\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2019 (2019), No. 17, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2019 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2019/17\hfil Bipolar Navier-Stokes-Poisson systems]
{Existence of global solutions to Cauchy problems for bipolar Navier-Stokes-Poisson
systems}

\author[J. Liu \hfil EJDE-2019/17\hfilneg]
{Jian Liu}

\address{Jian Liu \newline
College of Teacher Education,
Quzhou University, Quzhou 324000, China}
\email{liujian.maths@gmail.com}

\thanks{Submitted December 21, 2018. Published January 29, 2019.}
\subjclass[2010]{35Q35, 76N03}
\keywords{Cauchy problem; bipolar Navier-Stokes-Poisson system;
\hfill\break\indent global strong solution}

\begin{abstract}
 In this article, we consider the Cauchy problem for one-dimensional
 compressible bipolar Navier-Stokes-Poisson system with density-dependent
 viscosities. Under certain assumptions on the initial data, we prove
 the existence and uniqueness of a global strong solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction} \label{introduction}

 Bipolar Navier-Stokes-Poisson (BNSP) has been used to simulate the transport
of charged particles under the influence of electrostatic force governed
by the self-consistent Poisson equation. In this paper, we consider the
 Cauchy problem for one-dimensional isentropic compressible BNSP with
density-dependent viscosities,

\begin{equation} \label{1.1o}
\begin{gathered}
\rho_t+(\rho u)_x=0,\\
(\rho u)_t+(\rho u^2)_x+p(\rho)_x=\rho\Phi_x+(\mu(\rho)u_x)_x, \\
n_t+(nv)_x=0,\\
(nv)_t +(nv^2)_x+p(n)_x=-n\Phi_x+(\mu(n)v_x)_x,\\
\Phi_{xx}=\rho-n.
\end{gathered}
\end{equation}
Here $\rho(x,t)\geq0$, $n(x,t)\geq0$ denote the charge densities, $u$, $v$
the charge velocities, $\Phi$ the electrostatic potential,
$p(\rho)=\rho^{\gamma}$ and $p(n)=n^{\gamma}$, $\gamma>1$ are the pressure
of charge, such as electron and ion, and $\mu(\rho)$, $\mu(n)$ are the
viscosity coefficients.

There have been extensive studies on the  existence and asymptotic behavior of
global solutions to the unipolar Navier-Stokes-Poisson system (NSP).
The  existence of global weak solutions to NSP with general initial
data was proved in \cite{DD,KL,ZT}. The quasi-neutral and some related asymptotic
limits were studied in \cite{DJL, DM, JLL}. When the Poisson equation describes
the self-gravitonal force for stellar gases, the  existence of global
weak solution and asymptotic behavior were also investigated together with
the stability analysis, we refer the reader to \cite{DZ} and the references therein.
The results in \cite{DLUY, MN1980, P1985} imply that the electric field affects
the large time behavior of the solution and give rise to different asymptotic
 behaviors of Navier-Stokes and NSP. In addition, Hao-Li \cite{HL} proved the
 well-posedness of NSP in the Besov space. Li-Matsumura-Zhang \cite{LMZ}
 proved the  existence and the optimal time convergence rates of the
global classical solution. Recently, Bie-Wang-Yao in \cite{BWY} proved
optimal decay rate in the critical $L^p$ framework.

For bipolar Navier-Stokes-Poisson system \eqref{1.1o}, there are also abundant
results concerning the existence and asymptotic behavior of the global weak
solution. Li-Yang-Zou \cite{LYZ} proved optimal $L^2$ time convergence rate
for the global classical solution for a small initial perturbation of
the constant equilibrium state. The optimal time decay rate of global strong
solution is established in \cite{HLYZ, ZC}.
Liu-Lian in \cite{LL} proved global existence of solution to free boundary
value problem. Lin-Hao-Li \cite{LHL} studied the  existence and
uniqueness of global strong solutions in hybrid Besov spaces with the initial
data close to an equilibrium state. Wu-Wang  \cite{WW} proved pointwise
estimates for BNSP system. As a continuation of the study in this direction,
in this paper, we will study the Cauchy problem for BNSP in one-dimension.

The rest of this paper is as follows. In section 2, we state the main results
of this article. In section 3, we give some a-priori estimates for the solution.
In section 4, we prove the existence and uniqueness of global strong solutions.

\section{Main result}

In this article, we consider the existence and uniqueness of global solutions
for the  Cauchy problem \eqref{1.1o} in the whole space
$\mathbb{R}$. Assume $\mu(\rho)=\rho^{\alpha}$, $\mu(n)=n^{\alpha}$, then
\eqref{1.1o} can be rewritten as
\begin{equation} \label{2.1o}
\begin{gathered}
\rho_t+(\rho u)_x=0,\\
(\rho u)_t+(\rho u^2)_x+(\rho^{\gamma})_x=\rho\Phi_x+(\rho^{\alpha}u_x)_x,\\
n_t+(nv)_x=0,\\
(nv)_t +(nv^2)_x+(n^{\gamma})_x=-n\Phi_x+(n^{\alpha}v_x)_x,\\
\Phi_{xx}=\rho-n,\\
{\Phi_x}{(\pm\infty,t)}=0,\\
(\rho,u,n,v)(x,0)=(\rho_0,u_0,n_0,v_0)(x),x\in\mathbb{R},\\
(\rho_0,u_0,n_0,v_0){(\pm\infty)}=(\overline{\rho},0,\overline{n},0).
\end{gathered}
\end{equation}
We assume the initial data satisfy
\begin{equation} \label{2.1o1}
\begin{gathered}
(\rho_0-\overline{\rho},u_0,n_0-\overline{n},v_0)\in\,H^1(\mathbb{R}),\\
 0<\rho_1\leq\rho_0(x)\leq\rho_2,\quad 0<n_1\leq\,n_0(x)\leq n_2,\\
\Phi_{x0}=\int_{-\infty}^x(\rho_0-n_0)(y)\,\mathrm{d}y\in\,L^2(\mathbb{R}),
\end{gathered}
\end{equation}
where $\rho_1$,$\rho_2$,$n_1$,$n_2$,$\overline{\rho}$ and $\overline{n}$
are positive constants. We define
\begin{gather*}
E_{01}=:\frac12\int_{\mathbb{R}}\rho_0u_0^2\,\mathrm{d}x
+\frac{1}{\gamma-1}\int_{\mathbb{R}}
\Big((\rho_0^{\gamma}-\overline{\rho}^{\gamma})-
\gamma\overline{\rho}^{\gamma-1}(\rho_0-\overline{\rho})\Big)\,\mathrm{d}x, \\
E_{02}=:\frac12\int_{\mathbb{R}}n_0v_0^2\,\mathrm{d}x
+\frac{1}{\gamma-1}\int_{\mathbb{R}}
\Big((n_0^{\gamma}-\overline{n}^{\gamma})-
\gamma\overline{n}^{\gamma-1}(n_0-\overline{n})\Big)\,\mathrm{d}x, \\
E_{11}=:\frac12\int_{\mathbb{R}}\rho_0
\Big(u_0+\frac{1}{\alpha}\rho_0^{-1}(\rho_0^{\alpha})_x\Big)^2\,\mathrm{d}x
+\frac{1}{\gamma-1}\int_{\mathbb{R}}
\Big((\rho_0^{\gamma}-\overline{\rho}^{\gamma})
-\gamma\overline{\rho}^{\gamma-1}(\rho_0-\overline{\rho})\Big)\,\mathrm{d}x, \\
E_{12}=:\frac12\int_{\mathbb{R}}n_0
\Big(v_0+\frac{1}{\alpha}n_0^{-1}(n_0^{\alpha})_x\Big)^2\,\mathrm{d}x
+\frac{1}{\gamma-1}\int_{\mathbb{R}}
\Big((n_0^{\gamma}-\overline{n}^{\gamma})
-\gamma\overline{n}^{\gamma-1}(n_0-\overline{n})\Big)\,\mathrm{d}x, \\
E_0=:\frac12\int_{\mathbb{R}}\Phi_{x0}^2\,\mathrm{d}x+E_{01}+E_{02},\quad
E_1=:\frac12\int_{\mathbb{R}}\Phi_{x0}^2\,\mathrm{d}x+E_{11}+E_{12}.
\end{gather*}
Then, the main result of this paper can be stated as follows.

\begin{theorem}\label{thm2.1}
Let $\gamma>1$, $\alpha>0$ and $\alpha\neq1/2$.
Assume that the initial data satisfies \eqref{2.1o1} for $0<\alpha<1/2$, and
\eqref{2.1o1} with $E_0^{1/2}(E_0+E_1)^{1/2} <\frac{1}{2\alpha-1}
 \bar\rho^{\frac\gamma2+\alpha-\frac12}$ for $\alpha>1/2$.
Then there exist positive constants $\rho_\pm$ and $n_\pm$
with $\rho_-<\bar\rho<\rho_+$,  $n_-<\bar n<n_+$ so that the unique global strong
solution $(\rho, n, u, v, \Phi_x)$ to \eqref{2.1o} exists and satisfies
\begin{equation} \label{2.1o5}
\begin{gathered}
 0<\rho_-\leq\rho\leq\rho^+,\,0<n_-\leq\,n\leq\,n^+,\\
 u,v\in L^{\infty}([0,T];H^1(\mathbb{R}))\cap L^2([0,T];H^2(\mathbb{R})),\\
\rho_x,u_x,n_x,v_x\in L^{\infty}([0,T];L^2(\mathbb{R}))
 \cap L^2([0,T];L^2(\mathbb{R})),\\
\rho_t,u_t,n_t,v_t\in L^2([0,T];H^1(\mathbb{R})),
 \Phi_x \in L^{\infty}([0,T]; H^2(\mathbb{R})).
\end{gathered}
\end{equation}
\end{theorem}


\section{A-priori estimates}
The proof of Theorem \ref{thm2.1} consists of the
basic a-priori estimates and regular analysis. Using arguments similar
to those in \cite{LLLH}, we establish the following lemmas.

\begin{lemma}\label{lm3.1}
Let $T>0$, and $(\rho, n, u, v, \Phi_x)$ with $\rho>0$, $n>0$ be a solution
to \eqref{2.1o} for $t\in[0, T]$ under the conditions in
Theorem~\ref{thm2.1}. Then
\begin{equation}\label{3.1o1}
\begin{aligned}
&\int_{\mathbb{R}}\frac12\Big(\rho u^2+nv^2+\Phi_x^2\Big)\,\mathrm{d}x
+\frac{1}{\gamma-1}\int_\mathbb{R}\Big((\rho^{\gamma}-\overline{\rho}^{\gamma})
-\gamma\overline{\rho}^{\gamma-1}(\rho-\overline{\rho})\Big)\,\mathrm{d}x
 \\
&+\frac{1}{\gamma-1}\int_\mathbb{R}\Big((n^{\gamma}-\overline{n}^{\gamma})
-\gamma\overline{n}^{\gamma-1}(n-\overline{n})\Big)\,\mathrm{d}x
+\int_0^t\int_{\mathbb{R}}\Big(\rho^{\alpha}u_x^2
+n^{\alpha}v_x^2\Big)\,\mathrm{d}x\,\mathrm{d}s \\
&=E_0.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Taking the product of \eqref{2.1o}$_2$ with $u$ and integrating on $\mathbb{R}$,
and using \eqref{2.1o}$_1$ and integrating by parts, we have
\begin{equation}
\frac12\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{\mathbb{R}}\rho u^2\,\mathrm{d}x+
\int_{\mathbb{R}}(\rho^{\gamma})_x u\,\mathrm{d}x
+\int_{\mathbb{R}}\rho^{\alpha}u_x^2\,\mathrm{d}x
=\int_{\mathbb{R}}\rho_t\Phi\,\mathrm{d}x,
\end{equation}
where
\begin{equation}
\int_{\mathbb{R}}(\rho^{\gamma})_xu\,\mathrm{d}x
=\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_\mathbb{R}\Big(\frac{1}{\gamma-1}(\rho^{\gamma}
-\overline{\rho}^{\gamma})-\frac{\gamma}{\gamma-1}\overline{\rho}^{\gamma-1}
(\rho-\overline{\rho})\Big)\,\mathrm{d}x,
\end{equation}
integrating with respect to $t\in[0,T]$,  we have
\begin{equation}\label{3.1o2}
\begin{aligned}
&\int_{\mathbb{R}}\Big(\frac12\rho u^2
+\frac{1}{\gamma-1}(\rho^{\gamma}-\overline{\rho}^{\gamma})-
\frac{\gamma}{\gamma-1}\overline{\rho}^{\gamma-1}
 (\rho-\overline{\rho})\Big)\,\mathrm{d}x
+\int_0^t\int_{\mathbb{R}}\rho^{\alpha}u_x^2\,\mathrm{d}x\,\mathrm{d}s \\
&=\int_0^t\int_{\mathbb{R}}\rho_s\Phi\,\mathrm{d}x\,\mathrm{d}s+E_{01}.
\end{aligned}
\end{equation}
Meanwhile, we have
\begin{equation} \label{3.1o3}
\begin{aligned}
&\int_{\mathbb{R}}\Big(\frac12nv^2+\frac{1}{\gamma-1}(n^{\gamma}
 -\overline{n}^{\gamma})
-\frac{\gamma}{\gamma-1}\overline{n}^{\gamma-1}(n-\overline{n})\Big)\,\mathrm{d}x
+\int_0^t\int_{\mathbb{R}}n^{\alpha}v_x^2\,\mathrm{d}x\,\mathrm{d}s \\
&=-\int_0^t\int_{\mathbb{R}}n_s\Phi \,\mathrm{d}x\,\mathrm{d}s+E_{02}.
\end{aligned}
\end{equation}
Adding \eqref{3.1o2} to \eqref{3.1o3}, we obtain
\begin{equation} \label{3.1o4}
\int_{\mathbb{R}}(\rho_t-n_t)\Phi\,\mathrm{d}x
=\int_{\mathbb{R}}\Phi_{xxt}\Phi\,\mathrm{d}x
=-\int_{\mathbb{R}}\Phi_x\Phi_{xt}\,\mathrm{d}x
=-\frac12\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{\mathbb{R}}\Phi_x^2\,\mathrm{d}x.
\end{equation}
The combination of \eqref{3.1o2}, \eqref{3.1o3} and \eqref{3.1o4} gives rise to
\eqref{3.1o1}.
\end{proof}

\begin{lemma}\label{lm3.2}
Under the  assumptions in Lemma~\ref{lm3.1}, it holds
\begin{equation} \label{3.2o1}
\begin{aligned}
&\int_{\mathbb{R}}\Big(\frac12\rho(u+\frac{1}{\alpha}\rho^{-1}(\rho^{\alpha})_x)^2
+\frac{1}{\gamma-1}(\rho^{\gamma}-\overline{\rho}^{\gamma})
-\frac{\gamma}{\gamma-1}\overline{\rho}^{\gamma-1}
(\rho-\overline{\rho})\Big)\,\mathrm{d}x \\
&+\int_{\mathbb{R}}\Big(\frac12n(v+\frac{1}{\alpha}n^{-1}(n^{\alpha})_x)^2
+\frac{1}{\gamma-1}(n^{\gamma}-\overline{n}^{\gamma})
-\frac{\gamma}{\gamma-1}\overline{n}^{\gamma-1}(n-\overline{n})\Big)\,\mathrm{d}x\\
&+\frac12\int_{\mathbb{R}}\Phi_x^2\,\mathrm{d}x
 +\gamma\int_0^t\int_{\mathbb{R}}\Big(\rho^{\gamma+\alpha-3}\rho_x^2
 +n^{\gamma+\alpha-3}n_x^2\Big)\,\mathrm{d}x\,\mathrm{d}s \\
&+\int_0^t\int_{\mathbb{R}}(\rho^{\alpha}-n^{\alpha})(\rho-n)
\,\mathrm{d}x\,\mathrm{d}s
=E_1.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Multiplying \eqref{2.1o}$_1$ by $\rho^{\alpha-1}$, and then differentiting
with respect to  $x$, then using \eqref{2.1o}$_2$ and  direct computations,
we obtain
\begin{equation} \label{3.2o2}
\rho(u+\frac{1}{\alpha}\rho^{-1}(\rho^{\alpha})_x)_t
+\rho\,u(u+\frac{1}{\alpha}\rho^{-1}(\rho^{\alpha})_x)_x
+(\rho^{\gamma})_x=\rho\Phi_x.
\end{equation}
Then, multiplying \eqref{3.2o2} by $u+\frac{1}{\alpha}\rho^{-1}(\rho^{\alpha})_x$,
and integrating  over $\mathbb{R}$ (by parts), using \eqref{2.1o}$_1$
and the boundary conditions, after direct computations, we obtain
\begin{equation} \label{3.2o3}
\begin{aligned}
&\int_{\mathbb{R}}\Big(\frac12\rho(u+\frac{1}{\alpha}\rho^{-1}(\rho^{\alpha})_x)^2
+\frac{1}{\gamma-1}(\rho^{\gamma}-\overline{\rho}^{\gamma})
- \frac{\gamma}{\gamma-1}\overline{\rho}^{\gamma-1}
 (\rho-\overline{\rho})\Big)\,\mathrm{d}x \\
&+\gamma\int_0^t\int_{\mathbb{R}}\rho^{\gamma+\alpha-3}\rho_x^2\,\mathrm{d}x\,\mathrm{d}s\\
&=\int_0^t\int_{\mathbb{R}}\rho_s\Phi \,\mathrm{d}x\,\mathrm{d}s
+\frac{1}{\alpha}\int_0^t\int_{\mathbb{R}}
(\rho^{\alpha})_x\Phi_x\,\mathrm{d}x\,\mathrm{d}s+E_{11}.
\end{aligned}
\end{equation}
Similarly, we have
\begin{equation} \label{3.2o4}
\begin{aligned}
&\int_{\mathbb{R}}\Big(\frac12n(v+\frac{1}{\alpha}n^{-1}(n^{\alpha})_x)^2
+\frac{1}{\gamma-1}(n^{\gamma}-\overline{n}^{\gamma})
-\frac{\gamma}{\gamma-1}\overline{n}^{\gamma-1}(n-\overline{n})\Big)\,\mathrm{d}x \\
&+\gamma\int_0^t\int_{\mathbb{R}}n^{\gamma+\alpha-3}n_x^2\,\mathrm{d}x\,\mathrm{d}s\\
&=-\int_0^t\int_{\mathbb{R}}n_s\Phi \,\mathrm{d}x\,\mathrm{d}s
-\frac{1}{\alpha}\int_0^t\int_{\mathbb{R}}
(n^{\alpha})_x\Phi_x\,\mathrm{d}x\,\mathrm{d}s+E_{12}.
\end{aligned}
\end{equation}
Adding \eqref{3.2o3} to \eqref{3.2o4},  we obtain
\begin{equation} \label{3.2o5}
\begin{aligned}
\int_0^t\int_{\mathbb{R}}(\rho^{\alpha}-n^{\alpha})_x\Phi_x\,\mathrm{d}x\,\mathrm{d}s
&=-\int_0^t\int_{\mathbb{R}}(\rho^{\alpha}-n^{\alpha})\Phi_{xx}\,\mathrm{d}x\,\mathrm{d}s\\
&=-\int_0^t\int_{\mathbb{R}}(\rho^{\alpha}-n^{\alpha})(\rho-n)\,\mathrm{d}x\,\mathrm{d}s.
\end{aligned}
\end{equation}
Combining  \eqref{3.2o3}, \eqref{3.2o4}, \eqref{3.2o5} and \eqref{3.1o4} gives rise
to \eqref{3.2o1}.
\end{proof}


\begin{lemma}\label{lm3.3}
Under the  assumptions in Lemma~\ref{lm3.1}, we have
\begin{equation} \label{3.3o1}
0<\rho_-\leq\rho\leq\rho^+,\quad 0<n_-\leq\,n\leq n^+.
\end{equation}
\end{lemma}

\begin{proof}
Denote
 \begin{gather} \label{3.3o2}
  \varphi(\rho):=\frac{1}{\gamma-1}\Big(\rho^{\gamma}-\bar{\rho}^{\gamma}
  -\gamma\bar{\rho}^{\gamma-1}(\rho-\bar{\rho})\Big), \\
 \label{3.3o3}
      \psi(\rho):=\int_{\bar{\rho}}^{\rho}\varphi(\eta)^{\frac{1}{2}}
      \eta^{\alpha-\frac{3}{2}}\,\mathrm{d}\eta.
 \end{gather}
It is easy to verify that $\varphi(\rho)\geq0$ and $\psi'(\rho)\ge0$.
In addition,  as $\rho\to+\infty$ it holds
\begin{equation} \label{3.3o4}
\begin{aligned}
\lim_{\rho\to+\infty}\psi(\rho)
& \to (\gamma-1)^{-1/2}\lim_{\rho\to+
\infty}\int_{\bar\rho}^\rho \eta^\frac{\gamma+2\alpha-3}2\,\mathrm{d}\eta\\
&=\lim_{\rho\to+\infty}
 \frac{2}{(\gamma+2\alpha-1)\sqrt{\gamma-1}}(\rho^{\frac{\gamma+2\alpha-1}{2}}
-\bar\rho^{\frac{\gamma+2\alpha-1}{2}})\to+\infty,
\end{aligned}
\end{equation}
and as $\rho\to 0$,
\begin{equation} \label{3.3o5}
\begin{aligned}
\lim_{\rho\to0}\psi(\rho)
&\to \lim_{\rho\to0}
\int_{\bar\rho}^\rho\bar\rho^{\frac\gamma2}
\eta^{\alpha-\frac32}\,\mathrm{d}\eta
\\
&=\lim_{\rho\to0}\frac{2}{2\alpha-1}\bar\rho^{\frac\gamma2}(\rho^{\alpha-\frac12}
-\bar\rho^{\alpha-\frac12}),\\
&\to \begin{cases}
-\infty,&\text{if }0<\alpha<\frac12, \\
-\frac{2}{2\alpha-1}\bar\rho^{\frac\gamma2+\alpha- \frac{1}{2}}
&\text{if } \alpha>\frac12.
\end{cases}
 \end{aligned}
\end{equation}
 We can choose two constants $\rho_\pm>0$ with
$\rho_-<\bar\rho<\rho_+$ and $\rho_+=\psi^{-1}(-\psi(\rho_-))$ so that
\begin{equation} \label{3.18a}
\begin{gathered}
 2E_0^{1/2}(E_0+E_1)^{1/2} < -\psi(\rho_-), \quad  \alpha\in(0,\frac12),\\
 2E_0^{1/2}(E_0+E_1)^{1/2} < -\psi(\rho_-)<\frac{2}{2\alpha-1}
   \bar\rho^{\frac\gamma2+\alpha-\frac12},  \quad  \alpha>\frac12,
   \end{gathered}
\end{equation}
which obviously satisfies
\begin{equation}
  \psi(\rho_-) < - 2E_0^{1/2}(E_0+E_1)^{1/2}
  < 2E_0^{1/2}(E_0+E_1)^{1/2} < \psi(\rho_+).
\end{equation}
From \eqref{3.1o1} and \eqref{3.2o1} it follows that
\begin{equation}
\begin{aligned}
|\psi(\rho(x))|&\leq|\int_{\mathbb{R}}\partial_x\psi(\rho)\,\mathrm{d}x|
   \leq\big|\int_{\mathbb{R}}\varphi(\rho)^{1/2}
\rho_x\rho^{\alpha-\frac32}\,\mathrm{d}x\big|  \\
 &\leq\Big(\int_{\mathbb{R}}\varphi(\rho)\,\mathrm{d}x\Big)^{1/2}
\Big(\int_{\mathbb{R}}\big(\mbox{$\frac{2}{2\alpha-1}$}
(\rho^{\alpha-\frac12})_x\big)^2\,\mathrm{d}x\Big)^{1/2}  \\
 &\leq2E_0^{1/2}(E_0+E_1)^{1/2},
\end{aligned}
\end{equation}
from which we obtain the half of \eqref{3.3o1} with $\rho_-$ and $\rho^+$
determined as above.

Similarly, we have the another half of \eqref{3.3o1}.
The proof is complete.
\end{proof}

\begin{lemma}\label{lm3.4}
Under the same assumptions as in Lemma~\ref{lm3.1}, it holds that
\begin{equation}\label{3.4o1}
\begin{aligned}
&\int_\mathbb{R}u_x^2\,\mathrm{d}x
+\int_0^t\int_\mathbb{R}u_s^2\,\mathrm{d}x\,\mathrm{d}s
+\int_0^t\int_\mathbb{R}u_{xx}^2\,\mathrm{d}x\,\mathrm{d}s \\
&+\int_\mathbb{R}v_x^2\,\mathrm{d}x+\int_0^t\int_\mathbb{R}v_s^2\,\mathrm{d}x\,\mathrm{d}s
+\int_0^t\int_\mathbb{R}v_{xx}^2\,\mathrm{d}x\,\mathrm{d}s
\end{aligned} \leq C(T).
\end{equation}
\end{lemma}

\begin{proof}
First we estimate for $u$. Multiplying \eqref{2.1o}$_2$ by $\rho^{-\alpha}u_t$,
 and integrate over $\mathbb{R}$. With the help of \eqref{2.1o}$_1$ and the boundary
conditions, after direct computations, we obtain
\begin{equation} \label{3.4o2}
\begin{aligned}
&\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_\mathbb{R}\Big(\frac12u_x^2
 -\rho^{\gamma-\alpha}u_x\Big)\,\mathrm{d}x
 +\int_\mathbb{R}\rho^{1-\alpha}u_t^2\,\mathrm{d}x \\
&=-\int_\mathbb{R}\rho^{1-\alpha}u u_x u_t\,\mathrm{d}x
 -\alpha\int_\mathbb{R}\rho^{\gamma-\alpha-1}\rho_x u_t\,\mathrm{d}x
 +(\gamma-\alpha)\int_\mathbb{R}\rho^{\gamma-\alpha-1}\rho_x u u_x\,\mathrm{d}x \\
&\quad +(\gamma-\alpha)\int_\mathbb{R}\rho^{\gamma-\alpha}u_x^2\,\mathrm{d}x
 +\alpha\int_\mathbb{R}\rho^{-1}\rho_x u_x u_t \,\mathrm{d}x
 +\int_\mathbb{R}\rho^{1-\alpha}u_t\Phi_x \,\mathrm{d}x.
\end{aligned}
\end{equation}
Integrating \eqref{3.4o2} over $t\in[0,T]$ and direct computations yield
\begin{align*}
&\frac12\int_\mathbb{R}u_x^2\,\mathrm{d}x
+\int_0^t\int_\mathbb{R}\rho^{1-\alpha}u_s^2\,\mathrm{d}x\,\mathrm{d}s \\
&=\int_\mathbb{R}\rho^{\gamma-\alpha}u_x\,\mathrm{d}x
-\int_0^t\int_\mathbb{R}\rho^{1-\alpha}u u_x u_s \,\mathrm{d}x\,\mathrm{d}s
-\alpha\int_0^t\int_\mathbb{R}\rho^{\gamma-\alpha-1}\rho_x u_s\,\mathrm{d}x\,\mathrm{d}s \\
&\quad +(\gamma-\alpha)\int_0^t\int_\mathbb{R}
\rho^{\gamma-\alpha-1}u\rho_x u_x\,\mathrm{d}x\,\mathrm{d}s
+\alpha\int_0^t\int_\mathbb{R}\rho^{-1}\rho_x u_x u_s \,\mathrm{d}x\,\mathrm{d}s\\
&\quad +\int_0^t\int_\mathbb{R}\rho^{1-\alpha}u_s\Phi_x\,\mathrm{d}x\,\mathrm{d}s
+(\gamma-\alpha)\int_0^t\int_\mathbb{R}\rho^{\gamma-\alpha}u_x^2
 \,\mathrm{d}x\,\mathrm{d}s \\
&\quad +\int_\mathbb{R}\Big(\frac12u_{x0}^2
 -\rho_0^{\gamma-\alpha}u_{x0}\Big)\,\mathrm{d}x.
\end{align*}
With the help of \eqref{2.1o1}, Lemmas \ref{lm3.1}, \ref{lm3.2} and \ref{lm3.3},
and Young's inequality,  direct computation yield
\begin{equation} \label{3.4o3}
\begin{aligned}
&\frac12\int_\mathbb{R}u_x^2\,\mathrm{d}x
+\frac35\int_0^t\int_\mathbb{R}\rho^{1-\alpha}u_s^2\,\mathrm{d}x\,\mathrm{d}s \\
&\leq  C(T)+C\int_0^t\int_\mathbb{R}u^2u_x^2\,\mathrm{d}x\,\mathrm{d}s
+C\int_0^t\int_\mathbb{R}\rho_x^2u_x^2\,\mathrm{d}x\,\mathrm{d}s.
\end{aligned}
\end{equation}
Next, we estimate  $\int_0^t\int_\mathbb{R}u_{xx}^2\,\mathrm{d}x\,\mathrm{d}s$.
From \eqref{2.1o}$_1$ and \eqref{2.1o}$_2$, we have
\begin{equation}
u_{xx}=\rho^{1-\alpha}u_t+\rho^{1-\alpha}u u_x
+\gamma\rho^{-\alpha-\gamma-1}\rho_x
-\rho^{1-\alpha}\Phi_x-\alpha\rho^{-1}\rho_x u_x.
\end{equation}
Combination Lemma~\ref{lm3.2} and Young's inequality, we obtain
\begin{equation} \label{3.4o4}
\int_\mathbb{R}u_{xx}^2\,\mathrm{d}x
\leq \frac{1}{10}\int_\mathbb{R}\rho^{1-\alpha}u_t^2\,\mathrm{d}x
+C\int_\mathbb{R}u^2u_x^2\,\mathrm{d}x+C\int_\mathbb{R}\rho_x^2u_x^2\,\mathrm{d}x+C.
\end{equation}
Integrating \eqref{3.4o4} over $t\in[0,T]$, combining \eqref{3.4o3},
Lemmas \ref{lm3.1} and \ref{lm3.2}, and using Gagliardo-Nirenberg-Sobolev inequality,
we have
\begin{equation} \label{3.4o5}
\begin{aligned}
&\frac12\int_\mathbb{R}u_x^2\,\mathrm{d}x
+\frac12\int_0^t\int_\mathbb{R}\rho^{1-\alpha}u_s^2\,\mathrm{d}x\,\mathrm{d}s
+\int_0^t\int_\mathbb{R}u_{xx}^2\,\mathrm{d}x\,\mathrm{d}s \\
&\leq C\int_0^t\int_\mathbb{R}(u^2+\rho_x^2)u_{x}^2\,\mathrm{d}x\,\mathrm{d}s+C(T) \\
&\leq C\int_0^t\|u_x\|^2_{L^\infty}\,\mathrm{d}s+C(T) \\
&\leq \frac14\int_0^t\int_\mathbb{R}u_{xx}^2\,\mathrm{d}x\,\mathrm{d}s+C(T)\,.
\end{aligned}
\end{equation}
Then
\begin{equation} \label{3.4o6}
\int_\mathbb{R}u_x^2\,\mathrm{d}x+\int_0^t\int_\mathbb{R}u_s^2\,\mathrm{d}x\,\mathrm{d}s
+\int_0^t\int_\mathbb{R}u_{xx}^2\,\mathrm{d}x\,\mathrm{d}s\leq C(T).
\end{equation}
Applying  similar arguments we  obtain
\begin{equation} \label{3.4o7}
\int_\mathbb{R}v_x^2\,\mathrm{d}x+\int_0^t\int_\mathbb{R}v_s^2\,\mathrm{d}x\,\mathrm{d}s
+\int_0^t\int_\mathbb{R}v_{xx}^2\,\mathrm{d}x\,\mathrm{d}s\leq C(T);
\end{equation}
thus  \eqref{3.4o1} follows.
\end{proof}

\begin{lemma}\label{lm3.5}
Under the  assumptions  in Lemma~\ref{lm3.1}, the solution $(\rho,n,u,v,\Phi_x)$
satisfies
\begin{equation} \label{3.5o1}
\int_\mathbb{R}u_t^2\,\mathrm{d}x
+\int_0^t\int_\mathbb{R}u_{xs}^2\,\mathrm{d}x\,\mathrm{d}s
+\int_\mathbb{R}v_t^2\,\mathrm{d}x
+\int_0^t\int_\mathbb{R}v_{xs}^2\,\mathrm{d}x\,\mathrm{d}s
\leq C(T).
\end{equation}
\end{lemma}

\begin{proof}
Differentiating \eqref{2.1o}$_2$ with respect to $t$, then multiplying by
$u_t$ and integrating  over $\mathbb{R}$ (by parts), with \eqref{2.1o}$_1$
and Young's equality, we obtain
\begin{equation} \label{3.5o2}
\begin{aligned}
&\frac12\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_\mathbb{R}\rho u_t^2\,\mathrm{d}x
+\int_\mathbb{R}\rho^\alpha u_{xt}^2\,\mathrm{d}x \\
&=\frac12\int_\mathbb{R}\rho_t u_t^2\,\mathrm{d}x
-\int_\mathbb{R}(\rho u u_x)_t u_t\,\mathrm{d}x
+\int_\mathbb{R}(\rho^\gamma)_t u_{tx}\,\mathrm{d}x \\
&\quad -\int_\mathbb{R}(\rho^\alpha)_t u_x u_{xt}\,\mathrm{d}x
+\int_\mathbb{R}(\rho\Phi_x)_t u_t\,\mathrm{d}x \\
&\leq\frac12\int_\mathbb{R}\rho^\alpha u_{xt}^2\,\mathrm{d}x
+C\int_\mathbb{R}(u_x^2+u_t^2+\rho_x^2)\,\mathrm{d}x
+\int_\mathbb{R}(\rho\Phi_x)_t u_t\,\mathrm{d}x.
\end{aligned}
\end{equation}
Integrating \eqref{3.5o2} over $[0,t]$, we have
\begin{equation} \label{3.5o3}
\frac12\int_\mathbb{R}\rho u_t^2\,\mathrm{d}x
+\frac12\int_0^t\int_\mathbb{R}\rho^\alpha u_{xs}^2\,\mathrm{d}x\,\mathrm{d}s
\leq C(T)+\int_0^t\int_\mathbb{R}(\rho\Phi_x)_s u_s \,\mathrm{d}x\,\mathrm{d}s.
\end{equation}
Next we estimate $\int_0^t\int_\mathbb{R}(\rho\Phi_x)_s u_s \,\mathrm{d}x\,\mathrm{d}s.$
From \eqref{2.1o}$_1$ and \eqref{2.1o}$_5$, it follows that
\begin{equation} \label{3.5o4}
(\rho\Phi_x)_s u_s=-\rho_x u u_s\Phi_x-\rho u_x u_s\Phi_x-\rho u_s(\rho u-nv).
\end{equation}
Using \eqref{2.1o}$_5$, \eqref{2.1o}$_6$, we have
\[
\int_\mathbb{R}\Phi_{xxx}\Phi_x\,\mathrm{d}x
=-\int_\mathbb{R}\Phi_{xx}^2\,\mathrm{d}x
=\int_\mathbb{R}\Phi_x(\rho-n)_x\,\mathrm{d}x,
\]
which implies
\[
\int_\mathbb{R}\Phi_{xx}^2\,\mathrm{d}x
\leq\frac12\int_\mathbb{R}(2\Phi_x^2+\rho_x^2+n_x^2)\,\mathrm{d}x.
\]
Combining  Lemmas \ref{lm3.1} and \ref{lm3.2}, and Sobolev embedding theorem
yields
\begin{equation} \label{3.5o5}
\|\Phi_x\|_{L^\infty}\leq C.
\end{equation}
Then
\begin{align*}
&\int_\mathbb{R}(\rho\Phi_x)_s u_s\,\mathrm{d}x\\
&=-\int_\mathbb{R}\Big(\rho_x u u_s\Phi_x
+\rho u_x u_s\Phi_x+\rho u_s(\rho u-nv)\Big)\,\mathrm{d}x \\
&\leq |\int_\mathbb{R}\rho_xuu_s\Phi_x \,\mathrm{d}x|
 +|\int_\mathbb{R}\rho u_x u_s\Phi_x\,\mathrm{d}x|
+|\int_\mathbb{R}\rho u_s(\rho u-nv)\,\mathrm{d}x| \\
&\leq C(\|\rho_x\|_{L^2}^2+\|u_t\|_{L^2}^2)
 +C|\int_\mathbb{R}(u u_{tx}\Phi_x+uu_t\Phi_{xx})\,\mathrm{d}x|\\
&\quad +C\big(\|u_t\|_{L^2}^2  +\|u\|_{L^2}^2+\|v\|_{L^2}^2\big) \\
&\leq \frac14\|\rho^{\frac{\alpha}{2}}u_{tx}\|_{L^2}^2
 +C\Big(\|\rho_x\|_{L^2}^2+\|u_t\|_{L^2}^2+\|u\|_{L^2}^2
 +\|v\|_{L^2}^2+\|\Phi_x\|_{L^2}^2\Big)+C \\
&\leq \frac14\|\rho^{\frac{\alpha}{2}}u_{tx}\|_{L^2}^2+C\|u_t\|_{L^2}^2+C\,.
\end{align*}
Therefore,
\begin{equation} \label{3.5o6}
\begin{aligned}
\int_0^t\int_\mathbb{R}(\rho\Phi_x)_s u_s \,\mathrm{d}x\,\mathrm{d}s
&\leq C(T)+\frac14\int_0^t\int_\mathbb{R}\rho^{\alpha}u_{sx}^2\,\mathrm{d}x\,\mathrm{d}s
 +C\int_0^t\int_\mathbb{R}u_s^2\,\mathrm{d}x\,\mathrm{d}s \\
&\leq C(T)+\frac14\int_0^t\int_\mathbb{R}\rho^{\alpha}u_{sx}^2\,\mathrm{d}x\,\mathrm{d}s.
\end{aligned}
\end{equation}
Using \eqref{3.5o6} and \eqref{3.5o3}, we obtain
\begin{equation} \label{3.5o7}
\int_\mathbb{R}u_t^2\,\mathrm{d}x
+\int_0^t\int_\mathbb{R}u_{sx}^2\,\mathrm{d}x\,\mathrm{d}s\leq C(T).
\end{equation}
Applying  similar arguments we  obtain
\begin{equation} \label{3.5o8}
\int_\mathbb{R}v_t^2\,\mathrm{d}x
+\int_0^t\int_\mathbb{R}v_{sx}^2\,\mathrm{d}x\,\mathrm{d}s\leq C(T).
\end{equation}
Then \eqref{3.5o7} and \eqref{3.5o8} give rise to \eqref{3.5o1}.
\end{proof}

\section{Proof of  main results}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
We prove only the existence of the solution $(\rho, u)$; existence of
 $(n, v)$ can be proved by the same method.

Let $(\rho_0, u_0)$ be the initial data as described in the theorem,
and let $\rho_0^\delta:=j_\delta\ast\rho_0$,
$u_0^\delta:=j_\delta\ast u_0$, where $j_\delta=\delta^{-1}j(x/\delta)$
is the standard mollifier.
Then, for any $0<\beta<1$ we have $\rho_0^\delta\in C^{1+\beta}(\mathbb{R})$ and
$u_0^\delta\in C^{2+\beta}(\mathbb{R})$. This implies
$\rho_0^\delta\to \rho_0$ in
$W^{1,2}(\mathbb{R})$, and $u_0^\delta\to  u_0$
in $L^2(\mathbb{R})$,  as $\delta\to 0$.

Next, we consider the Cauchy problem \eqref{2.1o}$_1$ and \eqref{2.1o}$_2$
with the initial data $(\rho_0, u_0)$ replaced by
 $(\rho_0^\delta, u_0^\delta)$, $\Phi_x$ be regarded as external force.
For this problem we can apply the standard argument (the energy estimates
and the contraction mapping theorem) to obtain the existence of a unique
local solution $(\rho^\delta, u^\delta)$ with $\rho^\delta$, $\rho_x^\delta$,
$\rho_t^\delta$, $\rho_{tx}^\delta$, $u^\delta$, $u_x^\delta$, $u_t^\delta$,
$u_{xx}^\delta\in C^{\beta,\beta/2} (\mathbb{R}\times [0,T^\ast])$ for some
$T^\ast>0$. Furthermore, from Lemmas \ref{lm3.1}-\ref{lm3.5}, we see that
$\rho^\delta$ is pointwise bounded from below and above, $u^\delta$,
$\rho_x^\delta\in L^{\infty}([0,T];L^2(\mathbb{R}))$,
$u_x^\delta\in L^2([0,T];L^2(\mathbb{R}))$, $\rho^\delta$, $\rho_x^\delta$,
$\rho_t^\delta$, $\rho_{tx}^\delta$, $u^\delta$, $u_x^\delta$, $u_t^\delta$,
$u_{xx}^\delta\in C^{\beta,\beta/2}(\mathbb{R}\times[0,T])$ for any $T>0$.
Therefore, we can continue the local solution globally in time and deduce
that there exists a unique global solution $(\rho^\delta, u^\delta)$ of
the Cauchy problem \eqref{2.1o}$_1$ and \eqref{2.1o}$_2$ with
$(\rho_0, u_0)$ replaced by $(\rho_0^\delta, u_0^\delta)$, which is
carried out as in \cite{AKM}.

Thus, we extract a subsequence of $(\rho^\delta, u^\delta)$, still denoted
 by $(\rho^\delta, u^\delta)$, such that as $\delta\to 0$,
\begin{gather} \label{4.1}
u^\delta\rightharpoonup u\quad \text{weak$\ast$ in }
 L^{\infty}([0,T];L^2(\mathbb{R})), \\
 \label{4.2}
\rho^\delta\rightharpoonup \rho\quad\text{weak$\ast$ in }
 L^{\infty}([0,T];L^2(\mathbb{R})), \\
\label{4.3}
(\rho_t^\delta, u_x^\delta)\to (\rho_t, u_x)\quad\text{weak in }
L^2([0,T];L^2(\mathbb{R})).
\end{gather}
Moreover, from \eqref{3.1o1}, \eqref{3.2o1} and \eqref{3.3o1},
the  existence of  a global weak solution to the Cauchy problem \eqref{2.1o}$_1$
and \eqref{2.1o}$_2$ can be  proved directly as in \cite{JXZ}.
 As a matter of fact, because of \eqref{3.4o1} and \eqref{3.5o1},
 $(\rho,u)$ is also a global strong solution.
Uniqueness of this strong solution can be proved as in \cite{JXZ}.
We omit the details here.
\end{proof}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referees for the valuable
comments and suggestions that greatly improved  this article.
The author is grateful to Professor Hai-Liang Li for his helpful discussions
and suggestions about the problem.
This research was supported by NSFC Nos.11501323 and 11701323.

\begin{thebibliography}{99}

\bibitem{AKM} Antontsev, S.; Kazhikhov, A.; Monakhov, V.;
\emph{Boundary value problems in mechanics of
nonhomogeneous fluids}, North-Holland, Amsterdam, New York, 1990.

\bibitem{BWY} Bie, Q.; Wang, Q.; Yao, Z.;
\emph{Optimal decay rate for the compressible Navier-Stokes-Poisson system in the
critical $L^p$ framework},
J. Differ. Equ., \textbf{263} (2017), 8391-8417.

\bibitem{DJL} Degond, P.; Jin, S.; Liu, J.;
\emph{Mach-number uniform asymptotic-preserving gauge schemes for compressible flows},
Bull Inst. Math. Acad. Sin., \textbf{2} (4) (2007), 851-892.

\bibitem{DD} Donatelli, D.;
\emph{Local and global existence for the coupled Navier-Stokes-Poisson problem},
Quart. Appl. Math., \textbf{61} (2003), 345-361.

\bibitem{DM} Donatelli, D.; Marcati, P.;
\emph{A quasineutral type limit for the Navier-Stokes-Poisson system with large data},
Nonlinearity, \textbf{21} (1) (2008), 135-148.

\bibitem{DLUY} Duan, R.; Liu, H.; Ukai, S.; Yang, T.;
\emph{Optimal $L^p$-$L^q$ convergence rates for
the compressible Navier-Stokes equations with potential force},
J. Differ. Equ., \textbf{238} (1) (2007), 220-233.

\bibitem{DZ} Ducomet, B.; Zlotnik, A.;
\emph{Stabilization and stability for the spherically symmetric
Navier-Stokes-Poisson system},
Appl. Math. Let., \textbf{18} (10) (2005), 1190-1198.

\bibitem{HL} Hao, C.; Li, H.;
\emph{Global existence for compressible Navier-Stokes-Poisson equations
in three and higher dimensions},
J. Differ. Equ., \textbf{246} (2009), 4791-4812.

\bibitem{HLYZ} Hsiao, L.; Li, H.; Yang, T.; Zou, C.;
\emph{Compressible non-isentropic bipolar Navier-Stokes-Poisson system in
$\mathrm{R}^{3}$},
Acta Math. Sci., \textbf{31B} (6) (2011), 2169-2194.

\bibitem{JLL} Ju, Q.; Li, F.; Li, H.;
\emph{The quasineutral limit of Navier-Stokes-Poisson system with heat
conductivity and general initial data},
J. Differ. Equ., \textbf{247} (2009), 203-224.

\bibitem{JXZ} Jiang, S.; Xin, Z.; Zhang, P.;
\emph{Global weak solutions to 1D compressible isentropy Navier-Stokes
with density-dependent viscosity},
Methods Appl. Anal., \textbf{12} (2005), 239-252.

\bibitem{KL} Kong, H.; Li, H.;
\emph{Free boundary value problem to 3D spherically symmetric compressible
Navier-Stokes-Poisson equations},
\emph{Z. Angew. Math. Phys.}, \textbf{68} (21) (2017), 1-34.

\bibitem{LLLH} Lian, R.; Liu, J.; Li, H.; Hisao, L.;
\emph{Cauchy problem for one-dimensional compressible Navier-Stokes equations},
Acta Math. Sci., \textbf{32B} (1) (2012), 315-324.

\bibitem{LYZ} Li, H.; Yang, T.; Zou, C.;
\emph{Time asymptotic behavior of bipolar Navier-Stokes-Poisson system},
Acta Math. Sci., \textbf{29B} (6) (2009), 1721-1736.

\bibitem{LMZ} Li, H.; Matsumura, A.; Zhang, G.;
\emph{Optimal decay rate of the compressible Navier-Stokes-Poisson system in $R^{3}$},
Arch. Ration. Mech. Anal., \textbf{196} (2010), 681-713.

\bibitem{LHL} Lin, Y.; Hao, C.; Li, H.;
\emph{Global well-posedness of compressible bipolar compressible 
Navier-Stokes-Poisson equations}, 
Acta Math. Sini., \textbf{28} (5) (2012), 925-940.

\bibitem{LL} Liu, J.; Lian, R.;
\emph{Global existence of solution to free boundary value problem for
Bipolar Navier-Stokes-Poisson system},
Electronic J. Differ. Equ., \textbf{2013} (200) (2013), 1-15.

\bibitem{LLQ} Liu, J.; Lian, R.; Qian, M.;
\emph{Global existence of solution to Bipolar Navier-Stokes-Poisson system},
Acta Math. Sci., \textbf{34A} (4) (2014), 960-976.

\bibitem{MN1980} Matsumura, A.; Nishida, T.;
\emph{The initial value problem for the equation
of motion of viscous and heat-conductive gases},
J. Math. Kyoto. Univ., \textbf{20} (1980), 67-104.

\bibitem{MRS} Markowich, P.; Ringhofer, C.; Schmeiser, C.;
\emph{Semiconductor Equations}, Springer, 1990.

\bibitem{P1985} Ponce, G.;
\emph{Global existence of small solution to a class of nonlinear evolution equations},
Nonlinear Anal., \textbf{9} (1985), 339-418.

\bibitem{WW} Wu, Z.; Wang, W.;
\emph{Pointwise Estimates for Bipolar Compressible Navier-Stokes-Poisson System
in Dimension Three},
Arch. Ration. Mech. Anal., \textbf{226} (2017), 587-638.

\bibitem{ZT} Zhang, Y.; Tan, Z.;
\emph{On the existence of solutions to the Navier-Stokes-Poisson equations of
 a two-dimensional compressible flow},
Math. Methods Appl. Sci., \textbf{30} (3) (2007), 305-329.

\bibitem{ZC} Zou, C.;
\emph{Large time behaviors of the isentropic bipolar compressible
Navier-Stokes-Poisson system},
Acta Math. Sci., \textbf{31B} (5) (2011), 1725-1740.

\end{thebibliography}

\end{document}