Asymptotic expansion for a periodic boundary condition
type: Item
instance of: scholarly article
title: Asymptotic expansion for a periodic boundary condition
zbMATH Open document ID: 0849.35043
author: Ján Filo
author: Stephan Luckhaus
published in: Journal of Differential Equations
publication date: 1996-11-07T00:00:00Z
review text: The authors introduce an approximation to flow problems in porous media when the boundary of the flow domain is partially covered by fluid. If the domain is \(\Omega= (0, \pi)\times (0, \pi)\) with a typical point \((x, y)\), they write \(\alpha\) and \(\gamma\) for the subsets of \(\partial\Omega\) on which \(y= 0\) and \(y= \pi\), respectively, and \(\beta= \partial\Omega \backslash (\alpha\cup \gamma)\). Then their approximation can be written as \[ u_t- \Delta u= f(x, y, t)\text{ in } \Omega\times (0, T),\;a\Biggl({x\over \varepsilon}\Biggr) (u_y+ \sigma u+ \psi)+ \Biggl(1- a\Biggl({x\over \varepsilon}\Biggr)\Biggr) u= 0\text{ on } \alpha\times (0, T), \] \[ u_x= 0\text{ on } \beta\times (0, T),\;u= 0\text{ on } \gamma\times (0, T),\;u= \varphi\text{ on } \Omega\times \{0\}, \] where \(a\) is the \(2\pi\)-periodic function which is zero on intervals of length \(\delta< 2\pi\) centered at odd multiples of \(\pi\) and zero elsewhere, \(\psi\) and \(\sigma\) are functions of \((x, t)\), and \(\varepsilon\) is the reciprocal of an even integer. Under suitable regularity hypotheses on the functions \(f\), \(\psi\), \(\sigma\), and \(\varphi\), they study the behavior of the solution \(u_\varepsilon\) as \(\varepsilon\to 0\). Specifically, they show that \(u_\varepsilon= u+ \varepsilon u^1+ \varepsilon R_\varepsilon\), with \(u\) and \(u^1\) solutions of explicitly given initial-boundary value problems and \(R_\varepsilon\) an error term which is bounded in \(L^\infty\) and which tends to zero in any \(L^p\) (\(p\) finite) as \(\varepsilon\to 0\).
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 790886
zbMATH Keywords: flow in porous media
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1108
OpenAlex ID: W2072819769
DOI: 10.1006/JDEQ.1995.1108
description: scientific article; zbMATH DE number 790886