Structure of solutions of 3D axisymmetric Navier-Stokes equations near maximal points
type: Item
instance of: scholarly article
title: Structure of solutions of 3D axisymmetric Navier-Stokes equations near maximal points
zbMATH Open document ID: 1244.76008
author: Zhen Lei
author: Qi S. Zhang
published in: Pacific Journal of Mathematics
publication date: 2012-05-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.4172
full work available at URL: p05.xhtml
reviewed by: Keisuke Uchikoshi
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 6038887
zbMATH Keywords: axisymmetric Navier-Stokes equations
zbMATH Keywords: structure of singularities
MaRDI profile type: Publication
OpenAlex ID: W2021564702
arXiv ID: 1008.4172
DOI: 10.2140/PJM.2011.254.335
review text: The authors consider 3D incompressible Navier-Stokes equations: NEWLINE\[NEWLINE \partial_t v +v\cdot \nabla v+\nabla p=\rho \Delta v,\;\;\nabla\cdot v=0 NEWLINE\]NEWLINE for \(t\geq 0, x\in\mathbb{R}^3\). Introducing a cylindrical coordinate system \(x=(x_1 ,x_2 ,x_3 )=(r\cos\theta ,r\sin\theta ,z)\), they consider a \(z\)-axis symmetric solution of the form NEWLINE\[NEWLINE v (x)=v^r (r,z,t)e_r +v^{\theta} (r,z,t)e_{r,z,t} + v^z (r,z,t)e_z . NEWLINE\]NEWLINE Here \(e_r ={}^t (\cos\theta ,\sin\theta ,0),\;e_{\theta} ={}^t (-\sin\theta ,\cos\theta ,0),\;e_z =(0,0,1)\). They assume that at some point \((x_0 ,t_0 )\) the flow speed \(Q=|v|\) takes an almost maximum value in the following sense: \(4\max{x\in\mathbb{R}^3 ,\;t\leq t_0} Q(x ,t )\leq Q(x_0 ,t_0 )\). Then, they prove that the solution is close to a constant function on some cubic domain of the half region \(\{t\leq t_0 \}\) near \((x_0 ,t_0 )\).
description: scientific article; zbMATH DE number 6038887