#  Stability of strong viscous shock wave under periodic perturbation for 1-D
isentropic navier-stokes system in the half space

Lin Chang  School of Mathematical Sciences, Beihang University, Beijing, China
<EMAIL_ADDRESS>,  Lin He  College of Mathematics, Sichuan University,
Chengdu, China<EMAIL_ADDRESS>and  Jin Ma  School of Mathematics and
Information Science, Shandong Technology and Business University, Yantai
264005, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, a viscous shock wave under space-periodic perturbation of 1-D
isentropic Navier-Stokes system in the half space is investigated. It is shown
that if the initial periodic perturbation around the viscous shock wave is
small, then the solution time asymptotically tends to a viscous shock wave
with a shift partially determined by the periodic oscillations. Moreover, the
strength of the shock wave could be arbitrarily large. This result essentially
improves the previous work ”A. Matsumura, 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), no. 1, 1-22.” where the
strength of shock wave is sufficiently small and the initial periodic
oscillations vanish.

###### Key words and phrases:

Impermeable wall problem, Large amplitude shock, Space-periodic perturbation,
Asymptotic stability

##  1\. Introduction

We consider a one-dimensional isentropic Navier-Stokes system for a general
viscous gas, i.e.,

$\left\\{\begin{array}[]{ll}v_{t}-u_{x}=0,&\\\
u_{t}+p_{x}=(\mu(v)\frac{u_{x}}{v})_{x},&\end{array}\right.$ (1.1)

where $v(x,t)$ is the specific volume, $u(x,t)$ the fluid velocity and
$p=av^{-\gamma}$ is the pressure. Constant $a>0$, $\gamma>1$ are adiabatic
constants. $\mu(v)=\mu_{0}v^{-\alpha}$ is the viscosity coefficient with
$\alpha\geq 0$. Without loss of generality, we assume $\mu_{0}=1$ in what
follows.

The system (1.1) is a basic system of hydrodynamic equations, it has a variety
of wave phenomena, such as viscous shock waves and rarefaction waves. So it is
important to study the stability of the viscous shock wave for system (1.1).
The stability of viscous shock wave for the Cauchy problem has been
extensively studied in a large literature since the pioneer works of [5, 22],
see the other interesting works [4, 7, 9, 12, 13, 16, 17, 18, 20, 23, 27, 15]
as the shock wave is weak.

Physicists and engineers are more concerned with the stability of large
amplitude shock (strong shock). However, the stability of large amplitude
shock (strong shock) is challenging in mathematics. There have been no
research results of this area until the last few years.

In 2010, Matsumura-Wang [25] proved that the large amplitude shock wave is
asymptotically stable by a clever weighted energy method as
$\alpha>\frac{\gamma-1}{2}$. In 2016, Vasseur-Yao [28] successfully removed
the condition $\alpha>\frac{\gamma-1}{2}$ by introducing a new variable called
“effective velocity”. Recently, He-Huang [6] extended the result of [28] to
general pressure $p(v)$ and general viscosity $\mu(v)$, where $\mu(v)$ could
be any positive smooth function.

On the other hand, it is also interesting to investigate the stability of
viscous shock waves for the initial-boundary value problem. In this paper, we
considered an impermeable wall problem of (1.1) in the half space $x\geq 0$,
i.e.,

$\left\\{\begin{array}[]{ll}(v,u)(x,0)=(v_{0},u_{0})(x)\longrightarrow(v_{+}+\zeta(x),u_{+}+\varphi(x)),~{}x\rightarrow+\infty,&\\\
u(0,t)=0,~{}~{}t\in{R_{+}},&\end{array}\right.$ (1.2)

where $v_{+}>0,u_{+}<0$. And $\left(\zeta,\varphi\right)$ are periodic
functions with period $\pi>0$ and satisfy

$\int_{0}^{\pi}\left(\zeta,\varphi\right)(x)dx=0.$ (1.3)

When the periodic functions $\left(\zeta,\varphi\right)$ vanish, Matsumura-Mei
[21] considered the impermeable wall problem (1.1), (1.2) in 1999. And
recently, an interesting result by [3] considering the multi-dimensional case
of this problem.

The impermeable wall means that the velocity at the boundary $x=0$ must be
zero because there is no flow across the boundary. They showed in [21] that
when $\alpha=0$ the solution of (1.1),(1.2) tends to a 2-viscous shock wave
connecting the left state $(v_{-},0)$ and the right one $(v_{+},u_{+})$
provided that both the strength of shock and the initial perturbation are
small and the 2-viscous shock is initially far away from the boundary, where
$v_{-}$ is determined by the RH condition, i.e.,

$\displaystyle\left\\{\begin{array}[]{ll}-s_{2}(v_{+}-v_{-})-(u_{+}-u_{-})=0,&\\\
-s_{2}(u_{+}-u_{-})+(p(v_{+})-p(v_{-}))=0.&\end{array}\right.$ (1.6)

Moreover, we assume that $u_{-}=0$. The condition that the strength of shock
is small was removed from [2]. The condition that the shock is initially far
away from the boundary was removed from [24]. How to remove both these two
conditions mentioned above at the same time is still open. Let us briefly
recall the idea of [24].

Since $u(0,t)=0$ at the boundary, we can exchange the impermeable wall problem
(1.1) and (1.2) in the half space to the Cauchy problem in the whole space by
defining $({{\tilde{v}}}(x,t),\tilde{u}(x,t))=(v(-x,t),-u(-x,t))$ as $x<0$ so
that $(\tilde{v},\tilde{u})(x,t)$ still satisfies the system (1.1) in the
whole space, i.e.,

$\left\\{\begin{array}[]{ll}\tilde{v}_{t}-\tilde{u}_{x}=0,&\\\
\tilde{u}_{t}+p({\tilde{v}})_{x}=(\frac{{\tilde{u}}_{x}}{{\tilde{v}}^{\alpha+1}})_{x},~{}~{}x\in{R}&\end{array}\right.$
(1.7)

equipped with the initial data

$({\tilde{v}_{0}},{\tilde{u}_{0}})(x)=:({\tilde{v}},{\tilde{u}})(x,0)=\left\\{\begin{array}[]{ll}(v_{0}(-x),-u_{0}(-x)),&x\leq
0,\\\ (v_{0}(x),u_{0}(x)),&x\geq 0,\\\ \end{array}\right.$ (1.8)

satisfying

$(\tilde{v}_{0},{\tilde{u}_{0}})(x)\rightarrow\left\\{\begin{array}[]{ll}(v_{+},u_{+}),&(x\rightarrow+\infty),\\\
(v_{+},-u_{+}),&(x\rightarrow-\infty).\end{array}\right.$ (1.9)

(a) Combination of the two shock waves

(b)  The graphs of $V_{i}$ and $U_{i}$ ,i=1,2

It is obvious that the solution of the Cauchy problem (1.7)-(1.9) confined in
the half line $x>0$ is exactly the one of the impermeable wall problem
(1.1),(1.2). In view of the far field states at $x=\pm\infty$ given by (1.9),
it is expected that the solution to (1.7)-(1.9) asymptotically tends to a
composite wave consisting of 1-viscous shock wave connecting $(v_{+},-u_{+})$
at the left and an intermediate state $(v_{\star},u_{\star})$ at the right,
and 2-viscous shock wave connecting $(v_{\star},u_{\star})$ at the left and
$(v_{+},u_{+})$ at the right. Fortunately $(v_{\star},u_{\star})=(v_{-},0)$ by
the principle of RH condition and (1.9), see Fig (A), where
$S_{1}(v_{+},-u_{+})$ means the 1-shock curve in the phase plane $(v,u)$
starting from the left state $(v_{+},-u_{+})$ and $S_{2}(v_{-},0)$ means the
2-shock curve in the phase plane $(v,u)$ starting from the left state
$(v_{-},0)$. The Figure B contains the graphs of the shock waves in the planes
$(x,v)$ and $(x,u)$. The wall $x=0$ can be regarded as a mirror and the
1-viscous shock is a mirror image of the 2-viscous shock in the plane $(x,v)$,
and the interaction between the 2-shock and the boundary $x=0$ for the
impermeable wall problem (1.1)-(1.2) is replaced to consider the one between
the 2-shock and its mirrored shock for the Cauchy problem (1.7)-(1.9).

In this paper, we want to improve the work of [2] where $\zeta=\varphi=0$.
Motivated by [24], the extended initial data in (1.8) satisfies

$(\tilde{v}_{0},{\tilde{u}_{0}})(x)\rightarrow\left\\{\begin{array}[]{ll}(v_{+}+\zeta(x),u_{+}+\varphi(x)),&(x\rightarrow+\infty),\\\
(v_{+}+\zeta(-x),-u_{+}-\varphi(-x)),&(x\rightarrow-\infty).\end{array}\right.$
(1.10)

We outline the strategy as follows. We apply the anti-derivative method to
study the stability of the traveling wave solution
$(V_{2}^{S},U_{2}^{S})(x-s_{2}t)$, in which the anti-derivative of the
perturbation ($\tilde{v}-V_{2}^{S},\tilde{u}-U_{2}^{S}$), namely,
$(\phi,\psi)(x,t)=\int_{-\infty}^{x}(\tilde{v}-V_{2}^{S},\tilde{u}-U_{2}^{S})(y,t)dy$,
“should” belong to some Sobolev spaces like $H^{2}(\mathbb{R})$. However, the
method above can not be applicable directly in this paper since
($\tilde{v}-U_{2}^{S},\tilde{u}-U_{2}^{S}$) oscillates at the far field and
hence does not belong to any $L^{p}$ space for $p\geq 1$. Motivated by [30],
we introduce a suitable ansatz $(V,U)(x,t)$, which has the same oscillations
as the solution $(\tilde{v},\tilde{u})(x,t)$ at the far field, so that
$\int_{-\infty}^{x}(\tilde{v}-V,\tilde{u}-U)(x,t)dx$ belongs to some Sobolev
spaces and the anti-derivative method is still available.

The rest of the paper will be arranged as follows. In Section 2, a suitable
ansatz is constructed and the main results are stated. In Section 3, the
stability problem is reformulated to a perturbation equation around the
ansatz. In Section 4, the a priori estimates are established. In Section 5,
the main results are proved. In Section 6, some complementary proofs are
provided.

Notation. The functional $\|\cdot\|_{L^{p}(\Omega)}$ defined by
$\|f\|_{L^{p}(\Omega)}=(\int_{\Omega}|f|^{p}(\xi){d\xi})^{\frac{1}{p}}$. When
$\Omega=(-\infty,\infty)$, the symbol $\Omega$ is often omitted. As $p=2$, we
denote for simplicity,

$\|f\|=\left(\int_{-\infty}^{\infty}f^{2}(\xi){d\xi}\right)^{\frac{1}{2}}.$

In addition, $H^{m}$ denotes the $m$-th order Sobolev space of functions
defined by

$\|f\|_{m}=\left(\sum_{k=0}^{m}\|\partial^{k}_{\xi}f\|^{2}\right)^{\frac{1}{2}}.$

##  2\. Preliminaries and the Main Theorem

###  2.1. Preliminaries

As pointed out by [21, 2], when perturbation functions $\zeta,\varphi$ vanish,
the solution of the impermeable wall problem (1.1)-(1.2) is expected to tend
toward the outgoing viscous shock $(V_{2}^{S},U_{2}^{S})(\xi_{2})$ satisfying

$\left\\{\begin{array}[]{ll}{-s_{2}}(V_{2}^{S})^{\prime}-(U_{2}^{S})^{\prime}=0,&\\\
{-s_{2}}(U^{S}_{2})^{\prime}+p(V^{S}_{2})^{\prime}=\left(\frac{(U^{S}_{2})^{\prime}}{(V^{S}_{2})^{\alpha+1}}\right)^{\prime},&\\\
(V^{S}_{2},U^{S}_{2})(-\infty)=(v_{-},0),\quad(V^{S}_{2},U^{S}_{2})(+\infty)=(v_{+},u_{+}),&\end{array}\right.$
(2.1)

where ${}^{\prime}=d/d\xi_{2}$, $\xi_{2}=x-s_{2}t$, $s_{2}$ is the shock speed
determined by the R-H condition (1.6) and $v_{\pm}>0,u_{+}<0$ are given
constants. Using $(\ref{2.1})_{1}$ and $(\ref{2.1})_{2}$, it follows that

$\displaystyle\begin{split}&{s}_{2}^{2}(V^{S}_{2})^{\prime}+p(V^{S}_{2})^{\prime}=\left(\frac{-s_{2}(V^{S}_{2})^{\prime}}{(V^{S}_{2})^{\alpha+1}}\right)^{\prime}.\end{split}$
(2.2)

Integrate (2.2) over $(-\infty,\xi_{2})$. one has

$\displaystyle\begin{split}&\frac{s_{2}(V^{S}_{2})^{\prime}}{(V^{S}_{2})^{\alpha+1}}=-{s}_{2}^{2}(V^{S}_{2})-p(V^{S}_{2})-b:=h(V^{S}_{2}),\quad
V^{S}_{2}(\pm\infty)=v_{\pm},\end{split}$
$\displaystyle\begin{split}&U^{S}_{2}=-s_{2}(V^{S}_{2}-v_{-})=-s_{2}(V^{S}_{2}-v_{+})+u_{+},\end{split}$
(2.3)

where $b=-s_{2}^{2}v_{-}-p(v_{-})=-s_{2}^{2}v_{+}-p(v_{+})$. For abbreviation,
we denote $s_{2}$ by $s$. We have the following lemma.

######  Lemma 2.1 ([21]).

There exists a unique viscous shock $(V^{S}_{2},U^{S}_{2})(\xi_{2})$ up to a
shift satisfying

$\displaystyle 0<v_{-}<V^{S}_{2}(\xi_{2})<v_{+},\quad
h(V^{S}_{2})>0,\quad(U^{S}_{2})^{\prime}<0,$
$\displaystyle\left|V^{S}_{2}(\xi)-v_{\pm}\right|=O(1)\theta
e^{-c_{\pm}|\xi_{2}|},$ (2.4)

as $\xi_{2}\rightarrow\pm\infty,$ where $\theta=\left|v_{+}-v_{-}\right|$,
$c_{\pm}=\frac{v_{\pm}^{\alpha+1}}{s}|p^{\prime}(v_{\pm})+s^{2}|$,
$s=\frac{-u_{+}}{v_{+}-v_{-}}.$

The initial data are assumed to satisfied

$\displaystyle\begin{split}&v_{0}(x)-\zeta(x)-V_{2}^{S}(x-\beta_{1})\in
L^{1}\cap H^{1}(R_{+}),\\\ &u_{0}(x)-\varphi(x)-U_{2}^{S}(x-\beta_{1})\in
L^{1}\cap H^{1}(R_{+}),\end{split}$ (2.5)

and

$u_{0}(0)=0$ (2.6)

as compatibility condition, where $\beta_{1}>0$ is a constant. Set

$\displaystyle(A_{0},B_{0})(x):=-\int_{x}^{\infty}(v_{0}(y)-\zeta(y)-V_{2}^{S}(y-\beta_{1}),u_{0}(y)-\varphi(y)-U_{2}^{S}(y-\beta_{1}))dy.$

We further assume that

$\displaystyle(A_{0},B_{0})\in L^{2}(R_{+}).$ (2.7)

Borrowing from the idea of [24], we construct a composite wave. By [24], the
mirrored shock $(V^{S}_{1},U^{S}_{1})(\xi_{1}),\xi_{1}=x-s_{1}t,s_{1}=-s$,
satisfies

$\left\\{\begin{array}[]{ll}{s}(V^{S}_{1})^{\prime}-{U^{S}_{1}}^{\prime}=0,&\\\
{s}(U^{S}_{1})^{\prime}+p(V^{S}_{1})^{\prime}=(\frac{(U^{S}_{1})^{\prime}}{(V^{S}_{1})^{\alpha+1}})^{\prime},&\\\
(V^{S}_{1},U^{S}_{1})(-\infty)=(v_{+},-u_{+}),\quad(V^{S}_{1},U^{S}_{1})(+\infty)=(v_{-},0).&\end{array}\right.$
(2.8)

Thanks [21], one has

$V^{S}_{1}(\xi)=V^{S}_{2}(-\xi),\quad
U^{S}_{1}(\xi)=-U^{S}_{2}(-\xi),~{}~{}\forall\xi\in{R}.$ (2.9)

The composite wave by two viscous shock weaves is defined as

$\displaystyle\begin{split}\tilde{V}(x,t;\beta):=&V_{1}^{S}(x+st+\beta)+V^{S}_{2}(x-st-\beta)-v_{-},\\\
\tilde{U}(x,t;\beta):=&U_{1}^{S}(x+st+\beta)+U^{S}_{2}(x-st-\beta),\end{split}$
(2.10)

where $\beta$ is a constant. Motivated by [10, 8, 14, 11], we need two
periodic solutions to (1.1) to establish the ansatz. Some properties of the
solution are listed.

######  Lemma 2.2.

[11] Assume that $(v_{0},u_{0})(x)\in H^{k}(0,\pi)$ with $k\geq 2$ is periodic
with period $\pi>0$ and average $(\bar{v},\bar{u}).$ Then there exists
$\varepsilon_{0}>0$ such that if

$\varepsilon_{1}:=\|{(v_{0},u_{0})-({\bar{v}},{\bar{u}})}\|_{H^{k}(0,\pi)}\leq\varepsilon_{0},$

there exits a unique periodic solution

$(v,u)(x,t)\in C\big{(}0,+\infty;H^{k}(0,\pi)\big{)}$

to (1.1) with the initial data $(v,u)(x,0)=(v_{0},u_{0})(x),$ which has the
average $({\bar{v}},{\bar{u}}),$ and satisfies

$\|{(v,u)-({\bar{v}},{\bar{u}})}\|_{H^{k}(0,\pi)}(t)\leq
C\varepsilon_{1}e^{-2\sigma_{0}t},\quad t\geq 0,$

where the constants $C>0$ and $\sigma_{0}>0$ are independent of
$\varepsilon_{1}$ and $t.$

###  2.2. Ansatz

In order to make the anti-derivative method is available, we choose a suitable
pair of ansatz $(V,U)$ such that
$\lim_{x\rightarrow\pm\infty}(v-V,u-U)(x,t)=(0,0)$ for any $t\geq 0$.
Motivated by [29], we define that the periodic solutions
$\left(v_{l,r},u_{l,r}\right)$ of (1.1) as $x\rightarrow\mp\infty$ for all
$t\geq 0,$ which have the periodic initial data:

$\displaystyle\begin{split}&\left(v_{r},u_{r}\right)(x,0)=(v_{+},u_{+})+\left(\zeta,\varphi\right)(x),\\\
&\left(v_{l},u_{l}\right)(x,0)=(v_{+},-u_{+})+\left(\zeta,-\varphi\right)(-x).\end{split}$

For the viscous shocks $\left(V^{S}_{1},U^{S}_{1}\right)$ and
$\left(V^{S}_{2},U^{S}_{2}\right),$ define

$\displaystyle\begin{split}g_{1}(x)&:=\frac{V^{S}_{1}(x)-v_{+}}{v_{-}-v_{+}}=\frac{U^{S}_{1}(x)+u_{+}}{u_{+}},\\\
g_{2}(x)&:=\frac{V^{S}_{2}(x)-v_{-}}{v_{+}-v_{-}}=\frac{U^{S}_{2}(x)}{u_{+}},\end{split}$
(2.11)

where we have used the R-H condition (1.6). It is straightforward to check
that $0\leq g_{i}(x)\leq 1$ and $g_{i}^{\prime}(x)>0$ for $i=1,2.$ With
functions $v_{l,r},u_{l,r},g_{1,2}$ in hand, we are ready to construct the
ansatz. Let $\mathcal{X}(t),\mathcal{Y}(t)$ are two $C^{1}$ curves on
$[0,+\infty)$ which will be determined later. Set

$\displaystyle V(x,t):={}$ $\displaystyle
v_{l}(x,t)\left[1-g_{1}(x+st+{\mathcal{X}})\right]+v_{-}\left[g_{1}(x+st+{\mathcal{X}})-g_{2}(x-st-{\mathcal{X}})\right]$
$\displaystyle+v_{r}(x,t)g_{2}(x-st-{\mathcal{X}}),$ $\displaystyle
U(x,t):={}$ $\displaystyle
u_{l}(x,t)\left[1-g_{1}(x+st+{\mathcal{Y}})\right]+u_{r}(x,t)g_{2}(x-st-{\mathcal{Y}}).$

Plugging the ansatz $(V,U)$ into (1.1), we have

$\begin{cases}V_{t}-U_{x}=(F_{1,1}+F_{1,2}+\mathcal{X}^{\prime}F_{1,3})_{x},&\\\
U_{t}+p(V)_{x}-\mu\big{(}\frac{\partial_{x}U}{V}\big{)}_{x}=(F_{2,1}+F_{2,2}+\mathcal{Y}^{\prime}F_{2,3})_{x},&\end{cases}$
(2.12)

where

$F_{i,j}=\int_{-\infty}^{x}f_{i,j}(y,t)dy;i=1,2;\quad j=2,3,$
$\begin{cases}F_{1,1}=u_{l}\left[g_{1}(x+st+{\mathcal{Y}})-g_{1}(x+st+{\mathcal{X}})\right]\\\
\qquad\quad-
u_{r}\left[g_{2}(x-st-{\mathcal{Y}})-g_{2}(x-st-{\mathcal{X}})\right],\\\
f_{1,2}=\left[-s(v_{l}-v_{+})+(u_{l}+u_{+})\right]g_{1}^{\prime}(x+st+{\mathcal{X}})\\\
\qquad\quad-\left[s(v_{r}-v_{+})+(u_{r}-u_{+})\right]g_{2}^{\prime}(x-st-{\mathcal{X}}),\\\
f_{1,3}=(v_{-}-v_{l})g_{1}^{\prime}(x+st+{\mathcal{X}})\\\
\qquad\quad+(v_{-}-v_{r})g_{2}^{\prime}(x-st-{\mathcal{X}}),\end{cases}$

and

$\begin{cases}F_{2,1}=p(V)-p(v_{l})\left[1-g_{1}(x+st+{\mathcal{Y}})\right]-p(v_{r})g_{2}(x-st-{\mathcal{Y}})\\\
\qquad\quad-\big{[}\frac{U_{x}}{V^{\alpha+1}}-\frac{u_{lx}}{v_{l}^{\alpha+1}}(1-g_{1}(x+st+{\mathcal{Y}}))-\frac{u_{rx}}{v_{r}^{\alpha+1}}g_{2}(x-st-{\mathcal{Y}})\big{]},\\\
f_{2,2}=\big{[}-su_{l}-p(v_{l})+\frac{u_{lx}}{v_{l}^{\alpha+1}}\big{]}g^{\prime}_{1}(x+st+{\mathcal{Y}}))\\\
\qquad\quad-\big{[}su_{r}-p(v_{r})+\frac{u_{rx}}{v_{r}^{\alpha+1}}\big{]}g_{2}^{\prime}(x-st-{\mathcal{Y}}),\\\
f_{2,3}=-u_{l}g^{\prime}_{1}(x+st+{\mathcal{Y}}))-u_{r}g_{2}^{\prime}(x-st-{\mathcal{Y}}).\end{cases}$

###  2.3. Location of The Shift ${\mathcal{X}}(t)$ and ${\mathcal{Y}}(t)$.

To apply the anti-derivative method which is always used to study the
stability of viscous shock, introduced in [26], we expect that

$\displaystyle\begin{split}0=&\int_{-\infty}^{\infty}\left(\begin{array}[]{cccc}{\tilde{v}}(x,t)-V(x,t)\\\
\tilde{u}(x,t)-U(x,t)\end{array}\right),\quad\forall t\geq 0.\end{split}$

When $t=0$, the shifts $\mathcal{X}(0)$ and $\mathcal{Y}(0)$ should satisfy

$\displaystyle\begin{split}0=&\int_{-\infty}^{\infty}\left(\begin{array}[]{cccc}\tilde{v}_{0}(x)-V(x,0)\\\
\tilde{u}_{0}(x)-U(x,0)\end{array}\right)dx:=\left(\begin{array}[]{cccc}I_{1}(\mathcal{X}(0))\\\
I_{2}(\mathcal{Y}(0))\end{array}\right).\end{split}$ (2.13)

Our next task is to show $\mathcal{X}(t),\mathcal{Y}(t)$ when $t>0$. To make
the system (2.12) as a conservative form, the curves $\mathcal{X}(t)$ and
$\mathcal{Y}(t)$ should satisfy

$\displaystyle\mathcal{X}^{\prime}(t)=-\lim_{x\rightarrow\infty}\frac{F_{1,2}(x,t)}{F_{1,3}(x,t)},~{}\mathcal{Y}^{\prime}(t)=-\lim_{x\rightarrow\infty}\frac{F_{2,2}(x,t)}{F_{2,3}(x,t)},$
(2.14)

With the aid of (2.2), we know $F_{1,3}\neq 0,F_{2,3}\neq 0$, provided that
the initial periodic perturbations $\left(\zeta,\varphi\right)$ are small. Due
to (1.8) and (2.9), $\tilde{u}_{0}(x),U(x,0)$ are odd functions and
${\tilde{v}}_{0}(x),V(x,0)$ are even functions, thus $I_{02}=0$, i.e, we can
choose any $\mathcal{Y}_{0}$ to guarantee that $I_{2}(\mathcal{Y}_{0})=0$. For
$I_{2}(\mathcal{X}_{0})=0$, using (2.4) and (2.9), one gets that

$\displaystyle\begin{split}I_{1}(\omega)=2&\int_{0}^{\infty}[v_{0}(x)-\tilde{q}(x)-V^{S}_{2}(x-\omega)]+[v_{-}-V^{S}_{1}(x+\omega)]dx\\\
=2&\int_{0}^{\infty}[v_{0}(x)-\tilde{q}(x)-V^{S}_{2}(x-\omega)]dx-2\int_{0}^{\infty}\frac{1}{s}U^{S}_{1}(x+\omega)dx\\\
=2&\int_{0}^{\infty}[{v}_{0}(x)-\tilde{q}(x)-V^{S}_{2}(x-\omega)]dx+2\int_{0}^{\infty}\frac{1}{s}U^{S}_{2}(-x-\omega)dx\\\
=2&\int_{0}^{\infty}[{v}_{0}(x)-\tilde{q}(x)-V^{S}_{2}(x-\omega)]dx+2\int_{0}^{\infty}U^{S}_{2}(-st-\omega)dt.\end{split}$
(2.15)

where

$\displaystyle\begin{split}\tilde{q}(x)=&\zeta(-x)\left[1-g_{1}(x+\omega)\right]+\zeta(x)\left[g_{2}(x-\omega)\right].\end{split}$
(2.16)

By directly calculate, we have $I_{1}(\infty)=\infty$,
$I_{1}(-\infty)=-\infty$.

$\displaystyle\begin{split}I_{1}^{\prime}(\omega)&=2\left\\{(v_{+}-v_{-})-\int_{0}^{\infty}[\zeta(-x)g_{1}^{\prime}(x+\omega)+\zeta(x)g_{2}^{\prime}(x-\omega)]dx\right\\}\\\
&\geq
2\left\\{(v_{+}-v_{-})-\|\zeta\|_{L_{\infty}}\int_{0}^{\infty}[g_{1}^{\prime}(x+\omega)+g_{2}^{\prime}(x-\omega)]dx\right\\}\\\
&\geq 2\left\\{(v_{+}-v_{-})-2\varepsilon\right\\}.\end{split}$ (2.17)

Moreover, choosing $\varepsilon$ suitable small, we have $3(v_{+}-v_{-})\geq
I_{1}^{\prime}(\omega)\geq v_{+}-v_{-}>0$ Thus there exists a unique constant
$\mathcal{X}_{0}$ such that $I_{1}(\mathcal{X}_{0})=0$. Moreover, using
$I_{1}(\mathcal{X}_{0})=I_{1}(\beta_{1})+\int_{\beta_{1}}^{\mathcal{X}_{0}}I_{1}^{\prime}(s)ds$,
the constant $\mathcal{X}_{0}$ is between $\frac{1}{2}\tilde{M}+\beta_{1}$ and
$\frac{3}{2}\tilde{M}+\beta_{1}$, where

$\displaystyle\begin{split}\tilde{M}=&\frac{1}{v_{-}-v_{+}}\left(\int_{0}^{\infty}[{v}_{0}(x)-\tilde{q}(x)-V^{S}_{2}(x-\beta_{1})]dx+\int_{0}^{\infty}U^{S}_{2}(-st-\beta_{1})dt\right)\\\
=&\frac{1}{v_{-}-v_{+}}\left(\int_{0}^{\infty}[{v}_{0}(x)-\zeta(x)-V^{S}_{2}(x-\beta_{1})]dx+\int_{0}^{\infty}U^{S}_{2}(-st-\beta_{1})dt\right)\\\
&-\frac{1}{v_{-}-v_{+}}\left(\int_{0}^{\infty}\left[\zeta(x)[g_{2}(x-\beta_{1})-1\right]+\zeta(-x)\left[1-g_{1}(x+\beta_{1})\right]dx\right)\\\
\leq&\frac{1}{v_{-}-v_{+}}\left(\int_{0}^{\infty}[{v}_{0}(x)-\zeta(x)-V^{S}_{2}(x-\beta_{1})]dx+\int_{0}^{\infty}U^{S}_{2}(-st-\beta_{1})dt\right)+C\varepsilon,\end{split}$
(2.18)

where we have used the following inequality

$\displaystyle\begin{split}&\int_{0}^{\infty}\left[\zeta(x)[g_{2}(x-\beta_{1})-1\right]+\zeta(-x)\left[1-g_{1}(x+\beta_{1})\right]dx\\\
=&\frac{-1}{(v_{-}-v_{+})}\int_{0}^{\infty}\zeta(x)\left[V^{S}_{2}(x-\beta_{1})-v_{+}\right]+\zeta(-x)\left[V^{S}_{1}(x+\beta_{1})-v_{-}\right]dx\\\
\leq&C\|\zeta(x)\|_{L_{\infty}}\int_{0}^{\infty}\left|V^{S}_{2}(x-\beta_{1})-v_{+}\right|+\left|V^{S}_{1}(x+\beta_{1})-v_{-}\right|dx\leq
C\varepsilon.\end{split}$ (2.19)

By (2.5), we know $\tilde{M}$ exists. Thus we can obtain the curves
$\mathcal{X}(t)$ and $\mathcal{Y}(t)$. More precisely, it holds that

######  Lemma 2.3.

Assume that (1.3), (1.6) hold. Then there exists an $\varepsilon_{0}>0$ such
that if

$\|{\zeta}\|_{H^{2}(0,\pi)}+\|\varphi\|_{H^{2}(0,\pi)}<\varepsilon<\varepsilon_{0},$

there exists a constant pair $(\mathcal{X},\mathcal{Y})(0)$ satisfying (2.13)
where $\mathcal{X}(0)$ is uniquely determined and $\mathcal{Y}(0)$ can take
any constant. Moreover, there exists a unique solution
$(\mathcal{X},\mathcal{Y})(t)\in C^{1}(0,+\infty)$ to the system (2.14) with
the fixed initial data
$(\mathcal{X},\mathcal{Y})(0)=(\mathcal{X}_{0},\mathcal{Y}_{0}$) satisfying

$|{\left(\mathcal{X}^{\prime},\mathcal{Y}^{\prime}\right)(t)}|+|{\left(\mathcal{X},\mathcal{Y}\right)(t)-\left(\mathcal{X}_{\infty},\mathcal{Y}_{\infty}\right)}|\leq
C\varepsilon e^{-2\sigma_{0}t},\qquad t\geq 0.$

Moreover, the corresponding constant locations
$\mathcal{X}_{\infty},\mathcal{Y}_{\infty}$ as follows,

$\displaystyle\mathcal{X}_{\infty}=~{}$
$\displaystyle\mathcal{X}_{0}+\frac{1}{2(v_{+}-v_{-})\pi}\left\\{\int_{0}^{\pi}\int_{0}^{x}\zeta(y)-\zeta(-y)dydx\right.,$
(2.20)
$\displaystyle\left.+2\pi\int_{0}^{+\infty}\left[\zeta(x)\left(1-g_{2}(x+\mathcal{X}_{0})\right)-\zeta(x)\left(1-g_{1}(x-\mathcal{X}_{0})\right)\right]dx\right\\},$

and

$\displaystyle\mathcal{Y}_{\infty}=~{}$
$\displaystyle\mathcal{Y}_{0}+\frac{1}{2u_{+}\pi}\left\\{\int_{0}^{\pi}\int_{0}^{x}\varphi(y)+\varphi(-y)dydx\right.$
(2.21)
$\displaystyle-\int_{0}^{+\infty}\int_{0}^{\pi}\left[p(v_{l})-p({v}_{r})\right]dxdt$
$\displaystyle\left.+\int_{0}^{\pi}g(v_{+}+\zeta(-x))-g(v_{+}+\zeta(x))dx\right\\},$

where $g(v)=\frac{1}{\alpha}v^{-\alpha},$ if $\alpha\neq 0$; $g(v)=-\ln v$, if
$\alpha\neq 0$.

Since the proof of Lemma 2.3 is similar to that in [29, 11], we put it in
section 6.

###  2.4. the Main Result

We define

$\displaystyle\begin{split}&\phi_{0}(x)=-\int^{\infty}_{x}\widetilde{v}_{0}(y)-V(y,0)dy,\\\
&\psi_{0}(x)=-\int^{\infty}_{x}\widetilde{u}_{0}(y)-U(y,0)dy.\end{split}$

In view of (2.13), we further assume that

$\displaystyle(\phi_{0},\psi_{0})\in H^{2}(R).$ (2.22)

Using the arbitrariness of $\mathcal{Y}_{0}$, one can find a suitable
$\mathcal{Y}_{0}$, such that $\mathcal{X}_{\infty}=\mathcal{Y}_{\infty}$. From
now on, we denote
$\beta:=\mathcal{X}_{\infty}=\mathcal{Y}_{\infty},\tilde{V}(x,t;\beta)=\tilde{V}(x,t),\tilde{U}(x,t;\beta)=\tilde{U}(x,t)$
for simple.

######  Lemma 2.4.

Suppose that (2.22) holds, there exists a positive constant $\delta_{1}$ such
that if

$\|\phi_{0}\|_{2}+\|\psi_{0}\|_{2}+\beta_{1}^{-1}+\varepsilon\leq\delta_{1},$

then the Cauchy problem (1.7),(1.10) has a unique global solution
$(\widetilde{v},\widetilde{u})(x,t)$ satisfying

$\displaystyle\begin{split}&{\widetilde{v}(x,t)}-V(x,t)\in
C^{0}([0,+\infty);H^{1})\cap L^{2}([0,+\infty);H^{2}),\\\
&{\widetilde{u}(x,t)}-U(x,t)\in C^{0}([0,+\infty);H^{1})\cap
L^{2}([0,+\infty);H^{1}),\\\ \end{split}$ (2.23)

and

$\displaystyle\begin{split}&\sup_{x\in{R}}|{\widetilde{v}(x,t)}-V(x,t)|\rightarrow
0,\text{ as }t\rightarrow+\infty,\\\
&\sup_{x\in{R}}|{\widetilde{u}(x,t)}-U(x,t)|\rightarrow 0,\text{ as
}t\rightarrow+\infty.\end{split}$ (2.24)

Now, we turn to the original initial-value problem. Our main theorem is:

######  Theorem 2.1.

For any given constants $u_{+}<0$ and $v_{+}>0$, if (2.5)-(2.7) hold. There
exists a positive constant $\delta_{2}$ such that if

$\|A_{0}\|_{H^{2}({R}_{+})}+\|B_{0}\|_{H^{2}({R}_{+})}+\beta_{1}^{-1}+\varepsilon\leq\delta_{2},$

then the IBVP (1.1), (1.2) has a unique global solution $(v,u)(x,t)$,
satisfying

$\displaystyle\begin{split}&\sup_{x\in{R}_{+}}|{(v,u)(x,t)}-(V^{S}_{2},U^{S}_{2})(x-st-\beta)|\rightarrow
0,\text{as }t\rightarrow+\infty,\end{split}$

where $\beta$ is determined by (2.20).

##  3\. Reformulation of the Original Problem

Set

$\displaystyle\begin{split}\phi&(x,t):=\int^{x}_{-\infty}{(\tilde{v}-V)}(y,t){d}y,\\\
\psi&(x,t):=\int^{x}_{-\infty}{(\tilde{u}-U)}(y,t){d}y.\end{split}$

Thus $(\tilde{v},\tilde{u})(x,t)$ satisfy

$\displaystyle\tilde{v}$ $\displaystyle(x,t)=\phi_{x}(x,t)+V(x,t),$
$\displaystyle\tilde{u}$ $\displaystyle(x,t)=\psi_{x}(x,t)+U(x,t).$

From $(\ref{2.8})$, we know the ansazt $(V,U)$ satisfies

$\left\\{\begin{array}[]{ll}{V}_{t}-{U}_{x}=-F_{1x},&\\\
U_{t}+p(V)_{x}-\left(\frac{U_{x}}{V^{\alpha+1}}\right)_{x}=-F_{2x},&\\\
(V,U)(\pm\infty,t)=(v_{+},\pm u_{+}),&\end{array}\right.$ (3.1)

where

$\displaystyle F_{1}(x,t)$
$\displaystyle:=-F_{1,1}(x,t)-F_{1,2}(x,t)-\mathcal{X}^{\prime}(t)F_{1,3}(x,t),$
(3.2) $\displaystyle F_{2}(x,t)$
$\displaystyle:=-F_{2,1}(x,t)-F_{2,2}(x,t)-\mathcal{Y}^{\prime}(t)F_{2,3}(x,t).$

Motivated by [21] and [1], with the help of (3.1) and (1.1), it follows that

$\left\\{\begin{array}[]{ll}\phi_{t}-\psi_{x}=F_{1},&\\\
\psi_{t}-f(V,U_{x})\phi_{x}-\frac{\psi_{xx}}{V^{\alpha+1}}=F_{2}+J.&\end{array}\right.$
(3.3)

The initial condition satisfies

$\displaystyle\left(\phi_{0},\psi_{0}\right)$
$\displaystyle(x)\in{H}^{2},\quad x\in{R},$ (3.4)

where

$\displaystyle
f(V,U_{x})=-p^{\prime}(V)-(\alpha+1)\frac{U_{x}}{V^{\alpha+2}},$ (3.5)
$\displaystyle J$
$\displaystyle=\frac{{u}_{x}}{{v}^{\alpha+1}}-\frac{U_{x}}{V^{\alpha+1}}-\frac{\psi_{xx}}{V^{\alpha+1}}+(\alpha+1)\frac{U_{x}\phi_{x}}{V^{\alpha+2}}-\left[p({v})-p(V)-p^{\prime}(V)\phi_{x}\right].$
(3.6)

######  Lemma 3.1.

Under the assumptions of Theorem 2.1, the anti-derivative variables (3.2)
exist and satisfy that

$\|{F_{1}}\|_{2}\leq C\varepsilon e^{-\sigma_{0}t},\|F_{2}\|_{1}\leq
C\varepsilon e^{-\sigma_{0}t}+Ce^{-c_{-}\beta_{1}}e^{-sc_{-}t}.$

The proof is based on Lemma 2.2, Lemma 2.3 and Lemma 5.2 and we place it in
section 6 for brevity.

We will seek the solution in the functional space ${X}_{\delta}(0,T)$ for any
$0\leq T<+\infty$,

$\displaystyle\begin{split}{X}_{\delta}(0,T):=&\left\\{(\phi,\psi)\in
C([0,T];{H}^{2})|\phi_{x}\in{L}^{2}(0,T;{H}^{1}),\psi_{x}\in{L}^{2}(0,T;{H}^{2})\right.\\\
&\sup_{0\leq t\leq T}\|(\phi,\psi)(t)\|_{2}\leq\delta\\},\end{split}$

where ${\delta}\ll 1$ is small.

######  Remark 3.1.

The function space is well defined because the Dirac function will not appear
in $\phi,\phi_{x},\phi_{xx},\psi,\psi_{x},\psi_{xx},\psi_{xxx}$, which can be
guaranteed by $u(0)=0$.

######  Proposition 3.1.

(A priori estimate) For some time $T>0$, if $(\phi,\psi)\in{X}_{\delta}(0,T)$
is the solution of (3.3), (3.4). Then there exists a positive constant
$\delta_{0}$ independent of $T$, such that if

$\sup_{0\leq t\leq T}\|(\phi,\psi)(t)\|_{{2}}\leq\delta\leq\delta_{0},$

for $t\in[0,T]$, then

$\displaystyle\|(\phi,\psi)(t)\|_{{2}}^{2}+\int_{0}^{t}(\|\phi_{x}(t)\|^{2}_{1}+\|\psi_{x}(t)\|_{2}^{2}){d}t\leq
C_{0}(\|(\phi_{0},\psi_{0})\|_{{2}}^{2}+e^{-c_{-}\beta_{1}}+\varepsilon),$

where $C_{0}>1$ ia a constant independent of $T$.

Once Proposition 3.1 is obtained, the local solution $(\phi,\psi)$ can be
extend to $T=+\infty.$ See the following lemma.

######  Lemma 3.2.

If $(\phi_{0},\psi_{0})\in{H}^{2}$, there exists a positive constant
$\delta_{1}=\frac{\delta_{0}}{\sqrt{C_{0}}}$, such that if

$\|(\phi_{0},\psi_{0})\|_{{2}}^{2}+e^{-c_{-}\beta_{1}}+\varepsilon\leq\delta_{1}^{2},$

then the initial value problem (3.3), (3.4) has a unique global solution
$(\phi,\psi)\in{X}_{\delta_{0}}(0,\infty)$ satisfying

$\displaystyle\sup_{t\geq
0}\|(\phi,\psi)(t)\|_{{2}}^{2}+\int_{0}^{\infty}(\|\phi_{x}(t)\|^{2}_{1}+\|\psi_{x}(t)\|_{2}^{2}){d}t\leq
C_{0}(\|(\phi_{0},\psi_{0})\|_{{2}}^{2}+e^{-c_{-}\beta_{1}}+\varepsilon).$

##  4\. A Priori Estimate

For some $T>0$, the problem $(\ref{3.3}),(\ref{3.4})$ is assumed that has a
solution $(\phi,\psi)\in{X}_{\delta}(0,T)$ in this section.

$\displaystyle\sup_{0\leq t\leq T}\|(\phi,\psi)(t)\|_{2}\leq\delta.$ (4.1)

The Sobolev inequality gives that $\frac{1}{2}v_{-}\leq
v\leq\frac{3}{2}v_{+}$, and

$\displaystyle\sup_{0\leq t\leq
T}\\{\|(\phi,\psi)(t)\|_{{L}^{\infty}}+\|(\phi_{x},\psi_{x})(t)\|_{{L}^{\infty}}\\}\leq{\delta}.$

Motivated by [28], we introduce the new effective velocity
$\widetilde{h}=\widetilde{u}-\widetilde{v}^{-(\alpha+1)}\widetilde{v}_{x}$. It
holds that

$\left\\{\begin{array}[]{ll}\widetilde{v}_{t}-\widetilde{h}_{x}=(\frac{\widetilde{v}_{x}}{\widetilde{v}^{\alpha+1}})_{x},&\\\
\widetilde{h}_{t}+\widetilde{p}_{x}=0.&\end{array}\right.$ (4.2)

Similarly, we define $H=U-V^{-(\alpha+1)}V_{x}$, then (3.1) becomes

$\left\\{\begin{array}[]{ll}V_{t}-H_{x}=\left(\frac{V_{x}}{V^{\alpha+1}}\right)_{x}-F_{1,x},&\\\
H_{t}+p(V)_{x}=-F_{2,x}.&\end{array}\right.$ (4.3)

We define

$\displaystyle\int_{-\infty}^{x}(\widetilde{h}-H){dx}=\Psi.$ (4.4)

Substitute (4.3) from (4.2) and integrate the resulting system with respect to
$x$. Using (4.4), we have

$\left\\{\begin{array}[]{ll}\phi_{t}-\Psi_{x}-\frac{\phi_{xx}}{V^{\alpha+1}}=G+{F}_{1},&\\\
\Psi_{t}+p^{\prime}(V)\phi_{x}=-p(\tilde{v}|V)+F_{2}-\frac{F_{1x}}{V^{\alpha+1}},&\end{array}\right.$
(4.5)

where

$\displaystyle
G=\frac{\widetilde{v}_{x}}{\widetilde{v}^{\alpha+1}}-\frac{V_{x}}{V^{\alpha+1}}-\frac{\phi_{xx}}{V^{\alpha+1}},\quad
p(\widetilde{v}|V)=\left(p(\widetilde{v})-p(V)\right)-p^{\prime}(V)\phi_{x}.$

Now we give some lemmas that are useful in energy estimate.

######  Lemma 4.1.

([6, 21]) Under the assumption of (4.1), we have

$\displaystyle\begin{split}&p(\widetilde{v}|V)\leq
C\phi_{x}^{2},\quad|p(\widetilde{v}|V)_{x}|\leq
C(|\phi_{xx}\phi_{x}|+|V_{x}|\phi_{x}^{2}),\\\ &|G|\leq
C(|\phi_{xx}\phi_{x}|+|V_{x}|\phi_{x}),\\\ \end{split}$ (4.6)

and

$\displaystyle\begin{split}&|J|\leq C(\phi_{x}^{2}+|\phi_{x}\psi_{xx}|),\\\
&|J_{x}|\leq
C(\phi_{x}^{2}+|\phi_{x}\phi_{xx}|+|\psi_{xx}\phi_{xx}|+|\psi_{xxx}\phi_{x}|+|\phi_{x}\psi_{xx}|).\end{split}$
(4.7)

######  Lemma 4.2.

The error terms

$\displaystyle\begin{split}q(x,t):=V(x,t)-\tilde{V}(x,t);\quad
z(x,t):=U(x,t)-\tilde{U}(x,t),\end{split}$ (4.8)

satisfy

$\|(z,q)(.,t)\|_{2}\leq C\varepsilon e^{-2\sigma_{0}t}.$

###### Proof.

By direct calculate, one gets that

$\displaystyle\begin{split}q(x,t):=&(v_{l}-v_{+})(x,t)\left[1-g_{1}(x+st+{\mathcal{X}})\right]+(v_{r}-v_{+})(x,t)g_{2}(x-st-{\mathcal{X}})\\\
&+V_{1}^{S}(x+st+{\mathcal{X}})+V^{S}_{2}(x-st-{\mathcal{X}})-V_{1}^{S}(x+st+\beta)-V^{S}_{2}(x-st-\beta)\\\
\leq&C(|v_{l}-{v}_{+}|+|v_{r}-{v}_{+}|+|{\mathcal{X}}-\beta|),\end{split}$
(4.9)

and

$\displaystyle\begin{split}\frac{\partial^{k}q}{\partial x^{k}}\leq
C\left(\left|\frac{\partial^{k}v_{l}}{\partial
x^{k}}\right|+\left|\frac{\partial^{k}v_{r}}{\partial
x^{k}}\right|+\left|{\mathcal{X}}-\beta\right|\right),\quad k=1,2.\end{split}$
(4.10)

With the aid of Lemma 2.2 and Lemma 2.3, one gets that

$\|q\|_{2}\leq C\varepsilon e^{-2\sigma_{0}t}.$ (4.11)

Similar, we obtain

$\|z\|_{2}\leq C\varepsilon e^{-2\sigma_{0}t}.$ (4.12)

∎

###  4.1. Low Order Estimates.

######  Lemma 4.3.

Under the same assumptions of Proposition 3.1, we have

$\displaystyle\begin{split}&\|(\phi,\Psi)\|^{2}(t)+\int_{0}^{t}\int_{-\infty}^{\infty}\sqrt{-\tilde{U}_{x}}\Psi^{2}dxdt+\int_{0}^{t}\|\phi_{x}\|^{2}dt\\\
\leq&C\|(\phi_{0},\Psi_{0})\|^{2}+C{\delta}\int_{0}^{t}\|\phi_{xx}\|^{2}dt+Ce^{-c_{-}\beta_{1}}+C\varepsilon.\end{split}$

###### Proof.

We multiply $(\ref{4.5})_{1}$ and $(\ref{4.5})_{2}$ by $\phi$ and
$\frac{\Psi}{-p^{\prime}(V)}$, respectively, sum them up, and intergrading
result with respect to $t$ and $x$ over $[0,t]\times{R}$, we have

$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}\left(\phi^{2}-\frac{\Psi^{2}}{p^{\prime}(V)}\right)dx+\int_{0}^{t}\int_{-\infty}^{\infty}\frac{-p^{\prime\prime}(V)}{2(p^{\prime}(V))^{2}}\tilde{U}_{x}\Psi^{2}+\frac{\phi_{x}^{2}}{V^{\alpha+1}}dxdt$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left.\int_{-\infty}^{\infty}\left(\phi^{2}-\frac{\Psi^{2}}{p^{\prime}(V)}\right)dx\right|_{t=0}$
$\displaystyle+\int_{0}^{t}\int_{-\infty}^{\infty}\left[G+(\alpha+1)\frac{V_{x}\phi_{x}}{V^{\alpha+2}}\right]\phi+\frac{p(\tilde{v}|V)\Psi}{p^{\prime}(V)}dxdt$
(4.13)
$\displaystyle+\int_{0}^{t}\int_{-\infty}^{\infty}F_{1}\phi-\frac{\Psi}{p^{\prime}(V)}\left(F_{2}-\frac{F_{1x}}{V^{\alpha+1}}\right)dxdt$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\int_{-\infty}^{\infty}\frac{p^{\prime\prime}(V)}{p^{\prime}(V)^{2}}(z-F_{1})_{x}\Psi^{2}dxdt$
$\displaystyle:=$
$\displaystyle\frac{1}{2}\left.\int_{-\infty}^{\infty}\left(\phi^{2}-\frac{\Psi^{2}}{p^{\prime}(V)}\right)dx\right|_{t=0}+\sum_{i=1}^{3}A_{i}.$
(4.14)

By direct calculate, one gets that

$\left|G+(\alpha+1)\frac{V_{x}\phi_{x}}{V^{\alpha+2}}\right|\leq
C|\phi_{x}|(\phi_{x}^{2}+\phi_{xx}^{2}).$ (4.15)

Due to (4.6), (4.15), we can get

$\displaystyle\begin{split}A_{1}\leq&C\int_{0}^{t}\|\phi\|_{L^{\infty}}\left\|\phi_{x}(\phi_{x}^{2}+\phi_{xx}^{2})\right\|_{L^{1}}dt+C\int_{0}^{t}\|\Psi\|_{L^{\infty}}\|\phi_{x}^{2}\|_{L^{1}}dt\\\
\leq&C\delta\int_{0}^{t}\|\phi_{x}\|^{2}+\|\phi_{xx}\|^{2}dt.\end{split}$
(4.16)

With the aid of Lemma 3.1, H$\mathrm{\ddot{o}}$lder inequality, we have

$\begin{split}A_{2}\leq&C\int_{0}^{t}\|\phi,\Psi\|(\|F_{1}\|_{1}+\|F_{2}\|)dt\\\
\leq&C\sup_{\tau\in[0,t]}(\|\phi,\Psi\|^{2}+1)\int_{0}^{t}\|F_{1}\|_{1}+\|F_{2}\|dt\\\
\leq&C(\varepsilon+e^{-c_{-}\beta_{1}}).\end{split}$ (4.17)

Using H$\mathrm{\ddot{o}}$lder inequality, Sobolev inequality, combining Lemma
3.1, Lemma 4.2, one gets

$\displaystyle\begin{split}A_{3}\leq&C\int_{0}^{t}\|(z-F_{1})_{x}\|_{L^{\infty}}\|\Psi^{2}\|_{L^{1}}dt\\\
\leq&C\sup_{\tau\in[0,t]}\|\Psi(\tau)\|^{2}\int_{0}^{t}\|(z-F_{1})_{x}\|_{H^{1}}dt\leq
C\varepsilon.\end{split}$ (4.18)

Inserting (4.16)-(4.18) into (4.13), using the smallness of $\delta$, we
obtain the proof of Lemma 4.3. ∎

######  Lemma 4.4.

Under the same assumptions of Proposition 3.1, we have

$\displaystyle\|(\phi,\Psi)(t)\|_{1}^{2}+\int_{0}^{t}\|\phi_{x}\|_{1}^{2}{dt}\leq
C\|(\phi_{0},\Psi_{0})\|_{1}^{2}+Ce^{-c_{-}\beta_{1}}+C\varepsilon.$

###### Proof.

We multiply $(\ref{4.5})_{1}$ and $(\ref{4.5})_{2}$ by $-\phi_{xx}$ and
$\frac{\Psi_{xx}}{p^{\prime}(V)}$, respectively and sum over the result,
intergrade the result with respect to $t$ and $x$ over $[0,t]\times{R}$, we
have

$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}\left(\phi_{x}^{2}-\frac{\Psi_{x}^{2}}{p^{\prime}(V)}\right)dx+\int_{0}^{t}\int_{-\infty}^{\infty}\frac{-p^{\prime\prime}(V)}{2(p^{\prime}(V))^{2}}\tilde{U}_{x}\Psi_{x}^{2}+\frac{\phi^{2}_{xx}}{V^{\alpha+1}}dxdt$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}\left.\left(\phi_{x}^{2}-\frac{\Psi_{x}^{2}}{p^{\prime}(V)}\right)dx\right|_{t=0}+\int_{0}^{t}\int_{-\infty}^{\infty}\frac{p^{\prime\prime}(V)}{p^{\prime}(V)}\tilde{V}_{x}\Psi_{x}\phi_{x}dxdt$
$\displaystyle+\int_{0}^{t}\int_{-\infty}^{\infty}\frac{p(\tilde{v}|V)_{x}}{p^{\prime}(V)}\Psi_{x}-G\phi_{xx}dxdt$
$\displaystyle+\int_{0}^{t}\int_{-\infty}^{\infty}\frac{p^{\prime\prime}(V)}{2p^{\prime}(V)^{2}}(z-F_{1})_{x}\Psi_{x}^{2}+\frac{p^{\prime\prime}(V)}{p^{\prime}(V)}q_{x}\Psi_{x}\phi_{x}dxdt$
(4.19)
$\displaystyle-\int_{0}^{t}\int_{-\infty}^{\infty}F_{1}\phi_{xx}+\frac{\Psi_{x}}{p^{\prime}(V)}\left(F_{2}-\frac{F_{1x}}{V^{\alpha+1}}\right)_{x}dxdt$
$\displaystyle:=$
$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}\left.\left(\phi_{x}^{2}-\frac{\Psi_{x}^{2}}{p^{\prime}(V)}\right)dx\right|_{t=0}+\sum_{i=1}^{4}B_{i}.$

With the aid of the Cauchy inequality, we have

$\displaystyle\begin{split}B_{1}\leq&\frac{s}{4}\int_{0}^{t}\int_{-\infty}^{\infty}\frac{p^{\prime\prime}(V)}{(p^{\prime}(V))^{2}}|\tilde{V}_{x}|\Psi_{x}^{2}dxdt+C\int_{0}^{t}\int_{-\infty}^{\infty}{p^{\prime\prime}(V)}|\tilde{V}_{x}|\phi_{x}^{2}dxdt\\\
\leq&-\frac{1}{4}\int_{0}^{t}\int_{-\infty}^{\infty}\frac{p^{\prime\prime}(V)}{(p^{\prime}(V))^{2}}|\tilde{V}_{t}|{\Psi_{x}^{2}}\Psi_{x}^{2}dxdt+C\int_{0}^{t}\|\phi_{x}\|^{2}dt.\end{split}$
(4.20)

The last inequality is based on the following inequality

$\displaystyle\begin{split}-\tilde{V}_{t}=&-(V^{S}_{1}(x+st+\beta)+V^{S}_{2}(x-st-\beta))_{t}=-s(V^{S}_{1}(x+st+\beta)-V^{S}_{2}(x-st-\beta))_{x}\\\
>&s|V^{S}_{1x}(x+st+\beta)+V^{S}_{2x}(x-st-\beta)|=s|\tilde{V}_{x}|,\end{split}$

where we have used $(V^{S}_{1})^{\prime}<0$, $(V^{S}_{2})^{\prime}>0$, $s>0$.

The Cauchy inequality and the Sobolev inequality gives that

$\displaystyle\begin{split}B_{2}&\leq
C\int_{0}^{t}\int_{-\infty}^{\infty}(|\phi_{xx}\phi_{x}|+|V_{x}\phi_{x}|)|{\phi_{xx}}|+\left|\frac{1}{p^{\prime}(V)}p(\tilde{v}|V)_{x}\Psi_{x}\right|dxdt\\\
&\leq(C\delta+\eta)\int_{0}^{t}\|\phi_{xx}\|^{2}dt+(C_{\eta}+C\delta)\int_{0}^{t}\|\phi_{x}\|^{2}dt.\end{split}$

Similar like (4.17) and (4.18), the error terms $B_{3},B_{4}$ can be estimated
as

$B_{3}+B_{4}\leq Ce^{-c_{-}\beta_{1}}+C\varepsilon.$ (4.21)

Inserting (4.20)-(4.21) into (4.1), we get

$\displaystyle\begin{split}&\frac{1}{2}\int_{-\infty}^{\infty}\left(\phi_{x}^{2}-\frac{\Psi_{x}^{2}}{p^{\prime}(V)}\right)dx-\frac{1}{4}\int_{0}^{t}\int_{-\infty}^{\infty}\frac{p^{\prime\prime}(V)}{(p^{\prime}(V))^{2}}|\tilde{V}_{t}|{\Psi_{x}^{2}}\Psi_{x}^{2}dxdt+\int_{0}^{t}\int_{-\infty}^{\infty}\frac{\phi^{2}_{xx}}{V^{\alpha+1}}dxdt\\\
\leq&C\left(\|\phi_{0x}\|^{2}+\|\Psi_{0x}\|^{2}\right)+(C+C\delta+C_{\eta})\int_{0}^{t}\|\phi_{x}\|^{2}dt+(C\delta+\eta)\int_{0}^{t}\|\phi_{xx}\|^{2}dt\\\
&+Ce^{-c_{-}\beta_{1}}+C\varepsilon.\end{split}$

Choosing $\eta$ appropriately small and $\delta$ sufficient small, together
with Lemma 4.3 we get the proof of Lemma 4.4. ∎

######  Lemma 4.5.

Under the same assumptions of Proposition 3.1, we have

$\displaystyle\int_{0}^{t}\|\Psi_{x}(t)\|^{2}dt\leq
C\|(\phi_{0},\Psi_{0})\|_{1}^{2}+Ce^{-c_{-}\beta_{1}}+C\varepsilon.$

###### Proof.

We multiply $(\ref{4.5})_{1}$ by $\Psi_{x}$ and make use of $(\ref{4.5})_{2}$,
we get

$\displaystyle\Psi_{x}^{2}=-\Psi_{x}G-\Psi_{x}F_{1}-\frac{\Psi_{x}\phi_{xx}}{V^{\alpha+1}}+(\phi\Psi_{x})_{t}-\phi\left[(p(V)-p(\widetilde{v})+F_{2}-\frac{F_{1x}}{V^{\alpha+1}}\right]_{x}.$
(4.22)

Intergrade $(\ref{b6})$ with respect to $t$ and $x$ over $[0,t]\times{R}$, we
have

$\displaystyle\begin{split}&\int_{0}^{t}\int_{-\infty}^{\infty}\Psi_{x}^{2}dxdt\\\
=&-\int_{0}^{t}\int_{-\infty}^{\infty}\Psi_{x}Gdxdt+\left.\int_{-\infty}^{\infty}\phi\Psi_{x}dx-\int_{-\infty}^{\infty}\phi\Psi_{x}dx\right|_{t=0}\\\
&-\int_{0}^{t}\int_{-\infty}^{\infty}\frac{\Psi_{x}\phi_{xx}}{V^{\alpha+1}}dxdt-\int_{0}^{t}\int_{-\infty}^{\infty}\phi_{x}\left(p(\tilde{v})-p(V)\right)dxdt\\\
&+\int_{0}^{t}\int_{-\infty}^{\infty}\phi_{x}[F_{2}-\frac{F_{1x}}{V^{\alpha+1}}]-\Psi_{x}F_{1}dxdt:=\sum_{i=1}^{6}H_{i}.\end{split}$

We estimate $H_{i}$ term by term. By the Cauchy inequality, it follows that

$\displaystyle\begin{split}H_{1}&\leq
C\int_{0}^{t}\int_{-\infty}^{\infty}\Psi_{x}(|\phi_{x}\phi_{xx}|+|V_{x}\phi_{x}|)dxdt\\\
&\leq\eta\int_{0}^{t}\|\Psi_{x}\|^{2}dt+C_{\eta}\int_{0}^{t}(\|\phi_{xx}\|^{2}+\|\phi_{x}\|^{2})dt.\end{split}$
(4.23)

In addition, it is straightforward to imply that

$\displaystyle\begin{split}H_{2}+H_{3}=\int_{-\infty}^{\infty}\phi\Psi_{x}-\phi\Psi_{0x}dx\leq\|(\phi,\Psi_{x})\|^{2}+\|(\phi_{0},\Psi_{0,x})\|^{2},\end{split}$
(4.24)

$\displaystyle\begin{split}H_{4}\leq\eta\int_{0}^{t}\|\Psi_{x}\|^{2}{dt}+C_{\eta}\int_{0}^{t}\|\phi_{xx}\|^{2}dt,\quad
H_{5}\leq C\int_{0}^{t}\|\phi_{x}\|^{2}dt,\end{split}$ (4.25)

and

$\displaystyle\begin{split}H_{6}&\leq\eta\int_{0}^{t}\|\phi_{x},\Psi_{x}\|^{2}dt+C_{\eta}\int_{0}^{t}\|F_{2}\|^{2}+\|F_{1}\|_{1}^{2}dt\\\
&\leq\eta\int_{0}^{t}\|\phi_{x},\Psi_{x}\|^{2}dt+C_{\eta}(e^{-c_{-}\beta_{1}}+\varepsilon).\end{split}$
(4.26)

Thanks to (4.23)-(4.26) and Lemma 4.4, taking $\eta$ sufficient small, we
obtain the proof of Lemma 4.5. ∎

Combining Lemma 4.3-Lemma 4.5, we obtain the following low-order estimate

$\displaystyle\|(\phi,\Psi)\|_{1}^{2}(t)+\int_{0}^{t}\|\Psi_{x}\|^{2}+\|\phi_{x}\|_{1}^{2}dt\leq
C\|(\phi_{0},\Psi_{0})\|_{1}^{2}+Ce^{-c_{-}\beta}+C\varepsilon,$ (4.27)

###  4.2. High Order Estimates.

If we continue to get the estimates of second order derivative
$\phi_{xx},\Psi_{xx}$, new difficulties arise. In fact, in order to close the
a priori estimate, $\|\Psi_{xx}\|_{2}$ should be sufficiently small.
Unfortunately, it means that we have to add an additional condition
“$v^{\prime\prime}(0)=0$” which can guarantee that the Dirac function will not
appear. Next, we need change variables $(\phi,\Psi)$ to $(\phi,\psi)$.

######  Lemma 4.6.

Under the same assumptions of Proposition 3.1, for $0\leq t\leq T$, it holds
that:

$\displaystyle\begin{split}\|\Psi_{0}\|_{1}^{2}\leq&\|\psi_{0}\|_{1}^{2}+C\|\phi_{0}\|_{2}^{2},\quad\|\psi\|^{2}\leq\|\Psi\|^{2}+C\|\phi\|_{1}^{2},\\\
\|\psi_{x}\|^{2}\leq&\|\Psi_{x}\|^{2}+C\|\phi_{x}\|_{1}^{2}.\end{split}$

###### Proof.

This lemma is similar like [1] and the proof is omitted. ∎

Using this lemma, low order estimate (4.27) can be rewritten as

######  Lemma 4.7.

Under the same assumptions of Proposition 3.1, it holds that

$\displaystyle\begin{split}&(\|\phi\|_{1}^{2}+\|\psi\|^{2})(t)+\int_{0}^{t}\|\psi_{x}\|^{2}+\|\phi_{x}\|_{1}^{2}dt\leq
C\|\phi_{0}\|_{2}^{2}+C\|\psi_{0}\|_{1}^{2}+Ce^{-c_{-}\beta_{1}}+C\varepsilon.\end{split}$

Next, we turn to the original equation (3.3) to study the higher order
estimates.

######  Lemma 4.8.

Under the same assumptions of Proposition 3.1, it holds that

$\displaystyle\begin{split}&\|\psi_{x}\|^{2}(t)+\int_{0}^{t}\|\psi_{xx}\|^{2}dt\leq
C\|\phi_{0}\|_{2}^{2}+C\|\psi_{0}\|_{1}^{2}+Ce^{-c_{-}\beta_{1}}+C\varepsilon.\end{split}$
(4.28)

###### Proof.

Multiplying $(\ref{3.3})_{2}$ by $-\psi_{xx}$, integrating the result with
respect to $t$ and $x$ over $[0,t]\times{R}$ gives

$\displaystyle\begin{split}&\frac{1}{2}\|\psi_{x}\|^{2}(t)+\int_{0}^{t}\int_{-\infty}^{\infty}\frac{{\psi_{xx}^{2}}}{V^{\alpha+1}}dxdt\\\
=&\frac{1}{2}\|\psi_{0x}\|^{2}-\int_{0}^{t}\int_{-\infty}^{\infty}F_{2}\psi_{xx}dxdt\\\
&-\int_{0}^{t}\int_{-\infty}^{\infty}f(V,U_{x})\phi_{x}\psi_{xx}dxdt-\int_{0}^{t}\int_{-\infty}^{\infty}J\psi_{xx}dxdt\\\
=:&\frac{1}{2}\|\psi_{0x}\|^{2}+\sum_{i=1}^{3}M_{i}.\end{split}$ (4.29)

Making use of Lemma 3.1, we have

$\displaystyle\begin{split}M_{1}&\leq\eta\int_{0}^{t}\|\psi_{xx}\|^{2}dt+C_{\eta}\int_{0}^{t}\|F_{2}\|^{2}dt\\\
&\leq\eta\int_{0}^{t}\|\psi_{xx}\|^{2}dt+C_{\eta}(e^{-c_{-}\beta_{1}}+\varepsilon).\end{split}$
(4.30)

The Cauchy inequality implies that

$\displaystyle
M_{2}\leq\eta\int_{0}^{t}\|\psi_{xx}\|^{2}dt+C_{\eta}\int_{0}^{t}\|\phi_{x}\|^{2}dt.$
(4.31)

By $(\ref{4.7})_{1}$ and the Sobolev inequality, yields

$\displaystyle\begin{split}M_{3}&\leq
C\int_{0}^{t}\int_{-\infty}^{\infty}\left(\left|\phi_{x}\right|^{2}+\left|\phi_{x}\right|\left|\psi_{xx}\right|\right)\left|\psi_{xx}\right|dxdt\\\
&\leq
C\int_{0}^{t}\int_{-\infty}^{\infty}\left|\phi_{x}\right|\left(\left|\phi_{x}\right|^{2}+\left|\psi_{xx}\right|^{2}\right)dxdt\\\
&\leq
C\delta\int_{0}^{t}\left(\left\|\phi_{x}\right\|^{2}+\left\|\psi_{xx}\right\|^{2}\right)dt.\end{split}$
(4.32)

Substituting (4.30)-(4.32) into $(\ref{4.26})$ and using Lemma 4.7, we obtain
(4.28). ∎

######  Lemma 4.9.

Under the same assumptions of Proposition 3.1, it holds that

$\displaystyle\begin{split}&\|\phi_{xx}\|^{2}+\int_{0}^{t}\|\phi_{xx}\|^{2}dt\leq
C\|\phi_{0}\|_{2}^{2}+C\|\psi_{0}\|_{1}^{2}+C\delta\int_{0}^{t}\left\|\psi_{xxx}\right\|^{2}dt+Ce^{-c_{-}\beta_{1}}+C\varepsilon.\end{split}$
(4.33)

###### Proof.

Differentiating $(\ref{3.3})_{1}$ with respect to $x$, using
$(\ref{3.3})_{2}$, we have

$\displaystyle\begin{split}\frac{\phi_{xt}}{V^{\alpha+1}}+f(V,U_{x})\phi_{x}=\psi_{t}-J+\frac{F_{1x}}{V^{\alpha+1}}-F_{2}.\end{split}$
(4.34)

Differentiating $(\ref{4.31})$ in respect of $x$ and multiplying the
derivative by $\phi_{xx}$, integrating the result in respect of $t$ and $x$
over $[0,t]\times{R}$, one has

$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}\frac{\phi_{xx}^{2}}{V^{\alpha+1}}dx+\int_{0}^{t}\int_{-\infty}^{\infty}\left(f(\tilde{V},\tilde{U}_{x})+\frac{\alpha+1}{2}\frac{\tilde{U}_{x}}{\tilde{V}^{\alpha+2}}\right)\phi_{xx}^{2}dxdt$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left.\int_{-\infty}^{\infty}\frac{\phi_{xx}^{2}}{V^{\alpha+1}}dx\right|_{t=0}-\left.\int_{-\infty}^{\infty}\psi_{x}\phi_{xx}dx\right|_{t=0}+\int_{-\infty}^{\infty}\psi_{x}\phi_{xx}dx$
$\displaystyle+\int_{0}^{t}\int_{-\infty}^{\infty}\left\\{\frac{F_{1x}}{V^{\alpha+1}}-F_{2}\right\\}_{x}\phi_{xx}dxdt+\int_{0}^{t}\|\psi_{xx}\|^{2}dt-\int_{0}^{t}\int_{-\infty}^{\infty}J_{x}\phi_{xx}dxdt$
(4.35)
$\displaystyle+(\alpha+1)\int_{0}^{t}\int_{-\infty}^{\infty}\frac{V_{x}}{V^{\alpha+2}}\psi_{xx}\phi_{xx}dxdt-\int_{0}^{t}\int_{-\infty}^{\infty}f(V,U_{x})_{x}\phi_{x}\phi_{xx}dxdt$
$\displaystyle-\int_{0}^{t}\int_{-\infty}^{\infty}\left[f(V,U_{x})-f(\tilde{V},\tilde{U}_{x})+\frac{\alpha+1}{2}\left(\frac{U_{x}}{V^{\alpha+2}}-\frac{\tilde{U}_{x}}{\tilde{V}^{\alpha+2}}\right)\right]\phi_{xx}^{2}dxdt$
$\displaystyle=:$
$\displaystyle\frac{1}{2}\left.\int_{-\infty}^{\infty}\frac{\phi_{xx}^{2}}{V^{\alpha+1}}dx\right|_{t=0}-\left.\int_{-\infty}^{\infty}\psi_{x}\phi_{xx}\right|_{t=0}dx+\sum_{i=1}^{7}N_{i}.$

By $\tilde{U}_{x}<0$, one has

$\displaystyle\begin{split}&f(\tilde{V},\tilde{U}_{x})+\frac{\alpha+1}{2}\frac{\tilde{U}_{x}}{V^{\alpha+2}}\\\
=&-p^{\prime}(\tilde{V})-\frac{\alpha+1}{2}\frac{\tilde{U}_{x}}{\tilde{V}^{\alpha+2}}\geq-p^{\prime}(v_{-})>0.\end{split}$
(4.36)

The Cauchy inequality yields

$\displaystyle N_{1}\leq\eta\|\phi_{xx}\|^{2}+C_{\eta}\|\psi_{x}\|^{2}.$
(4.37)

Similar to (4.30), we get

$\displaystyle
N_{2}\leq\eta\int_{0}^{t}\|\phi_{xx}\|^{2}dt+C_{\eta}(e^{-c_{-}\beta_{1}}+\varepsilon).$
(4.38)

$N_{3}$ can be controlled by (4.28). Using $(\ref{4.7})_{2}$, and Cauchy
inequality, we have

$\displaystyle\begin{split}N_{4}\leq&\eta\int_{0}^{t}\|\phi_{xx}\|^{2}dt+C_{\eta}\int_{0}^{t}\left\|J_{x}\right\|^{2}dt\\\
\leq&\eta\int_{0}^{t}\|\phi_{xx}\|^{2}dt+C_{\eta}\delta\int_{0}^{t}\left(\left\|\phi_{x}\right\|_{1}^{2}+\left\|\psi_{x}\right\|_{2}^{2}\right)dt.\end{split}$

The Cauchy inequality yields

$\displaystyle\begin{split}N_{5}\leq
C&\int_{0}^{t}\int_{-\infty}^{\infty}\left|\psi_{xx}\phi_{xx}\right|dxdt\leq\eta\int_{0}^{t}\|\phi_{xx}\|^{2}dt+C_{\eta}\int_{0}^{t}\left\|\psi_{xx}\right\|^{2}dt.\end{split}$
(4.39)

With the help of

$f(V,U_{x})_{x}=-p^{\prime\prime}(V)V_{x}-(\alpha+1)\frac{U_{xx}}{V^{\alpha+2}}+(\alpha+1)(\alpha+2)\frac{U_{x}}{V^{\alpha+3}}V_{x}<C,$

one gets

$\displaystyle\begin{split}|N_{6}|\leq&\eta\int_{0}^{t}\|\phi_{xx}\|^{2}dt+C_{\eta}\int_{0}^{t}\left\|\phi_{x}\right\|^{2}dt.\end{split}$
(4.40)

Similar like (4.18), one gets that

$\displaystyle\begin{split}N_{7}\leq&C\int_{0}^{t}\|q+z_{x}\|_{L^{\infty}}\|\phi^{2}\|_{L^{1}}dt\\\
\leq&C\sup_{\tau\in[0,t]}\|\Psi(\tau)\|^{2}\int_{0}^{t}\|z_{x}+q\|_{H^{1}}dt\leq
C\varepsilon.\end{split}$ (4.41)

Choosing $\eta$ small, substituting (4.36)-(4.41) into (4.2) and using Lemma
4.7, Lemma 4.8, we have (4.33). ∎

On the other hand, differentiating the second equation of (3.3) with respect
to $x$, multiplying the derivative by $-\psi_{xxx}$, integrating the resulting
equality over $(-\infty,\infty){\times}[0,t]$, using Lemma 4.7 \- Lemma 4.9,
we can get the highest order estimate in the same way, which is listed as
follows and the proof is omitted.

######  Lemma 4.10.

Under the same assumptions of Proposition 3.1, it holds that

$\displaystyle\|\psi_{xx}(t)\|^{2}+\int_{0}^{t}\|\psi_{xxx}\|^{2}dt\leq
C\|(\phi_{0},\psi_{0})\|_{2}^{2}+Ce^{-c_{-}\beta_{1}}+C\varepsilon.$

Finally, Proposition 3.1 is obtained by Lemma 4.7-Lemma4.10.

##  5\. Proof of Theorem 2.1

It is straightforward to imply (2.23) from Lemma 3.2. It remains to show
(2.24). The following useful lemma will be used.

######  Lemma 5.1.

([22]) Assume that the function $f(t)\geq
0\in{L}^{1}(0,+\infty)\cap{BV}(0,+\infty)$, then it holds that
$f(t)\rightarrow 0$ as $t\rightarrow\infty$.

Let us turn to the system (3.3). Differentiating (3.3)1 with respect to $x$,
multiplying the resulting equation by $\phi_{x}$ and integrating it with
respect to ${x}$ on $(-\infty,\infty)$, we have

$\left|\frac{d}{dt}\left(\|\phi_{x}\|^{2}\right)\right|\leq
C(\|\phi_{x}\|^{2}+\|\psi_{xx}\|^{2}).$

With the aid of Lemma 3.2, we have

$\int_{-\infty}^{\infty}\left|\frac{d}{dt}\left(\|\phi_{xx}\|^{2}\right)\right|dt\leq
C,$

which implies $\|\phi_{x}\|^{2}\in{L}^{1}(0,+\infty)\cap{BV}(0,+\infty)$. By
Lemma 5.1, we have

$\|\phi_{x}\|\rightarrow 0\quad\text{as}\quad t\rightarrow+\infty.$

Since $\|\phi_{xx}\|$ is bounded, the Sobolev inequality implies that

$\displaystyle\|{\tilde{v}}-V\|_{\infty}^{2}=\|\phi_{x}\|_{\infty}^{2}\leq
2\|\phi_{x}(t)\|\|\phi_{xx}(t)\|\rightarrow 0.$

Similarly, we have

$\displaystyle\|{\tilde{u}}-U\|_{\infty}^{2}=\|\psi_{x}\|_{\infty}^{2}\leq
2\|\psi_{x}(t)\|\|\psi_{xx}(t)\|\rightarrow 0.$

Therefore, the proof of Lemma 2.4 is completed.

###  5.1. Proof of Theorem 2.1

######  Lemma 5.2.

Under the assumptions (2.5)-(2.7), when $\phi_{0},\psi_{0}$ and $\beta$
satisfy

$\quad\beta\rightarrow\beta_{1},\quad\|\phi_{0}\|_{H^{2}({R}_{+})}+\|\psi_{0}\|_{H^{2}({R}_{+})}\rightarrow
0,\quad\text{as}\quad\|A_{0},B_{0}\|_{H^{2}({R}_{+})}+\beta_{1}^{-1}+\varepsilon\rightarrow
0.$

###### Proof.

Using (2.5) and (2.7), we know $\left(A_{0},B_{0}\right)\in H^{2}(R_{+})$

With the aid of $0<$ $-U(-st-\beta_{1})\leq Ce^{-c-(st+\beta_{1})}$ (see Lemma
2.1) and $\beta_{1}>0$, it follows that
$\left|\int_{0}^{\infty}U(-st-\beta_{1})dt\right|\leq Ce^{-c-\beta_{1}}$. Thus
if $\beta_{1}^{-1}+\varepsilon\rightarrow 0$ and
$\left\|A_{0}\right\|_{H^{2}({R}_{+})}\rightarrow 0$, using (2.18), we obtain

$|\tilde{M}|\leq
C\left(\left\|A_{0}\right\|_{H^{2}({R}_{+})}+e^{-c-\beta_{1}}+\varepsilon\right)\rightarrow
0.$

Similar, with the help of (2.20), we have

$|\beta-\mathcal{X}_{0}|\rightarrow 0.$

Thus, it follows that

$|\beta-\beta_{1}|\leq|\beta-\mathcal{X}_{0}|+|\mathcal{X}_{0}-\beta_{1}|\leq|\beta-\mathcal{X}_{0}|+\frac{3}{2}|\tilde{M}|\rightarrow
0.$

Set

$\displaystyle\begin{split}(\tilde{A}_{0},\tilde{B}_{0})(x)&:=-\int_{x}^{\infty}(v_{0}(y)-\zeta(y)-V_{2}^{S}(y-\beta),u_{0}(y)-\varphi(y)-U_{2}^{S}(y-\beta))dy,\\\
\chi_{1}(x)&:=\int_{0}^{\beta_{1}-\beta}\left[v_{+}-V(x-\beta_{1}+\theta)\right]d\theta.\end{split}$
(5.1)

Make full use of (2.4), when $|\beta_{1}-\beta|<1$, we have

$\left|v_{+}-V(x-\beta_{1}+\theta)\right|\leq Ce^{-c+|x-\beta_{1}+\theta|}\leq
Ce^{-c_{+}|x-\beta_{1}|}e^{c_{+}|\beta_{1}-\beta|}\leq
Ce^{-c_{+}|x-\beta_{1}|}$

Thus, we have

$\displaystyle\left\|\chi_{1}\right\|^{2}_{({R}_{+})}$ $\displaystyle\leq
C\int_{0}^{\infty}(\beta_{1}-\beta)^{2}e^{-2c_{+}|x-\beta_{1}|}dx\leq
C(\beta_{1}-\beta)^{2}$

where $C$ is independent of $(\beta_{1}-\beta)$ and $\beta$. Similarly, we can
prove that $\left\|\chi_{1}^{\prime}\right\|_{({R}_{+})}^{2}\leq
C(\beta_{1}-\beta)^{2}$ and
$\left\|\chi_{1}^{\prime\prime}\right\|_{({R}_{+})}^{2}\leq
C(\beta_{1}-\beta)^{2}$. Thus, we proved
$\left\|\chi_{1}\right\|_{H^{2}(R_{+})}\leq C|(\beta_{1}-\beta)|$. In the same
way, we have that

$\|\chi_{2}\|_{H^{2}(R_{+})}:=\left\|\int_{0}^{\beta_{1}-\beta}\left[u_{+}-U(x+\theta-\beta)\right]d\theta\right\|_{H^{2}(R_{+})}\leq
C|\beta_{1}-\beta|.$

Thus, we obtain

$\displaystyle\begin{split}&\|(\tilde{A}_{0},\tilde{B}_{0})\|_{H^{2}(R_{+})}\leq\left\|\left(A_{0},B_{0}\right)\right\|_{H^{2}(R_{+})}+\left\|\left(\chi_{1},\chi_{2}\right)\right\|_{H^{2}(R_{+})}\\\
\leq&C(\left\|\left(A_{0},B_{0}\right)\right\|_{H^{2}(R_{+})}+|\beta_{1}-\beta|).\end{split}$
(5.2)

It follows from $|\phi_{0}|,|\psi_{0}|$ are all even functions that

$\|\phi_{0}\|_{H^{2}(R_{+})}=\frac{1}{2}\|\phi_{0}\|_{H^{2}(R)},\|\psi_{0}\|_{H^{2}(R_{+})}=\frac{1}{2}\|\psi_{0}\|_{H^{2}(R)}.$
(5.3)

Using (2.9), (4.8) and $(\ref{{2.6}})_{1}$, when $x\in R_{+}$, one gets

$\displaystyle V(x,0)-\zeta(x)-V^{S}_{2}(x-\beta)$ $\displaystyle=$
$\displaystyle[V(x,0)-\tilde{V}(x,0)-\zeta(x)]+[\tilde{V}(x,0)-V^{S}_{2}(x-\beta)]$
$\displaystyle=$ $\displaystyle q(x,0)-\zeta(x)+[V^{S}_{2}(-x-\beta)-v_{-}]$
$\displaystyle\leq$
$\displaystyle|\zeta(x)\left[g_{2}(x-\beta_{1})-g_{1}(x+\beta_{1})\right]|+|V^{S}_{1}(x-\beta)-V^{S}_{1}(x-\mathcal{X})|$
$\displaystyle+|V^{S}_{2}(x+\beta)-V^{S}_{2}(x+\mathcal{X})|+|V^{S}_{2}(-x-\beta)-v_{-}|.$

With the aid of (2.19), Lemma 2.1 and Lemma 2.3, it follows that

$\|\phi_{0}-\tilde{A}_{0}\|_{H^{2}(R_{+})}\leq\varepsilon+e^{-c_{-}\beta}.$
(5.4)

Similar, we have

$\|\psi_{0}-\tilde{B}_{0}\|_{H^{2}(R_{+})}\leq\varepsilon+e^{-c_{-}\beta}.$
(5.5)

Combining (5.2)-(5.5), one gets that
$\|\phi_{0}\|_{H^{2}({R}_{+})}+\|\psi_{0}\|_{H^{2}({R}_{+})}\rightarrow 0$. ∎

Once this lemma is proved , we begin the proof of our main result. Using
(2.9), (4.8) and $(\ref{{2.6}})_{1}$, when $x\in R_{+}$, one gets

$\displaystyle v(x,t)-V^{S}_{2}(x-st-\beta)$ $\displaystyle=$
$\displaystyle[v(x,t)-V(x,t)]+[V(x,t)-\tilde{V}(x,t)]+[\tilde{V}(x,t)-V^{S}_{2}(x-st-\beta)]$
$\displaystyle=$
$\displaystyle[{\widetilde{v}(x,t)}-V(x,t)]+q(x,t)+[V^{S}_{2}(-x-st-\beta)-v_{-}]$
$\displaystyle\leq$
$\displaystyle|{\widetilde{v}(x,t)}-V(x,t)|+|q(x,t)|+|V^{S}_{2}(-x-st-\beta)-v_{-}|.$

We obtain that

$\displaystyle\begin{split}&\|v(x,t)-V^{S}_{2}(x-st-\beta)\|_{L^{\infty}}\\\
\leq&\|{\widetilde{v}(x,t)}-V(x,t)\|_{L^{\infty}}+\|q(x,t)\|_{L^{\infty}}+\|V^{S}_{2}(-x-st-\beta)-v_{-}\|_{L^{\infty}}.\end{split}$

Together with (5.3), Lemma 2.1-Lemma 2.4 and Lemma 5.2 we obtain the proof of
Theorem 2.1.

##  6\. Proof of Lemma 2.3 and Lemma 3.1

###  6.1. Proof of Lemma 2.3

###### Proof.

By Lemma 2.2, we have
$|{\mathcal{X}^{\prime}(t)}|,|{\mathcal{Y}^{\prime}(t)}|\leq C\varepsilon
e^{-2\sigma_{0}t}$ for all $t>0$. Thus
$\lim\limits_{t\rightarrow+\infty}\mathcal{X}(t)$ and
$\lim\limits_{t\rightarrow+\infty}\mathcal{Y}(t)$ are all exist. In the
following part of this subsection, we compute the two limits. Motivated by
[11], we define the domain

$\begin{cases}\Omega_{y}^{N}(t):=\left\\{(x,\tau):0<\tau<t,\quad\Gamma_{l}^{N}(\tau)<x<\Gamma_{r}^{N}(t)\right\\},\\\
\Gamma_{l}^{N}(\tau):=-s\tau-\mathcal{X}(\tau)+(-N+y)\pi,\\\
\Gamma_{r}^{N}(\tau):=s\tau+\mathcal{X}(\tau)+(N+y)\pi,\end{cases}$ (6.1)

where $y\in[0,1],N\in N^{*}$. Using $(\ref{3.1})_{1}$, we have

$\lim_{N\rightarrow+\infty}\int_{0}^{1}\iint_{\Omega_{y}^{N}(t)}(V_{t}-U_{x})dxd\tau
dy=0.$

With the aid of Green formula, one gets

$\displaystyle\begin{split}\lim_{N\rightarrow+\infty}\int_{0}^{1}\mathfrak{f}(N,y)dy=0.\end{split}$
(6.2)

where

$\displaystyle\begin{split}\mathfrak{f}(N,y)=&\int_{\Gamma_{l}^{N}(0)}^{\Gamma_{r}^{N}(0)}V(x,0)dx+\int_{0}^{t}[(s+\mathcal{X}^{\prime})V+U](\Gamma_{r}^{N}(\tau),\tau)d\tau\\\
&-\int_{\Gamma_{l}^{N}(t)}^{\Gamma_{r}^{N}(t)}V(x,t)dx-\int_{0}^{t}[(-s-\mathcal{X}^{\prime})V+U](\Gamma_{l}^{N}(\tau),\tau)d\tau:=\sum_{i=1}^{4}S_{i}(N,y).\end{split}$

We rewrite $S_{1}+S_{3}$ as:

$\displaystyle S_{1}+S_{3}=\sum_{i=1}^{4}I_{i},$

where

$\displaystyle I_{1}$
$\displaystyle=\int_{\Gamma_{l}^{N}(0)}^{\Gamma_{r}^{N}(0)}\zeta(-x)(1-g_{1}(x+\mathcal{X}_{0}))+\zeta(x)g_{2}(x-\mathcal{X}_{0})dx,$
$\displaystyle I_{2}$
$\displaystyle=\int_{\Gamma_{l}^{N}(0)}^{\Gamma_{r}^{N}(0)}V^{S}_{1}(x+{\mathcal{X}_{0}})-{v}_{-}+V^{S}_{2}(x-{\mathcal{X}_{0}})dx,$
$\displaystyle I_{3}$
$\displaystyle=-\int_{\Gamma_{l}^{N}(t)}^{\Gamma_{r}^{N}(t)}\zeta_{l}(x,t)(1-g_{1}(x+st+{\mathcal{X}}))+\zeta_{r}(x,t)g_{2}(x-st-{\mathcal{X}})dx,$
$\displaystyle I_{4}$
$\displaystyle=-\int_{\Gamma_{l}^{N}(t)}^{\Gamma_{r}^{N}(t)}V^{S}_{1}(x+st+{{\mathcal{X}}})-{v}_{-}+V^{S}_{2}(x-st-{{\mathcal{X}}})dx.$

Here

$\displaystyle\zeta_{i}=v_{i}-\bar{v}_{i};\quad i=l,r.$

Moreover, $I_{1}$ can be rewrite as

$\displaystyle I_{1}=$
$\displaystyle\int_{0}^{\Gamma_{r}^{N}(0)}[\zeta(-x)(1-g_{1}(x+\mathcal{X}_{0}))-\zeta(x)(1-g_{2}(x-\mathcal{X}_{0}))]dx+\int_{0}^{\Gamma_{r}^{N}(0)}\zeta(x)dx,$
$\displaystyle+\int_{\Gamma_{l}^{N}(0)}^{0}[-\zeta(-x)g_{1}(x+\mathcal{X}_{0})+\zeta(x)g_{2}(x-\mathcal{X}_{0})]dx+\int_{\Gamma_{l}^{N}(0)}^{0}\zeta(-x)dx.$

Since $\int_{0}^{\pi}\zeta(x)dx=0$, then

$\displaystyle\int_{0}^{1}\int_{0}^{\mathcal{X}_{0}+y\pi}\zeta(x)dxdy$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{\mathcal{X}_{0}+z}\zeta(x)dxdz=\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{y}\zeta(x)dxdy,$
$\displaystyle\int_{0}^{1}\int_{\mathcal{X}_{0}+y\pi}^{0}\zeta(-x)dxdy$
$\displaystyle=-\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{\mathcal{X}_{0}+z}\zeta(-x)dxdz=-\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{y}\zeta(-x)dxdy.$

So we obtain

$\displaystyle\lim_{N\rightarrow+\infty}\int_{0}^{1}I_{1}dy=$
$\displaystyle\int_{0}^{\infty}\zeta(-x)(1-g_{1}(x+\mathcal{X}_{0}))-\zeta(x)(1-g_{2}(x-\mathcal{X}_{0}))dx,$
$\displaystyle+\int_{-\infty}^{0}-\zeta(-x)g_{1}(x+\mathcal{X}_{0})+\zeta(x)g_{2}(x-\mathcal{X}_{0})dx,$
$\displaystyle+\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{y}\zeta(x)-\zeta(-x)dxdy,$
$\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}\zeta(-x)(1-g_{1}(x+\mathcal{X}_{0}))-\zeta(x)(1-g_{2}(x-\mathcal{X}_{0}))dx,$
$\displaystyle+\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{y}\zeta(x)-\zeta(-x)dxdy,$
(6.3)

where we have used (2.9),(2.11)in the last equality. With the aid of Lemma
2.2, one gets that

$\displaystyle\left|\lim_{N\rightarrow+\infty}\int_{0}^{1}I_{3}dy\right|\leq
Ce^{-2\sigma_{0}t}.$ (6.4)

By directly calculate, we have

$\displaystyle I_{2}+I_{4}=$ $\displaystyle-
v_{-}[\Gamma_{r}^{N}(0)-\Gamma_{l}^{N}(0)]+v_{-}[\Gamma_{r}^{N}(t)-\Gamma_{l}^{N}(t)]$
$\displaystyle-$
$\displaystyle\int_{(N+y)\pi+2\mathcal{X}_{0}}^{2st+2{\mathcal{X}}+(N+y)\pi}V^{S}_{1}(x)dx+\int_{(-N+y)\pi-2\mathcal{X}_{0}}^{-2st-2{\mathcal{X}}+(-N+y)\pi}V^{S}_{2}(x)dx.$

Using (2.8) (2.9), one gets that

$\displaystyle\lim_{N\rightarrow+\infty}\int_{0}^{1}(I_{2}+I_{4})dy=-2v_{-}(st+{\mathcal{X}}-\mathcal{X}_{0}).$
(6.5)

The integral on $\Gamma_{r}^{N}$ in (6.2) satisfies that

$\displaystyle\lim_{N\rightarrow+\infty}\int_{0}^{1}S_{2}(N,y)dy$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow+\infty}\int_{0}^{t}\int_{0}^{1}[(s+\mathcal{X}^{\prime})v_{r}+u_{r}](\Gamma_{r}^{N}(\tau),\tau)dyd\tau$
$\displaystyle=$
$\displaystyle(st+{\mathcal{X}}-\mathcal{X}_{0})v_{+}+u_{+}t.$ (6.6)

Here we have used Since $(V,U)\rightarrow(v_{r},u_{r})$ as
$x\rightarrow+\infty$. By same method, we obtain

$\lim_{N\rightarrow+\infty}\int_{0}^{1}S_{4}(N,y)dy=-[(-st-{\mathcal{X}}+\mathcal{X}_{0})v_{+}-u_{+}t].$
(6.7)

Collecting (6.2)-(6.7), it follows that

$\displaystyle
2\int_{0}^{+\infty}\left[\zeta(-x)\left(1-g_{1}(x+\mathcal{X}_{0})\right)-\zeta(x)\left(1-g_{2}(x-\mathcal{X}_{0})\right)\right]dx$
$\displaystyle\quad+$
$\displaystyle\frac{1}{\pi}\int_{0}^{\pi}\int_{0}^{y}\zeta(x)-\zeta(-x)dxdy$
$\displaystyle\quad+$ $\displaystyle
2(v_{+}-v_{-})(st+\mathcal{X}-\mathcal{X}_{0})+2u_{+}t=O(e^{-2\sigma_{0}t}),$

Thus we obtain (2.20) where we have used R-H conditions (1.6)1. We omit the
proof of (2.21), since it is similar with (2.20). ∎

###  6.2. Proof of Lemma 3.1

We only give the proof of $F_{1}$, due to the fact that the proof of $F_{2}$
is similar.

Case 1. For $x<st,$ we rewrite $F_{1}(x,t)$ as follows.

$F_{1}(x,t):=D_{1,1}^{-}(x,t)+D_{1,2}^{-}(x,t),$

where

$\displaystyle D_{1,1}^{-}(x,t):=$
$\displaystyle\theta\left\\{(-\mathcal{X}^{\prime})g_{1}(x+st+{\mathcal{X}})+(-s)\left(g_{1}(x+st+{\mathcal{X}})-g_{1}(x+st+{\mathcal{Y}})\right)\right\\}$
$\displaystyle-\theta\left\\{(\mathcal{X}^{\prime})g_{2}(x-st-\mathcal{X})+s\left(g_{2}(x-st-\mathcal{X})-g_{2}(x-st-{\mathcal{Y}})\right)\right\\},$
$\displaystyle D_{1,2}^{-}(x,t):=$
$\displaystyle\zeta_{l}(x,t)\left[g_{1}(x+st+\mathcal{Y})-g_{1}(x+st+\mathcal{X})\right]+\int_{-\infty}^{x}\varphi_{l}(y,t)g^{\prime}_{1}(y+st+\mathcal{X})dy$
$\displaystyle+(s+\mathcal{X}^{\prime})\int_{-\infty}^{x}\zeta_{l}(y,t)g^{\prime}_{1}(y+st+\mathcal{X})dy$
$\displaystyle-\zeta_{r}(x,t)\left[g_{2}(x-st-\mathcal{Y})-g_{2}(x-st-\mathcal{X})\right]-\int_{-\infty}^{x}\varphi_{r}(y,t)g^{\prime}_{2}(y-st-\mathcal{X})dy$
$\displaystyle-(s+\mathcal{X}^{\prime})\int_{-\infty}^{x}\zeta_{r}(y,t)g^{\prime}_{2}(y-st-\mathcal{X})dy.$

It follows that

$\displaystyle\begin{split}&\sum_{k=0}^{2}\int_{-\infty}^{st}|\partial_{x}^{k}F_{1}(x,t)|^{2}dx\\\
\leq&C\varepsilon^{2}e^{-4\sigma_{0}t}\sum_{k=0}^{2}\int_{-\infty}^{st}\left|g_{1}^{(k)}(x+st+{\mathcal{X}})\right|^{2}+\left|{g_{2}^{(k)}}(x-st-{\mathcal{X}})\right|^{2}dx\\\
\leq&C\varepsilon^{2}e^{-4\sigma_{0}t}\int_{-\infty}^{-st}\theta(e^{-\theta|{x+st}|}+e^{-\theta|{x-st}|})dx+\int_{-st}^{st}(1+\theta
e^{-\theta|{x-st}|})dx\\\
\leq&C\varepsilon^{2}e^{-4\sigma_{0}t}\big{[}C+2st\big{]}\leq
C\varepsilon^{2}e^{-2\sigma_{0}t}.\end{split}$ (6.8)

Here we have used Lemma 2.2, (2.4), (2.9), (2.11). Moreover, $F_{2}$ can be
rewritten as as follows.

$\displaystyle\begin{split}F_{2}:=W+D_{2,1}^{-}(x,t)+D_{2,2}^{-}(x,t),,\end{split}$
(6.9)

where

$\displaystyle\begin{split}W:=&p(\tilde{V})+p(v_{-})-p(V^{S}_{1}(x+st+{\beta}))-p(V^{S}_{2}(x-st-{\beta}))\\\
&+\frac{U^{S}_{2x}(x-st-{\beta})}{V^{S}_{2}(x-st-\beta)^{\alpha+1}}+\frac{U^{S}_{1x}(x+st+{\beta})}{V^{S}_{1}(x+st+\beta)^{\alpha+1}}-\frac{\tilde{U}_{x}}{\tilde{V}^{\alpha+1}},\end{split}$
(6.10)

$\displaystyle D_{2,1}^{-}(x,t):=$
$\displaystyle-\left[p(V^{S}_{1}(x+st+{\mathcal{Y}}))-p(V^{S}_{1}(x+st+{\beta}))\right]$
$\displaystyle-\left[p(V^{S}_{2}(x-st-{\mathcal{Y}}))-p(V^{S}_{2}(x-st-{\beta}))\right]$
$\displaystyle+\left[\frac{U^{S}_{2x}(x-st-{\mathcal{Y}})}{V^{S}_{2}(x-st-{\mathcal{Y}})^{\alpha+1}}-\frac{U^{S}_{2x}(x-st-{\beta})}{V^{S}_{2}(x-st-\beta)^{\alpha+1}}\right]$
$\displaystyle+\left[\frac{U^{S}_{1x}(x+st+{\mathcal{Y}})}{V^{S}_{1}(x+st+{\mathcal{Y}})^{\alpha+1}}-\frac{U^{S}_{1x}(x+st+{\beta})}{V^{S}_{1}(x+st+\beta)^{\alpha+1}}\right]$
$\displaystyle-\left[\frac{U_{x}}{V^{\alpha+1}}-\frac{\tilde{U}_{x}}{\tilde{V}^{\alpha+1}}\right]+p(V)-p(\tilde{V}),$

and

$\displaystyle D_{2,2}^{-}(x,t):=$
$\displaystyle\int_{-\infty}^{x}\big{[}-s(u_{l}+u_{+})-\mathcal{Y}^{\prime}u_{l}+p(v_{+})-p(v_{l})+\frac{u_{lx}}{v_{l}^{\alpha+1}}\big{]}g^{\prime}_{1}(x+st+{\mathcal{Y}}))dx$
$\displaystyle-\int_{-\infty}^{x}\big{[}s(u_{r}-u_{+})+\mathcal{Y}^{\prime}u_{r}-p(v_{r})+p(v_{+})+\frac{u_{rx}}{v_{r}^{\alpha+1}}\big{]}g_{2}^{\prime}(x-st-{\mathcal{Y}})dx.$

$\displaystyle\begin{split}|W|\leq&\left|\left(\frac{1}{(V^{S}_{1}(x+st+\beta))^{\alpha+1}}-\frac{1}{\tilde{V}^{\alpha+1}}\right)U^{S}_{1x}(x+st+\beta)\right|\\\
&+\left|\left(\frac{1}{(V^{S}_{2}(x-st-\beta))^{\alpha+1}}-\frac{1}{\tilde{V}^{\alpha+1}}\right)U^{S}_{2x}(x-st-\beta)\right|\\\
&+\left|p(V^{S}_{1}(x+st+\beta)+V^{S}_{2}(x-st-\beta)-v_{-})-p(V^{S}_{1}(x+st+\beta))\right|\\\
&+\left|p(v_{-})-p(V^{S}_{2}(x-st-\beta))\right|\\\
\leq&C\\{|(V^{S}_{2}(x-st-\beta)-v_{-})|+|U^{S}_{2x}(x-st-\beta)|\\}.\end{split}$
(6.11)

By $(\ref{2.1})$, we get

$\displaystyle\begin{split}&\left|\frac{\partial^{j}U^{S}_{2}(x-st-\beta)}{\partial
x^{j}}\right|,\left|\frac{\partial^{j}(V^{S}_{2}(x-st-\beta)-v_{-})}{\partial
x^{j}}\right|\\\ \leq C&|V^{S}_{2}(x-st-\beta)-v_{-}|,\forall
j\in\mathbb{N}.\end{split}$ (6.12)

On the other hand, in the same way, it is still true to replace
($V^{S}_{2}(x-st-\beta),U^{S}_{2}(x-st-\beta)$) with
($V^{S}_{1}(x+st+\beta),U^{S}_{1}(x+st+\beta)$) in (6.12). We get
$\left|\frac{\partial^{n}W}{\partial x^{n}}\right|\leq
C|V_{i}^{S}(x+(-st-\beta)^{i+1})-v_{-}|,i=1,2;\forall n\in\mathbb{N}.$ If we
choose $\beta>0$ sufficiently large, for $n=0,1$, it follows that:

$\displaystyle\begin{split}\int_{-\infty}^{\infty}\left|\frac{\partial^{n}W}{\partial{x}^{n}}\right|^{2}dx=&\int_{-\infty}^{0}\left|\frac{\partial^{n}W}{\partial{x}^{n}}\right|^{2}\operatorname{d}x+\int_{0}^{\infty}\left|\frac{\partial^{n}W}{\partial{x}^{n}}\right|^{2}dx\\\
\leq&C\int_{-\infty}^{0}|V_{2}(x-st-\beta)-v_{-}|^{2}dx+C\int_{0}^{\infty}|V_{1}(x+st+\beta)-v_{-}|^{2}dx\\\
\leq&C\theta^{2}\int_{-\infty}^{0}\exp[2c_{-}(x-st-\beta)]dx+C\theta^{2}\int_{0}^{\infty}\exp[-2c_{-}(x+st+\beta)]dx\\\
\leq&Ce^{-2c_{-}st}e^{-2c_{-}\beta}=Ce^{-2c_{-}st}e^{-2c_{-}\beta_{1}}e^{-2c_{-}(\beta-\beta_{1})}\leq
Ce^{-2c_{-}st}e^{-2c_{-}\beta_{1}},\end{split}$

where we have used Lemma 2.1 in the second inequality and Lemma 5.2 in the
last inequality. Thus, we obtain that

$\displaystyle\|W\|_{{2}}\leq Ce^{-c_{-}\beta_{1}}e^{-sc_{-}t}.$

Similar like (6.8), one can get that

$\sum_{k=0}^{2}\sum_{i=1}^{2}\int_{-\infty}^{st}|\partial_{x}^{k}D_{2,i}^{-}(x,t)|^{2}dx\leq
C\varepsilon^{2}e^{-2\sigma_{0}t}.$ (6.13)

Case 2. If $x>st$, using (2.14), one can decompose $F_{1},F_{2}$ as

$F_{1}(x,t)=-F_{1,1}(x,t)+\int_{x}^{+\infty}f_{1,2}(y,t)dy+\mathcal{X}^{\prime}\int_{x}^{+\infty}F_{1,3}(y,t)dy,$
(6.14)
$F_{2}(x,t)=-F_{2,1}(x,t)+\int_{x}^{+\infty}f_{2,2}(y,t)dy+\mathcal{Y}^{\prime}\int_{x}^{+\infty}F_{2,3}(y,t)dy,$
(6.15)

and using similar arguments as in the case 1 to obtain that

$\sum_{k=0}^{2}\int_{st}^{+\infty}|\partial_{x}^{k}F_{i}(x,t)|^{2}dx\leq
C\varepsilon^{2}e^{-2\sigma_{0}t},\quad i=1,2.$ (6.16)

######  Remark 6.1.

If $\gamma=1$ (isothermal gas) in our equations, we can get the same result by
the same method.

######  Remark 6.2.

In our proof, we make the position of the shock is far away from the wall, is
this necessary?

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.tiffpng.pngconvert #1 .tiff

# Feature Disentanglement in generating three-dimensional structure from two-
dimensional slice with sliceGAN

Hyungjin Chung1 and Jong Chul Ye†,1

Department of Bio and Brain Engineering, KAIST, Daejeon, Korea

†Correspondence should be addressed to J.C.Y<EMAIL_ADDRESS>
Deep generative models are known to be able to model arbitrary probability
distributions. Among these, a recent deep generative model, dubbed sliceGAN
[1], proposed a new way of using the generative adversarial network (GAN) to
capture the micro-structural characteristics of a two-dimensional (2D) slice
and generate three-dimensional (3D) volumes with similar properties. While 3D
micrographs are largely beneficial in simulating diverse material behavior,
they are often much harder to obtain than their 2D counterparts. Hence,
sliceGAN opens up many interesting directions of research by learning the
representative distribution from 2D slices, and transferring the learned
knowledge to generate arbitrary 3D volumes.

In the original implementation of sliceGAN, the authors proposed training a
single 3D generator which takes in as input a noise vector sampled from a
uniform distribution, as well as three separate discriminators that enforce
the probability distributions of the generated slices in $x,y,$ and $z$
dimensions to be realistic. At every optimization step, the discriminator
takes in every 2D slice from the generated 3D volume as fake data, while the
same number of random patches from the real 2D slice image are fed as real
data. As the training progresses, the generator produces more and more
realistic 2D slices in every dimensions, and consequently, once the training
is complete, the generator is able to synthesize 3D volumes having the same
characteristics of the original 2D slice.

Notably, sliceGAN mostly focuses on the generation of $n$-phase materials,
which are implemented using $n$ different channel output and the final softmax
layer as shown in Figure 1(a). It is trivial that the discrete modeling used
in $n$-phase material modelling is easier than estimating the probability
distribution of a continuous data, since in the discrete case the voxel only
belongs to one of $n$ classes. Consistent with the authors’ claim [1], when we
tried reproducing the training step using both $n$-phase data and greyscale
data, we acquired considerably better results when modelling $n$-phase data.

Another limitation of the prior work [1] is that we cannot control the
characteristic of the generated samples at inference phase. Any noise vector
sampled from uniform random distribution is capable of generating 3D volumes
of the same characteristics, but latent space steering is not possible.
Consequently, even when different data share the same features except for one
feature such as grain size, sliceGAN needs to be trained for every specific
datum, which takes about 4 hours per single datum.

On the other hand, over the several years of advances in GAN, it has now
become common to be able to control the visual attributes of the sample being
generated from the input noise vector by steering feature vectors. One of the
most famous and widely used examples is styleGAN [2], where adaptive instance
normalization (AdaIN) [3] is adopted to inject style at certain layers of the
generator. This so-called style module supposedly disentangles the feature
space, and by constructing the network this way, each layer is able to control
different attributes of the data being generated. In the context of medical
imaging [4, 5], AdaIN was combined with cycleGAN [6] for continuous conversion
between two domains $\mathcal{X}$, and $\mathcal{Y}$. Leveraging the idea of
feature disentanglement, in this reusability report, we present a simple
extension of sliceGAN by endowing the model the ability to control the
attributes of the generated 3D volume. By this extension, sliceGAN can be
freed from having to be trained per every specific dataset, but can be trained
with different data from the same class. Then, once the network is trained,
controlled synthesis of diverse data in the same class is possible.

Figure 1: Overview of AdaIN-style generator for continuous feature
disentanglement of generated three-dimensional volume. (a) Generator
architecture, (b) basic building block of the generator, and (c) AdaIN code
generator.

## Feature disentanglement in SliceGAN using AdaIN

Consider the feature tensor at a specific layer:

${\bm{X}}=[{\bm{x}}_{1}\dots\bm{x}_{C}]\in{\mathbb{R}}^{HWD\times C},$ (1)

where ${\bm{x}}_{i}$ is the $i$-th channel feature vector of ${\bm{X}}$
containing a single-channel volume feature of size ${\mathbb{R}}^{H\times
W\times D}$. To the volume feature, we can utilize AdaIN to inject some
arbitrary information ${\bm{y}}$ at each channel dimension, formally described
as

${\bm{z}}_{i}=\mathcal{A}({\bm{x}}_{i},{\bm{y}}_{i}),\quad i=1,\dots,C,$ (2)

with

$\mathcal{A}({\bm{x}},{\bm{y}}):=\frac{\sigma({\bm{y}})}{\sigma({\bm{x}})}({\bm{x}}-\mu({\bm{x}})\mathbf{1})+\mu({\bm{y}})\mathbf{1}.$
(3)

Here, $\mu(\cdot)$ and $\sigma(\cdot)$ refers to mean and standard deviation,
respectively, and $\mathbf{1}$ is a vector filled with ones of size
${\mathbb{R}}^{HWD}$. In our extension of sliceGAN, this AdaIN transform is
performed after every convolution of the sliceGAN generator as shown in Figure
1(a)(b). Now, to inject meaningful feature to ${\bm{y}}$, we utilize AdaIN
code generator shown in Figure 1(c), which is a simple MLP with ReLU
nonlinearity, taking in a static vector representing certain features such as
grain size, and outputs a AdaIN feature vector to be used as ${\bm{y}}$. Once
the network is trained with AdaIN-style generator, we can simply change the
code vector at the inference phase while keeping the same noise vector.

Specifically, to control feature that represents the grain size, in our
implementation the code vector is defined as ${\bm{c}}=\alpha\mathbf{1}$,
where $\alpha$ is a scale parameter and $\mathbf{1}$ is a vector filled with
ones. Then, it is remarkable that feature interpolation is possible at
inference phase [4]. For example, in the generation of code vector, if a small
scale parameter $\alpha$ was used together with data of small grain size and a
larger scale parameter $\alpha^{\prime}$ was used together with data of large
grain size, then when we inject a value in between $\alpha$ and
$\alpha^{\prime}$ to interpolate the code vector, the generator synthesizes a
volume with medium grain size, which we further elaborate with empirical
results in the following section.

## Experiments and Implementation

The data that we chose as our proof-of-concept was acquired from an open
source database [7]. The database contains microstructures of three phase
synthetically generated with DREAM3D [8], with log-normal feature size
distribution for each phase. Three structures with size 75 $\times$ 75
$\times$ 15 $\mu$m3, 125 nm voxel size, mean feature diameter of 0.57 $\mu$m3
were acquired with varying scale parameter. More specifically, the scale
parameter was represented as the log of geometric standard deviation,
$\ln\sigma_{g}$, and the value for each data was specified to 0.15, 0.35, and
0.6. Consequently, the input code vector ${\bm{c}}$ represented in Figure 1(c)
has the shape $128\times 1$, with every element set to the same scale
parameter value $\alpha\in\\{0.15,0.35,0.6\\}$ for each target data.

For implementation, we make minimal changes to the codebase provided by the
authors of sliceGAN [1]. We keep the generator and the discriminator
architecture the same as in the original paper, only replacing the batch
normalization block with the AdaIN block as shown in Figure 1(a-c). Training
strategies, including the choice of generator/discriminator update steps and
respective batch size were deliberately kept identical to the original work,
and we found that it was sufficient to produce high-quality results.

Figure 2: (a) Random patches taken from the source target images. (b)
Synthesized results by the code vector ${\bm{c}}=\alpha\mathbf{1}$ from the
interpolated scale parameter $\alpha$. All results are generated from a single
generator. The first row shows the 3D rendered view of each data, and the
second and third rows show $x-y$, $y-z$ plane cutviews, respectively.

In Figure 2, we summarize the results. Although there are only three different
values of code vector used at the training stage as presented in Figure 2(a),
when we select any values within the range, the generator synthesizes feasible
results, as shown in Figure 2(b). Hence, now that the feature is disentangled,
we can choose any values of scale parameter $\alpha$ according to the desired
grain size. It should also be denoted that we also tried extrapolation outside
the range of the training set. However, values smaller than 0.15 did not
synthesize smaller grain, and values larger than 0.60 did not synthesize
larger grain.

## Discussion

SliceGAN [1] was verified to be well-applicable to synthesis of various
$n$-phase materials. Without any fine-tuning, we could acquire satisfactory
results as presented in Figure 2, showing the robustness of the work. Our
extension of disentangling the feature so that it is controllable at inference
phase was straightforward, and the extension did not require any change from
the original work on medical imaging [4, 5]. We believe that our approach
would be applicable to a different dataset with features other than the grain
size, which requires a curated dataset.

The training data acquired from [7] were not strictly a single slice of 2D
slice, but of multiple slices of large matrix size. We tried both training
sliceGAN from random patches of single slice, and training sliceGAN from
random patches cut from multiple slices, and found that both approaches
produce stable results without mode collapse. Nonetheless, we conjecture that
the model will benefit from using multiple slices for training data, but this
effect is not clearly observable in the case of simple $n$-phase materials
that are successfully trained from even a single slice.

## Future directions

First, extending the work of sliceGAN to the generation of continuous
greyscale microstructures is a promising direction of research. This can be
carried out along many directions. For example, the generator and
discriminator architectures used in sliceGAN are simplistic, having a single
convolutional block per each scale. Consequently, we believe that scaling the
architecture to a larger size would be a benefit, although it would require a
larger memory GPU (Original work [1] used 12GB GPU, and we used 11GB GPU).
Moreover, progressive learning strategy as in [9, 2] could also strengthen the
method. Once the generation of greyscale microstructures become robust, we
believe that the synthesis of RGB color volumes will also work easily, by
simply expanding the channel dimension.

Correspondence and requests for materials should be addressed to Jong Chul Ye.
(email: jong.ye@kaist.ac.kr).

This research was funded by the National Research Foundation (NRF) of Korea
grant NRF-2020R1A2B5B03001980.

H.C. performed all experiments, wrote the extended code, and prepared the
manuscript. J.C.Y. supervised the project in conception and discussion, and
prepared the manuscript.

The authors declare that they have no competing financial interests.

The data is accessible at ref.[7]

The code is available at the following link

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