A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part I: Structures and well-posedness
Vincent Duchêne, Tatsuo Iguchi
TL;DR
The paper analyzes the Kakinuma model for interfacial gravity waves, establishing a rigorous local well-posedness theory in Sobolev spaces under a stability condition, non-cavitation, and compatibility constraints. It proves that the model reproduces the full interfacial dispersion accurately in the shallow-water regime, and it derives a linear dispersion relation showing close agreement with the full model for suitable index choices, while highlighting deep-water limitations. The authors further demonstrate a Hamiltonian structure for the model, formulate canonical variables, and prove conservation laws for mass, energy, and momentum (flat bottom). The construction relies on a careful energy method, symmetric-positive systems, elliptic estimates for the associated operators, and parabolic regularization to obtain a robust existence theory. Overall, the work provides a solid mathematical foundation for the Kakinuma model as a stable, variationally consistent approximate description of two-layer interfacial gravity waves with conserved quantities and a clear Hamiltonian framework.
Abstract
We consider a model, which we named the Kakinuma model, for interfacial gravity waves. As is well-known, the full model for interfacial gravity waves has a variational structure whose Lagrangian is an extension of Luke's Lagrangian for surface gravity waves, that is, water waves. The Kakinuma model is a system of Euler-Lagrange equations for approximate Lagrangians, which are obtained by approximating the velocity potentials in the Lagrangian for the full model. In this paper, we first analyze the linear dispersion relation for the Kakinuma model and show that the dispersion curves highly fit that of the full model in the shallow water regime. We then analyze the linearized equations around constant states and derive a stability condition, which is satisfied for small initial data when the denser water is below the lighter water. We show that the initial value problem is in fact well-posed locally in time in Sobolev spaces under the stability condition, the non-cavitation assumption and intrinsic compatibility conditions in spite of the fact that the initial value problem for the full model does not have any stability domain so that its initial value problem is ill-posed in Sobolev spaces. Moreover, it is shown that the Kakinuma model enjoys a Hamiltonian structure and has conservative quantities: mass, total energy, and in the case of the flat bottom, momentum.