Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity
type: Item
instance of: scholarly article
title: Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity
zbMATH Open document ID: 1144.35044
DOI: 10.4171/ZAA/1348
author: Sergei A. Nazarov
author: Maria Specovius-Neugebauer
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2008-07-02T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: The subject of the paper is approximations to solutions of Stokes and Navier-Stokes equations in domains that are layer-like at infinity. Such a domain is \(\Omega\subset \mathbb{R}^3\) with a smooth boundary \(\partial\Omega\), and \(\Omega\) coincides with the layer \[ \Lambda=\{x=(y,z):\;y=(y_1,y_2)\in \mathbb{R}^2,\;| z| <\frac{1}{2}\} \] outside the ball \(B_{R_0}=\{x\in \mathbb{R}^3:\;| x| <R_0\}\). \((v^\infty,p^\infty)\) is the solution to the Stokes problem \[ -\nu \Delta v^\infty\,+\nabla p^\infty=f, \quad\text{div}\,v^\infty=0\quad \text{in}\;\Omega,\quad v^\infty=0 \quad \text{on}\;\partial\Omega. \] \((v^R,p^R)\) is the approximation to \((v^\infty,p^\infty)\). \((v^R,p^R)\) is the solution to the problem in a bounded domain \(\Omega_R=\{x=(r,\varphi,z)\in\Omega:\;r<R\}\) with boundary \(\partial\Omega_R=\Sigma_R\cup\Gamma_R\), \(\Sigma_R=\{x\in\partial\Omega:\;r<R\}\), \(\Gamma_R=\{x\in\Omega:\;r=R,| z| <\frac{1}{2}\}\) \[ \begin{aligned} -\nu \Delta v^R\,+\nabla p^R=f, &\quad\text{div}\,v^R=0\quad \text{in}\;\Omega_R,\quad v^R=0 \quad \text{on}\;\Sigma_R\\ &M_R(v^R,p^R)=0 \quad \text{on}\;\Gamma_R \end{aligned} \] Here \(M_R\) is a suitable artificial operator which involves the Steklov-Poincare operator. The norms \(\| v^\infty-v^R\|_{H^1(\Omega_R)}\) and \(\| p^\infty-p^R\|_{L^2(\Omega_R)}\) are estimated on the base of asymptotics of \((v^\infty,p^\infty)\). These norms vanish of order \(O(R^{-\alpha})\), where \(\alpha\) can be arbitrary large. Similar result is obtained to the Navier-Stokes equations.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35A35
zbMATH DE Number: 5296722
zbMATH Keywords: Stokes problem in layers
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: artificial boundary conditions
zbMATH Keywords: Steklov-Poincaré operator
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5296722