$H_{2}=1+\underline{h}_{2}^{-1}\zeta-\underline{h}_{2}^{-1}b$ satisfy

$\begin{cases}\underline{h}_{1}^{-1}\|\zeta\|_{H^{m-1}}+\underline{h}_{2}^{-1}\|\zeta\|_{H^{m-1}}+\underline{h}_{2}^{-1}\|b\|_{W^{m,\infty}}\leq
M,\\\ H_{1}(\bm{x})\geq c,\quad H_{2}(\bm{x})\geq
c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},\end{cases}$

and if $\bm{\varphi}_{1}$ and $\bm{\varphi}_{2}$ satisfy

$\begin{cases}\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})\bm{\varphi}_{1}=f_{1,i}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})\bm{\varphi}_{2}=f_{2,i}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\end{cases}$

then for any $k=0,\pm 1,\ldots,\pm(m-1)$ we have

$(\underline{h}_{\ell}\delta)^{-2}\|\bm{\varphi}_{\ell}^{\prime}\|_{H^{k}}\leq
C(\|\nabla\bm{\varphi}_{\ell}\|_{H^{k+1}}+\|\bm{\varphi}_{\ell}^{\prime}\|_{H^{k+1}}+\|\bm{f}_{\ell}^{\prime}\|_{H^{k}})\qquad(\ell=1,2).$

The next lemma gives an energy estimate of the solution to the Kakinuma model
(6.1) under appropriate assumptions on the solution. We remind that the
mathematical energy function $E_{m}(t)$ is defined by (5.3).

######  Lemma 6.4.

Let $c,M,M_{1},\underline{h}_{\mathrm{min}}$ be positive constants. There
exist two positive constants $C=C(c,M,\underline{h}_{\mathrm{min}})$ and
$C_{1}=C_{1}(c,M,M_{1},\underline{h}_{\mathrm{min}})$ such that for any
positive parameters
$\underline{\rho}_{1},\underline{\rho}_{2},\underline{h}_{1},\underline{h}_{2},\delta$
satisfying the natural restrictions (2.14),
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, and the condition
${\underline{h}_{\mathrm{min}}\leq\underline{h}_{1},\underline{h}_{2}}$, if a
regular solution $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ to the Kakinuma model
(6.1) with a bottom topography $b$ satisfies (6.12),
$\underline{h}_{2}^{-1}\bigl{(}\|b\|_{W^{m+1,\infty}}+(\underline{h}_{2}\delta)\|b\|_{W^{m+2,\infty}}\bigr{)}\leq
M_{1}$, and $E_{m}(t)\leq M_{1}$ for some time interval $[0,T]$, then we have
$E_{m}(t)\leq C\mathrm{e}^{C_{1}t}E_{m}(0)$ for $0\leq t\leq T$.

###### Proof.

Let $\beta$ be a multi-index such that $1\leq|\beta|\leq m$. Applying
$\partial^{\beta}$ to the Kakinuma model (6.1), after a tedious but
straightforward calculation, we obtain

(6.14)
$\begin{cases}\displaystyle{\bm{l}}_{1}(H_{1})(\partial_{t}+\bm{u}_{1}\cdot\nabla)\partial^{\beta}\zeta+\underline{h}_{1}L_{1}^{\mathrm{pr}}(H_{1},\delta,\underline{h}_{1})\partial^{\beta}\bm{\phi}_{1}=\bm{f}_{1,\beta},\\\
\displaystyle{\bm{l}}_{2}(H_{2})(\partial_{t}+\bm{u}_{2}\cdot\nabla)\partial^{\beta}\zeta-\underline{h}_{2}L_{2}^{\mathrm{pr}}(H_{2},\delta,\underline{h}_{2})\partial^{\beta}\bm{\phi}_{2}=\bm{f}_{2,\beta},\\\
\underline{\rho}_{1}{\bm{l}}_{1}(H_{1})\cdot(\partial_{t}+\bm{u}_{1}\cdot\nabla)\partial^{\beta}\bm{\phi}_{1}-\underline{\rho}_{2}{\bm{l}}_{2}(H_{2})\cdot(\partial_{t}+\bm{u}_{2}\cdot\nabla)\partial^{\beta}\bm{\phi}_{2}-a\partial^{\beta}\zeta=f_{0,\beta},\end{cases}$

where $L_{1}^{\mathrm{pr}}$ and $L_{2}^{\mathrm{pr}}$ are defined by (6.6),
the function $a$ by (3.9), and

(6.15) $\displaystyle\bm{f}_{1,\beta}$
$\displaystyle:=-[\partial^{\beta},\bm{l}_{1}(H_{1})]\partial_{t}\zeta+\underline{h}_{1}\bigl{\\{}[\partial^{\beta},A_{1}(H_{1})]\Delta\bm{\phi}_{1}-(\bm{l}_{1}(H_{1})\otimes\bm{l}_{1}(H_{1}))(\nabla
H_{1}\cdot\nabla)\partial^{\beta}\bm{\phi}_{1}$
$\displaystyle\qquad+[\partial^{\beta},\bm{l}_{1}(H_{1})\otimes\bm{u}_{1}]\nabla
H_{1}-(\underline{h}_{1}\delta)^{-2}[\partial^{\beta},C_{1}(H_{1})]\bm{\phi}_{1}\bigr{\\}},$
(6.16) $\displaystyle\bm{f}_{2,\beta}$
$\displaystyle:=-[\partial^{\beta},\bm{l}_{2}(H_{2})]\partial_{t}\zeta-\underline{h}_{2}\bigl{\\{}[\partial^{\beta},A_{2}(H_{2})]\Delta\bm{\phi}_{2}-(\bm{l}_{2}(H_{2})\otimes\bm{l}_{2}(H_{2}))(\nabla
H_{2}\cdot\nabla)\partial^{\beta}\bm{\phi}_{2}$
$\displaystyle\qquad+[\partial^{\beta},\bm{l}_{2}(H_{2})\otimes\bm{u}_{2}]\nabla
H_{2}-(\underline{h}_{2}\delta)^{-2}[\partial^{\beta},C_{2}(H_{2})]\bm{\phi}_{2}$
$\displaystyle\qquad-\bm{l}_{2}(H_{2})(\bm{u}_{2}\cdot\partial^{\beta}(\underline{h}_{2}^{-1}\nabla
b))-\partial^{\beta}\bigl{(}\tilde{B}_{2}(H_{2})(\underline{h}_{2}^{-1}\nabla
b\cdot\nabla)\bm{\phi}_{2}+\tilde{C}_{2}(H_{2},\underline{h}_{2}^{-1}b)\bm{\phi}_{2}\bigr{)}\bigr{\\}},$
(6.17) $\displaystyle f_{0,\beta}$
$\displaystyle:=-\underline{\rho}_{1}\bigl{\\{}\bigl{(}[\partial^{\beta},\bm{l}_{1}(H_{1})]-\bm{l}_{1}^{\prime}(H_{1})(\partial^{\beta}H_{1})\bigr{)}^{\mathrm{T}}\partial_{t}\bm{\phi}_{1}$
$\displaystyle\qquad+\tfrac{1}{2}[\partial^{\beta};\bm{u}_{1},\bm{u}_{1}]+\tfrac{1}{2}(\underline{h}_{1}\delta)^{-2}[\partial^{\beta};w_{1},w_{1}]$
$\displaystyle\qquad+\bm{u}_{1}\cdot\bigl{(}\bigl{(}[\partial^{\beta},\bm{l}_{1}(H_{1})]-\bm{l}_{1}^{\prime}(H_{1})(\partial^{\beta}H_{1})\bigr{)}\otimes\nabla\bigr{)}^{\mathrm{T}}\bm{\phi}_{1}$
$\displaystyle\qquad-(\underline{h}_{1}\delta)^{-2}w_{1}\bigl{(}\bigl{(}[\partial^{\beta},\bm{l}_{1}^{\prime}(H_{1})]-\bm{l}_{1}^{\prime\prime}(H_{1})(\partial^{\beta}H_{1})\bigr{)}^{\mathrm{T}}\bm{\phi}_{1}+\bm{l}_{1}^{\prime}(H_{1})\cdot\partial^{\beta}\bm{\phi}_{1}\bigr{)}\bigr{\\}}$
$\displaystyle\quad\;+\underline{\rho}_{2}\bigl{\\{}\bigl{(}[\partial^{\beta},\bm{l}_{2}(H_{2})]-\bm{l}_{2}^{\prime}(H_{2})(\partial^{\beta}H_{2})-\bm{l}_{2}^{\prime}(H_{2})(\partial^{\beta}(\underline{h}_{2}^{-1}b))\bigr{)}^{\mathrm{T}}\partial_{t}\bm{\phi}_{2}$
$\displaystyle\qquad+\tfrac{1}{2}[\partial^{\beta};\bm{u}_{2},\bm{u}_{2}]+\tfrac{1}{2}(\underline{h}_{2}\delta)^{-2}[\partial^{\beta};w_{2},w_{2}]$
$\displaystyle\qquad+\bm{u}_{2}\cdot\bigl{(}\bigl{(}[\partial^{\beta},\bm{l}_{2}(H_{2})]-\bm{l}_{2}^{\prime}(H_{2})(\partial^{\beta}H_{2})-\bm{l}_{2}^{\prime}(H_{2})(\partial^{\beta}(\underline{h}_{2}^{-1}b))\bigr{)}\otimes\nabla\bigr{)}^{\mathrm{T}}\bm{\phi}_{2}$
$\displaystyle\qquad-\bm{u}_{2}\cdot[\partial^{\beta},\underline{h}_{2}^{-1}\nabla
b\otimes\bm{\phi}_{2}]\bm{l}_{2}^{\prime}(H_{2})$
$\displaystyle\qquad-(\bm{u}_{2}\cdot\underline{h}_{2}^{-1}\nabla
b)\bm{\phi}_{2}\cdot\bigl{(}\partial^{\beta}\bm{l}_{2}^{\prime}(H_{2})-\bm{l}_{2}^{\prime\prime}(H_{2})(\partial^{\beta}H_{2})-\bm{l}_{2}^{\prime\prime}(H_{2})(\partial^{\beta}(\underline{h}_{2}^{-1}b))\bigr{)}$
$\displaystyle\qquad+(\underline{h}_{2}\delta)^{-2}w_{2}\bigl{(}\bigl{(}[\partial^{\beta},\bm{l}_{2}^{\prime}(H_{2})]-\bm{l}_{2}^{\prime\prime}(H_{2})(\partial^{\beta}H_{2})-\bm{l}_{2}^{\prime\prime}(H_{2})(\partial^{\beta}(\underline{h}_{2}^{-1}b))\bigr{)}^{\mathrm{T}}\bm{\phi}_{2}$
$\displaystyle\qquad\qquad+\bm{l}_{2}^{\prime}(H_{2})\cdot\partial^{\beta}\bm{\phi}_{2}\bigr{)}\bigr{\\}}.$

Here,
$[\partial^{\beta};u,v]=\partial^{\beta}(uv)-(\partial^{\beta}u)v-u(\partial^{\beta}v)$
is the symmetric commutator. For vector valued functions, it is defined by
$[\partial^{\beta};\bm{u},\bm{v}]=\partial^{\beta}(\bm{u}\cdot\bm{v})-(\partial^{\beta}\bm{u})\cdot\bm{v}-\bm{u}\cdot(\partial^{\beta}\bm{v})$.

On the other hand, by Lemma 5.2 we have the estimate (5.4) for time
derivatives of the solution. Particularly, we have

(6.18)
$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\partial_{t}\bm{u}_{\ell}\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}w_{\ell}\|_{H^{m-1}}^{2}+\|\partial_{t}\bm{\phi}_{\ell}^{\prime}\|_{H^{m}}^{2}+\|\partial_{t}^{2}\bm{\phi}_{\ell}^{\prime}\|_{H^{m-1}}^{2}\bigr{)}\lesssim
E_{m}.$

Note that we have also the estimate (5.8) for the velocities
$(\bm{u}_{\ell},w_{\ell})$ $(\ell=1,2)$. Moreover, it follows from Lemma 6.3
that
$\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{-4}\|\bm{\phi}_{\ell}^{\prime}\|_{H^{m-1}}^{2}\lesssim
E_{m}$ for $\ell=1,2$. In view of the definition (3.9) of the function $a$, it
is not difficult to check the estimate
$\|a-1\|_{H^{m}}^{2}+\|\partial_{t}a\|_{H^{m-1}}^{2}\lesssim E_{m}$.
Therefore, by the Sobolev imbedding theorem we see that all the assumptions in
Lemma 6.2 are satisfied, so that for the solution
$\bm{U}=(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})^{\mathrm{T}}$ we have

$\mathscr{E}(\partial^{\beta}{\bm{U}}(t))\leq
C\mathrm{e}^{C_{1}t}\mathscr{E}(\partial^{\beta}{\bm{U}}(0))+C_{1}\int_{0}^{t}\mathrm{e}^{C_{1}(t-\tau)}\mathscr{F}_{\beta}(\tau)\mathrm{d}\tau,$

where

$\displaystyle\mathscr{F}_{\beta}$
$\displaystyle=\|f_{0,\beta}\|_{H^{1}}(\|\partial_{t}\partial^{\beta}\zeta\|_{H^{-1}}+\|\partial^{\beta}\zeta\|_{L^{2}})$
$\displaystyle\quad\;+\sum_{\ell=1,2}\underline{\rho}_{\ell}(\|\bm{f}_{\ell,\beta}\|_{L^{2}}+\|\partial^{\beta}\zeta\|_{L^{2}})\|(\partial_{t}\partial^{\beta}\bm{\phi}_{\ell},\nabla\partial^{\beta}\bm{\phi}_{\ell})\|_{L^{2}}.$

In view of the estimates (5.4), (5.8), and (6.18) together with

$\|([\partial^{\beta},\bm{l}_{\ell}(H_{\ell})]-\bm{l}_{\ell}^{\prime}(H_{1})(\partial^{\beta}H_{\ell}))^{\mathrm{T}}\bm{\varphi}_{\ell}\|_{H^{1}}\lesssim\|\bm{\varphi}_{\ell}^{\prime}\|_{H^{m}}$

for $\ell=1,2$, we obtain $\mathscr{F}_{\beta}\lesssim E_{m}$. We note that
the multi-index $\beta$ is assumed to satisfy ${1\leq|\beta|\leq m}$. As for
the case $\beta=0$, in view of
$\frac{\mathrm{d}}{\mathrm{d}t}\mathscr{E}(\bm{U}(t))\lesssim E_{m}(t)$ we
infer the inequality
$\mathscr{E}({\bm{U}}(t))\leq\mathscr{E}({\bm{U}}(0))+C_{1}\int_{0}^{t}E_{m}(\tau)\mathrm{d}\tau$.
Summarizing the above estimates we obtain

$E_{m}(t)\leq
C\mathrm{e}^{C_{1}t}E_{m}(0)+C_{1}\int_{0}^{t}\mathrm{e}^{C_{1}(t-\tau)}E_{m}(\tau)\mathrm{d}\tau$

with constants $C=C(c,M,\underline{h}_{\mathrm{min}})$ and
$C_{1}=C_{1}(c,M,M_{1},\underline{h}_{\mathrm{min}})$. Therefore, Gronwall’s
inequality gives the desired estimate. ∎

Now, we are ready to prove Theorem 3.1. Suppose that the initial data
$(\zeta_{(0)},\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ and the bottom topography
$b$ satisfy (3.10)–(3.13). Let $C_{0}$ be a positive constant such that

$\sum_{\ell=1,2}(\|H_{\ell(0)}\|_{L^{\infty}}+\underline{\rho}_{\ell}\underline{h}_{\ell}\|\bm{u}_{\ell(0)}\|_{L^{\infty}}^{2})+\|a_{(0)}\|_{L^{\infty}}\leq
C_{0}.$

Such a constant $C_{0}$ exists as a constant depending on
$c_{0},M_{0},\underline{h}_{\mathrm{min}}$, and $m$. We will show that the
solution $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ satisfies (3.14), (3.15), and

(6.19)
$\sum_{\ell=1,2}(\|H_{\ell}(t)\|_{L^{\infty}}+\underline{\rho}_{\ell}\underline{h}_{\ell}\|\bm{u}_{\ell}(t)\|_{L^{\infty}}^{2})+\|a(t)\|_{L^{\infty}}\leq
2C_{0}$

for $0\leq t\leq T$ with a constant $M$ and a time $T$ which will be
determined below. We note that (3.14) is equivalent to $E_{m}(t)\leq M$. To
this end, we assume that the solution satisfies (3.14), (3.15), and (6.19) for
$0\leq t\leq T$. In the following, the constant depending on
$c_{0},C_{0},\underline{h}_{\mathrm{min}},m$ but not on $M$ is denoted by $C$
and the constant depending also on $M$ by $C_{1}$. These constants may change
from line to line. Then, it follows from Lemma 6.4 that $E_{m}(t)\leq
C\mbox{e}^{C_{1}t}M_{0}$ for $0\leq t\leq T$. Therefore, if we chose
$M=2CM_{0}$ and if $T$ is so small that $T\leq C_{1}^{-1}\log 2$, then (3.14)
holds in fact for $0\leq t\leq T$. It remains to show (3.15) and (6.19). As
before, we can check

$\begin{cases}\sum_{\ell=1,2}\bigl{(}\|\partial_{t}H_{\ell}(t)\|_{L^{\infty}}+\sqrt{\underline{\rho}_{\ell}\underline{h}_{\ell}}\|\partial_{t}\bm{u}_{\ell}(t)\|_{L^{\infty}}\bigr{)}+\|\partial_{t}a(t)\|_{L^{\infty}}\leq
C_{1},\\\
\displaystyle\|\partial_{t}\biggl{(}a(t)-\frac{\underline{\rho}_{1}\underline{\rho}_{2}}{\underline{\rho}_{1}H_{2}(t)\alpha_{2}+\underline{\rho}_{2}H_{1}(t)\alpha_{1}}|{\bm{u}}_{1}(t)-{\bm{u}}_{2}(t)|^{2}\biggr{)}\|_{L^{\infty}}\leq
C_{1}.\end{cases}$

Therefore, if $T$ is so small that $T\leq(2C_{1})^{-1}c_{0}$ and
$T\leq((2C_{0}^{1/2}+1)C_{1})^{-1}C_{0}$, then the lower bound (3.15) and the
upper bound (6.19) hold in fact for $0\leq t\leq T$. This completes the proof
of Theorem 3.1.

##  7 Approximation of solutions; proof of Theorem 3.8

In this section we prove Theorem 3.8, which gives a rigorous justification of
the Kakinuma model as a higher order shallow water approximation of the full
model for interfacial gravity waves under the hypothesis of the existence of
the solution to the full model with uniform bounds.

###  7.1 Supplementary estimate for the Dirichlet-to-Neumann map

In this subsection, we give a supplementary estimate to Lemma 4.2 for the
Dirichlet-to-Neumann map $\Lambda(\zeta,b,\delta)$ defined by (4.6) appearing
in the framework of surface waves. We recall the map
$\Lambda^{(N)}(\zeta,b,\delta)\colon\phi\mapsto\mathcal{L}_{0}(H,b,\delta)\bm{\phi}$,
where $\mathcal{L}_{0}(H,b,\delta)$ is defined by (4.3) and $\bm{\phi}$ is the
unique solution to (4.5). In this section we omit the dependence of $t$ in
notations.

######  Lemma 7.1.

Let $c,M$ be positive constants and $m,j$ integers such that
$m>\frac{n}{2}+1$, $m\geq 2(j+1)$, and $1\leq j\leq 2N+1$. We assume (H1) or
(H2). There exists a positive constant $C$ such that if $\zeta\in H^{m}$,
$b\in W^{m+1,\infty}$, and $H=1+\zeta-b$ satisfy (4.8), then for any
$\phi\in\mathring{H}^{k+2(j+1)}$ with $0\leq k\leq m-2(j+1)$ and any
$\delta\in(0,1]$ we have

$\|(-\Delta)^{-\frac{1}{2}}(\Lambda^{(N)}(\zeta,b,\delta)\phi-\Lambda(\zeta,b,\delta)\phi)\|_{H^{k}}\leq
C\delta^{2j}\|\nabla\phi\|_{H^{k+2j+1}}.$

###### Proof.

This lemma can be proved in a similar way to the proof of Lemma 4.2 with a
slight modification. For the completeness, we sketch the proof. By the duality
$(H^{k})^{*}=H^{-k}$ and the symmetry of the operator
$(-\Delta)^{-\frac{1}{2}}$, it is sufficient to show the estimate

$|((\Lambda-\Lambda^{(N)})\phi,\psi)_{L^{2}}|\lesssim\delta^{2j}\|\nabla\phi\|_{H^{k+2j+1}}\|\nabla\psi\|_{H^{-k}}$

for any $\phi\in\mathring{H}^{k+2(j+1)}$ and any $\psi\in H^{1-k}$. We
decompose it as

$\displaystyle((\Lambda-\Lambda^{(N)})\phi,\psi)_{L^{2}}$
$\displaystyle=((\Lambda-\Lambda^{(2N+2)})\phi,\psi)_{L^{2}}+((\Lambda^{(2N+2)}-\Lambda^{(N)})\phi,\psi)_{L^{2}}$
$\displaystyle=:I_{1}+I_{2}$

and evaluate the two components of the right-hand side separately.

We remind the definitions (4.1) of the $(N^{*}+1)$ vector-valued function
$\bm{l}(H)$ and (4.3) of the operator $\mathcal{L}_{i}(H,b,\delta)$, which
acts on $(N^{*}+1)$ vector-valued functions. These depend on $N$, so that we
denote them by $\bm{l}^{(N)}(H)$ and $\mathcal{L}_{i}^{(N)}(H,b,\delta)$,
respectively, in the following argument. Let $\Phi$ be the solution to the
boundary value problem (4.7) and let
$\bm{\phi}=(\phi_{0},\phi_{1},\ldots,\phi_{N^{*}})$,
$\tilde{\bm{\phi}}=(\tilde{\phi}_{0},\tilde{\phi}_{1},\ldots,\tilde{\phi}_{2N^{*}+2})$,
and $\bm{\psi}=(\psi_{0},\psi_{1},\ldots,\psi_{2N^{*}+2})$ be the solutions to
the problems

$\begin{cases}\mathcal{L}_{i}^{(N)}(H,b,\delta)\bm{\phi}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\ {\bm{l}}^{(N)}(H)\cdot\bm{\phi}=\phi,\end{cases}$
$\begin{cases}\mathcal{L}_{i}^{(2N+2)}(H,b,\delta)\tilde{\bm{\phi}}=0\quad\mbox{for}\quad
i=1,2,\ldots,2N^{*}+2,\\\
{\bm{l}}^{(2N+2)}(H)\cdot\tilde{\bm{\phi}}=\phi,\end{cases}$

and

$\begin{cases}\mathcal{L}_{i}^{(2N+2)}(H,b,\delta){\bm{\psi}}=0\quad\mbox{for}\quad
i=1,2,\ldots,2N^{*}+2,\\\
{\bm{l}}^{(2N+2)}(H)\cdot{\bm{\psi}}=\psi,\end{cases}$

respectively. Put

(7.1)
$\begin{cases}\displaystyle\tilde{\Phi}^{\mathrm{app}}(\bm{x},z):=\sum_{i=0}^{2N^{*}+2}(z+1-b(x))^{p_{i}}\tilde{\phi}_{i}(\bm{x}),\\\
\displaystyle\Psi(\bm{x},z):=\sum_{i=0}^{2N^{*}+2}(z+1-b(x))^{p_{i}}\psi_{i}(\bm{x}),\end{cases}$

and $\Phi^{\mathrm{res}}:=\Phi-\tilde{\Phi}^{\mathrm{app}}$. We note that
$\tilde{\Phi}^{\mathrm{app}}$ is a higher order approximation of the velocity
potential $\Phi$ and that it satisfies the boundary value problem (4.7)
approximately in the sense that

$\begin{cases}\Delta\tilde{\Phi}^{\mathrm{app}}+\delta^{-2}\partial_{z}^{2}\tilde{\Phi}^{\mathrm{app}}=R&\mbox{in}\quad-1+b(\bm{x})<z<\zeta(\bm{x}),\\\
\tilde{\Phi}^{\mathrm{app}}=\phi&\mbox{on}\quad z=\zeta(\bm{x}),\\\ \nabla
b\cdot\nabla\tilde{\Phi}^{\mathrm{app}}-\delta^{-2}\partial_{z}\tilde{\Phi}^{\mathrm{app}}=r_{B}&\mbox{on}\quad
z=-1+b(\bm{x}),\end{cases}$

where the residual $R$ can be written in the form

$R(\bm{x},z)=\sum_{i=0}^{2N^{*}+2}(z+1-b(\bm{x}))^{p_{i}}r_{i}(\bm{x}).$

Estimates for the residuals $(r_{0},r_{1},\ldots,r_{2N^{*}+2})$ and $r_{B}$
were given in [6, Lemmas 6.4 and 6.9]. In fact, we have
$\|(r_{0},r_{1},\ldots,r_{2N^{*}+2})\|_{H^{k}}+\|r_{B}\|_{H^{k}}\lesssim\delta^{2j}\|\nabla\phi\|_{H^{k+2j+1}}$
for $-m\leq k\leq m-2(j+1)$ and $0\leq j\leq 2N+1$.

Now, with a slight modification from the strategy in [6], we use the identity

$I_{1}=\int_{\Omega}I_{\delta}\nabla_{X}\Phi^{\mathrm{res}}\cdot
I_{\delta}\nabla_{X}\Psi\mathrm{d}X,$

where we denote $\Omega:=\\{X=(\bm{x},z)\,;\,-1+b(\bm{x})<z<\zeta(\bm{x})\\}$,
$I_{\delta}:=\operatorname{diag}(1,\ldots,1,\delta^{-1})$, and
$\nabla_{X}:=(\nabla,\partial_{z})=(\partial_{1},\ldots,\partial_{n},\partial_{z})$.
Indeed, we have on one hand

$(\Lambda\phi,\psi)_{L^{2}}=\int_{\Omega}I_{\delta}\nabla_{X}\Phi\cdot
I_{\delta}\nabla_{X}\Psi\mathrm{d}X$

as a consequence of (4.7), $\Psi(\bm{x},\zeta(\bm{x}))=\psi(\bm{x})$, and
Green’s identity, and on the other hand

$\displaystyle(\Lambda^{(2N+2)}\phi,\psi)_{L^{2}}$
$\displaystyle=(\mathcal{L}_{0}^{(2N+2)}\tilde{\bm{\phi}},{\bm{l}}^{(2N+2)}\cdot{\bm{\psi}})_{L^{2}}=\sum_{i=0}^{2N^{*}+2}(H^{p_{i}}\mathcal{L}_{0}^{(2N+2)}\tilde{\bm{\phi}},\psi_{i})_{L^{2}}$
$\displaystyle=\sum_{i,j=0}^{2N^{*}+2}(L_{ij}\tilde{\phi}_{j},\psi_{i})_{L^{2}}=\int_{\Omega}I_{\delta}\nabla_{X}\tilde{\Phi}^{\mathrm{app}}\cdot
I_{\delta}\nabla_{X}\Psi\mathrm{d}X,$

where the last identity follows from the expressions (4.2) and (7.1).

To evaluate $I_{1}$, it is convenient to transform the water region $\Omega$
into a simple flat domain $\Omega_{0}=\mathbf{R}^{n}\times(-1,0)$ by using a
diffeomorphism which simply stretches the vertical direction
$\Theta(\bm{x},z)=(\bm{x},\theta(\bm{x},z))\colon\Omega_{0}\to\Omega$, where
$\theta(\bm{x},z)=\zeta(\bm{x})(z+1)+(1-b(\bm{x}))z$. Put
$\tilde{\Phi}^{\mathrm{res}}=\Phi^{\mathrm{res}}\circ\Theta$ and
$\tilde{\Psi}=\Psi\circ\Theta$. Then, the above integral is transformed into

$I_{1}=\int_{\Omega_{0}}\mathcal{P}I_{\delta}\nabla_{X}\tilde{\Phi}^{\mathrm{res}}\cdot
I_{\delta}\nabla_{X}\tilde{\Psi}\mathrm{d}X,$

where

$\mathcal{P}=\det\biggl{(}\frac{\partial\Theta}{\partial
X}\biggr{)}I_{\delta}^{-1}\biggl{(}\frac{\partial\Theta}{\partial
X}\biggr{)}^{-1}I_{\delta}^{2}\biggl{(}\biggl{(}\frac{\partial\Theta}{\partial
X}\biggr{)}^{-1}\biggr{)}^{\mathrm{T}}I_{\delta}^{-1}.$

Therefore, under the restriction $|k|\leq m-1$ and using the hypothesis (4.8),
we have

$|I_{1}|\lesssim\|J^{k}I_{\delta}\nabla_{X}\tilde{\Phi}^{\mathrm{res}}\|_{L^{2}(\Omega_{0})}\|J^{-k}I_{\delta}\nabla_{X}\tilde{\Psi}\|_{L^{2}(\Omega_{0})},$

where $J=(1-\Delta)^{\frac{1}{2}}$. Moreover, $\tilde{\Phi}^{\mathrm{res}}$
satisfies the boundary value problem

$\begin{cases}\nabla_{X}\cdot
I_{\delta}\mathcal{P}I_{\delta}\nabla_{X}\tilde{\Phi}^{\mathrm{res}}=-\tilde{R}&\mbox{in}\quad\Omega_{0},\\\
\tilde{\Phi}^{\mathrm{res}}=0&\mbox{on}\quad z=0,\\\ \bm{e}_{z}\cdot
I_{\delta}\mathcal{P}I_{\delta}\nabla_{X}\tilde{\Phi}^{\mathrm{res}}=-r_{B}&\mbox{on}\quad
z=-1,\end{cases}$

where
$\tilde{R}=R\circ\Theta=\sum_{i=0}^{2N^{*}+2}(z+1)^{p_{i}}H^{p_{i}}r_{j}$ and
$\bm{e}_{z}=(0,\ldots,0,1)^{\mathrm{T}}$. By applying the standard theory of
elliptic partial differential equations to the above problem, for $0\leq k\leq
m-1$ we have

$\displaystyle\|J^{k}I_{\delta}\nabla_{X}\tilde{\Phi}^{\mathrm{res}}\|_{L^{2}(\Omega_{0})}$
$\displaystyle\lesssim\delta(\|J^{k}\tilde{R}\|_{L^{2}(\Omega_{0})}+\|r_{B}\|_{H^{k}})$
$\displaystyle\lesssim\delta(\|(r_{0},r_{1},\ldots,r_{2N^{*}+2})\|_{H^{k}}+\|r_{B}\|_{H^{k}}).$

Moreover, in view of
$\tilde{\Psi}=\sum_{i=0}^{2N^{*}+2}(z+1)^{p_{i}}H^{p_{i}}\psi_{j}$ and by
Lemma 4.1, we have

$\displaystyle\|J^{-k}I_{\delta}\nabla_{X}\tilde{\Psi}\|_{L^{2}(\Omega_{0})}$
$\displaystyle\lesssim\|\nabla\bm{\psi}\|_{H^{-k}}+\delta^{-1}\|\bm{\psi}^{\prime}\|_{H^{-k}}$
$\displaystyle\lesssim\|\nabla\psi\|_{H^{-k}}$

for $|k|\leq m-1$. Summarizing the above estimates we have
$|I_{1}|\lesssim\delta^{2j+1}\|\nabla\phi\|_{H^{k+2j+1}}\|\nabla\psi\|_{H^{-k}}$
for $0\leq k\leq m-2(j+1)$ and $0\leq j\leq 2N+1$.

As for the term $I_{2}$, the evaluation is exactly the same as in [6]. In
fact, the identities

$\displaystyle I_{2}$
$\displaystyle=\sum_{i,j=0}^{2N^{*}+2}(L_{ij}\tilde{\phi}_{j},\psi_{i})_{L^{2}}-\sum_{j=0}^{N^{*}}(L_{0j}\phi_{j},\psi)_{L^{2}}$
$\displaystyle=\sum_{j=0}^{N^{*}}\sum_{i=N^{*}+1}^{2N^{*}+2}((L_{ij}-H^{p_{i}}L_{0j})\varphi_{j},\psi_{i})_{L^{2}}-\sum_{i,j=N^{*}+1}^{2N^{*}+2}((L_{ij}-H^{p_{i}}L_{0j})\tilde{\phi}_{j},\psi_{i})_{L^{2}}$

were shown in [6, Equation (7.7)], where
$\bm{\varphi}:=(\varphi_{0},\varphi_{1},\ldots,\varphi_{N^{*}})$ was defined
by $\varphi_{i}:=\phi_{i}-\tilde{\phi}_{i}$ for $i=0,1,\ldots,N^{*}$. Now, we
decompose $j=j_{1}+j_{2}$ such that $1\leq j_{1}\leq N+1$ and $0\leq j_{2}\leq
N$. Then, by [6, Lemmas 5.2, 5.4, 6.2 and 6.7] we see that

$\displaystyle|I_{2}|$
$\displaystyle\lesssim\\{\|\bm{\varphi}\|_{H^{k+2j_{1}+1}}+\|(\tilde{\phi}_{N^{*}+1},\ldots,\tilde{\phi}_{2N^{*}+2})\|_{H^{k+2j_{1}+1}}$
$\displaystyle\quad\;+\delta^{-2}(\|\bm{\varphi}\|_{H^{k+2j_{1}-1}}+\|(\tilde{\phi}_{N^{*}+1},\ldots,\tilde{\phi}_{2N^{*}+2})\|_{H^{k+2j_{1}-1}})\\}\|(\psi_{N^{*}+1},\ldots,\psi_{2N^{*}+2})\|_{H^{-(k+2j_{1}-1)}}$
$\displaystyle\lesssim\delta^{2(j_{1}+j_{2})}\|\nabla\phi\|_{H^{k+2(j_{1}+j_{2})}}\|\nabla\psi\|_{H^{-k}}$

if $\max\\{|k|,|k+2j_{1}-2|,|k+2j_{1}+1|,|k+2(j_{1}+j_{2})|\\}\leq m-1$ and
$\max\\{|k|,|k+1|,|k+2j_{1}-1|\\}\leq m$. These conditions are satisfied under
the restriction $-m+1\leq k\leq m-2(j+1)$.

To summarize, we obtain as desired
$|((\Lambda-\Lambda^{(N)})\phi,\psi)_{L^{2}}|\lesssim\delta^{2j}\|\nabla\phi\|_{H^{k+2j+1}}\|\nabla\psi\|_{H^{-k}}$
for $0\leq k\leq m-2(j+1)$ and $1\leq j\leq 2N+1$. The proof is complete. ∎

This lemma and the scaling relations (4.15) imply immediately the following
lemma.

######  Lemma 7.2.

Let $c,M$ be positive constants and $m,j$ integers such that
$m>\frac{n}{2}+1$, $m\geq 2(j+1)$, and $1\leq j\leq 2N+1$. We assume (H1) or
(H2). There exists a positive constant $C$ such that for any positive
parameters $\underline{h}_{1},\underline{h}_{2},\delta$ satisfying
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, if $\zeta\in H^{m}$,
$b\in W^{m+1,\infty}$, $H_{1}=1-\underline{h}_{1}^{-1}\zeta$, and
$H_{2}=1+\underline{h}_{2}^{-1}\zeta-\underline{h}_{2}^{-1}b$ satisfy (4.14),
then for any $\phi_{1},\phi_{2}\in\mathring{H}^{k+2(j+1)}$ with $0\leq k\leq
m-2(j+1)$ we have

$\begin{cases}\|(-\Delta)^{-\frac{1}{2}}(\underline{h}_{1}\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})\phi_{1}-\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1})\|_{H^{k}}\leq
C\underline{h}_{1}(\underline{h}_{1}\delta)^{2j}\|\nabla\phi_{1}\|_{H^{k+2j+1}},\\\
\|(-\Delta)^{-\frac{1}{2}}(\underline{h}_{2}\Lambda_{2}^{(N)}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}-\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2})\|_{H^{k}}\leq
C\underline{h}_{2}(\underline{h}_{2}\delta)^{2j}\|\nabla\phi_{2}\|_{H^{k+2j+1}}.\end{cases}$

We remind also the estimate for the Dirichlet-to-Neumann map
$\Lambda(\zeta,b,\delta)$ itself. The following lemma is now standard. For
sharper estimates, we refer to T. Iguchi [4] and D. Lannes [14].

######  Lemma 7.3.

Let $c,M$ be positive constants $m$ an integer such that $m>\frac{n}{2}+2$.
There exists a positive constant $C$ such that if $\zeta\in H^{m}$, $b\in
W^{m,\infty}$, and $H=1+\zeta-b$ satisfy (4.4), then for any
$\phi\in\mathring{H}^{k+1}$ with $|k|\leq m-1$ and any $\delta\in(0,1]$ we
have $\|\Lambda(\zeta,b,\delta)\phi\|_{H^{k-1}}\leq C\|\nabla\phi\|_{H^{k}}$.

This lemma and the scaling relations (4.15) imply immediately the following
lemma.

######  Lemma 7.4.

Let $c,M$ be positive constants and $m$ an integer such that
$m>\frac{n}{2}+2$. There exists a positive constant $C$ such that for any
positive parameters $\underline{h}_{1},\underline{h}_{2},\delta$ satisfying
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, if $\zeta\in H^{m}$,
$b\in W^{m,\infty}$, $H_{1}=1-\underline{h}_{1}^{-1}\zeta$, and
$H_{2}=1+\underline{h}_{2}^{-1}\zeta-\underline{h}_{2}^{-1}b$ satisfy (4.11),
then for any $\phi_{1},\phi_{2}\in\mathring{H}^{k+1}$ with $|k|\leq m-1$ we
have

$\begin{cases}\|\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}\|_{H^{k-1}}\leq
C\underline{h}_{1}\|\nabla\phi_{1}\|_{H^{k}},\\\
\|\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}\|_{H^{k-1}}\leq
C\underline{h}_{2}\|\nabla\phi_{2}\|_{H^{k}}.\end{cases}$

###  7.2 Consistency of the Kakinuma model revisited

As we mentioned in Remark 3.7, the approximate solution to the Kakinuma model
made from the solution $(\zeta,\phi_{1},\phi_{2})$ to the full model can be
constructed as a solution to (3.20), that is,

(7.2)
$\begin{cases}\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})\tilde{\bm{\phi}}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})\tilde{\bm{\phi}}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}(H_{1},\delta,\underline{h}_{1})\tilde{\bm{\phi}}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}(H_{2},b,\delta,\underline{h}_{2})\tilde{\bm{\phi}}_{2}=0,\\\
\underline{\rho}_{2}\bm{l}_{2}(H_{2})\cdot\tilde{\bm{\phi}}_{2}-\underline{\rho}_{1}\bm{l}_{1}(H_{1})\cdot\tilde{\bm{\phi}}_{1}=\underline{\rho}_{2}\phi_{2}-\underline{\rho}_{2}\phi_{1},\end{cases}$

in place of (3.18), that is,

(7.3)
$\begin{cases}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}=\phi_{1},\quad\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})\bm{\phi}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}=\phi_{2},\quad\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})\bm{\phi}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*}.\end{cases}$

To show this fact, we need to guarantee that the difference between these two
solutions is of order
$O((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})$. The
following lemma gives such an estimate.

######  Lemma 7.5.

Let $c,M$ be positive constants and $m$ an integer such that $m>\frac{n}{2}+1$
and ${m\geq 4(N+1)}$. We assume (H1) or (H2). There exists a positive constant
$C$ such that for any positive parameters
$\underline{\rho}_{1},\underline{\rho}_{2},\underline{h}_{1},\underline{h}_{2},\delta$
satisfying $\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, if
$\zeta\in H^{m}$, $b\in W^{m+1,\infty}$,
$H_{1}=1-\underline{h}_{1}^{-1}\zeta$, and
$H_{2}=1+\underline{h}_{2}^{-1}\zeta-\underline{h}_{2}^{-1}b$ satisfy (4.14),
then for any $\phi_{1},\phi_{2}\in\mathring{H}^{k+4(N+1)}$ with $0\leq k\leq
m-4(N+1)$ satisfying the compatibility condition
$\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}+\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}=0$
the solution $(\bm{\phi}_{1},\bm{\phi}_{2})$ to (7.3) and the solution
$(\tilde{\bm{\phi}}_{1},\tilde{\bm{\phi}}_{2})$ to (7.2) satisfy

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla(\tilde{\bm{\phi}}_{\ell}-\bm{\phi}_{\ell})\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-4}\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k-1}}^{2})$
$\displaystyle\leq
C\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2(4N+2)}\|\nabla\phi_{\ell}\|_{H^{k+4N+3}}^{2}.$

###### Proof.

For simplicity, we write
$\mathcal{L}_{1,i}=\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})$,
$\bm{l}_{1}=\bm{l}_{1}(H_{1})$, and so on. We recall that
$\Lambda_{1}^{(N)}\colon\phi_{1}\mapsto\mathcal{L}_{1,0}\bm{\phi}_{1}$ and
$\Lambda_{2}^{(N)}\colon\phi_{2}\mapsto\mathcal{L}_{2,0}\bm{\phi}_{2}$. Notice
that $\tilde{\bm{\phi}}_{\ell}-\bm{\phi}_{\ell}$ for $\ell=1,2$ satisfy

$\begin{cases}\mathcal{L}_{1,i}(\tilde{\bm{\phi}}_{1}-\bm{\phi}_{1})=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(\tilde{\bm{\phi}}_{2}-\bm{\phi}_{2})=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}(\tilde{\bm{\phi}}_{1}-\bm{\phi}_{1})+\underline{h}_{2}\mathcal{L}_{2,0}(\tilde{\bm{\phi}}_{2}-\bm{\phi}_{2})=(\Lambda_{1}-\underline{h}_{1}\Lambda_{1}^{(N)})\phi_{1}+(\Lambda_{2}-\underline{h}_{2}\Lambda_{2}^{(N)})\phi_{2},\\\
\underline{\rho}_{2}\bm{l}_{2}\cdot(\tilde{\bm{\phi}}_{2}-\bm{\phi}_{2})-\underline{\rho}_{1}\bm{l}_{1}\cdot(\tilde{\bm{\phi}}_{1}-\bm{\phi}_{1})=0.\end{cases}$

Since the right-hand side of the third equation can be written as
$\nabla\cdot\bm{f}_{3}$ with

$\bm{f}_{3}=-\nabla(-\Delta)^{-1}\bigl{(}(\Lambda_{1}-\underline{h}_{1}\Lambda_{1}^{(N)})\phi_{1}-(\Lambda_{2}-\underline{h}_{2}\Lambda_{2}^{(N)})\phi_{2}\bigr{)},$

by Lemmas 5.1 and 7.2 we obtain

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla(\tilde{\bm{\phi}}_{\ell}-\bm{\phi}_{\ell})\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k}}^{2})$
$\displaystyle\makebox[100.00015pt]{}\lesssim\min\biggl{\\{}\frac{\underline{\rho}_{1}}{\underline{h}_{1}},\frac{\underline{\rho}_{2}}{\underline{h}_{2}}\biggr{\\}}\|\bm{f}_{3}\|_{H^{k}}^{2}$
$\displaystyle\makebox[100.00015pt]{}\lesssim\sum_{l=1,2}\frac{\underline{\rho}_{\ell}}{\underline{h}_{\ell}}\|(-\Delta)^{-\frac{1}{2}}(\Lambda_{\ell}-\underline{h}_{\ell}\Lambda_{\ell}^{(N)})\phi_{\ell}\|_{H^{k}}^{2}$
$\displaystyle\makebox[100.00015pt]{}\lesssim\sum_{l=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2(4N+2)}\|\nabla\phi_{\ell}\|_{H^{k+4N+3}}^{2}.$

Moreover, it follows from Lemma 6.3 that

$(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k-1}}\lesssim\|\nabla(\tilde{\bm{\phi}}_{\ell}-\bm{\phi}_{\ell})\|_{H^{k}}+(\underline{h}_{\ell}\delta)^{-1}\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k}}$

for $\ell=1,2$. This completes the proof. ∎

The following proposition gives another version of Theorem 3.5 for the
consistency of the Kakinuma model.

######  Proposition 7.6.

Let $c,M$ be positive constants and $m$ an integer such that $m\geq 4N+4$ and
$m>\frac{n}{2}+2$. We assume (H1) or (H2). There exists a positive constant
$C$ such that for any positive parameters
$\underline{\rho}_{1},\underline{\rho}_{2},\underline{h}_{1},\underline{h}_{2},\delta$
satisfying $\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, and for
any solution $(\zeta,\phi_{1},\phi_{2})$ to the full model for interfacial
gravity waves (2.17) on a time interval $[0,T]$ satisfying (3.17), if we
define $H_{1}$ and $H_{2}$ as in (2.19) and
$(\tilde{\bm{\phi}}_{1},\tilde{\bm{\phi}}_{2})$ as a solution to (7.2), then
$(\zeta,\tilde{\bm{\phi}}_{1},\tilde{\bm{\phi}}_{2})$ satisfy approximately
the Kakinuma model as

(7.4)
$\begin{cases}\displaystyle{\bm{l}}_{1}(H_{1})\underline{h}_{1}^{-1}\partial_{t}\zeta+L_{1}(H_{1},\delta,\underline{h}_{1})\tilde{\bm{\phi}}_{1}=\bm{\mathfrak{r}}_{1},\\\
\displaystyle{\bm{l}}_{2}(H_{2})\underline{h}_{2}^{-1}\partial_{t}\zeta-
L_{2}(H_{2},b,\delta,\underline{h}_{2})\tilde{\bm{\phi}}_{2}=\bm{\mathfrak{r}}_{2},\\\
\underline{\rho}_{1}\bigl{\\{}{\bm{l}}_{1}(H_{1})\cdot\partial_{t}\tilde{\bm{\phi}}_{1}+\frac{1}{2}\bigl{(}|\tilde{\bm{u}}_{1}|^{2}+(\underline{h}_{1}\delta)^{-2}\tilde{w}_{1}^{2}\bigr{)}\bigr{\\}}\\\
\quad-\underline{\rho}_{2}\bigl{\\{}{\bm{l}}_{2}(H_{2})\cdot\partial_{t}\tilde{\bm{\phi}}_{2}+\frac{1}{2}\bigl{(}|\tilde{\bm{u}}_{2}|^{2}+(\underline{h}_{2}\delta)^{-2}\tilde{w}_{2}^{2}\bigr{)}\bigr{\\}}-\zeta=\mathfrak{r}_{0},\end{cases}$

where $\tilde{\bm{u}}_{1},\tilde{\bm{u}}_{2},\tilde{w}_{1},\tilde{w}_{2}$ are
defined by (3.4) with $(\bm{\phi}_{1},\bm{\phi}_{2})$ replaced by
$(\tilde{\bm{\phi}}_{1},\tilde{\bm{\phi}}_{2})$, and the errors
$(\bm{\mathfrak{r}}_{1},\bm{\mathfrak{r}}_{2},\mathfrak{r}_{0})$ satisfy

(7.5)
$\begin{cases}\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\bm{\mathfrak{r}}_{\ell}(t)\|_{H^{m-(4N+5)}}^{2}\leq
C\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2(4N+2)}\|\nabla\phi_{\ell}(t)\|_{H^{m-1}}^{2},\\\
\displaystyle\|\mathfrak{r}_{0}(t)\|_{H^{m-4(N+1)}}\leq
C\bigl{(}(\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2}\bigr{)}(\underline{h}_{1}^{-1}+\underline{h}_{2}^{-1})\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}(t)\|_{H^{m-1}}^{2},\end{cases}$

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

###### Proof.

Let $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ be the unique solutions to (7.3), and
$(\tilde{\bm{\mathfrak{r}}}_{1},\tilde{\bm{\mathfrak{r}}}_{2},\tilde{\mathfrak{r}}_{0})$
the errors in Theorem 3.5. Then, the errors
$(\bm{\mathfrak{r}}_{1},\bm{\mathfrak{r}}_{2},\mathfrak{r}_{0})$ in the
proposition can be written as

$\begin{cases}\bm{\mathfrak{r}}_{1}=\tilde{\bm{\mathfrak{r}}}_{1}-L_{1}(H_{1},\delta,\underline{h}_{1})(\tilde{\bm{\phi}}_{1}-\bm{\phi}_{1}),\\\
\bm{\mathfrak{r}}_{2}=\tilde{\bm{\mathfrak{r}}}_{2}+L_{2}(H_{2},b,\delta,\underline{h}_{2})(\tilde{\bm{\phi}}_{2}-\bm{\phi}_{2}),\\\
\mathfrak{r}_{0}=\tilde{\mathfrak{r}}_{0}+\underline{\rho}_{1}\\{\underline{h}_{1}^{-1}(\partial_{t}\zeta)(\tilde{w}_{1}-w_{1})-\frac{1}{2}\bigl{(}(\tilde{\bm{u}}_{1}+\bm{u}_{1})\cdot(\tilde{\bm{u}}_{1}-\bm{u}_{1})+(\underline{h}_{1}\delta)^{-2}(\tilde{w}_{1}+w_{1})(\tilde{w}_{1}-w_{1})\bigr{)}\\}\\\
\qquad-\underline{\rho}_{2}\\{\underline{h}_{2}^{-1}(\partial_{t}\zeta)(\tilde{w}_{2}-w_{2})-\frac{1}{2}\bigl{(}(\tilde{\bm{u}}_{2}+\bm{u}_{2})\cdot(\tilde{\bm{u}}_{2}-\bm{u}_{2})+(\underline{h}_{2}\delta)^{-2}(\tilde{w}_{2}+w_{2})(\tilde{w}_{2}-w_{2})\bigr{)}\\}.\end{cases}$

Therefore, we have

$\|\bm{\mathfrak{r}}_{\ell}-\tilde{\bm{\mathfrak{r}}}_{\ell}\|_{H^{k}}\lesssim\|\nabla(\tilde{\bm{\phi}}_{\ell}-\bm{\phi}_{\ell})\|_{H^{k+1}}+\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k+1}}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k}}$

for $-m\leq k\leq m-1$ and $\ell=1,2$. Applying this estimate with
$k=m-(4N+5)$ and the estimate in Lemma 7.5 with $k=m-4(N+1)$ and using the
result in Theorem 3.5, we obtain the first estimate in (7.5). Since
$m-2>\frac{n}{2}$, we have

$\displaystyle\|\mathfrak{r}_{0}-\tilde{\mathfrak{r}}_{0}\|_{H^{k}}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\\{(\|\tilde{\bm{u}}_{\ell}\|_{H^{m-2}}+\|\bm{u}_{\ell}\|_{H^{m-2}})\|\tilde{\bm{u}}_{\ell}-\bm{u}_{\ell}\|_{H^{k}}$
$\displaystyle\quad\;+\bigl{(}\underline{h}_{\ell}^{-1}\|\partial_{t}\zeta\|_{H^{m-2}}+(\underline{h}_{\ell}\delta)^{-2}(\|\tilde{w}_{\ell}\|_{H^{m-2}}+\|w_{\ell}\|_{H^{m-2}})\bigr{)}\|\tilde{w}_{\ell}-w_{\ell}\|_{H^{k}}\\}$

for $|k|\leq m-2$. Here, it follows from Lemmas 4.4, 5.1, and 7.5 that

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\bm{u}_{\ell}\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|w_{\ell}\|_{H^{m-1}}^{2})$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\bm{\phi}_{\ell}\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\prime}\|_{H^{m-1}}^{2})$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}\|_{H^{m-1}}^{2},$
$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\tilde{\bm{u}}_{\ell}\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{w}_{\ell}\|_{H^{m-1}}^{2})$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\tilde{\bm{\phi}}_{\ell}\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\prime}\|_{H^{m-1}}^{2})$
$\displaystyle\lesssim\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}\|\nabla(\underline{\rho}_{2}\phi_{2}-\underline{\rho}_{1}\phi_{1})\|_{H^{m-1}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}\|_{H^{m-1}}^{2},$

and

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\tilde{\bm{u}}_{\ell}-\bm{u}_{\ell}\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{w}_{\ell}-w_{\ell}\|_{H^{k}}^{2})$
$\displaystyle\makebox[50.00008pt]{}\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla(\tilde{\bm{\phi}}_{\ell}-\bm{\phi}_{\ell})\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\prime}-\bm{\phi}_{\ell}^{\prime}\|_{H^{k}}^{2})$
$\displaystyle\makebox[50.00008pt]{}\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2(4N+2)}\|\nabla\phi_{\ell}\|_{H^{k+4N+3}}^{2}$

for $0\leq k\leq m-4(N+1)$. Moreover, it follows from Lemma 7.4 that
$\|\partial_{t}\zeta\|_{H^{m-2}}=\|\Lambda_{\ell}\phi_{\ell}\|_{H^{m-2}}\lesssim\underline{h}_{\ell}\|\nabla\phi_{\ell}\|_{H^{m-1}}$
for $\ell=1,2$. Summarizing the above estimates and using the result in
Theorem 3.5, we easily obtain the second estimate in (7.5). The proof is
complete. ∎

###  7.3 Completion of the proof of Theorem 3.8

Now we are ready to prove Theorem 3.8. Let $(\zeta^{\mbox{\rm\tiny
IW}},\phi_{1}^{\mbox{\rm\tiny IW}},\phi_{2}^{\mbox{\rm\tiny IW}})$ be the
solution to the full model for interfacial gravity waves (2.17) with uniform
bound stated in the theorem, and define ${\phi^{\mbox{\rm\tiny
IW}}:=\underline{\rho}_{2}\phi_{2}^{\mbox{\rm\tiny
IW}}-\underline{\rho}_{1}\phi_{1}^{\mbox{\rm\tiny IW}}}$, which is a canonical
variable of the full model. We first ensure a uniform bound on the time
derivative of the canonical variables $(\zeta^{\mbox{\rm\tiny
IW}},\phi^{\mbox{\rm\tiny IW}})$. It follows from the first and the second
equations in (2.17) that $\partial_{t}\zeta^{\mbox{\rm\tiny
IW}}=-\Lambda_{1}^{\mbox{\rm\tiny IW}}\phi_{1}^{\mbox{\rm\tiny
IW}}=\Lambda_{2}^{\mbox{\rm\tiny IW}}\phi_{2}^{\mbox{\rm\tiny IW}}$, where
$\Lambda_{1}^{\mbox{\rm\tiny IW}}=\Lambda_{1}(\zeta^{\mbox{\rm\tiny
IW}},\delta,\underline{h}_{1})$ and $\Lambda_{2}^{\mbox{\rm\tiny
IW}}=\Lambda_{2}(\zeta^{\mbox{\rm\tiny IW}},b,\delta,\underline{h}_{2})$.
Similar notations will be used in the following without any comment.
Therefore, by Lemma 7.4 we have

$\displaystyle\|\partial_{t}\zeta^{\mbox{\rm\tiny IW}}\|_{H^{m-1}}^{2}$
$\displaystyle=\min\\{\|\Lambda_{1}^{\mbox{\rm\tiny
IW}}\phi_{1}^{\mbox{\rm\tiny
IW}}\|_{H^{m-1}}^{2},\|\Lambda_{2}^{\mbox{\rm\tiny
IW}}\phi_{2}^{\mbox{\rm\tiny IW}}\|_{H^{m-1}}^{2}\\}$
$\displaystyle\lesssim\min\\{\underline{h}_{1}^{2}\|\nabla\phi_{1}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2},\underline{h}_{2}^{2}\|\nabla\phi_{2}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}\\}$
$\displaystyle\lesssim\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}$ $\displaystyle\leq
2\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2},$

where we used (2.15). It follows from the third equation in (2.17) that

$\displaystyle\partial_{t}\phi^{\mbox{\rm\tiny IW}}$
$\displaystyle=\underline{\rho}_{2}\partial_{t}\phi_{2}^{\mbox{\rm\tiny
IW}}-\underline{\rho}_{1}\partial_{t}\phi_{1}^{\mbox{\rm\tiny IW}}$
$\displaystyle=\frac{1}{2}\underline{\rho}_{1}\biggl{(}|\nabla\phi_{1}^{\mbox{\rm\tiny
IW}}|^{2}-\delta^{2}\frac{(\Lambda_{1}^{\mbox{\rm\tiny
IW}}\phi_{1}^{\mbox{\rm\tiny IW}}-\nabla\zeta^{\mbox{\rm\tiny
IW}}\cdot\nabla\phi_{1}^{\mbox{\rm\tiny
IW}})^{2}}{1+\delta^{2}|\nabla\zeta^{\mbox{\rm\tiny IW}}|^{2}}\biggr{)}$
$\displaystyle\quad\;-\frac{1}{2}\underline{\rho}_{2}\biggl{(}|\nabla\phi_{2}^{\mbox{\rm\tiny
IW}}|^{2}-\delta^{2}\frac{(\Lambda_{2}^{\mbox{\rm\tiny
IW}}\phi_{2}^{\mbox{\rm\tiny IW}}+\nabla\zeta^{\mbox{\rm\tiny
IW}}\cdot\nabla\phi_{2}^{\mbox{\rm\tiny
IW}})^{2}}{1+\delta^{2}|\nabla\zeta^{\mbox{\rm\tiny
IW}}|^{2}}\biggr{)}-\zeta^{\mbox{\rm\tiny IW}}.$

Here, we note that in view of the conditions
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$ and
$\underline{h}_{1}^{-1},\underline{h}_{2}^{-1}\lesssim 1$ we have
$\delta\lesssim 1$. Therefore, by Lemma 7.4 we have

$\displaystyle\|\partial_{t}\phi^{\mbox{\rm\tiny IW}}\|_{H^{m-1}}$
$\displaystyle\lesssim\|\zeta^{\mbox{\rm\tiny
IW}}\|_{H^{m-1}}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\\{\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m-1}}^{2}+\delta^{2}(\underline{h}_{\ell}^{2}\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}+\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m-1}}^{2})\\}$ $\displaystyle\lesssim\|\zeta^{\mbox{\rm\tiny
IW}}\|_{H^{m-1}}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}.$

Hence, we obtain $\|\partial_{t}\zeta^{\mbox{\rm\tiny
IW}}\|_{H^{m-1}}+\|\partial_{t}\phi^{\mbox{\rm\tiny IW}}\|_{H^{m-1}}\lesssim
1$.

Let $(\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$ be the solution to (7.2)
with $(\zeta,\phi)=(\zeta^{\mbox{\rm\tiny IW}},\phi^{\mbox{\rm\tiny IW}})$.
Then, Proposition 7.6 states that $(\zeta^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$ satisfy approximately the
Kakinuma model as (7.4) and the errors
$(\bm{\mathfrak{r}}_{1},\bm{\mathfrak{r}}_{2},\mathfrak{r}_{0})$ satisfy
(7.5). Moreover, it follows from Lemma 5.1 that

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}}^{2})$
$\displaystyle\lesssim\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}\|\nabla\phi^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}\lesssim 1,$

which yields

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{w}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-4}\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m-1}}^{2})\lesssim 1,$

where $\tilde{\bm{u}}_{1}^{\mbox{\rm\tiny
IW}},\tilde{\bm{u}}_{2}^{\mbox{\rm\tiny IW}},\tilde{w}_{1}^{\mbox{\rm\tiny
IW}},\tilde{w}_{2}^{\mbox{\rm\tiny IW}}$ are defined by (3.4) with
$(\bm{\phi}_{1},\bm{\phi}_{2})$ replaced by
$(\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$, and we used Lemma 6.3. We
proceed to evaluate $(\partial_{t}\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\partial_{t}\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$. To this end, we
derive equations for these time derivatives by differentiating (7.2) with
respect to $t$. The procedure is almost the same as in the proof of Lemma 5.2.
The only difference is the last equation in (5.5), especially, the expression
of $f_{4}$. In this case, $f_{4}$ has the form

$f_{4}=\partial_{t}\phi^{\mbox{\rm\tiny
IW}}+\underline{\rho}_{1}\tilde{w}_{1}^{\mbox{\rm\tiny
IW}}\underline{h}_{1}^{-1}\partial_{t}\zeta^{\mbox{\rm\tiny
IW}}-\underline{\rho}_{2}\tilde{w}_{2}^{\mbox{\rm\tiny
IW}}\underline{h}_{2}^{-1}\partial_{t}\zeta^{\mbox{\rm\tiny IW}},$

so that $\|f_{4}\|_{H^{m-1}}\lesssim 1$. Therefore, we obtain

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\partial_{t}\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m-2}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m-2}}^{2})\lesssim 1.$

Let $(\zeta^{\mbox{\rm\tiny K}},\bm{\phi}_{1}^{\mbox{\rm\tiny
K}},\bm{\phi}_{2}^{\mbox{\rm\tiny K}})$ be the solution to the initial value
problem for the Kakinuma model stated in the theorem, whose unique existence
is guaranteed by Theorem 3.1 and Proposition 3.3. Note also that the solution
satisfies the uniform bound (3.14) together with the stability and non-
cavitation conditions (3.15). It follows from Lemma 6.3 that
$\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{-4}\|\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}\,\prime}\|_{H^{m-1}}^{2}\lesssim 1$ for $\ell=1,2$. Moreover, the time
derivatives $(\partial_{t}\zeta^{\mbox{\rm\tiny
K}},\partial_{t}\bm{\phi}_{1}^{\mbox{\rm\tiny
K}},\partial_{t}\bm{\phi}_{2}^{\mbox{\rm\tiny K}})$ satisfy (5.4) and
$(\bm{u}_{\ell}^{\mbox{\rm\tiny K}},w_{\ell}^{\mbox{\rm\tiny K}})$
$(\ell=1,2)$, which are defined by (3.4) with $(\bm{\phi}_{1},\bm{\phi}_{2})$
replaced by $({\bm{\phi}}_{1}^{\mbox{\rm\tiny
K}},{\bm{\phi}}_{2}^{\mbox{\rm\tiny K}})$, satisfy (5.8). Putting

$\zeta^{\mathrm{res}}:=\zeta^{\mbox{\rm\tiny K}}-\zeta^{\mbox{\rm\tiny
IW}},\qquad\bm{\phi}_{\ell}^{\mathrm{res}}:=\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}}-\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny IW}}\quad(\ell=1,2),$

we will show that
$(\zeta^{\mathrm{res}},\bm{\phi}_{1}^{\mathrm{res}},\bm{\phi}_{2}^{\mathrm{res}})$
can be estimated by the errors
$(\bm{\mathfrak{r}}_{1},\bm{\mathfrak{r}}_{2},\mathfrak{r}_{0})$. To this end,
we are going to evaluate

$E_{k}^{\mathrm{res}}(t):=\|\zeta^{\mathrm{res}}(t)\|_{H^{k}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\bm{\phi}_{\ell}^{\mathrm{res}}(t)\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}(t)\|_{H^{k}}^{2})$

for an appropriate integer $k$ by making use of energy estimates similar to
the ones obtained in Sections 5 and 6 for the proof of the well-posedness of
the initial value problem for the Kakinuma model. Here, we note that
$E_{k}^{\mathrm{res}}(0)=0$.

As in the case of the energy estimate for the Kakinuma model, we first need to
evaluate times derivatives
$(\partial_{t}\zeta^{\mathrm{res}},\partial_{t}\bm{\phi}_{1}^{\mathrm{res}},\partial_{t}\bm{\phi}_{2}^{\mathrm{res}})$
in terms of $E_{k}^{\mathrm{res}}$. By taking difference between the first
components of the first two equations in (3.5) and (7.4),
$\partial_{t}\zeta^{\mathrm{res}}$ can be written in two way as

$\displaystyle\partial_{t}\zeta^{\mathrm{res}}$
$\displaystyle=-\underline{h}_{1}\\{\mathcal{L}_{1,0}^{\mbox{\rm\tiny
K}}\bm{\phi}_{1}^{\mathrm{res}}+(\mathcal{L}_{1,0}^{\mbox{\rm\tiny
K}}-\mathcal{L}_{1,0}^{\mbox{\rm\tiny
IW}})\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny IW}}+\mathfrak{r}_{1,0}\\}$
$\displaystyle=\underline{h}_{2}\\{\mathcal{L}_{2,0}^{\mbox{\rm\tiny
K}}\bm{\phi}_{2}^{\mathrm{res}}+(\mathcal{L}_{2,0}^{\mbox{\rm\tiny
K}}-\mathcal{L}_{2,0}^{\mbox{\rm\tiny
IW}})\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}}+\mathfrak{r}_{2,0}\\},$

where $\mathcal{L}_{1,0}^{\mbox{\rm\tiny
K}}=\mathcal{L}_{1,0}(H_{1}^{\mbox{\rm\tiny K}},\delta,\underline{h}_{1})$,
$H_{1}^{\mbox{\rm\tiny K}}=1-\underline{h}_{1}^{-1}\zeta^{\mbox{\rm\tiny K}}$,
and similar simplifications are used, and $\mathfrak{r}_{\ell,0}$ is the $0$th
component of the error $\bm{\mathfrak{r}}_{\ell}$ for $\ell=1,2$. Therefore,
we have

$\displaystyle\|\partial_{t}\zeta^{\mathrm{res}}\|_{H^{k-1}}$
$\displaystyle\lesssim\underline{h}_{\ell}\\{\|\nabla\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k}}+\|\bm{\phi}_{\ell}^{\mathrm{res\,\prime}}\|_{H^{k}}$
$\displaystyle\quad+\|\zeta^{\mathrm{res}}\|_{H^{k}}(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+\|\tilde{\bm{\phi}}_{\ell}^{\mathrm{\mbox{\rm\tiny
IW}\,\prime}}\|_{H^{m}})+\|\mathfrak{r}_{\ell,0}\|_{H^{k-1}}\\}$

for $\ell=1,2$ and $|k|\leq m$. Hence, by the technique used in the proof of
Lemma 5.2 we obtain

$\displaystyle\|\partial_{t}\zeta^{\mathrm{res}}\|_{H^{k-1}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\\{\|\nabla\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k}}^{2}+\|\bm{\phi}_{\ell}^{\mathrm{res\,\prime}}\|_{H^{k}}^{2}$
$\displaystyle\qquad+\|\zeta^{\mathrm{res}}\|_{H^{k}}^{2}(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}^{2}+\|\tilde{\bm{\phi}}_{\ell}^{\mathrm{\mbox{\rm\tiny
IW}\,\prime}}\|_{H^{m}}^{2})+\|\mathfrak{r}_{\ell,0}\|_{H^{k-1}}^{2}\\}$
$\displaystyle\lesssim
E_{k}^{\mathrm{res}}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\bm{\mathfrak{r}}_{\ell}\|_{H^{k-1}}^{2}$

for $|k|\leq m$. We proceed to evaluate
$(\partial_{t}\bm{\phi}_{1}^{\mathrm{res}},\partial_{t}\bm{\phi}_{2}^{\mathrm{res}})$.
We recall that $(\partial_{t}\bm{\phi}_{1}^{\mbox{\rm\tiny
K}},\partial_{t}\bm{\phi}_{2}^{\mbox{\rm\tiny K}})$ satisfy (5.5) with
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=(\zeta^{\mbox{\rm\tiny
K}},\bm{\phi}_{1}^{\mbox{\rm\tiny K}},\bm{\phi}_{2}^{\mbox{\rm\tiny K}})$ and
note that, differentiating the first three equations of (7.2) with respect to
$t$ and using the last equation in (7.4),
$(\partial_{t}\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\partial_{t}\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$ also satisfy
(5.5) with $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=(\zeta^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$ and $f_{4}$ added with the
error term $-\mathfrak{r}_{0}$. By taking the difference between these
equations, we have therefore

$\begin{cases}\mathcal{L}_{1,i}^{\mbox{\rm\tiny
IW}}\partial_{t}\bm{\phi}_{1}^{\mathrm{res}}=f_{1,i}^{\mathrm{res}}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\ \mathcal{L}_{2,i}^{\mbox{\rm\tiny
IW}}\partial_{t}\bm{\phi}_{2}^{\mathrm{res}}=f_{2,i}^{\mathrm{res}}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\ \underline{h}_{1}\mathcal{L}_{1,0}^{\mbox{\rm\tiny
IW}}\partial_{t}\bm{\phi}_{1}^{\mathrm{res}}+\underline{h}_{2}\mathcal{L}_{2,0}^{\mbox{\rm\tiny
IW}}\partial_{t}\bm{\phi}_{2}^{\mathrm{res}}=\nabla\cdot\bm{f}_{3}^{\mathrm{res}},\\\
-\underline{\rho}_{1}\bm{l}_{1}^{\mbox{\rm\tiny
IW}}\cdot\partial_{t}\bm{\phi}_{1}^{\mathrm{res}}+\underline{\rho}_{2}\bm{l}_{2}^{\mbox{\rm\tiny
IW}}\cdot\partial_{t}\bm{\phi}_{2}^{\mathrm{res}}=f_{4}^{\mathrm{res}},\end{cases}$

where

$\begin{cases}f_{1,i}^{\mathrm{res}}=f_{1,i}^{\mbox{\rm\tiny
K}}-\tilde{f}_{1,i}^{\mbox{\rm\tiny IW}}+(\mathcal{L}_{1,i}^{\mbox{\rm\tiny
IW}}-\mathcal{L}_{1,i}^{\mbox{\rm\tiny
K}})\partial_{t}\bm{\phi}_{1}^{\mbox{\rm\tiny K}}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\ f_{2,i}^{\mathrm{res}}=f_{2,i}^{\mbox{\rm\tiny
K}}-\tilde{f}_{2,i}^{\mbox{\rm\tiny IW}}+(\mathcal{L}_{2,i}^{\mbox{\rm\tiny
IW}}-\mathcal{L}_{2,i}^{\mbox{\rm\tiny
K}})\partial_{t}\bm{\phi}_{2}^{\mbox{\rm\tiny K}}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\ \bm{f}_{3}^{\mathrm{res}}=\bm{f}_{3}^{\mbox{\rm\tiny
K}}-\tilde{\bm{f}}_{3}^{\mbox{\rm\tiny
IW}}+\underline{h}_{1}((\bm{a}_{1,0}^{\mbox{\rm\tiny
K}}-\bm{a}_{1,0}^{\mbox{\rm\tiny
IW}})\otimes\nabla)^{\mathrm{T}}\partial_{t}\bm{\phi}_{1}^{\mbox{\rm\tiny
K}}\\\ \qquad\quad{}+\underline{h}_{2}\\{((\bm{a}_{2,0}^{\mbox{\rm\tiny
K}}-\bm{a}_{2,0}^{\mbox{\rm\tiny
IW}})\otimes\nabla)^{\mathrm{T}}\partial_{t}\bm{\phi}_{2}^{\mbox{\rm\tiny
K}}-((\bm{b}_{2,0}^{\mbox{\rm\tiny K}}-\bm{b}_{2,0}^{\mbox{\rm\tiny
IW}})\cdot\partial_{t}\bm{\phi}_{2}^{\mbox{\rm\tiny
K}})\underline{h}_{2}^{-1}\nabla b\\},\\\
f_{4}^{\mathrm{res}}=f_{4}^{\mbox{\rm\tiny K}}-\tilde{f}_{4}^{\mbox{\rm\tiny
IW}}+\mathfrak{r}_{0}-\underline{\rho}_{1}(\bm{l}_{1}^{\mbox{\rm\tiny
IW}}-\bm{l}_{1}^{\mbox{\rm\tiny
K}})\cdot\partial_{t}\bm{\phi}_{1}^{\mbox{\rm\tiny
K}}+\underline{\rho}_{2}(\bm{l}_{2}^{\mbox{\rm\tiny
IW}}-\bm{l}_{2}^{\mbox{\rm\tiny
K}})\cdot\partial_{t}\bm{\phi}_{2}^{\mbox{\rm\tiny K}}.\end{cases}$

Here, $f_{1,i}^{\mbox{\rm\tiny K}}$, $f_{2,i}^{\mbox{\rm\tiny K}}$,
$\bm{f}_{3}^{\mbox{\rm\tiny K}}$, $f_{4}^{\mbox{\rm\tiny K}}$ (respectively
$\tilde{f}_{1,i}^{\mbox{\rm\tiny IW}}$, $\tilde{f}_{2,i}^{\mbox{\rm\tiny
IW}}$, $\tilde{\bm{f}}_{3}^{\mbox{\rm\tiny IW}}$,
$\tilde{f}_{4}^{\mbox{\rm\tiny IW}}$) are those in (5.6) with
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=(\zeta^{\mbox{\rm\tiny
K}},\bm{\phi}_{1}^{\mbox{\rm\tiny K}},\bm{\phi}_{2}^{\mbox{\rm\tiny K}})$
(respectively $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=(\zeta^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$),
$\bm{a}_{\ell,0}^{\mbox{\rm\tiny K}}=\bm{a}_{\ell,0}(H_{\ell}^{\mbox{\rm\tiny
K}})$ and $\bm{b}_{2,0}^{\mbox{\rm\tiny K}}=\bm{b}_{2,0}(H_{2}^{\mbox{\rm\tiny
K}})$, where $\bm{a}_{\ell,0}(H_{\ell})$ and $\bm{b}_{2,0}(H_{2})$ are the
$0$th columns of the matrixes $A_{\ell}(H_{\ell})$ and $B_{2}(H_{2})$ defined
by (6.2) and (6.4), respectively, and so on. Note the relations
$\mathcal{L}_{1,0}\bm{\phi}_{1}=-\nabla\cdot((\bm{a}_{1,0}\otimes\nabla)^{\mathrm{T}}\bm{\phi}_{1})$
and
$\mathcal{L}_{2,0}\bm{\phi}_{2}=-\nabla\cdot((\bm{a}_{2,0}\otimes\nabla)^{\mathrm{T}}\bm{\phi}_{2}-(\bm{b}_{2,0}\cdot\bm{\phi}_{2})\underline{h}_{2}^{-1}\nabla
b)$. Therefore, by Lemma 5.1 we have, for $1\leq k\leq m+1$,

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\nabla\partial_{t}\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}^{2}\bigr{)}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2}\|\bm{f}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}^{2}+\min\biggl{\\{}\frac{\underline{\rho}_{1}}{\underline{h}_{1}},\frac{\underline{\rho}_{2}}{\underline{h}_{2}}\biggr{\\}}\|\bm{f}_{3}^{\mathrm{res}}\|_{H^{k-1}}^{2}+\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}\|f_{4}^{\mathrm{res}}\|_{H^{k}}^{2}.$

We will evaluate each term in the right-hand side. For $1\leq k\leq m-1$, we
see that

$\displaystyle\|\bm{f}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}$
$\displaystyle\lesssim\underline{h}_{\ell}^{-1}\|\zeta^{\mathrm{res}}\|_{H^{k}}(\|\nabla\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}\,\prime}\|_{H^{m}})\underline{h}_{\ell}^{-1}\|\partial_{t}\zeta^{\mbox{\rm\tiny
K}}\|_{H^{m-1}}$
$\displaystyle\quad\;+(\|\nabla\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k}}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k}})\underline{h}_{\ell}^{-1}\|\partial_{t}\zeta^{\mbox{\rm\tiny
K}}\|_{H^{m-1}}$
$\displaystyle\quad\;+(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}})\underline{h}_{\ell}^{-1}\|\partial_{t}\zeta^{\mathrm{res}}\|_{H^{k-1}}$
$\displaystyle\quad\;+\underline{h}_{\ell}^{-1}\|\zeta^{\mathrm{res}}\|_{H^{k}}(\|\nabla\partial_{t}{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m-1}}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
K}\,\prime}\|_{H^{m-1}})$

for $\ell=1,2$,

$\displaystyle\|\bm{f}_{3}^{\mathrm{res}}\|_{H^{k-1}}$
$\displaystyle\lesssim\sum_{\ell=1,2}\\{\|\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}-\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{k}}\|\partial_{t}\zeta^{\mbox{\rm\tiny
K}}\|_{H^{m-1}}+\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}\|\partial_{t}\zeta^{\mathrm{res}}\|_{H^{k-1}}$
$\displaystyle\qquad+\|\zeta^{\mathrm{res}}\|_{H^{k}}(\|\nabla\partial_{t}\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m-1}}+\|\partial_{t}\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}\,\prime}\|_{H^{m-1}})\\},$

and

$\displaystyle\|f_{4}^{\mathrm{res}}\|_{H^{k}}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\big{\\{}(\|\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}+\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}})\|\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}-\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny IW}}\|_{H^{k}}$
$\displaystyle\qquad+(\underline{h}_{\ell}\delta)^{-2}(\|w_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}+\|\tilde{w}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}})\|w_{\ell}^{\mbox{\rm\tiny K}}-\tilde{w}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{k}}$
$\displaystyle\qquad+\underline{h}_{\ell}^{-1}\|\zeta^{\mathrm{res}}\|_{H^{k}}\|\partial_{t}\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}\,\prime}\|_{H^{m-1}}\big{\\}}+\|\zeta^{\mathrm{res}}\|_{H^{k}}+\|\mathfrak{r}_{0}\|_{H^{k}}.$

Moreover, for any $0\leq k\leq m$ we have also

(7.6)
$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}-\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|w_{\ell}^{\mbox{\rm\tiny
K}}-\tilde{w}_{\ell}^{\mbox{\rm\tiny IW}}\|_{H^{k}}^{2})\lesssim
E_{k}^{\mathrm{res}}.$

Summarizing the above estimates and using
$\underline{h}_{1}^{-1},\underline{h}_{2}^{-1}\lesssim 1$ we obtain, for
$1\leq k\leq m-1$,

(7.7)
$\displaystyle\|\partial_{t}\zeta^{\mathrm{res}}\|_{H^{k-1}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\nabla\partial_{t}\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}^{2}\bigr{)}$
$\displaystyle\lesssim
E_{k}^{\mathrm{res}}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\bm{\mathfrak{r}}_{\ell}\|_{H^{k-1}}^{2}+\|\mathfrak{r}_{0}\|_{H^{k}}^{2}.$

We need also to evaluate
$\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{-4}\|\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}^{2}$
for $\ell=1,2$ in terms of $E_{k}^{\mathrm{res}}$. In view of

$\begin{cases}\mathcal{L}_{1,i}^{\mbox{\rm\tiny
IW}}\bm{\phi}_{1}^{\mathrm{res}}=\mathcal{L}_{1,i}^{\mbox{\rm\tiny
IW}}\bm{\phi}_{1}^{\mbox{\rm\tiny K}}=(\mathcal{L}_{1,i}^{\mbox{\rm\tiny
IW}}-\mathcal{L}_{1,i}^{\mbox{\rm\tiny K}})\bm{\phi}_{1}^{\mbox{\rm\tiny
K}}=:h_{1,i}^{\mathrm{res}}\quad\mbox{for}\quad i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}^{\mbox{\rm\tiny
IW}}\bm{\phi}_{2}^{\mathrm{res}}=\mathcal{L}_{2,i}^{\mbox{\rm\tiny
IW}}\bm{\phi}_{2}^{\mbox{\rm\tiny K}}=(\mathcal{L}_{2,i}^{\mbox{\rm\tiny
IW}}-\mathcal{L}_{2,i}^{\mbox{\rm\tiny K}})\bm{\phi}_{2}^{\mbox{\rm\tiny
K}}=:h_{2,i}^{\mathrm{res}}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\end{cases}$

Lemma 6.3 yields
$(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}\lesssim\|\nabla\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k}}+\|\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k}}+\|\bm{h}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}$
and we have
$\|\bm{h}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}\lesssim(\|\nabla\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}+\|\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}\,\prime}\|_{H^{m}}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\mbox{\rm\tiny
K}\,\prime}\|_{H^{m-1}})\|\zeta^{\mathrm{res}}\|_{H^{k}}$ for $1\leq k\leq m$.
Therefore, for $1\leq k\leq m$ we obtain

(7.8)
$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{-4}\|\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}^{2}\lesssim
E_{k}^{\mathrm{res}}.$

Now, by deriving equations for spatial derivatives of
$(\zeta^{\mathrm{res}},\bm{\phi}_{1}^{\mathrm{res}},\bm{\phi}_{2}^{\mathrm{res}})$
and applying the energy estimate obtained in Subsection 6.1 we will evaluate
$E_{k}^{\mathrm{res}}$. Let $\beta$ be a multi-index such that
$1\leq|\beta|\leq k$. Applying $\partial^{\beta}$ to the Kakinuma model (3.5)
for $(\zeta^{\mbox{\rm\tiny K}},\bm{\phi}_{1}^{\mbox{\rm\tiny
K}},\bm{\phi}_{2}^{\mbox{\rm\tiny K}})$ and to (7.4) for
$(\zeta^{\mbox{\rm\tiny IW}},\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}},\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}})$ and taking the difference
between the resulting equations, we obtain

$\begin{cases}\displaystyle{\bm{l}}_{1}^{\mbox{\rm\tiny
K}}(\partial_{t}+\bm{u}_{1}^{\mbox{\rm\tiny
K}}\cdot\nabla)\partial^{\beta}\zeta^{\mathrm{res}}+\underline{h}_{1}L_{1}^{\mbox{\rm\tiny
K},\mathrm{pr}}\partial^{\beta}\bm{\phi}_{1}^{\mathrm{res}}=\bm{f}_{1,\beta}^{\mathrm{res}},\\\
\displaystyle{\bm{l}}_{2}^{\mbox{\rm\tiny
K}}(\partial_{t}+\bm{u}_{2}^{\mbox{\rm\tiny
K}}\cdot\nabla)\partial^{\beta}\zeta^{\mathrm{res}}-\underline{h}_{2}L_{2}^{\mbox{\rm\tiny
K},\mathrm{pr}}\partial^{\beta}\bm{\phi}_{2}^{\mathrm{res}}=\bm{f}_{2,\beta}^{\mathrm{res}},\\\
\underline{\rho}_{1}{\bm{l}}_{1}^{\mbox{\rm\tiny
K}}\cdot(\partial_{t}+\bm{u}_{1}^{\mbox{\rm\tiny
K}}\cdot\nabla)\partial^{\beta}\bm{\phi}_{1}^{\mathrm{res}}-\underline{\rho}_{2}{\bm{l}}_{2}^{\mbox{\rm\tiny
K}}\cdot(\partial_{t}+\bm{u}_{1}^{\mbox{\rm\tiny
K}}\cdot\nabla)\partial^{\beta}\bm{\phi}_{2}^{\mathrm{res}}-a^{\mbox{\rm\tiny
K}}\partial^{\beta}\zeta^{\mathrm{res}}=f_{0,\beta}^{\mathrm{res}},\end{cases}$

where

$\begin{cases}\bm{f}_{1,\beta}^{\mathrm{res}}:=\bm{f}_{1,\beta}^{\mbox{\rm\tiny
K}}-\tilde{\bm{f}}_{1,\beta}^{\mbox{\rm\tiny
IW}}-\underline{h}_{1}\partial^{\beta}\bm{\mathfrak{r}}_{1}+\underline{h}_{1}(L_{1}^{\mbox{\rm\tiny
IW},\mathrm{pr}}-L_{1}^{\mbox{\rm\tiny
K},\mathrm{pr}})\partial^{\beta}\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny IW}}\\\
\qquad\quad{}+\bigl{(}\bm{l}_{1}^{\mbox{\rm\tiny
IW}}(\partial_{t}+\tilde{\bm{u}}_{1}^{\mbox{\rm\tiny
IW}}\cdot\nabla)-\bm{l}_{1}^{\mbox{\rm\tiny
K}}(\partial_{t}+\bm{u}_{1}^{\mbox{\rm\tiny
K}}\cdot\nabla)\bigr{)}\partial^{\beta}\zeta^{\mbox{\rm\tiny IW}},\\\
\bm{f}_{2,\beta}^{\mathrm{res}}:=\bm{f}_{2,\beta}^{\mbox{\rm\tiny
K}}-\tilde{\bm{f}}_{2,\beta}^{\mbox{\rm\tiny
IW}}-\underline{h}_{2}\partial^{\beta}\bm{\mathfrak{r}}_{2}-\underline{h}_{2}(L_{2}^{\mbox{\rm\tiny
IW},\mathrm{pr}}-L_{2}^{\mbox{\rm\tiny
K},\mathrm{pr}})\partial^{\beta}\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny IW}}\\\
\qquad\quad{}+\bigl{(}\bm{l}_{2}^{\mbox{\rm\tiny
IW}}(\partial_{t}+\tilde{\bm{u}}_{2}^{\mbox{\rm\tiny
IW}}\cdot\nabla)-\bm{l}_{2}^{\mbox{\rm\tiny
K}}(\partial_{t}+\bm{u}_{2}^{\mbox{\rm\tiny
K}}\cdot\nabla)\bigr{)}\partial^{\beta}\zeta^{\mbox{\rm\tiny IW}},\\\
f_{0,\beta}^{\mathrm{res}}:=f_{0,\beta}^{\mbox{\rm\tiny
K}}-\tilde{f}_{0,\beta}^{\mbox{\rm\tiny
IW}}-\partial^{\beta}\mathfrak{r}_{0}-(\tilde{a}^{\mbox{\rm\tiny
IW}}-a^{\mbox{\rm\tiny K}})\partial^{\beta}\zeta^{\mbox{\rm\tiny IW}}\\\
\qquad\quad{}+\underline{\rho}_{1}\bigl{(}{\bm{l}}_{1}^{\mbox{\rm\tiny
IW}}(\partial_{t}+\tilde{\bm{u}}_{1}^{\mbox{\rm\tiny
IW}}\cdot\nabla)-{\bm{l}}_{1}^{\mbox{\rm\tiny
K}}(\partial_{t}+\bm{u}_{1}^{\mbox{\rm\tiny
K}}\cdot\nabla)\bigr{)}\cdot\partial^{\beta}\tilde{\bm{\phi}}_{1}^{\mbox{\rm\tiny
IW}}\\\ \qquad\quad{}-\underline{\rho}_{2}\bigl{(}{\bm{l}}_{2}^{\mbox{\rm\tiny
IW}}(\partial_{t}+\tilde{\bm{u}}_{2}^{\mbox{\rm\tiny
IW}}\cdot\nabla)-{\bm{l}}_{2}^{\mbox{\rm\tiny
K}}(\partial_{t}+\bm{u}_{2}^{\mbox{\rm\tiny
K}}\cdot\nabla)\bigr{)}\cdot\partial^{\beta}\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny
IW}}.\end{cases}$

Here, $\bm{f}_{1,\beta}^{\mbox{\rm\tiny K}}$,
$\bm{f}_{2,\beta}^{\mbox{\rm\tiny K}}$, and $f_{0,\beta}^{\mbox{\rm\tiny K}}$
are those in (6.15)–(6.17) with
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=(\zeta^{\mbox{\rm\tiny
K}},\bm{\phi}_{1}^{\mbox{\rm\tiny K}},\bm{\phi}_{2}^{\mbox{\rm\tiny K}})$, and
so on. As we saw, all the assumptions in Lemma 6.2 are satisfied, so that we
have

$\mathscr{E}(\partial^{\beta}\bm{U}^{\mathrm{res}}(t))\lesssim\int_{0}^{t}\mathscr{F}_{\beta}^{\mathrm{res}}(\tau)\mathrm{d}\tau,$

where
$\bm{U}^{\mathrm{res}}:=(\zeta^{\mathrm{res}},\bm{\phi}_{1}^{\mathrm{res}},\bm{\phi}_{2}^{\mathrm{res}})^{\mathrm{T}}$,
$\mathscr{E}$ is defined in (6.11), and

$\displaystyle\mathscr{F}_{\beta}^{\mathrm{res}}$
$\displaystyle=\|f_{0,\beta}^{\mathrm{res}}\|_{H^{1}}(\|\partial_{t}\zeta^{\mathrm{res}}\|_{H^{k-1}}+\|\zeta^{\mathrm{res}}\|_{H^{k}})$
$\displaystyle\quad\;+\sum_{\ell=1,2}\underline{\rho}_{\ell}(\|\bm{f}_{\ell,\beta}^{\mathrm{res}}\|_{L^{2}}+\|\zeta^{\mathrm{res}}\|_{H^{k}})(\|\nabla\partial_{t}\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k-1}}+\|\nabla\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k}}).$

In view of $\|(\zeta^{\mbox{\rm\tiny IW}},\zeta^{\mbox{\rm\tiny
K}})\|_{H^{m}}\lesssim 1$, straightforward calculations yield

$\displaystyle\|\bm{f}_{\ell,\beta}^{\mathrm{res}}\|_{L^{2}}$
$\displaystyle\lesssim(\|\partial_{t}\zeta^{\mbox{\rm\tiny
IW}}\|_{H^{m-1}}+\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}})\|\zeta^{\mathrm{res}}\|_{H^{k}}$
$\displaystyle\quad\;+\underline{h}_{\ell}(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m-1}})\|\zeta^{\mathrm{res}}\|_{H^{k}}$
$\displaystyle\quad\;+\underline{h}_{\ell}(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}}\|_{H^{k}}+\|\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k}}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}})$
$\displaystyle\quad\;+\|\partial_{t}\zeta^{\mathrm{res}}\|_{H^{k-1}}+\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}-\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{k}}+\underline{h}_{\ell}\|\bm{\mathfrak{r}}_{\ell}\|_{H^{k}}$

for $\ell=1,2$ and $\frac{n}{2}<k\leq m-1$. As for
$f_{0,\beta}^{\mathrm{res}}$, we note the relation

$\displaystyle\bigl{\\{}\bigl{(}[\partial^{\beta},\bm{l}_{2}^{\mbox{\rm\tiny
K}}]-\bm{l}_{2}^{\prime}(H_{2}^{\mbox{\rm\tiny
K}})(\underline{h}_{2}^{-1}\partial^{\beta}\zeta^{\mbox{\rm\tiny
K}})\bigr{)}-\bigl{(}[\partial^{\beta},\bm{l}_{2}^{\mbox{\rm\tiny
IW}}]-\bm{l}_{2}^{\prime}(H_{2}^{\mbox{\rm\tiny
IW}})(\underline{h}_{2}^{-1}\partial^{\beta}\zeta^{\mbox{\rm\tiny
IW}})\bigr{)}\bigr{\\}}^{\mathrm{T}}\partial_{t}\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny
IW}}$
$\displaystyle\makebox[10.00002pt]{}=\int_{0}^{1}\bigl{\\{}[\partial^{\beta},\bm{l}_{2}^{\prime}(sH^{\mbox{\rm\tiny
IW}}+(1-s)H^{\mbox{\rm\tiny K}})]$
$\displaystyle\makebox[45.00006pt]{}-\bm{l}_{2}^{\prime\prime}(sH^{\mbox{\rm\tiny
IW}}+(1-s)H^{\mbox{\rm\tiny
K}})\underline{h}_{2}^{-1}\partial^{\beta}(s\zeta^{\mbox{\rm\tiny
IW}}+(1-s)\zeta^{\mbox{\rm\tiny
K}})\bigr{\\}}^{\mathrm{T}}(\underline{h}_{2}^{-1}\zeta^{\mathrm{res}})\partial_{t}\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny
IW}}$
$\displaystyle\makebox[10.00002pt]{}\phantom{=\int_{0}^{1}}+\bm{l}_{2}^{\prime}(sH^{\mbox{\rm\tiny
IW}}+(1-s)H^{\mbox{\rm\tiny
K}})\bigl{\\{}[\partial^{\beta},\underline{h}_{2}^{-1}\zeta^{\mathrm{res}}]-(\partial^{\beta}(\underline{h}_{2}^{-1}\zeta^{\mathrm{res}}))\bigr{\\}}^{\mathrm{T}}\partial_{t}\tilde{\bm{\phi}}_{2}^{\mbox{\rm\tiny
IW}}\mathrm{d}s.$

Therefore, straightforward calculations yield

$\displaystyle\|f_{0,\beta}^{\mathrm{res}}\|_{H^{1}}$
$\displaystyle\lesssim\sum_{l=1,2}\underline{\rho}_{\ell}\bigl{\\{}(\|\nabla\partial_{t}\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m-2}}+\|\partial_{t}\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m-2}})\|\zeta^{\mathrm{res}}\|_{H^{k}}$
$\displaystyle\quad\;+(\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+\|\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}})(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}})\|\zeta^{\mathrm{res}}\|_{H^{k}}$
$\displaystyle\quad\;+(\underline{h}_{\ell}\delta)^{-2}\|w_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}}\|\zeta^{\mathrm{res}}\|_{H^{k}}+\|\nabla\partial_{t}\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}}\|_{H^{k-1}}+\|\partial_{t}\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k-1}}$
$\displaystyle\quad\;+\|\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}(\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}}\|_{H^{k}}+\|\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k}})+(\underline{h}_{\ell}\delta)^{-2}\|w_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}\|\tilde{\bm{\phi}}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k}}$
$\displaystyle\quad\;+(\|\bm{u}_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}+\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+\|\nabla\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}})\|\tilde{\bm{u}}_{\ell}^{\mbox{\rm\tiny
IW}}-\bm{u}_{\ell}^{\mbox{\rm\tiny K}}\|_{H^{k}}$
$\displaystyle\quad\;+(\underline{h}_{\ell}\delta)^{-2}(\|w_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{m}}+\|\tilde{w}_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m}}+\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}})\|\tilde{w}_{\ell}^{\mbox{\rm\tiny
IW}}-w_{\ell}^{\mbox{\rm\tiny
K}}\|_{H^{k}}\bigr{\\}}+\|\mathfrak{r}_{0}\|_{H^{k+1}}$

for $\frac{n}{2}<k\leq m-2$. In view of the above estimates and (7.6)–(7.8) we
obtain $\mathscr{F}_{\beta}^{\mathrm{res}}\lesssim
E_{k}^{\mathrm{res}}+\mathfrak{R}_{k}$ with
$\mathfrak{R}_{k}:=\|\mathfrak{r}_{0}\|_{H^{k+1}}^{2}+\sum_{l=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\bm{\mathfrak{r}}_{\ell}\|_{H^{k}}^{2}$.
We note that the multi-index $\beta$ is assumed to satisfy $1\leq|\beta|\leq
k$. As for the case $\beta=0$, we have
$\frac{\mathrm{d}}{\mathrm{d}t}E_{0}^{\mathrm{res}}(t)\lesssim
E_{k}^{\mathrm{res}}(t)$, hence
${E_{0}^{\mathrm{res}}(t)\lesssim\int_{0}^{t}E_{k}^{\mathrm{res}}(\tau)\mathrm{d}\tau}$.
Summarizing the above estimates we obtain
$E_{k}^{\mathrm{res}}(t)\lesssim\int_{0}^{t}(E_{k}^{\mathrm{res}}(\tau)+\mathfrak{R}_{k}(\tau))\mathrm{d}\tau$
for $\frac{n}{2}<k\leq m-2$. Putting ${k=m-4(N+1)}$ and applying Gronwall’s
inequality and (7.5) in Proposition 7.6 we obtain
$E_{m-4(N+1)}^{\mathrm{res}}(t)\lesssim(\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2}$
for $0\leq t\leq\min\\{T,T^{\mbox{\tiny\rm IW}}\\}$.

It remains to evaluate $\phi_{\ell}^{\mbox{\rm\tiny
IW}}-\phi_{\ell}^{\mbox{\rm\tiny K}}$ for $\ell=1,2$. Let
$(\bm{\phi}_{1}^{\mbox{\rm\tiny IW}},\bm{\phi}_{2}^{\mbox{\rm\tiny IW}})$ be
the solution to (3.18) with $(\zeta,\phi_{1},\phi_{2})=(\zeta^{\mbox{\rm\tiny
IW}},\phi_{1}^{\mbox{\rm\tiny IW}},\phi_{2}^{\mbox{\rm\tiny IW}})$. Then, we
have $\phi_{\ell}^{\mbox{\rm\tiny K}}-\phi_{\ell}^{\mbox{\rm\tiny
IW}}=\bm{l}_{\ell}^{\mbox{\rm\tiny
K}}\cdot\bm{\phi}_{\ell}^{\mathrm{res}}+(\bm{l}_{\ell}^{\mbox{\rm\tiny
K}}-\bm{l}_{\ell}^{\mbox{\rm\tiny
IW}})\cdot\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}+\bm{l}_{\ell}^{\mbox{\rm\tiny
IW}}\cdot(\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}-\bm{\phi}_{\ell}^{\mbox{\rm\tiny IW}}),$ so that for any $0\leq k\leq
m-1$

$\displaystyle\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
K}}-\nabla\phi_{\ell}^{\mbox{\rm\tiny IW}}\|_{H^{k}}$
$\displaystyle\lesssim\|\nabla\bm{\phi}_{\ell}^{\mathrm{res}}\|_{H^{k}}+\|\bm{\phi}_{\ell}^{\mathrm{res}\,\prime}\|_{H^{k}}+\underline{h}_{\ell}^{-1}\|\zeta^{\mathrm{res}}\|_{H^{k+1}}\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}\|_{H^{m}}$
$\displaystyle\quad\;+\|\nabla(\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}}-\bm{\phi}_{\ell}^{\mbox{\rm\tiny
IW}})\|_{H^{k}}+\|\tilde{\bm{\phi}}_{\ell}^{\mbox{\rm\tiny
IW}\,\prime}-\bm{\phi}_{\ell}^{\mbox{\rm\tiny IW}\,\prime}\|_{H^{k}}.$

Therefore, the previous result together with Lemma 7.5 implies

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}^{\mbox{\rm\tiny
K}}-\nabla\phi_{\ell}^{\mbox{\rm\tiny
IW}}\|_{H^{m-(4N+5)}}^{2}\lesssim(\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2}.$

This completes the proof of Theorem 3.8.

##  8 Approximation of Hamiltonians; proof of Theorem 3.9

As was shown in the companion paper [3, Theorem 8.4], the Kakinuma model
(2.18) enjoys a Hamiltonian structure analogous to the one exhibited on the
full model for interfacial gravity waves by T. B. Benjamin and T. J. Bridges
in [1]. In this section, we will prove Theorem 3.9, which states that the
Hamiltonian $\mathscr{H}^{\mbox{\rm\tiny K}}(\zeta,\phi)$ of the Kakinuma
model approximates the Hamiltonian $\mathscr{H}^{\mbox{\rm\tiny
IW}}(\zeta,\phi)$ of the full model with an error of order
$O((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})$.

###  8.1 Preliminary elliptic estimates

We consider the following transmission problem

(8.1) $\begin{cases}\nabla_{X}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{\ell}=0&\mbox{in}\quad\Omega_{\ell}\qquad(\ell=1,2),\\\
\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{\ell}=0&\mbox{on}\quad\Sigma_{\ell}\qquad(\ell=1,2),\\\
\bm{n}\cdot I_{\delta}^{2}\nabla_{X}\Phi_{2}-\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{1}=r_{S}&\mbox{on}\quad\Gamma,\\\
\underline{\rho}_{2}\Phi_{2}-\underline{\rho_{1}}\Phi_{1}=\phi&\mbox{on}\quad\Gamma,\end{cases}$

where the rigid-lid $\Sigma_{1}$ of the upper layer $\Omega_{1}$, the bottom
$\Sigma_{2}$ of the lower layer $\Omega_{2}$, and the interface $\Gamma$ are
defined by $z=\underline{h}_{1}$, $z=-\underline{h}_{2}+b(\bm{x})$, and
$z=\zeta(\bm{x})$, respectively,
$I_{\delta}:=\operatorname{diag}(1,\ldots,1,\delta^{-1})$,
$\nabla_{X}:=(\nabla,\partial_{z})^{\mathrm{T}}=(\partial_{1},\ldots,\partial_{n},\partial_{z})$,
and $\bm{n}$ is an upward normal vector, specifically, $\bm{n}=\bm{e}_{z}$ on
$\Sigma_{1}$, $\bm{n}=(-\nabla b,1)^{\mathrm{T}}$ on $\Sigma_{2}$, and
$\bm{n}=(-\zeta,1)^{\mathrm{T}}$ on $\Gamma$.

######  Lemma 8.1.

Let $c,M$ be positive constants. There exists a positive constant $C$ such
that for any positive parameters
$\underline{\rho}_{1},\underline{\rho}_{2},\underline{h}_{1},\underline{h}_{2},\delta$
satisfying $\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, if
$\zeta,b\in W^{1,\infty}$, $H_{1}=1-\underline{h}_{1}^{-1}\zeta$, and
$H_{2}=1+\underline{h}_{2}^{-1}\zeta-\underline{h}_{2}^{-1}b$ satisfy

$\begin{cases}\underline{h}_{1}^{-1}\|\zeta\|_{W^{1,\infty}}+\underline{h}_{2}^{-1}\|\zeta\|_{W^{1,\infty}}+\underline{h}_{2}^{-1}\|b\|_{W^{1,\infty}}\leq
M,\\\ H_{1}(\bm{x})\geq c,\quad H_{2}(\bm{x})\geq
c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},\end{cases}$

then for any $(r_{S},\phi)$ satisfying $\nabla\phi\in H^{-\frac{1}{2}}$ and
$(-\Delta)^{-\frac{1}{2}}r_{S}\in H^{\frac{1}{2}}$ there exists a solution
$(\Phi_{1},\Phi_{2})$ to the transmission problem (8.1). The solution is
unique up to an additive constant of the form
$(\underline{\rho}_{2}\mathcal{C},\underline{\rho}_{1}\mathcal{C})$ and
satisfies

(8.2)
$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Phi_{\ell}\|_{L^{2}(\Omega_{\ell})}^{2}$
$\displaystyle\leq
C\bigl{(}\|((\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-1}\Lambda_{1,0}\Lambda_{2,0})^{\frac{1}{2}}\phi\|_{L^{2}}^{2}+\underline{\rho}_{1}\underline{\rho}_{2}\|(\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-\frac{1}{2}}r_{S}\|_{L^{2}}^{2}\bigr{)},$

where $\Lambda_{1,0}=\Lambda_{1}(0,\delta,\underline{h}_{1})$ and
$\Lambda_{2,0}=\Lambda_{2}(0,0,\delta,\underline{h}_{2})$ are Dirichlet-to-
Neumann maps in the case $\zeta(\bm{x})\equiv b(\bm{x})\equiv 0$.
Particularly, if we further impose $\phi\in\mathring{H}^{1}$,
$(-\Delta)^{-\frac{1}{2}}r_{S}\in H^{1}$, the natural restrictions (2.14), and
$\underline{h}_{\mathrm{min}}\leq\underline{h}_{1},\underline{h}_{2}$ with a
positive constant $\underline{h}_{\mathrm{min}}$, then we have

(8.3)
$\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Phi_{\ell}\|_{L^{2}(\Omega_{\ell})}^{2}\leq
C\|\nabla\phi\|_{L^{2}}^{2}+C\min_{\ell=1,2}\left\\{\frac{\underline{\rho}_{\ell}}{\underline{h}_{\ell}}\|((-\Delta)^{-\frac{1}{2}}+\underline{h}_{\ell}\delta)r_{S}\|_{L^{2}}^{2}\right\\},$

where the constant $C$ depends also on $\underline{h}_{\mathrm{min}}$.

###### Proof.

The existence and the uniqueness of the solution is standard, so that we focus
on deriving the uniform estimate of the solution. To this end, it is
convenient to transform the water regions $\Omega_{1}$ and $\Omega_{2}$ into
simple domains $\Omega_{1,0}=\mathbf{R}^{n}\times(0,\underline{h}_{1})$ and
$\Omega_{2,0}=\mathbf{R}^{n}\times(-\underline{h}_{2},0)$ by using
diffeomorphisms
$\Theta_{\ell}(\bm{x},z)=(\bm{x},\theta_{\ell}(\bm{x},z))\colon\Omega_{\ell,0}\to\Omega_{\ell}$
$(\ell=1,2)$, respectively, where
$\theta_{1}(\bm{x},z)=(1-\underline{h}_{1}^{-1}\zeta(\bm{x}))z+\zeta(\bm{x})$
and
$\theta_{2}(\bm{x},z)=(1+\underline{h}_{2}^{-1}(\zeta(\bm{x})-b(\bm{x})))z+\zeta(\bm{x})$.
Put $\tilde{\Phi}_{\ell}=\Phi_{\ell}\circ\Theta_{\ell}$ $(\ell=1,2)$. Then,
the transmission problem (8.1) is transformed into

$\begin{cases}\nabla_{X}\cdot
I_{\delta}\mathcal{P}_{\ell}I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}=0&\mbox{in}\quad\Omega_{\ell,0}\qquad(\ell=1,2),\\\
\bm{e}_{z}\cdot
I_{\delta}\mathcal{P}_{\ell}I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}=0&\mbox{on}\quad\Sigma_{\ell,0}\qquad(\ell=1,2),\\\
\bm{e}_{z}\cdot
I_{\delta}\mathcal{P}_{2}I_{\delta}\nabla_{X}\tilde{\Phi}_{2}-\bm{e}_{z}\cdot
I_{\delta}\mathcal{P}_{1}I_{\delta}\nabla_{X}\tilde{\Phi}_{1}=r_{S}&\mbox{on}\quad\Gamma_{0},\\\
\underline{\rho}_{2}\tilde{\Phi}_{2}-\underline{\rho_{1}}\tilde{\Phi}_{1}=\phi&\mbox{on}\quad\Gamma_{0},\end{cases}$

where $\Sigma_{1,0}$, $\Sigma_{2,0}$, and $\Gamma_{0}$ are represented as
$z=\underline{h}_{1}$, $z=-\underline{h}_{2}$, and $z=0$, respectively, and

$\mathcal{P}_{\ell}:=\det\biggl{(}\frac{\partial\Theta_{\ell}}{\partial
X}\biggr{)}I_{\delta}^{-1}\biggl{(}\frac{\partial\Theta_{\ell}}{\partial
X}\biggr{)}^{-1}I_{\delta}^{2}\biggl{(}\biggl{(}\frac{\partial\Theta_{\ell}}{\partial
X}\biggr{)}^{-1}\biggr{)}^{\mathrm{T}}I_{\delta}^{-1}\qquad(\ell=1,2).$

We note that
$\|I_{\delta}\nabla_{X}\Phi_{\ell}\|_{L^{2}(\Omega_{\ell})}\simeq\|I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}\|_{L^{2}(\Omega_{\ell,0})}$
$(\ell=1,2)$. Let $(\Psi_{1},\Psi_{2})$ be a solution to the transmission
problem

$\begin{cases}\nabla_{X}\cdot
I_{\delta}^{2}\nabla_{X}\Psi_{\ell}=0&\mbox{in}\quad\Omega_{\ell,0}\qquad(\ell=1,2),\\\
\bm{e}_{z}\cdot
I_{\delta}^{2}\nabla_{X}\Psi_{\ell}=0&\mbox{on}\quad\Sigma_{\ell,0}\qquad(\ell=1,2),\\\
\bm{e}_{z}\cdot I_{\delta}^{2}\nabla_{X}\Psi_{2}-\bm{e}_{z}\cdot
I_{\delta}^{2}\nabla_{X}\Psi_{1}=r_{S}&\mbox{on}\quad\Gamma_{0},\\\
\underline{\rho}_{2}\Psi_{2}-\underline{\rho_{1}}\Psi_{1}=\phi&\mbox{on}\quad\Gamma_{0},\end{cases}$

and put $\Phi_{\ell}^{\mathrm{res}}=\tilde{\Phi}_{\ell}-\Psi_{\ell}$
$(\ell=1,2)$. Then, we can decompose

$|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}|^{2}-I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot(I-\mathcal{P}_{\ell})I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}=\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot\\{(I_{\delta}\mathcal{P}_{\ell}I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}-I_{\delta}^{2}\nabla_{X}\Psi_{\ell})\\}$

for $\ell=1,2$ and
$\underline{\rho}_{1}\Phi_{1}^{\mathrm{res}}=\underline{\rho}_{2}\Phi_{2}^{\mathrm{res}}$
on $z=0$. Therefore, denoting the unit outward normal vector to
$\partial\Omega_{\ell,0}$ by $N_{\ell}$ $(\ell=1,2)$ we have

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell,0}}\bigl{(}|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}|^{2}-I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot(I-\mathcal{P}_{\ell})I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}\bigr{)}\mathrm{d}X$
$\displaystyle\quad=\sum_{\ell=1,2}\int_{\partial\Omega_{\ell,0}}\underline{\rho}_{\ell}\Phi_{\ell}^{\mathrm{res}}(N_{\ell}\cdot
I_{\delta}\mathcal{P}_{\ell}I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}-N_{\ell}\cdot
I_{\delta}^{2}\nabla_{X}\Psi_{\ell})\mathrm{d}S$
$\displaystyle\quad=\sum_{\ell=1,2}\int_{\mathbf{R}^{n}}\underline{\rho}_{1}\bigl{[}\Phi_{1}^{\mathrm{res}}\\{(\bm{e}_{z}\cdot
I_{\delta}\mathcal{P}_{2}I_{\delta}\nabla_{X}\tilde{\Phi}_{2}-\bm{e}_{z}\cdot
I_{\delta}^{2}\nabla_{X}\Psi_{2})$
$\displaystyle\makebox[70.0001pt]{}-(\bm{e}_{z}\cdot
I_{\delta}\mathcal{P}_{1}I_{\delta}\nabla_{X}\tilde{\Phi}_{1}-\bm{e}_{z}\cdot
I_{\delta}^{2}\nabla_{X}\Psi_{1})\\}\bigr{]}\bigr{|}_{z=0}\mathrm{d}\bm{x}$
$\displaystyle\quad=0,$

so that we obtain

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell,0}}|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}|^{2}\mathrm{d}X=\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell,0}}I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot(I-\mathcal{P}_{\ell})I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}\mathrm{d}X.$

Similarly, in view of the decomposition

$\displaystyle
I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot\mathcal{P}_{\ell}I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}-I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot(I-\mathcal{P}_{\ell})I_{\delta}\nabla_{X}\Psi_{\ell}$
$\displaystyle=\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot\\{(I_{\delta}\mathcal{P}_{\ell}I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}-I_{\delta}^{2}\nabla_{X}\Psi_{\ell})\\}$

for $\ell=1,2$, we obtain

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell,0}}I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot\mathcal{P}_{\ell}I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}=\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell,0}}I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot(I-\mathcal{P}_{\ell})I_{\delta}\nabla_{X}\Psi_{\ell}\mathrm{d}X.$

It follows from these two identities that

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\|_{L^{2}(\Omega_{\ell,0})}^{2}\lesssim\min\biggl{\\{}\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}\|_{L^{2}(\Omega_{\ell,0})}^{2},\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Psi_{\ell}\|_{L^{2}(\Omega_{\ell,0})}^{2}\biggr{\\}},$

which yields the equivalence

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}\|_{L^{2}(\Omega_{\ell,0})}^{2}\simeq\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Psi_{\ell}\|_{L^{2}(\Omega_{\ell,0})}^{2}.$

Therefore, it is sufficient to evaluate the right-hand side of the above
equation. In other words, the evaluation is reduced to the simple case
$\zeta(\bm{x})\equiv b(\bm{x})\equiv 0$.

Putting $\psi_{\ell}=\Psi_{\ell}|_{z=0}$ $(\ell=1,2)$, we see that

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Psi_{\ell}\|_{L^{2}(\Omega_{\ell,0})}^{2}=\sum_{\ell=1,2}\underline{\rho}_{\ell}(\Lambda_{\ell,0}\psi_{\ell},\psi_{\ell})_{L^{2}}$

and that

$\begin{cases}\Lambda_{1,0}\psi_{1}+\Lambda_{2,0}\psi_{2}=r_{S},\\\
\underline{\rho}_{2}\psi_{2}-\underline{\rho}_{1}\psi_{1}=\phi.\end{cases}$

Particularly, we have

$\begin{pmatrix}\psi_{1}\\\
\psi_{2}\end{pmatrix}=(\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-1}\begin{pmatrix}-\Lambda_{2,0}\phi+\underline{\rho}_{2}r_{S}\\\
\Lambda_{1,0}\phi+\underline{\rho}_{1}r_{S}\end{pmatrix}.$

Therefore,

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Psi_{\ell}\|_{L^{2}(\Omega_{\ell,0})}^{2}=\begin{cases}\|((\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-1}\Lambda_{1,0}\Lambda_{2,0})^{\frac{1}{2}}\phi\|_{L^{2}}^{2}&\mbox{if
}r_{S}=0,\\\
\underline{\rho}_{1}\underline{\rho}_{2}\|(\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-\frac{1}{2}}r_{S}\|_{L^{2}}^{2}&\mbox{if
}\phi=0.\\\ \end{cases}$

Hence, by the linearity of the problem we obtain (8.2).

Finally, in order to show (8.3) it is sufficient to evaluate the symbols of
the Fourier multipliers
$(\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-1}\Lambda_{1,0}\Lambda_{2,0}$
and
$\underline{\rho}_{1}\underline{\rho}_{2}(\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-1}$.
We remind that the symbol of the Dirichlet-to-Neumann map $\Lambda_{\ell,0}$
is given by
$\sigma(\Lambda_{\ell,0})=\delta^{-1}|\bm{\xi}|\tanh(\underline{h}_{\ell}\delta|\bm{\xi}|)$
for $\ell=1,2$. In view of $0\leq\tanh\xi\leq\xi$ for $\xi\geq 0$, we have

$\displaystyle\sigma((\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-1}\Lambda_{1,0}\Lambda_{2,0})$
$\displaystyle\leq\min\biggl{\\{}\frac{\sigma({\Lambda}_{1,0})}{\underline{\rho}_{1}},\frac{\sigma({\Lambda}_{2,0})}{\underline{\rho}_{2}}\biggr{\\}}$
$\displaystyle\leq\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}|\bm{\xi}|^{2}$
$\displaystyle\leq 2|\bm{\xi}|^{2},$

where we used (2.15). In view of $\tanh\xi\simeq(1+\xi)^{-1}\xi$ for $\xi\geq
0$ and the relation (2.14), we have

$\displaystyle\sigma(\underline{\rho}_{1}\underline{\rho}_{2}(\underline{\rho}_{1}\Lambda_{2,0}+\underline{\rho}_{2}\Lambda_{1,0})^{-1})$
$\displaystyle\simeq\frac{\underline{\rho}_{1}\underline{\rho}_{2}}{\underline{h}_{1}\underline{h}_{2}}\frac{(1+\underline{h}_{1}\delta|\bm{\xi}|)(1+\underline{h}_{2}\delta|\bm{\xi}|)}{(1+\delta|\bm{\xi}|)|\bm{\xi}|^{2}}$
$\displaystyle\lesssim\min\biggl{\\{}\frac{\underline{\rho}_{1}}{\underline{h}_{1}}\underline{\rho}_{2}\frac{1+\underline{h}_{1}\delta|\bm{\xi}|}{|\bm{\xi}|^{2}},\frac{\underline{\rho}_{2}}{\underline{h}_{2}}\underline{\rho}_{1}\frac{1+\underline{h}_{2}\delta|\bm{\xi}|}{|\bm{\xi}|^{2}}\biggr{\\}}$
$\displaystyle\lesssim\min\biggl{\\{}\frac{\underline{\rho}_{1}}{\underline{h}_{1}}(|\bm{\xi}|^{-1}+\underline{h}_{1}\delta)^{2},\frac{\underline{\rho}_{2}}{\underline{h}_{2}}(|\bm{\xi}|^{-1}+\underline{h}_{2}\delta)^{2}\biggr{\\}},$

where we used $1\lesssim\underline{h}_{1},\underline{h}_{2}$. These estimates
imply (8.3). The proof is complete. ∎

###  8.2 Completion of the proof of Theorem 3.9

Now we are ready to prove Theorem 3.9. We remind the definitions (3.3) of
$\bm{l}_{1}(H_{1})$, $\bm{l}_{2}(H_{2})$ and (3.6) of the operators
$\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})$ and
$\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})$. These depend on $N$, so
that we denote them by $\bm{l}_{1}^{(N)}(H_{1})$, $\bm{l}_{2}^{(N)}(H_{2})$
and $\mathcal{L}_{1,i}^{(N)}(H_{1},\delta,\underline{h}_{1})$ and
$\mathcal{L}_{2,i}^{(N)}(H_{2},b,\delta,\underline{h}_{2})$, respectively, in
the following argument. For given $(\zeta,\phi)$, let $\Phi$ be the solution
to the transmission problem (8.1) with $r_{S}=0$ and let
$(\bm{\phi}_{1},\bm{\phi}_{2})$ and
$(\tilde{\bm{\phi}}_{1},\tilde{\bm{\phi}}_{2})$ be the solutions to the
problems

$\begin{cases}\mathcal{L}_{1,i}^{(N)}(H_{1},\delta,\underline{h}_{1})\bm{\phi}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}^{(N)}(H_{2},b,\delta,\underline{h}_{2})\bm{\phi}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}^{(N)}(H_{1},\delta,\underline{h}_{1})\bm{\phi}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}^{(N)}(H_{2},b,\delta,\underline{h}_{2})\bm{\phi}_{2}=0,\\\
\underline{\rho}_{2}\bm{l}_{2}^{(N)}(H_{2})\cdot\bm{\phi}_{2}-\underline{\rho}_{1}\bm{l}_{1}^{(N)}(H_{1})\cdot\bm{\phi}_{1}=\phi\end{cases}$

and

$\begin{cases}\mathcal{L}_{1,i}^{(2N+2)}(H_{1},\delta,\underline{h}_{1})\tilde{\bm{\phi}}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,2N+2,\\\
\mathcal{L}_{2,i}^{(2N+2)}(H_{2},b,\delta,\underline{h}_{2})\tilde{\bm{\phi}}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,2N^{*}+2,\\\
\underline{h}_{1}\mathcal{L}_{1,0}^{(2N+2)}(H_{1},\delta,\underline{h}_{1})\tilde{\bm{\phi}}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}^{(2N+2)}(H_{2},b,\delta,\underline{h}_{2})\tilde{\bm{\phi}}_{2}=0,\\\
\underline{\rho}_{2}\bm{l}_{2}^{(2N+2)}(H_{2})\cdot\tilde{\bm{\phi}}_{2}-\underline{\rho}_{1}\bm{l}_{1}^{(2N+2)}(H_{1})\cdot\tilde{\bm{\phi}}_{1}=\phi,\end{cases}$

respectively, and define $(\Phi_{1}^{\mathrm{app}},\Phi_{2}^{\mathrm{app}})$
and $(\tilde{\Phi}_{1}^{\mathrm{app}},\tilde{\Phi}_{2}^{\mathrm{app}})$ by
(2.23) and

$\begin{cases}\displaystyle\tilde{\Phi}_{1}^{\mathrm{app}}(\bm{x},z)=\sum_{i=0}^{2N+2}(1-\underline{h}_{1}^{-1}z)^{2i}\tilde{\phi}_{1,i}(\bm{x}),\\\\[10.76385pt]
\displaystyle\tilde{\Phi}_{2}^{\mathrm{app}}(\bm{x},z)=\sum_{i=0}^{2N^{*}+2}(1+\underline{h}_{2}^{-1}(z-b(\bm{x})))^{p_{i}}\tilde{\phi}_{2,i}(\bm{x}),\end{cases}$

respectively. Then, by the definitions of the Hamiltonian functionals
$\mathscr{H}^{\mbox{\rm\tiny IW}}(\zeta,\phi)$ and
$\mathscr{H}^{\mbox{\rm\tiny K}}(\zeta,\phi)$ given in Section 2.3, we have

$\displaystyle 2(\mathscr{H}^{\mbox{\rm\tiny
IW}}(\zeta,\phi)-\mathscr{H}^{\mbox{\rm\tiny K}}(\zeta,\phi))$
$\displaystyle=\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell}}(|I_{\delta}\nabla_{X}\Phi_{\ell}|^{2}-|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{app}}|^{2})\mathrm{d}X$
$\displaystyle=\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell}}(|I_{\delta}\nabla_{X}\Phi_{\ell}|^{2}-|I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}^{\mathrm{app}}|^{2})\mathrm{d}X$
$\displaystyle\quad\;+\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell}}(|I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}^{\mathrm{app}}|^{2}-|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{app}}|^{2})\mathrm{d}X$
$\displaystyle=:I_{1}+I_{2}.$

We will evaluate $I_{1}$ and $I_{2}$, separately.

In order to evaluate $I_{1}$, we put
$\Phi_{\ell}^{\mathrm{res}}=\Phi_{\ell}-\tilde{\Phi}_{\ell}^{\mathrm{app}}$
$(\ell=1,2)$, so that

$\displaystyle|I_{1}|$
$\displaystyle=\biggl{|}\sum_{\ell=1,2}\underline{\rho}_{\ell}\int_{\Omega_{\ell}}I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\cdot
I_{\delta}\nabla_{X}(\Phi_{\ell}+\tilde{\Phi}_{\ell}^{\mathrm{app}})\mathrm{d}X\biggr{|}$
$\displaystyle\leq\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\|_{L^{2}(\Omega_{\ell})}(\|I_{\delta}\nabla_{X}\Phi_{\ell}\|_{L^{2}(\Omega_{\ell})}+\|I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}^{\mathrm{app}}\|_{L^{2}(\Omega_{\ell})}).$

It follows from Lemma 8.1 that
$\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Phi_{\ell}\|_{L^{2}(\Omega_{\ell})}^{2}\lesssim\|\nabla\phi\|_{L^{2}}^{2}$.
We see also that

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\tilde{\Phi}_{\ell}^{\mathrm{app}}\|_{L^{2}(\Omega_{\ell})}^{2}$
$\displaystyle=\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(L_{\ell}^{(2N+2)}\tilde{\bm{\phi}}_{\ell},\tilde{\bm{\phi}}_{\ell})_{L^{2}}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\tilde{\bm{\phi}}_{\ell}\|_{L^{2}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\tilde{\bm{\phi}}^{\prime}\|_{L^{2}}^{2})$
$\displaystyle\lesssim\|\nabla\phi\|_{L^{2}}^{2},$

where we used Lemma 5.1 and (2.15). In order to evaluate
$\|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}\|_{L^{2}(\Omega_{\ell})}$,
we first notice that $(\Phi_{1}^{\mathrm{res}},\Phi_{2}^{\mathrm{res}})$
satisfy

$\begin{cases}\nabla_{X}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{\ell}^{\mathrm{res}}=R_{\ell}&\mbox{in}\quad\Omega_{\ell}\qquad(\ell=1,2),\\\
\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{1}^{\mathrm{res}}=0&\mbox{on}\quad\Sigma_{1},\\\
\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{2}^{\mathrm{res}}=\underline{h}_{2}r_{B}&\mbox{on}\quad\Sigma_{2},\\\
\underline{\rho}_{2}\Phi_{2}^{\mathrm{res}}-\underline{\rho}_{1}\Phi_{1}^{\mathrm{res}}=0&\mbox{on}\quad\Gamma,\\\
\Lambda_{1}[\Phi_{1}^{\mathrm{res}}|_{z=\zeta}]+\Lambda_{2}[\Phi_{2}^{\mathrm{res}}|_{z=\zeta}]=r_{S},\end{cases}$

where

$\begin{cases}R_{\ell}=-\nabla_{X}\cdot
I_{\delta}^{2}\nabla_{X}\tilde{\Phi}_{\ell}^{\mathrm{app}}\qquad(\ell=1,2),\\\
r_{B}=-\underline{h}_{2}^{-1}(-\nabla b,1)^{\mathrm{T}}\cdot
I_{\delta}^{2}(\nabla_{X}\tilde{\Phi}_{\ell}^{\mathrm{app}})|_{z=-\underline{h}_{2}+b},\\\
\displaystyle
r_{S}=\sum_{\ell=1,2}(\underline{h}_{\ell}\Lambda_{\ell}^{(2N+2)}-\Lambda_{\ell})[\tilde{\Phi}_{\ell}^{\mathrm{app}}|_{z=\zeta}].\\\
\end{cases}$

Here, we note that $R_{\ell}$ $(\ell=1,2)$ can be written the form

$\begin{cases}\displaystyle
R_{1}(\bm{x},z)=\sum_{i=0}^{2N+2}(1-\underline{h}_{1}^{-1}z)^{2i}r_{1,i}(\bm{x}),\\\\[10.76385pt]
\displaystyle
R_{2}(\bm{x},z)=\sum_{i=0}^{2N^{*}+2}(1+\underline{h}_{2}^{-1}(z-b(\bm{x})))^{p_{i}}r_{2,i}(\bm{x}).\end{cases}$

Estimates for the residuals $(r_{1,0},r_{1,1},\ldots,r_{1,2N+2})$,
$(r_{2,0},r_{2,0},\ldots,r_{2,2N^{*}+2})$, and $r_{B}$ were given in [6,
Lemmas 6.4 and 6.9] and their proofs. In fact, we have

$\displaystyle\|(r_{1,0},r_{1,1},\ldots,r_{1,2N+2})\|_{L^{2}}$
$\displaystyle\lesssim\|\tilde{\phi}_{1,2N+2}\|_{H^{2}}$
$\displaystyle\lesssim(\underline{h}_{1}\delta)^{4N+2}\|\nabla\tilde{\bm{\phi}}_{1}\|_{H^{4N+3}}$

and

$\displaystyle\|(r_{2,0},r_{2,1},\ldots,r_{2,2N^{*}+2})\|_{L^{2}}+\|r_{B}\|_{L^{2}}$
$\displaystyle\lesssim\|(\tilde{\phi}_{2,2N^{*}+1},\tilde{\phi}_{2,2N^{*}+2})\|_{H^{2}}$
$\displaystyle\lesssim(\underline{h}_{2}\delta)^{4N+2}(\|\nabla\tilde{\bm{\phi}}_{2}\|_{H^{4N+3}}+\|\tilde{\bm{\phi}}_{2}^{\prime}\|_{H^{4N+3}}).$

We decompose
$\Phi_{\ell}^{\mathrm{res}}=\Phi_{\ell}^{\mathrm{res,1}}+\Phi_{\ell}^{\mathrm{res,2}}$,
where $(\Phi_{1}^{\mathrm{res,1}},\Phi_{2}^{\mathrm{res,1}})$ is a unique
solution to the problem

$\begin{cases}\nabla_{X}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{\ell}^{\mathrm{res,1}}=R_{\ell}&\mbox{in}\quad\Omega_{\ell}\qquad(\ell=1,2),\\\
\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{1}^{\mathrm{res,1}}=0&\mbox{on}\quad\Sigma_{1},\\\
\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{2}^{\mathrm{res,1}}=\underline{h}_{2}r_{B}&\mbox{on}\quad\Sigma_{2},\\\
\Phi_{\ell}^{\mathrm{res,1}}=0&\mbox{on}\quad\Gamma\qquad(\ell=1,2),\end{cases}$

so that $(\Phi_{1}^{\mathrm{res,2}},\Phi_{2}^{\mathrm{res,2}})$ satisfy

(8.4) $\begin{cases}\nabla_{X}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{\ell}^{\mathrm{res,2}}=0&\mbox{in}\quad\Omega_{\ell}\qquad(\ell=1,2),\\\
\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{\ell}^{\mathrm{res,2}}=0&\mbox{on}\quad\Sigma_{\ell}\qquad(\ell=1,2),\\\
\bm{n}\cdot I_{\delta}^{2}\nabla_{X}\Phi_{2}^{\mathrm{res,2}}-\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{1}^{\mathrm{res,2}}=r_{S}&\mbox{on}\quad\Gamma,\\\
\underline{\rho}_{2}\Phi_{2}^{\mathrm{res,2}}-\underline{\rho}_{1}\Phi_{1}^{\mathrm{res,2}}=0&\mbox{on}\quad\Gamma,\end{cases}$

where we used the relations
$\Lambda_{1}[\Phi_{1}^{\mathrm{res,2}}|_{z=\zeta}]=-\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{1}^{\mathrm{res,2}}$ and
$\Lambda_{2}[\Phi_{2}^{\mathrm{res,2}}|_{z=\zeta}]=\bm{n}\cdot
I_{\delta}^{2}\nabla_{X}\Phi_{2}^{\mathrm{res,2}}$ on $\Gamma$. It is easy to
see that

$\displaystyle\|I_{\delta}\nabla_{X}\Phi_{1}^{\mathrm{res,1}}\|_{L^{2}(\Omega_{1})}^{2}$
$\displaystyle\lesssim(\underline{h}_{1}\delta)^{2}\|R_{1}\|_{L^{2}(\Omega_{1})}^{2}$
$\displaystyle\lesssim\underline{h}_{1}(\underline{h}_{1}\delta)^{2}\|(r_{1,0},r_{1,1},\ldots,r_{1,2N+2})\|_{L^{2}}$
$\displaystyle\lesssim\underline{h}_{1}(\underline{h}_{1}\delta)^{2(4N+3)}\|\nabla\tilde{\bm{\phi}}_{1}\|_{H^{4N+3}}^{2}$

and that

$\displaystyle\|I_{\delta}\nabla_{X}\Phi_{2}^{\mathrm{res,1}}\|_{L^{2}(\Omega_{2})}^{2}$
$\displaystyle\lesssim\underline{h}_{2}(\underline{h}_{2}\delta)^{2}(\underline{h}_{2}^{-1}\|R_{2}\|_{L^{2}(\Omega_{2})}^{2}+\|r_{B}\|_{L^{2}}^{2})$
$\displaystyle\lesssim\underline{h}_{2}(\underline{h}_{2}\delta)^{2}(\|(r_{2,0},r_{2,1},\ldots,r_{2,2N^{*}+2})\|_{L^{2}}+\|r_{B}\|_{L^{2}}^{2})$
$\displaystyle\lesssim\underline{h}_{2}(\underline{h}_{2}\delta)^{2(4N+3)}(\|\nabla\tilde{\bm{\phi}}_{2}\|_{H^{4N+3}}+\|\tilde{\bm{\phi}}_{2}^{\prime}\|_{H^{4N+3}}).$

Therefore, by Lemma 5.1 together with (2.15) we have

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res,1}}\|_{L^{2}(\Omega_{\ell})}^{2}\lesssim((\underline{h}_{1}\delta)^{4N+3}+(\underline{h}_{2}\delta)^{4N+3})^{2}\|\nabla\phi\|_{H^{4N+3}}^{2}.$

On the other hand, it follows from Lemmas 8.1, 4.5, 7.2, and 5.1 that

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\|I_{\delta}\nabla_{X}\Phi_{\ell}^{\mathrm{res,2}}\|_{L^{2}(\Omega_{\ell})}^{2}$
$\displaystyle\lesssim\min_{\ell=1,2}\frac{\underline{\rho}_{\ell}}{\underline{h}_{\ell}}\|((-\Delta)^{-\frac{1}{2}}+\underline{h}_{\ell}\delta)r_{S}\|_{L^{2}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\frac{\underline{\rho}_{\ell}}{\underline{h}_{\ell}}\|((-\Delta)^{-\frac{1}{2}}+\underline{h}_{\ell}\delta)(\underline{h}_{\ell}\Lambda_{\ell}^{(2N+2)}-\Lambda_{\ell})[\tilde{\Phi}_{\ell}^{\mathrm{app}}|_{z=\zeta}]\|_{L^{2}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2(4N+2)}\|\nabla(\tilde{\Phi}_{\ell}^{\mathrm{app}}|_{z=\zeta})\|_{H^{4N+3}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2(4N+2)}(\|\nabla\tilde{\bm{\phi}}_{\ell}\|_{H^{4N+3}}^{2}+\|\tilde{\bm{\phi}}_{\ell}^{\prime}\|_{H^{4N+3}}^{2})$
$\displaystyle\lesssim((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})^{2}\|\nabla\phi\|_{H^{4N+3}}^{2}.$

Summarizing the above estimates, we obtain
$|I_{1}|\lesssim((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})\|\nabla\phi\|_{H^{4N+3}}\|\nabla\phi\|_{L^{2}}$.

We proceed to evaluate $I_{2}$, which can be written as

$\displaystyle I_{2}$
$\displaystyle=\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(L_{\ell}^{(2N+2)}\tilde{\bm{\phi}}_{\ell},\tilde{\bm{\phi}}_{\ell})_{L^{2}}-\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(L_{\ell}^{(N)}\bm{\phi}_{\ell},\bm{\phi}_{\ell})_{L^{2}}$
$\displaystyle=:I_{2,1}+I_{2,2}.$

In view of (3.8), we see that

$\displaystyle I_{2,1}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}(\mathcal{L}_{1,0}^{(2N+2)}\tilde{\bm{\phi}}_{1},\bm{l}_{1}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{1})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}(\mathcal{L}_{2,0}^{(2N+2)}\tilde{\bm{\phi}}_{2},\bm{l}_{2}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{2})_{L^{2}}$
$\displaystyle=(\underline{h}_{2}\mathcal{L}_{2,0}^{(2N+2)}\tilde{\bm{\phi}}_{2},\underline{\rho}_{2}\bm{l}_{2}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{2}-\underline{\rho}_{1}\bm{l}_{1}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{1})_{L^{2}}$
$\displaystyle=(\underline{h}_{2}\mathcal{L}_{2,0}^{(2N+2)}\tilde{\bm{\phi}}_{2},\phi)_{L^{2}}$
$\displaystyle=(\underline{h}_{2}\mathcal{L}_{2,0}^{(2N+2)}\tilde{\bm{\phi}}_{2},\underline{\rho}_{2}\bm{l}_{2}^{(N)}\cdot\bm{\phi}_{2}-\underline{\rho}_{1}\bm{l}_{1}^{(N)}\cdot\bm{\phi}_{1})_{L^{2}}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}(\mathcal{L}_{1,0}^{(2N+2)}\tilde{\bm{\phi}}_{1},\bm{l}_{1}^{(N)}\cdot\bm{\phi}_{1})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}(\mathcal{L}_{2,0}^{(2N+2)}\tilde{\bm{\phi}}_{2},\bm{l}_{2}^{(N)}\cdot\bm{\phi}_{2})_{L^{2}}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}\sum_{i=0}^{N}\sum_{j=0}^{2N+2}(L_{1,ij}\tilde{\phi}_{1,j},\phi_{1,i})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}\sum_{i=0}^{N^{*}}\sum_{j=0}^{2N^{*}+2}(L_{2,ij}\tilde{\phi}_{2,j},\phi_{2,i})_{L^{2}}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}\sum_{i=0}^{N}\sum_{j=0}^{2N+2}(L_{1,ji}\phi_{1,i},\tilde{\phi}_{1,j})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}\sum_{i=0}^{N^{*}}\sum_{j=0}^{2N^{*}+2}(L_{2,ji}\phi_{2,i},\tilde{\phi}_{2,j})_{L^{2}},$

where we used $L_{\ell,ij}^{*}=L_{\ell,ji}$. Similarly, we see also that

$\displaystyle I_{2,2}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}(\mathcal{L}_{1,0}^{(N)}\bm{\phi}_{1},\bm{l}_{1}^{(N)}\cdot\bm{\phi}_{1})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}(\mathcal{L}_{2,0}^{(N)}\bm{\phi}_{2},\bm{l}_{2}^{(N)}\cdot\bm{\phi}_{2})_{L^{2}}$
$\displaystyle=(\underline{h}_{2}\mathcal{L}_{2,0}^{(N)}\bm{\phi}_{2},\underline{\rho}_{2}\bm{l}_{2}^{(N)}\cdot\bm{\phi}_{2}-\underline{\rho}_{1}\bm{l}_{1}^{(N)}\cdot\bm{\phi}_{1})_{L^{2}}$
$\displaystyle=(\underline{h}_{2}\mathcal{L}_{2,0}^{(N)}\bm{\phi}_{2},\phi)_{L^{2}}$
$\displaystyle=(\underline{h}_{2}\mathcal{L}_{2,0}^{(N)}\bm{\phi}_{2},\underline{\rho}_{2}\bm{l}_{2}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{2}-\underline{\rho}_{1}\bm{l}_{1}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{1})_{L^{2}}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}(\mathcal{L}_{1,0}^{(N)}\bm{\phi}_{1},\bm{l}_{1}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{1})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}(\mathcal{L}_{2,0}^{(N)}\bm{\phi}_{2},\bm{l}_{2}^{(2N+2)}\cdot\tilde{\bm{\phi}}_{2})_{L^{2}}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}\sum_{j=0}^{2N+2}(H_{1}^{2j}\mathcal{L}_{1,0}^{(N)}\bm{\phi}_{1},\tilde{\phi}_{1,j})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}\sum_{j=0}^{2N^{*}+2}(H_{2}^{p_{j}}\mathcal{L}_{2,0}^{(N)}\bm{\phi}_{2},\tilde{\phi}_{2,j})_{L^{2}}.$

Here, it follows from (3.8) that
$H_{1}^{2j}\mathcal{L}_{1,0}^{(N)}\bm{\phi}_{1}=\sum_{i=0}^{N}L_{1,ji}\phi_{1,i}$
and
$H_{2}^{p_{j}}\mathcal{L}_{2,0}^{(N)}\bm{\phi}_{1}=\sum_{i=0}^{N^{*}}L_{2,ji}\phi_{2,i}$
hold only for $j=0,1,\ldots,N$ and for $j=0,1,\ldots,N^{*}$, respectively.
Therefore, we have

$\displaystyle I_{2,2}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}\sum_{i=0}^{N}\sum_{j=0}^{N}(L_{1,ji}\phi_{1,i},\tilde{\phi}_{1,j})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}\sum_{i=0}^{N^{*}}\sum_{j=0}^{N^{*}}(L_{2,ji}\phi_{2,i},\tilde{\phi}_{2,j})_{L^{2}}$
$\displaystyle\quad\;+\underline{\rho}_{1}\underline{h}_{1}\sum_{i=0}^{N}\sum_{j=N+1}^{2N+2}(H_{1}^{2j}L_{1,0i}\phi_{1,i},\tilde{\phi}_{1,j})_{L^{2}}+\underline{\rho}_{2}\underline{h}_{2}\sum_{i=0}^{N^{*}}\sum_{j=N^{*}+1}^{2N^{*}+2}(H_{2}^{p_{j}}L_{2,0i}\phi_{2,i},\tilde{\phi}_{2,j})_{L^{2}},$

so that

$\displaystyle I_{2}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}\sum_{i=0}^{N}\sum_{j=N+1}^{2N+2}((L_{1,ji}-H_{1}^{2j}L_{1,0i})\phi_{1,i},\tilde{\phi}_{1,j})_{L^{2}}$
$\displaystyle\quad\;+\underline{\rho}_{2}\underline{h}_{2}\sum_{i=0}^{N^{*}}\sum_{j=N^{*}+1}^{2N^{*}+2}((L_{2,ji}-H_{2}^{p_{j}}L_{2,0i})\phi_{2,i},\tilde{\phi}_{2,j})_{L^{2}}$
$\displaystyle=\underline{\rho}_{1}\underline{h}_{1}\sum_{i=0}^{N}\sum_{j=N+1}^{2N+2}((L_{1,ji}-H_{1}^{2j}L_{1,0i})(\phi_{1,i}-\tilde{\phi}_{1,i}),\tilde{\phi}_{1,j})_{L^{2}}$
$\displaystyle\quad\;+\underline{\rho}_{2}\underline{h}_{2}\sum_{i=0}^{N^{*}}\sum_{j=N^{*}+1}^{2N^{*}+2}((L_{2,ji}-H_{2}^{p_{j}}L_{2,0i})(\phi_{2,i}-\tilde{\phi}_{2,i}),\tilde{\phi}_{2,j})_{L^{2}}$
$\displaystyle\quad\;-\underline{\rho}_{1}\underline{h}_{1}\sum_{i=N+1}^{2N+2}\sum_{j=N+1}^{2N+2}((L_{1,ji}-H_{1}^{2j}L_{1,0i})\tilde{\phi}_{1,i},\tilde{\phi}_{1,j})_{L^{2}}$
$\displaystyle\quad\;-\underline{\rho}_{2}\underline{h}_{2}\sum_{i=N^{*}+1}^{2N^{*}+2}\sum_{j=N^{*}+1}^{2N^{*}+2}((L_{2,ji}-H_{2}^{p_{j}}L_{2,0i})\tilde{\phi}_{2,i},\tilde{\phi}_{2,j})_{L^{2}}.$

Hence, denoting by
$\bm{\varphi}_{1}=(\varphi_{1,0},\varphi_{1,1},\ldots,\varphi_{1,N})^{\mathrm{T}}$
and
$\bm{\varphi}_{2}=(\varphi_{2,0},\varphi_{2,1},\ldots,\varphi_{2,N^{*}})^{\mathrm{T}}$
with $\varphi_{\ell,i}=\phi_{\ell,i}-\tilde{\phi}_{\ell,i}$ we obtain

$\displaystyle|I_{2}|$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\bm{\varphi}_{\ell}\|_{L^{2}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\varphi}_{\ell}^{\prime}\|_{L^{2}}^{2})$
$\displaystyle\quad\;+\underline{\rho}_{1}\underline{h}_{1}\|(\tilde{\phi}_{1,N+1},\tilde{\phi}_{1,N+2},\ldots,\tilde{\phi}_{1,2N+2})\|_{H^{1}}^{2}$
$\displaystyle\quad\;+\underline{\rho}_{2}\underline{h}_{2}\|(\tilde{\phi}_{2,N^{*}+1},\tilde{\phi}_{2,N^{*}+2},\ldots,\tilde{\phi}_{2,2N^{*}+2})\|_{H^{1}}^{2}$
$\displaystyle\quad\;+\underline{\rho}_{1}\underline{h}_{1}(\underline{h}_{1}\delta)^{-2}\|(\tilde{\phi}_{1,N+1},\tilde{\phi}_{1,N+2},\ldots,\tilde{\phi}_{1,2N+2})\|_{L^{2}}^{2}$
$\displaystyle\quad\;+\underline{\rho}_{2}\underline{h}_{2}(\underline{h}_{2}\delta)^{-2}\|(\tilde{\phi}_{2,N^{*}+1},\tilde{\phi}_{2,N^{*}+2},\ldots,\tilde{\phi}_{2,2N^{*}+2})\|_{L^{2}}^{2}.$

Here, we note that $(\bm{\varphi}_{1},\bm{\varphi}_{2})$ satisfy

$\begin{cases}\mathcal{L}_{1,i}^{(N)}\bm{\varphi}_{1}=r_{1,i}\quad\mbox{for}\quad
i=0,1,\ldots,N,\\\
\mathcal{L}_{2,i}^{(N)}\bm{\varphi}_{2}=r_{2,i}\quad\mbox{for}\quad
i=0,1,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}^{(N)}\bm{\varphi}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}^{(N)}\bm{\varphi}_{2}=\nabla\cdot(\underline{h}_{1}\bm{r}_{3,1}+\underline{h}_{2}\bm{r}_{3,2}),\\\
\underline{\rho}_{2}\bm{l}_{2}^{(N)}\cdot\bm{\varphi}_{2}-\underline{\rho}_{1}\bm{l}_{1}^{(N)}\cdot\bm{\varphi}_{1}=\underline{\rho}_{1}r_{4,1}+\underline{\rho}_{2}r_{4,2},\end{cases}$

where

$\begin{cases}r_{1,i}=-\sum_{j=N+1}^{2N+2}(L_{1,ij}-H^{2i}L_{1,0j})\tilde{\phi}_{1,j}\quad\mbox{for}\quad
i=0,1,\ldots,N,\\\
r_{2,i}=-\sum_{j=N^{*}+1}^{2N^{*}+2}(L_{2,ij}-H^{p_{i}}L_{2,0j})\tilde{\phi}_{2,j}\quad\mbox{for}\quad
i=0,1,\ldots,N^{*},\\\
\nabla\cdot\bm{r}_{3,1}=\sum_{j=N+1}^{2N+2}L_{1,0j}\tilde{\phi}_{1,j},\quad\nabla\cdot\bm{r}_{3,2}=\sum_{j=N^{*}+1}^{2N^{*}+2}L_{2,0j}\tilde{\phi}_{2,j},\\\
r_{4,1}=\sum_{j=N+1}^{2N+2}H_{1}^{2j}\tilde{\phi}_{1,j},\quad
r_{4,2}=-\sum_{j=N^{*}+1}^{2N^{*}+2}H_{2}^{p_{j}}\tilde{\phi}_{2,j}.\end{cases}$

We put $\bm{r}_{1}^{\prime}=(0,r_{1,1},\ldots,r_{1,N})^{\mathrm{T}}$ and
$\bm{r}_{2}^{\prime}=(0,r_{2,1},\ldots,r_{2,N})^{\mathrm{T}}$. Then, with a
suitable decomposition
$\bm{r}_{\ell}=\bm{r}_{\ell}^{\mathrm{high}}+(\underline{h}_{\ell}\delta)^{-2}\bm{r}_{\ell}^{\mathrm{low}}$
for $\ell=1,2$, and using the linearity of (5.1), we see by Lemma 5.1 that

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\bm{\varphi}_{\ell}\|_{L^{2}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\varphi}_{\ell}^{\prime}\|_{L^{2}}^{2})$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\bm{r}_{\ell}^{\mathrm{high}}\|_{H^{-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{r}_{\ell}^{\mathrm{low}}\|_{L^{2}}^{2}+\|\bm{r}_{3,\ell}\|_{L^{2}}^{2}+\|r_{4,\ell}\|_{H^{1}}^{2})$
$\displaystyle\lesssim\underline{\rho}_{1}\underline{h}_{1}\|(\tilde{\phi}_{1,N+1},\tilde{\phi}_{1,N+2},\ldots,\tilde{\phi}_{1,2N+2})\|_{H^{1}}^{2}$
$\displaystyle\quad\;+\underline{\rho}_{2}\underline{h}_{2}\|(\tilde{\phi}_{2,N^{*}+1},\tilde{\phi}_{2,N^{*}+2},\ldots,\tilde{\phi}_{2,2N^{*}+2})\|_{H^{1}}^{2}$
$\displaystyle\quad\;+\underline{\rho}_{1}\underline{h}_{1}(\underline{h}_{1}\delta)^{-2}\|(\tilde{\phi}_{1,N+1},\tilde{\phi}_{1,N+2},\ldots,\tilde{\phi}_{1,2N+2})\|_{L^{2}}^{2}$
$\displaystyle\quad\;+\underline{\rho}_{2}\underline{h}_{2}(\underline{h}_{2}\delta)^{-2}\|(\tilde{\phi}_{2,N^{*}+1},\tilde{\phi}_{2,N^{*}+2},\ldots,\tilde{\phi}_{2,2N^{*}+2})\|_{L^{2}}^{2}.$

Moreover, it follows from [6, Lemmas 5.2 and 5.4] that

$\displaystyle\|(\tilde{\phi}_{1,N+1},\tilde{\phi}_{1,N+2},\ldots,\tilde{\phi}_{1,2N+2})\|_{H^{k}}\lesssim(\underline{h}_{1}\delta)^{2N+2-k}\|\nabla\tilde{\bm{\phi}}_{1}\|_{H^{2N+1}}$
$\displaystyle\|(\tilde{\phi}_{2,N^{*}+1},\tilde{\phi}_{2,N^{*}+2},\ldots,\tilde{\phi}_{2,2N^{*}+2})\|_{H^{k}}\lesssim(\underline{h}_{2}\delta)^{2N+2-k}(\|\nabla\tilde{\bm{\phi}}_{2}\|_{H^{2N+1}}+\|\tilde{\bm{\phi}}_{2}^{\prime}\|_{H^{2N+1}}),$

for $k=0,2$, and hence also for $k=1$ by interpolation, so that

$\displaystyle|I_{2}|$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{4N+2}(\|\nabla\tilde{\bm{\phi}}_{\ell}\|_{H^{2N+1}}^{2}+\|\tilde{\bm{\phi}}_{\ell}^{\prime}\|_{H^{2N+1}}^{2})$
$\displaystyle\lesssim((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})\|\nabla\phi\|_{H^{2N+1}}^{2}$
$\displaystyle\lesssim((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})\|\nabla\phi\|_{H^{4N+2}}\|\nabla\phi\|_{L^{2}},$

where we used Lemma 5.1 with (2.15), and interpolation. This completes the
proof of Theorem 3.9.

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Vincent Duchêne

Institut de Recherche Mathématique de Rennes

Univ Rennes, CNRS, IRMAR – UMR 6625

F-35000 Rennes, France

E-mail<EMAIL_ADDRESS>

Tatsuo Iguchi

Department of Mathematics

Faculty of Science and Technology, Keio University

3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

E-mail<EMAIL_ADDRESS>