## Abstract

We consider the Navier-Stokes equations on a thin domain of the form Ω_{ε} = {x ∈ ℝ^{3} | x_{1}, x _{2} ∈ (0, 1), 0 < x_{3} < εg {x_{1}, x_{2})} supplemented with the following mixed boundary conditions: periodic boundary conditions on the lateral boundary and Navier boundary conditions on the top and the bottom. Under the assumption that ∥u _{0}∥_{H1(Ωε} ≤ C_{ε} ^{1/2}, par;Mu_{0}^{i}∥_{L2(Omega;ε)} C for i ∈ {1,2} and similar assumptions on the forcing term, we show global existence of strong solutions; here u_{0}^{i} denotes the i-th component of the initial data u_{0} and M is the average in the vertical direction, that is, Mu_{0}^{i}(x_{1}, x_{2} = (εg)^{-1} ∫_{0}^{εg} u_{0} ^{i}(x_{1}, x_{2}, x_{3}) dx_{3} Moreover, if the initial data, respectively the forcing term, converge to a bidimensional vector field, respectively forcing term, as ε → 0, we prove convergence to a solution of a limiting system which is a Navier-Stokes-like equation where the function g plays an important role. Finally, we compare the attractor of the Navier-Stokes equations with the one of the limiting equation. Indiana University Mathematics Journal

Original language | English (US) |
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Pages (from-to) | 1083-1155 |

Number of pages | 73 |

Journal | Indiana University Mathematics Journal |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - 2007 |

## Keywords

- Global attractor
- Global existence
- Global regularity
- Navier-stokes equations
- Thin domain