Stationary Navier-Stokes equations under inhomogeneous boundary conditions in 3D exterior domains
type: Item
instance of: scholarly article
title: Stationary Navier-Stokes equations under inhomogeneous boundary conditions in 3D exterior domains
zbMATH Open document ID: 1479.35363
author: Hideo Kozono
author: Anton Seyfert
author: Senjo Shimizu
author: Taku Yanagisawa
author: Matthias Hieber
published in: Calculus of Variations and Partial Differential Equations
publication date: 2021-08-24T00:00:00Z
review text: This interesting recent work investigates the existence of weak solutions to the stationary Navier-Stokes equations in a 3D exterior domain \(\Omega\). The boundary \(\partial \Omega\) is a disjoint union of finitely many smooth, compact surfaces \(\{\Gamma_i\}_{i=1}^L\), and the boundary condition for \(v\) is that \(v|_{\Gamma_i}=\beta_i \in C^\infty(\Gamma_i)\) for each \(i \in \{0,1,\ldots, L\}\). This question is related to a famous open problem of Leray. In the paper, it is proved that if the \(L^3\)-norm of the harmonic part of \(\sum_{i=0}^L\int_{\Gamma_i}\beta_i \cdot\nu\,\mathrm{d}S\) is smaller than a universal constant times the viscosity, then there exists a weak solution to the stationary Navier-Stokes equations in the \(W^{1,2}\)-closure of smooth, divergence-free vector fields on \(\Omega\). The proof is based on the authors' works [J. Funct. Anal. 281, No. 8, Article ID 109144 (2021; Zbl 1472.35152); ``A characterization of harmonic \(L^r\)-vector fields in three dimensional exterior domains'', Preprint] on Hodge decomposition of vector fields on exterior domains: they first find a background solution, i.e., a divergence-free \(W^{1,2}\)-extension of the boundary data, and then apply a Galerkin approximation argument to the difference between the true solution and this background solution.
reviewed by: Siran Li
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 7384561
zbMATH Keywords: stationary Navier-Stokes equations
zbMATH Keywords: exterior domain
zbMATH Keywords: Hodge decomposition
zbMATH Keywords: weak solution
Wikidata QID: Q115386494
MaRDI profile type: Publication
full work available at URL: s00526-021-02050-1
OpenAlex ID: W3187488105
cites work: Q3964958
cites work: On the equations rot\ v\(=g\) and div\ u\(=f\) with zero boundary conditions
cites work: Leray's inequality in general multi-connected domains in \({\mathbb{R}^n}\)
cites work: Q3848500
cites work: An Introduction to the Mathematical Theory of the Navier-Stokes Equations
cites work: A characterization of harmonic \(L^r\)-vector fields in two-dimensional exterior domains
cites work: \(L^r\)-Helmholtz-Weyl decomposition for three dimensional exterior domains
cites work: Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains
cites work: Q4033097
cites work: REPRESENTATION FORMULA, NET FORCE AND ENERGY RELATION TO THE STATIONARY NAVIER-STOKES EQUATIONS IN 3-DIMENSIONAL EXTERIOR DOMAINS
cites work: $L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains
cites work: Leray's problem on the stationary Navier-Stokes equations with inhomogeneous boundary data
cites work: Q5576913
cites work: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique
cites work: A new approach to the existence of weak solutions of the steady Navier-Stokes system with inhomogeneous boundary data in domains with noncompact boundaries
cites work: Q3976328
cites work: Q3861771
DOI: 10.1007/S00526-021-02050-1
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description: scientific article; zbMATH DE number 7384561