A mathematical analysis of the Kakinuma model for interfacial gravity waves. I: Structures and well-posedness. (English) Zbl 1540.35306
Summary: 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 noncavitation 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 a flat bottom, momentum.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76B55 | Internal waves for incompressible inviscid fluids |
| 35L65 | Hyperbolic conservation laws |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
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| [33] | Received 19 January 2022; revised 4 November 2022; accepted 13 December 2022. Vincent Duchêne Institut de Recherche Mathématique de Rennes, Université de Rennes, CNRS, IRMAR -UMR 6625, 35000 Rennes, France; vincent.duchene@univ-rennes1.fr Tatsuo Iguchi Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan; iguchi@math.keio.ac.jp |
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