### 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.

Original language | English |
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Pages (from-to) | 631-653 |

Number of pages | 23 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 20 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 Jun 1 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Condensed Matter Physics
- Computational Mathematics
- Applied Mathematics

### Cite this

**Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves.** / Nemoto, Ryo; Iguchi, Tatsuo.

Research output: Contribution to journal › Article

*Journal of Mathematical Fluid Mechanics*, vol. 20, no. 2, pp. 631-653. https://doi.org/10.1007/s00021-017-0338-1

}

TY - JOUR

T1 - Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves

AU - Nemoto, Ryo

AU - Iguchi, Tatsuo

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85048067033&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048067033&partnerID=8YFLogxK

U2 - 10.1007/s00021-017-0338-1

DO - 10.1007/s00021-017-0338-1

M3 - Article

AN - SCOPUS:85048067033

VL - 20

SP - 631

EP - 653

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 2

ER -