# The limit to rarefaction wave with vacuum for 1D compressible fluids with
temperature-dependent viscosities

Mingjie Li, Teng Wang, and Yi Wang
∗College of Science, Minzu University of China,
Beijing 100081, P. R. China
† Institute of Applied Mathematics, AMSS,
CAS, Beijing 100190, P. R. China
‡ Institute of Applied Mathematics and NCMIS,
AMSS, CAS, Beijing 100190, P. R. China
The research of M. Li was supported by the NSFC Grant No. 11201503. E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>research of Y. Wang was
supported by the NSFC Grant No. 11171326 and the CAS Program for Cross $\&$
Cooperative Team of the Science $\&$ Technology Innovation. E-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper we study the zero dissipation limit of the one-dimensional full
compressible Navier-Stokes(CNS) equations with temperature-dependent viscosity
and heat-conduction coefficient. It is proved that given a rarefaction wave
with one-side vacuum state to the full compressible Euler equations, we can
construct a sequence of solutions to the full CNS equations which converge to
the above rarefaction wave with vacuum as the viscosity and the heat
conduction coefficient tend to zero. Moreover, the uniform convergence rate is
obtained. The main difficulty in our proof lies in the degeneracies of the
density, the temperature and the temperature-dependent viscosities at the
vacuum region in the zero dissipation limit.

Keywords: compressible Navier-Stokes equations, temperature-dependent
viscosities, zero dissipation limit, rarefaction wave, vacuum.

##  1 Introduction and main result

We consider the zero dissipation limit of the one-dimensional compressible
Navier-Stokes equations with heat-conduction in Eulerian coordinates which
read

$\left\\{\begin{array}[]{l}\rho_{t}+(\rho
u)_{x}=0,\quad\quad\qquad\qquad\qquad\quad
x\in\mathbf{R}=(-\infty,+\infty),t>0,\\\\[5.69054pt] (\rho u)_{t}+\big{(}\rho
u^{2}+p\big{)}_{x}=\big{(}\epsilon\mu(\theta)u_{x}\big{)}_{x},\\\\[5.69054pt]
\displaystyle[\rho(e+\frac{1}{2}u^{2})]_{t}+[\rho
u(e+\frac{1}{2}u^{2})+up]_{x}=(\epsilon\kappa(\theta)\theta_{x}+\epsilon\mu(\theta)uu_{x})_{x},\end{array}\right.$
(1.1)

where $\rho(x,t)\geq 0,~{}u(x,t),~{}p(x,t),~{}e(x,t)\geq 0$ and
$\theta(x,t)\geq 0$ represent the mass density, velocity, pressure, internal
energy and absolute temperature of the gas, respectively, and
$\epsilon\mu(\theta)$ and $\epsilon\kappa(\theta)$ denote the viscosity and
heat-conduction coefficient, respectively, with $\epsilon>0$ being positive
constant and

$\displaystyle\mu(\theta)=\mu_{1}\theta^{\alpha},\qquad{\rm
and}\qquad\kappa(\theta)=\kappa_{1}\theta^{\alpha}$ (1.2)

for positive constants $\mu_{1}$, $\kappa_{1}$ and $\alpha>0$. Without loss of
generality, it is assumed that $\mu_{1}=\kappa_{1}=1.$ Here we consider the
ideal polytropic gas, that is, the pressure $p$ and the internal energy $e$
are given respectively by

$\displaystyle p=R\rho\theta=A\rho^{\gamma}\exp(\frac{\gamma-1}{R}S),\quad
e=\frac{R\theta}{\gamma-1},$ (1.3)

satisfying the second law of thermodynamics

$\displaystyle de=\theta dS+\frac{p}{\rho^{2}}d\rho.$ (1.4)

In the state equations (1.3) and (1.4), $S=S(x,t)$ denotes the entropy of the
gas and $\gamma>1$ is the adiabatic exponent and $A,R$ are both positive
constants. For simplicity, it is normalized that

$A=R=\gamma-1.$

The compressible Navier-Stokes equations (1.1) with the temperature-dependent
viscosities (1.2) can be derived exactly from the Chapman-Enskog expansion for
the Boltzmann equation with respect to the Knudsen number, one can refer to
[2], [13] for the details. Formally, as $\epsilon\rightarrow 0+$, the system
(1.1) tends to the corresponding inviscid Euler equations

$\left\\{\begin{array}[]{l}\rho_{t}+(\rho u)_{x}=0,\\\\[5.69054pt] (\rho
u)_{t}+\big{(}\rho u^{2}+p\big{)}_{x}=0,\\\\[5.69054pt]
\displaystyle[\rho(e+\frac{1}{2}u^{2})]_{t}+[\rho
u(e+\frac{1}{2}u^{2})+up]_{x}=0.\end{array}\right.$ (1.5)

The Euler system (1.5) is a strictly hyperbolic one for $\rho>0$ whose first
and third characteristic fields are genuinely nonlinear and second
characteristic field is linearly degenerate, that is, in the equivalent system

$\left(\begin{array}[]{l}\displaystyle\rho\\\ \displaystyle u\\\ \displaystyle
S\end{array}\right)_{t}+\left(\begin{array}[]{ccc}\displaystyle
u&\quad\rho&\quad 0\\\ \frac{p_{\rho}}{\rho}&\quad
u&\quad\frac{p_{S}}{\rho}\\\ \displaystyle 0&\quad 0&\quad
u\end{array}\right)\left(\begin{array}[]{l}\displaystyle\rho\\\ \displaystyle
u\\\ \displaystyle S\end{array}\right)_{x}=0,$

the Jacobi matrix

$\left(\begin{array}[]{ccc}\displaystyle u&\quad\rho&\quad 0\\\
\frac{p_{\rho}}{\rho}&\quad u&\quad\frac{p_{S}}{\rho}\\\ \displaystyle 0&\quad
0&\quad u\end{array}\right)$

has three distinct eigenvalues

$\lambda_{1}(\rho,u,\theta)=u-\sqrt{p_{\rho}(\rho,S)},~{}\lambda_{2}(\rho,u,\theta)=u,~{}\lambda_{3}(\rho,u,\theta)=u+\sqrt{p_{\rho}(\rho,S)}$

with corresponding right eigenvectors

$r_{1}(\rho,u,S)=(-\rho,\sqrt{p_{\rho}(\rho,S)},0)^{t},~{}r_{2}(\rho,u,S)=(p_{S},0,-p_{\rho})^{t},~{}r_{3}(\rho,u,S)=(\rho,\sqrt{p_{\rho}(\rho,S)},0)^{t},$

such that

$r_{i}(\rho,u,S)\cdot\nabla_{(\rho,u,S)}\lambda_{i}(\rho,u,S)\neq 0,\quad
i=1,3,$

and

$r_{2}(\rho,u,S)\cdot\nabla_{(\rho,u,S)}\lambda_{2}(\rho,u,S)\equiv 0.$

Thus the two $i$-Riemann invariants $(i=1,3)$ can be defined by (cf. [20])

$\Sigma_{i}^{(1)}=u+(-1)^{\frac{i-1}{2}}\int^{\rho}\frac{\sqrt{p_{z}(z,S)}}{z}dz,\qquad\Sigma_{i}^{(2)}=S,$
(1.6)

such that

$\nabla_{(\rho,u,S)}\Sigma_{i}^{(j)}(\rho,u,S)\cdot r_{i}(\rho,u,S)\equiv
0,\quad i=1,3,~{}j=1,2.$

The study of the limiting process of viscous flows when the viscosity tends to
zero is one of the important problems in the theory of the compressible fluid.
When the solution of the inviscid flow is smooth, the zero dissipation limit
can be solved by classical scaling method. However, the inviscid compressible
flow contains singularities such as shock, contact discontinuity and the
vacuum in general. Therefore, determining how to justify the zero dissipation
limit to the Euler equations with basic wave patterns and the vacuum states is
a natural and difficult problem.

There have been many results on the zero dissipation limit of the compressible
fluid with basic wave patterns without vacuum. For the system of the
hyperbolic conservation laws with artificial viscosity

$u_{t}+f(u)_{x}=\varepsilon u_{xx},$

Goodman-Xin [4] first verified the viscous limit for piecewise smooth
solutions separated by non-interacting shock waves using a matched asymptotic
expansion method. Later Yu [25] proved it for the corresponding hyperbolic
conservation laws with both shock and initial layers. In 2005, important
progress made by Bianchini-Bressan[1] justifies the vanishing viscosity limit
in BV space even though the problem is still unsolved for the physical system
such as the compressible Navier-Stokes equations.

For the isentropic compressible Navier-Stokes equations with constant
viscosity where the conservation of energy in (1.1) is neglected, Hoff-Liu [5]
first proved the vanishing viscosity limit for piecewise constant shock even
with initial layer. Later Xin [23] obtained the zero dissipation limit for
rarefaction waves without vacuum for both rarefaction wave data and well-
prepared smooth data. Then Wang [21] generalized the result of Goodmann-Xin
[4] to the isentropic Navier-Stokes equations. Recently, Chen-Perepelitsa [3]
proved the vanishing viscosity to the compressible Euler equations for the
isentropic compressible Navier-Stokes equations (1.1) with constant viscosity
by compensated compactness method for the case that the far field of the
initial values of Euler system has no vacuums.

For the full Navier-Stokes equations (1.1) with constant viscosity, there are
also many results on the zero dissipation limit to the corresponding full
Euler system with basic wave patterns without vacuum. We refer to Jiang-Ni-Sun
[11] and Xin-Zeng [24] for the rarefaction wave, Wang [22] for the shock wave,
Ma [17] for the contact discontinuity, Huang-Wang-Yang [9] and Huang-Jiang-
Wang [7] for the superposition of two rarefaction waves and a contact
discontinuity, Huang-Wang-Yang [10] for the superposition of one shock and one
rarefaction wave and Zhang-Pan-Wang-Tan [26] for the superposition of two
shock waves with the initial layer. Recently, Huang-Wang-Wang-Yang [8] succeed
in justifies the vanishing viscosity limit of compressible Navier-Stokes
equations in the setting of Riemann solutions for the superposition of shock
wave, rarefaction wave and contact discontinuity.

It is well-known that the vacuum states are generic in inviscid compressible
Euler equations (1.5) since the vacuum states may occur in the Riemann
solutions instantaneously as $t>0$ even if the initial Riemann data are non-
vacuum states on both sides at $t=0$. Therefore, vacuum states are important
physical states in gas dynamics and often yield degeneracies and certain
singularities in the physical system, which cause some essential analytical
difficulties. For example, the velocity can not even be defined in the vacuum
region. In the setting of Riemann solutions, as pointed out by Liu-Smoller
[15], among the three elementary hyperbolic waves, i.e., shock and rarefaction
waves and contact discontinuities, to the one-dimensional isentropic
compressible Euler equations (1.5), only the rarefaction wave can be connected
to the vacuum states. There are some mathematical results on the time-
asymptotic stability and the vanishing viscosity limit to the rarefaction wave
with the vacuum. Perepelitsa [19] consider the time-asymptotic stability of
solutions to 1-d isentropic compressible Navier-Stokes equations with fixed
and constant viscosity toward rarefaction waves connected to vacuum in
Lagrangian coordinate. Then Jiu-Wang-Xin [12] study the large time asymptotic
behavior toward rarefaction wave with vacuum for solution to the one-
dimensional compressible Navier-Stokes equations with density-dependent
viscosity, which can be viewed as the compressible Navier-Stokes equations
(1.1) with fixed $\epsilon=1$ in isentropic regime. Note that in isentropic
regime, there is no energy equation $\eqref{ns}_{3}$ and the viscosity
coefficient in the momentum equation $\eqref{ns}_{2}$ transfer to depend on
the density. More recently, Huang-Li-Wang [6] justified the vanishing
viscosity limit of one-dimensional isentropic compressible Navier-Stokes
equations with constant viscosity to the rarefaction wave with one-side vacuum
state to the corresponding compressible Euler equations. However, for the full
compressible Navier-Stokes equations (1.1) with temperature-dependent
viscosities, as far as we know, there is no any result on the zero dissipation
limit to the rarefaction wave with the vacuum due to various difficulties
mentioned below.

Now we give a description of the rarefaction wave connected to the vacuum to
the full compressible Euler equations (1.5); see also the reference [20]. For
definiteness, 3-rarefaction wave will be considered. If we investigate the
compressible Euler system (1.5) with the Riemann initial data

$\left\\{\begin{array}[]{rr}\rho(0,x)=0,&x<0,\\\
(\rho,u,\theta)(0,x)=(\rho_{+},u_{+},\theta_{+}),&x>0,\end{array}\right.$
(1.7)

where the left side is the vacuum state and $\rho_{+}>0,u_{+},\theta_{+}>0$
are prescribed constants on the right state, then the Riemann problem (1.5),
(1.7) admits a $3-$rarefaction wave connected to the vacuum on the left hand
side. By the fact that along the $3-$rarefaction wave curve, $3-$Riemann
invariant $\Sigma^{(i)}_{3}(\rho,u,\theta),~{}(i=1,2)$ defined in (1.6) keeps
constant in $(x,t)$, one can get the velocity

$u_{-}=\Sigma_{3}^{(1)}(\rho_{+},u_{+},\theta_{+})$

being the speed of the gas coming into the vacuum from the 3-rarefaction wave.
On the other hand, in the vacuum region, the absolute temperature $\theta$
also becomes zero due to the state equation (1.3), that is,
$\theta=\rho^{\gamma-1}e^{S}$ and the fact that the entropy $S$ keeps constant
along the 3-rarefaction wave. Correspondingly, the main difficulty of the
present paper lies in how to deal with the degeneracies of the temperature-
dependent viscosities in the vacuum region in the zero dissipation limit
process.

As described above, the $3-$rarefaction wave connecting the vacuum state
$\rho=0$ to $(\rho_{+},u_{+},\theta_{+})$ is the self-similar solution
$(\rho^{r_{3}},u^{r_{3}},\theta^{r_{3}})(\xi),~{}(\xi=\frac{x}{t})$ of (1.5)
defined by

$\begin{array}[]{c}\lambda_{3}(\rho^{r_{3}}(\xi),u^{r_{3}}(\xi),\theta^{r_{3}}(\xi))=\left\\{\begin{array}[]{ll}\displaystyle\rho^{r_{3}}(\xi)\equiv
0,&\displaystyle{\rm if}~{}~{}\xi<\lambda_{3}(0,u_{-},0)=u_{-},\\\
\displaystyle\xi,&\displaystyle{\rm
if}~{}~{}u_{-}\leq\xi\leq\lambda_{3}(\rho_{+},u_{+},\theta_{+}),\\\
\displaystyle\lambda_{3}(\rho_{+},u_{+},\theta_{+}),&\displaystyle{\rm
if}~{}~{}\xi>\lambda_{3}(\rho_{+},u_{+},\theta_{+}),\end{array}\right.\end{array}$
(1.8)
$\Sigma_{3}^{(1)}(\rho^{r_{3}}(\xi),u^{r_{3}}(\xi),\theta^{r_{3}}(\xi))=\Sigma_{3}^{(1)}(0,u_{-},0)=\Sigma_{3}^{(1)}(\rho_{+},u_{+},\theta_{+}),$
(1.9)

and

$\Sigma_{3}^{(2)}=S^{r_{3}}=S_{+}:=-(\gamma-1)\log\rho_{+}+\log\theta_{+}.$
(1.10)

Thus one can define the momentum $m^{r_{3}}$ and the total internal energy
$n^{r_{3}}$ of 3-rarefaction wave by

$m^{r_{3}}(\xi):=\left\\{\begin{array}[]{ll}\rho^{r_{3}}(\xi)u^{r_{3}}(\xi),{}{}{}&{\rm
if}~{}~{}~{}\rho^{r_{3}}>0,\\\ 0,{}{}{}&{\rm
if}~{}~{}~{}\rho^{r_{3}}=0,\end{array}\right.$ (1.11)

and

$n^{r_{3}}(\xi):=\left\\{\begin{array}[]{ll}\rho^{r_{3}}(\xi)\theta^{r_{3}}(\xi),{}{}{}&{\rm
if}~{}~{}~{}\rho^{r_{3}}>0,\\\ 0,{}{}{}&{\rm
if}~{}~{}~{}\rho^{r_{3}}=0.\end{array}\right.$ (1.12)

In the present paper, we want to construct a sequence of global-in-time
solutions
$(\rho^{\epsilon},m^{\epsilon}:=\rho^{\epsilon}u^{\epsilon},n^{\epsilon}:=\rho^{\epsilon}\theta^{\epsilon})(x,t)$
to the compressible Navier-Stokes equations (1.1) with temperature-dependent
viscosities, which converge to the 3-rarefaction wave
$(\rho^{r_{3}},m^{r_{3}},n^{r_{3}})(x/t)$ with vacuum defined above as
$\epsilon$ tends to zero. The effects of initial layers will be ignored by
choosing the well-prepared initial data (3.1) depending on the viscosity for
the Navier-Stokes equations.

As mentioned before, the main novelty and difficulty here is determining how
to control the degeneracies caused by the vacuum in the rarefaction wave. To
overcome this difficulty, we first cut off the 3-rarefaction wave with vacuum
along the rarefaction wave curve suitably ($\nu$ is the cut-off parameter and
the details can be seen in Section 2), and use the fact that the viscosity
$\epsilon$ can control the degeneracies caused by the vacuum in rarefaction
waves by choosing suitably $\nu=\nu(\epsilon)$. In fact, we choose
$\nu=\epsilon^{a}|\ln\epsilon|$ with $a$ defined in (1.14) in the present
paper. The other observation is that we can carry out the energy estimates
under the a priori assumptions (3.10)-(3.11) such that the perturbation is
suitably small in $L^{\infty}(\mathbf{R})$ norm with some decay rate with
respect to $\epsilon$.

On the other hand, compared with the previous works [6] for the isentropic
Naiver-Stokes equations with the constant viscosity case, some new
difficulties occur for the full Navier-Stokes equations (1.1) with
temperature-dependent viscosities considered in the present paper. Firstly, in
order to overcome the difficulties caused by the non-isentropic regime, the
relative entropy-entropy flux pair $(\eta,q)$ defined in (3.17) is used as in
[16]. Secondly, since in the vacuum region, the temperature becomes zero due
to the entropy keeps constant by the structure of rarefaction wave. Therefore,
not only the density but also the temperature cause the degeneracies in the
vacuum region and so the temperature-dependent viscosities do. Thus, to deal
with the terms for the temperature-dependent viscosities, such as (3.35),
becomes subtle. In fact, the derivative estimates of the perturbation of the
density depends on the second order derivative estimates of velocity with some
degenerate coefficients (see (3.50)), which is quite different from the
constant viscosity case in [6] and [14]. By choosing the convergence rate $a$
suitably as in (1.14) and then the parameters $\nu,\delta$ as in (3.12) closes
the a priori estimates and yields the desired result.

Now we state our main result as follows.

######  Theorem 1.1.

Let $(\rho^{r_{3}},m^{r_{3}},n^{r_{3}})(x/t)$ be the 3-rarefaction wave with
one-side vacuum state defined by (1.8)-(1.12). Then there exists a small
positive constant $\epsilon_{0}$ such that for any
$\epsilon\in(0,\epsilon_{0})$, we can construct a family of global smooth
solutions
$(\rho^{\epsilon},m^{\epsilon}=\rho^{\epsilon}u^{\epsilon},n^{\epsilon}=\rho^{\epsilon}\theta^{\epsilon})(x,t)$
to the compressible Navier-Stokes equation (1.1) satisfying the following
properties.
(1)

$\begin{array}[]{rl}(\rho^{\epsilon}-\rho^{r_{3}},m^{\epsilon}-m^{r_{3}},n^{\epsilon}-n^{r_{3}}),(\rho^{\epsilon},m^{\epsilon},n^{\epsilon})_{x}&\displaystyle\in
C(0,+\infty;L^{2}(\mathbf{R})),\\\
u^{\epsilon}_{xx},\theta^{\epsilon}_{xx}&\displaystyle\in
L^{2}(0,+\infty;L^{2}(\mathbf{R})).\end{array}$

(2) As the viscosity $\epsilon\rightarrow 0$,
$(\rho^{\epsilon},m^{\epsilon},n^{\epsilon})(x,t)$ converges to
$(\rho^{r_{3}},m^{r_{3}},n^{r_{3}})(x/t)$ pointwisely except the original
point $(0,0)$. Furthermore, for any given positive constant $l$, there exists
a constant $C_{l}>0$, independent of $\epsilon$, such that

$\begin{array}[]{ll}\displaystyle\sup_{t\geq
l}\|\rho^{\epsilon}(\cdot,t)-\rho^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\leq
C_{l}~{}\epsilon^{a}|\ln\epsilon|,\\\ \displaystyle\sup_{t\geq
l}\|m^{\epsilon}(\cdot,t)-m^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\leq
C_{l}~{}\epsilon^{a}|\ln\epsilon|,\\\ \displaystyle\sup_{t\geq
l}\|n^{\epsilon}(\cdot,t)-n^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\leq
C_{l}~{}\epsilon^{a}|\ln\epsilon|,\end{array}$ (1.13)

with the positive constant $a$ given by

$\displaystyle a=\frac{1}{18\gamma+12\alpha(\gamma-1)}.$ (1.14)

######  Remark 1.2.

From (1.13) and (1.14), one can see that the decay rate
$\epsilon^{a}|\ln\epsilon|$ in Theorem 1.1 decreases monotonically with
respect to $\alpha$, which is consistent to the observation that the viscosity
becomes weaker as $\alpha$ becomes larger due to the vacuum in the rarefaction
wave, although the convergence rates in (1.13) and (1.14) may not be optimal.

######  Remark 1.3.

By using some ideas in the present paper, Li-Wang [14] generalize Huang-Li-
Wang [6]’s result to the non-isentropic compressible Navier-Stokes equations
(1.1) with constant viscosity.

The rest of the paper is organized as follows. In section 2, we construct a
smooth 3-rarefaction wave profile which approximates the cut-off rarefaction
wave for the Euler equations based on the inviscid Burgers equation. Then the
global-in-time solution to CNS (1.1) is obtained around the smooth
3-rarefaction wave profile and finally the proof the Theorem 1.1 is shown in
Section 3.

Throughout this paper, $H^{k}(\mathbf{R}),k=0,1,2,...$, denotes the $l$-th
order Sobolev space with its norm

$\|f\|_{k}=(\sum^{k}_{j=0}\|\partial^{j}_{y}f\|^{2})^{\frac{1}{2}},\quad{\rm
and}~{}\|\cdot\|:=\|\cdot\|_{L^{2}(dz)},$

while $L^{2}(dz)$ means the $L^{2}$ integral over $\mathbf{R}$ with respect to
the Lebesgue measure $dz$, and $z=x$ or $y$. For simplicity, we also write $C$
as generic positive constants which are independent of time $t$ or $\tau$ and
viscosity $\epsilon$ unless otherwise stated.

##  2 Rarefaction waves

Since there is no exact rarefaction wave profile for the Navier-Stokes
equations (1.1), the following approximate rarefaction wave profile satisfying
the Euler equations was motivated by Matsumura-Nishihara [18] and Xin [23].

Consider the Riemann problem for the inviscid Burgers equation:

$\displaystyle\left\\{\begin{array}[]{ll}w_{t}+ww_{x}=0,\\\
w(x,0)=\left\\{\begin{array}[]{ll}w_{-},&x<0,\\\
w_{+},&x>0.\end{array}\right.\end{array}\right.$ (2.5)

If $w_{-}<w_{+}$, then the Riemann problem $(\ref{bur})$ admits a rarefaction
wave solution $w^{r}(x,t)=w^{r}(\frac{x}{t})$ given by

$\displaystyle
w^{r}(\frac{x}{t})=\left\\{\begin{array}[]{lr}w_{-},&\frac{x}{t}\leq w_{-},\\\
\frac{x}{t},&w_{-}\leq\frac{x}{t}\leq w_{+},\\\ w_{+},&\frac{x}{t}\geq
w_{+}.\end{array}\right.$ (2.9)

Motivated by [18, 23], the approximate rarefaction wave to the compressible
Navier-Stokes equations (1.1) can be constructed by the Burgers equation

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle w_{t}+ww_{x}=0,\\\
\displaystyle
w(0,x)=w_{\delta}(x)=w(\frac{x}{\delta})=\frac{w_{+}+w_{-}}{2}+\frac{w_{+}-w_{-}}{2}\tanh\frac{x}{\delta},\end{array}\right.$
(2.12)

where $\delta>0$ is a small parameter to be determined and the hyperbolic
tangent function $\tanh x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$. In fact, we
choose $\delta=\epsilon^{a}$ in (3.12) with $a$ given by (1.14). Note that the
solution $w^{r}_{\delta}(t,x)$ of the problem (2.12) can be given explicitly
by

$w^{r}_{\delta}(t,x)=w_{\delta}(x_{0}(t,x)),\qquad
x=x_{0}(t,x)+w_{\delta}(x_{0}(t,x))t.$ (2.13)

And $w^{r}_{\delta}(t,x)$ has the following properties:

######  Lemma 2.1.

([23, 6]) The problem $(\ref{dbur})$ has a unique smooth global solution
$w_{\delta}^{r}(x,t)$ for each $\delta>0$ such that

  * (1)

$w_{-}<w_{\delta}^{r}(x,t)<w_{+},\ \partial_{x}w^{r}_{\delta}(x,t)>0,$ for
$x\in\mathbf{R},\ t\geq 0,\ \delta>0.$

  * (2)

The following estimates hold for all $\ t>0,\ \delta>0$ and p$\in[1,\infty]$:

$\displaystyle\|\partial_{x}w^{r}_{\delta}(\cdot,t)\|_{L^{p}}\leq$
$\displaystyle\displaystyle C(w_{+}-w_{-})^{1/p}(\delta+t)^{-1+1/p},$ (2.14)
$\displaystyle\|\partial^{2}_{x}w^{r}_{\delta}(\cdot,t)\|_{L^{p}}\leq$
$\displaystyle\displaystyle C(\delta+t)^{-1}\delta^{-1+1/p},$ (2.15)
$\displaystyle|\frac{\partial^{2}w^{r}_{\delta}(x,t)}{\partial
x^{2}}|\leq\frac{4}{\delta}\frac{\partial w^{r}_{\delta}(x,t)}{\partial x}.$
(2.16)

  * (3)

There exist a constant $\delta_{0}\in(0,1)$ such that for
$\delta\in(0,\delta_{0}],t>0$,

$\|w^{r}_{\delta}(\cdot,t)-w^{r}(\frac{\cdot}{t})\|_{L^{\infty}}\leq\displaystyle
C\delta t^{-1}\big{[}\ln(1+t)+|\ln\delta|\big{]}.$

Note that Lemma 2.1 is a little different from the one in [23] as stated in
[6]. For the detailed proof of Lemma 2.1, one can refer to [23] and [6] and we
omit it here for brevity.

As mentioned in the introduction, we will first cut off the 3-rarefaction wave
with vacuum along the wave curve in order to overcome the degeneracies caused
by the vacuum. More precisely, for any constant $\nu>0$ to be determined, we
can get a state $(\rho,u,\theta)=(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1})$
belonging to the 3-rarefaction wave curve, where
$\bar{S}=S_{+}=-(\gamma-1)\log\rho_{+}+\log\theta_{+}$. From the fact that
3-Riemann invariant $\Sigma_{3}^{(i)}(\rho,u,\theta),~{}(i=1,2)$ is constant
along the 3-rarefaction wave curve, $u_{\nu}$ can be computed explicitly by

$u_{\nu}=\Sigma_{3}^{(1)}(\rho_{+},u_{+},\theta_{+})+2\sqrt{\frac{\gamma}{\gamma-1}\nu^{\gamma-1}e^{S_{+}}}.$

Now we get a new cut-off 3-rarefaction wave
$(\rho_{\nu}^{r_{3}},u_{\nu}^{r_{3}},\theta_{\nu}^{r_{3}})(\xi),~{}(\xi=x/t)$
connecting the state $(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1})$ to the state
$(\rho_{+},u_{+},\theta_{+})$ which can be expressed explicitly by

$\displaystyle\lambda_{3}(\rho_{\nu}^{r_{3}},u_{\nu}^{r_{3}},\theta_{\nu}^{r_{3}})(\xi)=\left\\{\begin{array}[]{ll}\lambda_{3}(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1}),&\xi<\lambda_{3}(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1}),\\\
\xi,&\lambda_{3}(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1})\leq\xi\leq\lambda_{3}(\rho_{+},u_{+},\theta_{+}),\\\
\lambda_{3}(\rho_{+},u_{+},\theta_{+}),&\xi>\lambda_{3}(\rho_{+},u_{+},\theta_{+})\end{array}\right.$
(2.20)

and

$\displaystyle\Sigma_{3}^{(1)}(\rho_{\nu}^{r_{3}},u_{\nu}^{r_{3}},\theta_{\nu}^{r_{3}})=\Sigma_{3}^{(1)}(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1})=\Sigma_{3}^{(1)}(\rho_{+},u_{+},\theta_{+}).$
(2.21)

Correspondingly, we can define the momentum function and the total internal
energy $m^{r_{3}}_{\nu}:=\rho^{r_{3}}_{\nu}u^{r_{3}}_{\nu}$ and
$n^{r_{3}}_{\nu}:=\rho^{r_{3}}_{\nu}\theta^{r_{3}}_{\nu}$ respectively. It is
easy to show that the cut-off 3-rarefaction wave
$(\rho_{\nu}^{r_{3}},m_{\nu}^{r_{3}},n_{\nu}^{r_{3}})(x/t)$ converges to the
original 3-rarefaction wave with vacuum
$(\rho^{r_{3}},m^{r_{3}},n^{r_{3}})(x/t)$ in sup-norm with the convergence
rate $\nu$ as $\nu$ tends to zero. More precisely, it holds that

######  Lemma 2.2.

There exist a constant $\nu_{0}\in(0,1)$ such that for
$\nu\in(0,\nu_{0}],t>0$,

$\|(\rho_{\nu}^{r_{3}},m_{\nu}^{r_{3}},n_{\nu}^{r_{3}})(\cdot/t)-(\rho^{r_{3}},m^{r_{3}},n^{r_{3}})(\cdot/t)\|_{L^{\infty}}\leq\displaystyle
C\nu,$

where the positive constant $C$ is independent of $\nu$.

Now the approximate rarefaction wave
$(\bar{\rho}_{\nu,\delta},\bar{u}_{\nu,\delta},\bar{\theta}_{\nu,\delta})(x,t)$
of the cut-off 3-rarefaction wave
$(\rho_{\nu}^{r_{3}},u_{\nu}^{r_{3}},\theta_{\nu}^{r_{3}})(\frac{x}{t})$ to
compressible Euler equations $(\ref{eul})$ can be defined by

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
w_{+}=\lambda_{3}(\rho_{+},u_{+},\theta_{+}),\quad
w_{-}=\lambda_{3}(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1}),\\\ \displaystyle
w_{\delta}^{r}(x,t)=\lambda_{3}(\bar{\rho}_{\nu,\delta},\bar{u}_{\nu,\delta},\bar{\theta}_{\nu,\delta})(x,t),\\\
\displaystyle\Sigma_{3}^{(1)}(\bar{\rho}_{\nu,\delta},\bar{u}_{\nu,\delta},\bar{\theta}_{\nu,\delta})(x,t)=\Sigma_{3}^{(1)}(\rho_{+},u_{+},\theta_{+})=\Sigma_{3}^{(1)}(\nu,u_{\nu},e^{\bar{S}}\nu^{\gamma-1}),\end{array}\right.$
(2.25)

where $w_{\delta}^{r}(x,t)$ is the solution of Burger’s equation
$(\ref{dbur})$ defined in (2.13). From now on, the subscription of
$(\bar{\rho}_{\delta,\nu},\bar{u}_{\delta,\nu},\bar{\theta}_{\delta,\nu})(x,t)$
will be abbreviated as $(\bar{\rho},\bar{u},\bar{\theta})(x,t)$ for
simplicity. Then the approximate 3-rarefaction wave
$(\bar{\rho},\bar{u},\bar{\theta})$ defined above satisfies

$\left\\{\begin{array}[]{l}\bar{\rho}_{t}+(\bar{\rho}\bar{u})_{x}=0,\\\\[5.69054pt]
(\bar{\rho}\bar{u})_{t}+\big{(}\bar{\rho}\bar{u}^{2}+\bar{p}\big{)}_{x}=0,\\\\[5.69054pt]
\displaystyle\Big{[}\bar{\rho}(\bar{e}+\frac{\bar{u}^{2}}{2})\Big{]}_{t}+\Big{[}\bar{\rho}\bar{u}(\bar{e}+\frac{\bar{u}^{2}}{2})+\bar{u}\bar{p}\Big{]}_{x}=0,\end{array}\right.$
(2.26)

where

$\bar{p}=R\bar{\rho}\bar{\theta}=A\bar{\rho}^{\gamma}\exp{(\frac{\gamma-1}{R}\bar{S})},\quad{\rm
and}~{}~{}\bar{e}=\frac{R}{\gamma-1}\bar{\theta}.$

The properties of the approximate rarefaction wave
$(\bar{\rho},\bar{u},\bar{\theta})$ is listed without proof in the following
Lemma.

######  Lemma 2.3.

The approximate cut-off 3-rarefaction wave $(\bar{\rho},\bar{u},\bar{\theta})$
defined in (2.25) satisfies the following properties:

  * (i)

$\bar{u}_{x}(x,t)=\frac{2}{\gamma+1}(w_{\delta}^{r})_{x}>0,$ for
$x\in\mathbf{R},\ t\geq 0,$
$\bar{\rho}_{x}=\frac{1}{\sqrt{\gamma(\gamma-1)e^{S_{+}}}}\bar{\rho}^{\frac{3-\gamma}{2}}\bar{u}_{x}$,
and
$\bar{\rho}_{xx}=\frac{1}{\sqrt{\gamma(\gamma-1)e^{S_{+}}}}\bar{\rho}^{\frac{3-\gamma}{2}}\bar{u}_{xx}+\frac{3-\gamma}{2\gamma(\gamma-1)e^{S_{+}}}\bar{\rho}^{2-\gamma}(\bar{u}_{x})^{2}$,
$\bar{\theta}_{x}=\sqrt{\frac{\gamma-1}{\gamma}}\bar{\theta}^{\frac{1}{2}}\bar{u}_{x}$,
and
$\bar{\theta}_{xx}=\sqrt{\frac{\gamma-1}{\gamma}}\bar{\theta}^{\frac{1}{2}}\bar{u}_{xx}+\frac{\gamma-1}{2\gamma}(\bar{u}_{x})^{2}$.

  * (ii)

The following estimates hold for all $\ t>0,\ \delta>0$ and p$\in[1,\infty]$:

$\begin{array}[]{l}\|\bar{u}_{x}(\cdot,t)\|_{L^{p}}\leq
C(w_{+}-w_{-})^{1/p}(\delta+t)^{-1+1/p},\\\\[11.38109pt]
\|\bar{u}_{xx}(\cdot,t)\|_{L^{p}}\leq
C(\delta+t)^{-1}\delta^{-1+1/p}.\end{array}$

  * (iii)

There exist a constant $\delta_{0}\in(0,1)$ such that for
$\delta\in(0,\delta_{0}],t>0$,

$\begin{array}[]{ll}\|(\bar{\rho}-\rho^{r_{3}}_{\nu},\bar{u}-u^{r_{3}}_{\nu},\bar{\theta}-\theta^{r_{3}}_{\nu})(\cdot,t)\|_{L^{\infty}}\leq\displaystyle
C\delta t^{-1}\big{[}\ln(1+t)+|\ln\delta|\big{]}.\end{array}$

The proof of Lemma 2.3 can be got similarly as in [6] and will be omitted for
brevity.

##  3 Proof of Theorem 1.1

In order to prove Theorem 1.1, we construct the global smooth solution
$(\rho^{\epsilon},u^{\epsilon},\theta^{\epsilon})$ as the perturbation around
the approximate rarefaction wave $(\bar{\rho},\bar{u},\bar{\theta})$. Consider
the Cauchy problem (1.1) with the smooth initial data

$\displaystyle(\rho^{\epsilon},u^{\epsilon},\theta^{\epsilon})(x,t=0)=(\bar{\rho},\bar{u},\bar{\theta})(x,0).$
(3.1)

Then we introduce the perturbation

$(\phi,\psi,\zeta)(y,\tau)=(\rho^{\epsilon},u^{\epsilon},\theta^{\epsilon})(x,t)-(\bar{\rho},\bar{u},\bar{\theta})(x,t),$
(3.2)

where $y,\tau$ are the scaled variables as

$y=\frac{x}{\epsilon},\quad\tau=\frac{t}{\epsilon},$ (3.3)

and $(\rho^{\epsilon},u^{\epsilon},\theta^{\epsilon})$ is assumed to be the
solution to the problem $(\ref{ns})$. For the simplicity of the notation, we
will omit the superscription of
$(\rho^{\epsilon},u^{\epsilon},\theta^{\epsilon})$ as $(\rho,u,\theta)$ from
now on if there is no confusion of the notation. Substituting $(\ref{pert})$
and $(\ref{tao})$ into the system $(\ref{ns})$ and using the equations for
$(\bar{\rho},\bar{u},\bar{\theta})$, one can obtain

$\left\\{\begin{array}[]{ll}\displaystyle\phi_{\tau}+\rho\psi_{y}+u\phi_{y}=-f,\\\
\displaystyle\rho\psi_{\tau}+\rho
u\psi_{y}+(\gamma-1)(\theta\phi_{y}+\rho\zeta_{y})-\mu(\bar{\theta})\psi_{yy}=-g+\mu(\bar{\theta})_{y}u_{y}+\big{(}(\mu(\theta)-\mu(\bar{\theta}))u_{y}\big{)}_{y},\\\
\displaystyle\rho\zeta_{\tau}+\rho
u\zeta_{y}+(\gamma-1)\rho\theta\psi_{y}-\kappa(\bar{\theta})\zeta_{yy}=-h+\kappa(\bar{\theta})_{y}u_{y}+\big{(}(\kappa(\theta)-\kappa(\bar{\theta}))\theta_{y}\big{)}_{y}+\mu(\theta)u_{y}^{2},\end{array}\right.$
(3.4)

with the initial data

$(\phi,\psi,\zeta)(y,0)=0,$ (3.5)

where

$\left\\{\begin{array}[]{l}f={\bar{u}}_{y}\phi+{\bar{\rho}}_{y}\psi,\\\
\displaystyle
g=-\mu(\bar{\theta})\bar{u}_{yy}+\rho\psi\bar{u}_{y}+(\gamma-1)(\bar{\rho}_{y}\zeta-\frac{\bar{\rho}_{y}\bar{\theta}\phi}{\bar{\rho}}),\\\\[2.84526pt]
\displaystyle
h=\rho\psi\bar{\theta}_{y}+(\gamma-1)\rho\zeta\bar{u}_{y}-\kappa(\bar{\theta})\bar{\theta}_{yy}.\end{array}\right.$
(3.6)

We seek a global-in-time solution $(\phi,\psi,\zeta)$ to the problem
$(\ref{mass})-(\ref{g})$. To this end, the solution space for
$(\ref{mass})-(\ref{g})$ is defined by

$\displaystyle\chi(0,\tau_{1}(\epsilon))=\Big{\\{}(\phi,\psi,\zeta)\Big{|}$
$\displaystyle(\phi,\psi,\zeta)\in
C([0,\tau_{1}(\epsilon)];H^{1}(\mathbf{R})),\quad\phi_{y}\in
L^{2}(0,\tau_{1}(\epsilon);L^{2}(\mathbf{R})),$
$\displaystyle\psi_{y},\zeta_{y}\in
L^{2}(0,\tau_{1}(\epsilon);H^{1}(\mathbf{R}))\Big{\\}}$

with $0<\tau_{1}(\epsilon)\leq+\infty$.

######  Theorem 3.1.

There exist positive constants $\epsilon_{1}$ and $C$ independent of
$\epsilon$, such that if $0<\epsilon\leq\epsilon_{1}$, then the problem
$(\ref{mass})-(\ref{init})$ admits a unique global-in-time solution
$(\phi,\psi,\zeta)\in\chi(0,+\infty)$ satisfying

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,+\infty)}\int_{\mathbf{R}}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}(\tau,y)dy\\\\[8.53581pt]
&\displaystyle+\int^{+\infty}_{0}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}+\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}\Big{]}dyd\tau\leq\epsilon^{\frac{1}{3}},\end{array}$
(3.7)
$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,+\infty)}\int_{\mathbf{R}}\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}\phi_{y}^{2}dy+\int^{+\infty}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau\leq\epsilon^{\frac{1}{3}-3a\gamma}|\ln\epsilon|^{-3\gamma},\end{array}$
(3.8)

and

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,+\infty)}\int_{\mathbf{R}}(\psi_{y}^{2}+\zeta_{y}^{2})dy+\int^{+\infty}_{0}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}(\psi_{y}^{2}+\zeta_{y}^{2})+\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}(\psi_{yy}^{2}+\zeta_{yy}^{2})\Big{]}dyd\tau\leq\
\epsilon^{\frac{1}{9}},\end{array}$ (3.9)

where $a$ is given by (1.14).

In what follows, the analysis is always carried out under the a priori
assumptions

$\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\|\phi(\cdot,\tau)\|_{L^{\infty}}\leq\epsilon^{a},\quad\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\|\zeta(\cdot,\tau)\|_{L^{\infty}}\leq\epsilon^{(\gamma-1)a},$
(3.10)
$\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\|\psi(\cdot,\tau)\|_{L^{\infty}}\leq\epsilon^{a},\quad\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\|(\psi_{y},\zeta_{y})(\cdot,\tau)\|\leq
1,$ (3.11)

where $a$ is given by (1.14), $[0,\tau_{1}(\epsilon)]$ is the time interval in
which the solution exists and $\tau_{1}(\epsilon)$ may depend on $\epsilon$.

Take

$\nu=\epsilon^{a}|\ln\epsilon|,\qquad\delta=\epsilon^{a},$ (3.12)

in the sequel. Then it follows that $\nu\geq C\epsilon^{a}$ with
$C\geq\max\\{2,\big{(}2e^{-\bar{S}}\big{)}^{\frac{1}{\gamma-1}}\\}$ if
$\epsilon\ll 1$. Under the a priori assumption (3.10), one can get

$\displaystyle\frac{\bar{\rho}}{2}\leq\rho\leq\frac{3\bar{\rho}}{2},\quad{\rm
and}\quad\frac{\bar{\theta}}{2}\leq\theta\leq\frac{3\bar{\theta}}{2}.$ (3.13)

In fact, if $\epsilon\ll 1$, then one has

$\rho=\bar{\rho}+\phi\geq\bar{\rho}-\|\phi\|_{L^{\infty}}\geq\bar{\rho}-\epsilon^{a}\geq\bar{\rho}-\frac{1}{2}\nu\geq\frac{\bar{\rho}}{2},$

and

$\rho=\bar{\rho}+\phi\leq\bar{\rho}+\|\phi\|_{L^{\infty}}\leq\bar{\rho}+\epsilon^{a}\leq\bar{\rho}+\frac{1}{2}\nu\leq\frac{3\bar{\rho}}{2}.$

Similarly, note that
$\bar{\theta}=\bar{\rho}^{\gamma-1}e^{\bar{S}}\geq\nu^{\gamma-1}e^{\bar{S}}$
by the definition of the rarefaction wave profile defined in (2.25), it holds
that

$\theta=\bar{\theta}+\zeta\geq\bar{\theta}-\|\zeta\|_{L^{\infty}}\geq\bar{\theta}-\epsilon^{a(\gamma-1)}\geq\bar{\theta}-\frac{e^{\bar{S}}}{2}\nu^{\gamma-1}\geq\bar{\theta}-\frac{\bar{\theta}}{2}=\frac{\bar{\theta}}{2},$

and

$\theta=\bar{\theta}+\zeta\leq\bar{\theta}+\|\zeta\|_{L^{\infty}}\leq\bar{\theta}+\epsilon^{a(\gamma-1)}\leq\bar{\theta}+\frac{e^{\bar{S}}}{2}\nu^{\gamma-1}\leq\bar{\theta}+\frac{\bar{\theta}}{2}=\frac{3\bar{\theta}}{2}.$

Since the proof for the local existence of the solution to
$(\ref{mass})-(\ref{g})$ is standard, we omit it for brevity. Note that in
order to get the convergence rate of the local solution with respect to
$\epsilon$ as in (3.10), the local existence time interval, denoted by
$[0,\tau_{0}]$ where $\tau_{0}$ may depend on $\epsilon$, that is,
$\tau_{0}=\tau_{0}(\epsilon)$. The next step for the proof of Theorem 3.1 is
to extend the local solution to the global solution in $[0,\infty)$ for small
but fixed viscosity coefficient and heat conduction coefficient $\epsilon$. To
do so, it is sufficient to show the following a priori estimates for fixed
$\epsilon$ with $0<\epsilon\ll 1$.

######  Lemma 3.2.

(A priori estimates) Let $(\phi,\psi,\zeta)\in\chi(0,\tau_{1}(\epsilon))$ be a
solution to the problem $(\ref{mass})-(\ref{g})$, where $\tau_{1}(\epsilon)$
is the maximum existence time of the solution satisfying a priori assumptions
(3.10)-(3.11). Then there exists a positive constant $\epsilon_{2}$ such that
if $0<\epsilon\leq\epsilon_{2}$, then

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}(\tau,y)dy\\\\[8.53581pt]
&\displaystyle+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}+\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}\Big{]}dyd\tau\leq\epsilon^{\frac{1}{3}},\end{array}$
(3.14)
$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}\phi_{y}^{2}dy+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau\leq\epsilon^{\frac{1}{3}-3a\gamma}|\ln\epsilon|^{-3\gamma},\end{array}$
(3.15)

and

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}(\psi_{y}^{2}+\zeta_{y}^{2})dy+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}(\psi_{y}^{2}+\zeta_{y}^{2})+\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}(\psi_{yy}^{2}+\zeta_{yy}^{2})\Big{]}dyd\tau\leq\
\epsilon^{\frac{1}{9}}.\end{array}$ (3.16)

Proof of Lemma 3.2: The proof of Lemma 3.2 consists of the following steps.
Step 1.  First, as in [16], one can define the entropy-entropy flux pair
$(\eta,q)$ as

$\left\\{\begin{array}[]{l}\displaystyle\eta=-\bar{\theta}\\{\rho
S-\bar{\rho}\bar{S}-\nabla_{\textbf{X}}(\rho
S)|_{\textbf{X}=\overline{\textbf{X}}}\cdot(\textbf{X}-\overline{\textbf{X}})\\},\\\\[5.69054pt]
\displaystyle q=-\bar{\theta}\\{\rho
uS-\bar{\rho}\bar{u}\bar{S}-\nabla_{\textbf{X}}(\rho
S)|_{\textbf{X}=\overline{\textbf{X}}}\cdot(\textbf{Y}-\overline{\textbf{Y}})\\},\end{array}\right.$
(3.17)

where

$\begin{array}[]{ll}\textbf{X}=\big{(}\rho,\rho
u,\rho(\frac{1}{2}|u|^{2}+\theta)\big{)}^{t},\\\ \textbf{Y}=\big{(}\rho u,\rho
u^{2}+(\gamma-1)\rho\theta,\rho
u(\frac{1}{2}|u|^{2}+\gamma\theta)\big{)}^{t}.\end{array}$ (3.18)

Since

$\begin{array}[]{ll}(\rho S)_{\rho}=S+\frac{|u|^{2}}{2\theta}-\gamma,\\\ (\rho
S)_{m}=-\frac{u}{\theta},\\\ (\rho S)_{E}=\frac{1}{\theta},\end{array}$ (3.19)

with $m=\rho u$ and $E=\rho(\theta+\frac{|u|^{2}}{2})$, we get

$\left\\{\begin{array}[]{l}\displaystyle\eta=\rho\theta-\bar{\theta}\rho
S+\rho[(\bar{S}-\gamma)\bar{\theta}+\frac{1}{2}|u-\bar{u}|^{2}]+(\gamma-1)\bar{\rho}\bar{\theta},\\\\[5.69054pt]
~{}~{}~{}=\displaystyle(\gamma-1)\rho\bar{\theta}\Phi(\frac{\bar{\rho}}{\rho})+\frac{\rho\psi^{2}}{2}+\rho\bar{\theta}\Phi(\frac{\theta}{\bar{\theta}}),\\\
\displaystyle
q=u\eta+(\gamma-1)(u-\bar{u})(\rho\theta-\bar{\rho}\bar{\theta}),\end{array}\right.$
(3.20)

where

$\Phi(\eta)=\eta-\ln\eta-1.$ (3.21)

Direct computations yield

$\eta_{\tau}+q_{y}+{\bar{u}}_{y}H=\psi(\mu(\theta)u_{y})_{y}+\frac{\zeta}{\theta}\mu(\theta)u_{y}^{2}+\frac{\zeta}{\theta}(\kappa(\theta)\theta_{y})_{y},$
(3.22)

where

$\begin{array}[]{ll}H=\displaystyle\rho(u-\bar{u})^{2}+(\gamma-1)\rho\bar{\theta}\Phi(\frac{\theta}{\bar{\theta}})+(\gamma-1)^{2}\rho\bar{\theta}\Phi(\frac{\bar{\rho}}{\rho})+\sqrt{\frac{\gamma-1}{\gamma}}\bar{\theta}^{\frac{1}{2}}\rho(u-\bar{u})\big{(}(\gamma-1)\log\frac{\bar{\rho}}{\rho}+\log\frac{\theta}{\bar{\theta}}\big{)}\\\\[8.53581pt]
\quad\displaystyle\geq(1-\beta)\rho(u-\bar{u})^{2}+(\gamma-1)\rho\bar{\theta}\big{[}\Phi(\frac{\theta}{\bar{\theta}})+(\gamma-1)\Phi(\frac{\bar{\rho}}{\rho})-\frac{1}{4\beta\gamma}\big{(}(\gamma-1)\log\frac{\bar{\rho}}{\rho}+\log\frac{\theta}{\bar{\theta}}\big{)}^{2}\big{]}\\\\[8.53581pt]
\end{array}$ (3.23)

with $0<\beta<1$ is the positive constant to be determined. Let

$x_{1}=\frac{\theta}{\bar{\theta}},\qquad x_{2}=\frac{\bar{\rho}}{\rho},$

under the a priori assumptions (3.10), one has $x_{1},x_{2}\sim 1$ as
$\epsilon\rightarrow 0.$ Consider the following function

$f^{\beta}(x_{1},x_{2})=x_{1}-\log x_{1}-1+(\gamma-1)(x_{2}-\log
x_{2}-1)-\frac{1}{4\beta\gamma}\big{(}(\gamma-1)\log x_{2}+\log
x_{1}\big{)}^{2}.$

It is easy to check that

$f^{\beta}(1,1)=f^{\beta}_{x_{1}}(1,1)=f^{\beta}_{x_{2}}(1,1)=0,$

and the Hessian matrix of $f^{\beta}$ at point (1,1) is

$\nabla^{2}f^{\beta}(1,1)=\left(\begin{array}[]{cc}\displaystyle
1-\frac{1}{2\beta\gamma}&\quad\displaystyle-\frac{\gamma-1}{2\beta\gamma}\\\\[8.53581pt]
\displaystyle-\frac{\gamma-1}{2\beta\gamma}&\quad\displaystyle(\gamma-1)(1-\frac{\gamma-1}{2\beta\gamma})\end{array}\right).$

Thus the determinant of $\nabla^{2}f^{\beta}(1,1)$ is

$\det\nabla^{2}f^{\beta}(1,1)=(\gamma-1)(1-\frac{1}{2\beta}).$

Take $\beta=\frac{3}{4}$, it is easy to see that $\nabla^{2}f^{\beta}(1,1)$ is
definitely positive near the point $(1,1)$. So one has

$f^{\beta}(x_{1},x_{2})\geq C_{\beta}(x_{1}^{2}+x_{2}^{2}),\qquad{\rm
as}~{}~{}x_{1},x_{2}\sim 1.$

Therefore, under the a priori assumptions (3.10) and take $\beta=\frac{3}{4}$
in (3.23), one can get

$H\geq
C~{}[\bar{\rho}\psi^{2}+\frac{\bar{\theta}}{\bar{\rho}}\phi^{2}+\frac{\bar{\rho}}{\bar{\theta}}\zeta^{2}].$
(3.24)

Similarly, by the facts $\Phi(1)=\Phi^{\prime}(1)=0$, and
$\Phi^{\prime\prime}(1)=1>0$, one has

$\eta\geq
C~{}[\bar{\rho}\psi^{2}+\frac{\bar{\theta}}{\bar{\rho}}\phi^{2}+\frac{\bar{\rho}}{\bar{\theta}}\zeta^{2}]$

under a a priori assumptions (3.10). The right hand-side of (3.22) can be
rewritten as

$\begin{array}[]{ll}\displaystyle\big{[}\mu(\theta)\psi\psi_{y}+\frac{\kappa(\theta)}{\theta}\displaystyle\zeta\zeta_{y}\big{]}_{y}-\mu(\theta)\frac{\bar{\theta}}{\theta}\displaystyle\psi_{y}^{2}-\kappa(\theta)\frac{\bar{\theta}}{\theta^{2}}\displaystyle\zeta_{y}^{2}+\mu(\theta)\psi\bar{u}_{yy}+\frac{\kappa(\theta)}{\theta}\displaystyle\zeta\bar{\theta}_{yy}\\\\[8.53581pt]
\qquad+\displaystyle\frac{\mu(\theta)}{\theta}\zeta(2\psi_{y}\bar{u}_{y}+\bar{u}_{y}^{2})+\frac{\kappa(\theta)}{\theta^{2}}\displaystyle\zeta\zeta_{y}\bar{\theta}_{y}+\psi\mu(\theta)_{y}\bar{u}_{y}+\frac{1}{\theta}\displaystyle\kappa(\theta)_{y}\zeta\bar{\theta}_{y}.\end{array}$
(3.25)

Then integrating the equation (3.22) over ${\mathbf{R}}^{1}\times[0,\tau]$ and
using (3.13), (3.22)-(3.25) imply

$\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}(\tau,y)dy\\\\[8.53581pt]
&\displaystyle+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}+\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}\Big{]}dyd\tau\\\\[11.38109pt]
\leq&\displaystyle
C\int^{\tau}_{0}\int_{\mathbf{R}}|\bar{\theta}^{\alpha-1}\zeta||(\bar{\theta}_{yy},\bar{u}_{y}^{2})|+|\bar{\theta}^{\alpha}\psi\bar{u}_{yy}|+|\theta^{\alpha-1/2}\psi\bar{u}_{y}^{2}|+|\bar{u}_{y}(\bar{\theta}^{\alpha-1}\zeta\psi_{y},\bar{\theta}^{\alpha-\frac{3}{2}}\zeta\zeta_{y},\bar{\theta}^{\alpha-1}\psi\zeta_{y})|dyd\tau\\\\[11.38109pt]
:=&\displaystyle\sum_{i=1}^{4}I_{i}.\end{array}$ (3.26)

By Sobolev inequality and Lemma 2.3, we can obtain

$\begin{array}[]{ll}I_{1}&=\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}|\bar{\theta}^{\alpha-1}\zeta|~{}|(\bar{\theta}_{yy},\bar{u}_{y}^{2})|dyd\tau\displaystyle\leq
C\nu^{1-\gamma}\int^{\tau}_{0}\|(\bar{\theta}_{yy},\bar{u}_{y}^{2})\|_{L^{1}}\|\zeta\|^{1/2}\|\zeta_{y}\|^{1/2}d\tau\\\\[11.38109pt]
&\displaystyle\leq
C\nu^{-\frac{(4+\alpha)(\gamma-1)}{4}}\int^{\tau}_{0}\frac{1}{\tau+\delta/\epsilon}\|\zeta\|^{1/2}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{1/2}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{2}d\tau+C\nu^{-\frac{(4+\alpha)(\gamma-1)}{3}}\int^{\tau}_{0}\|\zeta\|^{\frac{2}{3}}\big{(}\frac{1}{\tau+\delta/\epsilon}\big{)}^{\frac{4}{3}}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{2}d\tau+C\nu^{-\frac{(4+\alpha)(\gamma-1)+2}{3}}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\sqrt{\bar{\rho}^{2-\gamma}}\zeta\|^{2/3}\int^{\tau}_{0}\big{(}\frac{1}{\tau+\delta/\epsilon}\big{)}^{\frac{4}{3}}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{2}d\tau+\frac{1}{16}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\sqrt{\bar{\rho}^{2-\gamma}}\zeta\|^{2}+C\nu^{-\frac{(4+\alpha)(\gamma-1)+2}{2}}(\frac{\epsilon}{\delta})^{1/2}\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{2}d\tau+\frac{1}{16}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\sqrt{\bar{\rho}^{2-\gamma}}\zeta\|^{2}+\epsilon^{\frac{1}{3}},\end{array}$
(3.27)

where we used the fact that

$\displaystyle
C\nu^{-\frac{(4+\alpha)(\gamma-1)+2}{2}}(\frac{\epsilon}{\delta})^{1/2}=C\epsilon^{\frac{1-a(4\gamma-1+\alpha(\gamma-1))}{2}}|\ln\epsilon|^{-\frac{(4+\alpha)(\gamma-1)+2}{2}}\leq\epsilon^{\frac{1}{3}},\qquad{\rm
if}~{}~{}\epsilon\ll 1.$

Similarly, it holds that

$\begin{array}[]{ll}I_{2}&=\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}|\bar{\theta}^{\alpha}\psi\bar{u}_{yy}|dyd\tau\displaystyle\leq
C\int^{\tau}_{0}\|\bar{u}_{yy}\|_{L^{1}}\|\psi\|^{1/2}\|\psi_{y}\|^{1/2}d\tau\\\\[11.38109pt]
&\displaystyle\leq
C\nu^{-\frac{\alpha(\gamma-1)}{4}}\int^{\tau}_{0}\frac{1}{\tau+\delta/\epsilon}\|\psi\|^{1/2}~{}\|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}\|^{1/2}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}\|^{2}d\tau+C\nu^{-\frac{\alpha(\gamma-1)}{3}}\int_{0}^{\tau}\big{(}\frac{1}{\tau+\delta/\epsilon}\big{)}^{\frac{4}{3}}\|\psi\|^{2/3}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}\|^{2}d\tau+C\nu^{-\frac{\alpha(\gamma-1)+1}{3}}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\sqrt{\bar{\rho}}\psi\|^{2/3}\int_{0}^{\tau}\big{(}\frac{1}{\tau+\delta/\epsilon}\big{)}^{\frac{4}{3}}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}\|^{2}d\tau+\frac{1}{16}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\sqrt{\bar{\rho}}\psi\|^{2}+C\nu^{-\frac{\alpha(\gamma-1)+1}{2}}(\frac{\epsilon}{\delta})^{1/2}\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}\|^{2}d\tau+\frac{1}{16}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\sqrt{\bar{\rho}}\psi\|^{2}+\epsilon^{\frac{1}{3}},\end{array}$
(3.28)

where we used the fact that

$\displaystyle
C\nu^{-\frac{\alpha(\gamma-1)+1}{2}}(\frac{\epsilon}{\delta})^{1/2}=C\epsilon^{\frac{1-a(\alpha(\gamma-1)+2)}{2}}|\ln\epsilon|^{-\frac{\alpha(\gamma-1)+1}{2}}\leq\epsilon^{\frac{1}{3}},\qquad{\rm
if}~{}~{}\epsilon\ll 1.$

By using Lemma 2.3 yields

$\begin{array}[]{ll}I_{3}&=\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}|\bar{\theta}^{\alpha-1/2}\psi\bar{u}_{y}^{2}|dyd\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{u}_{y}\bar{\rho}\psi^{2}dyd\tau+C\nu^{-\gamma}\int^{\tau}_{0}\|\bar{u}_{y}\|^{3}_{L^{3}}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{u}_{y}\bar{\rho}\psi^{2}dyd\tau+C\nu^{-\gamma}\int^{\tau}_{0}\big{(}\frac{1}{\tau+\delta/\epsilon}\big{)}^{2}d\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{u}_{y}\bar{\rho}\psi^{2}dyd\tau+C\nu^{-\gamma}(\frac{\varepsilon}{\delta}).\end{array}$
(3.29)

And

$\begin{array}[]{ll}I_{4}&\leq\displaystyle
C\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}^{-\frac{1}{2}}\bar{\theta}^{\frac{\alpha-1}{2}}|\bar{u}_{y}|\big{(}|\sqrt{\bar{\rho}^{2-\gamma}}\zeta||\bar{\theta}^{\frac{\alpha}{2}}\psi_{y}|,|\sqrt{\bar{\rho}^{2-\gamma}}\zeta||\bar{\theta}^{\frac{\alpha-1}{2}}\zeta_{y}|,|\sqrt{\bar{\rho}}\psi||\bar{\theta}^{\frac{\alpha-1}{2}}\zeta_{y}|\big{)}dyd\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\int_{\mathbf{R}}\big{(}\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}\big{)}dyd\tau+C\frac{\epsilon}{\nu^{\gamma}\delta}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{u}_{y}\big{(}\bar{\rho}^{2-\gamma}\zeta^{2}+\bar{\rho}\psi^{2}\big{)}dyd\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{16}\int^{\tau}_{0}\int_{\mathbf{R}}\big{(}\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}+\bar{u}_{y}\bar{\rho}^{2-\gamma}\zeta^{2}+\bar{u}_{y}\bar{\rho}\psi^{2}\big{)}dyd\tau,\end{array}$
(3.30)

where we used the fact that

$\displaystyle
C\frac{\epsilon}{\nu^{\gamma}\delta}=C\epsilon^{1-a-a\gamma}|\ln\epsilon|^{-\gamma}\leq
C\epsilon^{\frac{1}{2}}|\ln\epsilon|^{-\gamma}\leq\frac{1}{16},\qquad{\rm
if}~{}~{}\epsilon\ll 1.$

Combining (3.27)-(3.30) and recalling (3.12) yield that

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}(\tau,y)dy\\\\[8.53581pt]
&\displaystyle+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}+\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}\Big{]}dyd\tau\leq\epsilon^{\frac{1}{3}}.\end{array}$
(3.31)

Step 2.  Next we derive the estimation of $\phi_{y}$. Differentiating
$(\ref{mass})_{1}$ with respect to $y$ and then multiplying the resulted
equation by $\mu^{2}(\bar{\theta})\phi_{y}/\rho^{3}$ imply that

$\begin{array}[]{ll}&\displaystyle(\mu^{2}(\bar{\theta})\frac{\phi_{y}^{2}}{2\rho^{3}})_{\tau}+(\mu^{2}(\bar{\theta})\frac{u\phi_{y}^{2}}{2\rho^{3}})_{y}+\mu^{2}(\bar{\theta})\frac{\psi_{yy}\phi_{y}}{\rho^{2}}\\\\[11.38109pt]
=&\displaystyle-\mu^{2}(\bar{\theta})\frac{\phi_{y}}{\rho^{3}}({\bar{u}}_{yy}\phi+{\bar{\rho}}_{yy}\psi+2{\bar{\rho}}_{y}\psi_{y})+\frac{\phi_{y}^{2}}{2\rho^{3}}(\mu^{2}(\bar{\theta})_{\tau}+\mu^{2}(\bar{\theta})_{y}u).\end{array}$
(3.32)

Multiplying $(\ref{mass})_{2}$ by $\mu(\bar{\theta})\phi_{y}/\rho^{2}$ gives

$\begin{array}[]{ll}&\displaystyle(\mu(\bar{\theta})\frac{\psi\phi_{y}}{\rho})_{\tau}-(\mu(\bar{\theta})\frac{\psi\phi_{\tau}}{\rho})_{y}-\mu(\bar{\theta})\psi_{y}^{2}+(\gamma-1)\Big{(}\mu(\bar{\theta})\frac{\theta\phi_{y}^{2}}{\rho^{2}}+\mu(\bar{\theta})\frac{\phi_{y}\zeta_{y}}{\rho}\Big{)}\\\\[5.69054pt]
&\displaystyle-\mu^{2}(\bar{\theta})\frac{\psi_{yy}\phi_{y}}{\rho^{2}}+\mu(\bar{\theta})\Big{(}-{\bar{u}}_{y}\frac{\psi_{y}\phi}{\rho}+{\bar{\rho}}_{y}{\bar{u}}_{y}\frac{\psi\phi}{\rho^{2}}-\bar{\rho}{\bar{u}}_{y}\frac{\psi\phi_{y}}{\rho^{2}}\Big{)}+\mu(\bar{\theta})g\frac{\phi_{y}}{\rho^{2}}=\mu(\bar{\theta})\mu(\bar{\theta})_{y}\frac{u_{y}\phi_{y}}{\rho^{2}}\\\\[11.38109pt]
&\displaystyle+\mu(\bar{\theta})_{\tau}\frac{\psi\phi_{y}}{\rho}-\mu(\bar{\theta})_{y}\frac{\psi\phi_{\tau}}{\rho}-\mu(\bar{\theta})\frac{\psi^{2}}{\rho^{2}}\bar{\rho}_{y}(\phi_{y}+\bar{\rho}_{y})+((\mu(\theta)-\mu(\bar{\theta}))u_{y})_{y}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}.\end{array}$
(3.33)

Combining $(\ref{m})$ and $(\ref{v})$ together, then integrating the resulted
equation over ${\mathbf{R}}^{1}\times[0,\tau]$ imply

$\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\Big{(}\mu^{2}(\bar{\theta})\frac{\phi_{y}^{2}}{2\rho^{3}}+\mu(\bar{\theta})\frac{\psi\phi_{y}}{\rho}\Big{)}dy+(\gamma-1)\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\mu(\bar{\theta})\frac{\theta\phi_{y}^{2}}{\rho^{2}}+\mu(\bar{\theta})\frac{\phi_{y}\zeta_{y}}{\rho}\Big{)}dyd\tau\\\\[14.22636pt]
=&\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}\mu(\bar{\theta})\psi_{y}^{2}+\mu(\bar{\theta})\Big{(}{\bar{u}}_{y}\frac{\psi_{y}\phi}{\rho}-{\bar{\rho}}_{y}{\bar{u}}_{y}\frac{\psi\phi}{\rho^{2}}+\bar{\rho}{\bar{u}}_{y}\frac{\psi\phi_{y}}{\rho^{2}}\Big{)}dyd\tau\\\\[14.22636pt]
&\displaystyle-\mu^{2}(\bar{\theta})\frac{\phi_{y}}{\rho^{3}}\Big{(}{\bar{u}}_{yy}\phi+{\bar{\rho}}_{yy}\psi+2{\bar{\rho}}_{y}\psi_{y}\Big{)}-\mu(\bar{\theta})g\frac{\phi_{y}}{\rho^{2}}\Big{\\}}dyd\tau\\\\[14.22636pt]
&\displaystyle+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}\frac{\phi_{y}^{2}}{2\rho^{3}}\Big{(}\mu^{2}(\bar{\theta})_{\tau}+\mu^{2}(\bar{\theta})_{y}u\Big{)}+\mu(\bar{\theta})\mu(\bar{\theta})_{y}\frac{u_{y}\phi_{y}}{\rho^{2}}+\mu(\bar{\theta})_{\tau}\frac{\psi\phi_{y}}{\rho}-\mu(\bar{\theta})_{y}\frac{\psi\phi_{\tau}}{\rho}\\\\[14.22636pt]
&\displaystyle-\mu(\bar{\theta})\bar{\rho}_{y}\frac{\psi^{2}}{\rho^{2}}(\phi_{y}+\bar{\rho}_{y})+\big{(}(\mu(\theta)-\mu(\bar{\theta}))u_{y}\big{)}_{y}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}\Big{\\}}dyd\tau.\end{array}$
(3.34)

Combining (3.31) with (3.34), and using the equation $\eqref{mass}_{1}$ and
the fact that

$\begin{array}[]{ll}&\displaystyle\Big{(}\big{(}\mu(\theta)-\mu(\bar{\theta})\big{)}u_{y}\Big{)}_{y}\\\\[11.38109pt]
=&\displaystyle\big{(}(\mu(\theta)-\mu(\bar{\theta})\big{)}(\psi_{yy}+\bar{u}_{yy})+\alpha(\theta^{\alpha-1}\theta_{y}-\bar{\theta}^{\alpha-1}\bar{\theta}_{y})(\psi_{y}+\bar{u}_{y})\\\\[8.53581pt]
=&\displaystyle\big{(}(\mu(\theta)-\mu(\bar{\theta})\big{)}(\psi_{yy}+\bar{u}_{yy})+\alpha\big{[}\theta^{\alpha-1}\zeta_{y}\psi_{y}+\theta^{\alpha-1}\zeta_{y}\bar{u}_{y}\\\\[8.53581pt]
&\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(\theta^{\alpha-1}-\bar{\theta}^{\alpha-1})\bar{\theta}_{y}\psi_{y}+(\theta^{\alpha-1}-\bar{\theta}^{\alpha-1})\bar{\theta}_{y}\bar{u}_{y}\big{]},\end{array}$
(3.35)

it holds that

$\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\Big{(}\mu^{2}(\bar{\theta})\frac{\phi_{y}^{2}}{\bar{\rho}^{3}}+\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}(\tau,y)dy\\\\[14.22636pt]
+&\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}+\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}+\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}\Big{]}dyd\tau\\\\[11.38109pt]
\leq&\displaystyle
C|\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\mu(\bar{\theta})\bar{u}_{y}\frac{\psi_{y}\phi}{\rho}+\mu(\bar{\theta})\bar{u}_{y}\bar{\rho}\frac{\psi\phi_{y}}{\rho^{2}}-2\mu^{2}(\bar{\theta})\frac{\phi_{y}}{\rho^{3}}\bar{\rho}_{y}\psi_{y}\Big{)}dyd\tau|\\\\[14.22636pt]
&\displaystyle+C|\int_{0}^{\tau}\int_{\mathbf{R}}\mu^{2}(\bar{\theta})\frac{\phi_{y}}{\rho^{3}}\bar{u}_{yy}\phi+\mu^{2}(\bar{\theta})\frac{\phi_{y}}{\rho^{3}}\bar{\rho}_{yy}\psi+\mu(\theta)\frac{\phi_{y}}{\rho^{2}}\big{(}\mu(\theta)-\mu(\bar{\theta})\big{)}\bar{u}_{yy}dyd\tau|\\\\[14.22636pt]
&\displaystyle+C|\int_{0}^{\tau}\int_{\mathbf{R}}\mu(\bar{\theta})g\frac{\phi_{y}}{\rho^{2}}dyd\tau|+C|\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\phi_{y}^{2}}{2\rho^{3}}\big{(}\mu^{2}(\bar{\theta})_{\tau}+\mu^{2}(\bar{\theta})_{y}u\big{)}dyd\tau|\\\\[14.22636pt]
&\displaystyle+C|\int_{0}^{\tau}\int_{\mathbf{R}}\Big{\\{}\mu(\bar{\theta})\bar{\rho}_{y}\bar{u}_{y}\frac{\psi\phi}{\rho^{2}}+\mu(\bar{\theta})\mu(\bar{\theta})_{y}\frac{\psi_{y}\phi_{y}}{\rho^{2}}-\mu(\bar{\theta})_{y}\frac{\psi(\rho\psi_{y}+u\phi_{y}+\bar{u}_{y}\phi+\bar{\rho}_{y}\psi)}{\rho}\\\\[14.22636pt]
&\displaystyle-\mu(\bar{\theta})\bar{\rho}^{2}_{y}\frac{\psi^{2}}{\rho^{2}}+\alpha\big{(}\theta^{\alpha-1}\zeta_{y}\bar{u}_{y}+(\theta^{\alpha-1}-\bar{\theta}^{\alpha-1})\bar{\theta}_{y}\psi_{y}\big{)}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}\Big{\\}}~{}dyd\tau|\\\\[14.22636pt]
&\displaystyle+C|\int_{0}^{\tau}\int_{\mathbf{R}}\Big{(}\mu(\bar{\theta})_{\tau}\frac{\psi\phi_{y}}{\rho}-\mu(\bar{\theta})\bar{\rho}_{y}\frac{\psi^{2}}{\rho^{2}}\phi_{y}\Big{)}~{}dyd\tau|\\\\[14.22636pt]
&\displaystyle+C|\int_{0}^{\tau}\int_{\mathbf{R}}\Big{(}\mu(\bar{\theta})\mu(\bar{\theta})_{y}\frac{\bar{u}_{y}\phi_{y}}{\rho^{2}}+\alpha(\theta^{\alpha-1}-\bar{\theta}^{\alpha-1})\bar{\theta}_{y}\bar{u}_{y}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}\Big{)}~{}dyd\tau|\\\\[14.22636pt]
&\displaystyle+C|\int_{0}^{\tau}\int_{\mathbf{R}}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}\big{(}\mu(\theta)-\mu(\bar{\theta})\big{)}\psi_{yy}dyd\tau|+C|\int_{0}^{\tau}\int_{\mathbf{R}}\theta^{\alpha-1}\zeta_{y}\psi_{y}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}~{}dyd\tau|\\\\[14.22636pt]
&\displaystyle+C\epsilon^{\frac{1}{3}}:=\sum_{i=1}^{9}J_{i}+C\epsilon^{\frac{1}{3}}.\end{array}$
(3.36)

The terms on the right-hand side of (3.36) will be estimated one by one as
follows. By Lemma 2.3, (3.13), and Cauchy inequality, it holds that

$\begin{array}[]{ll}J_{1}&\leq\displaystyle
C|\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}\bar{\rho}^{\frac{\alpha(\gamma-1)-\gamma}{2}}\bar{u}_{y}\big{(}|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}||\sqrt{\bar{\rho}^{\gamma-2}}\phi|,|\sqrt{\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}}\phi_{y}||\sqrt{\bar{\rho}}\psi|\big{)}\\\\[8.53581pt]
&\displaystyle\quad+\bar{\rho}^{\alpha(\gamma-1)-\gamma}\bar{u}_{y}|\sqrt{\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}}\phi_{y}||\sqrt{\bar{\theta}^{\alpha}}\psi_{y}|\Big{\\}}dyd\tau|\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}(\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}+\bar{\theta}^{\alpha}\psi^{2}_{y})~{}dyd\tau+C\frac{\epsilon}{\nu^{\gamma}\delta}\int_{0}^{\tau}\int_{\mathbf{R}}\bar{u}_{y}(\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2})~{}dyd\tau,\end{array}$
(3.37)

Recalling (2.16) from Lemma 2.1 and the fact $(i)$ in Lemma 2.3, one can
arrive at

$\begin{array}[]{ll}|\bar{\rho}_{xx}|\leq
C(\bar{\rho}^{\frac{3-\gamma}{2}}\displaystyle\frac{\bar{u}_{x}}{\delta}+\bar{\rho}^{2-\gamma}\bar{u}^{2}_{x}).\end{array}$
(3.38)

Thus one has

$\begin{array}[]{ll}J_{2}&\leq\displaystyle
C|\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\rho}^{\frac{3\alpha(\gamma-1)-3\gamma}{2}}\frac{\epsilon}{\delta}\bar{u}_{y}|\sqrt{\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}}\phi_{y}|(|\sqrt{\bar{\rho}^{\gamma-2}}\phi|+|\sqrt{\bar{\rho}}\psi|+|\sqrt{\bar{\rho}^{2-\gamma}}\zeta|)dyd\tau|\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}~{}dyd\tau+C\big{(}\frac{\epsilon}{\nu^{\gamma}\delta}\big{)}^{3}\int_{0}^{\tau}\int_{\mathbf{R}}\bar{u}_{y}(\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2})~{}dyd\tau.\end{array}$
(3.39)

And

$\begin{array}[]{ll}J_{3}&=\displaystyle
C|\int^{\tau}_{0}\int_{\mathbf{R}}\mu(\bar{\theta})g\frac{\phi_{y}}{\rho^{2}}dyd\tau|\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}~{}dyd\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-2}g^{2}dyd\tau.\end{array}$
(3.40)

Recalling (3.6), (3.13) and $(i)$ in Lemma 2.3, one can get

$\begin{array}[]{ll}|g|&\leq\displaystyle
C\big{(}\bar{\theta}^{\alpha}|\bar{u}_{yy}|+|\bar{\rho}\bar{u}_{y}\psi|+|\bar{\rho}_{y}\zeta|+|\bar{\rho}_{y}\bar{\rho}^{\gamma-2}\phi|\big{)}\\\\[5.69054pt]
&\displaystyle\leq
C\big{(}\bar{\theta}^{\alpha}|\bar{u}_{yy}|+\bar{u}_{y}(|\bar{\rho}\psi|+|\bar{\rho}^{\frac{3-\gamma}{2}}\zeta|+|\bar{\rho}^{\frac{\gamma-1}{2}}\phi|)\big{)}.\end{array}$
(3.41)

Thus the last term in (3.40) can be estimated by

$\begin{array}[]{ll}&\displaystyle|\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-2}g^{2}dyd\tau|\\\\[5.69054pt]
&\displaystyle\leq
C\nu^{-1-\gamma}\int_{0}^{\tau}\|\bar{u}_{yy}\|^{2}d\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-1}\bar{u}^{2}_{y}(\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2})dyd\tau\\\\[8.53581pt]
&\displaystyle\leq
C\nu^{-1-\gamma}\big{(}\frac{\epsilon}{\delta}\big{)}^{2}+C\frac{\epsilon}{\nu^{\gamma}\delta}\int_{0}^{\tau}\int_{\mathbf{R}}\bar{u}_{y}(\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2})dyd\tau.\end{array}$
(3.42)

By Cauchy inequality, it holds that

$\begin{array}[]{ll}J_{4}&\displaystyle\leq
C|\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}~{}\bar{\theta}^{\alpha-\frac{3}{2}}\bar{\rho}^{-1}\bar{u}_{y}dyd\tau|\leq
C\nu^{\frac{1-3\gamma}{2}}\frac{\epsilon}{\delta}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau\qquad{\rm
if}~{}~{}\epsilon\ll 1.\end{array}$ (3.43)

Recalling (3.13) and $(i)$ in Lemma 2.3, one can get

$\begin{array}[]{ll}J_{5}&\displaystyle\leq
C|\int_{0}^{\tau}\int_{\mathbf{R}}\Big{\\{}\bar{\rho}^{\alpha(\gamma-1)-\gamma}\bar{u}_{y}|\sqrt{\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}}\phi_{y}|~{}|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}|+\bar{\rho}^{\alpha(\gamma-1)-\gamma}\bar{u}_{y}|\sqrt{\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}}\phi_{y}|~{}|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}|\\\\[8.53581pt]
&\displaystyle\quad+\bar{\rho}^{\frac{\alpha(\gamma-1)-\gamma}{2}}\bar{u}_{y}|\sqrt{\bar{\rho}}\psi|~{}|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}|+\bar{\rho}^{\frac{\alpha(\gamma-1)}{2}-\gamma}\bar{u}_{y}|\sqrt{\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}}\phi_{y}|~{}|\sqrt{\bar{\rho}}\psi|\\\\[8.53581pt]
&\displaystyle\quad+\bar{\rho}^{\alpha(\gamma-1)-\gamma}\bar{u}_{y}(|\sqrt{\bar{u}_{y}\bar{\rho}}\psi|~{}|\sqrt{\bar{u}_{y}\bar{\rho}^{\gamma-2}}\phi|+\bar{u}_{y}\bar{\rho}\psi^{2})\Big{\\}}dyd\tau|\\\\[8.53581pt]
&\displaystyle\leq(\frac{1}{32}+C\frac{\epsilon}{\nu^{\gamma}\delta})\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi^{2}_{y}dyd\tau+C\frac{\epsilon}{\nu^{2\gamma}\delta}\int_{0}^{\tau}\int_{\mathbf{R}}(\bar{\theta}^{\alpha}\psi^{2}_{y}+\bar{\theta}^{\alpha-1}\zeta^{2}_{y}+\bar{u}_{y}\bar{\rho}\psi^{2}+\bar{u}_{y}\bar{\rho}^{\gamma-2}\phi^{2})dyd\tau.\end{array}$
(3.44)

Similarly, $J_{6}$ can be estimated as

$\begin{array}[]{ll}J_{6}&\displaystyle\leq
C|\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha/2-1}\bar{\rho}^{-1/2}\bar{u}_{y}|\sqrt{\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}}\phi_{y}|~{}|\sqrt{\bar{\rho}}\psi|dyd\tau|\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\nu^{1-2\gamma}\frac{\epsilon}{\delta}\int_{0}^{\tau}\int_{\mathbf{R}}\bar{u}_{y}\bar{\rho}\psi^{2}dyd\tau.\end{array}$
(3.45)

Recalling (3.13) and Lemma 2.3, one can get

$\begin{array}[]{ll}J_{7}&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{3\alpha-2}\bar{\rho}^{-2}\bar{u}_{y}^{4}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\nu^{-2\gamma}\int_{0}^{\tau}\|\bar{u}_{y}\|^{4}_{L^{4}}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C(\frac{\epsilon}{\nu^{\gamma}\delta})^{2}.\end{array}$
(3.46)

Recalling (3.11) and Cauchy inequality, it holds that

$\begin{array}[]{ll}J_{8}&\displaystyle=C|\int_{0}^{\tau}\int_{\mathbf{R}}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}\big{(}\mu(\theta)-\mu(\bar{\theta})\big{)}\psi_{yy}dyd\tau|\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}\zeta^{2}~{}\bar{\theta}^{2\alpha-3}\bar{\rho}^{-1}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\nu^{2-3\gamma}\sup_{[0,\tau_{1}(\epsilon)]}\|\zeta\|\sup_{[0,\tau_{1}(\epsilon)]}\|\zeta_{y}\|\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\nu^{1-3\gamma}\sup_{[0,\tau_{1}(\epsilon)]}\|\sqrt{\bar{\rho}^{2-\gamma}}\zeta\|\sup_{[0,\tau_{1}(\epsilon)]}\|\zeta_{y}\|\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{16}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\nu^{1-3\gamma}\epsilon^{\frac{1}{6}}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau.\end{array}$
(3.47)

Similarly,

$\begin{array}[]{ll}J_{9}\displaystyle=C|\alpha\int_{0}^{\tau}\int_{\mathbf{R}}\theta^{\alpha-1}\zeta_{y}\psi_{y}\mu(\bar{\theta})\frac{\phi_{y}}{\rho^{2}}~{}dyd\tau|\\\\[8.53581pt]
\displaystyle\leq
C\nu^{-\frac{\gamma}{2}}\int_{0}^{t}\|\sqrt{\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}}\phi_{y}\|\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|\|\psi_{y}\|^{1/2}\|\psi_{yy}\|^{1/2}d\tau\\\\[8.53581pt]
\displaystyle\leq
C\nu^{-\frac{2\gamma+\alpha(\gamma-1)}{4}}\int_{0}^{t}\|\sqrt{\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}}\phi_{y}\|\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|\|\psi_{y}\|^{1/2}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\psi_{yy}\|^{1/2}d\tau\\\\[8.53581pt]
\displaystyle\leq\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau\\\\[8.53581pt]
\displaystyle\quad+C\nu^{-\frac{2\gamma+3\alpha(\gamma-1)}{3}}|\ln\epsilon|^{\frac{1}{3}}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}}\phi_{y}\|^{4/3}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{4/3}\|\psi_{y}\|^{2/3}d\tau\\\\[8.53581pt]
\displaystyle\leq\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\frac{1}{16}\sup_{[0,\tau_{1}(\epsilon)]}\|\sqrt{\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}}\phi_{y}\|^{2}\\\\[8.53581pt]
\displaystyle\quad+C\nu^{-2\gamma-3\alpha(\gamma-1)}|\ln\epsilon|\Big{(}\int_{0}^{\tau}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{4/3}\|\psi_{y}\|^{2/3}d\tau\Big{)}^{3}\\\\[8.53581pt]
\displaystyle\leq\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\frac{1}{16}\sup_{[0,\tau_{1}(\epsilon)]}\|\sqrt{\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}}\phi_{y}\|^{2}\\\\[8.53581pt]
\displaystyle\quad+C\nu^{-2\gamma-4\alpha(\gamma-1)}|\ln\epsilon|\Big{(}\int_{0}^{\tau}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{2}+\|\sqrt{\bar{\theta}^{\alpha}}\psi_{y}\|^{2}d\tau\Big{)}^{3}\\\\[8.53581pt]
\displaystyle\leq\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\frac{1}{16}\sup_{[0,\tau_{1}(\epsilon)]}\|\sqrt{\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}}\phi_{y}\|^{2}+C\nu^{-2\gamma-4\alpha(\gamma-1)}|\ln\epsilon|\epsilon\\\\[8.53581pt]
\displaystyle\leq\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\frac{1}{16}\sup_{[0,\tau_{1}(\epsilon)]}\|\sqrt{\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}}\phi_{y}\|^{2}+\epsilon^{\frac{1}{3}},\end{array}$
(3.48)

where we have used the fact that

$C\nu^{-2\gamma-4\alpha(\gamma-1)}|\ln\epsilon|\epsilon=C\epsilon^{1-(2\gamma+4\alpha(\gamma-1))a}|\ln\epsilon|^{1-2\gamma-4\alpha(\gamma-1)}\leq\epsilon^{\frac{1}{3}},~{}~{}~{}{\rm
if}~{}~{}\epsilon\ll 1.$

Combining (3.37)-(3.48) yields that

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\Big{(}\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}\phi_{y}^{2}+\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}(\tau,y)dy\\\\[14.22636pt]
+&\displaystyle\int_{0}^{\tau_{1}(\epsilon)}\int_{\mathbf{R}}\Big{[}\bar{u}_{y}\Big{(}\bar{\rho}^{\gamma-2}\phi^{2}+\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2}\Big{)}+\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2}+\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}\Big{]}dyd\tau\\\\[11.38109pt]
\leq&\displaystyle
C\big{(}\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}+\nu^{1-3\gamma}\epsilon^{\frac{1}{6}}\big{)}\int_{0}^{\tau_{1}(\epsilon)}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\epsilon^{\frac{1}{3}}.\end{array}$
(3.49)

In particular, it holds that

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}\phi_{y}^{2}dy+\int_{0}^{\tau_{1}(\epsilon)}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau\\\\[11.38109pt]
\leq&\displaystyle
C\big{(}\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}+\nu^{1-3\gamma}\epsilon^{\frac{1}{6}}\big{)}\int_{0}^{\tau_{1}(\epsilon)}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\epsilon^{\frac{1}{3}}.\end{array}$
(3.50)

Step 3.  In the following, we estimate $\displaystyle\sup_{\tau}\|\psi_{y}\|$.
For this, rewrite $\eqref{mass}_{2}$ as

$\rho\psi_{\tau}+\rho
u\psi_{y}+(\gamma-1)\big{(}\theta\phi_{y}+\rho\zeta_{y}\big{)}-\mu(\theta)\psi_{yy}=-\bar{g}+\mu(\theta)_{y}u_{y},$
(3.51)

where

$\bar{g}=-\mu(\theta)\bar{u}_{yy}+\rho\psi\bar{u}_{y}+(\gamma-1)\big{(}\bar{\rho}_{y}\zeta-\frac{\bar{\rho}_{y}\bar{\theta}\phi}{\bar{\rho}}\big{)}.$
(3.52)

Multiplying (3.51) by $-\psi_{yy}/\rho$ gives

$\begin{array}[]{l}\displaystyle(\frac{\psi^{2}_{y}}{2})_{\tau}-(\psi_{\tau}\psi_{y}+u\frac{\psi^{2}_{y}}{2})_{y}+\frac{1}{2}\bar{u}_{y}\psi^{2}_{y}+\frac{\mu(\theta)}{\rho}\psi^{2}_{yy}\\\
\displaystyle=-\frac{\psi^{3}_{y}}{2}+(\gamma-1)\big{(}\theta\phi_{y}+\rho\zeta_{y}\big{)}\frac{\psi_{yy}}{\rho}+\bar{g}\frac{\psi_{yy}}{\rho}-\mu(\theta)_{y}u_{y}\frac{\psi_{yy}}{\rho}.\end{array}$
(3.53)

Integrating the above equation over ${\mathbf{R}}^{1}\times[0,\tau]$ yields

$\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\frac{\psi^{2}_{y}}{2}dy+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\frac{{\bar{u}}_{y}\psi_{y}^{2}}{2}+\frac{\mu(\theta)}{\rho}\psi_{yy}^{2}\Big{)}dyd\tau\\\\[8.53581pt]
=&\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}\bar{g}\frac{\psi_{yy}}{\rho}-\frac{\psi_{y}^{3}}{2}+(\gamma-1)\big{(}\theta\phi_{y}+\rho\zeta_{y}\big{)}\frac{\psi_{yy}}{\rho}-\mu(\theta)_{y}u_{y}\frac{\psi_{yy}}{\rho}\Big{\\}}dyd\tau.\end{array}$
(3.54)

First, it follows from (3.42) that

$\begin{array}[]{ll}&\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{g}\frac{\psi_{yy}}{\rho}dyd\tau\Big{|}\\\\[8.53581pt]
\leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi^{2}_{yy}dyd\tau+C\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\theta}^{-\alpha}\bar{\rho}^{-1}\bar{g}^{2}dyd\tau\Big{|}\\\\[8.53581pt]
\leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi^{2}_{yy}dyd\tau+C\nu^{-2\alpha(\gamma-1)}\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-2}g^{2}dyd\tau\\\\[8.53581pt]
\leq&\displaystyle\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi^{2}_{yy}dyd\tau+C\nu^{-2\alpha(\gamma-1)-\gamma}\frac{\epsilon}{\delta}\epsilon^{\frac{1}{3}}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi^{2}_{yy}dyd\tau+\epsilon^{\frac{1}{3}},\end{array}$
(3.55)

where in the last inequality we used the fact that

$C\nu^{-2\alpha(\gamma-1)-\gamma}\frac{\epsilon}{\delta}=C\epsilon^{1-a(\gamma+2\alpha(\gamma-1)+1)}|\ln\epsilon|^{-2\alpha(\gamma-1)-\gamma}\leq
C\epsilon^{\frac{1}{2}}|\ln\epsilon|^{-2\alpha(\gamma-1)-\gamma}\leq
1,\quad{\rm if}~{}~{}\epsilon\ll 1.$

Furthermore, we can compute that

$\begin{array}[]{ll}\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi_{y}^{3}}{2}dyd\tau\Big{|}\leq
C\int^{\tau}_{0}\|\psi_{yy}\|^{\frac{1}{2}}\|\psi_{y}\|^{\frac{5}{2}}d\tau\leq
C\nu^{-\alpha(\gamma-1)/4}\int^{\tau}_{0}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\psi_{yy}\|^{\frac{1}{2}}\|\psi_{y}\|^{\frac{5}{2}}\\\\[8.53581pt]
\qquad\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{-\alpha(\gamma-1)/3}\int^{\tau}_{0}\|\psi_{y}\|^{\frac{10}{3}}d\tau\\\\[8.53581pt]
\qquad\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{-4\alpha(\gamma-1)/3}\sup_{\tau\in[0,\tau_{1}]}\|\psi_{y}\|^{\frac{4}{3}}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\psi_{y}^{2}dyd\tau\\\\[8.53581pt]
\qquad\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{-4\alpha(\gamma-1)/3}\epsilon^{\frac{1}{3}},\end{array}$
(3.56)

where in the last inequality we have used the a priori assumptions (3.11). By
Cauchy inequality, one has

$\begin{array}[]{ll}&\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}(\theta\phi_{y}+\rho\zeta_{y})\frac{\psi_{yy}}{\rho}dyd\tau\Big{|}\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}+C\int^{\tau}_{0}\int_{\mathbf{R}}\big{(}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta^{2}_{y}\big{)}\bar{\rho}^{\gamma-2\alpha(\gamma-1)}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}+C\nu^{-2\alpha(\gamma-1)}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau+C\nu^{-2\alpha(\gamma-1)}\epsilon^{\frac{1}{3}}.\end{array}$
(3.57)

Finally, one has

$\begin{array}[]{ll}&\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}\mu(\theta)_{y}u_{y}\frac{\psi_{yy}}{\rho}dyd\tau\Big{|}\\\\[8.53581pt]
\leq&\displaystyle
C\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-1}|\psi_{yy}|\big{(}|\psi_{y}\zeta_{y}|+\bar{u}_{y}(\bar{\theta}^{1/2}|\psi_{y}|+|\zeta_{y}|)+\bar{\theta}^{1/2}\bar{u}^{2}_{y}\big{)}dyd\tau:=\sum_{i=1}^{3}K_{i}.\end{array}$
(3.58)

The terms $K_{i}~{}(i=1,2,3)$ will be estimated as follows,

$\begin{array}[]{ll}K_{1}&\displaystyle=C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-1}|\psi_{yy}\psi_{y}\zeta_{y}|dyd\tau\leq
C\nu^{-\gamma/2}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\psi_{yy}\|\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|\|\psi_{y}\|^{\frac{1}{2}}\|\psi_{yy}\|^{\frac{1}{2}}d\tau\\\\[8.53581pt]
&\displaystyle\leq
C\nu^{-\frac{2\gamma+\alpha(\gamma-1)}{4}}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\psi_{yy}\|^{\frac{3}{2}}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|\|\psi_{y}\|^{\frac{1}{2}}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}+C\nu^{-2\gamma-\alpha(\gamma-1)}\int_{0}^{\tau}\|\sqrt{\bar{\theta}^{\alpha-1}}\zeta_{y}\|^{4}\|\psi_{y}\|^{2}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{1-3\gamma-\alpha(\gamma-1)}\sup_{[0,\tau_{1}(\epsilon)]}\|\zeta_{y}\|^{2}\sup_{[0,\tau_{1}(\epsilon)]}\|\psi_{y}\|^{2}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\zeta^{2}_{y}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{1-3\gamma-\alpha(\gamma-1)}\epsilon^{\frac{1}{3}}\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\nu^{1-3\gamma-2\alpha(\gamma-1)}\epsilon^{\frac{1}{3}}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\epsilon^{\frac{1}{6}+a},\end{array}$
(3.59)

where we used the fact that

$\nu^{1-3\gamma-2\alpha(\gamma-1)}\epsilon^{\frac{1}{6}}=\epsilon^{a}|\ln\epsilon|^{1-3\gamma-2\alpha(\gamma-1)}\leq\epsilon^{a},~{}~{}~{}{\rm
if}~{}~{}\epsilon\ll 1.$

By Cauchy inequality, it holds that

$\begin{array}[]{ll}K_{2}&\displaystyle=C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-1}|\psi_{yy}|\bar{u}_{y}(\bar{\theta}^{\frac{1}{2}}|\psi_{y}|+|\zeta_{y}|)dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}(\bar{\theta}^{\alpha}\psi_{y}^{2}+\bar{\theta}^{\alpha-1}\zeta_{y}^{2})(\bar{\theta}\bar{\rho})^{-1}\bar{u}_{y}^{2}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{-\gamma}(\frac{\epsilon}{\delta})^{2}\epsilon^{\frac{1}{3}}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\epsilon^{\frac{1}{3}}.\end{array}$
(3.60)

Recalling Lemma 2.3, one can get

$\begin{array}[]{ll}K_{3}&\displaystyle=C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-\frac{1}{2}}\bar{\rho}^{-1}|\psi_{yy}|\bar{u}_{y}^{2}dtd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-1}\bar{u}_{y}^{4}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{-\gamma}\int_{0}^{\tau}\|\bar{u}_{y}\|^{4}_{L^{4}}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\nu^{-\gamma}(\frac{\epsilon}{\delta})^{2}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\epsilon^{\frac{1}{3}}.\end{array}$
(3.61)

Substituting (3.55)-(3.61) into (3.54), it holds that

$\begin{array}[]{ll}\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\psi_{y}^{2}dy+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\Big{(}{\bar{u}}_{y}\psi_{y}^{2}+\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}\Big{)}dyd\tau\\\
\qquad\qquad\displaystyle\leq\
C\nu^{-2\alpha(\gamma-1)}\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi^{2}_{y}dyd\tau+\epsilon^{\frac{1}{6}+a}.\end{array}$
(3.62)

Then substituting (3.62) into (3.50), one can get

$\begin{array}[]{ll}\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}\phi_{y}^{2}dy+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau\\\\[11.38109pt]
\leq\displaystyle\big{(}\nu^{2\alpha(\gamma-1)}|\ln\epsilon|^{-1}+C\nu^{1-3\gamma}\epsilon^{\frac{1}{6}}\big{)}\int_{0}^{\tau_{1}(\epsilon)}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+C\epsilon^{\frac{1}{3}}\\\\[11.38109pt]
\leq\displaystyle
C\big{(}|\ln\epsilon|^{-1}+\epsilon^{a}\big{)}\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi^{2}_{y}dyd\tau+\nu^{2\alpha(\gamma-1)}\epsilon^{\frac{1}{6}+a}|\ln\epsilon|^{-1}+C\nu^{1-3\gamma}\epsilon^{\frac{1}{3}+a}+C\epsilon^{\frac{1}{3}}.\end{array}$
(3.63)

Note that

$\nu^{2\alpha(\gamma-1)}\epsilon^{\frac{1}{6}+a}|\ln\epsilon|^{-1}=\epsilon^{\frac{1}{3}-3a\gamma+a}|\ln\epsilon|^{2\alpha(\gamma-1)-1}\leq\epsilon^{\frac{1}{3}-3a\gamma}|\ln\epsilon|^{-3\gamma},\quad{\rm
if}~{}~{}\epsilon\ll 1,$

and

$C\nu^{1-3\gamma}\epsilon^{\frac{1}{3}+a}=C\epsilon^{\frac{1}{3}-3a\gamma+2a}|\ln\epsilon|^{1-3\gamma}\leq\epsilon^{\frac{1}{3}-3a\gamma}|\ln\epsilon|^{-3\gamma},\quad{\rm
if}~{}~{}\epsilon\ll 1,$

one can get from (3.62) and (3.63) that

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}\phi_{y}^{2}dy+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha+1}}{\bar{\rho}^{2}}\phi_{y}^{2}dyd\tau\leq\epsilon^{\frac{1}{3}-3a\gamma}|\ln\epsilon|^{-3\gamma},\quad{\rm
if}~{}~{}\epsilon\ll 1.\end{array}$ (3.64)

Meanwhile, it holds that

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\psi_{y}^{2}dy+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\Big{(}{\bar{u}}_{y}\psi_{y}^{2}+\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}\Big{)}dyd\tau\\\\[11.38109pt]
&\displaystyle\leq
C\nu^{-2\alpha(\gamma-1)}\epsilon^{\frac{1}{3}-3a\gamma}|\ln\epsilon|^{-3\gamma}+\epsilon^{\frac{1}{6}+a}=C\epsilon^{\frac{1}{6}}|\ln\epsilon|^{-3\gamma-2\alpha(\gamma-1)}+\epsilon^{\frac{1}{6}+a}\leq\epsilon^{\frac{1}{6}}.\end{array}$
(3.65)

Step 4.  Finally, we estimate $\displaystyle\sup_{\tau}\|\zeta_{y}\|$. For
this, rewrite $\eqref{mass}_{3}$ as

$\rho\zeta_{\tau}+\rho
u\zeta_{y}+(\gamma-1)\rho\theta\psi_{y}-\kappa(\theta)\zeta_{yy}=-\bar{h}+\kappa(\theta)_{y}\theta_{y}+\mu(\theta)u^{2}_{y},$
(3.66)

where

$\bar{h}=-\kappa(\theta)\bar{\theta}_{yy}+\rho\psi\bar{\theta}_{y}+(\gamma-1)\rho\zeta\bar{u}_{y}.$
(3.67)

Multiplying (3.66) by $-\zeta_{yy}/\rho$ gives

$\begin{array}[]{l}\displaystyle(\frac{\zeta^{2}_{y}}{2})_{\tau}-(\zeta_{\tau}\zeta_{y}+u\frac{\zeta^{2}_{y}}{2})_{y}+\frac{1}{2}\bar{u}_{y}\zeta^{2}_{y}+\frac{\kappa(\theta)}{\rho}\zeta^{2}_{yy}\\\
\displaystyle=\bar{h}\frac{\zeta_{yy}}{\rho}+(\gamma-1)\theta\psi_{y}\zeta_{yy}-\frac{\psi_{y}\zeta^{2}_{y}}{2}-\big{(}\kappa(\theta)_{y}\theta_{y}+\mu(\theta)u^{2}_{y}\big{)}\frac{\zeta_{yy}}{\rho}.\end{array}$
(3.68)

Integrating the above equation over ${\mathbf{R}}^{1}\times[0,\tau]$ yields

$\begin{array}[]{ll}&\displaystyle\int_{\mathbf{R}}\frac{\zeta^{2}_{y}}{2}dy+\int^{\tau}_{0}\int_{\mathbf{R}}\Big{(}\frac{{\bar{u}}_{y}\zeta_{y}^{2}}{2}+\frac{\kappa(\theta)}{\rho}\zeta_{yy}^{2}\Big{)}dyd\tau\\\\[8.53581pt]
=&\displaystyle\int^{\tau}_{0}\int_{\mathbf{R}}\Big{\\{}\bar{h}\frac{\zeta_{yy}}{\rho}+(\gamma-1)\theta\psi_{y}\zeta_{yy}-\frac{\psi_{y}\zeta^{2}_{y}}{2}-\big{(}\kappa(\theta)_{y}\theta_{y}+\mu(\theta)u^{2}_{y}\big{)}\frac{\zeta_{yy}}{\rho}\Big{\\}}dyd\tau.\end{array}$
(3.69)

First,

$\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}\bar{h}\frac{\zeta_{yy}}{\rho}dyd\tau\leq\frac{1}{8}\int_{0}^{\tau}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta^{2}_{yy}dyd\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{-\alpha}\bar{\rho}^{-1}|\bar{h}|^{2}dyd\tau.$
(3.70)

Recalling (3.13), (3.67), and $(i)$ in Lemma 2.3, one can get

$\begin{array}[]{ll}|\bar{h}|&\leq\displaystyle
C\big{(}\bar{\theta}^{\alpha}|\bar{\theta}_{yy}|+|\bar{\rho}\bar{\theta}_{y}\psi|+|\bar{u}_{y}\bar{\rho}\zeta|\big{)}\\\\[5.69054pt]
&\displaystyle\leq
C\Big{\\{}\bar{\theta}^{\alpha}(\bar{\theta}^{\frac{1}{2}}|\bar{u}_{yy}|+|\bar{u}_{y}|^{2})+\bar{u}_{y}(|\bar{\theta}^{\frac{1}{2}}\bar{\rho}\psi|+|\bar{\rho}\zeta|)\Big{\\}}.\end{array}$
(3.71)

So the last term in (3.70) can be estimated by

$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\bar{\rho}^{-1}|\bar{h}|^{2}dyd\tau\\\\[5.69054pt]
&\displaystyle\leq
C\nu^{-1}\int_{0}^{\tau}(\|\bar{u}_{yy}\|^{2}+\|\bar{u}_{y}\|^{4}_{L^{4}})d\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{u}_{y}(\bar{\rho}\psi^{2}+\bar{\rho}^{2-\gamma}\zeta^{2})\bar{\theta}^{1-\alpha}\bar{u}_{y}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq
C\nu^{-1}(\frac{\epsilon}{\delta})^{2}+C\nu^{-\alpha(\gamma-1)}\frac{\epsilon}{\delta}\epsilon^{\frac{1}{3}}\leq\epsilon^{\frac{1}{3}}.\end{array}$
(3.72)

By Cauchy inequality, it holds that

$\begin{array}[]{ll}\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}\theta\psi_{y}\zeta_{yy}dyd\tau\Big{|}\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}+C\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\psi^{2}_{y}\bar{\theta}^{2-2\alpha}\bar{\rho}dyd\tau\\\\[8.53581pt]
\qquad\qquad\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}+C\nu^{-2\alpha(\gamma-1)}\epsilon^{\frac{1}{3}}.\end{array}$
(3.73)

By Sobolev inequality and (3.62), it holds that

$\begin{array}[]{ll}&\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\psi_{y}\zeta^{2}_{y}}{2}dyd\tau\Big{|}\leq
C\int^{\tau}_{0}\|\psi_{y}\|\|\zeta_{y}\|^{2}_{L^{4}}d\tau\leq
C\int^{\tau}_{0}\|\psi_{y}\|\|\zeta_{y}\|^{\frac{3}{2}}\|\zeta_{yy}\|^{\frac{1}{2}}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{-\alpha(\gamma-1)/3}\int^{\tau}_{0}\|\psi_{y}\|^{\frac{4}{3}}\|\zeta_{y}\|^{2}d\tau\\\\[8.53581pt]
&\qquad\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{-4\alpha(\gamma-1)/3}\sup_{\tau\in[0,\tau_{1}]}\|\psi_{y}\|^{\frac{4}{3}}\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\zeta_{y}^{2}dyd\tau\\\\[11.38109pt]
&\qquad\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{-4\alpha(\gamma-1)/3}\epsilon^{\frac{1}{3}}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2}dyd\tau+\epsilon^{\frac{1}{9}},\end{array}$
(3.74)

where we have used the a priori assumptions (3.11), and in the last inequality
we have used the fact that

$C\nu^{-4\alpha(\gamma-1)/3}\epsilon^{\frac{1}{3}}=C\epsilon^{\frac{1-4\alpha(\gamma-1)a}{3}}|\ln\epsilon|^{-\frac{4\alpha(\gamma-1)}{3}}\leq\epsilon^{\frac{1}{9}},~{}~{}{\rm
if}~{}~{}\epsilon\ll 1.$

By Cauchy inequality, we have

$\begin{array}[]{ll}&\displaystyle\Big{|}\int^{\tau}_{0}\int_{\mathbf{R}}(\kappa(\theta)_{y}\theta_{y}+\mu(\theta)u^{2}_{y})\frac{\zeta_{yy}}{\rho}dyd\tau\Big{|}\\\\[8.53581pt]
\leq&\displaystyle
C\int^{\tau}_{0}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\bar{\rho}^{-1}|\zeta_{yy}|(\bar{\theta}^{-1}\zeta^{2}_{y}+\psi^{2}_{y}+\bar{u}^{2}_{y})dyd\tau:=\sum^{3}_{i=1}L_{i}.\end{array}$
(3.75)

Now we estimate the terms on the right-hand side of (3.75) one by one. By
Sobolev inequality, it holds that

$\begin{array}[]{ll}L_{1}&\displaystyle=\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\bar{\rho}^{-1}|\zeta_{yy}\zeta^{2}_{y}|dyd\tau\leq
C\nu^{1/2-\gamma}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\zeta_{yy}\|\|\zeta_{y}\|^{2}_{L^{4}}d\tau\\\\[8.53581pt]
&\displaystyle\leq
C\nu^{1/2-\gamma}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\zeta_{yy}\|\|\zeta_{yy}\|^{\frac{1}{2}}\|\zeta_{y}\|^{\frac{3}{2}}d\tau\leq
C\nu^{\frac{2-4\gamma-\alpha(\gamma-1)}{4}}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\zeta_{yy}\|^{\frac{3}{2}}\|\zeta_{y}\|^{\frac{3}{2}}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{2-4\gamma-\alpha(\gamma-1)}\int_{0}^{\tau}\|\zeta_{y}\|^{6}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{2-4\gamma-2\alpha(\gamma-1)}\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\|\zeta_{y}\|^{4}\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha-1}\zeta_{y}^{2}dyd\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{2-4\gamma-2\alpha(\gamma-1)}\epsilon^{\frac{1}{3}}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+\epsilon^{\frac{1}{9}},\end{array}$
(3.76)

where in the last inequality we have used the fact that

$C\nu^{2-4\gamma-2\alpha(\gamma-1)}\epsilon^{\frac{1}{3}}=C\epsilon^{\frac{1}{3}+a(2-4\gamma-2\alpha(\gamma-1))}|\ln\epsilon|^{2-4\gamma-2\alpha(\gamma-1)}\leq\epsilon^{\frac{1}{9}},~{}~{}{\rm
if}~{}~{}\epsilon\ll 1.$

Similarly, one has

$\begin{array}[]{ll}L_{2}&\displaystyle=\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\bar{\rho}^{-1}|\zeta_{yy}\psi^{2}_{y}|dyd\tau\leq
C\nu^{-\frac{1}{2}}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\zeta_{yy}\|\|\psi_{y}\|^{2}_{L^{4}}d\tau\\\\[8.53581pt]
&\displaystyle\leq
C\nu^{-\frac{1}{2}}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\zeta_{yy}\|\|\psi_{yy}\|^{\frac{1}{2}}\|\psi_{y}\|^{\frac{3}{2}}d\tau\leq
C\nu^{-\frac{2+\alpha(\gamma-1)}{4}}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\zeta_{yy}\|\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\psi_{yy}\|^{\frac{1}{2}}\|\psi_{y}\|^{\frac{3}{2}}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{-\frac{2+\alpha(\gamma-1)}{2}}\int_{0}^{\tau}\|\sqrt{\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}}\psi_{yy}\|\|\psi_{y}\|^{3}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}(\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}+\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\psi_{yy}^{2})dyd\tau+C\nu^{-2-\alpha(\gamma-1)}\int_{0}^{\tau}\|\psi_{y}\|^{6}d\tau\\\\[8.53581pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+\frac{1}{8}\epsilon^{\frac{1}{9}}+C\nu^{-2-2\alpha(\gamma-1)}\sup_{[0,\tau_{1}(\epsilon)]}\|\psi_{y}\|^{4}\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\psi_{y}dyd\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+\frac{1}{8}\epsilon^{\frac{1}{9}}+C\nu^{-2-2\alpha(\gamma-1)}\epsilon^{\frac{5}{9}}\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+\epsilon^{\frac{1}{9}}.\end{array}$
(3.77)

Recalling Lemma 2.3 and using Cauchy inequality, it holds that

$\begin{array}[]{ll}L_{3}&\displaystyle=\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\bar{\rho}^{-1}|\zeta_{yy}|\bar{u}_{y}^{2}dtd\tau\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\int_{0}^{\tau}\int_{\mathbf{R}}\bar{\theta}^{\alpha}\bar{\rho}^{-1}\bar{u}_{y}^{4}dyd\tau\\\\[11.38109pt]
&\displaystyle\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+C\nu^{-1}\int_{0}^{\tau}\|\bar{u}_{y}\|^{4}_{L^{4}}d\tau\leq\frac{1}{8}\int^{\tau}_{0}\int_{\mathbf{R}}\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}dyd\tau+\epsilon^{\frac{1}{9}}.\end{array}$
(3.78)

Substituting (3.70), (3.74)-(3.78) into (3.69), it holds that

$\begin{array}[]{ll}&\displaystyle\sup_{\tau\in[0,\tau_{1}(\epsilon)]}\int_{\mathbf{R}}\zeta_{y}^{2}dy+\int^{\tau_{1}(\epsilon)}_{0}\int_{\mathbf{R}}\Big{(}{\bar{u}}_{y}\psi_{y}^{2}+\frac{\bar{\theta}^{\alpha}}{\bar{\rho}}\zeta_{yy}^{2}\Big{)}dyd\tau\leq\epsilon^{\frac{1}{9}}.\end{array}$
(3.79)

Therefore, (3.14), (3.15) and (3.16) can be derived directly from
$(\ref{step1})$, (3.64), (3.65) and (3.79). It follows from (3.14)-(3.16) that
if $\epsilon$ is suitably small, then

$\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\phi(\cdot,\tau)\|_{L^{\infty}}&\displaystyle\leq\sqrt{2}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\phi(\cdot,\tau)\|^{1/2}\|\phi_{y}(\cdot,\tau)\|^{1/2}\\\\[8.53581pt]
&\displaystyle\leq
C\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\Big{(}\nu^{-\gamma-2\alpha(\gamma-1)}\int_{\mathbf{R}}\bar{\rho}^{\gamma-2}\phi^{2}dy\int_{\mathbf{R}}\frac{\bar{\theta}^{2\alpha}}{\bar{\rho}^{3}}\phi_{y}^{2}dy\Big{)}^{\frac{1}{4}}\\\\[14.22636pt]
&\displaystyle\leq
C\Big{(}\nu^{-\gamma-2\alpha(\gamma-1)}\epsilon^{\frac{1}{3}}\cdot\epsilon^{\frac{1}{3}-3a\gamma}|\ln\epsilon|^{-3\gamma}\Big{)}^{\frac{1}{4}}\\\\[14.22636pt]
&\displaystyle=C\epsilon^{\frac{4\gamma+3\alpha(\gamma-1)}{2(18\gamma+12\alpha(\gamma-1))}}|\ln\epsilon|^{-\frac{2\gamma+\alpha(\gamma-1)}{2}}\leq\epsilon^{\frac{1}{18\gamma+12\alpha(\gamma-1)}}=\epsilon^{a},\end{array}$
$\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\psi(\cdot,\tau)\|_{L^{\infty}}&\displaystyle\leq\sqrt{2}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\psi(\cdot,\tau)\|^{1/2}\|\psi_{y}(\cdot,\tau)\|^{1/2}\\\\[8.53581pt]
&\displaystyle\leq
C\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\Big{(}\nu^{-1}\int_{\mathbf{R}}\bar{\rho}\psi^{2}dy\int_{\mathbf{R}}\psi_{y}^{2}dy\Big{)}^{\frac{1}{4}}\\\\[14.22636pt]
&\displaystyle\leq
C\Big{(}\nu^{-1}\epsilon^{\frac{1}{3}}\cdot\epsilon^{\frac{1}{9}}\Big{)}^{\frac{1}{4}}=C\epsilon^{\frac{1}{9}-\frac{a}{4}}|\ln\epsilon|^{-\frac{1}{4}}\\\\[8.53581pt]
&\displaystyle=C\epsilon^{\frac{24\gamma+16\alpha(\gamma-1)-3}{12(18\gamma+12\alpha(\gamma-1))}}|\ln\epsilon|^{-\frac{1}{4}}\leq\epsilon^{\frac{1}{18\gamma+12\alpha(\gamma-1)}}=\epsilon^{a},\end{array}$

and

$\begin{array}[]{ll}\displaystyle\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\zeta(\cdot,\tau)\|_{L^{\infty}}&\displaystyle\leq\sqrt{2}\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\|\zeta(\cdot,\tau)\|^{1/2}\|\zeta_{y}(\cdot,\tau)\|^{1/2}\\\\[8.53581pt]
&\displaystyle\leq
C\sup_{0\leq\tau\leq\tau_{1}(\epsilon)}\Big{(}\nu^{-2}\int_{\mathbf{R}}\bar{\rho}^{2-\gamma}\zeta^{2}dy\int_{\mathbf{R}}\zeta_{y}^{2}dy\Big{)}^{\frac{1}{4}}\\\\[14.22636pt]
&\displaystyle\leq
C\Big{(}\nu^{-2}\epsilon^{\frac{1}{3}}\cdot\epsilon^{\frac{1}{9}}\Big{)}^{\frac{1}{4}}=C\epsilon^{\frac{1}{9}-\frac{a}{2}}|\ln\epsilon|^{-\frac{1}{2}}\\\\[8.53581pt]
&\displaystyle=C\epsilon^{\frac{12\gamma+8\alpha(\gamma-1)-3}{6(18\gamma+12\alpha(\gamma-1))}}|\ln\epsilon|^{-\frac{1}{2}}\leq\epsilon^{\frac{\gamma-1}{18\gamma+12\alpha(\gamma-1)}}=\epsilon^{(\gamma-1)a}.\end{array}$

Thus the a priori assumptions (3.10)-(3.11) are verified and the proof of
Lemma 3.2 is completed. $\Box$

It is note that the a priori estimates (3.14)-(3.16) are better than the a
priori assumptions (3.10)-(3.11) in the time interval $[0,\tau_{1}(\epsilon)]$
with $\tau_{1}(\epsilon)$ being the maximum existence time. Based on these a
priori estimates, we can claim $\tau_{1}(\epsilon)=\infty$. In fact, if
$\tau_{1}(\epsilon)<\infty$, then by again using the local existence at time
$\tau=\tau_{1}(\epsilon)$, we can find another time
$\tau_{2}(\epsilon)>\tau_{1}(\epsilon)$ so that the solution satisfies the
assumptions (3.10)-(3.11) in the time interval $[0,\tau_{2}(\epsilon)]$ which
contradicts the assumption that $\tau_{1}(\epsilon)$ is the maximum existence
time. Therefore we extend the local solution to the global one in $[0,\infty)$
for small but fixed $\epsilon$.

Proof of Theorem 1.1: It remains to prove (1.13) with $a$ given in (1.14).
From Lemma 2.2, Lemma 2.3 (iii), (3.7)-(3.9) and recalling that
$\nu=\epsilon^{a}|\ln\epsilon|,~{}~{}\delta=\epsilon^{a}$ in (3.12), it holds
that for any given positive constant $l$, there exists a constant $C_{l}>0$
which is independent of $\epsilon$ such that

$\begin{array}[]{ll}&\displaystyle\sup_{t\geq
l}\|\rho(\cdot,t)-\rho^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\\\
\leq&\displaystyle\sup_{\tau\in[0,+\infty)}\|\phi(\cdot,\tau)\|_{L^{\infty}}+\sup_{t\geq
l}\|\bar{\rho}(\cdot,t)-\rho^{r_{3}}_{\nu}(\frac{\cdot}{t})\|_{L^{\infty}}+\sup_{t\geq
l}\|\rho^{r_{3}}_{\nu}(\frac{\cdot}{t})-\rho^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\\\\[11.38109pt]
\leq&\displaystyle C_{l}\big{(}\epsilon^{a}+\delta|\ln\delta|+\nu\big{)}\leq
C_{l}~{}\epsilon^{a}|\ln\epsilon|.\end{array}$

About the convergence of the momentum, we have

$\begin{array}[]{ll}&\displaystyle\sup_{t\geq
l}\|m(\cdot,t)-m^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\\\ \leq&\displaystyle
C\sup_{\tau\in[0,+\infty)}\Big{(}\|\psi(\cdot,\tau)\|_{L^{\infty}}+\|\phi(\cdot,\tau)\|_{L^{\infty}}\Big{)}+\sup_{t\geq
l}\Big{(}\|\bar{m}(\cdot,t)-m^{r_{3}}_{\nu}(\frac{\cdot}{t})\|_{L^{\infty}}+\|m^{r_{3}}_{\nu}(\frac{\cdot}{t})-m^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\Big{)}\\\
\leq&\displaystyle C_{l}\big{(}\epsilon^{a}+\delta|\ln\delta|+\nu\big{)}\leq
C_{l}~{}\epsilon^{a}|\ln\epsilon|.\end{array}$

About the convergence of the total energy, we can get

$\begin{array}[]{ll}&\displaystyle\sup_{t\geq
l}\|n(\cdot,t)-n^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\\\ \leq&\displaystyle
C\sup_{\tau\in[0,+\infty)}\Big{(}\|\zeta(\cdot,\tau)\|_{L^{\infty}}+\|\phi(\cdot,\tau)\|_{L^{\infty}}\Big{)}+\sup_{t\geq
l}\Big{(}\|\bar{\rho}\bar{\theta}(\cdot,t)-n^{r_{3}}_{\nu}(\frac{\cdot}{t})\|_{L^{\infty}}+\|n^{r_{3}}_{\nu}(\frac{\cdot}{t})-n^{r_{3}}(\frac{\cdot}{t})\|_{L^{\infty}}\Big{)}\\\\[11.38109pt]
\leq&\displaystyle C_{l}\big{(}\epsilon^{a}+\delta|\ln\delta|+\nu\big{)}\leq
C_{l}~{}\epsilon^{a}|\ln\epsilon|,\end{array}$

where we have used the fact that

$\displaystyle\sup_{\tau\in[0,+\infty)}\|\zeta(\cdot,\tau)\|_{L^{\infty}}\leq
C\epsilon^{\frac{1}{9}-\frac{a}{2}}|\ln\epsilon|^{-\frac{1}{2}}\leq\epsilon^{a}.$

Then the proof of Theorem 1.1 is completed. $\Box$

## References

  * [1] S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2), 161 (2005), pp. 223-342.
  * [2] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, 3rd edition, 1990.
  * [3] G. Q. Chen, M. Perepelitsa, Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Comm. Pure Appl. Math., 63, (2010), pp. 1469-1504.
  * [4] J. Goodman, Z. P. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121 (1992), pp. 235-265.
  * [5] D. Hoff, T. P. Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J., 38 (1989), pp. 861-915.
  * [6] F. Huang, M. Li, Y. Wang, Zero dissipation limit to rarefaction wave with vacuum for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 44 (2012), pp. 1742-1759.
  * [7] F.M. Huang, S. Jiang and Y. Wang, Zero dissipation limit of full compressible Navier-Stokes equations with a Riemann initial data, Preprint.
  * [8] F. Huang, Y. Wang, Y. Wang, T. Yang, The Limit of the Boltzmann Equation to the Euler Equations for Riemann Problems, SIAM J. Math. Anal., 45 (2013), no. 3, pp.1741-1811.
  * [9] F. M. Huang, Y. Wang and T. Yang, Fluid Dynamic Limit to the Riemann Solutions of Euler Equations: I. Superposition of rarefaction waves and contact discontinuity, Kinetic and Related Models, 3 (2010), pp. 685-728.
  * [10] F. M. Huang, Y. Wang and T. Yang, Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to Riemann problem, Arch. Ration. Mech. Anal., 203 (2012), pp. 379-413.
  * [11] S. Jiang, G. X. Ni and W. J. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids, SIAM J. Math. Anal., 38 (2006), pp. 368-384.
  * [12] Q. S. Jiu, Y. Wang and Z. P. Xin, Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity, Preprint, 2011\.
  * [13] S. Kawashima, A. Matsumura, T. Nishida, On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys., 70 (1979), no. 2, pp. 97-124.
  * [14] M. Li, T. Wang, Zero dissipation limit to rarefaction wave with vacuum for one-dimensional full compressible Navier-Stokes equations, Preprint, 2013.
  * [15] T. P. Liu, J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Adv. in Appl. Math., 1 (1980), pp. 345-359.
  * [16] T. Liu, T. Yang, S. H. Yu and H. J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Rat. Mech. Anal., 181 (2006), 333–371.
  * [17] S. X. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), pp. 95–110.
  * [18] A. Matsumura, K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), pp. 1-13.
  * [19] M. Perepelitsa, Asymptotics toward rarefaction waves and vacuum for 1-D compressible Navier-Stokes equations, SIAM J. Math. Anal., 42 (2010), pp. 1404-1412.
  * [20] J. Smoller, Shock Waves and Reaction-Diffusion Equations. 2nd ed. Grundlehren der Mathematischen Wissenschaften. 258. New York: Springer-Verlag, xxii, 1994.
  * [21] H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), pp. 411-432.
  * [22] Y. Wang, Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock, Acta Mathematica Scientia, 28B (2008), pp. 727-748.
  * [23] Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), pp. 621-665.
  * [24] Z. P. Xin, H. H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations, J. Diff. Eqs., 249 (2010), pp. 827-871.
  * [25] S. H. Yu, Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws, Arch. Ration. Mech. Anal., 146 (1999), pp. 275-370.
  * [26] Y. Zhang, R. Pan, Y. Wang, Z. Tan, Zero dissipation limit with two interacting shocks of the 1D non-isentropic Navier-Stokes equations, to appear in Indiana Univ. Math. J..