Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. II: Linearization on a symmetric solution.
type: Item
instance of: scholarly article
title: Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. II: Linearization on a symmetric solution.
zbMATH Open document ID: 1043.76015
DOI: 10.1016/S0021-7824(01)01232-6
author: Sergei A. Nazarov
author: Adélia Sequeira
author: Juha H. Videman
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-09-07T00:00:00Z
review text: The steady flow of a Navier-Stokes fluid is analysed in a two-dimensional asymmetric unbounded domain \(\widetilde\Omega\), having two outlets to infinity: a half-plane \(K= \{(x_1, x_2)\in \mathbb{R}^2: x_1>0\}\) and a semi-strip \[ \Pi_-= \{(x_1, x_2)\in \mathbb{R}^2: x_1< 0,\,| x_2|< 1\}. \] Assuming that \(\widetilde\Omega\) can be obtained as a small perturbation of the symmetric domain \(\Omega\) (i.e. \(\widetilde\Omega=k(\Omega)\) and \(k: \mathbb{R}^2\to \mathbb{R}^2\) is a diffeomorphism such that \(k= I\) outside the ball \(B_R\) and \(k- I\) is small in a certain sense), the existence of a unique solution to the Navier-Stokes system in \(\widetilde\Omega\) is shown. The change of variables \(\widetilde\Omega\ni \widetilde x\to x= k^{-1}(\widetilde x)\in \Omega\) turns the Navier-Stokes problem in \(\widetilde\Omega\) into the perturbed Navier-Stokes problem in \(\Omega\), which later is considered as a perturbation of the linear problem. This linearization around the symmetric solution for non-zero flux leads to the absence of compatibility conditions for the convective term, and for negative flux to the domination of nonlinear terms by the linear ones. The paper is completed with examples of non-symmetric domains. The well-posedness and even existence of physically reasonable solutions in a non-symmetric domain with a prescribed positive flux remains still an open question.
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 1975952
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: weighted function spaces
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: unbounded domain
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
reviewed by: A. Cichocka
MaRDI profile type: Publication
cites work: Existence, uniqueness and asymptotic behaviour of solutions of steady-state Navier-Stokes equations in a plane aperture domain
cites work: Interpolation zwischen den Klassen 𝔖p von Operatoren in Hilberträumen
cites work: AN OPERATOR GENERALIZATION OF THE LOGARITHMIC RESIDUE THEOREM AND THE THEOREM OF ROUCHÉ
cites work: Q3874765
cites work: Q5544635
cites work: [Russian Text Ignored.]
cites work: Estimates in 𝐿_{𝑝} and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary
cites work: Q4715880
cites work: The operator of a boundary value problem with Chaplygin-Zhukovskij-Kutta type conditions on an edge of the boundary has the Fredholm property
cites work: The Navier-Stokes problem in a two-dimensional domain with angular outlets to infinity
cites work: Q4524845
cites work: On steady Stokes and Navier-Stokes problems with zero velocity at infinity in a three-dimensional exterior domain
cites work: Q4032162
cites work: Elliptic problems in domains with piecewise smooth boundaries
cites work: QUIET FLOWS FOR STOKES AND NAVIER-STOKES PROBLEMS IN DOMAINS WITH CYLINDRICAL OUTLETS TO INFINITY
cites work: Asymptotic behaviour at infinity of three-dimensional steady viscoelastic flows.
cites work: Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. I: Linearization on an antisymmetric solution.
cites work: Q4743200
cites work: Q5182723
full work available at URL: s0021-7824(01)01232-6
OpenAlex ID: W4235862060
description: scientific article; zbMATH DE number 1975952