### Abstract

In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the L_{p} in time L_{q} in space maximal regularity class, (2<p<∞, N<q<∞, and 2/p+N/q<1), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the L_{p}-L_{q} framework for the linearized equations with the help of maximal L_{p}-L_{q} regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

Original language | English |
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Pages (from-to) | 117-152 |

Number of pages | 36 |

Journal | Evolution Equations and Control Theory |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

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### Keywords

- Free boundary problems
- Global well-posedness
- Navier-stokes equations
- Surface tension

### ASJC Scopus subject areas

- Modelling and Simulation
- Control and Optimization
- Applied Mathematics

### Cite this

**Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface.** / Shibata, Yoshihiro.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface

AU - Shibata, Yoshihiro

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the Lp in time Lq in space maximal regularity class, (2p-Lq framework for the linearized equations with the help of maximal Lp-Lq regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

AB - In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the Lp in time Lq in space maximal regularity class, (2p-Lq framework for the linearized equations with the help of maximal Lp-Lq regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

KW - Free boundary problems

KW - Global well-posedness

KW - Navier-stokes equations

KW - Surface tension

UR - http://www.scopus.com/inward/record.url?scp=85041622259&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041622259&partnerID=8YFLogxK

U2 - 10.3934/eect.2018007

DO - 10.3934/eect.2018007

M3 - Article

AN - SCOPUS:85041622259

VL - 7

SP - 117

EP - 152

JO - Evolution Equations and Control Theory

JF - Evolution Equations and Control Theory

SN - 2163-2472

IS - 1

ER -