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arXiv · 1407.6942

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

Michel Chipot·Jerome Droniou·Gabriela Planas·James C. Robinson·Wei Xue
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Abstract

In this paper we treat three problems on a two-dimensional `punctured periodic domain': we take $Ω_r=(-L,L)^2\setminus D_r$, where $D_r=B(0,r)$ is the disc of radius $r$ centred at the origin. We impose periodic boundary conditions on the boundary of the box $Ω=(-L,L)^2$, and Dirichlet boundary conditions on the circumference of the disc. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier-Stokes equations, all with a fixed forcing function $f$ (which must satisfy $\int_Ωf=0$ for the stationary problems), and examine the behaviour of solutions as $r\to0$. 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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Michel Chipot, Jerome Droniou, Gabriela Planas, James C. Robinson, Wei Xue. 2017-11-10. Limits of the Stokes and Navier-Stokes equations in a punctured periodic domain. https://arxiv.org/abs/1407.6942

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