# Asymptotic stability of rarefaction waves for compressible Navier-Stokes
equations with relaxation

Yuxi Hu and Xuefang Wang

###### Abstract.

The asymptotic stability of rarefaction wave for 1-d relaxed compressible
isentropic Navier-Stokes equations is established. For initial data with
different far-field values, we show that there exists a unique global in time
solution. Moreover, as time goes to infinity, the obtained solutions are shown
to converge uniformly to rarefaction wave solution of $p$-system with
corresponding Riemann initial data. The proof is based on $L^{2}$ energy
methods.
Keywords: Compressible Navier-Stokes equations; Rarefaction waves; Stability
AMS classification code: 35B25, 76N10

##  1\.  Introduction

In this paper, we consider the system of one-dimensional isentropic
compressible Navier Stokes equations in Lagrange coordinates as

$\displaystyle\begin{cases}v_{t}=u_{x},\\\
u_{t}+p(v)_{x}=\tilde{S}_{x},\end{cases}$ (1.1)

with

$\displaystyle\tau\tilde{S}_{t}+\tilde{S}=\mu\frac{u_{x}}{v},$ (1.2)

where $v,u,p,\tilde{S}$ denote the specific volume per unit mass, fluid
velocity, pressure and stress tensor, respectively. The equations (1.1) are
the consequences of conservation of mass and balance of momentum,
respectively. The viscosity coefficient $\mu$ is assumed to be a positive
constant as well as the relaxation parameter $\tau$. Moreover, we assume the
pressure $p$ satisfy $p(v)=av^{-\gamma}$ with $\gamma>1$ being the adiabatic
index.

In the constitutive relation (1.2), $\tau$ is the relaxtion time describing
the time lag in the response of the stress tensor to the velocity gradient. In
fact, even in simple fluid, water for example, the“time lag” exists, but it is
very small ranging from 1 ps to 1 ns, see [14, 19]. However, Pelton et al.
[17] showed that such a“time lag” cannot be neglected, even for simple fluids,
in the experiments of high-frequency (20GHZ) vibration of nano-scale
mechanical devices immersed in water-glycerol mixtures. It turned out that,
cp. also [2], equation (1.2) provides a general formalism to characterize the
fluid-structure interaction of nano-scale mechanical devices vibrating in
simple fluids. A similar relaxed constitutive relation was already proposed by
Maxwell in [15], in order to describe the relation of stress tensor and
velocity gradient for a non-simple fluid.

The system (1.1) is coupled with the initial conditions

$\displaystyle(v,u,\tilde{S})(0,x)=(v_{0},u_{0},\tilde{S}_{0})(x),\quad
x\in\mathbb{R},$ (1.3)

with

$\lim\limits_{{x}\rightarrow{\pm\infty}}(v_{0},u_{0},\tilde{S}_{0})(x)=(v_{\pm},u_{\pm},0).$

Here, we assume, for the far field conditions, $v_{+}\neq v_{-},u_{+}\neq
u_{-}$ in general. Furthermore, $(v_{-},u_{-})$ and $(v_{+},u_{+})$ are
supposed to be the Riemann initial data which generates a centered rarefaction
wave for the following $p-$system

$\displaystyle\begin{cases}v_{t}=u_{x},\\\ u_{t}+p(v)_{x}=0.\end{cases}$ (1.4)

That means, there exists a continuous weak solution with the form
$(v^{r},u^{r})(x/t)$ of $p-$ system (1.4) with

$\displaystyle(v^{r},u^{r})(0,x)=\begin{cases}(v_{-},u_{-}),\quad x<0,\\\
(v_{+},u_{+}),\quad x>0\end{cases}$ (1.5)

The main purpose of this paper is devoted to establish the asymptotic
stability of the rarefaction wave $(v^{r},u^{r},0)$ defined above. More
precisely, we assume the initial data $(v_{0},u_{0},\tilde{S}_{0})$ is close
to $(v_{0}^{r},u_{0}^{r},0)$ in a suitable sense and the amplitude
$\delta=|v_{+}-v_{-}|+|u_{+}-u_{-}|$ of the rarefaction wave is sufficiently
small. Then, we will show that there exists a unique global defined solution
$(v,u,\tilde{S})$ to problem (1.1)-(1.3) which approaches the rarefaction wave
$(v^{r},u^{r},0)$ uniformly in $x$ as $t\rightarrow\infty$. That is, we have

$\|(v-v^{r},u-u^{r},\tilde{S}\|_{L^{\infty}}\rightarrow
0,\quad\mathrm{as}\quad t\rightarrow\infty.$

See Theorem 2.1 for more details.

Note that, when $\tau=0$, the system (1.1), (1.2) reduce to the classical
isentropic compressible Navier-Stokes equations, which read as

$\displaystyle\begin{cases}v_{t}=u_{x},\\\ u_{t}+p(v)_{x}=\left(\frac{\mu
u_{x}}{v}\right)_{x}.\end{cases}$ (1.6)

For such system, the well-posedness and asymptotic stability results have been
widely studied with various initial data, see [7, 8, 12, 13]. In particular,
Matsumura and Nishihara [12] first established the asymptotic stability of
rarefaction waves with small amplitude in 1986. Later, they [13] extended
their results to large data cases. The half-space problem was conducted by
Kawashima and Zhu [8]. A similar asymptotic stability result was also obtained
for the full system of the compressible Navier-Stokes equations, see [9].

The asymptotic stability of rarefaction waves were also shown for other
related system, see [11] for the Broadwell model, [10] for a model system of
radiating gas, [16, 1] for a model of hyperbolic balance law.

One should note that it is not obvious that the results which hold for the
classical systems also hold for the relaxed system. Indeed, and for example,
Hu and Wang [5] and Hu, Racke and Wang [6] showed that, for the one-
dimensional isentropic and/or non-isentropic compressible Navier-Stokes
system, solutions exist globally with arbitrary large initial data for
classical system, while solutions blow up in finite time with some large
initial data for the corresponding relaxed system. A similar qualitative
change was observed before for certain thermoelastic systems, where the non-
relaxed system is exponentially stable, while the relaxed one is not, see
Quintanilla and Racke resp. Fernández Sare and Muñoz Rivera [18, 3] for
plates, and Fernández Sare and Racke [4] for Timoshenko beams.

The paper is organized as follows. Some preliminaries and main theorem
(Theorem 2.1) are given in Section 2. In Section 3, following to [12, 13] , we
construct smooth approximations of rarefaction waves. The original problem is
reformulated in Section 4 and there the statements of asymptotic stability
results for the reformulated problems are given (Theorem 4.1). The a priori
estimates are also given in Section 4 (Proposition 4.2). Last, the proof of
the a priori estimates (Proposition 4.2) are given in Section 5.

Notations: $L^{p}(\mathbb{R})$ and $W^{s,p}(\mathbb{R})$ ($1\leq p\leq\infty$)
denote the usual Lebesgue and Sobolev spaces over $\mathbb{R}$ with the norm
$\|\cdot\|_{L^{p}}$ and $\|\cdot\|_{W^{s,p}}$, respectively. Note that, when
$s=0$, $W^{0,p}=L^{p}$. For $p=2$, $W^{s,2}$ are abbreviated to $H^{s}$ as
usual. Let $T$ and $B$ be a positive constant and a Banach space,
respectively. $C^{k}(0,T;B)(k\geq 0)$ denotes the space of $B$-valued
$k$-times continuously differentiable functions on $[0,T]$, and $L^{p}(0,T;B)$
denotes the space of $B$-valued $L^{p}$-functions on $[0,T]$. The
corresponding space $B$-valued functions on $[0,\infty)$ are defined
similarly.

##  2\.  Preliminaries and main results

Let’s first recall the definition of rarefaction waves of p-system (1.4) with
Riemann initial data (1.5). For fixed
$(v_{-},u_{-})(v_{-}>0,u_{-}\in\mathbb{R})$, we define a proper neighbourhood
$\omega\subset\mathbb{R}_{v,u}^{2}:=\mathbb{R}_{+}\times\mathbb{R}$ as

$\displaystyle
R_{1}(v_{-},u_{-}):=\left\\{(v,u)\in\omega|u=u_{-}-\int_{v_{-}}^{v}\lambda_{1}(s)\mathrm{d}s,u\geq
u_{-}\right\\},$ $\displaystyle
R_{2}(v_{-},u_{-}):=\left\\{(v,u)\in\omega|u=u_{-}-\int_{v_{-}}^{v}\lambda_{2}(s)\mathrm{d}s,u\geq
u_{-}\right\\}$

and

$\displaystyle RR(v_{-},u_{-}):=\left\\{(v,u)\in\omega|u\geq
u_{-}-\int_{v_{-}}^{v}\lambda_{1}(s)\mathrm{d}s,u\geq
u_{-}-\int_{v_{-}}^{v}\lambda_{2}(s)\mathrm{d}s\right\\},$

where $\lambda_{1}(v)=-\sqrt{-p^{\prime}(v)}$,
$\lambda_{2}(v)=\sqrt{-p^{\prime}(v)}$ are eigenvalues of the matrix

$\displaystyle\begin{pmatrix}0&-1\\\ p^{\prime}(v)&0\end{pmatrix}.$

If $(v_{+},u_{+})\in RR(v_{-},u_{-})$, then Riemann problem (1.4), (1.5)
admits a continuous weak solution of the form $(v^{r},u^{r})(x/t)$ which is
called centered rarefaction wave.

Let

$I_{0}:=\|(v_{0}-v_{0}^{r},u_{0}-u_{0}^{r})\|_{L^{2}}+\|((v_{0})_{x},(u_{0})_{x})\|_{H^{1}}+\|\tilde{S}_{0}\|_{H^{2}}.$

Our main result is stated as follows.

######  Theorem 2.1.

Let $(v_{+},u_{+})\in RR(v_{-},u_{-})$ and
$\delta=|v_{+}-v_{-}|+|u_{+}-u_{-}|$. Assume the initial data
$(v_{0},u_{0},\tilde{S}_{0})$ satisfy

$\displaystyle(v_{0}-v_{0}^{r},u_{0}-u_{0}^{r},\tilde{S}_{0})\in L^{2},$
$\displaystyle((v_{0})_{x},(u_{0})_{x})\in H^{1},\tilde{S}_{0}\in H^{2}$

and there exists a positive constant $\epsilon_{0}$ such that if
$I_{0}+\delta<\epsilon_{0}$, then the initial value problem (1.1)-(1.3) has a
unique global solution in time satisfying

$\displaystyle\begin{cases}(v-v_{0}^{r},u-u_{0}^{r},\tilde{S})\in
C^{0}(0,+\infty,L^{2}),\\\ (v_{x},u_{x})\in C^{0}(0,+\infty,H^{1})\cap
L^{2}(0,+\infty,H^{1}),S\in C^{0}(0,+\infty,H^{2})\cap
L^{2}(0,+\infty,H^{2}).\end{cases}$ (2.1)

Moreover, this solution approaches the rarefaction wave $(v^{r},u^{r},0)$
uniformly in $x\in\mathbb{R}$ as $t\rightarrow+\infty$:

$\displaystyle\|(v(t,x)-v^{r}(x/t),u(t,x)-u^{r}(x/t),\tilde{S}(t,x))\|_{L^{\infty}}\rightarrow
0,\quad\mathrm{as}\quad t\rightarrow\infty.$ (2.2)

##  3\. Smooth approximation of rarefaction waves

We start from the inviscid Burgers equation

$\displaystyle\begin{cases}w_{t}^{r}+w^{r}w_{x}^{r}=0,\\\
w^{r}(0,x)=w_{0}^{r}(x)=\begin{cases}w_{-},\quad x<0,\\\ w_{+},\quad
x>0.\end{cases}\end{cases}$ (3.1)

Let $w_{-}<w_{+}$, then (3.1) has a continuous weak solution $w^{r}(x/t)$ with

$\displaystyle w^{r}(x/t)=\begin{cases}w_{-},\quad x/t\leq w_{-},\\\ x/t,\quad
w_{-}\leq x/t\leq w_{+},\\\ w_{+},\quad\xi\geq w_{+}.\end{cases}$ (3.2)

This solution is called central rarefaction wave connecting the two constant
state $w_{-}$ and $w_{+}$. Next, we consider the following system whose
solutions approximate $w^{r}(x/t)$ smoothly.

$\displaystyle\begin{cases}w_{t}+ww_{x}=0,\\\
w(0,x)=w_{0}(x)=\hat{w}+\tilde{w}k_{q}\int_{0}^{\epsilon
x}(1+y^{2})^{-q}\mathrm{d}y,\end{cases}$ (3.3)

where $\hat{w}=\frac{w_{+}+w_{-}}{2},\tilde{w}=\frac{w_{+}-w_{-}}{2}$,
$\epsilon>0$ is a constant. $k_{q}$ is a constant satisfying
$k_{q}\int_{0}^{\infty}(1+y^{2})^{-q}\mathrm{d}y=1$ for $q>\frac{3}{2}$. Then
we have the following lemma

######  Lemma 3.1.

[12, 13] Let $w_{-}<w_{+}$, then there exists a smooth solution $w(t,x)$ of
system (3.3) satisfying

1) $w_{-}<w(t,x)<w_{+}$,
$w_{x}(t,x)>0,\quad\forall(t,x)\in\mathbb{R}_{+}\times\mathbb{R}$

2) $\forall 1\leq p\leq\infty$, there exists a constant $C_{p}$ such that

$\displaystyle\|w_{x}\|_{L^{p}}\leq
C_{p}\min\left(\epsilon^{1-\frac{1}{p}}\tilde{w},\tilde{w}^{\frac{1}{p}}t^{-1+\frac{1}{p}}\right)$
$\displaystyle\|\partial_{x}^{k}w\|_{L^{p}}\leq
C\min\left(\epsilon^{2-\frac{1}{p}}\tilde{w},\epsilon^{(1-\frac{1}{2q})(1-\frac{1}{p})}\tilde{w}^{-\frac{p-1}{2pq}}t^{-1-\frac{p-1}{2pq}}\right).$

3) if $w_{-}>0$, then

$\displaystyle|w(t,x)-w_{-}|\leq C\tilde{w}(1+(\epsilon
x)^{2})^{-\frac{q}{3}}(1+(\epsilon w_{-}t)^{2})^{-\frac{q}{3}},$
$\displaystyle|w_{x}|\leq C\epsilon\tilde{w}(1+(\epsilon
x)^{2})^{-\frac{q}{2}}(1+(\epsilon w_{-}t)^{2})^{-\frac{q}{2}}.$

4) if $w_{+}<0$, then

$\displaystyle|w(t,x)-w_{-}|\leq C\tilde{w}(1+(\epsilon
x)^{2})^{-\frac{q}{3}}(1+(\epsilon w_{+}t)^{2})^{-\frac{q}{3}},$
$\displaystyle|w_{x}|\leq C\epsilon\tilde{w}(1+(\epsilon
x)^{2})^{-\frac{q}{2}}(1+(\epsilon w_{+}t)^{2})^{-\frac{q}{2}}.$

5)
$\lim\limits_{{t}\rightarrow{\infty}}\sup_{\mathbb{R}}|w(t,x)-w^{r}(x/t)|=0$.

Next, we construct smooth solutions which approximate the weak solutions
$(v^{r},u^{r})(x/t)$ of system (1.4)-(1.5) by using $w(t,x)$. Firstly, suppose
$(v_{+},u_{+})\in RR(v_{-},u_{-})$, then there exists a unique pair
$(\bar{v},\bar{u})$ satisfying

$(\bar{v},\bar{u})\in R_{1}(v_{-},u_{-}),\quad(v_{+},u_{+})\in
R_{2}(\bar{v},\bar{u}).$

Moreover, the continuous weak solution of system (1.4)-(1.5) can be expressed
as

$\displaystyle(v^{r},u^{r})(\xi)=(v_{1}^{r}+v_{2}^{r}-\bar{v},u_{1}^{r}+u_{2}^{r}-\bar{u})(\xi),\quad,\xi:=x/t,$
(3.4)

where

$\displaystyle\lambda_{1}(v_{1}^{r}(\xi))=w_{1}^{r}(\xi),\lambda_{2}(v_{2}^{r}(\xi))=w_{2}^{r}(\xi),$
$\displaystyle
u_{1}^{r}(\xi)=u_{-}-\int_{v_{1}}^{v_{1}^{r}(\xi)}\lambda_{1}(s)\mathrm{d}s,u_{2}^{r}(\xi)=\bar{u}-\int_{\bar{v}}^{v_{2}^{r}(\xi)}\lambda_{2}(s)\mathrm{d}s,$

and $w_{1}^{r}(\xi),w_{2}^{r}(\xi)$ are given by (3.2) with

$\displaystyle
w_{1-}=\lambda_{1}(v_{-}),w_{1+}=\lambda_{1}(\bar{v}),w_{2-}=\lambda_{2}(\bar{v}),w_{2+}=\lambda_{2}(v_{+}).$

Now, we define $(V,U)$ as

$\displaystyle\begin{cases}(V,U)(t,x)=(V_{1}+V_{2}-\bar{v},U_{1}+U_{2}-\bar{u})(t,x),\\\
\lambda_{1}(V_{1})=w_{1}(t,x),\lambda_{2}(V_{2})=w_{2}(t,x),\\\
U_{1}=u_{1}-\int_{v_{1}}^{V_{1}}\lambda_{1}(s)\mathrm{d}s,U_{2}=\bar{u}-\int_{\bar{v}}^{V_{2}}\lambda_{2}(s)\mathrm{d}s\end{cases}$
(3.5)

where $w_{1}(t,x),w_{2}(t,x)$ are solutions of (3.3). Note that
$(V_{1},U_{1})$ and $(V_{2},U_{2})$ defined above are exact solution of
$p$-system $\eqref{1.4}$ and $(V,U)$ satisfies

$\displaystyle\begin{cases}V_{t}-U_{x}=0,\\\
U_{t}+p(V)_{x}=g(V)_{x},\end{cases}$ (3.6)

where $g(V)=p(V)-p(V_{1})-p(V_{2})+p(\bar{v})$.

By use of Lemma 3.1, we have

######  Lemma 3.2.

[12, 13, 16] (V, U) defined in (3.5) satisfy

1) $V_{t}>0$, $\forall(x,t)\in\mathbb{R}_{+}\times\mathbb{R}$,

2) There exists a constant $C$ such that

$\displaystyle|V_{x}|\leq CV_{t},\quad V_{t}\leq
C\epsilon\delta,\quad\forall(t,x)\in\mathbb{R}_{+}\times\mathbb{R}$ (3.7)

3) Denote $\tilde{w}_{i}=\frac{w_{i+}-w_{i-}}{2}$, then $\forall
t\in\mathbb{R}_{+}$,

$\displaystyle\|g(V)_{x}\|_{L^{p}}\leq
C\epsilon^{1-\frac{1}{p}}\tilde{w}_{1}\tilde{w}_{2}\left\\{(1+(\epsilon
w_{2-}t)^{2})^{-\frac{q}{3}}+(1+(\epsilon
w_{1+}t)^{2})^{-\frac{q}{3}}\right\\}$

and

$\int_{0}^{\infty}\|g(V)_{x}\|_{L^{p}}\mathrm{d}x\leq
C\delta^{2}\epsilon^{-\frac{1}{p}}.$

4)$\forall t\in\mathbb{R}_{+}$,

$\|V_{x}\|_{L^{p}},\|U_{x}\|_{L^{p}}\leq
C\min\\{\delta\epsilon^{1-\frac{1}{p}},\delta^{\frac{1}{p}}(1+t)^{-1+\frac{1}{p}}\\},$

5) $\forall t\in\mathbb{R}_{+}$ and $k=2,3,4$,

$\displaystyle\|\partial_{x}^{k}V\|_{L^{p}},\|\partial_{x}^{k}U\|_{L^{p}}\leq
C\left(\delta^{-\frac{p-1}{2pq}}\epsilon^{(1-\frac{1}{2q})(1-\frac{1}{p})}(1+t)^{-1-\frac{p-1}{2pq}}+\delta^{\frac{1}{p}}(1+t)^{-2+\frac{1}{p}}\right)$

and $\forall p>1$,

$\int_{0}^{\infty}(\|\partial_{x}^{k}V\|_{L^{p}}+\|\partial_{x}^{k}U\|_{L^{p}})\mathrm{d}t\leq
C\delta^{-\frac{p-1}{2pq}}.$

6)
$\lim\limits_{{t}\rightarrow{\infty}}\sup_{\mathbb{R}}|(V,U)(t,x)-(v^{r},u^{r})(x/t)|=0$.

##  4\. Reformulation of the problem

Let $(v^{r},u^{r})(x/t)$ be centered rarefaction wave for system (1.4)-(1.5)
which is given in (3.4), and let $(V,U)(t,x)$ be the smooth approximations of
$(v^{r},u^{r})(x/t)$. Note that $(V,U)$ are the functions defined in (3.5) and
satisfying (3.6). Now, let
$\varphi=v-V,\psi=u-U,S=\tilde{S}-\mu\frac{U_{x}}{V}$ be perturbations, then

$\displaystyle\begin{cases}\varphi_{t}=\psi_{x},\\\
\psi_{t}+[p(\varphi+V)-p(V)]_{x}-S_{x}=\mu\left(\frac{U_{x}}{V}\right)_{x}-g(V)_{x},\\\
\tau
S_{t}+S-\frac{\mu}{v}\psi_{x}=-\tau\mu\left(\frac{U_{x}}{V}\right)_{t}-\frac{U_{x}\cdot\varphi}{vV},\end{cases}$
(4.1)

with

$\displaystyle(\varphi,\psi,S)(x,0)=(\varphi_{0},\psi_{0},S_{0}),$ (4.2)

where

$\displaystyle\varphi_{0}=v_{0}-V_{0},\psi_{0}=u_{0}-U_{0},S_{0}=\tilde{S}_{0}-\mu\frac{(U_{0})_{x}}{V_{0}}.$
(4.3)

For this reformulated problem (4.1)-(4.2), we have the following theorem of
global existence and asymptotic stabiltiy.

######  Theorem 4.1.

Let $(v_{+},u_{+})\in RR(v_{-},u_{-})$ and
$\delta=|v_{+}-v_{-}|+|u_{+}-u_{-}|$. Assume the initial data
$(\varphi_{0},\psi_{0},S_{0})\in H^{2}$ and let
$E_{0}:=\|(\varphi_{0},\psi_{0},S_{0})\|_{H^{2}}$. Then, there exists a
positive constant $\delta_{1}$ such tht if $E_{0}+\delta<\delta_{1}$, the
initial value problem (4.1)-(4.2) has a unique global solution in time
satisfying

$\displaystyle(\varphi,\psi,S)(t,x)\in C^{0}(0,+\infty,H^{2}),$ (4.4)
$\displaystyle(\varphi_{x},\psi_{x})(t,x)\in L^{2}(0,+\infty,H^{1}),\quad
S(t,x)\in L^{2}(0,+\infty,H^{2}).$ (4.5)

Moreover, this solution decay to $(0,0,0)$ uniformly in $x$ as
$t\rightarrow+\infty$:

$\displaystyle\|(\varphi,\psi,S)\|_{W^{1,\infty}}\rightarrow
0,\quad\mathrm{as}\quad t\rightarrow+\infty.$ (4.6)

To prove Theorem 4.1, the key point is to show the a priori estimate of
solutions to the problem (4.1), (4.2). To do this, we introduce the energy
norm $E(t)$ as follows

$E(t)=\sup_{0\leq s\leq
t}\|(\varphi,\psi,S)(s,\cdot)\|_{H^{2}}+\int_{0}^{t}(\|(\varphi_{x},\psi_{x},S_{x})\|_{H^{1}}+\|\sqrt{V_{t}}\varphi\|^{2}+\|S\|^{2})\mathrm{d}t.$

The a priori estimate result is stated as follows.

######  Proposition 4.2.

Let $T>0$ and $(\varphi,\psi,S)(t,x)$ be a solution to the problem (4.1),
(4.2) such that

$(\varphi,\psi,S)(t,x)\in C^{0}([0,T],H^{2})\cap C^{1}([0,T],H^{1}).$

Then, there exists a positive constant $\delta_{2}$ which is independent of
$T$ such that if

$\displaystyle E(T)+\delta<\delta_{2},$ (4.7)

then the solution $(\varphi,\psi,S)$ satisfies

$\displaystyle E(t)\leq C(E_{0}+E^{\frac{3}{2}}(t)+\delta^{\theta})$ (4.8)

for $t\in(0,T)$ with $\theta=\min\\{\frac{1}{2},\frac{3}{2}-\frac{1}{q}\\}$.

We will give the proof of Proposition 4.2 in Section 5.

Proof of Theorem 4.1. Firstly, we choose $\delta_{2}$ such that
$C\delta_{2}^{\frac{1}{2}}\leq\frac{1}{2}$. Then, under the assumption (4.7)
and using (4.8), we have

$\displaystyle E(t)\leq 2C(E_{0}+\delta^{\theta}).$ (4.9)

Now, we choose $\epsilon_{0}$ small enough such that
$2C(\epsilon_{0}+\epsilon_{0}^{\theta})+\epsilon_{0}<\frac{\delta_{2}}{2}$.
Then, we get $E(t)+\delta<\frac{\delta_{2}}{2}$ which closes the assumption
(4.7) by noting that $E_{0}+\delta<\epsilon_{0}<\frac{\delta_{2}}{2}$.
Therefore, based on the local existence theorem and the a priori estimate
(4.9), we can get a global solution in time with regularity (4.4) and (4.5) by
the classical continuation methods.

Now, we show the convergence result (4.6). Firstly, by Sobolev interpolation
theorem, we have

$\displaystyle\|(\varphi,\psi,S)\|_{L^{\infty}}\leq
C\|(\varphi,\psi,S)\|_{L^{2}}^{\frac{1}{2}}\|(\varphi_{x},\psi_{x},S_{x})\|_{L^{2}}^{\frac{1}{2}},$
$\displaystyle\|(\varphi_{x},\psi_{x},S_{x})\|_{L^{\infty}}\leq
C\|(\varphi_{x},\psi_{x},S_{x})\|_{L^{2}}^{\frac{1}{2}}\|(\varphi_{xx},\psi_{xx},S_{xx})\|_{L^{2}}^{\frac{1}{2}},$

Thus, the convergence $\|(\varphi_{x},\psi_{x},S_{x})\|_{L^{2}}\rightarrow 0$
as $t\rightarrow+\infty$ implies (4.6) immediately. Let
$\Psi(t):=\|(\varphi_{x},\psi_{x},S_{x})(t,\cdot)\|_{L^{2}}^{2}$. Then, by
(4.5), $\Psi(t)\in L^{1}(0,+\infty)$. From the system (4.1), we can easily get
$\Psi^{\prime}(t)\in L^{1}(0,+\infty)$. Therefore, we get $\Psi(t)\in
W^{1,1}(0,+\infty)$ which implies $\Psi(t)\rightarrow 0$ as
$t\rightarrow+\infty$. Thus the proof of Theorem 4.1 is complete.

Finally, by using Theorem 4.1, we are able to show Theorem 2.1 hold.

Proof of Theorem 2.1. Assume that $I_{0}+\delta$ is suitable small where

$I_{0}=\|(v_{0}-v_{0}^{r},u_{0}-u_{0}^{r})\|_{L^{2}}+\|((v_{0})_{x},(u_{0})_{x})\|_{H^{1}}+\|\tilde{S}_{0}\|_{H^{2}},$

then we have

$\displaystyle\|(\varphi_{0},\psi_{0})\|_{L^{2}}\leq\|(v_{0}-v_{0}^{r},u_{0}-u_{0}^{r})\|_{L^{2}}+\|(v_{0}^{r}-V_{0},u_{0}^{r}-U_{0})\|_{L^{2}}\leq
I_{0}+C\delta,$
$\displaystyle\|\partial_{x}(\varphi_{0},\psi_{0})\|_{H^{1}}+\|S_{0}\|_{H^{2}}\leq\|\partial_{x}(v_{0},u_{0})\|_{H^{1}}+\|\tilde{S}_{0}\|_{H^{2}}+\|\partial_{x}(V_{0},U_{0})\|_{H^{1}}+\|\mu\frac{\partial_{x}U_{0}}{V_{0}}\|_{H^{2}}\leq
I_{0}+C\delta.$

Therefore, we derive that $E_{0}+\delta\leq I_{0}+C\delta$ which can be
sufficiently small. Therefore, by using Theorem 4.1, we get a unique global
solution $(\varphi,\psi,S)$ to the problem (4.1), (4.2). Then, the functions
defined by $(v,u,\tilde{S})=(\varphi+V,\psi+U,S+\mu\frac{U_{x}}{V})$ solves
the original problem (1.1)-(1.3). To show the convergence (2.2) hold, we note
that

$\|(v-v^{r},u-u^{r},\tilde{S})\|_{L^{\infty}}\leq\|(\varphi,\psi,S)\|_{L^{\infty}}+\|(V-v^{r},U-u^{r},\mu\frac{U_{x}}{V})\|_{L^{\infty}}\rightarrow
0$

as $t\rightarrow+\infty$, where we used the convergence result (4.6) and Lemma
3.2. This completes the proof of Theorem 2.1.

##  5\. Proof of Proposition 4.2

In the following lemmas, we always assume that $E(t)+\delta\leq\delta_{2}$ for
some small $\delta_{2}$ and $\epsilon=\delta^{3}$. In particular, we have
$0<V_{-}\leq V\leq V_{+}$ and $0<v_{-}\leq v\leq v_{+}$. Firstly, we get the
following $L^{2}$ estimates.

######  Lemma 5.1.

There exists a constant $C$ such that for $0\leq t\leq T$,

$\displaystyle\|\varphi\|_{L^{2}}^{2}+\|\psi\|_{L^{2}}^{2}+\|S\|_{L^{2}}^{2}+\int_{0}^{t}(\|\sqrt{V_{t}}\varphi\|_{L^{2}}^{2}+\|S\|_{L^{2}}^{2})\mathrm{d}t\leq
C(E_{0}+\delta^{\theta}),$ (5.1)

where $\theta=\min\\{\frac{1}{2},\frac{3}{2}-\frac{1}{q}\\}.$

###### Proof.

Multiplying the equations $\eqref{4.1}_{1},\eqref{4.1}_{2},\eqref{4.1}_{3}$ by
$p(V)-p(\varphi+V)$, $\psi$ and $\frac{v}{\mu}S$, respectively, and
integrating over $\mathbb{R}$ with respect to $x$, we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}E_{1}(\varphi,\psi,S)+E_{2}(\varphi,S)=\int_{\mathbb{R}}\left(\mu\psi\left(\frac{U_{x}}{V}\right)_{x}-\psi
g(V)_{x}+\tau vS\left(\frac{U_{x}}{V}\right)_{t}+\frac{\tau
v_{t}}{2\mu}S^{2}\right)\mathrm{d}x,$ (5.2)

where

$E_{1}(\varphi,\psi,S)=\int_{\mathbb{R}}\left(p(V)\varphi-\int_{V}^{V+\varphi}p(\xi)\mathrm{d}\xi+\frac{1}{2}\psi^{2}+\frac{\tau
V}{2\mu}S^{2}\right)\mathrm{d}x$

and

$E_{2}(\varphi,S)=\int_{\mathbb{R}}\left(\frac{v}{\mu}S^{2}-\frac{V_{t}\varphi
S}{V}+(p(V+\varphi)-p(V)-p^{\prime}(V)\varphi)V_{t}\right)\mathrm{d}x.$

Note that

$p(V)\varphi-\int_{V}^{V+\varphi}p(\xi)\mathrm{d}\xi\geq C\varphi^{2},$

since $p(\cdot)$ is a convex function on $(0,+\infty)$. Therefore, we derive
that

$\displaystyle E_{1}(\varphi,\psi,S)\geq C\|(\varphi,\psi,S)\|_{L^{2}}^{2}.$
(5.3)

On the other hand, let
$p(V+\varphi)-p(V)-p^{\prime}(V)\varphi=f(v,V)\varphi^{2}$, then $f(v,V)\geq
C>0$. Since $V_{t}\leq C\epsilon\delta$ where $\epsilon$ and $\delta$ are
sufficiently small, we have

$\displaystyle\frac{v}{\mu}S^{2}-\frac{V_{t}\varphi
S}{V}+(p(V+\varphi)-p(V)-p^{\prime}(V)\varphi)V_{t}\geq
C(S^{2}+V_{t}\varphi^{2}).$ (5.4)

Now, we estimate the right-hand-side of (5.2). Firstly, using Lemma 3.2, we
get

$\displaystyle\int_{\mathbb{R}}\left|\mu\psi\left(\frac{U_{x}}{V}\right)_{x}\right|\mathrm{d}x\leq
C\int_{\mathbb{R}}\left(|\psi U_{xx}|+|\psi U_{x}V_{x}|\right)\mathrm{d}x$
$\displaystyle\leq
C\|\psi\|_{L^{2}}\left(\|U_{xx}\|_{L^{2}}+\|U_{x}\|_{L^{4}}\|V_{x}\|_{L^{4}}\right)\leq
C\left(\delta^{\frac{3}{2}-\frac{1}{q}}(1+t)^{-1-\frac{1}{4q}}+\delta^{\frac{1}{2}}(1+t)^{-\frac{3}{2}}\right)\|\psi\|_{L^{2}}$

and

$\displaystyle\int_{\mathbb{R}}|\psi
g(V)_{x}|\mathrm{d}x\leq\|\psi\|_{L^{2}}\|g(V)_{x}\|_{L^{2}}.$

Moreover, using (3.6) and Lemma 3.2, we have

$\displaystyle\int_{\mathbb{R}}\left|\tau
vS\left(\frac{U_{x}}{V}\right)_{t}\right|\mathrm{d}x$ $\displaystyle\leq
C\int_{\mathbb{R}}S(|g(V)_{xx}|+|p(V)_{xx}|+U_{x}^{2})\mathrm{d}x$
$\displaystyle\leq\varepsilon\int_{\mathbb{R}}S^{2}\mathrm{d}x+C(\varepsilon)(\|V_{x}\|_{L^{4}}^{4}+\|V_{xx}\|_{L^{2}}^{2}+\|U_{x}\|_{L^{4}}^{4})$
$\displaystyle\leq
C\varepsilon\int_{\mathbb{R}}S^{2}\mathrm{d}x+C\delta(1+t)^{-3}+(\delta^{\frac{3}{2}-\frac{1}{q}}(1+t)^{-1-\frac{1}{4q}}+\delta^{\frac{1}{2}}(1+t)^{-\frac{3}{2}})^{2}.$

Besides, note that
$\|v_{t}\|_{L^{\infty}}=\|V_{t}+\varphi_{t}\|_{L^{\infty}}\leq
C(\epsilon\delta+\delta_{1})$, we derive that

$\displaystyle\int_{\mathbb{R}}\frac{\tau v_{t}}{2\mu}S^{2}\mathrm{d}x\leq
C(\delta^{4}+\delta_{2})\int_{\mathbb{R}}S^{2}\mathrm{d}x,$

which can be absorbed by $E_{2}(\varphi,S)$ for suitable small $\delta$ and
$\delta_{2}$. Combining the above estimates, and integrating (5.2) over
$(0,t)$, we derive that

$\displaystyle\|\varphi\|_{L^{2}}^{2}+\|\psi\|_{L^{2}}^{2}+\|S\|_{L^{2}}^{2}+\int_{0}^{t}(\|\sqrt{V_{t}}\varphi\|_{L^{2}}^{2}+\|S\|_{L^{2}}^{2})\mathrm{d}s$
$\displaystyle\leq
CE_{0}+C\int_{0}^{t}(\delta^{\frac{1}{2}}(1+t)^{-\frac{3}{2}}+\delta^{\frac{3}{2}-\frac{1}{q}}(1+t)^{-1-\frac{1}{4q}}+\|g(V)_{x}\|_{L^{2}})\|\psi\|_{L^{2}}\mathrm{d}s+C\delta+C\delta^{3-\frac{2}{q}}$
$\displaystyle\leq C(E_{0}+\delta^{\theta}),$

where we have used the fact that
$\int_{0}^{+\infty}\|g(V)_{x}\|_{L^{2}}\mathrm{d}t\leq
C\delta^{2}\epsilon^{-\frac{1}{2}}=C\delta^{\frac{1}{2}}$ from Lemma 3.2.
Therefore, the proof of this lemma is completed. ∎

######  Lemma 5.2.

There exists a constant $C$ such that for $0\leq t\leq T$, we have

$\displaystyle\|\varphi_{x}\|_{H^{1}}^{2}+\|\psi_{x}\|_{H^{1}}^{2}+\|S_{x}\|_{H^{1}}^{2}+\int_{0}^{t}\|S_{x}\|_{H^{1}}^{2}\mathrm{d}t\leq
C(E_{0}+\delta^{\theta}+E^{\frac{3}{2}}(t)).$ (5.5)

###### Proof.

By applying $\partial_{x}^{k}(k=1,2)$ to the equations (4.1), we derive the
following system

$\displaystyle\begin{cases}\partial_{t}\partial_{x}^{k}\varphi=\partial_{x}^{k+1}\psi,\\\
\partial_{t}\partial_{x}^{k}\psi+\partial_{x}^{k}(p^{\prime}(\varphi+V)\varphi_{x})+\partial_{x}^{k}((p^{\prime}(\varphi+V)-p^{\prime}(V))V_{x})-\partial_{x}^{k+1}S=\mu\partial_{x}^{k+1}\left(\frac{U_{x}}{V}\right)-\partial_{x}^{k+1}g(V),\\\
\tau\partial_{t}\partial_{x}^{k}S+\partial_{x}^{k}S-\partial_{x}^{k}\left(\frac{\mu}{v}\psi_{x}\right)=-\tau\mu\partial_{t}\partial_{x}^{k}\left(\frac{U_{x}}{V}\right)-\partial_{x}^{k}\left(\frac{U_{x}\varphi}{vV}\right).\end{cases}$
(5.6)

Multiplying the above equations by
$-p^{\prime}(\varphi+V)\partial_{x}^{k}\varphi$, $\partial_{x}^{k}\psi$,
$\frac{v}{\mu}\partial_{x}^{k}S$, respectively, and integrating over
$\mathbb{R}$, we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}}\left(-\frac{1}{2}p^{\prime}(\varphi+V)(\partial_{x}^{k}\varphi)^{2}+\frac{1}{2}(\partial_{x}^{k}\psi)^{2}+\frac{\tau
v}{2\mu}(\partial_{x}^{k}S)^{2}\right)\mathrm{d}x+\int_{\mathbb{R}}\left(\frac{v}{\mu}-\frac{\tau}{2\mu}\partial_{t}v\right)(\partial_{x}^{k}S)^{2}\mathrm{d}x=:\sum\limits_{{j}={1}}^{8}R_{k}^{j},$

where

$R_{k}^{1}=-\int_{\mathbb{R}}\frac{1}{2}p^{\prime\prime}(\varphi+V)((\varphi_{t}+V_{t})(\partial_{x}^{k}\varphi)^{2}+(\varphi_{x}+V_{x})\partial_{x}^{k}\psi\partial_{x}^{k}\varphi)\mathrm{d}x,$
$R_{k}^{2}=-\int_{\mathbb{R}}\left(\partial_{x}^{k}(p^{\prime}(\varphi+V)\varphi_{x})-p^{\prime}(\varphi+V)\partial_{x}^{k+1}\varphi\right)\partial_{x}^{k}\psi\mathrm{d}x,$
$R_{k}^{3}=-\int_{\mathbb{R}}\partial_{x}^{k}((p^{\prime}(\varphi+V)-p^{\prime}(V))V_{x})\partial_{x}^{k}\psi\mathrm{d}x,\quad
R_{k}^{4}=\int_{\mathbb{R}}\mu\partial_{x}^{k+1}\left(\frac{U_{x}}{V}\right)\partial_{x}^{k}\psi\mathrm{d}xV_{x},$
$R_{k}^{5}=-\int_{\mathbb{R}}\partial_{x}^{k+1}g(V)\partial_{x}^{k}\psi\mathrm{d}x,R_{k}^{6}=-\int_{\mathbb{R}}\tau\mu\partial_{t}\partial_{x}^{k}\left(\frac{U_{x}}{V}\right)\partial_{x}^{k}S\mathrm{d}x,R_{k}^{7}=\int_{\mathbb{R}}\partial_{x}^{k}\left(\frac{U_{x}\varphi}{vV}\right)\partial_{x}^{k}S\mathrm{d}x,$
$R_{k}^{8}=\int_{\mathbb{R}}(\partial_{x}^{k}(\frac{\mu}{v}\psi_{x})-\frac{\mu}{v}\partial_{x}^{k+1}\psi)\frac{v}{\mu}\partial_{x}^{k}S\mathrm{d}x.$

We shall show that for $1\leq j\leq 8,k=1,2$,

$\displaystyle\int_{0}^{t}R_{k}^{j}(t)\mathrm{d}t\leq
C(\delta^{\theta}+E^{\frac{3}{2}}(t))+\varepsilon\int_{0}^{t}\|S_{x}\|_{H^{1}}^{2}\mathrm{d}t+C\int_{0}^{t}(1+t)^{-d}(\|\varphi_{x}\|_{H^{1}}^{2}+\|\psi_{x}\|_{H^{1}}^{2})\mathrm{d}t.$
(5.7)

Firstly,

$\displaystyle\int_{0}^{t}R_{k}^{1}\mathrm{d}t\leq
C\int_{0}^{t}\int_{\mathbb{R}}(|\psi_{x}|+|\varphi_{x}|+|V_{x}|)((\partial_{x}^{k}\varphi)^{2}+(\partial_{x}^{k}\psi)^{2})\mathrm{d}x\mathrm{d}t\leq
C(E^{\frac{3}{2}}(t)+\delta^{4}).$

Secondly, we have for $k=1$ that

$\displaystyle\int_{0}^{t}R_{1}^{2}(t)\mathrm{d}t=\int_{0}^{t}\int_{\mathbb{R}}p^{\prime\prime}(\varphi+V)(\varphi_{x}+V_{x})\varphi_{x}\psi_{x}\mathrm{d}x\mathrm{d}t\leq
C(E^{\frac{3}{2}}(t)+\delta^{4}).$

For $k=2$,

$\displaystyle\int_{0}^{t}R_{2}^{2}(t)\mathrm{d}t$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\left(p^{\prime\prime\prime}(\varphi+V)(\varphi_{x}+V_{x})^{2}\varphi_{x}+p^{\prime\prime}(\varphi+V)(3\varphi_{xx}\varphi_{x}+V_{xx}\varphi_{x}+2V_{x}\varphi_{xx})\right)\psi_{xx}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{t}\|(\varphi_{x},V_{x},V_{xx})\|_{L^{\infty}}(\|\varphi_{x}\|_{H^{1}}^{2}+\|\psi_{x}\|_{H^{1}}^{2})\mathrm{d}t$
$\displaystyle\leq
C(E^{\frac{3}{2}}(t)+\delta^{4}+\delta^{3-\frac{2}{q}})+C\int_{0}^{t}(1+t)^{-2}(\|\varphi_{x}\|_{H^{1}}^{2}+\|\psi_{x}\|_{H^{1}}^{2})\mathrm{d}t.$

Similarly, for $k=1$, we have

$\displaystyle\int_{0}^{t}R_{1}^{3}(t)\mathrm{d}t$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\left((p^{\prime\prime}(\varphi+V)(\varphi_{x}+V_{x})-p^{\prime\prime}(V)V_{x})V_{x}+(p^{\prime}(\varphi+V)-p^{\prime}(V))V_{xx}\right)\psi_{x}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{t}\left(\|V_{x}\|_{L^{\infty}}(\|\varphi_{x}\|_{H^{1}}^{2}+\|\psi_{x}\|_{H^{1}}^{2})+(\|V_{x}\|_{L^{4}}^{2}+\|V_{xx}\|_{L^{2}})\|\partial_{x}\psi\|_{L^{2}}\right)\mathrm{d}t$
$\displaystyle\leq
C(\delta^{4}+\delta^{\frac{3}{2}-\frac{1}{q}}+\delta^{\frac{1}{2}})$

and for $k=2$,

$\displaystyle\int_{0}^{t}R_{2}^{3}(t)\mathrm{d}t$ $\displaystyle\leq
C\int_{0}^{t}\left((E^{\frac{1}{2}}(t)+\|V_{x}\|_{L^{\infty}}+\|V_{xx}\|_{L^{\infty}})(\|\varphi_{x}\|_{H^{1}}^{2}+\|\psi_{x}\|_{H^{1}}^{2})+C(\|(V_{x})^{3}\|_{L^{2}}+\|V_{xx}V_{x}\|_{L^{2}})\|\psi_{xx}\|_{L^{2}}\right)\mathrm{d}t$
$\displaystyle\leq
C(E^{\frac{3}{2}}(t)+\delta^{4}+\delta^{3-\frac{2}{q}}+\delta^{\frac{1}{2}}+\delta^{\frac{11}{2}-\frac{1}{q}}).$

Now, we estimate $R_{k}^{4}$. For $k=1$,

$\displaystyle\int_{0}^{t}R_{1}^{4}(t)\mathrm{d}t=\int_{0}^{t}\int_{\mathbb{R}}\mu\left(\frac{U_{x}}{V}\right)_{xx}\psi_{x}\mathrm{d}x$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\mu\left(\frac{U_{xxx}}{V}-\frac{2U_{xx}V_{x}}{V^{2}}-\frac{U_{x}V_{xx}}{V^{2}}+\frac{2V_{x}^{2}U_{x}}{V^{3}}\right)\psi_{x}\mathrm{d}x$
$\displaystyle\leq
C\int_{0}^{t}(\|U_{xxx}\|+\|U_{xx}\|_{L^{4}}\|V_{x}\|_{L^{4}}+\|V_{xx}\|_{L^{4}}\|U_{x}\|_{L^{4}}+\|V_{x}^{2}\|_{L^{4}}\|U_{x}\|_{L^{4}})\|\psi_{x}\|\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{t}(\delta^{\frac{3}{2}-\frac{1}{q}}(1+t)^{-1-\frac{1}{4q}}+\delta^{\frac{1}{2}}(1+t)^{-\frac{5}{2}})\|\psi_{x}\|\mathrm{d}t\leq
C(\delta^{\frac{3}{2}-\frac{1}{q}}+\delta^{\frac{1}{2}}).$

For $k=2$, some tedious calculations give

$\displaystyle\partial_{x}^{3}\left(\frac{U_{x}}{V}\right)$
$\displaystyle=\frac{\partial_{x}^{4}U}{V}-\frac{3\partial_{x}^{3}UV_{x}+3U_{xx}V_{xx}+U_{x}\partial_{x}^{3}V}{V^{2}}+\frac{6U_{xx}(V_{x})^{2}+6U_{x}V_{xx}V_{x}}{V^{3}}-\frac{6U_{x}(V_{x})^{3}}{V^{4}}.$

So, we have

$\displaystyle\int_{0}^{t}R_{2}^{4}(t)\mathrm{d}t=\int_{0}^{t}\int_{\mathbb{R}}\mu\partial_{x}^{3}\left(\frac{U_{x}}{V}\right)\psi_{xx}\mathrm{d}x$
$\displaystyle\leq
C\int_{0}^{t}\left(\|\partial_{x}^{4}U\|_{L^{2}}+\|\partial_{x}^{3}UV_{x}\|_{L^{2}}+\|U_{xx}V_{xx}\|_{L^{2}}+\|U_{x}\partial_{x}^{3}V\|_{L^{2}}+\|U_{x}V_{xx}V_{x}\|_{L^{2}}+\|U_{x}(V_{x})^{3}\|_{L^{2}}\right)\|\psi_{xx}\|_{L^{2}}\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{t}(\delta^{\frac{3}{2}-\frac{1}{q}}(1+t)^{-1-\frac{1}{4q}}+\delta^{\frac{1}{2}}(1+t)^{-\frac{3}{2}}+\delta^{\frac{9}{2}-\frac{3}{q}}(1+t)^{-2-\frac{3}{4q}}+\delta^{\frac{9}{2}}(1+t)^{-\frac{5}{2}})\|\psi_{xx}\|_{L^{2}}\mathrm{d}t$
$\displaystyle\leq C(\delta^{\frac{3}{2}-\frac{1}{q}}+\delta^{\frac{1}{2}}).$

Similarly, for $k=1,2$, we can get

$\displaystyle\int_{0}^{t}R_{k}^{5}\mathrm{d}t$ $\displaystyle\leq
C\int_{0}^{t}(\|((V_{x})^{2},(V_{x})^{3})\|_{L^{2}}+\|(V_{xx},\partial_{x}^{3}V)\|_{L^{2}}+\|V_{x}V_{xx}\|_{L^{2}})\|(\partial_{x},\partial_{x}^{2})\psi\|_{L^{2}}\mathrm{d}t$
$\displaystyle\leq
C\int_{0}^{t}(\delta^{\frac{3}{2}-\frac{1}{q}}(1+t)^{-1-\frac{1}{4q}}+\delta^{\frac{1}{2}}(1+t)^{-\frac{3}{2}})\|(\partial_{x},\partial_{x}^{2})\psi\|_{L^{2}}\mathrm{d}t\leq
C(\delta^{\frac{3}{2}-\frac{1}{q}}+\delta^{\frac{1}{2}}).$

On the other hand, $\int_{0}^{t}R_{k}^{6}(t)\mathrm{d}t$ and
$\int_{0}^{t}R_{k}^{7}(t)\mathrm{d}t$ are estimated as follows. Note that for
$k=1$, we have

$\displaystyle\left(\frac{U_{x}}{V}\right)_{tx}=\frac{(U_{xxt}V+U_{xt}V_{x}-U_{xx}V_{t}-U_{x}V_{tx})V^{2}-(U_{xt}V-U_{x}V_{t})2VV_{x}}{V^{4}},$
$\displaystyle\left(\frac{U_{x}\varphi}{Vv}\right)_{x}=\frac{(U_{xx}\varphi+U_{x}\varphi_{x})Vv-
U_{x}\varphi(V_{x}v+Vv_{x})}{V^{2}v^{2}}.$

So, we derive that

$\displaystyle\int_{0}^{t}R_{1}^{6}(t)\mathrm{d}t\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{R}}\left|\left(\frac{U_{x}}{V}\right)_{tx}^{2}\right|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{R}}|V_{xxx}+V_{xx}V_{x}+V_{x}^{3}+U_{xx}U_{x}+U_{x}^{2}V_{x}|^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)\int_{0}^{t}(\|V_{x}\|_{L^{6}}^{6}+\|V_{x}\|_{L^{4}}^{2}(\|V_{xx}\|_{L^{4}}^{2}+\|U_{xx}\|_{L^{4}}^{2}+\|U_{x}\|_{L^{8}}^{4})+\|V_{xxx}\|_{L^{2}}^{2})\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+\int_{0}^{t}(\delta(1+t)^{-3}+\delta^{5-\frac{3}{q}}(1+t)^{-\frac{7}{2}-\frac{3}{4q}}+\delta^{3-\frac{2}{q}}(1+t)^{-2-\frac{1}{2q}})\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+\delta+\delta^{3-\frac{2}{q}}.$

and

$\displaystyle\int_{0}^{t}R_{1}^{7}(t)\mathrm{d}t\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{R}}|U_{xx}\varphi+U_{x}\varphi_{x}+U_{x}V_{x}\varphi+U_{x}\varphi\varphi_{x}|^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)((E^{\frac{3}{2}}(t)+\delta^{\frac{1}{2}}+\delta^{3-\frac{2}{q}})+\int_{0}^{t}(1+t)^{-2}\|\varphi_{x}\|_{L^{2}}^{2}\mathrm{d}t).$

The estimates for $\int_{0}^{t}R_{2}^{6}(t)\mathrm{d}t$ and
$\int_{0}^{t}R_{2}^{7}(t)\mathrm{d}t$ can be done in a similar way, we omit
the details. Finally, let’s estimate the
term$\int_{0}^{t}R_{k}^{8}(t)\mathrm{d}t$. Firstly, for $k=1$, we have

$\displaystyle\int_{0}^{t}R_{1}^{8}(t)\mathrm{d}t$
$\displaystyle=-\int_{0}^{t}\int_{\mathbb{R}}\frac{v_{x}\psi_{x}S_{x}}{v}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{R}}(V_{x}^{2}+\varphi_{x}^{2})\psi_{x}^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{x}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)(E^{2}(t)+\delta^{8}).$

and for $k=2$,

$\displaystyle\int_{0}^{t}R_{2}^{8}(t)\mathrm{d}t$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\frac{2v_{x}^{2}\partial_{x}\psi-2vv_{x}\psi_{xx}}{v^{2}}S_{xx}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{xx}^{2}+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{R}}((|V_{x}|^{4}+|\varphi_{x}|^{4})|\psi_{x}|^{2}+(|V_{x}|^{2}+|\varphi_{x}|^{2})|\psi_{xx}|^{2})\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}\frac{v}{\mu}S_{xx}^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)(E^{2}(t)+\delta^{8}).$

Combining the above estimates and using the Gronwall’s inequality, we get the
desired results. ∎

The next lemma gives the dissipative estimates of $\varphi_{x}$ and
$\psi_{x}$.

######  Lemma 5.3.

There exists a constant $C$ such that for $0\leq t\leq T$, we have

$\displaystyle\int_{0}^{t}(\|\varphi_{x}\|_{H^{1}}^{2}+\|\psi_{x}\|_{H^{1}}^{2})\mathrm{d}t\leq
C(E_{0}+\delta^{\theta}+E^{\frac{3}{2}}(t)).$ (5.8)

###### Proof.

Multiplying the equation $\eqref{5.6}_{2}$ by $\partial_{x}^{k+1}\varphi$ for
$k=0$ or $1$, and integrating over $(0,t)\times\mathbb{R}$, we get

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}-p^{\prime}(\varphi+V)(\partial_{x}^{k+1}\varphi)^{2}\mathrm{d}x\mathrm{d}t=:\sum\limits_{{j}={1}}^{6}M_{k}^{j},$

where

$M_{k}^{1}=\int_{0}^{t}\int_{\mathbb{R}}\partial_{t}\partial_{x}^{k}\psi\cdot\partial_{x}^{k+1}\varphi\mathrm{d}x\mathrm{d}t,M_{k}^{2}=\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k}(p^{\prime}(\varphi+V)\varphi_{x})-p^{\prime}(\varphi+V)\partial_{x}^{k+1}\varphi)\cdot\partial_{x}^{k+1}\varphi\mathrm{d}x\mathrm{d}t,$
$M_{k}^{3}=\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{k}((p^{\prime}(\varphi+V)-p^{\prime}(V))V_{x})\cdot\partial_{x}^{k+1}\varphi\mathrm{d}x\mathrm{d}t,M_{k}^{4}=\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{k+1}S\cdot\partial_{x}^{k+1}\varphi\mathrm{d}x\mathrm{d}t,$
$M_{k}^{5}=-\int_{0}^{t}\int_{\mathbb{R}}\mu\partial_{x}^{k+1}\left(\frac{U_{x}}{V}\right)\partial_{x}^{k+1}\varphi\mathrm{d}x\mathrm{d}t,M_{k}^{6}=\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{k+1}g(V)\partial_{x}^{k+1}\varphi\mathrm{d}x\mathrm{d}t.$

We shall show that

$\displaystyle\sum\limits_{{j}={1}}^{6}M_{k}^{j}\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}\varphi)^{2}\mathrm{d}x\mathrm{d}t+C\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}\psi)^{2}\mathrm{d}x\mathrm{d}t+C(E^{\frac{3}{2}}(t)+E_{0}+\delta^{\theta}).$
(5.9)

Firstly, by doing integration by part and using equation $\eqref{5.6}_{1}$, we
get

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\partial_{t}\partial_{x}^{k}\psi\partial_{x}^{k+1}\varphi\mathrm{d}x\mathrm{d}t$
$\displaystyle=\int_{\mathbb{R}}\partial_{x}^{k}\psi(t)\partial_{x}^{k+1}\varphi(t)\mathrm{d}x-\int_{\mathbb{R}}\partial_{x}^{k}\psi_{0}\partial_{x}^{k+1}\varphi_{0}\mathrm{d}x+\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{k+1}\psi\partial_{t}\partial_{x}^{k}\varphi\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}\psi)^{2}\mathrm{d}x\mathrm{d}t+C(E^{\frac{3}{2}}(t)+\delta^{\theta}+E_{0}).$

Secondly, for $k=0$, $M_{0}^{2}$ vanishes and for $k=1$,

$\displaystyle M_{1}^{2}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}p^{\prime\prime}(\varphi+V)(\varphi_{x}+V_{x})\varphi_{x}\varphi_{xx}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{xx})^{2}\mathrm{d}x\mathrm{d}t+C\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{x}^{4}+V_{x}^{2}\varphi_{x}^{2})\mathrm{d}x$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{xx})^{2}\mathrm{d}x\mathrm{d}t+C(E^{2}(t)+\delta^{8}).$

Similarly, we get for $k=0$,

$\displaystyle M_{0}^{3}$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{x})^{2}\mathrm{d}x\mathrm{d}t+C\int_{0}^{t}\int_{\mathbb{R}}\varphi^{2}V_{x}^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{x})^{2}\mathrm{d}x\mathrm{d}t+\sup\|\varphi\|_{L^{2}}^{2}\int_{0}^{t}\|V_{x}\|_{L^{\infty}}^{2}\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{x})^{2}\mathrm{d}x\mathrm{d}t+C(E_{0}+\delta),$

and $k=1$,

$\displaystyle M_{1}^{3}$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{xx})^{2}\mathrm{d}x\mathrm{d}t+\int_{0}^{t}\int_{\mathbb{R}}(V_{xx}^{2}+V_{x}^{4}+\varphi_{x}^{2}V_{x}^{2})\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{xx})^{2}\mathrm{d}x\mathrm{d}t+C(\delta^{3-\frac{2}{q}}+\delta).$

By using the dissipation of $S_{x}$, we have

$\displaystyle M_{k}^{4}$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}\varphi)^{2}\mathrm{d}x\mathrm{d}t+C\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}S)^{2}\mathrm{d}\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}\varphi)^{2}\mathrm{d}x\mathrm{d}t+C(E^{\frac{3}{2}}(t)+E_{0}+\delta^{\theta}).$

$M_{k}^{5}$ and $M_{k}^{6}$ can be estimated in the same way as in Lemme 5.2.
Indeed, we have

$\displaystyle M_{0}^{5}$ $\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{R}}(|\partial_{xx}U|+|U_{x}||V_{x}|)|\varphi_{x}|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}|\varphi_{x}|^{2}\mathrm{d}x\mathrm{d}t+C\int_{0}^{t}(\|\partial_{xx}U\|_{L^{2}}^{2}+\|U_{x}|_{L^{4}}^{4}+|V_{x}|_{L^{4}}^{4})\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}|\varphi_{x}|^{2}\mathrm{d}x\mathrm{d}t+C(\delta^{3-\frac{2}{q}}+\delta),$

and

$\displaystyle M_{1}^{5}$ $\displaystyle\leq
C\int_{0}^{t}(\|U_{xxx}\|+\|U_{xx}\|_{L^{4}}\|V_{x}\|_{L^{4}}+\|V_{xx}\|_{L^{4}}\|U_{x}\|_{L^{4}}+\|V_{x}^{2}\|_{L^{4}}\|U_{x}\|_{L^{4}})\|\partial_{xx}\varphi\|\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\partial_{xx}\varphi)^{2}\mathrm{d}x\mathrm{d}t+C(\delta^{3-\frac{2}{q}}+\delta^{5-\frac{3}{q}}+\delta).$

Note that $\int_{0}^{t}\|\partial_{x}g(V)\|_{L^{2}}\mathrm{d}t\leq
C\delta^{\frac{1}{2}}$, we get

$M_{0}^{6}\leq\sup\|\varphi_{x}\|_{L^{2}}\int_{0}^{t}\|\partial_{x}g(V)\|_{L^{2}}\mathrm{d}t\leq
C\delta^{\frac{1}{2}}.$

Besides, we have

$\displaystyle
M_{1}^{6}\leq\int_{0}^{t}\int_{\mathbb{R}}(|V_{xx}|+|V_{x}|^{2})|\varphi_{xx}|\mathrm{d}x\mathrm{d}t\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\varphi_{xx})^{2}\mathrm{d}x\mathrm{d}t+C(\delta^{3-\frac{2}{q}}+\delta).$

Combining the above estimates, summing up $k$ from $0$ to $1$, we finally get

$\displaystyle\int_{0}^{t}\|\varphi_{x}\|_{H^{1}}^{2}\mathrm{d}t\leq
C\int_{0}^{t}\|\psi_{x}\|_{H^{1}}^{2}\mathrm{d}t+C(E^{\frac{3}{2}}(t)+E_{0}+\delta^{\theta}).$
(5.10)

Now, we derive the dissipative estimates of $\psi_{x}$. Multiplying the
equation by $\partial_{x}^{k+1}\psi$, we derive that

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}(\partial_{x}^{k+1}\psi)^{2}\mathrm{d}x\mathrm{d}t=:\sum\limits_{{j}={1}}^{5}N_{k}^{j},$

where

$N_{k}^{1}=\int_{0}^{t}\int_{\mathbb{R}}\tau\partial_{t}\partial_{x}^{k}S\partial_{x}^{k+1}\psi\mathrm{d}x\mathrm{d}t,N_{k}^{2}=\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{k}S\partial_{x}^{k+1}\psi\mathrm{d}x\mathrm{d}t,$
$N_{k}^{3}=-\int_{0}^{t}\int_{\mathbb{R}}\left(\partial_{x}^{k}\left(\frac{\mu}{v}\partial_{x}\psi\right)-\frac{\mu}{v}\partial_{x}^{k+1}\psi\right)\partial_{x}^{k+1}\psi\mathrm{d}x\mathrm{d}t,$
$N_{k}^{4}=\int_{0}^{t}\int_{\mathbb{R}}\tau\mu\partial_{t}\partial_{x}^{k}\left(\frac{U_{x}}{V}\right)\partial_{x}^{k+1}\psi\mathrm{d}x\mathrm{d}t,N_{k}^{5}=\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{k}\left(\frac{U_{x}\varphi}{vV}\right)\partial_{x}^{k+1}\psi\mathrm{d}x\mathrm{d}t.$

we shall show that

$\displaystyle\sum\limits_{{j}={1}}^{5}N_{k}^{j}\leq\frac{1}{2}\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}(\partial_{x}^{k+1}\psi)^{2}\mathrm{d}x\mathrm{d}t+\varepsilon\int_{0}^{t}\|\partial_{x}\varphi\|_{H^{1}}^{2}\mathrm{d}t+C(E^{\frac{3}{2}}+E_{0}+\delta^{\theta}),$
(5.11)

which implies that

$\displaystyle\int_{0}^{t}\|\psi_{x}\|_{H^{1}}^{2}\mathrm{d}t\leq\varepsilon\int_{0}^{t}\|\varphi_{x}\|_{H^{1}}^{2}\mathrm{d}t+C(E^{\frac{3}{2}}(t)+E_{0}+\delta^{\theta}).$
(5.12)

Combining (5.10) and (5.12) together, we get the desired result. Therefore, we
only need to show (5.11) hold.

Firstly, using Lemma 5.2 and integrating by part with respect to $x$ and $t$,
we have

$\displaystyle N_{k}^{1}\leq
C(E^{\frac{3}{2}}(t)+\delta^{\theta}+E_{0})+\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}^{k+1}S\partial_{t}\partial_{x}^{k}\psi\mathrm{d}x\mathrm{d}t,$

while, using equation $\eqref{5.6}_{2}$ and Lemma 5.2, and exploiting the same
methods as above (estimates of $M_{k}^{j},j=2,\dots 6$, for example), we get

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\tau\partial_{x}^{k+1}S\partial_{t}\partial_{x}^{k}\psi\mathrm{d}x\mathrm{d}t$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\tau\partial_{x}^{k+1}S\left(-\partial_{x}^{k}(p^{\prime}(\varphi+V)\varphi_{x})-\partial_{x}^{k}((p^{\prime}(\varphi+V)-p^{\prime}(V))V_{x})\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\tau\partial_{x}^{k+1}S+\mu\partial_{x}^{k+1}\left(\frac{U_{x}}{V}\right)-\partial_{x}^{k+1}g(V)\right)\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}\varphi)^{2}\mathrm{d}x\mathrm{d}t+C(\varepsilon)\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}S)^{2}\mathrm{d}x\mathrm{d}t+C(E^{\frac{3}{2}}(t)+\delta^{\theta}+E_{0})$
$\displaystyle\leq\varepsilon\int_{0}^{t}\int_{\mathbb{R}}(\partial_{x}^{k+1}\varphi)^{2}\mathrm{d}x\mathrm{d}t+C(E^{\frac{3}{2}}(t)+\delta^{\theta}+E_{0}),$

which gives the estimates of $N_{k}^{1}$. Moreover, it’s easy to see that

$\displaystyle N_{k}^{2}$
$\displaystyle\leq\frac{1}{8}\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}(\partial_{x}^{k+1}\psi)^{2}\mathrm{d}x\mathrm{d}t+C\int_{0}^{t}\|S\|_{H^{1}}^{2}\mathrm{d}t$
$\displaystyle\leq\frac{1}{8}\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}(\partial_{x}^{k+1}\psi)^{2}\mathrm{d}x\mathrm{d}t+C(E^{\frac{3}{2}}(t)+\delta^{\theta}+E_{0}).$

Now, we estimate the term $N_{k}^{j},j=3,4,5$. Note that, for $k=0$,
$N_{0}^{3}$ vanishes, and for $k=1$,

$\displaystyle N_{1}^{3}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu
v_{x}}{v^{2}}\psi_{x}\psi_{xx}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\frac{1}{8}\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}(\psi_{xx})^{2}\mathrm{d}x\mathrm{d}t+C\int_{0}^{t}\int_{\mathbb{R}}(|V_{x}|^{2}+|\varphi_{x}|^{2})|\psi_{x}|^{2}\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\frac{1}{8}\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}(\psi_{xx})^{2}\mathrm{d}x\mathrm{d}t+C(E^{2}(t)+\delta^{8}).$

On the other hand, for $k=0$, using the equation (3.6) and Lemma 3.2, we have

$\displaystyle N_{0}^{4}+N_{0}^{5}$ $\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{R}}\left(|g(V)_{xx}|+|p(V)_{xx}|+|U_{x}|^{2}+|U_{x}||\varphi|\right)|\psi_{x}|\mathrm{d}x\mathrm{d}t$
$\displaystyle\leq\frac{1}{8}\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}|\psi_{x}|^{2}\mathrm{d}x\mathrm{d}t+C(E_{0}+\delta+\delta^{3-\frac{2}{q}}),$

and for $k=1$, comparing with the estimates of $R_{1}^{6}$ and $R_{1}^{7}$, we
do have

$\displaystyle
N_{1}^{4}+N_{1}^{5}\leq\frac{1}{8}\int_{0}^{t}\int_{\mathbb{R}}\frac{\mu}{v}|\psi_{xx}|^{2}\mathrm{d}x\mathrm{d}t+C(E^{\frac{3}{2}}(t)+E_{0}+\delta^{\theta}).$

This finish the proof of the Lemma. ∎

Combining the results of Lemma 5.1-5.3, the proof of the Proposition 4.2 is
finished.

Acknowledgement: Yuxi Hu’s research is supported by NNSFC (Grant No. 11701556)
and Yue Qi Young Scholar project, China University of Mining and Technology
(Beijing).

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Yuxi Hu, Department of Mathematics, China University of Mining and Technology,
Beijing, 100083, P.R. China<EMAIL_ADDRESS>
Xuefang Wang, Tianjin Binhai New Area Dagang No.8 Middle school, Tianjin,
300270, P.R. China.