Solvability of the initial value problem to the Isobe-Kakinuma model for water waves

TL;DR

This work proves local-in-time solvability for the initial value problem of the Isobe–Kakinuma water-wave model by exploiting a compatibility condition induced by the t=0 characteristic surface and a generalized Rayleigh–Taylor sign condition. The authors reformulate the linearized system into a symmetric positive framework and construct a reduced, noncharacteristic system via a parabolic regularization, obtaining uniform energy estimates and passing to the limit. They show that the reduced solution satisfies the original IK equations and conserves an energy functional, thereby establishing well-posedness in Sobolev spaces for suitable initial data and bottom topography. Additionally, they analyze the model’s linear dispersion, relating the IK phase speed to Padé approximants of the classical water-wave speed, which supports the model’s accuracy in the shallow-water regime.

Abstract

We consider the initial value problem to the Isobe-Kakinuma model for water waves and the structure of the model. The Isobe-Kakinuma model is the 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. The Isobe-Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations. Since the hypersurface $t=0$ is characteristic for the Isobe-Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh-Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces. We also discuss the linear dispersion relation to the model.