### Abstract

We consider the two-phase problem for two-dimensional and irrotational motion of incompressible ideal fluids in the case that the fluids are separated into the lower and the upper parts by an almost horizontal interface and that there is an almost flat bottom below the lower fluid. It is proved that the Cauchy problem is well-posed, locally in time, in a Sobolev space of finite smoothness, if the surface tension is taken into account and the initial data are suitably close to the equilibrium rest state. The main part of the proof is the reduction of the problem to a quasi-linear system of integro-differential equations for the function defining the interface and the horizontal component of the velocity of the lower fluid on the interface.

Original language | English |
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Pages (from-to) | 791-821 |

Number of pages | 31 |

Journal | Mathematical Models and Methods in Applied Sciences |

Volume | 7 |

Issue number | 6 |

Publication status | Published - 1997 Sep |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Modelling and Simulation

### Cite this

**Two-phase problem for two-dimensional water waves of finite depth.** / Iguchi, Tatsuo.

Research output: Contribution to journal › Article

*Mathematical Models and Methods in Applied Sciences*, vol. 7, no. 6, pp. 791-821.

}

TY - JOUR

T1 - Two-phase problem for two-dimensional water waves of finite depth

AU - Iguchi, Tatsuo

PY - 1997/9

Y1 - 1997/9

N2 - We consider the two-phase problem for two-dimensional and irrotational motion of incompressible ideal fluids in the case that the fluids are separated into the lower and the upper parts by an almost horizontal interface and that there is an almost flat bottom below the lower fluid. It is proved that the Cauchy problem is well-posed, locally in time, in a Sobolev space of finite smoothness, if the surface tension is taken into account and the initial data are suitably close to the equilibrium rest state. The main part of the proof is the reduction of the problem to a quasi-linear system of integro-differential equations for the function defining the interface and the horizontal component of the velocity of the lower fluid on the interface.

AB - We consider the two-phase problem for two-dimensional and irrotational motion of incompressible ideal fluids in the case that the fluids are separated into the lower and the upper parts by an almost horizontal interface and that there is an almost flat bottom below the lower fluid. It is proved that the Cauchy problem is well-posed, locally in time, in a Sobolev space of finite smoothness, if the surface tension is taken into account and the initial data are suitably close to the equilibrium rest state. The main part of the proof is the reduction of the problem to a quasi-linear system of integro-differential equations for the function defining the interface and the horizontal component of the velocity of the lower fluid on the interface.

UR - http://www.scopus.com/inward/record.url?scp=0031534754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031534754&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031534754

VL - 7

SP - 791

EP - 821

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 6

ER -