Isolated singularity for the stationary Navier-Stokes system
type: Item
instance of: scholarly article
title: Isolated singularity for the stationary Navier-Stokes system
zbMATH Open document ID: 0970.35095
DOI: 10.1007/PL00000951
author: Hi Jun Choe
author: Hyunseok Kim
published in: Journal of Mathematical Fluid Mechanics
publication date: 2001-10-10T00:00:00Z
review text: The main aim of the paper is to find growth conditions near a singular point for a solution of the stationary Navier-Stokes system, in order to remove the singularity. The method is analogous to the one that is used to remove the isolated singularities of harmonic functions. In the first part of the paper a classical solution to the stationary Stokes system is given, near an isolated singular point, as a power series expansion in terms of homogeneous biharmonic polynomials. By means of this expansion, the authors find a growth condition on the solution for a singularity to be removable. Also, they obtain a new proof of a well-known Liouville-type theorem. In the second part they use the previous results to represent a solution to the stationary Navier-Stokes system as a power series expansion. The nonlinear part is treated as a non-homogeneous term. They are able then to deduce the following removable isolated singularity theorem for solutions to the stationary Navier-Stokes equations (\(d\geq 3\)): If \((u,p)\) is a classical solution in \(B_R-\{0\}\) such that \(|u(x)|=o(|x|^{-\frac{d-1}{2}})\) as \(|x|\to 0\) or \(u\in L^{\frac{2d}{d-1}}(B_R)\), then \((u,p)\) is a distributional solution in \(B_R\). Moreover if \(u\in L^q(B_R)\), with \(q>d\), then \((u,p)\) is smooth in \(B_R\). (A similar statement holds also when \(d=2\)).
reviewed by: Marco Romito
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35C10
zbMATH DE Number: 1484690
zbMATH Keywords: stationary Navier-Stokes system
zbMATH Keywords: homogeneous harmonic polynomials
zbMATH Keywords: power series expansion
zbMATH Keywords: removable isolated singularity
MaRDI profile type: Publication
full work available at URL: pl00000951
OpenAlex ID: W2019466102
Recommended article: A removable isolated singularity theorem for the stationary Navier-Stokes equations
Recommended article: On the existence of singular solutions of the stationary Navier-Stokes problem
Recommended article: Stability of singular solutions to the Navier-Stokes system
Recommended article: Singularities of solutions of the linearized Navier-Stokes system
Recommended article: Q2939764
Recommended article: ON A SINGULAR INITIAL‐VALUE PROBLEM FOR THE NAVIER–STOKES EQUATIONS
Recommended article: Singularities of certain finite energy solutions to the Navier-Stokes system
Recommended article: Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. I: One singularity
Recommended article: ON SINGULAR SOLUTIONS OF THE STATIONARY NAVIER-STOKES SYSTEM IN POWER CUSP DOMAINS
Recommended article: On the singular set in the Navier-Stokes equations
description: scientific article; zbMATH DE number 1484690