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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1803.09236 |
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| _version_ | 1866909478748684288 |
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| author | Iguchi, Tatsuo |
| author_facet | Iguchi, Tatsuo |
| contents | 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 $δ$, which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order $O(δ^{4N+2})$ in the case of the flat bottom and of order $O(δ^{4[N/2]+2})$ in the case of variable bottoms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_09236 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A mathematical justification of the Isobe-Kakinuma model for water waves with and without bottom topography Iguchi, Tatsuo Analysis of PDEs 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 $δ$, which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order $O(δ^{4N+2})$ in the case of the flat bottom and of order $O(δ^{4[N/2]+2})$ in the case of variable bottoms. |
| title | A mathematical justification of the Isobe-Kakinuma model for water waves with and without bottom topography |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1803.09236 |