Comparison with surface gravity waves.

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 $\mathscr {L}^\mathrm {app}(\boldsymbol {\phi },\zeta )=\mathscr {L}_\mathrm {Luke}(\Phi ^\mathrm {app},\zeta )$, where

(1.3)\begin{equation} \Phi^\mathrm{app}(\boldsymbol{x},z,t) = \sum_{i=0}^N Z_i(z;b(\boldsymbol{x}))\phi_{i}(\boldsymbol{x},t). \end{equation}

The Isobe–Kakinuma model was first proposed by Isobe [Reference Isobe21, Reference Isobe22] and then applied by Kakinuma to simulate numerically the water waves. Recently, this model was analysed from a mathematical point of view when the function system $\{Z_i\}$ is a set of polynomials in $z: Z_i(z;b(\boldsymbol {x}))=(z+h-b(\boldsymbol {x}))^{p_i}$ with integers $p_i$ satisfying $0=p_0< p_1<\cdots < p_N$. The initial value problem was analysed by Murakami and Iguchi [Reference Murakami and Iguchi35] in a special case and by Nemoto and Iguchi [Reference Nemoto and Iguchi36] 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 in the sense that the operator acting on time derivatives of the unknowns has a non-trivial kernel. As a consequence, 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, Iguchi [Reference Iguchi18, Reference Iguchi19] showed that under the choice of the function system

(1.4)\begin{equation} Z_i(z;b(\boldsymbol{x})) = \begin{cases} (z+h)^{2i} & \text{in the case of the flat bottom}, \\ (z+h-b(\boldsymbol{x}))^i & \text{in the case of a variable bottom}, \end{cases} \end{equation}

the Isobe–Kakinuma model is a higher order shallow water approximation for the water wave problem in the strongly non-linear regime. Furthermore, Duchêne and Iguchi [Reference Duchêne and Iguchi13] showed that the Isobe–Kakinuma model also enjoys a Hamiltonian structure analogous to the one exhibited by Zakharov [Reference Zakharov43] 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 [Reference Duchêne and Iguchi14] is to extend these results on surface gravity waves to the framework of interfacial gravity waves. With respect to surface gravity waves, our interfacial gravity waves framework brings two additional difficulties. The first one is that, due to the rigid-lid assumption, the full system for interfacial gravity waves described in § 2 features only one evolution equation for the two velocity potentials, and a constraint associated with the fixed fluid domain. From a physical perspective, the unknown velocity potential at the interface may be interpreted as a Lagrange multiplier associated with the constraint. A second important difference between water waves and interfacial gravity waves is that the latter suffer from Kelvin–Helmholtz instabilities. As a consequence, the initial value problem of the full model for interfacial gravity waves is ill-posed in Sobolev spaces; see Iguchi et al. [Reference Iguchi, Tanaka and Tani20], Kamotski and Lebeau [Reference Kamotski and Lebeau26]. This raises the question of the validity of any model for interfacial gravity waves. A partial answer is offered by the work of Lannes [Reference Lannes28], which proves the existence and uniqueness of solutions over large time intervals in the presence of interfacial tension. While interfacial tension effects are not expected to be the relevant regularization mechanism for the propagation of waves between, for instance, fresh and salted water, the key observation is that physical systems allow the propagation of waves with large amplitude and long wavelengths provided that some mechanism tames Kelvin–Helmholtz instabilities acting on the high-frequency component of the flow. This description is consistent with the fact that the initial value problem of the bi-layer shallow water system for the propagation of interfacial gravity waves in the hydrostatic framework is well-posed in Sobolev spaces under some hyperbolicity condition describing the absence of low-frequency Kelvin–Helmholtz instabilities, as proved by Bresch and Renardy [Reference Bresch and Renardy5]. Let us mention however that such a property is not automatic for higher order shallow water models. Specifically, we note that the Miyata–Choi–Camassa model derived by Miyata [Reference Miyata34] and Choi and Camassa [Reference Choi and Camassa8] and which can be regarded as a two-layer generalization of the Green–Naghdi equations for water waves turns out to overestimate Kelvin–Helmholtz estimates with respect to the full model; see Lannes and Ming [Reference Lannes and Ming30].

In [Reference Duchêne and Iguchi14], we analysed the initial value problem of the Kakinuma model when the approximated velocity potentials are defined by (1.2). We found that the Kakinuma model has a stability regime which can be expressed as

(1.5)\begin{equation} - \partial_z (P_2^\mathrm{app} - P_1^\mathrm{app} ) - \frac{\rho_1\rho_2}{\rho_1H_2\alpha_2 + \rho_2H_1\alpha_1} |\nabla\Phi_2^\mathrm{app} - \nabla\Phi_1^\mathrm{app}|^2 \geq c_0 > 0 \end{equation}

on the interface, where $H_1:= h_1 - \zeta$ and $H_2:=h_2 + \zeta - b$ are the depths of the upper and the lower layers, $P_1^\mathrm {app}$ and $P_2^\mathrm {app}$ are approximate pressures of the fluids 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. It is worth noticing that, consistently with the expectation that the Kakinuma model is a higher order model for the full system for interfacial gravity waves and that the latter suffers from Kelvin–Helmholtz instabilities, 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. When $N=N^*=0$, the Kakinuma model coincides with the aforementioned bi-layer shallow water system, and the stability regime coincides with the hyperbolic domain exhibited in [Reference Bresch and Renardy5]. Moreover, when the motion of the fluids 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)\begin{equation} (\rho_2-\rho_1)g>0. \end{equation}

In [Reference Duchêne and Iguchi14], we showed that under the stability condition (1.5), the non-cavitation assumptions

(1.7)\begin{equation} H_1\geq c_0>0, \quad H_2\geq c_0>0, \end{equation}

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. We also showed in [Reference Duchêne and Iguchi14] that the Kakinuma model enjoys a Hamiltonian structure analogous to the one exhibited by Benjamin and Bridges [Reference Benjamin and Bridges3] on the full model for interfacial gravity waves.

Comparison with other higher order models.

The Isobe–Kakinuma and the Kakinuma models belong to higher order models for the water waves and for the full interfacial gravity waves, respectively. By this we mean a family of systems of equations parametrized by nonnegative integers describing the order of the system within the family, that is $N$ for the Isobe–Kakinuma model, and whose solutions are expected to approach solutions to the full system as the order increases. Several such models have been introduced in the literature, mostly in the water waves framework, and we will restrict the discussion to water waves in this paragraph.

Based on a Taylor expansion of the Dirichlet-to-Neumann operator at stake in the water waves system with respect to the shape of the domain, Dommermuth and Yue [Reference Dommermuth and Yue10], West et al. [Reference West, Brueckner, Janda, Milder and Milton41] and Craig and Sulem [Reference Craig and Sulem9] have proposed the so-called high order spectral (HOS) models. While these models have been successfully employed in efficient numerical schemes (see recent accounts by Wilkening and Vasan [Reference Wilkening and Vasan42], Nicholls [Reference Nicholls37] and Guyenne [Reference Guyenne16]), the equations feature Fourier multipliers which prevent their direct use in situations involving non-trivial geometries such as horizontal boundaries. Moreover, the rigorous justification of HOS models is challenged by well-posedness issues; see the discussion in Ambrose et al. [Reference Ambrose, Bona and Nicholls1], and Duchêne and Melinand [Reference Duchêne and Melinand15].

A second class of higher order models originate from formal shallow water expansions put forward by Boussinesq [Reference Boussinesq4] and Rayleigh [Reference Rayleigh39]. A systematic derivation procedure has been described by Friedrichs in the appendix to [Reference Stoker40]. Recently, these higher order shallow water models have been described and discussed by Matsuno in [Reference Matsuno32, Reference Matsuno33] and Choi in [Reference Choi6, Reference Choi7]. The derivation procedure displays formula for approximate velocity potentials under form (1.3)–(1.4) (in particular, only even powers appear in the flat bottom case), with the important difference that the functions $\phi _{i}$ ($i=0,\ldots,N$) are prescribed through explicit recursion relations. An important consequence of this derivation is that the resulting systems of equations involve only standard differential operators. However, the order of the differential operators at stake augments with the order of the system, which renders such models impractical for numerical simulations.

By contrast, the Isobe–Kakinuma model features only differential operators of order at most two acting on the variables $\phi _{i}$ ($i=0,\ldots,N$) which are unknowns of the system. Notice that the size of the system augments with its order, $N$. However, the degrees of freedom do not augment with the order since, as mentioned above, some compatibility conditions must be satisfied. In fact all quantities are uniquely determined by two scalar functions which represent the canonical variables in the Hamiltonian formulation of the water waves system. Let us mention that function systems different from (1.4) have been considered by Athanassoulis and Belibassakis [Reference Athanassoulis and Belibassakis2], Klopman et al. [Reference Klopman, van Groesen and Dingemans27] and Papoutsellis and Athanassoulis [Reference Papoutsellis and Athanassoulis38] (see also references therein). While the systems obtained in these works have a similar nature, they are all different. We let the reader refer to Duchêne [Reference Duchêne11, chapter D] for an extended discussion and comparison of these models.

The choice of the function systems in (1.4) is motivated by the aforementioned Friedrichs expansion and is essential in the analysis of Iguchi [Reference Iguchi18, Reference Iguchi19] proving that the Isobe–Kakinuma model is a higher order shallow water approximation for the water wave problem in the strongly non-linear regime. We note that one may modify (1.4) by putting all odd and even terms $(z+h)^{i}$ for $i=0,1,\ldots$ in the case of the flat bottom. However, in that case, one needs to use the terms up to order $2N$ to keep the same precision of the approximation. Therefore, such a choice increases the number of unkonwns and equations by $N$ so that it is undesirable for practical application. In other words, one can save memories in numerical simulations by using only even terms in the case of the flat bottom. On the contrary, if we put only odd terms $(z+h-b(\boldsymbol {x}))^{2i}$ for $i=0,1,2,\ldots$ in the case of a non-flat bottom, then the corresponding Isobe–Kakinuma model does not give any good approximation even if we take $N$ a sufficiently large number, because the corresponding approximate velocity potential $\Phi ^\mathrm {app}$ cannot approximate the boundary condition on the bottom so well due to the lack of odd order terms $(z+h-b(\boldsymbol {x}))^{2i+1}$ for $i=0,1,2,\ldots$.

Following this discussion, the choice of the function systems (1.2) with (H1) or (H2) in our interfacial waves framework is very natural. In particular, the rigid-lid is assumed to be flat so that we do not need to use odd order terms $(-z+h_1)^{2i+1}$ for $i=0,1,2,\ldots$, in the approximate velocity potential $\Phi _1^\mathrm {app}$ to obtain a good approximation, because $\Phi _1^\mathrm {app}$ can approximate the boundary condition on the rigid-lid without such terms.

Description of the results.

In the present paper, we show that the Kakinuma model obtained through the approximated potentials (1.2) with

1. (H1) $N^*=N$ and $p_i=2i$ $(i=0,1,\ldots,N)$ in the case of the flat bottom $b(\boldsymbol {x})\equiv 0$,

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 non-linear regime. Our results apply to the dimensionless Kakinuma model obtained after suitable rescaling. The system of equations then depend on the positive dimensionless parameters $\delta _1$ and $\delta _2$ which 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 $\lambda$. The shallow water regime is described through the smallness of the parameters $\delta _1$ and $\delta _2$. What is more, our results are uniform with respect to parameters satisfying either $\rho _2\lesssim \rho _1<\rho _2$, or $\rho _1\ll \rho _2$ and $h_2\lesssim h_1$. We notice that the rigid-lid framework is expected to be invalid in the regime $\rho _1\ll \rho _2$ and $h_1\ll h_2$ which is excluded in this paper; see Duchêne [Reference Duchêne12].

Our first result extends the result of [Reference Duchêne and Iguchi14] on the well-posedness of the initial value problem by showing that solutions to the dimensionless Kakinuma model are defined on a time interval which does not vanish for arbitrarily small values of $\delta _1$ and $\delta _2$.

While the non-cavitation assumption and the stability condition are automatically satisfied for small initial data and small bottom topography $b$, an arrangement of non-trivial initial data satisfying the compatibility conditions with suitable bounds is a non-trivial issue, and demands a specific analysis.

Then, we show that under the special choice of the indices $p_0,p_1,\ldots,p_{N^*}$ as in (H1) or (H2), the dimensionless Kakinuma model is consistent with the full model for interfacial gravity waves with an error of order $O(\delta _1^{4N+2}+\delta _2^{4N+2})$.

In the last result, we assume the existence of a solution to the full model with a uniform bound since for general initial data in Sobolev spaces, one cannot expect to construct a solution to the initial value problem, due to the ill-posedness of the problem discussed previously. The same issue arises for the full justification of the Kakinuma model.

Theorem 1.4 Full justification

Assuming the existence of a solution to the dimensionless full interfacial gravity waves system with a uniform bound and satisfying initially the (dimensionless) stability condition (1.5) and (dimensionless) non-cavitation assumptions (1.7), then the Kakinuma model with (H1) or (H2) and appropriate initial data produces an approximate solution with the error estimate

\[ |\zeta^{{\rm K}}(\boldsymbol{x},t)-\zeta^{{\rm IW}}(\boldsymbol{x},t)| \lesssim \delta_1^{4N+2}+\delta_2^{4N+2} \]

on some time interval independent of $\delta _1\in (0,1]$ and $\delta _2\in (0,1]$, where $\zeta ^{{\rm K}}$ and $\zeta ^{{\rm IW}}$ are solutions to the dimensionless Kakinuma model and to the full model, respectively.

In our last main result, we show that the Hamiltonian structure of the Kakinuma model is a shallow water approximation of the Hamiltonian structure of the full interfacial gravity waves model.

Theorem 1.5 Hamiltonians

Assume (H1) or (H2). Under appropriate assumptions on the canonical variables $(\zeta,\phi )$, we have

\[ |\mathscr{H}^{{\rm K}}(\zeta,\phi)-\mathscr{H}^{{\rm IW}}(\zeta,\phi)| \lesssim \delta_1^{4N+2}+\delta_2^{4N+2}, \]

where $\mathscr {H}^{{\rm K}}$ and $\mathscr {H}^{{\rm IW}}$ are the Hamiltonians of the dimensionless Kakinuma model and of the dimensionless full interfacial gravity waves model, respectively.

Structures of the Kakinuma model.

In order to obtain our main results, we exploit several structures of the Kakinuma model. The Kakinuma model can be written compactly as

(1.8)\begin{equation} \begin{cases} {\boldsymbol l}_1(H_1)\partial_t\zeta + L_1(H_1){\boldsymbol \phi}_1 = {\boldsymbol 0}, \\ {\boldsymbol l}_2(H_2)\partial_t\zeta - L_2(H_2,b){\boldsymbol \phi}_2 = {\boldsymbol 0}, \\ {\rho}_1\left\{ {\boldsymbol l}_1(H_1) \cdot \partial_t{\boldsymbol \phi}_1 + \frac12\left( |{\boldsymbol u}_1|^2 + w_1^2 \right) \right\} \\ \quad - {\rho}_2\left\{ {\boldsymbol l}_2(H_2) \cdot \partial_t{\boldsymbol \phi}_2 + \frac12\left(|{\boldsymbol u}_2|^2 + w_2^2 \right) \right\} + (\rho_1-\rho_2)g\zeta = 0, \end{cases} \end{equation}

where we denote ${\boldsymbol \phi }_1 := (\phi _{1,0},\phi _{1,1},\ldots,\phi _{1,N})^\mathrm {T}$, ${\boldsymbol \phi }_2 := (\phi _{2,0},\phi _{2,1},\ldots,\phi _{2,N^*})^\mathrm {T}$, put ${\boldsymbol l}_1(H_1) := (1,H_1^2,H_1^4,\ldots,H_1^{2N})^\mathrm {T}$, ${\boldsymbol l}_2(H_2) := (1,H_2^{p_1},H_2^{p_2},\ldots,H_2^{p_{N^*}})^\mathrm {T}$, and the linear operators $L_\ell$, and functions ${\boldsymbol u}_\ell$ and $w_\ell$ for $\ell =1,2$ are defined (after non-dimensionalization) in § 3. Here we recognize the fact that the hypersurface $t=0$ in the space-time $\mathbf {R}^n\times \mathbf {R}$ is characteristic for the Kakinuma model, since the system of evolution equations is overdetermined for the variable $\zeta$, and underdetermined for the variables ${\boldsymbol \phi }_1$ and ${\boldsymbol \phi }_2$. As a consequence, solutions to the Kakinuma model must satisfy some compatibility conditions. Introducing linear operators $\mathcal {L}_{1,i}$ $(i=0,\ldots,N)$ acting on ${\boldsymbol \varphi }_1 = (\varphi _{1,0},\ldots,\varphi _{1,N})^\mathrm {T}$ and $\mathcal {L}_{2,i}$ $(i=0,\ldots,N^*)$ acting on ${\boldsymbol \varphi }_2 = (\varphi _{2,0},\ldots,\varphi _{2,N^*})^\mathrm {T}$ by

\begin{align*} \begin{cases} \mathcal{L}_{1,0}(H_1) {\boldsymbol \varphi}_1 := \sum_{j=0}^{N} L_{1,0j}(H_1)\varphi_{1,j}, \\ \mathcal{L}_{1,i}(H_1) {\boldsymbol \varphi}_1 := \sum_{j=0}^N ( L_{1,ij}(H_1)\varphi_{1,j} - H_1^{2i}L_{1,0j}(H_1)\varphi_{1,j})\quad\text{for}\ i=1,2,\ldots,N, \\ \mathcal{L}_{2,0}(H_2,b) {\boldsymbol \varphi}_2 := \sum_{j=0}^{N^*} L_{2,0j}(H_2,b)\varphi_{2,j}, \\ \mathcal{L}_{2,i}(H_2,b) {\boldsymbol \varphi}_2 := \sum_{j=0}^{N^*} ( L_{2,ij}(H_2,b)\varphi_{2,j}\\ \quad - H_2^{p_i}L_{2,0j}(H_2,b)\varphi_{2,j} ) \quad\text{for}\ i=1,2,\ldots,N^*, \end{cases} \end{align*}

the necessary conditions can be written simply as

(1.9)\begin{equation} \begin{cases} \mathcal{L}_{1,i}(H_1) {\boldsymbol \phi}_1 = 0 \quad\text{for}\ i=1,2,\ldots,N, \\ \mathcal{L}_{2,i}(H_2,b) {\boldsymbol \phi}_2 = 0 \quad\text{for}\ i=1,2,\ldots,N^*, \\ \mathcal{L}_{1,0}(H_1) {\boldsymbol \phi}_1 + \mathcal{L}_{2,0}(H_2,b) {\boldsymbol \phi}_2 = 0. \end{cases} \end{equation}

The first two vectorial identities are analogous to the compatibility conditions of the Isobe–Kakinuma model for water waves, while the last identity is specific to the bi-layer framework and is related to the continuity of the normal component of the velocity at the interface.

A first key ingredient of the analysis is the fact that for sufficiently regular functions $\zeta$, $b$ and $\phi _1$ (respectively $\phi _2$), there exists a unique solution ${\boldsymbol \phi }_1$ (respectively ${\boldsymbol \phi }_2$) to the problems

(1.10)\begin{equation} \begin{cases} \boldsymbol{l}_1(H_1)\cdot\boldsymbol{\phi}_1=\phi_1, \quad \mathcal{L}_{1,i}(H_1)\boldsymbol{\phi}_1=0\quad\text{for}\ i=1,2,\ldots,N, \\ \boldsymbol{l}_2(H_2)\cdot\boldsymbol{\phi}_2=\phi_2, \quad \mathcal{L}_{2,i}(H_2,b)\boldsymbol{\phi}_2=0\quad\text{for}\ i=1,2,\ldots,N^* \end{cases} \end{equation}

satisfying suitable elliptic estimates. What is more, the well-defined linear operators

\begin{align*} & \Lambda_1^{(N)}(\zeta) \colon \phi_1 \mapsto \mathcal{L}_{1,0}(H_1)\boldsymbol{\phi}_1, \\ & \Lambda_2^{(N^*)}(\zeta,b) \colon \phi_2 \mapsto \mathcal{L}_{2,0}(H_2,b)\boldsymbol{\phi}_2, \end{align*}

are found to approximate the corresponding Dirichlet-to-Neumann maps $\Lambda _1(\zeta )$ and $\Lambda _2(\zeta,b)$ defined by

\begin{align*} & \Lambda_1(\zeta)\phi_1 := \left( -\partial_z\Phi_1+\nabla\Phi_1 \cdot \nabla\zeta \right)\bigr\vert_{z=\zeta(\boldsymbol{x},t)}, \\ & \Lambda_2(\zeta,b)\phi_2 := \left( \partial_z\Phi_2-\nabla\Phi_2 \cdot \nabla\zeta \right)\bigr\vert_{z=\zeta(\boldsymbol{x},t)}, \end{align*}

where $\Phi _1$ and $\Phi _2$ are the unique solutions to Laplace's equations

\[ \begin{cases} \Delta\Phi_1 + \partial_z^2\Phi_1 = 0 & \text{in}\quad \Omega_1(t), \\ \Phi_1=\phi_1 & \text{on}\quad \Gamma(t), \\ \partial_z\Phi_1 = 0 & \text{on}\quad \Sigma_1, \end{cases} \text{ and } \quad \begin{cases} \Delta\Phi_2 + \partial_z^2\Phi_2 = 0 & \text{in}\quad \Omega_2(t), \\ \Phi_2=\phi_2 & \text{on}\quad \Gamma(t), \\ \nabla\Phi_2 \cdot \nabla b - \partial_z\Phi_2 = 0 & \text{on}\quad \Sigma_2, \end{cases} \]

where 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. Specifically, it is proved that, under the special choice of the indices $p_0,p_1,\ldots,p_{N^*}$ in (H1) or (H2) and after suitable rescaling, the difference between the dimensionless operators is of size $O(\delta _1^{4N+2}+\delta _2^{4N+2})$. This analysis, which follows directly from the corresponding analysis for surface waves developed in [Reference Iguchi19] and scaling arguments, provides the key argument in the proof of the consistency result described in theorem 1.3.

In order to study the Kakinuma model, we also need to analyse the full elliptic system

(1.11)\begin{equation} \begin{cases} \mathcal{L}_{1,i}(H_1) {\boldsymbol \phi}_1 = f_{1,i} \quad\text{for}\ i=1,2,\ldots,N, \\ \mathcal{L}_{2,i}(H_2,b) {\boldsymbol \phi}_2 = f_{2,i} \quad\text{for}\ i=1,2,\ldots,N^*, \\ \mathcal{L}_{1,0}(H_1) {\boldsymbol \phi}_1 + \mathcal{L}_{2,0}(H_2,b) {\boldsymbol \phi}_2 = \nabla\cdot {\boldsymbol f}_3,\ - \boldsymbol{l}_1(H_1)\cdot\boldsymbol{\phi}_1+\boldsymbol{l}_2(H_2)\cdot\boldsymbol{\phi}_2=f_4, \end{cases} \end{equation}

for sufficiently regular functions $\zeta$, $b$ and ${\boldsymbol f}_1=(f_{1,1},\ldots f_{1,N})^\mathrm {T},{\boldsymbol f}_2=(f_{2,1},\ldots, f_{2,N^*})^\mathrm {T},{\boldsymbol f}_3,f_4$. The ellipticity of the problem relies on the coercivity of the corresponding operators $L_1(H_1)$ and $L_2(H_2)$. The solvability of (1.11) is essential in several directions. Firstly, it provides an alternative consistency result, where solutions to the full interfacial gravity waves system produce approximate solutions to the Kakinuma model but satisfying exactly and not approximately the necessary conditions (1.9). In turn, this provides a crucial ingredient to the full justification of the Kakinuma model described in theorem 1.4. Furthermore, the arrangement of initial data satisfying the compatibility conditions as stated in proposition 1.2 amounts to solving (1.11) with ${\boldsymbol f}_1={\boldsymbol 0}$, ${\boldsymbol f}_2={\boldsymbol 0}$, ${\boldsymbol f}_3={\boldsymbol 0}$ and $f_4=\phi$. Similarly, our result on the Hamiltonians $\mathscr {H}^{\text {K}}$ and $\mathscr {H}^{\text {IW}}$ described in theorem 1.5 relies on a comparison of solutions to (1.11) with ${\boldsymbol f}_1={\boldsymbol 0}$, ${\boldsymbol f}_2={\boldsymbol 0}$, ${\boldsymbol f}_3={\boldsymbol 0}$ and $f_4=\phi$ and solutions to

\[ \begin{cases} \Delta\Phi_1 + \partial_z^2\Phi_1 = 0 & \text{in}\quad \Omega_1(t), \\ \Delta\Phi_2 + \partial_z^2\Phi_2 = 0 & \text{in}\quad \Omega_2(t), \\ \partial_z\Phi_1 = 0 & \text{on}\quad \Sigma_1, \\ \nabla\Phi_2 \cdot \nabla b - \partial_z\Phi_2 = 0 & \text{on}\quad \Sigma_2,\\ (\nabla\Phi_1 \cdot \nabla \zeta - \partial_z\Phi_1) -(\nabla\Phi_2 \cdot \nabla \zeta - \partial_z\Phi_2)=0 & \text{on}\quad \Gamma(t), \\ \rho_2\Phi_2-\rho_1\Phi_1 = \phi & \text{on}\quad \Gamma(t), \\ \end{cases} \]

thus extending to the interfacial gravity waves framework the analysis in [Reference Duchêne and Iguchi13]. Finally, the solvability of (1.11) allows to determine and control time derivatives $\partial _t{\boldsymbol \phi }_1$ and $\partial _t{\boldsymbol \phi }_2$ of sufficiently regular solutions to the Kakinuma model (1.8) by using the equations obtained when differentiating with respect to time the compatibility conditions (1.9) combined with the last equation of (1.8). This is a crucial ingredient for the analysis of the initial value problem.

Another crucial ingredient for the analysis of the initial value problem concerns uniform energy estimates on the linearized Kakinuma system. To this end, we write the linearized system under the form

(1.12)\begin{equation} \mathscr{A}_1(\partial_t+\boldsymbol{u}\cdot\nabla)\dot{\boldsymbol{U}} + \mathscr{A}_0^\mathrm{mod}\dot{\boldsymbol{U}} = \dot{\boldsymbol{F}}, \end{equation}

where $\dot {\boldsymbol {U}}:=(\dot {\zeta }, \dot {\boldsymbol {\phi }}_1, \dot {\boldsymbol {\phi }}_2)^\mathrm {T}$ is the deviation from the reference state $\boldsymbol {U}:=({\zeta }, {\boldsymbol {\phi }}_1, {\boldsymbol {\phi }}_2)^\mathrm {T}$, $\boldsymbol {u}$ is a suitable velocity which is a convex combination of $\boldsymbol {u}_1$ and $\boldsymbol {u}_2$ whose weights depend on $\rho _\ell$, $H_\ell$ as well as $\alpha _\ell$ ($\ell =1,2$) the positive constants mentioned previously, $\dot {\boldsymbol {F}}$ represents lower order terms and $\mathscr {A}_1:=\mathscr {A}_1(\boldsymbol {U})$ is a skew-symmetric matrix and $\mathscr {A}_0^\mathrm {mod}:=\mathscr {A}_0^\mathrm {mod}(\boldsymbol {U})$ is a linear operator symmetric in $L^2$. The energy function associated to (1.12) is given by $(\mathscr {A}_0^\mathrm {mod}\dot {\boldsymbol {U}},\dot {\boldsymbol {U}})_{L^2}$, and we prove that

\[ (\mathscr{A}_0^\mathrm{mod}\dot{U},\dot{U})_{L^2} \simeq \mathscr{E}(\dot{\boldsymbol{U}}) := \|\dot{\zeta}\|_{L^2}^2 + \sum_{\ell=1,2}\rho_\ell( \|\nabla\dot{\boldsymbol{\phi}}_\ell\|_{L^2}^2 + \|\dot{\boldsymbol{\phi}}_\ell'\|_{L^2}^2) \]

under the non-cavitation assumption (1.7) and the stability condition (1.5). Because the structure of (1.12) is not standard, the control of the energy function is obtained by testing (1.12) with the time derivatives, $\partial _t\dot {\boldsymbol {U}}$. This, together with suitable product and commutator estimates in Sobolev spaces, provides the a priori control of the energy function for solutions to the Kakinuma model and their derivatives, and we show that this control is uniform in the shallow water regime after suitable rescaling. Since the construction and uniqueness of a solution was obtained in the companion paper [Reference Duchêne and Iguchi14], the uniform estimates provide the proof of the long-time well-posedness of the initial value problem for the Kakinuma model result stated in theorem 1.1. Furthermore, using the aforementioned consistency result, we prove that the difference between solutions to the full interfacial gravity waves system and corresponding solutions to the Kakinuma model satisfy an identity analogous to (1.12), and hence infer a control of the energy function of the difference and its derivatives, which yields the full justification of the Kakinuma model stated in theorem 1.4.

Outline.

The contents of this paper are as follows. In § 2 we first recall the basic equations governing the interfacial gravity waves and write down the Kakinuma model that we are going to analyse in this paper, and then rewrite them in a non-dimensional form by introducing several non-dimensional parameters. Hamiltonians of the full model and of the Kakinuma model in the non-dimensional variables are also provided. In § 3 we first introduce some differential operators, which enable us to write the Kakinuma model simply in form (1.8), and then we present the precise statements of our main results in this paper. In § 4 we first recall results in the framework of surface waves related to the consistency of the Isobe–Kakinuma model, and then prove theorems 3.5 and 3.6 concerning the consistency of the Kakinuma model by a simple scaling argument. In § 5 we first derive an elliptic estimate related to the compatibility conditions for the Kakinuma model, which explains how to prepare the initial data, as stated in proposition 3.4. Then we give uniform a priori bounds on regular solutions to the Kakinuma model, especially, a priori bounds of time derivatives. In § 6 we provide uniform energy estimates for the solution to the Kakinuma model and prove theorem 3.1, which 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. In § 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 3.6, 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 3.9 which provides a conditional 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, in § 8 we prove theorem 3.10 which gives an error estimate between the Hamiltonian of the Kakinuma model and that of the full model. For the convenience of the reader, the structure of the paper and proofs dependencies are sketched in figure 2.

Figure 2.

Articulation of the proofs.

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 $\boldsymbol{\phi}=(\phi _0,\phi _1,\ldots,\phi _N)^\mathrm {T}$ we denote the last $N$ components by $\boldsymbol{\phi'}=(\phi _1,\ldots,\phi _N)^\mathrm {T}$. $f \lesssim g$ 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.