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A Mathematical Justification of the Isobe–Kakinuma Model for Water Waves with and without Bottom Topography

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Abstract

We consider the Isobe–Kakinuma model for water waves in both cases of the flat and the variable bottoms. The Isobe–Kakinuma model is a system of 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 appropriately. The Isobe–Kakinuma model consists of \((N+1)\) second order and a first order partial differential equations, where N is a nonnegative integer. We justify rigorously the Isobe–Kakinuma model as a higher order shallow water approximation in the strongly nonlinear regime by giving an error estimate between the solutions of the Isobe–Kakinuma model and of the full water wave problem in terms of the small nondimensional parameter \(\delta \), which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order \(O(\delta ^{4N+2})\) in the case of the flat bottom and of order \(O(\delta ^{4[N/2]+2})\) in the case of variable bottoms.

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Acknowledgements

This work was carried out when the author was visiting Université de Bordeaux on his sabbatical leave during the 2017 academic year. He is very grateful to the member of Institut de Mathḿatiques de Bordeaux, especially, David Lannes for their kind hospitalities and for fruitful discussions. This work was partially supported by JSPS KAKENHI Grant Numbers JP17K18742 and JP17H02856.

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  1. Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Tatsuo Iguchi

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  1. Tatsuo Iguchi
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Correspondence to Tatsuo Iguchi.

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Communicated by A. Constantin

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Iguchi, T. A Mathematical Justification of the Isobe–Kakinuma Model for Water Waves with and without Bottom Topography. J. Math. Fluid Mech. 20, 1985–2018 (2018). https://doi.org/10.1007/s00021-018-0398-x

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