# Large-time behavior of composite waves of viscous shocks for the barotropic
Navier-Stokes equations

Sungho Han
Department of Mathematical Sciences
Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of
Korea<EMAIL_ADDRESS>,  Moon-Jin Kang
Department of Mathematical Sciences
Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of
Korea<EMAIL_ADDRESS>and  Jeongho Kim
School of Mathematics,
Korea Institute for Advanced Study, 85 Hoegiro, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

We study the large-time behavior of the 1D barotropic Navier-Stokes flow
perturbed from Riemann data generating a composition of two shock waves with
small amplitudes. We prove that the perturbed Navier-Stokes flow converges,
uniformly in space, towards a composition of two viscous shock waves as time
goes to infinity, up to dynamical shifts. Especially, the strengths of the two
waves can be chosen independently. This is the first result for the
convergence to a composite wave of two viscous shocks with independently small
amplitudes.

###### Key words and phrases:

barotropic Navier-Stokes equations; composition of two viscous shock waves;
long-time stability; $a$-contraction with shifts

Acknowledgment. S. Han is supported by the NRF-2019R1C1C1009355. M.-J. Kang is
partially supported by the NRF-2019R1C1C1009355 and the POSCO Science
Fellowship of POSCO TJ Park Foundation. J. Kim is supported by a KIAS
Individual Grant (SP087401) via the Center for Mathematical Challenges at
Korea Institute for Advanced Study.

###### Contents

  1. 1 Introduction
  2. 2 Main ideas of the proof and useful lemmas
    1. 2.1 Main ideas and the method of $a$-contraction with shifts
    2. 2.2 Viscous shock waves
    3. 2.3 Estimate on the relative quantities
  3. 3 Proof of Theorem 1.1
    1. 3.1 Local existence
    2. 3.2 Construction of shifts
    3. 3.3 Main proposition for a priori estimates
    4. 3.4 Global-in-time existence for perturbation
    5. 3.5 Time-asymptotic behavior
    6. 3.6 Notations
  4. 4 Estimates on weighted relative entropy with shifts
    1. 4.1 Construction of weight functions
    2. 4.2 Estimate on shifts
    3. 4.3 Relative entropy method
    4. 4.4 Maximization on $h-\widetilde{h}$
    5. 4.5 Decompositions
    6. 4.6 Estimate of the main part $\mathcal{R}_{1}$
    7. 4.7 Estimate of the remaining terms
    8. 4.8 Estimate in small time
    9. 4.9 Proof of Lemma 4.1
  5. 5 Proof of Proposition 3.2
    1. 5.1 Zeroth-order estimate
    2. 5.2 First-order estimate
  6. A Proof of Lemma 3.2
  7. B Proof of Lemma 3.3

##  1\. Introduction

We consider the one-dimensional barotropic compressible Navier-Stokes
equations in the Lagrangian mass coordinates:

$\displaystyle\begin{aligned} &v_{t}-u_{x}=0,\\\
&u_{t}+p(v)_{x}=\left(\mu\frac{u_{x}}{v}\right)_{x},\quad x\in\mathbb{R},\quad
t>0\end{aligned}$ (1.1)

where $v=v(t,x)>0$ and $u=u(t,x)$ denote the specific volume and the velocity
of the fluid, respectively. The pressure $p=p(v)$ is given by the $\gamma$-law
$p(v)=bv^{-\gamma}$, with $b>0$ and $\gamma>1$ and $\mu>0$ denotes the
viscosity coefficient of the fluid. For simplicity, we normalize the
coefficients as $b=1$ and $\mu=1$.
Consider initial data of the system (LABEL:eq:NS) given by $(v_{0},u_{0})$,
which connects prescribed far-field constant states:

$\lim_{x\to\pm\infty}(v_{0}(x),u_{0}(x))=(v_{\pm},u_{\pm}).$ (1.2)

A heuristic argument (see e.g. [16]) describes that large-time behavior of
solutions $(v,u)$ to the Navier-Stokes equations (LABEL:eq:NS) has a close
relationship with the Riemann problem of the associated Euler equations:

$\displaystyle\begin{aligned} &v_{t}-u_{x}=0,\\\ &u_{t}+p(v)_{x}=0,\quad
x\in\mathbb{R},\quad t>0,\end{aligned}$ (1.3)

subject to the Riemann initial data

$(v(t,x),u(t,x))|_{t=0}=\begin{cases}(v_{-},u_{-}),\quad x<0,\\\
(v_{+},u_{+}),\quad x>0.\end{cases}$ (1.4)

We consider the end states $(v_{\pm},u_{\pm})$ such that there exists a unique
intermediate state $(v_{m},u_{m})$ which is connected with $(v_{-},u_{-})$ by
1-shock curve and with $(v_{+},u_{+})$ by 2-shock curve. That is, there exists
a unique $(v_{m},u_{m})$ such that the following Rankine-Hugoniot condition
and Lax entropy condition hold:

$\displaystyle\begin{aligned}
&\begin{cases}-\sigma_{1}(v_{m}-v_{-})-(u_{m}-u_{-})=0,\\\
-\sigma_{1}(u_{m}-u_{-})+(p(v_{m})-p(v_{-}))=0,\end{cases}\quad\sigma_{1}:=-\sqrt{-\frac{p(v_{m})-p(v_{-})}{v_{m}-v_{-}}},\quad
v_{-}>v_{m},\quad u_{-}>u_{m};\\\
&\begin{cases}-\sigma_{2}(v_{+}-v_{m})-(u_{+}-u_{m})=0,\\\
-\sigma_{2}(u_{+}-u_{m})+(p(v_{+})-p(v_{m}))=0,\end{cases}\quad\sigma_{2}:=\sqrt{-\frac{p(v_{+})-p(v_{m})}{v_{+}-v_{m}}},\quad
v_{m}<v_{+},\quad u_{m}>u_{+}.\end{aligned}$ (1.5)

Then, the Euler equations (LABEL:eq:Euler) with (1.4)-(LABEL:RH-condition)
admit a unique self-similar solution, the so-called Riemann solution
$(\bar{v},\bar{u})$, represented by the composition
$(\bar{v},\bar{u})=(v^{s}_{1},u^{s}_{1})+(v^{s}_{2},u^{s}_{2})-(v_{m},u_{m})$
of 1-shock wave $(v^{s}_{1},u^{s}_{1})$ and 2-shock wave
$(v^{s}_{2},u^{s}_{2})$ defined as (see e.g. [23])

$(v^{s}_{1},u^{s}_{1})(t,x)=\begin{cases}(v_{-},u_{-}),\quad x<\sigma_{1}t,\\\
(v_{m},u_{m}),\quad
x>\sigma_{1}t,\end{cases},\quad(v^{s}_{2},u^{s}_{2})(t,x)=\begin{cases}(v_{m},u_{m}),\quad
x<\sigma_{2}t,\\\ (v_{+},u_{+}),\quad x>\sigma_{2}t.\end{cases}$

The viscous counterpart of the Riemann solution $(\bar{v},\bar{u})$ is given
by the composite wave:

$(\widetilde{v}(t,x),\widetilde{u}(t,x)):=\Big{(}\widetilde{v}_{1}(x-\sigma_{1}t),\widetilde{u}_{1}(x-\sigma_{1}t)\Big{)}+\Big{(}\widetilde{v}_{2}(x-\sigma_{2}t),\widetilde{u}_{2}(x-\sigma_{2}t)\Big{)}-(v_{m},u_{m}),$
(1.6)

which is composed of 1-viscous shock
$(\widetilde{v}_{1},\widetilde{u}_{1})(x-\sigma_{1}t)$ and 2-viscous shock
$(\widetilde{v}_{2},\widetilde{u}_{2})(x-\sigma_{2}t)$ satisfying: for each
$i=1,2,$

$\begin{cases}-\sigma_{i}(\widetilde{v}_{i})^{\prime}-(\widetilde{u}_{i})^{\prime}=0,\\\
-\sigma_{i}(\widetilde{u}_{i})^{\prime}+p(\widetilde{v}_{i})^{\prime}=\left(\frac{(\widetilde{u}_{i})^{\prime}}{\widetilde{v}_{i}}\right)^{\prime},\\\
(\widetilde{v}_{1},\widetilde{u}_{1})(-\infty)=(v_{-},u_{-}),\quad(\widetilde{v}_{1},\widetilde{u}_{1})(+\infty)=(v_{m},u_{m}),\\\
(\widetilde{v}_{2},\widetilde{u}_{2})(-\infty)=(v_{m},u_{m}),\quad(\widetilde{v}_{2},\widetilde{u}_{2})(+\infty)=(v_{+},u_{+}).\end{cases}$
(1.7)

Notice that each viscous shock
$(\widetilde{v}_{i}(x-\sigma_{i}t),\widetilde{u}_{i}(x-\sigma_{i}t))$ is a
traveling wave solution to (LABEL:eq:NS).

In this paper, we aims to prove that solutions to the Navier-Stokes system
(LABEL:eq:NS)-(1.2) with (LABEL:RH-condition) converge to the composite wave
$(\bar{v},\bar{u})$ up to shifts, uniformly in $x$ as $t\to\infty$.

As previous results on time-asymptotic stability for (LABEL:eq:NS)-(1.2) when
the end states are connected by a single shock, Matsumura and Nishihara [17]
first proved the convergence of solutions toward a single viscous shock with
small amplitude, uniformly in $x$, using the anti-derivative method under the
zero-mass condition. Later on, this zero-mass condition is removed by
introducing a constant shift with diffusion wave by Liu [13], Liu-Zeng [15]
and Szepessy-Xin [25]. On the other hand, Masica and Zumbrun [21] showed the
spectral stability of viscous shock wave under the weaker condition compared
to the zero-mass condition, called a spectral condition. Recently, Wang-Wang
[28] studied on a planar shock wave for the three-dimensional barotropic
Navier-Stokes equations, by utilizing a new method called “$a$-contraction
with shifts”.

For the composition of two shocks as in our setting, Huang-Matsumura [3]
showed the convergence toward a composite wave composed of two viscous shocks
for the Navier-Stokes-Fourier system when the strengths of two viscous shocks
are small with the same order. As mentioned in [16], the same result also
could be obtained for the barotropic case (LABEL:eq:NS), using the parallel
argument as in [3, 17].
In this paper, however, we do not assume the same order smallness, that is,
our result provides the uniform convergence toward a composition of two
viscous shocks with independently small amplitudes.

On the other hand, when the initial data (1.4) generates rarefaction waves,
the time-asymptotic stability of the rarefaction wave has been proven by
Matsumura-Nishihara [18, 19]. Similar stability results are also shown for the
Navier-Stokes-Fourier system in [14, 22].

However, all the mentioned literature treated either shocks or rarefaction
waves, but not the composition of them. Indeed, the time-asymptotic stability
of the composition of shock and rarefaction waves is a challenging problem
[17]. This open problem is recently solved by the second author, Vasseur and
Wang [11], using the method of $a$-contraction with shift for the shock wave,
combining with the energy method for the rarefaction wave.

We will apply the method of $a$-contraction with shifts to prove the time-
asymptotic behavior as a composition of two viscous shock waves in the
following theorem.

######  Theorem 1.1.

For a given constant state $(v_{+},u_{+})\in\mathbb{R}_{+}\times\mathbb{R}$,
there exist positive constants $\delta_{0},\varepsilon_{0}$ such that the
following holds.
For any constant states $(v_{m},u_{m})$ and $(v_{-},u_{-})$ satisfying
(LABEL:RH-condition) with

$|v_{+}-v_{m}|+|v_{m}-v_{-}|<\delta_{0},$ (1.8)

let $(\widetilde{v}_{i},\widetilde{u}_{i})(x-\sigma_{i}t)$ be the $i$-viscous
shock wave satisfying (1.7). In addition, let $(v_{0},u_{0})$ be any initial
data satisfying

$\sum_{\pm}\left(\|v_{0}-v_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}+\|u_{0}-u_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}\right)+\|\partial_{x}v_{0}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}u_{0}\|_{L^{2}(\mathbb{R})}<\varepsilon_{0},$

where $\mathbb{R}_{+}:=(0,+\infty)$ and $\mathbb{R}_{-}:=(-\infty,0)$. Then,
the compressible Navier-Stokes system (LABEL:eq:NS)-(1.2) with (LABEL:RH-
condition) admits a unique global-in-time solution $(v,u)$ in the following
sense: there exist absolutely continuous shift functions $X_{1}(t),X_{2}(t)$
such that

$\displaystyle
v(t,x)-(\widetilde{v}_{1}(x-\sigma_{1}t-X_{1}(t))+\widetilde{v}_{2}(x-\sigma_{2}t-X_{2}(t))-v_{m})\in
C(0,+\infty;H^{1}(\mathbb{R})),$ $\displaystyle
u(t,x)-(\widetilde{u}_{1}(x-\sigma_{1}t-X_{2}(t))+\widetilde{u}_{2}(x-\sigma_{2}t-X_{2}(t))-u_{m})\in
C(0,+\infty;H^{1}(\mathbb{R})).$

Moreover, we have the large-time behavior:

$\displaystyle\lim_{t\to+\infty}\sup_{x\in\mathbb{R}}\Big{|}v(t,x)-\big{(}\widetilde{v}_{1}(x-\sigma_{1}t-X_{1}(t))+\widetilde{v}_{2}(x-\sigma_{2}t-X_{2}(t))-v_{m}\big{)}\Big{|}=0,$
$\displaystyle\lim_{t\to+\infty}\sup_{x\in\mathbb{R}}\Big{|}u(t,x)-\big{(}\widetilde{u}_{1}(x-\sigma_{1}t-X_{1}(t))+\widetilde{u}_{2}(x-\sigma_{2}t-X_{2}(t))-u_{m}\big{)}\Big{|}=0,$

where

$\lim_{t\to+\infty}|\dot{X}_{i}(t)|=0,\quad\mbox{for}\quad i=1,2.$ (1.9)

Especially, the shifts are well-separated in the following sense:

$X_{1}(t)+\sigma_{1}t\leq\frac{\sigma_{1}}{2}t<0<\frac{\sigma_{2}}{2}t\leq
X_{2}(t)+\sigma_{2}t,\quad t>0.$ (1.10)

######  Remark 1.1.

1\. It follows from the Rankine-Hugoniot condition (LABEL:RH-condition) that
the strength of 1-shock (resp. 2-shock) can be measured by $|v_{-}-v_{m}|$
(resp. $|v_{m}-v_{+}|$). Thus, by the smallness condition (1.8), the strengths
of the two shocks can be chosen independently. This removes the same order
assumption, i.e.
$|v_{-}-v_{m}|+|v_{-}-v_{m}|\lesssim\min(|v_{-}-v_{m}|,|v_{-}-v_{m}|)\ll 1$
that is a crucial assumption in [3].
2\. The property (1.9) implies that

$\lim_{t\to+\infty}\frac{X_{i}(t)}{t}=0,\quad\mbox{for}\quad i=1,2.$

Thus, the shift functions $X_{i}(t)$ grow at most sub-linearly w.r.t. $t$, by
which the shifted composite wave

$\big{(}\widetilde{v}_{1},\widetilde{u}_{1}\big{)}\big{(}x-\sigma_{1}t-X_{1}(t)\big{)}+\big{(}\widetilde{v}_{2},\widetilde{u}_{2}\big{)}\big{(}x-\sigma_{2}t-X_{2}(t)\big{)}-(v_{m},u_{m})$

time-asymptotically keeps the original composite wave.

The remaining part of the paper is organized as follows. In Section 2, we
present a main idea of the proof, and useful estimates for the shock waves.
Elementary estimates for the pressure and relative entropy are also presented.
In Section 3, we present the proof of the main theorem under the a priori
estimates. The a priori estimates will be proved in the following two
sections. In Section 4, we use the method of $a$-contraction with shifts to
estimate the $L^{2}$-perturbation. Then, the remaining estimates will be
obtained in Section 5.

##  2\. Main ideas of the proof and useful lemmas

In this section, we present a main idea of the proof, and useful estimates for
the shock waves and relative quantities.

###  2.1. Main ideas and the method of $a$-contraction with shifts

The main tool to obtain the desired nonlinear stability is the method of
$a$-contraction with shifts, which was introduced by the second author and
Vasseur in [5] (see also [27]) to study the stability of extremal shocks for
the hyperbolic system of conservation laws such as the Euler system
(LABEL:eq:Euler).

This method has been extended to studying viscous models as follows. For the
one-dimensional barotropic Navier-Stokes system, the contraction property of
any large perturbations for a single viscous shock is proved in [8], and for a
composite wave of two shocks in [7]. Those results play a crucial role to
prove the stability of entropy Riemann shocks of the isentropic Euler system
in the class of inviscid limits from the Navier-Stokes system in [8, 9]. As
mentioned in Introduction, this method was also used in [11, 28] to show the
long-time behavior of the barotropic Navier-Stokes system for the composition
of shock and rarefaction in 1D, and for a single shock in multi-D. As
applications of the method to other viscous hyperbolic systems, we also refer
to [4, 6, 10] for viscous scalar conservation laws, [12] for the stability of
a planar contact discontinuity of the 3D full Euler system in the class of
zero dissipation limits, and [1, 2] for the viscous hyperbolic system arising
from a chemotaxis model.

To illustrate a key idea of the method for the viscous system (LABEL:eq:NS),
consider the entropy $\eta$ of the Euler system (LABEL:eq:Euler) defined as

$\eta(U):=\frac{u^{2}}{2}+Q(v),\quad Q(v):=\frac{1}{(\gamma-1)v^{\gamma-1}},$

where $U=(v,u)$. Then, the relative entropy $\eta(U|\overline{U})$ between two
states $U$ and $\overline{U}$ is defined as

$\eta(U|\overline{U}):=\eta(U)-\eta(\overline{U})-D\eta(\overline{U})(U-\overline{U})=\frac{|u-\overline{u}|^{2}}{2}+Q(v|\overline{v}),$

where
$Q(v|\overline{v}):=Q(v)-Q(\overline{v})-Q^{\prime}(\overline{v})(v-\overline{v})$.
Since $Q(v)$ is a strictly convex function in $v$, $Q(\cdot|\cdot)$ is locally
quadratic in the sense that for any $0<a<b$, there exists $C>0$ such that

$C^{-1}|v_{1}-v_{2}|^{2}\leq Q(v_{1}|v_{2})\leq
C|v_{1}-v_{2}|^{2},\qquad\forall v_{1},v_{2}\in[a,b].$

When $\overline{U}(x-\sigma t)$ is a single viscous shock, the method is to
find a weight function $a:\mathbb{R}\to\mathbb{R}_{+}$ and a time-dependent
shift function $X:\mathbb{R}_{+}\to\mathbb{R}$ such that the weighted relative
entropy with shift is not increasing in time:

$\frac{d}{dt}\int_{\mathbb{R}}a(x-\sigma
t-{X}(t))\eta(U(t,x)|\overline{U}(x-\sigma t-{X}(t)))\,dx\leq 0.$

First, by a standard computation based on the relative entropy method, the
left-hand side can be decomposed into three parts (as in Lemma 4.2 for
example):

$\mbox{LHS}=X^{\prime}(t)Y(U)+\mathcal{J}^{\rm bad}(U)-\mathcal{J}^{\rm
good}(U),$

where $\mathcal{J}^{\rm bad}(U)$ and $\mathcal{J}^{\rm good}(U)$ consist of
the all bad terms and all good terms respectively. To make the right-hand side
non-positive, we might use the typical energy method for parabolic equations.
However, since the barotropic Naiver-Stokes system has the dissipation in one
variable only (more precisely, in the $u$ variable for (LABEL:eq:NS)), the
weight function $a$ would be found to provide an additional good term in terms
of the $v$ variable, by which the bad terms could be represented only by the
$u$ variables. Indeed, since $\sigma$ is a non-zero constant, constructing a
monotone function $a$ satisfying $\sigma a^{\prime}>0$, we have a good term

$-\sigma\int_{\mathbb{R}}a^{\prime}(x-\sigma
t-{X}(t))\eta(U(t,x)|\overline{U}(x-\sigma t-{X}(t)))\,dx.$

In fact, the weight function $a$ will be defined by the first component
$\overline{v}$ of the viscous shock such that $a^{\prime}$ localizes the
perturbation in space as done by $\overline{v}^{\prime}$, and the image of $a$
is a bounded open interval. Using the above term, we maximize all bad terms in
terms of the $v$ variable, from which the remaining bad terms are related to
the $u$ variable only, and localized by $a^{\prime}$ or
$\overline{v}^{\prime}$. To absorb the remaining bad terms by the diffusion
term, we may use the following Poincaré-type inequality.

######  Lemma 2.1.

[8, Lemma 2.9] For any $f:[0,1]\to\mathbb{R}$ satisfying
$\int_{0}^{1}y(1-y)|f^{\prime}|^{2}dy<\infty$,

$\int_{0}^{1}\left|f-\int_{0}^{1}fdy\right|^{2}dy\leq\frac{1}{2}\int_{0}^{1}y(1-y)|f^{\prime}|^{2}dy.$
(2.1)

However, to apply Lemma 2.1, we may need the other good term as an average of
linear perturbation on $u$-variable, which together with the bad terms would
give a variance of the perturbation that could be absorbed by the dissipation
as in the Poincaré-type inequality. Here, the desired good term would be
extracted from the shift part $X^{\prime}(t)Y(U)$ by a sophisticated
construction of the shift $X(t)$. This gives the desired contraction estimate.

However, the composition (1.6) of two viscous shocks leads to several
problematic issues. Since the composition of two viscous shocks is not an
exact solution to the Navier-Stokes equations, unlike the case of a single
viscous shock, due to the nonlinear terms in the equations, we need to control
some error terms on the nonlinear interactions between two shock waves.
Second, as we consider two shock waves, we should construct two shift
functions, say $X_{1}$ and $X_{2}$, and two weight functions to control
perturbation near each shock wave, based on the above method for a single
wave. On top of that, since all the bad terms are localized by the derivative
of each weight or shock, whereas the diffusion term is not localized, we need
to localize the diffusion term by introducing some auxiliary localization
functions. The localization functions will be defined by two shift functions
as in (3.11), by constructing the two shift functions to be well-separated
like (1.10).

###  2.2. Viscous shock waves

We review the useful estimates of viscous shocks and their derivatives that
will be used throughout the paper.

######  Lemma 2.2.

[7, Lemma 5.1] For a given constant
$U_{*}:=(v_{*},u_{*})\in\mathbb{R}_{+}\times\mathbb{R}$, there exist positive
constants $\delta_{0}$, $C$, $C_{1}$ and $C_{2}$ such that the following
holds. Let $U_{-}:=(v_{-},u_{-})$, $U_{m}:=(v_{m},u_{m})$, and
$U_{+}:=(v_{+},u_{+})\in\mathbb{R}_{+}\times\mathbb{R}$ be any constants such
that $U_{-},U_{m},U_{+}\in B_{\delta_{0}}(U_{*})$, and
$|p(v_{-})-p(v_{m})|=:\delta_{1}<\delta_{0}$ and
$|p(v_{m})-p(v_{+})|=:\delta_{2}<\delta_{0}$. Let
$\widetilde{U}_{1}=(\widetilde{v}_{1},\widetilde{u}_{1})$ and
$\widetilde{U}_{2}=(\widetilde{v}_{2},\widetilde{u}_{2})$ be the 1- and
2-shocks connecting from $U_{-}$ to $U_{m}$ and from $U_{m}$ to $U_{+}$
respectively, satisfying $\widetilde{v}_{1}(0)=\frac{v_{-}+v_{m}}{2}$ and
$\widetilde{v}_{2}(0)=\frac{v_{m}+v_{+}}{2}$ without loss of generality. Then,
the following estimates hold: for each $i=1,2$,

$\displaystyle\begin{aligned} &v_{i}^{\prime}\sim
u_{i}^{\prime}\quad\mbox{i.e.,}\quad
C^{-1}\widetilde{v}_{i}^{\prime}(x-\sigma_{i}t)\leq\widetilde{u}_{i}^{\prime}(x-\sigma_{i}t)\leq
C\widetilde{v}_{i}^{\prime}(x-\sigma_{i}t),\quad x\in\mathbb{R},\,t>0,\\\
&C^{-1}\delta_{i}e^{-C_{1}\delta_{i}|x-\sigma_{i}t|}\leq\widetilde{v}_{i}(x-\sigma_{i}t)-v_{m}\leq
C\delta_{i}e^{-C_{2}\delta_{i}|x-\sigma_{i}t|},\quad(-1)^{i}(x-\sigma_{i}t)\leq
0,\\\
&-C^{-1}\delta_{i}^{2}e^{-C_{1}\delta_{i}|x-\sigma_{i}t|}\leq\widetilde{v}_{i}^{\prime}(x-\sigma_{i}t)\leq-C\delta_{i}^{2}e^{-C_{2}\delta_{i}|x-\sigma_{i}t|},\quad
x\in\mathbb{R},\quad t>0,\\\ \end{aligned}$ (2.2)

in addition,

$|(\widetilde{v}_{i}^{\prime\prime}(x-\sigma_{i}t),\widetilde{u}_{i}^{\prime\prime}(x-\sigma_{i}t))|\leq
C\delta_{i}|(\widetilde{v}_{i}^{\prime}(x-\sigma_{i}t),\widetilde{u}_{i}^{\prime}(x-\sigma_{i}t))|.$
(2.3)

######  Remark 2.1.

Throughout the paper, we will use Lemma 2.2 with $U_{*}=U_{+}$. Thus, the
constants $\delta_{0}$, $C$, $C_{1}$ and $C_{2}$ depend only on $U_{+}$.

###  2.3. Estimate on the relative quantities

As our method is based on the estimate of relative entropy, we need to control
the relative quantities, such as the relative pressure $p(v|w)$ or the
relative internal energy $Q(v|w)$. Recall that the relative quantities are
defined as

$p(v|w)=p(v)-p(w)-p^{\prime}(w)(v-w),\quad
Q(v|w)=Q(v)-Q(w)-Q^{\prime}(w)(v-w).$

Considering the Taylor expansion, it is expected that they are almost
quadratic quantities at least locally. The following estimates show the exact
estimate on this locally quadratic behavior of the relative quantities. The
proof of the lemma can be found in [8].

######  Lemma 2.3.

Let $\gamma>1$ and $v_{+}$ be given constants. Then, there exist constants
$C,\delta_{*}>0$ such that the following assertions hold:

  1. (1)

For any $v,w$ satisfying $0<w<2v_{+}$ and $0<v<3v_{+}$,

$|v-w|^{2}\leq CQ(v|w),\quad|v-w|^{2}\leq Cp(v|w).$ (2.4)

  2. (2)

For any $v,w$ satisfying $v,w>v_{+}/2$,

$|p(v)-p(w)|\leq C|v-w|.$ (2.5)

  3. (3)

For any $0<\delta<\delta_{*}$ and any $(v,w)\in\mathbb{R}_{+}^{2}$ satisfying
$|p(v)-p(w)|<\delta$ and $|p(w)-p(v_{+})|<\delta$,

$\displaystyle\begin{aligned}
&p(v|w)\leq\left(\frac{\gamma+1}{2\gamma}\frac{1}{p(w)}+C\delta\right)|p(v)-p(w)|^{2},\\\
&Q(v|w)\geq\frac{p(w)^{-\frac{1}{\gamma}-1}}{2\gamma}|p(v)-p(w)|^{2}-\frac{1+\gamma}{3\gamma^{2}}p(w)^{-\frac{1}{\gamma}-2}(p(v)-p(w))^{3},\\\
&Q(v|w)\leq\left(\frac{p(w)^{-\frac{1}{\gamma}-1}}{2\gamma}+C\delta\right)|p(v)-p(w)|^{2}.\end{aligned}$
(2.6)

##  3\. Proof of Theorem 1.1

In this section, we present a main part of proof for Theorem 1.1.

###  3.1. Local existence

In order to get the desired result on time-asymptotic stability of
$H^{1}$-perturbations of a composite wave connecting two different constant
states at far-fields, we need to recall the well-known result on local-in-time
existence of solutions to (LABEL:eq:NS) connecting the two different constant
states in $H^{1}$-norm.

######  Proposition 3.1.

[20, 24] Let $\underline{v}$ and $\underline{u}$ be smooth monotone functions
such that

$\underline{v}(x)=v_{\pm},\quad\underline{u}(x)=u_{\pm},\quad\mbox{for}\quad\pm
x\geq 1.$

Then, for any constants $M_{0}$, $M_{1}$, $\underline{\kappa}_{0}$,
$\overline{\kappa}_{0}$, $\underline{\kappa}_{1}$, $\overline{\kappa}_{1}$
with

$0<M_{0}<M_{1},\quad\mbox{and}\quad
0<\underline{\kappa}_{1}<\underline{\kappa}_{0}<\overline{\kappa}_{0}<\overline{\kappa}_{1},$

there exists a constant $T_{0}>0$ such that if the initial data
$(v_{0},u_{0})$ satisfy

$\|v_{0}-\underline{v}\|_{H^{1}(\mathbb{R})}+\|u_{0}-\underline{u}\|_{H^{1}(\mathbb{R})}\leq
M_{0},\quad\mbox{and}\quad\underline{\kappa}_{0}\leq
v_{0}(x)\leq\overline{\kappa}_{0},\quad\forall x\in\mathbb{R},$

then the Navier-Stokes equations (LABEL:eq:NS) admit a unique solution $(v,u)$
on $[0,T_{0}]$ satisfying

$\displaystyle v-\underline{v}\in C([0,T_{0}];H^{1}(\mathbb{R})),$
$\displaystyle u-\underline{u}\in C([0,T_{0}];H^{1}(\mathbb{R}))\cap
L^{2}([0,T_{0}];H^{2}(\mathbb{R})),$

together with

$\|v-\underline{v}\|_{L^{\infty}([0,T_{0}];H^{1}(\mathbb{R}))}+\|u-\underline{u}\|_{L^{\infty}([0,T_{0}];H^{1}(\mathbb{R}))}\leq
M_{1},$

and

$\underline{\kappa}_{1}\leq
v(t,x)\leq\overline{\kappa}_{1},\quad\forall(t,x)\in[0,T_{0}]\times\mathbb{R}.$
(3.1)

###  3.2. Construction of shifts

As desired, we should show that a small $H^{1}$-perturbation of a composite
wave of two viscous shocks is orbitally stable for a large time, more
precisely, the perturbation uniformly converges to the composite wave up to
shifts where each shock is shifted by $X_{i}(t)$:

$\displaystyle\begin{aligned}
&(\widetilde{v}^{X_{1},X_{2}},\widetilde{u}^{X_{1},X_{2}})(t,x)\\\
&\qquad:=\left(\widetilde{v}_{1}^{X_{1}}(x-\sigma_{1}t)+\widetilde{v}_{2}^{X_{2}}(x-\sigma_{2}t)-v_{m},\widetilde{u}_{1}^{X_{1}}(x-\sigma_{1}t)+\widetilde{u}_{2}^{X_{2}}(x-\sigma_{2}t)-u_{m}\right),\end{aligned}$
(3.2)

where $F^{X_{i}}$ denotes a function $F$ shifted by $X_{i}$, that is,
$F^{X_{i}}(x):=F(x-X_{i}(t))$ for any function $F$. This notation will be used
throughout the paper.
We here introduce the shift functions explicitly, from which we could obtain a
bound of derivative of shifts (at least locally-in-time) in Lemma 3.1, and
obtain the desired a priori estimates in Proposition 3.2. Those will be used
for the continuation argument in Section 3.4. We define a pair of shifts
$(X_{1},X_{2})$ as a solution to the system of ODEs:

$\left\\{\begin{array}[]{ll}\displaystyle\dot{X}_{1}(t)=-\frac{M}{\delta_{1}}\bigg{[}\int_{\mathbb{R}}\frac{a^{X_{1},X_{2}}}{\sigma_{1}}(\widetilde{h}^{X_{1}}_{1})_{x}(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))\,dx-\int_{\mathbb{R}}a^{X_{1},X_{2}}(p(\widetilde{v}^{X_{1}}_{1}))_{x}(v-\widetilde{v}^{X_{1},X_{2}})\,dx\bigg{]},\\\\[11.38109pt]
\displaystyle\dot{X}_{2}(t)=-\frac{M}{\delta_{2}}\bigg{[}\int_{\mathbb{R}}\frac{a^{X_{1},X_{2}}}{\sigma_{2}}(\widetilde{h}^{X_{2}}_{2})_{x}(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))\,dx-\int_{\mathbb{R}}a^{X_{1},X_{2}}(p(\widetilde{v}^{X_{2}}_{2}))_{x}(v-\widetilde{v}^{X_{1},X_{2}})\,dx\bigg{]},\\\\[11.38109pt]
\displaystyle X_{1}(0)=X_{2}(0)=0,\end{array}\right.$ (3.3)

where $a^{X_{1},X_{2}}$ is the shifted weight function defined in (4.7),
$\widetilde{h}_{i}:=\widetilde{u}_{i}-(\ln\widetilde{v}_{i})_{x}$, and $M$ is
the specific constant chosen as $M:=\frac{5(\gamma+1)}{8\gamma
p(v_{m})}\big{(}-p^{\prime}(v_{m})\big{)}^{\frac{3}{2}}$, which will be used
in the proof of Proposition 3.2. However, for well-posedness of the above ODEs
in Lemma 3.1, we only need the following assumption for the shifted weight
function at this point: $a^{X_{1},X_{2}}$ is a $C^{1}$-function of
$(x,X_{1},X_{2})$ with finite $C^{1}$-norm. This is verified by the explicit
one defined in (4.7).
The following lemma ensures that (3.3) has a unique absolutely continuous
solution at least for the lifespan $[0,T_{0}]$ of solution $v$ satisfying
(3.1).

######  Lemma 3.1.

For any $c_{1},c_{2}>0$, there exists a constant $C>0$ such that the following
is true. For any $T>0$, and any function $v\in
L^{\infty}((0,T)\times\mathbb{R})$ verifying

$c_{1}\leq v(t,x)\leq c_{2},\qquad\forall(t,x)\in[0,T]\times\mathbb{R},$ (3.4)

the system (3.3) has a unique absolutely continuous solution $(X_{1},X_{2})$
on $[0,T]$. Moreover,

$|X_{1}(t)|+|X_{2}(t)|\leq Ct,\quad\forall t\leq T.$ (3.5)

###### Proof.

Let $F(t,X_{1},X_{2})$ denote the right-hand side of the system (3.3).
Using (3.4) and the facts that the $C^{1}$-norm of $a^{X_{1},X_{2}}$ w.r.t.
$(x,X_{1},X_{2})$ is finite, and

$\|\widetilde{h}_{i}\|_{C^{2}}<\infty,\quad\|\widetilde{v}_{i}\|_{C^{2}}<\infty,\quad\|(\widetilde{h}_{i})_{x}\|_{L^{1}}\leq
C\delta_{i},\quad\|(\widetilde{v}_{i})_{x}\|_{L^{1}}\leq C\delta_{i},$

we have

$\displaystyle\begin{aligned}
&\sup_{X_{1},X_{2}\in\mathbb{R}}|F(t,X_{1},X_{2})|\\\ &\quad\leq
C\|a\|_{C^{1}}\sum_{i=1}^{2}\frac{1}{\delta_{i}}\Big{(}\|(p(\widetilde{v}_{i})\|_{L^{\infty}}+\|p(v)\|_{L^{\infty}}+\|\widetilde{v}_{i}\|_{L^{\infty}}+\|v\|_{L^{\infty}}\Big{)}(\|(\widetilde{h}_{i})_{x}\|_{L^{1}}+\|(\widetilde{v}_{i})_{x}\|_{L^{1}})\leq
C,\end{aligned}$ (3.6)

where the constant $C$ is independent of $t$. Likewise, we have

$\sup_{X_{1},X_{2}\in\mathbb{R}}|\nabla_{X_{1},X_{2}}F(t,X_{1},X_{2})|\leq C.$

Therefore, (3.3) has a unique absolutely continuous solution by a simple
adaptation of Cauchy-Lipschitz theorem (for example, by [1, Lemma A.1] and [7,
Lemma C.1]).
Since $|\dot{X_{1}}(t)|+|\dot{X_{2}}(t)|\leq C$ by (LABEL:temco), we have
(3.5). ∎

###  3.3. Main proposition for a priori estimates

Based on the previous setting, we present the main proposition for a priori
estimate.

######  Proposition 3.2.

For a given constant $U_{+}:=(v_{+},u_{+})\in\mathbb{R}_{+}\times\mathbb{R}$,
there exist positive constants $\delta_{0}$, $C_{0}$, and $\varepsilon_{1}$
such that the following holds:
For any constant states $U_{m}:=(v_{m},u_{m})\in S_{2}(v_{+},u_{+})$ and
$U_{+}:=(v_{-},u_{-})\in S_{1}(v_{m},u_{m})$ satisfying
$|p(v_{-})-p(v_{m})|=:\delta_{1}<\delta_{0}$ and
$|p(v_{m})-p(v_{+})|=:\delta_{2}<\delta_{0}$, let
$(\widetilde{v}^{X_{1},X_{2}},\widetilde{u}^{X_{1},X_{2}})$ denote the
composite wave of two shifted shocks as in (LABEL:composite_wave), where
$(X_{1},X_{2})$ solves (3.3). Suppose that $(v,u)$ is the solution to
$\eqref{eq:NS}$ on $[0,T]$ for some $T>0$, and satisfy

$\displaystyle v-\widetilde{v}^{X_{1},X_{2}}\in C([0,T];H^{1}(\mathbb{R})),$
$\displaystyle u-\widetilde{u}^{X_{1},X_{2}}\in C([0,T];H^{1}(\mathbb{R}))\cap
L^{2}(0,T;H^{2}(\mathbb{R})),$

and

$\|v-\widetilde{v}^{X_{1},X_{2}}\|_{L^{\infty}(0,T;H^{1}(\mathbb{R}))}+\|u-\widetilde{u}^{X_{1},X_{2}}\|_{L^{\infty}(0,T;H^{1}(\mathbb{R}))}\leq\varepsilon_{1}.$
(3.7)

Then, for all $t\in[0,T]$,

$\displaystyle\begin{aligned}
\sup_{t\in[0,T]}&\left(\|v-\widetilde{v}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}+\|u-\widetilde{u}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}\right)+\sqrt{\int_{0}^{t}\sum_{i=1}^{2}\delta_{i}|X_{i}|^{2}\,ds}\\\
&\quad+\sqrt{\int_{0}^{t}\left(\mathcal{G}^{S}(U)+{D}(U)+D_{1}(U)+D_{2}(U)\right)\,ds}\\\
&\leq
C_{0}\left(\|v_{0}-\widetilde{v}(0,\cdot)\|_{H^{1}(\mathbb{R})}+\|u_{0}-\widetilde{u}(0,\cdot)\|_{H^{1}(\mathbb{R})}\right)+C_{0}\delta_{0}^{1/4}.\end{aligned}$
(3.8)

In particular,

$|\dot{X}_{1}(t)|+|\dot{X}_{2}(t)|\leq
C_{0}\|(v-\widetilde{v}^{X_{1},X_{2}})(t,\cdot)\|_{L^{\infty}(\mathbb{R})},\qquad
t\leq T.$ (3.9)

Here, the constant $C_{0}$ is independent of $T$ and

$\displaystyle\mathcal{G}^{S}(U)$
$\displaystyle:=\sum_{i=1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(v-\widetilde{v}^{X_{1},X_{2}})|^{2}\,dx,$
(3.10) $\displaystyle D(U)$
$\displaystyle:=\int_{\mathbb{R}}|\partial_{x}(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))|^{2}\,dx,$
$\displaystyle D_{1}(U)$
$\displaystyle:=\int_{\mathbb{R}}|(u-\widetilde{u}^{X_{1},X_{2}})_{x}|^{2}\,dx,$
$\displaystyle D_{2}(U)$
$\displaystyle:=\int_{\mathbb{R}}|(u-\widetilde{u}^{X_{1},X_{2}})_{xx}|^{2}\,dx,$

where $\phi_{1},\phi_{2}$ are cutoff functions defined by

$\phi_{1}(t,x):=\begin{cases}1&\quad\mbox{if}\quad
x<\frac{X_{1}(t)+\sigma_{1}t}{2},\\\ 0&\quad\mbox{if}\quad
x>\frac{X_{2}(t)+\sigma_{2}t}{2},\\\ \mbox{linearly decreasing $1$ to
$0$}&\quad\mbox{if}\quad\frac{X_{1}+\sigma_{1}t}{2}\leq
x\leq\frac{X_{2}+\sigma_{2}t}{2},\end{cases}\quad\phi_{2}(t,x):=1-\phi_{1}(t,x).$
(3.11)

######  Remark 3.1.

Since the (shifted) composite wave
$(\widetilde{v}^{X_{1},X_{2}},\widetilde{u}^{X_{1},X_{2}})$ is not a solution
to the Navier-Stokes equation as in (3.25), we need to control the interaction
term of the 1- and 2-waves to get the desired results of Proposition 3.2. For
that, we localize the perturbation near each wave by using the above cutoff
functions. More specifically, $\phi_{1}$ (resp. $\phi_{2}$) localizes the
perturbation near the 1-wave (resp. 2-wave) shifted by $X_{1}$ (resp.
$X_{2}$). Notice from (3.22) that

$X_{2}(t)+\sigma_{2}t\geq\frac{\sigma_{2}}{2}t>0>\frac{\sigma_{1}}{2}t\geq
X_{1}(t)+\sigma_{1}t,\qquad t>0,$

and so the functions $\phi_{1}$ and $\phi_{2}$ are well-separated as time
goes.

###  3.4. Global-in-time existence for perturbation

We here prove the global existence of solution by using the continuation
argument based on Proposition 3.1 and Proposition 3.2. We first assume that
the constants $\delta_{0},C$, and $\varepsilon_{1}$ are given by Proposition
3.2, and consider the end states $(v_{-},u_{-})$ and $(v_{+},u_{+})$
satisfying the conditions in Proposition 3.2. First of all, the smooth
monotone functions $\underline{v}(x)$ and $\underline{u}(x)$ given in
Proposition 3.1 satisfy

$\displaystyle\begin{aligned}
\sum_{\pm}&\left(\|\underline{v}-v_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}+\|\underline{u}-u_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}\right)+\|\partial_{x}\underline{v}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\underline{u}\|_{L^{2}(\mathbb{R})}\\\
&\leq C(|v_{+}-v_{-}|+|u_{+}-u_{-}|)\leq
C_{1}(\delta_{1}+\delta_{2}).\end{aligned}$ (3.12)

Then, we estimate the $H^{1}$-perturbation between the monotone functions
$\underline{v},\underline{u}$ and the initial composite waves
$\widetilde{v}(0,\cdot),\widetilde{u}(0,\cdot)$ in (1.6) as

$\displaystyle\begin{aligned}
&\|\underline{v}(\cdot)-\widetilde{v}(0,\cdot)\|_{H^{1}(\mathbb{R})}+\|\underline{u}(\cdot)-\widetilde{u}(0,\cdot)\|_{H^{1}(\mathbb{R})}\\\
&\leq\|\underline{v}-\widetilde{v}(0,\cdot)\|_{L^{2}(\mathbb{R})}+\|\partial_{x}(\underline{v}-\widetilde{v}(0,\cdot))\|_{L^{2}(\mathbb{R})}+\|\underline{u}-\widetilde{u}(0,\cdot)\|_{L^{2}(\mathbb{R})}+\|\partial_{x}(\underline{u}-\widetilde{u}(0,\cdot))\|_{L^{2}(\mathbb{R})}\\\
&\leq\sum_{\pm}\left(\|\underline{v}-v_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}+\|\underline{u}-u_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}\right)\\\
&\quad+\|\widetilde{v}_{1}-v_{m}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{v}_{1}-v_{-}\|_{L^{2}(\mathbb{R}_{-})}+\|\widetilde{v}_{2}-v_{+}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{v}_{2}-v_{m}\|_{L^{2}(\mathbb{R}_{-})}\\\
&\quad+\|\partial_{x}\underline{v}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{v}_{1}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{v}_{2}\|_{L^{2}(\mathbb{R})}\\\
&\quad+\|\widetilde{u}_{1}-u_{m}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{u}_{1}-u_{-}\|_{L^{2}(\mathbb{R}_{-})}+\|\widetilde{u}_{2}-u_{+}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{u}_{2}-u_{m}\|_{L^{2}(\mathbb{R}_{-})}\\\
&\quad+\|\partial_{x}\underline{u}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{u}_{1}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{u}_{2}\|_{L^{2}(\mathbb{R})}\\\
&\leq C_{2}(\sqrt{\delta_{1}}+\sqrt{\delta_{2}}),\end{aligned}$ (3.13)

where we used the shock estimates (LABEL:shock-est) and (3.12) in the last
inequality. Now, for the constant $\varepsilon_{1}$, we choose $\delta_{0}$
small enough so that the following condition holds for any
$\delta_{1},\delta_{2}<\delta_{0}$:

$\frac{\frac{\varepsilon_{1}}{4}-C_{0}\delta_{0}^{1/4}}{C_{0}+1}-C_{1}(\delta_{1}+\delta_{2})-C_{2}(\sqrt{\delta_{1}}+\sqrt{\delta_{2}})\geq
0.$

Consider the two positive constants $\varepsilon_{*}$ and $\varepsilon_{0}$
defined as

$\varepsilon_{*}:=\frac{\frac{\varepsilon_{1}}{2}-C_{0}\delta_{0}^{1/4}}{C_{0}+1}-C_{2}(\sqrt{\delta_{1}}+\sqrt{\delta_{2}}),\qquad\varepsilon_{0}:=\frac{\varepsilon_{1}}{4(C_{0}+1)}.$

We now consider any initial data $(v_{0},u_{0})$ satisfying

$\sum_{\pm}(\|v_{0}-v_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}+\|u_{0}-u_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})})+\|\partial_{x}v_{0}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}u_{0}\|_{L^{2}(\mathbb{R})}<\varepsilon_{0},$

which implies that the initial $H^{1}$-perturbation from the monotone
functions $\underline{v},\underline{u}$ is small:

$\|v_{0}-\underline{v}\|_{H^{1}(\mathbb{R})}+\|u_{0}-\underline{u}\|_{H^{1}(\mathbb{R})}\leq\varepsilon_{0}+C_{1}(\delta_{1}+\delta_{2})\leq\varepsilon_{*}.$
(3.14)

In particular, Sobolev embedding implies
$\|v_{0}-\underline{v}\|_{L^{\infty}(\mathbb{R})}\leq C\varepsilon_{*}$, and
therefore,

$\frac{v_{-}}{2}<v_{0}(x)<2v_{+},\quad\forall x\in\mathbb{R}.$

Since $0<\varepsilon_{*}<\frac{\varepsilon_{1}}{2}$, we use Proposition 3.1 to
guarantee the local existence of the unique solution $(v,u)$ on $[0,T_{0}]$
satisfying

$\|v-\underline{v}\|_{L^{\infty}(0,T_{0};H^{1}(\mathbb{R}))}+\|u-\underline{u}\|_{L^{\infty}(0,T_{0};H^{1}(\mathbb{R}))}\leq\frac{\varepsilon_{1}}{2},$
(3.15)

and

$\frac{v_{-}}{3}<v(t,x)<3v_{+},\quad\forall(t,x)\in[0,T_{0}]\times\mathbb{R}.$

We now use a similar argument as in (LABEL:H1-perterb-1) to obtain

$\displaystyle\|\underline{v}-\widetilde{v}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}+\|\underline{u}-\widetilde{u}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}$
$\displaystyle\leq\sum_{\pm}\left(\|\underline{v}-v_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}+\|\underline{u}-u_{\pm}\|_{L^{2}(\mathbb{R}_{\pm})}\right)$
$\displaystyle\quad+\|\widetilde{v}_{1}^{X_{1}}-v_{m}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{v}_{1}^{X_{1}}-v_{-}\|_{L^{2}(\mathbb{R}_{-})}+\|\widetilde{v}_{2}^{X_{2}}-v_{+}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{v}_{2}^{X_{2}}-v_{m}\|_{L^{2}(\mathbb{R}_{-})}$
$\displaystyle\quad+\|\partial_{x}\underline{v}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{v}_{1}^{X_{1}}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{v}_{2}^{X_{2}}\|_{L^{2}(\mathbb{R})}$
$\displaystyle\quad+\|\widetilde{u}_{1}^{X_{1}}-u_{m}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{u}_{1}^{X_{1}}-u_{-}\|_{L^{2}(\mathbb{R}_{-})}+\|\widetilde{u}_{2}^{X_{2}}-u_{+}\|_{L^{2}(\mathbb{R}_{+})}+\|\widetilde{u}_{2}^{X_{2}}-u_{m}\|_{L^{2}(\mathbb{R}_{-})}$
$\displaystyle\quad+\|\partial_{x}\underline{u}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{u}_{1}^{X_{1}}\|_{L^{2}(\mathbb{R})}+\|\partial_{x}\widetilde{u}_{2}^{X_{2}}\|_{L^{2}(\mathbb{R})}$
$\displaystyle\leq
C\sqrt{\delta_{1}}(1+\sqrt{X_{1}(t)})+C\sqrt{\delta_{2}}(1+\sqrt{X_{2}(t)}),$

where we used

$\displaystyle\int_{0}^{\infty}$
$\displaystyle|\widetilde{v}_{1}(x-\sigma_{1}t-X_{1}(t))-v_{m}|^{2}\,dx$
$\displaystyle=\int_{0}^{\infty}|\widetilde{v}_{1}(x-\sigma_{1}t)-v_{m}|^{2}\,dx+\int_{-X_{1}(t)}^{0}|\widetilde{v}_{1}(x-\sigma_{1}t)-v_{m}|^{2}\,dx$
$\displaystyle\leq C\delta_{1}(1+|X_{1}(t)|),$

and a similar estimates for $\widetilde{v}_{2}$. Then, using Lemma 3.1, we
have

$\|\underline{v}-\widetilde{v}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}+\|\underline{u}-\widetilde{u}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}\leq
C\sqrt{\delta_{0}}(1+\sqrt{t}).$

Therefore, taking $0<T_{1}<T_{0}$ small enough such that
$C\sqrt{\delta_{0}}(1+\sqrt{T_{1}})<\frac{\varepsilon_{1}}{2}$, we obtain

$\|\underline{v}-\widetilde{v}^{X_{1},X_{2}}\|_{L^{\infty}(0,T_{1};H^{1}(\mathbb{R}))}+\|\underline{u}-\widetilde{u}^{X_{1},X_{2}}\|_{L^{\infty}(0,T_{1};H^{1}(\mathbb{R}))}\leq\frac{\varepsilon_{1}}{2}.$
(3.16)

We now combine (3.15) and (3.16) to obtain the following $H^{1}$-perturbation
estimate

$\|v-\widetilde{v}^{X_{1},X_{2}}\|_{L^{\infty}(0,T_{1};H^{1}(\mathbb{R}))}+\|u-\widetilde{u}^{X_{1},X_{2}}\|_{L^{\infty}(0,T_{1};H^{1}(\mathbb{R}))}<\varepsilon_{1}.$
(3.17)

In particular, since the shifts $X_{i}(t)$ are absolutely continuous, we have

$v-\widetilde{v}^{X_{1},X_{2}},u-\widetilde{u}^{X_{1},X_{2}}\in
C([0,T_{1}];H^{1}(\mathbb{R})).$

In order to extend the solution globally-in-time, we consider the maximal time
$T_{*}$ defined as

$T_{*}:=\sup\left\\{t>0~{}\Bigg{|}~{}\sup_{[0,t]}\left(\|v-\widetilde{v}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}+\|u-\widetilde{u}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}\right)\leq\varepsilon_{1}\right\\}.$

Suppose $T_{*}<+\infty$. Then, by the continuation argument,

$\sup_{[0,T_{*}]}\left(\|v-\widetilde{v}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}+\|u-\widetilde{u}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}\right)=\varepsilon_{1}.$
(3.18)

However, by (LABEL:H1-perterb-1) and (3.14),

$\|v_{0}-\widetilde{v}(0,\cdot)\|_{H^{1}(\mathbb{R})}+\|u_{0}-\widetilde{u}(0,\cdot)\|_{H^{1}(\mathbb{R})}\leq\frac{\frac{\varepsilon_{1}}{2}-C_{0}\delta_{0}^{1/4}}{C_{0}+1},$

which together with Proposition 3.2, implies that

$\displaystyle\sup_{[0,T_{*}]}$
$\displaystyle\left(\|v-\widetilde{v}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}+\|u-\widetilde{u}^{X_{1},X_{2}}\|_{H^{1}(\mathbb{R})}\right)$
$\displaystyle\leq
C_{0}\frac{\frac{\varepsilon_{1}}{2}-C_{0}\delta_{0}^{1/4}}{C_{0}+1}+C_{0}\delta_{0}^{1/4}$
$\displaystyle\leq\left(\frac{\varepsilon_{1}}{2}-C_{0}\delta_{0}^{1/4}\right)+C_{0}\delta_{0}^{1/4}=\frac{\varepsilon_{1}}{2}.$

This contradicts to (3.18) and therefore, we conclude that $T_{*}=+\infty$.
Then, it again follows from Proposition 3.2 that

$\displaystyle\sup_{t>0}$ $\displaystyle\left(\left\lVert
v-\widetilde{v}^{X_{1},X_{2}}\right\rVert_{H^{1}(\mathbb{R})}+\left\lVert
u-\widetilde{u}^{X_{1},X_{2}}\right\rVert_{H^{1}(\mathbb{R})}\right)+\sqrt{\int_{0}^{t}\sum_{i=1}^{2}\delta_{i}|X_{i}|^{2}\,ds}$
(3.19)
$\displaystyle\quad+\sqrt{\int_{0}^{\infty}\left(\mathcal{G}^{S}(U)+D(U)+D_{1}(U)+D_{2}(U)\right)\,ds}$
$\displaystyle\leq
C_{0}\left(\|v_{0}-\widetilde{v}(0,\cdot)\|_{H^{1}(\mathbb{R})}+\|u_{0}-\widetilde{u}(0,\cdot)\|_{H^{1}(\mathbb{R})}\right)+C_{0}\delta_{0}^{1/4},$

and

$|\dot{X}_{1}(t)|+|\dot{X}_{2}(t)|\leq
C_{0}\|(v-\widetilde{v}^{X_{1},X_{2}})(t,\cdot)\|_{L^{\infty}(\mathbb{R})},\qquad
t>0.$ (3.20)

Especially, since the right-hand side of (3.19) is small enough, we find that

$\frac{v_{-}}{3}<v(t,x)<3v_{+},\quad\forall(t,x)\in[0,\infty)\times\mathbb{R}.$
(3.21)

###  3.5. Time-asymptotic behavior

Based on the global estimates (3.19)-(3.21), we will show the time-asymptotic
behavior in Theorem 1.1. The proof mainly follows the typical argument, but we
will crucially use the following two lemmas based on the wave separation by
shifts, since the energy functional $\mathcal{G}^{S}$ is localized by the
cutoff functions $\phi_{i}$.
First, the estimate (3.19) with the Sobolev embedding implies

$\|v-\widetilde{v}^{X_{1},X_{2}}\|_{L^{\infty}(\mathbb{R}_{+}\times\mathbb{R})}\leq
C(\varepsilon_{0}+\delta_{0}^{1/4}).$

Thus, (3.20) and the smallness of $\delta_{0}$ and $\varepsilon_{0}$ imply

$X_{1}(t)\leq-\frac{\sigma_{1}}{2}t,\quad
X_{2}(t)\geq-\frac{\sigma_{2}}{2}t,\quad t>0,$

or equivalently,

$X_{1}(t)+\sigma_{1}t\leq\frac{\sigma_{1}}{2}t,\quad
X_{2}(t)+\sigma_{2}t\geq\frac{\sigma_{2}}{2}t,\quad t>0,$ (3.22)

which proves (1.10). Thus, the waves $\widetilde{v}_{1}^{X_{1}}$ and
$\widetilde{v}_{2}^{X_{2}}$ are well-separated, and so are
$\widetilde{v}_{1}^{X_{1}}$ and $\phi_{2}$. Using this property, we have the
following lemmas on the wave separation.

######  Lemma 3.2.

Assume (3.22). Given $v_{+}>0$, there exist positive constants $\delta_{0},C$
such that for any $\delta_{1},\delta_{2}\in(0,\delta_{0})$, the following
estimates hold. For each $i=1,2$,

$\displaystyle|(\widetilde{v}_{i})^{X_{i}}_{x}||\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}^{X_{i}}_{i}|\leq
C\delta_{i}\delta_{1}\delta_{2}\exp\left(-C\min\\{\delta_{1},\delta_{2}\\}t\right),\quad
t>0,\quad x\in\mathbb{R},$
$\displaystyle\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}^{X_{i}}_{i}|\,dx\leq
C\delta_{1}\delta_{2}\exp\left(-C\min\\{\delta_{1},\delta_{2}\\}t\right),\quad
t>0,$
$\displaystyle\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}|\,dx\leq
C\delta_{1}\delta_{2}\exp\left(-C\min\\{\delta_{1},\delta_{2}\\}t\right),\quad
t>0.$

######  Lemma 3.3.

Assume (3.22). Let $\phi_{i}$ be the functions defined in (3.11). Given
$v_{+}>0$, there exist positive constants $\delta_{0},C$ such that for any
$\delta_{1},\delta_{2}\in(0,\delta_{0})$, the following estimates hold.

$\displaystyle\phi_{2}|(\widetilde{v}_{1})^{X_{1}}_{x}|\leq
C\delta_{1}^{2}\exp(-C\delta_{1}t),\quad\phi_{1}|(\widetilde{v}_{2})^{X_{2}}_{x}|\leq
C\delta_{2}^{2}\exp(-C\delta_{2}t),\quad t>0,\quad x\in\mathbb{R},$
$\displaystyle\int_{\mathbb{R}}|(\widetilde{v}_{1})_{x}^{X_{1}}|\phi_{2}\,dx\leq
C\delta_{1}\exp(-C\delta_{1}t),\quad\int_{\mathbb{R}}|(\widetilde{v}_{2})_{x}^{X_{2}}|\phi_{1}\,dx\leq
C\delta_{2}\exp(-C\delta_{2}t),\quad t>0.$

The proofs of the above lemmas are straightforward from the ones of [7, Lemma
7.3] and [7, Lemma 7.4] respectively. However, we present the proofs of the
above lemmas in the appendices for readers’ convenience.

To get the desired result, we follow the classical method as in [11]. But, we
here present its details for readers. Consider a function $g$ defined by

$g(t):=\|(v-\widetilde{v}^{X_{1},X_{2}})_{x}\|_{L^{2}(\mathbb{R})}^{2}+\|(u-\widetilde{u}^{X_{1},X_{2}})_{x}\|_{L^{2}(\mathbb{R})}^{2}.$

We will prove that $g\in W^{1,1}(\mathbb{R}_{+})$, which then implies

$\lim_{t\to\infty}g(t)=0,$

which together with the interpolation inequlaity and (3.19) yields

$\lim_{t\to\infty}(\|v-\widetilde{v}^{X_{1},X_{2}}\|_{L^{\infty}(\mathbb{R})}+\|u-\widetilde{u}^{X_{1},X_{2}}\|_{L^{\infty}(\mathbb{R})})=0.$
(3.23)

Moreover, (3.20) and (3.23) imply that

$\lim_{t\rightarrow+\infty}|\dot{X}_{i}(t)|\leq
C_{0}\|(v-\widetilde{v}^{X_{1},X_{2}})(t)\|_{L^{\infty}(\mathbb{R})}=0,\quad
i=1,2.$

Those give the desired result. Hence, it remains to show that $g\in
W^{1,1}(\mathbb{R}_{+})$.
Since

$\displaystyle(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))_{x}$
$\displaystyle=p^{\prime}(v)(v-\widetilde{v}^{X_{1},X_{2}})_{x}+(\widetilde{v}_{1}^{X_{1}}+\widetilde{v}_{2}^{X_{2}})_{x}(p^{\prime}(v)-p^{\prime}(\widetilde{v}^{X_{1},X_{2}})),$

the uniform bound (3.21) yields

$|(v-\widetilde{v}^{X_{1},X_{2}})_{x}|\leq
C|(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))_{x}|+C\left(|(\widetilde{v}_{1})^{X_{1}}_{x}|+|(\widetilde{v}_{2})^{X_{2}}_{x}|\right)|v-\widetilde{v}^{X_{1},X_{2}}|.$
(3.24)

Using (3.24) with $\phi_{1}+\phi_{2}=1$ (defined as in (3.11)), and then Lemma
3.3 and (3.19), we have

$\displaystyle\int_{0}^{\infty}|g(t)|\,dt$ $\displaystyle\leq
C\int_{0}^{\infty}\left(\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}||\phi_{1}(v-\widetilde{v}^{X_{1},X_{2}})|^{2}\,dx+\int_{\mathbb{R}}|(\widetilde{v}_{2})^{X_{2}}_{x}||\phi_{2}(v-\widetilde{v}^{X_{1},X_{2}})|^{2}\,dx\right.$
$\displaystyle\qquad+\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}||\phi_{2}(v-\widetilde{v}^{X_{1},X_{2}})|^{2}\,dx+\int_{\mathbb{R}}|(\widetilde{v}_{2})^{X_{2}}_{x}||\phi_{1}(v-\widetilde{v}^{X_{1},X_{2}})|^{2}\,dx$
$\displaystyle\qquad\left.+\int_{\mathbb{R}}|\partial_{x}(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))|^{2}\,dx+\int_{\mathbb{R}}|(u-\widetilde{u}^{X_{1},X_{2}})_{x}|^{2}\,dx\right)\,dt$
$\displaystyle\leq C\int_{0}^{\infty}(\mathcal{G}^{S}(U)+{D}(U)+D_{1}(U))\,dt$
$\displaystyle\qquad+C\left\lVert
v-\widetilde{v}^{X_{1},X_{2}}\right\rVert_{L^{\infty}(0,\infty;L^{2}(\mathbb{R}))}\int_{0}^{\infty}\int_{\mathbb{R}}(|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}+|(\widetilde{v}_{2})^{X_{2}}_{x}|\phi_{1})\,dxdt,$
$\displaystyle<\infty.$

This implies $g\in L^{1}(\mathbb{R}_{+})$. To show $g^{\prime}\in
L^{1}(\mathbb{R}_{+})$, we will use the following system satisfied by the
shifted composite wave :

$\begin{cases}\widetilde{v}^{X_{1},X_{2}}_{t}-\widetilde{u}^{X_{1},X_{2}}_{x}=-\sum_{i=1}^{2}\dot{X}_{i}(t)(\widetilde{v}_{i})^{X_{i}}_{x}\\\
\widetilde{u}^{X_{1},X_{2}}_{t}+p(\widetilde{v}^{X_{1},X_{2}})_{x}=\left(\frac{\widetilde{u}^{X_{1},X_{2}}_{x}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}-\sum_{i=1}^{2}\dot{X_{i}}(\widetilde{u}_{i})^{X_{i}}_{x}+E_{2}+E_{3},\end{cases}$
(3.25)

where

$E_{2}:=p(\widetilde{v}^{X_{1},X_{2}})_{x}-\sum_{i=1}^{2}p(\widetilde{v}_{i}^{X_{i}})_{x},\quad
E_{3}:=-\left(\frac{\widetilde{u}^{X_{1},X_{2}}_{x}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}+\sum_{i=1}^{2}\left(\frac{(\widetilde{u}_{i})^{X_{i}}_{x}}{\widetilde{v}_{i}^{X_{i}}}\right)_{x}.$

Then it follows from (LABEL:eq:NS) and (3.25) that

$\displaystyle(v-\widetilde{v}^{X_{1},X_{2}})_{t}-\sum_{i=1}^{2}\dot{X}_{i}(\widetilde{v}_{i})^{X_{i}}_{x}-(u-\widetilde{u}^{X_{1},X_{2}})_{x}=0,$
$\displaystyle(u-\widetilde{u}^{X_{1},X_{2}})_{t}-\sum_{i=1}^{2}\dot{X}_{i}(\widetilde{u}_{i})^{X_{i}}_{x}+(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))_{x}=\left(\frac{u_{x}}{v}-\frac{\widetilde{u}^{X_{1},X_{2}}_{x}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}-E_{2}-E_{3}.$

This with integration by parts yields

$\displaystyle\int_{0}^{\infty}|g^{\prime}(t)|\,dt=\int_{0}^{\infty}2\left|\int_{\mathbb{R}}(v-\widetilde{v}^{X_{1},X_{2}})_{x}(v-\widetilde{v}^{X_{1},X_{2}})_{xt}\,dx+\int_{\mathbb{R}}(u-\widetilde{u}^{X_{1},X_{2}})_{x}(u-\widetilde{u}^{X_{1},X_{2}})_{xt}\,dx\right|\,dt$
$\displaystyle\leq
2\int_{0}^{\infty}\int_{\mathbb{R}}\left|(v-\widetilde{v}^{X_{1},X_{2}})_{x}\left[\sum_{i=1}^{2}\dot{X}_{i}(t)(\widetilde{v}_{i})^{X_{i}}_{xx}+(u-\widetilde{u}^{X_{1},X_{2}})_{xx}\right]+(u-\widetilde{u}^{X_{1},X_{2}})_{x}\left[\sum_{i=1}^{2}\dot{X}_{i}(t)(\widetilde{u}_{i})^{X_{i}}_{xx}\right]\right|dxdt$
$\displaystyle\qquad+2\int_{0}^{\infty}\int_{\mathbb{R}}\left|(u-\widetilde{u}^{X_{1},X_{2}})_{xx}\left[-(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))_{x}+\left(\frac{u_{x}}{v}-\frac{\widetilde{u}_{x}^{X_{1},X_{2}}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}-E_{2}-E_{3}\right]\right|dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\left(\int_{\mathbb{R}}|(v-\widetilde{v}^{X_{1},X_{2}})_{x}|^{2}+|(u-\widetilde{u}^{X_{1},X_{2}})_{x}|^{2}+|(u-\widetilde{u}^{X_{1},X_{2}})_{xx}|^{2}+|\partial_{x}(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))|^{2}dx\right.$
$\displaystyle\quad\left.+\int_{\mathbb{R}}\sum_{i=1}^{2}|\dot{X}_{i}(t)|^{2}\left[|(\widetilde{v}_{i})^{X_{i}}_{xx}|^{2}+|(\widetilde{u}_{i})^{X_{i}}_{xx}|^{2}\right]dx+\int_{\mathbb{R}}\left[\left|\left(\frac{u_{x}}{v}-\frac{\widetilde{u}_{x}^{X_{1},X_{2}}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}\right|^{2}+|E_{2}|^{2}+|E_{3}|^{2}\right]dx\right)dt$

Using the same estimates as before, (3.19) and Lemma 2.2, we first have

$\displaystyle\int_{0}^{\infty}|g^{\prime}(t)|\,dt$ $\displaystyle\leq
C\int_{0}^{\infty}\left(\sum_{i=1}^{2}|\dot{X}_{i}(t)|^{2}+\mathcal{G}^{S}(U)+{D}(U)+D_{1}(U)+D_{2}(U)\right)\,dt+C$
$\displaystyle\qquad+\int_{\mathbb{R}}\left[\left|\left(\frac{u_{x}}{v}-\frac{\widetilde{u}_{x}^{X_{1},X_{2}}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}\right|^{2}+|E_{2}|^{2}+|E_{3}|^{2}\right]dxdt$

For the last three terms above, we get further estimate as follows:
Using (3.21) and (3.24) with Lemma 2.2, we get

$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}\left|\left(\frac{u_{x}}{v}-\frac{\widetilde{u}_{x}^{X_{1},X_{2}}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}\right|^{2}\,dxdt$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}}\left|\frac{1}{v}(u-\widetilde{u}^{X_{1},X_{2}})_{xx}+(\widetilde{u}^{X_{1},X_{2}})_{xx}\left(\frac{1}{v}-\frac{1}{\widetilde{v}^{X_{1},X_{2}}}\right)-\frac{(u-\widetilde{u}^{X_{1},X_{2}})_{x}}{v^{2}}(v-\widetilde{v}^{X_{1},X_{2}})_{x}\right.$
$\displaystyle\qquad\left.-\frac{\widetilde{u}^{X_{1},X_{2}}_{x}}{v^{2}}(v-\widetilde{v}^{X_{1},X_{2}})_{x}-\frac{\widetilde{v}_{x}^{X_{1},X_{2}}}{v^{2}}(u-\widetilde{u}^{X_{1},X_{2}})_{x}+(\widetilde{v}^{X_{1},X_{2}})_{x}(\widetilde{u}^{X_{1},X_{2}})_{x}\left(\frac{1}{(\widetilde{v}^{X_{1},X_{2}})^{2}}-\frac{1}{v^{2}}\right)\right|^{2}\,dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\int_{\mathbb{R}}\bigg{[}|(u-\widetilde{u}^{X_{1},X_{2}})_{xx}|^{2}+(|(\widetilde{v}_{1})^{X_{1}}_{x}|^{2}+|(\widetilde{v}_{2})^{X_{2}}_{x}|^{2})|v-\widetilde{v}^{X_{1},X_{2}}|^{2}+|\partial_{x}(p(v)-p(\widetilde{v}^{X_{1},X_{2}}))|^{2}$
$\displaystyle\qquad+|(u-\widetilde{u}^{X_{1},X_{2}})_{x}|^{2}|(v-\widetilde{v}^{X_{1},X_{2}})_{x}|^{2}+|(u-\widetilde{u}^{X_{1},X_{2}})_{x}|^{2}\bigg{]}\,dxdt.$

Using (3.19) with the same estimates as before, we have

$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}\left|\left(\frac{u_{x}}{v}-\frac{\widetilde{u}_{x}^{X_{1},X_{2}}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}\right|^{2}\,dxdt$
$\displaystyle\leq
C+C\int_{0}^{\infty}(\mathcal{G}^{S}(U)+{D}(U)+D_{1}(U)+D_{2}(U))\,dt$
$\displaystyle\qquad+C\int_{0}^{\infty}\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{x}\right\rVert_{L^{\infty}(\mathbb{R})}^{2}\int_{\mathbb{R}}|(v-\widetilde{v}^{X_{1},X_{2}})_{x}|^{2}\,dxdt.$

The last term is estimated by using the interpolation inequality with (3.19)
as

$\displaystyle\int_{0}^{\infty}\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{x}\right\rVert_{L^{\infty}(\mathbb{R})}^{2}\int_{\mathbb{R}}|(v-\widetilde{v}^{X_{1},X_{2}})_{x}|^{2}\,dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{x}\right\rVert_{L^{2}(\mathbb{R})}\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{xx}\right\rVert_{L^{2}(\mathbb{R})}\left\lVert(v-\widetilde{v}^{X_{1},X_{2}})_{x}\right\rVert_{L^{2}(\mathbb{R})}^{2}\,dt$
$\displaystyle\leq
C\int_{0}^{\infty}\left[\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{x}\right\rVert_{L^{2}(\mathbb{R})}^{2}+\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{xx}\right\rVert_{L^{2}(\mathbb{R})}^{2}\left\lVert(v-\widetilde{v}^{X_{1},X_{2}})_{x}\right\rVert_{L^{2}(\mathbb{R})}^{4}\right]\,dt$
$\displaystyle\leq
C\int_{0}^{\infty}\left[\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{x}\right\rVert_{L^{2}(\mathbb{R})}^{2}+\left\lVert(u-\widetilde{u}^{X_{1},X_{2}})_{xx}\right\rVert_{L^{2}(\mathbb{R})}^{2}\right]\,dt$
$\displaystyle\leq C\int_{0}^{\infty}\left(D_{1}(U)+D_{2}(U)\right)\,dt.$

Similarly, using Lemma 3.2, we have

$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}|E_{3}|^{2}\,dxdt=\int_{0}^{\infty}\int_{\mathbb{R}}\left|\sum_{i=1}^{2}\left(\frac{(\widetilde{u}_{i}^{X_{i}})_{x}}{\widetilde{v}_{i}^{X_{i}}}\right)_{x}-\left(\frac{\widetilde{u}_{x}^{X_{1},X_{2}}}{\widetilde{v}^{X_{1},X_{2}}}\right)_{x}\right|^{2}\,dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\int_{\mathbb{R}}\Big{(}(|(\widetilde{u}_{1})^{X_{1}}_{xx}|+|(\widetilde{u}_{1})^{X_{1}}_{x}||(\widetilde{v}_{1})^{X_{1}}_{x}|)|\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{1}^{X_{1}}|$
$\displaystyle\hskip
28.45274pt+(|(\widetilde{u}_{2})^{X_{2}}_{xx}|+|(\widetilde{u}_{2})^{X_{2}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}|)|\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{2}^{X_{2}}|+|(\widetilde{u}_{1})^{X_{1}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}|+|(\widetilde{u}_{2})^{X_{2}}_{x}||(\widetilde{v}_{1})^{X_{1}}_{x}|\Big{)}^{2}\,dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\int_{\mathbb{R}}\Big{(}(|(\widetilde{v}_{1})^{X_{1}}_{xx}|+|(\widetilde{v}_{1})^{X_{1}}_{x}|^{2})|\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{1}^{X_{1}}|+(|(\widetilde{v}_{2})^{X_{2}}_{xx}|+|(\widetilde{v}_{2})^{X_{2}}_{x}|^{2})|\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{2}^{X_{2}}|$
$\displaystyle\qquad+|(\widetilde{v}_{1})^{X_{1}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}|\Big{)}^{2}dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\left(\||(\widetilde{v}_{1})^{X_{1}}_{x}|\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{1}^{X_{1}}|+|(\widetilde{v}_{2})^{X_{2}}_{x}||\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{2}^{X_{2}}|+|(\widetilde{v}_{1})^{X_{1}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}|\|_{L^{2}(\mathbb{R})}^{2}\right)\,dt<\infty,$

and

$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}|E_{2}|^{2}\,dxdt$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}}\left|\sum_{i=1}^{2}p(\widetilde{v}_{i}^{X_{i}})_{x}-p(\widetilde{v}^{X_{1},X_{2}})_{x}\right|^{2}\,dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\int_{\mathbb{R}}\left(|p^{\prime}(\widetilde{v}^{X_{1},X_{2}})-p^{\prime}(\widetilde{v}_{1}^{X_{1}})||(\widetilde{v}_{1})^{X_{1}}_{x}|+|p^{\prime}(\widetilde{v}^{X_{1},X_{2}})-p^{\prime}(\widetilde{v}_{2}^{X_{2}})||(\widetilde{v}_{2})^{X_{2}}_{x}|\right)^{2}dxdt$
$\displaystyle\leq
C\int_{0}^{\infty}\||(\widetilde{v}_{1})^{X_{1}}_{x}||\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{1}^{X_{1}}|+|(\widetilde{v}_{2})^{X_{2}}_{x}||\widetilde{v}^{X_{1},X_{2}}-\widetilde{v}_{2}^{X_{2}}|\|_{L^{2}(\mathbb{R})}^{2}\,dt<\infty.$

Hence, $g^{\prime}\in L^{1}(\mathbb{R}_{+})$ which completes the proof.

Therefore, according to Section 3.4 and Section 3.5, it remains to prove
Proposition 3.2 to obtain global-in-time well-posedness and the time-
asymptotic behavior, as in Theorem 1.1. In the following two sections, we
mainly focus on proving Proposition 3.2, by using the method of
$a$-contraction with shifts.

###  3.6. Notations

In what follows, we use the following notations for simplicity.
1\. $C$ denotes a positive $O(1)$-constant that may change from line to line,
but is independent of the small constants
$\delta_{0},\varepsilon_{1},\delta_{1},\delta_{2}$, $\lambda$ (to be
introduced below) and the time $T$.
2\. We omit the dependence on the pair of shifts $(X_{1},X_{2})$ of the
composite wave (LABEL:composite_wave) without confusion as:

$(\widetilde{v},\widetilde{u})(t,x):=(\widetilde{v}^{X_{1},X_{2}},\widetilde{u}^{X_{1},X_{2}})(t,x).$

##  4\. Estimates on weighted relative entropy with shifts

We will use the method of $a$-contraction with shifts to obtain the
$L^{\infty}(0,T;L^{2}(\mathbb{R}))$-bounds of perturbations in (3.8). For
simplicity of our analysis, we consider the effective velocity $h:=u-(\ln
v)_{x}$ as in [11] (see also [8, 9]) to transform the Navier-Stokes equations
(LABEL:eq:NS) to the system:

$\displaystyle\begin{aligned} &v_{t}-h_{x}=\left(\ln v\right)_{xx},\\\
&h_{t}+p(v)_{x}=0.\end{aligned}$ (4.1)

Then, it follows from (1.7) that the associated viscous shocks
$\widetilde{v}_{i}$ and
$\widetilde{h}_{i}:=\widetilde{u}_{i}-(\ln\widetilde{v}_{i})_{x}$ satisfy the
following ODEs:

$\begin{cases}-\sigma_{i}(\widetilde{v}_{i})^{\prime}-(\widetilde{h}_{i})^{\prime}=(\ln\widetilde{v}_{i})^{\prime\prime},\\\
-\sigma_{i}(\widetilde{h}_{i})^{\prime}+(p(\widetilde{v}_{i}))^{\prime}=0,\\\
(\widetilde{v}_{1},\widetilde{h}_{1})(-\infty)=(v_{-},u_{-}),\quad(\widetilde{v}_{1},\widetilde{h}_{1})(+\infty)=(v_{m},u_{m}),\\\
(\widetilde{v}_{2},\widetilde{h}_{2})(-\infty)=(v_{m},u_{m}),\quad(\widetilde{v}_{2},\widetilde{h}_{2})(+\infty)=(v_{+},u_{+}).\end{cases}$
(4.2)

Let $(\widetilde{v},\widetilde{h})$ denote the shifted composite wave as

$(\widetilde{v},\widetilde{h})(t,x):=\left(\widetilde{v}_{1}^{X_{1}}(x-\sigma_{1}t)+\widetilde{v}_{2}^{X_{2}}(x-\sigma_{2}t)-v_{m},\widetilde{h}_{1}^{X_{1}}(x-\sigma_{1}t)+\widetilde{h}_{2}^{X_{2}}(x-\sigma_{2}t)-u_{m}\right),$

which satisfies

$\begin{cases}\widetilde{v}_{t}-\widetilde{h}_{x}=-\sum_{i=1}^{2}\dot{X}_{i}(t)(\widetilde{v}_{i})^{X_{i}}_{x}+E_{1}\\\
\widetilde{h}_{t}+p(\widetilde{v})_{x}=-\sum_{i=1}^{2}\dot{X_{i}}(\widetilde{h}_{i})^{X_{i}}_{x}+E_{2},\end{cases}$
(4.3)

where $E_{1}$ and $E_{2}$ are the interaction terms defined as

$E_{1}:=-(\ln\widetilde{v})_{xx}+\sum_{i=1}^{2}(\ln\widetilde{v}^{X_{i}}_{i})_{xx},\quad
E_{2}:=p(\widetilde{v})_{x}-\sum_{i=1}^{2}p(\widetilde{v}_{i}^{X_{i}})_{x}.$

######  Remark 4.1.

The structure of system (LABEL:eq:NS-h) would be better for our analysis than
using the original system (LABEL:eq:NS), since (LABEL:eq:NS-h) is linear in
the $h$-variable, but nonlinear in the $v$-variable that would be controlled
by the parabolic term $(\ln v)_{xx}$. However, this transformation would not
be crucial in our setting for small perturbation, but just for simplicity.

The goal of this section is to obtain the following control on the relative
entropy.

######  Lemma 4.1.

There exists a positive constant $C$ such that for all $t\in[0,T]$,

$\displaystyle\begin{aligned}
&\int_{\mathbb{R}}\left(\frac{|h-\widetilde{h}|^{2}}{2}+Q(v|\widetilde{v})\right)\,dx+\int_{0}^{t}\left(\sum_{i=1}^{2}\delta_{i}|\dot{X}_{i}|^{2}+{G}_{1}+\mathcal{G}^{S}+D\right)\,ds\\\
&\leq
C\int_{\mathbb{R}}\left(\frac{|h_{0}(x)-\widetilde{h}(0,x)|^{2}}{2}+Q(v_{0}(x)|\widetilde{v}(0,x))\right)\,dx+C\delta_{0},\end{aligned}$
(4.4)

where

$\displaystyle{G}_{1}(U):=\sum_{i=1}^{2}\int_{\mathbb{R}}|(a_{i})^{X_{i}}_{x}|\left|h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right|^{2}\,dx,$
$\displaystyle\mathcal{G}^{S}(U):=\sum_{i=1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}dx,$
$\displaystyle{D}(U):=\int_{\mathbb{R}}|\partial_{x}(p(v)-p(\widetilde{v}))|^{2}\,dx.$

###  4.1. Construction of weight functions

In order to deal with the two shock waves, we first introduce two weight
functions $a_{1},a_{2}$ associated with 1-shock and 2-shock respectively: for
each $i=1,2,$ we define

$a_{i}(x-\sigma_{i}t)=1+\frac{\lambda(p(v_{m})-p(\widetilde{v_{i}}(x-\sigma_{i}t)))}{\delta_{i}},$

where $\delta_{i}$ denotes the strength of $i$-shock. Here, $\lambda$ is a
small constant chosen as $\delta_{i}\ll\lambda<C\sqrt{\delta_{i}}$ for $i=1,2$
so that it is large enough compared to the strengths of shocks.
Notice that $\frac{1}{2}<1-\lambda\leq a_{i}\leq 1$ and

$(a_{i})_{x}=-\frac{\lambda}{\delta_{i}}(p(\widetilde{v_{i}}))_{x},$ (4.5)

from which we have

$|(a_{i})_{x}|\sim\frac{\lambda}{\delta_{i}}|(\widetilde{v_{i}})_{x}|,\quad\mbox{and
so,}\quad\|(a_{i})_{x}\|_{L^{\infty}(\mathbb{R})}\leq\lambda\delta_{i},\quad\|(a_{i})_{x}\|_{L^{1}(\mathbb{R})}=\lambda.$
(4.6)

To handle the shifted composite wave, we consider the following composition of
shifted weight functions as

$a^{X_{1},X_{2}}(t,x):=a_{1}^{X_{1}}(x-\sigma_{1}t)+a_{2}^{X_{2}}(x-\sigma_{2}t)-1=a_{1}(x-\sigma_{1}t-X_{1}(t))+a_{2}(x-\sigma_{2}t-X_{2}(t))-1.$
(4.7)

For notational simplicity as before, we will omit the dependence on shifts:

$a(t,x):=a^{X_{1},X_{2}}(t,x).$

###  4.2. Estimate on shifts

We here show the estimate (3.9) on the derivative of shifts. First, take
$\varepsilon_{1}$ and $\delta_{0}$ small enough such that
$\varepsilon_{1},\delta_{0}\in(0,\delta_{*})$, where $\delta_{*}$ is the
constant as in Lemma 2.3 so that we can apply Lemma 2.3 to the proof of the
main theorem.
We use the assumption (3.7) with the Sobolev inequality and (2.5) to have

$\|p(v)-p(\widetilde{v})\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq
C\|v-\widetilde{v}\|_{L^{\infty}((0,T)\times\mathbb{R})}\leq
C\varepsilon_{1}.$ (4.8)

Then, using (4.8) with
$\sigma_{i}(\widetilde{h}_{i})^{\prime}=p(\widetilde{v}_{i})^{\prime}$ from
(4.2), it follows from (3.3) that for each $i$,

$|\dot{X}_{i}(t)|\leq\frac{C}{\delta_{i}}\||p(v)-p(\widetilde{v})|+|v-\widetilde{v}|\|_{L^{\infty}(\mathbb{R})}\int_{\mathbb{R}}(\widetilde{v}_{i})^{X_{i}}_{x}dx\leq
C\|v-\widetilde{v}\|_{L^{\infty}(\mathbb{R})},\quad t\leq T.$ (4.9)

This gives the desired estimate (3.9). As in (3.22), the above estimates imply

$X_{1}(t)+\sigma_{1}t\leq\frac{\sigma_{1}}{2}t,\quad
X_{2}(t)+\sigma_{2}t\geq\frac{\sigma_{2}}{2}t,\quad t\leq T.$ (4.10)

###  4.3. Relative entropy method

First of all, we rewrite the system (LABEL:eq:NS-h) in the abstract form:

$\partial_{t}U+\partial_{x}A(U)=\partial_{x}\Big{(}M(U)\partial_{x}D\eta(U)\Big{)},$
(4.11)

where

$U:={v\choose h},\quad A(U):={-h\choose p(v)},\quad M(U):={\frac{1}{\gamma
p(v)}\quad 0\choose\quad\ 0\quad\ 0},$

and

$D\eta(U)={-p(v)\choose h}.$ (4.12)

Indeed, by $-p^{\prime}(v)v=\gamma p(v)$, one has

$\big{(}\ln v\big{)}_{xx}=\left(\frac{(-p(v))_{x}}{\gamma p(v)}\right)_{x}.$

Then, using the nonnegative matrix $M(U)$ and the gradient (4.12) of the
entropy $\eta(U):=\frac{h^{2}}{2}+Q(v)$ of (LABEL:eq:NS-h), where
$Q(v)=\frac{v^{-\gamma+1}}{\gamma-1}$, i.e., $Q^{\prime}(v)=-p(v)$, we have

${\big{(}\ln v\big{)}_{xx}\choose
0}=\partial_{x}\Big{(}M(U)\partial_{x}D\eta(U)\Big{)}.$

Let $\widetilde{U}_{1}(x-\sigma_{1}t)$ and $\widetilde{U}_{2}(x-\sigma_{2}t)$
be the viscous 1-shock and 2-shock of the system (4.11). Then, the system
(4.2) can be cast in the following form: for each $i=1,2,$

$-\sigma_{i}(\widetilde{U}_{i})_{x}+A(\widetilde{U}_{i})_{x}=\partial_{x}\left(M(\widetilde{U}_{i})\partial_{x}D\eta(\widetilde{U}_{i})\right).$
(4.13)

Then, from (4.3), the shifted composite wave
$\widetilde{U}:=(\widetilde{v},\widetilde{h})$ satisfies

$\partial_{t}\widetilde{U}+A(\widetilde{U})_{x}=\partial_{x}(M(\widetilde{U})\partial_{x}D\eta(\widetilde{U}))-\sum_{i=1}^{2}\dot{X}_{i}(\widetilde{U}_{i})^{X_{i}}_{x}+\begin{pmatrix}E_{1}\\\
E_{2}\end{pmatrix}.$ (4.14)

For the relative entropy method, we use the relative entropy functional
$\eta(U|V)$ and relative flux $A(U|V)$ defined as

$\eta(U|V):=\eta(U)-\eta(V)-D\eta(V)(U-V)$

and

$A(U|V):=A(U)-A(V)-DA(V)(U-V).$

We also define the relative entropy flux $G(U;V)$ as

$G(U;V):=G(U)-G(V)-D\eta(V)(A(U)-A(V)),$

where $G$ is the entropy flux for $\eta$ satisfying
$\partial_{i}G(U)=\sum_{k=1}^{2}\partial_{k}\eta(U)+\partial_{i}A_{k}(U)$ for
$i=1,2$. The relative quantities for the system (LABEL:eq:NS-h) can be
explicitly written as follows:

$\displaystyle\eta(U|\widetilde{U})=\frac{|h-\widetilde{h}|^{2}}{2}+Q(v|\widetilde{v}),$
$\displaystyle A(U|\widetilde{U})=\begin{pmatrix}0\\\
p(v|\widetilde{v})\end{pmatrix},$ $\displaystyle
G(U;\widetilde{U})=(p(v)-p(\widetilde{v}))(h-\widetilde{h}).$

Here, the relative internal energy $Q(v|\widetilde{v})$ and the relative
pressure $p(v|\widetilde{v})$ are defined as

$Q(v|\widetilde{v})=Q(v)-Q(\widetilde{v})-Q^{\prime}(\widetilde{v})(v-\widetilde{v}),\quad\mbox{and}\quad
p(v|\widetilde{v})=p(v)-p(\widetilde{v})-p^{\prime}(\widetilde{v})(v-\widetilde{v}).$

In order to obtain the desired estimate in Lemma 4.1, we need to control the
relative entropy between the solution $U$ to (4.11) and the shifted composite
wave $\widetilde{U}$. However, as we mentioned, the direct estimate of the
relative entropy would fail to control it, and therefore, we instead estimate
the following weighted relative entropy:

$\int_{\mathbb{R}}a(t,x)\eta\left(U(t,x)\Big{|}\widetilde{U}(t,x)\right)\,dx.$

It follows from the choice of the weight function that the global weight
$a(t,x)$ satisfies $1/2\leq a\leq 1$ by the smallness of $\lambda$, which
recovers the control of the relative entropy. Following the estimate in [7,
Lemma 5.2] (see also [11, Lemma 4.3] and [26, Lemma 4]), the time-derivative
of the weighted relative entropy can be computed as follows.

######  Lemma 4.2.

Let $a:=a^{X_{1},X_{2}}$ be the weight function defined in (4.7) and $X_{1}$,
$X_{2}$ be any absolutely continuous function. Let $U$ be a solution to system
(4.11) and $\widetilde{U}$ be the shifted composition wave satisfying (4.14).
Then, we have

$\frac{d}{dt}\int_{\mathbb{R}}a(t,x)\eta\left(U(t,x)\Big{|}\widetilde{U}(t,x)\right)\,dx=\sum_{i=1}^{2}(\dot{X}_{i}(t)Y_{i}(U))+\mathcal{J}^{\rm
bad}(U)-\mathcal{J}^{\rm good}(U),$

where

$Y_{i}(U):=-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\eta(U|\widetilde{U})\,dx+\int_{\mathbb{R}}a(\widetilde{U}_{i})^{X_{i}}_{x}D^{2}\eta(\widetilde{U})(U-\widetilde{U})\,dx,$
(4.15)

and

$\displaystyle\mathcal{J}^{\rm
bad}(U):=\sum_{i=1}^{2}\Bigg{[}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}(p(v)-p(\widetilde{v}))(h-\widetilde{h})\,dx+\sigma_{i}\int_{\mathbb{R}}a(\widetilde{v}_{i})^{X_{i}}_{x}p(v|\widetilde{v})\,dx$
$\displaystyle\hskip
85.35826pt-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\frac{p(v)-p(\widetilde{v})}{\gamma
p(v)}\partial_{x}(p(v)-p(\widetilde{v}))\,dx$ $\displaystyle\hskip
85.35826pt-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}(p(v)-p(\widetilde{v}))^{2}\frac{\partial_{x}p(\widetilde{v})}{\gamma
p(v)p(\widetilde{v})}\,dx\Bigg{]}$ $\displaystyle\hskip
56.9055pt-\int_{\mathbb{R}}a\partial_{x}(p(v)-p(\widetilde{v}))\frac{p(v)-p(\widetilde{v})}{\gamma
p(v)p(\widetilde{v})}\partial_{x}p(\widetilde{v})\,dx$ $\displaystyle\hskip
56.9055pt+\int_{\mathbb{R}}a(p(v)-p(\widetilde{v}))E_{1}\,dx-\int_{\mathbb{R}}a(h-\widetilde{h})E_{2}\,dx,$
$\displaystyle\mathcal{J}^{\rm
good}(U):=\sum_{i=1}^{2}\left[\frac{\sigma_{i}}{2}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left|h-\widetilde{h}\right|^{2}\,dx+\sigma_{i}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}Q(v|\widetilde{v})\,dx\right]$
$\displaystyle\hskip 85.35826pt+\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}|\partial_{x}(p(v)-p(\widetilde{v}))|^{2}\,dx.$

######  Remark 4.2.

We note that the definition of the weight $a_{i}$ implies

$\sigma_{i}(a_{i})^{X_{i}}_{x}=-\frac{\lambda\sigma_{i}}{\delta_{i}}(p(\widetilde{v_{i}}^{X_{i}}))_{x}=\frac{\gamma\lambda\sigma_{i}}{\delta_{i}}(\widetilde{v_{i}}^{X_{i}})^{-\gamma-1}(\widetilde{v}_{i}^{X_{i}})_{x}>0,$

from which we observe that $\mathcal{J}^{\rm good}$ consists of good terms.

###  4.4. Maximization on $h-\widetilde{h}$

Among the terms in $\mathcal{J}^{\rm bad}(U)$, a primary bad term is

$\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}(p(v)-p(\widetilde{v}))(h-\widetilde{h})\,dx$

where the perturbations for $p(v)$ and $h$ are coupled. In order to exploit
the parabolic term on $v$-variable and hence use the Poincaré-type inequality,
we separate $h-\widetilde{h}$ from $p(v)-p(\widetilde{v})$ by using the
quadratic structure of $h-\widetilde{h}$. Precisely, since

$\displaystyle(a_{i})^{X_{i}}_{x}$
$\displaystyle(p(v)-p(\widetilde{v}))(h-\widetilde{h})-\frac{\sigma_{i}}{2}(a_{i})^{X_{i}}_{x}|h-\widetilde{h}|^{2}$
$\displaystyle=-\frac{\sigma_{i}(a_{i})^{X_{i}}_{x}}{2}\left|h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right|^{2}+\frac{(a_{i})^{X_{i}}_{x}}{2\sigma_{i}}|p(v)-p(\widetilde{v})|^{2},$

we rewrite the terms $\mathcal{J}^{\rm bad}(U)-\mathcal{J}^{\rm good}(U)$ in
the above lemma as

$\mathcal{J}^{\text{bad}}(U)-\mathcal{J}^{\text{good}}(U)=\mathcal{B}(U)-\mathcal{G}(U),$

where

$\displaystyle\mathcal{B}(U)$
$\displaystyle:=\sum_{i=1}^{2}\Bigg{[}\frac{1}{2\sigma_{i}}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}|p(v)-p(\widetilde{v})|^{2}\,dx+\sigma_{i}\int_{\mathbb{R}}a(\widetilde{v}_{i})^{X_{i}}_{x}p(v|\widetilde{v})\,dx$
$\displaystyle\quad\qquad-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\frac{p(v)-p(\widetilde{v})}{\gamma
p(v)}\partial_{x}(p(v)-p(\widetilde{v}))\,dx$
$\displaystyle\quad\qquad-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}(p(v)-p(\widetilde{v}))^{2}\frac{\partial_{x}p(\widetilde{v})}{\gamma
p(v)p(\widetilde{v})}\,dx\Bigg{]}$
$\displaystyle\quad-\int_{\mathbb{R}}a\partial_{x}(p(v)-p(\widetilde{v}))\frac{p(v)-p(\widetilde{v})}{\gamma
p(v)p(\widetilde{v})}\partial_{x}p(\widetilde{v})\,dx$
$\displaystyle\quad+\int_{\mathbb{R}}a(p(v)-p(\widetilde{v}))E_{1}\,dx-\int_{\mathbb{R}}a(h-\widetilde{h})E_{2}\,dx,$
$\displaystyle\mathcal{G}(U)$
$\displaystyle:=\sum_{i=1}^{2}\left[\frac{\sigma_{i}}{2}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left|h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right|^{2}\,dx+\sigma_{i}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}Q(v|\widetilde{v})\,dx\right]$
$\displaystyle\quad+\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}|\partial_{x}(p(v)-p(\widetilde{v}))|^{2}\,dx.$

Therefore, we will estimate the right-hand side of the following equation:

$\frac{d}{dt}\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx=\sum_{i=1}^{2}(\dot{X}_{i}Y_{i}(U))+\mathcal{B}(U)-\mathcal{G}(U).$
(4.16)

###  4.5. Decompositions

We first name each term of $\mathcal{B}(U)$ and $\mathcal{G}(U)$ as follows:

$\displaystyle\mathcal{B}(U)=\sum_{i=1}^{5}\mathcal{B}_{i}(U)+\mathcal{S}_{1}+\mathcal{S}_{2},$
$\displaystyle\mathcal{G}(U)=\mathcal{G}_{1}(U)+\mathcal{G}_{2}(U)+\mathcal{D}(U),$

where

$\displaystyle\mathcal{B}_{1}(U):=\sum_{i=1}^{2}\frac{1}{2\sigma_{i}}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}|p(v)-p(\widetilde{v})|^{2}\,dx,$
$\displaystyle\mathcal{B}_{2}(U):=\sum_{i=1}^{2}\sigma_{i}\int_{\mathbb{R}}a(\widetilde{v}_{i})^{X_{i}}_{x}p(v|\widetilde{v})\,dx,$
$\displaystyle\mathcal{B}_{3}(U):=-\sum_{i=1}^{2}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\frac{p(v)-p(\widetilde{v})}{\gamma
p(v)}\partial_{x}(p(v)-p(\widetilde{v}))\,dx,$
$\displaystyle\mathcal{B}_{4}(U):=-\sum_{i=1}^{2}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}(p(v)-p(\widetilde{v}))^{2}\frac{\partial_{x}p(\widetilde{v})}{\gamma
p(v)p(\widetilde{v})}\,dx,$
$\displaystyle\mathcal{B}_{5}(U):=-\int_{\mathbb{R}}a\partial_{x}(p(v)-p(\widetilde{v}))\frac{p(v)-p(\widetilde{v})}{\gamma
p(v)p(\widetilde{v})}\partial_{x}p(\widetilde{v})\,dx,$
$\displaystyle\mathcal{S}_{1}:=\int_{\mathbb{R}}a(p(v)-p(\widetilde{v}))E_{1}\,dx,\quad\mathcal{S}_{2}:=-\int_{\mathbb{R}}a(h-\widetilde{h})E_{2}\,dx,$

and

$\displaystyle\mathcal{G}_{1}(U):=\sum_{i=1}^{2}\frac{\sigma_{i}}{2}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left|h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right|^{2}\,dx,$
$\displaystyle\mathcal{G}_{2}(U):=\sum_{i=1}^{2}\sigma_{i}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}Q(v|\widetilde{v})\,dx,$
$\displaystyle\mathcal{D}(U):=\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}|\partial_{x}(p(v)-p(\widetilde{v}))|^{2}\,dx.$

For each $Y_{i}$ in (4.15), we first write $Y_{i}$ more explicitly as follows:

$\displaystyle Y_{i}(U)$
$\displaystyle=-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left(\frac{|h-\widetilde{h}|^{2}}{2}+Q(v|\widetilde{v})\right)\,dx+\int_{\mathbb{R}}a(\widetilde{h}_{i})^{X_{i}}_{x}(h-\widetilde{h})\,dx$
$\displaystyle\quad-\int_{\mathbb{R}}ap^{\prime}(\widetilde{v})(\widetilde{v}_{i})^{X_{i}}_{x}(v-\widetilde{v})\,dx.$

Since a key idea in our analysis is to apply Poincaré-type inequality of Lemma
2.1 by extracting a good term on an average of perturbation
$p(v)-p(\widetilde{v})$ from the shift part
$\sum_{i=1}^{2}(\dot{X}_{i}Y_{i}(U))$, we decompose $Y_{i}$ as follows: for
each $i$,

$Y_{i}=\sum_{j=1}^{6}Y_{ij},$

where

$\displaystyle
Y_{i1}:=\int_{\mathbb{R}}\frac{a}{\sigma_{i}}(\widetilde{h}_{i})^{X_{i}}_{x}(p(v)-p(\widetilde{v}))\,dx,$
$\displaystyle
Y_{i2}:=-\int_{\mathbb{R}}ap^{\prime}(\widetilde{v}^{X_{i}}_{i})(\widetilde{v}_{i})^{X_{i}}_{x}(v-\widetilde{v})\,dx=-\int_{\mathbb{R}}a(p(\widetilde{v}^{X_{i}}_{i}))_{x}(v-\widetilde{v})\,dx,$
$\displaystyle
Y_{i3}:=\int_{\mathbb{R}}a(\widetilde{h}_{i})^{X_{i}}_{x}\left(h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right)\,dx,$
$\displaystyle
Y_{i4}:=-\int_{\mathbb{R}}a(p^{\prime}(\widetilde{v})-p^{\prime}(\widetilde{v}^{X_{i}}_{i}))(\widetilde{v}_{i})^{X_{i}}_{x}(v-\widetilde{v})\,dx,$
$\displaystyle
Y_{i5}:=-\frac{1}{2}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left(h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right)\left(h-\widetilde{h}+\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right)\,dx,$
$\displaystyle
Y_{i6}:=-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}Q(v|\widetilde{v})\,dx-\int_{\mathbb{R}}\frac{(a_{i})^{X_{i}}_{x}}{2\sigma_{i}^{2}}(p(v)-p(\widetilde{v}))^{2}\,dx.$

Observe that it follows from our construction (3.3) on shifts $X_{i}$ that

$\dot{X}_{i}=-\frac{M}{\delta_{i}}(Y_{i1}+Y_{i2}),$

which yields

$\dot{X}_{i}Y_{i}(U)=-\frac{\delta_{i}}{M}|\dot{X_{i}}|^{2}+\dot{X}_{i}\sum_{j=3}^{6}Y_{ij}.$
(4.17)

Here, the good term $-\frac{\delta_{i}}{M}|\dot{X_{i}}|^{2}$ would give an
average of linear perturbation on $v$-variable as mentioned above, while the
remaining part would be controlled by the good terms in $\mathcal{G}(U)$. To
show it, we combine (4.16) and (4.17) to have

$\displaystyle\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx=\mathcal{R},\quad\mbox{where}\\\
&\mathcal{R}:=-\sum_{i=1}^{2}\frac{\delta_{i}}{M}|\dot{X}_{i}|^{2}+\sum_{i=1}^{2}\left(\dot{X_{i}}\sum_{j=3}^{6}Y_{ij}\right)+\sum_{i=1}^{5}\mathcal{B}_{i}+\mathcal{S}_{1}+\mathcal{S}_{2}-\mathcal{G}_{1}-\mathcal{G}_{2}-\mathcal{D}.\\\
&\quad=\underbrace{-\sum_{i=1}^{2}\frac{\delta_{i}}{2M}|\dot{X}_{i}|^{2}+\mathcal{B}_{1}+\mathcal{B}_{2}-\mathcal{G}_{2}-\frac{3}{4}\mathcal{D}}_{=:\mathcal{R}_{1}}\\\
&\quad\quad\underbrace{-\sum_{i=1}^{2}\frac{\delta_{i}}{2M}|\dot{X}_{i}|^{2}+\sum_{i=1}^{2}\left(\dot{X_{i}}\sum_{j=3}^{6}Y_{ij}\right)+\sum_{i=3}^{5}\mathcal{B}_{i}+\mathcal{S}_{1}+\mathcal{S}_{2}-\mathcal{G}_{1}-\frac{1}{4}\mathcal{D}}_{=:\mathcal{R}_{2}}.\end{aligned}$
(4.18)

A motivation of the decomposition (LABEL:est) is that the two bad terms
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are the main bad terms, which should
be controlled by using the sharp Poincaré-type inequality of Lemma 2.1. The
remaining terms in $\mathcal{R}_{2}$ can be controlled in a rather rough way.
We first focus on the estimate of $\mathcal{R}_{1}$.

###  4.6. Estimate of the main part $\mathcal{R}_{1}$

A key idea for estimates of $\mathcal{R}_{1}$ is to use the Poincaré-type
inequality (2.1) in Lemma 2.1. To apply the Poincaré-type inequality, we need
to localize the perturbation near each wave by using the cutoff functions
$\phi_{1},\phi_{2}$ defined in (3.11) (see Remark 3.1), and then change of
variables from whole space $\mathbb{R}$ to a bounded interval $(0,1)$ for each
wave. For that, we will consider the following change of variables in space:

$y_{1}:=1-\frac{p(v_{m})-p(\widetilde{v}_{1}(x-\sigma_{1}t))}{\delta_{1}},\qquad
y_{2}:=\frac{p(v_{m})-p(\widetilde{v}_{2}(x-\sigma_{2}t))}{\delta_{2}}.$

Indeed, for each $i$, $y_{i}:\mathbb{R}\to(0,1)$ is monotone in
$\xi_{i}:=x-\sigma_{i}t$, since

$\frac{dy_{1}}{d\xi_{1}}=\frac{1}{\delta_{1}}p(\widetilde{v}_{1})^{\prime}>0,\quad\frac{dy_{2}}{d\xi_{2}}=-\frac{1}{\delta_{2}}p(\widetilde{v}_{2})^{\prime}>0,$

and

$\lim_{\xi_{i}\to-\infty}y_{i}=0,\quad\lim_{\xi_{i}\to+\infty}y_{i}=1.$

In terms of the new variables, we will apply the Poincaré-type inequality to
each perturbation $w_{i}$ localized by $\phi_{i}$ respectively:

$\displaystyle
w_{1}:=\phi_{1}(x+X_{1}(t))\Bigg{(}p(v(t,x+X_{1}(t)))-p\Big{(}\widetilde{v}_{1}(x-\sigma_{1}t)+\widetilde{v}_{2}(x-\sigma_{2}t-X_{2}(t)+X_{1}(t))-v_{m}\Big{)}\Bigg{)}\circ
y_{1}^{-1},$ $\displaystyle
w_{2}:=\phi_{2}(x+X_{2}(t))\Bigg{(}p(v(t,x+X_{2}(t)))-p\Big{(}\widetilde{v}_{1}(x-\sigma_{1}t-X_{1}(t)+X_{2}(t))+\widetilde{v}_{2}(x-\sigma_{2}t)-v_{m}\Big{)}\Bigg{)}\circ
y_{2}^{-1}.$

In what follows, for simplicity, we use the following notations to denote
constants of $O(1)$-scale:

$\sigma_{m}:=\sqrt{-p^{\prime}(v_{m})},\quad\alpha_{m}:=\frac{\gamma+1}{2\gamma\sigma_{m}p(v_{m})},$

which are indeed independent of the small constants $\delta_{i}$ since
$v_{+}/2\leq v_{m}\leq v_{+}$.
Since the shock strengths $\delta_{i}$ are bounded by $\delta_{0}$
respectively, the following estimates on the $O(1)$-constants hold:

$|\sigma_{1}-(-\sigma_{m})|\leq C\delta_{1},\quad|\sigma_{2}-\sigma_{m}|\leq
C\delta_{2},$ (4.19)

and

$\|\sigma_{m}^{2}-|p^{\prime}(\widetilde{v}_{i})|\|_{L^{\infty}}\leq
C\delta_{i},\quad\left\|\frac{1}{\sigma_{m}^{2}}-\frac{p(\widetilde{v}_{i})^{-\frac{1}{\gamma}-1}}{\gamma}\right\|_{L^{\infty}}\leq
C\delta_{i},\quad\left\|\frac{1}{\sigma_{m}^{2}}-\frac{p(\widetilde{v})^{-\frac{1}{\gamma}-1}}{\gamma}\right\|_{L^{\infty}}\leq
C\delta_{0}.$ (4.20)

We are now ready to estimate the terms in $\mathcal{R}_{1}$. As mentioned, we
need to extract a good term on an average of the perturbation $w_{i}$ from the
shift part $\frac{\delta_{i}}{2M}|\dot{X}_{i}|^{2}$ as follows, so that we
could apply Lemma 2.1.

$\bullet$ (Estimate of shift part $\frac{\delta_{i}}{2M}|\dot{X}_{i}|^{2}$):
We will show that: for each $i$,

$\displaystyle\begin{aligned}
-\frac{\delta_{i}}{2M}|\dot{X}_{i}|^{2}&\leq-\frac{M\delta_{i}}{\sigma_{m}^{4}}\left(\int_{0}^{1}w_{i}\,dy_{i}\right)^{2}+C\delta_{i}(\lambda+\delta_{0}+\varepsilon_{1})^{2}\int_{0}^{1}|w_{i}|^{2}\,dy_{i}\\\
&\quad+C\delta_{i}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.\end{aligned}$
(4.21)

As the estimates of $\frac{\delta_{1}}{2M}|\dot{X}_{1}|^{2}$ and
$\frac{\delta_{2}}{2M}|\dot{X}_{2}|^{2}$ are the same, we only handle the case
of $X_{1}$. Since $\dot{X}_{1}=-\frac{M}{\delta_{1}}(Y_{11}+Y_{12})$, we first
estimate $Y_{11}$ and $Y_{12}$.
Using the relation $\phi_{1}+\phi_{2}=1$, we have

$\displaystyle Y_{11}$
$\displaystyle=\int_{\mathbb{R}}\frac{a}{\sigma_{1}}(\widetilde{h}_{1})^{X_{1}}_{x}\phi_{1}(p(v)-p(\widetilde{v}))\,dx+\int_{\mathbb{R}}\frac{a}{\sigma_{1}}(\widetilde{h}_{1})^{X_{1}}_{x}\phi_{2}(p(v)-p(\widetilde{v}))\,dx$
$\displaystyle=\frac{1}{\sigma_{1}^{2}}\int_{\mathbb{R}}ap(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{1}(p(v)-p(\widetilde{v}))\,dx+\int_{\mathbb{R}}\frac{a}{\sigma_{1}^{2}}p(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{2}(t,x)(p(v)-p(\widetilde{v}))\,dx.$

The first term is a main part for applying the Poincaré inequality. We use
change of variable $x\to x+X_{1}(t)$ and apply the change of variables for
$y_{1},w_{1}$ to observe that

$\frac{1}{\sigma_{1}^{2}}\int_{\mathbb{R}}ap(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{1}(p(v)-p(\widetilde{v}))\,dx=\frac{\delta_{1}}{\sigma_{1}^{2}}\int_{0}^{1}a(t,x+X_{1}(t))w_{1}\,dy_{1}.$

Then, using (4.19) and
$\|a-1\|_{L^{\infty}(\mathbb{R}_{+}\times\mathbb{R})}\leq\lambda$, we have

$\left|Y_{11}-\frac{\delta_{1}}{\sigma_{m}^{2}}\int_{0}^{1}w_{1}\,dy_{1}\right|\leq
C\delta_{1}(\lambda+\delta_{0})\int_{0}^{1}|w_{1}|\,dy_{1}+C\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|p(v)-p(\widetilde{v})|\,dx.$

To estimate $Y_{12}$, we first use the equation of state $v=p(v)^{-1/\gamma}$
and Taylor expansion to have

$\left|v-\widetilde{v}-\left(-\frac{p(\widetilde{v})^{-\frac{1}{\gamma}-1}}{\gamma}(p(v)-p(\widetilde{v}))\right)\right|\leq
C|p(v)-p(\widetilde{v})|^{2},$

which together with the estimates (4.20) and (4.8) yields

$\left|v-\widetilde{v}-\left(-\frac{1}{\sigma_{m}^{2}}(p(v)-p(\widetilde{v}))\right)\right|\leq
C(\delta_{0}+\varepsilon_{1})|p(v)-p(\widetilde{v})|.$

This with the same argument as above implies

$\left|Y_{12}-\frac{\delta_{1}}{\sigma_{m}^{2}}\int_{0}^{1}w_{1}\,dy_{1}\right|\leq
C\delta_{1}(\lambda+\delta_{0}+\varepsilon_{1})\int_{0}^{1}|w_{1}|\,dy_{1}+C\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|p(v)-p(\widetilde{v})|\,dx.$

We combine the estimates for $Y_{11}$ and $Y_{12}$ to obtain

$\displaystyle\left|\dot{X}_{1}+\frac{2M}{\sigma_{m}^{2}}\int_{0}^{1}w_{1}\,dy_{1}\right|$
$\displaystyle\leq\frac{M}{\delta_{1}}\left(\left|Y_{11}-\frac{\delta_{1}}{\sigma_{m}^{2}}\int_{0}^{1}w_{1}\,dy_{1}\right|+\left|Y_{12}-\frac{\delta_{1}}{\sigma_{m}^{2}}\int_{0}^{1}w_{1}\,dy_{1}\right|\right)$
$\displaystyle\leq
C(\lambda+\delta_{0}+\varepsilon_{1})\int_{0}^{1}|w_{1}|\,dy_{1}+\frac{C}{\delta_{1}}\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|p(v)-p(\widetilde{v})|dx,$

which implies

$\left(\left|\frac{2M}{\sigma_{m}^{2}}\int_{0}^{1}w_{1}\,dy_{1}\right|-|\dot{X}_{1}|\right)^{2}\leq
C(\lambda+\delta_{0}+\varepsilon_{1})^{2}\int_{0}^{1}|w_{1}|^{2}\,dy_{1}+\frac{C}{\delta_{1}^{2}}\left(\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|(p(v)-p(\widetilde{v}))|dx\right)^{2}.$

We use an elementary inequality $\frac{x^{2}}{2}-y^{2}\leq(x-y)^{2}$ for
$x,y\geq 0$ to obtain

$\displaystyle\begin{aligned}
\frac{2M^{2}}{\sigma_{m}^{4}}\left(\int_{0}^{1}w_{1}\,dy_{1}\right)^{2}-|\dot{X}_{1}|^{2}&\leq
C(\lambda+\delta_{0}+\varepsilon_{1})^{2}\int_{0}^{1}|w_{1}|^{2}\,dy_{1}\\\
&\quad+\frac{C}{\delta_{1}^{2}}\left(\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|(p(v)-p(\widetilde{v}))|dx\right)^{2}.\end{aligned}$
(4.22)

The last term can be estimated as follows: by using Lemmas 2.2, 2.3, 3.3,

$\displaystyle\frac{C}{\delta_{1}^{2}}\left(\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|(p(v)-p(\widetilde{v}))|\,dx\right)^{2}$
$\displaystyle\leq\frac{C}{\delta_{1}^{2}}\int_{\mathbb{R}}(|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2})^{2}dx\int_{\mathbb{R}}|(p(v)-p(\widetilde{v}))|^{2}\,dx$
$\displaystyle\leq\frac{C}{\delta_{1}^{2}}\sup_{t,x}\big{(}\phi_{2}^{2}|(\widetilde{v}_{1})^{X_{1}}_{x}|\big{)}\|(\widetilde{v}_{1})^{X_{1}}_{x}\|_{L^{1}(\mathbb{R})}\int_{\mathbb{R}}Q(v|\widetilde{v})\,dx$
$\displaystyle\leq
C\delta_{1}\exp(-C\delta_{1}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Therefore, we substitute the above estimate to (4.22) to derive the following
estimate on $\dot{X}_{1}$:

$\displaystyle-\frac{\delta_{1}}{2M}|\dot{X}_{1}|^{2}$
$\displaystyle\leq-\frac{M\delta_{1}}{\sigma_{m}^{4}}\left(\int_{0}^{1}w_{1}\,dy_{1}\right)^{2}+C\delta_{1}(\lambda+\delta_{0}+\varepsilon_{1})^{2}\int_{0}^{1}|w_{1}|^{2}\,dy_{1}+C\delta_{1}^{2}\exp(-C\delta_{1}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx,$

which is the desired estimate (4.21).

$\bullet$ (Estimate of the bad term $\mathcal{B}_{1}$ and good term
$\mathcal{G}_{2}$): Recall

$\displaystyle\mathcal{B}_{1}(U):=\sum_{i=1}^{2}\underbrace{\frac{1}{2\sigma_{i}}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}|p(v)-p(\widetilde{v})|^{2}\,dx}_{=:\mathcal{B}_{i1}},$
$\displaystyle\mathcal{G}_{2}(U):=\sum_{i=1}^{2}\underbrace{\sigma_{i}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}Q(v|\widetilde{v})\,dx}_{=:\mathcal{G}_{i2}}.$

Since the estimates for the two cases $i=1,2$ are the same, we only handle the
case of $i=2$ for simplicity.
First, we use the estimate on $Q(v|\widetilde{v})$ in Lemma 2.3 to obtain

$\displaystyle\mathcal{G}_{22}$
$\displaystyle\geq\sigma_{2}\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}\frac{p(\widetilde{v}^{X_{2}}_{2})^{-\frac{1}{\gamma}-1}}{2\gamma}|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\quad-\sigma_{2}\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}\frac{1+\gamma}{3\gamma^{2}}p(\widetilde{v})^{-\frac{1}{\gamma}-2}(p(v)-p(\widetilde{v}))^{3}\,dx$
$\displaystyle\quad+\frac{\sigma_{2}}{2\gamma}\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}\left(p(\widetilde{v})^{-\frac{1}{\gamma}-1}-p(\widetilde{v}^{X_{2}}_{2})^{-\frac{1}{\gamma}-1}\right)|p(v)-p(\widetilde{v})|^{2}\,dx.$

For simplicity, let $\widehat{\mathcal{G}}_{22}$ denote the good term given as

$\widehat{\mathcal{G}}_{22}:=\sigma_{2}\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}\frac{p(\widetilde{v}^{X_{2}}_{2})^{-\frac{1}{\gamma}-1}}{2\gamma}|p(v)-p(\widetilde{v})|^{2}\,dx.$

Using (4.19) and (4.20), we have

$\displaystyle\mathcal{B}_{21}\leq\frac{1}{2\sigma_{m}}\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}|p(v)-p(\widetilde{v})|^{2}\,dx+\frac{C\delta_{2}}{2\sigma_{m}}\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}|p(v)-p(\widetilde{v})|^{2}\,dx$

and

$\displaystyle\widehat{\mathcal{G}}_{22}\geq\frac{1}{2\sigma_{m}}(1-C\delta_{2})\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}|p(v)-p(\widetilde{v})|^{2}\,dx.$

Then, using $\phi_{1}+\phi_{2}=1$ and (4.5), we estimate

$\displaystyle\begin{aligned} \mathcal{B}_{21}-\widehat{\mathcal{G}}_{22}&\leq
C\delta_{2}\int_{\mathbb{R}}(a_{2})^{X_{2}}_{x}|p(v)-p(\widetilde{v})|^{2}\,dx\\\
&\leq
C\delta_{2}\lambda\int_{\mathbb{R}}\frac{|p(\widetilde{v}_{2}^{X_{2}})_{x}|}{\delta_{2}}|\phi_{2}(p(v)-p(\widetilde{v}))|^{2}\,dx+C\lambda\int_{\mathbb{R}}|(\widetilde{v}_{2})^{X_{2}}_{x}|\phi_{1}^{2}|p(v)-p(\widetilde{v})|^{2}\,dx.\end{aligned}$
(4.23)

By change of variable $x\to x+X_{2}(t)$, the first term of the right-hand side
of (4.23) is rewritten in the new variables $w_{2},y_{2}$:

$C\delta_{2}\lambda\int_{\mathbb{R}}\frac{|p(\widetilde{v}_{2}^{X_{2}})_{x}|}{\delta_{2}}|\phi_{2}(p(v)-p(\widetilde{v}))|^{2}\,dx=C\delta_{2}\lambda\int_{0}^{1}|w_{2}|^{2}\,dy_{2}.$

Using Lemma 3.3, the last term of (4.23) can be estimated as

$\displaystyle
C\lambda\int_{\mathbb{R}}|(\widetilde{v}_{2})^{X_{2}}_{x}|\phi_{1}^{2}|p(v)-p(\widetilde{v})|^{2}\,dx\leq
C\lambda\delta^{2}_{2}\exp(-C\delta_{2}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Hence we have

$\mathcal{B}_{1}-\widehat{\mathcal{G}}_{2}\leq\sum_{i=1}^{2}C\lambda\delta_{i}\int_{0}^{1}|w_{i}|^{2}\,dy_{i}+C\lambda\delta_{i}^{2}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

$\bullet$ (Estimate of the bad term $\mathcal{B}_{2}$): Recall

$\displaystyle\mathcal{B}_{2}(U):=\sum_{i=1}^{2}\underbrace{\sigma_{i}\int_{\mathbb{R}}a(\widetilde{v}^{X_{i}}_{i})_{x}p(v|\widetilde{v})\,dx}_{=:\mathcal{B}_{i2}}.$

Again, we only handle the case of $i=2$ for simplicity. First, we have

$\displaystyle\mathcal{B}_{22}=\sigma_{2}\int_{\mathbb{R}}a(\widetilde{v}_{2})^{X_{2}}_{x}\phi_{2}^{2}p(v|\widetilde{v})\,dx+\sigma_{2}\int_{\mathbb{R}}a(\widetilde{v}_{2})^{X_{2}}_{x}(1-\phi_{2}^{2})p(v|\widetilde{v})\,dx.$

For the first term, using Lemma 2.3, (4.19), (4.20) and then the change of
variables, we obtain

$\displaystyle\sigma_{2}\int_{\mathbb{R}}a(\widetilde{v}_{2})^{X_{2}}_{x}\phi_{2}^{2}p(v|\widetilde{v})\,dx=\sigma_{2}\int_{\mathbb{R}}a\frac{p(\widetilde{v}_{2}^{X_{2}})_{x}}{p^{\prime}(\widetilde{v}_{2}^{X_{2}})}\phi_{2}^{2}p(v|\widetilde{v})\,dx$
$\displaystyle\leq\sigma_{2}\int_{\mathbb{R}}a\frac{|p(\widetilde{v}_{2}^{X_{2}})_{x}|}{|p^{\prime}(\widetilde{v}_{2}^{X_{2}})|}\phi_{2}^{2}\left(\frac{\gamma+1}{2\gamma
p(\widetilde{v})}+C\varepsilon_{1}\right)|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{\gamma+1}{2\gamma\sigma_{m}p(v_{m})}(1+C(\delta_{0}+\lambda+\varepsilon_{1}))\int_{\mathbb{R}}|p(\widetilde{v}_{2}^{X_{2}})_{x}||\phi_{2}(p(v)-p(\widetilde{v}))|^{2}\,dx$
$\displaystyle\leq\delta_{2}\alpha_{m}(1+C(\delta_{0}+\lambda+\varepsilon_{1}))\int_{0}^{1}|w_{2}|^{2}\,dy_{2},$

where we used the simple notation
$\alpha_{m}=\frac{\gamma+1}{2\gamma\sigma_{m}p(v_{m})}$.
Again, the last interaction term of $\mathcal{B}_{22}$ can be estimated as

$\displaystyle\sigma_{2}\int_{\mathbb{R}}a(\widetilde{v}^{X_{2}}_{2})_{x}(1+\phi_{2})\phi_{1}p(v|\widetilde{v})\,dx$
$\displaystyle\leq
C\int_{\mathbb{R}}|(\widetilde{v}_{2})^{X_{2}}_{x}|\phi_{1}|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq
C\delta_{2}^{2}\exp(-C\delta_{2}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Therefore, we have

$\displaystyle\mathcal{B}_{22}\leq\delta_{2}\alpha_{m}(1+C(\delta_{0}+\lambda+\varepsilon_{1}))\int_{0}^{1}|w_{2}|^{2}\,dy_{2}+C\delta_{2}^{2}\exp(-C\delta_{2}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx,$

which yields

$\mathcal{B}_{2}\leq\sum_{i=1}^{2}\delta_{i}\alpha_{m}(1+C(\delta_{0}+\lambda+\varepsilon_{1}))\int_{0}^{1}|w_{i}|^{2}\,dy_{i}+C\delta_{i}^{2}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Hence, combining the above estimates on $\mathcal{B}_{1},\mathcal{G}_{2}$ and
$\mathcal{B}_{2}$, we have

$\displaystyle\begin{aligned}
\mathcal{B}_{1}-\mathcal{G}_{2}+\mathcal{B}_{2}&\leq\sum_{i=1}^{2}\delta_{i}\alpha_{m}(1+C(\delta_{0}+\lambda+\varepsilon_{1}))\int_{0}^{1}|w_{i}|^{2}\,dy_{i}+C\delta_{i}^{2}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx\\\
&\quad+\sum_{i=1}^{2}\Bigg{[}-\sigma_{i}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\frac{1+\gamma}{3\gamma^{2}}p(\widetilde{v})^{-\frac{1}{\gamma}-2}(p(v)-p(\widetilde{v}))^{3}\,dx\\\
&\hskip
56.9055pt+\frac{\sigma_{i}}{2\gamma}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left(p(\widetilde{v})^{-\frac{1}{\gamma}-1}-p(\widetilde{v}_{i}^{X_{i}})^{-\frac{1}{\gamma}-1}\right)|p(v)-p(\widetilde{v})|^{2}\,dx\Bigg{]}.\end{aligned}$
(4.24)

$\bullet$ (Estimate of the diffusion term $\mathcal{D}(U)$): First of all,
using the fact that $\phi_{1}+\phi_{2}=1$ and
$1\geq\phi_{i}\geq\phi_{i}^{2}\geq 0$ for any $i$, we separate
$\mathcal{D}(U)$ into

$\mathcal{D}(U)=\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}(\phi_{1}+\phi_{2})|\partial_{x}\big{(}p(v)-p(\widetilde{v})\big{)}|^{2}dx\geq\sum_{i=1}^{2}\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}\phi_{i}^{2}|\partial_{x}\big{(}p(v)-p(\widetilde{v})\big{)}|^{2}dx.$

Since Young’s inequality yields: for any $\delta_{*}>0$ small enough,

$\displaystyle\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}|\partial_{x}\big{(}\phi_{i}(p(v)-p(\widetilde{v}))\big{)}|^{2}dx$
$\displaystyle\leq(1+\delta_{*})\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}\phi_{i}^{2}|\partial_{x}\big{(}p(v)-p(\widetilde{v})\big{)}|^{2}dx$
$\displaystyle\quad+\frac{C}{\delta_{*}}\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}|\partial_{x}\phi_{i}|^{2}|p(v)-p(\widetilde{v})|^{2}dx,$

we have

$\displaystyle-\mathcal{D}(U)$
$\displaystyle\leq-\frac{1}{1+\delta_{*}}\sum_{i=1}^{2}\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}|\partial_{x}\big{(}\phi_{i}(p(v)-p(\widetilde{v}))\big{)}|^{2}dx+\frac{C}{\delta_{*}}\int_{\mathbb{R}}\frac{a}{\gamma
p(v)}|\partial_{x}\phi_{i}|^{2}|p(v)-p(\widetilde{v})|^{2}dx$
$\displaystyle=:J_{1}+J_{2}.$

To write $J_{1}$ in terms of the variables $y_{i},w_{i}$, we use the following
estimates in the proof of [11, Lemma 4.5]:

$\left|\frac{1}{y_{i}(1-y_{i})}\frac{1}{\gamma
p(\widetilde{v}_{i})}\frac{dy_{i}}{dx}-\frac{\delta_{i}p^{\prime\prime}(v_{m})}{2|p^{\prime}(v_{m})|^{2}\sigma_{m}}\right|\leq
C\delta_{i}^{2}.$

This with
$\left\|a\frac{p(\widetilde{v}_{i})}{p(v)}-1\right\|_{L^{\infty}}\leq
C(\delta_{0}+\varepsilon_{1}+\lambda)$ yields

$\displaystyle J_{1}$
$\displaystyle=-\frac{1}{1+\delta_{*}}\sum_{i=1}^{2}\int_{0}^{1}a\frac{p(\widetilde{v}_{i})}{p(v)}\frac{1}{\gamma
p(\widetilde{v}_{i})}|\partial_{y_{i}}w_{i}|^{2}\left(\frac{dy_{i}}{dx}\right)\,dy_{i}$
$\displaystyle\leq-\sum_{i=1}^{2}(1-C(\delta_{0}+\varepsilon_{1}+\lambda+\delta_{*}))\left(\frac{\delta_{i}p^{\prime\prime}(v_{m})}{2|p^{\prime}(v_{m})|^{2}\sigma_{m}}-C\delta_{i}^{2}\right)\int_{0}^{1}y_{i}(1-y_{i})|\partial_{y_{i}}w_{i}|^{2}\,dy_{i}$
$\displaystyle\leq-\sum_{i=1}^{2}\delta_{i}\alpha_{m}(1-C(\delta_{0}+\varepsilon_{1}+\lambda+\delta_{*}))\int_{0}^{1}y_{i}(1-y_{i})|\partial_{y_{i}}w_{i}|^{2}\,dy_{i},$

where we used
$\frac{p^{\prime\prime}(v_{m})}{2|p^{\prime}(v_{m})|^{2}\sigma_{m}}=\frac{\gamma+1}{2\gamma\sigma_{m}p(v_{m})}=\alpha_{m}$.
To estimate the term $J_{2}$, we use the following estimate: for each $i=1,2$,

$|\partial_{x}\phi_{i}(t,x)|\leq\frac{4}{\sigma_{2}-\sigma_{1}}\frac{1}{t},\quad\forall
x\in\mathbb{R},\quad t\in(0,T].$ (4.25)

Indeed, (4.10) yields

$\frac{1}{2}\Big{(}(X_{2}(t)+\sigma_{2}t)-(X_{1}(t)+\sigma_{1}t)\Big{)}\geq\frac{\sigma_{2}-\sigma_{1}}{4}t>0,\quad
t\in(0,T],$

which implies (4.25).
Thus,

$J_{2}\leq\frac{C}{\delta_{*}t^{2}}\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx,$

which implies that for any $0<\delta_{*}<1$ small enough (to be determined
below),

$\displaystyle\begin{aligned}
-\mathcal{D}(U)\leq-\sum_{i=1}^{2}\delta_{i}\alpha_{m}(1-C(\delta_{0}+\lambda+\varepsilon_{1}+\delta_{*}))\int_{0}^{1}y_{i}(1-y_{i})|\partial_{y_{i}}w_{i}|^{2}\,dy_{i}+\frac{C}{\delta_{*}t^{2}}\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.\end{aligned}$
(4.26)

$\bullet$ (Conclusion): Combining the estimates (4.24) and (4.26), we have

$\displaystyle\mathcal{B}_{1}$
$\displaystyle+\mathcal{B}_{2}-\widehat{\mathcal{G}_{2}}-\frac{3}{4}\mathcal{D}$
$\displaystyle\leq\sum_{i=1}^{2}\delta_{i}\alpha_{m}\left((1+C(\delta_{0}+\lambda+\varepsilon_{1}))\int_{0}^{1}|w_{i}|^{2}\,dy_{i}-\frac{3}{4}(1-C_{0}(\delta_{0}+\lambda+\varepsilon_{1}+\delta_{*}))\int_{0}^{1}y_{i}(1-y_{i})|\partial_{y_{i}}w_{i}|^{2}\,dy_{i}\right)$
$\displaystyle\quad+C\left(\sum_{i=1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)+\frac{1}{\delta_{*}t^{2}}\right)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Now, choose $\delta_{*}$ as

$\delta_{*}=\frac{1}{12C_{0}},$

which together with the smallness of $\delta_{0},\lambda,\varepsilon_{1}$
yields

$C_{0}(\delta_{0}+\lambda+\varepsilon_{1}+\delta_{*})<\frac{1}{6},$

and

$\displaystyle\mathcal{B}_{1}$
$\displaystyle+\mathcal{B}_{2}-\widehat{\mathcal{G}_{2}}-\frac{3}{4}\mathcal{D}$
$\displaystyle\leq\sum_{i=1}^{2}\delta_{i}\alpha_{m}\left(\frac{9}{8}\int_{0}^{1}|w_{i}|^{2}\,dy_{i}-\frac{5}{8}\int_{0}^{1}y_{i}(1-y_{i})|\partial_{y_{i}}w_{i}|^{2}\,dy_{i}\right)$
$\displaystyle\quad+C\left(\sum_{i=1}^{2}\delta_{i}\exp(-C\delta_{i}t)+\frac{1}{t^{2}}\right)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Then, using Lemma 2.1 with the identity:

$\int_{0}^{1}|w-\bar{w}|^{2}dy=\int_{0}^{1}w^{2}dy-{\bar{w}}^{2},\qquad\bar{w}:=\int_{0}^{1}wdy,$

we have

$\displaystyle\mathcal{B}_{1}+\mathcal{B}_{2}-\widehat{\mathcal{G}_{2}}-\frac{3}{4}\mathcal{D}$
$\displaystyle\leq\sum_{i=1}^{2}\left[-\frac{\delta_{i}\alpha_{m}}{8}\int_{0}^{1}|w_{i}|^{2}\,dy_{i}+\frac{5\delta_{i}\alpha_{m}}{4}\left(\int_{0}^{1}w_{i}\,dy_{i}\right)^{2}\right]$
$\displaystyle\quad+C\left(\sum_{i=1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)+\frac{1}{t^{2}}\right)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Finally, using (4.21) with the choice $M=\frac{5}{4}\sigma_{m}^{4}\alpha_{m}$,
we have

$\displaystyle\begin{aligned}
&-\sum_{i=1}^{2}\frac{\delta_{i}}{2M}|\dot{X}_{i}|^{2}+\mathcal{B}_{1}+\mathcal{B}_{2}-\mathcal{G}_{2}-\frac{3}{4}\mathcal{D}\\\
&\leq\sum_{i=1}^{2}\Bigg{[}-\frac{\delta_{i}\alpha_{m}}{16}\int_{0}^{1}|w_{i}|^{2}\,dy_{i}+\sigma_{i}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\frac{1+\gamma}{3\gamma^{2}}p(\widetilde{v})^{-\frac{1}{\gamma}-2}(p(v)-p(\widetilde{v}))^{3}\,dx\\\
&\hskip
56.9055pt-\frac{\sigma_{i}}{2\gamma}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left(p(\widetilde{v})^{-\frac{1}{\gamma}-1}-p(\widetilde{v}^{X_{i}}_{i})^{-\frac{1}{\gamma}-1}\right)|p(v)-p(\widetilde{v})|^{2}\,dx\Bigg{]}\\\
&\quad+C\left(\sum_{i=1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)+\frac{1}{t^{2}}\right)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx,\end{aligned}$

which implies

$\displaystyle\begin{aligned}
\mathcal{R}_{1}&\leq\sum_{i=1}^{2}\Bigg{[}-C_{1}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}\,dx+C\int_{\mathbb{R}}|(a_{i})^{X_{i}}_{x}||p(v)-p(\widetilde{v})|^{3}\,dx\\\
&\hskip
42.67912pt+C\int_{\mathbb{R}}|(a_{i})^{X_{i}}_{x}||\widetilde{v}-\widetilde{v}_{i}^{X_{i}}||p(v)-p(\widetilde{v})|^{2}\,dx\Bigg{]}\\\
&\quad+C\left(\sum_{i=1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)+\frac{1}{t^{2}}\right)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.\end{aligned}$
(4.27)

###  4.7. Estimate of the remaining terms

We substitute (4.27) to (LABEL:est) and use the Young’s inequality

$\sum_{i=1}^{2}\left(\dot{X}_{i}\sum_{j=3}^{6}Y_{ij}\right)\leq\sum_{i=1}^{2}\frac{\delta_{i}}{4M}|\dot{X}_{i}|^{2}+\sum_{i=1}^{2}\frac{C}{\delta_{i}}\sum_{j=3}^{6}|Y_{ij}|^{2}$
(4.28)

to have

$\displaystyle\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx\\\ &\leq-
C_{1}\mathcal{G}^{S}+\mathcal{K}_{1}+\mathcal{K}_{2}\\\
&\quad+C\left(\sum_{i=1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)+\frac{1}{t^{2}}\right)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx\\\
&\quad-\sum_{i=1}^{2}\frac{\delta_{i}}{4M}|\dot{X}_{i}|^{2}+\sum_{i=1}^{2}\frac{C}{\delta_{i}}\sum_{j=3}^{6}|Y_{ij}|^{2}+\sum_{i=3}^{5}\mathcal{B}_{i}+\mathcal{S}_{1}+\mathcal{S}_{2}-\mathcal{G}_{1}-\frac{1}{4}\mathcal{D},\end{aligned}$
(4.29)

where

$\displaystyle\mathcal{G}^{S}:=\sum_{i=1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}dx$
$\displaystyle\mathcal{K}_{1}:=\sum_{i=1}^{2}\underbrace{C\int_{\mathbb{R}}|(a_{i})^{X_{i}}_{x}||p(v)-p(\widetilde{v})|^{3}dx}_{=:\mathcal{K}_{i1}}$
$\displaystyle\mathcal{K}_{2}:=\sum_{i=1}^{2}\underbrace{C\int_{\mathbb{R}}|(a_{i})^{X_{i}}_{x}||\widetilde{v}-\widetilde{v}_{i}^{X_{i}}||p(v)-p(\widetilde{v})|^{2}\,dx}_{=:\mathcal{K}_{i2}}$

In what follows, to control the remaining terms, we will use the good terms
$\mathcal{G}_{1}$,$\mathcal{G}^{S}$ and the diffusion term $\mathcal{D}$.

$\bullet$ (Estimate of $\mathcal{K}_{1}$): To control the term
$\mathcal{K}_{i1}$ by the above good terms, we localize it via $\phi_{i}$ as

$\mathcal{K}_{i1}\leq
C\frac{\lambda}{\delta_{i}}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|\phi_{i}|p(v)-p(\widetilde{v})|^{3}\,dx+C\frac{\lambda}{\delta_{i}}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|(1-\phi_{i})|p(v)-p(\widetilde{v})|^{3}\,dx.$
(4.30)

Using the Gagliardo-Nirenberg interpolation inequality and
$\delta_{i}\ll\lambda\leq C\sqrt{\delta_{i}}$, the first term is controlled by
the good terms as

$\displaystyle
C\frac{\lambda}{\delta_{i}}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|\phi_{i}|p(v)-p(\widetilde{v})|^{3}\,dx$
$\displaystyle\leq C\frac{\lambda}{\delta_{i}}\left\lVert
p(v)-p(\widetilde{v})\right\rVert^{2}_{L^{\infty}(\mathbb{R})}\sqrt{\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}\,dx}\sqrt{\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|\,dx}$
$\displaystyle\leq
C\frac{\lambda}{\sqrt{\delta_{i}}}\left\lVert\partial_{x}(p(v)-p(\widetilde{v}))\right\rVert_{L^{2}(\mathbb{R})}\left\lVert
p(v)-p(\widetilde{v})\right\rVert_{L^{2}(\mathbb{R})}\sqrt{\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}\,dx}$
$\displaystyle\leq
C\varepsilon_{1}\left\lVert\partial_{x}(p(v)-p(\widetilde{v}))\right\rVert_{L^{2}(\mathbb{R})}\sqrt{\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}\,dx}$
$\displaystyle\leq
C\varepsilon_{1}\left\lVert\partial_{x}(p(v)-p(\widetilde{v}))\right\rVert^{2}_{L^{2}(\mathbb{R})}+C\varepsilon_{1}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}\,dx\leq\frac{1}{48}(\mathcal{D}+C_{1}\mathcal{G}^{S})$

Using Lemma 3.3, the second term in (4.30) is estimated as

$\displaystyle
C\frac{\lambda}{\delta_{i}}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|(1-\phi_{i})|p(v)-p(\widetilde{v})|^{3}\,dx$
$\displaystyle\leq
C\varepsilon_{1}\lambda\delta_{i}\exp(-C\delta_{i}t)\int_{\mathbb{R}}|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq
C\varepsilon_{1}\lambda\delta_{i}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Combining the estimates for $i=1,2$, we derive

$\displaystyle\mathcal{K}_{1}\leq\frac{1}{24}(\mathcal{D}+C_{1}\mathcal{G}^{S})+C\sum_{i=1}^{2}\varepsilon_{1}\lambda\delta_{i}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

$\bullet$ (Estimate of $\mathcal{K}_{2}$): To estimate $\mathcal{K}_{2}$, we
use Lemma 3.2 to obtain

$\displaystyle\mathcal{K}_{i2}$
$\displaystyle=C\int_{\mathbb{R}}|(a_{i})^{X_{i}}_{x}||\widetilde{v}-\widetilde{v}^{X_{i}}_{i}||p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq
C\frac{\lambda}{\delta_{i}}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\widetilde{v}-\widetilde{v}^{X_{i}}_{i}||p(v)-p(\widetilde{v})|^{2}dx$
$\displaystyle\leq
C\lambda\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx,$

which completes the estimate on $\mathcal{K}_{2}$:

$\mathcal{K}_{2}\leq
C\lambda\delta_{1}\delta_{2}\exp(-C\min\\{\delta_{1},\delta_{2}\\}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

$\bullet$ (Estimate of $\frac{C}{\delta_{i}}|Y_{ij}|^{2}$ for
$i=1,2,j=3,\ldots,6$): Using (4.2)2 and (4.6), we estimate $Y_{i3}$ as

$\displaystyle|Y_{i3}|$
$\displaystyle=\left|\frac{1}{\sigma_{i}}\int_{\mathbb{R}}ap(\widetilde{v}_{i}^{X_{i}})_{x}\left(h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right)\,dx\right|$
$\displaystyle\leq
C\frac{\delta_{i}}{\lambda}\int_{\mathbb{R}}\left|(a_{i})^{X_{i}}_{x}\right|\left|h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right|\,dx\leq
C\frac{\delta_{i}}{\sqrt{\lambda}}\sqrt{\mathcal{G}_{1}},$

which gives

$\frac{C}{\delta_{i}}|Y_{i3}|^{2}\leq\frac{C\delta_{i}}{\lambda}\mathcal{G}_{1}.$

Using Lemma 3.2, we control $Y_{i4}$ as

$\displaystyle|Y_{i4}|$ $\displaystyle\leq
C\int_{\mathbb{R}}|\widetilde{v}-\widetilde{v}^{X_{i}}_{i}||(\widetilde{v}_{i})^{X_{i}}_{x}||p(v)-p(\widetilde{v})|\,dx$
$\displaystyle\leq
C\sqrt{\int_{\mathbb{R}}(|\widetilde{v}-\widetilde{v}^{X_{1}}_{1}||(\widetilde{v}_{1})^{X_{1}}_{x}|)^{2}\,dx}\sqrt{\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx}$
$\displaystyle\leq
C\sqrt{\delta_{i}}\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})t)\sqrt{\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx},$

which yields

$\frac{C}{\delta_{i}}|Y_{i4}|^{2}\leq
C\delta_{1}^{2}\delta_{2}^{2}\exp(-C\min(\delta_{1},\delta_{2})t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

For $Y_{i5}$, we first estimate $h-\widetilde{h}$ in terms of
$u-\widetilde{u}$ and $v-\widetilde{v}$ as follows. Using
$\widetilde{v}_{x}=(\widetilde{v}_{1})^{X_{1}}_{x}+(\widetilde{v}_{2})^{X_{2}}_{x}$
and $C^{-1}\leq v,\widetilde{v}^{X_{1}}_{1},\widetilde{v}^{X_{2}}_{2}\leq C$,
we have

$\displaystyle\begin{aligned} |h-\widetilde{h}|&\leq|u-\widetilde{u}|+|(\ln
v)_{x}+(\ln\widetilde{v}^{X_{1}}_{1})_{x}-(\ln\widetilde{v}^{X_{2}}_{2})_{x}|\\\
&=|u-\widetilde{u}|+\left|\frac{v_{x}}{v}-\frac{(\widetilde{v}_{1})^{X_{1}}_{x}}{\widetilde{v}^{X_{1}}_{1}}-\frac{(\widetilde{v}_{2})^{X_{2}}_{x}}{\widetilde{v}^{X_{2}}_{2}}\right|\\\
&\leq|u-\widetilde{u}|+\left|\frac{v_{x}-\widetilde{v}_{x}}{v}\right|+\left|\left(\frac{1}{v}-\frac{1}{\widetilde{v}}\right)\widetilde{v}_{x}\right|+\left|\left(\frac{1}{\widetilde{v}}-\frac{1}{\widetilde{v}^{X_{1}}_{1}}\right)(\widetilde{v}_{1})^{X_{1}}_{x}\right|+\left|\left(\frac{1}{\widetilde{v}}-\frac{1}{\widetilde{v}^{X_{2}}_{2}}\right)(\widetilde{v}_{2})^{X_{2}}_{x}\right|\\\
&\leq|u-\widetilde{u}|+C(|(v-\widetilde{v})_{x}|+|v-\widetilde{v}||\widetilde{v}_{x}|+|\widetilde{v}-\widetilde{v}^{X_{1}}_{1}||(\widetilde{v}_{1})^{X_{1}}_{x}|+|\widetilde{v}-\widetilde{v}^{X_{2}}_{2}||(\widetilde{v}_{2})^{X_{2}}_{x}|).\end{aligned}$
(4.31)

Then, Lemma 3.2 implies

$\displaystyle\lVert h-\widetilde{h}\rVert_{L^{2}(\mathbb{R})}$
$\displaystyle\leq C\Bigg{(}\left\lVert
u-\widetilde{u}\right\rVert_{L^{2}(\mathbb{R})}+\left\lVert
v-\widetilde{v}\right\rVert_{H^{1}(\mathbb{R})}$ $\displaystyle\hskip
42.67912pt+\sqrt{\int_{\mathbb{R}}(|\widetilde{v}-\widetilde{v}^{X_{1}}_{1}||(\widetilde{v}_{1})^{X_{1}}_{x}|)^{2}\,dx}+\sqrt{\int_{\mathbb{R}}(|\widetilde{v}-\widetilde{v}^{X_{2}}_{2}||(\widetilde{v}_{2})^{X_{2}}_{x}|)^{2}\,dx}\Bigg{)}$
$\displaystyle\leq C\left(\left\lVert
u-\widetilde{u}\right\rVert_{L^{2}(\mathbb{R})}+\left\lVert
v-\widetilde{v}\right\rVert_{H^{1}(\mathbb{R})}+C\delta_{1}^{3/2}\delta_{2}+C\delta_{1}\delta_{2}^{3/2}\right),$

and therefore,

$\lVert h-\widetilde{h}\rVert_{L^{\infty}(0,T;L^{2}(\mathbb{R}))}\leq
C(\varepsilon_{1}+C\delta_{1}^{3/2}\delta_{2}+C\delta_{1}\delta_{2}^{3/2}).$
(4.32)

Hence, we estimate $Y_{i5}$ as

$\displaystyle|Y_{i5}|^{2}$ $\displaystyle\leq
C\left(\frac{\sigma_{i}}{2}\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left|h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right|^{2}\,dx\right)\left(\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\left|h-\widetilde{h}+\frac{p(v)-p(\widetilde{v})}{\sigma_{i}}\right|^{2}\,dx\right)$
$\displaystyle\leq
C\mathcal{G}_{1i}\left\lVert(a_{i})^{X_{i}}_{x}\right\rVert_{L^{\infty}}\left(\lVert{h-\widetilde{h}\rVert}_{L^{\infty}(0,T;L^{2}(\mathbb{R}))}+\lVert{v-\widetilde{v}\rVert}_{L^{\infty}(0,T;L^{2}(\mathbb{R}))}\right)^{2}$
$\displaystyle\leq
C\lambda\delta_{i}(\varepsilon_{1}+C\delta_{1}\delta_{2})^{2}\mathcal{G}_{1},$

which yields

$\frac{C}{\delta_{i}}|Y_{i5}|^{2}\leq
C\lambda(\varepsilon_{1}+C\delta_{1}\delta_{2})^{2}\mathcal{G}_{1}.$

Finally, using Lemma 2.3 (3), $\lambda\leq C\sqrt{\delta_{i}}$ and Lemma 3.3,
we estimate $Y_{i6}$ as

$\displaystyle\frac{C}{\delta_{i}}|Y_{i6}|^{2}$
$\displaystyle\leq\frac{C}{\delta_{i}}\left(\int_{\mathbb{R}}|(a_{i})^{X_{i}}_{x}||p(v)-p(\widetilde{v})|^{2}\,dx\right)^{2}\leq\frac{C\lambda^{2}}{\delta_{i}^{3}}\left(\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||p(v)-p(\widetilde{v})|^{2}\,dx\right)^{2}$
$\displaystyle\leq\frac{C\lambda^{2}}{\delta_{i}}\left\lVert
p(v)-p(\widetilde{v})\right\rVert^{2}_{L^{2}(\mathbb{R})}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq
C\varepsilon_{1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle=C\varepsilon_{1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}\,dx+C\varepsilon_{1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|(1-\phi_{i}^{2})|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{C_{1}}{48}\mathcal{G}^{S}_{i}+C\varepsilon_{1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|(1-\phi_{i})|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{C_{1}}{48}\mathcal{G}^{S}_{i}+C\varepsilon_{1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Combining all the estimates for $Y_{ij}$ and use the smallness of the
parameters, we conclude that

$\displaystyle\sum_{i=1}^{2}\left(\frac{C}{\delta_{i}}\sum_{j=3}^{6}|Y_{ij}|^{2}\right)$
$\displaystyle\leq\sum_{i=1}^{2}\bigg{(}\frac{C\delta_{i}}{\lambda}\mathcal{G}_{1}+C\lambda(\varepsilon_{1}+C\delta_{1}\delta_{2})^{2}\mathcal{G}_{1}$
$\displaystyle\quad+\frac{C_{1}}{48}\mathcal{G}^{S}+C\Big{(}\delta_{1}^{2}\delta_{2}^{2}\exp(-C\min(\delta_{1},\delta_{2})t)+\varepsilon_{1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)\Big{)}\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx$
$\displaystyle\leq\frac{1}{4}\mathcal{G}_{1}+\frac{C_{1}}{24}\mathcal{G}^{S}$
$\displaystyle\quad+C\Big{(}\delta_{1}^{2}\delta_{2}^{2}\exp(-C\min(\delta_{1},\delta_{2})t)+\varepsilon_{1}^{2}\sum_{i=1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)\Big{)}\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

$\bullet$ (Estimate of $\mathcal{B}_{3}$):  Recall
$\mathcal{B}_{3}:=\mathcal{B}_{13}+\mathcal{B}_{23}$ where

$\mathcal{B}_{i3}:=-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}\frac{p(v)-p(\widetilde{v})}{\gamma
p(v)}\partial_{x}(p(v)-p(\widetilde{v}))\,dx.$

As done before, using Lemma 3.3, we estimate $\mathcal{B}_{i3}$ as

$\displaystyle|\mathcal{B}_{i3}|$
$\displaystyle\leq\frac{1}{48}\mathcal{D}+C\left(\frac{\lambda}{\delta_{i}}\right)^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|^{2}|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{1}{48}\mathcal{D}+C\lambda^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\phi_{i}(p(v)-p(\widetilde{v}))|^{2}\,dx+C\lambda^{2}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|(1-\phi_{i})|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{1}{48}(\mathcal{D}+C_{1}\mathcal{G}^{S})+C\lambda^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Therefore, we have

$\displaystyle\mathcal{B}_{3}\leq\frac{1}{24}(\mathcal{D}+C_{1}\mathcal{G}^{S})+C\lambda^{2}\sum_{i=1}^{2}\delta_{i}^{2}\exp(-C\delta_{i}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

$\bullet$ (Estimate of $\mathcal{B}_{4}$):  Recall
$\mathcal{B}_{4}:=\mathcal{B}_{14}+\mathcal{B}_{24}$ where

$\mathcal{B}_{i4}:=-\int_{\mathbb{R}}(a_{i})^{X_{i}}_{x}(p(v)-p(\widetilde{v}))^{2}\frac{\partial_{x}p(\widetilde{v})}{\gamma
p(v)p(\widetilde{v})}\,dx.$

Likewise, we estimate $\mathcal{B}_{i4}$ by using Lemma 3.3 with (4.6) and
$|\partial_{x}p(\widetilde{v})|\leq
C\left(|(\widetilde{v}_{1})^{X_{1}}_{x}|+|(\widetilde{v}_{2})^{X_{2}}_{x}|\right)$
as

$\displaystyle|\mathcal{B}_{i4}|$ $\displaystyle\leq
C\lambda\delta_{i}\sum_{j=1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{j})^{X_{j}}_{x}||p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq
C\lambda\delta_{i}\mathcal{G}^{S}+C\lambda\delta_{i}\sum_{j=1}^{2}\int_{\mathbb{R}}|(\widetilde{v}_{j})^{X_{j}}_{x}||1-\phi_{j}||p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{C_{1}}{12}\mathcal{G}^{S}+C\lambda\delta_{i}\sum_{j=1}^{2}\delta_{j}^{2}\exp(-C\delta_{j}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

Therefore, we bound $\mathcal{B}_{4}$ as

$\displaystyle|\mathcal{B}_{4}|\leq\frac{C_{1}}{6}\mathcal{G}^{S}+C\sum_{j=1}^{2}\delta_{j}^{2}\exp(-C\delta_{j}t)\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

$\bullet$(Estimate for $\mathcal{B}_{5}$):  As in $\mathcal{B}_{i3}$, we have

$\displaystyle|\mathcal{B}_{5}|$
$\displaystyle\leq\frac{1}{48}\mathcal{D}+C\int_{\mathbb{R}}\left(|(\widetilde{v}_{1})^{X_{1}}_{x}|^{2}+|(\widetilde{v}_{2})^{X_{2}}_{x}|^{2}\right)|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{1}{48}\mathcal{D}+\frac{C_{1}}{12}\mathcal{G}^{S}+C(\delta_{1}^{2}\exp(-C\delta_{1}t)+\delta_{2}^{2}\exp(-C\delta_{2}t))\int_{\mathbb{R}}\eta(U|\widetilde{U})\,dx.$

$\bullet$(Estimate for the interaction terms $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$):  We first estimate $E_{1}$ and $E_{2}$. We use
$(\widetilde{v})_{x}=(\widetilde{v}_{1})^{X_{1}}_{x}+(\widetilde{v}_{2})^{X_{2}}_{x}$
to estimate $E_{1}$ as

$\displaystyle E_{1}$
$\displaystyle=\partial_{x}\left(\frac{(\widetilde{v})_{x}}{\widetilde{v}}-\frac{(\widetilde{v}_{1})^{X_{1}}_{x}}{\widetilde{v}^{X_{1}}_{1}}-\frac{(\widetilde{v}_{2})^{X_{2}}_{x}}{\widetilde{v}^{X_{2}}_{2}}\right)=\partial_{x}\left(\left(\frac{1}{\widetilde{v}}-\frac{1}{\widetilde{v}_{1}^{X_{1}}}\right)(\widetilde{v}_{1})^{X_{1}}_{x}+\left(\frac{1}{\widetilde{v}}-\frac{1}{\widetilde{v}^{X_{2}}_{2}}\right)(\widetilde{v}_{2})^{X_{2}}_{x}\right)$
$\displaystyle=\left(\frac{1}{\widetilde{v}}-\frac{1}{\widetilde{v}^{X_{1}}_{1}}\right)(\widetilde{v}_{1})^{X_{1}}_{xx}+\left(\frac{1}{\widetilde{v}}-\frac{1}{\widetilde{v}^{X_{2}}_{2}}\right)(\widetilde{v}_{2})^{X_{2}}_{xx}$
$\displaystyle\quad+\left(\frac{1}{(\widetilde{v}^{X_{1}}_{1})^{2}}-\frac{1}{(\widetilde{v})^{2}}\right)|(\widetilde{v}_{1})^{X_{1}}_{x}|^{2}+\left(\frac{1}{(\widetilde{v}^{X_{2}}_{2})^{2}}-\frac{1}{(\widetilde{v})^{2}}\right)|(\widetilde{v}_{2})^{X_{2}}_{x}|^{2}-2\frac{(\widetilde{v}_{1})^{X_{1}}_{x}(\widetilde{v}_{2})^{X_{2}}_{x}}{(\widetilde{v})^{2}}.$

Therefore, thanks to (2.3), $E_{1}$ is bounded as

$\displaystyle|E_{1}|$ $\displaystyle\leq
C\delta_{1}|(\widetilde{v}_{1})^{X_{1}}_{x}||\widetilde{v}-\widetilde{v}^{X_{1}}_{1}|+C\delta_{2}|(\widetilde{v}_{2})^{X_{2}}_{x}||\widetilde{v}-\widetilde{v}^{X_{2}}_{2}|+C|(\widetilde{v}_{1})^{X_{1}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}|.$

Similarly, we estimate $E_{2}$ as

$\displaystyle|E_{2}|$
$\displaystyle\leq|p^{\prime}(\widetilde{v})-p^{\prime}(\widetilde{v}^{X_{1}}_{1})||(\widetilde{v}_{1})^{X_{1}}_{x}|+|p^{\prime}(\widetilde{v})-p^{\prime}(\widetilde{v}^{X_{2}}_{2})||(\widetilde{v}_{2})^{X_{2}}_{x}|$
$\displaystyle\leq
C|(\widetilde{v}_{1})^{X_{1}}_{x}||\widetilde{v}-\widetilde{v}_{1}^{X_{1}}|+C|(\widetilde{v}_{2})^{X_{2}}_{x}||\widetilde{v}-\widetilde{v}_{2}^{X_{2}}|.$

Therefore, the above estimates yield

$\displaystyle|\mathcal{S}_{1}|+|\mathcal{S}_{2}|$ $\displaystyle\leq
C\sum_{i=1}^{2}\bigg{(}\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}||\widetilde{v}-\widetilde{v}^{X_{i}}_{i}|\big{(}|p(v)-p(\widetilde{v})|+|h-\widetilde{h}|\big{)}\,dx\bigg{)}$
$\displaystyle\quad+C\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}||p(v)-p(\widetilde{v})|\,dx.$

Using (4.32), Lemma 3.2 and the assumption (3.7) with (4.8), we have

$\displaystyle|\mathcal{S}_{1}|+|\mathcal{S}_{2}|$ $\displaystyle\leq
C\sum_{i=1}^{2}\bigg{(}(\|v-\widetilde{v}\|_{L^{\infty}(0,T;L^{2}(\mathbb{R}))}+\lVert
h-\widetilde{h}\rVert_{L^{\infty}(0,T;L^{2}(\mathbb{R}))})\sqrt{\int_{\mathbb{R}}|(\widetilde{v}_{i})^{X_{i}}_{x}|^{2}|\widetilde{v}-\widetilde{v}^{X_{i}}_{i}|^{2}dx}\bigg{)}$
$\displaystyle\quad+C\varepsilon_{1}\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}||(\widetilde{v}_{2})^{X_{2}}_{x}|\,dx$
$\displaystyle\leq
C\varepsilon_{1}\delta_{1}\delta_{2}(\delta_{1}^{1/2}+\delta_{2}^{1/2})\exp(-C\min(\delta_{1},\delta_{2})t)+C\delta_{1}^{2}\delta_{2}^{2}(\delta_{1}+\delta_{2})\exp(-C\min(\delta_{1},\delta_{2})t)$
$\displaystyle\quad+C\varepsilon_{1}\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})t).$

Hence we finally estimate $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ as

$|\mathcal{S}_{1}|+|\mathcal{S}_{2}|\leq
C\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})t).$ (4.33)

###  4.8. Estimate in small time

Note that the estimate (4.27) on $\mathcal{R}_{1}$ contains the coefficient
$\frac{1}{t^{2}}$, which is not integrable near $t=0$. Thus, in order to get
the desired result, we would find a rougher estimate in a short time. To this
end, we return to the previous right-hand side $\mathcal{R}$ in (LABEL:est):

$\displaystyle\mathcal{R}=-\sum_{i=1}^{2}\frac{\delta_{i}}{M}|\dot{X}_{i}|^{2}+\sum_{i=1}^{2}\left(\dot{X_{i}}\sum_{j=3}^{6}Y_{ij}\right)+\sum_{i=1}^{5}\mathcal{B}_{i}+\mathcal{S}_{1}+\mathcal{S}_{2}-\mathcal{G}_{1}-\mathcal{G}_{2}-\mathcal{D}.$

By the Young’s inequality (4.28), we first have

$\displaystyle\mathcal{R}+\sum_{i=1}^{2}\frac{\delta_{i}}{4M}|\dot{X}_{i}|^{2}+\mathcal{G}_{1}+\mathcal{D}+\mathcal{G}^{S}\leq\sum_{i=1}^{2}\left(\frac{C}{\delta_{i}}\sum_{j=3}^{6}|Y_{ij}|^{2}\right)+\sum_{i=1}^{5}\mathcal{B}_{i}+\mathcal{S}_{1}+\mathcal{S}_{2}+\mathcal{G}^{S}.$

Using (3.7) and Lemma 2.2 with (4.6), we have

$\displaystyle\sum_{j=3}^{6}|Y_{ij}|$ $\displaystyle\leq
C\left\lVert(\widetilde{v}_{i})^{X_{i}}_{x}\right\rVert_{L^{2}(\mathbb{R})}(\|h-\widetilde{h}\|_{L^{2}(\mathbb{R})}+\left\lVert
p(v)-p(\widetilde{v})\right\rVert_{L^{2}(\mathbb{R})})$
$\displaystyle+\left\lVert(a_{i})^{X_{i}}_{x}\right\rVert_{L^{\infty}(\mathbb{R})}(\|h-\widetilde{h}\|_{L^{2}(\mathbb{R})}^{2}+\left\lVert
p(v)-p(\widetilde{v})\right\rVert_{L^{2}(\mathbb{R})}^{2})$ $\displaystyle\leq
C\delta_{i}\varepsilon_{1},$

which yields

$\sum_{i=1}^{2}\left(\frac{C}{\delta_{i}}\sum_{j=3}^{6}|Y_{ij}|^{2}\right)\leq
C\varepsilon_{1}^{2}\sum_{i=1}^{2}\delta_{i}.$

Likewise, we have

$\sum_{i=1}^{5}\mathcal{B}_{i}\leq
C\sum_{i=1}^{2}\Big{(}\left\lVert(a_{i})^{X_{i}}_{x}\right\rVert_{L^{\infty}(\mathbb{R})}+\left\lVert(\widetilde{v}_{i})_{x}\right\rVert_{L^{\infty}(\mathbb{R})}\Big{)}\left\lVert
v-\widetilde{v}\right\rVert_{H^{1}(\mathbb{R})}^{2}\leq
C\varepsilon_{1}^{2}\sum_{i=1}^{2}\delta_{i},$

and

$\mathcal{G}^{S}\leq
C\sum_{i=1}^{2}\left\lVert(\widetilde{v}_{i})^{X_{i}}_{x}\right\rVert_{L^{\infty}(\mathbb{R})}\left\lVert
p(v)-p(\widetilde{v})\right\rVert_{L^{2}(\mathbb{R})}^{2}\leq
C\varepsilon_{1}^{2}\sum_{i=1}^{2}\delta_{i}^{2}.$

Hence, the above estimates and (4.33) provides a rough bound: for any
$\delta_{1},\delta_{2}\in(0,\delta_{0})$,

$\mathcal{R}+\sum_{i=1}^{2}\frac{\delta_{i}}{4M}|\dot{X}_{i}|^{2}+\mathcal{G}_{1}+\mathcal{D}+\mathcal{G}^{S}\leq
C\delta_{0},\quad t>0.$ (4.34)

###  4.9. Proof of Lemma 4.1

We here complete the proof of Lemma 4.1. First of all, from (4.34) with
(LABEL:est), we have a rough estimate for a short time $t\leq 1$ as follows:

$\displaystyle\frac{d}{dt}\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx$
$\displaystyle\leq C\delta_{0},$

which implies

$\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx\Bigg{|}_{t=1}+\int_{0}^{1}\bigg{(}\sum_{i=1}^{2}\frac{\delta_{i}}{4M}|\dot{X}_{i}|^{2}+\mathcal{G}_{1}+\mathcal{D}+\mathcal{G}^{S}\bigg{)}dt\leq\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx\Bigg{|}_{t=0}+C\delta_{0}.$
(4.35)

On the other hand, for $t\geq 1$, we combine all the estimates in Section 4.6
and Section 4.7 to derive

$\displaystyle\frac{d}{dt}$
$\displaystyle\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx+\sum_{i=1}^{2}\frac{\delta_{i}}{4M}|\dot{X}_{i}|^{2}+\frac{1}{2}\mathcal{G}_{1}+\frac{C_{1}}{2}\mathcal{G}^{S}+\frac{1}{8}\mathcal{D}$
$\displaystyle\leq
C\left(\sum_{i=1}^{2}\delta_{i}\exp(-C\delta_{i}t)+\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})t)+\frac{1}{t^{2}}\right)\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx$
$\displaystyle\quad+C\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})t).$

Hence, we use Grönwall inequality to conclude that for all $t\geq 1$,

$\displaystyle\begin{aligned}
\int_{\mathbb{R}}a&\eta(U(t,x)|\widetilde{U}(t,x))\,dx+\int_{1}^{t}\left(\sum_{i=1}^{2}\frac{\delta_{i}}{4M}|\dot{X}_{i}|^{2}+\frac{1}{2}\mathcal{G}_{1}+\frac{C_{1}}{2}\mathcal{G}^{S}+\frac{1}{8}\mathcal{D}\right)\,ds\\\
&\leq\left(\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx\Bigg{|}_{t=1}+\frac{C\delta_{1}\delta_{2}}{\min(\delta_{1},\delta_{2})}\right)\\\
&\quad\times\exp\left(\int_{1}^{t}\sum_{i=1}^{2}\delta_{i}\exp(-C\delta_{i}s)+\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})s)+\frac{1}{s^{2}}\,ds\right)\\\
&\leq
C\int_{\mathbb{R}}a\eta(U|\widetilde{U})\,dx\Bigg{|}_{t=1}+C\max(\delta_{1},\delta_{2}).\end{aligned}$
(4.36)

Finally, we combine the estimates (4.35) and (4.36) to conclude that

$\displaystyle\int_{\mathbb{R}}a$
$\displaystyle\eta(U(t,x)|\widetilde{U}(t,x))\,dx+\int_{0}^{t}\left(\sum_{i=1}^{2}\delta_{i}|\dot{X}_{i}|^{2}+\mathcal{G}_{1}+\mathcal{G}^{S}+\mathcal{D}\right)\,ds$
$\displaystyle\leq
C\int_{\mathbb{R}}a(0,x)\eta(U_{0}(x)|\widetilde{U}(0,x))\,dx+C\delta_{0},$

which together with $\frac{1}{2}\leq a\leq 1$, $D(U)\leq C\mathcal{D}(U)$ and
$G_{1}(U)\leq C\mathcal{G}_{1}(U)$, completes the proof of Lemma 4.1.

##  5\. Proof of Proposition 3.2

In this section, we complete the proof of Proposition 3.2.

###  5.1. Zeroth-order estimate

We first estimate the zeroth-order term of $\left\lVert
u-\widetilde{u}\right\rVert_{L^{2}(\mathbb{R})}$.

######  Lemma 5.1.

Under the hypotheses of Proposition 3.2, there exists a positive constant
$C>0$, that is independent of $\delta_{1},\delta_{2},\varepsilon_{1},T$, such
that for all $t\in(0,T]$,

$\displaystyle\left\lVert
v-\widetilde{v}\right\rVert_{H^{1}(\mathbb{R})}^{2}+\left\lVert
u-\widetilde{u}\right\rVert_{L^{2}(\mathbb{R})}^{2}+\int_{0}^{t}\sum_{i=1}^{2}\delta_{i}|\dot{X}_{i}|^{2}\,ds+\int_{0}^{t}\left(\mathcal{G}^{S}+D+D_{1}\right)\,ds$
(5.1) $\displaystyle\qquad\leq C\left(\left\lVert
v_{0}-\widetilde{v}(0,\cdot)\right\rVert_{H^{1}(\mathbb{R})}^{2}+\left\lVert
u_{0}-\widetilde{u}(0,\cdot)\right\rVert_{L^{2}(\mathbb{R})}^{2}\right)+C\delta_{0}^{1/2}$

where $\mathcal{G}^{S}$ and $D$ are the good terms of Lemma 4.1, and

${D}_{1}:=\int_{\mathbb{R}}|(u-\widetilde{u})_{x}|^{2}\,dx.$ (5.2)

###### Proof.

As in Section 4.3, the system (LABEL:eq:NS) can be written as:

$U_{t}+A(U)_{x}=\partial_{x}(M(U)\partial_{x}D\eta(U)),$

where

$U:=\begin{pmatrix}v\\\ u\end{pmatrix},\quad A(U):=\begin{pmatrix}-u\\\
p(v)\end{pmatrix},\quad\eta(U):=\frac{u^{2}}{2}+Q(v)=\frac{u^{2}}{2}+\frac{v^{-\gamma+1}}{\gamma-1},\quad
M(U):=\begin{pmatrix}0&0\\\ 0&\frac{1}{v}\end{pmatrix}.$

Then, the shifted composite wave $\widetilde{U}=(\widetilde{v},\widetilde{u})$
satisfies

$\partial_{t}\widetilde{U}+A(\widetilde{U})_{x}=\partial_{x}(M(\widetilde{U})\partial_{x}D\eta(\widetilde{U}))-\sum_{i=1}^{2}\dot{X}_{i}(\widetilde{U}_{i})^{X_{i}}_{x}+\begin{pmatrix}0\\\
E_{2}+E_{3}\end{pmatrix},$ (5.3)

where

$E_{2}:=p(\widetilde{v})_{x}-\sum_{i=1}^{2}p(\widetilde{v}^{X_{i}}_{i})_{x},\quad
E_{3}:=-\left(\frac{\widetilde{u}_{x}}{\widetilde{v}}\right)_{x}+\sum_{i=1}^{2}\left(\frac{(\widetilde{u}_{i})^{X_{i}}_{x}}{\widetilde{v}^{X_{i}}_{i}}\right)_{x}.$

Then, the relative entropy method (see e.g. [11, Lemma 5.1]) implies that

$\frac{d}{dt}\int_{\mathbb{R}}\eta(U(t,x)|\widetilde{U}(t,x))\,dx=\sum_{i=1}^{2}(\dot{X}_{i}(t)\mathcal{Y}_{i}(t))+\sum_{i=1}^{6}\mathcal{I}_{i}(U),$

where

$\displaystyle\mathcal{Y}_{i}$
$\displaystyle:=\int_{\mathbb{R}}(\widetilde{U}_{i})^{X_{i}}_{x}D^{2}\eta(\widetilde{U})(U-\widetilde{U})\,dx=-\int_{\mathbb{R}}p^{\prime}(\widetilde{v})(\widetilde{v}_{i})^{X_{i}}_{x}(v-\widetilde{v})\,dx+\int_{\mathbb{R}}(\widetilde{u}_{i})^{X_{i}}_{x}(u-\widetilde{u})\,dx=:\mathcal{Y}_{i1}+\mathcal{Y}_{i2},$
$\displaystyle\mathcal{I}_{1}$
$\displaystyle:=-\int_{\mathbb{R}}\partial_{x}G(U;\widetilde{U})\,dx=-\int_{\mathbb{R}}\partial_{x}\left((p-\widetilde{p})(u-\widetilde{u})\right)\,dx=0,$
$\displaystyle\mathcal{I}_{2}$
$\displaystyle:=-\int_{\mathbb{R}}(\partial_{x}D\eta(\widetilde{U}))A(U|\widetilde{U})\,dx=-\sum_{i=1}^{2}\int_{\mathbb{R}}(\widetilde{u}_{i})^{X_{i}}_{x}p(v|\widetilde{v})\,dx=:\mathcal{I}_{21}+\mathcal{I}_{22},$
$\displaystyle\mathcal{I}_{3}$
$\displaystyle:=\int_{\mathbb{R}}\left(D\eta(U)-D\eta(\widetilde{U})\right)\partial_{x}\left(M(U)\partial_{x}(D\eta(U)-D\eta(\widetilde{U}))\right)\,dx=-\underbrace{\int_{\mathbb{R}}\frac{1}{v}|\partial_{x}(u-\widetilde{u})|^{2}\,dx}_{=:\mathbf{D}_{1}},$
$\displaystyle\mathcal{I}_{4}$
$\displaystyle:=\int_{\mathbb{R}}\left(D\eta(U)-D\eta(\widetilde{U})\right)\partial_{x}\left((M(U)-M(\widetilde{U}))\partial_{x}D\eta(\widetilde{U})\right)\,dx=\int_{\mathbb{R}}(u-\widetilde{u})\left(\left(\frac{1}{v}-\frac{1}{\widetilde{v}}\right)\widetilde{u}_{x}\right)_{x}\,dx,$
$\displaystyle\mathcal{I}_{5}$
$\displaystyle:=\int_{\mathbb{R}}(D\eta)(U|\widetilde{U})\partial_{x}\left(M(\widetilde{U})\partial_{x}D\eta(\widetilde{U})\right)\,dx=\int_{\mathbb{R}}\begin{pmatrix}-p(v|\widetilde{v})\\\
0\end{pmatrix}\cdot\begin{pmatrix}0\\\
(\frac{\widetilde{u}_{x}}{\widetilde{v}})_{x}\end{pmatrix}\,dx=0,$
$\displaystyle\mathcal{I}_{6}$
$\displaystyle:=-\int_{\mathbb{R}}D^{2}\eta(\widetilde{U})(U-\widetilde{U})\begin{pmatrix}0\\\
E_{2}+E_{3}\end{pmatrix}\,dx=-\int_{\mathbb{R}}(u-\widetilde{u})(E_{2}+E_{3})\,dx$
$\displaystyle=-\int_{\mathbb{R}}(u-\widetilde{u})\left(\sum_{i=1}^{2}p(\widetilde{v}^{X_{i}}_{i})_{x}-p(\widetilde{v})_{x}+\sum_{i=1}^{2}\left(\frac{(\widetilde{u}_{i})^{X_{i}}_{x}}{\widetilde{v}^{X_{i}}_{i}}\right)_{x}-\left(\frac{\widetilde{u}_{x}}{\widetilde{v}}\right)_{x}\right)\,dx.$

In the following, we estimate each term above one by one.

$\bullet$ (Estimate of $\mathcal{Y}_{i1}$): We use $\phi_{1}+\phi_{2}=1$ to
have

$\mathcal{Y}_{11}=-\int_{\mathbb{R}}p^{\prime}(\widetilde{v})(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{1}(v-\widetilde{v})\,dx-\int_{\mathbb{R}}p^{\prime}(\widetilde{v})(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{2}(v-\widetilde{v})\,dx.$

Since

$\left|\int_{\mathbb{R}}p^{\prime}(\widetilde{v})(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{1}(v-\widetilde{v})\,dx\right|^{2}\leq
C\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\,dx\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}||\phi_{1}(v-\widetilde{v})|^{2}\,dx\leq
C\delta_{1}\mathcal{G}^{S},$

and (by Lemma 3.3)

$\left|\int_{\mathbb{R}}p^{\prime}(\widetilde{v})(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{2}(v-\widetilde{v})\,dx\right|^{2}\leq
C\int_{\mathbb{R}}\left|(\widetilde{v}_{1})^{X_{1}}_{x}\phi_{2}\right|^{2}dx\int_{\mathbb{R}}\left|v-\widetilde{v}\right|^{2}dx\leq
C\delta_{1}^{3}\exp(-C\delta_{1}t)\left\lVert
v-\widetilde{v}\right\rVert^{2}_{L^{2}(\mathbb{R})},$

we have

$\displaystyle|\dot{X}_{1}\mathcal{Y}_{11}|$
$\displaystyle\leq\frac{\delta_{1}}{4}|\dot{X}_{1}|^{2}+\frac{C}{\delta_{1}}|\mathcal{Y}_{11}|^{2}\leq\frac{\delta_{1}}{4}|\dot{X}_{1}|^{2}+C\mathcal{G}^{S}+C\delta_{1}^{2}\exp(-C\delta_{1}t)\left\lVert
v-\widetilde{v}\right\rVert^{2}_{L^{2}(\mathbb{R})}.$

Likewise, we estimate $\dot{X}_{2}\mathcal{Y}_{21}$ as

$|\dot{X}_{2}\mathcal{Y}_{21}|\leq\frac{\delta_{2}}{4}|\dot{X}_{2}|^{2}+C\mathcal{G}^{S}+C\delta_{2}^{2}\exp(-C\delta_{2}t)\left\lVert
v-\widetilde{v}\right\rVert^{2}_{L^{2}(\mathbb{R})}.$

$\bullet$ (Estimate of $\mathcal{Y}_{i2}$): We use a similar estimate as in
(4.31) to obtain

$\displaystyle|u-\widetilde{u}|\leq|h-\widetilde{h}|+C(|(v-\widetilde{v})_{x}|+|\widetilde{v}_{x}||v-\widetilde{v}|+|(\widetilde{v}_{1})^{X_{1}}_{x}||\widetilde{v}-\widetilde{v}^{X_{1}}_{1}|+|(\widetilde{v}_{2})^{X_{2}}_{x}||\widetilde{v}-\widetilde{v}^{X_{2}}_{2}|).$

On the other hand, since

$(p(v)-p(\widetilde{v}))_{x}=p^{\prime}(v)(v-\widetilde{v})_{x}+\widetilde{v}_{x}(p^{\prime}(v)-p^{\prime}(\widetilde{v})),$

we have

$|(v-\widetilde{v})_{x}|\leq
C|(p(v)-p(\widetilde{v}))_{x}|+C|\widetilde{v}_{x}||v-\widetilde{v}|.$

Then, using the localizations $\phi_{i}$, and then the good term $G_{1}$ of
Lemma 4.1, and Lemma 3.2 and Lemma 3.3 with (2.5), we have

$\displaystyle|\mathcal{Y}_{12}|$ $\displaystyle\leq
C\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\bigg{(}\left|h-\widetilde{h}-\frac{p(v)-p(\widetilde{v})}{\sigma_{1}}\right|+|p(v)-p(\widetilde{v})|$
$\displaystyle\hskip
85.35826pt+|(p(v)-p(v))_{x}|+|\widetilde{v}_{x}||v-\widetilde{v}|+|(\widetilde{v}_{1})^{X_{1}}_{x}||\widetilde{v}-\widetilde{v}^{X_{1}}_{1}|+|(\widetilde{v}_{2})^{X_{2}}_{x}||\widetilde{v}-\widetilde{v}^{X_{1}}_{2}|\bigg{)}\,dx$
$\displaystyle\leq
C\left(\frac{\delta_{1}}{\sqrt{\lambda}}\sqrt{{G}_{1}}+\sqrt{\delta_{1}}\sqrt{\mathcal{G}^{S}}+\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}||\phi_{2}(p(v)-p(\widetilde{v}))|\,dx+\delta_{1}\sqrt{D}+\delta_{1}\delta_{2}\exp(-C\min(\delta_{1},\delta_{2})t)\right).$

This yields

$\displaystyle|\dot{X_{1}}\mathcal{Y}_{12}|$
$\displaystyle\leq\frac{\delta_{1}}{4}|\dot{X}_{1}|^{2}+C\frac{\delta_{1}}{\lambda}{G}_{1}+C\mathcal{G}^{S}+C\delta_{1}D$
$\displaystyle\quad+C\delta_{1}^{2}\exp(-C\delta_{1}t)\left\lVert
v-\widetilde{v}\right\rVert^{2}_{L^{2}(\mathbb{R})}+C\delta_{1}\delta_{2}^{2}\exp(-C\min(\delta_{1},\delta_{2})t).$

A similar estimate holds for $|\dot{X}_{2}\mathcal{Y}_{22}|$.

$\bullet$ (Estimate of $\mathcal{I}_{2}$): We use the estimate (LABEL:est-
rel-3)1 and the localization, and then use Lemma 3.3 with (2.5) to obtain

$\displaystyle|\mathcal{I}_{21}|$ $\displaystyle\leq
C\mathcal{G}^{S}+C\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq
C\mathcal{G}^{S}+C\sqrt{\int_{\mathbb{R}}|(v_{1})^{X_{1}}_{x}\phi_{2}|^{2}\,dx}\sqrt{\int_{\mathbb{R}}|p(v)-p(\widetilde{v})|^{4}\,dx}$
$\displaystyle\leq
C\mathcal{G}^{S}+C\delta_{1}^{3/2}\exp(-C\delta_{1}t)\left\lVert
v-\widetilde{v}\right\rVert_{L^{\infty}(\mathbb{R})}\left\lVert
v-\widetilde{v}\right\rVert_{L^{2}(\mathbb{R})},$

and the same estimate holds for $\mathcal{I}_{22}$.

$\bullet$ (Estimate of $\mathcal{I}_{4}$): To control $\mathcal{I}_{4}$, we
will use the good term $\mathbf{D}_{1}$ generated by $\mathcal{I}_{3}$. We use
the localization, and then use Lemma 3.3 with (2.5) to derive

$\displaystyle|\mathcal{I}_{4}|$
$\displaystyle\leq\int_{\mathbb{R}}|(u-\widetilde{u})_{x}||v-\widetilde{v}|(|(\widetilde{u}_{1})^{X_{1}}_{x}|+|(\widetilde{u}_{2})^{X_{2}}_{x}|)\,dx$
$\displaystyle\leq\frac{1}{4}\mathbf{D}_{1}+C(\delta_{1}+\delta_{2})\mathcal{G}^{S}$
$\displaystyle\quad+C\delta_{1}\int_{\mathbb{R}}|(\widetilde{v}_{1})^{X_{1}}_{x}|\phi_{2}|p(v)-p(\widetilde{v})|^{2}\,dx+C\delta_{2}\int_{\mathbb{R}}|(\widetilde{v}_{2})^{X_{2}}_{x}|\phi_{1}|p(v)-p(\widetilde{v})|^{2}\,dx$
$\displaystyle\leq\frac{1}{4}\mathbf{D}_{1}+C(\delta_{1}+\delta_{2})\mathcal{G}^{S}+C\sum_{i=1}^{2}\delta_{i}^{5/2}\exp(-C\delta_{i}t)\left\lVert
v-\widetilde{v}\right\rVert_{L^{\infty}(\mathbb{R})}\left\lVert
v-\widetilde{v}\right\rVert_{L^{2}(\mathbb{R})}.$