A mathematical analysis of the Kakinuma model
for interfacial gravity waves.
Part II: Justification as a shallow water approximation

Vincent Duchêne and Tatsuo Iguchi

###### Abstract

We consider the Kakinuma model for the motion of interfacial gravity waves.
The Kakinuma model is a system of Euler–Lagrange equations for an approximate
Lagrangian, which is obtained by approximating the velocity potentials in the
Lagrangian of the full model. Structures of the Kakinuma model and the well-
posedness of its initial value problem were analyzed in the companion paper
[3]. In this present paper, we show that the Kakinuma model is a higher order
shallow water approximation to the full model for interfacial gravity waves
with an error of order $O(\delta_{1}^{4N+2}+\delta_{2}^{4N+2})$ in the sense
of consistency, where $\delta_{1}$ and $\delta_{2}$ are shallowness
parameters, which are the ratios of the mean thicknesses of the upper and the
lower layers to the typical horizontal wavelength, respectively, and $N$ is,
roughly speaking, the size of the Kakinuma model and can be taken an
arbitrarily large number. Moreover, under a hypothesis of the existence of the
solution to the full model with a uniform bound, a rigorous justification of
the Kakinuma model is proved by giving an error estimate between the solution
to the Kakinuma model and that of the full model. An error estimate between
the Hamiltonian of the Kakinuma model and that of the full model is also
provided.

##  1 Introduction

We will consider the motion of the interfacial gravity waves at the interface
between two layers of immiscible waters in $(n+1)$-dimensional Euclidean
space. Let $t$ be the time, $\bm{x}=(x_{1},\ldots,x_{n})$ the horizontal
spatial coordinates, and $z$ the vertical spatial coordinate. We assume that
the layers are infinite in the horizontal directions, bounded from above by a
flat rigid-lid, and from below by a time-independent variable topography. The
interface, the rigid-lid, and the bottom are represented as
$z=\zeta(\bm{x},t)$, $z=h_{1}$, and $z=-h_{2}+b(\bm{x})$, respectively, where
$\zeta=\zeta(\bm{x},t)$ is the elevation of the interface, $h_{1}$ and $h_{2}$
are mean thicknesses of the upper and lower layers, and $b=b(\bm{x})$
represents the bottom topography. See Figure 1.1.

$\bm{x}$$z$$\zeta(\bm{x},t)$$h_{1}$$h_{2}$$\rho_{1}$$\rho_{2}$

Figure 1.1: Internal gravity waves

We assume that the waters in the upper and the lower layers are both
incompressible and inviscid fluids with constant densities $\rho_{1}$ and
$\rho_{2}$, respectively, and that the flows are both irrotational. Then, the
motion of the waters is described by the velocity potentials
$\Phi_{1}(\bm{x},z,t)$ and $\Phi_{2}(\bm{x},z,t)$ and the pressures
$P_{1}(\bm{x},z,t)$ and $P_{2}(\bm{x},z,t)$ in the upper and the lower layers.
We recall the governing equations, referred as the full model for interfacial
gravity waves, in Section 2 below. Generalizing the work of J. C. Luke [15],
these equations can be obtained as the Euler–Lagrange equations associated
with the Lagrangian density $\mathscr{L}(\Phi_{1},\Phi_{2},\zeta)$ given by
the vertical integral of the pressure in both water regions. Building on this
variational structure, T. Kakinuma [9, 10, 11] proposed and studied
numerically the model obtained as the Euler–Lagrange equations for an
approximated Lagrangian density,
$\mathscr{L}(\Phi_{1}^{\mathrm{app}},\Phi_{2}^{\mathrm{app}},\zeta)$, where

(1.1)
$\Phi_{\ell}^{\mathrm{app}}(\bm{x},z,t)=\sum_{i=0}^{N_{\ell}}Z_{\ell,i}(z;\tilde{h}_{\ell}(\bm{x}))\phi_{\ell,i}(\bm{x},t)$

for $\ell=1,2$, and $\\{Z_{1,i}\\}$ and $\\{Z_{2,i}\\}$ are appropriate
function systems in the vertical coordinate $z$ and may depend on
$\tilde{h}_{1}(\bm{x})$ and $\tilde{h}_{2}(\bm{x})$, respectively, which are
thickness of the upper and the lower layers in the rest state, whereas
$\bm{\phi}_{\ell}=(\phi_{\ell,0},\phi_{\ell,1},\ldots,\phi_{\ell,N_{\ell}})^{\mathrm{T}}$,
$\ell=1,2$, are unknown variables. This yields a coupled system of equations
for $\bm{\phi}_{1}$, $\bm{\phi}_{2}$, and $\zeta$, depending on the function
systems $\\{Z_{1,i}\\}$ and $\\{Z_{2,i}\\}$, which we named Kakinuma model.
Note that in our setting of the problem we have $\tilde{h}_{1}(\bm{x})=h_{1}$
and $\tilde{h}_{2}(\bm{x})=h_{2}-b(\bm{x})$.

The Kakinuma model is an extension to interfacial gravity waves of the so-
called Isobe–Kakinuma model for surface gravity waves, that is, water waves,
in which Luke’s Lagrangian density $\mathscr{L}_{\mathrm{Luke}}(\Phi,\zeta)$,
where $\zeta$ is the surface elevation and $\Phi$ is the velocity potential of
the water, is approximated by a density
$\mathscr{L}^{\mathrm{app}}(\bm{\phi},\zeta)=\mathscr{L}_{\mathrm{Luke}}(\Phi^{\mathrm{app}},\zeta)$,
where

(1.2)
$\Phi^{\mathrm{app}}(\bm{x},z,t)=\sum_{i=0}^{N}Z_{i}(z;b(\bm{x}))\phi_{i}(\bm{x},t).$

The Isobe–Kakinuma model was first proposed by M. Isobe [7, 8] and then
applied by T. Kakinuma to simulate numerically the water waves. Recently, this
model was analyzed from a mathematical point of view when the function system
$\\{Z_{i}\\}$ is a set of polynomials in $z$:
$Z_{i}(z;b(\bm{x}))=(z+h-b(\bm{x}))^{p_{i}}$ with integers $p_{i}$ satisfying
$0=p_{0}<p_{1}<\cdots<p_{N}$. The initial value problem was analyzed by Y.
Murakami and T. Iguchi [16] in a special case and by R. Nemoto and T. Iguchi
[17] in the general case. The hypersurface $t=0$ in the space-time
$\mathbf{R}^{n}\times\mathbf{R}$ is characteristic for the Isobe–Kakinuma
model, so that one needs to impose some compatibility conditions on the
initial data for the existence of the solution. Under these compatibility
conditions, the non-cavitation condition, and a Rayleigh–Taylor type condition
$-\partial_{z}P^{\mathrm{app}}\geq c_{0}>0$ on the water surface, where
$P^{\mathrm{app}}$ is an approximate pressure in the Isobe–Kakinuma model
calculated from Bernoulli’s equation, they showed the well-posedness of the
initial value problem in Sobolev spaces locally in time. Moreover, T. Iguchi
[5, 6] showed that under the choice of the function system

(1.3) $Z_{i}(z;b(\bm{x}))=\begin{cases}(z+h)^{2i}&\mbox{in the case of the
flat bottom},\\\ (z+h-b(\bm{x}))^{i}&\mbox{in the case of a variable
bottom},\end{cases}$

the Isobe–Kakinuma model is a higher order shallow water approximation for the
water wave problem in the strongly nonlinear regime. Furthermore, V. Duchêne
and T. Iguchi [2] showed that the Isobe–Kakinuma model also enjoys a
Hamiltonian structure analogous to the one exhibited by V. E. Zakharov [19] on
the full water wave problem and that the Hamiltonian of the Isobe–Kakinuma
model is a higher order shallow water approximation to the one of the full
water wave problem.

Our aim in the present paper and the companion paper [3] is to extend these
results on surface gravity waves to the framework of interfacial gravity
waves. In [3], we analyzed the Cauchy problem for Kakinuma model when the
approximated velocity potentials are defined by

(1.4)
$\begin{cases}\displaystyle\Phi_{1}^{\mathrm{app}}(\bm{x},z,t):=\sum_{i=0}^{N}(-z+h_{1})^{2i}\phi_{1,i}(\bm{x},t),\\\\[10.76385pt]
\displaystyle\Phi_{2}^{\mathrm{app}}(\bm{x},z,t):=\sum_{i=0}^{N^{*}}(z+h_{2}-b(\bm{x}))^{p_{i}}\phi_{2,i}(\bm{x},t),\end{cases}$

where $N,N^{*}$, and $p_{0},p_{1},\ldots,p_{N^{*}}$ are nonnegative integers
satisfying ${0=p_{0}<p_{1}<\cdots<p_{N^{*}}}$. We found that contrary to the
full model for interfacial gravity waves, the Kakinuma model has a stability
regime which can be expressed as

(1.5)
$-\partial_{z}(P_{2}^{\mathrm{app}}-P_{1}^{\mathrm{app}})-\frac{\rho_{1}\rho_{2}}{\rho_{1}H_{2}\alpha_{2}+\rho_{2}H_{1}\alpha_{1}}|\nabla\Phi_{2}^{\mathrm{app}}-\nabla\Phi_{1}^{\mathrm{app}}|^{2}\geq
c_{0}>0$

on the interface, where $P_{1}^{\mathrm{app}}$ and $P_{2}^{\mathrm{app}}$ are
approximate pressures of the waters in the upper and the lower layers,
$\alpha_{1}$ and $\alpha_{2}$ are positive constants depending only on $N$ and
on $p_{0},p_{1},\ldots,p_{N^{*}}$, respectively. This is a generalization of
the aforementioned Rayleigh–Taylor type condition for the Isobe–Kakinuma
model. Moreover, when the motion of the waters together with the motion of the
interface is in the rest state, the above stability condition is reduced to
the well-known stable stratification condition

(1.6) $(\rho_{2}-\rho_{1})g>0.$

In [3], we showed that under the stability condition (1.5), the non-cavitation
assumptions, and intrinsic compatibility conditions on the initial data, the
initial value problem for the Kakinuma model is well-posed in Sobolev spaces
locally in time. It is worth noticing that the constants $\alpha_{1}$ and
$\alpha_{2}$ converge to $0$ as $N$ and $N^{*}$ go to infinity so that the
stability condition becomes more and more stringent as $N$ and $N^{*}$ grow.
However, this is consistent with the fact that the initial value problem for
the full model for interfacial gravity waves is ill-posed in Sobolev spaces;
see T. Iguchi, N. Tanaka, and A. Tani [18], V. Kamotski and G. Lebeau [12] and
D. Lannes [13]. We also showed in [3] that the Kakinuma model enjoys a
Hamiltonian structure analogous to the one exhibited by T. B. Benjamin and T.
J. Bridges [1] on the full model for interfacial gravity waves. In the present
paper we will complete the analysis by showing that the Kakinuma model
obtained through the approximated potentials (1.4) with

  1. (H1)

$N^{*}=N$ and $p_{i}=2i$ $(i=0,1,\ldots,N)$ in the case of the flat bottom
$b(\bm{x})\equiv 0$,

  2. (H2)

$N^{*}=2N$ and $p_{i}=i$ $(i=0,1,\ldots,2N)$ in the case with general bottom
topographies,

provides a higher order shallow water approximation to the full model for
interfacial gravity waves in the strongly nonlinear regime. More precisely, we
will show that, after suitable rescaling, the dimensionless Kakinuma model is
consistent with the full model for interfacial gravity waves with an error of
order $O(\delta_{1}^{4N+2}+\delta_{2}^{4N+2})$, where $\delta_{1}$ and
$\delta_{2}$ are shallowness parameters related to the upper and the lower
layers, respectively, that is, $\delta_{\ell}=\frac{h_{\ell}}{\lambda}$
$(\ell=1,2)$ with the typical horizontal wavelength $\lambda$. A full
justification of the Kakinuma model as shallow water approximations is not
straightforward, because one cannot expect to construct a solution to the
initial value problem for the full model with a uniform bound for general
initial data in Sobolev spaces due to the ill-posedness of the problem.
Nevertheless, if we assume the existence of a solution to the full model with
a uniform bound and the stability condition (1.5) for the initial data, then
we can show the existence of a corresponding solution to the Kakinuma model
with appropriate initial data and the error estimate

$|\zeta^{\mbox{\rm\tiny K}}(\bm{x},t)-\zeta^{\mbox{\rm\tiny
IW}}(\bm{x},t)|\lesssim\delta_{1}^{4N+2}+\delta_{2}^{4N+2}$

on some time interval independent of $\delta_{1}$ and $\delta_{2}$, where
$\zeta^{\mbox{\rm\tiny K}}$ and $\zeta^{\mbox{\rm\tiny IW}}$ are solutions to
the dimensionless Kakinuma model and to the full model, respectively.
Moreover, under an appropriate assumption on the canonical variables
$(\zeta,\phi)$, we show the error estimate

$|\mathscr{H}^{\mbox{\rm\tiny K}}(\zeta,\phi)-\mathscr{H}^{\mbox{\rm\tiny
IW}}(\zeta,\phi)|\lesssim\delta_{1}^{4N+2}+\delta_{2}^{4N+2},$

where $\mathscr{H}^{\mbox{\rm\tiny K}}$ and $\mathscr{H}^{\mbox{\rm\tiny IW}}$
are Hamiltonians of the Kakinuma model and of the full model, respectively.
Our error bounds in this paper are uniform with respect to the positive
densities $\rho_{1}$ and $\rho_{2}$ satisfying the stable stratification
condition $(\rho_{2}-\rho_{1})g>0$ and positive mean thicknesses of the upper
layer $h_{1}$ and of the lower layer $h_{2}$ in the following two regimes: (i)
$\rho_{1}\simeq\rho_{2}$; (ii) $\rho_{1}\ll\rho_{2}$ and $h_{2}\lesssim
h_{1}$. In other words, in addition to assuming the stable stratification
condition, the regime (iii) $\rho_{1}\ll\rho_{2}$ and $h_{1}\ll h_{2}$ will be
excluded in this paper.

The contents of this paper are as follows. In Section 2 we first recall the
basic equations governing the interfacial gravity waves and write down the
Kakinuma model that we are going to analyze in this paper, and then rewrite
them in a nondimensional form by introducing several nondimensional
parameters. Hamiltonians of the full model and of the Kakinuma model in the
nondimensional variables are also provided. In Section 3 we first introduce
some differential operators, which enable us to write the Kakinuma model in a
simple form, and then we present our main results in this paper: Theorem 3.1
ensures the existence of the solution to the initial value problem for the
Kakinuma model on a time interval independent of parameters, especially,
$\delta_{1}$ and $\delta_{2}$, under the stability condition, the non-
cavitation assumptions, and intrinsic compatibility conditions on the initial
data, together with a uniform bound of the solution; Remark 3.2 and
Proposition 3.3 explain how to prepare the initial data for the Kakinuma
model, which have to satisfy the compatibility conditions; Theorems 3.4 and
3.5 ensure the consistency of the Kakinuma model to the full model; More
precisely, Theorem 3.4 states that the solutions to the Kakinuma model satisfy
approximately the full model with an error of order
$O(\delta_{1}^{4N+2}+\delta_{2}^{4N+2})$; Conversely, Theorem 3.5 states that
the solutions to the full model satisfy approximately the Kakinuma model with
an error of the same order; Theorem 3.8 gives conditionally a rigorous
justification of the Kakinuma model, that is, assuming the existence of a
solution to the full model with a uniform bound we derive an error estimate
between a corresponding solution to the Kakinuma model and that of the full
model; Finally, Theorem 3.9 gives an error estimate between the Hamiltonian of
the Kakinuma model and that of the full model. In Section 4 we first remind
results in the framework of surface waves related to the consistency of the
Isobe–Kakinuma model, and then prove Theorems 3.4 and 3.5 by a simple scaling
argument. In Section 5 we first derive an elliptic estimate related to the
compatibility conditions, and then give uniform a priori bounds on regular
solutions to the Kakinuma model, especially, a priori bounds of time
derivatives. In Section 6 we provide uniform energy estimates for the solution
to the Kakinuma model and prove Theorem 3.1. In Section 7 we first give a
supplementary estimate on an approximation of the Dirichlet-to-Neumann map,
and then revisit the consistency of the Kakinuma model. We prove Proposition
7.6, which is another version of the consistency given in Theorem 3.5, where
we adopt a different construction of an approximate solution to the Kakinuma
model from the solution to the full model. Then, by making use of the well-
posedness of the initial value problem for the Kakinuma model we prove Theorem
3.8 a conditional rigorous justification of the Kakinuma model. Finally, in
Section 8 we prove Theorem 3.9 on the approximation of the Hamiltonian.

Notation. We denote by $W^{m,p}$ the $L^{p}$ Sobolev space of order $m$ on
$\mathbf{R}^{n}$ and $H^{m}=W^{m,2}$. We put
$\mathring{H}^{m}=\\{\phi\,;\,\nabla\phi\in H^{m-1}\\}$. The norm of a Banach
space $B$ is denoted by $\|\cdot\|_{B}$. The $L^{2}$-inner product is denoted
by $(\cdot,\cdot)_{L^{2}}$. We put $\partial_{t}=\frac{\partial}{\partial t}$,
$\partial_{j}=\partial_{x_{j}}=\frac{\partial}{\partial x_{j}}$, and
$\partial_{z}=\frac{\partial}{\partial z}$. $[P,Q]=PQ-QP$ denotes the
commutator and $[P;u,v]=P(uv)-(Pu)v-u(Pv)$ denotes the symmetric commutator.
For a matrix $A$ we denote by $A^{\mathrm{T}}$ the transpose of $A$. $O$
denotes a zero matrix. For a vector
$\mbox{\boldmath$\phi$}=(\phi_{0},\phi_{1},\ldots,\phi_{N})^{\mathrm{T}}$ we
denote the last $N$ components by
$\mbox{\boldmath$\phi$}^{\prime}=(\phi_{1},\ldots,\phi_{N})^{\mathrm{T}}$. We
use the notational convention $\frac{0}{0}=0$. $f\lesssim g$ means that there
exists a non-essential positive constant $C$ such that $f\leq Cg$ holds.
$f\simeq g$ means that $f\lesssim g$ and $g\lesssim f$ hold.

Acknowledgement
T. I. was partially supported by JSPS KAKENHI Grant Number JP17K18742 and
JP22H01133. V. D. thanks the Centre Henri Lebesgue ANR-11-LABX-0020-01 for
creating an attractive mathematical environment.

##  2 The basic equations and the Kakinuma model

###  2.1 Equations with physical variables

We first recall the equations governing potential flows for two layers of
immiscible, incompressible, homogeneous, and inviscid fluids, and then write
down the Kakinuma model at stake in this work. In the following, we denote the
upper layer, the lower layer, the interface, the rigid-lid, and the bottom at
time t by $\Omega_{1}(t)$, $\Omega_{2}(t)$, $\Gamma(t)$, $\Sigma_{1}$, and
$\Sigma_{2}$, respectively. The velocity potentials $\Phi_{1}(\bm{x},z,t)$ and
$\Phi_{2}(\bm{x},z,t)$ in the upper and lower layers, respectively, satisfy
Laplace’s equations

(2.1)
$\displaystyle\Delta\Phi_{1}+\partial_{z}^{2}\Phi_{1}=0\quad\mbox{in}\quad\Omega_{1}(t),$
(2.2)
$\displaystyle\Delta\Phi_{2}+\partial_{z}^{2}\Phi_{2}=0\quad\mbox{in}\quad\Omega_{2}(t),$

where $\Delta=\partial_{1}^{2}+\cdots+\partial_{n}^{2}$ is the Laplacian with
respect to the horizontal space variables $\bm{x}=(x_{1},\ldots,x_{n})$.
Bernoulli’s laws of each layers have the form

(2.3)
$\displaystyle\rho_{1}\left(\partial_{t}\Phi_{1}+\frac{1}{2}(|\nabla\Phi_{1}|^{2}+(\partial_{z}\Phi_{1})^{2})+gz\right)+P_{1}=0\quad\mbox{in}\quad\Omega_{1}(t),$
(2.4)
$\displaystyle\rho_{2}\left(\partial_{t}\Phi_{2}+\frac{1}{2}(|\nabla\Phi_{2}|^{2}+(\partial_{z}\Phi_{2})^{2})+gz\right)+P_{2}=0\quad\mbox{in}\quad\Omega_{2}(t),$

where $\nabla=(\partial_{1},\ldots,\partial_{n})$, the positive constant $g$
is the acceleration due to gravity, and $P_{1}(\bm{x},z,t)$ and
$P_{2}(\bm{x},z,t)$ are pressures in the upper and lower layers, respectively.
The dynamical boundary condition on the interface is given by

(2.5) $P_{1}=P_{2}\quad\mbox{on}\quad\Gamma(t).$

The kinematic boundary conditions on the interface, the rigid-lid, and the
bottom are given by

(2.6)
$\displaystyle\partial_{t}\zeta+\nabla\Phi_{1}\cdot\nabla\zeta-\partial_{z}\Phi_{1}=0\quad\mbox{on}\quad\Gamma(t),$
(2.7)
$\displaystyle\partial_{t}\zeta+\nabla\Phi_{2}\cdot\nabla\zeta-\partial_{z}\Phi_{2}=0\quad\mbox{on}\quad\Gamma(t),$
(2.8)
$\displaystyle\partial_{z}\Phi_{1}=0\makebox[78.79143pt]{}\quad\mbox{on}\quad\Sigma_{1},$
(2.9) $\displaystyle\nabla\Phi_{2}\cdot\nabla
b-\partial_{z}\Phi_{2}=0\makebox[27.12492pt]{}\quad\mbox{on}\quad\Sigma_{2}.$

These are the basic equations for interfacial gravity waves. It follows from
Bernoulli’s laws (2.3)–(2.4) and the dynamical boundary condition (2.5) that

(2.10)
$\displaystyle\rho_{1}\left(\partial_{t}\Phi_{1}+\frac{1}{2}(|\nabla\Phi_{1}|^{2}+(\partial_{z}\Phi_{1})^{2})\right)$
$\displaystyle-\rho_{2}\left(\partial_{t}\Phi_{2}+\frac{1}{2}(|\nabla\Phi_{2}|^{2}+(\partial_{z}\Phi_{2})^{2})\right)=(\rho_{2}-\rho_{1})g\zeta\quad\mbox{on}\quad\Gamma(t).$

We will always assume the stable stratification condition
$(\rho_{2}-\rho_{1})g>0$. As in the case of the surface water waves, the basic
equations have a variational structure and the corresponding Luke’s Lagrangian
is given, up to terms which do not contribute to the variation of the
Lagrangian, by the vertical integral of the pressure in the water regions.
After using Bernoulli’s laws (2.3)–(2.4) we can find the Lagrangian density

(2.11) $\displaystyle\mathscr{L}(\Phi_{1},\Phi_{2},\zeta)$
$\displaystyle=-\rho_{1}\int_{\zeta}^{h_{1}}\left(\partial_{t}\Phi_{1}+\frac{1}{2}(|\nabla\Phi_{1}|^{2}+(\partial_{z}\Phi_{1})^{2})\right){\rm
d}z$
$\displaystyle\quad\;-\rho_{2}\int_{-h_{2}+b}^{\zeta}\left(\partial_{t}\Phi_{2}+\frac{1}{2}(|\nabla\Phi_{2}|^{2}+(\partial_{z}\Phi_{2})^{2})\right){\rm
d}z-\frac{1}{2}(\rho_{2}-\rho_{1})g\zeta^{2}.$

In fact, one checks readily that (2.1)–(2.2) and (2.6)–(2.10) are
Euler–Lagrange equations associated with the action function

$\mathscr{J}(\Phi_{1},\Phi_{2},\zeta):=\int_{t_{0}}^{t_{1}}\\!\\!\\!\int_{\mathbf{R}^{n}}\mathscr{L}(\Phi_{1},\Phi_{2},\zeta)\,\mathrm{d}\bm{x}\,\mathrm{d}t.$

We proceed to the Kakinuma model. Let $N$ and $N^{*}$ be nonnegative integers.
In view of the analysis for the Isobe–Kakinuma model for surface water waves,
we approximate the velocity potentials $\Phi_{1}$ and $\Phi_{2}$ in the
Lagrangian by

(2.12)
$\begin{cases}\displaystyle\Phi_{1}^{\mathrm{app}}(\bm{x},z,t)=\sum_{i=0}^{N}(-z+h_{1})^{2i}\phi_{1,i}(\bm{x},t),\\\\[10.76385pt]
\displaystyle\Phi_{2}^{\mathrm{app}}(\bm{x},z,t)=\sum_{i=0}^{N^{*}}(z+h_{2}-b(\bm{x}))^{p_{i}}\phi_{2,i}(\bm{x},t),\end{cases}$

where $p_{0},p_{1},\ldots,p_{N^{*}}$ are nonnegative integers satisfying
$0=p_{0}<p_{1}<\cdots<p_{N^{*}}$. Plugging (2.12) into the Lagrangian density
(2.11), we obtain an approximate Lagrangian density

$\mathscr{L}^{\mathrm{app}}(\bm{\phi}_{1},\bm{\phi}_{2},\zeta):=\mathscr{L}(\Phi_{1}^{\mathrm{app}},\Phi_{2}^{\mathrm{app}},\zeta),$

where $\bm{\phi}_{1}:=(\phi_{1,0},\phi_{1,1},\ldots,\phi_{1,N})^{\mathrm{T}}$
and
$\bm{\phi}_{2}:=(\phi_{2,0},\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\mathrm{T}}$.
The corresponding Euler–Lagrange equation is the Kakinuma model, which has the
form

(2.13) $\begin{cases}\displaystyle
H_{1}^{2i}\partial_{t}\zeta-\sum_{j=0}^{N}\biggl{\\{}\nabla\cdot\biggr{(}\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,j}\biggr{)}-\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}\phi_{1,j}\biggr{\\}}=0\\\
\makebox[270.00041pt]{}\mbox{for}\quad i=0,1,\ldots,N,\\\ \displaystyle
H_{2}^{p_{i}}\partial_{t}\zeta+\sum_{j=0}^{N^{*}}\biggl{\\{}\nabla\cdot\biggl{(}\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,j}-\frac{p_{j}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\phi_{2,j}\nabla
b\biggr{)}\\\
\displaystyle\phantom{H_{2}^{p_{i}}\partial_{t}\zeta+\sum_{j=0}^{N}}+\frac{p_{i}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\nabla
b\cdot\nabla\phi_{2,j}-\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}(1+|\nabla
b|^{2})\phi_{2,j}\biggr{\\}}=0\\\ \makebox[270.00041pt]{}\mbox{for}\quad
i=0,1,\ldots,N^{*},\\\
\displaystyle\rho_{1}\biggl{\\{}\sum_{j=0}^{N}H_{1}^{2j}\partial_{t}\phi_{1,j}+g\zeta+\frac{1}{2}\biggl{(}\biggl{|}\sum_{j=0}^{N}H_{1}^{2j}\nabla\phi_{1,j}\biggr{|}^{2}+\biggl{(}\sum_{j=0}^{N}2jH_{1}^{2j-1}\phi_{1,j}\biggr{)}^{2}\biggr{)}\biggr{\\}}\\\
\displaystyle\quad-\rho_{2}\biggl{\\{}\sum_{j=0}^{N^{*}}H_{2}^{p_{j}}\partial_{t}\phi_{2,j}+g\zeta\\\
\displaystyle\qquad\quad+\frac{1}{2}\biggl{(}\biggl{|}\sum_{j=0}^{N^{*}}(H_{2}^{p_{j}}\nabla\phi_{2,j}-p_{j}H_{2}^{p_{j}-1}\phi_{2,j}\nabla
b)\biggr{|}^{2}+\biggl{(}\sum_{j=0}^{N^{*}}p_{j}H_{2}^{p_{j}-1}\phi_{2,j}\biggr{)}^{2}\biggr{)}\biggr{\\}}=0,\end{cases}$

where $H_{1}$ and $H_{2}$ are thicknesses of the upper and the lower layers,
that is,

$H_{1}(t,\bm{x}):=h_{1}-\zeta(\bm{x},t),\qquad
H_{2}(\bm{x},t):=h_{2}+\zeta(\bm{x},t)-b(\bm{x}).$

Here and in what follows we use the notational convention $0/0=0$.

###  2.2 The dimensionless equations

In order to rigorously validate the Kakinuma model (2.13) as a higher order
shallow water approximation of the full model for interfacial gravity waves
(2.1)–(2.9), we first introduce nondimensional parameters and then non-
dimensionalize the equations, through a convenient rescaling of variables. Let
$\lambda$ be a typical horizontal wavelength. Following D. Lannes [13], we
introduce a nondimensional parameter $\delta$ by

$\delta:=\frac{h}{\lambda}\qquad\mbox{with}\quad
h:=\frac{h_{1}h_{2}}{\underline{\rho}_{1}h_{2}+\underline{\rho}_{2}h_{1}},$

where $\underline{\rho}_{1}$ and $\underline{\rho}_{2}$ are relative
densities. We also need to use relative thicknesses $\underline{h}_{1}$ and
$\underline{h}_{2}$ of the layers. These nondimensional parameters are defined
by

$\underline{\rho}_{\ell}:=\frac{\rho_{\ell}}{\rho_{1}+\rho_{2}},\qquad\underline{h}_{\ell}:=\frac{h_{\ell}}{h}\qquad(\ell=1,2),$

which satisfy the relations

(2.14)
$\underline{\rho}_{1}+\underline{\rho}_{2}=1,\qquad\frac{\underline{\rho}_{1}}{\underline{h}_{1}}+\frac{\underline{\rho}_{2}}{\underline{h}_{2}}=1.$

Note also that $\min\\{h_{1},h_{2}\\}\leq h\leq\max\\{h_{1},h_{2}\\}$. It
follows from the second relation in (2.14) that

(2.15)
$1<\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}\leq
2.$

Here, we note that the standard shallowness parameters
$\delta_{1}:=\frac{h_{1}}{\lambda}$ and $\delta_{2}:=\frac{h_{2}}{\lambda}$
relative to the upper and the lower layers, respectively, are related to the
above parameters by $\delta_{\ell}=\underline{h}_{\ell}\delta$ for $\ell=1,2$.
In many results of this paper, we restrict our consideration to the parameter
regime

(2.16) $\underline{h}_{1}^{-1},\underline{h}_{2}^{-1}\lesssim 1.$

To understand this restriction, it is convenient to use nondimensional
parameters $\gamma:=\frac{\rho_{1}}{\rho_{2}}$ and
$\theta:=\frac{h_{1}}{h_{2}}$. In terms of these parameters,
$\underline{h}_{\ell}^{-1}$ $(\ell=1,2)$ can be represented as

$\underline{h}_{1}^{-1}=\frac{\gamma+1}{\gamma+\theta},\qquad\underline{h}_{2}^{-1}=\frac{\gamma^{-1}+1}{\gamma^{-1}+\theta^{-1}}.$

Therefore, the only cases that (2.16) excludes are the case $\gamma,\theta\ll
1$ and the case $\gamma,\theta\gg 1$. Since we shall also assume the stable
stratification condition $(\rho_{2}-\rho_{1})g>0$, we can describe the two
regimes considered in this paper as

  1. (i)

$\gamma\simeq 1$, i.e., $\rho_{1}\simeq\rho_{2}$,

  2. (ii)

$\gamma\ll 1$ and $\theta\gtrsim 1$, i.e., $\rho_{1}\ll\rho_{2}$ and
$h_{2}\lesssim h_{1}$.

Introducing $c_{\mbox{\rm\tiny
SW}}:=\sqrt{(\underline{\rho}_{2}-\underline{\rho}_{1})gh}$ the speed of
infinitely long and small interfacial gravity waves, we rescale the
independent and the dependent variables by

$\bm{x}=\lambda\tilde{\bm{x}},\quad z=h\tilde{z},\quad
t=\frac{\lambda}{c_{\mbox{\rm\tiny
SW}}}\tilde{t},\quad\zeta=h\tilde{\zeta},\quad
b=h\tilde{b},\quad\Phi_{\ell}=\lambda c_{\mbox{\rm\tiny
SW}}\tilde{\Phi}_{\ell}\quad(\ell=1,2).$

Plugging these into the full model (2.1)–(2.2) and (2.6)–(2.10) and dropping
the tilde sign in the notation we obtain

$\begin{cases}\Delta\Phi_{1}+\delta^{-2}\partial_{z}^{2}\Phi_{1}=0&\mbox{in}\quad\Omega_{1}(t),\\\
\Delta\Phi_{2}+\delta^{-2}\partial_{z}^{2}\Phi_{2}=0&\mbox{in}\quad\Omega_{2}(t),\\\
\partial_{t}\zeta+\nabla\Phi_{1}\cdot\nabla\zeta-\delta^{-2}\partial_{z}\Phi_{1}=0&\mbox{on}\quad\Gamma(t),\\\
\partial_{t}\zeta+\nabla\Phi_{2}\cdot\nabla\zeta-\delta^{-2}\partial_{z}\Phi_{2}=0&\mbox{on}\quad\Gamma(t),\\\
\partial_{z}\Phi_{1}=0&\mbox{on}\quad\Sigma_{1},\\\ \nabla\Phi_{2}\cdot\nabla
b-\delta^{-2}\partial_{z}\Phi_{2}=0&\mbox{on}\quad\Sigma_{2},\\\
\underline{\rho}_{1}\Bigl{(}\partial_{t}\Phi_{1}+\frac{1}{2}|\nabla\Phi_{1}|^{2}+\frac{1}{2}\delta^{-2}(\partial_{z}\Phi_{1})^{2}\Bigr{)}\\\
\quad-\underline{\rho}_{2}\Bigl{(}\partial_{t}\Phi_{2}+\frac{1}{2}|\nabla\Phi_{2}|^{2}+\frac{1}{2}\delta^{-2}(\partial_{z}\Phi_{2})^{2}\Bigr{)}-\zeta=0&\mbox{on}\quad\Gamma(t),\end{cases}$

where in this scaling the upper layer $\Omega_{1}(t)$, the lower layer
$\Omega_{2}(t)$, the interface $\Gamma(t)$, the rigid-lid $\Sigma_{1}$, and
the bottom $\Sigma_{2}$ are written as

$\displaystyle\Omega_{1}(t)=\\{X=(\bm{x},z)\in\mathbf{R}^{n+1}\,;\,\zeta(\bm{x},t)<z<\underline{h}_{1}\\},$
$\displaystyle\Omega_{2}(t)=\\{X=(\bm{x},z)\in\mathbf{R}^{n+1}\,;\,-\underline{h}_{2}+b(\bm{x})<z<\zeta(\bm{x},t)\\},$
$\displaystyle\Gamma(t)=\\{X=(\bm{x},z)\in\mathbf{R}^{n+1}\,;\,z=\zeta(\bm{x},t)\\},$
$\displaystyle\Sigma_{1}=\\{X=(\bm{x},z)\in\mathbf{R}^{n+1}\,;\,z=\underline{h}_{1}\\},$
$\displaystyle\Sigma_{2}=\\{X=(\bm{x},z)\in\mathbf{R}^{n+1}\,;\,z=-\underline{h}_{2}+b(\bm{x})\\}.$

Denoting

$\phi_{\ell}(\bm{x},t):=\Phi_{\ell}(\bm{x},\zeta(\bm{x},t),t)\qquad(\ell=1,2)$

and using the chain rule, the above system can be written in a more compact
and closed form as

(2.17)
$\begin{cases}\partial_{t}\zeta+\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}=0,\\\
\partial_{t}\zeta-\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}=0,\\\
\displaystyle\underline{\rho}_{1}\biggl{(}\partial_{t}\phi_{1}+\frac{1}{2}|\nabla\phi_{1}|^{2}-\frac{1}{2}\delta^{2}\frac{(\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}-\nabla\zeta\cdot\nabla\phi_{1})^{2}}{1+\delta^{2}|\nabla\zeta|^{2}}\biggr{)}\\\
\displaystyle\quad-\underline{\rho}_{2}\biggl{(}\partial_{t}\phi_{2}+\frac{1}{2}|\nabla\phi_{2}|^{2}-\frac{1}{2}\delta^{2}\frac{(\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}+\nabla\zeta\cdot\nabla\phi_{2})^{2}}{1+\delta^{2}|\nabla\zeta|^{2}}\biggr{)}-\zeta=0,\end{cases}$

where $\Lambda_{1}(\zeta,\delta,\underline{h}_{1})$ and
$\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})$ are the Dirichlet-to-Neumann
maps for Laplace’s equations. More precisely, these are defined by

$\displaystyle\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}:=\bigl{(}-\delta^{-2}\partial_{z}\Phi_{1}+\nabla\Phi_{1}\cdot\nabla\zeta\bigr{)}\bigr{|}_{z=\zeta(\bm{x},t)},$
$\displaystyle\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}:=\bigl{(}\delta^{-2}\partial_{z}\Phi_{2}-\nabla\Phi_{2}\cdot\nabla\zeta\bigr{)}\bigr{|}_{z=\zeta(\bm{x},t)},$

where $\Phi_{1}$ and $\Phi_{2}$ are unique solutions to the boundary value
problems

$\begin{cases}\Delta\Phi_{1}+\delta^{-2}\partial_{z}^{2}\Phi_{1}=0&\mbox{in}\quad\Omega_{1}(t),\\\
\Phi_{1}=\phi_{1}&\mbox{on}\quad\Gamma(t),\\\
\partial_{z}\Phi_{1}=0&\mbox{on}\quad\Sigma_{1},\end{cases}\text{ and
}\quad\begin{cases}\Delta\Phi_{2}+\delta^{-2}\partial_{z}^{2}\Phi_{2}=0&\mbox{in}\quad\Omega_{2}(t),\\\
\Phi_{2}=\phi_{2}&\mbox{on}\quad\Gamma(t),\\\ \nabla\Phi_{2}\cdot\nabla
b-\delta^{-2}\partial_{z}\Phi_{2}=0&\mbox{on}\quad\Sigma_{2}.\end{cases}$

As for the Kakinuma model, we introduce additionally the rescaled variables

$\phi_{1,i}:=\frac{\lambda c_{\mbox{\rm\tiny
SW}}}{h_{1}^{2i}}\tilde{\phi}_{1,i},\qquad\phi_{2,i}:=\frac{\lambda
c_{\mbox{\rm\tiny SW}}}{h_{2}^{p_{i}}}\tilde{\phi}_{2,i}.$

Plugging these and the previous scaling into the Kakinuma model (2.13) and
dropping the tilde sign in the notation we obtain the Kakinuma model in the
nondimensional form, which is written as

(2.18) $\begin{cases}\displaystyle
H_{1}^{2i}\partial_{t}\zeta-\underline{h}_{1}\sum_{j=0}^{N}\biggl{\\{}\nabla\cdot\biggr{(}\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\phi_{1,j}\biggr{)}-\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}(\underline{h}_{1}\delta)^{-2}\phi_{1,j}\biggr{\\}}=0\\\
\makebox[290.00044pt]{}\mbox{for}\quad i=0,1,\ldots,N,\\\ \displaystyle
H_{2}^{p_{i}}\partial_{t}\zeta+\underline{h}_{2}\sum_{j=0}^{N^{*}}\biggl{\\{}\nabla\cdot\biggl{(}\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\phi_{2,j}-\frac{p_{j}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\phi_{2,j}\underline{h}_{2}^{-1}\nabla
b\biggr{)}\\\
\displaystyle\qquad+\frac{p_{i}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\underline{h}_{2}^{-1}\nabla
b\cdot\nabla\phi_{2,j}-\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}((\underline{h}_{2}\delta)^{-2}+\underline{h}_{2}^{-2}|\nabla
b|^{2})\phi_{2,j}\biggr{\\}}=0\\\ \makebox[290.00044pt]{}\mbox{for}\quad
i=0,1,\ldots,N^{*},\\\
\displaystyle\underline{\rho}_{1}\biggl{\\{}\sum_{j=0}^{N}H_{1}^{2j}\partial_{t}\phi_{1,j}+\frac{1}{2}\biggl{(}\biggl{|}\sum_{j=0}^{N}H_{1}^{2j}\nabla\phi_{1,j}\biggr{|}^{2}+(\underline{h}_{1}\delta)^{-2}\biggl{(}\sum_{j=0}^{N}2jH_{1}^{2j-1}\phi_{1,j}\biggr{)}^{2}\biggr{)}\biggr{\\}}\\\
\displaystyle\qquad-\underline{\rho}_{2}\biggl{\\{}\sum_{j=0}^{N^{*}}H_{2}^{2j}\partial_{t}\phi_{2,j}+\frac{1}{2}\biggl{(}\biggl{|}\sum_{j=0}^{N^{*}}(H_{2}^{p_{j}}\nabla\phi_{2,j}-p_{j}H_{2}^{p_{j}-1}\phi_{2,j}\underline{h}_{2}^{-1}\nabla
b)\biggr{|}^{2}\\\
\displaystyle\makebox[170.00026pt]{}+(\underline{h}_{2}\delta)^{-2}\biggl{(}\sum_{j=0}^{N^{*}}p_{j}H_{2}^{p_{j}-1}\phi_{2,j}\biggr{)}^{2}\biggr{)}\biggr{\\}}-\zeta=0,\end{cases}$

where

(2.19) $H_{1}(\bm{x},t):=1-\underline{h}_{1}^{-1}\zeta(\bm{x},t),\qquad
H_{2}(\bm{x},t):=1+\underline{h}_{2}^{-1}\zeta(\bm{x},t)-\underline{h}_{2}^{-1}b(\bm{x}).$

We impose the initial conditions to the Kakinuma model of the form

(2.20)
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})=(\zeta_{(0)},\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})\makebox[30.00005pt]{at}t=0.$

###  2.3 Hamiltonian structures

It is well-known that the full model for interfacial gravity waves has a
conserved energy

$\displaystyle\mathscr{E}$
$\displaystyle:=\sum_{\ell=1,2}\iint_{\Omega_{\ell}(t)}\frac{1}{2}\underline{\rho}_{\ell}\bigl{(}|\nabla\Phi_{\ell}(\bm{x},z,t)|^{2}+\delta^{-2}(\partial_{z}\Phi_{\ell}(\bm{x},z,t))^{2}\bigr{)}\mathrm{d}\bm{x}\mathrm{d}z+\int_{\mathbf{R}^{n}}\frac{1}{2}\zeta(\bm{x},t)^{2}\mathrm{d}\bm{x}$
$\displaystyle=\sum_{\ell=1,2}\frac{1}{2}\underline{\rho}_{\ell}(\Lambda_{\ell}(\zeta)\phi_{\ell}(t),\phi_{\ell}(t))_{L^{2}}+\frac{1}{2}\|\zeta(t)\|_{L^{2}}^{2},$

which is the total energy, that is, the sum of the kinetic energies of the
waters in the upper and the lower layers and the potential energy due to the
gravity. Here and in what follows, we denote simply
$\Lambda_{1}(\zeta)=\Lambda_{1}(\zeta,\delta,\underline{h}_{1})$ and
$\Lambda_{2}(\zeta)=\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})$. Moreover,
T. B. Benjamin and T. J. Bridges [1] found that the full model can be written
in Hamilton’s canonical form

$\partial_{t}\zeta=\frac{\delta\mathscr{H}^{\mbox{\rm\tiny
IW}}}{\delta\phi},\qquad\partial_{t}\phi=-\frac{\delta\mathscr{H}^{\mbox{\rm\tiny
IW}}}{\delta\zeta},$

where the canonical variable $\phi$ is defined by

(2.21) $\phi=\underline{\rho}_{2}\phi_{2}-\underline{\rho}_{1}\phi_{1}$

and the Hamiltonian $\mathscr{H}^{\mbox{\rm\tiny IW}}$ is the total energy
$\mathscr{E}$ written in terms of the canonical variables $(\zeta,\phi)$. It
follows from the kinematic boundary conditions on the interface that
$\Lambda_{1}(\zeta)\phi_{1}+\Lambda_{2}(\zeta)\phi_{2}=0$, so that $\phi_{1}$
and $\phi_{2}$ can be written in terms of the canonical variables
$(\zeta,\phi)$ as

$\begin{cases}\phi_{1}=-(\underline{\rho}_{1}\Lambda_{2}(\zeta)+\underline{\rho}_{2}\Lambda_{1}(\zeta))^{-1}\Lambda_{2}(\zeta)\phi,\\\
\phi_{2}=(\underline{\rho}_{1}\Lambda_{2}(\zeta)+\underline{\rho}_{2}\Lambda_{1}(\zeta))^{-1}\Lambda_{1}(\zeta)\phi.\end{cases}$

Therefore, the Hamiltonian $\mathscr{H}^{\mbox{\rm\tiny IW}}(\zeta,\phi)$ of
the full model for interfacial gravity waves is given explicitly by

$\mathscr{H}^{\mbox{\rm\tiny
IW}}(\zeta,\phi)=\frac{1}{2}((\underline{\rho}_{1}\Lambda_{2}(\zeta)+\underline{\rho}_{2}\Lambda_{1}(\zeta))^{-1}\Lambda_{1}(\zeta)\phi,\Lambda_{2}(\zeta)\phi)_{L^{2}}+\frac{1}{2}\|\zeta\|_{L^{2}}^{2}.$

As was shown in the companion paper [3], the Kakinuma model (2.18) also enjoys
a Hamiltonian structure analogous to that of the full model for interfacial
gravity waves. The canonical variables are the elevation of the interface
$\zeta$ and $\phi$ defined by

(2.22) $\displaystyle\phi(\bm{x},t)$
$\displaystyle:=\underline{\rho}_{2}\Phi_{2}^{\mathrm{app}}(\bm{x},\zeta(\bm{x},t),t)-\underline{\rho}_{1}\Phi_{1}^{\mathrm{app}}(\bm{x},\zeta(\bm{x},t),t)$
$\displaystyle=\underline{\rho}_{2}\sum_{i=0}^{N^{*}}H_{2}(\bm{x},t)^{p_{i}}\phi_{2,i}(\bm{x},t)-\underline{\rho}_{1}\sum_{i=0}^{N}H_{1}(\bm{x},t)^{2i}\phi_{1,i}(\bm{x},t),$

where $\Phi_{\ell}^{\mathrm{app}}$ $(\ell=1,2)$ are nondimensional versions of
the approximate velocity potentials, which are defined by

(2.23)
$\begin{cases}\displaystyle\Phi_{1}^{\mathrm{app}}(\bm{x},z,t):=\sum_{i=0}^{N}(1-\underline{h}_{1}^{-1}z)^{2i}\phi_{1,i}(\bm{x},t),\\\\[10.76385pt]
\displaystyle\Phi_{2}^{\mathrm{app}}(\bm{x},z,t):=\sum_{i=0}^{N^{*}}(1+\underline{h}_{2}^{-1}(z-b(\bm{x})))^{p_{i}}\phi_{2,i}(\bm{x},t),\end{cases}$

and $H_{\ell}$ $(\ell=1,2)$ are thicknesses of the upper and lower layers
defined by (2.19). We note that if the canonical variables $(\zeta,\phi)$ are
given, then the Kakinuma model (2.18) determines
$\bm{\phi}_{1}=(\phi_{1,0},\phi_{1,1},\ldots,\phi_{1,N})^{\mathrm{T}}$ and
$\bm{\phi}_{2}=(\phi_{2,0},\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\mathrm{T}}$,
which are unique up to an additive constant of the form
$(\mathcal{C}\underline{\rho}_{1},\mathcal{C}\underline{\rho}_{2})$ to
$(\phi_{1,0},\phi_{2,0})$. For details, we refer to [3, Subsection 8.1] and
Lemma 5.1 in Section 5. Then, the Hamiltonian $\mathscr{H}^{\mbox{\rm\tiny
K}}(\zeta,\phi)$ of the Kakinuma model is given by

$\displaystyle\mathscr{H}^{\mbox{\rm\tiny K}}(\zeta,\phi)$
$\displaystyle:=\sum_{\ell=1,2}\iint_{\Omega_{\ell}}\frac{1}{2}\underline{\rho}_{\ell}\bigl{(}|\nabla\Phi_{\ell}^{\mathrm{app}}(\bm{x},z,t)|^{2}+\delta^{-2}(\partial_{z}\Phi_{\ell}^{\mathrm{app}}(\bm{x},z,t))^{2}\bigr{)}\mathrm{d}\bm{x}\mathrm{d}z+\int_{\mathbf{R}^{n}}\frac{1}{2}\zeta(\bm{x},t)^{2}\mathrm{d}\bm{x}.$

##  3 Statements of the main results

Before stating the main results in this paper, let us introduce some notations
which allow in particular to rewrite (2.18) in a compact form. We introduce
second order differential operators
$L_{1,ij}=L_{1,ij}(H_{1},\delta,\underline{h}_{1})$ $(i,j=0,1,\ldots,N)$ and
$L_{2,ij}=L_{2,ij}(H_{2},b,\delta,\underline{h}_{2})$ $(i,j=0,1,\ldots,N^{*})$
by

(3.1) $\displaystyle L_{1,ij}\varphi_{1,j}$
$\displaystyle:=-\nabla\cdot\biggl{(}\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\nabla\varphi_{1,j}\biggr{)}+\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}(\underline{h}_{1}\delta)^{-2}\varphi_{1,j},$
(3.2) $\displaystyle L_{2,ij}\varphi_{2,j}$
$\displaystyle:=-\nabla\cdot\biggl{(}\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\nabla\varphi_{2,j}-\frac{p_{j}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\varphi_{2,j}\underline{h}_{2}^{-1}\nabla
b\biggr{)}$
$\displaystyle\quad\,-\frac{p_{i}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\underline{h}_{2}^{-1}\nabla
b\cdot\nabla\varphi_{2,j}$
$\displaystyle\quad\,+\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}((\underline{h}_{2}\delta)^{-2}+\underline{h}_{2}^{-2}|\nabla
b|^{2})\varphi_{2,j}.$

Notice that we have $(L_{\ell,ij})^{*}=L_{\ell,ji}$ for $\ell=1,2$, where
$(L_{\ell,ij})^{*}$ is the adjoint operator of $L_{\ell,ij}$ in
$L^{2}(\mathbf{R}^{n})$. We put
${\bm{\phi}}_{1}:=(\phi_{1,0},\phi_{1,1},\ldots,\phi_{1,N})^{\mathrm{T}}$,
${\bm{\phi}}_{2}:=(\phi_{2,0},\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\mathrm{T}}$,
and

(3.3)
$\begin{cases}{\bm{l}}_{1}(H_{1}):=(1,H_{1}^{2},H_{1}^{4},\ldots,H_{1}^{2N})^{\mathrm{T}},\\\
{\bm{l}}_{1}^{\prime}(H_{1}):=(0,2H_{1},\ldots,2NH_{1}^{2N-1})^{\mathrm{T}},\\\
{\bm{l}}_{1}^{\prime\prime}(H_{1}):=(0,2,\ldots,2N(2N-1)H_{1}^{2N-2})^{\mathrm{T}},\\\
{\bm{l}}_{2}(H_{2}):=(1,H_{2}^{p_{1}},H_{2}^{p_{2}},\ldots,H_{2}^{p_{N^{*}}})^{\mathrm{T}},\\\
{\bm{l}}_{2}^{\prime}(H_{2}):=(0,p_{1}H_{2}^{p_{1}-1},\ldots,p_{N^{*}}H_{2}^{p_{N^{*}}})^{\mathrm{T}},\\\
{\bm{l}}_{2}^{\prime\prime}(H_{2}):=(0,p_{1}(p_{1}-1)H_{2}^{p_{1}-2},\ldots,p_{N^{*}}(p_{N^{*}}-1)H_{2}^{p_{N^{*}}})^{\mathrm{T}},\end{cases}$

and define ${\bm{u}}_{\ell}$ and $w_{\ell}$ for $\ell=1,2$, which represent
approximately the horizontal and the vertical components of the velocity field
on the interface from the water region $\Omega_{\ell}(t)$, by

(3.4)
$\left\\{\begin{array}[]{ll}\displaystyle{\bm{u}}_{1}:=({\bm{l}}_{1}(H_{1})\otimes\nabla)^{\mathrm{T}}{\bm{\phi}}_{1},&\displaystyle
w_{1}:=-{\bm{l}}_{1}^{\prime}(H_{1})\cdot{\bm{\phi}}_{1},\\\
\displaystyle{\bm{u}}_{2}:=({\bm{l}}_{2}(H_{2})\otimes\nabla)^{\mathrm{T}}{\bm{\phi}}_{2}-({\bm{l}}_{2}^{\prime}(H_{2})\cdot{\bm{\phi}}_{2})\underline{h}_{2}^{-1}\nabla
b,&\displaystyle
w_{2}:={\bm{l}}_{2}^{\prime}(H_{2})\cdot{\bm{\phi}}_{2}.\end{array}\right.$

Then, denoting $L_{1}:=(L_{1,ij})_{0\leq i,j\leq N}$ and
$L_{2}:=(L_{2,ij})_{0\leq i,j\leq N^{*}}$ we can write the Kakinuma model
(2.18) more compactly as

(3.5)
$\begin{cases}\displaystyle{\bm{l}}_{1}(H_{1})\partial_{t}\zeta+\underline{h}_{1}L_{1}(H_{1},\delta,\underline{h}_{1}){\bm{\phi}}_{1}={\bm{0}},\\\
\displaystyle{\bm{l}}_{2}(H_{2})\partial_{t}\zeta-\underline{h}_{2}L_{2}(H_{2},b,\delta,\underline{h}_{2}){\bm{\phi}}_{2}={\bm{0}},\\\
\underline{\rho}_{1}\bigl{\\{}{\bm{l}}_{1}(H_{1})\cdot\partial_{t}{\bm{\phi}}_{1}+\frac{1}{2}\bigl{(}|{\bm{u}}_{1}|^{2}+(\underline{h}_{1}\delta)^{-2}w_{1}^{2}\bigr{)}\bigr{\\}}\\\
\quad-\underline{\rho}_{2}\bigl{\\{}{\bm{l}}_{2}(H_{2})\cdot\partial_{t}{\bm{\phi}}_{2}+\frac{1}{2}\bigl{(}|{\bm{u}}_{2}|^{2}+(\underline{h}_{2}\delta)^{-2}w_{2}^{2}\bigr{)}\bigr{\\}}-\zeta=0.\end{cases}$

By eliminating $\partial_{t}\zeta$ from the Kakinuma model, we obtain
$N+N^{*}+1$ scalar relations which are necessary conditions for the existence
of the solution to the Kakinuma model, as stated below. Introducing linear
operators
$\mathcal{L}_{1,i}:=\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})$
$(i=0,\ldots,N)$ acting on
${\bm{\varphi}}_{1}=(\varphi_{1,0},\ldots,\varphi_{1,N})^{\mathrm{T}}$ and
$\mathcal{L}_{2,i}:=\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})$
$(i=0,\ldots,N^{*})$ acting on
${\bm{\varphi}}_{2}=(\varphi_{2,0},\ldots,\varphi_{2,N^{*}})^{\mathrm{T}}$ by

(3.6)
$\begin{cases}\displaystyle\mathcal{L}_{1,0}{\bm{\varphi}}_{1}:=\sum_{j=0}^{N}L_{1,0j}\varphi_{1,j},\\\
\displaystyle\mathcal{L}_{1,i}{\bm{\varphi}}_{1}:=\sum_{j=0}^{N}(L_{1,ij}\varphi_{1,j}-H_{1}^{2i}L_{1,0j}\varphi_{1,j})\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\displaystyle\mathcal{L}_{2,0}{\bm{\varphi}}_{2}:=\sum_{j=0}^{N^{*}}L_{2,0j}\varphi_{2,j},\\\
\displaystyle\mathcal{L}_{2,i}{\bm{\varphi}}_{2}:=\sum_{j=0}^{N^{*}}(L_{2,ij}\varphi_{2,j}-H_{2}^{p_{i}}L_{2,0j}\varphi_{2,j})\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\end{cases}$

the necessary conditions can be written simply as

(3.7)
$\begin{cases}\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1}){\bm{\phi}}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2}){\bm{\phi}}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}(H_{1},\delta,\underline{h}_{1}){\bm{\phi}}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}(H_{2},b,\delta,\underline{h}_{2}){\bm{\phi}}_{2}=0.\end{cases}$

Hereafter, these necessary conditions will be referred to as the compatibility
conditions. Notice that under these compatibility conditions we have for
$\ell=1,2$

(3.8)
$L_{\ell}\bm{\phi}_{\ell}=\bm{l}_{\ell}\mathcal{L}_{\ell,0}\bm{\phi}_{\ell},$

where $\bm{l}_{\ell}=\bm{l}_{\ell}(H_{\ell})$ and similar simplifications of
notations will be used in the following without any comments. In connection
with the stability condition (1.5), we introduce a function

(3.9) $\displaystyle a:=1$
$\displaystyle+\underline{\rho}_{1}\underline{h}_{1}^{-1}\\{\bm{l}_{1}^{\prime}(H_{1})\cdot(\partial_{t}+\bm{u}_{1}\cdot\nabla)\bm{\phi}_{1}-(\underline{h}_{1}\delta)^{-2}w_{1}\bm{l}_{1}^{\prime\prime}(H_{1})\cdot\bm{\phi}_{1}\\}$
$\displaystyle+\underline{\rho}_{2}\underline{h}_{2}^{-1}\\{\bm{l}_{2}^{\prime}(H_{2})\cdot(\partial_{t}+\bm{u}_{2}\cdot\nabla)\bm{\phi}_{2}+\bigl{(}(\underline{h}_{2}\delta)^{-2}w_{2}-\underline{h}_{2}^{-1}\nabla
b\cdot\bm{u}_{2}\bigr{)}\bm{l}_{2}^{\prime\prime}(H_{2})\cdot\bm{\phi}_{2}\\},$

which corresponds to
$-(\partial_{z}(P_{2}^{\mathrm{app}}-P_{1}^{\mathrm{app}}))|_{\Gamma(t)}$ in
the stability condition.

Our first main result in this paper is the existence of the solution to the
initial value problem (2.18)–(2.20) for the Kakinuma model on a time interval
independent of parameters, especially, the shallowness parameters
$\delta_{1}=\underline{h}_{1}\delta$ and $\delta_{2}=\underline{h}_{2}\delta$
together with a uniform bound of the solution. For simplicity, we denote
$H_{\ell(0)}:=H_{\ell}|_{t=0}$, $\bm{u}_{\ell(0)}:=\bm{u}_{\ell}|_{t=0}$ for
$\ell=1,2$, and $a_{(0)}:=a|_{t=0}$, which can be written in terms of the
initial data according to the initial condition (2.20). Although the function
$a$ includes the terms $(\partial_{t}\bm{\phi}_{\ell}^{\prime})|_{t=0}$ for
$\ell=1,2$, where
$\bm{\phi}_{1}^{\prime}=(\phi_{1,1},\ldots,\phi_{1,N})^{\mathrm{T}}$ and
$\bm{\phi}_{2}^{\prime}=(\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\mathrm{T}}$, and
the hypersurface $t=0$ is characteristic for the Kakinuma model, we can
uniquely determine them in terms of the initial data. For details, we refer to
Remark 5.3.

######  Theorem 3.1.

Let $c_{0},M_{0},\underline{h}_{\mathrm{min}}$ be positive constants and $m$
an integer such that ${m>\frac{n}{2}+1}$. There exist a time $T>0$ and a
constant $M>0$ 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),
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, as well as the
condition
${\underline{h}_{\mathrm{min}}\leq\underline{h}_{1},\underline{h}_{2}}$, if
the initial data $(\zeta_{(0)},\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ and the
bottom topography $b$ satisfy

(3.10)
$\begin{cases}\displaystyle\|\zeta_{(0)}\|_{H^{m}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\nabla\bm{\phi}_{\ell(0)}\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell(0)}^{\prime}\|_{H^{m}}^{2}\bigr{)}\leq
M_{0},\\\
\underline{h}_{2}^{-1}\bigl{(}\|b\|_{W^{m+1,\infty}}+(\underline{h}_{2}\delta)\|b\|_{W^{m+2,\infty}}\bigr{)}\leq
M_{0},\end{cases}$

the non-cavitation assumption

(3.11) $H_{1(0)}(\bm{x})\geq c_{0},\quad H_{2(0)}(\bm{x})\geq
c_{0}\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},$

the stability condition

(3.12)
$a_{(0)}(\bm{x})-\frac{\underline{\rho}_{1}\underline{\rho}_{2}}{\underline{\rho}_{1}\underline{h}_{2}H_{2(0)}(\bm{x})\alpha_{2}+\underline{\rho}_{2}\underline{h}_{1}H_{1(0)}(\bm{x})\alpha_{1}}|{\bm{u}}_{1(0)}(\bm{x})-{\bm{u}}_{2(0)}(\bm{x})|^{2}\geq
c_{0}\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},$

with positive constants $\alpha_{1}$ and $\alpha_{2}$ defined by (6.9), and
the compatibility conditions

(3.13)
$\begin{cases}\mathcal{L}_{1,i}(H_{1(0)},\delta,\underline{h}_{1}){\bm{\phi}}_{1(0)}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2(0)},b,\delta,\underline{h}_{2}){\bm{\phi}}_{2(0)}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}(H_{1(0)},\delta,\underline{h}_{1}){\bm{\phi}}_{1(0)}+\underline{h}_{2}\mathcal{L}_{2,0}(H_{2(0)},b,\delta,\underline{h}_{2}){\bm{\phi}}_{2(0)}=0,\end{cases}$

then the initial value problem (2.18)–(2.20) has a unique solution
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ on the time interval $[0,T]$ satisfying

$\begin{cases}\zeta,\nabla\phi_{1,0},\nabla\phi_{2,0}\in C([0,T];H^{m})\cap
C^{1}([0,T];H^{m-1}),\\\ {\bm{\phi}}_{1}^{\prime},{\bm{\phi}}_{2}^{\prime}\in
C([0,T];H^{m+1})\cap C^{1}([0,T];H^{m}),\end{cases}$

where we recall the notation
${\bm{\phi}}_{1}^{\prime}=(\phi_{1,1},\phi_{1,2},\ldots,\phi_{1,N})^{\mathrm{T}}$
and
${\bm{\phi}}_{2}^{\prime}=(\phi_{2,1},\phi_{2,2},\ldots,\phi_{2,N^{*}})^{\mathrm{T}}$.
Moreover, the solution satisfies the uniform bound

(3.14)
$\|\zeta(t)\|_{H^{m}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\nabla\bm{\phi}_{\ell}(t)\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\prime}(t)\|_{H^{m}}^{2}\bigr{)}\leq
M$

for $t\in[0,T]$ together with

(3.15) $\begin{cases}\displaystyle
a(\bm{x},t)-\frac{\underline{\rho}_{1}\underline{\rho}_{2}}{\underline{\rho}_{1}\underline{h}_{2}H_{2}(\bm{x},t)\alpha_{2}+\underline{\rho}_{2}\underline{h}_{1}H_{1}(\bm{x},t)\alpha_{1}}|{\bm{u}}_{1}(\bm{x},t)-{\bm{u}}_{2}(\bm{x},t)|^{2}\geq
c_{0}/2,\\\ \displaystyle H_{1}(\bm{x},t)\geq c_{0}/2,\quad
H_{2}(\bm{x},t)\geq
c_{0}/2\qquad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},t\in[0,T].\end{cases}$

######  Remark 3.2.

It is easy to check that the non-cavitation assumption (3.11) and the
stability condition (3.12) are automatically satisfied for small initial data
$(\zeta_{(0)},\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ and small bottom topography
$b$, whereas an arrangement of nontrivial initial data satisfying the
compatibility conditions (3.13) together with the uniform bound (3.10) is a
non-trivial issue. To this end, we use the canonical variable $\phi$ defined
by (2.22), which can be written as

(3.16)
$\phi=\underline{\rho}_{2}\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}-\underline{\rho}_{1}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}.$

Given the initial data $(\zeta_{(0)},\phi_{(0)})$ for the canonical variables
$(\zeta,\phi)$ of the Kakinuma model and the bottom topography $b$, the
necessary conditions (3.7) and the above relation (3.16) determine the initial
data $(\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ for the Kakinuma model satisfying
the compatibility conditions (3.13) and the uniform bound (3.10), which is
unique up to an additive constant of the form
$(\mathcal{C}\underline{\rho}_{2},\mathcal{C}\underline{\rho}_{1})$ to
$(\phi_{1,0(0)},\phi_{2,0(0)})$. In fact, we have the following proposition,
which is a simple corollary of Lemma 5.1 given in Section 5.

######  Proposition 3.3.

Let $c_{0},M_{0}$ be positive constants and $m$ an integer such that
$m>\frac{n}{2}+1$. 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 the natural restrictions (2.14) and
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, if the initial data
$(\zeta_{(0)},\phi_{(0)})\in H^{m}\times\mathring{H}^{m}$ of the canonical
variables, the bottom topography $b\in W^{m,\infty}$, and initial depths
$H_{1(0)}:=1-\underline{h}_{1}^{-1}\zeta_{(0)}$ and
$H_{2(0)}:=1+\underline{h}_{2}^{-1}\zeta_{(0)}-\underline{h}_{2}^{-1}b$
satisfy

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

then there exist initial data $(\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ satisfying
the compatibility conditions (3.13) as well as
$\phi_{(0)}=\underline{\rho}_{2}\bm{l}_{2}(H_{2(0)})\cdot\bm{\phi}_{2(0)}-\underline{\rho}_{1}\bm{l}_{1}(H_{1(0)})\cdot\bm{\phi}_{1(0)}$.
Moreover, we have

$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\nabla\bm{\phi}_{\ell(0)}\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell(0)}^{\prime}\|_{H^{m-1}}^{2}\bigr{)}\leq
C\|\nabla\phi_{(0)}\|_{H^{m-1}}^{2}.$

The next theorem shows that the Kakinuma model (2.18)–(2.19) is consistent
with the full model for interfacial gravity waves (2.17) at order
$O((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})$ under
the special choice of the indices $p_{0},p_{1},\ldots,p_{N^{*}}$ as

  1. (H1)

$N^{*}=N$ and $p_{i}=2i$ $(i=0,1,\ldots,N)$ in the case of the flat bottom
$b(\bm{x})\equiv 0$,

  2. (H2)

$N^{*}=2N$ and $p_{i}=i$ $(i=0,1,\ldots,2N)$ in the case with general bottom
topographies.

######  Theorem 3.4.

Let $c,M$ be positive constants and $m$ an integer such that $m\geq 4(N+1)$
and $m>\frac{n}{2}+1$. 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 $\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$ and for any
solution $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ to the Kakinuma model
(2.18)–(2.19) on a time interval $[0,T]$ with a bottom topography $b\in
W^{m+1,\infty}$ satisfying

(3.17)
$\begin{cases}\underline{h}_{1}^{-1}\|\zeta(t)\|_{H^{m}}+\underline{h}_{2}^{-1}\|\zeta(t)\|_{H^{m}}+\underline{h}_{2}^{-1}\|b\|_{W^{m+1,\infty}}\leq
M,\\\ H_{1}(\bm{x},t)\geq c,\quad H_{2}(\bm{x},t)\geq
c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},t\in[0,T],\end{cases}$

if we define $\phi_{\ell}:=\bm{l}_{\ell}(H_{\ell})\cdot\bm{\phi}_{\ell}$ for
$\ell=1,2$, then $(\zeta,\phi_{1},\phi_{2})$ satisfy approximately the full
model for interfacial gravity waves as

$\begin{cases}\partial_{t}\zeta+\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}=\mathfrak{r}_{1},\\\
\partial_{t}\zeta-\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}=\mathfrak{r}_{2},\\\
\displaystyle\underline{\rho}_{1}\biggl{(}\partial_{t}\phi_{1}+\frac{1}{2}|\nabla\phi_{1}|^{2}-\frac{1}{2}\delta^{2}\frac{(\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}-\nabla\zeta\cdot\nabla\phi_{1})^{2}}{1+\delta^{2}|\nabla\zeta|^{2}}\biggr{)}\\\
\displaystyle\quad-\underline{\rho}_{2}\biggl{(}\partial_{t}\phi_{2}+\frac{1}{2}|\nabla\phi_{2}|^{2}-\frac{1}{2}\delta^{2}\frac{(\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}+\nabla\zeta\cdot\nabla\phi_{2})^{2}}{1+\delta^{2}|\nabla\zeta|^{2}}\biggr{)}-\zeta=\mathfrak{r}_{0}.\end{cases}$

Here, the errors $(\mathfrak{r}_{1},\mathfrak{r}_{2},\mathfrak{r}_{0})$
satisfy

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

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

Particularly, we see that under the special choice of indices (H1) or (H2) the
solutions to the Kakinuma model constructed in Theorem 3.1 satisfy
approximately the full model for interfacial gravity waves with the choice
$\phi_{\ell}=\bm{l}_{\ell}(H_{\ell})\cdot\bm{\phi}_{\ell}$ $(\ell=1,2)$ and
that the error is of order
$O((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})$.

Conversely, the next theorem shows that the full model for interfacial gravity
waves is consistent with the Kakinuma model at order
$O((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2})$ under
the special choice of indices (H1) or (H2).

######  Theorem 3.5.

Let $c,M$ be positive constants and $m$ an integer such that $m\geq 4(N+1)$
and $m>\frac{n}{2}+1$. 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 $\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$ and for any
solution $(\zeta,\phi_{1},\phi_{2})$ to the full model for interfacial gravity
waves (2.17) on a time interval $[0,T]$ with a bottom topography $b\in
W^{m+1,\infty}$ satisfying (3.17), if we define $H_{1}$ and $H_{2}$ as in
(2.19) and $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ as the unique solutions to the
problems

(3.18)
$\begin{cases}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}=\phi_{1},\quad\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})\bm{\phi}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}=\phi_{2},\quad\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})\bm{\phi}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\end{cases}$

then $(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ satisfy approximately the Kakinuma
model as

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

Here, the errors
$(\tilde{\bm{\mathfrak{r}}}_{1},\tilde{\bm{\mathfrak{r}}}_{2},\tilde{\mathfrak{r}}_{0})$
satisfy

(3.19)
$\begin{cases}\|\tilde{\bm{\mathfrak{r}}}_{\ell}(t)\|_{H^{m-4(N+1)}}\leq
C(\underline{h}_{\ell}\delta)^{4N+2}\|\nabla\phi_{\ell}(t)\|_{H^{m-1}}\quad(\ell=1,2),\\\
\displaystyle\|\tilde{\mathfrak{r}}_{0}(t)\|_{H^{m-4(N+1)}}\leq
C\sum_{\ell=1,2}\underline{\rho}_{\ell}(\underline{h}_{\ell}\delta)^{4N+2}\|\nabla\phi_{\ell}(t)\|_{H^{m-1}}^{2}\end{cases}$

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

######  Remark 3.6.

The unique existence of the solutions $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ to
the problems (3.18) is guaranteed by Lemma 4.4 below under an additional
assumption $\phi_{1}(\cdot,t),\phi_{2}(\cdot,t)\in\mathring{H}^{m}$. Lemma 4.4
is essentially a simple corollary of [6, Lemma 3.4].

######  Remark 3.7.

In order to define the approximate solution $(\bm{\phi}_{1},\bm{\phi}_{2})$ to
the Kakinuma model from the solution $(\zeta,\phi_{1},\phi_{2})$ to the full
model, we can use, in place of (3.18), the following system of equations

(3.20)
$\begin{cases}\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})\bm{\phi}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{1},b,\delta,\underline{h}_{2})\bm{\phi}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}(H_{1},\delta,\underline{h}_{1}){\bm{\phi}}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}(H_{2},b,\delta,\underline{h}_{2}){\bm{\phi}}_{2}=0,\\\
\underline{\rho}_{2}\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}-\underline{\rho}_{1}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}=\phi,\end{cases}$

where $\phi=\underline{\rho}_{2}\phi_{2}-\underline{\rho}_{1}\phi_{1}$ is the
canonical variable for the full model for interfacial gravity waves. The above
system is nothing but the compatibility conditions (3.7) together with the
definition (3.16) of the canonical variable for the Kakinuma model. The
existence of the approximate solution $(\bm{\phi}_{1},\bm{\phi}_{2})$ is
guaranteed by Lemma 5.1 given in Section 5. Then, we have similar error
estimates to (3.19). For details, we refer to Proposition 7.6.

The above Theorems 3.4 and 3.5 concern essentially the approximation of the
equations. To give a rigorous justification of the Kakinuma model as a higher
order shallow water approximation, one needs to give an error estimate between
solutions to the Kakinuma model and that to the full model. However, we cannot
expect to construct general solutions to the initial value problem for the
full model for interfacial gravity waves because the initial value problem is
ill-posed. Nevertheless, if we assume the existence of a solution to the full
model with a uniform bound with respect to the shallowness parameters
$\delta_{1}=\underline{h}_{1}\delta$ and $\delta_{2}=\underline{h}_{2}\delta$,
then we can give an error estimate with respect to a solution to the Kakinuma
model by making use of the well-posedness of the initial value problem for the
Kakinuma model as we can see in the following theorem.

######  Theorem 3.8.

Let $c,M,\underline{h}_{\mathrm{min}}$ be positive constants and $m$ an
integer such that ${m>\frac{n}{2}+4(N+1)}$. We assume (H1) or (H2). Then,
there exist a time $T>0$ and a constant $C>0$ such that the following holds
true. Let
$\underline{\rho}_{1},\underline{\rho}_{2},\underline{h}_{1},\underline{h}_{2},\delta$
be positive parameters satisfying the natural restrictions (2.14),
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, and the condition
${\underline{h}_{\mathrm{min}}\leq\underline{h}_{1},\underline{h}_{2}}$, and
let $b\in W^{m+2,\infty}$ such that
$\underline{h}_{2}^{-1}\|b\|_{W^{m+2,\infty}}\leq M$. Suppose that the full
model for interfacial gravity waves (2.17) possesses a solution
$(\zeta^{\mbox{\tiny\rm IW}},\phi_{1}^{\mbox{\tiny\rm
IW}},\phi_{2}^{\mbox{\tiny\rm IW}})\in C([0,T^{\mbox{\tiny\rm
IW}}];H^{m+1}\times\mathring{H}^{m+1}\times\mathring{H}^{m+1})$ satisfying a
uniform bound

$\begin{cases}\displaystyle\|\zeta^{\mbox{\tiny\rm
IW}}(t)\|_{H^{m+1}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\nabla\phi_{\ell}^{\mbox{\tiny\rm
IW}}(t)\|_{H^{m}}^{2}\leq M,\\\ H_{1}^{\mbox{\tiny\rm IW}}(\bm{x},t)\geq
c,\quad H_{2}^{\mbox{\tiny\rm IW}}(\bm{x},t)\geq
c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},t\in[0,T^{\mbox{\tiny\rm
IW}}],\end{cases}$

where we denote $H_{1}^{\mbox{\tiny\rm
IW}}:=1-\underline{h}_{1}^{-1}\zeta^{\mbox{\tiny\rm IW}}$ and
$H_{2}^{\mbox{\tiny\rm IW}}:=1+\underline{h}_{2}^{-1}\zeta^{\mbox{\tiny\rm
IW}}-\underline{h}_{2}^{-1}b$. Let $\zeta_{(0)}:=\zeta^{\mbox{\tiny\rm
IW}}|_{t=0}$ and $\phi_{(0)}:=(\underline{\rho}_{2}\phi_{2}^{\mbox{\tiny\rm
IW}}-\underline{\rho}_{1}\phi_{1}^{\mbox{\tiny\rm IW}})|_{t=0}$ be the initial
data for the canonical variables, and let
$(\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ be the initial data to the Kakinuma
model constructed from $(\zeta_{(0)},\phi_{(0)})$ by Proposition 3.3. Assume
moreover that the initial data
$(\zeta_{(0)},\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ satisfy the stability
condition (3.12), let $(\zeta^{\mbox{\tiny\rm
K}},\bm{\phi}_{1}^{\mbox{\tiny\rm K}},\bm{\phi}_{2}^{\mbox{\tiny\rm K}})$ be
the solution to the initial value problem for the Kakinuma model (2.18)–(2.20)
on the time interval $[0,T]$ whose unique existence is guaranteed by Theorem
3.1, and put $\phi_{\ell}^{\mbox{\tiny\rm
K}}=\bm{l}_{\ell}(H_{\ell})\cdot\bm{\phi}_{\ell}^{\mbox{\tiny\rm K}}$ for
$\ell=1,2$. Then, we have the error bound

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

for $0\leq t\leq\min\\{T,T^{\mbox{\tiny\rm IW}}\\}$.

The next theorem is the final main result in this paper and states the
consistency of the Hamiltonian $\mathscr{H}^{\mbox{\rm\tiny K}}(\zeta,\phi)$
of the Kakinuma model with respect to the Hamiltonian
$\mathscr{H}^{\mbox{\rm\tiny IW}}(\zeta,\phi)$ of the full model for
interfacial gravity waves exhibited by T. B. Benjamin and T. J. Bridges [1].

######  Theorem 3.9.

Let $c,M,\underline{h}_{\mathrm{min}}$ 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 the natural restrictions (2.14),
$\underline{h}_{1}\delta,\underline{h}_{2}\delta\leq 1$, and the condition
${\underline{h}_{\mathrm{min}}\leq\underline{h}_{1},\underline{h}_{2}}$, and
for any $(\zeta,\phi)\in H^{m}\times\mathring{H}^{4(N+1)}$ and $b\in
W^{m+1,\infty}$ satisfying

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

with $H_{1}$ and $H_{2}$ defined by (2.19), we have

$|\mathscr{H}^{\mbox{\rm\tiny K}}(\zeta,\phi)-\mathscr{H}^{\mbox{\rm\tiny
IW}}(\zeta,\phi)|\leq
C\|\nabla\phi\|_{H^{4N+3}}\|\nabla\phi\|_{L^{2}}((\underline{h}_{1}\delta)^{4N+2}+(\underline{h}_{2}\delta)^{4N+2}).$

##  4 Consistency of the Kakinuma model; proof of Theorems 3.4 and 3.5

In this section we show that under the special choice of the indices
$p_{0},p_{1},\ldots,p_{N^{*}}$ as

  1. (H1)

$N^{*}=N$ and $p_{i}=2i$ $(i=0,1,\ldots,N)$ in the case of the flat bottom
$b(\bm{x})\equiv 0$,

  2. (H2)

$N^{*}=2N$ and $p_{i}=i$ $(i=0,1,\ldots,2N)$ in the case with general bottom
topographies,

the Kakinuma model (2.18)–(2.19) is a higher order model to the full model for
interfacial gravity waves (2.17) in the limit
${\delta_{1}=\underline{h}_{1}\delta\to 0}$,
${\delta_{2}=\underline{h}_{2}\delta\to 0}$, in the sense of consistency.
Specifically, we prove Theorems 3.4 and 3.5. Our proof relies essentially on
results obtained in the framework of surface waves in [6], which are recalled
in Subsection 4.1. The extension to the framework of interfacial waves and the
completion of the proof are provided in Subsection 4.2.

###  4.1 Results in the framework of surface waves

In this subsection, we consider the case of surface waves where the water
surface and the bottom of the water are represented as $z=\zeta(\bm{x})$ and
$z=-1+b(\bm{x})$, respectively. Here, the time $t$ is fixed arbitrarily, so
that we omit the dependence of $t$ in notations. Let
$H(\bm{x})=1+\zeta(\bm{x})-b(\bm{x})$ be the water depth. For a nonnegative
integer $N$, let $N^{*}$ and $p_{0},p_{1},\ldots,p_{N^{*}}$ be nonnegative
integers satisfying the condition (H1) or (H2). Put

(4.1) ${\bm{l}}(H):=(1,H^{p_{1}},\dots,H^{p_{N^{*}}})^{\mathrm{T}}$

and define $L_{ij}=L_{ij}(H,b,\delta)$ $(i,j=0,1,\ldots,N^{*})$ by

(4.2) $\displaystyle L_{ij}\varphi_{j}:=$
$\displaystyle-\nabla\cdot\biggl{(}\frac{1}{p_{i}+p_{j}+1}H^{p_{i}+p_{j}+1}\nabla\varphi_{j}-\frac{p_{j}}{p_{i}+p_{j}}H^{p_{i}+p_{j}}\varphi_{j}\nabla
b\biggr{)}$ $\displaystyle-\frac{p_{i}}{p_{i}+p_{j}}H^{p_{i}+p_{j}}\nabla
b\cdot\nabla\varphi_{j}+\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H^{p_{i}+p_{j}-1}(\delta^{-2}+|\nabla
b|^{2})\varphi_{j}.$

Introduce linear operators $\mathcal{L}_{i}=\mathcal{L}_{i}(H,b,\delta)$
$(i=0,1,\ldots,N^{*})$ acting on
${\bm{\varphi}}=(\varphi_{0},\ldots,\varphi_{N^{*}})^{\mathrm{T}}$ by

(4.3)
$\begin{cases}\displaystyle\mathcal{L}_{0}{\bm{\varphi}}:=\sum_{j=0}^{N^{*}}L_{0j}\varphi_{j},\\\
\displaystyle\mathcal{L}_{i}{\bm{\varphi}}:=\sum_{j=0}^{N^{*}}(L_{ij}\varphi_{j}-H^{p_{i}}L_{0j}\varphi_{j})\quad\mbox{for}\quad
i=1,2,\ldots,N^{*}.\end{cases}$

The following Lemma has been proved in [6, Lemmas 3.2 and 3.4].

######  Lemma 4.1.

Let $c,M$ be positive constants and $m$ an integer such that
$m>\frac{n}{2}+1$. 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) $\begin{cases}\|\zeta\|_{H^{m}}+\|b\|_{W^{m,\infty}}\leq M,\\\
H(\bm{x})\geq c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},\end{cases}$

then for any $k=\pm 0,\dots,\pm(m-1)$, any $\delta\in(0,1]$, and any
$\phi\in\mathring{H}^{k+1}$ there exists a unique solution
$\bm{\phi}=(\phi_{0},\phi_{1},\ldots,\phi_{N^{*}})=(\phi_{0},\bm{\phi}^{\prime})\in\mathring{H}^{k+1}\times(H^{k+1})^{N^{*}}$
to the problem

(4.5)
$\begin{cases}\mathcal{L}_{i}(H,b,\delta){\bm{\phi}}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\ {\bm{l}}(H)\cdot{\bm{\phi}}=\phi.\end{cases}$

Moreover, the solution satisfies
$\|\nabla\bm{\phi}\|_{H^{k}}+\delta^{-1}\|\bm{\phi}^{\prime}\|_{H^{k}}\leq
C\|\nabla\phi\|_{H^{k}}$.

As a corollary of this lemma, under the assumptions of Lemma 4.1

$\Lambda^{(N)}(\zeta,b,\delta)\colon\phi\mapsto\mathcal{L}_{0}(H,b,\delta)\bm{\phi},$

where $\bm{\phi}$ is the unique solution to (4.5), is defined as a bounded
linear operator from $\mathring{H}^{k+1}$ to $H^{k-1}$ for any $k=\pm
0,\dots,\pm(m-1)$. A key result is that the operator
$\Lambda^{(N)}(\zeta,b,\delta)$ provide good approximations in the shallow
water regime $\delta\ll 1$ to the corresponding Dirichlet-to-Neumann map
$\Lambda(\zeta,b,\delta)$, which is defined by

(4.6)
$\Lambda(\zeta,b,\delta)\phi:=\big{(}\delta^{-2}\partial_{z}\Phi-\nabla\zeta\cdot\nabla\Phi\big{)}\big{|}_{z=\zeta},$

where $\Phi$ is the unique solution to the boundary value problem

(4.7)
$\begin{cases}\Delta\Phi+\delta^{-2}\partial_{z}^{2}\Phi=0&\mbox{in}\quad-1+b(\bm{x})<z<\zeta(\bm{x}),\\\
\Phi=\phi&\mbox{on}\quad z=\zeta(\bm{x}),\\\ \nabla
b\cdot\nabla\Phi-\delta^{-2}\partial_{z}\Phi=0&\mbox{on}\quad
z=-1+b(\bm{x}).\end{cases}$

More precisely, we have the following Lemma.

######  Lemma 4.2.

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 if $\zeta\in H^{m}$,
$b\in W^{m+1,\infty}$, and $H=1+\zeta-b$ satisfy

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

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

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

###### Proof.

We observe that the bound on
$\mathfrak{r}_{1}:=\Lambda^{(N)}(\zeta,b,\delta)\phi-\Lambda(\zeta,b,\delta)\phi$
in the case $j=2N+1$ and $k=m-4(N+1)$ is given in [6, Theorem 2.2] and proved
in [6, Sections 8.1 and 8.2]. The proof is also valid in the case $1\leq j\leq
2N+1$ and $0\leq k\leq m-2(j+1)$. ∎

The above estimate allows us to obtain the desired consistency result on the
equations describing the conservation of mass. We need a similar estimate for
the contributions of Bernoulli’s equation. To this end, we denote

(4.9)
$B(\phi;\zeta,b,\delta):=\frac{1}{2}|\nabla\phi|^{2}-\frac{1}{2}\delta^{2}\frac{(\Lambda(\zeta,b,\delta)\phi+\nabla\zeta\cdot\nabla\phi)^{2}}{1+\delta^{2}|\nabla\zeta|^{2}}$

and

(4.10)
$B^{(N)}(\phi;\zeta,b,\delta):=\frac{1}{2}\bigl{(}|{\bm{u}}|^{2}+\delta^{-2}w^{2}\bigr{)}-w\Lambda^{(N)}(\zeta,b,\delta)\phi$

with

$\begin{cases}\displaystyle\bm{u}:=({\bm{l}}(H)\otimes\nabla)^{\mathrm{T}}{\bm{\phi}}-({\bm{l}}^{\prime}(H)\cdot{\bm{\phi}})\nabla
b,\\\ \displaystyle w:={\bm{l}}^{\prime}(H)\cdot{\bm{\phi}},\end{cases}$

where
${\bm{l}}^{\prime}(H):=(0,p_{1}H^{p_{1}-1},\ldots,p_{N^{*}}H^{p_{N^{*}}-1})^{\mathrm{T}}$
and ${\bm{\phi}}:=(\phi_{0},\phi_{1},\ldots,\phi_{N^{*}})^{\mathrm{T}}$ is the
solution to (4.5), whose unique existence is guaranteed by Lemma 4.1. Then,
the following lemma shows that $B^{(N)}(\phi;\zeta,b,\delta)$ is a higher
order approximation to $B(\phi;\zeta,b,\delta)$ in the shallow water regime
$\delta\ll 1$.

######  Lemma 4.3.

Let $c,M$ be positive constants and $m$ an integer such that $m\geq 4(N+1)$
and $m>\frac{n}{2}+1$. 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 $\phi\in\mathring{H}^{m}$ and any
$\delta\in(0,1]$ we have

$\|B^{(N)}(\phi;\zeta,b,\delta)-B(\phi;\zeta,b,\delta)\|_{H^{m-4(N+1)}}\leq
C\delta^{4N+2}\|\nabla\phi\|_{H^{m-1}}^{2}.$

###### Proof.

Notice first that differentiating $\phi=\bm{l}(H)\cdot\bm{\phi}$ we have
$\nabla\phi={\bm{u}}+w\nabla\zeta$, so that

$\displaystyle B^{(N)}(\phi;\zeta,b,\delta)=$
$\displaystyle\frac{1}{2}\bigl{(}|\nabla\phi|^{2}+\delta^{-2}w^{2}(1+\delta^{2}|\nabla\zeta|^{2})\bigr{)}-w\bigl{(}\nabla\zeta\cdot\nabla\phi+\Lambda^{(N)}(\zeta,b,\delta)\phi\bigr{)}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}|\nabla\phi|^{2}+\delta^{-2}w^{2}(1+\delta^{2}|\nabla\zeta|^{2})\bigr{)}-w\bigl{(}\Lambda(\zeta,b,\delta)\phi+\nabla\zeta\cdot\nabla\phi\bigr{)}$
$\displaystyle+w\bigl{(}\Lambda(\zeta,b,\delta)\phi-\Lambda^{(N)}(\zeta,b,\delta)\phi\bigr{)}.$

If we introduce a residual $\mathfrak{r}$ by

$\mathfrak{r}=(\delta^{-2}\partial_{z}\Phi^{\mathrm{app}}-\nabla\zeta\cdot\nabla\Phi^{\mathrm{app}})|_{z=\zeta}-(\delta^{-2}\partial_{z}\Phi-\nabla\zeta\cdot\nabla\Phi)|_{z=\zeta},$

where $\Phi$ is the solution to the boundary value problem (4.7) and
$\Phi^{\mathrm{app}}$ is an approximate velocity potential defined by

$\Phi^{\mathrm{app}}(\bm{x},z)=\sum_{i=0}^{N^{*}}(z+1-b(\bm{x}))^{p_{i}}\phi_{i}(\bm{x}),$

then we have
$\mathfrak{r}=\delta^{-2}w-\nabla\zeta\cdot\bm{u}-\Lambda(\zeta,b,\delta)\phi=\delta^{-2}w(1+\delta^{2}|\nabla\zeta|^{2})-\nabla\zeta\cdot\nabla\phi-\Lambda(\zeta,b,\delta)\phi$.
Therefore, we obtain

$B^{(N)}(\phi;\zeta,b,\delta)-B(\phi;\zeta,b,\delta)=\frac{1}{2}\delta^{2}\frac{\mathfrak{r}^{2}}{1+\delta^{2}|\nabla\zeta|^{2}}+w\bigl{(}\Lambda(\zeta,b,\delta)\phi-\Lambda^{(N)}(\zeta,b,\delta)\phi\bigr{)}.$

The desired estimate for the second term readily follows from Lemma 4.1 and
Lemma 4.2. As for the first term, in view of $m>\frac{n}{2}$ we can use a
calculus inequality
$\|\mathfrak{r}^{2}\|_{H^{k}}\lesssim\|\mathfrak{r}\|_{H^{(m+k)/2}}^{2}$ for
$k\in\\{0,1,\dots,m\\}$. Particularly, we have
$\|\mathfrak{r}^{2}\|_{H^{m-4(N+1)}}\lesssim\|\mathfrak{r}\|_{H^{m-2(N+1)}}^{2}$.
The last term can be evaluated by estimates in [6, Sections 8.1 and 8.2]. ∎

###  4.2 Results in the framework of interfacial waves

In this section, we prove Theorems 3.4 and 3.5. To this end, we first rewrite
the Kakinuma model (2.18) using a formulation which allows a direct comparison
with the full model for interfacial gravity waves (2.17), thanks to the
following Lemma.

######  Lemma 4.4.

Let $c,M$ be positive constants and $m$ an integer such that
$m>\frac{n}{2}+1$. 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)
$\begin{cases}\underline{h}_{1}^{-1}\|\zeta\|_{H^{m}}+\underline{h}_{2}^{-1}\|\zeta\|_{H^{m}}+\underline{h}_{2}^{-1}\|b\|_{W^{m,\infty}}\leq
M,\\\ H_{1}(\bm{x})\geq c,\quad H_{2}(\bm{x})\geq
c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},\end{cases}$

then for any $k=0,\pm 1,\ldots,\pm(m-1)$ and any
$\phi_{1},\phi_{2}\in\mathring{H}^{k+1}$ there exists a unique solution
$\bm{\phi}_{1}=(\phi_{1,0},\bm{\phi}_{1}^{\prime})\in\mathring{H}^{k+1}\times(H^{k+1})^{N}$,
$\bm{\phi}_{2}=(\phi_{2,0},\bm{\phi}_{2}^{\prime})\in\mathring{H}^{k+1}\times(H^{k+1})^{N^{*}}$
to the problem

(4.12)
$\begin{cases}\bm{l}_{1}(H_{1})\cdot\bm{\phi}_{1}=\phi_{1},\quad\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})\bm{\phi}_{1}=0\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\bm{l}_{2}(H_{2})\cdot\bm{\phi}_{2}=\phi_{2},\quad\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})\bm{\phi}_{2}=0\quad\mbox{for}\quad
i=1,2,\ldots,N^{*}.\end{cases}$

Moreover, the solution satisfies
$\|\nabla\bm{\phi}_{\ell}\|_{H^{k}}+(\underline{h}_{\ell}\delta)^{-1}\|\bm{\phi}_{\ell}^{\prime}\|_{H^{k}}\leq
C\|\nabla\phi_{\ell}\|_{H^{k}}$ for $\ell=1,2$.

###### Proof.

Notice that we have identities

$L_{1,ij}(H_{1},\delta,\underline{h}_{1})=L_{ij}(H_{1},0,\underline{h}_{1}\delta),\quad
L_{2,ij}(H_{2},b,\delta,\underline{h}_{2})=L_{ij}(H_{2},\underline{h}_{2}^{-1}b,\underline{h}_{2}\delta)$

with suitable choices of indices $\\{p_{i}\\}$. Hence, Lemma 4.1 gives the
desired result. ∎

As a corollary of this lemma, under the assumptions of Lemma 4.4

$\displaystyle\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})\colon\phi_{1}\mapsto\mathcal{L}_{1,0}(H_{1},\underline{h}_{1},\delta)\bm{\phi}_{1},$
$\displaystyle\Lambda_{2}^{(N)}(\zeta,b,\delta,\underline{h}_{2})\colon\phi_{2}\mapsto\mathcal{L}_{2,0}(H_{2},b,\underline{h}_{2},\delta)\bm{\phi}_{2},$

where $(\bm{\phi}_{1},\bm{\phi}_{2})$ is the unique solution to (4.12), are
defined as bounded linear operators from $\mathring{H}^{k+1}$ to $H^{k-1}$ for
any $k=\pm 0,\dots,\pm(m-1)$. Using these definitions and noting the relations
(3.8) and
$\bm{l}_{\ell}(H_{\ell})\cdot\partial_{t}\bm{\phi}_{\ell}=\partial_{t}(\bm{l}_{\ell}(H_{\ell})\cdot\bm{\phi}_{\ell})-w_{\ell}\underline{h}_{\ell}^{-1}\partial_{t}\zeta$,
we can transform the Kakinuma model (2.18)–(2.19) equivalently as

(4.13)
$\begin{cases}\partial_{t}\zeta+\underline{h}_{1}\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})\phi_{1}=0,\\\
\partial_{t}\zeta-\underline{h}_{2}\Lambda_{1}^{(N)}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}=0,\\\
\underline{\rho}_{1}\bigl{\\{}\partial_{t}\phi_{1}+\frac{1}{2}\bigl{(}|{\bm{u}}_{1}|^{2}+(\underline{h}_{1}\delta)^{-2}w_{1}^{2}\bigr{)}+w_{1}\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})\phi_{1}\bigr{\\}}\\\
\quad-\underline{\rho}_{2}\bigl{\\{}\partial_{t}\phi_{2}+\frac{1}{2}\bigl{(}|{\bm{u}}_{2}|^{2}+(\underline{h}_{2}\delta)^{-2}w_{2}^{2}\bigr{)}-w_{2}\Lambda_{2}^{(N)}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}\bigr{\\}}-\zeta=0,\end{cases}$

where we recall that ${\bm{u}}_{1}$, ${\bm{u}}_{2}$, $w_{1}$, and $w_{2}$ are
uniquely determined from $\phi_{1}$ and $\phi_{2}$ by (3.4), wherein
${\bm{\phi}}_{1}$ and ${\bm{\phi}}_{2}$ are defined as the solutions to
(4.12).

We further introduce notations, which are contributions of Bernoulli’s
equation and interfacial versions of $B$ and $B^{(N)}$ defined by (4.9) and
(4.10). We denote

$\begin{cases}\displaystyle
B_{1}(\phi_{1};\zeta,\delta,\underline{h}_{1}):=\frac{1}{2}|\nabla\phi_{1}|^{2}-\frac{1}{2}\delta^{2}\frac{(\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}-\nabla\zeta\cdot\nabla\phi_{1})^{2}}{1+\delta^{2}|\nabla\zeta|^{2}},\\\
\displaystyle
B_{2}(\phi_{2};\zeta,b,\delta,\underline{h}_{2}):=\frac{1}{2}|\nabla\phi_{2}|^{2}-\frac{1}{2}\delta^{2}\frac{(\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}+\nabla\zeta\cdot\nabla\phi_{2})^{2}}{1+\delta^{2}|\nabla\zeta|^{2}},\end{cases}$

and

$\begin{cases}\displaystyle
B_{1}^{(N)}(\phi_{1};\zeta,\delta,\underline{h}_{1}):=\frac{1}{2}\bigl{(}|\bm{u}_{1}|^{2}+(\underline{h}_{1}\delta)^{-2}w_{1}^{2}\bigr{)}+w_{1}\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})\phi_{1},\\\
\displaystyle
B_{2}^{(N)}(\phi_{2};\zeta,b,\delta,\underline{h}_{2}):=\frac{1}{2}\bigl{(}|\bm{u}_{2}|^{2}+(\underline{h}_{2}\delta)^{-2}w_{2}^{2}\bigr{)}-w_{2}\Lambda_{2}^{(N)}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}.\end{cases}$

Then, the full model for interfacial gravity waves (2.17) and the Kakinuma
model (4.13) can be written simply as

$\begin{cases}\partial_{t}\zeta+\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}=0,\\\
\partial_{t}\zeta-\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}=0,\\\
\displaystyle\underline{\rho}_{1}\bigl{(}\partial_{t}\phi_{1}+B_{1}(\phi_{1};\zeta,\delta,\underline{h}_{1})\bigr{)}-\underline{\rho}_{2}\bigl{(}\partial_{t}\phi_{2}+B_{2}(\phi_{2};\zeta,b,\delta,\underline{h}_{2})\bigr{)}-\zeta=0,\end{cases}$

and

$\begin{cases}\partial_{t}\zeta+\underline{h}_{1}\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})\phi_{1}=0,\\\
\partial_{t}\zeta-\underline{h}_{2}\Lambda_{1}^{(N)}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}=0,\\\
\underline{\rho}_{1}\bigl{(}\partial_{t}\phi_{1}+B_{1}^{(N)}(\phi_{1};\zeta,\delta,\underline{h}_{1})\bigr{)}-\underline{\rho}_{2}\bigl{(}\partial_{t}\phi_{2}+B_{2}^{(N)}(\phi_{2};\zeta,b,\delta,\underline{h}_{2})\bigr{)}-\zeta=0,\end{cases}$

respectively. The following lemmas show that
$\underline{h}_{1}\Lambda_{1}^{(N)}$, $\underline{h}_{2}\Lambda_{2}^{(N)}$,
$B_{1}^{(N)}$, and $B_{2}^{(N)}$ are higher order approximations in the
shallow water regime $\delta_{1}=\underline{h}_{1}\delta\ll 1$ and
$\delta_{2}=\underline{h}_{2}\delta\ll 1$ to $\Lambda_{1}$, $\Lambda_{2}$,
$B_{1}$, and $B_{2}$, respectively.

######  Lemma 4.5.

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

then for any $\phi_{1},\phi_{2}\in\mathring{H}^{k+2(j+1)}$ with $0\leq k\leq
m-2(j+1)$ we have

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

###### Proof.

By simple scaling arguments, we have

(4.15)
$\begin{cases}\Lambda_{1}(\zeta,\delta,\underline{h}_{1})=\underline{h}_{1}\Lambda(-\underline{h}_{1}^{-1}\zeta,0,\underline{h}_{1}\delta),\\\
\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})=\underline{h}_{2}\Lambda(\underline{h}_{2}^{-1}\zeta,\underline{h}_{2}^{-1}b,\underline{h}_{2}\delta),\\\
\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})=\Lambda^{(N)}(-\underline{h}_{1}^{-1}\zeta,0,\underline{h}_{1}\delta),\\\
\Lambda_{2}^{(N)}(\zeta,b,\delta,\underline{h}_{2})=\Lambda^{(N)}(\underline{h}_{2}^{-1}\zeta,\underline{h}_{2}^{-1}b,\underline{h}_{2}\delta).\end{cases}$

Therefore, the results follow from Lemma 4.2. ∎

######  Lemma 4.6.

Let $c,M$ be positive constants and $m$ an integer such that $m\geq 4(N+1)$
and $m>\frac{n}{2}+1$. 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 $\phi_{1},\phi_{2}\in\mathring{H}^{m}$ we have

$\begin{cases}\|B_{1}^{(N)}(\phi_{1};\zeta,\delta,\underline{h}_{1})-B_{1}(\phi_{1};\zeta,\delta,\underline{h}_{1})\|_{H^{m-4(N+1)}}\leq
C\|\nabla\phi_{1}\|_{H^{m-1}}^{2}(\underline{h}_{1}\delta)^{4N+2},\\\
\|B_{2}^{(N)}(\phi_{2};\zeta,b,\delta,\underline{h}_{2})-B_{2}(\phi_{2};\zeta,b,\delta,\underline{h}_{2})\|_{H^{m-4(N+1)}}\leq
C\|\nabla\phi_{2}\|_{H^{m-1}}^{2}(\underline{h}_{2}\delta)^{4N+2}.\end{cases}$

###### Proof.

By simple scaling arguments, we have

$\begin{cases}B_{1}(\phi_{1};\zeta,\delta,\underline{h}_{1})=B(\phi_{1};-\underline{h}_{1}^{-1}\zeta,0,\underline{h}_{1}\delta),\\\
B_{2}(\phi_{2};\zeta,b,\delta,\underline{h}_{1})=B(\phi_{2};\underline{h}_{2}^{-1}\zeta,\underline{h}_{2}^{-1}b,\underline{h}_{2}\delta),\\\
B_{1}^{(N)}(\phi_{1};\zeta,\delta,\underline{h}_{1})=B^{(N)}(\phi_{1};-\underline{h}_{1}^{-1}\zeta,0,\underline{h}_{1}\delta),\\\
B_{2}^{(N)}(\phi_{2};\zeta,b,\delta,\underline{h}_{1})=B^{(N)}(\phi_{2};\underline{h}_{2}^{-1}\zeta,\underline{h}_{2}^{-1}b,\underline{h}_{2}\delta).\end{cases}$

Therefore, the results follow from Lemma 4.3. ∎

We can now prove Theorems 3.4 and 3.5. In view of (3.8) the errors
$(\mathfrak{r}_{1},\mathfrak{r}_{2},\mathfrak{r}_{0})$ and
$(\tilde{\bm{\mathfrak{r}}}_{1},\tilde{\bm{\mathfrak{r}}}_{2},\tilde{\mathfrak{r}}_{0})$
can be written explicitly as

$\begin{cases}\mathfrak{r}_{1}=\Lambda_{1}(\zeta,\delta,\underline{h}_{1})\phi_{1}-\underline{h}_{1}\Lambda_{1}^{(N)}(\zeta,\delta,\underline{h}_{1})\phi_{1},\\\
\mathfrak{r}_{2}=\underline{h}_{2}\Lambda_{2}^{(N)}(\zeta,b,\delta,\underline{h}_{2})\phi_{2}-\Lambda_{2}(\zeta,b,\delta,\underline{h}_{2})\phi_{2},\\\
\mathfrak{r}_{0}=\frac{1}{2}\underline{\rho}_{1}\bigl{(}B_{1}(\phi_{1};\zeta,\delta,\underline{h}_{1})-B_{1}^{(N)}(\phi_{1};\zeta,\delta,\underline{h}_{1})\bigr{)}\\\
\qquad-\frac{1}{2}\underline{\rho}_{2}\bigl{(}B_{2}(\phi_{2};\zeta,b,\delta,\underline{h}_{2})-B_{2}^{(N)}(\phi_{2};\zeta,b,\delta,\underline{h}_{2})\bigr{)},\\\
\tilde{\bm{\mathfrak{r}}}_{1}=-\underline{h}_{1}^{-1}\bm{l}_{1}(H_{1})\mathfrak{r}_{1},\quad\tilde{\bm{\mathfrak{r}}}_{2}=-\underline{h}_{2}^{-1}\bm{l}_{2}(H_{2})\mathfrak{r}_{2},\quad\tilde{\mathfrak{r}}_{0}=-\mathfrak{r}_{0}.\end{cases}$

Therefore, the theorems are simple corollaries of the above Lemmas 4.5 and
4.6.

##  5 Elliptic estimates and time derivatives

In this section we derive useful uniform a priori bounds on regular solutions
to the Kakinuma model (2.18)–(2.19). Firstly, due to the fact that the
hypersurface $t=0$ in the space-time $\mathbf{R}^{n}\times\mathbf{R}$ is
characteristic for the Kakinuma model, we need the following key elliptic
estimate in order to be able to estimate time derivatives of the solution. Let
us recall that the operators $\mathcal{L}_{1,i}$ for $i=0,1,\ldots,N$ and
$\mathcal{L}_{2,i}$ for $i=0,1,\ldots,N^{*}$ are defined by (3.6), and the
vectors $\bm{l}_{1}(H_{1})$ and $\bm{l}_{2}(H_{2})$ are defined by (3.3). We
recall the convention that for a vector
$\mbox{\boldmath$\phi$}=(\phi_{0},\phi_{1},\ldots,\phi_{N})^{\mathrm{T}}$ we
denote the last $N$ components by
$\mbox{\boldmath$\phi$}^{\prime}=(\phi_{1},\ldots,\phi_{N})^{\mathrm{T}}$.

######  Lemma 5.1.

Let $c,M$ be positive constants and $m$ an integer such that
$m>\frac{n}{2}+1$. 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
$\zeta\in H^{m}$, $b\in W^{m,\infty}$, $H_{1}=1-\underline{h}_{1}^{-1}\zeta$,
and $H_{2}=1+\underline{h}_{2}^{-1}\zeta-\underline{h}_{2}^{-1}b$ satisfy
(4.11), then for any
$\bm{f}_{1}^{\prime}=(f_{1,1},\ldots,f_{1,N})^{\mathrm{T}}\in(H^{k})^{N}$,
$\bm{f}_{2}^{\prime}=(f_{2,1},\ldots,$
$f_{2,N^{*}})^{\mathrm{T}}\in(H^{k})^{N^{*}}$, $\bm{f}_{3}\in(H^{k})^{n}$, and
$f_{4}\in\mathring{H}^{k+1}$ with $k\in\\{0,1,\ldots,m-1\\}$, there exists a
solution $(\bm{\varphi}_{1},\bm{\varphi}_{2})$ to

(5.1)
$\begin{cases}\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1}){\bm{\varphi}}_{1}=f_{1,i}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2}){\bm{\varphi}}_{2}=f_{2,i}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}(H_{1}\delta,\underline{h}_{1}){\bm{\varphi}}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}(H_{2},b,\delta,\underline{h}_{2}){\bm{\varphi}}_{2}=\nabla\cdot\bm{f}_{3},\\\
-\underline{\rho}_{1}{\bm{l}}_{1}(H_{1})\cdot{\bm{\varphi}}_{1}+\underline{\rho}_{2}{\bm{l}}_{2}(H_{2})\cdot{\bm{\varphi}}_{2}=f_{4},\end{cases}$

satisfying

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\bigl{(}\|\nabla\bm{\varphi}_{\ell}\|_{H^{k}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\varphi}_{\ell}^{\prime}\|_{H^{k}}^{2}\bigr{)}$
$\displaystyle\leq
C\biggl{(}\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\min\bigl{\\{}\|\bm{f}_{\ell}^{\prime}\|_{H^{k-1}}^{2},(\underline{h}_{\ell}\delta)^{2}\|\bm{f}_{\ell}^{\prime}\|_{H^{k}}^{2}\bigr{\\}}$
$\displaystyle\qquad+\min\biggl{\\{}\frac{\underline{\rho}_{1}}{\underline{h}_{1}},\frac{\underline{\rho}_{2}}{\underline{h}_{2}}\biggr{\\}}\|\bm{f}_{3}\|_{H^{k}}^{2}+\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}\|\nabla
f_{4}\|_{H^{k}}^{2}\biggr{)}.$

Moreover, the solution is unique up to an additive constant of the form
$(\mathcal{C}\underline{\rho}_{2},\mathcal{C}\underline{\rho}_{1})$ to
$(\varphi_{1,0},\varphi_{2,0})$.

###### Proof.

The existence and uniqueness up to an additive constant of the solution has
been given in the companion paper [3, Lemma 6.4]. We focus here on the
derivation of uniform estimates. By direct rescaling within the proof of [3,
Lemma 6.1], we infer that

$(L_{\ell}\bm{\varphi}_{\ell},\bm{\varphi}_{\ell})_{L^{2}}\simeq\|\nabla\bm{\varphi}_{\ell}\|_{L^{2}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\varphi}_{\ell}^{\prime}\|_{L^{2}}^{2}$

for $\ell=1,2$. We note the identities

$\begin{cases}L_{1}\bm{\varphi}_{1}=\bm{l}_{1}\mathcal{L}_{1,0}\bm{\varphi}_{1}+(0,\mathcal{L}_{1,1}\bm{\varphi}_{1},\ldots,\mathcal{L}_{1,N}\bm{\varphi}_{1})^{\mathrm{T}},\\\
L_{2}\bm{\varphi}_{2}=\bm{l}_{2}\mathcal{L}_{2,0}\bm{\varphi}_{2}+(0,\mathcal{L}_{2,1}\bm{\varphi}_{2},\ldots,\mathcal{L}_{2,N^{*}}\bm{\varphi}_{2})^{\mathrm{T}},\end{cases}$

so that for the solution $(\bm{\varphi}_{1},\bm{\varphi}_{2})$ to (5.1) we
have

(5.2)
$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(L_{\ell}\bm{\varphi}_{\ell},\bm{\varphi}_{\ell})_{L^{2}}$
$\displaystyle=\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\mathcal{L}_{\ell,0}\bm{\varphi}_{\ell},\bm{l}_{\ell}\cdot\bm{\varphi}_{\ell})_{L^{2}}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\bm{f}_{\ell}^{\prime},\bm{\varphi}_{\ell}^{\prime})_{L^{2}}$
$\displaystyle=:I_{1}+I_{2}.$

Therefore, it is sufficient to evaluate $I_{1}$ and $I_{2}$. As for the term
$I_{2}$ we have

$\displaystyle|(\bm{f}_{\ell}^{\prime},\bm{\varphi}_{\ell}^{\prime})_{L^{2}}|$
$\displaystyle\leq\min\\{\|\bm{f}_{\ell}^{\prime}\|_{H^{-1}}\|\bm{\varphi}_{\ell}^{\prime}\|_{H^{1}},\|\bm{f}_{\ell}^{\prime}\|_{L^{2}}\|\bm{\varphi}_{\ell}^{\prime}\|_{L^{2}}\\}$
$\displaystyle\leq\min\\{\|\bm{f}_{\ell}^{\prime}\|_{H^{-1}},(\underline{h}_{\ell}\delta)\|\bm{f}_{\ell}^{\prime}\|_{L^{2}}\\}(\|\nabla\bm{\varphi}_{\ell}\|_{L^{2}}+(\underline{h}_{\ell}\delta)^{-1}\|\bm{\varphi}_{\ell}^{\prime}\|_{L^{2}}).$

As for the term $I_{1}$, we note the trivial identities

$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\mathcal{L}_{\ell,0}\bm{\varphi}_{\ell},\bm{l}_{\ell}\cdot\bm{\varphi}_{\ell})_{L^{2}}$
$\displaystyle=\begin{cases}(\underline{h}_{1}\mathcal{L}_{1,0}\bm{\varphi}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}\bm{\varphi}_{2},\underline{\rho}_{1}\bm{l}_{1}\cdot\bm{\varphi}_{1})_{L^{2}}+(\underline{h}_{2}\mathcal{L}_{2,0}\bm{\varphi}_{2},\underline{\rho}_{2}\bm{l}_{2}\cdot\bm{\varphi}_{2}-\underline{\rho}_{1}\bm{l}_{1}\cdot\bm{\varphi}_{1})_{L^{2}},\\\
(\underline{h}_{1}\mathcal{L}_{1,0}\bm{\varphi}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}\bm{\varphi}_{2},\underline{\rho}_{2}\bm{l}_{2}\cdot\bm{\varphi}_{2})_{L^{2}}+(\underline{h}_{1}\mathcal{L}_{1,0}\bm{\varphi}_{1},\underline{\rho}_{1}\bm{l}_{1}\cdot\bm{\varphi}_{1}-\underline{\rho}_{2}\bm{l}_{2}\cdot\bm{\varphi}_{2})_{L^{2}}.\end{cases}$

Therefore, the term $I_{1}$ in (5.2) can be expressed in two ways as

$I_{1}=\begin{cases}\underline{\rho}_{1}(\nabla\cdot\bm{f}_{3},\bm{l}_{1}\cdot\bm{\varphi}_{1})_{L^{2}}+\underline{h}_{2}(\mathcal{L}_{2,0}\bm{\varphi}_{2},f_{4})_{L^{2}},\\\
\underline{\rho}_{2}(\nabla\cdot\bm{f}_{3},\bm{l}_{2}\cdot\bm{\varphi}_{2})_{L^{2}}-\underline{h}_{1}(\mathcal{L}_{1,0}\bm{\varphi}_{1},f_{4})_{L^{2}}.\end{cases}$

By the linearity of (5.1) it is sufficient to evaluate it in the case
$f_{4}=0$ and in the case $\bm{f}_{3}=\bm{0}$, separately. In the case
$f_{4}=0$, we evaluate it as

$\displaystyle|I_{1}|$
$\displaystyle\leq\min\\{\underline{\rho}_{1}\|\bm{f}_{3}\|_{L^{2}}\|\nabla(\bm{l}_{1}\cdot\bm{\varphi}_{1})\|_{L^{2}},\underline{\rho}_{2}\|\bm{f}_{3}\|_{L^{2}}\|\nabla(\bm{l}_{2}\cdot\bm{\varphi}_{2})\|_{L^{2}}\\}$
$\displaystyle=\min\biggl{\\{}\sqrt{\frac{\underline{\rho}_{1}}{\underline{h}_{1}}}\|\bm{f}_{3}\|_{L^{2}}\sqrt{\underline{\rho}_{1}\underline{h}_{1}}\|\nabla(\bm{l}_{1}\cdot\bm{\varphi}_{1})\|_{L^{2}},\sqrt{\frac{\underline{\rho}_{2}}{\underline{h}_{2}}}\|\bm{f}_{3}\|_{L^{2}}\sqrt{\underline{\rho}_{2}\underline{h}_{2}}\|\nabla(\bm{l}_{2}\cdot\bm{\varphi}_{2})\|_{L^{2}}\biggr{\\}}$
$\displaystyle\lesssim\min\biggl{\\{}\sqrt{\frac{\underline{\rho}_{1}}{\underline{h}_{1}}},\sqrt{\frac{\underline{\rho}_{2}}{\underline{h}_{2}}}\biggr{\\}}\|\bm{f}_{3}\|_{L^{2}}\sum_{\ell=1,2}\sqrt{\underline{\rho}_{\ell}\underline{h}_{\ell}}(\|\nabla\bm{\varphi}_{\ell}\|_{L^{2}}+\|\bm{\varphi}_{\ell}^{\prime}\|_{L^{2}}).$

In the case $\bm{f}_{3}=\bm{0}$ we evaluate it as

$\displaystyle|I_{1}|$
$\displaystyle\lesssim\min\\{\underline{h}_{1}\|\nabla\bm{\varphi}_{1}\|_{L^{2}}\|\nabla
f_{4}\|_{L^{2}},\underline{h}_{2}(\|\nabla\bm{\varphi}_{2}\|_{L^{2}}+\|\bm{\varphi}_{2}^{\prime}\|_{L^{2}})\|\nabla
f_{4}\|_{L^{2}}\\}$
$\displaystyle=\min\biggl{\\{}\sqrt{\frac{\underline{h}_{1}}{\underline{\rho}_{1}}}\|\nabla
f_{4}\|_{L^{2}}\sqrt{\underline{\rho}_{1}\underline{h}_{1}}\|\nabla\bm{\varphi}_{1}\|_{L^{2}},\sqrt{\frac{\underline{h}_{2}}{\underline{\rho}_{2}}}\|\nabla
f_{4}\|_{L^{2}}\sqrt{\underline{\rho}_{2}\underline{h}_{2}}(\|\nabla\bm{\varphi}_{1}\|_{L^{2}}+\|\bm{\varphi}_{2}^{\prime}\|_{L^{2}})\biggr{\\}}$
$\displaystyle\leq\min\biggl{\\{}\sqrt{\frac{\underline{h}_{1}}{\underline{\rho}_{1}}},\sqrt{\frac{\underline{h}_{2}}{\underline{\rho}_{2}}}\biggr{\\}}\|\nabla
f_{4}\|_{L^{2}}\sum_{\ell=1,2}\sqrt{\underline{\rho}_{\ell}\underline{h}_{\ell}}(\|\nabla\bm{\varphi}_{\ell}\|_{L^{2}}+\|\bm{\varphi}_{\ell}^{\prime}\|_{L^{2}}).$

From the above estimates we deduce immediately the desired inequality for
$k=0$.

In order to obtain the desired inequality on derivatives, we let
$k\in\\{1,2,\ldots,m-1\\}$ and $\beta$ be a multi-index such that
$1\leq|\beta|\leq k$. Applying the differential operator $\partial^{\beta}$ to
(5.1), we have

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

where

$\begin{cases}f_{1,i,\beta}:=-[\partial^{\beta},\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})]{\bm{\varphi}}_{1}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
f_{2,i,\beta}:=-[\partial^{\beta},\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})]{\bm{\varphi}}_{2}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\nabla\cdot\bm{f}_{3,1,\beta}:=-[\partial^{\beta},\mathcal{L}_{1,0}(H_{1},\delta,\underline{h}_{1})]{\bm{\varphi}}_{1},\\\
\nabla\cdot\bm{f}_{3,2,\beta}:=-[\partial^{\beta},\mathcal{L}_{2,0}(H_{2},b,\delta,\underline{h}_{2})]{\bm{\varphi}}_{2},\\\
f_{4,1,\beta}:=[\partial^{\beta},{\bm{l}}_{1}(H_{1})]\cdot{\bm{\varphi}}_{1},\\\
f_{4,2,\beta}:=-[\partial^{\beta},{\bm{l}}_{2}(H_{2})]\cdot{\bm{\varphi}}_{2}.\end{cases}$

We put $\bm{f}_{1,\beta}=(0,f_{1,1,\beta},\ldots,f_{1,N,\beta})$ and
$\bm{f}_{2,\beta}=(0,f_{2,1,\beta},\ldots,f_{2,N^{*},\beta})$. Then, with a
suitable decomposition
$\bm{f}_{\ell,\beta}=\bm{f}_{\ell,\beta}^{\mathrm{high}}+\bm{f}_{\ell,\beta}^{\mathrm{low}}$
for $\ell=1,2$, we see that

$\displaystyle\|\bm{f}_{\ell,\beta}^{\mathrm{high}}\|_{H^{-1}}+(\underline{h}_{\ell}\delta)\|\bm{f}_{\ell,\beta}^{\mathrm{low}}\|_{L^{2}}+\|\bm{f}_{3,\ell,\beta}\|_{L^{2}}+\|\nabla
f_{4,\ell}\|_{L^{2}}\lesssim\|\nabla\bm{\varphi}_{\ell}\|_{H^{k-1}}+(\underline{h}_{\ell}\delta)^{-1}\|\bm{\varphi}_{\ell}^{\prime}\|_{H^{k-1}}$

for $\ell=1,2$. Therefore, in view of the linearity of (5.1) the desired
inequality for $k\geq 1$ follows by induction on $k$. ∎

From the above elliptic estimates we deduce the following bounds on time
derivatives of regular solutions to the Kakinuma model (2.18)–(2.19). We
introduce a mathematical energy $E_{m}(t)$ for a solution
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ to the Kakinuma model by

(5.3)
$E_{m}(t):=\|\zeta(t)\|_{H^{m}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\bm{\phi}_{\ell}(t)\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\bm{\phi}_{\ell}^{\prime}(t)\|_{H^{m}}^{2}),$

where $\bm{\phi}_{1}^{\prime}=(\phi_{1,1},\ldots,\phi_{1,N})^{\mathrm{T}}$ and
$\bm{\phi}_{2}^{\prime}=(\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\mathrm{T}}$.

######  Lemma 5.2.

Let $c,M_{1},\underline{h}_{\mathrm{min}}$ be positive constants and $m$ an
integer such that $m>\frac{n}{2}+1$. There exists a positive constant $C_{1}$
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
(2.18)–(2.19) with bottom topography $b\in W^{m+1,\infty}$ satisfy

$\begin{cases}E_{m}(t)+\underline{h}_{2}^{-1}\|b\|_{W^{m+1,\infty}}\leq
M_{1},\\\ H_{1}(\bm{x},t)\geq c,\quad H_{2}(\bm{x},t)\geq
c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},0\leq t\leq T,\end{cases}$

then we have

(5.4)
$\displaystyle\|\partial_{t}\zeta(t)\|_{H^{m-1}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\partial_{t}\bm{\phi}_{\ell}(t)\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}\bm{\phi}_{\ell}^{\prime}(t)\|_{H^{m-1}}^{2})$
$\displaystyle+\|\partial_{t}^{2}\zeta(t)\|_{H^{m-2}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\partial_{t}^{2}\bm{\phi}_{\ell}(t)\|_{H^{m-2}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}^{2}\bm{\phi}_{\ell}^{\prime}(t)\|_{H^{m-2}}^{2})\leq
C_{1}E_{m}(t)$

for $0\leq t\leq T$.

###### Proof.

First, we remind that the Kakinuma model (2.18) can be written compactly as
(3.5). It follows from the first component of the first two equations in (3.5)
that $\partial_{t}\zeta$ can be written in two way as
$\partial_{t}\zeta=-\underline{h}_{1}\mathcal{L}_{1,0}\bm{\phi}_{1}=\underline{h}_{2}\mathcal{L}_{2,0}\bm{\phi}_{2}$,
so that

$\displaystyle\|\partial_{t}\zeta\|_{H^{m-1}}^{2}$
$\displaystyle=\min\\{\underline{h}_{1}^{2}\|\mathcal{L}_{1,0}\bm{\phi}_{1}\|_{H^{m-1}}^{2},\underline{h}_{2}^{2}\|\mathcal{L}_{2,0}\bm{\phi}_{2}\|_{H^{m-1}}^{2}\\}$
$\displaystyle\lesssim\min\\{\underline{h}_{1}^{2}\|\nabla\bm{\phi}_{1}\|_{H^{m}}^{2},\underline{h}_{2}^{2}(\|\nabla\bm{\phi}_{2}\|_{H^{m}}^{2}+\|\bm{\phi}_{2}^{\prime}\|_{H^{m}}^{2})\\}$
$\displaystyle\leq\min\biggl{\\{}\frac{\underline{h}_{1}}{\underline{\rho}_{1}},\frac{\underline{h}_{2}}{\underline{\rho}_{2}}\biggr{\\}}E_{m}\leq
2E_{m},$

where we used (2.15).

As for the estimate of
$(\partial_{t}\bm{\phi}_{1},\partial_{t}\bm{\phi}_{2})$, we differentiate the
compatibility conditions (3.7) with respect to time and use the last equation
in (3.5). Then, we have

(5.5)
$\begin{cases}\mathcal{L}_{1,i}\partial_{t}\bm{\phi}_{1}=f_{1,i}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
\mathcal{L}_{2,i}\partial_{t}\bm{\phi}_{2}=f_{2,i}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\underline{h}_{1}\mathcal{L}_{1,0}\partial_{t}\bm{\phi}_{1}+\underline{h}_{2}\mathcal{L}_{2,0}\partial_{t}\bm{\phi}_{2}=\nabla\cdot\bm{f}_{3},\\\
-\underline{\rho}_{1}\bm{l}_{1}\cdot\partial_{t}\bm{\phi}_{1}+\underline{\rho}_{2}\bm{l}_{2}\cdot\partial_{t}\bm{\phi}_{2}=f_{4},\end{cases}$

where

(5.6)
$\begin{cases}f_{1,i}:=-[\partial_{t},\mathcal{L}_{1,i}(H_{1},\delta,\underline{h}_{1})]\bm{\phi}_{1}\quad\mbox{for}\quad
i=1,2,\ldots,N,\\\
f_{2,i}:=-[\partial_{t},\mathcal{L}_{2,i}(H_{2},b,\delta,\underline{h}_{2})]\bm{\phi}_{2}\quad\mbox{for}\quad
i=1,2,\ldots,N^{*},\\\
\bm{f}_{3}:=(\bm{u}_{2}-\bm{u}_{1})\partial_{t}\zeta,\\\
f_{4}:=\frac{1}{2}\underline{\rho}_{1}\bigl{(}|{\bm{u}}_{1}|^{2}+(\underline{h}_{1}\delta)^{-2}w_{1}^{2}\bigr{)}-\frac{1}{2}\underline{\rho}_{2}\bigl{(}|{\bm{u}}_{2}|^{2}+(\underline{h}_{2}\delta)^{-2}w_{2}^{2}\bigr{)}-\zeta.\end{cases}$

Therefore, by Lemma 5.1 we have

(5.7)
$\displaystyle\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\partial_{t}\bm{\phi}_{\ell}\|_{H^{m-1}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\partial_{t}\bm{\phi}_{\ell}^{\prime}\|_{H^{m-1}}^{2})$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2}\|\bm{f}_{\ell}^{\prime}\|_{H^{m-1}}^{2}+\min\biggl{\\{}\frac{\underline{\rho}_{1}}{\underline{h}_{1}},\frac{\underline{\rho}_{2}}{\underline{h}_{2}}\biggr{\\}}\|\bm{f}_{3}\|_{H^{m-1}}^{2}+\|f_{4}\|_{H^{m}}^{2},$

where $\bm{f}_{1}^{\prime}=(f_{1,1},\ldots,f_{1,N})^{\mathrm{T}}$,
$\bm{f}_{2}^{\prime}=(f_{2,1},\ldots,f_{2,N^{*}})^{\mathrm{T}}$, and we used
(2.15). We proceed to evaluate the right-hand side. By writing down the
operators $\mathcal{L}_{\ell,i}$ explicitly, we see that the operators do not
include any derivatives of $H_{\ell}$. Therefore, we can write $f_{\ell,i}$ as

$f_{1,i}=\biggl{(}\biggl{(}\frac{\partial}{\partial
H_{1}}\mathcal{L}_{1,i}\biggr{)}\bm{\phi}_{1}\biggr{)}\underline{h}_{1}^{-1}\partial_{t}\zeta,\quad
f_{2,i}=-\biggl{(}\biggl{(}\frac{\partial}{\partial
H_{2}}\mathcal{L}_{2,i}\biggr{)}\bm{\phi}_{2}\biggr{)}\underline{h}_{2}^{-1}\partial_{t}\zeta.$

We note also that the differential operators $\frac{\partial}{\partial
H_{\ell}}\mathcal{L}_{\ell,i}$ have a similar structure as
$\mathcal{L}_{\ell,i}$. Therefore,

$\displaystyle\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2}\|\bm{f}_{\ell}^{\prime}\|_{H^{m-1}}^{2}$
$\displaystyle\lesssim\underline{\rho}_{\ell}\underline{h}_{\ell}(\underline{h}_{\ell}\delta)^{2}(\|\nabla\bm{\phi}_{\ell}\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-4}\|\bm{\phi}_{\ell}^{\prime}\|_{H^{m-1}}^{2})\|\underline{h}_{\ell}^{-1}\partial_{t}\zeta\|_{H^{m-1}}^{2}$
$\displaystyle\lesssim E_{m}^{2}\quad\mbox{for}\quad\ell=1,2,$

where, here and henceforth, we utilize fully our restriction
$\underline{h}_{1}^{-1},\underline{h}_{2}^{-1}\lesssim 1$. In view of the
definition (3.4) of $\bm{u}_{1},\bm{u}_{2},w_{1}$, and $w_{2}$, we see easily
that

(5.8)
$\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\bm{u}_{\ell}\|_{H^{m}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|w_{\ell}\|_{H^{m}}^{2})\lesssim
E_{m}.$

We evaluate the term on $\bm{f}_{3}$ as

$\displaystyle\min\biggl{\\{}\frac{\underline{\rho}_{1}}{\underline{h}_{1}},\frac{\underline{\rho}_{2}}{\underline{h}_{2}}\biggr{\\}}\|\bm{f}_{3}\|_{H^{m-1}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\frac{\underline{\rho}_{\ell}}{\underline{h}_{\ell}}\|\bm{u}_{\ell}\partial_{t}\zeta\|_{H^{m-1}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\|\bm{u}_{\ell}\|_{H^{m-1}}^{2}\|\underline{h}_{\ell}^{-1}\partial_{t}\zeta\|_{H^{m-1}}^{2}$
$\displaystyle\lesssim E_{m}^{2}.$

Similarly, we have

$\displaystyle\|f_{4}\|_{H^{m}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{\rho}_{\ell}^{2}(\|{\bm{u}}_{\ell}\|_{H^{m}}^{2}+(\underline{h}_{1}\delta)^{-2}\|w_{\ell}\|_{H^{m}}^{2})^{2}+\|\zeta\|_{H^{m}}^{2}$
$\displaystyle\lesssim\sum_{\ell=1,2}\underline{h}_{\ell}^{-2}\\{\underline{\rho}_{\ell}\underline{h}_{\ell}(\|{\bm{u}}_{\ell}\|_{H^{m}}^{2}+(\underline{h}_{1}\delta)^{-2}\|w_{\ell}\|_{H^{m}}^{2})\\}^{2}+\|\zeta\|_{H^{m}}^{2}$
$\displaystyle\lesssim E_{m}^{2}+E_{m}.$

Plugging in (5.7) the above estimates, we obtain the desired estimate for
$(\partial_{t}\bm{\phi}_{1},\partial_{t}\bm{\phi}_{2})$.

Finally, the estimate of $\partial_{t}^{2}\zeta$ can be obtained by
differentiating
$\partial_{t}\zeta=-\underline{h}_{1}\mathcal{L}_{1,0}\bm{\phi}_{1}=\underline{h}_{2}\mathcal{L}_{2,0}\bm{\phi}_{2}$
with respect to time. Then, the estimate of
$(\partial_{t}^{2}\bm{\phi}_{1},\partial_{t}^{2}\bm{\phi}_{2})$ can be
obtained by differentiating (5.5) with respect to time once more and applying
Lemma 5.1. ∎

######  Remark 5.3.

In view of the above arguments, we see easily that for the Kakinuma model
(2.18)–(2.19), $(\partial_{t}\bm{\phi}_{1},\partial_{t}\bm{\phi}_{2})|_{t=0}$
can be determined from the initial data
$(\zeta_{(0)},\bm{\phi}_{1(0)},\bm{\phi}_{2(0)})$ and the bottom topography
$b$, although the hypersurface $t=0$ is characteristic for the model. They are
unique up to an additive constant of the form
$(\mathcal{C}\underline{\rho}_{2},\mathcal{C}\underline{\rho}_{1})$ to
$(\partial_{t}\phi_{1,0},\partial_{t}\phi_{2,0})|_{t=0}$. Particularly,
$(\partial_{t}\bm{\phi}_{1}^{\prime},\partial_{t}\bm{\phi}_{2}^{\prime})|_{t=0}$
and hence $a|_{t=0}$ with the function $a$ given in (3.9) can be uniquely
determined from the data.

##  6 Uniform energy estimates; proof of Theorem 3.1

In this section we provide uniform energy estimates for solutions to the
Kakinuma model. Consequently, we prove Theorem 3.1. We remind that the
Kakinuma model (2.18)–(2.19) can be written compactly as

(6.1)
$\begin{cases}\displaystyle{\bm{l}}_{1}(H_{1})\partial_{t}\zeta+\underline{h}_{1}L_{1}(H_{1},\delta,\underline{h}_{1}){\bm{\phi}}_{1}={\bm{0}},\\\
\displaystyle{\bm{l}}_{2}(H_{2})\partial_{t}\zeta-\underline{h}_{2}L_{2}(H_{2},b,\delta,\underline{h}_{2}){\bm{\phi}}_{2}={\bm{0}},\\\
\underline{\rho}_{1}\bigl{\\{}{\bm{l}}_{1}(H_{1})\cdot\partial_{t}{\bm{\phi}}_{1}+\frac{1}{2}\bigl{(}|{\bm{u}}_{1}|^{2}+(\underline{h}_{1}\delta)^{-2}w_{1}^{2}\bigr{)}\bigr{\\}}\\\
\quad-\underline{\rho}_{2}\bigl{\\{}{\bm{l}}_{2}(H_{2})\cdot\partial_{t}{\bm{\phi}}_{2}+\frac{1}{2}\bigl{(}|{\bm{u}}_{2}|^{2}+(\underline{h}_{2}\delta)^{-2}w_{2}^{2}\bigr{)}\bigr{\\}}-\zeta=0,\end{cases}$

where we recall that $H_{1}:=1-\underline{h}_{1}^{-1}\zeta$,
$H_{2}:=1+\underline{h}_{2}^{-1}\zeta-\underline{h}_{2}^{-1}b$,
${\bm{\phi}}_{1}:=(\phi_{1,0},\phi_{1,1},\ldots,\phi_{1,N})^{\mathrm{T}}$,
${\bm{\phi}}_{2}:=(\phi_{2,0},\phi_{2,1},\ldots,\phi_{2,N^{*}})^{\mathrm{T}}$,
and ${\bm{l}}_{1}$, ${\bm{l}}_{2}$, $L_{1}$, $L_{2}$, ${\bm{u}}_{1}$,
${\bm{u}}_{2}$, $w_{1}$, $w_{2}$ are defined in Section 3.

###  6.1 Analysis of linearized equations

Before deriving linearized equations to the Kakinuma model (6.1), we introduce
some more notations. For $\ell=1,2$, the coefficient matrices of the principal
part and the singular part with respect to the small parameter
$\delta_{\ell}=\underline{h}_{\ell}\delta$ of the operator $L_{\ell}$ are
denoted by $A_{\ell}(H_{\ell})$ and $C_{\ell}(H_{\ell})$, respectively, that
is,

(6.2) $\begin{cases}\displaystyle
A_{1}(H_{1}):=\left(\frac{1}{2(i+j)+1}H_{1}^{2(i+j)+1}\right)_{0\leq i,j\leq
N},\\\ \displaystyle
A_{2}(H_{2}):=\left(\frac{1}{p_{i}+p_{j}+1}H_{2}^{p_{i}+p_{j}+1}\right)_{0\leq
i,j\leq N^{*}},\end{cases}$

and

(6.3) $\begin{cases}\displaystyle
C_{1}(H_{1}):=\left(\frac{4ij}{2(i+j)-1}H_{1}^{2(i+j)-1}\right)_{0\leq i,j\leq
N},\\\ \displaystyle
C_{2}(H_{2}):=\left(\frac{p_{i}p_{j}}{p_{i}+p_{j}-1}H_{2}^{p_{i}+p_{j}-1}\right)_{0\leq
i.j\leq N^{*}}.\end{cases}$

We put also

(6.4) $\begin{cases}\displaystyle
B_{2}(H_{2}):=\left(\frac{p_{j}}{p_{i}+p_{j}}H_{2}^{p_{i}+p_{j}}\right)_{0\leq
i,j\leq N^{*}},\\\
\tilde{B}_{2}(H_{2}):=B_{2}(H_{2})-B_{2}(H_{2})^{\mathrm{T}},\\\
\tilde{C}_{2}(H_{2},\underline{h}_{2}^{-1}b):=|\underline{h}_{2}^{-1}\nabla
b|^{2}C_{2}(H_{2})+\underline{h}_{2}^{-1}(\Delta b)B_{2}(H_{2}).\end{cases}$

Then, the operators $L_{1}$ and $L_{2}$ can also be written as

(6.5)
$\begin{cases}L_{1}\bm{\phi}_{1}=-A_{1}\Delta\bm{\phi}_{1}-\bm{l}_{1}(\bm{u}_{1}\cdot\nabla
H_{1})+(\underline{h}_{1}\delta)^{-2}C_{1}\bm{\phi}_{1},\\\
L_{2}\bm{\phi}_{2}=-A_{2}\Delta\bm{\phi}_{2}-\bm{l}_{2}(\bm{u}_{2}\cdot\nabla
H_{2})+(\underline{h}_{2}\delta)^{-2}C_{2}\bm{\phi}_{2}+\tilde{B}_{2}(\underline{h}_{2}^{-1}\nabla
b\cdot\nabla)\bm{\phi}_{2}+\tilde{C}_{2}\bm{\phi}_{2}.\end{cases}$

For $\ell=1,2$, we decompose the operator $L_{\ell}$ as
$L_{\ell}=L_{\ell}^{\mathrm{pr}}+L_{\ell}^{\mathrm{low}}$, where

(6.6)
$L_{\ell}^{\mathrm{pr}}(H_{\ell})\bm{\varphi}_{\ell}:=-\sum_{l=1}^{n}\partial_{l}(A_{\ell}(H_{\ell})\partial_{l}\bm{\varphi}_{\ell})+(\underline{h}_{\ell}\delta)^{-2}C_{\ell}(H_{\ell})\bm{\varphi}_{\ell}.$

We now linearize the Kakinuma model (6.1) around an arbitrary flow
$(\zeta,\bm{\phi}_{1},\bm{\phi}_{2})$ and denote the variation by
$(\dot{\zeta},\dot{\bm{\phi}}_{1},\dot{\bm{\phi}}_{2})$. After neglecting
lower order terms, the linearized equations have the form

(6.7)
$\begin{cases}\displaystyle{\bm{l}}_{1}(H_{1})(\partial_{t}+\bm{u}_{1}\cdot\nabla)\dot{\zeta}+\underline{h}_{1}L_{1}^{\mathrm{pr}}(H_{1},\delta,\underline{h}_{1})\dot{\bm{\phi}}_{1}=\dot{{\bm{f}}}_{1},\\\
\displaystyle{\bm{l}}_{2}(H_{2})(\partial_{t}+\bm{u}_{2}\cdot\nabla)\dot{\zeta}-\underline{h}_{2}L_{2}^{\mathrm{pr}}(H_{2},\delta,\underline{h}_{2})\dot{\bm{\phi}}_{2}=\dot{{\bm{f}}}_{2},\\\
\underline{\rho}_{1}{\bm{l}}_{1}(H_{1})\cdot(\partial_{t}+\bm{u}_{1}\cdot\nabla)\dot{\bm{\phi}}_{1}-\underline{\rho}_{2}{\bm{l}}_{2}(H_{2})\cdot(\partial_{t}+\bm{u}_{1}\cdot\nabla)\dot{\bm{\phi}}_{2}-a\dot{\zeta}=\dot{f}_{0},\end{cases}$

where the function $a$ is defined by (3.9). In order to derive a good
symmetric structure of the equations, following the companion paper [3] we
introduce

(6.8)
$\theta_{1}:=\frac{\underline{\rho}_{2}\underline{h}_{1}H_{1}\alpha_{1}}{\underline{\rho}_{1}\underline{h}_{2}H_{2}\alpha_{2}+\underline{\rho}_{2}\underline{h}_{1}H_{1}\alpha_{1}},\qquad\theta_{2}:=\frac{\underline{\rho}_{1}\underline{h}_{2}H_{2}\alpha_{2}}{\underline{\rho}_{1}\underline{h}_{2}H_{2}\alpha_{2}+\underline{\rho}_{2}\underline{h}_{1}H_{1}\alpha_{1}},$

where

(6.9) $\alpha_{\ell}:=\frac{\det
A_{\ell,0}}{\det\tilde{A}_{\ell,0}},\qquad\tilde{A}_{\ell,0}:=\begin{pmatrix}0&\bm{1}^{\mathrm{T}}\\\
-\bm{1}&A_{\ell,0}\end{pmatrix},\qquad A_{\ell,0}:=A_{\ell}(1)$

for $\ell=1,2$ and $\bm{1}:=(1,\ldots,1)^{\mathrm{T}}$. Then, we have
$\theta_{1}+\theta_{2}=1$. We remind that $\alpha_{1}$ and $\alpha_{2}$ are
positive constants depending only on $N$ and $N^{*}$, respectively, and go to
$0$ as $N,N^{*}\to\infty$. We also introduce

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

Then, we have $\bm{u}_{1}=\bm{u}-\theta_{1}\bm{v}$ and
$\bm{u}_{2}=\bm{u}+\theta_{2}\bm{v}$. Plugging these into the linearized
equations (6.7), we can write them in a matrix form as

(6.10)
$\mathscr{A}_{1}(\partial_{t}+\bm{u}\cdot\nabla)\dot{\bm{U}}+\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}}=\dot{\bm{F}},$

where

$\dot{\bm{U}}:=\begin{pmatrix}\dot{\zeta}\\\ \dot{\bm{\phi}}_{1}\\\
\dot{\bm{\phi}}_{2}\end{pmatrix},\qquad\dot{\bm{F}}:=\begin{pmatrix}\dot{f}_{0}\\\
\underline{\rho}_{1}(\dot{\bm{f}}_{1}-(\nabla\cdot(\theta_{1}\bm{l}_{1}\otimes\bm{v}))\dot{\zeta}\\\
\underline{\rho}_{2}(\dot{\bm{f}}_{2}-(\nabla\cdot(\theta_{2}\bm{l}_{2}\otimes\bm{v}))\dot{\zeta}\end{pmatrix},$

and

$\displaystyle\mathscr{A}_{1}:=\begin{pmatrix}0&-\underline{\rho}_{1}\bm{l}_{1}^{\mathrm{T}}&\underline{\rho}_{2}\bm{l}_{2}^{\mathrm{T}}\\\
\underline{\rho}_{1}\bm{l}_{1}&O&O\\\
-\underline{\rho}_{2}\bm{l}_{2}&O&O\end{pmatrix},$
$\displaystyle\mathscr{A}_{0}^{\mathrm{mod}}:=\begin{pmatrix}a&\underline{\rho}_{1}\theta_{1}\bm{l}_{1}^{\mathrm{T}}(\bm{v}\cdot\nabla)&\underline{\rho}_{2}\theta_{2}\bm{l}_{2}^{\mathrm{T}}(\bm{v}\cdot\nabla)\\\
(\bm{v}\cdot\nabla)^{*}(\underline{\rho}_{1}\theta_{1}\bm{l}_{1}\,\cdot\,)&\underline{\rho}_{1}\underline{h}_{1}L_{1}^{\mathrm{pr}}&O\\\
(\bm{v}\cdot\nabla)^{*}(\underline{\rho}_{2}\theta_{1}\bm{l}_{2}\,\cdot\,)&O&\underline{\rho}_{2}\underline{h}_{2}L_{2}^{\mathrm{pr}}\end{pmatrix}.$

Here, $(\bm{v}\cdot\nabla)^{*}$ denotes the adjoint operator of
$\bm{v}\cdot\nabla$ in $L^{2}$, that is,
$(\bm{v}\cdot\nabla)^{*}f=-\nabla\cdot(f\bm{v})$. We note that
$\mathscr{A}_{1}$ is a skew-symmetric matrix and
$\mathscr{A}_{0}^{\mathrm{mod}}$ is symmetric in $L^{2}$. Therefore, the
corresponding energy function is given by
$(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$. We put

(6.11)
$\mathscr{E}(\dot{\bm{U}}):=\|\dot{\zeta}\|_{L^{2}}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\nabla\dot{\bm{\phi}}_{\ell}\|_{L^{2}}^{2}+(\underline{h}_{\ell}\delta)^{-2}\|\dot{\bm{\phi}}_{\ell}^{\prime}\|_{L^{2}}^{2}).$

The following lemma shows that
$(\mathscr{A}_{0}^{\mathrm{mod}}\dot{U},\dot{U})_{L^{2}}\simeq\mathscr{E}(\dot{\bm{U}})$
under the non-cavitation assumption and the stability condition.

######  Lemma 6.1.

Let $c,M,\underline{h}_{\mathrm{min}}$ 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 the condition
$\underline{h}_{\mathrm{min}}\leq\underline{h}_{1},\underline{h}_{2}$, if
$H_{1},H_{2},\bm{u}_{1},\bm{u}_{2}$, and the function $a$ satisfy

(6.12)
$\begin{cases}\displaystyle\sum_{\ell=1,2}\bigl{(}\|H_{\ell}\|_{L^{\infty}}+\sqrt{\underline{\rho}_{\ell}\underline{h}_{\ell}}\|\bm{u}_{\ell}\|_{L^{\infty}}\bigr{)}+\|a\|_{L^{\infty}}\leq
M,\\\ \displaystyle
a(\bm{x})-\frac{\underline{\rho}_{1}\underline{\rho}_{2}}{\underline{\rho}_{1}\underline{h}_{2}H_{2}(\bm{x})\alpha_{2}+\underline{\rho}_{2}\underline{h}_{1}H_{1}(\bm{x})\alpha_{1}}|\bm{u}_{2}(\bm{x})-\bm{u}_{1}(\bm{x})|^{2}\geq
c,\\\ H_{1}(\bm{x})\geq c,\quad H_{2}(\bm{x})\geq
c\quad\mbox{for}\quad\bm{x}\in\mathbf{R}^{n},\end{cases}$

then for any
$\dot{\bm{U}}=(\dot{\zeta},\dot{\bm{\phi}}_{1},\dot{\bm{\phi}}_{2})^{\mathrm{T}}\in
L^{2}\times(\mathring{H}^{1}\times(H^{1})^{N})\times(\mathring{H}^{1}\times(H^{1})^{N^{*}})$
we have

$C^{-1}\mathscr{E}(\dot{\bm{U}})\leq(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}\leq
C\mathscr{E}(\dot{\bm{U}}).$

###### Proof.

This lemma can be shown along with the proof of [3, Lemma 7.4]. For the sake
of completeness, we sketch the proof. We first note that

$\displaystyle(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$
$\displaystyle=(a\dot{\zeta},\dot{\zeta})_{L^{2}}+\sum_{\ell=1,2}\\{\underline{\rho}_{\ell}\underline{h}_{\ell}(L_{\ell}^{\mathrm{pr}}\dot{\bm{\phi}}_{\ell},\dot{\bm{\phi}}_{\ell})_{L^{2}}+2\underline{\rho}_{\ell}(\theta_{\ell}\bm{l}_{\ell}\cdot(\bm{v}\cdot\nabla)\dot{\bm{\phi}}_{\ell},\dot{\zeta})_{L^{2}}\\}$
$\displaystyle=(a\dot{\zeta},\dot{\zeta})_{L^{2}}+\sum_{\ell=1,2}\biggl{\\{}\underline{\rho}_{\ell}\underline{h}_{\ell}\biggl{(}\sum_{l=1}^{n}(A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell},\partial_{l}\dot{\bm{\phi}}_{\ell})_{L^{2}}+(\underline{h}_{\ell}\delta)^{-2}(C_{\ell}\dot{\bm{\phi}}_{\ell},\dot{\bm{\phi}}_{\ell})_{L^{2}}\biggr{)}$
$\displaystyle\makebox[90.00014pt]{}+2\underline{\rho}_{\ell}(\theta_{\ell}\bm{v}\cdot(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell},\dot{\zeta})_{L^{2}}\biggr{\\}},$

where we used the identity
$\bm{a}\cdot(\bm{v}\cdot\nabla)\bm{\varphi}=\bm{v}\cdot(\bm{a}\otimes\nabla)^{\mathrm{T}}\bm{\varphi}$.
On the other hand, we can put

$\begin{pmatrix}q_{\ell}(H_{\ell})&\bm{q}_{\ell}(H_{\ell})^{\mathrm{T}}\\\
-\bm{q}_{\ell}(H_{\ell})&Q_{\ell}(H_{\ell})\end{pmatrix}:=\begin{pmatrix}0&\bm{l}_{\ell}(H_{\ell})^{\mathrm{T}}\\\
-\bm{l}_{\ell}(H_{\ell})&A_{\ell}(H_{\ell})\end{pmatrix}^{-1}$

for $\ell=1,2$. Then, we see that $q_{\ell}(H_{\ell})=H_{\ell}\alpha_{\ell}$
and that $Q_{\ell}(H_{\ell})$ is nonnegative. Moreover, the identity

(6.13)
$A_{\ell}(H_{\ell})\bm{\varphi}_{\ell}\cdot\bm{\varphi}_{\ell}=q_{\ell}(H_{\ell})(\bm{l}_{\ell}(H_{\ell})\cdot\bm{\varphi}_{\ell})^{2}+Q_{\ell}(H_{\ell})A_{\ell}(H_{\ell})\bm{\varphi}_{\ell}\cdot
A_{\ell}(H_{\ell})\bm{\varphi}_{\ell}$

holds for any $\bm{\varphi}_{\ell}$. Therefore,

$\displaystyle\sum_{l=1}^{n}(A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell},\partial_{l}\dot{\bm{\phi}}_{\ell})_{L^{2}}$
$\displaystyle=\sum_{l=1}^{n}\\{(q_{\ell}\bm{l}_{\ell}\cdot\partial_{l}\dot{\bm{\phi}}_{\ell},\bm{l}_{\ell}\cdot\partial_{l}\dot{\bm{\phi}}_{\ell})_{L^{2}}+(Q_{\ell}A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell},A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell})_{L^{2}}\\}$
$\displaystyle=(H_{\ell}\alpha_{\ell}(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell},(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell})_{L^{2}}+\sum_{l=1}^{n}(Q_{\ell}A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell},A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell})_{L^{2}},$

so that

$\displaystyle(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$
$\displaystyle=(a\dot{\zeta},\dot{\zeta})_{L^{2}}+\sum_{\ell=1,2}\\{\underline{\rho}_{\ell}\underline{h}_{\ell}(H_{\ell}\alpha_{\ell}(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell},(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell})_{L^{2}}$
$\displaystyle\makebox[80.00012pt]{}+2\underline{\rho}_{\ell}(\theta_{\ell}\bm{v}\cdot(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell},\dot{\zeta})_{L^{2}}\\}$
$\displaystyle\quad\;+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\biggl{\\{}\sum_{l=1}^{n}(Q_{\ell}A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell},A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell})_{L^{2}}+(\underline{h}_{\ell}\delta)^{-2}(C_{\ell}\dot{\bm{\phi}}_{\ell},\dot{\bm{\phi}}_{\ell})_{L^{2}}\biggr{\\}}$
$\displaystyle=:I_{1}+I_{2}.$

We proceed to evaluate $I_{1}$.

$\displaystyle I_{1}$
$\displaystyle\geq\int_{\mathbf{R}^{n}}\biggl{\\{}a\dot{\zeta}^{2}+\sum_{\ell=1,2}\bigl{(}\underline{\rho}_{\ell}\underline{h}_{\ell}H_{\ell}\alpha_{\ell}|(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell}|^{2}-2\underline{\rho}_{\ell}\theta_{\ell}|\bm{v}||(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell}||\dot{\zeta}|\bigr{)}\biggr{\\}}\mathrm{d}\bm{x}$
$\displaystyle=\int_{\mathbf{R}^{n}}\mathfrak{A}_{0}\begin{pmatrix}\dot{\zeta}\\\
\sqrt{\underline{\rho}_{1}\underline{h}_{1}}|(\bm{l}_{1}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{1}|\\\
\sqrt{\underline{\rho}_{2}\underline{h}_{2}}|(\bm{l}_{2}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{2}|\end{pmatrix}\cdot\begin{pmatrix}\dot{\zeta}\\\
\sqrt{\underline{\rho}_{1}\underline{h}_{1}}|(\bm{l}_{1}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{1}|\\\
\sqrt{\underline{\rho}_{2}\underline{h}_{2}}|(\bm{l}_{2}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{2}|\end{pmatrix}\mathrm{d}\bm{x},$

where the matrix $\mathfrak{A}_{0}$ is given by

$\mathfrak{A}_{0}=\begin{pmatrix}a&-\sqrt{\underline{\rho}_{1}/\underline{h}_{1}}\theta_{1}|\bm{v}|&-\sqrt{\underline{\rho}_{2}/\underline{h}_{2}}\theta_{2}|\bm{v}|\\\
-\sqrt{\underline{\rho}_{1}/\underline{h}_{1}}\theta_{1}|\bm{v}|&H_{1}\alpha_{1}&0\\\
-\sqrt{\underline{\rho}_{2}/\underline{h}_{2}}\theta_{2}|\bm{v}|&0&H_{2}\alpha_{2}\end{pmatrix}.$

Here, we see that

$\displaystyle\det\mathfrak{A}_{0}=H_{1}H_{2}\alpha_{1}\alpha_{2}\biggl{(}a-\frac{\underline{\rho}_{1}\underline{\rho}_{2}}{\underline{\rho}_{1}\underline{h}_{2}H_{2}\alpha_{2}+\underline{\rho}_{2}\underline{h}_{1}H_{1}\alpha_{1}}|\bm{v}|^{2}\biggr{)}\geq
c^{3}\alpha_{1}\alpha_{2}>0,$

so that $\mathfrak{A}_{0}$ is positive definite by Sylvester’s criterion.
Moreover,
$\operatorname{tr}\mathfrak{A}_{0}\leq\max\\{1,\alpha_{1},\alpha_{2}\\}\,M\lesssim
1$ and the minimal eigenvalue of the matrix $\mathfrak{A}_{0}$ is bounded from
below by $4\det\mathfrak{A}_{0}/(\operatorname{tr}\mathfrak{A}_{0})^{2}\gtrsim
1$. Therefore, we obtain

$I_{1}\gtrsim\int_{\mathbf{R}^{n}}\biggl{(}\dot{\zeta}^{2}+\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}H_{\ell}\alpha_{\ell}|(\bm{l}_{\ell}\otimes\nabla)^{\mathrm{T}}\dot{\bm{\phi}}_{\ell}|^{2}\biggr{)}\mathrm{d}\bm{x}.$

As for $I_{2}$, it is easy to see that
$(C_{\ell}\dot{\bm{\phi}}_{\ell},\dot{\bm{\phi}}_{\ell})_{L^{2}}\simeq\|\dot{\bm{\phi}}_{\ell}^{\prime}\|_{L^{2}}^{2}$
for $\ell=1,2$. Summarizing the above estimates and using the decomposition
(6.13) again, we obtain
$(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}\gtrsim\mathscr{E}(\dot{\bm{U}})$.

In order to obtain the estimate of
$(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$ from
above, it is sufficient to show that each element of the matrix
$\mathfrak{A}_{0}$ is uniformly bounded. Since $\theta_{1}+\theta_{2}=1$, we
have

$\begin{cases}\sqrt{\underline{\rho}_{1}/\underline{h}_{1}}\theta_{1}|\bm{v}|\leq\underline{h}_{1}^{-1}\sqrt{\underline{\rho}_{1}\underline{h}_{1}}|\bm{u}_{1}|+\sqrt{\underline{\rho}_{1}/\underline{h}_{1}}\theta_{1}|\bm{u}_{2}|,\\\
\sqrt{\underline{\rho}_{2}/\underline{h}_{2}}\theta_{2}|\bm{v}|\leq\sqrt{\underline{\rho}_{2}/\underline{h}_{2}}\theta_{2}|\bm{u}_{1}|+\underline{h}_{2}^{-1}\sqrt{\underline{\rho}_{2}\underline{h}_{2}}|\bm{u}_{2}|.\end{cases}$

Here, we see that

$\displaystyle\sqrt{\underline{\rho}_{1}/\underline{h}_{1}}\theta_{1}|\bm{u}_{2}|$
$\displaystyle=\frac{1}{\underline{h}_{2}}\sqrt{\frac{H_{1}\alpha_{1}}{H_{2}\alpha_{2}}}\frac{\sqrt{(\underline{\rho}_{1}\underline{h}_{2}H_{2}\alpha_{2})(\underline{\rho}_{2}\underline{h}_{1}H_{1}\alpha_{1})}}{\underline{\rho}_{1}\underline{h}_{2}H_{2}\alpha_{2}+\underline{\rho}_{2}\underline{h}_{1}H_{1}\alpha_{1}}\sqrt{\underline{\rho}_{2}\underline{h}_{2}}|\bm{u}_{2}|$
$\displaystyle\leq\frac{1}{2\underline{h}_{2}}\sqrt{\frac{H_{1}\alpha_{1}}{H_{2}\alpha_{2}}}\sqrt{\underline{\rho}_{2}\underline{h}_{2}}|\bm{u}_{2}|$
$\displaystyle\leq\frac{1}{2\underline{h}_{\mathrm{min}}\sqrt{\frac{M\alpha_{1}}{c\alpha_{2}}}M}\lesssim
1.$

Similarly, we have
$\sqrt{\underline{\rho}_{2}/\underline{h}_{2}}\theta_{2}|\bm{u}_{1}|\lesssim
1$. Therefore, we obtain
$(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}\lesssim\mathscr{E}(\dot{\bm{U}})$.
∎

In the following Lemma we provide uniform energy estimates for regular
solutions to the linearized Kakinuma model (6.7).

######  Lemma 6.2.

Let $c,M,M_{1},\underline{h}_{\mathrm{min}}$ be positive constants. There
exist 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) and the condition
${\underline{h}_{\mathrm{min}}\leq\underline{h}_{1},\underline{h}_{2}}$, if
$H_{1},H_{2},\bm{u}_{1},\bm{u}_{2}$, and the function $a$ satisfy (6.12) and

$\sum_{\ell=1,2}\bigl{(}\|\partial_{t}H_{\ell}\|_{L^{\infty}}+\|\nabla
H_{\ell}\|_{L^{\infty}}+\underline{\rho}_{\ell}\underline{h}_{\ell}(\|\partial_{t}\bm{u}_{\ell}\|_{L^{\infty}}^{2}+\|\nabla\bm{u}_{\ell}\|_{L^{\infty}}^{2})\bigr{)}+\|\partial_{t}a\|_{L^{\infty}}+\|\nabla
a\|_{L^{\infty}}\leq M_{1},\\\ $

then for any regular solution
$\dot{\bm{U}}=(\dot{\zeta},\dot{\bm{\phi}}_{1},\dot{\bm{\phi}}_{2})^{\mathrm{T}}$
to the linearized Kakinuma model (6.7) we have

$\mathscr{E}(\dot{\bm{U}}(t))\leq
C\mathrm{e}^{C_{1}t}\mathscr{E}(\dot{\bm{U}}(0))+C_{1}\int_{0}^{t}\mathrm{e}^{C_{1}(t-\tau)}\biggl{\\{}\|\dot{f}_{0}(\tau)\|_{H^{1}}(\|\partial_{t}\dot{\zeta}(\tau)\|_{H^{-1}}+\|\dot{\zeta}(\tau)\|_{L^{2}})\\\
+\sum_{\ell=1,2}\underline{\rho}_{\ell}(\|\dot{\bm{f}}_{\ell}(\tau)\|_{L^{2}}+\|\dot{\zeta}(\tau)\|_{L^{2}})\|(\partial_{t}\dot{\bm{\phi}}_{\ell}(\tau),\nabla\dot{\bm{\phi}}_{\ell}(\tau))\|_{L^{2}}\biggr{\\}}\mathrm{d}\tau.$

###### Proof.

We deduce from (6.10) that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$
$\displaystyle=([\partial_{t},\mathscr{A}_{0}^{\mathrm{mod}}]\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}+2(\mathscr{A}_{0}^{\mathrm{mod}}\partial_{t}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$
$\displaystyle=([\partial_{t},\mathscr{A}_{0}^{\mathrm{mod}}]\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}+2((\partial_{t}+\bm{u}\cdot\nabla)\dot{\bm{U}},\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}})_{L^{2}}-2((\bm{u}\cdot\nabla)\dot{\bm{U}},\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}})_{L^{2}}$
$\displaystyle=([\partial_{t},\mathscr{A}_{0}^{\mathrm{mod}}]\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}-2((\bm{u}\cdot\nabla)\dot{\bm{U}},\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}})_{L^{2}}+2((\partial_{t}+\bm{u}\cdot\nabla)\dot{\bm{U}},\dot{\bm{F}})_{L^{2}}$
$\displaystyle=:I_{1}+I_{2}+I_{3},$

where we used the fact that $\mathscr{A}_{0}^{\mathrm{mod}}$ is a symmetric
operator in $L^{2}$ and that $\mathscr{A}_{1}$ is a skew-symmetric matrix. As
for $I_{1}$, we have

$\displaystyle
I_{1}=((\partial_{t}a)\dot{\zeta},\dot{\zeta})_{L^{2}}+\sum_{\ell=1,2}\biggl{\\{}\underline{\rho}_{\ell}\underline{h}_{\ell}\biggl{(}\sum_{l=1}^{n}((\partial_{t}A_{\ell})\partial_{l}\dot{\bm{\phi}}_{\ell},\partial_{l}\dot{\bm{\phi}}_{\ell})_{L^{2}}+(\underline{h}_{\ell}\delta)^{-2}((\partial_{t}C_{\ell})\dot{\bm{\phi}}_{\ell},\dot{\bm{\phi}}_{\ell})_{L^{2}}\biggr{)}$
$\displaystyle+2\underline{\rho}_{\ell}([\partial_{t},\theta_{\ell}\bm{l}_{\ell}^{\mathrm{T}}(\bm{v}\cdot\nabla)]\dot{\bm{\phi}}_{\ell},\dot{\zeta})_{L^{2}}\biggr{\\}}.$

Here, as in the proof of Lemma 6.1 we have
$\sqrt{\underline{\rho}_{\ell}/\underline{h}_{\ell}}\theta_{\ell}(|\bm{v}|+|\partial_{t}\bm{v}|)\lesssim
1$ for $\ell=1,2$. In view of the relations
$\partial_{t}\theta_{1}=-\partial_{t}\theta_{2}=\theta_{1}\theta_{2}(H_{1}^{-1}\partial_{t}H_{1}-H_{2}^{-1}\partial_{t}H_{2})$,
we have $|\partial_{t}\theta_{\ell}|\lesssim\theta_{1}\theta_{2}$ for
$\ell=1,2$. Therefore, we obtain $|I_{1}|\lesssim\mathscr{E}(\dot{\bm{U}})$.
As for $I_{2}$, by integration by parts we have

$\displaystyle I_{2}$
$\displaystyle=((\nabla\cdot(a\bm{u}))\dot{\zeta},\dot{\zeta})_{L^{2}}$
$\displaystyle\quad\;-\sum_{\ell=1,2}\underline{\rho}_{\ell}\underline{h}_{\ell}\biggl{\\{}\sum_{l=1}^{n}\bigl{\\{}2(A_{\ell}\partial_{l}\dot{\bm{\phi}}_{\ell},((\partial_{l}\bm{u})\cdot\nabla)\dot{\bm{\phi}}_{\ell})_{L^{2}}+(((\bm{u}\cdot\nabla)^{*}A_{\ell})\partial_{l}\dot{\bm{\phi}}_{\ell},\partial_{\ell}\dot{\bm{\phi}}_{\ell})_{L^{2}}\bigr{\\}}$
$\displaystyle\makebox[70.0001pt]{}+(\underline{h}_{\ell}\delta)^{-2}(((\bm{u}\cdot\nabla)^{*}C_{\ell})\dot{\bm{\phi}}_{\ell},\dot{\bm{\phi}}_{\ell})_{L^{2}}\biggr{\\}}$
$\displaystyle\quad\;+2\sum_{\ell=1,2}\underline{\rho}_{\ell}\bigl{\\{}((\nabla\cdot\bm{u})\dot{\zeta},\theta_{\ell}\bm{l}_{\ell}^{\mathrm{T}}(\bm{v}\cdot\nabla)\dot{\bm{\phi}}_{\ell})_{L^{2}}+(\dot{\zeta},[\bm{u}\cdot\nabla,\theta_{\ell}\bm{l}_{\ell}^{\mathrm{T}}(\bm{v}\cdot\nabla)]\dot{\bm{\phi}}_{\ell})_{L^{2}}.$

By using (2.14), we see that

$\theta_{1}\simeq\frac{\underline{\rho}_{2}\underline{h}_{1}}{\underline{\rho}_{1}\underline{h}_{2}+\underline{\rho}_{2}\underline{h}_{1}}=\frac{\underline{\rho}_{2}}{\underline{h}_{2}},\qquad\theta_{2}\simeq\frac{\underline{\rho}_{1}\underline{h}_{2}}{\underline{\rho}_{1}\underline{h}_{2}+\underline{\rho}_{2}\underline{h}_{1}}=\frac{\underline{\rho}_{1}}{\underline{h}_{1}}.$

Therefore, we have
$|\bm{u}|\leq\theta_{2}|\bm{u}_{1}|+\theta_{1}|\bm{u}_{2}|\lesssim 1$. In view
of $|\nabla\theta_{\ell}|\lesssim\theta_{1}\theta_{2}$ for $\ell=1,2$, we have
also $|\nabla\bm{u}|\lesssim 1$ and
$\sqrt{\rho_{\ell}/\underline{h}_{\ell}}\theta_{l}|\nabla\bm{v}|\lesssim 1$
for $\ell=1,2$. Hence, we obtain $|I_{2}|\lesssim\mathscr{E}(\dot{\bm{U}})$.
Finally, as for $I_{3}$, we have

$\displaystyle I_{3}$
$\displaystyle=2(\partial_{t}\dot{\zeta},\dot{f}_{0})_{L^{2}}-2(\dot{\zeta},\nabla\cdot(\bm{u}\dot{f}_{0}))_{L^{2}}$
$\displaystyle\quad\;+2\sum_{\ell=1,2}\underline{\rho}_{\ell}((\partial_{t}+\bm{u}\cdot\nabla)\dot{\bm{\phi}}_{\ell},\dot{\bm{f}}_{\ell}-(\nabla\cdot(\theta_{\ell}\bm{l}_{\ell}\otimes\bm{v}))\dot{\zeta})_{L^{2}}$
$\displaystyle\lesssim\|\dot{f}_{0}\|_{H^{1}}(\|\partial_{t}\dot{\zeta}\|_{H^{-1}}+\|\dot{\zeta}\|_{L^{2}})+\sum_{\ell=1,2}\underline{\rho}_{\ell}(\|\dot{\bm{f}}_{\ell}\|_{L^{2}}+\|\dot{\zeta}\|_{L^{2}})\|(\partial_{t}\dot{\bm{\phi}}_{\ell},\nabla\dot{\bm{\phi}}_{\ell})\|_{L^{2}}.$

Summarizing the above estimates we obtain

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}(\mathscr{A}_{0}^{\mathrm{mod}}\dot{\bm{U}},\dot{\bm{U}})_{L^{2}}$
$\displaystyle\lesssim\mathscr{E}(\dot{\bm{U}})+\|\dot{f}_{0}\|_{H^{1}}(\|\partial_{t}\dot{\zeta}\|_{H^{-1}}+\|\dot{\zeta}\|_{L^{2}})$
$\displaystyle\quad\;+\sum_{\ell=1,2}\underline{\rho}_{\ell}(\|\dot{\bm{f}}_{\ell}\|_{L^{2}}+\|\dot{\zeta}\|_{L^{2}})\|(\partial_{t}\dot{\bm{\phi}}_{\ell},\nabla\dot{\bm{\phi}}_{\ell})\|_{L^{2}}.$

This together with Lemma 6.1 and Gronwall’s inequality gives the desired
estimate. ∎

###  6.2 Energy estimates

In this subsection, we will complete the proof of Theorem 3.1. The existence
and the uniqueness of the solution to the initial value problem for the
Kakinuma model (6.1) has already been established in the companion paper [3],
so that it is sufficient to derive the uniform bound (3.14) of the solution
for some time interval $[0,T]$ independent of parameters. The following lemma
can be shown in the same way as the proof of [6, Lemma 4.2].

######  Lemma 6.3.

Let $c,M$ be positive constants and $m$ an integer such that
$m>\frac{n}{2}+1$. 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 $\zeta\in
H^{m-1}$, $b\in W^{m,\infty}$, $H_{1}=1-\underline{h}_{1}^{-1}\zeta$, and