## 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 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics