# Global regular solutions for 1-D degenerate compressible Navier-Stokes
equations with large data and far field vacuum

Yue Cao  School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong
University, Shanghai 200240, P. R. China<EMAIL_ADDRESS>,  Hao Li
School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R.
China<EMAIL_ADDRESS>and  Shengguo Zhu  School of Mathematical Sciences,
CMA-Shanghai, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, P.
R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, the Cauchy problem for the one-dimensional (1-D) isentropic
compressible Navier-Stokes equations (CNS) is considered. When the viscosity
$\mu(\rho)$ depends on the density $\rho$ in a sublinear power law
($\rho^{\delta}$ with $0<\delta\leq 1$), based on an elaborate analysis of the
intrinsic singular structure of this degenerate system, we prove the global-
in-time well-posedness of regular solutions with conserved total mass,
momentum, and finite total energy in some inhomogeneous Sobolev spaces.
Moreover, the solutions we obtained satisfy that $\rho$ keeps positive for all
point $x\in\mathbb{R}$ but decays to zero in the far field, which is
consistent with the facts that the total mass of the whole space is conserved,
and CNS is a model of non-dilute fluids where $\rho$ is bounded below away
from zero. The key to the proof is the introduction of a well-designed
reformulated structure by introducing some new variables and initial
compatibility conditions, which, actually, can transfer the degeneracies of
the time evolution and the viscosity to the possible singularity of some
special source terms. Then, combined with the BD entropy estimates and
transport properties of the so-called effective velocity
$v=u+\varphi(\rho)_{x}$ ($u$ is the velocity of the fluid, and $\varphi(\rho)$
is a function of $\rho$ defined by
$\varphi^{\prime}(\rho)=\mu(\rho)/\rho^{2}$), one can obtain the required
uniform a priori estimates of corresponding solutions. Moreover, in contrast
to the classical theory in the case of the constant viscosity, one can show
that the $L^{\infty}$ norm of $u$ of the global regular solution we obtained
does not decay to zero as time $t$ goes to infinity. It is worth pointing out
that the well-posedness theory established here can be applied to the viscous
Saint-Venant system for the motion of shallow water.

###### Key words and phrases:

Degenerate compressible Navier-Stokes equations, One-dimension, Regular
solution, Far field vacuum, Large data, Global-in-time well-posedness.

###### 2010 Mathematics Subject Classification:

35A01, 35Q30, 35B65, 35A09.

##  1\. introduction

The time evolution of the mass density $\rho\geq 0$ and the velocity
$u=\left(u^{(1)},u^{(2)},\cdots,u^{(d)}\right)^{\top}$ $\in\mathbb{R}^{d}$ of
a general viscous isentropic compressible fluid occupying a spatial domain
$\Omega\subset\mathbb{R}^{d}$ is governed by the following isentropic
compressible Navier-Stokes equations (ICNS):

$\begin{cases}\rho_{t}+\text{div}(\rho u)=0,\\\\[4.0pt] (\rho
u)_{t}+\text{div}(\rho u\otimes u)+\nabla P=\text{div}\mathbb{T}.\end{cases}$
(1.1)

Here, $x=(x_{1},x_{2},\cdots,x_{d})^{\top}\in\Omega$, $t\geq 0$ are the space
and time variables, respectively. For the polytropic gases, the constitutive
relation is given by

$P=A\rho^{\gamma},\quad A>0,\quad\gamma>1,$ (1.2)

where $A$ is an entropy constant and $\gamma$ is the adiabatic exponent.
$\mathbb{T}$ denotes the viscous stress tensor with the form

$\begin{split}&\mathbb{T}=2\mu(\rho)D(u)+\lambda(\rho)\text{div}u\,\mathbb{I}_{d},\end{split}$
(1.3)

where $D(u)=\frac{1}{2}\big{(}\nabla u+(\nabla u)^{\top}\big{)}$ is the
deformation tensor, $\mathbb{I}_{d}$ is the $d\times d$ identity matrix,

$\mu(\rho)=\alpha\rho^{\delta},\quad\lambda(\rho)=\beta\rho^{\delta},$ (1.4)

for some constant $\delta\geq 0$, $\mu(\rho)$ is the shear viscosity
coefficient, $\lambda(\rho)+\frac{2}{d}\mu(\rho)$ is the bulk viscosity
coefficient, $\alpha$ and $\beta$ are both constants satisfying

$\alpha>0\quad\text{and}\quad 2\alpha+d\beta\geq 0.$ (1.5)

In the theory of gas dynamics, the CNS can be derived from the Boltzmann
equations through the Chapman-Enskog expansion, cf. Chapman-Cowling [9] and
Li-Qin [30]. Under some proper physical assumptions, one can find that the
viscosity coefficients and heat conductivity coefficient $\kappa$ are not
constants but functions of the absolute temperature $\theta$ such as:

$\begin{split}\mu(\theta)=&a_{1}\theta^{\frac{1}{2}}F(\theta),\quad\lambda(\theta)=a_{2}\theta^{\frac{1}{2}}F(\theta),\quad\kappa(\theta)=a_{3}\theta^{\frac{1}{2}}F(\theta),\end{split}$
(1.6)

for some constants $a_{i}$ $(i=1,2,3)$ (see [9]). Actually for the cut-off
inverse power force models, if the intermolecular potential varies as
$r^{-\nu}$, where $r$ is intermolecular distance, then in (1.6):

$F(\theta)=\theta^{b}\quad\text{with}\quad b=\frac{2}{\nu}\in[0,\infty).$

In particular (see §10 of [9]), for ionized gas, $\nu=1$ and $b=2$; for
Maxwellian molecules, $\nu=4$ and $b=\frac{1}{2}$; while for rigid elastic
spherical molecules, $\nu=\infty$ and $b=0$.

According to Liu-Xin-Yang [34], for isentropic and polytropic fluids, such a
dependence is inherited through the laws of Boyle and Gay-Lussac:

$P=R\rho\theta=A\rho^{\gamma}\quad\text{for \ constant}\quad R>0,$

i.e., $\theta=AR^{-1}\rho^{\gamma-1}$, and one can see that the viscosity
coefficients are functions of $\rho$ of the form $(\ref{fandan})$. Actually,
there do exist some physical models that satisfy the density-dependent
viscosities assumption (1.4), such as Korteweg system, Shallow water
equations, lake equations and quantum Navier-Stokes system (see [2, 3, 4, 5,
6, 36, 46]).

In the current paper, the following 1-D degenerate ICNS is considered:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\\[4.0pt] &(\rho
u)_{t}+(\rho u^{2})_{x}+P(\rho)_{x}=(\mu(\rho)u_{x})_{x},\end{aligned}\right.$
(1.7)

where the pressure $P$ and the viscosity coefficient $\mu(\rho)$ are given by
(1.2) and (1.4), respectively. We are concerned with the global well-posedness
of regular solutions with large data and far field vacuum to the Cauchy
problem of (1.7) with the following initial data and far field behavior:

$\begin{split}\displaystyle(\rho(0,x),u(0,x))=(\rho_{0}(x),u_{0}(x))\quad&\text{for}\quad
x\in\mathbb{R},\\\\[4.0pt]
\displaystyle\left(\rho(t,x),u(t,x)\right)\to\left(0,0\right)\quad\text{as}\quad\left|x\right|\to\infty\quad&\text{for}\quad
t\geq 0.\end{split}$ (1.8)

Throughout this paper, we adopt the following simplified notations, most of
them are for the standard homogeneous and inhomogeneous Sobolev spaces:

$\begin{split}\displaystyle&\|f\|_{s}=\|f\|_{H^{s}(\mathbb{R}^{d})},\quad|f|_{p}=\|f\|_{L^{p}(\mathbb{R}^{d})},\quad\|f\|_{m,p}=\|f\|_{W^{m,p}(\mathbb{R}^{d})},\\\\[6.0pt]
\displaystyle&D^{k,r}=\\{f\in
L^{1}_{loc}(\mathbb{R}^{d}):|f|_{D^{k,r}}=|\nabla^{k}f|_{r}<\infty\\},\quad
D^{k}=D^{k,2},\\\\[6.0pt] \displaystyle&D^{1}_{*}=\\{f\in
L^{6}(\mathbb{R}^{3}):|f|_{D^{1}_{*}}=|\nabla f|_{2}<\infty\\},\\\\[6.0pt]
\displaystyle&|f|_{D^{1}_{*}}=\|f\|_{D^{1}_{*}(\mathbb{R}^{3})},\quad\|f\|_{L^{p}([0,T];L^{q})}=\|f\|_{L^{p}([0,T];L^{q}(\mathbb{R}^{d}))},\\\\[6.0pt]
\displaystyle&\|f\|_{X_{1}\cap X_{2}}=\|f\|_{X_{1}}+\|f\|_{X_{2}},\quad\int
f\text{d}x=\int_{\mathbb{R}^{d}}f\text{d}x,\\\\[6.0pt]
&X([0,T];Y)=X([0,T];Y(\mathbb{R}^{d})),\quad\|(f,g)\|_{X}=\|f\|_{X}+\|g\|_{X}.\end{split}$

We will clearly indicate the value of $d$ where the above notations are used.
A detailed study of homogeneous Sobolev spaces can be found in [15].

For the flows of constant viscosities ($\delta=0$ in (1.4)), there is a lot of
literature on the well-posedness of strong/classical solutions to the system
(1.1). When $\inf_{x}{\rho_{0}(x)}>0$, the local well-posedness of three-
dimensonal (3-D) classical solutions to the Cauchy problem of (1.1) follows
from the standard symmetric hyperbolic-parabolic structure satisfying the
well-known Kawashima’s condition, cf. [25, 40, 42], which has been extended to
be a global one by Matsumura-Nishida [37] for initial data close to a non-
vacuum equilibrium in some Sobolev space $H^{s}(\mathbb{R}^{3})$
($s>\frac{5}{2}$). In $\mathbb{R}$, the global well-posedness of strong
solutions with arbitrarily large data in some bounded domains has been proven
by Kazhikhov-Shelukhin [27], and later, Kawashima-Nishida [26] extended this
theory to the unbounded domains. However, these approaches do not work when
$\inf_{x}{\rho_{0}(x)}=0$, which occurs when some physical requirements are
imposed, such as finite total initial mass and energy in the whole space. One
of the main issues in the presence of vacuum is the degeneracy of the time
evolution operator, which makes it hard to understand the behavior of the
velocity field near the vacuum. A remedy was suggested by Cho-Choe-Kim [11],
where they imposed initially a compatibility condition:

$-\text{div}\mathbb{T}_{0}+\nabla P(\rho_{0})=\sqrt{\rho_{0}}g\quad\text{for\
\ some}\ \ g\in L^{2}(\mathbb{R}^{3}),$

which leads to $\sqrt{\rho}u_{t}\in L^{\infty}([0,T_{*}];L^{2})$ for a short
time $T_{*}>0$. Then they established successfully the local well-posedness of
smooth solutions with vacuum in $\mathbb{R}^{3}$ (see also Cho-Kim [12]),
which, recently, has been shown to be a global one with small energy by Huang-
Li-Xin [21] in $\mathbb{R}^{3}$. Later, Jiu-Li-Ye [22] proved that the 1-D
Cauchy problem of (1.1) admits a unique global classical solution with
arbitrarily large data and vacuum.

For density-dependent viscosities ($\delta>0$ in (1.4)) which degenerate at
the vacuum, system (1.1) has received extensive attentions in recent years.
When $\inf_{x}{\rho_{0}(x)}>0$, some important progresses on the global well-
posedness of classical solutions to the Cauchy problem of (1.1) have been
obtained, which include Burtea-Haspot [8], Constantin-Drivas-Nguyen-
Pasqualotto [13], Haspot [20], Kang-Vasseur [24], Mellet-Vasseur [39] for 1-D
flow with arbitrarily large data, and Sundbye [44] and Wang-Xu [47] for two-
dimensional (2-D) flow with initial data close to a non-vacuum equilibrium in
some Sobolev spaces. When vacuum appears, instead of the uniform elliptic
structure, the viscosity degenerates when density vanishes, which raises the
difficulty of the problem to another level. Actually, in [11, 21, 22], the
uniform ellipticity of the Lamé operator $L$ defined by

$Lu=-\alpha\triangle u-(\alpha+\beta)\nabla\mathtt{div}u$

($Lu=-\alpha u_{xx}$ in $\mathbb{R}$) plays an essential role in the high
order regularity estimates on $u$. One can use the standard elliptic theory to
estimate $|u|_{D^{k+2}}$ by the $D^{k}$-norm of all other terms in momentum
equations. However, for $\delta>0$, viscosity coefficients vanish as the
density function connects to vacuum continuously. This degeneracy makes it
difficult to adapt the approach of the constant viscosity case in [11, 21, 22]
to the current case. A remarkable discovery of a new mathematical entropy
function was made by Bresch-Desjardins [5] for $\lambda(\rho)$ and $\mu(\rho)$
satisfying the relation

$\lambda(\rho)=2(\mu^{\prime}(\rho)\rho-\mu(\rho)),$ (1.9)

which offers an estimate $\mu^{\prime}(\rho)\nabla\sqrt{\rho}\in
L^{\infty}([0,T];L^{2}(\mathbb{R}^{d}))$ provided that
$\mu^{\prime}(\rho_{0})\nabla\sqrt{\rho_{0}}\in L^{2}(\mathbb{R}^{d})$ for any
$d\geq 1$. This observation plays an important role in the development of the
global existence of weak solutions with vacuum for system (1.1) and some
related models, see Bresch-Desjardins [2], Bresch-Vasseur-Yu [7], Jiu-Xin
[23], Li-Xin [29], Mellet-Vasseur [38], Vasseur-Yu [45] and so on. However,
the regularities and uniquness of such weak solutions with vacuum remain open.

Notice that, for the cases $\delta\in(0,\infty)$, if $\rho>0$,
$\eqref{eq:1.1}_{2}$ can be formally rewritten as

$\begin{split}u_{t}+u\cdot\nabla
u+\frac{A\gamma}{\gamma-1}\nabla\rho^{\gamma-1}+\rho^{\delta-1}Lu=\psi\cdot
Q(u),\end{split}$ (1.10)

where the quantities $\psi$ and $Q(u)$ are given by

$\begin{split}\psi\triangleq&\nabla\ln\rho\quad\text{when}\quad\delta=1;\\\
\psi\triangleq&\frac{\delta}{\delta-1}\nabla\rho^{\delta-1}\quad\text{when}\quad\delta\in(0,1)\cup(1,\infty);\\\
Q(u)\triangleq&\alpha(\nabla u+(\nabla
u)^{\top})+\beta\mathtt{div}u\mathbb{I}_{d}.\end{split}$ (1.11)

When $\delta=1$, from (1.10)-(1.11), the degeneracy of time evolution and
elliptic structure in momentum equations caused by the vacuum has been
transferred to the possible singularity of the term $\nabla\ln\rho$, which
actually can be controlled by a symmetric hyperbolic system with a source term
$\nabla\text{div}u$ in Li-Pan-Zhu [31]. Then via establishing the uniform a
priori estimates in $L^{6}\cap D^{1}\cap D^{2}$ for $\nabla\ln\rho$, the
existence of 2-D local classical solution with far field vacuum to (1.1) has
been obtained in [31], which also applies to the 2-D Shallow water equations.
When $\delta>1$, (1.10)-(1.11) imply that actually the velocity $u$ can be
governed by a nonlinear degenerate parabolic system without singularity near
the vacuum region. Based on this observation, by using some hyperbolic
approach which bridges the parabolic system (1.10) when $\rho>0$ and the
hyperbolic one

$u_{t}+u\cdot\nabla u=0\quad\text{when}\quad\rho=0,$

the existence of 3-D local classical solutions with vacuum to (1.1) was
established in Li-Pan-Zhu [32]. The corresponding global well-posedness in
some homogeneous Sobolev spaces has been established by Xin-Zhu [48] under
some initial smallness assumptions. Recently, for the case $0<\delta<1$, via
introducing an elaborate elliptic approach on the operators
$L(\rho^{\delta-1}u)$ and some initial compatibility conditions, the existence
of 3-D local regular solution with far field vacuum has been obtained by Xin-
Zhu [49]. Some other interesting results and discussions can also be seen in
Chen-Chen-Zhu [10], Ding-Zhu [14], Geng-Li-Zhu [16], Germain-Lefloch [18],
Guo-Li-Xin [19], Lions [33], Luo-Xin-Zeng [35], Yang-Zhao [50], Yang-Zhu [51],
Zhu [52] and so on.

However, due to the complicated mathematical structure of the degenerate ICNS
and lack of smooth effect on solutions when vacuum appears, even many
important progresses have been obtained, a lot of fundamental questions still
remain open, such as which kind of initial data can cause the finite time
blow-up of smooth solutions, or provide the global existence of smooth
solutions with conserved total mass, momentum, and finite total energy. In the
current paper, we identify a class of 1-D initial data, which admits a global
classical solution with far field vacuum to the Cauchy problem for the cases
$0<\delta\leq 1$ in some inhomogeneous Sobolev spaces.

###  1.1. Global-in-time well-posedness of regular solutions

In order to present our results clearly, we first introduce the following two
definitions for the case $0<\delta<1$ and the case $\delta=1$ of regular
solutions with far field vacuum to the Cauchy problem (1.7)-(1.8),
respectively.

######  Definition 1.1.

Assume $0<\delta<1$ and $T>0$. The pair $(\rho,u)$ is called a regular
solution to the Cauchy problem (1.7)-(1.8) in $[0,T]\times\mathbb{R}$, if
$(\rho,u)$ satisfies this problem in the sense of distributions and:

$\displaystyle(1)\ \inf_{(t,x)\in[0,T]\times\mathbb{R}}\rho(t,x)=0,\quad
0<\rho^{\gamma-1}\in C([0,T];H^{2}),$ $\displaystyle\quad\ \
(\rho^{\delta-1})_{x}\in C([0,T];H^{1});$ $\displaystyle(2)\ u\in
C([0,T];H^{2})\cap L^{2}([0,T];H^{3}),\quad\rho^{\frac{\delta-1}{2}}u_{x}\in
C([0,T];L^{2}),$ $\displaystyle\quad\ \ u_{t}\in C([0,T];L^{2})\cap
L^{2}([0,T];H^{1}),\quad\rho^{\frac{\delta-1}{2}}u_{tx}\in
L^{2}([0,T];L^{2}),$ $\displaystyle\quad\ \ \rho^{\delta-1}u_{xx}\in
C([0,T];L^{2})\cap L^{2}([0,T];H^{1}).$

######  Definition 1.2.

Assume $\delta=1$ and $T>0$. The pair $(\rho,u)$ is called a regular solution
to the Cauchy problem (1.7)-(1.8) in $[0,T]\times\mathbb{R}$, if $(\rho,u)$
satisfies this problem in the sense of distributions and:

$\displaystyle(1)\ \inf_{(t,x)\in[0,T]\times\mathbb{R}}\rho(t,x)=0,\quad
0<\rho^{\gamma-1}\in C([0,T];H^{2}),$ $\displaystyle\quad\ \ (\ln\rho)_{x}\in
C([0,T];H^{1});$ $\displaystyle(2)\ u\in C([0,T];H^{2})\cap
L^{2}([0,T];H^{3}),\quad u_{t}\in C([0,T];L^{2})\cap L^{2}([0,T];H^{1}).$

######  Remark 1.1.

First, it follows from the Definitions 1.1-1.2 that $(\rho^{\delta-1})_{x}\in
L^{\infty}$ for the case $0<\delta<1$, and $(\ln\rho)_{x}\in L^{\infty}$ for
the case $\delta=1$, which, along with $\rho^{\gamma-1}\in C([0,T];H^{2})$,
means that the vacuum occurs if and only if in the far field.

Second, we introduce some physical quantities that will be used in this paper:

$\begin{split}m(t)=&\int\rho(t,x){\rm{d}}x\quad\textrm{(total mass)},\\\
\mathbb{P}(t)=&\int\rho(t,x)u(t,x){\rm{d}}x\quad\textrm{(momentum)},\\\
E_{k}(t)=&\frac{1}{2}\int\rho(t,x)u^{2}(t,x){\rm{d}}x\quad\textrm{ (total
kinetic energy)},\\\
E(t)=&E_{k}(t)+\int\frac{P(t,x)}{\gamma-1}{\rm{d}}x\quad\textrm{ (total
energy)}.\end{split}$

Actually, it follows from the definitions that the regular solution satisfies
the conservation of total mass (see Lemma 3.3) and momentum (see Lemma 4.1).
Furthermore, it satisfies the energy equality (see Lemma 3.1 and its proof in
Appendix B). Note that the conservation of momentum is not clear for the
strong solution with vacuum to the flows of constant viscosities Jiu-Li-Ye
[22] (see an example on non-conservation of momentum in [49] ). In this sense,
the definitions of regular solutions here are consistent with the physical
background of CNS.

The regular solutions select velocity in a physically reasonable way when the
density approaches the vacuum at far fields. Under the help of this notion of
solutions, the momentum equations can be reformulated into a special quasi-
linear parabolic system with some possible singular source terms near the
vacuum, and the coefficients in front of $u_{xx}$ will tend to $\infty$ as
$\rho\rightarrow 0$ in the far field for $0<\delta<1$. However, the problems
become trackable through some elaborate arguments on the energy estimates.

For the case $0<\delta<1$, the global-in-time well-posedness result reads as:

######  Theorem 1.1.

Assume $\delta$ and $\gamma$ satisfy

$0<\delta<1,\quad\gamma>1,\quad\gamma\geq\delta+\frac{1}{2}.$ (1.12)

If the initial data $(\rho_{0},u_{0})$ satisfy

$0<\rho_{0}\in L^{1},\quad(\rho_{0}^{\gamma-1},u_{0})\in
H^{2},\quad(\rho_{0}^{\delta-1})_{x}\in H^{1},$ (1.13)

and the initial compatibility conditions

$\partial_{x}u_{0}=\rho_{0}^{\frac{1-\delta}{2}}g_{1},\quad\alpha\partial_{x}^{2}u_{0}=\rho_{0}^{1-\delta}g_{2},$
(1.14)

for some $g_{i}\in L^{2}(i=1,2)$, then the Cauchy problem (1.7)-(1.8) admits a
unique global classical solution $(\rho,u)$ in $(0,\infty)\times\mathbb{R}$
satisfying that, for any $0<T<\infty$, $(\rho,u)$ is a regular one in
$[0,T]\times\mathbb{R}$ as defined in Definition 1.1, and

$\begin{split}&\rho\in C([0,T];L^{1}),\quad(\rho^{\gamma-1})_{t}\in
C([0,T];H^{1}),\quad(\rho^{\delta-1})_{tx}\in C([0,T];L^{2}),\\\
&t^{\frac{1}{2}}\rho^{\frac{\delta-1}{2}}u_{tx}\in L^{\infty}([0,T];L^{2})\cap
L^{2}([0,T];D^{1}),\quad t^{\frac{1}{2}}u_{tt}\in
L^{2}([0,T];L^{2}).\end{split}$ (1.15)

######  Remark 1.2.

First, observe that there are no smallness conditions imposed on
$(\rho_{0},u_{0})$.

Second, one can find the following class of initial data $(\rho_{0},u_{0})$
satisfying the conditions (1.13)-(1.14):

$\rho_{0}(x)=\frac{1}{1+x^{2\sigma}},\quad u_{0}(x)\in C_{0}^{2}(\mathbb{R}),$

where
$\max\big{\\{}\frac{3}{4},\frac{1}{4(\gamma-1)}\big{\\}}<\sigma<\frac{1}{4(1-\delta)}.$
Particularly, when $\partial_{x}u_{0}$ is compactly supported, the
compatibility conditions (1.14) are satisfied automatically.

Moreover, the compatibility conditions (1.14) are also necessary for the
existence of the unique regular solution $(\rho,u)$ obtained in Theorem 1.1.
In particular, the one shown in $\eqref{e1.8}_{2}$ plays a key role in the
derivation of $u_{t}\in L^{\infty}([0,T_{*}];L^{2}(\mathbb{R}))$, which will
be used in the uniform estimates for $|u|_{D^{2}}$.

For the case $\delta=1$, the global-in-time well-posedness result reads as:

######  Theorem 1.2.

Assume $\delta=1$ and $\gamma>\frac{3}{2}$. If the initial data
$(\rho_{0},u_{0})$ satisfy

$0<\rho_{0}\in L^{1},\quad(\rho_{0}^{\gamma-1},u_{0})\in
H^{2},\quad(\ln\rho_{0})_{x}\in H^{1},$ (1.16)

then the Cauchy problem (1.7)-(1.8) admits a unique global classical solution
$(\rho,u)$ in $(0,\infty)\times\mathbb{R}$ satisfying that, for any
$0<T<\infty$, $(\rho,u)$ is a regular one in $[0,T]\times\mathbb{R}$ as
defined in Definition 1.2, and

$\begin{split}&\rho\in C([0,T];L^{1}),\quad(\rho^{\gamma-1})_{t}\in
C([0,T];H^{1}),\quad(\ln\rho)_{tx}\in C([0,T];L^{2}),\\\
&t^{\frac{1}{2}}u_{tx}\in L^{\infty}([0,T];L^{2})\cap L^{2}([0,T];D^{1}),\quad
t^{\frac{1}{2}}u_{tt}\in L^{2}([0,T];L^{2}).\end{split}$ (1.17)

######  Remark 1.3.

First, observe that there are no smallness conditions imposed on
$(\rho_{0},u_{0})$.

Second, one can find the following class of initial data $(\rho_{0},u_{0})$
satisfying (1.16):

$\rho_{0}(x)=\frac{1}{1+x^{2\sigma}},\quad u_{0}(x)\in C_{0}^{2}(\mathbb{R}),$

where $\sigma>\max\big{\\{}\frac{3}{4},\frac{1}{4(\gamma-1)}\big{\\}}.$

Moreover, it should be pointed out that the theory established in Theorem 1.2
can be applied to the viscous Saint-Venant system for the motion of shallow
water (i.e., $\delta=1$, $\gamma=2$, see Gerbeau-Perthame [17]).

######  Remark 1.4.

It is worth pointing out that in the domain $\Omega=[0,1]$, Li-Li-Xin [28]
proved that the entropy weak solutions for general large initial data
satisfying finite entropy to the corresponding initial-boundary value problem
of the system (1.7) with $\delta>\frac{1}{2}$ exist globally in time.
Moreover, the authors showed that for more regular initial data, there is a
global entropy weak solution which is unique and regular with well-defined
velocity field for short time, and the interface of initial vacuum propagates
along the particle path during this time period. Furthermore, the dynamics of
weak solutions and vacuum states are investigated rigorously. It is shown that
for any global entropy weak solution, any (possibly existing) vacuum state
must vanish within finite time, which means that there exist some time
$T_{0}>0$ (depending on initial data) and a constant $\rho_{-}$ such that

$\begin{split}\inf_{x\in\overline{\Omega}}\rho(t,x)\geq\rho_{-}>0\quad\text{for}\quad
t\geq T_{0}.\end{split}$ (1.18)

The velocity (even if regular enough and well-defined) blows up in finite time
as the vacuum states vanish, and after the vanishing of vacuum states, the
global entropy weak solution becomes a strong solution and tends to the non-
vacuum equilibrium state exponentially in time.

According to the conclusions obtained in Theorems 1.1-1.2, one finds that if
we replaced the domain $\Omega=[0,1]$ by the whole space $\mathbb{R}$, at
least for sufficiently smooth initial values, the phenomena on vacuum states
observed in (1.18) does not happen again. Actually, for the global regular
solutions obtained in Theorems 1.1-1.2 to the Cauchy problem (1.7)-(1.8), it
is obvious that

$\inf_{x\in\mathbb{R}}\rho(t,x)=0\quad\text{for any }\quad t\geq 0.$

###  1.2. Remarks on the asymptotic behavior of $u$

Moreover, in contrast to the classical theory in the case of the constant
viscosity [21, 37], one can show that the $L^{\infty}$ norm of $u$ of the
global regular solution we obtained does not decay to zero as time $t$ goes to
infinity. First, based on the physical quantities introduced in Remark 1.1, we
define a solution class as follows:

######  Definition 1.3.

Let $T>0$ be any constant. For the Cauchy problem (1.7)-(1.8), a classical
solution $(\rho,u)$ is said to be in $D(T)$ if $(\rho,u)$ satisfies the
following conditions:

$\begin{split}&(\textrm{A})\quad\text{Conservation of total mass:}\\\
&\qquad\qquad\qquad 0<m(0)=m(t)<\infty\ \text{for any}\ t\in[0,T];\\\
&(\textrm{B})\quad\text{Conservation of momentum:}\\\ &\qquad\qquad\qquad
0<|\mathbb{P}(0)|=|\mathbb{P}(t)|<\infty\ \text{for any}\ t\in[0,T];\\\
&(\textrm{C})\quad\text{Finite kinetic energy:}\\\ &\qquad\qquad\qquad
0<E_{k}(t)<\infty\ \text{for any}\ t\in[0,T].\end{split}$

Then one has:

######  Theorem 1.3.

Assume $\gamma>1$ and $\delta\geq 0$. Then for the Cauchy problem (1.7)-(1.8),
there is no classical solution $(\rho,u)\in D(\infty)$ satisfying

$\limsup_{t\rightarrow\infty}|u(t,\cdot)|_{\infty}=0.$ (1.19)

According to Theorems 1.1-1.3 and Remark 1.1, one shows that

######  Corollary 1.1.

Assume that $|\mathbb{P}(0)|>0$. Then for the Cauchy problem (1.7)-(1.8), the
global regular solutions $(\rho,u)$ obtained in Theorems 1.1-1.2 do not
satisfy (1.19).

The rest of the paper is organized as follows: §$2$-§$3$ are devoted to
establishing the global well-posedness of regular solutions with large data
stated in Theorems 1.1-1.2. We start with the reformulation of the original
problem (1.7)-(1.8) as (2.2) (resp. (2.14)) in terms of the new variables for
the case $0<\delta<1$ (resp. $\delta=1$), and establish the local-in-time
well-posedness of smooth solutions to the reformulated problems under the
assumption that the initial density keeps positive for all point
$x\in\mathbb{R}$ but decays to zero as $|x|\rightarrow\infty$ in §2. The local
smooth solutions to (2.2) and (2.14) are shown to be global ones in §3 by
deriving the global-in-time a priori estimates through energy methods. Here,
we have employed some arguments due to Bresch-Desjardins [2, 3] and Bresch-
Desjardins-Lin [4] to deal with the strong degeneracy of the density-dependent
viscosity $\mu(\rho)=\alpha\rho^{\delta}$, which include the well-known B-D
entropy estimates (see Lemma 3.2) and the effective velocity (see Lemma 3.7).
Finally, in §4, we give the proofs for the non-existence theories of global
regular solutions with $L^{\infty}$ decay on $u$ shown in Theorem 1.3 and
Corollary 1.1. Furthermore, we give several appendixes to list some lemmas
that are frequently used in our proof, and review some related local-in-time
well-posedness theories for multi-dimensional flows of degenerate viscosities.

##  2\. Local-in-time well-posedness

This section will be devoted to providing the local-in-time well-posedenss of
regular solutions with far field vacuum to the Cauchy problem (1.7)-(1.8) for
the cases $0<\delta\leq 1$.

###  2.1. Case 1: $0<\delta<1$

For the case $0<\delta<1$, the corresponding local-in-time well-posedenss can
be achieved by the following four steps.

####  2.1.1. Reformulation

Via introducing the following new variables

$\phi=\frac{A\gamma}{\gamma-1}\rho^{\gamma-1},\quad\psi=\frac{\delta}{\delta-1}(\rho^{\delta-1})_{x},$
(2.1)

the Cauchy problem (1.7)-(1.8) can be rewritten as

$\left\\{\begin{aligned} &\ \phi_{t}+u\phi_{x}+(\gamma-1)\phi
u_{x}=0,\\\\[6.0pt] &\ u_{t}+uu_{x}+\phi_{x}-a\alpha\phi^{2e}u_{xx}=\alpha\psi
u_{x},\\\\[6.0pt] &\ \psi_{t}+(u\psi)_{x}+(\delta-1)\psi
u_{x}+a\delta\phi^{2e}u_{xx}=0,\\\\[6.0pt] &\
(\phi,u,\psi)|_{t=0}=(\phi_{0},u_{0},\psi_{0})\\\\[6.0pt]
=&\Big{(}\frac{A\gamma}{\gamma-1}\rho^{\gamma-1}_{0}(x),u_{0}(x),\frac{\delta}{\delta-1}(\rho^{\delta-1}_{0}(x))_{x}\Big{)}\quad\text{for}\quad
x\in\mathbb{R},\\\\[6.0pt] &\
(\phi,u,\psi)\to(0,0,0)\quad\text{as}\quad|x|\to\infty\quad\text{for}\quad
t\geq 0,\end{aligned}\right.$ (2.2)

where the constants $a$ and $e$ are given by

$a=\Big{(}\frac{A\gamma}{\gamma-1}\Big{)}^{\frac{1-\delta}{\gamma-1}}\quad\text{and}\quad
e=\frac{\delta-1}{2(\gamma-1)}<0.$

####  2.1.2. Discussion on $\nabla\rho_{0}^{\frac{\delta-1}{2}}\in L^{4}$ in
[49]

In [49], for the 3-D degenerate system (1.1)-(1.5), via introducing the
following new variables

$\phi=\frac{A\gamma}{\gamma-1}\rho^{\gamma-1},\quad\psi=\frac{\delta}{\delta-1}\nabla\rho^{\delta-1},$

Xin-Zhu studied the following reformulated problem:

$\left\\{\begin{aligned} &\
\phi_{t}+u\cdot\nabla\phi+(\gamma-1)\phi\text{div}u=0,\\\\[6.0pt] &\
u_{t}+u\cdot\nabla u+\nabla\phi+a\phi^{2e}Lu=\psi\cdot Q(u),\\\\[6.0pt] &\
\psi_{t}+\nabla(u\cdot\psi)+(\delta-1)\psi\text{div}u+a\delta\phi^{2e}\nabla\text{div}u=0,\\\\[6.0pt]
&\ (\phi,u,\psi)|_{t=0}=(\phi_{0},u_{0},\psi_{0})\\\\[6.0pt]
=&\Big{(}\frac{A\gamma}{\gamma-1}\rho^{\gamma-1}_{0}(x),u_{0}(x),\frac{\delta}{\delta-1}\nabla\rho^{\delta-1}_{0}(x)\Big{)}\quad\text{for}\quad
x\in\mathbb{R}^{3},\\\\[6.0pt] &\
(\phi,u,\psi)\rightarrow(0,0,0)\quad\text{as}\quad|x|\rightarrow\infty\quad\text{for}\quad
t\geq 0.\end{aligned}\right.$ (2.3)

For the above problem, the local-in-time existence of the unique regular
solution with far field vacuum has been proven in [49] (see Lemma C.1 in
Appendix C). It is worth pointing out that in the initial assumptions of Lemma
C.1, Xin-Zhu required that

$\nabla\rho_{0}^{\frac{\delta-1}{2}}\in L^{4},$ (2.4)

which, actually, is only used in the initial data’s approximation process from
the non-vacuum flow to the flow with far field vacuum, and does not appear in
the corresponding energy estimates process in the proof for the local
existence of the regular solution shown in [49]. Therefore, in Lemma C.1, the
regularity of the quantity $\nabla\rho^{\frac{\delta-1}{2}}$ in positive time
has not been mentioned.

Actually, in Section 3.6 (page 136) of [49], in order to approximate the flow
with far field vacuum by the non-vacuum flow, for any $\eta\in(0,1)$, one sets

$\phi^{\eta}_{0}=\phi_{0}+\eta,\quad\psi^{\eta}_{0}=\frac{a\delta}{\delta-1}\nabla(\phi_{0}+\eta)^{2e},\quad
h^{\eta}_{0}=(\phi_{0}+\eta)^{2e}.$

Then the initial compatibility conditions of the perturbed initial data can be
given as

$\begin{cases}\displaystyle\nabla u_{0}=(\phi_{0}+\eta)^{-e}g^{\eta}_{1},\quad
aLu_{0}=(\phi_{0}+\eta)^{-2e}g^{\eta}_{2},\\\\[12.0pt]
\displaystyle\nabla\Big{(}a(\phi_{0}+\eta)^{2e}Lu_{0}\Big{)}=(\phi_{0}+\eta)^{-e}g^{\eta}_{3},\end{cases}$

where $g^{\eta}_{i}$ ($i=1,2,3$) are given as

$\begin{cases}\displaystyle
g^{\eta}_{1}=\frac{\phi^{-e}_{0}}{(\phi_{0}+\eta)^{-e}}g_{1},\quad
g^{\eta}_{2}=\frac{\phi^{-2e}_{0}}{(\phi_{0}+\eta)^{-2e}}g_{2},\\\\[8.0pt]
\displaystyle
g^{\eta}_{3}=\frac{\phi^{-3e}_{0}}{(\phi_{0}+\eta)^{-3e}}\Big{(}g_{3}-\frac{a\eta\nabla\phi^{2e}_{0}}{\phi_{0}+\eta}\phi^{e}_{0}Lu_{0}\Big{)}.\end{cases}$
(2.5)

According to the definition of $\psi_{0}^{\eta}$, one has

$\begin{split}\psi_{0}^{\eta}=&\frac{a\delta}{\delta-1}\Big{(}\frac{\phi_{0}}{\phi_{0}+\eta}\Big{)}^{1-2e}\nabla\phi_{0}^{2e},\\\
\nabla\psi_{0}^{\eta}=&\frac{a\delta}{\delta-1}\Big{(}\Big{(}\frac{\phi_{0}}{\phi_{0}+\eta}\Big{)}^{1-2e}\nabla^{2}\phi_{0}^{2e}+\frac{(1-2e)\eta\nabla\phi_{0}\otimes\nabla\phi_{0}^{2e}}{(\phi_{0}+\eta)^{2}}\Big{(}\frac{\phi_{0}}{\phi_{0}+\eta}\Big{)}^{-2e}\Big{)}\\\
=&\frac{a\delta}{\delta-1}\Big{(}\Big{(}\frac{\phi_{0}}{\phi_{0}+\eta}\Big{)}^{1-2e}\Big{(}\nabla^{2}\phi_{0}^{2e}+\frac{2(1-2e)}{e}\frac{\eta}{\phi_{0}+\eta}\nabla\phi_{0}^{e}\otimes\nabla\phi_{0}^{e}\Big{)}\Big{)},\\\
\nabla^{2}\psi_{0}^{\eta}=&\frac{a\delta}{\delta-1}\Big{(}\Big{(}\frac{\phi_{0}}{\phi_{0}+\eta}\Big{)}^{1-2e}\Big{(}\nabla^{3}\phi_{0}^{2e}+\frac{1-2e}{e}\frac{\eta\phi_{0}^{-e}}{\phi_{0}+\eta}\nabla\phi_{0}^{e}\otimes\nabla^{2}\phi_{0}^{2e}\Big{)}\\\
&+\frac{2(1-2e)}{e}\Big{(}\frac{-\eta\phi_{0}^{1-e}}{e(\phi_{0}+\eta)^{2}}(\nabla\phi_{0}^{e})^{3}+\frac{2\eta}{\phi_{0}+\eta}\nabla\phi_{0}^{e}\otimes\nabla^{2}\phi_{0}^{e}\Big{)}\Big{)}.\end{split}$
(2.6)

From the formula (2.6), one can find that the condition (2.4) is only used to
guarantee the uniform boundedness of $|\psi^{\eta}_{0}|_{D^{1}_{*}}$ with
respect to $\eta$. Then it follows from the initial assumptions of Lemma C.1
that, there exists a $\eta_{1}>0$ such that for any $0<\eta\leq\eta_{1}$,

$\begin{split}1+\eta+\|\phi^{\eta}_{0}-\eta\|_{3}+|\psi^{\eta}_{0}|_{D^{1}_{*}\cap
D^{2}}+\|u_{0}\|_{3}+|g^{\eta}_{1}|_{2}+|g^{\eta}_{2}|_{2}+|g^{\eta}_{3}|_{2}\\\
+\|(h^{\eta}_{0})^{-1}\|_{L^{\infty}\cap D^{1,6}\cap D^{2,3}\cap
D^{3}}+\|\nabla h^{\eta}_{0}/h^{\eta}_{0}\|_{L^{\infty}\cap L^{6}\cap
D^{1,3}\cap D^{2}}&\leq c_{0},\end{split}$

where $c_{0}$ is a positive constant independent of $\eta$. Based on such kind
of initial approximation process, the Cauchy problem (2.3) can be solved
locally in time by studying the corresponding Cauchy problem away from the
vacuum and then passing to the limit as $\eta\to 0$. The details can be found
in [49].

For 1-D case, the corresponding initial data’s approximation process from the
non-vacuum flow to the flow with far field vacuum can also be achieved from
the above process with some minor modifications. Similarly, for any
$\eta\in(0,1)$, one sets

$\phi^{\eta}_{0}=\phi_{0}+\eta,\quad\psi^{\eta}_{0}=\frac{a\delta}{\delta-1}\Big{(}(\phi_{0}+\eta)^{2e}\Big{)}_{x},\quad
h^{\eta}_{0}=(\phi_{0}+\eta)^{2e}.$

Then the initial compatibility conditions of the perturbed initial data can be
given as

$\displaystyle\partial_{x}u_{0}=(\phi_{0}+\eta)^{-e}g^{\eta}_{1},\quad
a\alpha\partial^{2}_{x}u_{0}=(\phi_{0}+\eta)^{-2e}g^{\eta}_{2},$

where $g^{\eta}_{i}$ ($i=1,2$) are given in the first line of (2.5). Then it
follows from (1.13)-(1.14) that there also exists a $\eta_{1}>0$ such that for
any $0<\eta<\eta_{1}$,

$\begin{split}1+\eta+\|\phi_{0}^{\eta}-\eta\|_{2}+\|\psi_{0}^{\eta}\|_{1}+\|u_{0}\|_{2}+|g_{1}^{\eta}|_{2}&\\\
+|g_{2}^{\eta}|_{2}+\|(h^{\eta}_{0})^{-1}\|_{L^{\infty}\cap D^{1}\cap
D^{2}}+\|(\ln h^{\eta}_{0})_{x}\|_{1}&\leq c_{0}.\end{split}$

However, compared with the 3D case in [49], it is worth pointing out that the
following initial assumption

$\partial_{x}\rho_{0}^{\frac{\delta-1}{2}}\in L^{4}$ (2.7)

is no longer required for the current 1D problem. Actually, one easily has

$\partial_{x}\rho_{0}^{\frac{\delta-1}{2}}=\frac{1}{2}\rho_{0}^{\frac{1-\delta}{2}}\partial_{x}\rho_{0}^{\delta-1},$

which, along with the fact $\partial_{x}\rho_{0}^{\delta-1}\in H^{1}$, and the
Sobolev embedding theorem, implies that (2.7) holds automatically.

####  2.1.3. Local-in-time well-posedness

Since the mathematical structure of the system considered in problem (2.2) is
exactly the same as the one in problem (2.3), then based on the discussion in
Section 2.1.2 and Lemma C.1, one can obtain the following local-in-time well-
posedness theory in 1-D space.

######  Lemma 2.1.

Assume that $0<\delta<1$ and $\gamma>1$. If the initial data
$(\rho_{0},u_{0})$ satisfy (1.13)-(1.14) except $\rho_{0}\in L^{1}$, then
there exist a time $T_{*}>0$ and a unique regular solution $(\rho,u)$ in
$[0,T_{*}]\times\mathbb{R}$ to the Cauchy problem (1.7)-(1.8) satisfying
(1.15) with $T$ replaced by $T_{*}$ except $\rho\in C([0,T_{*}];L^{1})$.

####  2.1.4. $\rho\in C([0,T_{*}];L^{1}$)

It should be emphasized that the initial assumption $\rho_{0}\in L^{1}$ in
Theorem 1.1 is only used in the proof of global-in-time existence of the
unique regular solution satisfying (1.15), which is not required in the proof
for the local-in-time existence of the general regular solution of [49]. Thus,
in the following lemma, we show that actually, the solution obtained in Lemma
2.1 still satisfies:

######  Lemma 2.2.

Let $(\rho,u)$ in $[0,T_{*}]\times\mathbb{R}$ be the solution to the Cauchy
problem (1.7)-(1.8) obtained in Lemma 2.1. Then

$\rho\in C([0,T_{*}];L^{1})\quad\text{if}\quad\rho_{0}\in
L^{1}\quad\text{additionally}.$ (2.8)

###### Proof.

First, one proves that

$\rho\in L^{\infty}([0,T_{*}];L^{1}).$ (2.9)

For this purpose, let $f:\mathbb{R}^{+}\to\mathbb{R}$ satisfy

$\displaystyle f(s)=\begin{cases}1,&s\in[0,\frac{1}{2}]\\\ \text{non-negative
polynomial},&s\in[\frac{1}{2},1]\\\ e^{-s},&s\in[1,\infty)\end{cases}$

such that $f\in C^{2}$. Then there exists a generic constant $C_{*}>0$ such
that $|f^{\prime}(s)|\leq C_{*}f(s)$. Define, for any $R>0$,
$f_{R}(x)=f(\frac{|x|}{R})$. Then, according to Lemma 2.1, one can obtain that
for any given $R>0$,

$\displaystyle\int\Big{(}(|\rho_{t}|+|\rho_{x}u|+|\rho u_{x}|)f_{R}(x)+\rho
f_{R}(x)+|\rho u\partial_{x}f_{R}(x)|\Big{)}\text{d}x<$ $\displaystyle\infty.$

Multiplying $\eqref{e1.1}_{1}$ by $f_{R}(x)$ and integrating over
$\mathbb{R}$, one has

$\begin{split}\frac{\text{d}}{\text{d}t}\int\rho
f_{R}(x)\text{d}x=&-\int(\rho_{x}u+\rho u_{x})f_{R}(x)\text{d}x\\\ =&\int\rho
uf^{\prime}_{R}(x)\text{d}x\leq C|u|_{\infty}\int\rho
f_{R}(x)\text{d}x\end{split}$

for some positive constant $C$ depending only on the initial data, $A$,
$\gamma$, $\delta$, $C_{*}$ and $T_{*}$ but independent of $R$, which, along
with the Gronwall inequality, implies that

$\int\rho f_{R}(x)\text{d}x\leq C\quad\text{for}\quad t\in[0,T_{*}].$ (2.10)

Since $\rho f_{R}(x)\to\rho$ almost everywhere as $R\to\infty$, then it
follows from the Fatou lemma (see Lemma A.2 in Appendix A) that

$\int\rho\text{d}x\leq\liminf\limits_{R\to\infty}\int\rho
f_{R}(x)\text{d}x\leq C\quad\text{for}\quad t\in[0,T_{*}].$ (2.11)

Second, according to $\eqref{e1.1}_{1}$, one has

$|\rho_{t}|_{1}\leq C(|\rho_{x}u|_{1}+|\rho u_{x}|_{1})\leq
C(|\psi|_{2}|u|_{2}|\rho|_{\infty}^{2-\delta}+|\rho|_{2}|u_{x}|_{2})\leq C,$
(2.12)

which, along with (2.9) and the Sobolev embedding theorem, implies that (2.8)
holds.

The proof of Lemma 2.2 is complete. ∎

###  2.2. Case 2: $\delta=1$

For the case $\delta=1$, the corresponding local-in-time well-posedenss can be
achieved by the following three steps.

####  2.2.1. Reformulation

Via introducing the following new variables

$\phi=\frac{A\gamma}{\gamma-1}\rho^{\gamma-1},\quad\psi=\frac{1}{\gamma-1}\frac{\phi_{x}}{\phi}=(\ln\rho)_{x},$
(2.13)

the Cauchy problem (1.7)-(1.8) can be rewritten as

$\left\\{\begin{aligned} &\ \phi_{t}+u\phi_{x}+(\gamma-1)\phi
u_{x}=0,\\\\[6.0pt] &\ u_{t}+uu_{x}+\phi_{x}-\alpha u_{xx}=\alpha\psi
u_{x},\\\\[6.0pt] &\ \psi_{t}+(u\psi)_{x}+u_{xx}=0,\\\\[6.0pt] &\
(\phi,u,\psi)|_{t=0}=(\phi_{0},u_{0},\psi_{0})\\\\[6.0pt]
=&\Big{(}\frac{A\gamma}{\gamma-1}\rho^{\gamma-1}_{0}(x),u_{0}(x),(\ln\rho_{0}(x))_{x}\Big{)}\quad\text{for}\quad
x\in\mathbb{R},\\\\[6.0pt] &\
(\phi,u,\psi)\to(0,0,0)\quad\text{as}\quad|x|\to\infty\quad\text{for}\quad
t\geq 0.\end{aligned}\right.$ (2.14)

####  2.2.2. Local-in-time existence

In [31], for the 2-D degenerate system (1.1)-(1.5), via introducing the
following new variables

$\tilde{\phi}=\rho^{\frac{\gamma-1}{2}},\quad\psi=\frac{2}{\gamma-1}\frac{\nabla\tilde{\phi}}{\tilde{\phi}}=\nabla\ln\rho,$

Li-Pan-Zhu studied the following reformulated problem:

$\left\\{\begin{aligned} &\
\tilde{\phi_{t}}+u\cdot\nabla\tilde{\phi}+\frac{\gamma-1}{2}\tilde{\phi}\operatorname{\mathrm{div}}u=0,\\\\[2.0pt]
&\ u_{t}+u\cdot\nabla
u+\frac{2A\gamma}{\gamma-1}\tilde{\phi}\nabla\tilde{\phi}+Lu=\psi\cdot
Q(u),\\\\[2.0pt] &\
\psi_{t}+\nabla(u\cdot\psi)+\nabla\operatorname{\mathrm{div}}u=0,\\\\[2.0pt]
&(\tilde{\phi},u,\psi)|_{t=0}=(\tilde{\phi}_{0},u_{0},\psi_{0})\\\\[2.0pt]
=&\Big{(}\rho^{\frac{\gamma-1}{2}}_{0}(x),u_{0}(x),\nabla\ln\rho_{0}(x)\Big{)}\quad\text{for}\quad
x\in\mathbb{R}^{2},\\\\[2.0pt] &\
(\tilde{\phi},u,\psi)\to(0,0,0)\quad\text{as}\quad|x|\to\infty\quad\text{for}\quad
t\geq 0.\end{aligned}\right.$ (2.15)

Since the mathematical structure of the system considered in problem (2.14) is
exactly the same as the one in problem (2.15), then based on Lemma C.2, one
can obtain the following local-in-time well-posedness theory in 1-D space.

######  Lemma 2.3.

Assume that $\delta=1$ and $\gamma>1$. If the initial data $(\rho_{0},u_{0})$
satisfy (1.16) except $\rho_{0}\in L^{1}$, then there exist a time $T_{*}>0$
and a unique regular solution $(\rho,u)$ in $[0,T_{*}]\times\mathbb{R}$ to the
Cauchy problem (1.7)-(1.8) satisfying (1.17) with $T$ replaced by $T_{*}$
except $\rho\in C([0,T_{*}];L^{1})$.

####  2.2.3. $\rho\in C([0,T_{*}];L^{1})$

As mentioned in Section 2.1.4, one also needs to show that

######  Lemma 2.4.

Let $(\rho,u)$ in $[0,T_{*}]\times\mathbb{R}$ be the solution to the Cauchy
problem (1.7)-(1.8) obtained in Lemma 2.3. Then

$\rho\in C([0,T_{*}];L^{1})\quad\text{if}\quad\rho_{0}\in
L^{1}\quad\text{additionally}.$ (2.16)

The proof of this lemma is similar to that of Lemma 2.2. Here we omit it.

##  3\. Global-in-time well-posedness

This section is devoted to proving Theorems 1.1-1.2, and we always assume
$0<\delta\leq 1$ and $\gamma>1$. Let $T>0$ be some time and $(\rho,u)$ be
regular solutions to the problem (1.7)-(1.8) in $[0,T]\times\mathbb{R}$
obtained in Lemmass 2.1-2.2 and Lemmas 2.3-2.4. The main aim in the rest of
this section is to establish the global-in-time a priori estimates for these
solutions. Hereinafter, we denote $C_{0}$ (resp. $C$) a generic positive
constant depending only on $(\rho_{0},u_{0},A,\gamma,\alpha,\delta)$ (resp.
$(C_{0},T)$), which may be different from line to line.

###  3.1. The $L^{\infty}$ estimate of $\rho$

We consider the upper bound of the mass density $\rho$ in
$[0,T]\times\mathbb{R}$. First, the standard energy estimates yields that

######  Lemma 3.1.

For any $T>0$, it holds that

$\int\left(\frac{1}{2}\rho
u^{2}+\frac{A}{\gamma-1}\rho^{\gamma}\right)(t,\cdot){\rm{d}}x+\alpha\int_{0}^{t}\int\rho^{\delta}u_{x}^{2}{\rm{d}}x{\rm{d}}s\leq
C_{0}\quad{\rm{for}}\ \ 0\leq t\leq T.$

Second, we give the well-known B-D entropy estimate.

######  Lemma 3.2 ([5]).

For any $T>0$, it holds that

$\int\left(\frac{1}{2}\rho\left|u+\varphi(\rho)_{x}\right|^{2}+\frac{A}{\gamma-1}\rho^{\gamma}\right)(t,\cdot){\rm{d}}x+\alpha
A\gamma\int_{0}^{t}\int\rho^{\gamma+\delta-3}\rho_{x}^{2}{\rm{d}}x{\rm{d}}s\leq
C_{0}$

for $0\leq t\leq T$, where
$\varphi^{\prime}(\rho)=\frac{\mu(\rho)}{\rho^{2}}$.

The conclusions obtained in Lemmas 3.1-3.2 are classical, and the
corresponding proofs can be found in Appendix B.

Next we show the regular solution $(\rho,u)$ keeps the conservation of total
mass.

######  Lemma 3.3.

For any $T>0$, it holds that

$m(t)=m(0)\quad{\rm{for}}\ \ 0\leq t\leq T.$

###### Proof.

It follows from $\eqref{e1.1}_{1}$ that

$\frac{\text{d}}{\text{d}t}\int\rho(t,\cdot)\text{d}x=-\int(\rho
u)_{x}(t,\cdot)\text{d}x=0,$

where one has used fact that

$\rho\in C([0,T];L^{1})\quad\text{and}\quad\rho u(t,\cdot)\in
W^{1,1}(\mathbb{R}).$

The proof of Lemma 3.3 is complete. ∎

Now we are ready to give the uniform upper bound of the density.

######  Lemma 3.4.

For any $T>0$, it holds that

$|\rho(t,\cdot)|_{\infty}\leq C_{0}\quad{\rm{for}}\ \ 0\leq t\leq T.$

###### Proof.

First, integrating the equation $\eqref{e1.1}_{2}$ over $(-\infty,x]$ with
respect to $x$, one has

$\frac{\text{d}}{\text{d}t}\int_{-\infty}^{x}\rho u(t,z){\rm{d}}z+\rho
u^{2}+A\rho^{\gamma}=\alpha\rho^{\delta}u_{x},$ (3.1)

where one has used the far field behavior $\eqref{e1.3}_{2}$.

Denote $\displaystyle\xi(t,x)=\int_{-\infty}^{x}\rho u(t,z){\rm{d}}z$. Then it
follows from (3.1) that

$\xi_{t}+u\xi_{x}+\alpha\rho^{\delta-1}(\rho_{t}+u\rho_{x})+A\rho^{\gamma}=0.$
(3.2)

Second, let $y(t;x)$ be the solution of the following problem

$\left\\{\begin{aligned} &\frac{\text{d}y(t;x)}{\text{d}t}=u(t,y(t;x)),\\\
&y(0;x)=x.\end{aligned}\right.$ (3.3)

It follows from (3.2) that

$\frac{\text{d}}{\text{d}t}\left(\xi+\frac{\alpha}{\delta}\rho^{\delta}\right)(t,y(t;x))\leq
0,$

which, along with the fact

$\xi(t,y(t;x))\leq|\sqrt{\rho}u(t,\cdot)|_{2}|\rho(t,\cdot)|_{1}^{\frac{1}{2}}\leq
C_{0}\quad{\rm{for}}\ \ 0\leq t\leq T,$ (3.4)

implies that

$\left(\xi+\frac{\alpha}{\delta}\rho^{\delta}\right)(t,y(t;x))\leq\xi(0,y(0;x))+\frac{\alpha}{\delta}\rho_{0}^{\delta}(x)\leq
C_{0}.$ (3.5)

Finally, according to (3.4)-(3.5), one can obtain that

$|\rho^{\delta}(t,\cdot)|_{\infty}\leq C_{0}\quad{\rm{for}}\ \ 0\leq t\leq T.$

The proof of Lemma 3.4 is complete.

∎

###  3.2. The $L^{2}$ estimate of $u$

We consider the $L^{2}$ estimate of $u$. For this purpose, one first needs to
show the following several auxiliary lemmas. The first one is on the $L^{q}$
estimate of $\rho^{\frac{1}{q}}u$ for any $2\leq q<\infty$.

######  Lemma 3.5.

Assume $2\leq q<\infty$. Then for any $T>0$, it holds that

$|\rho^{\frac{1}{q}}u(t,\cdot)|_{q}\leq C\quad{\rm{for}}\ \ 0\leq t\leq T,$

where the constant $C$ depends on $q$.

###### Proof.

First, multiplying $\eqref{e1.1}_{2}$ by $|u|^{p}u$ $(p\geq 2)$ and
integrating over $\mathbb{R}$, one has

$\begin{split}&\frac{1}{p+2}\frac{\text{d}}{\text{d}t}\big{|}\rho^{\frac{1}{p+2}}u\big{|}_{p+2}^{p+2}+\alpha(p+1)\big{|}\rho^{\frac{\delta}{2}}u^{\frac{p}{2}}u_{x}\big{|}_{2}^{2}\\\
=&A(p+1)\int\rho^{\gamma}|u|^{p}u_{x}{\rm{d}}x\\\
\leq&\frac{\alpha(p+1)}{2}\int\rho^{\delta}|u|^{p}u_{x}^{2}{\rm{d}}x+\frac{A^{2}(p+1)}{2\alpha}\int\rho^{2\gamma-\delta}|u|^{p}{\rm{d}}x.\end{split}$
(3.6)

When $p=2$, it follows from (3.6), Lemmas 3.1 and 3.3-3.4 that

$\begin{split}&\frac{1}{4}\frac{\text{d}}{\text{d}t}\big{|}\rho^{\frac{1}{4}}u\big{|}_{4}^{4}+\frac{3\alpha}{2}\big{|}\rho^{\frac{\delta}{2}}uu_{x}\big{|}_{2}^{2}\\\
\leq&C|\rho|_{\infty}^{2\gamma-\delta-1}|\rho^{\frac{1}{4}}u|_{4}|\rho^{\frac{1}{2}}u|_{2}|\rho|_{1}^{\frac{1}{4}}\leq
C(1+|\rho^{\frac{1}{4}}u|_{4}^{4}).\end{split}$ (3.7)

When $p>2$, it follows from (3.6), Lemmas 3.1 and 3.3-3.4 that

$\begin{split}&\frac{1}{p+2}\frac{\text{d}}{\text{d}t}\big{|}\rho^{\frac{1}{p+2}}u\big{|}_{p+2}^{p+2}+\frac{\alpha(p+1)}{2}\big{|}\rho^{\frac{\delta}{2}}u^{\frac{p}{2}}u_{x}\big{|}_{2}^{2}\\\
\leq&C|\rho|_{\infty}^{2\gamma-\delta-1}|\rho^{\frac{p-2}{p}}u^{\frac{(p-2)(p+2)}{p}}|_{\frac{p}{p-2}}|\rho^{\frac{2}{p}}u^{\frac{4}{p}}|_{\frac{p}{2}}\\\
\leq&C|\rho|_{\infty}^{2\gamma-\delta-1}|\rho^{\frac{1}{p+2}}u|_{p+2}^{\frac{(p-2)(p+2)}{p}}|\rho^{\frac{1}{2}}u|_{2}^{\frac{4}{p}}\\\
\leq&C|\rho^{\frac{1}{p+2}}u|_{p+2}^{\frac{(p-2)(p+2)}{p}}\leq
C\big{(}1+|\rho^{\frac{1}{p+2}}u|_{p+2}^{p+2}\big{)}.\end{split}$ (3.8)

Second, integrating (3.7) and (3.8) over $[0,t]$, one can obtain that for any
$p\geq 2$,

$\begin{split}&\big{|}\rho^{\frac{1}{p+2}}u\big{|}_{p+2}^{p+2}+\int_{0}^{t}\big{|}\rho^{\frac{\delta}{2}}u^{\frac{p}{2}}u_{x}\big{|}_{2}^{2}\text{d}s\\\
\leq&\big{|}\rho_{0}^{\frac{1}{p+2}}u_{0}\big{|}_{p+2}^{p+2}+C\Big{(}t+\int_{0}^{t}\big{|}\rho^{\frac{1}{p+2}}u\big{|}_{p+2}^{p+2}\text{d}s\Big{)},\end{split}$

which, along with the Gronwall inequality, yields that

$\begin{split}&\big{|}\rho^{\frac{1}{p+2}}u(t,\cdot)\big{|}_{p+2}^{p+2}+\int_{0}^{t}\big{|}\rho^{\frac{\delta}{2}}u^{\frac{p}{2}}u_{x}(s,\cdot)\big{|}_{2}^{2}{\rm{d}s}\\\
\leq&C\Big{(}1+\big{|}\rho_{0}^{\frac{1}{p+2}}u_{0}\big{|}_{p+2}^{p+2}\Big{)}\leq
C\Big{(}1+|\rho_{0}|_{1}|u_{0}|_{\infty}^{p+2}\Big{)}\leq C\quad{\rm{for}}\ \
0\leq t\leq T.\end{split}$ (3.9)

Finally, the desired conclusion can be achieved from (3.9) and Lemma 3.1.

The proof of Lemma 3.5 is complete. ∎

The following lemma gives the estimate of
$\int_{0}^{t}|\rho^{\iota}u(s,\cdot)|_{\infty}{\rm{d}}s$
$(\frac{1}{2}\leq\iota<1)$, which plays a crucial role in obtaining the
$L^{\infty}$ estimate of the so-called effective velocity $v$ (see (3.13)).

######  Lemma 3.6.

Assume

$\iota=\frac{1}{2}\quad\rm{for}\quad
0<\delta<1;\quad\frac{1}{2}<\iota<1\quad\rm{for}\quad\delta=1.$ (3.10)

Then for any $T>0$, it holds that

$\int_{0}^{t}|\rho^{\iota}u(s,\cdot)|_{\infty}{\rm{d}}s\leq C\quad{\rm{for}}\
\ 0\leq t\leq T.$

###### Proof.

Case 1: $0<\delta<1$. First, it follows direct calculations that

$\partial_{x}(\rho^{\frac{1}{2}}u)=\frac{1}{2}\rho^{\delta-\frac{3}{2}}\rho_{x}\rho^{\frac{1-\delta}{2}}\rho^{\frac{1-\delta}{2}}u+\rho^{\frac{1-\delta}{2}}\rho^{\frac{\delta}{2}}u_{x}.$

According to Lemmas 3.1-3.3 and 3.5, one has

$|\rho^{\delta-\frac{3}{2}}\rho_{x}|_{2}+|\rho^{\frac{1-\delta}{2}}u|_{\frac{2}{1-\delta}}+|\rho^{\frac{1-\delta}{2}}|_{\frac{2}{1-\delta}}+\int_{0}^{t}|\rho^{\frac{\delta}{2}}u_{x}|_{2}^{2}{\rm{d}}s\leq
C,$

which, along with Lemma 3.4, yields that

$\int_{0}^{t}|\partial_{x}(\rho^{\frac{1}{2}}u)(s,\cdot)|_{\frac{2}{2-\delta}}^{2}\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.11)

Second, notice that $\frac{2}{2-\delta}\in(1,2)$, then by the Sobolev
embedding theorem (or see Lemma A.1), one has

$|\rho^{\frac{1}{2}}u|_{\infty}\leq
C|\rho^{\frac{1}{2}}u|_{2}^{1-\aleph}|\partial_{x}(\rho^{\frac{1}{2}}u)|^{\aleph}_{\frac{2}{2-\delta}}\quad\text{with}\quad\aleph=\frac{1}{1+\delta}\in(0,1),$

which, along with Lemma 3.1 and (3.11), yields that

$\int_{0}^{t}|\rho^{\frac{1}{2}}u(s,\cdot)|_{\infty}{\rm{d}}s\leq
C\int_{0}^{t}\Big{(}1+|\partial_{x}(\rho^{\frac{1}{2}}u)|^{2}_{\frac{2}{2-\delta}}\Big{)}(s,\cdot)\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

Case 2: $\delta=1$. First, it follows direct calculations that

$\partial_{x}(\rho^{\iota}u)=\iota\rho^{-\frac{1}{2}}\rho_{x}\rho^{\iota-\frac{1}{2}}u+\rho^{\iota-\frac{1}{2}}\rho^{\frac{1}{2}}u_{x}.$

According to Lemmas 3.1-3.3, 3.5 and $\frac{1}{2}<\iota<1$, one has

$|\rho^{-\frac{1}{2}}\rho_{x}|_{2}+|\rho^{\iota-\frac{1}{2}}u|_{\frac{2}{2\iota-1}}+|\rho^{\iota-\frac{1}{2}}|_{\frac{2}{2\iota-1}}+\int_{0}^{t}|\rho^{\frac{1}{2}}u_{x}|_{2}^{2}{\rm{d}}s\leq
C,$

which yields that

$\int_{0}^{t}|\partial_{x}(\rho^{\iota}u)(s,\cdot)|_{\frac{1}{\iota}}^{2}\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.12)

Second, notice that $\frac{1}{\iota}\in(1,2)$, then by Sobolev embedding
theorem (or see Lemma A.1), one has

$|\rho^{\iota}u|_{\infty}\leq
C|\rho^{\iota}u|_{2}^{1-\Xi}|\partial_{x}(\rho^{\iota}u)|^{\Xi}_{\frac{1}{\iota}}\quad\text{with}\quad\Xi=\frac{1}{3-2\iota}\in(0,1),$

which, along with Lemmas 3.1, 3.4 and (3.12), yields that

$\int_{0}^{t}|\rho^{\iota}u(s,\cdot)|_{\infty}{\rm{d}}s\leq
C\int_{0}^{t}\Big{(}1+|\partial_{x}(\rho^{\iota}u)|^{2}_{\frac{1}{\iota}}\Big{)}(s,\cdot)\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

The proof of Lemma 3.6 is complete. ∎

Next we show the $L^{\infty}$ estimate of the effective velocity:

$v=u+\varphi(\rho)_{x}=u+\alpha\rho^{\delta-2}\rho_{x}.$ (3.13)

######  Lemma 3.7.

Assume additionally

$\gamma\geq\delta+\frac{1}{2}\quad\rm{for}\quad
0<\delta<1;\quad\gamma>\frac{3}{2}\quad\rm{for}\quad\delta=1.$ (3.14)

Then for any $T>0$, it holds that

$|v(t,\cdot)|_{\infty}\leq C\quad{\rm{for}}\ \ 0\leq t\leq T.$

###### Proof.

First, it follows from the system $\eqref{e1.1}$ and the definition of $v$
that

$v_{t}+uv_{x}+\frac{A\gamma}{\alpha}\rho^{\gamma-\delta}v-\frac{A\gamma}{\alpha}\rho^{\gamma-\delta}u=0.$
(3.15)

Here, the detail derivation of the above equation for $v$ can be found in
Appendix B.

Via the standard characteristic method, one can obtain

$v=\left(v_{0}+\int_{0}^{t}\frac{A\gamma}{\alpha}\rho^{\gamma-\delta}u\
\exp\Big{(}\int_{0}^{s}\frac{A\gamma}{\alpha}\rho^{\gamma-\delta}{\rm{d}}\tau\Big{)}{\rm{d}}s\right)\exp\left(-\int_{0}^{t}\frac{A\gamma}{\alpha}\rho^{\gamma-\delta}{\rm{d}}s\right).$

Then according to Lemmas 3.4 and 3.6, one has

$\begin{split}|v|_{\infty}\leq&C\Big{(}|v_{0}|_{\infty}+\int_{0}^{t}|\rho^{\gamma-\delta}u|_{\infty}{\rm{d}}s\Big{)}\\\
\leq&C\Big{(}1+\int_{0}^{t}|\rho|^{\gamma-\delta-\frac{1}{2}}_{\infty}|\rho^{\frac{1}{2}}u|_{\infty}{\rm{d}}s\Big{)}\\\
\leq&C\quad\text{for}\quad 0<\delta<1,\quad\gamma\geq\delta+\frac{1}{2},\\\
|v|_{\infty}\leq&C\Big{(}|v_{0}|_{\infty}+\int_{0}^{t}|\rho^{\gamma-1}u|_{\infty}{\rm{d}}s\Big{)}\\\
\leq&C\Big{(}1+\int_{0}^{t}|\rho|^{\gamma-1-\iota}_{\infty}|\rho^{\iota}u|_{\infty}{\rm{d}}s\Big{)}\\\
\leq&C\quad\text{for}\quad\delta=1,\quad\gamma\geq
1+\iota>\frac{3}{2}.\end{split}$ (3.16)

The proof of Lemma 3.7 is complete. ∎

Now we show the $L^{2}$ estimate of velocity $u$.

######  Lemma 3.8.

Assume (3.14) additionally. Then for any $T>0$, it holds that

$|u(t,\cdot)|_{2}^{2}+\int_{0}^{t}|\rho^{\frac{\delta-1}{2}}u_{x}(s,\cdot)|_{2}^{2}{\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

###### Proof.

Case 1: $0<\delta<1$. From $\eqref{e2.2}_{2}$, one has

$u_{t}+uu_{x}+A\gamma\rho^{\gamma-2}\rho_{x}=\alpha(\rho^{\delta-1}u_{x})_{x}+\alpha\rho^{\delta-2}\rho_{x}u_{x}.$
(3.17)

Multiplying (3.17) by $u$ and integrating over $\mathbb{R}$, one has

$\begin{split}&\frac{1}{2}\frac{\text{d}}{\text{d}t}|u|_{2}^{2}+\alpha|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{2}\\\
=&\int\big{(}-uu_{x}-A\gamma\rho^{\gamma-2}\rho_{x}+\alpha\rho^{\delta-2}\rho_{x}u_{x}\big{)}u{\rm{d}}x\triangleq\sum_{i=1}^{3}\text{J}_{i}.\end{split}$
(3.18)

According to Hölder’s inequality, Young’s inequality and Lemmas 3.1-3.7, one
can obtain

$\displaystyle\text{J}_{1}=$ $\displaystyle-\int u^{2}u_{x}{\rm{d}}x=0,$
(3.19) $\displaystyle\text{J}_{2}=$
$\displaystyle-A\gamma\int\rho^{\gamma-2}\rho_{x}u{\rm{d}}x=-A\gamma\int\rho^{\gamma-\delta}\rho^{\delta-2}\rho_{x}u{\rm{d}}x$
$\displaystyle=$
$\displaystyle-\frac{A\gamma}{\alpha}\int\rho^{\gamma-\delta}(v-u)u{\rm{d}}x$
$\displaystyle\leq$ $\displaystyle
C\Big{(}|\sqrt{\rho}|_{2}|u|_{2}|v|_{\infty}|\rho|_{\infty}^{\gamma-\delta-\frac{1}{2}}+|\rho|_{\infty}^{\gamma-\delta}|u|_{2}^{2}\Big{)}\leq
C|u|_{2}^{2}+C,$ $\displaystyle\text{J}_{3}=$
$\displaystyle\alpha\int\rho^{\delta-2}\rho_{x}u_{x}u{\rm{d}}x=\int\left(v-u\right)u_{x}u{\rm{d}}x$
$\displaystyle\leq$ $\displaystyle
C|v|_{\infty}|\rho|_{\infty}^{\frac{1-\delta}{2}}|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}|u|_{2}\leq
C|u|_{2}^{2}+\frac{\alpha}{2}|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{2}.$

It follows from (3.18)-(3.19) and the Gronwall inequality that

$|u(t,\cdot)|_{2}^{2}+\int_{0}^{t}|\rho^{\frac{\delta-1}{2}}u_{x}(s,\cdot)|_{2}^{2}{\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

Case 2: $\delta=1$. Multiplying $\eqref{e2.1}_{2}$ by $u$ and integrating over
$\mathbb{R}$, one has

$\frac{1}{2}\frac{\text{d}}{\text{d}t}|u|_{2}^{2}+\alpha|u_{x}|_{2}^{2}=\int\big{(}-uu_{x}-A\gamma\rho^{\gamma-2}\rho_{x}+\alpha\psi
u_{x}\big{)}u{\rm{d}}x.$

Via the similar argument for estimating $\text{J}_{1}$-$\text{J}_{3}$, one can
obtain

$|u(t,\cdot)|_{2}^{2}+\int_{0}^{t}|u_{x}(s,\cdot)|_{2}^{2}{\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

The proof of Lemma 3.8 is complete. ∎

###  3.3. The first order estimates of $u$

Now we are ready to consider the first order estimates of $u$.

######  Lemma 3.9.

Assume (3.14) additionally. Then for any $T>0$, it holds that

$|\rho^{\frac{\delta-1}{2}}u_{x}(t,\cdot)|_{2}^{2}+\int_{0}^{t}\big{(}|u_{t}|_{2}^{2}+|\rho^{\delta-1}u_{xx}|_{2}^{2}+|u_{x}|_{\infty}\big{)}(s,\cdot){\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

###### Proof.

Case 1: $0<\delta<1$. Multiplying (3.17) by $u_{t}$ and integrating over
$\mathbb{R}$, one has

$\begin{split}&\frac{\alpha}{2}\frac{\text{d}}{\text{d}t}|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{2}+|u_{t}|_{2}^{2}\\\
=&\int\Big{(}\frac{\alpha}{2}(\rho^{\delta-1})_{t}u_{x}^{2}-\big{(}uu_{x}+A\gamma\rho^{\gamma-2}\rho_{x}-\alpha\rho^{\delta-2}\rho_{x}u_{x}\big{)}u_{t}\Big{)}{\rm{d}}x\triangleq\sum_{i=4}^{7}\text{J}_{i}.\end{split}$
(3.20)

According to the equation $\eqref{e1.1}_{1}$, Hölder’s inequality, Lemma A.1
and Lemmas 3.1-3.8, one can obtain that

$\begin{split}\text{J}_{4}=&\frac{\alpha}{2}\int(\rho^{\delta-1})_{t}u_{x}^{2}{\rm{d}}x=-\frac{\alpha}{2}\int\big{(}(\rho^{\delta-1})_{x}u+(\delta-1)\rho^{\delta-1}u_{x}\big{)}u_{x}^{2}{\rm{d}}x\\\
=&-\frac{\alpha(\delta-1)}{2}\int\rho^{\delta-2}\rho_{x}uu_{x}^{2}{\rm{d}}x-\frac{\alpha(\delta-1)}{2}\int\rho^{\delta-1}u_{x}^{3}{\rm{d}}x\\\
=&-\frac{\delta-1}{2}\int(v-u)uu_{x}^{2}{\rm{d}}x-\frac{\alpha(\delta-1)}{2}\int\rho^{\delta-1}u_{x}^{3}{\rm{d}}x\\\
=&-\frac{\delta-1}{2}\int vuu_{x}^{2}{\rm{d}}x+\frac{\delta-1}{2}\int
u^{2}u_{x}^{2}{\rm{d}}x-\frac{\alpha(\delta-1)}{2}\int\rho^{\delta-1}u_{x}^{3}{\rm{d}}x\\\
\leq&-\frac{\delta-1}{2}\int
vuu_{x}^{2}{\rm{d}}x-\frac{\alpha(\delta-1)}{2}\int\rho^{\delta-1}u_{x}^{3}{\rm{d}}x\\\
\leq&C\big{(}|v|_{\infty}|u|_{\infty}|\rho|_{\infty}^{1-\delta}+|u_{x}|_{\infty}\big{)}|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{2}\\\
\leq&C\big{(}1+|u_{xx}|_{2}^{\frac{1}{2}}\big{)}|u_{x}|_{2}^{\frac{1}{2}}|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{2}.\end{split}$
(3.21)

Then we need to consider $|u_{xx}|_{2}$. According to (3.17), one has

$\displaystyle|u_{xx}|_{2}\leq$ $\displaystyle
C|\rho|_{\infty}^{1-\delta}|\rho^{\delta-1}u_{xx}|_{2}$ $\displaystyle\leq$
$\displaystyle
C\big{(}|u_{t}|_{2}+|uu_{x}|_{2}+|\rho^{\delta-2}\rho_{x}u_{x}|_{2}+|\rho^{\gamma-2}\rho_{x}|_{2}\big{)}$
$\displaystyle\leq$ $\displaystyle
C\big{(}|u_{t}|_{2}+|u|_{2}|u_{x}|_{\infty}+|(v-u)u_{x}|_{2}+|\rho^{\gamma-\delta}(v-u)|_{2}\big{)}$
$\displaystyle\leq$ $\displaystyle
C\big{(}|u_{t}|_{2}+|u|_{2}|u_{x}|_{2}^{\frac{1}{2}}|u_{xx}|_{2}^{\frac{1}{2}}+(|v|_{\infty}+|u|_{\infty})|u_{x}|_{2}$
$\displaystyle+|\rho^{\gamma-\delta}|_{2}|v|_{\infty}+|\rho|_{\infty}^{\gamma-\delta}|u|_{2}\big{)}$
$\displaystyle\leq$ $\displaystyle
C\big{(}1+|u_{t}|_{2}+|u_{x}|_{2}^{\frac{3}{2}}\big{)}+\frac{1}{2}|u_{xx}|_{2},$

which implies that

$|u_{xx}|_{2}\leq C\Big{(}1+|u_{t}|_{2}+|u_{x}|_{2}^{\frac{3}{2}}\Big{)}.$
(3.22)

Consequently, according to (3.21)-(3.22), Lemma 3.4 and Young’s inequality,
one can obtain

$\text{J}_{4}\leq
C\Big{(}1+|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{\frac{10}{3}}\Big{)}+\frac{1}{8}|u_{t}|_{2}^{2}.$
(3.23)

For the terms $\text{J}_{i}\ (i=5,6,7)$, one can similarly obtain

$\begin{split}\text{J}_{5}=&-\int uu_{x}u_{t}{\rm{d}}x\leq
C|u|_{\infty}|u_{x}|_{2}|u_{t}|_{2}\\\
\leq&C|u|_{2}^{\frac{1}{2}}|u_{x}|_{2}^{\frac{1}{2}}|\rho|_{\infty}^{\frac{1-\delta}{2}}|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}|u_{t}|_{2}\\\
\leq&C|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{3}+\frac{1}{8}|u_{t}|_{2}^{2},\\\
\text{J}_{6}=&-A\gamma\int\rho^{\gamma-2}\rho_{x}u_{t}{\rm{d}}x=-\frac{A\gamma}{\alpha}\int\rho^{\gamma-\delta}(v-u)u_{t}{\rm{d}}x\\\
\leq&C\Big{(}|\rho^{\gamma-\delta}|_{2}|v|_{\infty}+|\rho|_{\infty}^{\gamma-\delta}|u|_{2}\Big{)}|u_{t}|_{2}\leq
C+\frac{1}{8}|u_{t}|^{2}_{2},\\\
\text{J}_{7}=&\alpha\int\rho^{\delta-2}\rho_{x}u_{x}u_{t}{\rm{d}}x=\int(v-u)u_{x}u_{t}{\rm{d}}x\\\
\leq&C\left(|v|_{\infty}+|u|_{\infty}\right)|u_{x}|_{2}|u_{t}|_{2}\\\
\leq&C(1+|u_{x}|_{2})|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{2}+\frac{1}{8}|u_{t}|_{2}^{2}.\end{split}$
(3.24)

Substituting (3.21)-(3.24) into (3.20), one gets

$\frac{\text{d}}{\text{d}t}|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{2}+|u_{t}|_{2}^{2}\leq
C\Big{(}1+|\rho^{\frac{\delta-1}{2}}u_{x}|_{2}^{\frac{10}{3}}\Big{)},$ (3.25)

which, along with the Gronwall inequality and (3.22), yields that

$|\rho^{\frac{\delta-1}{2}}u_{x}(t,\cdot)|_{2}^{2}+\int_{0}^{t}\big{(}|u_{t}|_{2}^{2}+|\rho^{\delta-1}u_{xx}|_{2}^{2}\big{)}(s,\cdot){\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.26)

Case 2: $\delta=1$. First, similarly to the derivation of (3.22), one still
has

$|u_{xx}|_{2}\leq C\Big{(}1+|u_{t}|_{2}+|u_{x}|_{2}^{\frac{3}{2}}\Big{)}.$
(3.27)

Second, multiplying $\eqref{e2.1}_{2}$ by $u_{t}$ and integrating over
$\mathbb{R}$, one has

$\displaystyle\frac{\alpha}{2}\frac{\text{d}}{\text{d}t}|u_{x}|_{2}^{2}+|u_{t}|_{2}^{2}$
$\displaystyle=$ $\displaystyle-\int
uu_{x}u_{t}{\rm{d}}x-A\gamma\int\rho^{\gamma-2}\rho_{x}u_{t}{\rm{d}}x+\alpha\int\rho^{-1}\rho_{x}u_{x}u_{t}{\rm{d}}x.$

Via the similar arguments for estimating $\text{J}_{5}$-$\text{J}_{7}$, one
can get

$|u_{x}(t,\cdot)|_{2}^{2}+\int_{0}^{t}(|u_{t}(s,\cdot)|_{2}^{2}+|u_{xx}(s,\cdot)|_{2}^{2}){\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.28)

Finally, it follows from (3.26), (3.28) and Sobolev embedding theorem that

$\begin{split}\int_{0}^{t}|u_{x}(s,\cdot)|_{\infty}{\rm{d}}s\leq
C\int_{0}^{t}|u_{x}(s,\cdot)|_{2}^{\frac{1}{2}}|u_{xx}(s,\cdot)|_{2}^{\frac{1}{2}}{\rm{d}}s\leq
C\quad{\rm{for}}&\ \ 0\leq t\leq T.\end{split}$

The proof of Lemma 3.9 is complete. ∎

###  3.4. The second order estimates of $u$

We consider the second order estimates of $u$. For this purpose, we first give
the $L^{2}$ and $L^{4}$ estimates for $\psi$.

######  Lemma 3.10.

Assume (3.14) additionally. Then for any $T>0$, it holds that

$|\psi(t,\cdot)|_{2}+|\psi(t,\cdot)|_{4}\leq C\quad{\rm{for}}\ \ 0\leq t\leq
T.$

###### Proof.

Multiplying (3.15) by $v$ and integrating the resulting equation over
$\mathbb{R}$, one has

$\begin{split}\frac{\text{d}}{\text{d}t}|v|_{2}^{2}\leq
C\left(|u_{x}|_{\infty}|v|_{2}^{2}+|\rho|^{\gamma-\delta}_{\infty}|v|_{2}^{2}+|\rho|^{\gamma-\delta}_{\infty}|u|_{2}|v|_{2}\right)\leq
C(1+|u_{x}|_{\infty})|v|_{2}^{2}+C,\end{split}$

which, along with Lemma 3.9 and the Gronwall inequality, yields that

$|v(t,\cdot)|_{2}\leq C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.29)

It thus follows from (3.29), Lemmas 3.7-3.9 and the Sobolev embedding theorem
that

$\begin{split}|\psi(t,\cdot)|_{2}\leq
C\left(|v(t,\cdot)|_{2}+|u(t,\cdot)|_{2}\right)\leq C\quad{\rm{for}}&\ \ 0\leq
t\leq T,\\\ |\psi(t,\cdot)|_{4}\leq
C\left(|v(t,\cdot)|_{4}+|u(t,\cdot)|_{4}\right)\leq C\quad{\rm{for}}&\ \ 0\leq
t\leq T.\end{split}$

The proof of Lemma 3.10 is complete. ∎

Now we are ready to show the second order estimates of $u$.

######  Lemma 3.11.

Assume (3.14) additionally. Then for any $T>0$, it holds that

$|u_{t}(t,\cdot)|^{2}_{2}+|u_{xx}(t,\cdot)|_{2}^{2}+|\rho^{\delta-1}u_{xx}(t,\cdot)|_{2}^{2}+\int_{0}^{t}|\rho^{\frac{\delta-1}{2}}u_{tx}(s,\cdot)|^{2}_{2}{\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

###### Proof.

Case 1: $0<\delta<1$. First, according to Lemmas 3.4, 3.8-3.10 and the Sobolev
embedding theorem, one has

$\begin{split}&|\phi_{x}|_{4}=|A\gamma\rho^{\gamma-\delta}\rho^{\delta-2}\rho_{x}|_{4}\\\
\leq&C|\rho|^{\gamma-\delta}_{\infty}|\rho^{\delta-2}\rho_{x}|_{4}\leq
C(|v|_{4}+|u|_{4})\leq C.\end{split}$

Then it follows from $\eqref{e2.2}_{1}$ that

$|\phi_{t}|_{2}\leq
C\left(|u|_{4}|\phi_{x}|_{4}+|\phi|_{\infty}|u_{x}|_{2}\right)\leq C.$

According to (3.22) and Lemma 3.9, one has

$\begin{split}|u_{xx}|_{2}\leq&C\big{(}1+|u_{t}|_{2}+|u_{x}|_{2}^{\frac{3}{2}}\big{)}\leq
C\big{(}1+|u_{t}|_{2}\big{)},\\\
|\rho^{\delta-1}u_{xx}|_{2}\leq&C\big{(}1+|u_{t}|_{2}+|u_{x}|_{2}^{\frac{3}{2}}\big{)}+\frac{1}{2}|u_{xx}|_{2}\leq
C\big{(}1+|u_{t}|_{2}\big{)}.\end{split}$ (3.30)

Second, differentiating $(\ref{e2.2})_{2}$ with respect to $t$, multiplying
the resulting equation by $u_{t}$ and integrating over $\mathbb{R}$, one has

$\begin{split}&\frac{1}{2}\frac{\text{d}}{\text{d}t}|u_{t}|^{2}_{2}+\alpha|\rho^{\frac{\delta-1}{2}}u_{tx}|^{2}_{2}\\\
=&\int\Big{(}\alpha\big{(}\rho^{\delta-1}\big{)}_{t}u_{xx}-\alpha\big{(}\rho^{\delta-1}\big{)}_{x}u_{tx}+\alpha(\psi
u_{x})_{t}-(uu_{x})_{t}-\phi_{tx}\Big{)}u_{t}{\rm{d}}x\triangleq\sum_{i=8}^{12}\text{J}_{i}.\end{split}$
(3.31)

According to the equations $\eqref{e1.1}_{1}$ and $\eqref{e2.2}_{3}$, Hölder’s
inequality, Young’s inequality and Lemmas 3.4 and 3.8-3.10, one gets

$\begin{split}\text{J}_{8}=&\alpha\int\big{(}\rho^{\delta-1}\big{)}_{t}u_{xx}u_{t}{\rm{d}}x=\alpha(\delta-1)\int\rho^{\delta-2}\rho_{t}u_{xx}u_{t}{\rm{d}}x\\\
=&-\alpha(\delta-1)\int\Big{(}\rho^{\delta-2}\rho_{x}u+\rho^{\delta-1}u_{x}\Big{)}u_{xx}u_{t}{\rm{d}}x\\\
\leq&C\Big{(}|\rho^{\delta-2}\rho_{x}|_{4}|u|_{4}|u_{xx}|_{2}+\big{|}\rho^{\frac{\delta-1}{2}}u_{x}\big{|}_{2}\big{|}\rho^{\frac{\delta-1}{2}}u_{xx}\big{|}_{2}\Big{)}|u_{t}|_{\infty}\\\
\leq&C\Big{(}|u_{xx}|_{2}+\big{|}\rho^{\frac{\delta-1}{2}}u_{xx}\big{|}_{2}\Big{)}|u_{t}|_{2}^{\frac{1}{2}}|u_{tx}|_{2}^{\frac{1}{2}}\\\
\leq&C\Big{(}|u_{t}|_{2}^{2}+|\rho^{\frac{\delta-1}{2}}u_{xx}|_{2}^{2}\Big{)}+\frac{\alpha}{8}|\rho^{\frac{\delta-1}{2}}u_{tx}|^{2}_{2},\\\
\text{J}_{9}=&-\alpha\int\big{(}\rho^{\delta-1}\big{)}_{x}u_{tx}u_{t}{\rm{d}}x=-\alpha(\delta-1)\int\rho^{\delta-2}\rho_{x}u_{tx}u_{t}{\rm{d}}x\\\
\leq&C|\rho^{\delta-2}\rho_{x}|_{4}|u_{tx}|_{2}|u_{t}|_{4}\\\
\leq&C|u_{tx}|_{2}^{\frac{5}{4}}|u_{t}|_{2}^{\frac{3}{4}}\leq
C|u_{t}|_{2}^{2}+\frac{\alpha}{8}|\rho^{\frac{\delta-1}{2}}u_{tx}|^{2}_{2},\end{split}$
(3.32)

where one has used the facts

$|u_{t}|_{\infty}\leq
C|u_{t}|_{2}^{\frac{1}{2}}|u_{tx}|_{2}^{\frac{1}{2}}\quad\text{and}\quad|u_{t}|_{4}\leq
C|u_{tx}|_{2}^{\frac{1}{4}}|u_{t}|_{2}^{\frac{3}{4}}.$

Similarly, via the integration by parts, one has

$\begin{split}\text{J}_{10}=&\alpha\int(\psi u_{x})_{t}u_{t}{\rm{d}}x\\\
=&\alpha\int\Big{(}\psi u_{tx}u_{t}-\big{(}(u\psi)_{x}+(\delta-1)\psi
u_{x}+\delta\rho^{\delta-1}u_{xx}\big{)}u_{x}u_{t}\Big{)}{\rm{d}}x\\\
\leq&C\Big{(}|\psi|_{4}\big{(}|u_{tx}|_{2}|u_{t}|_{4}+|u|_{\infty}|u_{xx}|_{2}|u_{t}|_{4}+|u|_{4}|u_{x}|_{\infty}|u_{tx}|_{2}\\\
&+|u_{x}|_{2}|u_{t}|_{4}|u_{x}|_{\infty}\big{)}+|\rho^{\delta-1}u_{xx}|_{2}|u_{x}|_{2}|u_{t}|_{\infty}\Big{)}\\\
\leq&C\Big{(}1+|u_{t}|_{2}^{2}+|\rho^{\delta-1}u_{xx}|_{2}^{2}\Big{)}+\frac{\alpha}{8}|\rho^{\frac{\delta-1}{2}}u_{tx}|^{2}_{2},\\\
\text{J}_{11}=&-\int(uu_{x})_{t}u_{t}{\rm{d}}x=-\int(uu_{tx}u_{t}+u_{t}^{2}u_{x}){\rm{d}}x\\\
\leq&C\big{(}|u|_{\infty}|u_{tx}|_{2}|u_{t}|_{2}+|u_{x}|_{\infty}|u_{t}|_{2}^{2}\big{)}\\\
\leq&C\Big{(}1+|u_{xx}|_{2}^{\frac{1}{2}}\Big{)}|u_{t}|_{2}^{2}+\frac{\alpha}{8}|\rho^{\frac{\delta-1}{2}}u_{tx}|^{2}_{2},\\\
\text{J}_{12}=&-\int\phi_{tx}u_{t}{\rm{d}}x=\int\phi_{t}u_{tx}{\rm{d}}x\\\
\leq&C|\phi_{t}|_{2}|u_{tx}|_{2}\leq
C+\frac{\alpha}{8}|\rho^{\frac{\delta-1}{2}}u_{tx}|^{2}_{2}.\end{split}$
(3.33)

Substituting (3.32)-(3.33) into (3.31), one can obtain

$\frac{\text{d}}{\text{d}t}|u_{t}|^{2}_{2}+|\rho^{\frac{\delta-1}{2}}u_{tx}|^{2}_{2}\leq
C\Big{(}1+\big{(}1+|u_{xx}|_{2}^{\frac{1}{2}}\big{)}|u_{t}|^{2}_{2}+|\rho^{\delta-1}u_{xx}|_{2}^{2}\Big{)}.$
(3.34)

Integrating (3.34) over $(\tau,t)(\tau\in(0,t))$, one has

$\begin{split}&|u_{t}(t,\cdot)|_{2}^{2}+\int_{\tau}^{t}|\rho^{\frac{\delta-1}{2}}u_{tx}(s,\cdot)|_{2}^{2}{\rm{d}}s\\\
\leq&|u_{t}(\tau,\cdot)|_{2}^{2}+C\int_{\tau}^{t}\Big{(}1+\big{(}1+|u_{xx}|_{2}^{\frac{1}{2}}\big{)}|u_{t}|^{2}_{2}+|\rho^{\delta-1}u_{xx}|_{2}^{2}\Big{)}(s,\cdot){\rm{d}}s.\end{split}$
(3.35)

It follows from $\eqref{e2.2}_{2}$ that

$|u_{t}(\tau,\cdot)|_{2}\leq
C\Big{(}|u|_{\infty}|u_{x}|_{2}+|\phi_{x}|_{2}+|\rho^{\delta-1}u_{xx}|_{2}+|\psi|_{2}|u_{x}|_{\infty}\Big{)}(\tau),$
(3.36)

which, together with the time continuity of $(\rho,u)$ and (1.13)-(1.14),
implies that

$\lim\sup_{\tau\to 0}|u_{t}(\tau,\cdot)|_{2}\leq
C\big{(}|u_{0}|_{\infty}|\partial_{x}u_{0}|_{2}+|\partial_{x}\phi_{0}|_{2}+|g_{2}|_{2}+|\psi_{0}|_{2}|\partial_{x}u_{0}|_{\infty}\big{)}\leq
C.$

Letting $\tau\to 0$ in (3.35) and using the Gronwall inequality, one can
obtain

$|u_{t}(t,\cdot)|^{2}_{2}+\int_{0}^{t}|\rho^{\frac{\delta-1}{2}}u_{tx}(s,\cdot)|^{2}_{2}\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

It thus follows from (3.30) that

$|u_{xx}(t,\cdot)|_{2}+|\rho^{\delta-1}u_{xx}(t,\cdot)|_{2}\leq
C\big{(}1+|u_{t}(t,\cdot)|_{2}\big{)}\leq C\quad{\rm{for}}\ \ 0\leq t\leq T.$

Case 2: $\delta=1$. First, according to $(\ref{e2.1})_{2}$, Hölder’s
inequality, Young’s inequality and Lemmas 3.8-3.10, one has

$\begin{split}|u_{xx}|_{2}=&\frac{1}{\alpha}\Big{|}u_{t}+uu_{x}+\phi_{x}-\alpha\psi
u_{x}\Big{|}_{2}\\\
\leq&C\big{(}|u_{t}|_{2}+|u|_{\infty}|u_{x}|_{2}+|\phi_{x}|_{2}+|\psi|_{4}|u_{x}|_{4}\big{)}\\\
\leq&C\big{(}1+|u_{t}|_{2}+|u_{x}|_{2}^{\frac{3}{4}}|u_{xx}|_{2}^{\frac{1}{4}}\big{)}\\\
\leq&C\big{(}1+|u_{t}|_{2}\big{)}+\frac{1}{2}|u_{xx}|_{2},\end{split}$

which implies that

$\begin{split}|u_{xx}|_{2}\leq C(1+|u_{t}|_{2}).\end{split}$ (3.37)

Second, differentiating $(\ref{e2.1})_{2}$ with respect to $t$, multiplying
the resulting equation by $u_{t}$ and integrating over $\mathbb{R}$, one has

$\frac{1}{2}\frac{\text{d}}{\text{d}t}|u_{t}|^{2}_{2}+\alpha|u_{tx}|^{2}_{2}=-\int\Big{(}uu_{x}+\phi_{x}-\alpha\psi
u_{x}\Big{)}_{t}u_{t}\text{d}x\triangleq\sum_{i=13}^{15}\text{J}_{i}.$ (3.38)

Then according to Hölder’s inequality, Young’s inequality, the Sobolev
embedding theorem and Lemmas 3.1-3.10, one has

$\begin{split}\text{J}_{13}=&-\int(uu_{x})_{t}u_{t}\text{d}x=-\int\big{(}|u_{t}|^{2}u_{x}+uu_{tx}u_{t}\big{)}\text{d}x\\\
\leq&C\big{(}|u_{x}|_{\infty}|u_{t}|^{2}_{2}+|u|_{\infty}|u_{tx}|_{2}|u_{t}|_{2}\big{)}\\\
\leq&C\big{(}1+|u_{x}|_{\infty})|u_{t}|^{2}_{2}+\frac{\alpha}{8}|u_{tx}|_{2}^{2},\\\
\text{J}_{14}=&-\int\phi_{tx}u_{t}\text{d}x=\int\phi_{t}u_{tx}\text{d}x\\\
\leq&C|\phi_{t}|_{2}|u_{tx}|_{2}\leq C+\frac{\alpha}{8}|u_{tx}|_{2}^{2},\\\
\text{J}_{15}=&\alpha\int(\psi u_{x})_{t}u_{t}\text{d}x=\alpha\int\Big{(}\psi
u_{tx}u_{t}+\psi_{t}u_{x}u_{t}\Big{)}\text{d}x\\\ =&\alpha\int\Big{(}\psi
u_{tx}u_{t}-(u\psi)_{x}u_{x}u_{t}-u_{xx}u_{x}u_{t}\Big{)}\text{d}x\\\
\leq&C\Big{(}|\psi|_{2}|u_{tx}|_{2}|u_{t}|_{\infty}+|u|_{\infty}|\psi|_{2}\big{(}|u_{xx}|_{2}|u_{t}|_{\infty}+|u_{x}|_{\infty}|u_{tx}|_{2}\big{)}\\\
&+|u_{x}|_{\infty}|u_{xx}|_{2}|u_{t}|_{2}\Big{)}\\\
\leq&C\Big{(}|u_{t}|_{2}^{\frac{1}{2}}|u_{tx}|_{2}^{\frac{3}{2}}+|u_{xx}|_{2}|u_{t}|_{2}^{\frac{1}{2}}|u_{tx}|_{2}^{\frac{1}{2}}+|u_{xx}|_{2}^{\frac{1}{2}}|u_{tx}|_{2}+|u_{xx}|_{2}^{\frac{3}{2}}|u_{t}|_{2}\Big{)}\\\
\leq&C\big{(}1+|u_{xx}|_{2}^{2}\big{)}\big{(}1+|u_{t}|^{2}_{2}\big{)}+\frac{\alpha}{8}|u_{tx}|_{2}^{2},\end{split}$

which, along with (3.37)-(3.38), yields that

$\begin{split}&\frac{\text{d}}{\text{d}t}|u_{t}|^{2}_{2}+|u_{tx}|^{2}_{2}\leq
C\big{(}1+|u_{xx}|_{2}^{2}\big{)}\big{(}1+|u_{t}|^{2}_{2}\big{)}\leq
C\big{(}1+|u_{t}|^{2}_{2}\big{)}^{2}.\end{split}$ (3.39)

Integrating (3.39) over $(\tau,t)$ $(\tau\in(0,t))$, one has

$\begin{split}&|u_{t}(t,\cdot)|^{2}_{2}+\int_{\tau}^{t}|u_{tx}(s,\cdot)|^{2}_{2}\text{d}s\leq|u_{t}(\tau,\cdot)|^{2}_{2}+C\int_{\tau}^{t}\big{(}1+|u_{t}(s,\cdot)|^{2}_{2}\big{)}^{2}\text{d}s.\end{split}$
(3.40)

According to $(\ref{e2.1})_{2}$, one can obtain

$\begin{split}|u_{t}(\tau,\cdot)|_{2}\leq&C\big{(}|uu_{x}|_{2}+|\phi_{x}|_{2}+|u_{xx}|_{2}+|\psi
u_{x}|_{2}\big{)}(\tau)\\\
\leq&C\big{(}|u|_{\infty}|u_{x}|_{2}+|\phi_{x}|_{2}+|u_{xx}|_{2}+|\psi|_{2}|u_{x}|_{\infty}\big{)}(\tau),\end{split}$

which, along with the time continuity of $(\rho,u)$ and (1.16), yields that

$\begin{split}&\lim\sup_{\tau\rightarrow 0}|u_{t}(\tau,\cdot)|_{2}\leq
C\big{(}|u_{0}|_{\infty}\|u_{0}\|_{1}+\|\phi_{0}\|_{1}+\|u_{0}\|_{2}+|\psi_{0}|_{2}\|u_{0}\|_{2}\big{)}\leq
C.\end{split}$

Letting $\tau\rightarrow 0$ in (3.40), it follows from the Gronwall inequality
that

$|u_{t}(t,\cdot)|^{2}_{2}+|u_{xx}(t,\cdot)|_{2}^{2}+\int_{0}^{t}|u_{tx}(s,\cdot)|^{2}_{2}\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$

The proof of Lemma 3.11 is complete. ∎

###  3.5. The high order estimates of $\phi$ and $\psi$

The following lemma provides the high order estimates of $\phi$ and $\psi$.

######  Lemma 3.12.

Assume (3.14) additionally. Then for any $T>0$, it holds that

$\begin{split}&|\phi_{xx}(t,\cdot)|^{2}_{2}+|\phi_{tx}(t,\cdot)|^{2}_{2}+|\psi_{x}(t,\cdot)|_{2}^{2}+|\psi_{t}(t,\cdot)|^{2}_{2}\\\
&+\int_{0}^{t}\big{(}|u_{xxx}|^{2}_{2}+|\rho^{\frac{\delta-1}{2}}u_{xxx}|_{2}^{2}+|\rho^{\delta-1}u_{xxx}|_{2}^{2}+|\phi_{tt}|^{2}_{2}\big{)}(s,\cdot){\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.\end{split}$

###### Proof.

Case 1: $0<\delta<1.$ The proof will be divided into two steps.

Step 1: the estimate on $u_{xxx}$. First, it follows from $\eqref{e2.2}_{2}$,
Hölder’s inequality, Young’s inequality and Lemmas 3.1-3.11 that

$\begin{split}|u_{xxx}|_{2}=&|\rho^{1-\delta}\rho^{\delta-1}u_{xxx}|_{2}\leq
C|\rho|_{\infty}^{1-\delta}|\rho^{\delta-1}u_{xxx}|_{2}\leq
C|\rho^{\delta-1}u_{xxx}|_{2}\\\
\leq&C\big{(}|u_{tx}|_{2}+|(uu_{x})_{x}|_{2}+|\phi_{xx}|_{2}+|(\psi
u_{x})_{x}|_{2}+|(\rho^{\delta-1})_{x}u_{xx}|_{2}\big{)}\\\
\leq&C\big{(}|u_{tx}|_{2}+|u|_{\infty}|u_{xx}|_{2}+|u_{x}|_{\infty}|u_{x}|_{2}+|\phi_{xx}|_{2}+|\psi|_{\infty}|u_{xx}|_{2}+|\psi_{x}|_{2}|u_{x}|_{\infty}\big{)}\\\
\leq&C\big{(}1+|u_{tx}|_{2}+|\psi_{x}|_{2}+|\phi_{xx}|_{2}\big{)}.\end{split}$
(3.41)

Second, we consider the last two terms on the right-hand side of (3.41). For
$|\psi_{x}|_{2}$, differentiating $(\ref{e2.2})_{3}$ with respect to $x$,
multiplying the resulting equation by $\psi_{x}$ and integrating over
$\mathbb{R}$, one can obtain

$\begin{split}\frac{\text{d}}{\text{d}t}|\psi_{x}|^{2}_{2}\leq&C\int\big{(}|u_{x}\psi_{x}\psi_{x}|+|u_{xx}\psi\psi_{x}|+|\rho^{\delta-1}u_{xxx}\psi_{x}|\big{)}\text{d}x\\\
\leq&C\big{(}1+|\psi_{x}|_{2}^{2}+|\rho^{\delta-1}u_{xxx}|_{2}^{2}\big{)}.\end{split}$
(3.42)

For $|\phi_{xx}|_{2}$. Applying $\partial_{x}^{2}$ to $\eqref{e2.2}_{1}$,
multiplying the resulting equation by $\phi_{xx}$ and integrating over
$\mathbb{R}$, one has

$\begin{split}\frac{\text{d}}{\text{d}t}|\phi_{xx}|^{2}_{2}\leq&C\int\big{(}|u_{xx}\phi_{x}|+|u_{x}\phi_{xx}|+|\phi
u_{xxx}|\big{)}|\phi_{xx}|{\rm{d}}x\\\
\leq&C\big{(}|u_{xx}|_{2}|\phi_{x}|_{\infty}+|u_{x}|_{\infty}|\phi_{xx}|_{2}+|\phi|_{\infty}|u_{xxx}|_{2}\big{)}|\phi_{xx}|_{2}\\\
\leq&C\big{(}1+|\phi_{xx}|_{2}^{2}+|u_{xxx}|_{2}^{2}\big{)}.\end{split}$
(3.43)

According to (3.41)-(3.43), one can obtain

$\frac{\text{d}}{\text{d}t}\big{(}|\psi_{x}|^{2}_{2}+|\phi_{xx}|^{2}_{2}\big{)}\leq
C\big{(}1+|u_{tx}|_{2}^{2}+|\psi_{x}|_{2}^{2}+|\phi_{xx}|_{2}^{2}\big{)},$

which, along with the Gronwall inequality and Lemma 3.11, yields that

$|\psi_{x}(t,\cdot)|_{2}+|\phi_{xx}(t,\cdot)|_{2}\leq C\quad{\rm{for}}\ \
0\leq t\leq T.$

At last, it thus follows from (3.41) and Lemma 3.11 that

$\begin{split}&\int_{0}^{t}\big{(}|u_{xxx}|^{2}_{2}+|\rho^{\delta-1}u_{xxx}|_{2}^{2}+|\rho^{\frac{\delta-1}{2}}u_{xxx}|_{2}^{2}\big{)}(s,\cdot)\text{d}s\\\
\leq&C\int_{0}^{t}\big{(}1+|\psi_{x}|_{2}^{2}+|u_{tx}|_{2}^{2}+|\phi_{xx}|_{2}^{2}\big{)}(s,\cdot)\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.\end{split}$ (3.44)

Step 2: estimates on time derivatives of the density. First, according to
$\eqref{e2.2}_{3}$, Hölder’s inequality and Lemma 3.11, one has

$\begin{split}|\psi_{t}|_{2}\leq&C\big{(}|\rho^{\delta-1}u_{xx}|_{2}+|u\psi_{x}|_{2}+|u_{x}\psi|_{2}\big{)}\\\
\leq&C\big{(}1+|u|_{\infty}|\psi_{x}|_{2}+|u_{x}|_{\infty}|\psi|_{2}\big{)}\leq
C.\end{split}$

Second, according to $\eqref{e2.2}_{1}$ and Hölder’s inequality, one has

$\begin{split}|\phi_{tx}|_{2}\leq&C\big{(}|u\phi_{xx}|_{2}+|u_{x}\phi_{x}|_{2}+|\phi
u_{xx}|_{2}\big{)}\\\
\leq&C\big{(}|u|_{\infty}|\phi_{xx}|_{2}+|u_{x}|_{\infty}|\phi_{x}|_{2}+|\phi|_{\infty}|u_{xx}|_{2}\big{)}\leq
C,\\\
|\phi_{tt}|_{2}\leq&C(|u_{t}\phi_{x}|_{2}+|u\phi_{tx}|_{2}+|\phi_{t}u_{x}|_{2}+|\phi
u_{tx}|_{2})\\\
\leq&C(|u_{t}|_{2}|\phi_{x}|_{\infty}+|u|_{\infty}|\phi_{tx}|_{2}+|\phi_{t}|_{2}|u_{x}|_{\infty}+|\phi|_{\infty}|u_{tx}|_{2})\\\
\leq&C\big{(}1+|u_{tx}|_{2}\big{)},\end{split}$

which, along with Lemma 3.11, yields that

$\int_{0}^{t}|\phi_{tt}(s,\cdot)|^{2}_{2}\text{d}s\leq C\quad{\rm{for}}\ \
0\leq t\leq T.$

Case 2: $\delta=1$. The proof will be divided into two steps.

Step 1: the estimate on $u_{xxx}$. First, it follows from $\eqref{e2.1}_{2}$,
Hölder’s inequality and Lemmas 3.1-3.11 that

$\begin{split}|u_{xxx}|_{2}\leq&C\big{(}|u_{tx}|_{2}+|(uu_{x})_{x}|_{2}+|\phi_{xx}|_{2}+|(\psi
u_{x})_{x}|_{2}\big{)}\\\
\leq&C\big{(}|u_{tx}|_{2}+|u|_{\infty}|u_{xx}|_{2}+|u_{x}|_{\infty}|u_{x}|_{2}+|\phi_{xx}|_{2}\\\
&+|\psi|_{\infty}|u_{xx}|_{2}+|\psi_{x}|_{2}|u_{x}|_{\infty}\big{)}\\\
\leq&C\big{(}1+|u_{tx}|_{2}+|\psi_{x}|_{2}+|\phi_{xx}|_{2}\big{)}.\end{split}$
(3.45)

Second, for $|\psi_{x}|_{2}$, differentiating $(\ref{e2.1})_{3}$ with respect
to $x$, multiplying the resulting equation by $\psi_{x}$ and using the
integration by part, one can obtain

$\begin{split}\frac{\text{d}}{\text{d}t}|\psi_{x}|^{2}_{2}\leq&C\int\big{(}|u_{x}\psi_{x}\psi_{x}|+|u_{xx}\psi\psi_{x}|+|u_{xxx}\psi_{x}|\big{)}\text{d}x\\\
\leq&C\big{(}|u_{x}|_{\infty}|\psi_{x}|_{2}^{2}+|u_{xx}|_{2}|\psi|_{\infty}|\psi_{x}|_{2}+|u_{xxx}|_{2}|\psi_{x}|_{2}\big{)}\\\
\leq&C\big{(}1+|\psi_{x}|_{2}^{2}+|u_{xxx}|_{2}^{2}\big{)}.\end{split}$ (3.46)

For $|\phi_{xx}|_{2}$, by the definition of $\phi$ and $\psi$, one has

$\begin{split}|\phi_{xx}|_{2}=&(\gamma-1)|(\psi\phi)_{x}|_{2}\\\
\leq&C\big{(}|\psi_{x}|_{2}|\phi|_{\infty}+|\psi|_{\infty}|\phi_{x}|_{2}\big{)}\leq
C(1+|\psi_{x}|_{2}).\end{split}$ (3.47)

Thus, combining (3.45)-(3.47) together, one deduces that

$\frac{\text{d}}{\text{d}t}|\psi_{x}|^{2}_{2}\leq
C\big{(}1+|\psi_{x}|_{2}^{2}+|u_{tx}|_{2}^{2}\big{)},$ (3.48)

which, together with the Gronwall inequality and Lemma 3.11, yields that

$|\psi_{x}(t,\cdot)|_{2}\leq C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.49)

It thus follows from (3.45), (3.47) and (3.49) that

$\begin{split}|\phi_{xx}(t,\cdot)|_{2}\leq C(1+|\psi_{x}(t,\cdot)|_{2})\leq
C\quad{\rm{for}}&\ \ 0\leq t\leq T,\\\
\int_{0}^{t}|u_{xxx}(s,\cdot)|^{2}_{2}\text{d}s\leq
C\int_{0}^{t}\big{(}1+|u_{tx}(s,\cdot)|_{2}^{2}\big{)}\text{d}s\leq
C\quad{\rm{for}}&\ \ 0\leq t\leq T.\end{split}$

Step 2: estimates on time derivatives of the density. The boundedness of
$|\psi_{t}(t,\cdot)|_{2}$, $|\phi_{tx}(t,\cdot)|_{2}$ and
$\int_{0}^{t}|\phi_{tt}(s,\cdot)|_{2}^{2}{\rm{d}}s$ can be shown via similar
argument as in the case $0<\delta<1$.

The proof of Lemma 3.12 is complete. ∎

###  3.6.  Time-weighted energy estimates of $u$

At last, we establish the time-weighted energy estimates of $u$.

######  Lemma 3.13.

Assume (3.14) additionally. Then for any $T>0$, it holds that

$\begin{split}&t|\rho^{\frac{\delta-1}{2}}u_{tx}(t,\cdot)|^{2}_{2}+\int_{0}^{t}s\big{(}|u_{tt}|^{2}_{2}+|u_{txx}|_{2}^{2}\big{)}(s,\cdot){\rm{d}}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.\end{split}$

###### Proof.

Case 1: $0<\delta<1$. First, differentiating $(\ref{e2.2})_{2}$ with respect
to $t$, multiplying the resulting equation by $u_{tt}$ and integrating over
$\mathbb{R}$, one arrives at

$\begin{split}&\frac{\alpha}{2}\frac{\text{d}}{\text{d}t}|\rho^{\frac{\delta-1}{2}}u_{tx}|_{2}^{2}+|u_{tt}|_{2}^{2}\\\
=&\int\Big{(}\alpha(\rho^{\delta-1})_{t}u_{xx}-\alpha(\rho^{\delta-1})_{x}u_{tx}-(uu_{x})_{t}\\\
&-\phi_{tx}+\alpha(\psi
u_{x})_{t}\Big{)}u_{tt}\text{d}x+\frac{\alpha}{2}\int(\rho^{\delta-1})_{t}u_{tx}^{2}\text{d}x\triangleq\sum_{i=16}^{21}\text{J}_{i}.\end{split}$
(3.50)

According to the equation $\eqref{e1.1}_{1}$, Hölder’s inequality, Lemma A.1
and Lemmas 3.1-3.12, one can obtain that

$\begin{split}\text{J}_{16}=&\alpha\int(\rho^{\delta-1})_{t}u_{xx}u_{tt}\text{d}x\\\
=&-\alpha\int\big{(}(\rho^{\delta-1})_{x}u+(\delta-1)\rho^{\delta-1}u_{x}\big{)}u_{xx}u_{tt}\text{d}x\\\
\leq&C\big{(}|\psi|_{\infty}|u|_{\infty}|u_{xx}|_{2}+|\rho^{\delta-1}u_{xx}|_{2}|u_{x}|_{\infty}\big{)}|u_{tt}|_{2}\leq
C+\frac{1}{8}|u_{tt}|_{2}^{2},\\\
\sum_{i=17}^{20}\text{J}_{i}=&\int\Big{(}-\alpha(\rho^{\delta-1})_{x}u_{tx}-(uu_{x})_{t}-\phi_{tx}+\alpha(\psi
u_{x})_{t}\Big{)}u_{tt}\text{d}x\\\
\leq&C\big{(}|u_{x}|_{2}|u_{t}|_{\infty}+|u|_{\infty}|u_{tx}|_{2}+|\phi_{tx}|_{2}+|\psi|_{\infty}|u_{tx}|_{2}+|\psi_{t}|_{2}|u_{x}|_{\infty}\big{)}|u_{tt}|_{2}\\\
\leq&C(1+|\rho^{\frac{\delta-1}{2}}u_{tx}|_{2}^{2})+\frac{1}{8}|u_{tt}|_{2}^{2},\\\
\text{J}_{21}=&\frac{\alpha}{2}\int(\rho^{\delta-1})_{t}u_{tx}^{2}\text{d}x\\\
\leq&C\big{(}|\psi|_{\infty}|u|_{\infty}|u_{tx}|_{2}^{2}+|\rho^{\frac{\delta-1}{2}}u_{tx}|_{2}^{2}|u_{x}|_{\infty}\big{)}\leq
C|\rho^{\frac{\delta-1}{2}}u_{tx}|_{2}^{2},\end{split}$

which, along with (3.50), implies that

$\frac{\text{d}}{\text{d}t}|\rho^{\frac{\delta-1}{2}}u_{tx}|_{2}^{2}+|u_{tt}|_{2}^{2}\leq
C\big{(}1+|\rho^{\frac{\delta-1}{2}}u_{tx}|_{2}^{2}\big{)}.$ (3.51)

Second, multiplying (3.51) by $s$ and integrating with respect to $s$ over
$[\tau,t]$ for $\tau\in(0,t)$, one has

$t|\rho^{\frac{\delta-1}{2}}u_{tx}(t,\cdot)|_{2}^{2}+\int_{\tau}^{t}s|u_{tt}(s,\cdot)|_{2}^{2}\text{d}s\leq\tau|\rho^{\frac{\delta-1}{2}}u_{tx}(\tau,\cdot)|_{2}^{2}+C.$
(3.52)

Recalling the definition of regular solutions in Definition 1.1, one has

$\rho^{\frac{\delta-1}{2}}u_{tx}\in L^{2}([0,T];L^{2}),$

which, along with the Lemma A.3 (see in Appendix A), implies that there exists
a sequence $s_{k}$ such that

$s_{k}\to 0,\quad\text{and}\quad
s_{k}|\rho^{\frac{\delta-1}{2}}u_{tx}(s_{k},\cdot)|_{2}^{2}\to
0\quad\text{as}\quad k\to\infty.$ (3.53)

After choosing $\tau=s_{k}\to 0$ in (3.52), one can obtain

$t|\rho^{\frac{\delta-1}{2}}u_{tx}(t,\cdot)|_{2}^{2}+\int_{0}^{t}s|u_{tt}(s,\cdot)|_{2}^{2}\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.54)

Second, according to $(\ref{e2.2})_{2}$ and the estimates in Lemmas 3.1-3.12,
one has

$\begin{split}|u_{txx}|_{2}=&|\rho^{1-\delta}\rho^{\delta-1}u_{txx}|_{2}\leq|\rho|_{\infty}^{1-\delta}|\rho^{\delta-1}u_{txx}|_{2}\\\
\leq&C\big{(}|(\rho^{\delta-1})_{t}u_{xx}|_{2}+|u_{tt}|_{2}+|(uu_{x})_{t}|_{2}+|\phi_{tx}|_{2}+|(\psi
u_{x})_{t}|_{2}\big{)}\\\
\leq&C\big{(}1+|\rho^{\delta-1}u_{x}u_{xx}|_{2}+|\rho^{\delta-2}\rho_{x}uu_{xx}|_{2}+|u_{tt}|_{2}+|u_{t}|_{2}|u_{x}|_{\infty}\\\
&+|u|_{\infty}|u_{tx}|_{2}+|\psi|_{\infty}|u_{tx}|_{2}+|\psi_{t}|_{2}|u_{x}|_{\infty}\big{)}\\\
\leq&C\big{(}1+|u_{tt}|_{2}+|u_{tx}|_{2}\big{)},\end{split}$

which, along with (3.54), implies that

$\int_{0}^{t}s|u_{txx}(s,\cdot)|_{2}^{2}{\rm{d}}s\leq C\quad{\rm{for}}\ \
0\leq t\leq T.$

Case 2: $\delta=1$. Differentiating $(\ref{e2.1})_{2}$ with respect to $t$,
multiplying the resulting equation by $u_{tt}$ and integrating over
$\mathbb{R}$, one has

$\frac{\alpha}{2}\frac{\text{d}}{\text{d}t}|u_{tx}|_{2}^{2}+|u_{tt}|_{2}^{2}=\int\Big{(}-(uu_{x})_{t}-\phi_{tx}+\alpha(\psi
u_{x})_{t}\Big{)}u_{tt}\text{d}x.$

Via the similar argument for estimating $\text{J}_{18}$-$\text{J}_{20}$, one
can get

$t|u_{tx}(t,\cdot)|_{2}^{2}+\int_{0}^{t}s\big{(}|u_{tt}|_{2}^{2}+|u_{txx}|_{2}^{2}\big{)}(s,\cdot)\text{d}s\leq
C\quad{\rm{for}}\ \ 0\leq t\leq T.$ (3.55)

The proof of Lemma 3.13 is complete. ∎

###  3.7. Proof of Theorem 1.1

Based on the local-in-time well-posedness obtained in Lemmas 2.1-2.2 and the
global-in-time a priori estimates established in Lemmas 3.1-3.13, now we are
ready to give the proof of Theorem 1.1.

Step 1: the global well-posedness of regular solutions. First, under the
initial assumption (1.13)-(1.14), according to Lemmas 2.1-2.2, there exists a
unique regular solution $(\rho,u)$ to the Cauchy problem (1.7)-(1.8) in
$[0,T_{*}]\times\mathbb{R}$ for some $T_{*}>0$, which satisfies (1.15) with
$T$ replaced by $T_{*}$. Let $T^{*}>0$ be the life span of the regular
solution satisfying (1.15) obtained in Lemmas 2.1-2.2. It is obvious that
$T^{*}\geq T_{*}$. Then we claim that $T^{*}=\infty$. Otherwise, if
$T^{*}<\infty$, according to the uniform a priori estimates obtained in Lemmas
3.1-3.13 and the standard weak compactness theory, one can know that for any
sequence $\\{t_{k}\\}_{k=1}^{\infty}$ satisfying $0<t_{k}<T^{*}$ and

$t_{k}\to T^{*}\quad\text{as}\quad k\to\infty,$

there exists a subsequence $\\{t_{1k}\\}_{k=1}^{\infty}$ and functions
$(\phi,u,\psi)(T^{*},x)$ such that

$\displaystyle\phi(t_{1k},x)\rightharpoonup\phi(T^{*},x)\quad\text{in\,\,}H^{2}\quad\text{as}\,\,k\to\infty,$
(3.56) $\displaystyle u(t_{1k},x)\rightharpoonup
u(T^{*},x)\quad\text{in\,\,}H^{2}\quad\text{as}\,\,k\to\infty,$
$\displaystyle\psi(t_{1k},x)\rightharpoonup\psi(T^{*},x)\quad\text{in\,\,}H^{1}\quad\text{as}\,\,k\to\infty.$

Second, we want to show that functions $(\phi,u,\psi)(T^{*},x)$ satisfy all
the initial assumptions shown in Lemma 2.1, which include (1.13)-(1.14) except
$\rho_{0}\in L^{1}$ and the following relationship between $\phi$ and $\psi$:

$\psi=\frac{a\delta}{\delta-1}(\phi^{2e})_{x}.$ (3.57)

It follows from (3.56) that (1.13) except $\rho_{0}\in L^{1}$ still holds at
the time $t=T^{*}$.

Next for the relation in (3.57), we need to consider the following equation

$\phi_{t}+u\phi_{x}+(\gamma-1)\phi u_{x}=0,$ (3.58)

which holds in $[0,T^{*})\times\mathbb{R}$ in the classical sense. Actually,
if we regard $(\phi(T^{*},x),u(T^{*},x),\\\ \psi(T^{*},x))$ and

$\phi_{t}(T^{*},x)=-u(T^{*},x)\phi_{x}(T^{*},x)-(\gamma-1)\phi(T^{*},x)u_{x}(T^{*},x),$

as the extended definitions of the regular solution
$(\phi(t,x),u(t,x),\psi(t,x))$ at the time $t=T^{*}$, then one has

$\text{ess}\sup_{0\leq t\leq
T^{*}}(\|\phi(t,\cdot)\|_{2}+\|\phi_{t}(t,\cdot)\|_{1})\leq C<\infty,$

which, together with the Sobolev embedding theorem, implies that

$\phi(t,x)\in C([0,T^{*}];H^{1}).$

It follows from the first line in $\eqref{f2}$ and the consistency of weak
convergence and strong convergence in $H^{1}$ space that

$\phi(t_{1k},x)\to\phi(T^{*},x)\quad\text{in}\quad H^{1}\quad\text{as}\quad
k\to\infty.$ (3.59)

Notice that for any $0<R<\infty$, there exists a generic constant $C(T^{*},R)$
such that

$\phi(t,x)\geq
C(T^{*},R)\quad\text{for\,\,any}\quad(t,x)\in[0,T^{*}]\times[-R,R],$ (3.60)

which, along with the last line in $\eqref{f2}$ and (3.59), implies that
(3.57) holds.

Moreover, it follows from Lemmas 3.1-3.13 and (3.56)-(3.57) that

$\sup_{\tau\leq t\leq
T^{*}}(|\rho^{\frac{\delta-1}{2}}u_{x}(t,\cdot)|_{2}+|\rho^{\delta-1}u_{xx}(t,\cdot)|_{2})\leq
C,$ (3.61)

for any $\tau\in(0,T^{*}]$, which implies that $(\rho(T^{*},x),u(T^{*},x))$
satisfy all the initial assumptions on the initial data of Lemma 2.1.

Finally, if we solve the system (1.7) with the initial time $T^{*}$, then
Lemma 2.1 ensures that for some constant $T_{0}>0$, $(\rho,u)(t,x)$ is the
unique regular solution in $[T^{*},T^{*}+T_{0}]\times\mathbb{R}$ to this
problem, which satisfies (1.15) with $[0,T]$ replaced by $[T^{*},T^{*}+T_{0}]$
except $\rho\in C([T^{*},T^{*}+T_{0}];L^{1})$. It follows from the boundedness
of all required norms of the solution $(\rho,u)(t,x)$ in
$[0,T^{*}+T_{0}]\times\mathbb{R}$ and the standard arguments for proving the
time continuity of the regular solution (see Section 3.6 (pages 138-139) of
[49] ) that, $(\rho,u)(t,x)$ is actually the unique regular solution in
$[0,T^{*}+T_{0}]\times\mathbb{R}$ to the Cauchy problem (1.7)-(1.8), which
satisfies (1.15) with $[0,T_{*}]$ replaced by $[0,T^{*}+T_{0}]$ except
$\rho\in C([0,T^{*}+T_{0}];L^{1})$. Furthermore, according to Lemma 2.2, one
has that the solution constructed above also satisfies $\rho\in
C([0,T^{*}+T_{0}];L^{1})$. Then the above conclusions contradict to the fact
that $0<T^{*}<\infty$ is the maximal existence time of the unique regular
solution $(\rho,u)(t,x)$ satisfying (1.15).

######  Remark 3.1.

According to the proof in this subsection, the definitions of
$(\phi(T^{*},x)$,
$u(T^{*},x),\psi(T^{*},x))$ do not depend on the choice of the time sequences
$\\{t_{k}\\}_{k=1}^{\infty}$.

Step 2: the global well-posedness of classical solutions. Now we show that the
regular solution obtained above is indeed a classical one in
$(0,T]\times\mathbb{R}$ for any $T>0$.

First, due to $\phi>0$ and

$\rho=\big{(}\frac{\gamma-1}{A\gamma}\phi\big{)}^{\frac{1}{\gamma-1}},$

one has

$(\rho,\rho_{t},\rho_{x},u,u_{x})\in C([0,T]\times\mathbb{R}).$ (3.62)

Second, for the term $u_{t}$, according to Lemma 3.13, one has

$\begin{split}&t^{\frac{1}{2}}u_{tx}\in L^{\infty}([0,T];L^{2}),\quad
t^{\frac{1}{2}}u_{tt}\in L^{2}([0,T];L^{2}),\\\ &t^{\frac{1}{2}}u_{txx}\in
L^{2}([0,T];L^{2}),\quad t^{\frac{1}{2}}u_{ttx}\in
L^{2}([0,T];H^{-1}),\end{split}$ (3.63)

which, along with the classical Sobolev embedding theorem:

$\begin{split}&L^{2}([0,T];H^{1})\cap W^{1,2}([0,T];H^{-1})\hookrightarrow
C([0,T];L^{2}),\end{split}$ (3.64)

yields that for any $0<\tau<T$,

$tu_{t}\in C([0,T];H^{1})\quad\text{and}\quad u_{t}\in
C([\tau,T]\times\mathbb{R}).$ (3.65)

Finally, it remains to show that

$u_{xx}\in C([\tau,T]\times\mathbb{R})$

for any $0<\tau<T$. Actually, it follows from the fact $\rho>0$, (3.62) and
(3.65) that

$u_{xx}=\frac{1}{\alpha}\rho^{1-\delta}\big{(}u_{t}+uu_{x}+\phi_{x}-\alpha\psi
u_{x}\big{)}\in C([\tau,T]\times\mathbb{R}).$

Until now, we have completed the proof of Theorem 1.1.

###  3.8. Proof of Theorem 1.2

The proof of Theorem 1.2 can be achieved via the similar argument used in
Section 3.7.

##  4\. Non-existence of global solutions with $L^{\infty}$ decay on $u$

This section will be devoted to providing the proofs for the non-existence
theories of global regular solutions with $L^{\infty}$ decay on $u$ shown in
Theorem 1.3 and Corollary 1.1.

###  4.1. Proof of Theorem 1.3

Now we are ready to prove Theorem 1.3. Let $T>0$ be any constant, and
$(\rho,u)\in D(T)$. It follows from the definitions of $m(t)$, $\mathbb{P}(t)$
and $E_{k}(t)$ that

$|\mathbb{P}(t)|\leq\int\rho(t,x)|u(t,x)|\text{d}x\leq\sqrt{2m(t)E_{k}(t)},$

which, together with the definition of the solution class $D(T)$, implies that

$0<\frac{|\mathbb{P}(0)|^{2}}{2m(0)}\leq
E_{k}(t)\leq\frac{1}{2}m(0)|u(t,\cdot)|^{2}_{\infty}\quad\text{for}\quad
t\in[0,T].$

Then one obtains that there exists a positive constant

$C_{u}=\frac{|\mathbb{P}(0)|}{m(0)}$

such that

$|u(t,\cdot)|_{\infty}\geq C_{u}\quad\text{for}\quad t\in[0,T].$

Thus one obtains the desired conclusion as shown in Theorem 1.3.

###  4.2. Proof of Corollary 1.1

Let $(\rho,u)(t,x)$ in $[0,T]\times\mathbb{R}$ be the global regular solution
obtained in Theorem 1.1 or 1.2. Next we just need to show that $(\rho,u)\in
D(T)$.

Actually, the conservation of the total mass and the energy equality have been
shown in Lemma 3.3 and Appendix B, and the conservation of the total momentum
can be verified as follows.

######  Lemma 4.1.

For any $T>0$, it holds that

$\mathbb{P}(t)=\mathbb{P}(0)\quad{\rm{for}}\quad t\in[0,T].$

###### Proof.

First, one can obtain that

$\begin{split}\mathbb{P}(t)=\int\rho u\text{d}x\leq
C|\rho|_{1}|u|_{\infty}<&\infty.\end{split}$ (4.1)

Second, the momentum equation $\eqref{e1.1}_{2}$ implies that

$\mathbb{P}_{t}=-\int(\rho u^{2})_{x}\text{d}x-\int
P_{x}\text{d}x+\int(\mu(\rho)u_{x})_{x}\text{d}x=0,$ (4.2)

where one has used the fact that

$\rho
u^{2}(t,\cdot),\quad\rho^{\gamma}(t,\cdot)\quad\text{and}\quad\rho^{\delta}u_{x}(t,\cdot)\in
W^{1,1}(\mathbb{R}).$

∎

According to Theorem 1.3 and Lemma 4.1, the proof of Corollary 1.1 is
complete.

##  Appendix A  Basic lemmas

In this appendix, we list some useful lemmas which were used frequently in the
previous sections. The first one is the well-known Gagliardo-Nirenberg
inequality.

######  Lemma A.1 ([41]).

Let $1\leq q,\ r\leq\infty$, and $p$ satisfies

$\frac{1}{p}=\frac{j}{d}+\zeta\Big{(}\frac{1}{r}-\frac{m}{d}\Big{)}+(1-\zeta)\frac{1}{q},\quad\text{and}\quad\frac{j}{m}\leq\zeta\leq
1.$

Then there exists a generic constant $C>0$ depends only on $j,m,d,q,r$ and
$\zeta$ such that for $f\in L^{q}\cap D^{m,r}(\mathbb{R}^{d})$,

$|D^{j}f|_{p}\leq C|D^{m}f|_{r}^{\zeta}|f|_{q}^{1-\zeta},$ (A.1)

with the exceptions that if $j=0,rm<d,q=\infty$ we assume that $f$ vanishes at
infinity or $f\in L^{\tilde{q}}$ for some finite $\tilde{q}>0$, while if
$1<r<\infty$ and $m-j-d/r$ is a non-negative integer we take $j/m\leq\zeta<1$.

In particular, letting $d=1$, $j=0$ and $p=\infty$ in (A.1), one has

$|f|_{\infty}\leq C|f|_{2}^{\frac{1}{2}}|f_{x}|_{2}^{\frac{1}{2}}.$

The second one is Fatou’s lemma.

######  Lemma A.2.

Given a measure space $(V,\mathcal{F},\nu)$ and a set $X\in\mathcal{F}$, let
$\\{f_{n}\\}$ be a sequence of $(\mathcal{F},\mathcal{B}_{\mathbb{R}_{\geq
0}})$-measurable non-negative functions $f_{n}:X\rightarrow[0,\infty]$. Define
the function $f:X\rightarrow[0,\infty]$ by setting

$f(x)=\liminf_{n\rightarrow\infty}f_{n}(x),$

for every $x\in X$. Then $f$ is $(\mathcal{F},\mathcal{B}_{\mathbb{R}_{\geq
0}})$-measurable, and

$\int_{X}f(x)\text{\rm
d}\nu\leq\liminf_{n\rightarrow\infty}\int_{X}f_{n}(x)\text{\rm d}\nu.$

The following lemma is used to obtain the time-weighted estimates of the
velocity $u$.

######  Lemma A.3.

[1] If $f(t,x)\in L^{2}([0,T];L^{2})$, then there exists a sequence $s_{k}$
such that

$s_{k}\rightarrow 0\quad\text{and}\quad
s_{k}|f(s_{k},\cdot)|^{2}_{2}\rightarrow 0\quad\text{as}\quad
k\rightarrow\infty.$

The last one shows some compactness results obtained via the Aubin-Lions
Lemma.

######  Lemma A.4 ([43]).

Let $X_{0}\subset X\subset X_{1}$ be three Banach spaces. Suppose that $X_{0}$
is compactly embedded in $X$ and $X$ is continuously embedded in $X_{1}$. Then
the following statements hold

  1. (1)

If $J$ is bounded in $L^{p}([0,T];X_{0})$ for $1\leq p<\infty$, and
$\frac{\partial J}{\partial t}$ is bounded in $L^{1}([0,T];X_{1})$, then $J$
is relatively compact in $L^{p}([0,T];X)$.

  2. (2)

If $J$ is bounded in $L^{\infty}([0,T];X_{0})$ and $\frac{\partial J}{\partial
t}$ is bounded in $L^{p}([0,T];X_{1})$ for $p>1$, then $J$ is relatively
compact in $C([0,T];X)$.

##  Appendix B  Proofs of Lemmas 3.1-3.2 and the equation (3.15)

In this appendix, for the sake of completeness, we will give the proof of
Lemmas 3.1-3.2 and the derivation of the equation (3.15).

Proof of Lemma 3.1. It follows from $\eqref{e1.1}_{1}$ that

$\frac{P(\rho)_{t}}{\gamma-1}+\frac{(P(\rho)u)_{x}}{\gamma-1}+P(\rho)u_{x}=0.$
(B.1)

Then multiplying $\eqref{e1.1}_{2}$ by $u$ and adding the resulting equation
to (B.1), and integrating over $\mathbb{R}$, one arrives at

$\frac{\text{d}}{\text{d}t}\int\left(\frac{1}{2}\rho
u^{2}+\frac{P(\rho)}{\gamma-1}\right)\text{d}x+\alpha\int\rho^{\delta}u_{x}^{2}\text{d}x=0,$
(B.2)

where one has used the fact that

$\rho
uu^{2}(t,\cdot),\quad\rho^{\gamma}u(t,\cdot)\quad\text{and}\quad\rho^{\delta}u_{x}u(t,\cdot)\in
W^{1,1}(\mathbb{R}).$

Finally, integrating (B.2) over $[0,t]$, one can get the desired estimate.

The proof of Lemma 3.1 is complete.

Proof of Lemma 3.2. According to $\eqref{e1.1}_{1}$ and $\eqref{e1.1}_{2}$,
one has

$\displaystyle\varphi(\rho)_{tx}+(\varphi(\rho)_{x}u)_{x}+(\rho\varphi^{\prime}(\rho)u_{x})_{x}=0,$
(B.3) $\displaystyle\rho(u_{t}+uu_{x})+P(\rho)_{x}=(\mu(\rho)u_{x})_{x}.$
(B.4)

Multiplying (B.3) by $\rho$, one has

$\rho\varphi(\rho)_{tx}+\rho
u\varphi(\rho)_{xx}+(\rho^{2}\varphi^{\prime}(\rho)u_{x})_{x}=0.$ (B.5)

Then adding (B.5) to (B.4), by the definition of $v$, one can obtain

$\rho(v_{t}+uv_{x})+P(\rho)_{x}-\left((\mu(\rho)-\rho^{2}\varphi^{\prime}(\rho))u_{x}\right)_{x}=0,$
(B.6)

which, along with

$\varphi^{\prime}(\rho)=\mu(\rho)/\rho^{2}$

in (B.6), yields that

$\rho(v_{t}+uv_{x})+P(\rho)_{x}=0.$ (B.7)

Multiplying (B.7) by $v$ and integrating over $\mathbb{R}$, one has

$\frac{\text{d}}{\text{d}t}\int\left(\frac{1}{2}\rho
v^{2}+\frac{A}{\gamma-1}\rho^{\gamma}\right){\rm{d}}x+\int
P(\rho)_{x}\varphi(\rho)_{x}{\rm{d}}x=0.$ (B.8)

Integrating (B.8) over $[0,T]$, one can obtain the desired conclusion.

The proof of Lemma 3.2 is complete.

Proof of (3.15). It follows from (B.7) that

$\begin{split}v_{t}+uv_{x}=&-\frac{P(\rho)_{x}}{\rho}=-A\gamma\rho^{\gamma-2}\rho_{x}\\\
=&-A\gamma\rho^{\gamma-\delta}\rho^{\delta-2}\rho_{x}=-\frac{A\gamma}{\alpha}\rho^{\gamma-\delta}(v-u),\end{split}$
(B.9)

which implies (3.15).

##  Appendix C  Local-in-time well-posedness for multi-dimensional case

In this appendix, for the sake of completeness, we will recall the local-in-
time well-posedness theory (obtained in [31, 49]) of regular solutions with
far field vacuum for the corresponding 3-D and 2-D Cauchy problems of the
degenerate viscous flows.

###  C.1. $0<\delta<1$

For the case $0<\delta<1$, we first give the definition of regular solutions
with far field vacuum to the 3-D Cauchy problem (1.1)-(1.5) with the following
initial data and far field behavior:

$\displaystyle(\rho(0,x),u(0,x))=(\rho_{0}(x),u_{0}(x))$
$\displaystyle\text{for}\quad x\in\mathbb{R}^{3},$ (C.1)
$\displaystyle\left(\rho(t,x),u(t,x)\right)\to\left(0,0\right)\quad\text{as}\quad\left|x\right|\to\infty$
$\displaystyle\text{for}\quad t\geq 0.$

######  Definition C.1 ([49]).

Assume $0<\delta<1$ and $T>0$. The pair $(\rho,u)$ is called a regular
solution to the Cauchy problem (1.1)-(1.5) with (C.1) in
$[0,T]\times\mathbb{R}^{3}$, if $(\rho,u)$ satisfies this problem in the sense
of distributions and:

$\displaystyle(1)\ \rho>0,\quad\rho^{\gamma-1}\in
C([0,T];H^{3}),\quad\nabla\rho^{\delta-1}\in L^{\infty}([0,T];L^{\infty}\cap
D^{2});$ $\displaystyle(2)\ u\in C([0,T];H^{3})\cap L^{2}([0,T];H^{4}),\quad
u_{t}\in C([0,T];H^{1})\cap L^{2}([0,T];D^{2}),$ $\displaystyle\ \
\quad\rho^{\frac{\delta-1}{2}}\nabla u\in
C([0,T];L^{2}),\quad\rho^{\frac{\delta-1}{2}}\nabla u_{t}\in
L^{\infty}([0,T];L^{2}),$ $\displaystyle\ \ \quad\rho^{\delta-1}\nabla u\in
L^{\infty}([0,T];D^{1}_{*}),\quad\rho^{\delta-1}\nabla^{2}u\in
C([0,T];H^{1})\cap L^{2}([0,T];D^{2}).$

The corresponding local-in-time well-posedness can be stated as follows.

######  Lemma C.1 ([49]).

Let parameters $\gamma$, $\delta$, $\alpha$ and $\beta$ satisfy

$\gamma>1,\quad 0<\delta<1,\quad\alpha>0,\quad 2\alpha+3\beta\geq 0.$

Assume the initial data $(\rho_{0},u_{0})$ satisfy

$\rho_{0}>0,\quad(\rho^{\gamma-1}_{0},u_{0})\in
H^{3},\quad\nabla\rho^{\delta-1}_{0}\in D^{1}_{*}\cap
D^{2},\quad\nabla\rho^{\frac{\delta-1}{2}}_{0}\in L^{4},$

and the initial compatibility conditions

$\nabla u_{0}=\rho^{\frac{1-\delta}{2}}_{0}g_{1},\quad\ \ \
Lu_{0}=\rho^{1-\delta}_{0}g_{2},\quad\ \ \
\nabla\left(\rho^{\delta-1}_{0}Lu_{0}\right)=\rho^{\frac{1-\delta}{2}}_{0}g_{3},$

for some $g_{i}(i=1,2,3)\in L^{2}$, then there exist a time $T_{*}>0$ and a
unique regular solution $(\rho,u)$ in $[0,T_{*}]\times\mathbb{R}^{3}$ to the
Cauchy problem (1.1)-(1.5) with (C.1) satisfying

$\begin{split}&\rho^{\frac{\delta-1}{2}}\nabla u_{t}\in
L^{\infty}([0,T_{*}];L^{2}),\quad\rho^{\delta-1}\nabla^{2}u_{t}\in
L^{2}([0,T_{*}];L^{2}),\\\ &t^{\frac{1}{2}}u\in
L^{\infty}([0,T_{*}];D^{4}),\quad t^{\frac{1}{2}}u_{t}\in
L^{\infty}([0,T_{*}];D^{2})\cap L^{2}([0,T_{*}];D^{3}),\\\ &u_{tt}\in
L^{2}([0,T_{*}];L^{2}),\quad t^{\frac{1}{2}}u_{tt}\in
L^{\infty}([0,T_{*}];L^{2})\cap L^{2}([0,T_{*}];D^{1}_{*}),\\\
&\rho^{1-\delta}\in L^{\infty}([0,T_{*}];L^{\infty}\cap D^{1,6}\cap
D^{2,3}\cap D^{3}),\\\ &\nabla\rho^{\delta-1}\in C([0,T_{*}];D^{1}_{*}\cap
D^{2}),\quad\nabla\ln\rho\in L^{\infty}([0,T_{*}];L^{\infty}\cap L^{6}\cap
D^{1,3}\cap D^{2}).\end{split}$

###  C.2. $\delta=1$

For the case $\delta=1$, we first give the definition of regular solutions
with far field vacuum to the 2-D Cauchy problem of (1.1)-(1.5) with the
following initial data and far field behavior:

$\displaystyle(\rho(0,x),u(0,x))=(\rho_{0}(x),u_{0}(x))$
$\displaystyle\text{for}\quad x\in\mathbb{R}^{2},$ (C.2)
$\displaystyle\left(\rho(t,x),u(t,x)\right)\to\left(0,0\right)\quad\text{as}\quad\left|x\right|\to\infty$
$\displaystyle\text{for}\quad t\geq 0.$

######  Definition C.2 ([31]).

Assume $\delta=1$ and $T>0$. The pair $(\rho,u)$ is called a regular solution
to the Cauchy problem (1.1)-(1.5) with (C.2) in $[0,T]\times\mathbb{R}^{2}$,
if $(\rho,u)$ satisfies this problem in the sense of distributions and:

$\displaystyle(1)\ 0<\rho\in C^{1}([0,T]\times\mathbb{R}^{2}),\
\rho^{\frac{\gamma-1}{2}}\in C([0,T];H^{3}),\ \nabla\ln\rho\in
C([0,T];L^{6}\cap D^{1}\cap D^{2});$ $\displaystyle(2)\ u\in
C([0,T];H^{3})\cap L^{2}([0,T];H^{4}),\quad u_{t}\in C([0,T];H^{1})\cap
L^{2}([0,T];D^{2});$ $\displaystyle(3)\
\lim_{|x|\to\infty}\left(u_{t}+u\cdot\nabla
u+Lu\right)=\lim_{|x|\to\infty}\left(\nabla\ln\rho\cdot
Q(u)\right)\quad{\rm{for}}\quad t>0.$

The corresponding local-in-time well-posedness can be stated as follows.

######  Lemma C.2 ([31]).

Let parameters $\gamma$, $\delta$, $\alpha$ and $\beta$ satisfy

$\gamma>1,\quad\delta=1,\quad\alpha>0,\quad\alpha+\beta\geq 0.$

Assume the initial data $(\rho_{0},u_{0})$ satisfy

$\rho_{0}>0,\quad(\rho^{\frac{\gamma-1}{2}}_{0},u_{0})\in
H^{3},\quad\nabla\ln\rho_{0}\in L^{6}\cap D^{1}\cap D^{2},$

then there exist a time $T_{*}>0$ and a unique regular solution $(\rho,u)$ in
$[0,T_{*}]\times\mathbb{R}^{2}$ to the Cauchy problem (1.1)-(1.5) with (C.2)
satisfying

$\begin{split}&(\rho^{\frac{\gamma-1}{2}})_{t}\in
C([0,T_{*}];H^{2}),\quad(\nabla\ln\rho)_{t}\in C([0,T_{*}];H^{1}),\\\
&u_{tt}\in L^{2}([0,T_{*}];L^{2}),\quad t^{\frac{1}{2}}u\in
L^{\infty}([0,T_{*}];D^{4}),\\\ &t^{\frac{1}{2}}u_{t}\in
L^{\infty}([0,T_{*}];D^{2})\cap L^{2}([0,T_{*}];D^{3}),\ \
t^{\frac{1}{2}}u_{tt}\in L^{\infty}([0,T_{*}];L^{2})\cap
L^{2}([0,T_{*}];D^{1}).\end{split}$

Acknowledgement: The research was supported in part by National Natural
Science Foundation of China under Grants 12101395 and 12161141004. The
research of S. Zhu was also supported partially by The Royal Society under
Grants: Newton International Fellowships NF170015, Newton International
Fellowships Alumni AL/201021 and AL/211005.

Conflict of Interest: The authors declare that they have no conflict of
interest. The authors also declare that this manuscript has not been
previously published, and will not be submitted elsewhere before your
decision.

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