\section{Introduction}

 Ferromagnetic materials can attain a large magnetization under the action of a small applied magnetic field.
 To explain this phenomenon, in $1907$, Weiss suggested that a \textit{spontaneous magnetization} occurs.
 In $1928$ Heisenberg explained the spontaneous magnetization postulated by Weiss in terms of the \textit{exchange energy}.
  In $1935$ Landau and Lifschitz (cf. \cite{LL}) proposed a quantitative theory, now known as \textit{micromagnetics}.
For a  piece of ferromagnet-which is supposed to be a regular bounded open set $\Omega$ in ${\mathbb R}^3$
with a smooth boundary, and locally on one side of $\Gamma$-
  the magnetic state at a point $x \in \Omega$ at time  $t$ is given by a
  vector $u(t,x) \in {\mathbb R}^3$ which belongs to the unit sphere of ${\mathbb R}^3$,
   called the  \textit{magnetic moment}.  The Landau-Lifschitz equations  read:
\begin{eqnarray}
\label{LL1}
 \partial_t{u^\varepsilon } =   u^\varepsilon  \wedge ( \cH(u^\varepsilon ) + \varepsilon^2 \Delta u^\varepsilon  )
- u^\varepsilon  \wedge \big( u^\varepsilon  \wedge ( \cH(u^\varepsilon ) + \varepsilon^2 \Delta u^\varepsilon  ) \big) \quad
 \mbox{ in  }  \Omega  ,
 \end{eqnarray}
where $\varepsilon >0$ is the exchange coefficient.
We denote ${\cH}(u) := H_{|\Omega }  \in L^2(\Omega;{\mathbb R}^3 )$
where the magnetic field $ H \in L^2({\mathbb R}^3;{\mathbb R}^3)$,
is the unique solution of the following elliptic problem,

\begin{equation}
\label{H}
\left\{\begin{array}{l}
H \in L^2( {\mathbb R}^3;{\mathbb R}^3) \ ,\\
\\
\mbox{{\rm curl}}\; H =0\mbox { in }{\mathbb R}^3 \ ,\\
\\
\mbox{{\rm div}}\; (H +\overline{u})=0\mbox { in }{\mathbb R}^3 \ ,
\end{array} \right.
\end{equation}

where $\overline{u}$ means the extension of $u$
by $0$ outside of the set $\Omega$.
The equations (\ref{LL1}) are supplemented by the homogeneous Neumann boundary condition:
\begin{eqnarray}
 \label{LL2}
\partial _{{\bf n}} u^\varepsilon  = 0 \,  \mbox{ in \  }  \Gamma   ,
\end{eqnarray}
where ${\bf n}$ is the unitary outward normal at the boundary $\Gamma$, and by an initial condition:
\begin{eqnarray}
\label{LL3}
u^\varepsilon _{| t=0}  = u_0 .
\end{eqnarray}
The solution
must also satisfy the constraint
\begin{equation} \label{unit}
| u^\varepsilon (t,x) | = 1 , \quad  \mbox{ for \  }  x\in \Omega , \ t \geq 0
\end{equation}
which is obviously propagated from the initial data
as soon as it is satisfied at $t=0$.

In this paper we study the asymptotic behaviour of the solutions of the Landau-Lifschitz
equations (\ref{LL1})-(\ref{LL2})-(\ref{LL3}) as the exchange coefficient
$\varepsilon$ tends to zero.
From a formal point of view, when $\varepsilon=0$, the system (\ref{LL1})-(\ref{LL2})-(\ref{LL3})
becomes
\begin{equation}
\label{LL0}
\left\{
\begin{array}{l}
 \partial_t{u^0} =   u^0 \wedge \cH(u^0)
- u^0 \wedge ( u^0 \wedge  \cH(u^0)  ) \quad
 \mbox{ in  } \Omega  \\
\\
u^0_{| t=0}  = u_0 \ ,
\end{array}
\right.
\end{equation}
where no boundary condition is needed.
In the paper \cite{CFG1} it is proved that, for
\emph{smooth enough solutions} the system (\ref{LL0})
is a "good approximation" of the full system (\ref{LL1})-(\ref{LL2})-(\ref{LL3})
in the sense that the solution $u^0$
of (\ref{LL0}) is indeed limit in $L^2([0,T]\times \Omega)$
 of solutions $u^\varepsilon$ of (\ref{LL1})-(\ref{LL2})-(\ref{LL3})
as $\varepsilon \rightarrow 0$. However, this result holds under
the assumption that   $u^0$ belongs
to the space $\cC\big( [0,T],H^5(\Omega)\big)$ where $H^5(\Omega)$
is the usual Sobolev space.
In particular this assumption excludes the case where $u^0$
is \emph{discontinuous } across
an hypersurface contained in $\Omega$
and it was one  motivation  behind this paper to treat that case.

First of all, let us observe that the system (\ref{LL0})
actually admits discontinuous solutions. To simplify,
we will restrict the analysis to piecewise smooth solutions.
We assume that   $\Sigma$ is  a smooth
compact hypersurface contained in $\Omega$.
For $0 \leq   s < \infty$ call $H^s_\Sigma(\Omega)$ the set of functions
$u\in L^2(\Omega)$ such that $u_{|\Omega_\pm} \in H^s( \Omega_\pm )$
where $H^s(\Omega_\pm)$ is the usual Sobolev space on $L^2$.
We endow  $H^s_\Sigma(\Omega)$  with the norm
$$
\| u \|_{H^s_\Sigma} := \| u_{|\Omega_-}  \|_{H^s(\Omega_-)}
+ \| u_{|\Omega_+}  \|_{H^s(\Omega_+)}
$$
This definition
extends to the case when $s=\infty$: the space $H^\infty_\Sigma(\Omega)$  is
the natural Fr\'echet space.
We get the following result of global existence of solution of
$(\ref{LL0})$ discontinuous through the hypersurface  $\Sigma$.
 \begin{theorem}
\label{hyp}
Let $s \in \,]\frac{3}{2} , \infty ]$ and $u_0 \in H^s_\Sigma(\Omega) $.
Then there  exists a unique $u^0 \in \cC^\infty\big({\mathbb R}, H^s_\Sigma(\Omega) \big)$
solution of the Cauchy problem $(\ref{LL0})$.
\end{theorem}
\begin{proof}
This result can be easily
obtained by following the proof of Proposition $4.1$ of \cite{CFG1}, with
only a few adaptations.
By the way in the closer setting of semilinear symmetric hyperbolic system, it is
well known since the works of M\'etivier \cite{m1}
that there exist some local piecewise regular solutions
discontinuous across a smooth hypersurface which is a characteristic
hypersurface of constant multiplicity for this hyperbolic system.
In the present setting, the proof is in fact
simpler since the hypersurface $\Sigma$ is totally characteristic.
Moreover thanks to $(\ref{unit}) $ and since the operator $\mathcal{H}$ satisfies the
transmission property,  our setting allows to conclude to a global
existence.
\end{proof}

Let us now claim a first theorem about the asymptotic behaviour of the solutions of the Landau-Lifschitz
equations (\ref{LL1})-(\ref{LL2})-(\ref{LL3}) as the exchange coefficient
$\varepsilon$ tends to zero.

\begin{theorem}
\label{main}
Let $u^0 \in \cC^\infty\big({\mathbb R}, H^\infty_\Sigma(\Omega) \big) $
be a solution of $(\ref{LL0})$.
 There exist $T>0$ and a family
of solutions $u^\varepsilon \in \cC^\infty([0,T] \times \Omega)$,
$\varepsilon \in ]0,1]$,
of the equation $(\ref{LL1})$ on $[0,T] \times \Omega$,
of the equation $(\ref{LL2})$ on $[0,T] \times \Gamma$,
such that  there exist $C >0$ and $\varepsilon_0$  such that for all $\varepsilon \in
]0,\varepsilon_0 ]$,
\begin{eqnarray*}
|| u^\varepsilon - u^0 ||_{L^2([0,T] \times \Omega)} \leqslant C \varepsilon^\frac{1}{2} .
\end{eqnarray*}

\end{theorem}

To begin with some comments about Theorem $\ref{main}$ let us stress that
we do not prescribe the initial data $(\ref{LL3})$ for the $u^\varepsilon$.
Thus the traces of the  $u^\varepsilon$ at $t=0$ are not equal in general to  the trace of  $u^0$ at $t=0$.
So Theorem $\ref{main}$ claims the existence of  local in time solutions  $u^\varepsilon  \in \cC^\infty$,
of the equation $(\ref{LL1})$ on $ \Omega$,
of the equation $(\ref{LL2})$ on $ \Gamma$, which converge to $u^0$ as
$\varepsilon$ tends to zero in $L^2$, with a rate of convergence in $\varepsilon^\frac{1}{2}$.

Indeed, in this paper, we will claim a more accurate result in  Theorem
$\ref{main2}$ by showing that the $u^\varepsilon$ can be described with a WKB expansion
which involves some boundary layers profiles.
On one hand, a boundary layer appears near the boundary to compensate the lost of the Neumann condition
 from the complete model (\ref{LL1})-(\ref{LL2})-(\ref{LL3}) to
the limite model  (\ref{LL0})  ($\varepsilon = 0$).
Such a boundary layer was already studied in paper \cite{CFG1}.
 The amplitude of this boundary is
weak and its behaviour is linear.
On another hand, there are some boundary layers on each side of the hypersurface $\Sigma$.
Their task is to compensate the lost of transmission conditions across $\Sigma$
from the complete model  (\ref{LL1})-(\ref{LL2})-(\ref{LL3}) to
the limit model  (\ref{LL0})  ($\varepsilon = 0$).

\begin{remark}
\rm
Such an analysis is inspired by the paper \cite{S1} where we show that discontinuous solutions of
 multidimensional semilinear symmetric hyperbolic systems,
 which are regular outside of a smooth hypersurface characteristic of constant multiplicity, are
  limits, when $\varepsilon \rightarrow 0$, of solutions  $(u^\varepsilon)_{\varepsilon \in ]0,1]}$ of the system perturbated
  by a viscosity of size $\varepsilon $.
In this paper, we adapt the method to the ferromagnetism quasi-static
model,
where in particular the non local operator ${\cH}$ occurs.
We point out that for the limit model ($\varepsilon=0$), the hypersurface is totally characteristic.
As a consequence, the analysis involves only characteristic boundary layers.
On the opposite, \cite{S1} stresses the occurrence of characteristic and non characteristic boundary layers.
 It could be also possible -as in \cite{S1}- to study the case where the singularity  is weaker than a jump of the function $u^0$
as a jump of a derivative of the function $u^0$.
Then we can take $T$ as big as we want and the quality of the approximation is as better
as the jump concerns a higher order
derivative. We also refer to papers \cite{GW}, \cite{GR}, \cite{S1} for the use of boundary layers in transmission strategy.
\end{remark}

\begin{remark}
\rm
It could be interesting to know if it is possible to obtain such a result
in the non static case for which the Landau-Lifschitz
equation is coupled with the Maxwell system of electromagnetic.
For such a model an
analysis of the boundary layer induced by the Neumann boundary condition
on $\Gamma$ is performed in \cite{CFG2}.
\end{remark}

\begin{remark}
\rm
With the same method than the one used in this paper, it is
possible to get global in time
$O(\varepsilon^s)$ approximation  for all $s < \frac{1}{2}$.
More precisely  for any $s <  \frac{1}{2}$ there exists a family
of solutions $u^\varepsilon  \in \cC^\infty([0,T_\varepsilon] \times \Omega)$, of the equation $(\ref{LL1})$ on $[0,T_\varepsilon] \times \Omega$,
of the equation $(\ref{LL2})$ on $[0,T_\varepsilon] \times \Gamma$,
$\varepsilon \in ]0,1]$, with lim$_{\varepsilon \rightarrow 0^+} \, T_\varepsilon = \infty$,
such that  for all $T>0$,  there exists $C >0$ and $\varepsilon_0$  such that for all $\varepsilon \in
]0,\varepsilon_0 ]$, there holds
$|| u^\varepsilon - u^0 ||_{L^2([0,T] \times \Omega)} \leqslant C \varepsilon^s$ .

\end{remark}

\section{Asymptotic expansion}

Let us fix some notations.
We will use the letter $\mathcal{S}$ to denote the Schwarz space of
rapidly decreasing functions.
We  define  the boundary layer profile spaces
\begin{eqnarray*}
  \mathcal{N}_{\pm} (T) :=    H^\infty ( [0,T] \times \Omega, \mathcal{S}( {\mathbb R}^\pm ) ).
\end{eqnarray*}
Since we will need an equation of
the boundary $\Gamma$, we fix once for all a function
$\Phi \in \cC^{\infty}({\mathbb R}^3, {\mathbb R})$ and we assume that
$\Omega = \{ \Phi > 0 \}$, $ \Gamma = \{ \Phi = 0 \}$
and  $| \nabla \Phi (x) | = 1 $ in an open neighborhood $\mathcal{V}_\Gamma $
of $\Gamma $ \footnote{ Hence for $x \in \Omega \cap \mathcal{V}_\Gamma$:
$\Phi(x) = dist(x,\Gamma )$.}.
Let us also  fix a function
$\Psi \in \cC^{\infty}({\mathbb R}^3, {\mathbb R})$ such that
$\Sigma = \{ \Psi=0 \} $
 and such that  $| \nabla \Psi (x) | = 1 $ in an open neighborhood $\mathcal{V}_\Sigma$
of $\Sigma $ \footnote{ Hence for $x \in \Omega \cap \mathcal{V}_\Sigma$:
$\psi (x) = dist(x,\Sigma)$.}.
We assume that the neighborhoods   $\mathcal{V}_\Gamma$ and  $\mathcal{V}_\Sigma$ have
been fixed small enough in order that $\cV_\Gamma \cap \cV_\Sigma =
\varnothing$.
 We will denote
$\Omega_+ := \Omega \cap \{ \Psi > 0 \}$ and
$\Omega_- := \Omega \cap \{ \Psi < 0 \}$.
We consider a $C^\infty$ unit vector field  $\partial_{\bf n} $ which coincides on
$\mathcal{V}_\Gamma$ with  $- \nabla _x \Phi \cdot \nabla_x $ and on
$\mathcal{V}_\Sigma$ with  $- \nabla _x \Psi \cdot \nabla_x $.

\input epsf
$$
\epsfxsize3.2in
\epsffile{ferrop.eps}
$$

In the easier case where $u^0$ is continuous across the hypersurface
$\Sigma$, paper \cite{CFG1} shows the existence of solutions $u^\varepsilon $,
$\varepsilon \in ]0,1]$,
of the equation $(\ref{LL1})$ in $\Omega$,
of the equation $(\ref{LL2})$ on $ \Gamma$,
of the form
\begin{eqnarray*}
u^{\varepsilon}  (t,x) &:=& u^0   (t,x) +  \varepsilon \Big(
\mathfrak{U} (t,x, \frac{\Phi(x)}{\varepsilon} )  +  \mathbf{w}^{\varepsilon}  (t,x)\Big)
\end{eqnarray*}
where the function $\mathfrak{U}$ is in $\mathcal{N}_{+} (\infty)$ and
satisfies $\mathfrak{U}(t,x,z)=0$ for $x \notin \mathcal{V}_\Gamma$.
The function $\mathfrak{U} $ describes a boundary layer which appears near the boundary
to compensate the lost of the Neumann condition
 from the complete model (\ref{LL1})-(\ref{LL2})-(\ref{LL3}) to
the limit model  (\ref{LL0})  ($\varepsilon = 0$).
 The amplitude of this boundary is
weak and its behaviour is linear.
For sake of completeness we will state this in
section $\ref{deja}$.
The  functions  $\mathbf{w}^{\varepsilon} $ can be seen as remainders.

Here since we deal with a ground state $ u^0 $ which is  discontinuous across the  hypersurface
$\Sigma$,
we look for solutions $u^\varepsilon $,
$\varepsilon \in ]0,1]$,
of the equation $(\ref{LL1})$ in $\Omega$,
of the equation $(\ref{LL2})$ on $ \Gamma$,
of the form
\begin{eqnarray}
\label{decomp}
u^{\varepsilon}  (t,x) :=  \mathcal{U}  (t,x, \frac{\Psi(x)}{\varepsilon})
+  \varepsilon \Big(\mathfrak{U} (t,x, \frac{\Phi(x)}{\varepsilon} )  +  \mathbf{w}^{\varepsilon}  (t,x)  \Big).
\end{eqnarray}
The function $\mathcal{U}$
describes a large amplitude  internal layer
profile i.e. a sharp
transition  in the neighborhood of the hypersurface $\Sigma$ of width
$\sim \varepsilon$.
More precisely the function $\mathcal{U}$ is $\cC^\infty$ and satisfies
\begin{eqnarray}
\label{sharp1}
\lim_{y\rightarrow \pm \infty}   \mathcal{U}  (t,x, y) &=&
  u^0  (t,x ) \quad \text{for} \ x \in \mathcal{V}_\Sigma \cap \Omega_\pm
  \\ \label{sharp2}
  \mathcal{U}  (t,x, y)&=& u^0  (t,x ) \quad \text{for} \ x \notin
  \mathcal{V}_\Sigma \ \text{and} \ y \in \mathbb{R}
\end{eqnarray}
The profile $\mathfrak{U}$, as we have already said it above, was  constructed in  \cite{CFG1}.
The  functions  $ \mathbf{w}^{\varepsilon}  $ can still be seen as remainders.
Let us explain this time more precisely what we mean by remainders.
Let us fix a finite set of smooth vectors fields $\cT_0 = \{
\cZ_i(x;\partial_x) ; i=1, \cdots ,\mu \}$ on ${\mathbb R}^3$, tangent to the
surfaces $\Gamma$ and $\Sigma$ (that is satisfying $\cZ_i(x;\partial_x)
\Phi =0$ on $\Gamma$ and $\cZ_i(x;\partial_x)\Psi = 0$ on $\Sigma$, for
all $i\in \{1, \cdots , \mu\}$), and generating the algebra of
smooth vector fields tangent to $\Gamma \cup \Sigma$. These vector
fields can be viewed as vector fields on ${\mathbb R}^{4}$ tangent to ${\mathbb R}
\times \Gamma$ and to ${\mathbb R} \times \Sigma$. By adding the vector
field $\partial_t$ to the family, one gets the set $\cT := \{ \partial_t \} \cup
\cT_0  $ which generates the set of smooth vector fields in ${\mathbb R}^4 $
tangent to $({\mathbb R} \times \Gamma) \cup ({\mathbb R} \times \Sigma)$. We denote
$\cZ_0 := \partial_t$. For all multi-index $\alpha \in
{\mathbb N}^{1+\mu}$ we note $\cZ^\alpha = \partial_t^{\alpha_0}\cZ^{\alpha_1}_1.
\cdots . \cZ^{\alpha_\mu}_\mu$, with $\alpha = (\alpha_0, \alpha_1,
\cdots , \alpha_\mu)$.
Let us introduce the usual  norm:
$$
\| u \|_{m} := \sum_{|\alpha| \leq m \, , \, \alpha \in
{\mathbb N}^{1+\mu}} | \cZ^\alpha u
\|_{L^2(]0,T[ \times \Omega)},
$$
and note $H^m_{co}(]0,T[\times \Omega)$  the space of $u\in
L^2(]0,T[ \times \Omega)$ such that this norm is finite.
We introduce the set $E$ of the family $( \mathbf{w}^\varepsilon  )
_{0<\varepsilon\leqslant 1})$ of functions  in $L^2(]0,T[ \times \Omega)$ such that for all $m \in \mathbb{N}$, there exists $\varepsilon_0 >0$ such that
\begin{eqnarray}
\label{estiti}
\text{sup}_{0<\varepsilon\leqslant \varepsilon_0} \, (|| \mathbf{w}^{\varepsilon} ||_{m} + ||\varepsilon \partial_{\bf n}
\, \mathbf{w}^{\varepsilon} ||_{m} +  \varepsilon (|| \mathbf{w}^{\varepsilon} ||_{\infty}  +
||\cZ \mathbf{w}^{\varepsilon}
||_{\infty} + ||\varepsilon \partial_{\bf n} \,  \mathbf{w}^{\varepsilon}||_{\infty} ))<
\infty .
\end{eqnarray}

In fact Theorem \ref{main} is the straightforward consequence of the
following result.

\begin{theorem}
\label{main2}

Let $u^0 \in \cC^\infty\big({\mathbb R}, H^\infty_\Sigma(\Omega) \big) $
be a solution of $(\ref{LL0})$.
There exist $T>0$, a profile $\mathcal{U}$ in $\cC^\infty ((0,T) \times
\Omega \times \mathbb{R} )$ which  satisfies
$(\ref{sharp1})-(\ref{sharp2})$
and a family $( \mathbf{w}^\varepsilon)$ in $E$
 such that the function $u^\varepsilon$ given by the formula (\ref{decomp}) are solutions in  $C^\infty$
 of the equation  $(\ref{LL1})$ on $[0,T] \times \Omega$,
of the equation $(\ref{LL2})$ on $[0,T] \times \Gamma$.
\end{theorem}
Theorem \ref{main2}  exhibits some large variations solutions of the Landau-Lifschitz
equations as the exchange coefficient $\varepsilon^2$ tends to zero,
by means of the  asymptotic expansions (\ref{decomp}).
The remainder of the paper is devoted to the proof of Theorem \ref{main2}.
 As in \cite{CFG1}, since the magnetic moment $u$ is unimodular, the equation (\ref{LL1}) is equivalent
 for smooth solutions to the following
one:
\begin{eqnarray}
\label{LL1new}
\cL^\varepsilon( u^\varepsilon, \partial )\, u^\varepsilon =
    {\bf F}\big(u^\varepsilon, \varepsilon \partial_x u^\varepsilon,
     \cH(u^\varepsilon)\big)
\end{eqnarray}
where we have noted
$$
\cL^\varepsilon(v,\partial) := \partial_t - \varepsilon^2 \Delta_x - \varepsilon^2 v \wedge \Delta_x,
$$
and
$$
{\bf F}(u,V,H):= |V|^2u + u\wedge H - u\wedge (u\wedge H),
$$
for all $u\in {\mathbb R}^3$, $V\in \cM({\mathbb R}^3,{\mathbb R}^3)$, $H\in {\mathbb R}^3$.
From now on we will  deal with equation (\ref{LL1new}) rather than
(\ref{LL1}).
We will proceed in three steps.
In subsection $\ref{s1}$ we will define the profile $\mathcal{U}$  as a
local in time solution of a pair of
nonlinear equations in $\Omega \times \mathbb{R}_\pm $ coupled by some
transmissions conditions on $\{y=0\}$.
In  subsection $\ref{deja}$  we  will recall the results of \cite{CFG1} about the profile $\mathfrak{U}$.
In subsection $\ref{s3}$  we  will  prove the existence of some remainders $\mathbf{w}^\varepsilon$
 till the
lifetime $T$ of the profile $\mathcal{U}$.
Eventually we will show that the remainders $\mathbf{w}^\varepsilon$ satisfy the
uniform  estimates uniform (\ref{estiti}).

\subsection{Construction of the internal layers }
\label{s1}

Despite that $\pm \frac{\Psi(x)}{\varepsilon}>0$ when  $x \in \Omega_\pm$ we will define $\mathcal{U}$ for all $(x,z) \in \Omega \times
\mathbb{R}_\pm $ since this will not cause any additional difficulty.
An Uryshon argument yields the existence of two functions  $u^{0}_\pm$  in $H^\infty ( (0,\infty) \times \Omega )$ such
that $u^{0}_\pm = u^{0}$ for all $x \in \Omega_\pm \cup  (\Omega_\mp -
\mathcal{V}_\Sigma)$.
We look for a viscous internal layer
profile $\mathcal{U}$ of the form
\begin{eqnarray}
\label{R1}
\mathcal{U}(t,x,y) :=
 \left\{
 \begin{array}{cc}
  u^{0}_+  (t,x) + \mathcal{U}_{+}  (t,x, y)  \quad
  &\text{if} \ y>0,
 \vspace{0.3cm}\\ u^{0}_-  (t,x) + \mathcal{U}_{-}  (t,x, y)  \quad
  &\text{if} \ y<0 .
 \end{array}
 \right.
\end{eqnarray}
The functions $\mathcal{U}_\pm$ are in $\mathcal{N}_{\pm} (T)$.
These functions  describe some internal large amplitude boundary layers,
 on each side of the hypersurface $\Sigma$.
To insure that the function $\mathcal{U}$ is in $C^1 ( (0,T) \times \Omega  \times  \mathbb{R})$
it is necessary to impose the transmission conditions:
\begin{eqnarray}
 \label{p2}
\left.
\begin{array}{c}
 \mathcal{U}_{+} -  \mathcal{U}_{- } = - u^0_{+} + u^0_{- } ,
\\ \partial_y \mathcal{U}_{+} - \partial_y \mathcal{U} _{- } =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \}.
\end{eqnarray}

In Theorem $\ref{profil0}$ we will define the profiles $\mathcal{U}_\pm$
as local solutions of
nonlinear equations in $\Omega \times \mathbb{R}_\pm $ coupled by some
transmissions conditions on $\{y=0\}$.
Let us look for convenient  equations.
We will plug the functions $u^{\varepsilon,0}$ defined by
$u^{\varepsilon,0} (t,x) := \mathcal{U}(t,x, \frac{\Psi(x)}{\varepsilon})$
instead of  $u^\varepsilon$ in $(\ref{LL1new})$.
In general it is not possible to verify  $(\ref{LL1new}) $ but we will try
to choose the functions  $\mathcal{U}_\pm$ such that the error term is as
small as possible.
Let us begin to look at the left side of $(\ref{LL1new})$.
With (\ref{R1})
  we get  in $L^\infty$
  %
\begin{eqnarray}
\label{M1}
\mathcal{L}^{\varepsilon} (u^{\varepsilon,0} , \partial ) u^{\varepsilon,0}  =
\partial_t \, u_{\pm}^0  +
\Big( L(\mathcal{U} , \partial_t ,  \partial_y^2 )  \mathcal{U}_{\pm}   \Big)| + O(\varepsilon) \quad   \mathrm{for} \
  x \in  \Omega_\pm  ,
\end{eqnarray}
where the vertical bar $|$ means that $y$ is evaluated in $y=
\frac{\Psi(x)}{\varepsilon}$ and
\begin{eqnarray*}
L(U,\partial_{t}, \partial_{y}^2) := \partial_t - \partial_y^2 - U \wedge \partial_y^2 .
\end{eqnarray*}
We now turn to the right side of $(\ref{LL1new})$.
We first look at  the
action of $\cH$ on the family $u^{\varepsilon,0}$:
\begin{eqnarray*}
\cH(u^{\varepsilon,0} ) = \cH(u^0_\pm ) - (\mathcal{U}_\pm  . n)| \, n + O(\varepsilon).
\end{eqnarray*}
Then
\begin{eqnarray}
\label{M2}
 {\bf F}\big(u^\varepsilon, \varepsilon \partial_x u^\varepsilon,    \cH(u^\varepsilon)\big) :=
  {\bf F}\big(u^0_\pm, 0,   \cH(u^0_\pm)\big)
 + F_\pm ( \mathcal{U}_\pm , \partial_y  \mathcal{U}_\pm  )|  + O(\varepsilon) \quad   \mathrm{for} \
  x \in  \Omega_\pm   ,
 \end{eqnarray}
 with for all $U\in {\mathbb R}^3$, $V\in \cM({\mathbb R}^3,{\mathbb R}^3)$,
\begin{eqnarray*}
 F_\pm (  U, V  )  &:=&
 | V |^2 \, ( u^0_\pm  + U )
+ U \wedge  \cH (u^0_\pm ) - (U  . n) ( u^0_\pm  +  U)  \wedge   n
\\ \nonumber  && + U \wedge (  u^0_\pm +U)
 \wedge (   \cH (u^0_\pm ) - (U  . n) n  )
 +  u^0_\pm  \wedge (  U  \wedge  (   \cH (u^0_\pm )   - (U  . n) n ) )
\\ \nonumber  && -  (U  . n) u^0_\pm  \wedge (  u^0_\pm   \wedge    n )
\end{eqnarray*}
Thanks to (\ref{M1}) and (\ref{M2})  we get by looking at the terms at order $0$
\begin{eqnarray*}
\partial_t \,   u_{\pm}^0+
 L(\mathcal{U} , \partial_t ,  \partial_y^2 )  \mathcal{U}_\pm = {\bf F}\big(u^0_\pm, 0,   \cH(u^0_\pm)\big)
 + F_\pm ( \mathcal{U}_\pm , \partial_y  \mathcal{U}_\pm  ) .
\end{eqnarray*}
Since for $x \in  \Omega_\pm$, the functions $u^0_\pm$ satisfies
$(\ref{LL0})$  we could simplify and we get
the nonlinear equations
\begin{eqnarray}
\label{p1} L(  u_{\pm}^0 + \mathcal{U}_\pm ,\partial_{t}, \partial_{y}^2 ) \mathcal{U}_\pm =
 F_\pm ( \mathcal{U}_\pm , \partial_y  \mathcal{U}_\pm  ) .
 \end{eqnarray}
The equations  $(\ref{p1})$ are parabolic with respect to $t,y$, the
variable $x$ can be seen as a parameter.
The following theorem claims that it is possible to find some solutions
$\mathcal{U}_{\pm} \in \mathcal{N}_{\pm} (T)$ of these equations even for all $x \in   \Omega  $.

\begin{theorem}
\label{profil0}
There exists $T > 0$ and there exist some functions $\mathcal{U}_{\pm} \in \mathcal{N}_{\pm} (T)$
 which verify the equations $(\ref{p1})$ when $(t,x,y) \in  (0,T) \times \Omega \times
 \mathbb{R}_\pm $ and  the transmission conditions $(\ref{p2})$.
 Moreover precisely for all $x \notin \mathcal{V}_\Sigma$ and $y \in  \mathbb{R}_\pm $
there holds $\mathcal{U}_{\pm}  (t,x, y)= 0$.
\end{theorem}

\begin{proof}

We will proceed in four steps.

\begin{stepbd}
We begin to reduce the problem to homogeneous boundary conditions.
\end{stepbd}

We introduce the functions $V_{\pm} $ and $ \mathcal{U}_{\pm} $ given by
 the formula
\begin{eqnarray*}
V_{\pm} (t,x,y) &:=& (1-\frac{e^{\mp y}}{2}) u^0_\pm (t,x) + \frac{e^{\mp y}}{2}
u^0_\mp (t,x) ,
\\ \mathbf{W}_{\pm} (t,x,y)&:=& \mathcal{U}_{\pm} (t,x,y) \pm  \frac{1}{2} (u^0_+  (t,x)-
u^0_- (t,x)) e^{\mp y} .
\end{eqnarray*}
Thus the transmission conditions $(\ref{p2})$ reads:
\begin{eqnarray}
 \label{np2}
\left.
\begin{array}{c}
 \mathbf{W}_{+} - \mathbf{W}_{-} =  0 ,
\\ \partial_y \mathbf{W}_{+} - \partial_y  \mathbf{W}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \}.
\end{eqnarray}
Moreover the equations  (\ref{p1})-(\ref{p2}) read for $ (t,x,y) \in
(0,T) \times \Omega \times \mathbb{R}_\pm$:
\begin{eqnarray}
\label{++}
 L (V_{\pm} + \mathbf{W}_{\pm} ,\partial_{t},  \partial_{y}^2 )
\mathbf{W}_{\pm} &=& \hat{F}_{\pm} (t,x,y,\mathbf{W}_{\pm} , \partial_y
\mathbf{W}_{\pm}) ,
\end{eqnarray}
where $\hat{F}_{\pm}$ are $C^\infty$ functions such that the functions
$\hat{F}_{\pm}(t,x,y, 0,0) $ are rapidly decreasing with respect to $y$.
\begin{stepbd}
We prove the existence of compatible initial data.
\end{stepbd}
Let us to explain why the initial values $\mathbf{W}_{0,+}$ must satisfy some compatibility conditions
at the corner $\{ t= y=0 \} $ are required in order to obtain smooth
solutions $\mathbf{W}_{\pm}$ of the problem  (\ref{++})-(\ref{np2}) with
$\mathbf{W}_{\pm} |_{t=0}:= \mathbf{W}_{0,\pm} $.
We start with the condition of order $0$.
Set $t=0$ in the transmission conditions (\ref{np2}) to see that  $\mathbf{W}_{0,+}$ must satisfy  the relation
\begin{eqnarray}
 \label{comp0}
\left.
\begin{array}{c}
 \mathbf{W}_{+} - \mathbf{W}_{0,-} =  0 ,
\\ \partial_y \mathbf{W}_{0,+} - \partial_y  \mathbf{W}_{0,-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (x,y) \in  \Omega \times \{ 0 \}.
\end{eqnarray}
Now, for each $k \geqslant 1$, apply the derivative $\partial_t^k $ to the  transmission conditions (\ref{p2}).
We get
\begin{eqnarray*}
\left.
\begin{array}{c}
\partial_t^k \mathbf{W}_{+} - \partial_t^k\mathbf{W}_{-} =  0 ,
\\ \partial_y \partial_t^k \mathbf{W}_{+} - \partial_y \partial_t^k  \mathbf{W}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \}.
\end{eqnarray*}
Now remark that, by iteration, we can extirpate $\partial_t^k \mathbf{W}_{\pm}$ by the interior equations (\ref{p1}) in terms of
 derivatives with respect to $y$. More precisely there exists some smooth functions $C^k_{\pm} $ such that
 $\partial_t^k \mathbf{W}_{\pm} = C^k_{\pm} ( (\partial_y^l
 \mathbf{W}_{\pm} )_{l \leqslant 2k})$.
 Thus the following $k$th order compatibility condition must hold:
 %
 \begin{eqnarray}
 \label{compk}
\left.
\begin{array}{c}
C^k_{+} ( (\partial_y^l \mathbf{W}_{\pm} )_{l \leqslant 2k})
- C^k_{-} ( (\partial_y^l \mathbf{W}_{\pm} )_{l \leqslant 2k}) =0 ,
\\ \partial_y C^k_{+} ( (\partial_y^l \mathbf{W}_{\pm} )_{l \leqslant 2k})
- \partial_y C^k_{-} ( (\partial_y^l \mathbf{W}_{\pm} )_{l \leqslant 2k}) =0 .
\end{array}
\right\}
\quad \mathrm{when} \ (x,y) \in  \Omega \times \{ 0 \}.
\end{eqnarray}

\begin{lemma}

There exist some initial values $\mathbf{W}_{0,\pm}$ in $H^\infty (\Omega , \mathcal{S} (\mathbb{R}_\pm ))$
which  satisfy the relation (\ref{comp0}) and (\ref{compk}) for all $k \geqslant 1$.

\end{lemma}

\begin{proof}

As we will follows the method of \cite{S1}, we only sketch the proof for sake of completeness.
We start by analyzing more accurately the compatibility conditions and more especially the way the functions  $C^k_{\pm} $
depend on the derivatives with respect to $y$. Indeed they exists some functions $\tilde{C}^k_{\pm} $ such that
 %
 \begin{eqnarray*}
C^k_{\pm} ( (\partial_y^l \mathbf{W}_{\pm} )_{l \leqslant 2k}) =   \tilde{C}^k_{\pm} ( (\partial_y^l \mathbf{W}_{\pm} )_{l \leqslant
2k-1})
+ (\partial_y^{2k} + (V_{\pm} + \mathbf{W}_{\pm} )  \wedge  \partial_y^{2k} ) \mathbf{W}_{\pm} .
\end{eqnarray*}
Since   given two functions $ \mathbf{W}^{(0)}_{\pm}$  in $H^\infty (\Omega )$ the applications
 %
 \begin{eqnarray*}
 \mathbf{W}_{\pm} \mapsto \mathbf{W}_{\pm} + ( V_{\pm}  +  \mathbf{W}^{(0)}_{\pm} ) \wedge   \mathbf{W}_{\pm}
\end{eqnarray*}
are two automorphisms of $H^\infty (\Omega )$
and an iteration, we deduce by iteration that there exists
a family $(\mathbf{W}^{(k)}_{\pm})_{k \in \mathbb{N}}$ in $H^\infty (\Omega )$ such that
 \begin{eqnarray*}
 \left.
\begin{array}{c}
C^k_{+} ( (\partial_y \mathbf{W}^{(l)}_{\pm} )_{l \leqslant 2k})
- C^k_{-} ( (\partial_y \mathbf{W}^{(l)}_{\pm} )_{l \leqslant 2k}) =0 ,
\\ \partial_y C^k_{+} ( (\partial_y \mathbf{W}^{(l)}_{\pm} )_{l \leqslant 2k})
- \partial_y C^k_{-} ( (\partial_y \mathbf{W}^{(l)}_{\pm} )_{l \leqslant 2k}) =0 .
\end{array}
\right.
 \end{eqnarray*}
 We end the proof by a classical Borel argument.
\end{proof}

As a consequence, we will assume in the rest of the proof that the functions $\mathbf{W}_{\pm}$ vanish for $t\leqslant 0$.
\begin{stepbd}
We look for linear estimates.
\end{stepbd}
In order to use an iterative scheme, we look at the linear problem
\begin{eqnarray}
\label{l1}
L (\mathfrak{W}_{\pm}, \partial_{t}, \partial_{y}^2 )
\mathbf{W}_{\pm} = f_{\pm} \quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \mathbb{R}_\pm ,
\\ \label{l2}
\left.
\begin{array}{c}
 \mathbf{W}_{+} - \mathbf{W}_{-} =  0 ,
\\ \partial_y \mathbf{W}_{+} - \partial_y  \mathbf{W}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \}.
\end{eqnarray}
For all real $\lambda \geqslant 1$, the space $L^2 ((0,T) \times \Omega \times \mathbb{R}_\pm)$
is endowed with the scalar product associated
to the Euclidean norm
 \begin{eqnarray*}
 || \mathbf{W}_{\pm} ||_{0,\lambda,T} := || e^{- \lambda t} \, W_{\pm} ||_{L^2 ((0,T) \times \Omega \times \mathbb{R}_\pm)  }
 \end{eqnarray*}
In order to avoid heavy notations, we will denote $W:=(\mathbf{W}_+ , \mathbf{W}_- )$, $f:=(f_+ , f_- )$ and $ \mathfrak{W} :=(
\mathfrak{W}_+ , \mathfrak{W}_- )$.
We endow the space $ L^2 ((0,T) \times \Omega \times \mathbb{R}_+) \times L^2 ((0,T) \times \Omega \times \mathbb{R}_-)  $ with the scalar product
associated to the Euclidean norm
 \begin{eqnarray*}
 || W ||_{0,\lambda,T} :=  || \mathbf{W}_+ ||_{+,0,\lambda,T} + || \mathbf{W}_- ||_{-,0,\lambda,T} .
 \end{eqnarray*}
For $m \in \mathbb{N}$, we introduce the following weighted norms:
 \begin{eqnarray*}
 || W ||_{m,\lambda,T} :=
 \sum_{|\alpha| \leqslant m} \, || \partial_{t,x}^\alpha \, W ||_{0,\lambda,T} ,
 \ \text{and} \  | W |_{m,\lambda,T} :=
 \sum_{|\alpha| \leqslant m} \, || \partial_{t,x}^\alpha \,
 \partial_{y}^{\alpha_4} \, W ||_{0,\lambda,T},
\end{eqnarray*}
where $\alpha:=(\alpha_0 ,..., \alpha_3) \in \mathbb{N}^4$ and
$\partial_{t,x}^\alpha :=\partial_{t}^{\alpha_0} \partial_{1}^{\alpha_1}
\partial_{2}^{\alpha_2} \partial_{3}^{\alpha_3}.$
\begin{proposition}

Let $R>0$. If $ \mathfrak{W}_\pm$  verify the following estimates
 \begin{eqnarray*}
 || \mathfrak{W}_+ ||_{Lip ((0,T) \times \Omega \times \mathbb{R}_+)} + ||\mathfrak{W}_+ ||_{Lip ((0,T) \times \Omega \times \mathbb{R}_+)}
 + | \mathfrak{W} |_{m,\lambda,T} < R ,
\end{eqnarray*}
  and the following  boundary conditions
 \begin{eqnarray}
  \label{l3}
\left.
\begin{array}{c}
 \mathfrak{W}_{+} - \mathfrak{W}_{-} =  0 ,
\\ \partial_y \mathfrak{W}_{+} - \partial_y  \mathfrak{W}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \},
\end{eqnarray}
 then there exist $\lambda_m >0$ and for all $k \in \mathbb{N}$, $\mu_{k,m} > 0$, such that for all $\lambda \geqslant \lambda_m$,
 %
 \begin{eqnarray}
 \label{esti1}
 | W |_{m,\lambda,T}   \leqslant \frac{\lambda_m}{\lambda}  | f |_{m,\lambda,T}
\end{eqnarray}
and for all $ \mu \geqslant \mu_{k,m} $,
 %
 \begin{eqnarray}
   \label{esti2}
    |y^k \, W |_{m,\lambda,T} \leqslant \frac{\mu_{k,m}}{\mu}  \sum_{j=0}^k | y^j \, f |_{m,\mu,T} .
\end{eqnarray}
\end{proposition}

\begin{proof}

We  multiply the equation (\ref{l1}) by $\mathbf{W}_{\pm}$ and integrate for $(x,y) \in   \Omega \times \mathbb{R}_\pm$.
Hence
\begin{eqnarray}
\label{estim}
(1/2) \partial_t \int_{\Omega  \times \mathbb{R}_\pm} | \mathbf{W}_{\pm} |^2 - J_{1,\pm}- J_{2,\pm}
 =  \int_{\Omega  \times \mathbb{R}_\pm} f_{\pm} . \mathbf{W}_{\pm}
\\ \nonumber \mathrm{where} \ J_{1,\pm} := \int_{\Omega  \times \mathbb{R}_\pm} \mathbf{W}_{\pm} . \partial_y^2 \mathbf{W}_{\pm}
\ \mathrm{and} \ J_{2,\pm} := \int_{\Omega  \times \mathbb{R}_\pm} \mathbf{W}_{\pm} .  (
\mathfrak{W}_{\pm} \wedge \partial_y^2)  \mathbf{W}_{\pm} .
\end{eqnarray}
 Integrating by parts, we get
\begin{eqnarray*}
J_{1,\pm} =
- \int_{\Omega \times \mathbb{R}_\pm } | \partial_y \mathbf{W}_{\pm} |^2
- I_{1,\pm} ,
\ \mathrm{and} \ J_{2,\pm} =
- \int_{\Omega \times \mathbb{R}_\pm } \mathbf{W}_{\pm} .  ( \partial_y \mathfrak{W}_{\pm} \wedge \partial_y)  \mathbf{W}_{\pm}
- I_{2,\pm} ,
\\ \nonumber
 \mathrm{where} \ I_{1,\pm} := \int_{\Omega } ( \mathbf{W}_{\pm}. \partial_y \mathbf{W}_{\pm} )|_{y=0} ,
\ \mathrm{and} \   I_{2,\pm} := \int_{\Omega } (\mathbf{W}_{\pm} .  (  \mathfrak{W}_{\pm} \wedge \partial_y  \mathbf{W}_{\pm}))|_{y=0} .
\end{eqnarray*}
Using the boundary conditions (\ref{l2}) and (\ref{l3}), we get $I_{1,+} - I_{1,-} =  I_{2,+} - I_{2,-} =  0$.
Taking that into account we add the two estimates in (\ref{estim}).
Then we multiply by $e^{-2 \lambda t}$ and integrate in time.
By a Gronwall lemma we get  that there exists $c >0$ such that for all $\lambda \geqslant c$,
\begin{eqnarray}
\label{estiL2}
 | \partial_y W |^2_{0,\lambda,T} + \lambda | W |^2_{0,\lambda,T} \leqslant c  |<f,W>_{\lambda,T}| .
\end{eqnarray}

We go on with  estimates tangential to $\{y=0\}$.
To do this we apply the derivative $\partial_{t,x}^{\alpha} \, $ to the equations (\ref{l1})-(\ref{l2}).
So we get that $\partial_{t,x}^{\alpha} \, \mathbf{W}_{\pm} $ verify
 %
\begin{eqnarray}
\label{l1tan}
L ( \mathfrak{W}_{\pm} , \partial_{t},\partial_{y}^2 )
\partial_{t,x}^{\alpha} \, \mathbf{W}_{\pm} = \tilde{f}_{\pm}  \quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \mathbb{R}_\pm ,
\\ \label{l2tan}
\left.
\begin{array}{c}
\partial_{t,x}^{\alpha} \,  \mathbf{W}_{+} - \partial_{t,x}^{\alpha} \, \mathbf{W}_{-} =  0 ,
\\ \partial_y \partial_{t,x}^{\alpha} \, \mathbf{W}_{+} - \partial_y  \partial_{t,x}^{\alpha} \, \mathbf{W}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \},
\end{eqnarray}
where
\begin{eqnarray}
\label{source}
\tilde{f}_{\pm} := \partial_{t,x}^{\alpha} \, f_{\pm}
+ \sum_{|\alpha_1 | + |\alpha_2 | = |\alpha|, |\alpha_2 | < |\alpha|}
 \partial_{t,x}^{\alpha_1} \, \mathfrak{W}_{\pm}  \wedge  \partial_y^2 \, \partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm} .
\end{eqnarray}
We apply the tangential derivative $\partial_{t,x}^{\alpha} \, $ to the boundary conditions (\ref{l3}) and get
 \begin{eqnarray}
  \label{l3tan}
\left.
\begin{array}{c}
\partial_{t,x}^{\alpha} \, \mathfrak{W}_{+} - \partial_{t,x}^{\alpha} \,  \mathfrak{W}_{-} =  0 ,
\\ \partial_y \partial_{t,x}^{\alpha} \,  \mathfrak{W}_{+} - \partial_y \partial_{t,x}^{\alpha} \,  \mathfrak{W}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \},
\end{eqnarray}
Using the estimate (\ref{estiL2}), we get, for all $\lambda \geqslant c$,
\begin{eqnarray*}
| \partial_y \partial_{t,x}^{\alpha} \, W |^2_{0,\lambda,T}
 + \lambda | \partial_{t,x}^{\alpha} \, W |^2_{0,\lambda,T} \leqslant
 c  |<\tilde{f} , \partial_{t,x}^{\alpha} \, W  >_{\lambda,T}|.
\end{eqnarray*}
Thanks to (\ref{source}), we get
\begin{eqnarray}
<\tilde{f} , \partial_{t,x}^{\alpha} \, W  >_{\lambda,T} =
< \partial_{t,x}^{\alpha} \, f  , \partial_{t,x}^{\alpha} \, W  >_{\lambda,T}
+ \sum_{|\alpha_1 | + |\alpha_2 | = |\alpha|, |\alpha_2 | < |\alpha|}  I_{\alpha_1 , \alpha_2 } ,
\end{eqnarray}
where $ I_{\alpha_1 , \alpha_2 } := I_{+,\alpha_1 , \alpha_2 } + I_{-,\alpha_1 , \alpha_2 }$ with
\begin{eqnarray*}
I_{ \pm,\alpha_1 , \alpha_2 } := < \partial_{t,x}^{\alpha_1} \, \mathfrak{W}_{\pm}  \wedge  \partial_y^2 \,
\partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm} , \partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm} >_{\lambda,T} .
\end{eqnarray*}
Using Cauchy-Schwarz inequality, we get
\begin{eqnarray*}
|< \partial_{t,x}^{\alpha} \, f  , \partial_{t,x}^{\alpha} \, W  >_{\lambda,T} |
\leqslant | f|_{0,\lambda,T} \, .| W|_{0,\lambda,T} .
\end{eqnarray*}
  We are going to estimate, for all $ \alpha_1 , \alpha_2 $
such that $|\alpha_1 | + |\alpha_2 | = |\alpha|, |\alpha_2 | < |\alpha|$, the term $ I_{\alpha_1 , \alpha_2 }$.
Integrating by parts, we get
$I_{ \pm,\alpha_1 , \alpha_2 } :=  \sum_{l=1}^3 \, I^l_{ \pm,\alpha_1 ,
\alpha_2 },$
with
\begin{eqnarray*}
 I^1_{ \pm,\alpha_1 , \alpha_2 } &:=& -
 < \partial_{t,x}^{\alpha_1} \, \partial_y \mathfrak{W}_{\pm}  \wedge  \partial_y \, \partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm}
 , \partial_{t,x}^{\alpha} \, \mathbf{W}_{\pm} >_{\lambda,T} ,
\\  I^2_{ \pm,\alpha_1 , \alpha_2 } &:=& -
 < \partial_{t,x}^{\alpha_1} \,  \mathfrak{W}_{\pm}  \wedge  \partial_y \, \partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm} ,
\partial_{t,x}^{\alpha} \, \partial_y \mathbf{W}_{\pm}>_{\lambda,T} ,
\\ I^3_{ \pm,\alpha_1 , \alpha_2 } &:=& \mp   <<
\{  (\partial_{t,x}^{\alpha_1} \, \mathfrak{W}_{\pm} \wedge  \partial_y \, \partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm} )\}|_{y=0},
\{ \partial_{t,x}^{\alpha} \, \partial_y \mathbf{W}_{\pm} \}|_{y=0} >>_{\lambda,T} ,
\end{eqnarray*}
where $<<.,.>>_{\lambda,T}$ denotes the scalar product of $L^2 ((0,T) \times \Omega )$ associated to the mesure $e^{-\lambda t} dt
dx$.
 Thanks to the boundary conditions   (\ref{l2tan})-(\ref{l3tan}),
we get $ I^3_{ +,\alpha_1 , \alpha_2 } - I^3_{ -,\alpha_1 , \alpha_2 } = 0$.
Using Cauchy-Schwarz inequality, we get
\begin{eqnarray*}
| I^1_{ \pm,\alpha_1 , \alpha_2 }| &\leqslant&
| \partial_{t,x}^{\alpha_1} \, \partial_y \mathfrak{W}_{\pm}  \wedge  \partial_y \, \partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm}
|_{0,\lambda,T} \, . ||  \mathbf{W}_{\pm}||_{m,\lambda,T} ,
\\ | I^2_{ \pm,\alpha_1 , \alpha_2 }| &\leqslant& |  \partial_{t,x}^{\alpha_1} \,  \mathfrak{W}_{\pm}  \wedge  \partial_y \,
\partial_{t,x}^{\alpha_2} \, \mathbf{W}_{\pm} |_{0,\lambda,T} \, . || \partial_y  \mathbf{W}_{\pm}||_{m,\lambda,T} ,
 \end{eqnarray*}
Using Gargliardo-Nirenberg inequalities, we get
\begin{eqnarray*}
| I^1_{ \pm,\alpha_1 , \alpha_2 } |   &\leqslant
 & \quad c ( ||\partial_y \mathfrak{W}_{\pm} ||_{m,\lambda,T} . ||  \mathbf{W}_{\pm}   ||_{Lip}
 +   || \mathfrak{W}_{\pm} ||_{Lip} . || \partial_y \mathbf{W}_{\pm} ||_{m,\lambda,T} ). ||\mathbf{W}_{\pm}||_{m,\lambda,T} ,
\\  | I^2_{ \pm,\alpha_1 , \alpha_2 } | &\leqslant
 & \quad c ( || \mathfrak{W}_{\pm} ||_{m-1,\lambda,T} . ||  \mathbf{W}_{\pm}   ||_{Lip}
 +   | \mathfrak{W}_{\pm} |_{Lip} . || \partial_y \mathbf{W}_{\pm} ||_{m-1,\lambda,T} ). || \partial_y  \mathbf{W}_{\pm}||_{m,\lambda,T} .
\end{eqnarray*}
Hence we get
\begin{eqnarray*}
|I_{\alpha_1 , \alpha_2 } | &\leqslant& \frac{1}{2} || \partial_y  \mathbf{W}_{\pm}||^2_{m,\lambda,T}
+ C ( || \mathbf{W}_{\pm}||^2_{m,\lambda,T} + || \partial_y  \mathbf{W}_{\pm}||^2_{m-1,\lambda,T}).
\end{eqnarray*}
We deduce that there exists $\lambda_m >0$
such that for all $\lambda \geqslant \lambda_m$, there holds
$ || W ||_{m,\lambda,T}  \leqslant \frac{\lambda_m}{\lambda}  || f
||_{m,\lambda,T} $.

To prove  the estimates (\ref{esti1}), it remains to get
 normal estimates.
The cases $\alpha_4 =0$ or $1$ are already treated in the tangential estimates.
If  $\alpha_4 \geqslant 2$, we proceed by iteration, extirpating $\partial_y^2 \mathbf{W}_{\pm}$ from the equations.

It remains to get the estimates (\ref{esti2}).
First we notice that  for  $p \geqslant 1$ the function $y^p \, \mathbf{W}_{\pm}$ verify the initial boundary value problem
\begin{eqnarray*}
L (\mathfrak{W}_{\pm}, \partial_{t}, \partial_{y}^2 )
\mathbf{W}_{\pm}^{[p]} = f_{\pm}^{[p]} \quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \mathbb{R}_\pm ,
\\ \left.
\begin{array}{c}
 \mathbf{W}^{[p]}_{+} - \mathbf{W}^{[p]}_{-} =  0 ,
\\ \partial_y \mathbf{W}^{[p]}_{+} - \partial_y  \mathbf{W}^{[p]}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \},
\\  \mathbf{W}^{[p]}_{\pm} = 0 \quad \mathrm{when} \ (t,x,y) \in  \{ 0 \} \times \Omega \times \mathbb{R}_\pm  ,
\end{eqnarray*}
where
\begin{eqnarray*}
f_{\pm}^{[p]} &=& y^p \, f_{\pm}
+ \sum_{j=0}^{p-1} \, ( q^1_j \partial_y  \mathbf{W}^{[j]}_{\pm} +  q^2_j \, \mathfrak{W}_{\pm}  \wedge \partial_y
\mathbf{W}^{[j]}_{\pm}),
\end{eqnarray*}
where the $q^1_j$ and the $q^2_j$ are in $\mathbb{N}$.
Thus we prove, by iteration on $p$  and thanks to the inequality
(\ref{esti1}),  the estimate
\begin{eqnarray*}
\sqrt{\mu} || \partial_y (y^p \, W)||_{m,\mu,T} + \mu || y^p \, W||_{m,\mu,T}  \leqslant \sum_{j=0}^p || y^j \,
f||_{m,\mu,T}
\end{eqnarray*}
which implies the estimate (\ref{esti2}).

\end{proof}
\begin{stepbd}
We use an iterative scheme.
\end{stepbd}
We define the iterative scheme $(\mathbf{W}^\nu_{\pm})_{\nu \in \mathbb{N}}$ by setting
$ \mathbf{W}^0_{\pm}$ equal to zero
and, by iteration, when $\mathbf{W}^\nu_{\pm}$ is defined, we take $\mathbf{W}^{\nu+1}_{\pm}$ as solution of
\begin{eqnarray*}
  L (V_{\pm} + \mathbf{W}^\nu_{\pm} , \partial_{t}, \partial_{y}^2 )
\mathbf{W}^{\nu+1}_{\pm} = \hat{F}(t,x,y,  \mathbf{W}^\nu_{\pm} , \partial_y
\mathbf{W}^\nu_{\pm}) \quad  \mathrm{when} \ (t,x,y) \in  (0,\infty) \times \Omega \times \mathbb{R}_\pm ,
\\  \left.
\begin{array}{c}
 \mathbf{W}^{\nu+1}_{+} - \mathbf{W}^{\nu+1}_{-} =  0 ,
\\ \partial_y \mathbf{W}^{\nu+1}_{+} - \partial_y  \mathbf{W}^{\nu+1}_{-} =0
\end{array}
\right\}
\quad \mathrm{when} \ (t,x,y) \in  (0,T) \times \Omega \times \{ 0 \} ,
\\ \mathbf{W}^{\nu+1}_{\pm} = 0 \quad \mathrm{when} \ (t,x,y) \in  \{ 0 \} \times \Omega \times \mathbb{R}_\pm .
\end{eqnarray*}

Thanks to the linear estimates, to a Sobolev embedding and to some Gargliardo-Nirenberg inequalities, we show that the
iterative scheme $(\mathbf{W}^\nu_{\pm})_{\nu \in \mathbb{N}}$ converge, when $\nu \rightarrow +\infty $  toward
 some  solutions $\mathbf{W}_{\pm}  \in \mathcal{N}_{\pm} (T)$ of the
 problem (\ref{++})-(\ref{np2}). By going back to the original problem
 (\ref{p1})-(\ref{p2}), the first sentence of Theorem $\ref{profil0}$ is
 now proved. When $x \notin \mathcal{V}_\Sigma$, the function $u^0_+ -
 u^0_-$ in the right hand side of  (\ref{p1}) vanishes and so do the
 functions  $\mathcal{U}_{\pm}  $.

\end{proof}
\begin{remark}
\rm
Notice that the possibility of a blow-up  can be controlled with Lipschitz norm in a very classical way.
However we do not know whether the solutions $\mathcal{U}$ actually
blow-up or globally exist.
\end{remark}
\subsection{Construction of  $\mathfrak{U}$}
\label{deja}

In this section we define the  boundary layer profile
$\mathfrak{U}$ as a solution of a linear boundary value problem.
Let us recall that this function describes a boundary layer which appears near the boundary
to compensate the lost of the Neumann condition
 from the complete model (\ref{LL1})-(\ref{LL2})-(\ref{LL3}) to
the limit model  (\ref{LL0})  ($\varepsilon = 0$).
Such a boundary layer was already mentioned in paper \cite{CFG1}.
Let $\Theta$  be a $C^\infty$  function on
$\Omega$ such that $\Theta =1$   in a neighborhood $\mathcal{W}_\Gamma$
 of $\Gamma$ such that
$\mathcal{W}_\Gamma \subset \subset  \mathcal{V}_\Gamma$
 and  $\Theta =0$  in $\Omega -
 \mathcal{W}_\Gamma$.

\begin{theorem}
There exists $\mathfrak{U}  \in \mathcal{N}_{+} (T)$ which verifies
\begin{eqnarray*}
 L ( u^0 , \partial_{t}, \partial_{z}^2 ) \mathfrak{U}
& = & -( \mathfrak{U}. n) u^0 \wedge   n
 + \mathfrak{U}  \wedge   \cH(u^0 )
\\ \nonumber && + \mathfrak{U} \wedge ( u^0 \wedge \cH(u^0 ) )
-( \mathfrak{U}. n)   u^0 \wedge ( u^0  \wedge  n )
 +u^0 \wedge ( \mathfrak{U}  \wedge \cH(u^0 ) ,
 \end{eqnarray*}
when $(t,x,z) \in  (0,T) \times \Omega \times \mathbb{R}_+ $,
\begin{eqnarray}
  \partial_z \mathfrak{U}  = \Theta (x) \partial_\mathfrak{n} u^0
  \quad \mathrm{when} \ (t,x,z) \in  (0,T) \times \Omega \times \{ 0 \}.
\end{eqnarray}
Moreover there holds $\mathfrak{U}(t,x,z)=0$ for $x  \notin \mathcal{V}_\Sigma$.
\end{theorem}

\begin{proof}
Proceeding as in the proof of Theorem $\ref{profil0}$, we prove the existence of compatible initial data.
Then we follow the proof of Proposition $4.2$ of \cite{CFG1}.
\end{proof}

\subsection{Construction of $\mathbf{w}^\varepsilon$ }
\label{s3}
In this section, we look at the remainder $\mathbf{w}^\varepsilon$.
We will proceed in four steps.
First in section \ref{c1} we  will reduce the initial problem  (\ref{LL1})-(\ref{LL2})-(\ref{LL3})
for the unknown  $u^\varepsilon$ to a problem for $\mathbf{w}^\varepsilon$.
Indeed in order  to get a homogeneous boundary problem,  we will add a
corrector to $\mathbf{w}^\varepsilon$ and rather work with the resulting term $w^\varepsilon$.
Some Borel classical arguments will insure the existence of convenient
initial data for the resulting reduced problem which means that
compatibility conditions either on $\Gamma$ and on $\Sigma$ are satisfied.
We will  prove that the solutions of this nonlinear problems
exist not only for a common non trivial time, in fact even till the
lifetime $T$ of the profiles $\mathcal{U}$.
Moreover these  solutions satisfy some estimates uniform with respect to
$\varepsilon$.
 The method lies on  a simple Picard iterative scheme  (cf. section
 \ref{c2}) and on linear estimates (cf. section
 \ref{c3}).
More precisely we will use  $L^2$-type conormal estimates of  only the
two first normal derivatives, and some Lipschitz estimates.
A few carefulness reveals that the presence of the operator $\mathcal{H}$ does not
cause any loss of factor $\varepsilon$ or any loss of derivatives.

\subsubsection{A reduced problem}
\label{c1}

Since we look for  solutions  $u^\varepsilon$ of (\ref{LL1})-(\ref{LL2})-(\ref{LL3})
 of the form (\ref{decomp}) where the functions
 \begin{eqnarray*}
\label{decomp2}
a^{\varepsilon}  (t,x) &:=&  \mathcal{U}  (t,x, \frac{\Psi(x)}{\varepsilon})
+  \varepsilon \Big(\mathfrak{U} (t,x, \frac{\Phi(x)}{\varepsilon} )  +  \mathbf{w}^{\varepsilon}  (t,x)  \Big)
\end{eqnarray*}
 have been constructed above,
we look for a problem in term of the  remainder $\mathbf{w}^\varepsilon$.
In fact, in order to get a homogeneous boundary problem,  we choose a function $\rho(t,x) \in H^\infty$ such that
\begin{eqnarray}
\label{B1}
  \partial _{{\bf n}} \rho\, |_\Gamma  &=& - \partial _{{\bf n}}   \mathfrak{U} (t,x,0)|_{\Gamma}.
 \end{eqnarray}
 and will look for remainders $\mathbf{w}^\varepsilon$
 of the form $\mathbf{w} ^\varepsilon =  \rho + w^\varepsilon $.
 Let us explain why.
On the boundary $\Gamma$, the function  $a^\varepsilon $ satisfies:
\begin{equation}
\label{B2}
\partial _{{\bf n}} a^\varepsilon|_{\Gamma} = \varepsilon \, \partial _{{\bf n}} \mathfrak{U} (t,x,0)|_{\Gamma},
\end{equation}
Hence in general $a^\varepsilon$ does not satisfy the homogeneous Neumann
boundary condition on $  \Gamma$.
 We  define the function $ \tilde{a}^\varepsilon := a^\varepsilon + \varepsilon \rho$.
Thus we look for solutions  $u^\varepsilon$ of (\ref{LL1})-(\ref{LL2})-(\ref{LL3})
 of the form $ u^\varepsilon =   a^\varepsilon + \varepsilon \, \mathbf{w}^\varepsilon     =    \tilde{a}^\varepsilon + \varepsilon \, w^\varepsilon $.
 Combine (\ref{LL2}), (\ref{B1}) and (\ref{B2}) to find a homogeneous
 Neumann boundary condition on $\Gamma $ for $w^\varepsilon$:
\begin{eqnarray}
\label{5}
\partial _{{\bf n}} w^\varepsilon  &=& 0  \quad  \mathrm{on} \ ]0,T[ \times \Gamma.
\end{eqnarray}

We now look for an equation on the unknown $w^\varepsilon$.
The function $\tilde{a}^\varepsilon$ belongs to  $\cC^1((0,T)\times \Omega)$ and
to $H^\infty_\Sigma(\Omega)$.
Moreover, $\tilde{a}^\varepsilon$ satisfies the equation
\begin{equation}\label{2}
    \cL(\tilde{a}^\varepsilon,\partial)\, \tilde{a}^\varepsilon =
    {\bf F}\big(\tilde{a}^\varepsilon, \varepsilon \partial_x \tilde{a}^\varepsilon,
     \cH(\tilde{a}^\varepsilon)\big) + \varepsilon  r^\varepsilon
\end{equation}
where the family $(r^\varepsilon )_\varepsilon$ lies in the set $E$ (defined above
Theorem $\ref{main2}$).
The system for the unknown $w^\varepsilon(t,x)$ writes
\begin{equation}
    \begin{aligned}
    \label{4} \cL ( \tilde{a}^\varepsilon + \varepsilon w^\varepsilon  , \partial) w^\varepsilon =
K(\varepsilon,\tilde{a}^\varepsilon,&\varepsilon \partial_x \tilde{a}^\varepsilon,\cH(\tilde{a}^\varepsilon),
w^\varepsilon,\varepsilon\partial_x w^\varepsilon,\cH(w^\varepsilon)) +  r^\varepsilon
 \quad  \mathrm{in } \  ]0,T[ \times \Omega
    \end{aligned}
\end{equation}
where $K$ is a smooth function of its arguments. Let us use more
concise notations, and note
\begin{eqnarray}
A^\varepsilon := \big(\, \tilde{a}^\varepsilon,\varepsilon \partial_x
\tilde{a}^\varepsilon,\cH(\tilde{a}^\varepsilon) \, \big)
\quad \text{and} \quad
W^\varepsilon := \big(\,  w^\varepsilon,\varepsilon\partial_x w^\varepsilon,\cH(w^\varepsilon)  \, \big).
\end{eqnarray}
Then, the Taylor formula shows that the function $K$ has the
following form:
\begin{equation}
\nonumber
K(\varepsilon,A^\varepsilon, W^\varepsilon) = G(\varepsilon, A^\varepsilon, \varepsilon  W^\varepsilon) W^\varepsilon
\end{equation}
where $G$ depends smoothly on its arguments (including $\varepsilon$),
which will be useful in the sequel.

 Following \cite{S1} there exist a family $(w^\varepsilon_{\text{init}})_\varepsilon$
 of  compatible  initial conditions for the problem (\ref{4})-(\ref{5})
 which verifies suitable uniform estimates with respect to $\varepsilon$.
We choose such a family.

\subsubsection{The iterative scheme}
\label{c2}

We want to solve the problem (\ref{4}),(\ref{5}). We use
a simple Picard(-Banach-Caccioppoli)  iterative scheme defining a sequence $w^{\varepsilon,\nu}$
which will converge to the solution of the problem. For clarity, we
adopt the following more concise notations
$$
A^\varepsilon :=\big(\, \tilde{a}^\varepsilon,\varepsilon \partial_x
\tilde{a}^\varepsilon,\cH(\tilde{a}^\varepsilon)\, \big) \quad    \text{and}   \quad   W^{\varepsilon,\nu} :=
\big(\, w^{\varepsilon,\nu},\varepsilon\partial_x w^{\varepsilon,\nu}, \cH(w^{\varepsilon,\nu}) \,
\big).
$$
With these notations, the iterative scheme writes
\begin{equation}
    \begin{aligned}
    \label{50} \cL ( \tilde{a}^\varepsilon + \varepsilon w^{\varepsilon, \nu}  , \partial)
    w^{\varepsilon,\nu+1} = f^{\varepsilon, \nu}
 \quad  \mathrm{in } \  ]0,T[ \times \Omega
    \end{aligned}
\end{equation}
where
\begin{equation}\label{91}
   f^{\varepsilon,\nu} := G(\varepsilon,A^\varepsilon,
\varepsilon W^{\varepsilon,\nu}) W^{\varepsilon,\nu} +  r^\varepsilon
\end{equation}
This equation
is coupled with the initial and boundary conditions:
\begin{eqnarray}\label{51}
  \partial _{{\bf n}} w^{\varepsilon, \nu +1} &=& 0 \ \mathrm{on} \ ]0,T[ \times \Gamma \\
  w^{\varepsilon, \nu +1}|_{t=0} &=& w^\varepsilon_{\text{init}}.\label{52}
\end{eqnarray}
The iterative scheme  is initialized with $w^{\varepsilon,0}(t,x) :=
w^\varepsilon_{\text{init}}(x)$.

\subsubsection{Estimates for a linear parabolic system}
\label{c3}

Consider the linear problem
\begin{eqnarray}
\label{num1}
  \cL(\tilde{a}^\varepsilon + \varepsilon {\bf b}, \partial) {\bf u}  &=& f \
  \mathrm{on} \ ]0,T[ \times \Omega\\
\label{num2}  \partial _{{\bf n}} {\bf u} &=& 0 \ \mathrm{on} \ ]0,T[ \times \Gamma ,
\end{eqnarray}
We endow the space $H^m_{co}(]0,T[\times \Omega)$  with the usual weighted norm with $\lambda \geq 1$:
$$
\|  {\bf u} \|_{m,\lambda} := \sum_{|\alpha| \leq m \, , \, \alpha \in
{\mathbb N}^{1+\mu}} \lambda ^{m-|\alpha|}\| e^{-\lambda t} \cZ^\alpha {\bf u}
\|_{L^2(]0,T[ \times \Omega)}.
$$
 In order to estimate the initial data, we introduce the similar norms
built with the set $\cT_0$ instead of $\cT$, integrating on $\Omega$
instead of $[0,T]\times \Omega$:
$$
| {\bf u} |_{m,\lambda} := \sum_{|\alpha| \leq m \, , \, \alpha_0 = 0\, ,
\, \alpha \in {\mathbb N}^{1+\mu}} \lambda ^{m-|\alpha|}\|  \cZ^\alpha {\bf u}
\|_{L^2(\Omega)}.
$$
We will use the following classical
Gagliardo-Moser-Nirenberg estimates  for conormal derivatives (see
\cite{G2}).
\begin{lemma}\label{moser}
Let $m\in {\mathbb N}$. There is $c_m>0$ such that, for any $a_1, \dots, a_k
\in H^m_{co}(]0,T[\times \Omega) \cap L^\infty(]0,T[\times \Omega)$,
 for all multi-index $\alpha_1 \in {\mathbb N}^{\mu +1}, \dots , \alpha_k
\in {\mathbb N}^{\mu +1}$, with $|\alpha_1|+\dots +|\alpha_k| \leq m$, for all $\lambda \geq 1$:
\begin{equation}\label{89}
\| \cZ^{\alpha_1} a_1 \dots \cZ^{\alpha_k}a_k \|_{0,\lambda} \ \leq
\ c_m \ \sum_{1 \leq j\leq k} \Big( \| a_j\|_{m,\lambda} \prod_{i
\ne j} \|a_i\|_\infty \Big).
\end{equation}
\end{lemma}

The following proposition gives some $\varepsilon$-conormal estimates for the
two first normal derivatives of the solutions of the problem
(\ref{num1})-(\ref{num2}).

\begin{proposition}
\label{22}
Let $R>0$ be an arbitrary constant and $m \geqslant 3$.  There exist
$C_m(R)>0$ and $\lambda_m>0$ such that for $\sigma$ fixed constant large enough, depending only on
the choices of the vector fields $\cZ_j$, the following holds true. Assume that
\begin{equation}\label{44}
\varepsilon  \, ( \ \|{\bf b}\|_\infty +\sum_{0\leq j\leq \mu} \|\cZ_j
{\bf b}\|_\infty+\|\varepsilon \partial_x {\bf b}\|_\infty \ ) \, \leq \, R,
\end{equation}
then, for all $\lambda \geq \lambda_m$,  the following estimates hold:
\begin{equation}\label{20}
\begin{aligned}
   \|\varepsilon  \partial_x {\bf u}\|_{m,\lambda} +
   \lambda \|{\bf u}\|_{m,\lambda}
    \ &\leq \ C_m(R) \ \big[ \ \lambda^{-1}\ \| f \|_{m,\lambda} + \ I_{m,\lambda}({\bf u})\\
    &+ \ \varepsilon \, (\, \|\varepsilon \partial_x {\bf b} \|_{m,\lambda} +
    \| {\bf b}\|_{m,\lambda}\, )\ (\, \|{\bf u}\|_\infty + \|\varepsilon\partial_x {\bf u}\|_\infty \, )\ \big],
    \end{aligned}
\end{equation}
where
$$
I_{m,\lambda}({\bf u}) := \sum _{0 \leq k \leq m}|(\partial_t^k {\bf u})_{|t=0}
|_{m-k,\lambda}.
$$
and
\begin{equation}\label{43}
\begin{aligned}
\|(\varepsilon \partial _{{\bf n}})^2 {\bf u} \|_{m,\lambda} \ \leq \ C_{m}(R) \big[ \
\|f\|_{m,\lambda} + \| {\bf u} \|_{m+1,\lambda}
+ \varepsilon \|{\bf b}\|_{m +1,\lambda}\,(\|{\bf u}\|_{\infty} +
\|f\|_\infty) \  \ \\
+ \varepsilon \| \varepsilon \partial _{{\bf n}} {\bf u} \|_{m +1,\lambda} + \varepsilon^2 \| {\bf u}
\|_{m+2,\lambda} \ \big].
\end{aligned}
\end{equation}
\end{proposition}

\begin{proof}

\emph{Step 1}.
Let us note ${\bf v}:= e^{-\lambda t} \, {\bf u}$, which satisfies
\begin{eqnarray}
  \cL(\tilde{a}^\varepsilon_{app} + \varepsilon {\bf b}, \partial) {\bf v} + \lambda {\bf v} &=&
   e^{-\lambda t} f \label{10}\
  \mathrm{on} \ ]0,T[ \times \Omega\\
  \partial _{{\bf n}} {\bf v} &=& 0 \ \mathrm{on} \ ]0,T[ \times \Gamma.\\
  {\bf v} &=& w^\varepsilon_{\text{init}} \ \mathrm{on} \ t=0.
\end{eqnarray}
 Let us note $\| . \|_{L^2}$ the $L^2$ norm in
 $[0,T]\times \Omega$, and $|.|_{L^2}$
 the $L^2$ norm in $\Omega$. Multiplying  (\ref{10}) by ${\bf v}$ and integrating
on $]0,T[ \times \Omega$ gives the following estimate,
integrating by parts
the $\varepsilon^2 \Delta_x$ with Green's formula in $\Omega$:
\begin{equation}\label{11}
    \varepsilon^2 \|\nabla_x {\bf v} \|^2_{L^2}+ \lambda \| {\bf v} \|^2_{L^2}
    \leq  2 \ |(( e^{-\lambda t}f, {\bf v} ))_{L^2}|
    + |{\bf v}(0)|_{L^2} ,
\end{equation}
for all $\lambda \geq \lambda_0$ if $\lambda_0$ is fixed
large enough, and for all $\varepsilon >0$. In terms of ${\bf u}$ it writes
\begin{equation}\label{11'}
    \varepsilon^2 \|\nabla_x {\bf u} \|^2_{0,\lambda}+ \lambda \| {\bf u} \|^2_{0,\lambda}
    \leq  2 \ |(( f, {\bf u} ))_{L^2_\lambda}|+ |{\bf u}(0)|_{L^2},
\end{equation}
where $L^2_\lambda $ is the Hilbert space $L^2(]0,T[ \times \Omega, d\mu)$
with the measure $d\mu := e^{-2\lambda t} dt dx$.

Using now the Cauchy-Schwarz inequality in the right hand side, and absorbing in the left hand side the term
in $\|v\|_{L^2}^2$ yields the desired estimate
for $m=0$ and some constant $c_0 >0$.
\\
\\ \emph{Step 2}. We show the inequality by induction on $m$.
Assume it for $m-1$. We apply a tangential operator $\cZ^\alpha$
with fields $\cZ_i \in \cT$ to the system, and $|\alpha| = m$. The function
$\cZ^\alpha {\bf u}$ satisfies the same boundary conditions.
The $L^2$ estimate (\ref{11'}) gives, for $\lambda \geq \lambda_0$:
\begin{equation}\label{21}
\begin{aligned}
     \varepsilon^2 \|\nabla_x \cZ^\alpha {\bf u} \|^2_{L^2}+ \lambda \| \cZ^\alpha {\bf u} \|^2_{L^2}
    \leq&  2 |(( e^{-\lambda t}\cZ^\alpha f  +  [(\tilde{a}^\varepsilon_{app} + \varepsilon {\bf b})\varepsilon^2\Delta_x,\cZ^\alpha ]\wedge
    {\bf u}   , \cZ^\alpha{\bf u} ))_{L^2_\lambda}|.
\end{aligned}
\end{equation}
where $[.,.]$  denotes the commutator.
Using Cauchy-Schwarz inequality and $2ab\leq 2\lambda^{-1} \, a^2 + \lambda b^2/2$ yields:
\begin{equation}\label{12}
\begin{aligned}
     \varepsilon^2 \|\nabla_x \cZ^\alpha {\bf u} \|^2_{L^2}+ \frac{\lambda}{2} \| \cZ^\alpha {\bf u} \|^2_{L^2}\
    \leq&  \ \frac{2}{\lambda}\ \|e^{-\lambda t}\cZ^\alpha f\|^2_{L^2}\\
    + \ 2 \ |(([(\tilde{a}^\varepsilon_{app} &+ \varepsilon {\bf b})\varepsilon^2\Delta_x,\cZ^\alpha ]\wedge {\bf u} , \cZ^\alpha {\bf u}))_{L^2_\lambda}|.
\end{aligned}
\end{equation}
We need to control the second term in the right hand side of (\ref{12}). The commutator
$[\ \tilde{a}^\varepsilon_{app}\, \varepsilon^2\Delta_x,\cZ^\alpha ]$
writes as a finite sum
\begin{equation}\label{13}
    \varepsilon ^2  \sum_{|\beta|\leq m+1} a_\beta^\varepsilon(t,x) \cZ^\beta +
    \varepsilon   \sum_{|\gamma|\leq m} b_\gamma^\varepsilon(t,x) \varepsilon \partial _{{\bf n}} \cZ^\gamma
    +   \sum_{|\delta|\leq m-1} c_\delta^\varepsilon(t,x) (\varepsilon\partial _{{\bf n}})^2
     \cZ^\delta
\end{equation}
where the coefficients $a_\beta^\varepsilon$ , $b_\gamma^\varepsilon$,
$c_\delta^\varepsilon$ are bounded functions satisfying
\begin{equation}
\label{14}    \sup _{\varepsilon \in ]0,1]}\|\varepsilon \partial _{{\bf n}} a_\beta^\varepsilon\|_{L^\infty(\Omega)}
    + \|\varepsilon \partial _{{\bf n}} b_\gamma^\varepsilon\|_{L^\infty(\Omega)} + \|\varepsilon \partial _{{\bf n}} c_\delta^\varepsilon \|_{L^\infty(\Omega)} < \infty
\end{equation}
for all $\beta, \gamma, \delta$, because (\ref{14})
holds clearly if we replace  $L^\infty(\Omega)$ by
 $L^\infty(\Omega_+)$ or by $L^\infty(\Omega_-)$,
 and because $\tilde{a}^\varepsilon_{app}$ is in
$H^1(\Omega)$ for all $\varepsilon >0$.
Hence we are led to control the corresponding three sort of terms:
\begin{eqnarray}
\label{15}
\varepsilon^2 (( \ a_\beta^\varepsilon \cZ^\beta {\bf u} \, ,
    \, \cZ^\alpha {\bf u} \ ))_{L^2_\lambda} ,  \
    \varepsilon (( \ b_\gamma^\varepsilon (\varepsilon \partial _{{\bf n}}) \cZ^\gamma {\bf u} \, ,
    \, \cZ^\alpha {\bf u} \ ))_{L^2_\lambda} , \
    (( \ c_\delta^\varepsilon (\varepsilon \partial _{{\bf n}})^2 \cZ^\delta {\bf u} \, ,
    \, \cZ^\alpha {\bf u} \ ))_{L^2_\lambda},
\end{eqnarray}
where $|\beta| \leq m+1$, $|\gamma| \leq m$, $|\delta| \leq m-1$.
The first two terms in (\ref{15}) are simply controlled by
$\delta  \| \varepsilon \nabla_x {\bf u} \|^2_{m,\lambda} +
C_\delta \, \delta^{-1} \,  \| {\bf u} \|^2_{m,\lambda}$
for $\delta$ arbitrarily small, and $C_\delta$ being a constant
depending on $\delta$, but independent of $\varepsilon$.
For the third term one uses an integration by parts (by Green's formula) of the field $\partial _{{\bf n}}$ to show that this term writes as a sum of terms of the form
$$
d^\varepsilon \
\varepsilon^{2-j-j'}(( (\varepsilon \partial _{{\bf n}})^j\cZ^{\delta}  {\bf u}, (\varepsilon \partial _{{\bf n}})^{j'} \cZ^{\alpha} {\bf u} ))_{L^2_\lambda}
$$
where $|\delta| \leq m-1$, $j,j'\in \{0,1\}$,
and $d^\varepsilon$ is a bounded function (uniformly in $\varepsilon$)
since all the boundary terms terms vanishes:
$\partial _{{\bf n}} \cZ^\alpha {\bf u}|_{\partial \Omega} = 0$,  for all $\alpha \in {\mathbb R}^\mu.$
It follows that the third term in (\ref{15}) is controlled by
$C\lambda^{-1} \| \varepsilon\nabla_x {\bf u} \|^2_{m,\lambda}
+ C\| {\bf u} \|^2_{m,\lambda}$
for a constant $C$ independent of $\varepsilon$, and all $\lambda \geq 1$.
Hence, by choosing  a $\delta >0$ arbitrarily small, and $\lambda_1>0$ large enough, there holds
\begin{equation}
\nonumber
     | \ (( \ [\tilde{a}^\varepsilon_{app} \varepsilon^2\Delta_x,\cZ^\alpha ]\wedge {\bf u} \, ,
    \, \cZ^\alpha {\bf u} \ ))_{L^2_\lambda} \ | \ \leq \
    \delta  \| \varepsilon \nabla_x {\bf u} \|^2_{m,\lambda} +
c_m \| {\bf u} \|^2_{m,\lambda}
\end{equation}
for all $\lambda \geq \lambda_1$, and for all $\varepsilon\in ]0,1]$,
with a constant $c_m$ independent of $\varepsilon$.

We need now to estimate the term
\begin{equation}\label{24}
(( [\ \varepsilon {\bf b}\, \varepsilon^2\Delta_x,\cZ^\alpha ]\wedge {\bf u} , \cZ^\alpha {\bf u}))_{L^2_\lambda}.
\end{equation}
The commutator $[\ {\bf b}\, \varepsilon^2\Delta_x,\cZ^\alpha ]$
writes as a finite sum
\begin{eqnarray}
\nonumber
\varepsilon ^2  \sum_{|\beta |\leq m,|\beta '|\leq m+1, |\beta| + |\beta'| \leq m+2} a_{\beta,\beta '}
(\cZ^\beta{\bf b} )\cZ^{\beta'} \label{25}\\
\nonumber
   + \ \varepsilon   \sum_{|\gamma |\leq m,|\gamma '|\leq m, |\gamma|+|\gamma'| \leq m+1} b_{\gamma,\gamma '}(\cZ^{\gamma }{\bf b} )(\varepsilon \partial _{{\bf n}}) \cZ^{\gamma '} \label{26}\\
 \nonumber
   +  \  \sum_{|\delta |\leq m,|\delta '|\leq m-1, |\delta| + |\delta'| \leq m } c_{\delta, \delta '}
    (\cZ^{\delta }{\bf b})(\varepsilon\partial _{{\bf n}})^2
     \cZ^{\delta '} \label{27}
\end{eqnarray}
where $a_{\beta,\beta '}, b_{\gamma,\gamma '},
c_{\delta, \delta '}$ are smooth fonctions on $\overline{\Omega}$. Hence to control the term (\ref{24}) we are led to
estimate tri-linear terms in $({\bf b},{\bf u},{\bf u})$ of the following form
(where $d\mu := e^{-2\lambda t}dt dx$):
\begin{eqnarray}
\varepsilon ^2 \int _{]0,T[\times \Omega} a_{\beta, \beta '}
\cZ^\beta{\bf b} .\cZ^{\beta'}{\bf u}_i . \cZ^\alpha {\bf u}_j \ d\mu ,
\quad |\beta|\leq m, |\beta '|\leq m+1, |\beta| + |\beta'| \leq m+2 \label{28}\\
\varepsilon \int _{]0,T[\times \Omega} b_{\gamma,\gamma '}\ \cZ^{\gamma }{\bf b} \ . \ \varepsilon \partial _{{\bf n}} \cZ^{\gamma '}{\bf u}_i \ . \ \cZ^\alpha {\bf u}_j \ d\mu , \quad |\gamma|\leq m, |\gamma '|\leq m, |\gamma|+|\gamma'| \leq m+1 \label{29}\\
\int _{]0,T[\times \Omega} c_{\delta,\delta '}\ \cZ^{\delta }{\bf b} . \,
(\varepsilon \partial _{{\bf n}})^2 \cZ^{\delta '}{\bf u}_i \, .  \cZ^\alpha {\bf u}_j \ d\mu , \quad
|\delta|\leq m, |\delta '|\leq m-1, |\delta|+|\delta'| \leq m  ,   \label{30}
\end{eqnarray}
where the ${\bf u}_i$ are the components of the vector ${\bf u}$.
Let us treat the term (\ref{30}). By the green formula, the integral can be written as a sum of integrals of the form
\begin{eqnarray}
     \int _{]0,T[\times \Omega} c_{\delta,\delta '}\ \cZ^{\delta }\varepsilon \partial _{{\bf n}}{\bf b} . \cZ^{\delta '}\varepsilon \partial _{{\bf n}}{\bf u}_i .  \cZ^\alpha {\bf u}_j \ d\mu \label{31}\\
    \int _{]0,T[\times \Omega} c_{\delta,\delta '}\ \cZ^{\delta }{\bf b} . \, \cZ^{\delta'}\varepsilon\partial _{{\bf n}}{\bf u}_i .\,  \cZ^\alpha\varepsilon\partial _{{\bf n}}{\bf u}_j \ d\mu ,\label{32}\\
    \varepsilon \ \int _{]0,T[\times \Omega} d_{\delta,\delta '}\ \cZ^{\delta }{\bf b} . \, \cZ^{\delta '}\varepsilon \partial _{{\bf n}}{\bf u}_i \, . \cZ^\alpha {\bf u}_j \ d\mu ,\label{33}
\end{eqnarray}
and other terms involving lower order derivatives easy to control.
The term (\ref{31}) is controlled by
$$
c \| \varepsilon \partial _{{\bf n}} \cZ^\delta {\bf b} \varepsilon \partial _{{\bf n}} \cZ^{\delta '}{\bf u}_i\|_{0,\lambda}
\ \| {\bf u}_j \|_{m,\lambda},
$$
which is bounded by using the Gagliargo-Nirenberg-Moser estimate by
\begin{equation}
\nonumber
    c\big( \ \| \varepsilon \partial _{{\bf n}} {\bf b} \|_{m,\lambda} |\varepsilon \partial _{{\bf n}} {\bf u}|_\infty +
    \| \varepsilon \partial _{{\bf n}} {\bf u} \|_{m,\lambda} |\varepsilon \partial _{{\bf n}} {\bf b}|_\infty \ \big)\
    \|{\bf u}\|_{m,\lambda}
\end{equation}
and hence by
\begin{equation}
\nonumber
    c(1+R) \big( \ \| \varepsilon \partial _{{\bf n}} {\bf b} \|_{m,\lambda} |\varepsilon \partial _{{\bf n}} {\bf u}|_\infty +
    \| \varepsilon \partial _{{\bf n}} {\bf u} \|_{m,\lambda}  \big)\
    \|{\bf u}\|_{m,\lambda}.
\end{equation}
For the term (\ref{32}) there are two cases. The first case is when $\delta = 0$.
In that case the integral is bounded by
$$
c \ \| \varepsilon \partial _{{\bf n}} {\bf u}_i \|_{m-1,\lambda} \
\| {\bf u}_j \|_{m, \lambda} \ \leq \
\lambda ^{-1} \| \varepsilon \partial _{{\bf n}} {\bf u} \|_{m,\lambda} ^2.
$$
The second case is when $|\delta| \geq 1$. In that case we write
$\cZ^\delta {\bf b} = \cZ^{\delta"} \cZ_k{\bf b}$ and apply the Gagliardo-Nirenberg-Moser inequality with $\cZ{\bf b}$ in $L^\infty$.
The term in bounded by
$$
c\big( \ \| \cZ{\bf b} \|_{m-1,\lambda} |\varepsilon \partial _{{\bf n}} {\bf u}|_\infty +
    \| \varepsilon \partial _{{\bf n}} {\bf u} \|_{m-1,\lambda} |\cZ {\bf b}|_\infty \ \big)\
    \|\varepsilon \partial _{{\bf n}} {\bf u}\|_{m,\lambda}
$$
and hence by
\begin{equation}
\nonumber
  c \ \| {\bf b} \|_{m,\lambda} |\varepsilon \partial _{{\bf n}} {\bf u}|_\infty  \|\varepsilon \partial _{{\bf n}} {\bf u}\|_{m,\lambda}
    +
    c R \lambda ^{-1}  \| \varepsilon \partial _{{\bf n}} {\bf u} \|_{m,\lambda}^2
\end{equation}
The next terms like (\ref{33}) are easier to treat in the same way,
and are bounded by the same terms.
The term (\ref{30}) was the more delicate to
estimate. The terms (\ref{29}) and (\ref{28}) are
simpler and can be treated in a similar way.
Replacing in the right hand side of (\ref{21}) and summing over all the possible
operators $\cZ^\alpha$ gives the desired estimate, and the proposition
is proved.
\end{proof}

\subsubsection{Iteration}
\label{c4}

Now classical arguments show the convergence of the iterative scheme
if $\varepsilon \in ]0,\varepsilon_0]$ and $\varepsilon_0$ is small enough. We describe
the main lines (see \cite{S1}). Let us fix an integer $m
 > 4$, and note
$$
R := 1 + \sup_{0<\varepsilon<1}  \{ \varepsilon  \, ( \ \| w^{\varepsilon, 0}\|_\infty
+\sum_{0\leq j\leq \mu} \|\cZ_j w^{\varepsilon, 0}\|_\infty+\|\varepsilon \partial_x
w^{\varepsilon, 0}\|_\infty \ ) \}.
$$

\begin{proposition}

Let be given $\lambda >1 $.
Then there exists  $h>1$ such that  for  $\varepsilon_0>0$ small enough,  for
all $\nu \in {\mathbb N}$, for all $\varepsilon \in ]0,\varepsilon_0]$, there hold
\begin{equation}\label{53}
 \| w^{\varepsilon, \nu}\|_\infty +\sum_{0\leq j\leq \mu}
\|\cZ_j w^{\varepsilon, \nu}\|_\infty+\|\varepsilon \partial_x w^{\varepsilon, \nu}\|_\infty  <
R\varepsilon ^{-1}
\end{equation}
and
\begin{equation}\label{54}
 \| w^{\varepsilon,\nu}\|_{m,\lambda} +
 \| \varepsilon \partial_{\bf n} w^{\varepsilon,\nu}\|_{m,\lambda} <
h.
\end{equation}
\end{proposition}

\begin{proof}

For  $h$ large enough,
the inequalities  (\ref{53}) and (\ref{54}) are satisfied for $\nu=0$.
Now suppose that $w^{\varepsilon,\nu}$ satisfies (\ref{53}), (\ref{54}).
We want to prove that  $w^{\varepsilon,\nu+1}$ also satisfies (\ref{53}), (\ref{54}).
The proposition \ref{22} gives a constant $C_m(R)$ and the inequality
(\ref{20}) holds with ${\bf u} = w^{\varepsilon,\nu+1}$, ${\bf b} = w^{\varepsilon,\nu}$,
and $f=f^{\varepsilon,\nu}$ defined in (\ref{91}).
 In order to control the right hand side of (\ref{50}), we need a
control of $\|\cH(w^{\varepsilon,\nu})\|_\infty$ and of
$\|\cH(w^{\varepsilon,\nu})\|_{m,\lambda}$, which is a consequence of the
following lemma.
\begin{lemma}\label{94} Let $m\in {\mathbb N}$. There exists $c>0$ such that for
all $\lambda \geq 1$,
\begin{equation}
\| \cH(v)\|_{m,\lambda} + \| \varepsilon \partial_{\bf n} \cH(v)\|_{m-1,\lambda} \leq c (\|
v\|_{m,\lambda} + \|\varepsilon  \partial_{\bf n} v\|_{m-1,\lambda} ).
\end{equation}
\end{lemma}

\begin{proof}
 We note $E(\partial) := ( \mbox{{\rm div}}\; , \mbox{{\rm curl}}\; )$ the operator from $[\cS'({\mathbb R}^3)]^3$ to $[\cS'({\mathbb R}^3)]^4$. We denote
by $E^{-1}(\partial)$ the inverse operator. Then $u = E^{-1}(\partial) f$, is
defined by $\hat{u}(\xi) = -i |\xi|^{-2} \big(\hat{a}(\xi) \xi - \xi \wedge
\hat{b}(\xi) \big) $ where $\hat{f}(\xi)=( \hat{a} (\xi),
      \hat{b}(\xi)) \in {\mathbb R} \times {\mathbb R}^3$. Thus
$\hat{u}(\xi) = M (\xi) \hat{f}(\xi)$,  where  $M(\xi)$ is a $3\times 4$ matrix whose entries are
\emph{rational functions of $\xi$ homogeneous of degree $-1$}. Let
us fix $\chi \in \cC^\infty_0({\mathbb R}^3, {\mathbb R})$ such that $\chi(\xi)=0$
when $|\xi| \leq 1$ and $\chi(\xi)=1$ when $|\xi|\geq 2$, and call
$P(D)$ and $R(D)$ the operators from  $[\cS'({\mathbb R}^3)]^4$ to
$[\cS'({\mathbb R}^3)]^3$ defined by
$P(D)f := \cF^{-1}\big(\, \chi M \hat{f}\, \big)$ and $ R(D)f :=
\cF^{-1}\big(\, (1-\chi) \hat{f}\, \big)$
where $\cF^{-1}$ means the inverse Fourier transform. In the
sequel we will simply note $\cS'({\mathbb R}^3)$ and $L^2(\tilde{\Omega})$ instead
of $\big[\cS'({\mathbb R}^3)\big]^4$ and $\big[L^2(\tilde{\Omega})\big]^4$, meaning
that we talk about the \emph{components} of the vector valued
functions, the (finite) number of components being understood. We
have $E^{-1}(\partial) = P(D) + R(D).$
The operator $P(D)$ is a special case of classical
pseudo-differential operator of class $S^{1}_{-1,0}({\mathbb R}^3 \times
{\mathbb R}^3)$, elliptic, and $R(D)$ is an infinitely smoothing operator of
class $S^{-\infty}_{1,0}({\mathbb R}^3 \times {\mathbb R}^3)$.

Let us now take into account the $t$ coordinate. Let us note
$\tilde{\Omega}= ]0,T[\times \Omega$,  $\tilde{\Gamma} = ]0,T[\times \Gamma$ and
$\tilde{\Sigma} = ]0,T[\times \Sigma$. We extend the actions of $P$ and $R$
to the spaces of functions or distributions which depend also on $t$
like $L^2(\tilde{\Omega})$ or $C\big( [0,T],\cS'({\mathbb R}^3)\big)$, by
considering $t$ as a parameter so that $Pu(t,x) := P(D)u(t,.)(x)$.
Let $v \in H^m_{co}(\tilde{\Omega}; {\mathbb R}^4)$ such that $\partial _{{\bf n}} v \in
H^{m-1}_{co}(\tilde{\Omega}; {\mathbb R}^4)$.   Then $\cH(v)= u_{|\tilde{\Omega}}$ where
$u\in L^2([0,T]\times {\mathbb R}^3)$ is defined by
$   E(\partial)u = \overline{E(\partial)v} +
    (v_{|\tilde{\Gamma}}.{\bf n}) \otimes \delta_{\tilde{\Gamma}} $,
where the notation $\overline{V}$ means the extension of $V$ by $0$
to $[0,T]\times {\mathbb R}^3$ (or to ${\mathbb R}^3$, depending on the context).

Let us note
$f :=\overline{E(\partial)v}$, which is in $H^{m-1}_{co}(]0,T[\times {\mathbb R}^3),$
and $g=(v.{\bf n})_{|\Gamma}$. This trace is well defined since by
assumption $v\in H^1(\tilde{\Omega})$, and using local coordinates patches
one sees that
$g \in H^{m-\frac{1}{2}}(\tilde{\Gamma})$,
the usual Sobolev spaces. The operator $P(D)$ satisfies the
\emph{transmission property} (introduced by Boutet de Monvel
\cite{BMV}, \cite{BMV2}) on $\Omega$ and on ${\mathbb R}^3\setminus \Omega$
because its symbol is a rational function of $\xi$, which is a
sufficient condition to satisfy the transmission condition. The
transmission property has been also studied and used by  Grubb,
and we also refer to papers
\cite{Grubb1} and \cite{Grubb2}. To avoid many repetitions, we will
note in what follows $\Omega_1 := \Omega$ and
$\Omega_2={\mathbb R}^3\setminus \Omega$. Since $P(D)$ is elliptic of order
1, the transmission property implies (see \cite{Grubb1} and
\cite{Grubb2}) that if $ v \in H^s(\Omega)$ then for $j=1,2$,
    $\big( P(D) \overline{v} \big)_{|\Omega_j}
    \in H^{s+1}(\Omega_j)$.
Let us note note $u_{(j)} = u_{|\tilde{\Omega}_j}$, for $j=1, 2$, so that
$\cH(v)= u_{(1)}\in L^2(\tilde{\Omega})$. Using the notations of
\cite{BMV}, \cite{Grubb1}, \cite{Grubb2},
\begin{equation}\label{102}
u _{(j)}= \big( E^{-1}(D)\overline{v}\big)_{|\tilde{\Omega}_{(j)}} =
P(D)^{(j)} f + K^{(j)}_\Gamma (g) + R(D)^{(j)}E(\partial)\overline{v},
\end{equation}
where
 $P(D)^{(j)} f = (P(D)f)_{|\tilde{\Omega}_{(j)}}$,
 $R(D)^{(j)}\overline{v}= (R(D)\overline{v})_{|\tilde{\Omega}_{(j)}}$ and where
$K^{(j)}_\Gamma (g) = \big(\,P(D)\big( g \otimes \delta_\Gamma)\,
\big)_{|\tilde{\Omega}_{(j)}}$  is the "Poisson operator":
\begin{equation}\label{103}
K_\Gamma^{(j)} : H^s(\Gamma) \rightarrow H^{s+1/2}(\Omega^{(j)}),
\quad
\end{equation}
(linear continuous), extended to functions depending on $t$ as a
parameter. (See theorems 2.4 and 2.5 of \cite{Grubb2}).

Let us now prove the lemma. First of all, $\partial_t^m\cH(v)=\cH(\partial_t^m
v)$ is in $L^2(\tilde{\Omega})$ because $\partial_t^m v\in L^2(\tilde{\Omega})$ and
$\cH$ acts on $L^2(\tilde{\Omega})$. It is also easy to show that
$\partial_t^{m-1}\cH(v) \in H^1(\tilde{\Omega})$: by assumption, for any $t\in
[0,T]$, $\partial_t^{m-1}v(t,.) \in H^1(\Omega)$, hence
$\cH(\partial_t^{m-1}\overline{v})(t,.) \in H^1(\Omega)$ because
$\partial_t^{m-1}\overline{v}(t,.)$ is piecewise-$H^1$ and because of the
properties of $\cH$.
 Hence $\partial_x\partial_t^{m-1}\cH(v)
\in L^2(\tilde{\Omega})$ and since we already know that $\partial_t^m\cH(v) \in
L^2(\tilde{\Omega})$ we have proved that $\partial_t^{m-1}\cH(v)\in
H^1(\tilde{\Omega})$.

Let us show now that $\cZ_j\partial_t^{m-2}\cH(v) \in H^1(\tilde{\Omega})$ for
$j=1, \dots , \mu$. We have
$    E(\partial)u = f + g \otimes \delta_\Gamma $.
Since $E(\partial)$ is elliptic (as an operator in $\cS'({\mathbb R}^3)$, but not
in $\cS'({\mathbb R}^4)$), we can express the normal derivatives of $u$ in
term of tangential derivatives and of $E(\partial)u$, and this implies
that the commutator $[E(\partial),\cZ_j]u$ writes
\begin{equation}\label{107}
[E(\partial), \cZ_j ]u = \sum_1^\mu A_j \cZ_j u + A_0 f + B g\otimes
\delta_\Gamma
\end{equation}
where $A_j,B$ are matrices with $\cC_b^\infty$ entries (depending on
the fields $\cZ_j$). It follows that
$$
E(\partial)\cZ_j u = \sum_{|\alpha|\leq 1} M_\alpha \cZ^\alpha f +
\sum_{|\alpha|\leq 1} N_\alpha (\cZ^\alpha g) \otimes \delta_\Gamma
$$
with $\cC_b^\infty({\mathbb R}^3)$ matrices $M_\alpha, N_\alpha$, and
applying $\partial_t^{m-2}$ gives:
\begin{equation}\label{108}
E(\partial)\cZ_j \partial_t^{m-2} u = \sum_{|\alpha|\leq 1} M_\alpha .
\cZ^\alpha \partial_t^{m-2}f + \sum_{|\alpha|\leq 1} N_\alpha .
(\cZ^\alpha\partial_t^{m-2} g) \otimes \delta_\Gamma
\end{equation}
Now $\cZ^\alpha \partial_t^{m-2}f \in L^2(\tilde{\Omega})$, because $f=
\overline{E(\partial)v}$, and the transmission property implies that for every
$t\in [0,T]$, the function
  $  P(D)^{(j)}\big(\, \cZ^\alpha\partial_t^{m-1}f \, \big)(t,.)$
  is in $  H^1\big( \, \Omega_{(j)} \, \big)$. This implies that $\partial_x P(D)^{(j)}\big(\,
\cZ^\alpha\partial_t^{m-2}f \, \big)\in L^2(\tilde{\Omega})$ and since we already
know that $\partial_t P(D)^{(j)}\big(\, \cZ^\alpha\partial_t^{m-2}f \, \big)\in
L^2(\tilde{\Omega})$ from the previous case, we deduce that for $j=1, 2$ the functions
    $P(D)^{(j)}\big(\, \cZ^\alpha\partial_t^{m-2}f \, \big)$ is in
    $H^1\big( \, \tilde{\Omega}_{(j)} \, \big)$.
Concerning the boundary term in (\ref{108}), since $g \in
H^{m-\frac{1}{2}}(\tilde{\Gamma})$ we know that
 $\cZ^\alpha\partial_t^{m-2} g \in H^{1/2}(\Gamma)$ and
the property (\ref{103}) implies that, for all $t\in [0,T]$, the
functions $K_\Gamma^{(j)} \big( \, \cZ^\alpha\partial_t^{m-2} g \, \big)(t,.) $
is in
$H^1\big(\Omega_{(j)}\big)$.
By the same way as before we deduce that for $j=1, 2$ the functions
$K_\Gamma^{(j)} \big( \, \cZ^\alpha\partial_t^{m-2} g \, \big) $ is in
$H^1\big(\tilde{\Omega}_{(j)}\big) $.
Now, applying $E(\partial)^{-1} = P(D) + R(D)$ to the equation
(\ref{108}) gives $\cZ_j
\partial_t^{m-2}u_{(j)} \in H^1\big(\tilde{\Omega}_{(j)}\big)$ as claimed. Then,
the proof can be continued by induction in the same way.
\end{proof}

The lemma \ref{94},
together with the Gagliardo-Nirenberg-Moser estimates and the
induction assumption, implies that (like the majoration of the term
(5.25) in paper \cite{S1}):
\begin{equation}\label{95}
    \| f^{\varepsilon,\nu} \|_{m,\lambda} \leq c (R)
    (\|w^{\varepsilon,\nu}\|_{m,\lambda}+ \| \varepsilon \partial_x
    w^{\varepsilon,\nu}\|_{m,\lambda}) < c(R) \rho(\lambda).
\end{equation}
Hence, the proposition \ref{22} implies that
\begin{equation}\label{96}
\begin{aligned}
\|\varepsilon \partial_x w^{\varepsilon,\nu+1}\|_{m,\lambda}+ \lambda
\|w^{\varepsilon,\nu+1}\|_{m,\lambda} &\leq C_m(R) \big[ \lambda^{-1} c(R)
\rho(\lambda) \\
&+ R (\|w^{\varepsilon,\nu+1}\|_\infty + \| \varepsilon
\partial_xw^{\varepsilon,\nu+1}\|_\infty)  + I_{m,\lambda}
(w^{\varepsilon,\lambda})\big].
\end{aligned}
\end{equation}
We now use the following Sobolev inequalities:
\begin{eqnarray*}
    \varepsilon^{1/2} \| {\bf u} \|_\infty &\leq&  e^{\sigma \lambda }
    ( \| {\bf u}\|_{m,\lambda} +
    \|\varepsilon\partial _{{\bf n}} {\bf u}\|_{m,\lambda}),
\\   \varepsilon^{1/2} \| \varepsilon\partial _{{\bf n}} {\bf u} \|_\infty &\leq&  e^{\sigma \lambda }
    ( \| \varepsilon \partial _{{\bf n}}{\bf u}\|_{m,\lambda} +
    \|(\varepsilon\partial _{{\bf n}})^2{\bf u}\|_{m,\lambda}).
\end{eqnarray*}
By taking $\lambda $ large enough et $\varepsilon >0$
small enough
the inequality (\ref{53}) is also satisfied for $w^{\varepsilon, \nu +1}$
and the proof by induction is complete.
\end{proof}

Now by extracting a convergent subsequence it is a classical
argument to show the convergence in $L^2(]0,T[\times \Omega)$ of
$w^{\varepsilon,\nu}$ to a solution $w^\varepsilon$ of the non linear problem
which satisfies the same estimates $(\ref{53})$, $(\ref{54})$. This
concludes the proof of  Theorem $\ref{main2}$. \hfill $\Box$