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\newcommand{\e}{\varepsilon}
\newcommand{\rie}{\rho_i^{\e}}
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\newcommand{\ri}[1]{\rho_i^{(#1)}}
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\begin{document}
\markboth{G. Manzi (et al.)}
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\title{A kinetic model of interface motion.\footnote{dedicated to Francesco Guerra on his sixtieth
birthday.}}
\author{Guido Manzi$^\dag$}
\author{Rossana Marra$^*$}
\address{Dipartimento di Fisica, Universit\`a di Roma Tor Vergata\\ Via della Ricerca Scientifica 1
00133 Roma, Italy\\$^\dag$ manzi@roma2.infn.it\\$^*$ marra@roma2.infn.it}
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\maketitle
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\begin{abstract}
We study a kinetic model for a system of two species of particles interacting via a repulsive long
range potential and with a reservoir at fixed temperature. The
interaction between the particles is modeled by a Vlasov term and the thermal bath by a Fokker-Plank
term. We show that in the diffusive and sharp interface limit the motion of the interfaces at low
temperature is described by a Stefan problem or a Mullins-Sekerka motion, depending on the time
scale.
\end{abstract}
\keywords{segregation; interface motion; sharp interface limit; kinetic models.}
\section{Introduction}
Systems of particles interacting via a weak long range
potential on the lattice have been introduced in a series of papers [1],[2],[3] to study
segregation phenomena and their behavior has been widely investigated. The macroscopic evolution of
the conserved order parameter is ruled by a nonlinear nonlocal integral differential equation
having non homogeneous stationary solutions at low temperature, corresponding to the presence of
two different phases separated by interfaces. When the phase domains are very large compared to
the size of the interfacial region (so-called sharp interface limit) the interface motion is
described in terms of a Stefan-like problem or the Mullins-Sekerka motion depending on the time
scale [2].
It
is well known that systems of interacting particles in the real space (as opposite to the lattice)
are much more difficult to study. We propose to start this analysis by considering kinetic models of
particles. We study in this paper a kinetic model of a binary mixture of particles interacting
via a weak long range potential and in contact with a reservoir at fixed temperature. We show that
this model undergoes
phase segregation and that in the sharp interface limit the evolution is again given in terms of a
Stefan-like problem or the Mullins-Sekerka model.
The equations for the one-particle distributions $f_i(x,v,\tau)$
are
\begin{equation}\partial_{\tau} f_{i} +v\cdot \nabla_xf_{i} +F_{i}\cdot \nabla_v
f_{i} =L_\beta f_{i}
\quad i=1,2,
\quad i\neq j,
\label{0.2}
\end{equation}
where $\beta=$ is the inverse temperature of the heat reservoir modeled by
the Fokker-Plank
operator on the velocity space $\mathbb{R}^3$
$$L_\beta f_{i}:=\nabla_v\cdot\biggl(M_\beta\nabla_v\biggl({f_i\over M_\beta}
\biggr)\bigg),\quad
M_\beta(v) = ({2\pi\over\beta } )^{-3/2}\exp(-\beta|v|^2/2)$$
and $F_{i}$ are the self-consistent
forces, whose potential has inverse range $\gamma$, representing the
repulsion between
particles of different species:
$$F_{i}(x,\tau)= -\nabla_x\int d\/x' \gamma^3
U(\gamma|x-x'|)\int {\rm d}vf_{j}
(x',v,\tau),\quad i=1,2,
\quad i\neq j.\label{0.3}$$
Our system is contained in a $3$-dimensional torus (to avoid boundary
effects) and
$U(r)$ is a non negative, smooth function on $\mathbb{R}_+$ with compact
support. This evolution conserves the densities of the two species.
Beyond the spatially constant equilibria, there may be other spatially
non homogeneous stationary
solutions.
Indeed, general entropy
arguments show that
the stationary solutions of these equations are local Maxwellians
with mean value $u=0$, variance $T=\beta^{-1}$, and
densities $\rho_i=\int dv f_i(x,v,\tau)$ satisfying
\begin{equation}
T\log \rho_i(x)+\int dx' \gamma^3U(\gamma |x-x'|) \rho_j(x')=C_i, \quad i=1,2, i\ne j.
\label{0.1}
\end{equation}
Moreover, it is proved in [4], under the assumption of a monotone potential, that at low
temperature there are non homogeneous solutions to (\ref{0.1}), stable in the sense that they minimize the
macroscopic
free energy functional
\begin{equation}
{\cal F}(\rho_1,\rho_2) =
T\int\left[ (\rho_1\ln\rho_1)(x) +
(\rho_2\ln\rho_2)(x)\right]{\rm d }x +
\int U(x-y)\rho_1(x)\rho_2(y){\rm d}x{\rm d}y\ .
\label{free)}
\end{equation}
For example, in $d=1$, these solutions are called fronts and have monotonicity properties. The asymptotic values
at $\pm\infty$ are the values $\bar\rho_i^\pm$ of the densities corresponding at equilibrium to two coexisting
different phases, one reach in species $1$ and the other reach in species $2$. Since these
solutions are unique up to a translation we fix a solution by imposing that
$\rho_1(0)=\rho_2(0)$.
A different kinetic model with conservation also
of momentum and energy, to take into account effects of
variations of temperature and hydrodynamical flows, has been investigated in recent papers
[5], [6]. The interface motion in this case is driven by the hydrodynamic velocity field.
The macroscopic equations for this model are obtained in the diffusive limit: they describe the behaviour of the
system on lenght scales of order
$\e^{-1}$ and time scales of order $\e^{-2}$ in the limit of vanishing $\e$, where $\e$ is the ratio between
the kinetic and the macroscopic scale. Moreover, we choose $\gamma=\e$ so that the range of the potential is finite on
the macroscopic scale. It can be proved that in this limit the equations become a set of two coupled parabolic
equations for the densities $\rho_i(x,\tau)$
\begin{equation}\beta^2\partial_{\tau} \rho_{i}=\Delta \rho_i+ \beta\nabla(\rho_i\nabla U\star
\rho_j),\quad i=1,2,\ i\ne j
\label{0.4}
\end{equation}
where $(U\star g)(x,\tau)=\int dy U(x-y) g(y,\tau)$. These equations can be rewritten in the form of a
gradient flux for the free energy
functional ${\cal F}$
$$
\partial_{\tau} \bar {\rho}=\nabla\cdot \Bigg({\cal M}\nabla {\delta{\cal
F}\over\delta\bar {\rho}
}\Bigg),\quad {\cal M}=
\beta^{-1}\left(
\matrix{
\rho_1&0\cr
0&\rho_2\cr}
\right).
\label{0.5}
$$
where $\bar {\rho}=(\rho_1,\rho_2)$, $\delta{\cal F}\over\delta \rho_i$
denotes the
functional derivative of
${\cal F}$ with respect to $\rho_i$ and
${\cal M}$ is the $2\times 2$ mobility matrix.
This form of the equation is very important to study the stability
properties of the stationary solutions. Since we know that the stationary solutions are
minimizers of the functional ${\cal F}$, we expect to be able to prove that the system relaxes to
that stationary state asymptotically in time, for example using the approach developed in [1] for
a nonconservative equation.
In this paper we study explicitly the sharp interface limit for our model. First, we proceed in
the usual way by investigating the behavior of the macroscopic equations (\ref{0.4}) in the limit $L\to
\infty$ where $L$ is the typical size of the domain. The time has to be scaled as $L^q$ as well.
Using formal matched asymptotic expansions we show that for $q=2$ the limiting equation is a
nonlinear diffusion equation with Dirichlet boundary conditions on the interface. For $q=3$ the
motion of the interface is given in terms of a quasi static free-boundary problem and depends on both
the mean curvature of the interface and the surface tension.
Next, we perform the macroscopic limit at the same time of the interface limit, namely we start from
the kinetic model (\ref{0.1}) and scale space
as
$\epsilon^{-1}$ and time as $\epsilon^{-q}, \ q=2,3$, by keeping fixed $\gamma$, and let $\e$ go to $0$.
Surprisingly enough, in this limit we get the same equations as before. We like to stress that
this second kind of scaling limit does not make sense for lattice models.
\section{Sharp interface limit.}
\setcounter{equation}{0}
In this section we start from the macroscopic equations (\ref{0.4}) and investigate the limit in which the linear
dimension $L$ of the domain goes to infinity, which is the same as sending to zero as $\e=L^{-1}$ the width of the interface (sharp
interface limit). We will consider the evolution equation (\ref{0.4}) with $x\in \e^{-1}\Omega$, where $\Omega$ is a
$3$-dimensional
torus, and
on a time scale of order
$\e^{-q}, q=2,3$. Since we want to study the motion of an interface separating domains of the two different phases,
we consider an initial condition for (\ref{0.4}) in which an interface is present. The initial datum is chosen to be
very close to a profile such that in the bulk its values are
$\bar\rho_i^{\pm}$ and the interpolation between them on the interface is realized along the normal direction in each
point by the fronts. Consider a smooth surface $\Gamma_0\subset \Omega$. Let
$\phi(r,\Gamma_0)$
be the signed distance of the point $r\in \Omega$ from the interface. Consider an
initial profile for the densities
$\rho_i$ of the following type: at distance greater than $O(\e)$ from the interface
(in the bulk) the density profiles $\rho^\e_i(r)$ are almost constant equal to $\bar\rho_i^\pm$; at distance $O(\e)$
(near the interface) we choose
\begin{equation}\rho^\e_i(r)= w_i(\e^{-1}\phi(r,\Gamma_0))+O(\e)
\label{datum}
\end{equation}
where $w_i(z)$ are the fronts solutions of (\ref{0.1}) in $d=1$ and $\gamma=1$, with asymptotic values $\bar\rho_i^\pm$. We remark
that these fronts are also stationary solutions of (\ref{0.4}).
The rescaled densities $\rho_i^{\e,q}(r,t)=\rho_i(\e^{-1}r,\e^{-q}t)$ are solutions of
\begin{equation}\label{scal}
\e^{q-2}\partial_{t}\rho_i^{\e,q}=\nabla_r\cdot\left(\rho_i^{\e,q}T\nabla_r\mu^{\e,q}_i\right)
\end{equation}
\[
\mu^{\e,q}_i(r,t)=T\ln\rho_i^{\e,q}(r,t)+U_{\e}\ast\rho_j^{\e,q}(r,t) ,\quad
U_{\e}(|r-r'|)=\e^{-3}U(\e^{-1}|r-r'|)
\]
We look for a solution to (\ref{scal}) in the form of a Taylor series in $\e$. The presence of the interface forces us to use two
different expansions in the bulk (outer expansion) and close to the interface (inner expansion). Far from
$\Gamma_{t}$ we write
\begin{equation}
\rho_i^{\e,q}(r,t)=\ri{ 0,q}(r,t)+\e\ri{ 1,q}(r,t)+\ldots
\end{equation}
while near the interface
\begin{equation}
\rho_i^{\e,q}(r,t)= \tilde\rho_i^{\e,q}=\tri{ 0,q}(z,r,t)+\e\tri{ 1,q}(z,r,t)+\ldots
\end{equation}
The functions $\tri{i}(z,r,t)$ depend also on a fast variable
$z$, defined as $z=\e^{-1}\phi(r,\Gamma_{t})$, where $\Gamma_t\subset\Omega$ is the interface at time $t$.
The variable $z$ takes into account the orthogonal displacement from the interface and has to be thought as a function of $r$ and
$t$.
The outer normal $\nu(r,t)$ to $\Gamma_{\tau}$
in the point $x(r)$ where the perpendicular through $r$ intersects the interface is the gradient of $\phi$ and the velocity $V(r,t)$ of
the interface is given by
$V=\partial_{t}\phi$.
The matching between the outer and inner expansion takes place in a neighborhood of the interface at a distance that goes to zero
when $\e$ tends to zero .
We require that $\rho_i^{\e,q}(x(r)+\e z\nu(r),t)\approx\tilde\rho_i^{\e,q}(z,r,t)$ for $z=\e^{-1+a}$, $a\in (0,1)$. Moreover,
we require that $\tilde\rho_i^{\e,q}$ satisfy at any order the following front centering condition:
\begin{equation}
\frac{d}{dh}\int_{-\infty}^{+\infty} dz |\tilde\rho_i^{\e,q}(z,r,t)-
w_i(z+h)|^2=0
\label{h}
\end{equation}
We plug
the
$\e$-series in (\ref{scal}) and equate order by order, by taking into account the matching conditions. We do not give any detail in
this section for lack of space and simply
state the result. We will explain how the method works starting from the kinetic model in the next section. The limiting equations are:
Case $q=2$.
\begin{equation}
\left\{
\begin{array}{ll}
\displaystyle{\partial_{ t}\ri{0}=\nabla_r\cdot\left(T\ri{0}\nabla_r\mi{0}\right)}&\mbox{for $r\in \Omega\setminus\Gamma_t$}\\
\displaystyle{\lim_{h\rightarrow 0^{\pm}}\ri{0}(r+h\nu,t)=\bar\rho_i^{\pm}}&\mbox{for $r\in\Gamma_t$}\\
\ri{0}(r,0)=\hat\rho_i(r)&\mbox{for all $r$ in $\Omega$}
\label{q2}
\end{array}
\right.
\end{equation}
where $
\mu_i^{(0)}=T\ln\rho_i^{(0)}+\rho_i^{(0)}\int drU(r)
$, $\bar\rho_i^{\pm}$ are the asymptotic values of the fronts $w_i$ and $\hat\rho_i$ is a suitable initial datum.
The interface moves with velocity given by
\begin{equation}
V=\frac{\left[T \rho_i^{(0)}\nu\cdot\nabla_r\mu_i^{(0)}\right]_-^+}{[w_i]_{-\infty}^{+\infty}}
\label{V2}
\end{equation}
where the symbol $[g]_-^+$ denotes the difference $\lim_{h\to 0^+}g(r+\nu h)-\lim_{h\to 0^-}g(r+\nu h)$ while
$[w_i]_{-\infty}^{+\infty}$ is the difference between the asymptotic values $\bar\rho^\pm$. Though $V$ seems to depend on the index $i$, in
virtue of the symmetry which links together
$\lim_{z\rightarrow\pm\infty} w_1$ and $\lim_{z\rightarrow\pm\infty} w_2$, the velocity of the
interface is well defined and indeed does not depend on the index $i$. The problem (\ref{q2})-(\ref{V2}) is a free-boundary partial
differential
equation, sometimes called a Stefan problem. The interface moves on this scale of time because of the difference of density on the two
sides of the interface. The evolution of the densities is ruled by a diffusive equation with Dirichlet boundary conditions on a boundary
moving according to the densities.
\vskip.2cm
Case $q=3$.
\begin{equation}
\left\{
\begin{array}{ll}
\triangle_r\mi{1}(r, t)=0&\mbox{for $r\in \Omega\setminus\Gamma_t$}\\
\displaystyle{\mi{1}(r, t)=-\frac{K(r, t)S}{[w_i]_{-\infty}^{+\infty}}}&\mbox{for $r\in\Gamma_t$}
\label{q3}
\end{array}
\right.
\end{equation}
and the interface moves with velocity
\begin{equation}
V =\frac{\left[T\rho_i^{(0)}\nu\cdot\nabla_r\mu_i^{(1)}\right]_-^+}{[w_i]_{-\infty}^{+\infty}}
\label{V3}
\end{equation}
where $K$ is the mean curvature and $S$ is the surface tension given by
\begin{equation} \frac{1}{2}\int dz dz' (z-z')\sum_{i\neq j}[ w'_i(z)\tilde U(z-z')w_j(z')],\
\tilde U=\int dy_1dy_2U(z,y_1,y_2)
\label{tension}
\end{equation}
The problem (\ref{q3})-(\ref{V3}) describes the evolution of the so-called Mullins-Sekerka flow. The interface on the slower scale of time
$\e^{-3}$ feels the surface tension effects.
\section{Sharp interface limit and diffusive limit.}
In this section we come back to the kinetic model, described by equations (\ref{0.2}) and perform the macroscopic limit
in such a way that at the same time the interface becomes sharp, namely
we scale position and time as $\e^{-1}$ and $\e^{-q}$ respectively, while keeping fixed (equal to $1$)
$\gamma$. We recall that here $\e$ has the meaning of ratio between the kinetic and macroscopic scales. The width of
the interface on the macroscopic scale is then of order
$\e$, so that in the limit $\e\to 0$ the interface becomes sharp. The rescaled density distributions
$
f_i^{\e}(r,v,t)=f_i(\e^{-1}r,v,\e^{-q}t)
$, with $f_i(x,v,\tau)$ solutions of (\ref{0.2}),
are solutions of
\begin{equation}\label{maineq}
\e^q\partial_{t}f_i^{\e}+\e v\cdot\nabla_r f_i^{\e}+\e F_i^{\e}\cdot\nabla_v f_i^{\e}=L_{\beta}f_i^{\e}.
\end{equation}
\[
F_i^{\e}(r,\tau)=-\nabla_r\int dr'\e^{-3}U(\e^{-1}|r-r'|)\int dv'f_j^{\e}(r',v',t)=:-\nabla_r g_i^{\e}.
\]
We consider a situation in which initially an interface is present, as in the previous section. Since the stationary non homogeneous
solution of (\ref{0.2})
is given by the Maxwellian multiplied by the front density profile $w_i$ we let our system start initially close to that stationary
solution and choose as initial datum $f_i^\e(r,v)=M_\beta(v)\rho_i^\e$, with the density profiles as in the previous section
(\ref{datum}). We use also in this case the method of the matching expansions and look for a solution to (\ref{maineq}) in the form
\begin{eqnarray*}
&f_i^{\e}(r,v,t)=f_i^{(0)}(r,v,t)+\e f_i^{(1)}(r,v,t)+\ldots&\\
&f_i^{\e}(r,v,t)=\tilde f_i^{\e}(z,r,v,t)=\tilde f_i^{(0)}(z,r,v,t)+\e\tilde f_i^{(1)}(z,r,v,t)+\ldots&
\end{eqnarray*}
where the first line is the expansion in the bulk (outer) while the second one is close to the
interface (inner). The outer expansion and the inner one are matched at a distance $\e^a, a\in(0,1)$. In this way one obtains links
between the values of $f_i^{(k)}$ almost next to $\Gamma_{\tau}$ and those of $\tilde f_i^{(h)}$ very far away from $\Gamma_{\tau}$ with
respect to the $z$ variable. In other words, we have the following
{\it matching conditions}
\goodbreak
\begin{eqnarray*}
&f_i^{(0)}(r,v,t)=\tilde f_i^{(0)}(z,r,v,t)&\mbox{(for $r$ next to $\Gamma_{t}$ and $z$ very large)}\\
&f_i^{(1)}+z\nu\cdot\nabla_r f_i^{(0)}=\tilde f_i^{(1)}&( \qquad\qquad\qquad\quad''\qquad\qquad\qquad\quad)\\
&\displaystyle{f_i^{(2)}+z\nu\cdot\nabla_r f_i^{(1)}+\frac{z^2}{2}(\nu\cdot\nabla_r)^2f_i^{(0)}=\tilde
f_i^{(2)}}&(\qquad\qquad\qquad\quad ''\qquad\qquad\qquad\quad)\\
&\ldots&
\label{match}
\end{eqnarray*}
Finally, we impose the front centering condition (\ref{h}), where now the density $\tilde\rho^{\e,q}$ is given by $\int dv
\tilde f_i^\e(z,r,v,t)$.
Let us start by the case $q=2$.
By inserting the outer expansion in the equation we get for the three lowest orders
\begin{eqnarray*}
\e^{-2})&L_{\beta}f_i^{(0)}=0&\\
\e^{-1})&v\cdot\nabla_r f_i^{(0)}+F_i^{(0)}\cdot\nabla_v f_i^{(0)}=L_{\beta}f_i^{(1)}&\\
\e^0\ \ )&\partial_{t}f_i^{(0)}+v\cdot\nabla_r f_i^{(1)}+F_i^{(0)}\cdot\nabla_v f_i^{(1)}+F_i^{(1)}\cdot\nabla_v
f_i^{(0)}=L_{\beta}f_i^{(2)}&
\end{eqnarray*}
and we readily deduce from the first equation (order $\e^{-2})$) that $f_i^{(0)}(r,v,t)=\rho_i^{(0)}(r,t)M_{\beta}(v)$. In view of that,
the equation at order
$\e^{-1}$ is rewritten in this way:
\[
M_{\beta}v\cdot(\nabla_r\rho_i^{(0)}+\beta\rho_i^{(0)}\nabla_r g_i^{(0)})=L_{\beta}f_i^{(1)}
\]
that gives
$
f_i^{(1)}=-T M_{\beta}v\cdot(\nabla_r\rho_i^{(0)}+\beta\rho_i^{(0)}\nabla_r g_i^{(0)})
$, where $g_i^{(0)}=\rho_i^{(0)}\int U(r) dr$.
By integrating the $\e^0$ order equation over the velocity all the terms vanish, but the first two on the
l. h. s.:
\[
\partial_{t}\rho_i^{(0)}-{T^2}(\triangle_r\rho_i^{(0)}+\beta\nabla_r\cdot(\rho_i^{(0)}\nabla_r g_i^{(0)}))=0
\]
This equation is exactly the first of (\ref{q2})
with $\mu_i^{(0)}$ the order zero chemical potential.
In order to go on with the inner expansion, we replace the derivative operators that act on $f_i^{\e}$ with
those corresponding to $\tilde f_i^{\e}$. For a function $ h(r,t)=\tilde h(z,r,t)$ we have
\begin{equation}
\nabla_r h=\e^{-1}\nu\partial_z\tilde h+\tilde \nabla_r \tilde h; \quad
\partial_{t}h=\e^{-1}V\partial_z\tilde h+\partial_{t}\tilde h
\end{equation}
where $\tilde\nabla_r$ is the gradient with respect only to the $r$ coordinate. Moreover, thanks to the fact that
$\nu\cdot\tnr\tw=0$, the following holds too
\[
\triangle_r h=\e^{-2}\partial_z^2\tilde h+\e^{-1}(\nabla_r\cdot\nu)\partial_z\tilde h+\tilde\triangle_r\tilde h
\]
We write down the orders $\e^{-2}$ and $\e^{-1}$:
\begin{eqnarray*}
\e^{-2})&v\cdot\nu\partial_z\tilde f_i^{(0)}-\nu\cdot\nabla_v\tilde f_i^{(0)}\partial_z\tilde g_i^{(0)}=L_{\beta}\tilde f_i^{(0)}&\\
\e^{-1})&V\partial_z\tilde f_i^{(0)}+v\cdot\nu\partial_z\tilde f_i^{(1)}+v\cdot\tilde\nabla_r\tilde f_i^{(0)}-
\nu\cdot\nabla_v\tilde f_i^{(0)}\partial_z\tilde g_i^{(1)}-&\\
&\nu\cdot\nabla_v\tilde f_i^{(1)}\partial_z\tilde g_i^{(0)}-\nabla_v\tilde f_i^{(0)}\cdot\tilde\nabla_r\tilde
g_i^{(0)}=L_{\beta}\tilde f_i^{(1)}&
\end{eqnarray*}
One can show that a solution of the first equation is necessarily of the form
$
\tilde f_i^{(0)}=\tilde\rho_i^{(0)}M_{\beta}
$.
By replacing this expression in the order $\e^{-2})$ one gets
\begin{equation}\label{istantone}
v\cdot\nu M_{\beta}(\partial_z\tilde\rho_i^{(0)}+\beta\rho_i^{(0)}\partial_z\tilde g_i^{(0)})=0\;\;\Longleftrightarrow\;\;\partial_z\tilde\mu_i^{(0)}=0,
\end{equation}
where $\tilde\mu_i^{(0)}$ is defined as $\mu_i^{(0)}$. (\ref{istantone}) is nothing but (\ref{0.1}), the relation defining a front. The first of the matching conditions imposes that at infinity the
density profiles have to be
$\bar\rho^\pm_i$. Hence the only solution of (\ref{istantone}) with these boundary conditions,
after taking into account the front centering condition (\ref{h}), is exactly
$w_i(z,t)$. Since $w_i$ are functions only of the variable $z$, we have $\tilde\nabla_r\tilde\rho_i^{(0)}=0$ and
$\tilde\nabla_r\tilde g_i^{(0)}=0$, too, because, as we will see later, $\tilde g_i^{(0)}=\tilde U\ast\tilde\rho_j^{(0)}$. Thus in the
$\e^{-1}$ order there are some simplifications and, by integrating over
$v$, we obtain
\begin{equation}\label{match1}
V\partial_z\tilde\rho_i^{(0)}+\int dv v\cdot\nu\partial_z\tilde f_i^{(1)}=0
\end{equation}
By integrating (\ref{match1}) over $z$ ,
$$V[\tilde\rho_i^{(0)}]^{+\infty}_{-\infty} =\Big[\int dv(\nu\cdot v)\tilde f_i^{(1)}(z,v,r,t)\Big]^{+\infty}_{-\infty}$$
By the second of the matching conditions for $z$ very large we have that
\[
\nu\cdot\int dvv\tilde f_i^{(1)}=\nu\cdot\int dvv(f_i^{(1)}+z\nu\cdot\nabla_r f_i^{(0)})=\nu\cdot\int dvv f_i^{(1)}
\]
The second equality comes from the fact that $f_i^{(0)}$ is an even function of $v$. By using the explicit expression of
$f_i^{(1)}$ we get that the above integral is equal to $
-\frac{\rho_i^{(0)}}{\beta}\nu\cdot\nabla_r \mu_i^{(0)}$. On the other hand,
the first of the matching conditions implies
$[\tilde\rho_i^{(0)}]^{+\infty}_{-\infty}=[\bar\rho^+-\bar\rho^-]$, so that
from (\ref{match1}) the expression for the velocity of the interface is given by \ref{V2}. This completes the analysis of the case
$q=2$.
We now examine the $q=3$ case. Replacing the outer expansion in the rescaled expansion and
equating order by order, one gets:
\begin{eqnarray*}
\e^{-3})&L_{\beta} f_i^{(0)}=0&\\
\e^{-2})&v\cdot\nabla_r f_i^{(0)}+F_i^{(0)}\cdot\nabla_v f_i^{(0)}=L_{\beta}f_i^{(1)}&\\
\e^{-1})&v\cdot\nabla_r f_i^{(1)}+F_i^{(0)}\cdot\nabla_v f_i^{(1)}+F_i^{(1)}\cdot\nabla_v f_i^{(0)}=L_{\beta} f_i^{(2)}&
\end{eqnarray*}
From the first one, we deduce that $f_i^{(0)}=\rho_i^{(0)}M_{\beta}$; thus, the solution of the order $\e^{-2}$ equation can be computed:
\[
f_i^{(1)}=\rho_i^{(1)}M_{\beta}-TM_{\beta}v\cdot(\nabla_r\rho_i^{(0)}+\beta\rho_i^{(0)}\nabla_rg_i^{(0)})=\rho_i^{(1)}M_{\beta}-M_{\beta}\rho_i^{(0)}v\cdot\nabla_r\mu_i^{(0)}
\]
The situation is quite different from the case $q=2$
because, as we will see next, at the order zero the density is constant so that the first relevant term is
$\rho_i^{(1)}$. Let us integrate the order $\e^{-1}$ equation in the velocities
\[
-\sum_k\sum_h\int
dvv_k(\partial_{r_k}\rho_i^{(0)}\partial_{r_h}\mu_i^{(0)})v_hM_{\beta}=0\;\;\Longleftrightarrow\;\;-T\nabla_r\cdot(\rho_i^{(0)}\nabla_r\mu_i^{(0)})=0
\]
The choice of the initial data implies that the only solution of that equation is the constant one;
consequently $\partial_{t}\rho_i^{(0)}=0$ and $\nabla_r\mu_i^{(0)}=0$. Thus $f_i^{(1)}$ becomes simply $\rho_i^{(1)}M_{\beta}$. As
usual, $f_i^{(2)}$ is determined by the $\e^{-1})$ equation, which becomes by the constancy of $\rho_i^{(0)}$ and the
explicit version of $F_i^{(1)}$ that we will see later,
\[
M_{\beta}v\cdot\nabla_r(\rho_i^{(1)}+\beta\rho_i^{(0)}g_i^{(1)})=L_{\beta}f_i^{(2)}
\]
So, discarding the term in the kernel of $L_{\beta}$, one has
\begin{equation}\label{f2}
f_i^{(2)}=-T M_{\beta}v\cdot\nabla_r(\rho_i^{(1)}+\beta\rho_i^{(0)}g_i^{(1)})
\end{equation}
We need the next $\e^0$ order equation:
\[
\e^0)\;\;\partial_{t}f_i^{(0)}+v\cdot\nabla_rf_i^{(2)}+F_i^{(0)}\cdot\nabla_vf_i^{
(2)}+F_i^{(1)}\cdot\nabla_vf_i^{(1)}+F_i^{(2)}\cdot\nabla_vf_i^{(0)}=L_{\beta}f_i^{(3)}
\]
Integrating in $v$ and using (\ref{f2}), we obtain
\[
-{T^2}\triangle_r(\rho_i^{(1)}+T\rho_i^{(0)}g_i^{(1)})=-T\rho_i^{(0)}\triangle_r\left(\rho_i^{(1)}T[\rho_i^{(0)}]^{-1}+g_i^{(1)}\right)=0\;\;\Longleftrightarrow\;\;\triangle_r\mu_i^{(1)}=0
\]
where we introduced $\mu_i^{(1)}=\rho_i^{(1)}/\beta\rho_i^{(0)}+g_i^{(1)}$, the order one correction to the chemical potential. This is
the first equation of the limiting problem (\ref{q3}).
Let us turn to the inner expansion.
\begin{eqnarray*}
\e^{-3})&v\cdot\nu\partial_z\tilde f_i^{(0)}-\nu\cdot\nabla_v\tilde f_i^{(0)}\partial_z\tilde g_i^{(0)}=L_{\beta}\tilde f_i^{(0)}&\\
\e^{-2})&v\cdot\nu\partial_z\tilde f_i^{(1)}+v\cdot\tilde\nabla_r\tilde f_i^{(0)}-\nu\cdot\nabla_v\tilde f_i^{(0)}\partial_z\tilde g_i^{(1)}
-\nu\cdot\nabla_v\tilde f_i^{(1)}\partial_z\tilde g_i^{(0)}\cr &-\nabla_v\tilde f_i^{(0)}\cdot\tilde\nabla_r\tilde
g_i^{(0)}=L_{\beta}\tilde f_i^{(1)}
\end{eqnarray*}
The first equation is exactly the same as that in the case $q=2$. We deduce that $\tilde\mu_i^{(0)}=cost$. In other words $\tilde
f_i^{(0)}=\tilde\rho_i^{(0)}M_{\beta}$, where $\tilde\rho_i^{(0)} =w_i$. Thus
$\tilde\nabla_r\tilde\rho_i^{(0)}=0$ and $\tilde\nabla_r\tilde g_i^{(0)}=0$. This provides some simplifications in the second equation.
If we now look for a solution in the form $\tilde f_i^{(1)}=\tilde\rho_i^{(1)}M_{\beta}$, we are led to the following equation:
\begin{equation}\label{e-2}
v\cdot\nu M_{\beta}(\partial_z\tilde\rho_i^{(1)}+\beta\tilde\rho_i^{(0)}\partial_z\tilde g_i^{(1)}+\beta\tilde\rho_i^{(1)}\partial_z\tilde g_i^{(0)})=0
\end{equation}
Taking into account that $-\beta\partial_z\tilde g_i^{(0)}=\partial_z\ln\tilde\rho_i^{(0)}$, because
$\partial_z\tilde\mu_i^{(0)}=0$, we get
\begin{equation}
v\cdot\nu
M_{\beta}\beta\tilde\rho_i^{(0)}\partial_z\left(T\tilde\rho_i^{(1)}(\tilde\rho_i^{(0)})^{-1}+\tilde g_i^{(1)}\right)=0
\Longleftrightarrow\;\;\partial_z\tilde\mu_i^{(1)}=0
\end{equation}
where $\tilde\mu_i^{(1)}$ is defined as $\mu_i^{(1)}$ above.
Let us go to order $\e^{-1}$ to investigate the expression for the velocity of the interface:
\begin{eqnarray*}
&V\partial_z\tilde f_i^{(0)}+v\cdot\nu\partial_z\tilde f_i^{(2)}+v\cdot\tilde\nabla_r\tilde f_i^{(1)}-\nu\cdot\nabla_v\tilde f_i^{(0)}\partial_z\tilde g_i^{(2)}-&\\
&\nu\cdot\tilde f_i^{(1)}\partial_z\tilde g_i^{(1)}-\nu\cdot\nabla_v\tilde f_i^{(2)}\partial_z\tilde g_i^{(0)}-\nabla_v\tilde f_i^{(0)}\cdot\tilde\nabla_r\tilde g_i^{(1)}-\nabla_v\tilde f_i^{(1)}\cdot\tilde\nabla_r\tilde g_i^{(0)}=L_{\beta}\tilde f_i^{(2)}&
\end{eqnarray*}
Integrating over $v$, only two terms survive, due to the presence of the velocity gradient, the constancy of $\tilde\mu_i^{(0)}$ and
the explicit form of $\tilde f_i^{(1)}$. Thus we are left with
\[
V\partial_z\tilde\rho_i^{(0)}+\int dvv\cdot\nu\partial_z\tilde f_i^{(2)}=0
\]
The next step consists in integrating in $z$, but before that we apply the matching conditions for $\tilde f_i^{(2)}$. One is reduced to
computing the following integral:
\[
\int dvv\cdot\nu\left(f_i^{(2)}+z\nu\cdot\nabla_r f_i^{(1)}+\frac{1}{2}z^2\sum_k\sum_h\nu_k\nu_h\partial_{r_k}\partial_{r_h}f_i^{(0)}\right)=\int dvv\cdot\nu f_i^{(2)}
\]
because we recall that $\rho_i^{(0)}$ is constant and the maxwellian is an even function. Thus, in view of (\ref{f2}), one has
\begin{eqnarray*}
&\displaystyle{\int dvv\cdot\nu f_i^{(2)}=-T\int dvM_{\beta}\sum_k\sum_h\nu_k v_kv_h(\partial_{r_h}
\rho_i^{(1)}+\beta\rho_i^{(0)}\partial_{r_h}g_i^{(1)})=}&\\
&\displaystyle{-{T^2}\nu\cdot\left[\beta\rho_i^{(0)}\nabla_r\left(T\rho_i^{(1)}(\rho_i^{(0)})^{-1}+g_i^{(1)}\right)\right]=-T\rho_i^{(0)}\nu\cdot\nabla_r\mu_i^{(1)}}&
\end{eqnarray*}
As a consequence, the velocity of the interface is given by (\ref{V3}). We are left with finding the expression of the term $g_i^{(1)}$
involving the long range potential $U$, which is defined in the second of (\ref{maineq} ), in order to determine the function
$\mu_i^{(1)}(r,t)$ on the interface. We start from
$$\tilde\mu_i^{(1)}=\tilde\rho_i^{(1)}/\beta\tilde\rho_i^{(0)}+\tilde g_i^{(1)},\quad \tilde g_i^{(\e)}=\int
dr'\e^{-3}U(\e^{-1}|r-r'|)\rho_j^{\e}(r',t)$$
It is possible to show (along the lines in [2]) that
$$\tilde\mu_i^{(1)}=\tilde\rho_i^{(1)}/\beta\tilde\rho_i^{(0)}+\tilde U\ast\tilde \rho_i^{(1)}+{K}\int dz' (z-z')\tilde
U(z-z')\tilde\rho^{(0)}_{j}(z')$$ Multiplying both side by $w'_i$ (we remember that $\tilde\rho^{(0)}_{j}=w_j$ ),
integrating over $z$ and noticing that the first two terms sum to zero because $w$ is solution of the front equation (\ref{0.4}), we
get
$$\tilde\mu_i^{(1)}[w_i]^{+\infty}_{-\infty}=\int dz dz' (z-z')[w_i'(z)\tilde U(z-z')w_j(z')]$$
To reconstruct the surface tension we use the symmetry $w'_1(z)=-w'_2(-z)$ [4].
Finally, by using the matching conditions involving $\tilde \mu^{(1)}$ and (\ref{tension}) we get the second of (\ref{q3}), so
completing the investigation of the case $q=3$.
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\end{document})