Solvability of a free noncompact boundary problem for a stationary Navier-Stokes system. I
type: Item
instance of: scholarly article
title: Solvability of a free noncompact boundary problem for a stationary Navier-Stokes system. I
zbMATH Open document ID: 0727.35108
DOI: 10.1007/BF00966633
published in: Lithuanian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The auxiliary results which have been used by the first author [Litov. Mat. Sb. 28, No. 2, 315--333 (1988; Zbl 0657.76030)] are proved. Namely, in the plane domain \(\Omega\) bounded in the lines \(S_ 1=\{\bar x\in {\mathbb R}^ 2:\) \(x_ 2=0\}\), \(S_ 2=\{\bar x\in {\mathbb R}^ 2:\) \(x_ 1<0\), \(x_ 2=h>0\}\), \(\Gamma =\{\bar x\in {\mathbb R}^ 2:\) \(x_ 1=x(s)\), \(x_ 2=y(s)\}\) the following boundary value problem is considered \[ - \nu\nabla^ 2\bar v+ (\bar v\cdot\nabla)\bar v+ \nabla p= -\nabla G,\;\nabla\cdot\bar v=0,\;\bar x\in \Omega, \] \[ \bar v|_{S_ 1\cup S_ 2}=0,\;\bar v\cdot \bar n|_{\Gamma}=0,\;{\bar \tau}\cdot T(\bar v)\bar n|_{\Gamma}=0, \] where \(\nu=\)const\(>0\) is the viscosity coefficient, \(\bar n\) and \({\bar \tau}\) are the normal and the tangential unit vectors to \(\Gamma\) respectively, \(T(\bar v)\) is the tensor with components \(T_{ij}=\partial v_ i/\partial x_ j+\partial v_ j/\partial x_ i\) \((i,j=1,2)\), \(G=g(-x_ 1\sin \alpha +x_ 2\cos \alpha)\), \(g=\)const\(>0\), \(\alpha=\)const\(>0\), \(\nabla=(\partial /\partial x_ 1,\partial /\partial x_ 2).\) The solvability of this problem in weighted Hölder spaces is proved and the estimates of solutions are established. The solvability of analogous boundary value problem for the Stokes system is proved too.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 4200789
zbMATH Keywords: boundary value problem
zbMATH Keywords: weighted Hölder spaces
zbMATH Keywords: Stokes system
author: K. Pileckas
author: Maria Specovius-Neugebauer
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4200789