A well-posed problem for the exterior Stokes equations in two and three dimensions
type: Item
instance of: scholarly article
title: A well-posed problem for the exterior Stokes equations in two and three dimensions
zbMATH Open document ID: 0731.35078
DOI: 10.1007/BF00376137
author: V. Girault
author: Adélia Sequeira
published in: Archive for Rational Mechanics and Analysis
publication date: 1991-01-01T00:00:00Z
review text: Let \(\Omega\) be a bounded domain of \(R^ n\) \((n=2\) or \(n=3)\) with boundary \(\Gamma\) and let \(\Omega '\) denote the complement of \({\bar \Omega}\). The authors consider the following exterior problem for the steady-state nonhomogeneous Stokes flow: \[ (*)\quad -\nu \Delta \vec u+\nabla p=\vec f,\quad div \vec u=0\text{ in } \Omega ',\quad \vec u|_{\Gamma}=\vec g, \] with a condition on \(\vec u\) at infinity expressed by \[ \int_{\Omega '}| \vec u|^ 2dx<+\infty,\quad \int_{\Omega '}(1/\omega^ 2)| \vec u|^ 2dx<+\infty. \] Here \(\omega\) is a weight function depending upon the dimension, \(\vec u\) is the velocity, p is the pressure, \(\nu\) is the coefficient of viscosity. Using the weighted Sobolev spaces of Hanouzet (in \(R^ 3)\) and Giroire (in \(R^ 2)\), the authors prove the following result: if \(\Omega \subset R^ n\) has a Lipschitz-continuous boundary \(\Gamma\) that is not necessarily connected, but has no interior connected component and \(\vec f\in (W_ 0^{-1}(\Omega '))^ n\), \(\vec g\in (H^{1/2}(\Omega '))^ n\), then the problem (*) has a unique solution (\(\vec u,p)\in (W^ 1_ 0(\Omega '))^ n\times L^ 2(\Omega ')\), which depends continuously on the data \(\vec f,\vec g\). Moreover, they showed that if the boundary and data are smoother, then so is the solution (\(\vec u,p)\) of the Stokes problem.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35Dxx
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 4208683
zbMATH Keywords: exterior Stokes equations
zbMATH Keywords: continuous dependence
zbMATH Keywords: weighted Sobolev spaces
reviewed by: A. A. Papin
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00376137
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description: scientific article; zbMATH DE number 4208683