On the existence of stationary solutions to compressible Navier-Stokes equations
type: Item
instance of: scholarly article
title: On the existence of stationary solutions to compressible Navier-Stokes equations
zbMATH Open document ID: 0627.76080
DOI: 10.1016/S0294-1449(16)30374-2
author: Alberto Valli
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78126
review text: The author considers compressible forms of the Navier-Stokes equations (not necessarily barotropic) in a bounded domain \(\Omega \in {\mathbb{R}}^ 3\) under homogeneous boundary conditions in a stationary (sufficiently small) external force field. In an earlier work [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 607-647 (1983; Zbl 0542.35062)] he proved the existence of stationary solutions by a limiting procedure, starting from the time dependent case. The present paper offers a new proof, proceeding through a linearization and using the Schauder fixed point theorem. To that extent the proof is analogous to the original one given by \textit{J. Leray} [J. Math. Pures Appl., IX. Seŕ. 12, 1-82 (1933; Zbl 0006.16702)] in the incompressible case. The reviewer remarks that since that case required no smallness restriction it seems reasonable to ask whether that restriction can ultimately be dispensed with here also. The author points out that still another proof was found concomitantly and independently by \textit{H. Beirão da Veiga} [Houston J. Math. (to appear)].
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 4020722
zbMATH Keywords: compressible forms of the Navier-Stokes equations
zbMATH Keywords: bounded domain
zbMATH Keywords: homogeneous boundary conditions
zbMATH Keywords: linearization
zbMATH Keywords: Schauder fixed point theorem
MaRDI profile type: Publication
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OpenAlex ID: W2492185385
description: scientific article; zbMATH DE number 4020722