The Stokes and Navier-Stokes equations in an aperture domain
type: Item
instance of: scholarly article
title: The Stokes and Navier-Stokes equations in an aperture domain
zbMATH Open document ID: 1171.35097
author: Takayuki Kubo
published in: Journal of the Mathematical Society of Japan
publication date: 2007-09-12T00:00:00Z
review text: The author deals with the nonstationary Navier-Stokes initial boundary value problem (IBVP) \[ \begin{gathered} \partial_t u-\Delta u+(u,\nabla) u+\nabla\pi= 0\quad\text{in }\Omega\times (0,\infty),\\ \nabla\cdot u= 0\quad\text{in }\Omega\times (0,\infty),\\ u(x,t)= 0\quad\text{on }\partial\Omega\times (0,\infty),\\ u(x,0)= u_0(x)\quad\text{in }\Omega,\end{gathered}\tag{1} \] where \(u\) is the unknown field, \(u= (u_1,\dots, u_n)\in (W^{2,p}(\Omega))^n\) and the unknown scalar pressure term \(\nabla\pi\in (L^p(\Omega))^n\), \(1< p<\infty\) and \(\Omega\) is an aperture domain. The author proves the global existence of a unique solution to the (1) with the vanishing flux condition and some decay properties as \(t\to\infty\), when the initial velocity is sufficiently small in \(L^n\) space. Moreover, he can prove the time-local existence of an unique solution to the (1) with the non-trivial flux condition.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 5190375
zbMATH Keywords: nonstationary Navier-Stokes
zbMATH Keywords: aperture domain
zbMATH Keywords: global existence
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05930837
OpenAlex ID: W1974609211
DOI: 10.2969/JMSJ/05930837
description: scientific article; zbMATH DE number 5190375