Isobe-Kakinuma model for water waves as a higher order shallow water approximation

TL;DR

This work rigorously justifies the Isobe–Kakinuma model as a higher-order shallow-water approximation to the full water-wave problem on a flat bottom. By deriving a variationally consistent IK model with Φ ≈ φ_0 + (h+z)^2 φ_1 and nondimensionalizing with the shallowness parameter $\delta$, the authors prove δ-uniform local well-posedness, establish a nonlinear symmetric structure for energy estimates, and show that the model is consistent with the full equations to $O(\delta^6)$. They further provide an explicit initial-data mapping from the WW problem to the IK variables, and prove a rigorous convergence result: the WW solution remains within $C\delta^6$ of the IK solution on a time interval, thereby justifying the IK model as a high-order shallow-water approximation with practical computational advantages due to its second-order PDE form. Key techniques include a symmetric-energy approach, careful elliptic estimates, and a detailed expansion of the Dirichlet-to-Neumann map to control remainder terms.

Abstract

We justify rigorously an Isobe-Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order $O(δ^2)$, where $δ$ is a small nondimensional parameter defined as the ratio of the mean depth to the typical wavelength. The Green-Naghdi equations are known as higher order approximate equations to the water wave equations with an error of order $O(δ^4)$. In this paper we show that the Isobe-Kakinuma model is a much higher order approximation to the water wave equations with an error of order $O(δ^6)$.