### Abstract

In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain RN (N 2). The velocity field is obtained in the maximal regularity class W2;1 q;p (0 T)) = Lp((0; T);Wq ()N) W^{1} p ((0; T);Lq()N) (2<1 and N <q <1) for any initial data satisfying certain compatibility conditions. The assumption of the domain is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of. A bounded domain, a perturbed half space, and a perturbed layer satisfy the conditions for the domain, and therefore drop problems and ocean problems are treated in the uniform manner. Our method is based on the maximal Lp-Lq regularity theorem of a linearized problem in a general domain.

Original language | English |
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Pages (from-to) | 315-342 |

Number of pages | 28 |

Journal | Discrete and Continuous Dynamical Systems - Series S |

Volume | 9 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2016 Feb 1 |

### Fingerprint

### Keywords

- Free boundary problems
- Gravity force
- Local well-posedness
- Navier-Stokes equations
- Surface tension

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

**Local well-posedness of free surface problems for the navier-stokes equations in a general domain.** / Shibata, Yoshihiro.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Local well-posedness of free surface problems for the navier-stokes equations in a general domain

AU - Shibata, Yoshihiro

PY - 2016/2/1

Y1 - 2016/2/1

N2 - In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain RN (N 2). The velocity field is obtained in the maximal regularity class W2;1 q;p (0 T)) = Lp((0; T);Wq ()N) W1 p ((0; T);Lq()N) (2<1 and N

AB - In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain RN (N 2). The velocity field is obtained in the maximal regularity class W2;1 q;p (0 T)) = Lp((0; T);Wq ()N) W1 p ((0; T);Lq()N) (2<1 and N

KW - Free boundary problems

KW - Gravity force

KW - Local well-posedness

KW - Navier-Stokes equations

KW - Surface tension

UR - http://www.scopus.com/inward/record.url?scp=84958750863&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958750863&partnerID=8YFLogxK

U2 - 10.3934/dcdss.2016.9.315

DO - 10.3934/dcdss.2016.9.315

M3 - Article

AN - SCOPUS:84958750863

VL - 9

SP - 315

EP - 342

JO - Discrete and Continuous Dynamical Systems - Series S

JF - Discrete and Continuous Dynamical Systems - Series S

SN - 1937-1632

IS - 1

ER -