### Abstract

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier-Stokes equations in a three-dimensional domain Ω_{ε}, whose thickness is of order O(ε) as ε → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ω_{ε}, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ε^{3/4}) as ε → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H ^{1}(Ω_{ε}), respectively, L ^{2}(Ω_{ε}), then the strong solution of the Navier-Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ε, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.

Original language | English |
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Pages (from-to) | 435-472 |

Number of pages | 38 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2010 |

### Keywords

- Navier friction boundary condition
- Navier-Stokes equations
- strong solution
- thin domain