## 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 |

## Keywords

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

## ASJC Scopus subject areas

- Modelling and Simulation
- Control and Optimization
- Applied Mathematics