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

TL;DR

The paper proves that the Kakinuma model for interfacial gravity waves provides a rigorously justified, higher order shallow-water approximation to the full two-layer Euler system. By exploiting the model’s variational and Hamiltonian structure, the authors establish uniform energy estimates, elliptic controls, and a conditional full- model justification with error terms of order $O((\underline{h}_1\delta)^{4N+2}+(\underline{h}_2\delta)^{4N+2})$. They also show that the Kakinuma Hamiltonian accurately approximates the full Hamiltonian within the same error bound, reinforcing the model’s physical fidelity. The results rely on careful analysis of Dirichlet–to–Neumann maps, compatibility conditions, and elliptic solvability in the bilayer domain, yielding long-time well-posedness in the shallow water regime under stability and non-cavitation conditions. Overall, the work provides a solid mathematical justification for using the Kakinuma model as a practical, higher order shallow-water approximation for interfacial gravity waves with an explicit, tunable accuracy dependent on the model order $N$ and the shallowness parameters $\delta$.

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 [arXiv:2103.12392]. 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 $O(δ_1^{4N+2}+δ_2^{4N+2})$ in the sense of consistency, where $δ_1$ and $δ_2$ are shallowness parameters, which are the ratios of the mean depths 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.

Figures (2)

• Figure 1.1: Internal gravity waves
• Figure 1.2: Articulation of the proofs.

Theorems & Definitions (14)

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