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

Michel Chipot, Jérôme Droniou, Gabriela Planas, James C. Robinson, Wei Xue

Research output: Contribution to journalArticleResearchpeer-review

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 languageEnglish
JournalAnalysis and Applications
DOIs
Publication statusAccepted/In press - 1 Jan 2019

Keywords

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

Cite this

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title = "Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain",
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.",
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year = "2019",
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Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain. / Chipot, Michel; Droniou, Jérôme; Planas, Gabriela; Robinson, James C.; Xue, Wei.

In: Analysis and Applications, 01.01.2019.

Research output: Contribution to journalArticleResearchpeer-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

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U2 - 10.1142/S0219530519500118

DO - 10.1142/S0219530519500118

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JO - Analysis and Applications

JF - Analysis and Applications

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