{xxtt}^{2}+{\theta}_{xxxx}^{2}+{\theta}_{t}^{2}+{\theta}_{xt}^{2}\right)\leq
c,$

which proves the existence of the solutions as in Theorem 1.2. The uniqueness
of the solutions can be proved by the standard method like in [4], we omit it
for brevity. The proof of Theorem 1.2 is complete. $\Box$

Acknowledgment.

The first author was supported by the National Basic Research Program of China
(973 Program) No. 2010CB808002, and by the National Natural Science Foundation
of China No. 11071086. The second author was supported by the National Natural
Science Foundation of China $\\#$10625105, $\\#$11071093, the PhD specialized
grant of the Ministry of Education of China $\\#$20100144110001 and the self-
determined research funds of CCNU from the colleges’basic research and
operation of MOE.

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