# Large-time behavior of solutions to the outflow problem for the compressible
Navier-Stokes-Maxwell equations

Huancheng Yao  School of Mathematics, South China University of Technology,
Guangzhou 510641, P.R. China  Changjiang Zhu Corresponding author.
E-mail addresses<EMAIL_ADDRESS>(Yao<EMAIL_ADDRESS>(Zhu).
School of Mathematics, South China University of Technology,
Guangzhou 510641, P.R. China

###### Abstract

We investigate the large-time behavior of solutions toward the combination of
the boundary layer and 3-rarefaction waves to the outflow problem for the
compressible non-isentropic Navier-Stokes equations coupling with the Maxwell
equations through the Lorentz force (called the Navier-Stokes-Maxwell
equations) on the half line $\mathbb{R}_{+}$. It includes the electrodynamic
effects into the dissipative structure of the hyperbolic-parabolic system and
turns out to be more complicated than that in the simpler compressible Navier-
Stokes equations. We prove that this typical composite wave pattern is time-
asymptotically stable with the composite boundary condition of the
electromagnetic fields, under some smallness conditions and the assumption
that the dielectric constant is bounded. This can be viewed as the first
result about the nonlinear stability of the combination of two different wave
patterns for the IBVP of the non-isentropic Navier-Stokes-Maxwell equations.

2021 Mathematics Subject Classification: 35Q30, 76N06, 76N30, 35Q61.

Keywords: Non-isentropic Navier-Stokes-Maxwell equations, electromagnetic
fields, dielectric constant, boundary layer, rarefaction wave.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and main results
    1. 2.1 Boundary layer
    2. 2.2 Rarefaction wave
    3. 2.3 Superposition of the boundary layer and rarefaction wave
  3. 3 Proofs of the theorems
    1. 3.1 Zero-order energy estimates
    2. 3.2 High-order energy estimates
  4. 4 Appendix: Derivation of 1-D models
  5. References

##  1 Introduction

Plasma dynamics is a field of studying flow problems of electrically
conducting fluids. A complete analysis in this broad field includes the study
of the gasdynamic field, the electromagnetic fields and the radiation field
simultaneously in [44]. In this paper, we consider the motion of an
electrically conducting fluid in the presence of electric field and magnetic
field. At the macroscopic level, the flow of this electrically conducting
fluid such as the movement in the electromagnetic fields generated by itself
is described by hydrodynamics equations, for example the compressible Navier-
Stokes equations. Since the dynamic motion of the fluid and the
electromagnetic fields couple strongly, the governing system in the non-
isentropic case is derived from fluid mechanics with appropriate modifications
to take account of the electromagnetic effects, which consists of the laws of
conservation of mass, momentum and energy, Maxwell’s law, and the law of
conservation of electric charge (see [15], [18]). In this paper, we shall
restrict ourselves to the one-dimensional motion (see [5], [51]) on the half
line $\mathbb{R}_{+}$:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\
&\rho(u_{t}+uu_{x})+p_{x}=\mu u_{xx}-(E+ub)b,\\\
&\frac{R}{\gamma-1}\rho(\theta_{t}+u\theta_{x})+pu_{x}=\mu
u_{x}^{2}+\kappa\theta_{xx}+(E+ub)^{2},\\\ &\varepsilon E_{t}-b_{x}+E+ub=0,\\\
&b_{t}-E_{x}=0,\end{aligned}\right.$ (1.1)

where $(x,t)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$. The detailed mathematical
derivation of system (1.1) will be given in Appendix. Here, $\rho(x,t)>0$
denotes the mass density; $u(x,t)$ is the fluid velocity; $\theta(x,t)>0$ is
the absolute temperature; $E(x,t)$ and $b(x,t)$ denote the electric field and
the magnetic field, respectively. The pressure $p$ is expressed by the
equations of states. For the sake of simplicity, we will focus on only
polytropic fluids throughout this paper, namely

$p=R\rho\theta=A\rho^{\gamma}\mathrm{exp}\left(\frac{\gamma-1}{R}s\right),$
(1.2)

where $s$ is the entropy. The parameters in the above equations, respectively,
$R>0$ is the gas constant and $\gamma>1$ is the adiabatic exponent; $\mu>0$ in
$\eqref{yuanfangcheng}_{2}$ and $\eqref{yuanfangcheng}_{3}$ is the viscosity
coefficient of the fluid; the heat conductivity coefficient $\kappa$ in
$\eqref{yuanfangcheng}_{3}$ is assumed to be a positive constant. Moreover,
$\varepsilon>0$ in $\eqref{yuanfangcheng}_{4}$ denotes the dielectric
constant.

The system (1.1) is obtained from the Navier-Stokes equations coupling with
the Maxwell equations through the Lorentz force. Thus it is usually called the
Navier-Stokes-Maxwell equations. Notice that the same terminology was used by
Masmoudi in [34] and Duan in [3] but for the different models. In this paper,
we consider the initial-boundary value problem for the compressible non-
isentropic Navier-Stokes-Maxwell equations on a half line. The initial data
for the system (1.1) is given by

$(\rho,u,\theta,E,b)(x,0)=(\rho_{0},u_{0},\theta_{0},E_{0},b_{0})(x),\quad\inf_{x\in\mathbb{R}_{+}}\rho_{0}(x)>0,\quad\inf_{x\in\mathbb{R}_{+}}\theta_{0}(x)>0.$
(1.3)

We assume that the initial data in the far field $x=+\infty$ is constant,
namely

$\lim_{x\rightarrow+\infty}(\rho_{0},u_{0},\theta_{0},E_{0},b_{0})(x)=(\rho_{+},u_{+},\theta_{+},E_{+},b_{+}),$

and the boundary data for $u$ and $\theta$ at $x=0$ is given by the following
constants

$(u,\theta)(0,t)=(u_{-},\theta_{-}),\quad\forall\;t\geq 0,$

where $\rho_{+}>0$, $\theta_{\pm}>0$ and $u_{-}<0$. The following
compatibility conditions hold as well

$u_{0}(0)=u_{-},\quad\theta_{0}(0)=\theta_{-}.$

In particular, we suppose the boundary values for $E$ and $b$ satisfy the
following condition:

$\left(\sqrt{\varepsilon}E-b\right)(0,t)=0.$ (1.4)

Actually, setting $V=\left(\varepsilon E,\;b\right)^{T}$, equations
$\eqref{yuanfangcheng}_{4}$ and $\eqref{yuanfangcheng}_{5}$ can be transformed
as the following form

$\displaystyle V_{t}+AV_{x}+BV=0,\quad\mathrm{for}\quad
A=\left(\begin{array}[]{cc}\displaystyle 0&\displaystyle-1\\\\[5.69054pt]
\displaystyle-1/\varepsilon&\displaystyle 0\end{array}\right),\quad
B=\left(\begin{array}[]{cc}\displaystyle 1/\varepsilon&\displaystyle
u\\\\[5.69054pt] \displaystyle 0&\displaystyle 0\end{array}\right).$ (1.9)

Here Matrix $A$ has two eigenvalues $\lambda_{1}=1/\sqrt{\varepsilon}$ and
$\lambda_{2}=-1/\sqrt{\varepsilon}$. Direct calculations show that the pair of
Riemann invariants $\left\\{W_{1},\;W_{2}\right\\}$ associated with the
eigenvalues $\lambda_{1}$ and $\lambda_{2}$ can be taken as

$\left\\{W_{1},\;W_{2}\right\\}=\frac{\sqrt{\varepsilon}}{2}\left\\{\sqrt{\varepsilon}E-b,\;\sqrt{\varepsilon}E+b\right\\}.$
(1.10)

Using this pair of Riemann invariants, we can diagonalize the equations in
(1.9) as

$W_{t}+\Lambda W_{x}+DW=0,$

where $W:=\left(W_{1},\;W_{2}\right)^{T}$ and
$\Lambda:=\mathrm{diag}(\lambda_{1},\;\lambda_{2})$. The basic theory of
hyperbolic systems of conservation laws (for example, see [48]) shows that we
must specify the boundary value $W_{1}(0,t)$ since $\lambda_{1}>0$; and
$W_{2}(0,t)$ is determined by the initial data since $\lambda_{2}<0$. Now each
$W_{i}\;(i=1\;\mathrm{or}\;2)$ is a linear combination of the
$V_{i}\;(i=1\;\mathrm{and}\;2)$, so we must specify only one condition on the
components of $V$ at the boundary $x=0$. Due to (1.10), we specify the
boundary value on $\sqrt{\varepsilon}E-b$ satisfying (1.4). It should be
pointed out that this type of boundary condition has ever been mentioned by
Chen-Jerome-Wang in [2].

Let us recall some known results about the Navier-Stokes-Maxwell equations.
There have been some research on the existence and large-time behavior of
solutions, and the vanishing dielectric constant limit problem to the
compressible Navier-Stokes-Maxwell equations. In [23] and [24], Kawashima and
Shizuta established the global existence of smooth solutions for small data
and studied its zero dielectric constant limit in the whole space
$\mathbb{R}^{2}$. Li and Mu [27] studied the low Mach number limit problem for
the solution of the full compressible Navier-Stokes-Maxwell equations
converging to that of the incompressible system in $\mathbb{R}^{3}$. Later,
Jiang and Li in [16] studied the vanishing dielectric constant limit and
obtained the convergence of the 3-D Navier-Stokes-Maxwell equations to the
full compressible magnetohydrodynamic equations in the torus $\mathbb{T}^{3}$.
Recently, Xu in [49] studied the large-time behavior of the classical solution
toward some given constant states and obtained the time-decay estimates in the
whole space $\mathbb{R}^{3}$ with small initial perturbation in $H^{3}\cap
L^{1}$. For the one-dimensional non-isentropic model, Fan and Hu in [5]
obtained the uniform estimates with respect to the dielectric constant and the
global-in-time existence in a bounded interval without vacuum. Furthermore,
Fan and Ou in [6] considered the one-dimensional full equations for a thermo-
radiative electromagnetic fluid in a form similar to that in (1.1); and
established the similar result to [5].

However, for the one-dimensional compressible Navier-Stokes-Maxwell equations,
there are few results about the large-time behavior of the solution toward
some non-constant states, especially wave patterns. To the authors’ best
knowledge, there are only four relevant results. To the Cauchy problem, Luo-
Yao-Zhu [33] and Yao-Zhu [51] established the stability of rarefaction wave
for the compressible isentropic and non-isentropic Navier-Stokes-Maxwell
equations under suitable smallness conditions, respectively. Huang-Liu in [14]
consider the stability of rarefaction wave for a macroscopic model derived
from the Vlasov-Maxwell-Boltzmann system, in which the model they consider is
obviously different from this in our paper, except for the similar dissipative
term $E+ub$. Recently, Yao-Zhu in [50] study the asymptotic stability of the
superposition of viscous contact wave with rarefaction waves for the
compressible Navier-Stokes-Maxwell equations, which is the first result on the
combination of two different wave patterns of this complex coupled model. But
for the large-time behavior of solutions to an IBVP of the non-isentropic
Navier-Stokes-Maxwell equations, as far as we know there are still few
results. Here, we will partly give a positive answer for this important
problem.

In fact, equations (1.1) reduce to the classical Navier-Stokes equations if we
ignore the effects of the electromagnetic fields. Motivated by the
relationship between the Navier-Stokes-Maxwell equations and Navier-Stokes
equations, we temporarily assume that $E_{+}=b_{+}=0$, namely, the initial-
boundary values satisfying

$\lim_{x\rightarrow+\infty}(\rho_{0},u_{0},\theta_{0},E_{0},b_{0})(x)=(\rho_{+},u_{+},\theta_{+},0,0),$
(1.11)
$(u,\theta,\sqrt{\varepsilon}E-b)(0,t)=(u_{-},\theta_{-},0),\quad\forall\;t\geq
0,$ (1.12)

and can consider the large-time behavior of solutions to the outflow problem
(1.1)-(1.3) and (1.11)-(1.12) in the setting of $E(x,t)=b(x,t)=0$. Then the
above outflow problem is reduced to consider the corresponding outflow problem
of the Navier-Stokes equations:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\
&\rho(u_{t}+uu_{x})+p_{x}=\mu u_{xx},\\\
&\frac{R}{\gamma-1}\rho(\theta_{t}+u\theta_{x})+pu_{x}=\mu
u_{x}^{2}+\kappa\theta_{xx},\end{aligned}\right.$

with the initial-boundary values

$(\rho,u,\theta)(x,0)=(\rho_{0},u_{0},\theta_{0})(x)\rightarrow(\rho_{+},u_{+},\theta_{+}),\quad\mathrm{as}\quad
x\rightarrow+\infty,$ $(u,\theta)(0,t)=(u_{-},\theta_{-}),\quad\forall\;t\geq
0.$

Hence, under the assumption $E_{+}=b_{+}=0$, when time tends to infinity, it
is reasonable for us to expect that the solutions to the outflow problem
(1.1)-(1.3) and (1.11)-(1.12) asymptotically converge to the profiles the same
as that of the Navier-Stokes equations. Moreover, the cases for $E_{+}\neq 0$
and $b_{+}\neq 0$ which lead to more complex structures are left for study in
future.

In the past three decades, there have been many works on the large-time
behavior of solutions to the Cauchy problem of 1-D compressible Navier-Stokes
equations (including its isentropic case) with some end constant states at far
fields $x=\pm\infty$ of initial data. We refer interested readers to [19, 20,
28, 31, 38, 43, 11, 13, 9, 4, 7] and some references therein. The above
literatures show that the large-time behavior of solutions to the Cauchy
problem with the far field constant states of initial data is basically
governed by its corresponding Riemann solutions to the compressible Euler
equations, just as contact discontinuity and shock wave are replaced by the
corresponding viscous contact wave and (shifted) viscous shock wave,
respectively.

However, for the large-time behavior of solutions to an IBVP of the Navier-
Stokes equations, there exists a different wave phenomenon from the Cauchy
problem. In fact, the authors of [29, 30, 32] found a new wave phenomenon
while studying the IBVP for scalar viscous conservation law. This phenomenon
appeared due to the boundary effect, and they named it boundary layer. Since
this wave’s form is the stationary solution, other people also call it
stationary solution. From then on, the investigation of the existence and
stability of the boundary layer, including the stability of its combinations
with viscous hyperbolic waves has aroused many researchers’ interests. Later,
Matsumura in [35] gave the complete classification of the large-time behavior
of the solutions for the compressible isentropic Navier-Stokes equations in
terms of the far field states and the boundary data. According to the sign of
$u_{-}$, i.e. the value of fluid velocity at the boundary $x=0$, the IBVP of
the Navier-Stokes equations can be divided into three cases: the outflow
problem $(u_{-}<0)$, the inflow problem $(u_{-}>0)$ and the impermeable wall
problem $(u_{-}=0)$. Since then, some conjectures in [35] have been
extensively investigated and verified for the isentropic and non-isentropic
Navier-Stokes equations by many authors. Here, we mention several works on the
asymptotic stability analysis of wave patterns to the IBVP: [22, 25, 12, 26,
21, 45, 41] for the outflow problem, [39, 10, 47, 46, 40] for the inflow
problem and [36, 37, 8] for the impermeable wall problem.

These three kinds of IBVP are still important topics in the theory of fluid
dynamics and plasma physics. So it is meaningful and interesting to study the
corresponding problems for the Navier-Stokes-Maxwell equations. In the present
paper, we only discuss the outflow problem. The outflow boundary value
$u_{-}<0$ means that fluid blows out from the boundary $x=0$ with the velocity
$u_{-}$. Thus this problem is called the outflow problem (see [35]). The
outflow boundary condition implies that the characteristic of the hyperbolic
equation $\eqref{yuanfangcheng}_{1}$ for the density $\rho$ is negative around
the boundary so that boundary conditions on $u$ and $\theta$ to parabolic
equations $\eqref{yuanfangcheng}_{2}$ and $\eqref{yuanfangcheng}_{3}$ are
necessary and sufficient for the wellposedness of the hydrodynamic parts.
Motivated by [45, 12, 21], we will consider asymptotic stability of solutions
towards the superposition of the boundary layer (including the nondegenerate
case) and the 3-rarefaction wave under some smallness conditions and with the
composite boundary condition of the electromagnetic fields. To our knowledge,
this can be viewed as the first result for the Navier-Stokes-Maxwell equations
in this direction.

Here, we briefly give some remarks on our problem and review some key
analytical techniques. Compared with the result of [45] and [21] for
compressible Navier-Stokes equations, the outflow problem for compressible
non-isentropic Navier-Stokes-Maxwell equations is more complicated.

Due to the strong interaction between the fluid motion and the electromagnetic
fields, the main difficulties to prove the nonlinear stability of wave
patterns lie in the additional terms produced by the electrodynamic effects.
The first bad term about the electric field $E$ and the magnetic field $b$ we
suffered is $-\int_{0}^{t}\int_{\mathbb{R}_{+}}(E+\psi b+\hat{u}b)\psi
b\,{\rm{d}}x{\rm{d}}\tau$. But the lack of damping decay mechanism of the
magnetic field $b$ hinders us from obtaining the time-space integrable good
term $\int_{0}^{t}\int_{\mathbb{R}_{+}}b^{2}\,\mathrm{d}x\mathrm{d}\tau$,
which is disadvantageous to the derivation of the zero-order energy estimates.
To overcome this obstacle, we try to use the structure of the Maxwell
equations and package extra terms together with the bad term
$-\int_{0}^{t}\int_{\mathbb{R}_{+}}(E+\psi b+\hat{u}b)\psi
b\,{\rm{d}}x{\rm{d}}\tau$ to produce a compound time-space integrable good
term $\int_{0}^{t}\int_{\mathbb{R}_{+}}(E+\psi
b+\hat{u}b)^{2}\,\mathrm{d}x{\rm{d}}\tau$, which is crucial to obtain the
zero-order energy estimates and is essential to get high-order energy
estimates. One can see Lemma 3.3 of the zero-order energy estimates for
details.

Secondly, we would encounter some obstacles under the composite boundary
condition of the electromagnetic fields: $(\sqrt{\varepsilon}E-b)(0,t)=0$. For
example, once using the Poincar$\acute{\rm e}$ type inequality (3.14) to
estimate
$\int_{0}^{t}\int_{\mathbb{R_{+}}}\tilde{u}_{x}^{2}b^{2}\,\mathrm{d}x\mathrm{d}\tau$,
then the bad term $\delta^{3}\int_{0}^{t}b^{2}(0,\tau)\,\mathrm{d}\tau$ would
arise. But the absence of good term
$\int_{0}^{t}\int_{\mathbb{R}_{+}}b^{2}\,\mathrm{d}x\mathrm{d}\tau$ makes it
invalid to apply $L^{\infty}_{x}$ Sobolev inequality (3.10) to estimate
$b^{2}(0,t)$. This requires a good term produces from the boundary estimates
so that we can absorb the corresponding bad term by choosing $\delta$ suitably
small. In addition, to treat the boundary term
$\int_{0}^{t}\int_{\mathbb{R_{+}}}-\left(\frac{1}{2}\hat{u}\varepsilon
E^{2}+\frac{1}{2}\hat{u}b^{2}+Eb\right)_{x}\,\mathrm{d}x\mathrm{d}\tau$ in
(3.1), the sign of $u_{-}$ is bad for this term. This fact also urges us to
use the boundary condition $(\sqrt{\varepsilon}E-b)(0,t)=0$ and the additional
technical condition (3.21) to produce a boundary good term
$\frac{3}{4}\int_{0}^{t}\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau$ in
(3.1) such that we can employ it to absorb the former bad term
$\delta^{3}\int_{0}^{t}b^{2}(0,\tau)\,\mathrm{d}\tau$ indeed. On the other
hand, for boundary terms
$\int_{0}^{t}\int_{\mathbb{R_{+}}}-\left(E_{x}b_{x}\right)_{x}\,\mathrm{d}x\mathrm{d}\tau$
and
$\int_{0}^{t}\int_{\mathbb{R_{+}}}2\varepsilon\left(EE_{x}\right)_{x}\,\mathrm{d}x\mathrm{d}\tau$,
we try to use the specific structure of the Maxwell equations and the
composite boundary condition $(\sqrt{\varepsilon}E-b)(0,t)=0$ to transform
$E_{x}(0,t)$ and $b_{x}(0,t)$ into some suitable forms and get desired
estimates eventually; see (3.42)-(3.50) and (3.2) for details.

Thirdly, for the boundary layer $\tilde{u}$, we have to estimate the terms
including the weight $\tilde{u}_{x}$ such as
$-\int_{0}^{t}\int_{\mathbb{R}_{+}}\frac{1}{2}\tilde{u}_{x}\left(\varepsilon
E^{2}+b^{2}\right)\,\mathrm{d}x\mathrm{d}\tau$ on the right-hand side of the
inequality (see in (3.1)). It is not workable to apply usual method to
estimate this term directly. Motivated by the idea of [45]: for each
$(\tilde{u},\tilde{\theta})(x)$ there exists a constant $M_{0}\geq 1$ just
depending on $u_{-},\theta_{-},\rho_{*},u_{*},\theta_{*}$ such that
$\tilde{u}_{x}\geq 0$ and $\tilde{\theta}_{x}\geq 0$ on $[M_{0},+\infty)$, we
divide the integral into two parts:
$-\left\\{\int_{0}^{t}\int_{0}^{M_{0}}+\int_{0}^{t}\int_{M_{0}}^{+\infty}\right\\}\frac{1}{2}\tilde{u}_{x}\left(\varepsilon
E^{2}+b^{2}\right)\,\mathrm{d}x\mathrm{d}\tau$. This treatment avoids us to
estimate the bad term
$\int_{0}^{t}\int_{M_{0}}^{+\infty}\frac{1}{2}\tilde{u}_{x}\left(\varepsilon
E^{2}+b^{2}\right)\,\mathrm{d}x\mathrm{d}\tau$. Then together with the
boundary good term
$\frac{3}{4}\int_{0}^{t}\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau$ in
(3.1), we can obtain the desired estimates; see the proof of (3.24)-(3.27) for
details.

Fourthly, in order to absorb some nonlinear bad terms by some good terms
concerning the electric field $E$ or the magnetic field $b$, we require a
technical condition (2.31) that $\varepsilon$ is bounded for some specific
positive constants $\bar{C}$. Through some elaborate analysis, we can finally
determine the value of the constant $\bar{C}$. One can see the discussion
about $\varepsilon$ in (3.21), (3.31), (3.35), (3.47), (3.57) and (3.59) for
details. So far it is unclear how to remove such restriction for the nonlinear
stability of wave patterns on the Navier-Stokes-Maxwell equations.

In the appendix, we provide a mathematical derivation for the one-dimensional
model (1.1). We also obtain additional four 1-D models, which remains to be
studied in mathematics and physics in future.

Notations: Throughout this paper, we denote positive constants generally large
(respectively, generally small) independent of $x$ and $t$ by $C$
(respectively, by $c$). And the character ‘C’ and ‘c’ may vary from line to
line. $\|\cdot\|_{L^{p}}$ stands the $L^{p}$-norm on the Lebesgue space
${L^{p}}(\mathbb{R_{+}})\;(1\leq p\leq\infty)$. For the sake of convenience,
we always denote $\|\cdot\|=\|\cdot\|_{L^{2}}$. What’s more, $H^{k}$ will be
used to denote the usual Sobolev space
$W^{k,2}(\mathbb{R_{+}})\;(k\in\mathbb{Z}_{+})$ with respect to variable $x$.

##  2 Preliminaries and main results

Let

$c(\rho,s):=\sqrt{p_{\rho}(\rho,s)}=\sqrt{R\gamma\theta}=:c(\rho,\theta),\qquad
M(\rho,u,\theta):=\frac{\left|u\right|}{c},$

which are called the local sound speed and the local Mach number. Let

$c_{+}:=c(\rho_{+},\theta_{+})=\sqrt{R\gamma\theta_{+}},\qquad
M_{+}:=M(\rho_{+},u_{+},\theta_{+})=\frac{\left|u_{+}\right|}{c_{+}},$

which are called the sound speed and the Mach number at the far field
$x=+\infty$, respectively. We divide the state space: the quarter 3D space
$\\{(\rho,u,\theta)\;|\;\rho>0,\;\theta>0\\}$ into three parts:

$\left\\{\begin{aligned}
&\Omega_{\mathrm{sub}}:=\left\\{(\rho,u,\theta)\left|\;\left|u\right|<\sqrt{R\gamma\theta}\right.\right\\},\\\\[2.84526pt]
&\Gamma_{\mathrm{tran}}:=\left\\{(\rho,u,\theta)\left|\;\left|u\right|=\sqrt{R\gamma\theta}\right.\right\\},\\\\[2.84526pt]
&\Omega_{\mathrm{super}}:=\left\\{(\rho,u,\theta)\left|\;\left|u\right|>\sqrt{R\gamma\theta}\right.\right\\},\end{aligned}\right.$

which are called subsonic, transonic and supersonic regions, respectively. If
we add the alternative condition $u<0$ or $u\geq 0$ into these regions, then
we have six connected subsets $\Omega_{\mathrm{sub}}^{\pm}$,
$\Gamma_{\mathrm{tran}}^{\pm}$ and $\Omega_{\mathrm{super}}^{\pm}$.

###  2.1 Boundary layer

It is known that the corresponding hyperbolic system of (1.1) with (1.2) has
three characteristic values

$\lambda_{1}(\rho,u,\theta)=u-\sqrt{R\gamma\theta},\qquad\lambda_{2}(\rho,u,\theta)=u,\qquad\lambda_{3}(\rho,u,\theta)=u+\sqrt{R\gamma\theta}.$

For the outflow problem $u_{-}<0$, one easily knows
$\lambda_{1}<\lambda_{2}<0$ at the boundary $x=0$. So if $u_{+}<0$ and $u_{-}$
is sufficiently close to $u_{+}$ such that $u_{-}<0$ also holds, then a
stationary solution
$(\tilde{\rho},\tilde{u},\tilde{\theta},\tilde{E},\tilde{b})(x)$ to the
outflow problem (1.1)-(1.3) and (1.11)-(1.12) is expected:

$\left\\{\begin{aligned} &(\tilde{\rho}\tilde{u})_{x}=0,\qquad
x\in\mathbb{R}_{+},\\\\[2.84526pt]
&\tilde{\rho}\tilde{u}\tilde{u}_{x}+\tilde{p}_{x}=\mu\tilde{u}_{xx},\\\\[2.84526pt]
&\frac{R}{\gamma-1}\tilde{\rho}\tilde{u}\tilde{\theta}_{x}+\tilde{p}\tilde{u}_{x}=\mu\tilde{u}_{x}^{2}+\kappa\tilde{\theta}_{xx},\\\\[2.84526pt]
&\tilde{u}(0)=u_{-},\quad\tilde{\theta}(0)=\theta_{-},\quad(\tilde{\rho},\tilde{u},\tilde{\theta})(+\infty)=\left(\rho_{+},u_{+},\theta_{+}\right),\\\\[2.84526pt]
&\inf_{x\in\mathbb{R}_{+}}\tilde{\rho}(x)>0,\quad\inf_{x\in\mathbb{R}_{+}}\tilde{\theta}(x)>0,\end{aligned}\right.$
(2.1)

with $\tilde{E}=\tilde{b}=0$, where
$\tilde{p}:=p(\tilde{\rho},\tilde{\theta})=R\tilde{\rho}\tilde{\theta}$. The
stationary solution is usually called the boundary layer, see [35], [45] for
example. From the fact that $\tilde{\rho}(x)>0$ and $u_{-}<0$, we have

$\rho_{-}:=\tilde{\rho}(0)=\frac{\rho_{+}u_{+}}{u_{-}},\qquad\tilde{\rho}(x)=\frac{\rho_{+}u_{+}}{\tilde{u}(x)},\qquad\tilde{u}(x)<0.$
(2.2)

Thus, (2.1) is equivalent to the coupling of (2.2) and the following ordinary
differential equations:

$\left\\{\begin{aligned}
&\tilde{u}_{x}=\frac{\rho_{+}u_{+}}{\mu}\left[\left(\tilde{u}-u_{+}\right)+R\left(\frac{\tilde{\theta}}{\tilde{u}}-\frac{\theta_{+}}{u_{+}}\right)\right],\qquad
x\in\mathbb{R}_{+},\\\\[5.69054pt]
&\tilde{\theta}_{x}=\frac{\rho_{+}u_{+}}{\kappa}\left[\frac{R\theta_{+}}{u_{+}}\left(\tilde{u}-u_{+}\right)+\frac{R}{\gamma-1}\left(\tilde{\theta}-\theta_{+}\right)-\frac{1}{2}\left(\tilde{u}-u_{+}\right)^{2}\right],\\\\[5.69054pt]
&(\tilde{u},\tilde{\theta})(0)=(u_{-},\theta_{-}),\qquad(\tilde{u},\tilde{\theta})(+\infty)=\left(u_{+},\theta_{+}\right),\end{aligned}\right.$
(2.3)

where $p_{+}:=p(\rho_{+},\theta_{+})=R\rho_{+}\theta_{+}$. The strength of the
boundary layer is measured by

$\delta=\left|u_{+}-u_{-}\right|+\left|\theta_{+}-\theta_{-}\right|.$ (2.4)

Then we have the following lemma.

######  Lemma 2.1.

$($Existence of the boundary layer$)$ Suppose that the boundary data
$(u_{-},\theta_{-})$ satisfy

$\left(u_{-},\theta_{-}\right)\in\mathcal{M}^{+}:=\left\\{(u,\theta)\in\mathbb{R}^{2};\;\left|\left(u-u_{+},\theta-\theta_{+}\right)\right|<\delta_{0}\right\\}$
(2.5)

for a certain positive constant $\delta_{0}$. Notice that (2.5) is equivalent
to the inequality $\delta<\delta_{0}$.

  * $\mathrm{(i)}$

For the supersonic case $M_{+}>1$, there exists a unique smooth solution
$(\tilde{u},\tilde{\theta})(x)$ to the problem (2.3) satisfying

$\left|\left(\tilde{u}(x)-u_{+},\tilde{\theta}(x)-\theta_{+}\right)^{(k)}\right|\leq
C\delta\mathrm{e}^{-cx},\quad^{(k)}:=\frac{\mathrm{d}^{k}}{\mathrm{d}x^{k}},\quad
k=0,1,2,\cdots,$ (2.6)

where $c$ and $C$ are positive constants.

  * $\mathrm{(ii)}$

For the transonic case $M_{+}=1$, there exists a centre-stable manifold
$\mathcal{M}\subset\mathcal{M}^{+}$ consisting of two trajectories
$\Gamma_{i}:=\left(M_{i1},M_{i2}\right)(\xi),\;i=1,2,\;\xi\in\mathbb{R}_{+}$,
tangent to the line $\mu u_{+}(u-u_{+})-(\gamma-1)\kappa(\theta-\theta_{+})=0$
on the opposite directions at $(u_{+},\theta_{+})$. Depending on the location
of $(u_{-},\theta_{-})$, this case is divided into three subcases:

Subcase 1. For each $(u_{-},\theta_{-})\in\mathcal{M}^{+}$,
$(\tilde{u},\tilde{\theta})\subset\mathcal{M}$, it holds that

$\left|\left(\tilde{u}(x)-u_{+},\tilde{\theta}(x)-\theta_{+}\right)^{(k)}\right|\leq
C\delta\mathrm{e}^{-cx},\quad k=0,1,2,\cdots.$ (2.7)

Subcase 2. For each $(u_{-},\theta_{-})\in\mathcal{M}^{+}$ satisfying
$\frac{\mu
u_{+}(u_{+}-u_{-})}{(\gamma-1)\kappa}-(\theta_{+}-\theta_{-})<M_{i2}(\xi)$,
where $\xi$ is determined uniquely by
$M_{i1}(\xi)=\frac{(\gamma-1)^{2}\kappa\left(u_{-}-u_{+}\right)}{R\mu\gamma+(\gamma-1)^{2}\kappa}+\frac{(\gamma-1)R\gamma\kappa\left(\theta_{-}-\theta_{+}\right)}{\left[R\mu\gamma+(\gamma-1)^{2}\kappa\right]u_{+}}$,
$i=1$ or $2$, there exists a unique solution
$(\tilde{u},\tilde{\theta})\subset\mathcal{M}^{+}$ satisfying

$\left|\left(\tilde{u}(x)-u_{+},\tilde{\theta}(x)-\theta_{+}\right)^{(k)}\right|\leq
C\frac{\delta^{k+1}}{(1+\delta x)^{k+1}},\quad k=0,1,2,\cdots,$ (2.8)

and

$\tilde{u}_{x}>0,\quad\tilde{\theta}_{x}>0\quad\;\;\;\mathrm{for}\;\;x\gg 1.$
(2.9)

Subcase 3. For each $(u_{-},\theta_{-})\in\mathcal{M}^{+}$, if it does not
belong to Subcases 1 or 2, then there exists no solution.

  * $\mathrm{(iii)}$

For the subsonic case $M_{+}<1$, there exists a curve such that the unique
smooth solution $(\tilde{u},\tilde{\theta})(x)$ to the problem (2.3)
satisfying (2.6).

######  Remark 1.

For the transonic case $M_{+}=1$, $\tilde{u}_{x}>0$ and $\tilde{\theta}_{x}>0$
when $x\gg 1$ will be fundamental to obtain some energy estimates in the proof
of Theorem 2.1 and 2.3. See Section 3 for details. The result of Case
$\mathrm{(ii)}$ is borrowed from [45] and we skip the proof for brevity. We
should mention that Lemma 2.1 was first obtained by Zhu et al. in [21] by
using the central manifold theorem and Qin in [45] gave another proof of Case
$\mathrm{(ii)}$ by employing the qualitative theory of ordinary differential
equations.

The asymptotic stability of the boundary layer
$(\tilde{\rho},\tilde{u},\tilde{\theta},0,0)$ is stated in the following
theorem.

######  Theorem 2.1.

Assume that the boundary layer $(\tilde{\rho},\tilde{u},\tilde{\theta},0,0)$
exists under one of the following three conditions: $\mathrm{(i)}$ $M_{+}>1$;
$\mathrm{(ii)}$ $M_{+}=1$; $\mathrm{(iii)}$ $M_{+}<1$. In addition, the
dielectric constant $\varepsilon$ satisfies

$0<\varepsilon<\bar{C}$ (2.10)

for some positive constant $\bar{C}$ $($depending only on $|u_{-}|$ and
$|u_{+}|$$)$. Then there exist two small positive constants $\delta_{1}$ and
$\varepsilon_{1}$ which are independent of $T$, such that if
$0<\delta<\min\\{\delta_{0},\delta_{1}\\}$ and the initial data satisfies

$\|(\rho_{0},u_{0},\theta_{0},E_{0},b_{0})-(\tilde{\rho},\tilde{u},\tilde{\theta},0,0)\|^{2}_{H^{1}}\leq\varepsilon_{1},$
(2.11)

then the initial-boundary value problem (1.1)-(1.3) and (1.11)-(1.12) has a
unique global solution $(\rho,u,\theta,E,b)$. Moreover, the solution
$(\rho,u,\theta,E,b)$ converges to the boundary layer
$(\tilde{\rho},\tilde{u},\tilde{\theta},0,0)$ uniformly as time tends to
infinity in the sense that:

$\lim_{t\rightarrow+\infty}\sup_{x\in\mathbb{R}_{+}}\left|(\rho,u,\theta,E,b)(x,t)-(\tilde{\rho},\tilde{u},\tilde{\theta},0,0)(x)\right|=0.$
(2.12)

######  Remark 2.

Motivated by [21] and [52], we expect to study the convergence rate of the
solutions towards the _non-degenerate_ boundary layer for the 1-D Navier-
Stokes-Maxwell equations. The main difficulty in the analysis lies that we
can’t get the weighted time-space integrable good terms
$\int_{0}^{t}\int_{\mathbb{R_{+}}}(1+\tau)^{\xi}W_{\hat{\nu},\beta}E^{2}\,\mathrm{d}x\mathrm{d}\tau$
and
$\int_{0}^{t}\int_{\mathbb{R_{+}}}(1+\tau)^{\xi}W_{\hat{\nu},\beta}b^{2}\,\mathrm{d}x\mathrm{d}\tau$
simultaneously, and only can obtain the dissipation good term
$\int_{0}^{t}\int_{\mathbb{R_{+}}}(1+\tau)^{\xi}W_{\hat{\nu},\beta}(E+ub)^{2}\,\mathrm{d}x\mathrm{d}\tau$
instead, which is a typical feature of the Navier-Stokes-Maxwell equations
with regularity-loss property. Precisely, obstacle occurs in the zero-order
weighted energy estimates on $E$ and $b$. By employing some similar argument
as that of Lemma 3.3, then choosing $\beta$, $\varepsilon$ and $\delta$
suitably small, it holds that

$\displaystyle\quad(1+t)^{\xi}\int_{\mathbb{R_{+}}}W_{\hat{\nu},\beta}\left(\rho\eta(x,t)+\frac{1}{2}\varepsilon
E^{2}+\frac{1}{2}b^{2}+\varepsilon E\tilde{u}b\right)\,\mathrm{d}x$
$\displaystyle\quad+c\int_{0}^{t}(1+\tau)^{\xi}\phi^{2}(0,\tau)\,\mathrm{d}\tau+\left(1-\left|u_{-}\right|\sqrt{\varepsilon}\right)\int_{0}^{t}(1+\tau)^{\xi}\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau$
$\displaystyle\quad+c\beta\int_{0}^{t}\int_{\mathbb{R_{+}}}(1+\tau)^{\xi}W_{\hat{\nu}-1,\beta}\left|(\phi,\psi,\zeta)\right|^{2}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\quad+c\int_{0}^{t}\int_{\mathbb{R_{+}}}(1+\tau)^{\xi}W_{\hat{\nu},\beta}\left|(\psi_{x},\zeta_{x},E+ub)\right|^{2}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq
C\int_{\mathbb{R_{+}}}W_{\hat{\nu},\beta}\left|(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})\right|^{2}\,\mathrm{d}x+C\delta\int_{0}^{t}(1+\tau)^{\xi}\,\varepsilon
E^{2}(0,\tau)\,\mathrm{d}\tau$
$\displaystyle+C\delta\int_{0}^{t}(1+\tau)^{\xi}\|(\phi_{x},\sqrt{\varepsilon}E_{x},b_{x})\|^{2}\,\mathrm{d}\tau+C\int_{0}^{t}\int_{\mathbb{R_{+}}}\xi(1+\tau)^{\xi-1}W_{\hat{\nu},\beta}\left|(\phi,\psi,\zeta)\right|^{2}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle-\beta\hat{\nu}\int_{0}^{t}\int_{\mathbb{R_{+}}}(1+\tau)^{\xi}W_{\hat{\nu}-1,\beta}\left(\frac{1}{2}\tilde{u}\varepsilon
E^{2}+\frac{1}{2}\tilde{u}b^{2}+Eb\right)\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R_{+}}}\xi(1+\tau)^{\xi-1}W_{\hat{\nu},\beta}\left(\frac{1}{2}\varepsilon
E^{2}+\frac{1}{2}b^{2}+\varepsilon
E\tilde{u}b\right)\,\mathrm{d}x\mathrm{d}\tau.$ (2.13)

Here, $W_{\hat{\nu},\beta}:=(1+\beta x)^{\hat{\nu}}$ is an algebraic weight
and $\eta(x,t)$ is defined in Lemma 3.3. In order to not only obtain the
weighted space integrable good term
$(1+t)^{\xi}\int_{\mathbb{R_{+}}}W_{\hat{\nu},\beta}\left(\varepsilon
E^{2}+b^{2}\right)\,\mathrm{d}x$ from the first term on the left-hand side of
(2), but also apply an induction to $\xi$ of the last term on the right-hand
side of (2), we hope that there exist two positive constants $m$ and $M$ such
that

$\left(\frac{1}{2}\varepsilon-
a_{1}\right)E^{2}+\left(\frac{1}{2}-a_{2}\right)b^{2}+\varepsilon
E\tilde{u}b\geq m\left(\varepsilon E^{2}+b^{2}\right)$ (2.14)

and

$\left(\frac{1}{2}\varepsilon+k_{1}\right)E^{2}+\left(\frac{1}{2}+k_{2}\right)b^{2}+\varepsilon
E\tilde{u}b\leq M\left(E+ub\right)^{2}$ (2.15)

for some given constants $0\leq a_{1}<\frac{1}{2}\varepsilon$, $0\leq
a_{2}<\frac{1}{2}$ and $k_{1},\,k_{2}\geq 0$. However, by means of positive
semidefinite quadratic form, it is not hard to deduce
$1>\varepsilon\tilde{u}^{2}$ from (2.14) and $\varepsilon\tilde{u}^{2}>1$ from
(2.15), respectively. This implies (2.14) and (2.15) can not be established
simultaneously. Thus it seems unable to apply an induction to $\xi$ of (2) to
get the desired convergence rate just depending only on the weighted time-
space integrable compound good term
$\int_{0}^{t}\int_{\mathbb{R_{+}}}(1+\tau)^{\xi}W_{\hat{\nu},\beta}\left(E+ub\right)^{2}\,\mathrm{d}x\mathrm{d}\tau$.

###  2.2 Rarefaction wave

It is well known that the 3-rarefaction wave curve through the right-hand side
state $(\rho_{+},u_{+},\theta_{+})$ is

$R_{3}\left(\rho_{+},u_{+},\theta_{+}\right):=\left\\{(\rho,u,\theta)\;\left|\begin{aligned}
&\;0<\rho<\rho_{+},\;\;\rho^{1-\gamma}\theta=\rho_{+}^{1-\gamma}\theta_{+},\\\\[5.69054pt]
&\;u=u_{+}+\int_{\rho_{+}}^{\rho}\sqrt{R\gamma\rho_{+}^{1-\gamma}\theta_{+}}\;\xi^{\frac{\gamma-3}{2}}\,\mathrm{d}\xi\end{aligned}\right.\right\\}.$
(2.16)

So for each pair of data $(u_{-},\theta_{-})$ with the restriction condition

$u_{-}=u_{+}+\int_{\rho_{+}}^{\left(\theta_{-}/\theta_{+}\right)^{\frac{1}{\gamma-1}}\rho_{+}}\sqrt{R\gamma\rho_{+}^{1-\gamma}\theta_{+}}\;\xi^{\frac{\gamma-3}{2}}\,\mathrm{d}\xi,$
(2.17)

there exists a unique $\rho_{-}$ such that $(\rho_{-},u_{-},\theta_{-})\in
R_{3}(\rho_{+},u_{+},\theta_{+})$. The 3-rarefaction wave
$(\rho^{r},u^{r},\theta^{r})(x/t)$ connecting $(\rho_{-},u_{-},\theta_{-})$
and $(\rho_{+},u_{+},\theta_{+})$ is the unique weak solution globally in time
to the following Riemann problem:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\qquad
x\in\mathbb{R},\;\;t>0,\\\\[5.69054pt] &(\rho u)_{t}+\left(\rho
u^{2}+p\right)_{x}=0,\\\\[5.69054pt]
&\left[\rho\left(\frac{R}{\gamma-1}\theta+\frac{1}{2}u^{2}\right)\right]_{t}+\left[\rho
u\left(\frac{R}{\gamma-1}\theta+\frac{1}{2}u^{2}\right)+pu\right]_{x}=0,\\\\[5.69054pt]
&(\rho,u,\theta)(x,0)=\left\\{\begin{array}[]{ll}\left(\rho_{-},u_{-},\theta_{-}\right),&x<0,\\\\[5.69054pt]
\left(\rho_{+},u_{+},\theta_{+}\right),&x>0,\end{array}\right.\end{aligned}\right.$
(2.18)

with the electric field rarefaction wave $\tilde{E}=0$ and the magnetic field
rarefaction wave $\tilde{b}=0$. Here $\theta_{-}<\theta_{+}$ (or equivalently,
$\rho_{-}<\rho_{+}$, $u_{-}<u_{+}$). In addition, if
$\lambda_{3}(\rho_{-},u_{-},\theta_{-})\geq 0$, then the rarefaction wave is
constant on $(x,t)\in\mathbb{R}_{-}\times[0,+\infty)$. For the outflow problem
when $u_{-}\in\Omega_{\mathrm{sub}}^{-}\cup\Gamma_{\mathrm{tran}}^{-}$, then
$\lambda_{3}(\rho_{-},u_{-},\theta_{-})\geq 0$. Thus in this situation, one
can expect that the solutions to the outflow problem converge towards
3-rarefaction wave which is similar to the Cauchy problem of (1.1). To give
the details of the large-time behavior of the solutions to the outflow
problem, it is necessary to construct a smooth approximation
$(\bar{\rho},\bar{u},\bar{\theta})(x,t)$ of
$(\rho^{r},u^{r},\theta^{r})(x/t)$. To this end, we borrow the idea from [12]
and [38].

Consider the solution to the following Cauchy problem:

$\left\\{\begin{aligned} &w_{t}+ww_{x}=0,\qquad
x\in\mathbb{R},\;\;t>0,\\\\[5.69054pt]
&w(x,0)=\left\\{\begin{array}[]{ll}\displaystyle w_{-},&\;x<0,\\\\[5.69054pt]
\displaystyle w_{-}+C_{q}\delta_{r}\int_{0}^{\alpha
x}y^{q}\mathrm{e}^{-y}\mathrm{~{}d}y,&\;x\geq
0.\end{array}\right.\end{aligned}\right.$ (2.19)

Here $\delta_{r}:=w_{+}-w_{-}>0$, $0<\alpha<1\leq q$ are two constants to be
determined later, and $C_{q}$ is a constant such that
$C_{q}\int_{0}^{+\infty}y^{q}\mathrm{e}^{-y}\,\mathrm{d}y=1$. Let
$w_{\pm}=\lambda_{3}(\rho_{\pm},u_{\pm},\theta_{\pm})$ and
$(\bar{\rho},\bar{u},\bar{\theta},\bar{E},\bar{b})(x,t)$ be defined as

$\left\\{\begin{aligned}
&\left(\bar{u}+\sqrt{R\gamma\bar{\theta}}\right)(x,t)=w(x,1+t),\qquad
x\in\mathbb{R},\;\;t>0,\\\\[5.69054pt]
&\left(\bar{\rho}^{1-\gamma}\bar{\theta}\right)(x,t)=\rho_{+}^{1-\gamma}\theta_{+},\\\\[5.69054pt]
&\bar{u}(x,t)=u_{+}+\int_{\rho_{+}}^{\bar{\rho}(x,t)}\sqrt{R\gamma\rho_{+}^{1-\gamma}\theta_{+}}\;\xi^{\frac{\gamma-3}{2}}\,\mathrm{d}\xi,\\\\[5.69054pt]
&\bar{E}(x,t)=\bar{b}(x,t)=0.\end{aligned}\right.$ (2.20)

Due to $w_{-}\geq 0$ and (2.19), one has $w(x,t)\equiv w_{-}$ on
$\mathbb{R}_{-}\times[0,+\infty)$. From (2.20), one easily knows that
$(\bar{\rho},\bar{u},\bar{\theta})(x,t)$ is constant on
$\mathbb{R}_{-}\times[0,+\infty)$ too. Here we restrict
$(\bar{\rho},\bar{u},\bar{\theta})(x,t)$ in the half space $\\{x\geq 0\\}$ and
still use $(\bar{\rho},\bar{u},\bar{\theta})(x,t)$ to represent
$(\bar{\rho},\bar{u},\bar{\theta})(x,t)|_{x\geq 0}$. Then one easily has

$\left\\{\begin{aligned} &\bar{\rho}_{t}+(\bar{\rho}\bar{u})_{x}=0,\qquad
x\in\mathbb{R}_{+},\;\;t>0,\\\\[5.69054pt]
&(\bar{\rho}\bar{u})_{t}+\left(\bar{\rho}\bar{u}^{2}+\bar{p}\right)_{x}=0,\\\\[5.69054pt]
&\left[\bar{\rho}\left(\frac{R}{\gamma-1}\bar{\theta}+\frac{1}{2}\bar{u}^{2}\right)\right]_{t}+\left[\bar{\rho}\bar{u}\left(\frac{R}{\gamma-1}\bar{\theta}+\frac{1}{2}\bar{u}^{2}\right)+\bar{p}\bar{u}\right]_{x}=0,\\\\[5.69054pt]
&(\bar{\rho},\bar{u},\bar{\theta})(0,t)=\left(\rho_{-},u_{-},\theta_{-}\right),\quad(\bar{\rho},\bar{u},\bar{\theta})(x,0)\rightarrow\left\\{\begin{array}[]{ll}\left(\rho_{-},u_{-},\theta_{-}\right),&\;x\rightarrow
0^{+},\\\\[5.69054pt]
\left(\rho_{+},u_{+},\theta_{+}\right),&\;x\rightarrow+\infty,\end{array}\right.\end{aligned}\right.$
(2.21)

where $\bar{p}:=p(\bar{\rho},\bar{\theta})=R\bar{\rho}\bar{\theta}$. Then the
following lemma holds.

######  Lemma 2.2.

$($see [12] $)$ $(\bar{\rho},\bar{u},\bar{\theta})(x,t)$ satisfies

  * $\mathrm{(i)}$

$0\leq\bar{\rho}_{x},\,\bar{\theta}_{x}\leq C\bar{u}_{x}$,
$\left|(\bar{\rho}_{xx},\bar{\theta}_{xx})\right|\leq
C(\left|\bar{u}_{xx}\right|+\bar{u}_{x}^{2})$,
$\forall\;(x,t)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$;

  * $\mathrm{(ii)}$

For any $p\;(1\leq p\leq+\infty)$, there exists a constant $C_{pq}$ such that

$\displaystyle\|(\bar{\rho}_{x},\bar{u}_{x},\bar{\theta}_{x})(t)\|_{L^{p}}\leq
C_{pq}\min\\{\delta_{r}\alpha^{1-\frac{1}{p}},\delta_{r}^{\frac{1}{p}}(1+t)^{-1+\frac{1}{p}}\\},$
(2.22)
$\displaystyle\|(\bar{\rho}_{xx},\bar{u}_{xx},\bar{\theta}_{xx})(t)\|_{L^{p}}\leq
C_{pq}\min\\{(\delta_{r}+\delta_{r}^{2})\alpha^{2-\frac{1}{p}},(\delta_{r}^{\frac{1}{p}}+\delta_{r}^{\frac{1}{q}})(1+t)^{-1+\frac{1}{q}}\\};$
(2.23)

  * $\mathrm{(iii)}$

If $x\leq(u_{-}+\sqrt{R\gamma\theta_{-}})(1+t)$, then
$(\bar{\rho},\bar{u},\bar{\theta})(x,t)-(\rho_{-},u_{-},\theta_{-})\equiv 0$;

  * $\mathrm{(iv)}$

$\lim_{t\rightarrow+\infty}\sup_{x\in\mathbb{R}_{+}}\left|(\bar{\rho},\bar{u},\bar{\theta})(x,t)-(\rho^{r},u^{r},\theta^{r})(\frac{x}{1+t})\right|=0$.

Now we can give the local stability of the 3-rarefaction wave:

######  Theorem 2.2.

Assume $(\rho_{+},u_{+},\theta_{+})\in\Omega^{-}_{\mathrm{sub}}\cup\\{u\geq
0\\}$, $\theta_{-}<\theta_{+}$ and

$u_{-}=u_{+}+\int_{\rho_{+}}^{(\theta_{-}/\theta_{+})^{\frac{1}{\gamma-1}}\rho_{+}}\sqrt{R\gamma\rho_{+}^{1-\gamma}\theta_{+}}\;\xi^{\frac{\gamma-3}{2}}\,\mathrm{d}\xi\geq-\sqrt{R\gamma\theta_{-}}\,.$

In addition, the dielectric constant $\varepsilon$ satisfies

$0<\varepsilon<\bar{C}$ (2.24)

for some positive constant $\bar{C}$ $($depending only on $|u_{\pm}|$ and
$\theta_{\pm}$$)$. There is a suitably small constant $\varepsilon_{2}>0$
which is independent of $T$, such that if

$\alpha+\|(\rho_{0},u_{0},\theta_{0},E_{0},b_{0})-(\bar{\rho}_{0},\bar{u}_{0},\bar{\theta}_{0},0,0)\|_{H^{1}}\leq\varepsilon_{2},$
(2.25)

then the outflow problem (1.1)-(1.3) and (1.11)-(1.12) has a unique global
solution $(\rho,u,\theta,E,b)(x,t)$. Furthermore,

$\lim_{t\rightarrow+\infty}\sup_{x\in\mathbb{R}_{+}}\left|(\rho,u,\theta,E,b)(x,t)-(\rho^{r},u^{r},\theta^{r},0,0)(x/t)\right|=0.$
(2.26)

###  2.3 Superposition of the boundary layer and rarefaction wave

Now let $(\rho_{+},u_{+},\theta_{+})\in\Omega^{-}_{\mathrm{sub}}\cup\\{u\geq
0\\}$. For $(\rho_{*},u_{*},\theta_{*})\in
R_{3}(\rho_{+},u_{+},\theta_{+})\cap(\Omega^{-}_{\mathrm{sub}}\cup\Gamma^{-}_{\mathrm{tran}})$,
let $S_{*}:=\\{(\rho,u,\theta)\,|\,\rho u=\rho_{*}u_{*}\\}$ be a family of
surfaces. From Section 2.2, we know that for each point
$(\rho_{*},u_{*},\theta_{*})$ there exists a uniquely determined 3-rarefaction
wave connecting it and $(\rho_{+},u_{+},\theta_{+})$. Among the three
variables, $\rho_{*}$, $u_{*}$ and $\theta_{*}$, just one is independent, the
other two can be determined accordingly. Precisely speaking, if let $\rho_{*}$
be independent, then

$\rho_{*}<\rho_{+},\qquad\rho_{*}^{1-\gamma}\theta_{*}=\rho_{+}^{1-\gamma}\theta_{+},\qquad
u_{*}=u_{+}+\int_{\rho_{+}}^{\rho_{*}}\sqrt{R\gamma\rho_{+}^{1-\gamma}\theta_{+}}\;\xi^{\frac{\gamma-3}{2}}\,\mathrm{d}\xi.$
(2.27)

Obviously, both $u_{*}$ and $\theta_{*}$ are strictly increasing and
continuously differentiable with respect to $\rho_{*}$. From Section 2.1, we
can easily know that each boundary layer belongs to one surface of the family.
Consider the family $S_{*}$ to be a function of $\rho_{*}$, then

$\frac{\mathrm{d}S_{*}}{\mathrm{d}\rho_{*}}=u_{*}+\rho_{*}\frac{\mathrm{d}u_{*}}{\mathrm{d}\rho_{*}}=u_{*}+\sqrt{R\gamma\theta_{*}}.$
(2.28)

Due to
$(\rho_{*},u_{*},\theta_{*})\in\Omega^{-}_{\mathrm{sub}}\cup\Gamma^{-}_{\mathrm{tran}}$,
we know that $R_{3}(\rho_{+},u_{+},\theta_{+})$ and each one of $S_{*}$ owns a
unique intersection point, i.e., $(\rho_{*},u_{*},\theta_{*})$. Moreover, all
of $S_{*}$ never intersect each other, especially when
$(\rho_{*},u_{*},\theta_{*})\in\Gamma^{-}_{\mathrm{tran}}$. For Case 1: if

$0\neq\left|u_{-}-u_{+}-\int_{\rho_{+}}^{(\theta_{-}/\theta_{+})^{\frac{1}{\gamma-1}}\rho_{+}}\sqrt{R\gamma\rho_{+}^{1-\gamma}\theta_{+}}\;\xi^{\frac{\gamma-3}{2}}\,\mathrm{d}\xi\right|\ll
1,$ (2.29)

then it is expected that there exists a unique point
$(\rho_{*},u_{*},\theta_{*})\in
R_{3}(\rho_{+},u_{+},\theta_{+})\cap\Omega^{-}_{\mathrm{sub}}$ such that
$\rho_{*}$, $u_{*}$, $\theta_{*}$, $u_{-}$ and $\theta_{-}$ satisfy (2.1) just
when $(\rho_{+},u_{+},\theta_{+})$ is replaced by
$(\rho_{*},u_{*},\theta_{*})$ there. For Case 2: when
$(\rho_{*},u_{*},\theta_{*})\in
R_{3}(\rho_{+},u_{+},\theta_{+})\cap\Gamma^{-}_{\mathrm{tran}}$, it holds that
$u_{*}=-\sqrt{R\gamma\theta_{*}}$, which means $(\rho_{*},u_{*},\theta_{*})$
is unique too.

Let

$(\hat{\rho},\hat{u},\hat{\theta})=(\tilde{\rho},\tilde{u},\tilde{\theta})+(\bar{\rho},\bar{u},\bar{\theta})-\left(\rho_{*},u_{*},\theta_{*}\right),$
(2.30)

with $\hat{E}=\hat{b}=0$, and the strength of boundary layer denoted by
$\delta=\left|(u_{*}-u_{-},\theta_{*}-\theta_{-})\right|$. Under the
preliminaries above, we can state the third result.

######  Theorem 2.3.

Assume $(\rho_{+},u_{+},\theta_{+})\in\Omega^{-}_{\mathrm{sub}}\cup\\{u\geq
0\\}$, $(\rho_{*},u_{*},\theta_{*})\in
R_{3}(\rho_{+},u_{+},\theta_{+})\cap(\Omega^{-}_{\mathrm{sub}}\cup\Gamma^{-}_{\mathrm{tran}})$
and $\rho_{*}$, $u_{*}$, $\theta_{*}$, $u_{-}$, $\theta_{-}$ satisfy (2.1)
just when $(\rho_{+},u_{+},\theta_{+})$ there is replaced by
$(\rho_{*},u_{*},\theta_{*})$. In addition, the dielectric constant
$\varepsilon$ satisfies

$0<\varepsilon<\bar{C}$ (2.31)

for some positive constant $\bar{C}$ $($depending only on $|u_{\pm}|$ and
$\theta_{\pm}$$)$. There exist two small positive constants $\delta_{2}$ and
$\varepsilon_{3}$ which are independent of $T$, such that if
$0<\delta<\min\\{\delta_{0},\delta_{2}\\}$ and

$\alpha+\|(\rho_{0},u_{0},\theta_{0},E_{0},b_{0})-(\hat{\rho}_{0},\hat{u}_{0},\hat{\theta}_{0},0,0)\|_{H^{1}(\mathbb{R}_{+})}\leq\varepsilon_{3},$
(2.32)

then the outflow problem (1.1)-(1.3) and (1.11)-(1.12) has a unique global
solution $(\rho,u,\theta,E,b)(x,t)$. Furthermore,

$\lim_{t\rightarrow+\infty}\sup_{x\in\mathbb{R}_{+}}\left|(\rho,u,\theta)(x,t)-(\tilde{\rho},\tilde{u},\tilde{\theta})(x)-(\rho^{r},u^{r},\theta^{r})({x}/{t})+(\rho_{*},u_{*},\theta_{*})\right|=0,$
(2.33)

and

$\lim_{t\rightarrow+\infty}\sup_{x\in\mathbb{R}_{+}}\left|(E,b)(x,t)-(0,0)\right|=0.$
(2.34)

######  Remark 3.

From (3.2) for the case $M_{+}=1$, we can take the constant $\bar{C}$ in
(2.31) as

$\bar{C}=\frac{1}{64\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}\cdot\left(\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}+\sqrt{R\gamma\max\\{\theta_{-},\theta_{+}\\}}\right)}.$
(2.35)

Then for each given $\varepsilon$ satisfying the condition (2.31), our system
(1.1) is explicitly well-defined. On the one hand, when we take
$\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}$ suitably small, the
dielectric constant $\varepsilon$ can be large enough, which can be seen from
the conditions (2.31) and (2.35) directly. This fact can relax the requirement
of smallness of $\varepsilon$. On the other hand, the conditions (2.31) and
(2.35) together can relax the restriction on
$\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}$ as long as the dielectric
constant $\varepsilon$ is suitably small. Thus, an interesting problem occurs,
that is how to remove the technical condition (2.31) in future.

######  Remark 4.

For the compressible non-isentropic Navier-Stokes-Maxwell equations, the
asymptotic stability of the wave patterns to the inflow problem and the
impermeable wall problem can also be taken into account and remains to be
studied in future.

##  3 Proofs of the theorems

It is easy to know that the proof of Theorems 2.1, 2.2 is similar to and
simpler than that of Theorem 2.3 below, thus the details of Theorems 2.1, 2.2
are omitted here. And it is noted that Theorem 2.3 concerns two cases of the
boundary layer: one is non-degenerate, the other is degenerate. If the
boundary layer is not degenerate, i.e. decays exponentially, then employing a
Poincar$\acute{\rm e}$-type inequality, one easily knows that all the terms
concerning the boundary layer are easier to be controlled than the
counterparts of the degenerate case. Hence, we only consider the proof of
Theorem 2.3 concerning the superposition of the degenerate boundary layer and
the 3-rarefaction wave.

Recall

$(\hat{\rho},\hat{u},\hat{\theta})(x,t)=(\tilde{\rho},\tilde{u},\tilde{\theta})(x)+(\bar{\rho},\bar{u},\bar{\theta})(x,t)-\left(\rho_{*},u_{*},\theta_{*}\right).$

After some simple calculations, we can obtain

$\left\\{\begin{aligned}
&\hat{\rho}_{t}+\hat{u}\hat{\rho}_{x}+\hat{\rho}\hat{u}_{x}=\hat{f},\qquad
x\in\mathbb{R}_{+},\;\;t>0,\\\\[5.69054pt]
&\hat{\rho}\left(\hat{u}_{t}+\hat{u}\hat{u}_{x}\right)+\hat{p}_{x}=\mu\tilde{u}_{xx}+\hat{g},\\\\[5.69054pt]
&\frac{R}{\gamma-1}\hat{\rho}\left(\hat{\theta}_{t}+\hat{u}\hat{\theta}_{x}\right)+\hat{p}\hat{u}_{x}=\kappa\tilde{\theta}_{xx}+\mu\tilde{u}_{x}^{2}+\hat{h},\\\\[5.69054pt]
&(\hat{\rho},\hat{u},\hat{\theta})(0,t)=\left(\rho_{-},u_{-},\theta_{-}\right),\qquad(\hat{\rho},\hat{u},\hat{\theta})(+\infty,t)=\left(\rho_{+},u_{+},\theta_{+}\right),\end{aligned}\right.$

where $\rho_{-}:=(\rho_{*}u_{*})/u_{-}$, and $\rho_{*}$, $u_{*}$,
$\theta_{*}$, $u_{-}$, $\theta_{-}$ satisfy (2.1). Here
$\hat{p}:=p(\hat{\rho},\hat{\theta})=R\hat{\rho}\hat{\theta}$ and

$\left\\{\begin{aligned}
&\hat{f}=\left(\bar{u}-u_{*}\right)\tilde{\rho}_{x}+\left(\bar{\rho}-\rho_{*}\right)\tilde{u}_{x}+\left(\tilde{u}-u_{*}\right)\bar{\rho}_{x}+\left(\tilde{\rho}-\rho_{*}\right)\bar{u}_{x},\\\\[2.84526pt]
&\hat{g}=\hat{\rho}\left[\left(\bar{u}-u_{*}\right)\tilde{u}_{x}+\left(\tilde{u}-u_{*}\right)\bar{u}_{x}\right]+\left(\bar{\rho}-\rho_{*}\right)\tilde{u}\tilde{u}_{x}+(\hat{p}-\tilde{p}-\bar{p})_{x}-\frac{\tilde{\rho}-\rho_{*}}{\bar{\rho}}\bar{p}_{x},\\\\[2.84526pt]
&\hat{h}=\frac{R}{\gamma-1}\hat{\rho}\left[\left(\bar{u}-u_{*}\right)\tilde{\theta}_{x}+\left(\tilde{u}-u_{*}\right)\bar{\theta}_{x}\right]+\frac{R}{\gamma-1}\left(\bar{\rho}-\rho_{*}\right)\tilde{u}\tilde{\theta}_{x}\\\\[2.84526pt]
&\qquad+(\hat{p}-\tilde{p})\tilde{u}_{x}+(\hat{p}-\bar{p})\bar{u}_{x}-R\bar{\theta}\left(\tilde{\rho}-\rho_{*}\right)\bar{u}_{x}.\end{aligned}\right.$

Combining $\bar{u}_{x}\geq 0$ and (2.1), we obtain that

$|\hat{f}|+|\hat{g}|+|\hat{h}|\leq
C\left((\bar{u}-u_{*})\left|\tilde{u}_{x}\right|+\left|\tilde{u}-u_{*}\right|\bar{u}_{x}\right).$
(3.1)

Define the perturbation as

$(\phi,\psi,\zeta,E,b)(x,t)=(\rho-\hat{\rho},u-\hat{u},\theta-\hat{\theta},E,b)(x,t).$
(3.2)

Then we transform the initial-boundary value problem (1.1)-(1.3) and
(1.11)-(1.12) as

$\left\\{\begin{aligned} &\phi_{t}+u\phi_{x}+\rho\psi_{x}=f,\qquad
x\in\mathbb{R}_{+},\;\;t>0,\\\\[5.69054pt]
&\rho\left(\psi_{t}+u\psi_{x}\right)+(p-\hat{p})_{x}=\mu\psi_{xx}-(E+\psi
b+\hat{u}b)b+g,\\\\[5.69054pt]
&\frac{R}{\gamma-1}\rho\left(\zeta_{t}+u\zeta_{x}\right)+p\psi_{x}=\kappa\zeta_{xx}+\mu\psi_{x}^{2}+(E+\psi
b+\hat{u}b)^{2}+h,\\\\[5.69054pt] &\varepsilon E_{t}-b_{x}+E+\psi
b+\hat{u}b=0,\\\\[5.69054pt] &b_{t}-E_{x}=0,\end{aligned}\right.$ (3.3)

with the initial data

$(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})(x):=(\phi,\psi,\zeta,E,b)(x,0)\rightarrow(0,0,0,0,0),\quad\text{
as }x\rightarrow+\infty,$ (3.4)

and the boundary condition

$(\phi,\psi,\zeta,\sqrt{\varepsilon}E-b)(0,t)=(\rho(0,t)-\rho_{-},0,0,0),$
(3.5)

where

$\left\\{\begin{aligned}
&f=-\hat{u}_{x}\phi-\hat{\rho}_{x}\psi-\hat{f},\\\\[2.84526pt]
&g=-\rho\hat{u}_{x}\psi+\hat{p}_{x}\frac{\phi}{\hat{\rho}}-\mu\tilde{u}_{xx}\frac{\phi}{\hat{\rho}}+\mu\bar{u}_{xx}-\frac{\rho}{\hat{\rho}}\hat{g},\\\\[2.84526pt]
&h=-\frac{R}{\gamma-1}\rho\hat{\theta}_{x}\psi-R\rho\hat{u}_{x}\zeta-(\kappa\tilde{\theta}_{xx}+\mu\tilde{u}_{x}^{2})\frac{\phi}{\hat{\rho}}\\\\[2.84526pt]
&\quad\;\;\,+\kappa\bar{\theta}_{xx}+2\mu\hat{u}_{x}\psi_{x}+2\mu\tilde{u}_{x}\bar{u}_{x}+\mu\bar{u}_{x}^{2}-\frac{\rho}{\hat{\rho}}\hat{h}.\end{aligned}\right.$
(3.6)

For interval $I\subset[0,\infty)$, we define a function space $X(I)$ as

$X(I):=\left\\{(\phi,\psi,\zeta,E,b)\;\left|\,\begin{aligned}
&\;\left(\phi,\psi,\zeta,E,b\right)\in
L^{\infty}\left(I;H^{1}(\mathbb{R}_{+})\right),\\\\[5.69054pt]
&\;\left(\phi_{x},E_{x},b_{x}\right)\in
L^{2}\left(I;L^{2}(\mathbb{R}_{+})\right),\\\\[5.69054pt]
&\;\left(\psi_{x},\zeta_{x}\right)\in
L^{2}\left(I;H^{1}(\mathbb{R}_{+})\right)\end{aligned}\right.\right\\}.$

To prove Theorem 2.3 for brevity, we only devote ourselves to deriving the
uniform a priori estimates of the perturbation from the superposition of the
degenerate boundary layer and the 3-rarefaction wave to the initial-boundary
value problem (3.3)-(3.6).

######  Proposition 3.1.

$($A priori estimates$)$ Suppose that the boundary layer in Theorem 2.3 is
degenerate. Let $(\phi,\psi,\zeta,E,b)$ $\in X(0,T)$ be a smooth solution to
the problem (3.3)-(3.6) on $0\leq t\leq T$ for $T>0$. There exist a positive
constant $\bar{C}$ $($depending only on $|u_{\pm}|$ and $\theta_{\pm}$$)$ and
two suitably small positive constants $\delta_{3}$ and $\varepsilon_{0}$ such
that if the dielectric constant $\varepsilon$ and the strength of boundary
layer $\delta$ satisfy $\varepsilon<\bar{C}$,
$\delta<\min\\{\delta_{0},\delta_{3}\\}$ and the following a priori assumption
holds:

$\sup_{0\leq t\leq T}\|(\phi,\psi,\zeta,E,b)(t)\|_{H^{1}}\leq\varepsilon_{0},$
(3.7)

then $(\phi,\psi,\zeta,E,b)(x,t)$ satisfies

$\displaystyle\quad\sup_{0\leq t\leq
T}\left[\|(\phi,\psi,\zeta,\sqrt{\varepsilon}E,b)\|^{2}_{H^{1}}+\sqrt{\varepsilon}E^{2}(0,t)\right]+\int_{0}^{T}\|(\phi_{x},\psi_{x},\zeta_{x},E_{x},b_{x},\psi_{xx},\zeta_{xx})\|^{2}\,\mathrm{d}\tau$
$\displaystyle\quad+\int_{0}^{T}\left[\phi^{2}(0,\tau)+\phi^{2}_{x}(0,\tau)+\sqrt{\varepsilon}E^{2}(0,\tau)+\varepsilon^{\frac{3}{2}}E_{\tau}^{2}(0,\tau)\right]\,\mathrm{d}\tau$
$\displaystyle\quad+\int_{0}^{T}\left[\|E+\psi
b+\hat{u}b\|^{2}+\|\sqrt{\bar{u}_{x}}(\phi,\psi,\zeta,\sqrt{\varepsilon}E,b)\|^{2}\right]\,\mathrm{d}\tau$
$\displaystyle\leq
C\left(\|(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})\|^{2}_{H^{1}}+\delta+\alpha^{\frac{1}{10}}\right).$
(3.8)

Owing to a smallness assumption (3.7) on
$\|(\phi,\psi,\zeta,E,b)(t)\|_{H^{1}}$, a quantity
$\|(\phi,\psi,\zeta,E,b)(t)\|_{L^{\infty}}$ is also sufficiently small, i.e.

$\|(\phi,\psi,\zeta,E,b)(t)\|_{L^{\infty}}\leq\sqrt{2}\varepsilon_{0},$ (3.9)

where we have used the following Sobolev inequality

$\|f\|_{L^{\infty}}\leq\sqrt{2}\|f\|^{\frac{1}{2}}\|f_{x}\|^{\frac{1}{2}},\qquad{\rm
for}\,\,f(x)\in H^{1}(\mathbb{R_{+}}).$ (3.10)

Once Proposition 3.1 is proved, we can close the a priori assumption (3.7).
The global existence of the solution to the initial-boundary value problem
(3.3)-(3.6) then follows from the standard continuation argument based on the
local existence and the a priori estimates. For $0<\varepsilon<\bar{C}$, the
estimate (3.1) and the equations (3.3) imply that

$\int_{0}^{\infty}\left(\|(\phi_{x},\psi_{x},\zeta_{x},E_{x},b_{x})(t)\|^{2}+\left|\frac{\rm
d}{{\rm
d}t}\|(\phi_{x},\psi_{x},\zeta_{x},E_{x},b_{x})(t)\|^{2}\right|\right){\rm{d}}t<\infty,$

which easily leads to

$\lim_{t\to{+\infty}}\|(\phi_{x},\psi_{x},\zeta_{x},E_{x},b_{x})(t)\|^{2}=0.$
(3.11)

Then using the Sobolev inequality (3.10), together with (3.11), directly
implies the large time behavior of the solutions: (2.33) and (2.34).

Firstly, due to (3.9) and the smallness of $\varepsilon_{0}$, it is easy to
deduce the following properties, which will be frequently used in the sequel.

######  Lemma 3.1.

If the strength of boundary layer
$\delta=\left|(u_{*}-u_{-},\;\theta_{*}-\theta_{-})\right|$ is small enough,
then

  * $\mathrm{(i)}$

$\tilde{u}(x)$, $\bar{u}(x,t)$, $\hat{u}(x,t)$ and $u(x,t)$ satisfy

$\displaystyle\frac{3}{2}u_{-}<\tilde{u}(x)<\frac{1}{2}u_{-}<0,\qquad\frac{5}{4}u_{-}<\bar{u}(x,t)\leq
u_{+},$
$\displaystyle\left|\hat{u}(x,t)\right|<\frac{3}{2}\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\},\qquad\left|u(x,t)\right|<2\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}.$

  * $\mathrm{(ii)}$

$\tilde{\theta}(x)$, $\bar{\theta}(x,t)$, $\hat{\theta}(x,t)$ and
$\theta(x,t)$ satisfy

$\displaystyle
0<\frac{1}{2}\theta_{-}<\tilde{\theta}(x)<\frac{3}{2}\theta_{-},\qquad
0<\frac{3}{4}\theta_{-}<\theta_{*}<\bar{\theta}(x,t)\leq\theta_{+},$
$\displaystyle\frac{1}{2}\min\\{\theta_{-},\theta_{+}\\}<\hat{\theta}(x,t)<\frac{5}{4}\max\\{\theta_{-},\theta_{+}\\},$
$\displaystyle\frac{1}{4}\min\\{\theta_{-},\theta_{+}\\}<\theta(x,t)<\frac{3}{2}\max\\{\theta_{-},\theta_{+}\\}.$

  * $\mathrm{(iii)}$

$\rho_{*}$, $\tilde{\rho}(x)$, $\bar{\rho}(x,t)$, $\hat{\rho}(x,t)$ and
$\rho(x,t)$ satisfy

$\displaystyle
0<\rho_{+}\left(\frac{3}{4}\frac{\theta_{-}}{\theta_{+}}\right)^{\frac{1}{\gamma-1}}<\rho_{*}\leq\rho_{+},$
$\displaystyle\frac{1}{2}\rho_{+}\left(\frac{3}{4}\frac{\theta_{-}}{\theta_{+}}\right)^{\frac{1}{\gamma-1}}<\tilde{\rho}(x)<\frac{3}{2}\rho_{+},\qquad\rho_{+}\left(\frac{3}{4}\frac{\theta_{-}}{\theta_{+}}\right)^{\frac{1}{\gamma-1}}<\bar{\rho}(x,t)\leq\rho_{+},$
$\displaystyle\frac{1}{2}\rho_{+}\left(\frac{3}{4}\frac{\theta_{-}}{\theta_{+}}\right)^{\frac{1}{\gamma-1}}<\hat{\rho}(x,t)<\frac{3}{2}\rho_{+},\qquad\frac{1}{4}\rho_{+}\left(\frac{3}{4}\frac{\theta_{-}}{\theta_{+}}\right)^{\frac{1}{\gamma-1}}<\rho(x,t)<\frac{7}{4}\rho_{+}.$

Next, the following useful lemma plays an important role in the proof of the a
priori estimates.

######  Lemma 3.2.

Assume functions $z(x,t)\in H_{x}^{1}(\mathbb{R}_{+})$, then

  * $\mathrm{(i)}$

for the boundary layer $\tilde{u}(x)$ satisfying Subcase 2 of the transonic
case $M_{+}=1$, it holds

$\int_{\mathbb{R_{+}}}\tilde{u}_{x}^{2}z^{2}\,\mathrm{d}x\leq
C\delta^{3}z^{2}(0,t)+C\delta^{2}\|z_{x}\|^{2}.$ (3.12)

  * $\mathrm{(ii)}$

for the smooth approximate rarefaction wave $\bar{u}(x,t)$, it holds

$\int_{\mathbb{R_{+}}}\bar{u}_{x}^{2}z^{2}\,\mathrm{d}x\leq
C\alpha^{\frac{1}{3}}\|z\|\left(\|z_{x}\|^{2}+(1+t)^{-\frac{4}{3}}\right).$
(3.13)

###### Proof.

(3.12) follows easily by employing the following Poincar$\acute{\rm e}$ type
inequality used by Nikkuni [42]:

$\left|z(x,t)\right|\leq\left|z(0,t)\right|+x^{\frac{1}{2}}\|z_{x}\|,\quad
z(x,t)\in H_{x}^{1}(\mathbb{R}_{+})$ (3.14)

and the decay rate of the degenerate boundary layer. For (3.13), by using the
Sobolev inequality (3.10) and (2.22) in Lemma 2.2, we have

$\displaystyle\int_{\mathbb{R_{+}}}\bar{u}_{x}^{2}z^{2}\,\mathrm{d}x$
$\displaystyle\leq\|z\|^{2}_{L^{\infty}_{x}}\|\bar{u}_{x}\|^{2}\leq
2\|z\|\cdot\|z_{x}\|\cdot\|\bar{u}_{x}\|^{2}\leq
C\|z\|\cdot\|z_{x}\|\cdot\alpha^{\frac{1}{3}}(1+t)^{-\frac{2}{3}}$
$\displaystyle\leq
C\alpha^{\frac{1}{3}}\|z\|\left(\|z_{x}\|^{2}+(1+t)^{-\frac{4}{3}}\right).$

∎

###  3.1 Zero-order energy estimates

######  Lemma 3.3.

Suppose that the conditions in Proposition 3.1 hold. Then for all $0<t<T$, we
have the following zero-order energy estimates

$\displaystyle\|(\phi,\psi,\zeta,\sqrt{\varepsilon}E,b)\|^{2}+\int_{0}^{t}\left[\phi^{2}(0,\tau)+\sqrt{\varepsilon}E^{2}(0,\tau)+\left\|\left(\psi_{x},\zeta_{x},E+\psi
b+\hat{u}b\right)\right\|^{2}\right]\,\mathrm{d}\tau$
$\displaystyle\quad+\int_{0}^{t}\|\sqrt{\bar{u}_{x}}(\phi,\psi,\zeta,\sqrt{\varepsilon}E,b)\|^{2}\,\mathrm{d}\tau\leq
C\|(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})\|^{2}+C(\delta+\alpha^{\frac{1}{10}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C(\delta+\alpha^{\frac{1}{10}})\int_{0}^{t}\|(\phi_{x},\sqrt{\varepsilon}E_{x},b_{x})\|^{2}\,\mathrm{d}\tau.$
(3.15)

###### Proof.

The proof of the zero-order energy estimates in Lemma 3.3 includes the
following two steps.

Step 1: Inspired by the work of the compressible Navier-Stokes equations in
[20] and [45], we take $\Phi(s)=s-1-\ln s$ for $s>0$, and set

$\eta(x,t)=\frac{1}{2}\psi^{2}+R\hat{\theta}\Phi\left(\frac{\hat{\rho}}{\rho}\right)+\frac{R}{\gamma-1}\hat{\theta}\Phi\left(\frac{\theta}{\hat{\theta}}\right).$

Tedious calculations give rise to

$(\rho\cdot\eta(x,t))_{t}+H_{x}+\mu\frac{\hat{\theta}}{\theta}\psi_{x}^{2}+\kappa\frac{\hat{\theta}}{\theta^{2}}\zeta_{x}^{2}=Q+(E+\psi
b+\hat{u}b)^{2}\frac{\zeta}{\theta}-(E+\psi b+\hat{u}b)\psi b,$ (3.16)

where

$\displaystyle H$ $\displaystyle=\rho
u\cdot\eta(x,t)+(p-\hat{p})\psi-\mu\psi\psi_{x}-\kappa\frac{\zeta\zeta_{x}}{\theta},$
$\displaystyle Q$
$\displaystyle=-\frac{R\hat{\theta}\hat{\rho}_{x}\phi\psi}{\hat{\rho}}-\frac{R\hat{\theta}\hat{f}\phi}{\hat{\rho}}+g\psi+\left(2\mu\hat{u}_{x}\psi_{x}+2\mu\tilde{u}_{x}\bar{u}_{x}+\mu\bar{u}_{x}^{2}+\kappa\bar{\theta}_{xx}\right)\frac{\zeta}{\theta}$
$\displaystyle\quad-\left(\frac{p\hat{\theta}_{x}\psi}{(\gamma-1)\hat{\theta}}+\frac{\kappa\tilde{\theta}_{xx}(p-\hat{p})+\mu\tilde{u}_{x}^{2}(p-\hat{p})+p\hat{h}}{\hat{p}}\right)\frac{\zeta}{\theta}+\kappa\frac{\zeta\zeta_{x}\hat{\theta}_{x}}{\theta^{2}}$
$\displaystyle\quad+\rho\left(\frac{R}{\gamma-1}\hat{\theta}_{x}\psi+\frac{\kappa\tilde{\theta}_{xx}+\mu\tilde{u}_{x}^{2}-\hat{p}\hat{u}_{x}+\hat{h}}{\hat{\rho}}\right)\left[(\gamma-1)\Phi\left(\frac{\hat{\rho}}{\rho}\right)+\Phi\left(\frac{\theta}{\hat{\theta}}\right)\right].$

Next, we use the specific structure of the Maxwell equations to treat the
terms about electromagnetic fields on the right-hand side of (3.16).
Multiplying $\eqref{raodong-fuhebo}_{4}$ by $E$ and $\eqref{raodong-
fuhebo}_{5}$ by $b$ respectively, then summing them up gives

$\frac{1}{2}(\varepsilon E^{2}+b^{2})_{t}-(Eb)_{x}+(E+\psi b+\hat{u}b)E=0.$
(3.17)

Multiplying $\eqref{raodong-fuhebo}_{4}$ by $\hat{u}b$ and applying
$\eqref{raodong-fuhebo}_{5}$ deduces

$\displaystyle(\varepsilon E\hat{u}b)_{t}-\frac{1}{2}\left(\hat{u}(\varepsilon
E^{2}+b^{2})\right)_{x}+\frac{1}{2}\bar{u}_{x}(\varepsilon
E^{2}+b^{2})+(E+\psi b+\hat{u}b)\hat{u}b$
$\displaystyle=-\frac{1}{2}\tilde{u}_{x}\left(\varepsilon
E^{2}+b^{2}\right)+\varepsilon Eb\hat{u}_{t}.$ (3.18)

Then summing (3.17), (3.1) and (3.16) up gives

$\displaystyle\left(\rho\cdot\eta(x,t)+\frac{1}{2}\varepsilon
E^{2}+\frac{1}{2}b^{2}+\varepsilon
E\hat{u}b\right)_{t}+\left(H-\left(\frac{1}{2}\hat{u}\varepsilon
E^{2}+\frac{1}{2}\hat{u}b^{2}+Eb\right)\right)_{x}+\mu\frac{\hat{\theta}}{\theta}\psi_{x}^{2}$
$\displaystyle+\kappa\frac{\hat{\theta}}{\theta^{2}}\zeta_{x}^{2}+\frac{1}{2}\bar{u}_{x}(\varepsilon
E^{2}+b^{2})+\left(E+\psi
b+\hat{u}b\right)^{2}=Q-\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})+\varepsilon Eb\bar{u}_{t}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\left(E+\psi
b+\hat{u}b\right)^{2}\frac{\zeta}{\theta}.$ (3.19)

Set $\beta_{1}:=\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}$. The boundary
condition $\sqrt{\varepsilon}E(0,t)=b(0,t)$ implies the following boundary
estimate

$\displaystyle\quad\int_{0}^{t}\int_{\mathbb{R_{+}}}-\left(\frac{1}{2}\hat{u}\varepsilon
E^{2}+\frac{1}{2}\hat{u}b^{2}+Eb\right)_{x}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}\left(\frac{1}{2}u_{-}\varepsilon
E^{2}(0,\tau)+\frac{1}{2}u_{-}b^{2}(0,\tau)+E(0,\tau)b(0,\tau)\right)\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}\left(u_{-}\varepsilon
E^{2}(0,\tau)+\sqrt{\varepsilon}E^{2}(0,\tau)\right)\,\mathrm{d}\tau=\int_{0}^{t}\left(1-\left|u_{-}\right|\sqrt{\varepsilon}\right)\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau$
$\displaystyle\geq\int_{0}^{t}\left(1-\beta_{1}\sqrt{\varepsilon}\right)\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau\geq\frac{3}{4}\int_{0}^{t}\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau,$
(3.20)

where in the last inequality we have taken

$1-\beta_{1}\sqrt{\varepsilon}\geq\frac{3}{4},\quad\mathrm{i.e.}\quad\beta_{1}^{2}\varepsilon\leq\frac{1}{16}.$
(3.21)

Moreover, using the boundary condition (3.5), thanks to the good sign of
$u_{-}$, we can get

$\displaystyle\int_{0}^{t}\int_{\mathbb{R_{+}}}H_{x}\,\mathrm{d}x\mathrm{d}\tau=-\int_{0}^{t}H(0,\tau)\,\mathrm{d}\tau$
$\displaystyle=-\,u_{-}R\rho_{-}\theta_{-}\int_{0}^{t}\Phi\left(\frac{\hat{\rho}}{\rho}\right)(0,\tau)\,\mathrm{d}\tau$
$\displaystyle\geq c\int_{0}^{t}\phi^{2}(0,\tau)\,\mathrm{d}\tau.$ (3.22)

Integrating (3.1) with respect to $x$ and $t$, then plugging (3.1) and (3.1)
into the resulting equality, and employing (3.9), we obtain

$\displaystyle\quad\int_{\mathbb{R_{+}}}\left(\rho\cdot\eta(x,t)+\frac{1}{2}\varepsilon
E^{2}+\frac{1}{2}b^{2}\right)\,\mathrm{d}x+\frac{1}{2}\int_{0}^{t}\|\sqrt{\bar{u}_{x}}(\sqrt{\varepsilon}E,b)\|^{2}\,\mathrm{d}\tau$
$\displaystyle\quad+\frac{3}{4}\int_{0}^{t}\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau+c_{1}\int_{0}^{t}\left[\phi^{2}(0,\tau)+\left\|\left(\psi_{x},\zeta_{x},E+\psi
b+\hat{u}b\right)\right\|^{2}\right]\,\mathrm{d}\tau$ $\displaystyle\leq
C\|(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})\|^{2}+\int_{0}^{t}\int_{\mathbb{R_{+}}}Q\,\mathrm{d}x\mathrm{d}\tau+\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon
Eb\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\quad-\int_{0}^{t}\int_{\mathbb{R_{+}}}\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau+\int_{\mathbb{R_{+}}}\left|\varepsilon
E\hat{u}b\right|\,\mathrm{d}x.$ (3.23)

Step 2: First, we will estimate the terms
$-\int_{0}^{t}\int_{\mathbb{R_{+}}}\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau$ and
$\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon
Eb\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau$ on the right-hand side of (3.1). For
$-\int_{0}^{t}\int_{\mathbb{R_{+}}}\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau$, motivated by the work of [45], we
note that in Subcase 2 of Case (ii) $(\tilde{u},\tilde{\theta})(x)$ is
parallel to the vector $(-\mu u_{*},(\gamma-1)\kappa)$ at
$(u_{*},\theta_{*})$. Hence, for each $(\tilde{u},\tilde{\theta})(x)$ there
exists a constant $M_{0}\geq 1$ just depending on
$u_{-},\theta_{-},\rho_{*},u_{*},\theta_{*}$ such that if $x>M_{0}$, then
$(\tilde{u},\tilde{\theta})(x)\nearrow(u_{*},\theta_{*})$ as
$x\rightarrow+\infty$. This implies $\tilde{u}_{x}\geq 0$ and
$\tilde{\theta}_{x}\geq 0$ on $[M_{0},+\infty)$. Thanks to this important
observation, we divide the integral into two parts:

$\displaystyle-\int_{0}^{t}\int_{\mathbb{R_{+}}}\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau=-\left\\{\int_{0}^{t}\int_{0}^{M_{0}}+\int_{0}^{t}\int_{M_{0}}^{+\infty}\right\\}\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau=:I_{1}+I_{2}.$ (3.24)

For $I_{1}$, employing the Poincar$\acute{\rm e}$ type inequality (3.14) on
$E$ and $b$, then using the boundary condition
$\sqrt{\varepsilon}E(0,t)=b(0,t)$, we can derive

$\displaystyle I_{1}$ $\displaystyle\leq
C\int_{0}^{t}\int_{0}^{M_{0}}\frac{\delta^{2}}{(1+\delta x)^{2}}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau$ $\displaystyle\leq
C\int_{0}^{t}\left(\varepsilon
E^{2}(0,\tau)+b^{2}(0,\tau)\right)\left(\int_{0}^{M_{0}}\frac{\delta^{2}}{(1+\delta
x)^{2}}\,\mathrm{d}x\right)\,\mathrm{d}\tau$
$\displaystyle\quad+C\int_{0}^{t}\left(\varepsilon\|E_{x}\|^{2}+\|b_{x}\|^{2}\right)\left(\int_{0}^{M_{0}}\frac{\delta^{2}x}{(1+\delta
x)^{2}}\,\mathrm{d}x\right)\mathrm{d}\tau$ $\displaystyle\leq
C\delta\int_{0}^{t}\varepsilon
E^{2}(0,\tau)\,\mathrm{d}\tau+C\delta\int_{0}^{t}\left(\varepsilon\|E_{x}\|^{2}+\|b_{x}\|^{2}\right)\mathrm{d}\tau,$
(3.25)

where in the last inequality we have used the following simple inequality:

$0\leq\ln(1+x)\leq x,\quad x\in\mathbb{R}_{+}.$

Moreover, $\tilde{u}_{x}\geq 0\;(x>M_{0})$ implies

$I_{2}=-\int_{0}^{t}\int_{M_{0}}^{+\infty}\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau\leq 0.$ (3.26)

Combining (3.1) and (3.26), we can get

$\displaystyle-\int_{0}^{t}\int_{\mathbb{R_{+}}}\frac{1}{2}\tilde{u}_{x}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x\mathrm{d}\tau\leq C\delta\int_{0}^{t}\varepsilon
E^{2}(0,\tau)\,\mathrm{d}\tau+C\delta\int_{0}^{t}\|(\sqrt{\varepsilon}E_{x},b_{x})\|^{2}\,\mathrm{d}\tau.$
(3.27)

For $\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon
Eb\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau$, we have

$\displaystyle\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon
Eb\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau=\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon(E+ub-
ub)b\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle=-\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon
ub^{2}\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau+\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon(E+ub)b\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau=:J_{1}+J_{2}.$
(3.28)

Set $\beta_{2}:=\max\\{\theta_{-},\theta_{+}\\}$,
$\beta_{3}:=\beta_{1}+\sqrt{R\gamma\beta_{2}}$ and recall
$\beta_{1}=\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}$. Due to (i) of
Lemma 3.1, elementary calculations give rise to

$\displaystyle\left|\bar{u}_{t}\right|$
$\displaystyle=\left|-\bar{u}\bar{u}_{x}-\frac{\bar{p}_{x}}{\bar{\rho}}\right|=\left|-\left(\bar{u}+\sqrt{R\gamma\bar{\theta}}\right)\bar{u}_{x}\right|\leq\left(\frac{5}{4}\beta_{1}+\sqrt{R\gamma\beta_{2}}\right)\bar{u}_{x}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\leq\frac{5}{4}\left(\beta_{1}+\sqrt{R\gamma\beta_{2}}\right)\bar{u}_{x}=\frac{5}{4}\beta_{3}\bar{u}_{x}.$
(3.29)

Then using (i) of Lemma 3.1 and (3.1) gives that

$\left|J_{1}\right|\leq\frac{5}{2}\beta_{1}\beta_{3}\varepsilon\int_{0}^{t}\int_{\mathbb{R_{+}}}\bar{u}_{x}b^{2}\,\mathrm{d}x\mathrm{d}\tau\leq\frac{5}{128}\int_{0}^{t}\int_{\mathbb{R_{+}}}\bar{u}_{x}b^{2}\,\mathrm{d}x\mathrm{d}\tau,$
(3.30)

where in the last inequality we have chosen

$\frac{5}{2}\beta_{1}\beta_{3}\varepsilon\leq\frac{5}{128},\quad\mathrm{i.e.}\quad\beta_{1}\beta_{3}\varepsilon\leq\frac{1}{64}.$
(3.31)

By using the Cauchy inequality and the decay rate of smooth approximate
rarefaction wave, we can deduce

$\displaystyle\left|J_{2}\right|$
$\displaystyle\leq\int_{0}^{t}\int_{\mathbb{R_{+}}}\left|E+ub\right|\cdot\left|\frac{5}{4}\beta_{3}\varepsilon\bar{u}_{x}b\right|\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{2}c_{1}\int_{0}^{t}\int_{\mathbb{R_{+}}}\left(E+ub\right)^{2}\,\mathrm{d}x\mathrm{d}\tau+\frac{1}{2c_{1}}\int_{0}^{t}\int_{\mathbb{R_{+}}}\frac{25}{16}\beta_{3}^{2}\varepsilon^{2}\bar{u}_{x}^{2}b^{2}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{2}c_{1}\int_{0}^{t}\int_{\mathbb{R_{+}}}\left(E+ub\right)^{2}\,\mathrm{d}x\mathrm{d}\tau+\frac{25}{32c_{1}}\beta_{3}^{2}\varepsilon^{2}\int_{0}^{t}\int_{\mathbb{R_{+}}}\|\bar{u}_{x}\|_{L^{\infty}_{x}}\bar{u}_{x}b^{2}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{2}c_{1}\int_{0}^{t}\int_{\mathbb{R_{+}}}\left(E+ub\right)^{2}\,\mathrm{d}x\mathrm{d}\tau+\frac{25}{32c_{1}}\beta_{3}^{2}\varepsilon^{2}\alpha\int_{0}^{t}\int_{\mathbb{R_{+}}}\bar{u}_{x}b^{2}\,\mathrm{d}x\mathrm{d}\tau,$

where $c_{1}$ is a constant the same as that in (3.1). Then choosing $\alpha$
suitably small leads to

$\left|J_{2}\right|\leq\frac{1}{2}c_{1}\int_{0}^{t}\int_{\mathbb{R_{+}}}\left(E+ub\right)^{2}\,\mathrm{d}x\mathrm{d}\tau+\frac{3}{128}\int_{0}^{t}\int_{\mathbb{R_{+}}}\bar{u}_{x}b^{2}\,\mathrm{d}x\mathrm{d}\tau.$
(3.32)

Putting (3.30) and (3.32) into (3.1), we have

$\displaystyle\left|\int_{0}^{t}\int_{\mathbb{R_{+}}}\varepsilon
Eb\bar{u}_{t}\,\mathrm{d}x\mathrm{d}\tau\right|\leq\frac{1}{16}\int_{0}^{t}\int_{\mathbb{R_{+}}}\bar{u}_{x}b^{2}\,\mathrm{d}x\mathrm{d}\tau+\frac{1}{2}c_{1}\int_{0}^{t}\int_{\mathbb{R_{+}}}\left(E+ub\right)^{2}\,\mathrm{d}x\mathrm{d}\tau.$
(3.33)

Next, recalling $\left|\hat{u}(x,t)\right|<\frac{3}{2}\beta_{1}$ and using the
Cauchy inequality, the space integration term
$\int_{\mathbb{R_{+}}}\left|\varepsilon E\hat{u}b\right|\,\mathrm{d}x$ in
(3.1) can be estimated as follows:

$\displaystyle\int_{\mathbb{R_{+}}}\left|\varepsilon
E\hat{u}b\right|\,\mathrm{d}x\leq\int_{\mathbb{R_{+}}}\varepsilon\left|E\cdot\frac{3}{2}\beta_{1}b\right|\,\mathrm{d}x\leq\frac{1}{4}\int_{\mathbb{R_{+}}}\varepsilon
E^{2}\,\mathrm{d}x+\frac{9}{4}\beta_{1}^{2}\varepsilon\int_{\mathbb{R_{+}}}b^{2}\,\mathrm{d}x\leq\frac{1}{4}\int_{\mathbb{R_{+}}}(\varepsilon
E^{2}+b^{2})\,\mathrm{d}x,$ (3.34)

where in the last inequality we have chosen

$\beta_{1}^{2}\varepsilon\leq\frac{1}{9}.$ (3.35)

Combining (3.27), (3.33), (3.34) and (3.1), then choosing $\delta$ suitably
small, we can obtain

$\displaystyle\quad\int_{\mathbb{R_{+}}}\left(\rho\cdot\eta(x,t)+\frac{1}{4}\varepsilon
E^{2}+\frac{1}{4}b^{2}\right)\,\mathrm{d}x+\frac{7}{16}\int_{0}^{t}\|\sqrt{\bar{u}_{x}}(\sqrt{\varepsilon}E,b)\|^{2}\,\mathrm{d}\tau$
$\displaystyle\quad+\frac{1}{2}\int_{0}^{t}\sqrt{\varepsilon}E^{2}(0,\tau)\,\mathrm{d}\tau+c_{2}\int_{0}^{t}\left[\phi^{2}(0,\tau)+\left\|\left(\psi_{x},\zeta_{x},E+\psi
b+\hat{u}b\right)\right\|^{2}\right]\,\mathrm{d}\tau$ $\displaystyle\leq
C\|(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})\|^{2}+C\delta\int_{0}^{t}\|(\sqrt{\varepsilon}E_{x},b_{x})\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\int_{\mathbb{R_{+}}}Q\,\mathrm{d}x\mathrm{d}\tau.$
(3.36)

Finally, for the terms
$\int_{0}^{t}\int_{\mathbb{R_{+}}}Q\,\mathrm{d}x\mathrm{d}\tau$, by performing
some similar computations as in [45] and choosing $\delta$ and $\alpha$
suitably small, we obtain

$\displaystyle\quad\int_{0}^{t}\int_{\mathbb{R_{+}}}Q\,\mathrm{d}x\mathrm{d}\tau+c_{3}\int_{0}^{t}\int_{\mathbb{R_{+}}}\bar{u}_{x}\left|\left(\phi,\psi,\zeta\right)\right|^{2}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{4}c_{2}\int_{0}^{t}\|(\psi_{x},\zeta_{x})\|^{2}\,\mathrm{d}\tau+C\delta\int_{0}^{t}\phi^{2}(0,\tau)\,\mathrm{d}\tau+C(\delta+\alpha^{\frac{1}{10}})\left(1+\int_{0}^{t}\|(\phi_{x},\psi_{x},\zeta_{x})\|^{2}\,\mathrm{d}\tau\right),$
(3.37)

where $c_{2}$ is a constant the same as that in (3.1). Putting (3.1) into
(3.1), then taking $\delta$ and $\alpha$ small enough, we can arrive at (3.3).

This completes the proof of Lemma 3.3.

∎

###  3.2 High-order energy estimates

In the following lemma, we will control the energy
$\|(\sqrt{\varepsilon}E_{x},b_{x})\|^{2}$.

######  Lemma 3.4.

Suppose that the conditions in Proposition 3.1 hold. Then for all $0<t<T$, we
have the following energy estimate

$\displaystyle\quad\|(\sqrt{\varepsilon}E_{x},b_{x})\|^{2}+\sqrt{\varepsilon}E^{2}(0,t)+\int_{0}^{t}\|(E_{x},b_{x})\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\varepsilon^{\frac{3}{2}}E_{\tau}^{2}(0,\tau)\,\mathrm{d}\tau$
$\displaystyle\leq
C\left(\|(\phi_{0},\psi_{0},\zeta_{0})\|^{2}+\|(E_{0},b_{0})\|^{2}_{H^{1}}\right)+C(\delta+\alpha^{\frac{1}{10}})\left(1+\int_{0}^{t}\|\phi_{x}\|^{2}\,\mathrm{d}\tau\right).$
(3.38)

###### Proof.

Firstly, taking the derivative of $\eqref{raodong-fuhebo}_{4}$ with respect to
$x$ and multiplying it by $E_{x}$, then integrating the resulting equality
with respect to $x$, we obtain

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R_{+}}}\frac{1}{2}\varepsilon
E_{x}^{2}\,\mathrm{d}x-\int_{\mathbb{R_{+}}}b_{xx}E_{x}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}E^{2}_{x}\,\mathrm{d}x=-\int_{\mathbb{R_{+}}}(\psi
b)_{x}E_{x}\,\mathrm{d}x-\int_{\mathbb{R_{+}}}(\hat{u}b)_{x}E_{x}\,\mathrm{d}x.$
(3.39)

Secondly, taking the derivative of $\eqref{raodong-fuhebo}_{5}$ with respect
to $x$ and multiplying $b_{x}$, then integrating the resulting equality with
respect to $x$, we get

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R_{+}}}\frac{1}{2}b^{2}_{x}\,\mathrm{d}x-\int_{\mathbb{R_{+}}}E_{xx}b_{x}\,\mathrm{d}x=0.$
(3.40)

Combining $\eqref{Exbx1}$ and $\eqref{Exbx2}$, we obtain

$\displaystyle\quad\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R_{+}}}\frac{1}{2}\left(\varepsilon
E_{x}^{2}+b_{x}^{2}\right)\,\mathrm{d}x-\int_{\mathbb{R_{+}}}\left(E_{x}b_{x}\right)_{x}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}E_{x}^{2}\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R_{+}}}(\psi
b)_{x}E_{x}\,\mathrm{d}x-\int_{\mathbb{R_{+}}}(\hat{u}b)_{x}E_{x}\,\mathrm{d}x.$
(3.41)

Now we estimate the boundary term in (3.2). The boundary condition
$\sqrt{\varepsilon}E(0,t)=b(0,t)$ implies

$\sqrt{\varepsilon}E_{t}(0,t)=b_{t}(0,t).$ (3.42)

Moreover, taking $x=0$ for $\eqref{raodong-fuhebo}_{5}$ leads to

$b_{t}(0,t)=E_{x}(0,t).$ (3.43)

Combining (3.42) and (3.43), we obtain

$E_{x}(0,t)=b_{t}(0,t)=\sqrt{\varepsilon}E_{t}(0,t).$ (3.44)

On the other hand, taking $x=0$ for $\eqref{raodong-fuhebo}_{4}$, together
with the boundary condition $\sqrt{\varepsilon}E(0,t)=b(0,t)$, we have

$b_{x}(0,t)=\varepsilon E_{t}(0,t)+E(0,t)+u_{-}b(0,t)=\varepsilon
E_{t}(0,t)+\left(1+u_{-}\sqrt{\varepsilon}\right)E(0,t).$ (3.45)

Using (3.44) and (3.45), we can derive

$\displaystyle\quad-\int_{0}^{t}\int_{\mathbb{R_{+}}}\left(E_{x}b_{x}\right)_{x}\,\mathrm{d}x\mathrm{d}\tau=\int_{0}^{t}\left(E_{x}b_{x}\right)(0,\tau)\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}\sqrt{\varepsilon}E_{\tau}(0,\tau)\cdot\left[\varepsilon
E_{\tau}(0,\tau)+\left(1+u_{-}\sqrt{\varepsilon}\right)E(0,\tau)\right]\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}\varepsilon^{\frac{3}{2}}E^{2}_{\tau}(0,\tau)\,\mathrm{d}\tau+\sqrt{\varepsilon}\left(1+u_{-}\sqrt{\varepsilon}\right)\int_{0}^{t}(E_{\tau}E)(0,\tau)\,\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}\varepsilon^{\frac{3}{2}}E^{2}_{\tau}(0,\tau)\,\mathrm{d}\tau+\frac{1}{2}\sqrt{\varepsilon}\left(1+u_{-}\sqrt{\varepsilon}\right)E^{2}(0,t)-\frac{1}{2}\sqrt{\varepsilon}\left(1+u_{-}\sqrt{\varepsilon}\right)E^{2}(0,0).$
(3.46)

Due to $u_{-}<0$, we have

$1+u_{-}\sqrt{\varepsilon}=1-\left|u_{-}\right|\sqrt{\varepsilon}\geq
1-\beta_{1}\sqrt{\varepsilon}.$

By choosing

$1-\beta_{1}\sqrt{\varepsilon}\geq\frac{1}{2},\quad\mathrm{i.e.}\quad\beta_{1}^{2}\varepsilon\leq\frac{1}{4},$
(3.47)

we can get

$\frac{1}{2}\sqrt{\varepsilon}\left(1+u_{-}\sqrt{\varepsilon}\right)E^{2}(0,t)\geq\frac{1}{4}\sqrt{\varepsilon}E^{2}(0,t).$
(3.48)

Furthermore, using the Sobolev inequality implies

$\left|-\frac{1}{2}\sqrt{\varepsilon}\left(1+u_{-}\sqrt{\varepsilon}\right)E^{2}(0,0)\right|\leq\left|\frac{1}{2}\sqrt{\varepsilon}\left(1+u_{-}\sqrt{\varepsilon}\right)\right|\cdot\|E_{0}\|^{2}_{L^{\infty}}\leq
C\|E_{0}\|^{2}_{H^{1}}.$

It follows that

$-\frac{1}{2}\sqrt{\varepsilon}\left(1+u_{-}\sqrt{\varepsilon}\right)E^{2}(0,0)\geq-\,C\|E_{0}\|^{2}_{H^{1}}.$
(3.49)

By plugging (3.48) and (3.49) into (3.2), we can deduce the following boundary
estimate

$-\int_{0}^{t}\int_{\mathbb{R_{+}}}\left(E_{x}b_{x}\right)_{x}\,\mathrm{d}x\mathrm{d}\tau\geq\int_{0}^{t}\varepsilon^{\frac{3}{2}}E^{2}_{\tau}(0,\tau)\,\mathrm{d}\tau+\frac{1}{4}\sqrt{\varepsilon}E^{2}(0,t)-C\|E_{0}\|^{2}_{H^{1}}.$
(3.50)

Next, by using (3.7), $\left|\hat{u}(x,t)\right|<\frac{3}{2}\beta_{1}$, and
the Cauchy inequality, we have

$\displaystyle\quad-\int_{\mathbb{R_{+}}}(\psi
b)_{x}E_{x}\,\mathrm{d}x-\int_{\mathbb{R_{+}}}(\hat{u}b)_{x}E_{x}\,\mathrm{d}x$
$\displaystyle\leq\int_{\mathbb{R_{+}}}\left(\left|\psi_{x}bE_{x}\right|+\left|\psi
b_{x}E_{x}\right|+\left|\hat{u}b_{x}E_{x}\right|+\left|\hat{u}_{x}bE_{x}\right|\right)\,\mathrm{d}x$
$\displaystyle\leq
C(\|b\|_{{L^{\infty}}}+\|\psi\|_{{L^{\infty}}})\cdot(\|\psi_{x}\|^{2}+\|E_{x}\|^{2}+\|b_{x}\|^{2})$
$\displaystyle\quad+\frac{1}{4}\int_{\mathbb{R_{+}}}E_{x}^{2}\,\mathrm{d}x+\frac{9}{4}\beta_{1}^{2}\int_{\mathbb{R_{+}}}b_{x}^{2}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}\left|\hat{u}_{x}bE_{x}\right|\,\mathrm{d}x$
$\displaystyle\leq
C\varepsilon_{0}\|(\psi_{x},E_{x},b_{x})\|^{2}+\frac{1}{4}\|E_{x}\|^{2}+\frac{9}{4}\beta_{1}^{2}\|b_{x}\|^{2}+\int_{\mathbb{R_{+}}}\left|\hat{u}_{x}bE_{x}\right|\,\mathrm{d}x.$
(3.51)

Moreover, using the Cauchy inequality, (3.12), (3.13) and
$\sqrt{\varepsilon}E(0,t)=b(0,t)$ leads to

$\displaystyle\int_{\mathbb{R_{+}}}\left|\hat{u}_{x}bE_{x}\right|\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{8}\int_{\mathbb{R_{+}}}E^{2}_{x}\,\mathrm{d}x+C\int_{\mathbb{R_{+}}}\hat{u}^{2}_{x}b^{2}\,\mathrm{d}x\leq\frac{1}{8}\|E_{x}\|^{2}+C\int_{\mathbb{R_{+}}}\tilde{u}^{2}_{x}b^{2}\,\mathrm{d}x+C\int_{\mathbb{R_{+}}}\bar{u}^{2}_{x}b^{2}\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{8}\|E_{x}\|^{2}+C\delta^{3}b^{2}(0,t)+C\delta^{2}\|b_{x}\|^{2}+C\varepsilon_{0}\alpha^{\frac{1}{3}}\left(\|b_{x}\|^{2}+(1+t)^{-\frac{4}{3}}\right)$
$\displaystyle\leq\frac{1}{8}\|E_{x}\|^{2}+C\delta^{3}\varepsilon
E^{2}(0,t)+C\left(\delta^{2}+\varepsilon_{0}\alpha^{\frac{1}{3}}\right)\|b_{x}\|^{2}+C\varepsilon_{0}\alpha^{\frac{1}{3}}(1+t)^{-\frac{4}{3}}.$
(3.52)

Thus putting (3.2)-(3.2) into (3.2), and integrating the resulting inequality
with respect to $t$, then employing (3.50) and choosing $\varepsilon_{0}$,
$\alpha$ and $\delta$ suitably small gives

$\displaystyle\qquad\|(\sqrt{\varepsilon}E_{x},b_{x})\|^{2}+\frac{1}{4}\sqrt{\varepsilon}E^{2}(0,t)+\int_{0}^{t}\|E_{x}\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\varepsilon^{\frac{3}{2}}E_{\tau}^{2}(0,\tau)\,\mathrm{d}\tau$
$\displaystyle\leq
C\left(\|E_{0}\|^{2}_{H^{1}}+\|b_{0x}\|^{2}\right)+C\alpha^{\frac{1}{3}}+C\varepsilon_{0}\int_{0}^{t}\|\psi_{x}\|^{2}\,\mathrm{d}\tau+C\delta^{3}\int_{0}^{t}\varepsilon
E^{2}(0,\tau)\,\mathrm{d}\tau+5\beta_{1}^{2}\int_{0}^{t}\|b_{x}\|^{2}\,\mathrm{d}\tau.$
(3.53)

Now we turn to estimate the term
$5\beta_{1}^{2}\int_{0}^{t}\|b_{x}\|^{2}\,\mathrm{d}\tau$ in (3.2).
Multiplying $\eqref{raodong-fuhebo}_{4}$ by $-b_{x}$ and integrating the
resulting equation with respect to $x$, together with $\eqref{raodong-
fuhebo}_{5}$, we can deduce

$\displaystyle\quad-\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R_{+}}}\varepsilon
Eb_{x}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}b_{x}^{2}\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R_{+}}}\varepsilon
Eb_{tx}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}(E+\psi
b+\hat{u}b)b_{x}\,\mathrm{d}x=-\int_{\mathbb{R_{+}}}\varepsilon
EE_{xx}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}(E+\psi
b+\hat{u}b)b_{x}\,\mathrm{d}x$
$\displaystyle=-\int_{\mathbb{R_{+}}}\varepsilon(EE_{x})_{x}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}\varepsilon
E^{2}_{x}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}(E+\psi
b+\hat{u}b)b_{x}\,\mathrm{d}x$
$\displaystyle\leq-\int_{\mathbb{R_{+}}}\varepsilon(EE_{x})_{x}\,\mathrm{d}x+\int_{\mathbb{R_{+}}}\varepsilon
E^{2}_{x}\,\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R_{+}}}(E+\psi
b+\hat{u}b)^{2}\,\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R_{+}}}b^{2}_{x}\,\mathrm{d}x,$

i.e.

$-\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R_{+}}}2\varepsilon
Eb_{x}\,\mathrm{d}x+\|b_{x}\|^{2}\leq-\int_{\mathbb{R_{+}}}2\varepsilon(EE_{x})_{x}\,\mathrm{d}x+2\varepsilon\|E_{x}\|^{2}+\|E+\psi
b+\hat{u}b\|^{2}.$ (3.54)

For the first term on the right-hand side of (3.54), using (3.44) gives the
following boundary estimate

$\displaystyle-\int_{0}^{t}\int_{\mathbb{R_{+}}}2\varepsilon\left(EE_{x}\right)_{x}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle=\int_{0}^{t}2\varepsilon(EE_{x})(0,\tau)\,\mathrm{d}\tau=\int_{0}^{t}2\varepsilon^{\frac{3}{2}}(EE_{\tau})(0,\tau)\,\mathrm{d}\tau$
$\displaystyle=\varepsilon^{\frac{3}{2}}E^{2}(0,t)-\varepsilon^{\frac{3}{2}}E^{2}(0,0)\leq\varepsilon^{\frac{3}{2}}E^{2}(0,t).$
(3.55)

Multiplying (3.54) by $8\beta_{1}^{2}$ and integrating the resulting
inequality with respect to $t$, then adding it to (3.2) and employing (3.2)
leads to

$\displaystyle\quad\|(\sqrt{\varepsilon}E_{x},b_{x})\|^{2}+\left(\frac{1}{4}-8\beta_{1}^{2}\varepsilon\right)\sqrt{\varepsilon}E^{2}(0,t)+3\beta_{1}^{2}\int_{0}^{t}\|b_{x}\|^{2}\,\mathrm{d}\tau$
$\displaystyle\quad+(1-16\beta_{1}^{2}\varepsilon)\int_{0}^{t}\|E_{x}\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\varepsilon^{\frac{3}{2}}E^{2}_{\tau}(0,\tau)\,\mathrm{d}\tau$
$\displaystyle\leq
C\left(\|E_{0}\|^{2}_{H^{1}}+\|b_{0x}\|^{2}\right)+C\alpha^{\frac{1}{3}}+8\beta_{1}^{2}\int_{0}^{t}\|E+\psi
b+\hat{u}b\|^{2}\,\mathrm{d}\tau$
$\displaystyle\quad+C\varepsilon_{0}\int_{0}^{t}\|\psi_{x}\|^{2}\,\mathrm{d}\tau+C\delta^{3}\int_{0}^{t}\varepsilon
E^{2}(0,\tau)\,\mathrm{d}\tau+16\beta_{1}^{2}\int_{\mathbb{R_{+}}}\varepsilon\left|Eb_{x}\right|\,\mathrm{d}x.$
(3.56)

In order to absorb the bad term
$16\beta_{1}^{2}\int_{\mathbb{R_{+}}}\varepsilon\left|Eb_{x}\right|\,\mathrm{d}x$
and hold the time-space integrable good term
$\int_{0}^{t}\|E_{x}\|^{2}\,\mathrm{d}\tau$, we choose
$\beta_{1}^{2}\varepsilon$ satisfying

$1-16\beta_{1}^{2}\varepsilon\geq\frac{1}{2},\quad\mathrm{i.e.}\quad\beta_{1}^{2}\varepsilon\leq\frac{1}{32}.$
(3.57)

Hence,

$16\beta_{1}^{2}\int_{\mathbb{R_{+}}}\varepsilon\left|Eb_{x}\right|\,\mathrm{d}x\leq
16\beta_{1}^{2}\varepsilon\int_{\mathbb{R_{+}}}\left(\frac{1}{2}b^{2}_{x}+\frac{1}{2}E^{2}\right)\,\mathrm{d}x\leq\frac{1}{4}\|b_{x}\|^{2}+8\beta_{1}^{2}\|\sqrt{\varepsilon}E\|^{2}.$
(3.58)

What’s more, taking

$\frac{1}{4}-8\beta_{1}^{2}\varepsilon\geq\frac{1}{8},\quad\mathrm{i.e.}\quad\beta_{1}^{2}\varepsilon\leq\frac{1}{64},$
(3.59)

so that we can transform the term
$\left(1/4-8\beta_{1}^{2}\varepsilon\right)\sqrt{\varepsilon}E^{2}(0,t)$ in
(3.2) into a good term. Putting (3.58) into (3.2) and using (3.3), then
choosing $\varepsilon_{0},\,\delta$ and $\alpha$ suitably small, we can
conclude (3.4).

Recalling $\beta_{1}=\max\\{\left|u_{-}\right|,\left|u_{+}\right|\\}$,
$\beta_{2}=\max\\{\theta_{-},\theta_{+}\\}$ and
$\beta_{3}=\beta_{1}+\sqrt{R\gamma\beta_{2}}$, from the discussion about
$\varepsilon$ in (3.21), (3.31), (3.35), (3.47), (3.57) and (3.59), we can
determine the constant $\bar{C}$ in the following manner:

$\displaystyle\varepsilon$
$\displaystyle<\min\left\\{\frac{1}{16\beta_{1}^{2}},\;\;\frac{1}{64\beta_{1}\beta_{3}},\;\;\frac{1}{9\beta_{1}^{2}},\;\;\frac{1}{4\beta_{1}^{2}},\;\;\frac{1}{32\beta_{1}^{2}},\;\;\frac{1}{64\beta_{1}^{2}}\right\\}$
$\displaystyle=\frac{1}{64\beta_{1}}\min\left\\{\frac{1}{\beta_{3}},\;\;\frac{1}{\beta_{1}}\right\\}=\frac{1}{64\beta_{1}\beta_{3}}=\frac{1}{64\beta_{1}\left(\beta_{1}+\sqrt{\gamma
R\beta_{2}}\right)}=:\bar{C},$ (3.60)

such that $\varepsilon<\bar{C}$. This completes the proof of Lemma 3.4.

∎

In the following two lemmas, we show the estimates for the first-order
derivative $(\phi_{x},\psi_{x},\zeta_{x})$.

######  Lemma 3.5.

Suppose that the conditions in Proposition 3.1 hold. Then for all $0<t<T$, we
have the following energy estimate

$\|\phi_{x}\|^{2}+\int_{0}^{t}\left(\phi_{x}^{2}(0,\tau)+\|\phi_{x}\|^{2}\right)\,\mathrm{d}\tau\leq
C\left(\|(\psi_{0},\zeta_{0})\|^{2}+\|(\phi_{0},E_{0},b_{0})\|^{2}_{H^{1}}\right)+C(\delta+\alpha^{\frac{1}{10}}).$
(3.61)

###### Proof.

Motivated by the work of [20] and [45], firstly, differentiating
$\eqref{raodong-fuhebo}_{1}$ with respect to $x$ and multiplying the resulting
equation by $\frac{\phi_{x}}{\rho^{3}}$ yields

$\left(\frac{\phi_{x}^{2}}{2\rho^{3}}\right)_{t}+\left(\frac{u\phi_{x}^{2}}{2\rho^{3}}\right)_{x}-\hat{u}_{x}\frac{\phi_{x}^{2}}{\rho^{3}}+\hat{\rho}_{x}\frac{\phi_{x}\psi_{x}}{\rho^{3}}+\frac{\phi_{x}\psi_{xx}}{\rho^{2}}=f_{x}\frac{\phi_{x}}{\rho^{3}}.$
(3.62)

To remove the high-order spatial derivative term
$\frac{\phi_{x}\psi_{xx}}{\rho^{2}}$ of (3.62), we need to employ the equation
$\eqref{raodong-fuhebo}_{2}$. Multiplying $\eqref{raodong-fuhebo}_{2}$ by
$\frac{\phi_{x}}{\rho^{2}}$, after some elementary computations, we can get

$\displaystyle\left(\frac{\phi_{x}\psi}{\rho}\right)_{t}-\left(\frac{\phi_{t}\psi}{\rho}+\frac{\hat{\rho}_{x}\psi^{2}}{\rho}\right)_{x}+\frac{\hat{\rho}_{x}\hat{u}_{x}\phi\psi}{\rho^{2}}+\frac{\hat{\rho}_{xx}\psi^{2}}{\rho}+\frac{\left(\hat{\rho}_{x}\hat{u}-\bar{\rho}\bar{u}_{x}-\bar{\rho}_{x}\bar{u}\right)\psi\phi_{x}}{\rho^{2}}-\psi_{x}^{2}$
$\displaystyle+\frac{\left(2\hat{\rho}_{x}\psi-\hat{u}_{x}\phi\right)\psi_{x}}{\rho}+\frac{(p-\hat{p})_{x}\phi_{x}}{\rho^{2}}-\frac{\hat{f}\psi_{x}}{\rho}+\frac{\hat{\rho}_{x}\hat{f}\psi}{\rho^{2}}=\mu\frac{\phi_{x}\psi_{xx}}{\rho^{2}}+g\frac{\phi_{x}}{\rho^{2}}-(E+\psi
b+\hat{u}b)b\frac{\phi_{x}}{\rho^{2}}.$ (3.63)

$\mu\times\eqref{guji-phix1}+\eqref{guji-varphix2}$ gives

$\left(\frac{\mu\phi_{x}^{2}}{2\rho^{3}}+\frac{\phi_{x}\psi}{\rho}\right)_{t}+\left(\frac{\mu
u\phi_{x}^{2}}{2\rho^{3}}-\frac{\phi_{t}\psi}{\rho}-\frac{\hat{\rho}_{x}\psi^{2}}{\rho}\right)_{x}+\frac{R\theta}{\rho^{2}}\phi_{x}^{2}=Q_{4}-(E+\psi
b+\hat{u}b)b\frac{\phi_{x}}{\rho^{2}},$ (3.64)

where

$\displaystyle Q_{4}$
$\displaystyle=-\frac{2\mu\hat{\rho}_{x}\phi_{x}\psi_{x}}{\rho^{3}}-\frac{\mu\left(\hat{u}_{xx}\phi+\hat{\rho}_{xx}\psi+\hat{f}_{x}\right)\phi_{x}}{\rho^{3}}-\frac{\hat{\rho}_{x}\hat{u}_{x}\psi\phi}{\rho^{2}}-\frac{\hat{\rho}_{xx}\psi^{2}}{\rho}$
$\displaystyle\quad-\frac{\left[\tilde{\rho}_{x}\hat{u}-\bar{\rho}\bar{u}_{x}+\bar{\rho}_{x}\left(\tilde{u}-u_{*}\right)\right]\psi\phi_{x}}{\rho^{2}}-2\frac{\hat{\rho}_{x}\psi\psi_{x}}{\rho}+\psi_{x}^{2}$
$\displaystyle\quad+\frac{\hat{u}_{x}\phi\psi_{x}}{\rho}-\frac{\left(R\hat{\rho}_{x}\zeta+R\rho\zeta_{x}+R\hat{\theta}_{x}\phi\right)\phi_{x}}{\rho^{2}}+\frac{\hat{f}\psi_{x}}{\rho}-\frac{\hat{\rho}_{x}\hat{f}\psi}{\rho^{2}}$
$\displaystyle\quad-\left(\rho\hat{\rho}\hat{u}_{x}\psi-\hat{p}_{x}\phi+\mu\tilde{u}_{xx}\phi-\mu\hat{\rho}\bar{u}_{xx}+\rho\hat{g}\right)\frac{\phi_{x}}{\hat{\rho}\rho^{2}}$
$\displaystyle\leq\frac{R\theta}{2\rho^{2}}\phi_{x}^{2}+C\left(\psi_{x}^{2}+\zeta_{x}^{2}\right)+C\left(\hat{f}^{2}+\hat{f}_{x}^{2}+\hat{g}^{2}+\bar{u}_{xx}^{2}\right)$
$\displaystyle\quad+C\left[\left(\tilde{u}_{x}^{2}+\left|\tilde{u}_{xx}\right|\right)+\left(\bar{u}_{x}^{2}+\left|\bar{u}_{xx}\right|\right)\right]\left(\phi^{2}+\psi^{2}+\zeta^{2}\right).$
(3.65)

Integrating (3.64) with respect to $x$ and $t$, due to the good sign of
$u_{-}$ again, we obtain

$\displaystyle\|\phi_{x}\|^{2}+\int_{0}^{t}\left(\phi_{x}^{2}(0,\tau)+\|\phi_{x}\|^{2}\right)\,\mathrm{d}\tau\leq
C\|(\psi_{0},\phi_{0x})\|^{2}+\sum_{i=1}^{3}K_{i}$
$\displaystyle\qquad\quad+C\left(\|\psi\|^{2}+\int_{0}^{t}\|(\psi_{x},\zeta_{x},E+\psi
b+\hat{u}b)\|^{2}\,\mathrm{d}\tau\right),$ (3.66)

where

$\displaystyle
K_{1}:=C\int_{0}^{t}\int_{\mathbb{R}_{+}}\left(\hat{f}^{2}+\hat{f}_{x}^{2}+\hat{g}^{2}+\bar{u}_{xx}^{2}\right)\,\mathrm{d}x\mathrm{d}\tau,$
$\displaystyle
K_{2}:=C\int_{0}^{t}\int_{\mathbb{R}_{+}}\left(\tilde{u}_{x}^{2}+\left|\tilde{u}_{xx}\right|\right)\left(\phi^{2}+\psi^{2}+\zeta^{2}\right)\,\mathrm{d}x\mathrm{d}\tau,$
$\displaystyle
K_{3}:=C\int_{0}^{t}\int_{\mathbb{R}_{+}}\left(\bar{u}_{x}^{2}+\left|\bar{u}_{xx}\right|\right)\left(\phi^{2}+\psi^{2}+\zeta^{2}\right)\,\mathrm{d}x\mathrm{d}\tau.$

Recalling $\bar{u}(0,t)=u_{*}$ and using the following elementary inequality
on $\bar{u}-u_{*}$:

$\left|z(x,t)\right|=\left|z(0,t)+\int_{0}^{x}z_{y}(y,t)\,\mathrm{d}y\right|\leq\left|z(0,t)\right|+x\|z_{x}\|_{L^{\infty}_{x}},$
(3.67)

we can derive

$\displaystyle K_{1}$ $\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left[(\bar{u}-u_{*})^{2}\tilde{u}_{x}^{2}+(\tilde{u}-u_{*})^{2}\bar{u}_{x}^{2}+\bar{u}_{xx}^{2}\right]\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left[x^{2}\|\bar{u}_{x}\|^{2}_{L^{\infty}}\tilde{u}_{x}^{2}+(\tilde{u}-u_{*})^{2}\bar{u}_{x}^{2}+\bar{u}_{xx}^{2}\right]\,\mathrm{d}x\mathrm{d}\tau\leq
C(\delta+\alpha).$ (3.68)

Moreover, it is easy to deduce

$\displaystyle K_{2}\leq
C\delta\int_{0}^{t}\left(\phi^{2}(0,\tau)+\|(\phi_{x},\psi_{x},\zeta_{x})\|^{2}\right)\,\mathrm{d}\tau,$
(3.69) $\displaystyle K_{3}\leq
C\alpha^{\frac{1}{3}}\left(1+\int_{0}^{t}\|(\phi_{x},\psi_{x},\zeta_{x})\|^{2}\,\mathrm{d}\tau\right).$
(3.70)

Plugging (3.2)-(3.70) into (3.2) then employing (3.3) and (3.4), together with
taking $\delta$ and $\alpha$ small enough, we can conclude (3.61). This
completes the proof of Lemma 3.5.

∎

######  Lemma 3.6.

Suppose that the conditions in Proposition 3.1 hold. Then for all $0<t<T$, we
have the following energy estimate

$\|(\psi_{x},\zeta_{x})\|^{2}+\int_{0}^{t}\|(\psi_{xx},\zeta_{xx})\|^{2}\,\mathrm{d}\tau\leq
C\left(\|(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})\|^{2}_{H^{1}}+\delta+\alpha^{\frac{1}{10}}\right).$
(3.71)

###### Proof.

Motivated by the treatment can be done for the Navier-Stokes equations, we
multiply $\eqref{raodong-fuhebo}_{2}$ by $\frac{\psi_{xx}}{\rho}$ to obtain

$\displaystyle\left(\frac{1}{2}\psi_{x}^{2}\right)_{t}-\left(\psi_{t}\psi_{x}+\frac{1}{2}u\psi_{x}^{2}\right)_{x}+\mu\frac{\psi_{xx}^{2}}{\rho}$
$\displaystyle=-\frac{1}{2}u_{x}\psi_{x}^{2}+(p-\hat{p})_{x}\frac{\psi_{xx}}{\rho}-g\frac{\psi_{xx}}{\rho}$
$\displaystyle\quad+(E+\psi b+\hat{u}b)b\frac{\psi_{xx}}{\rho}.$

Integrating the above equality over $\mathbb{R}_{+}\times[0,t]$ gives

$\displaystyle\|\psi_{x}\|^{2}+\int_{0}^{t}\|\psi_{xx}\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\|\sqrt{\bar{u}_{x}}\psi_{x}\|^{2}\,\mathrm{d}\tau\leq
C\|\psi_{0x}\|^{2}+C\int_{0}^{t}\psi_{x}^{2}(0,\tau)\,\mathrm{d}\tau+C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left|\psi_{x}\right|^{3}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\qquad\qquad\qquad\qquad+C\int_{0}^{t}\int_{\mathbb{R_{+}}}|\tilde{u}_{x}|\psi_{x}^{2}\,\mathrm{d}x\mathrm{d}\tau+C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left[(p-\hat{p})_{x}^{2}+(E+\psi
b+\hat{u}b)^{2}+g^{2}\right]\,\mathrm{d}x\mathrm{d}\tau.$ (3.72)

Utilizing the Sobolev inequality yields

$\displaystyle C\int_{0}^{t}\psi_{x}^{2}(0,\tau)\,\mathrm{d}\tau\leq
C\int_{0}^{t}\|\psi_{x}\|^{2}_{L^{\infty}_{x}}\,\mathrm{d}\tau\leq
C\int_{0}^{t}\|\psi_{x}\|\|\psi_{xx}\|\,\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{4}\int_{0}^{t}\|\psi_{xx}\|^{2}\,\mathrm{d}\tau+C\int_{0}^{t}\|\psi_{x}\|^{2}\,\mathrm{d}\tau,$
(3.73)

and

$\displaystyle\quad
C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left|\psi_{x}\right|^{3}\,\mathrm{d}x\mathrm{d}\tau\leq
C\int_{0}^{t}\|\psi_{x}\|_{L^{\infty}_{x}}\|\psi_{x}\|^{2}\,\mathrm{d}\tau\leq
C\int_{0}^{t}\|\psi_{xx}\|^{\frac{1}{2}}\|\psi_{x}\|^{\frac{5}{2}}\,\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{4}\int_{0}^{t}\|\psi_{xx}\|^{2}\,\mathrm{d}\tau+C\sup_{0\leq\tau\leq
t}\|\psi_{x}\|^{\frac{4}{3}}\int_{0}^{t}\|\psi_{x}\|^{2}\,\mathrm{d}\tau\leq\frac{1}{4}\int_{0}^{t}\|\psi_{xx}\|^{2}\,\mathrm{d}\tau+C\varepsilon_{0}^{\frac{4}{3}}\int_{0}^{t}\|\psi_{x}\|^{2}\,\mathrm{d}\tau.$
(3.74)

Similar to the treatment of (3.2), by employing (3.12) and (3.13) in Lemma
3.2, we have

$\displaystyle\quad
C\int_{0}^{t}\int_{\mathbb{R_{+}}}|\tilde{u}_{x}|\psi_{x}^{2}\,\mathrm{d}x\mathrm{d}\tau+C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left[(p-\hat{p})_{x}^{2}+(E+\psi
b+\hat{u}b)^{2}+g^{2}\right]\,\mathrm{d}x\mathrm{d}\tau$ $\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left[|\tilde{u}_{x}|\psi_{x}^{2}+(\phi_{x}^{2}+\zeta_{x}^{2}+\phi^{2}\zeta_{x}^{2}+\zeta^{2}\phi_{x}^{2})+(\hat{\theta}_{x}^{2}\phi^{2}+\hat{\rho}_{x}^{2}\zeta^{2})+g^{2}+(E+\psi
b+\hat{u}b)^{2}\right]\,\mathrm{d}x\mathrm{d}\tau$ $\displaystyle\leq
C\left\\{\int_{0}^{t}\left[\|(\phi_{x},\psi_{x},\zeta_{x})\|^{2}+\phi^{2}(0,\tau)+\|E+\psi
b+\hat{u}b\|^{2}\right]\,\mathrm{d}\tau+\alpha^{\frac{1}{3}}+\delta\right\\}.$
(3.75)

Plugging (3.2)-(3.2) into (3.2), then employing (3.3), (3.4) and (3.61), we
can obtain

$\|\psi_{x}\|^{2}+\int_{0}^{t}\|\psi_{xx}\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\|\sqrt{\bar{u}_{x}}\psi_{x}\|^{2}\,\mathrm{d}\tau\leq
C\left(\|\zeta_{0}\|^{2}+\|(\phi_{0},\psi_{0},E_{0},b_{0})\|^{2}_{H^{1}}+\delta+\alpha^{\frac{1}{10}}\right).$
(3.76)

On the other hand, multiplying $\eqref{raodong-fuhebo}_{3}$ by
$\frac{\zeta_{xx}}{\rho}$, we can get

$\displaystyle\quad\frac{R}{\gamma-1}\left[\frac{1}{2}\left(\zeta_{x}^{2}\right)_{t}-\left(\zeta_{t}\zeta_{x}+\frac{1}{2}u\zeta_{x}^{2}\right)_{x}\right]+\frac{R}{2(\gamma-1)}\bar{u}_{x}\zeta_{x}^{2}+\kappa\frac{\zeta_{xx}^{2}}{\rho}$
$\displaystyle=R\theta\psi_{x}\zeta_{xx}-\frac{R}{2(\gamma-1)}(\psi_{x}+\tilde{u}_{x})\zeta_{x}^{2}-\mu\psi_{x}^{2}\frac{\zeta_{xx}}{\rho}-h\frac{\zeta_{xx}}{\rho}-(E+\psi
b+\hat{u}b)^{2}\frac{\zeta_{xx}}{\rho}.$

Integrating the above equality over $\mathbb{R}_{+}\times[0,t]$ gives

$\displaystyle\|\zeta_{x}\|^{2}+\int_{0}^{t}\|\zeta_{xx}\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\|\sqrt{\bar{u}_{x}}\zeta_{x}\|^{2}\,\mathrm{d}\tau\leq
C\|\zeta_{0x}\|^{2}+C\int_{0}^{t}\zeta_{x}^{2}(0,\tau)\,\mathrm{d}\tau$
$\displaystyle\quad+C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left\\{\left|\psi_{x}+\tilde{u}_{x}\right|\zeta_{x}^{2}+\left[\left|\psi_{x}\right|+\psi_{x}^{2}+\left|h\right|+(E+\psi
b+\hat{u}b)^{2}\right]\cdot\left|\zeta_{xx}\right|\right\\}\,\mathrm{d}x\mathrm{d}\tau.$
(3.77)

Analogously, we can also obtain the following estimates:

$\displaystyle\quad
C\int_{0}^{t}\zeta^{2}_{x}(0,\tau)\,\mathrm{d}\tau+C\int_{0}^{t}\int_{\mathbb{R_{+}}}\left\\{\left|\psi_{x}+\tilde{u}_{x}\right|\zeta_{x}^{2}+\left(\left|\psi_{x}\right|+\psi_{x}^{2}+\left|h\right|\right)\left|\zeta_{xx}\right|\right\\}\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{4}\int_{0}^{t}\|\zeta_{xx}\|^{2}\,\mathrm{d}\tau+C\left\\{\int_{0}^{t}\left[\|(\phi_{x},\psi_{x},\zeta_{x},\psi_{xx})\|^{2}+\phi^{2}(0,\tau)\right]\,\mathrm{d}\tau+\delta+\alpha^{\frac{1}{3}}\right\\},$
(3.78)

and

$\displaystyle\int_{0}^{t}\int_{\mathbb{R_{+}}}(E+\psi
b+\hat{u}b)^{2}\left|\zeta_{xx}\right|\,\mathrm{d}x\mathrm{d}\tau\leq
C\varepsilon_{0}\int_{0}^{t}\|E+\psi
b+\hat{u}b\|\|\zeta_{xx}\|\,\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{4}\int_{0}^{t}\|\zeta_{xx}\|^{2}\,\mathrm{d}\tau+C\int_{0}^{t}\|E+\psi
b+\hat{u}b\|^{2}\,\mathrm{d}\tau.$ (3.79)

Combining (3.2)-(3.2), then employing (3.3), (3.4), (3.61) and (3.76) yields

$\|\zeta_{x}\|^{2}+\int_{0}^{t}\|\zeta_{xx}\|^{2}\,\mathrm{d}\tau+\int_{0}^{t}\|\sqrt{\bar{u}_{x}}\zeta_{x}\|^{2}\,\mathrm{d}\tau\leq
C\left(\|(\phi_{0},\psi_{0},\zeta_{0},E_{0},b_{0})\|^{2}_{H^{1}}+\delta+\alpha^{\frac{1}{10}}\right).$
(3.80)

Lemma 3.6 thus follows easily from (3.76) and (3.80).

∎

###### Proof of Proposition 3.1:.

We combine Lemma 3.3-Lemma 3.6, then choose $\varepsilon_{0}$, $\delta$ and
$\alpha$ small enough to finish the proof of Proposition 3.1. ∎

##  4 Appendix: Derivation of 1-D models

In this appendix, we will give a mathematical derivation of system (1.1) for
the completeness. As a by-product, we also obtain some additional 1-D models.

NSM concerns the motion of conducting fluids (gases) in the electromagnetic
fields with a very broad range of applications. Since the dynamic motion of
the fluid and the electromagnetic fields interact strongly on each other, we
must take into account the hydrodynamic and electrodynamic effects for the
governing system. The equations of NSM flows have the following form ([15],
[18], [49]):

$\left\\{\begin{aligned}
&\partial_{t}\rho+\operatorname{div}(\rho\boldsymbol{u})=0,\\\\[2.84526pt]
&\rho\left(\partial_{t}\boldsymbol{u}+\boldsymbol{u}\cdot\nabla\boldsymbol{u}\right)+\nabla
p=\mu^{\prime}\Delta\boldsymbol{u}+(\lambda+\mu^{\prime})\nabla(\operatorname{div}\boldsymbol{u})+\rho_{e}\boldsymbol{E}+\boldsymbol{J}\times\boldsymbol{B},\\\\[2.84526pt]
&\rho\frac{\partial
e}{\partial\theta}(\partial_{t}\theta+\boldsymbol{u}\cdot\nabla\theta)+\theta\frac{\partial
p}{\partial\theta}\operatorname{div}\boldsymbol{u}=\operatorname{div}(\kappa\nabla\theta)+\mathbb{N}(\boldsymbol{u})+(\boldsymbol{J}-\rho_{e}\boldsymbol{u})\cdot(\boldsymbol{E}+\boldsymbol{u}\times\boldsymbol{B}),\\\\[2.84526pt]
&\varepsilon\partial_{t}\boldsymbol{E}-\frac{1}{\mu_{0}}\operatorname{curl}\boldsymbol{B}+\boldsymbol{J}=0,\\\\[2.84526pt]
&\partial_{t}\boldsymbol{B}+\operatorname{curl}\boldsymbol{E}=0,\\\\[2.84526pt]
&\partial_{t}\rho_{e}+\operatorname{div}\boldsymbol{J}=0,\quad\varepsilon\operatorname{div}\boldsymbol{E}=\rho_{e},\quad\operatorname{div}\boldsymbol{B}=0,\end{aligned}\right.$
(4.1)

where $(\boldsymbol{x},t)\in\mathbb{R}^{3}\times\mathbb{R}_{+}$. Here,
$\rho(\boldsymbol{x},t)>0$ denotes the mass density,
$\boldsymbol{u}=\left(u_{1},u_{2},u_{3}\right)\in\mathbb{R}^{3}$ the fluid
velocity, $\theta(\boldsymbol{x},t)>0$ the absolute temperature,
$\boldsymbol{E}=\left(E_{1},E_{2},E_{3}\right)\in\mathbb{R}^{3}$ the electric
field, $\boldsymbol{B}=\left(B_{1},B_{2},B_{3}\right)\in\mathbb{R}^{3}$ the
magnetic field, and $\rho_{e}(\boldsymbol{x},t)$ the electric charge density.
The pressure $p$ and the internal energy $e$ are expressed by the equations of
states for polytropic fluids: $p=R\rho\theta$, $e=\frac{R}{\gamma-1}\theta$.
$R>0$ is the gas constant and $\gamma>1$ is the adiabatic exponent.
$\mathbb{N}(\boldsymbol{u})$ in $\eqref{chushifangcheng}_{3}$ denotes the
viscous dissipation function:
$\mathbb{N}(\boldsymbol{u})=\sum_{i,j=1}^{3}\frac{\mu^{\prime}}{2}\left(\frac{\partial
u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial
x_{i}}\right)^{2}+\lambda\left(\operatorname{div}\boldsymbol{u}\right)^{2}$.
$\mu^{\prime}$ and $\lambda$ are the viscosity coefficients of the fluid which
satisfy $\mu^{\prime}>0$ and $2\mu^{\prime}+3\lambda>0$. The electric current
density $\boldsymbol{J}$ can be expressed by Ohm’s law:
$\boldsymbol{J}=\rho_{e}\boldsymbol{u}+\sigma(\boldsymbol{E}+\boldsymbol{u}\times\boldsymbol{B})$.
$\sigma>0$ denotes the electic conductivity coefficient. The heat conductivity
coefficient $\kappa$ in $\eqref{chushifangcheng}_{3}$ and the magnetic
permeability $\mu_{0}$ in $\eqref{chushifangcheng}_{4}$ are positive
constants. Finally, $\varepsilon>0$ in $\eqref{chushifangcheng}_{4}$ is the
dielectric constant.

There is quite limited mathematical progress for the original nonlinear system
since as pointed out by Kawashima in [18], the system
$\eqref{chushifangcheng}$ is neither symmetric hyperbolic nor strictly
hyperbolic. Because of this, the classical local well-posedness theorem (cf.
[17]) is invalid to the system $\eqref{chushifangcheng}$. In addition, the
tightly coupled hydrodynamic and electrodynamic effects produce strong
nonlinearities, which leads to many difficulties. Owing to the mathematically
complicate structure of the original nonlinear system
$\eqref{chushifangcheng}$, some simplified models are derived according to the
actual physical application. As it was pointed out by Imai in [15], the
assumption that the electric charge density $\rho_{e}\simeq 0$ is physically
beneficial to the research of plasmas. Here, we would mention that the quasi-
neutrality assumption $\rho_{e}\simeq 0$ is different from the assumption of
exact neutrality $\rho_{e}=0$ since the latter would lead to the superfluous
condition $\operatorname{div}\boldsymbol{E}=0$. According to this quasi-
neutrality assumption, we can eliminate the terms involving $\rho_{e}$ in the
system $\eqref{chushifangcheng}$ and derive the following simplified system
(cf. [49]):

$\left\\{\begin{aligned}
&\partial_{t}\rho+\operatorname{div}(\rho\boldsymbol{u})=0,\\\\[2.84526pt]
&\rho\left(\partial_{t}\boldsymbol{u}+\boldsymbol{u}\cdot\nabla\boldsymbol{u}\right)+\nabla
p=\mu^{\prime}\Delta\boldsymbol{u}+(\lambda+\mu^{\prime})\nabla(\operatorname{div}\boldsymbol{u})+\boldsymbol{J}\times\boldsymbol{B},\\\\[2.84526pt]
&\frac{R}{\gamma-1}\rho(\partial_{t}\theta+\boldsymbol{u}\cdot\nabla\theta)+p\operatorname{div}\boldsymbol{u}=\operatorname{div}(\kappa\nabla\theta)+\mathbb{N}(\boldsymbol{u})+\boldsymbol{J}\cdot(\boldsymbol{E}+\boldsymbol{u}\times\boldsymbol{B}),\\\\[2.84526pt]
&\varepsilon\partial_{t}\boldsymbol{E}-\operatorname{curl}\boldsymbol{B}+\boldsymbol{J}=0,\\\\[2.84526pt]
&\partial_{t}\boldsymbol{B}+\operatorname{curl}\boldsymbol{E}=0,\quad\operatorname{div}\boldsymbol{B}=0,\end{aligned}\right.$
(4.2)

with the electric current density
$\boldsymbol{J}=\boldsymbol{E}+\boldsymbol{u}\times\boldsymbol{B}$. Here we
take $\mu_{0}=\sigma=1$ for simplicity. The system (4.2) is usually called the
Navier-Stokes-Maxwell equations (NSM) because it is obtained from the Navier-
Stokes equations coupling with the Maxwell equations through the Lorentz
force.

We shall derive the one-dimensional motion on a spatial domain. Without loss
of generality, consider a three-dimensional NSM flow with spatial variables
$\boldsymbol{x}=(x_{1},x_{2},x_{3})$ which is moving only in the longitudinal
direction $x_{1}$ and uniform in the transverse directions $(x_{2},x_{3})$.
This means that all the quantities
$(\rho,\boldsymbol{u},\theta,\boldsymbol{E},\boldsymbol{B})$ appearing in
(4.2) are independent of the second and the third component of space variable
$(x_{1},x_{2},x_{3})$. According to the location of the dependent spatial
variable, for example $x_{1}$ (below $x_{1}$ will be denoted by $x$), we set
$\boldsymbol{u}:=(u,\;0,\;0)$ for the sake of the hydrodynamic feature of this
one-dimensional flow. If the dependent spatial variable takes $x_{2}$
(respectively, $x_{3}$), we can set $\boldsymbol{u}:=(0,\;u,\;0)$
(respectively, $\boldsymbol{u}:=(0,\;0,\;u)$) correspondingly. And the
discussion is analogous. For this reason we have only to consider the
following nine different cases.

$\bullet$ Case 1:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(0,\;0,\;E),\quad\boldsymbol{B}:=(0,\;b,\;0).$

$\bullet$ Case 2:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(0,\;E,\;0),\quad\boldsymbol{B}:=(0,\;0,\;\tilde{b}).$

Denote $\mu:=\lambda+2\mu^{\prime}$. For Case 1, by employing direct
calculations for (4.2), we can deduce the following one-dimensional system:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\
&\rho(u_{t}+uu_{x})+p_{x}=\mu u_{xx}-(E+ub)b,\\\
&\frac{R}{\gamma-1}\rho(\theta_{t}+u\theta_{x})+pu_{x}=\mu
u_{x}^{2}+\kappa\theta_{xx}+(E+ub)^{2},\\\ &\varepsilon E_{t}-b_{x}+E+ub=0,\\\
&b_{t}-E_{x}=0.\end{aligned}\right.$ (4.3)

For Case 2, we introduce a new dependent variable: $b=-\,\tilde{b}$. Then the
resulting system is the same as (4.3). We should mention that Fan et al. in
[5] chose the component of $\boldsymbol{u}$, $\boldsymbol{E}$ and
$\boldsymbol{B}$ the same as that of Case 1 and first obtained system (4.3).

$\bullet$ Case 3:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(0,\;0,\;E),\quad\boldsymbol{B}:=(0,\;0,\;b).$

$\bullet$ Case 4:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(0,\;E,\;0),\quad\boldsymbol{B}:=(0,\;b,\;0)$.

For these two cases, we can deduce the following system:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\
&\rho(u_{t}+uu_{x})+p_{x}=\mu u_{xx}-ub^{2},\\\
&\frac{R}{\gamma-1}\rho(\theta_{t}+u\theta_{x})+pu_{x}=\mu
u_{x}^{2}+\kappa\theta_{xx}+E^{2}+(ub)^{2},\\\ &\varepsilon E_{t}+E=0,\quad
E_{x}=0,\\\ &b_{x}-ub=0,\quad b_{t}=0.\end{aligned}\right.$ (4.4)

From $\eqref{system-2}_{4}$ and $\eqref{system-2}_{5}$, we can obtain

$E=E(t)=E(0)\,{\mathop{\mathrm{e}}}^{-\frac{t}{\varepsilon}},\quad
b=b(x)=b(0)\,{\mathop{\mathrm{e}}}^{\int_{0}^{x}u(y,0)\,\mathrm{d}y}.$

$\bullet$ Case 5:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(0,\;0,\;E),\quad\boldsymbol{B}:=(b,\;0,\;0).$

$\bullet$ Case 6:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(0,\;E,\;0),\quad\boldsymbol{B}:=(b,\;0,\;0)$.

For these two cases, we can deduce the following system:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\
&\rho(u_{t}+uu_{x})+p_{x}=\mu u_{xx},\\\
&\frac{R}{\gamma-1}\rho(\theta_{t}+u\theta_{x})+pu_{x}=\mu
u_{x}^{2}+\kappa\theta_{xx}+E^{2},\\\ &Eb=0,\\\ &\varepsilon E_{t}+E=0,\quad
E_{x}=0,\\\ &b_{t}=0,\quad b_{x}=0.\end{aligned}\right.$ (4.5)

$\eqref{system-3}_{5}$ and $\eqref{system-3}_{6}$ lead to

$E=E(t)=E(0)\,{\mathop{\mathrm{e}}}^{-\frac{t}{\varepsilon}},\quad
b=\mathrm{constant}.$

The equation $\eqref{system-3}_{4}$ permits at least one zero-solution, i.e.

$E=E(t)=E(0)\,{\mathop{\mathrm{e}}}^{-\frac{t}{\varepsilon}},\quad b=0;$

or

$E=0,\quad b=\mathrm{constant}.$

$\bullet$ Case 7:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(E,\;0,\;0),\quad\boldsymbol{B}:=(0,\;b,\;0)$.

$\bullet$ Case 8:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(E,\;0,\;0),\quad\boldsymbol{B}:=(0,\;0,\;b).$

For these two cases, we can deduce the following system:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\
&\rho(u_{t}+uu_{x})+p_{x}=\mu u_{xx}-ub^{2},\\\
&\frac{R}{\gamma-1}\rho(\theta_{t}+u\theta_{x})+pu_{x}=\mu
u_{x}^{2}+\kappa\theta_{xx}+E^{2}+(ub)^{2},\\\ &Eb=0,\quad\varepsilon
E_{t}+E=0,\\\ &b_{x}-ub=0,\quad b_{t}=0.\end{aligned}\right.$ (4.6)

By a reasoning similar to the above, we obtain that

$E=E(x,t)=E(x,0)\,{\mathop{\mathrm{e}}}^{-\frac{t}{\varepsilon}},\quad b=0;$

or

$E=0,\quad
b=b(x)=b(0)\,{\mathop{\mathrm{e}}}^{\int_{0}^{x}u(y,0)\,\mathrm{d}y}.$

$\bullet$ Case 9:
$\boldsymbol{u}:=(u,\;0,\;0),\quad\boldsymbol{E}:=(E,\;0,\;0),\quad\boldsymbol{B}:=(b,\;0,\;0).$

For this case, we can deduce the following system:

$\left\\{\begin{aligned} &\rho_{t}+(\rho u)_{x}=0,\\\
&\rho(u_{t}+uu_{x})+p_{x}=\mu u_{xx},\\\
&\frac{R}{\gamma-1}\rho(\theta_{t}+u\theta_{x})+pu_{x}=\mu
u_{x}^{2}+\kappa\theta_{xx}+E^{2},\\\ &\varepsilon E_{t}+E=0,\\\
&b_{t}=0,\quad b_{x}=0.\end{aligned}\right.$ (4.7)

From $\eqref{system-5}_{4}$ and $\eqref{system-5}_{5}$, we can obtain

$E=E(x,t)=E(x,0)\,{\mathop{\mathrm{e}}}^{-\frac{t}{\varepsilon}},\quad
b=\mathrm{constant}.$

As a result, for the above-mentioned cases, we obtain five 1-D compressible
non-isentropic models: (4.3)-(4.7). We can see that system (4.3) (i.e., system
(1.1)) includes the electrodynamic effects into the dissipative structure of
the hydrodynamic equations and turns out to be more complicated than that in
the other four models. Moreover, it may be interesting to note that the
electromagnetic fields $E$ and $b$ in systems (4.4)-(4.7) can in principle be
determined explicitly by the dielectric constant $\varepsilon$, the boundary
data for $b$ at $x=0$, and the initial data for $E$ and $u$.

Further discussion about these five one-dimensional models in mathematics and
physics would be meaningful and interesting.

## Acknowledgments

The research was supported by the National Natural Science Foundation of China
$\\#$11771150, 11831003 and Guangdong Basic and Applied Basic Research
Foundation $\\#$2020B1515310015.

## References

  * [1]
  * [2] G.Q. Chen, J.W. Jerome and D.H. Wang,  _Compressible Euler-Maxwell equations_ ,  Transport Theory Statist. Phys., 29(2000), 311-331.
  * [3] R.J. Duan,  _Green’s function and large time behavior of the Navier-Stokes-Maxwell system_ ,  Anal. Appl. (Singap.), 10(2012), 133-197.
  * [4] R. Duan, H.X. Liu and H.J. Zhao,  _Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation_ ,  Trans. Amer. Math. Soc., 361(2009), 453-493.
  * [5] J.S. Fan and Y.X. Hu,  _Uniform existence of the 1-d complete equations for an electromagnetic fluid_ ,  J. Math. Anal. Appl., 419(2014), 1-9.
  * [6] J.S. Fan and Y.B. Ou,  _Uniform existence of the 1-D full equations for a thermo-radiative electromagnetic fluid_ ,  Nonlinear Anal., 106(2014), 151-158.
  * [7] F.M. Huang, J. Li and A. Matsumura,  _Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system_ ,  Arch. Ration. Mech. Anal., 197(2010), 89-116.
  * [8] F.M. Huang, J. Li and X.D. Shi,  _Asymptotic behavior of solutions to the full compressible Navier-Stokes equations in the half space_ ,  Commun. Math. Sci., 8(2010), 639-654.
  * [9] F.M. Huang and A. Matsumura,  _Stability of a composite wave of two viscous shock waves for the full compressible Navier-Stokes equation_ ,  Comm. Math. Phys., 289(2009), 841-861.
  * [10] F.M. Huang, A. Matsumura and X.D. Shi,  _Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas_ ,  Comm. Math. Phys., 239(2003), 261-285.
  * [11] F.M. Huang, A. Matsumura and Z.P. Xin,  _Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations_ ,  Arch. Ration. Mech. Anal., 179(2006), 55-77.
  * [12] F.M. Huang and X.H. Qin,  _Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation_ ,  J. Differential Equations, 246(2009), 4077-4096.
  * [13] F.M. Huang, Z.P. Xin and T. Yang,  _Contact discontinuity with general perturbations for gas motions_ ,  Adv. Math., 219(2008), 1246-1297.
  * [14] Y.T. Huang and H.X. Liu,  _Stability of rarefaction wave for a macroscopic model derived from the Vlasov-Maxwell-Boltzmann system_ ,  Acta Math. Sci., Ser. B, 38(2018), 857-888.
  * [15] I. Imai,  _General Principles of Magneto-Fluid Dynamics. In: Magneto-Fulid Dynamics_ ,  Suppl. Prog. Theor. Phys., 24(1962), 1-34.
  * [16] S. Jiang and F. C. Li,  _Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations_ ,  Asymptot. Anal., 95(2015), pp. 161-185.
  * [17] T. Kato,  _The Cauchy problem for quasi-linear symmetric hyperbolic systems_ ,  Arch. Ration. Mech. Anal., 58 (1975), 181-205.
  * [18] S. Kawashima,  _Smooth global solutions for two-dimensional equations of electromagnetofluid dynamics_ ,  Japan J. Appl. Math., 1(1984), 207-222.
  * [19] S. Kawashima and A. Matsumura,  _Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion_ ,  Comm. Math. Phys., 101(1985), 97-127.
  * [20] S. Kawashima, A. Matsumura and K. Nishihara,  _Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas_ ,  Proc. Japan Acad. Ser. A Math. Sci., 62(1986), 249-252.
  * [21] S. Kawashima, T. Nakamura, S. Nishibata and P.C. Zhu,  _Stationary waves to viscous heat-conductive gases in half space: existence, stability and convergence rate_ ,  Math. Models Methods Appl. Sci., 20(2010), 2201-2235.
  * [22] S. Kawashima, S. Nishibata and P.C. Zhu,  _Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space_ ,  Comm. Math. Phys., 240(2003), 483-500.
  * [23] S. Kawashima and Y. Shizuta,  _Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid_ ,  Tsukuba J. Math., 10(1986), pp. 131-149.
  * [24] S. Kawashima and Y. Shizuta,  _Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, II_ ,  Proc. Japan Acad. Ser. A Math. Sci., 62(1986), pp. 181-184.
  * [25] S. Kawashima and P.C. Zhu,  _Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space_ ,  J. Differential Equations, 244(2008), 3151-3179.
  * [26] S. Kawashima and P.C. Zhu,  _Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space_ ,  Arch. Ration. Mech. Anal., 194(2009), 105-132.
  * [27] F.C. Li and Y.M. Mu,  _Low Mach number limit of the full compressible Navier-Stokes-Maxwell system_ ,  J. Math. Anal. Appl., 412(2014), 334-344.
  * [28] T.P. Liu,  _Shock waves for compressible Navier-Stokes equations are stable_ ,  Comm. Pure Appl. Math., 39(1986), 565-594.
  * [29] T.P. Liu, A. Matsumura and K. Nishihara,  _Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves_ ,  SIAM J. Math. Anal., 29(1998), 293-308.
  * [30] T.P. Liu and K. Nishihara,  _Asymptotic behavior for scalar viscous conservation laws with boundary effect_ ,  J. Differential Equations, 133(1997), 296-320.
  * [31] T.P. Liu and Z.P. Xin,  _Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations_ ,  Comm. Math. Phys., 118(1988), 451-465.
  * [32] T.P. Liu and S.H. Yu,  _Propagation of a stationary shock layer in the presence of a boundary_ ,  Arch. Ration. Mech. Anal., 139(1997), 57-82.
  * [33] F.Q. Luo, H.C. Yao and C.J. Zhu,  _Stability of rarefaction wave for isentropic compressible Navier-Stokes-Maxwell equations_ ,  Nonlinear Anal. Real World Appl., 59(2021), 103234. 14 pp.
  * [34] N. Masmoudi,  _Global well posedness for the Maxwell-Navier-Stokes system in 2D_ ,  J. Math. Pures Appl., 93(2010), 559-571.
  * [35] A. Matsumura,  _Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas_ ,  Methods Appl. Anal., 8(2001), 645-666.
  * [36] A. Matsumura and M. Mei,  _Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary_ ,  Arch. Ration. Mech. Anal., 146(1999), 1-22.
  * [37] A. Matsumura and K. Nishihara,  _Global asymptotics toward the rarefaction wave for solutions of viscous p-system with boundary effect_ ,  Quart. Appl. Math., 58(2000), 69-83.
  * [38] A. Matsumura and K. Nishihara,  _Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas_ ,  Comm. Math. Phys., 144(1992), 325-335.
  * [39] A. Matsumura and K. Nishihara,  _Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas_ ,  Comm. Math. Phys., 222(2001), 449-474.
  * [40] T. Nakamura and S. Nishibata,  _Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas_ ,  J. Hyperbolic Differ. Equ., 8(2011), 651-670.
  * [41] T. Nakamura, S. Nishibata and T. Yuge,  _Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line_ ,  J. Differential Equations, 241(2007), 94-111.
  * [42] Y. Nikkuni,  _Stability of stationary solutions to some discrete velocity model of the Boltzmann equation in the half-space_ ,  Funkcial. Ekvac., 42(1999), 413-434.
  * [43] K. Nishihara, T. Yang and H.J. Zhao,  _Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations_ ,  SIAM J. Math. Anal., 35(2004), 1561-1597.
  * [44] I.S. Pai,  _Magnetogasdynamics and Plasma Dynamics_ ,  Springer-Verlag, 1962.
  * [45] X.H. Qin,  _Large-time behaviour of solutions to the outflow problem of full compressible Navier-Stokes equations_ ,  Nonlinearity, 24(2011), 1369-1394.
  * [46] X.H. Qin and Y. Wang,  _Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations_ ,  SIAM J. Math. Anal., 43(2011), 341-366.
  * [47] X.H. Qin and Y. Wang,  _Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations_ ,  SIAM J. Math. Anal., 41(2009), 2057-2087.
  * [48] J. Smoller,  _Shock Waves and Reaction-Diffusion Equations_ ,  2nd edition, Springer-Verlag, New York, 1994.
  * [49] X. Xu,  _Asymptotic behavior of solutions to an electromagnetic fluid model_ ,  Z. Angew. Math. Phys., 69(2018), pp. 1-19.
  * [50] H.C. Yao and C.J. Zhu,  _Asymptotic stability of the superposition of viscous contact wave with rarefaction waves for the compressible Navier-Stokes-Maxwell equations_ ,  2021\. arXiv:2108.01824v1 [math.AP].
  * [51] H.C. Yao and C.J. Zhu,  _Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations_ ,  Commun. Pure Appl. Anal., 20(2021), 1297-1317.
  * [52] H.Y. Yin,  _Converge rates towards stationary solutions for the outflow problem of planar magnetohydrodynamics on a half line_ ,  Proc. Roy. Soc. Edinburgh Sect. A, 149(2019), 1291-1322.