# sharp interface limit for compressible non-isentropic phase-field model

Yazhou Chen  College of Mathematics and Physics, Beijing University of
Chemical Technology, Beijing 100029, China<EMAIL_ADDRESS>,  Qiaolin
He  College of Mathematics, Sichuan University, Sichuan 610065, China
<EMAIL_ADDRESS>,  Xiaoding Shi  College of Mathematics and Physics,
Beijing University of Chemical Technology, Beijing 100029, China
<EMAIL_ADDRESS>and  Xiaoping Wang  Department of Mathematics, Hong
Kong University of Science and Technology, Hong Kong, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, the sharp interface limit for the compressible non-isentropic
Navier-Stokes/Allen-Cahn system is derived by the method of matched asymptotic
expansion. We show that the leading order problem satisfies the compressible
Navier-Stokes equations with the interface being a free boundary. We discuss
two cases in terms of different phase field diffusion coefficients. One is
$M_{\epsilon}=O(1)$ and $M_{\epsilon}=O(\frac{1}{\epsilon})$, where $\epsilon$
is the interface thickness. We have observed that the velocity and the
temperature of the compressible immiscible two-phase fluids continuously
through the interface. There is a jump for the tension tensor at the
interface, this jump depends on the surface tension and the mean curvature of
the interface. In particular, for the first case $M_{\epsilon}=O(1)$, no
matter how the density changes through the interface, the velocity of the
interface in the normal direction is the same as the normal velocity of the
fluid along the interface. But for the second case
$M_{\epsilon}=O(\frac{1}{\epsilon})$, This phenomenon cann’t occur where the
density passes continuously through the interface. In fact, on this part of
the interface, the normal velocity of the interface is determined by the mean
curvature of the interface, the velocity and the density of the compressible
immiscible two-phase fluids. That’s where the phase transition happens.

###### Key words and phrases:

compressible, non-isothermal, Navier-Stokes equation, Allen-Cahn equation,
sharp interface.

###### 2010 Mathematics Subject Classification:

35Q30, 76T30, 35C20

##  1\. Introduction

Understanding the geometry and distribution of the interface is very important
for determining the immiscible two-phase flow. The treatment of such two-phase
flow’s interface is derived from the idea of physicist J.D. Van der Waals
[13], who regarded the interface of immiscible two-phase flow as a region with
a certain thickness. Mathematical models based on this idea are often called
diffusion interface models, such as the famous Navier-Stokes/Allen-Cahn
system, which can be used to study the immiscible two-phase flow, such as
phase transformation, chemical reactions, etc., see[2]-[10] and the references
therein. In these literatures, by introducing diffusion interface instead of
sharp interface, the authors overcome the difficulties caused by the boundary
condition of interface.

For compressible immiscible two-phase flow, taking any one of the volume
particles in the flow, we assume $M_{i}$ the mass of the components in the
representative material volume $V$, $\phi_{i}=\frac{\rho_{i}}{\rho}$ the mass
concentration, $\rho_{i}=\frac{M_{i}}{V}$ the apparent mass density of the
fluid $i~{}(i=1,2)$. The total density is given by $\rho=\rho_{1}+\rho_{2}$
and $\phi=\phi_{1}-\phi_{2}$. We call $\phi$ the difference of the two
components for the fluid mixture. Obviously, $\phi$ describes the distribution
of the interface. The compressible heat-conducting Navier-Stokes/Allen-Cahn
system derived by Heida-M$\mathrm{\acute{a}}$lek-Rajagopal [8] is as following

$\left\\{\begin{array}[]{llll}\displaystyle\partial_{t}\rho+\textrm{div}(\rho\mathbf{u})=0,\\\
\displaystyle\partial_{t}(\rho\mathbf{u})+\mathrm{div}\big{(}\rho\mathbf{u}\otimes\mathbf{u}\big{)}=\mathrm{div}\mathbb{T},\\\
\displaystyle\partial_{t}(\rho\phi)+\mathrm{div}\big{(}\rho\phi\mathbf{u}\big{)}=-M_{\epsilon}\mu,\\\
\displaystyle\rho\mu=\rho\frac{\partial
f}{\partial\phi}-\mathrm{div}\big{(}\rho\frac{\partial
f}{\partial\nabla\phi}\big{)},\\\ \displaystyle\partial_{t}(\rho
E)+\mathrm{div}(\rho
E\mathbf{u})=\mathrm{div}\big{(}\mathbb{T}\mathbf{u}+\mathrm{k}\nabla\theta-
M_{\epsilon}\mu\frac{\partial
f}{\partial\nabla\phi}\big{)},\end{array}\right.$ (1.1)

where $\mathbf{x}\in\Omega\subset\mathds{R}^{N}$, $N$ is spatial dimension,
$t>0$. The unknown functions $\rho(\mathbf{x},t)$, $\mathbf{u}(\mathbf{x},t)$,
$\phi(\mathbf{x},t)$, $\theta(\mathbf{x},t)$ denote the total density, the
velocity, the difference of the two components for the fluid mixture, and the
absolute temperature, respectively. $\mu(\mathbf{x},t)$ is the chemical
potential of the fluid. $\epsilon>0$ is the thickness of the diffuse
interface. $\mathrm{k}>0$ is the coefficient of heat conduction.
$M_{\epsilon}=M(\epsilon)>0$ is the mobility coefficient. The Cauchy stress-
tensor is represented by

$\mathbb{T}=2\nu\mathbb{D}(\mathbf{u})+\lambda(\mathrm{div}\mathbf{u})\mathbb{I}-p\mathbb{I}-\rho\nabla\phi\otimes\frac{\partial
f}{\partial\nabla\phi},$ (1.2)

where $\mathbb{D}\mathbf{u}$ is the deformation tensor

$\mathbb{D}\mathbf{u}=\frac{1}{2}\big{(}\nabla\mathbf{u}+\nabla^{\top}\mathbf{u}\big{)},$
(1.3)

and $\mathbb{I}$ is the unit matrix, $\top$ means the transpose of the matrix.
$\nu>0,\lambda>0$ are viscosity coefficients, satisfying

$\nu>0,\ \ \lambda+\frac{2}{N}\nu\geq 0.$ (1.4)

The total energy density $\rho E$ is given by

$\rho E=\rho\big{(}e+f+\frac{1}{2}\mathbf{u}^{2}\big{)},$ (1.5)

where $\rho e$ is the internal energy, $\frac{\rho\mathbf{u}^{2}}{2}$ is the
kinetic energy, $f$ is the fluid-fluid interfacial free energy density, and it
has the following form (refer to Heida-M$\mathrm{\acute{a}}$lek-Rajagopal [8]
and Lowengrub-Truskinovsky [12]):

$f(\rho,\phi,\nabla\phi)\overset{\text{def}}{=}\frac{1}{4\epsilon\rho}(1-\phi^{2})^{2}+\frac{\epsilon}{2\rho}|\nabla\phi|^{2}.$
(1.6)

Here $p=p(\rho,\theta)$, $e=e(\rho,\theta)$ and $f=f(\rho,\phi,\nabla\phi)$
obey the second law of thermodynamics (see Lions [11]),

$ds=\frac{1}{\theta}\big{(}d(e+f)+pd(\frac{1}{\rho})\big{)},$ (1.7)

where $s$ is the thermodynamic entropy. Then deduced from (1.7), we have

$\frac{\partial
s}{\partial\theta}=\frac{1}{\theta}\frac{\partial(e+f)}{\partial\theta},\ \
\frac{\partial
s}{\partial\rho}=\frac{1}{\theta}\big{(}\frac{\partial(e+f)}{\partial\rho}-\frac{p}{\rho^{2}}\big{)},$
(1.8)

which implies the following compatibility equation

$\displaystyle p$ $\displaystyle=$
$\displaystyle\rho^{2}\frac{\partial(e+f)}{\partial\rho}+\theta\frac{\partial
p}{\partial\theta}$ (1.9) $\displaystyle=$
$\displaystyle\rho^{2}\frac{\partial
e(\rho,\theta)}{\partial\rho}-\frac{1}{4\epsilon}(\phi^{2}-1)^{2}-\frac{\epsilon}{2}|\nabla\phi|^{2}+\theta\frac{\partial
p}{\partial\theta}.$

Therefore, we have

$\left\\{\begin{array}[]{llll}\displaystyle\partial_{t}\rho+\textrm{div}(\rho\mathbf{u})=0,\\\
\displaystyle\rho\partial_{t}\mathbf{u}+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}-2\nu\mathrm{div}\mathbb{D}\mathbf{u}-\lambda\nabla\mathrm{div}\mathbf{u}+\nabla
p(\rho,\theta)=-\epsilon\mathrm{div}\big{(}\nabla\phi\otimes\nabla\phi\big{)},\\\
\displaystyle\rho\partial_{t}\phi+\rho\mathbf{u}\nabla\phi=-M_{\epsilon}\mu,\\\
\displaystyle\rho\mu=\frac{1}{\epsilon}(\phi^{3}-\phi)-\epsilon\Delta\phi,\\\
\displaystyle
e_{\theta}\big{(}\rho\partial_{t}\theta+\rho\mathbf{u}\cdot\nabla\theta\big{)}+\theta
p_{\theta}\mathrm{div}\mathbf{u}-\mathrm{k}\Delta\theta=2\nu|\mathbb{D}\mathbf{u}|^{2}+\lambda(\mathrm{div}\mathbf{u})^{2}+M_{\epsilon}\mu^{2}.\end{array}\right.$
(1.10)

About the study of compressible immiscible two-phase flow, most of the works
focused on isentropic compressible problems. Feireisl-
Petzeltov$\acute{a}$-Rocca-Schimperna [7] established the global existence of
finite energy weak solutions in 3-D by using the framework which was
introduced by Lions [11]. Ding-Li-Lou [5] proved the global existence of the
strong solutions in 1-D with large initial data. Chen-Guo [4] generalized the
result of [5] to the case that the initial vacuum is allowed. Abels-Liu [1]
proved the convergence of the solutions for the incompressible Stokes/Allen-
Cahn system to solutions of a sharp interface model for sufficiently small
times. Witterstein [15] showed that the sharp-interface limit of the
isentropic phase-field model is the standard two-phase compressible Navier-
Stokes equations by the method of asymptotic analysis. Wang-Wang [14], Xu-Di-
Yu [16] investigated the sharp-interface limits of the incompressible phase-
field model with a generalized Navier slip boundary condition. There is not
much work for non-isentropic case. Kotschote [10] obtained the local existence
and uniqueness result for strong solutions in 3-D for compressible non-
isothermal phase-field model.

The main purpose of this paper is to derive the sharp interface limit of the
non-isentropic compressible system (1.10) for two cases $(C1)$ and $(C2)$ by

$\displaystyle\displaystyle(C1)\quad M_{\epsilon}=1,\quad$
$\displaystyle\quad(C2)\quad M_{\epsilon}=\frac{1}{\epsilon}.$ (1.11)

The physical meaning of the second case $(C2)$ is that, the phase field
mobility rate is accelerated. Before we introduce our main result, let us
summarize the common symbols used in this paper. For $\forall t>0$, suppose
that unknown two-phase free interface is given by

$\Gamma(t):=\big{\\{}\mathbf{x}\in\Omega\big{|}\phi_{\epsilon}(\mathbf{x},t)=0\big{\\}}.$
(1.12)

Obviously, $\Gamma(t)$ divides the whole domain $\Omega$ into two separated
domain $\Omega^{-}(t)$ and $\Omega^{+}(t)$ which represented the domains
occupied by fluid 1 and fluid 2 respectively, more precisely

$\Omega^{-}(t)=\big{\\{}\mathbf{x}\in\Omega\big{|}\phi_{\epsilon}(\mathbf{x},t)<0\big{\\}},\qquad\Omega^{+}(t)=\big{\\{}\mathbf{x}\in\Omega\big{|}:\phi_{\epsilon}(\mathbf{x},t)>0\big{\\}},$

and

$\Omega=\Omega^{-}(t)\cup\Gamma(t)\cup\Omega^{+}(t),\quad\forall t>0.$

Therefore, we have the following definition

$\Omega^{-}=\big{\\{}(\mathbf{x},t)\big{|}\mathbf{x}\in\Omega^{-}(t)\big{\\}},\quad\Omega^{+}=\big{\\{}(\mathbf{x},t)\big{|}:\mathbf{x}\in\Omega^{+}(t)\big{\\}},\quad\Gamma=\big{\\{}(\mathbf{x},t)\big{|}\mathbf{x}\in\Gamma(t)\big{\\}}.$
(1.13)

We use the method of matched asymptotic expansion to determine the sharp
interface limit as $\epsilon\rightarrow 0$. Unlike incompressible fluids, the
density of compressible fluids varies. Especially for immiscible two-phase
flow, the change of density may cause phase transition. To illustrate the mass
density properties near the interface in more detail, we define the set
$S\subset\Gamma$ by

$S=\\{(\mathbf{x},t)\in\Gamma|\rho^{+}(\mathbf{x},t)=\rho^{-}(\mathbf{x},t)\\},$
(1.14)

which implies that the jump of the density $\rho$ can’t occur on this part of
the interface $\Gamma$. Now, we give the main theorem of sharp interface
limit.

######  Theorem 1.1.

We assume (1.12)-(1.13). Let $(\rho,\mathbf{u},\phi,\mu,\theta)$ be a solution
of the Navier-Stokes-Allen-Cahn system (1.10). We assume that an outer
asymptotic expansion, that is

$\displaystyle\rho=\rho_{0}+\epsilon\rho_{1}+\cdots,\quad\mathbf{u}=\mathbf{u}_{0}+\epsilon\mathbf{u}_{1}+\cdots,\quad\phi=\phi_{0}+\epsilon\phi_{1}+\cdots,$
$\displaystyle\mu=\epsilon^{-1}\mu_{0}+\mu_{1}+\cdots,\quad\theta=\theta_{0}+\epsilon\theta_{1}+\cdots,$

and an inner asymptotic expansion, that is

$\displaystyle\rho=\widetilde{\rho}_{0}+\epsilon\widetilde{\rho_{1}}+\cdots,\quad\mathbf{u}=\widetilde{\mathbf{u}}_{0}+\epsilon\widetilde{\mathbf{u}}_{1}+\cdots,\quad\phi=\widetilde{\phi}_{0}+\epsilon\widetilde{\phi}_{1}+\cdots,$
$\displaystyle\widetilde{\mu}=\epsilon^{-1}\widetilde{\mu}_{0}+\widetilde{\mu}_{1}+\cdots,\quad\widetilde{\theta}=\widetilde{\theta}_{0}+\epsilon\widetilde{\theta}_{1}+\cdots,$

for $\rho,\mathbf{u},\phi,\mu,\theta$. We suppose that $\rho_{0}>0$ and
$\widetilde{\rho}_{0}>0$. Then as $\epsilon\rightarrow 0$, the system (1.10)
converges to the sharp interface problem

$\left\\{\begin{array}[]{llll}\displaystyle\partial_{t}\rho+\mathrm{div}(\rho\mathbf{u})=0,&\mathrm{in}\
\Omega^{\pm},\\\
\displaystyle\rho\partial_{t}\mathbf{u}+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}-2\nu\mathrm{div}\mathbb{D}\mathbf{u}-\lambda\nabla\mathrm{div}\mathbf{u}+\nabla\big{(}\rho^{2}\frac{\partial
e}{\partial\rho}+\theta\frac{\partial
p}{\partial\theta}\big{)}=0,&\mathrm{in}\ \Omega^{\pm},\\\
e_{\theta}\big{(}\rho\partial_{t}\theta+\rho\mathbf{u}\cdot\nabla\theta\big{)}+\theta
p_{\theta}\mathrm{div}\mathbf{u}-\mathrm{k}\Delta\theta=2\nu|\mathbb{D}\mathbf{u}|^{2}+\lambda(\mathrm{div}\mathbf{u})^{2},&\mathrm{in}\
\Omega^{\pm},\\\ \displaystyle\phi=\pm 1,&\mathrm{in}\
\Omega^{\pm},\end{array}\right.$ (1.15)

with the jump conditions

$(C1)\quad M_{\epsilon}=1:\
\left\\{\begin{array}[]{llll}\displaystyle[\mathbf{u}]_{\Gamma}=0,\quad[\theta]_{\Gamma}=0,&\mathrm{on}~{}~{}\Gamma,\\\
\displaystyle
V_{\mathbf{n}}-\mathbf{u}\cdot\mathbf{n}=0,&\mathrm{on}~{}~{}\Gamma,\\\
\displaystyle\big{[}\big{(}2\nu\mathbb{D}\mathbf{u}+\lambda\mathrm{div}\mathbf{u}\mathbb{I}-\big{(}\rho^{2}\frac{\partial
e}{\partial\rho}+\theta\frac{\partial
p}{\partial\theta}\big{)}\mathbb{I}\big{)}\mathbf{n}\big{]}_{\Gamma}=\sigma\kappa\mathbf{n},&\mathrm{on}~{}~{}\Gamma,\end{array}\right.$
(1.16) $\ (C2)\quad M_{\epsilon}=\frac{1}{\epsilon}:\
\left\\{\begin{array}[]{llll}\displaystyle[\mathbf{u}]_{\Gamma}=0,\quad[\theta]_{\Gamma}=0,&\mathrm{on}~{}~{}\Gamma,\\\
\displaystyle[\rho]_{\Gamma}=0,\quad\rho^{2}(V_{\mathbf{n}}-\mathbf{u}\cdot\mathbf{n})=-\kappa,&\mathrm{on}~{}~{}S,\\\
\displaystyle
V_{\mathbf{n}}-\mathbf{u}\cdot\mathbf{n}=0=\kappa,&\mathrm{on}~{}~{}\Gamma\setminus
S,\\\
\displaystyle\big{[}\big{(}2\nu\mathbb{D}\mathbf{u}+\lambda\mathrm{div}\mathbf{u}\mathbb{I}-\big{(}\rho^{2}\frac{\partial
e}{\partial\rho}+\theta\frac{\partial
p}{\partial\theta}\big{)}\mathbb{I}\big{)}\mathbf{n}\big{]}_{\Gamma}=\sigma\kappa\mathbf{n},&\mathrm{on}~{}~{}\Gamma,\end{array}\right.$
(1.17)

on the interface $\Gamma$, where $\Gamma$ is defined as (1.13) and $S$ defined
as (1.14). $[\cdot]_{\Gamma}=\cdot^{+}-\cdot^{-}$ denotes the jump of limiting
values across the interface, $\sigma$ is the constant coefficient of surface
tension (see (2.84)), $\kappa$ the curvature of the interface $\Gamma$,
$\mathbf{n}$ the unit normal of the interface pointing to $\Omega^{+}$,
$V_{\mathbf{n}}$ denotes the normal velocity of the interface $\Gamma$.

######  Remark 1.1.

Theorem 1.1 shows that the leading order problem satisfies the compressible
Navier-Stokes equations with the interface being a free boundary. Whether the
diffusion coefficient $M_{\epsilon}=O(1)$ or
$M_{\epsilon}=O(\frac{1}{\epsilon})$, the velocity and the temperature of the
flow continuously through the interface. There is a jump for the tension
tensor at the interface, and this jump depends on the surface tension and the
mean curvature of the interface. In particular, for the first case
$M_{\epsilon}=O(1)$, no matter how the density changes through the interface,
the velocity of the interface in the normal direction is the same as the
normal velocity of the fluid along the interface. But for the second case
$M_{\epsilon}=O(\frac{1}{\epsilon})$, this phenomenon can only occur at the
part of the interface where the density is discontinuous, and the mean
curvature for this part of the interface is zero. In other parts of the
interface, the normal velocity of the interface is determined by the velocity,
density of the fluid and the curvature of the surface, and that’s where the
phase transition happens.

##  2\. Asymptotic expansion for non-isentropic compressible phase-field model

In this section we consider the asymptotic analysis for (1.10) for two cases
$(C1)$ and $(C2)$. The analysis process is as follows, firstly we consider the
outer expansions far from the interface $\Gamma$, and then we do the inner
expansions near the interface $\Gamma$, and finally we combine them together
to obtain the sharp interface limit of the system (1.10) in $\Omega$. To make
the presentation in this section clear, we use $\rho_{\epsilon}$,
$\mathbf{u}_{\epsilon}$, $\phi_{\epsilon}$, $\mu_{\epsilon}$ and
$\theta_{\epsilon}$ instead of $\rho$, $\mathbf{u}$, $\phi$, $\mu$ and
$\theta$ in the system (1.10), to show explicitly that these functions depend
on $\epsilon$.

###  2.1. Outer expansion

Far from the two-phase interface $\Gamma$, we use the following ansatz,

$\begin{array}[]{l}\displaystyle\rho_{\epsilon}^{\pm}=\rho_{0}^{\pm}+\epsilon\rho_{1}^{\pm}+\epsilon^{2}\rho_{2}^{\pm}+\cdots,\\\
\displaystyle\mathbf{u}_{\epsilon}^{\pm}=\mathbf{u}_{0}^{\pm}+\epsilon\mathbf{u}_{1}^{\pm}+\epsilon^{2}\mathbf{u}_{2}^{\pm}+\cdots,\\\
\displaystyle\phi_{\epsilon}^{\pm}=\phi_{0}^{\pm}+\epsilon\phi_{1}^{\pm}+\epsilon^{2}\phi_{2}^{\pm}+\cdots,\\\
\displaystyle\mu_{\epsilon}^{\pm}=\epsilon^{-1}\mu_{0}^{\pm}+\mu_{1}^{\pm}+\epsilon\mu_{2}^{\pm}+\cdots,\\\
\displaystyle\theta_{\epsilon}^{\pm}=\theta_{0}^{\pm}+\epsilon\theta_{1}^{\pm}+\epsilon^{2}\theta_{2}^{\pm}+\cdots.\end{array}$
(2.1)

Here $g^{\pm}$ denotes the restriction of a function $g$ in $\Omega^{+}$ and
$\Omega^{-}$ respectively. Since $\rho>0,\theta>0$, we presume that
$\rho_{0}^{\pm}>0,\theta>0$.

Combined with the above analysis, let us first plug ansatz (2.1) in the mass
conservation equation (1.10)1 and compare the coefficients of terms with the
same power of $\epsilon$, and we achieve on the lowest order

$\displaystyle\displaystyle\mathcal{O}(1):\quad\partial_{t}\rho_{0}^{\pm}+\textrm{div}(\rho_{0}^{\pm}\mathbf{u}_{0}^{\pm})=0.$
(2.2)

Similarly, plugging ansatz (2.1) in the momentum conservation equation
(1.10)2, we obtain

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad\nabla(\frac{1}{4}(1-(\phi_{0}^{\pm})^{2})^{2})=0,$ (2.3)
$\displaystyle\displaystyle\mathcal{O}(1):$
$\displaystyle\quad\rho_{0}^{\pm}\partial_{t}\mathbf{u}_{0}^{\pm}+\rho_{0}^{\pm}(\mathbf{u}_{0}^{\pm}\cdot\nabla)\mathbf{u}_{0}^{\pm}-\nabla(((\phi_{0}^{\pm})^{3}-\phi_{0}^{\pm})\phi_{1}^{\pm})=\textrm{div}\mathbb{S}_{0}^{\pm}-\nabla
p_{0}^{\pm},$ (2.4)

where

$\mathbb{S}_{0}^{\pm}=2\nu\mathbb{D}\mathbf{u}_{0}^{\pm}+\lambda\mathrm{div}\mathbf{u}_{0}^{\pm}\mathbb{I},$
(2.5)

and

$p_{0}^{\pm}=(\rho_{0}^{\pm})^{2}e_{\rho}(\rho_{0}^{\pm},\theta_{0}^{\pm})+\theta_{0}^{\pm}p_{\theta}(\rho_{0}^{\pm},\theta_{0}^{\pm}).$
(2.6)

Next, we plug ansatz (2.1) in conservation equation of phase field (1.10)3 to
achieve

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{\chi_{M_{\epsilon}}-2}):$
$\displaystyle\quad\mu_{0}^{\pm}=0,$ (2.7)
$\displaystyle\displaystyle\mathcal{O}(\epsilon^{\chi_{M_{\epsilon}}-1}):$
$\displaystyle\quad\chi_{M_{\epsilon}}(\rho_{0}^{\pm}\partial_{t}\phi_{0}^{\pm}+\rho_{0}^{\pm}\mathbf{u}_{0}^{\pm}\cdot\nabla\phi_{0}^{\pm})=-\mu_{1}^{\pm},$
(2.8)

where

$\chi_{M_{\epsilon}}=\left\\{\begin{array}[]{llll}\displaystyle 1,&\
\mathrm{for~{}~{}(C1)}\ M_{\epsilon}=1,\\\ \displaystyle 0,&\
\mathrm{for~{}~{}(C2)}\ M_{\epsilon}=\frac{1}{\epsilon}.\end{array}\right.$
(2.9)

Plugging ansatz (2.1) in potential equation (1.10)4 and comparing the
coefficients of terms with the same power of $\epsilon$, we obtain

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):\quad$
$\displaystyle\rho_{0}^{\pm}\mu_{0}^{\pm}=\phi_{0}^{\pm}\big{(}(\phi_{0}^{\pm})^{2}-1\big{)}.$
(2.10)

Finally, plugging ansatz (2.1) in energy equation (1.10)5, we obtain

$\displaystyle\displaystyle\mathcal{O}(1):e_{\theta}(\rho_{0}^{\pm},\theta_{0}^{\pm})(\rho_{0}^{\pm}\partial_{t}\theta_{0}^{\pm}+\rho_{0}^{\pm}\mathbf{u}_{0}^{\pm}\cdot\nabla\theta_{0}^{\pm})+\theta_{0}^{\pm}p_{\theta}(\rho_{0}^{\pm},\theta_{0}^{\pm})\mathrm{div}\mathbf{u}_{0}^{\pm}-\mathrm{k}\Delta\theta_{0}^{\pm}$
$\displaystyle=2\nu|\mathbb{D}\mathbf{u}_{0}^{\pm}|^{2}\\!+\lambda(\mathrm{div}\mathbf{u}_{0}^{\pm})^{2}\\!+\chi_{M_{\epsilon}}\big{(}(\mu_{1}^{\pm})^{2}\\!+2\mu_{0}^{\pm}\mu_{2}^{\pm}\big{)}\\!+(1\\!-\\!\chi_{M_{\epsilon}})\big{(}2\mu_{1}^{\pm}\mu_{2}^{\pm}\\!+\\!2\mu_{0}^{\pm}\mu_{3}^{\pm}\big{)},$
(2.11)

where $\chi_{M_{\epsilon}}$ is defined in (2.9). Then from the equation (2.3),
(2.7), (2.8) and (2.10), we have

$\displaystyle\displaystyle\phi_{0}^{\pm}=\pm
1,\quad\mu_{0}^{\pm}=0,\quad\mu_{1}^{\pm}=0,\qquad\text{in}~{}~{}\Omega^{\pm}.$
(2.12)

Combining (2.4) and (2.10), it follows that

$\displaystyle\displaystyle\rho_{0}^{\pm}\partial_{t}\mathbf{u}_{0}^{\pm}+\rho_{0}^{\pm}(\mathbf{u}_{0}^{\pm}\cdot\nabla)\mathbf{u}_{0}^{\pm}+\nabla
p_{0}^{\pm}=\textrm{div}\mathbb{S}_{0}^{\pm}.$ (2.13)

Moreover, (2.11) and (2.12) derive that

$\displaystyle
e_{\theta}(\rho_{0}^{\pm},\theta_{0}^{\pm})(\rho_{0}^{\pm}\partial_{t}\theta_{0}^{\pm}+\rho_{0}^{\pm}\mathbf{u}_{0}^{\pm}\cdot\nabla\theta_{0}^{\pm})+\theta_{0}^{\pm}p_{\theta}(\rho_{0}^{\pm},\theta_{0}^{\pm})\mathrm{div}\mathbf{u}_{0}^{\pm}-\mathrm{k}\Delta\theta_{0}^{\pm}$
$\displaystyle=2\nu|\mathbb{D}\mathbf{u}_{0}^{\pm}|^{2}+\lambda(\mathrm{div}\mathbf{u}_{0}^{\pm})^{2}.$
(2.14)

By using (2.2), (2.12), (2.13) and (2.14), we can state the following lemma.

######  Lemma 2.1.

Assume $\rho_{0}^{\pm}>0$. Letting $\epsilon\rightarrow 0$ in ansatz (2.1), we
derive from (2.2), (2.4), (2.7), (2.8), (2.10) and (2.11), in the regions
outside the transition layer, to the problem

$\displaystyle\partial_{t}\rho^{\pm}+\mathrm{div}(\rho^{\pm}\mathbf{u}^{\pm})=0,$
$\displaystyle\mathrm{in}\ \Omega^{\pm},$ (2.15)
$\displaystyle\rho^{\pm}\partial_{t}\mathbf{u}^{\pm}+\rho^{\pm}(\mathbf{u}^{\pm}\cdot\nabla)\mathbf{u}^{\pm}-\mathrm{div}(2\nu\mathbb{D}\mathbf{u}^{\pm}+\lambda\mathrm{div}\mathbf{u}^{\pm}\mathbb{I})$
$\displaystyle\qquad+\nabla\big{(}(\rho^{\pm})^{2}(e_{\rho}(\rho^{\pm},\theta^{\pm})+\theta^{\pm}p_{\theta}(\rho^{\pm},\theta^{\pm})\big{)}=0,$
$\displaystyle\mathrm{in}\ \Omega^{\pm},$ (2.16) $\displaystyle
e_{\theta}(\rho^{\pm},\theta^{\pm})(\rho^{\pm}\partial_{t}\theta^{\pm}+\rho^{\pm}\mathbf{u}^{\pm}\cdot\nabla\theta^{\pm})+\theta^{\pm}p_{\theta}(\rho^{\pm},\theta^{\pm})\mathrm{div}\mathbf{u}^{\pm}-\mathrm{k}\Delta\theta^{\pm}$
$\displaystyle\quad=2\nu|\mathbb{D}\mathbf{u}^{\pm}|^{2}+\lambda(\mathrm{div}\mathbf{u}^{\pm})^{2},$
$\displaystyle\mathrm{in}\ \Omega^{\pm},$ (2.17)

and $\phi_{0}^{\pm}=\pm 1$ in $\Omega^{\pm}$.

###  2.2. Inner expansion

In this subsection, we propose to analysis the inner expansion near the
interface $\Gamma$. Let $d(\mathbf{x},t)$ be signed distance to $\Gamma$,
which is well-defined near the interface. Then the unit normal of the
interface pointing to $\Omega^{+}$ is given by $\mathbf{n}=\nabla d$ and the
normal velocity of the interface $\Gamma$ is given by
$V_{\mathbf{n}}=-\partial_{t}d$. We introduce a new rescaled variable

$\xi=\frac{d(\mathbf{x},t)}{\epsilon}.$ (2.18)

For any function $g(\mathbf{x},t)$ (e.g.
$g=\rho_{\epsilon},\mathbf{u}_{\epsilon},\phi_{\epsilon},\mu_{\epsilon},\theta_{\epsilon}$),
we can rewrite it as

$g(\mathbf{x},t)=\widetilde{g}(\mathbf{x},t,\xi).$ (2.19)

Then we have

$\begin{array}[]{l}\nabla
g=\nabla\widetilde{g}+\epsilon^{-1}\partial_{\xi}\widetilde{g}\mathbf{n},\\\
\Delta
g=\Delta\widetilde{g}+\epsilon^{-1}\partial_{\xi}\widetilde{g}\kappa+2\epsilon^{-1}(\mathbf{n}\cdot\nabla)\partial_{\xi}\widetilde{g}+\epsilon^{-2}\partial_{\xi\xi}\widetilde{g},\\\
\partial_{t}g=\partial_{t}\widetilde{g}-\epsilon^{-1}\partial_{\xi}\widetilde{g}V_{\mathbf{n}}.\end{array}$
(2.20)

Here we use the fact that $\nabla\cdot\mathbf{n}=\kappa$, the mean curvature
of the interface. $\kappa(\mathbf{x})$ for $\mathbf{x}\in\Gamma(t)$ is
positive (resp. negative) if the domain $\Omega_{-}$ is convex (resp. concave)
near $\mathbf{x}$.

In the inner region, we assume that

$\begin{array}[]{l}\widetilde{\rho}_{\epsilon}=\widetilde{\rho}_{0}+\epsilon\widetilde{\rho}_{1}+\epsilon^{2}\widetilde{\rho}_{2}+\cdots,\\\
\widetilde{\mathbf{u}}_{\epsilon}=\widetilde{\mathbf{u}}_{0}+\epsilon\widetilde{\mathbf{u}}_{1}+\epsilon^{2}\widetilde{\mathbf{u}}_{2}+\cdots,\\\
\widetilde{\phi}_{\epsilon}=\widetilde{\phi}_{0}+\epsilon\widetilde{\phi}_{1}+\epsilon^{2}\widetilde{\phi}_{2}+\cdots,\\\
\widetilde{\mu}_{\epsilon}=\epsilon^{-1}\widetilde{\mu}_{0}+\widetilde{\mu}_{1}+\epsilon\widetilde{\mu}_{2}+\cdots,\\\
\widetilde{\theta}_{\epsilon}=\widetilde{\theta}_{0}+\epsilon\widetilde{\theta}_{1}+\epsilon^{2}\widetilde{\theta}_{2}+\cdots.\end{array}$
(2.21)

In the next we represent the system (1.10) in the new coordinates and compare
the order of the $\epsilon$ coefficients. Let us first plug ansatz (2.21) in
the mass conservation equation (1.10)1 to infer the following equation in new
coordinates

$\displaystyle\displaystyle\partial_{t}\widetilde{\rho}_{0}-\epsilon^{-1}\partial_{\xi}\widetilde{\rho}_{0}V_{n}+\textrm{div}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0})+\epsilon^{-1}\partial_{\xi}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0})\cdot\mathbf{n}+\mathcal{O}(\epsilon)=0.$
(2.22)

By comparing the coefficients of same $\epsilon$-order, we obtain

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad-\partial_{\xi}\widetilde{\rho}_{0}V_{\mathbf{n}}+\partial_{\xi}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0})\cdot\mathbf{n}=0,$
(2.23) $\displaystyle\displaystyle\mathcal{O}(1):$
$\displaystyle\quad\partial_{t}\widetilde{\rho}_{0}+\textrm{div}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0})-\partial_{\xi}\widetilde{\rho}_{1}V_{\mathbf{n}}+\partial_{\xi}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{1}+\widetilde{\rho}_{1}\widetilde{\mathbf{u}}_{0})\cdot\mathbf{n}=0.$
(2.24)

Next, we evaluate the right-hand-side of the momentum conservation equation
(1.10)2. By using (2.20), through differentiation, we obtain

$\displaystyle\mathop{\mathrm{div}}\nolimits(\nu(\nabla\mathbf{u}+\nabla^{T}\mathbf{u}))+\mathop{\mathrm{div}}\nolimits(\lambda\mathop{\mathrm{div}}\nolimits\mathbf{u}\mathbb{I})$
$\displaystyle=\frac{1}{\epsilon^{2}}\partial_{\xi}\big{(}(\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}\mathbf{n}+\frac{1}{\epsilon^{2}}\partial_{\xi}\big{(}\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\big{)}+\frac{1}{\epsilon}\partial_{\xi}\big{(}\nu(\nabla\widetilde{\mathbf{u}}_{0}+\nabla^{T}\widetilde{\mathbf{u}}_{0})\big{)}\cdot\mathbf{n}$
$\displaystyle\qquad+\frac{1}{\epsilon}\mathop{\mathrm{div}}\nolimits(\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\otimes\mathbf{n})+\frac{1}{\epsilon}\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\kappa+\frac{1}{\epsilon}\partial_{\xi}\big{(}\lambda\mathop{\mathrm{div}}\nolimits\widetilde{\mathbf{u}}_{0}\big{)}\cdot\mathbf{n}$
$\displaystyle\qquad+\frac{1}{\epsilon}\nabla\big{(}\lambda\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}+\frac{1}{\epsilon}(\nu+\lambda)(\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\nabla)\mathbf{n}+\mathcal{O}(1),$
(2.25)

and

$\displaystyle\nabla
p(\rho,\theta)+\epsilon\mathop{\mathrm{div}}\nolimits(\nabla\phi\otimes\nabla\phi)$
$\displaystyle=\frac{1}{\epsilon}\partial_{\xi}p_{0}\mathbf{n}-\frac{1}{\epsilon}\nabla({\tilde{\phi}_{0}}^{3}-\widetilde{\phi}_{0})-\frac{1}{4\epsilon^{2}}\partial_{\xi}({\tilde{\phi}_{0}}^{2}-1)^{2}\mathbf{n}+\frac{1}{2\epsilon^{2}}\partial_{\xi}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}$
$\displaystyle\quad+\frac{1}{\epsilon}\mathbf{n}\cdot\nabla|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}+\frac{1}{\epsilon}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\kappa\mathbf{n}+\frac{1}{\epsilon}\partial_{\xi\xi}\widetilde{\phi}_{0}\nabla\widetilde{\phi}_{0}+\mathcal{O}(1),$
(2.26)

where
$p_{0}=\tilde{\rho}_{0}^{2}e_{\rho}(\tilde{\rho}_{0},\tilde{\theta}_{0})+\tilde{\theta}_{0}p_{\theta}(\tilde{\rho}_{0},\tilde{\theta}_{0})$.
Substituting (2.2) and (2.2) into (1.10)2, we have

$\displaystyle-\frac{1}{\epsilon}\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\mathbf{u}}_{0}V_{n}+\frac{1}{\epsilon}\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\mathbf{u}}_{0}$
$\displaystyle=\frac{1}{\epsilon^{2}}\partial_{\xi}\big{(}(\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}\mathbf{n}+\frac{1}{\epsilon^{2}}\partial_{\xi}\big{(}\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\big{)}+\frac{1}{\epsilon}\partial_{\xi}\big{(}\nu(\nabla\widetilde{\mathbf{u}}_{0}+\nabla^{T}\widetilde{\mathbf{u}}_{0})\big{)}\cdot\mathbf{n}$
$\displaystyle\quad+\frac{1}{\epsilon}\mathop{\mathrm{div}}\nolimits(\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\otimes\mathbf{n})+\frac{1}{\epsilon}\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\kappa+\frac{1}{\epsilon}\partial_{\xi}\big{(}\lambda\mathop{\mathrm{div}}\nolimits\widetilde{\mathbf{u}}_{0}\big{)}\cdot\mathbf{n}+\frac{1}{\epsilon}\nabla\big{(}\lambda\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}$
$\displaystyle\quad+\frac{1}{\epsilon}(\nu+\lambda)(\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\nabla)\mathbf{n}-\frac{1}{\epsilon}\partial_{\xi}p_{0}\mathbf{n}+\frac{1}{4\epsilon}\nabla({\tilde{\phi}_{0}}^{2}-1)^{2}+\frac{1}{4\epsilon^{2}}\partial_{\xi}({\tilde{\phi}_{0}}^{2}-1)^{2}\mathbf{n}$
$\displaystyle\quad-\frac{1}{2\epsilon^{2}}\partial_{\xi}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}-\frac{1}{\epsilon}\mathbf{n}\cdot\nabla|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}-\frac{1}{\epsilon}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\kappa\mathbf{n}-\frac{1}{\epsilon}\partial_{\xi\xi}\widetilde{\phi}_{0}\nabla\widetilde{\phi}_{0}+\mathcal{O}(1).$
(2.27)

By comparing the coefficients of same $\epsilon$-order, we achieve

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-2}):$
$\displaystyle\quad\partial_{\xi}\big{(}(\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}\mathbf{n}+\partial_{\xi}\big{(}\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\big{)}+\frac{1}{4}\partial_{\xi}({\tilde{\phi}_{0}}^{2}-1)^{2}\mathbf{n}-\frac{1}{2}\partial_{\xi}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}=0,$
(2.28) $\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad-\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\mathbf{u}}_{0}V_{n}+\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\mathbf{u}}_{0}+\partial_{\xi}p_{0}\mathbf{n}$
$\displaystyle=\partial_{\xi}\big{(}\nu(\nabla\widetilde{\mathbf{u}}_{0}+\nabla^{T}\widetilde{\mathbf{u}}_{0})\big{)}\cdot\mathbf{n}+\mathop{\mathrm{div}}\nolimits(\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\otimes\mathbf{n})+\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\kappa$
$\displaystyle+\partial_{\xi}\big{(}\lambda\mathop{\mathrm{div}}\nolimits\widetilde{\mathbf{u}}_{0}\big{)}\cdot\mathbf{n}+\nabla\big{(}\lambda\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}+(\nu+\lambda)(\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\nabla)\mathbf{n}$
$\displaystyle+\frac{1}{4}\nabla({\tilde{\phi}_{0}}^{2}-1)^{2}+\partial_{\xi}(({\tilde{\phi}_{0}}^{3}-\widetilde{\phi}_{0})\widetilde{\phi}_{1})\mathbf{n}-\mathbf{n}\cdot\nabla|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}-|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\kappa\mathbf{n}$
$\displaystyle-\partial_{\xi\xi}\widetilde{\phi}_{0}\nabla\widetilde{\phi}_{0}+\partial_{\xi}\big{(}(\nu+\lambda)\partial_{\xi}\hat{\mathbf{u}}_{1}\cdot\mathbf{n}\big{)}\mathbf{n}+\partial_{\xi}\big{(}\nu\partial_{\xi}\hat{\mathbf{u}}_{1}\big{)}-\partial_{\xi}\big{(}\partial_{\xi}\widetilde{\phi}_{0}\partial_{\xi}\widetilde{\phi}_{1}\big{)}\mathbf{n}.$
(2.29)

Similarly, equation (1.10)3 can be rewritten in the new coordinates as

$\displaystyle-\frac{1}{\epsilon}\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\phi}_{0}V_{\mathbf{n}}+\frac{1}{\epsilon}\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\phi}_{0}=-M_{\epsilon}\frac{1}{\epsilon}\widetilde{\mu}_{0}+\mathcal{O}(1).$
(2.30)

For (C1) $M_{\epsilon}=1$, we compare the coefficients of same
$\epsilon$-order to get

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad-\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\phi}_{0}V_{n}+\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\phi}_{0}=-\widetilde{\mu}_{0}.$
(2.31)

For (C2) $M_{\epsilon}=\frac{1}{\epsilon}$, we have

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-2}):$
$\displaystyle\quad\widetilde{\mu}_{0}=0,$ (2.32)
$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad-\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\phi}_{0}V_{n}+\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\phi}_{0}=-\widetilde{\mu}_{1}.$
(2.33)

Plugging ansatz (2.21) in the potential equation (1.10)4, we obtain

$\displaystyle\frac{1}{\epsilon}\widetilde{\rho}_{0}\widetilde{\mu}_{0}=-2(\mathbf{n}\cdot\nabla)\partial_{\xi}\widetilde{\phi}_{0}-\partial_{\xi}\widetilde{\phi}_{0}\kappa-\frac{1}{\epsilon}\partial_{\xi\xi}\widetilde{\phi}_{0}+\frac{1}{\epsilon}({\tilde{\phi}_{0}}^{3}-\widetilde{\phi}_{0})+\mathcal{O}(\epsilon),$
(2.34)

comparing the coefficients of same $\epsilon$-order, we achieve

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad\widetilde{\rho}_{0}\widetilde{\mu}_{0}=-\partial_{\xi\xi}\widetilde{\phi}_{0}+{\tilde{\phi}_{0}}^{3}-\widetilde{\phi}_{0},$
(2.35) $\displaystyle\mathcal{O}(1):$
$\displaystyle\quad\widetilde{\rho}_{0}\widetilde{\mu}_{1}+\widetilde{\rho}_{1}\widetilde{\mu}_{0}=-2(\mathbf{n}\cdot\nabla)\partial_{\xi}\widetilde{\phi}_{0}-\partial_{\xi}\widetilde{\phi}_{0}\kappa-\partial_{\xi\xi}\widetilde{\phi}_{1}+(3{\tilde{\phi}_{0}}^{2}-1)\widetilde{\phi}_{1}.$
(2.36)

Finally, plugging ansatz (2.21) in the energy equation (1.10)5. For (C1)
$M_{\epsilon}=1$, we compare the coefficients of same $\epsilon$-order to get

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-2}):$
$\displaystyle\quad\mathrm{k}\partial_{\xi\xi}\widetilde{\theta}_{0}+\nu|\partial_{\xi}\widetilde{\mathbf{u}}_{0}|^{2}+(\nu+\lambda)|\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}|^{2}+\widetilde{\mu}_{0}^{2}=0,$
(2.37) $\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad
e_{\theta}(\widetilde{\rho}_{0},\widetilde{\theta}_{0})(-\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\theta}_{0}V_{n}+\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\theta}_{0})+\widetilde{\theta}_{0}p_{\theta}(\widetilde{\rho}_{0},\widetilde{\theta}_{0})\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}$
$\displaystyle\quad=2\mathrm{k}(\mathbf{n}\cdot\nabla)\partial_{\xi}\widetilde{\theta}_{0}+\mathrm{k}\partial_{\xi}\widetilde{\theta}_{0}\kappa+\mathrm{k}\partial_{\xi\xi}\widetilde{\theta}_{1}+2\nu\mathbb{D}\widetilde{\mathbf{u}}_{0}:\partial_{\xi}\widetilde{\mathbf{u}}_{0}\otimes\mathbf{n}+2\lambda\mathop{\mathrm{div}}\nolimits\widetilde{\mathbf{u}}_{0}\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}$
$\displaystyle\qquad+2\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\partial_{\xi}\widetilde{\mathbf{u}}_{1}+2(\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\mathbf{u}}_{1}\cdot\mathbf{n}+2\widetilde{\mu}_{0}\widetilde{\mu}_{1}.$
(2.38)

For (C2) $M_{\epsilon}=\frac{1}{\epsilon}$, we have

$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-3}):$
$\displaystyle\quad\widetilde{\mu}_{0}=0,$ (2.39)
$\displaystyle\displaystyle\mathcal{O}(\epsilon^{-2}):$
$\displaystyle\quad\mathrm{k}\partial_{\xi\xi}\widetilde{\theta}_{0}+\nu|\partial_{\xi}\widetilde{\mathbf{u}}_{0}|^{2}+(\nu+\lambda)|\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}|^{2}+2\widetilde{\mu}_{0}\widetilde{\mu}_{1}=0,$
(2.40) $\displaystyle\displaystyle\mathcal{O}(\epsilon^{-1}):$
$\displaystyle\quad
e_{\theta}(\widetilde{\rho}_{0},\widetilde{\theta}_{0})(-\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\theta}_{0}V_{n}+\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\theta}_{0})+\widetilde{\theta}_{0}p_{\theta}(\widetilde{\rho}_{0},\widetilde{\theta}_{0})\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}$
$\displaystyle\quad=2\mathrm{k}(\mathbf{n}\cdot\nabla)\partial_{\xi}\widetilde{\theta}_{0}+\mathrm{k}\partial_{\xi}\widetilde{\theta}_{0}\kappa+\mathrm{k}\partial_{\xi\xi}\widetilde{\theta}_{1}+2\nu\mathbb{D}\widetilde{\mathbf{u}}_{0}:\partial_{\xi}\widetilde{\mathbf{u}}_{0}\otimes\mathbf{n}+2\lambda\mathop{\mathrm{div}}\nolimits\widetilde{\mathbf{u}}_{0}\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}$
$\displaystyle\qquad+2\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\partial_{\xi}\widetilde{\mathbf{u}}_{1}+2(\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\partial_{\xi}\widetilde{\mathbf{u}}_{1}\cdot\mathbf{n}+\widetilde{\mu}_{1}^{2}+2\widetilde{\mu}_{0}\widetilde{\mu}_{2}.$
(2.41)

Noting that the matching conditions for inner and outer expansions are as
follows,

$\displaystyle\lim_{\xi\rightarrow\pm\infty}\hat{g}_{0}(\mathbf{x},\xi)=g_{0}^{\pm}(\mathbf{x}),$
(2.42)
$\displaystyle\lim_{\xi\rightarrow\pm\infty}(\nabla_{\mathbf{x}}\hat{g}_{0}(\mathbf{x},\xi)+\partial_{\xi}\hat{g}_{1}(\mathbf{x},\xi)\mathbf{n})=\nabla
g_{0}^{\pm}(\mathbf{x}).$ (2.43)

Zeroth approximation.  We consider the zeroth approximation in the inner
expansion:

$\displaystyle-\partial_{\xi}\widetilde{\rho}_{0}V_{\mathbf{n}}+\partial_{\xi}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0})\cdot\mathbf{n}=0,$
(2.44)
$\displaystyle\partial_{\xi}\big{(}(\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}\mathbf{n}+\partial_{\xi}\big{(}\nu\partial_{\xi}\widetilde{\mathbf{u}}_{0}\big{)}+\frac{1}{4}\partial_{\xi}({\tilde{\phi}_{0}}^{2}-1)^{2}\mathbf{n}-\frac{1}{2}\partial_{\xi}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}=0,$
(2.45)
$\displaystyle\chi_{M_{\epsilon}}\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\phi}_{0}(V_{\mathbf{n}}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})=\widetilde{\mu}_{0},$
(2.46)
$\displaystyle\widetilde{\rho}_{0}\widetilde{\mu}_{0}=-\partial_{\xi\xi}\widetilde{\phi}_{0}+{\tilde{\phi}_{0}}^{3}-\widetilde{\phi}_{0},$
(2.47)
$\displaystyle\mathrm{k}\partial_{\xi\xi}\widetilde{\theta}_{0}+\nu|\partial_{\xi}\widetilde{\mathbf{u}}_{0}|^{2}+(\nu+\lambda)|\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}|^{2}+\chi_{M_{\epsilon}}\widetilde{\mu}_{0}^{2}=0,$
(2.48)

with boundary conditions:

$\displaystyle\widetilde{\rho}_{0}(\mathbf{x},t,\pm\infty)=\rho_{0}^{\pm}(\mathbf{x},t),$
(2.49)
$\displaystyle\widetilde{\mathbf{u}}_{0}(\mathbf{x},t,\pm\infty)=\mathbf{u}_{0}^{\pm}(\mathbf{x},t),$
(2.50) $\displaystyle\widetilde{\phi}_{0}(\mathbf{x},t,\pm\infty)=\pm 1,$
(2.51)
$\displaystyle\widetilde{\theta}_{0}(\mathbf{x},t,\pm\infty)=\theta_{0}^{\pm}(\mathbf{x},t),$
(2.52)

where $\chi_{M_{\epsilon}}$ is defined by (2.9).

######  Lemma 2.2.

Let
$(\widetilde{\rho}_{0},\widetilde{\mathbf{u}}_{0},\widetilde{\phi}_{0},\widetilde{\theta}_{0})$
be a solution of (2.44)-(2.51). It holds

$\displaystyle\displaystyle\widetilde{\rho}_{0}(t,x,\xi)(V_{n}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})(t,x,\xi)=\rho_{0}^{\pm}(\mathbf{x},t)(V_{n}-\mathbf{u}_{0}^{\pm}\cdot\mathbf{n})(\mathbf{x},t),$
(2.53)
$\displaystyle\widetilde{\phi}_{0}(t,x,\xi)=\tanh(\frac{\xi}{\sqrt{2}}),$
(2.54)
$\displaystyle\partial_{\xi}\widetilde{\mathbf{u}}_{0}=0,\qquad\partial_{\xi}\widetilde{\theta}_{0}=0,$
(2.55)
$\displaystyle\left\\{\begin{array}[]{llll}V_{\mathbf{n}}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}=0,&\
\mathrm{for~{}~{}(C1)}\ M_{\epsilon}=1,\\\
\partial_{\xi}\widetilde{\rho}_{0}(V_{\mathbf{n}}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})=0,&\
\mathrm{for~{}~{}(C2)}\ M_{\epsilon}=\frac{1}{\epsilon}.\end{array}\right.$
(2.58)

###### Proof.

From equation (2.44) we have

$\displaystyle\displaystyle\partial_{\xi}(\widetilde{\rho}_{0}(V_{\mathbf{n}}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}))=0,$
(2.59)

and by integration we obtain (2.53). By multiplying (2.46) with
$\partial_{\xi}\widetilde{\phi}_{0}$, and combining with (2.47), we obtain

$\displaystyle\displaystyle-\chi_{M_{\epsilon}}(V_{n}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})|\partial_{\xi}\widetilde{\phi}_{0}|^{2}=\frac{1}{\widetilde{\rho}_{0}^{2}}\big{(}\frac{1}{2}\partial_{\xi}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}-\frac{1}{4}\partial_{\xi}(\tilde{\phi}_{0}^{2}-1)^{2}\big{)}.$
(2.60)

Multiplying equation (2.45) by the norm vector $\mathbf{n}$, we infer that

$\displaystyle\displaystyle\partial_{\xi}\big{(}(\lambda+2\nu)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}\big{)}+\frac{1}{4}\partial_{\xi}(\tilde{\phi}_{0}^{2}-1)^{2}-\frac{1}{2}\partial_{\xi}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}=0.$
(2.61)

By integrating equation (2.61) from $-\infty$ to $\xi$, we obtain

$\displaystyle\displaystyle(\lambda+2\nu)\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}+\frac{1}{4}(\tilde{\phi}_{0}^{2}-1)^{2}-\frac{1}{2}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}=0.$
(2.62)

Multiplying equation (2.62) by
$\frac{\partial}{\partial\xi}(\frac{1}{\widetilde{\rho}_{0}^{2}})$, combining
with (2.44), we infer that

$\displaystyle\displaystyle-2(\lambda+2\nu)\widetilde{\rho}_{0}^{-4}(V_{n}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})|\partial_{\xi}\widetilde{\rho}_{0}|^{2}=\big{(}\frac{1}{2}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}-\frac{1}{4}(\tilde{\phi}_{0}^{2}-1)^{2}\big{)}\frac{\partial}{\partial\xi}(\frac{1}{\widetilde{\rho}_{0}^{2}}).$
(2.63)

Combining (2.60) with (2.63), and integrating them from $-\infty$ to
$+\infty$, we obtain

$\displaystyle\displaystyle\int_{-\infty}^{+\infty}(V_{n}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})\big{(}\chi_{M_{\epsilon}}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}+2(\lambda+2\nu)\widetilde{\rho}_{0}^{-4}|\partial_{\xi}\widetilde{\rho}_{0}|^{2}\big{)}d\xi=0.$
(2.64)

Since $\widetilde{\rho}_{0}>0$, $\lambda+2\nu>0$, and (2.9), combining (2.64)
and (2.53) we obtain (2.58). Then from (2.46) and (2.60), for both (C1) and
(C2), we obtain

$\displaystyle\displaystyle\widetilde{\mu}_{0}=0,\quad\frac{1}{2}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}=\frac{1}{4}(\widetilde{\phi}_{0}^{2}-1)^{2}.$
(2.65)

Then we have

$\displaystyle\displaystyle\partial_{\xi}\widetilde{\phi}_{0}=\frac{1}{\sqrt{2}}(1-\widetilde{\phi}_{0}^{2}).$
(2.66)

Noticing $\widetilde{\phi}_{0}(\mathbf{x},t,0)=0$, we obtain (2.54). Since
$\lambda+2\nu>0$, combining (2.62) and (2.65), we have

$\displaystyle\displaystyle\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n}=0.$
(2.67)

Multiplying equation (2.29) by the tangential vector $\mathbf{\tau}$, we infer
that

$\displaystyle\displaystyle\partial_{\xi}(\lambda\partial_{\xi}\widetilde{\mathbf{u}}_{0})\cdot\mathbf{\tau}=0.$
(2.68)

Since $\lambda>0$, by integration from $-\infty$ to $\xi$, we have

$\displaystyle\displaystyle\partial_{\xi}\widetilde{\mathbf{u}}_{0}\cdot\mathbf{\tau}=0.$
(2.69)

By integrating equation (2.48) from $-\infty$ to $\xi$, combining with (2.67)
and (2.65), we obtain

$\displaystyle\displaystyle\mathrm{k}\partial_{\xi}\widetilde{\theta}_{0}=0.$
(2.70)

Since $\mathrm{k}>0$, combining (2.67) and (2.69), we derive (2.55). The proof
is completed. ∎

######  Remark 2.1.

For $(C1)~{}M_{\epsilon}=1$, we have $V_{n}-\mathbf{u}_{0}\cdot\mathbf{n}=0$
on $\Gamma$. For $(C2)~{}M_{\epsilon}=\frac{1}{\epsilon}$ we introduce the set
$S\subset\Gamma$ by

$\displaystyle\displaystyle\rho^{+}(\mathbf{x},t)=\rho^{-}(\mathbf{x},t),$
$\displaystyle\quad\text{on}~{}S\times(-\infty,+\infty).$ (2.71)

From (2.58), it implies that

$\displaystyle\displaystyle\rho^{+}(\mathbf{x},t)=\rho^{-}(\mathbf{x},t),\quad
V_{n}-\mathbf{u}_{0}\cdot\mathbf{n}=0,$
$\displaystyle\quad\text{on}~{}\Gamma\setminus S\times(-\infty,+\infty).$
(2.72)

First approximation.  We derive the system from the first order asymptotic
analysis by the equations (2.24), (2.29), (2.36) and (2.38). By (2.55), the
equations are simplified to the following system:

$\displaystyle\partial_{t}\widetilde{\rho}_{0}+\textrm{div}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{0})-\partial_{\xi}\widetilde{\rho}_{1}(V_{n}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})+\partial_{\xi}(\widetilde{\rho}_{0}\widetilde{\mathbf{u}}_{1}\cdot\mathbf{n})=0,$
(2.73)
$\displaystyle-\partial_{\xi}p_{0}\mathbf{n}+\frac{1}{4}\nabla(\tilde{\phi}_{0}^{2}-1)^{2}+\partial_{\xi}((\tilde{\phi}_{0}^{3}-\widetilde{\phi}_{0})\widetilde{\phi}_{1})\mathbf{n}+\partial_{\xi}((\nu+\lambda)\partial_{\xi}\hat{\mathbf{u}}_{1}\cdot\mathbf{n})\mathbf{n}+\partial_{\xi}(\nu\partial_{\xi}\hat{\mathbf{u}}_{1})$
$\displaystyle~{}~{}~{}\qquad=\partial_{\xi}\big{(}\partial_{\xi}\widetilde{\phi}_{0}\partial_{\xi}\widetilde{\phi}_{1}\big{)}\mathbf{n}+\mathbf{n}\cdot\nabla|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\mathbf{n}+|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\kappa\mathbf{n}+\partial_{\xi\xi}\widetilde{\phi}_{0}\nabla\widetilde{\phi}_{0},$
(2.74)
$\displaystyle\widetilde{\rho}_{0}\widetilde{\mu}_{1}=-2(\mathbf{n}\cdot\nabla)\partial_{\xi}\widetilde{\phi}_{0}-\partial_{\xi}\widetilde{\phi}_{0}\kappa-\partial_{\xi\xi}\widetilde{\phi}_{1}+(3{\tilde{\phi}_{0}}^{2}-1)\widetilde{\phi}_{1},$
(2.75) $\displaystyle\mathrm{k}\partial_{\xi\xi}\widetilde{\theta}_{1}=0,$
(2.76)

and for (C2) $M_{\epsilon}=\frac{1}{\epsilon}$ we have

$\displaystyle\widetilde{\rho}_{0}\partial_{\xi}\widetilde{\phi}_{0}(V_{n}-\widetilde{\mathbf{u}}_{0}\cdot\mathbf{n})=\widetilde{\mu}_{1}.$
(2.77)

######  Lemma 2.3.

It holds

$\displaystyle\displaystyle[(\mathbb{S}_{0}-p_{0}\mathbb{I})\mathbf{n}]_{\Gamma}=\sigma\kappa\mathbf{n},\quad\text{on}~{}\Gamma,$
(2.78)

where
$\mathbb{S}_{0}^{\pm}=2\nu\mathbb{D}\mathbf{u}_{0}^{\pm}+\lambda\mathrm{div}\mathbf{u}_{0}^{\pm}\mathbb{I}$,
$p_{0}^{\pm}=(\rho_{0}^{\pm})^{2}e_{\rho}(\rho_{0}^{\pm},\theta_{0}^{\pm})+\theta_{0}^{\pm}p_{\rho}(\rho_{0}^{\pm},\theta_{0}^{\pm})$.

###### Proof.

Since $\widetilde{\phi}_{0}$ is given by (2.54), then we simplify equation
(2.74) to

$\displaystyle-\partial_{\xi}p_{0}\mathbf{n}+\partial_{\xi}\big{(}(\lambda+\nu)\partial_{\xi}\widetilde{\mathbf{u}}_{1}+\nu(\partial_{\xi}\widetilde{\mathbf{u}}_{1}\cdot\mathbf{n})\mathbf{n}\big{)}$
$\displaystyle~{}~{}~{}=\partial_{\xi}\big{(}\partial_{\xi}\widetilde{\phi}_{0}\partial_{\xi}\widetilde{\phi}_{1}\big{)}\mathbf{n}+|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\kappa\mathbf{n}-\partial_{\xi}((\tilde{\phi}_{0}^{3}-\widetilde{\phi}_{0})\widetilde{\phi}_{1})\mathbf{n}.$
(2.79)

We integrate (2.79) from $\xi=-\infty$ to $\xi=+\infty$, and the first and
third summand on the right-hand-side vanish. The first summand on the left-
hand-side of (2.79) turns into

$\displaystyle\displaystyle-\int_{-\infty}^{\infty}\partial_{\xi}p_{0}\mathbf{n}d\xi=-(p(\rho_{0}^{+},\theta_{0}^{+})-p(\rho_{0}^{-},\theta_{0}^{-}))\mathbf{n}=-[p(\rho_{0},\theta_{0})]\mathbf{n}.$
(2.80)

From the matching condition (2.42)-(2.43), we obtain

$\displaystyle\displaystyle\lim_{\xi\rightarrow\pm\infty}\partial_{\xi}\mathbf{u}_{1}=\mathbf{n}\cdot\nabla\mathbf{u}_{0}.$
(2.81)

For the second summand on the left-hand-side of (2.79) we have

$\displaystyle\int_{-\infty}^{\infty}\partial_{\xi}\big{(}((\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{1}\cdot\mathbf{n})\mathbf{n}+\nu\partial_{\xi}\widetilde{\mathbf{u}}_{1}\big{)}d\xi$
$\displaystyle=\big{(}((\nu+\lambda)\partial_{\xi}\widetilde{\mathbf{u}}_{1}\cdot\mathbf{n})\mathbf{n}+\nu\partial_{\xi}\widetilde{\mathbf{u}}_{1}\big{)}|^{\xi=+\infty}_{\xi=-\infty}$
$\displaystyle=[\lambda\mathop{\mathrm{div}}\nolimits\widetilde{\mathbf{u}}_{0}\mathbf{n}+\nu(\nabla\widetilde{\mathbf{u}}_{0}+\nabla^{\top}\widetilde{\mathbf{u}}_{0})\mathbf{n}]$
$\displaystyle=[\mathbb{S}_{0}]\mathbf{n}.$ (2.82)

Combining the equations above together we conclude

$\displaystyle\displaystyle-[p(\rho_{0},\theta_{0})\mathbf{n}]+[\mathbb{S}_{0}\mathbf{n}]=\int_{-\infty}^{\infty}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\kappa\mathbf{n}d\xi=\sigma\kappa\mathbf{n},$
(2.83)

where

$\displaystyle\displaystyle\sigma=\int_{-\infty}^{\infty}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}d\xi=\frac{2}{3}\sqrt{2},$
(2.84)

then it follows (2.78). ∎

For $(C1)~{}M_{\epsilon}=1$, from Lemma 2.2 and Lemma 2.3, we obtain the
following jump conditions on the free interface $\Gamma$.

######  Lemma 2.4.

Letting $\epsilon\rightarrow 0$ in ansatz (2.21) for $(C1)~{}M_{\epsilon}=1$,
it holds

$\displaystyle[\mathbf{u}]_{\Gamma}=0,\quad[\theta]_{\Gamma}=0,$
$\displaystyle\text{on}~{}~{}\Gamma,$ (2.85) $\displaystyle
V_{\mathbf{n}}-\mathbf{u}\cdot\mathbf{n}=0,$
$\displaystyle\text{on}~{}~{}\Gamma,$ (2.86)
$\displaystyle\big{[}\big{(}2\nu\mathbb{D}\mathbf{u}+\lambda\mathrm{div}\mathbf{u}\mathbb{I}-\big{(}\rho^{2}\frac{\partial
e}{\partial\rho}+\theta\frac{\partial
p}{\partial\theta}\big{)}\mathbb{I}\big{)}\mathbf{n}\big{]}_{\Gamma}=\sigma\kappa\mathbf{n},$
$\displaystyle\text{on}~{}~{}\Gamma.$ (2.87)

For $(C2)~{}M_{\epsilon}=\frac{1}{\epsilon}$, we derive the equation (2.77) to
obtain the following lemma.

######  Lemma 2.5.

It holds

$\displaystyle\widetilde{\rho}_{0}(V_{\mathbf{n}}-{\mathbf{u}}_{0}\cdot\mathbf{n})\int_{-\infty}^{+\infty}\widetilde{\rho}_{0}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}d\xi+\sigma\kappa=0.$
(2.88)

###### Proof.

Combining (2.75) and (2.77), we have

$\displaystyle\displaystyle-\widetilde{\rho}_{0}^{2}(V_{\mathbf{n}}-{\mathbf{u}}_{0}\cdot\mathbf{n})\partial_{\xi}\widetilde{\phi}_{0}=2(\mathbf{n}\cdot\nabla)\partial_{\xi}\widetilde{\phi}_{0}+\partial_{\xi}\widetilde{\phi}_{0}\kappa+\partial_{\xi\xi}\widetilde{\phi}_{1}-(3\tilde{\phi}_{0}^{2}-1)\widetilde{\phi}_{1},$
(2.89)

multiplying (2.89) by $\partial_{\xi}\widetilde{\phi}_{0}$ and integrating
from $\xi=-\infty$ to $\xi=+\infty$ , we obtain

$\displaystyle-\int_{-\infty}^{+\infty}\widetilde{\rho}_{0}^{2}(V_{n}-{\mathbf{u}}_{0}\cdot\mathbf{n})|\partial_{\xi}\widetilde{\phi}_{0}|^{2}d\xi$
$\displaystyle=\int_{-\infty}^{+\infty}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}\kappa+\partial_{\xi\xi}\widetilde{\phi}_{1}\partial_{\xi}\widetilde{\phi}_{0}-(3\tilde{\phi}_{0}^{2}-1)\widetilde{\phi}_{1}\partial_{\xi}\widetilde{\phi}_{0}d\xi.$
(2.90)

From (2.35), (2.84) and (2.53), we have

$\displaystyle-\widetilde{\rho}_{0}(V_{\mathbf{n}}-{\mathbf{u}}_{0}\cdot\mathbf{n})\int_{-\infty}^{+\infty}\widetilde{\rho}_{0}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}d\xi$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}|\partial_{\xi}\widetilde{\phi}_{0}|^{2}d\xi\kappa+\int_{-\infty}^{+\infty}\big{(}\partial_{\xi\xi}\widetilde{\phi}_{1}\partial_{\xi}\widetilde{\phi}_{0}-(3\tilde{\phi}_{0}^{2}-1)\partial_{\xi}\widetilde{\phi}_{0}\widetilde{\phi}_{1}\big{)}d\xi$
$\displaystyle=$
$\displaystyle\sigma\kappa-\int_{-\infty}^{+\infty}\partial_{\xi}\widetilde{\phi}_{1}\big{(}-\partial_{\xi\xi}\widetilde{\phi}_{0}+(\tilde{\phi}_{0}^{3}-\widetilde{\phi}_{0})\big{)}d\xi=\sigma\kappa,$
(2.91)

that is (2.88). ∎

For $(C2)~{}M_{\epsilon}=\frac{1}{\epsilon}$, from Lemma 2.2, Lemma 2.3, Lemma
2.5 and Remark 2.1, we obtain the following jump conditions on the free
interface $\Gamma$.

######  Lemma 2.6.

Letting $\epsilon\rightarrow 0$ in ansatz (2.21) for
$(C2)~{}M_{\epsilon}=\frac{1}{\epsilon}$, it holds

$\left\\{\begin{array}[]{llll}\displaystyle[\mathbf{u}]_{\Gamma}=0,\quad[\theta]_{\Gamma}=0,\quad[\rho]_{\Gamma}=0,\\\
\displaystyle\big{[}\big{(}2\nu\mathbb{D}\mathbf{u}+\lambda\mathrm{div}\mathbf{u}\mathbb{I}-\big{(}\rho^{2}\frac{\partial
e}{\partial\rho}+\theta\frac{\partial
p}{\partial\theta}\big{)}\mathbb{I}\big{)}\mathbf{n}\big{]}_{\Gamma}=\sigma\kappa\mathbf{n},\\\
\displaystyle\rho^{2}(V_{\mathbf{n}}-\mathbf{u}\cdot\mathbf{n})=-\kappa,\end{array}\right.\text{on}~{}~{}S,$
(2.92)

$\left\\{\begin{array}[]{llll}\displaystyle[\mathbf{u}]_{\Gamma}=0,\quad[\theta]_{\Gamma}=0,\\\
\displaystyle V_{\mathbf{n}}-\mathbf{u}\cdot\mathbf{n}=0,\\\
\displaystyle\big{[}\big{(}2\nu\mathbb{D}\mathbf{u}+\lambda\mathrm{div}\mathbf{u}\mathbb{I}-\big{(}\rho^{2}\frac{\partial
e}{\partial\rho}+\theta\frac{\partial
p}{\partial\theta}\big{)}\mathbb{I}\big{)}\mathbf{n}\big{]}_{\Gamma}=0,\end{array}\right.\text{on}~{}~{}\Gamma\setminus
S.$ (2.93)

Proof of Theorem 1.1.  We analyze the limit process by the matched asymptotic
expansion. By applying the results of Lemma 2.1, Lemma 2.4 and Lemma 2.6, the
system (1.10) converges to the sharp interface problem (1.15) with the free
interface (1.16)-(1.17).

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