$L_{1}^{\rm low}(H_{1})=(L_{1,ij}^{\rm low}(H_{1}))_{0\leq i,j\leq N}$ and
$L_{2}(H_{2},b)=(L_{2,ij}^{\rm low}(H_{2},b))_{0\leq i,j\leq N^{*}}$ are given
by

$\displaystyle L_{1,ij}^{\rm
low}(H_{1})\varphi_{1,j}=\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}\varphi_{1,j},$
$\displaystyle L_{2,ij}^{\rm low}(H_{2},b)\varphi_{2,j}=\nabla
b\cdot(H_{2}^{p_{i}+p_{j}}\nabla\varphi_{2,j}-p_{j}H_{2}^{p_{i}+p_{j}-1}\varphi_{2,j}\nabla
b)+\frac{p_{j}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\nabla\cdot(\varphi_{2,j}\nabla
b)$
$\displaystyle\makebox[77.49976pt]{}-\frac{p_{i}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\nabla
b\cdot\nabla\varphi_{2,j}+\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}(1+|\nabla
b|^{2})\varphi_{2,j}.$

Let us recall the definitions of $\bm{u}$ in (5.5), and $\theta_{1}$ and
$\theta_{2}$ in (5.12), so that

$\bm{u}_{1}=\bm{u}-\theta_{1}\bm{v},\qquad\bm{u}_{2}=\bm{u}+\theta_{2}\bm{v}.$

Therefore, we can rewrite the first two equations in (7.3) as

$\begin{cases}\bm{l}_{1}(H_{1})(\partial_{t}\zeta-\varepsilon\Delta\zeta+(\bm{u}-\theta_{1}\bm{v})\cdot\nabla\zeta)-A_{1}(H_{1})\Delta\bm{\phi}_{1}+L_{1}^{\rm
low}(H_{1})\bm{\phi}_{1}=\bm{0},\\\
-\bm{l}_{2}(H_{2})(\partial_{t}\zeta-\varepsilon\Delta\zeta+(\bm{u}+\theta_{2}\bm{v})\cdot\nabla\zeta)-A_{2}(H_{2})\Delta\bm{\phi}_{2}+L_{2}^{\rm
low}(H_{2},b)\bm{\phi}_{2}=\bm{0}.\end{cases}$

Let $\beta=(\beta_{1},\ldots,\beta_{n})$ be a multi-index satisfying
$|\beta|\leq m$. Applying the differential operator $\partial^{\beta}$ to
these equations and noting the relation
$(\bm{v}\cdot\nabla)=-(\bm{v}\cdot\nabla)^{*}-(\nabla\cdot\bm{v})$, we have

(7.15)
$\begin{cases}\displaystyle\rho_{1}\bm{l}_{1}(\partial_{t}\zeta^{\beta}-\varepsilon\Delta\zeta^{\beta}+\bm{u}\cdot\nabla\zeta^{\beta})+(\bm{v}\cdot\nabla)^{*}(\rho_{1}\theta_{1}\bm{l}_{1}\zeta^{\beta})-\sum_{l=1}^{n}\partial_{l}(\rho_{1}A_{1}\partial_{l}{\bm{\phi}}_{1}^{\beta})=\bm{F}_{1,\beta},\\\
\displaystyle-\rho_{2}\bm{l}_{2}(\partial_{t}\zeta^{\beta}-\varepsilon\Delta\zeta^{\beta}+\bm{u}\cdot\nabla\zeta^{\beta})+(\bm{v}\cdot\nabla)^{*}(\rho_{2}\theta_{2}\bm{l}_{2}\zeta^{\beta})-\sum_{l=1}^{n}\partial_{l}(\rho_{2}A_{2}\partial_{l}\bm{\phi}_{2}^{\beta})=\bm{F}_{2,\beta},\end{cases}$

where $\zeta^{\beta}=\partial^{\beta}\zeta$,
$\bm{\phi}_{k}^{\beta}=\partial^{\beta}\bm{\phi}_{k}$ for $k=1,2$, and

$\begin{cases}\bm{F}_{1,\beta}=\rho_{1}\biggl{\\{}-[\partial^{\beta},\bm{l}_{1}]G_{0}-[\partial^{\beta},\bm{l}_{1}\bm{u}_{1}^{\rm
T}]\nabla\zeta-(\nabla\cdot\bm{v})\theta_{1}\bm{l}_{1}\zeta^{\beta}+[\bm{v}\cdot\nabla,\theta_{1}\bm{l}_{1}]\zeta^{\beta}\\\
\displaystyle\makebox[47.36096pt]{}-\sum_{l=1}^{n}(\partial_{l}A_{1})\partial_{l}\bm{\phi}_{1}^{\beta}+[\partial^{\beta},A_{1}]\Delta\bm{\phi}_{1}-\partial^{\beta}L_{1}^{\rm
low}(H_{1})\bm{\phi}_{1}\biggr{\\}},\\\
\bm{F}_{2,\beta}=\rho_{2}\biggl{\\{}[\partial^{\beta},\bm{l}_{2}]G_{0}+[\partial^{\beta},\bm{l}_{2}\bm{u}_{2}^{\rm
T}]\nabla\zeta-(\nabla\cdot\bm{v})\theta_{2}\bm{l}_{2}\zeta^{\beta}+[\bm{v}\cdot\nabla,\theta_{2}\bm{l}_{2}]\zeta^{\beta}\\\
\displaystyle\makebox[47.36096pt]{}-\sum_{l=1}^{n}(\partial_{l}A_{2})\partial_{l}\bm{\phi}_{2}^{\beta}+[\partial^{\beta},A_{2}]\Delta\bm{\phi}_{2}-\partial^{\beta}L_{2}^{\rm
low}(H_{2},b)\bm{\phi}_{2}\biggr{\\}}.\end{cases}$

In the above calculation, we used (7.7). Similarly, applying the differential
operator $\partial^{\beta}$ to the last equation in (7.3), we have

(7.16)
$\displaystyle-\rho_{1}\bm{l}_{1}\cdot(\partial_{t}\bm{\phi}_{1}^{\beta}-\varepsilon\Delta\bm{\phi}_{1}^{\beta}+(\bm{u}\cdot\nabla)\bm{\phi}_{1}^{\beta})+\rho_{1}\theta_{1}\bm{l}_{1}\cdot(\bm{v}\cdot\nabla)\bm{\phi}_{1}^{\beta}$
$\displaystyle\quad+\rho_{2}\bm{l}_{2}\cdot(\partial_{t}\bm{\phi}_{2}^{\beta}-\varepsilon\Delta\bm{\phi}_{2}^{\beta}+(\bm{u}\cdot\nabla)\bm{\phi}_{2}^{\beta})+\rho_{2}\theta_{2}\bm{l}_{2}\cdot(\bm{v}\cdot\nabla)\bm{\phi}_{2}^{\beta}+a\zeta^{\beta}=F_{0,\beta},$

where

$\displaystyle a$
$\displaystyle=\rho_{2}\left(\sum_{i=0}^{N^{*}}p_{i}H_{2}^{p_{i}-1}(G_{2,i}+\bm{u}_{2}\cdot\nabla\phi_{2,i})+\sum_{i=0}^{N^{*}}p_{i}(p_{i}-1)H_{2}^{p_{i}-2}(w_{2}-\bm{u}_{2}\cdot\nabla
b)\phi_{2,i}+g\right)$
$\displaystyle\quad+\rho_{1}\left(\sum_{i=0}^{N}2iH_{1}^{2i-1}(G_{1,i}+\bm{u}_{1}\cdot\nabla\phi_{1,i})-w_{1}\sum_{i=0}^{N}2i(2i-1)H^{2(i-1)}\phi_{1,i}-g\right),$
$\displaystyle F_{0,\beta}=$
$\displaystyle\rho_{1}\Biggl{\\{}\bigl{(}\partial^{\beta}\bm{l}_{1}(H_{1})-(\partial_{H_{1}}\bm{l}_{1}(H_{1}))\partial^{\beta}H_{1}\bigr{)}\cdot\bm{G}_{1}+[\partial^{\beta};\bm{l}_{1}(H_{1})^{\rm
T},\bm{G}_{1}]$
$\displaystyle\qquad+\bm{u}_{1}\cdot\sum_{j=0}^{N}\bigl{(}[\partial^{\beta};l_{1,j}(H_{1}),\nabla\phi_{1,j}]+(\partial^{\beta}l_{1,j}(H_{1})-(\partial_{H_{1}}l_{1,j}(H_{1}))\partial^{\beta}H_{1})\nabla\phi_{1,j}\bigr{)}$
$\displaystyle\qquad-
w_{1}\sum_{j=0}^{N}\bigl{(}[\partial^{\beta},\phi_{1,j}]\partial_{H_{1}}l_{1,j}(H_{1})+(\partial^{\beta}\partial_{H_{1}}l_{1,j}(H_{1})-(\partial_{H_{1}}^{2}l_{1,j}(H_{1}))\partial^{\beta}H_{1})\phi_{1,j}\bigr{)}$
$\displaystyle\qquad+\frac{1}{2}\bigl{(}[\partial^{\beta};\bm{u}_{1}^{\rm
T},\bm{u}_{1}]+[\partial^{\beta};w_{1},w_{1}]\bigr{)}\Biggr{\\}}$
$\displaystyle-\rho_{2}\Biggl{\\{}\bigl{(}\partial^{\beta}\bm{l}_{2}(H_{2})-(\partial_{H_{2}}\bm{l}_{2}(H_{2}))\partial^{\beta}\zeta\bigr{)}\cdot\bm{G}_{2}+[\partial^{\beta};\bm{l}_{2}(H_{2})^{\rm
T},\bm{G}_{2}]$
$\displaystyle\qquad+\bm{u}_{2}\cdot\sum_{j=0}^{N^{*}}\Bigl{(}[\partial^{\beta};l_{2,j}(H_{2}),\nabla\phi_{2,j}]+(\partial^{\beta}l_{2,j}(H_{2})-(\partial_{H_{2}}l_{2,j}(H_{2}))\partial^{\beta}H_{2})\nabla\phi_{2,j}$
$\displaystyle\qquad\qquad-[\partial^{\beta},\partial_{H_{2}}l_{2,j}(H_{2})]\phi_{2,j}\nabla
b-(\partial^{\beta}\partial_{H_{2}}l_{2,j}(H_{2})-(\partial_{H_{2}}^{2}l_{2,j}(H_{2}))\partial^{\beta}H_{2})\phi_{2,j}\nabla
b\Bigr{)}$
$\displaystyle\qquad+w_{2}\sum_{j=0}^{N^{*}}\bigl{(}[\partial^{\beta},\phi_{2,j}]\partial_{H_{2}}l_{2,j}(H_{2})+(\partial^{\beta}\partial_{H_{2}}l_{2,j}(H_{2})-(\partial_{H_{2}}^{2}l_{2,j}(H_{2}))\partial^{\beta}H_{2})\phi_{2,j}\bigr{)}$
$\displaystyle\qquad+\frac{1}{2}\bigl{(}[\partial^{\beta};\bm{u}_{2}^{\rm
T},\bm{u}_{2}]+[\partial^{\beta};w_{2},w_{2}]\bigr{)}\Biggr{\\}}.$

In the above calculation, we used (7.9) and the notation

$\begin{cases}\bm{l}_{1}(H_{1})=(l_{1,0}(H_{1}),l_{1,1}(H_{1}),\ldots,l_{1,N}(H_{1}))^{\rm
T},\\\
\bm{l}_{2}(H_{2})=(l_{2,0}(H_{2}),l_{2,1}(H_{2}),\ldots,l_{2,N^{*}}(H_{2}))^{\rm
T},\end{cases}$

and the notation for the symmetric commutator
$[\partial^{\beta};u,v]=\partial^{\beta}(uv)-(\partial^{\beta}u)v-u(\partial^{\beta}v)$.
We can rewrite (7.15) and (7.16) in a matrix form as

(7.17)
$\mathscr{A}_{1}\left(\partial_{t}\bm{U}^{\beta}-\varepsilon\Delta\bm{U}^{\beta}+(\bm{u}\cdot\nabla)\bm{U}^{\beta}\right)+\mathscr{A}_{0}^{\rm
mod}\bm{U}^{\beta}=\bm{F}_{\beta},$

where

$\bm{U}^{\beta}=\begin{pmatrix}\zeta^{\beta}\\\ \bm{\phi}_{1}^{\beta}\\\
\bm{\phi}_{2}^{\beta}\end{pmatrix},\quad\bm{F}_{\beta}=\begin{pmatrix}F_{0,\beta}\\\
\bm{F}_{1,\beta}\\\ \bm{F}_{2,\beta}\end{pmatrix},$

and

$\displaystyle\mathscr{A}_{1}=\begin{pmatrix}0&-\rho_{1}\bm{l}_{1}^{\rm
T}&\rho_{2}\bm{l}_{2}^{\rm T}\\\ \rho_{1}\bm{l}_{1}&O&O\\\
-\rho_{2}\bm{l}_{2}&O&O\end{pmatrix},$ $\displaystyle\mathscr{A}_{0}^{\rm
mod}=\begin{pmatrix}a&\rho_{1}\theta_{1}\bm{l}_{1}^{\rm
T}(\bm{v}\cdot\nabla)&\rho_{2}\theta_{2}\bm{l}_{2}^{\rm
T}(\bm{v}\cdot\nabla)\\\
(\bm{v}\cdot\nabla)^{*}(\rho_{1}\theta_{1}\bm{l}_{1}\,\cdot\,)&\displaystyle-\sum_{l=1}^{n}\partial_{l}(\rho_{1}A_{1}\partial_{l}\,\cdot\,)&O\\\
(\bm{v}\cdot\nabla)^{*}(\rho_{2}\theta_{2}\bm{l}_{2}\,\cdot\,)&O&\displaystyle-\sum_{l=1}^{n}\partial_{l}(\rho_{2}A_{2}\partial_{l}\,\cdot\,)\end{pmatrix}.$

Here, we note that $\mathscr{A}_{1}$ is a skew-symmetric matrix and
$\mathscr{A}_{0}^{\rm mod}$ is symmetric in $L^{2}(\mathbf{R}^{n})$.
Concerning the positivity of $\mathscr{A}_{0}^{\rm mod}$, we have the
following lemma.

######  Lemma 7.4.

Let $c_{0}$ and $C_{0}$ be positive constants. Then, there exists
$C=C(c_{0},C_{0})>0$ such that if $a,H_{1},H_{2}$, and $\bm{v}$ satisfy

(7.18)
$\begin{cases}\|a\|_{L^{\infty}}+\|(H_{1},H_{2})\|_{L^{\infty}}+\|\bm{v}\|_{L^{\infty}}\leq
C_{0},\\\ H_{1}(\bm{x})\geq c_{0},\quad H_{2}(\bm{x})\geq
c_{0}\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},\end{cases}$

and the stability condition

(7.19)
$a(\bm{x})-\frac{\rho_{1}\rho_{2}}{\rho_{1}H_{2}(\bm{x})\alpha_{2}+\rho_{2}H_{1}(\bm{x})\alpha_{1}}|\bm{v}(\bm{x})|^{2}\geq
c_{0}>0\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},$

then for any
$\dot{\bm{U}}=(\dot{\zeta},\dot{\bm{\phi}}_{1},\dot{\bm{\phi}}_{2})^{\rm T}$,
we have the equivalence

$\displaystyle
C^{-1}\|(\dot{\zeta},\nabla\dot{\bm{\phi}}_{1},\nabla\dot{\bm{\phi}}_{2})\|_{L^{2}}^{2}\leq(\mathscr{A}_{0}^{\rm
mod}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}\leq
C\|(\dot{\zeta},\nabla\dot{\bm{\phi}}_{1},\nabla\dot{\bm{\phi}}_{2})\|_{L^{2}}^{2}.$

###### Proof..

Introducing diagonal matrices $D_{1}(H_{1})$ and $D_{2}(H_{2})$ by

$\begin{cases}D_{1}(H_{1})=\mbox{diag}(1,H_{1}^{2},H_{1}^{4},\ldots,H_{1}^{2N}),\\\
D_{2}(H_{2})=\mbox{diag}(1,H_{2}^{p_{1}},H_{2}^{p_{2}},\ldots,H_{2}^{p_{N^{*}}}),\end{cases}$

we have

$A_{k}(H_{k})=H_{k}D_{k}(H_{k})A_{k,0}D_{k}(H_{k}),\quad k=1,2,$

where $A_{1,0}$ and $A_{2,0}$ are constant matrices defined by

$A_{1,0}=\biggl{(}\frac{1}{2(i+j)+1}\biggr{)}_{0\leq i,j\leq N},\quad
A_{2,0}=\biggl{(}\frac{1}{p_{i}+p_{j}+1}\biggr{)}_{0\leq i,j\leq N^{*}}.$

We have also

$\bm{1}\cdot
D_{k}(H_{k})\bm{\phi}_{k}=\bm{l}_{k}(H_{k})\cdot\bm{\phi}_{k},\quad k=1,2.$

Therefore,

$\displaystyle(\mathscr{A}_{0}^{\rm mod}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$
$\displaystyle=(a\dot{\zeta},\dot{\zeta})_{L^{2}}+\sum_{l=1}^{n}\sum_{k=1,2}(\rho_{k}H_{k}A_{k,0}D_{k}\partial_{l}\dot{\bm{\phi}}_{k},D_{k}\partial_{l}\dot{\bm{\phi}}_{k})_{L^{2}}$
$\displaystyle\quad\;+2\sum_{k=1,2}(\rho_{k}\theta_{k}\bm{l}_{k}\cdot(\bm{v}\cdot\nabla)\dot{\bm{\phi}}_{k},\dot{\zeta})_{L^{2}}$
$\displaystyle=\sum_{l=1}^{n}\sum_{k=1,2}(\rho_{k}H_{k}Q_{k,0}A_{k,0}D_{k}\partial_{l}\dot{\bm{\phi}}_{k},A_{k,0}D_{k}\partial_{l}\dot{\bm{\phi}}_{k})_{L^{2}}$
$\displaystyle\quad\;+(a\dot{\zeta},\dot{\zeta})_{L^{2}}+\sum_{k=1,2}\bigl{\\{}(\rho_{k}H_{k}\alpha_{k}(\bm{l}_{k}\otimes\nabla)^{\rm
T}\dot{\bm{\phi}}_{k},(\bm{l}_{k}\otimes\nabla)^{\rm
T}\dot{\bm{\phi}}_{k})_{L^{2}}$
$\displaystyle\makebox[100.00015pt]{}+2(\rho_{k}\theta_{k}\bm{v}\cdot(\bm{l}_{k}\otimes\nabla)^{\rm
T}\dot{\bm{\phi}}_{k},\dot{\zeta})_{L^{2}}\bigr{\\}}$
$\displaystyle=:I_{1}+I_{2},$

where we used the identity (5.7). Since $Q_{1,0}$ and $Q_{2,0}$ are
nonnegative and in view of

$\displaystyle I_{2}$
$\displaystyle\geq\int_{\mathbf{R}^{n}}\biggl{\\{}a\dot{\zeta}^{2}+\sum_{k=1,2}\bigl{\\{}\rho_{k}H_{k}\alpha_{k}|(\bm{l}_{k}\otimes\nabla)^{\rm
T}\dot{\bm{\phi}}_{k}|^{2}-2\rho_{k}\theta_{k}|\bm{v}||(\bm{l}_{k}\otimes\nabla)^{\rm
T}\dot{\bm{\phi}}_{k}||\dot{\zeta}|\bigr{\\}}\biggr{\\}}{\rm d}\bm{x}$

and the analysis in Section 5.1, we can show the desired equivalence. ∎

######  Lemma 7.5.

Let $g,\rho_{1},\rho_{2},h_{1},h_{2},c_{0},M_{0}$ be positive constants and
$m$ an integer such that $m>\frac{n}{2}+1$. There exist a positive time $T$
and a positive constant $C$ such that if initial data
$(\zeta_{(0)},\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ and bottom topography $b$
satisfy

$\begin{cases}\|(\zeta_{(0)},\nabla\phi_{1,0(0)},\nabla\phi_{2,0(0)})\|_{H^{m}}+\|(\bm{\phi}_{1(0)}^{\prime},\bm{\phi}_{2(0)}^{\prime})\|_{H^{m+1}}+\|b\|_{W^{m+2,\infty}}\leq
M_{0},\\\ h_{1}-\zeta_{(0)}(\bm{x})\geq 2c_{0},\quad
h_{2}+\zeta_{(0)}(\bm{x})-b(\bm{x})\geq
2c_{0}\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},\end{cases}$

the stability condition (7.19) with $c_{0}$ replaced by $2c_{0}$, and the
compatibility conditions (7.5), then for any $\varepsilon\in(0,1]$ the
solution
$(\zeta^{\varepsilon},\bm{\phi}_{1}^{\varepsilon},\bm{\phi}_{2}^{\varepsilon})$
constructed in Lemmas 7.2 and 7.3 satisfies

$\displaystyle\sup_{0\leq t\leq
T}\left(\|(\zeta^{\varepsilon}(t),\nabla\phi_{1,0}^{\varepsilon}(t),\nabla\phi_{2,0}^{\varepsilon}(t))\|_{H^{m}}^{2}+\|(\bm{\phi}_{1}^{\varepsilon\prime},\bm{\phi}_{2}^{\varepsilon\prime})\|_{H^{m+1}}^{2}\right)$
$\displaystyle\makebox[107.63855pt]{}+\varepsilon\int_{0}^{T}\|(\zeta^{\varepsilon}(t),\nabla\bm{\phi}_{1}^{\varepsilon}(t),\nabla\bm{\phi}_{2}^{\varepsilon}(t))\|_{H^{m+1}}^{2}{\rm
d}t\leq C.$

###### Proof..

Once again we simply write $\bm{U}=(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})^{\rm
T}$ in place of
$(\zeta^{\varepsilon},\bm{\phi}_{1}^{\varepsilon},\bm{\phi}_{2}^{\varepsilon})^{\rm
T}$. We define an energy function $\mathscr{E}_{m}(t)$ by

$\mathscr{E}_{m}(t)=\sum_{|\beta|\leq m}\left\\{(\mathscr{A}_{0}^{\rm
mod}\partial^{\beta}\bm{U}(t),\partial^{\beta}\bm{U}(t))_{L^{2}}+\|(\partial^{\beta}\bm{\phi}_{1}^{\prime}(t),\partial^{\beta}\bm{\phi}_{2}^{\prime}(t))\|_{L^{2}}^{2}\right\\}.$

We assume that the solution $(\zeta(t),\bm{\phi}_{1}(t),\bm{\phi}_{2}(t))$
satisfies (7.18) and the stability condition (7.19) for $0\leq t\leq T$. Then,
the energy function $\mathscr{E}_{m}(t)$ is equivalent to

$E_{m}(t)=\|(\zeta(t),\nabla\phi_{1,0}(t),\nabla\phi_{2,0}(t))\|_{H^{m}}^{2}+\|(\bm{\phi}_{1}^{\prime}(t),\bm{\phi}_{2}^{\prime}(t))\|_{H^{m+1}}^{2}.$

Furthermore, we assume that

(7.20) $E_{m}(t)+\varepsilon\int_{0}^{t}E_{m+1}(\tau){\rm d}\tau\leq M_{1}$

for $0\leq t\leq T$, where the constant $M_{1}$ and the time $T$ will be
determined later. In the following we simply write the constants depending
only on $(g,\rho_{1},\rho_{2},h_{1},h_{2},c_{0},C_{0},M_{0})$ by $C_{1}$ and
the constants depending also on $M_{1}$ by $C_{2}$. They may change from line
to line. Then, it holds that

$C_{1}^{-1}E_{j}(t)\leq\mathscr{E}_{j}(t)\leq C_{1}E_{j}(t)$

for $j=0,1,2,\ldots$. We are going to evaluate the evolution of the energy
function $\mathscr{E}_{m}(t)$. To this end, we take the $L^{2}$-inner product
of (7.17) with
$\partial_{t}\bm{U}^{\beta}-\varepsilon\Delta\bm{U}^{\beta}+(\bm{u}\cdot\nabla)\bm{U}^{\beta}$
and use integration by parts to get

$\displaystyle\frac{1}{2}\frac{\rm d}{{\rm d}t}(\mathscr{A}_{0}^{\rm
mod}\bm{U}^{\beta},\bm{U}^{\beta})_{L^{2}}+\varepsilon\sum_{l=1}^{n}(\mathscr{A}_{0}^{\rm
mod}\partial_{l}\bm{U}^{\beta},\partial_{l}\bm{U}^{\beta})_{L^{2}}$
$\displaystyle=\frac{1}{2}([\partial_{t},\mathscr{A}_{0}^{\rm
mod}]\bm{U}^{\beta},\bm{U}^{\beta})_{L^{2}}-\varepsilon\sum_{l=1}^{n}([\partial_{l},\mathscr{A}_{0}^{\rm
mod}]\bm{U}^{\beta},\partial_{l}\bm{U}^{\beta})_{L^{2}}-(\mathscr{A}_{0}^{\rm
mod}\bm{U}^{\beta},(\bm{u}\cdot\nabla)\bm{U}^{\beta})_{L^{2}}$
$\displaystyle\quad\;+(F_{0,\beta},\partial^{\beta}G_{0}+(\bm{u}\cdot\nabla)\zeta^{\beta})_{L^{2}}+\sum_{k=1,2}(\bm{F}_{k,\beta},\partial^{\beta}\bm{G}_{k}+(\bm{u}\cdot\nabla)\bm{\phi}_{k}^{\beta})_{L^{2}}.$

Here, we see that

$\displaystyle([\partial_{t},\mathscr{A}_{0}^{\rm
mod}]\bm{U}^{\beta},\bm{U}^{\beta})_{L^{2}}=((\partial_{t}a)\zeta^{\beta},\zeta^{\beta})_{L^{2}}$
$\displaystyle\quad+2\sum_{k=1,2}\rho_{k}([\partial_{t},\theta_{k}\bm{l}_{k}^{\rm
T}(\bm{v}\cdot\nabla)]\bm{\phi}_{k}^{\beta},\zeta^{\beta})_{L^{2}}+\sum_{l=1}^{n}\sum_{k=1,2}\rho_{k}((\partial_{t}A_{k})\partial_{l}\bm{\phi}_{k}^{\beta},\partial_{l}\bm{\phi}_{k}^{\beta})_{L^{2}},$
$\displaystyle([\partial_{l},\mathscr{A}_{0}^{\rm
mod}]\bm{U}^{\beta},\partial_{l}\bm{U}^{\beta})_{L^{2}}=((\partial_{l}a)\zeta^{\beta},\zeta^{\beta})_{L^{2}}$
$\displaystyle\quad+\sum_{k=1,2}\rho_{k}\left\\{([\partial_{l},\theta_{k}\bm{l}_{k}^{\rm
T}(\bm{v}\cdot\nabla)]\bm{\phi}_{k}^{\beta},\partial_{l}\zeta^{\beta})_{L^{2}}+(\zeta^{\beta},[\partial_{l},\theta_{k}\bm{l}_{k}^{\rm
T}(\bm{v}\cdot\nabla)]\partial_{l}\bm{\phi}_{k}^{\beta})_{L^{2}}\right\\}$
$\displaystyle\quad+\sum_{k=1,2}\sum_{j=1}^{n}\rho_{k}((\partial_{j}A_{k})\partial_{l}\bm{\phi}_{k}^{\beta},\partial_{j}\partial_{l}\bm{\phi}_{k}^{\beta})_{L^{2}},$
$\displaystyle(\mathscr{A}_{0}^{\rm
mod}\bm{U}^{\beta},(\bm{u}\cdot\nabla)\bm{U}^{\beta})_{L^{2}}=-\frac{1}{2}((\nabla\cdot(a\bm{u}))\zeta^{\beta},\zeta^{\beta})_{L^{2}}$
$\displaystyle\quad-\sum_{k=1,2}\rho_{k}\left\\{((\nabla\cdot\bm{u})\zeta^{\beta},\theta_{k}\bm{l}_{k}\cdot(\bm{v}\cdot\nabla)\bm{\phi}_{k}^{\beta})_{L^{2}}+(\zeta^{\beta},[(\bm{u}\cdot\nabla),\theta_{k}\bm{l}_{k}^{\rm
T}(\bm{v}\cdot\nabla)]\bm{\phi}_{k}^{\beta})_{L^{2}}\right\\}$
$\displaystyle\quad-\sum_{k=1,2}\sum_{l=1}^{n}\rho_{k}\left\\{(A_{k}\partial_{l}\bm{\phi}_{k}^{\beta},((\partial_{l}\bm{u})\cdot\nabla)\bm{\phi}_{k}^{\beta})_{L^{2}}+\frac{1}{2}(((\bm{u}\cdot\nabla)^{*}A_{k})\partial_{l}\bm{\phi}_{k}^{\beta},\partial_{l}\bm{\phi}_{k}^{\beta})_{L^{2}}\right\\},$

so that for $1\leq|\beta|\leq m$ we have

$\displaystyle\frac{1}{2}\frac{\rm d}{{\rm d}t}(\mathscr{A}_{0}^{\rm
mod}\bm{U}^{\beta},\bm{U}^{\beta})_{L^{2}}+\varepsilon\sum_{l=1}^{n}(\mathscr{A}_{0}^{\rm
mod}\partial_{l}\bm{U}^{\beta},\partial_{l}\bm{U}^{\beta})_{L^{2}}$
$\displaystyle\leq C_{2}(1+\varepsilon
E_{m+1}(t)^{\frac{1}{2}})+\|F_{0,\beta}\|_{H^{1}}\|\partial^{\beta}G_{0}+(\bm{u}\cdot\nabla)\zeta^{\beta}\|_{H^{-1}}$
$\displaystyle\quad\;+\sum_{k=1,2}\|\bm{F}_{k,\beta}\|_{L^{2}}\|\partial^{\beta}\bm{G}_{k}+(\bm{u}\cdot\nabla)\bm{\phi}_{k}^{\beta}\|_{L^{2}}$
$\displaystyle\leq C_{2}\left(1+\varepsilon E_{m+1}(t)^{\frac{1}{2}}\right).$

Similar estimate can be obtained in the case $|\beta|=0$ more directly. On the
other hand, it follows from (7.9) that

$\displaystyle\frac{1}{2}\frac{\rm d}{{\rm
d}t}\|(\bm{\phi}_{1}^{\beta\prime},\bm{\phi}_{2}^{\beta\prime})\|_{L^{2}}^{2}+\varepsilon\|(\nabla\bm{\phi}_{1}^{\beta\prime},\nabla\bm{\phi}_{2}^{\beta\prime})\|_{L^{2}}^{2}=\sum_{k=1,2}(\partial^{\beta}\bm{G}_{k}^{\prime},\bm{\phi}_{k}^{\beta\prime})_{L^{2}}^{2}\leq
C_{2}.$

Therefore, we obtain

$\frac{\rm d}{{\rm d}t}\mathscr{E}_{m}(t)+\varepsilon E_{m+1}(t)\leq
C_{2}\left(1+\varepsilon E_{m+1}(t)^{\frac{1}{2}}\right),$

which yields

$E_{m}(t)+\varepsilon\int_{0}^{t}E_{m+1}(\tau){\rm d}\tau\leq C_{1}+C_{2}t.$

Putting $M_{1}=2C_{1}$ and taking $T>0$ so that $C_{2}T\leq C_{1}$, we obtain
by a continuity argument that (7.20) holds for $0\leq t\leq T$.

It remains to show that $(\zeta(t),\bm{\phi}_{1}(t),\bm{\phi}_{2}(t))$
satisfies (7.18) and the stability condition (7.19) for $0\leq t\leq T$. By
the Sobolev embedding theorem, (7.7), and (7.9), we see that

(7.21)
$\displaystyle|\zeta(\bm{x},t)-\zeta_{(0)}(\bm{x})|+\sum_{k=1,2}\left(|\nabla\bm{\phi}_{k}(\bm{x},t)-\nabla\bm{\phi}_{k(0)}(\bm{x})|+|\bm{\phi}_{k}^{\prime}(\bm{x},t)-\bm{\phi}_{k(0)}^{\prime}(\bm{x})|\right)$
$\displaystyle\leq
C_{1}\biggl{(}\|\zeta(t)-\zeta_{(0)}\|_{H^{m-1}}+\sum_{k=1,2}\left(\|\nabla\bm{\phi}_{k}(t)-\nabla\bm{\phi}_{k(0)}\|_{H^{m-1}}+\|\bm{\phi}_{k}^{\prime}(t)-\bm{\phi}_{k(0)}^{\prime}\|_{H^{m-1}}\right)\biggr{)}$
$\displaystyle\leq
C_{1}\int_{0}^{t}\biggl{(}\|\partial_{t}\zeta(\tau)\|_{H^{m-1}}+\sum_{k=1,2}\left(\|\nabla\partial_{t}\bm{\phi}_{k}(\tau)\|_{H^{m-1}}+\|\partial_{t}\bm{\phi}_{k}^{\prime}(\tau)\|_{H^{m-1}}\right)\biggr{)}{\rm
d}\tau$ $\displaystyle\leq C_{1}\int_{0}^{t}\left(\|(G_{0},\nabla
G_{1,0},\nabla
G_{2,0})(\tau)\|_{H^{m-1}}+\|(\bm{G}_{1}^{\prime},\bm{G}_{2}^{\prime})(\tau)\|_{H^{m}}+\varepsilon
E_{m+1}(\tau)^{\frac{1}{2}}\right){\rm d}\tau$ $\displaystyle\leq
C_{2}\left(t+\sqrt{\varepsilon t}\right),$

which yields (7.18), except for the estimate for $a$, by taking $T>0$
sufficiently small. We now turn to the stability condition (7.19). In order to
evaluate $\partial_{t}a$, we need to obtain estimates for
$\partial_{t}\bm{G}_{k}^{\prime}$ for $k=1,2$. Differentiating (7.10) with
respect to $t$, we have

$\begin{cases}\mathcal{L}_{1,i}(H_{1})\partial_{t}\bm{G}_{1}=g_{1,i}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2},b)\partial_{t}\bm{G}_{2}=g_{2,i}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\mathcal{L}_{1,0}(H_{1})\partial_{t}\bm{G}_{1}+\mathcal{L}_{2,0}(H_{2},b)\partial_{t}\bm{G}_{2}=\nabla\cdot\bm{g}_{3},\\\
-\rho_{1}\bm{l}_{1}(H_{1})\cdot\partial_{t}\bm{G}_{1}+\rho_{2}\bm{l}_{2}(H_{2})\cdot\partial_{t}\bm{G}_{2}=g_{4},\end{cases}$

where

$\begin{cases}g_{1,i}=-[\partial_{t},\mathcal{L}_{1,i}(H_{1})]\bm{G}_{1}-\partial_{t}\bigl{(}f_{1,i}(\zeta,\bm{\phi}_{1})G_{0}-\varepsilon\widetilde{f}_{1,i}(\zeta,\bm{\phi}_{1})\bigr{)}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
g_{2,i}=-[\partial_{t},\mathcal{L}_{2,i}(H_{2},b)]\bm{G}_{2}-\partial_{t}\bigl{(}f_{2,i}(\zeta,\bm{\phi}_{2},b)G_{0}-\varepsilon\widetilde{f}_{2,i}(\zeta,\bm{\phi}_{2},b)\bigr{)}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\displaystyle\bm{g}_{3}=(\partial_{t}\zeta)\left(-\sum_{j=0}^{N}H_{1}^{2j}\nabla
G_{1,j}+\sum_{j=0}^{N^{*}}(H_{2}^{p_{j}}\nabla
G_{2,j}-p_{j}H_{2}^{p_{j}-1}G_{2,j}\nabla b)\right)\\\
\makebox[22.3888pt]{}+\partial_{t}\bigl{(}\bm{v}G_{0}+\varepsilon\bm{f}_{3}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2},b)\bigr{)},\\\
g_{4}=\rho_{1}[\partial_{t},\bm{l}_{1}(H_{1})^{\rm
T}]\bm{G}_{1}-\rho_{2}[\partial_{t},\bm{l}_{2}(H_{2})^{\rm
T}]\bm{G}_{2}+\partial_{t}F.\end{cases}$

Therefore, by Lemma 6.4 with $k=m-2$ we obtain

$\displaystyle\|(\nabla\partial_{t}G_{1,0},\nabla\partial_{t}G_{2,0})\|_{H^{m-2}}+\|(\partial_{t}\bm{G}_{1}^{\prime},\partial_{t}\bm{G}_{2}^{\prime})\|_{H^{m-1}}$
$\displaystyle\leq
C_{2}\left(\|(\bm{g}_{1},\bm{g}_{2})\|_{H^{m-3}}+\|(\bm{g}_{3},\nabla
g_{4})\|_{H^{m-2}}\right)$ $\displaystyle\leq
C_{2}\left(\|(\partial_{t}\zeta,\nabla\partial_{t}\phi_{1,0},\nabla\partial_{t}\phi_{2,0})\|_{H^{m-1}}+\|(\partial_{t}\bm{\phi}_{1}^{\prime},\partial_{t}\bm{\phi}_{2}^{\prime})\|_{H^{m}}\right).$

On the other hand, it follows from (7.7) and (7.9) that

$\displaystyle\|(\partial_{t}\zeta,\nabla\partial_{t}\phi_{1,0},\nabla\partial_{t}\phi_{2,0})\|_{H^{m-1}}+\|(\partial_{t}\bm{\phi}_{1}^{\prime},\partial_{t}\bm{\phi}_{2}^{\prime})\|_{H^{m}}$
$\displaystyle\leq\|(G_{0},\nabla G_{1,0},\nabla
G_{2,0})\|_{H^{m-1}}+\|(\bm{G}_{1}^{\prime},\bm{G}_{2}^{\prime})\|_{H^{m}}$
$\displaystyle\quad\;+\varepsilon\left(\|(\zeta,\nabla\phi_{1,0},\nabla\phi_{2,0})\|_{H^{m+1}}+\|({\bm{\phi}}_{1}^{\prime},{\bm{\phi}}_{2}^{\prime})\|_{H^{m+2}}\right)$
$\displaystyle\leq C_{2}\left(1+\varepsilon E_{m+1}(t)^{\frac{1}{2}}\right).$

Thus,

$\displaystyle\|\partial_{t}a\|_{H^{m-1}}$ $\displaystyle\leq
C_{2}\left(\|(\partial_{t}\zeta,\nabla\partial_{t}\phi_{1,0},\nabla\partial_{t}\phi_{2,0},\partial_{t}\bm{G}_{1}^{\prime},\partial_{t}\bm{G}_{2}^{\prime})\|_{H^{m-1}}+\|(\partial_{t}{\bm{\phi}}_{1}^{\prime},\partial_{t}{\bm{\phi}}_{2}^{\prime})\|_{H^{m}}\right)$
$\displaystyle\leq C_{2}\left(1+\varepsilon E_{m+1}(t)^{\frac{1}{2}}\right),$

so that

$|a(x,t)-a(x,0)|\leq C_{1}\int_{0}^{t}\|\partial_{t}a(\tau)\|_{H^{m-1}}{\rm
d}\tau\leq C_{2}\left(t+\sqrt{\varepsilon t}\right).$

This together with (7.21) yields (7.18) and the stability condition (7.19) by
taking $T>0$ sufficiently small. This completes the proof. ∎

Once we obtain this kind of uniform estimates, compactness arguments allow to
pass to the limit $\varepsilon\to+0$ in the regularized problem (7.3) and
(7.4). Moreover, using the Bona–Smith technique, we obtain the strong
continuity in time of the solution and the continuity of the flow map, and the
corresponding strong continuity of the time derivatives proceeds from (7.7)
and (7.9) with $\varepsilon=0$. Thus, Theorem 2.1 follows.

##  8 Hamiltonian structure

In this section, we will show that the Kakinuma model (2.14)–(2.16) also
enjoys a Hamiltonian structure analogous to the one exhibited by T. B.
Benjamin and T. J. Bridges [1] on the full interfacial gravity waves. We
remind again that the Kakinuma model can be written simply as

(8.1)
$\begin{cases}\displaystyle\bm{l}_{1}(H_{1})\partial_{t}\zeta+L_{1}(H_{1})\bm{\phi}_{1}=\bm{0},\\\
\displaystyle-\bm{l}_{2}(H_{2})\partial_{t}\zeta+L_{2}(H_{2},b)\bm{\phi}_{2}=\bm{0},\\\
\displaystyle-\rho_{1}\bm{l}_{1}(H_{1})\cdot\partial_{t}\bm{\phi}_{1}+\rho_{2}\bm{l}_{2}(H_{2})\cdot\partial_{t}\bm{\phi}_{2}=F,\end{cases}$

where $\bm{\phi}_{1}=(\phi_{1,0},\phi_{1,1},\ldots,\phi_{1,N})^{\rm T}$,
$\bm{\phi}_{2}=(\phi_{2,0},\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\rm T}$,
$\bm{l}_{k}$ and $L_{k}$ for $k=1,2$ are defined by (6.3) and $F$ is defined
by

(8.2) $\displaystyle F$
$\displaystyle=\rho_{1}\left\\{g\zeta+\frac{1}{2}\left(|(\nabla\Phi_{1}^{\rm
app})|_{z=\zeta}|^{2}+((\partial_{z}\Phi_{1}^{\rm
app})|_{z=\zeta})^{2}\right)\right\\}$
$\displaystyle\quad\;-\rho_{2}\left\\{g\zeta+\frac{1}{2}\left(|(\nabla\Phi_{2}^{\rm
app})|_{z=\zeta}|^{2}+((\partial_{z}\Phi_{2}^{\rm
app})|_{z=\zeta})^{2}\right)\right\\}.$

Here, $\Phi_{1}^{\rm app}$ and $\Phi_{2}^{\rm app}$ are approximate velocity
potentials defined by (1.4).

###  8.1 Hamiltonian

As was expected, the Hamiltonian would be the total energy. In terms of our
variables $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$, the total energy
$\mathscr{E}^{\rm K}$ is given by

(8.3) $\displaystyle\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=\int_{\mathbf{R}^{n}}e^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}){\rm d}\bm{x},$

where the density of the energy $e^{\rm K}=e^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ is given by

(8.4) $\displaystyle e^{\rm K}$
$\displaystyle=\int_{\zeta}^{h_{1}}\frac{1}{2}\rho_{1}\bigl{(}|\nabla\Phi_{1}^{\rm
app}|^{2}+(\partial_{z}\Phi_{1}^{\rm app})^{2}\bigr{)}{\rm
d}z+\int_{-h_{2}+b}^{\zeta}\frac{1}{2}\rho_{2}\bigl{(}|\nabla\Phi_{2}^{\rm
app}|^{2}+(\partial_{z}\Phi_{2}^{\rm app})^{2}\bigr{)}{\rm d}z$
$\displaystyle\qquad+\frac{1}{2}(\rho_{2}-\rho_{1})g\zeta^{2}\biggr{\\}}{\rm
d}\bm{x},$
$\displaystyle=\frac{1}{2}\rho_{1}\sum_{i,j=0}^{N}\biggl{(}\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,i}\cdot\nabla\phi_{1,j}+\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}\phi_{1,i}\phi_{1,j}\biggr{)}$
$\displaystyle\phantom{=\int_{\mathbf{R}^{n}}}+\frac{1}{2}\rho_{2}\sum_{i,j=0}^{N^{*}}\biggl{(}\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,i}\cdot\nabla\phi_{2,j}-\frac{2p_{i}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\phi_{2,i}\nabla
b\cdot\nabla\phi_{2,j}$
$\displaystyle\phantom{\qquad+\frac{1}{2}\rho_{2}\sum}+\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}(1+|\nabla
b|^{2})\phi_{2,i}\phi_{2,j}\biggr{)}+\frac{1}{2}(\rho_{2}-\rho_{1})g\zeta^{2}.$

By integration by parts, we also have

$\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=\int_{\mathbf{R}^{n}}\biggl{(}\frac{1}{2}\rho_{1}L_{1}(H_{1}){\bm{\phi}}_{1}\cdot{\bm{\phi}}_{1}+\frac{1}{2}\rho_{2}L_{2}(H_{2},b){\bm{\phi}}_{2}\cdot{\bm{\phi}}_{2}+\frac{1}{2}(\rho_{2}-\rho_{1})g\zeta^{2}\biggr{)}{\rm
d}\bm{x}.$

In view of the symmetry of the operators $L_{1}(H_{1})$ and $L_{2}(H_{2},b)$,
we can easily calculate the variational derivatives of this energy functional
and obtain

(8.5) $\begin{cases}\delta_{\zeta}\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=-F,\\\
\delta_{\bm{\phi}_{1}}\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=\rho_{1}L_{1}(H_{1})\bm{\phi}_{1},\\\
\delta_{\bm{\phi}_{2}}\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=\rho_{2}L_{2}(H_{2},b)\bm{\phi}_{2}.\end{cases}$

Therefore, the Kakinuma model (8.1) can be written as

(8.6) $\begin{pmatrix}0&\rho_{1}{\bm{l}}_{1}(H_{1})^{\rm
T}&-\rho_{2}{\bm{l}}_{2}(H_{2})^{\rm T}\\\ -\rho_{1}{\bm{l}}_{1}(H_{1})&O&O\\\
\rho_{2}{\bm{l}}_{2}(H_{2})&O&O\end{pmatrix}\partial_{t}\begin{pmatrix}\zeta\\\
{\bm{\phi}}_{1}\\\
{\bm{\phi}}_{2}\end{pmatrix}=\begin{pmatrix}\delta_{\zeta}\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})\\\
\delta_{\bm{\phi}_{1}}\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})\\\
\delta_{\bm{\phi}_{2}}\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})\end{pmatrix}.$

As we will see later, the canonical variables of the Kakinuma model are the
surface elevation $\zeta$ and $\phi$ given by

(8.7) $\phi=\rho_{2}\Phi_{2}^{\rm app}|_{z=\zeta}-\rho_{1}\Phi_{1}^{\rm
app}|_{z=\zeta}=\rho_{2}{\bm{l}}_{2}(H_{2})\cdot\bm{\phi}_{2}-\rho_{1}{\bm{l}}_{1}(H_{1})\cdot\bm{\phi}_{1},$

which is the canonical variable for the full interfacial gravity waves found
by T. B. Benjamin and T. J. Bridges [1] with $(\Phi_{1},\Phi_{2})$ replaced by
$(\Phi_{1}^{\rm app},\Phi_{2}^{\rm app})$. Then, the compatibility conditions
(2.18)–(2.20) and (8.7) are written in the form

(8.8)
$\begin{cases}\mathcal{L}_{1,i}(H_{1})\bm{\phi}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2},b)\bm{\phi}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1}+\mathcal{L}_{2,0}(H_{2},b)\bm{\phi}_{2}=0,\\\
-\rho_{1}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}+\rho_{2}\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}=\phi.\end{cases}$

Therefore, it follows from Lemma 6.4 that once the canonical variables
$(\zeta,\phi)$ are given in an appropriate class of functions,
$\bm{\phi}_{1}^{\prime}=(\phi_{1,1},\ldots,\phi_{1,N})^{\rm
T},\bm{\phi}_{2}^{\prime}=(\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\rm
T},\nabla\phi_{1,0},\nabla\phi_{2,0}$ can be determined uniquely. In other
words, these variables depend on the canonical variables $(\zeta,\phi)$ and
$b$, and furthermore they depend on $\phi$ linearly. Although the solution
$({\bm{\phi}}_{1},{\bm{\phi}}_{2})$ to the above equations is not unique, we
will denote the solution by

$\bm{\phi}_{1}=\bm{S}_{1}(\zeta,b)\phi,\quad\bm{\phi}_{2}=\bm{S}_{2}(\zeta,b)\phi.$

This abbreviation causes no confusion in the following calculations. Since we
will fix $b$, we simply write $\bm{S}_{1}(\zeta)$ and $\bm{S}_{2}(\zeta)$ in
place of $\bm{S}_{1}(\zeta,b)$ and $\bm{S}_{2}(\zeta,b)$ for simplicity. Now,
we define the Hamiltonian to the Kakinuma model by

(8.9) $\mathscr{H}^{\rm K}(\zeta,\phi)=\mathscr{E}^{\rm
K}(\zeta,\bm{S}_{1}(\zeta)\phi,\bm{S}_{2}(\zeta)\phi),$

which is uniquely determined from $(\zeta,\phi)$.

###  8.2 Hamilton’s canonical form

We proceed to show that the Kakinuma model (8.1) is equivalent to Hamilton’s
canonical form with the Hamiltonian defined by (8.9). In the following, we fix
$b\in W^{m,\infty}$ with $m>\frac{n}{2}+1$ and put

$U_{b}^{m}=\\{\zeta\in
H^{m}\,;\,\inf_{\bm{x}\in\mathbf{R}^{n}}(h_{1}-\zeta(\bm{x}))>0\mbox{ and
}\inf_{\bm{x}\in\mathbf{R}^{n}}(h_{2}+\zeta(\bm{x})-b(\bm{x}))>0\\},$

which is an open set in $H^{m}$. We also use the function space
$\mathring{H}^{k}=\\{\phi\,;\,\nabla\phi\in H^{m-1}\\}$. For Banach spaces
$\mathscr{X}$ and $\mathscr{Y}$, we denote by $B(\mathscr{X};\mathscr{Y})$ the
set of all linear and bounded operators from $\mathscr{X}$ into $\mathscr{Y}$.
By Lemma 6.4, we see easily the following lemma.

######  Lemma 8.1.

Let $m$ be an integer such that $m>\frac{n}{2}+1$ and $b\in W^{m,\infty}$. For
each $\zeta\in U^{m}_{b}$ and for $k=1,2,\ldots,m$, the linear operators

$\begin{cases}{\bf
S}_{1}(\zeta):\mathring{H}^{k}\ni\phi\mapsto\bm{\phi}_{1}\in\mathring{H}^{k}\times(H^{k})^{N},\\\
{\bf
S}_{2}(\zeta):\mathring{H}^{k}\ni\phi\mapsto\bm{\phi}_{2}\in\mathring{H}^{k}\times(H^{k})^{N^{*}},\end{cases}$

where $(\bm{\phi}_{1},\bm{\phi}_{2})$ is the solution to (8.8), are defined.
Moreover, we have $\bm{S}_{1}(\zeta)\in
B(\mathring{H}^{k};\mathring{H}^{k}\times(H^{k})^{N})$ and
$\bm{S}_{2}(\zeta)\in
B(\mathring{H}^{k};\mathring{H}^{k}\times(H^{k})^{N^{*}})$.

Formally, $\dot{\bm{\psi}}_{k}=D_{\zeta}\bm{S}_{k}(\zeta)[\dot{\zeta}]\phi$,
the Fréchet derivative of $\bm{S}_{k}(\zeta)\phi$ with respect to $\zeta$
applied to $\dot{\zeta}$ for $k=1,2$ satisfy

(8.10)
$\begin{cases}\mathcal{L}_{1,i}(H_{1})\dot{\bm{\psi}}_{1}=D_{H_{1}}\mathcal{L}_{1,i}(H_{1})[\dot{\zeta}]\bm{\phi}_{1}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2},b)\dot{\bm{\psi}}_{2}=-D_{H_{2}}\mathcal{L}_{2,i}(H_{2},b)[\dot{\zeta}]\bm{\phi}_{2}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\mathcal{L}_{1,0}(H_{1})\dot{\bm{\psi}}_{1}+\mathcal{L}_{2,0}(H_{2},b)\dot{\bm{\psi}}_{2}=D_{H_{1}}\mathcal{L}_{1,0}(H_{1})[\dot{\zeta}]\bm{\phi}_{1}-D_{H_{2}}\mathcal{L}_{2,0}(H_{2},b)[\dot{\zeta}]\bm{\phi}_{2},\\\
-\rho_{1}\bm{l}_{1}(H_{1})\cdot\dot{\bm{\psi}}_{1}+\rho_{2}\bm{l}_{2}(H_{2})\cdot\dot{\bm{\psi}}_{2}=-\bigl{(}\rho_{1}(\partial_{H_{1}}\bm{l}_{1}(H_{1}))\cdot\bm{\phi}_{1}+\rho_{2}(\partial_{H_{2}}\bm{l}_{2}(H_{2}))\cdot\bm{\phi}_{2}\bigr{)}\dot{\zeta}\end{cases}$

with $\bm{\phi}_{j}=\bm{S}_{j}(\zeta)\phi$ for $j=1,2$, where for
$i=1,\ldots,N$,

$\displaystyle
D_{H_{1}}\mathcal{L}_{1,i}(H_{1})[\dot{\zeta}]\bm{\phi}_{1}=\sum_{j=0}^{N}\left(D_{H_{1}}L_{1,ij}(H_{1})[\dot{\zeta}]-H_{1}^{2i}D_{H_{1}}L_{1,0j}(H_{1})[\dot{\zeta}]-2iH_{1}^{2i-1}\dot{\zeta}L_{1,0j}(H_{1})\right)\phi_{1,j},$
$\displaystyle
D_{H_{1}}L_{1,ij}(H_{1})[\dot{\zeta}]\phi_{1,j}=-\nabla\cdot(\dot{\zeta}H_{1}^{2(i+j)}\nabla\phi_{1,j})+4ij\dot{\zeta}H_{1}^{2(i+j-1)}\phi_{1,j},$

and so on. By using these equations together with Lemma 6.4 and standard
arguments, we can justify the Fréchet differentiability of $\bm{S}_{k}(\zeta)$
with respect to $\zeta$ for $k=1,2$. More precisely, we have the following
lemma.

######  Lemma 8.2.

Let $m$ be an integer such that $m>\frac{n}{2}+1$ and $b\in W^{m,\infty}$.
Then, the maps $U^{m}_{b}\ni\zeta\mapsto\bm{S}_{1}(\zeta)\in
B(\mathring{H}^{k};\mathring{H}^{k}\times(H^{k})^{N})$ and
$U^{m}_{b}\ni\zeta\mapsto\bm{S}_{2}(\zeta)\in
B(\mathring{H}^{k};\mathring{H}^{k}\times(H^{k})^{N^{*}})$ are Fréchet
differentiable for $k=1,2,\ldots,m$, and (8.10) holds.

We proceed to calculate the variational derivatives of the Hamiltonian
$\mathscr{H}^{\rm K}(\zeta,\phi)$, which are given by the following lemma.

######  Lemma 8.3.

Let $m$ be an integer such that $m>\frac{n}{2}+1$ and $b\in W^{m,\infty}$.
Then, the map
$U^{m}_{b}\times\mathring{H}^{1}\ni(\zeta,\phi)\mapsto\mathscr{H}^{\rm
K}(\zeta,\phi)\in\mathbf{R}$ is Fréchet differentiable and the variational
derivatives of the Hamiltonian are

$\begin{cases}\delta_{\phi}\mathscr{H}^{\rm
K}(\zeta,\phi)=-\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1},\\\
\delta_{\zeta}\mathscr{H}^{\rm K}(\zeta,\phi)=(\delta_{\zeta}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})+(\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1})\bigl{(}\rho_{1}(\partial_{H_{1}}\bm{l}_{1})(H_{1})\cdot\bm{\phi}_{1}+\rho_{2}(\partial_{H_{2}}\bm{l}_{2})(H_{2})\cdot\bm{\phi}_{2}\bigr{)},\end{cases}$

where $\bm{\phi}_{k}=\bm{S}_{k}(\zeta)$ for $k=1,2$.

###### Proof..

Let us calculate Fréchet derivatives of the Hamiltonian $\mathscr{H}^{\rm
K}(\zeta,\phi)$. Let us consider first $U^{m}_{b}\times
H^{2}\ni(\zeta,\phi)\mapsto\mathscr{H}^{\rm K}(\zeta,\phi)$. For any
$\dot{\phi}\in H^{2}$, we see that

$\displaystyle D_{\phi}\mathscr{H}^{\rm K}(\zeta,\phi)[\dot{\phi}]$
$\displaystyle=(D_{\bm{\phi}_{1}}\mathscr{E}^{\rm
K})(\zeta,\bm{S}_{1}(\zeta)\phi,\bm{S}_{2}(\zeta)\phi)[\bm{S}_{1}(\zeta)\dot{\phi}]+(D_{\bm{\phi}_{2}}\mathscr{E}^{\rm
K})(\zeta,\bm{S}_{1}(\zeta)\phi,\bm{S}_{2}(\zeta)\phi)[\bm{S}_{2}(\zeta)\dot{\phi}]$
$\displaystyle=((\delta_{\bm{\phi}_{1}}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}),\bm{S}_{1}(\zeta)\dot{\phi})_{L^{2}}+((\delta_{\bm{\phi}_{2}}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}),\bm{S}_{2}(\zeta)\dot{\phi})_{L^{2}}$
$\displaystyle=(\rho_{1}L_{1}(H_{1})\bm{\phi}_{1},\bm{S}_{1}(\zeta)\dot{\phi})_{L^{2}}+(\rho_{2}L_{2}(H_{2},b)\bm{\phi}_{2},\bm{S}_{2}(\zeta)\dot{\phi})_{L^{2}}$
$\displaystyle=(\rho_{1}\bm{l}_{1}(H_{1})\bigl{(}\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1}\bigr{)},\bm{S}_{1}(\zeta)\dot{\phi})_{L^{2}}-(\rho_{2}\bm{l}_{2}(H_{2})\bigl{(}\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1}\bigr{)},\bm{S}_{2}(\zeta)\dot{\phi})_{L^{2}}$
$\displaystyle=(\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1},\rho_{1}\bm{l}_{1}(H_{1})\cdot\bm{S}_{1}(\zeta)\dot{\phi}-\rho_{2}\bm{l}_{2}(H_{2})\cdot\bm{S}_{2}(\zeta)\dot{\phi})_{L^{2}}$
$\displaystyle=-(\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1},\dot{\phi})_{L^{2}},$

where we used (8.5) and Lemma 8.1. The above calculations are also valid when
$(\phi,\dot{\phi})\in\mathring{H}^{1}\times\mathring{H}^{1}$, provided we
replace the $L^{2}$ inner products with the
$\mathscr{X}^{\prime}$–$\mathscr{X}$ duality product where
$\mathscr{X}=\mathring{H}^{1}\times(H^{1})^{N}$ or
$\mathscr{X}=\mathring{H}^{1}\times(H^{1})^{N^{*}}$ for the first lines, and
$\mathscr{X}=\mathring{H}^{1}$ for the last line. This gives the first
equation of the lemma.

Similarly, for any $(\zeta,\phi)\in U_{b}^{m}\times\mathring{H}^{2}$ and
$\dot{\zeta}\in H^{m}$ we see that

$\displaystyle D_{\zeta}\mathscr{H}^{\rm K}(\zeta,\phi)[\dot{\zeta}]$
$\displaystyle=(D_{\zeta}\mathscr{E}^{\rm
K})(\zeta,\bm{S}_{1}(\zeta)\phi,\bm{S}_{2}(\zeta)\phi)[\dot{\zeta}]+(D_{\bm{\phi}_{1}}\mathscr{E}^{\rm
K})(\zeta,\bm{S}_{1}(\zeta)\phi,\bm{S}_{2}(\zeta)\phi)[D_{\zeta}\bm{S}_{1}(\zeta)[\dot{\zeta}]\phi]$
$\displaystyle\quad\;+(D_{\bm{\phi}_{2}}\mathscr{E}^{\rm
K})(\zeta,\bm{S}_{1}(\zeta)\phi,\bm{S}_{2}(\zeta)\phi)[D_{\zeta}\bm{S}_{2}(\zeta)[\dot{\zeta}]\phi]$
$\displaystyle=((\delta_{\zeta}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}),\dot{\zeta})_{L^{2}}+((\delta_{\bm{\phi}_{1}}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}),D_{\zeta}\bm{S}_{1}(\zeta)[\dot{\zeta}]\phi)_{L^{2}}$
$\displaystyle\quad\;+((\delta_{\bm{\phi}_{2}}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}),D_{\zeta}\bm{S}_{2}(\zeta)[\dot{\zeta}]\phi)_{L^{2}}.$

Here, we have

$\displaystyle((\delta_{\bm{\phi}_{1}}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}),D_{\zeta}\bm{S}_{1}(\zeta)[\dot{\zeta}]\phi)_{L^{2}}+((\delta_{\bm{\phi}_{2}}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1},\bm{\phi}_{2}),D_{\zeta}\bm{S}_{2}(\zeta)[\dot{\zeta}]\phi)_{L^{2}}$
$\displaystyle=(\rho_{1}L_{1}(H_{1})\bm{\phi}_{1},D_{\zeta}\bm{S}_{1}(\zeta)[\dot{\zeta}]\phi)_{L^{2}}+(\rho_{2}L_{2}(H_{2},b)\bm{\phi}_{2},D_{\zeta}\bm{S}_{2}(\zeta)[\dot{\zeta}]\phi)_{L^{2}}$
$\displaystyle=(\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1},\rho_{1}\bm{l}_{1}(H_{1})\cdot
D_{\zeta}\bm{S}_{1}(\zeta)[\dot{\zeta}]\phi-\rho_{2}\bm{l}_{2}(H_{2})\cdot
D_{\zeta}\bm{S}_{2}(\zeta)[\dot{\zeta}]\phi)_{L^{2}}$
$\displaystyle=(\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1},(\rho_{1}(\partial_{H_{1}}\bm{l}_{1})(H_{1})\cdot\bm{\phi}_{1}+\rho_{2}(\partial_{H_{2}}\bm{l}_{2})(H_{2})\cdot\bm{\phi}_{2})\dot{\zeta})_{L^{2}}$
$\displaystyle=((\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1})(\rho_{1}(\partial_{H_{1}}\bm{l}_{1})(H_{1})\cdot\bm{\phi}_{1}+\rho_{2}(\partial_{H_{2}}\bm{l}_{2})(H_{2})\cdot\bm{\phi}_{2}),\dot{\zeta})_{L^{2}},$

where we used the identity

$\displaystyle\rho_{1}\bm{l}_{1}(H_{1})\cdot
D_{\zeta}\bm{S}_{1}(\zeta)[\dot{\zeta}]\phi-\rho_{2}\bm{l}_{2}(H_{2})\cdot
D_{\zeta}\bm{S}_{2}(\zeta)[\dot{\zeta}]\phi$
$\displaystyle=\left(\rho_{1}(\partial_{H_{1}}\bm{l}_{1})(H_{1})\cdot\bm{\phi}_{1}+\rho_{2}(\partial_{H_{2}}\bm{l}_{2})(H_{2})\cdot\bm{\phi}_{2}\right)\dot{\zeta},$

stemming from (8.10). Again, the above identities are still valid for
$(\zeta,\phi)\in U_{b}^{m}\times\mathring{H}^{1}$ provided we replace the
$L^{2}$ inner products with suitable duality products. This concludes the
proof of the Fréchet differentiability, and the second equation of the lemma.
∎

Now, we are ready to show another main result in this paper.

######  Theorem 8.4.

Let $m$ be an integer such that $m>\frac{n}{2}+1$ and $b\in W^{m,\infty}$.
Then, the Kakinuma model (2.14)–(2.16) is equivalent to Hamilton’s canonical
equations

(8.11) $\partial_{t}\zeta=\frac{\delta\mathscr{H}^{\rm
K}}{\delta\phi},\quad\partial_{t}\phi=-\frac{\delta\mathscr{H}^{\rm
K}}{\delta\zeta},$

with $\mathscr{H}^{\rm K}$ defined by (8.9) as long as $\zeta(\cdot,t)\in
U^{m}_{b}$ and $\phi(\cdot,t)\in\mathring{H}^{1}$. More precisely, for any
regular solution $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ to the Kakinuma model
(2.14)–(2.16), if we define $\phi$ by (8.7), then $(\zeta,\phi)$ satisfies
Hamilton’s canonical equations (8.11). Conversely, for any regular solution
$(\zeta,\phi)$ to Hamilton’s canonical equations (8.11), if we define
$\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ by $\bm{\phi}_{k}=\bm{S}_{k}(\zeta)\phi$
for $k=1,2$, then $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ satisfies the Kakinuma
model (2.14)–(2.16).

###### Proof..

Suppose that $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ is a solution to the
Kakinuma model (2.14)–(2.16). Then, it satisfies (8.6), and in particular

(8.12) $\partial_{t}\zeta=-\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1}.$

Moreover, it follows from (8.7) and (8.6) that

$\displaystyle\partial_{t}\phi$
$\displaystyle=\rho_{2}\bm{l}_{2}(H_{2})\cdot\partial_{t}\bm{\phi}_{2}-\rho_{1}\bm{l}_{1}(H_{1})\cdot\partial_{t}\bm{\phi}_{1}+\left(\rho_{2}(\partial_{H_{2}}\bm{l}_{2}(H_{2}))\cdot\bm{\phi}_{2}+\rho_{1}(\partial_{H_{1}}\bm{l}_{1}(H_{1}))\cdot\bm{\phi}_{1}\right)\partial_{t}\zeta$
$\displaystyle=-(\delta_{\zeta}\mathscr{E}^{\rm
K})(\zeta,\bm{\phi}_{1}\bm{\phi}_{2})-(\mathcal{L}_{1,0}(H_{1})\bm{\phi}_{1})\left(\rho_{1}(\partial_{H_{1}}\bm{l}_{1}(H_{1}))\cdot\bm{\phi}_{1}+\rho_{2}(\partial_{H_{2}}\bm{l}_{2}(H_{2}))\cdot\bm{\phi}_{2}\right).$

These equations together with Lemma 8.3 show that $(\zeta,\phi)$ satisfies
(8.11).

Conversely, suppose that $(\zeta,\phi)$ satisfies Hamilton’s canonical
equations (8.11) and put $\bm{\phi}_{k}=\bm{S}_{k}(\zeta)\phi$ for $k=1,2$.
Then, it follows from (8.11) and Lemma 8.3 that we have (8.12). This fact and
Lemma 8.1 imply the equations

$\begin{cases}\bm{l}_{1}(H_{1})\partial_{t}\zeta+L_{1}(H_{1})\bm{\phi}_{1}=\bm{0},\\\
-\bm{l}_{2}(H_{2})\partial_{t}\zeta+L_{2}(H_{2},b)\bm{\phi}_{2}=\bm{0}.\end{cases}$

We see also that

$\displaystyle-\rho_{1}\bm{l}_{1}(H_{1})\cdot\partial_{t}\bm{\phi}_{1}+\rho_{2}\bm{l}_{2}(H_{2})\cdot\partial_{t}\bm{\phi}_{2}$
$\displaystyle=\partial_{t}\phi-\left(\rho_{1}(\partial_{H_{1}}\bm{l}_{1})(H_{1})\cdot\bm{\phi}_{1}+\rho_{2}(\partial_{H_{2}}\bm{l}_{2})(H_{2})\cdot\bm{\phi}_{2}\right)\partial_{t}\zeta$
$\displaystyle=-\delta_{\zeta}\mathscr{E}^{\rm
K}(\zeta,\bm{\phi}_{1}\bm{\phi}_{2})=F,$

where we used (8.11), (8.12), Lemma 8.3 and (8.5). Therefore,
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ satisfies (8.1), that is, the Kakinuma
model (2.14)–(2.16). ∎

##  9 Conservation laws

The Kakinuma model (2.14)–(2.16) has conservative quantities: the excess of
mass $\int_{\mathbf{R}^{n}}\zeta{\rm d}\bm{x}$ and the total energy
$\mathscr{E}^{\rm K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ given by (8.3).
Moreover, in the case of the flat bottom in the lower layer, the momentum
given by

$\displaystyle\mathscr{M}^{\rm K}(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$
$\displaystyle=\int\\!\\!\\!\int_{\Omega_{1}(t)}\rho_{1}\nabla\Phi_{1}^{\rm
app}{\rm d}\bm{x}{\rm
d}z+\int\\!\\!\\!\int_{\Omega_{2}(t)}\rho_{2}\nabla\Phi_{2}^{\rm app}{\rm
d}\bm{x}{\rm d}z$
$\displaystyle=\int_{\mathbf{R}^{n}}\zeta\nabla(-\rho_{1}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}+\rho_{2}\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}){\rm
d}\bm{x}$ $\displaystyle=\int_{\mathbf{R}^{n}}\zeta\nabla\phi{\rm d}\bm{x}$

is also conserved for the Kakinuma model. Here, we give also the corresponding
flux functions to these conservative quantities.

We have two forms of conservation of mass by (2.14) and (2.15) with $i=0$,
that is,

(9.1)
$\displaystyle\partial_{t}\zeta+\nabla\cdot\sum_{j=0}^{N}\left(-\frac{1}{2j+1}H_{1}^{2j+1}\nabla\phi_{1,j}\right)=0,$
(9.2)
$\displaystyle\partial_{t}\zeta+\nabla\cdot\sum_{j=0}^{N^{*}}\left(\frac{1}{p_{j}+1}H_{2}^{p_{j}+1}\nabla\phi_{2,j}-\frac{p_{j}}{p_{j}}H_{2}^{p_{j}}\phi_{2,j}\nabla
b\right)=0.$

######  Proposition 9.1.

Any regular solution $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ to the Kakinuma
model (2.14)–(2.16) satisfies the conservation of energy

$\partial_{t}e^{\rm K}+\nabla\cdot\bm{f}_{e}^{\rm K}=0,$

where the energy density $e^{\rm K}$ is defined by (8.4) and the corresponding
flux $\bm{f}_{e}^{\rm K}$ is given by

$\displaystyle\bm{f}_{e}^{\rm K}$
$\displaystyle=\rho_{1}\sum_{i,j=0}^{N}\left(-\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,j}\right)(\partial_{t}\phi_{1,i})$
$\displaystyle\quad\;+\rho_{2}\sum_{i,j=0}^{N^{*}}\left(-\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,j}+\frac{p_{j}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\phi_{2,j}\nabla
b\right)(\partial_{t}\phi_{2,i}).$

###### Proof..

By using $F$ defined by (8.2), we see that

$\displaystyle\partial_{t}e^{\rm K}$ $\displaystyle=-F\partial_{t}\zeta$
$\displaystyle\quad\;+\rho_{1}\sum_{i,j}^{N}\left(\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,j}\cdot\nabla\partial_{t}\phi_{1,i}+\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}\phi_{1,j}\partial_{t}\phi_{1,i}\right)$
$\displaystyle\quad\;+\rho_{2}\sum_{i,j=0}^{N^{*}}\left\\{\left(\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,j}-\frac{p_{j}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\phi_{2,j}\nabla
b\right)\cdot\nabla\partial_{t}\phi_{2,i}\right.$
$\displaystyle\qquad\quad+\left.\left(-\frac{p_{i}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\nabla
b\cdot\nabla\phi_{2,j}+\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}(1+|\nabla
b|^{2})\phi_{2,j}\right)\partial_{t}\phi_{2,i}\right\\}$
$\displaystyle=-F\partial_{t}\zeta-\nabla\cdot\bm{f}_{e}^{\rm
K}+\rho_{1}L_{1}(H_{1})\bm{\phi}_{1}\cdot\partial_{t}\bm{\phi}_{1}+\rho_{2}L_{2}(H_{2},b)\bm{\phi}_{2}\cdot\partial_{t}\bm{\phi}_{2},$

so that, by (8.1),

$\displaystyle\partial_{t}e^{\rm K}+\nabla\cdot\bm{f}_{e}^{\rm K}$
$\displaystyle=-F\partial_{t}\zeta+\rho_{1}L_{1}(H_{1})\bm{\phi}_{1}\cdot\partial_{t}\bm{\phi}_{1}+\rho_{2}L_{2}(H_{2},b)\bm{\phi}_{2}\cdot\partial_{t}\bm{\phi}_{2}$
$\displaystyle=\left(-F-\rho_{1}\bm{l}_{1}(H_{1})\cdot\partial_{t}\bm{\phi}_{1}+\rho_{2}\bm{l}_{2}(H_{2})\cdot\partial_{t}\bm{\phi}_{2}\right)\partial_{t}\zeta$
$\displaystyle=0,$

which is the desired identity. ∎

######  Proposition 9.2.

Suppose that the bottom in the lower layer is flat, that is, $b=0$. Then, any
regular solution $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ to the Kakinuma model
(2.14)–(2.16) satisfies the conservation of momentum

$\partial_{t}\bm{m}^{\rm K}+\nabla\cdot F_{\bm{m}}^{\rm K}=0,$

where the momentum density $\bm{m}^{\rm K}$ and the corresponding flux matrix
$F_{m}^{\rm K}$ are given by

$\displaystyle\bm{m}^{\rm K}$
$\displaystyle=\zeta\nabla\phi=\zeta\nabla(\rho_{2}\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}-\rho_{1}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}),$
$\displaystyle F_{\bm{m}}^{\rm K}$
$\displaystyle=-\left(\zeta\partial_{t}(\rho_{2}\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}-\rho_{1}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1})+e^{\rm
K}\right){\rm Id}$
$\displaystyle\quad\;+\rho_{1}\sum_{i,j=0}^{N}\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,i}\otimes\nabla\phi_{1,j}$
$\displaystyle\quad\;+\rho_{1}\sum_{i,j=0}^{N^{*}}\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,i}\otimes\nabla\phi_{2,j}.$

###### Proof..

For $l=1,2,\ldots,n$, we see by (8.1) that

$\displaystyle\partial_{t}(\zeta\partial_{l}\phi)-\partial_{l}(\zeta\partial_{t}\phi)$
$\displaystyle=(\partial_{t}\zeta)\left(\rho_{2}\bm{l}_{2}(H_{2})\cdot\partial_{l}\bm{\phi}_{2}-\rho_{1}\bm{l}_{1}(H_{1})\cdot\partial_{l}\bm{\phi}_{1}\right)$
$\displaystyle\quad\;-(\partial_{l}\zeta)\left(\rho_{2}\bm{l}_{2}(H_{2})\cdot\partial_{t}\bm{\phi}_{2}-\rho_{1}\bm{l}_{1}(H_{1})\cdot\partial_{t}\bm{\phi}_{1}\right)$
$\displaystyle=\rho_{2}L_{2}(H_{2},0)\bm{\phi}_{2}\cdot\partial_{l}\bm{\phi}_{2}+\rho_{1}L_{1}(H_{1})\bm{\phi}_{1}\cdot\partial_{l}\bm{\phi}_{1}-(\partial_{l}\zeta)F$
$\displaystyle=-\nabla\cdot\left\\{\rho_{1}\sum_{i,j=0}^{N}\left(\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,i}\right)\partial_{l}\phi_{1,j}\right.$
$\displaystyle\qquad\qquad\left.+\rho_{2}\sum_{i,j=0}^{N^{*}}\left(\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,i}\right)\partial_{l}\phi_{2,j}\right\\}+R_{1},$

where $F$ is given by (8.2) and

$\displaystyle R_{1}$
$\displaystyle=\rho_{1}\sum_{i,j=0}^{N}\left(\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,i}\cdot\nabla\partial_{l}\phi_{1,j}+\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}\phi_{1,i}\partial_{l}\phi_{1,j}\right)$
$\displaystyle\quad\;+\rho_{2}\sum_{i,j=0}^{N^{*}}\left(\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,i}\cdot\nabla\partial_{l}\phi_{2,j}+\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}\phi_{2,i}\partial_{l}\phi_{2,j}\right)$
$\displaystyle\quad\;-(\partial_{l}\zeta)F$
$\displaystyle=\partial_{l}\left\\{\frac{1}{2}\rho_{1}\sum_{i,j=0}^{N}\left(\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,i}\cdot\nabla\phi_{1,j}+\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}\phi_{1,i}\phi_{1,j}\right)\right.$
$\displaystyle\qquad\left.+\frac{1}{2}\rho_{2}\sum_{i,j=0}^{N^{*}}\left(\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,i}\cdot\nabla\phi_{2,j}+\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}\phi_{2,i}\phi_{2,j}\right)\right\\}$
$\displaystyle\quad\;+R_{2}.$

Here, we have

$\displaystyle R_{2}$
$\displaystyle=\frac{1}{2}\rho_{1}\sum_{i,j=0}^{N}\left(H_{1}^{2(i+j)}\nabla\phi_{1,i}\cdot\nabla\phi_{1,j}+4ijH_{1}^{2(i+j-1)}\phi_{1,i}\phi_{1,j}\right)\partial_{l}\zeta$
$\displaystyle\quad\;-\frac{1}{2}\rho_{2}\sum_{i,j=0}^{N^{*}}\left(H_{2}^{p_{i}+p_{j}}\nabla\phi_{2,i}\cdot\nabla\phi_{2,j}+p_{i}p_{j}H_{2}^{p_{i}+p_{j}-2}\phi_{2,i}\phi_{2,j}\right)\partial_{l}\zeta-F\partial_{l}\zeta$
$\displaystyle=(\rho_{2}-\rho_{1})g\zeta\partial_{l}\zeta=\partial_{l}\left(\frac{1}{2}(\rho_{2}-\rho_{1})g\zeta^{2}\right),$

so that $R_{1}=\partial_{l}e^{\rm K}$. These identities yield the desired one.
∎

## References

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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>


# Timing and Synchronization of the
DUNE Neutrino Detector

D. Cussans  on behalf of the DUNE collaboration.  University of Bristol, UK

###### Abstract

The DUNE neutrino experiment far detector has a fiducial mass of
$40\text{\,}\mathrm{kt}$. The O(1 M) readout channels are distributed over the
4 x 10kt modules and need to be synchronized to O( $10\text{\,}\mathrm{ns}$ )
with a reliable, simple, affordable system. For the majority of channels a
simple DC-balanced protocol is used, with clock and synchronization
information encoded on the same fibre. The remaining channels use “White
Rabbit” (IEEE-1588). Small scale tests show a timing jitter of
<$100\text{\,}\mathrm{ps}$. The DUNE timing system has been successfully
prototyped at the ProtoDUNE-SP detector at CERN.

###### keywords:

timing , synchronization , neutrino detection , IEEE-1588

††journal: Nuclear Instruments and Methods in Physics Research, Section A

##  1 The DUNE Experiment

The Deep Underground Neutrino Experiment (DUNE)[1] will detect neutrinos
generated 1,300km away in Fermilab, near Chicago. The far detector will
consist of four modules, each with 10kt fiducial mass of liquid Argon, located
1.5km underground at the Sanford Underground Research Facility (SURF).

The energy deposited by the products of neutrino interactions will be detected
in two ways: ionization drifted by an applied electric field to pick-up
electrodes, forming a time projection chamber (TPC), and scintillation light.

In order to reconstruct events with the photon detection system readout
channels must be synchronized to each other with a precision of nanoseconds.
To correlate events in the far detector with neutrino generation at Fermilab
the entire detector must be synchronized to GPS time to a precision of O(
$100\text{\,}\mathrm{ns}$ ).

The first module will be a “single phase” design where charge is drifted
directly onto an anode plane consisting of a grid of wires. Single phase
module(s) will use the synchronization protocol described here and tested at
protoDUNE-SP.

At least one additional subsequent module will be a “dual phase” design where
charge is drifted upwards through the liquid into the gas phase where charge
multiplication is used to amplify the signal. Dual phase modules will use the
White Rabbit implementation of IEEE-1588[2].

##  2 protocol

A continuous stream of 8b/10b[3] encoded serial data is transmitted over a
1000Base-Bx[4] optical link from the timing master to timing “end points”. In
DUNE a 62.5MHz clock will be derived from a
$312\text{\,}\mathrm{M}\mathrm{bit}\mathrm{/}\mathrm{s}$ data stream. In
ProtoDUNE a 50MHz clock and
$250\text{\,}\mathrm{M}\mathrm{bit}\mathrm{/}\mathrm{s}$ data stream were
used. The data carries commands that are used to synchronize a 64-bit time-
stamp counter in the end-point and propagate commands and synchronization
messages. This allows the readout clocks, calibration signals, etc. to be
synchronized throughout each module.

Each link from the timing master is bidirectional with data transmitted in
both directions down a single fibre with different light wavelengths.
Normally, data are only transmitted from the timing master to the endpoints.
However, the data can be echoed by the end points and transmitted back to the
timing master. This allows the timing master to measure the latency between
timing master and end point and adjust the clock phase and time-stamp at the
end point. This allows all end points to operate with the same time-stamp and
clock phase. For links where a single fibre from the timing master is
passively split to many end points only one end point at a time can transmit
back to the timing master. The end points can also report their status back to
the timing master.

The protocol allows all endpoints to be addressed, groups of endpoints
(partitions) or individual endpoints

Two types of messages are defined, fixed length messages, which have fixed
latency transmission from timing master to end-point, and variable length
messages. Figure 1 shows examples of fixed and variable length messages.

Figure 1: Example DUNE timing system messages. Fixed length and variable
length

The DUNE timing system protocol has been designed to be simple and robust. It
allows the use of passive optical splitting, which reduces cost by allowing a
single fibre to serve multiple end-points. It also allows passive optical
combining which permits hot-swap redundant timing masters.

Because of its simplicity, the end-point circuitry can be easily implemented
with an FPGA and few external components. The FPGA logic resources used are
very much smaller than are needed for timing protocols that require the use of
a micro-processor, such as IEEE-1588.

##  3 Laboratory Tests

Timing jitter performance was measured in the laboratory by connecting a
prototype timing master with a prototype endpoint by optical fibre. The clock
signals in the master and endpoint were compared. Timing jitter depends on the
bandwidth programmed into the PLL clock generators used but is always less
than $100\text{\,}\mathrm{ps}$ and typically less than
$20\text{\,}\mathrm{ps}$ . A block diagram of the apparatus used to test
cycle-to-cycle timing jitter is shown in figure 2. The distribution of the
time difference between clock edges at the timing master and timing endpoint
is shown in figure 3. In other tests over 100 endpoints were synchronized.

Figure 2: Block diagram of jitter performance test setup Figure 3:
Distribution of time difference between clock edges at timing master and
endpoint

##  4 Beam Test (ProtoDUNE)

The DUNE timing system was used to synchronize the ProtoDUNE-SP experiment
performed at the CERN Neutrino Platform. An AIDA-2020[5] Trigger Logic Unit
was used to accept accelerator synchronization signals and generate a timing
data stream that was then fanned out to the TPC and photon detector readout
systems. Figure 4 shows the timing system hardware installed in ProtoDUNE.

Figure 4: ProtoDUNE Timing System hardware installed at the CERN Neutrino
Platform and waiting cabling. The AIDA-2020 TLU with blue front panel is above
three active fanout units with red front panels.

As will be done in DUNE, active fanout units were followed by passive optical
splitter/combiner units allowing each fibre from the master timing system to
connect to multiple timing end-points.

Features such as measuring the master to endpoint latency were also tested.

##  5 Implementation at DUNE

The DUNE timing system will be made largely out of commercially available
components. MicroTCA crates will house commercial double width AMC boards each
carrying two custom FMCs. In each crate the link to the GPS receivers will be
done by a single custom AMC. Figure 5 illustrates the overall layout of the
DUNE timing system. Figure 6 shows the arrangement of modules in each of the
two microTCA crates.

The ability of the DUNE timing system protocol to be used with passive optical
splitting and combining will be used to provide hot-swap redundancy to the
system. The two completely independent master timing systems will allow
firmware and software updates without interrupting system running. Two
independent antennae will be used: One at the top each of the two access
shafts to SURF. This, together with a rubidium atomic clock in the master
timing system will enable failure of the link to GPS time to be detected.

Figure 5: Schematic view of the DUNE timing system. Figure 6: Block diagram of
timing system uTCA crate.

## References

## References

  * [1] The DUNE far detector interim design report, volumes 1–3arXiv:1807.10334,1807.10327,1807.10340.
  * [2] 1588-2008 - IEEE standard for a precision clock synchronization protocol for networked measurement and control systemsdoi:10.1109/IEEESTD.2008.4579760.
  * [3] A. X. Widmer, P. A. Franaszek, A DC-balanced, partitioned-block, 8b/10b transmission code., IBM Journal of Research and Development. 27 (5) (1983) 440––451.  doi:10.1147/rd.275.0440.
  * [4] 802.3-2018 - IEEE standard for ethernetdoi:10.1109/IEEESTD.2018.8457469.
  * [5] AIDA-2020: Advanced European Infrastructures for Detectors at Accelerators (European Union Horizon 2020 research and innovation programme, grant agreement no. 654168.).