### Abstract

We treat three problems on a two-dimensional "punctured periodic domain": we take ωr = (-L,L)2rK, where r > 0 and K is the closure of an open connected set that is star-shaped with respect to 0 and has a C1 boundary. We impose periodic boundary conditions on the boundary of ω = (-L,L)2, and Dirichlet boundary conditions on (rK). In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier-Stokes equations, all with a fixed forcing function f, and examine the behavior of solutions as r → 0. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of ω with periodic boundary conditions.

Original language | English |
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Pages (from-to) | 211-235 |

Number of pages | 25 |

Journal | Analysis and Applications |

Volume | 18 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2020 |

### Keywords

- asymptotic behavior
- Flow around vanishing obstacle
- Navier-Stokes equations
- Poisson problem

## Cite this

*Analysis and Applications*,

*18*(2), 211-235. https://doi.org/10.1142/S0219530519500118