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

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.

References in corpus (1)

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

Cited by in corpus (6)

• Modeling shallow water waves
• A mathematical justification of the Isobe-Kakinuma model for water waves with and without bottom topography
• A Hamiltonian structure of the Isobe-Kakinuma model for water waves
• Solitary wave solutions to the Isobe-Kakinuma model for water waves
• A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: Justification as a shallow water approximation
• A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part I: Structures and well-posedness