A mathematical justification of the Isobe-Kakinuma model for water waves with and without bottom topography

TL;DR

This work provides a rigorous justification of the Isobe–Kakinuma model as a higher-order shallow-water approximation to the full water-wave problem in both flat and bottom-topography settings. By exploiting Luke’s variational structure and carefully chosen vertical basis functions, the authors derive uniform in $\delta$ energy estimates for the IK system and prove two levels of consistency with the full water-wave equations, accompanied by precise error bounds $O(\delta^{4N+2})$ for flat bottoms and $O(\delta^{4[ N/2 ]+2})$ when the bottom varies. A pivotal step is the construction of a modified approximate velocity potential $\widetilde{\Phi}^{\text{app}}$ and the use of elliptic estimates (including a transformed-domain analysis) to relate the IK variables to the Dirichlet-to-Neumann map and to the WW problem. The paper culminates in a rigorous justification theorem showing that the IK solution tracks the WW solution with the stated rates on a time interval independent of $\delta$, thereby validating IK as a high-order, computationally appealing shallow-water model without introducing high-order derivatives. This provides a robust theoretical foundation for using IK in strongly nonlinear regimes and complex bottom topographies.

Abstract

We consider the Isobe-Kakinuma model for water waves in both cases of the flat and the variable bottoms. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian appropriately. The Isobe-Kakinuma model consists of $(N+1)$ second order and a first order partial differential equations, where $N$ is a nonnegative integer. We justify rigorously the Isobe-Kakinuma model as a higher order shallow water approximation in the strongly nonlinear regime by giving an error estimate between the solutions of the Isobe-Kakinuma model and of the full water wave problem in terms of the small nondimensional parameter $δ$, which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order $O(δ^{4N+2})$ in the case of the flat bottom and of order $O(δ^{4[N/2]+2})$ in the case of variable bottoms.