### 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 |
---|---|

Journal | Analysis and Applications |

DOIs | |

Publication status | Accepted/In press - 1 Jan 2019 |

### Keywords

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

### Cite this

*Analysis and Applications*. https://doi.org/10.1142/S0219530519500118

}

*Analysis and Applications*. https://doi.org/10.1142/S0219530519500118

**Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain.** / Chipot, Michel; Droniou, Jérôme; Planas, Gabriela; Robinson, James C.; Xue, Wei.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain

AU - Chipot, Michel

AU - Droniou, Jérôme

AU - Planas, Gabriela

AU - Robinson, James C.

AU - Xue, Wei

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - asymptotic behavior

KW - Flow around vanishing obstacle

KW - Navier-Stokes equations

KW - Poisson problem

UR - http://www.scopus.com/inward/record.url?scp=85069932689&partnerID=8YFLogxK

U2 - 10.1142/S0219530519500118

DO - 10.1142/S0219530519500118

M3 - Article

JO - Analysis and Applications

JF - Analysis and Applications

SN - 0219-5305

ER -