On stability of Leray's stationary solutions of the Navier-Stokes system in exterior domains
type: Item
instance of: scholarly article
title: On stability of Leray's stationary solutions of the Navier-Stokes system in exterior domains
zbMATH Open document ID: 1368.35208
author: Hajime Koba
published in: Journal of the Mathematical Society of Japan
publication date: 2017-04-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1484730029
review text: Let \(\Omega \subset \mathbb{R}^3\) be an exterior domain with smooth boundary \(\partial \Omega\). Consider the system \[ \begin{aligned} & u_t-\nu\Delta u+(u,\nabla)u+\nabla\Pi=\nabla \cdot F \;\text{ in } \Omega \times (0, \infty) , \\ & \nabla \cdot u =0 \;\text{ in } \Omega \times (0, \infty), \\ & u|_{\partial \Omega}=0, \;\lim_{|x|\rightarrow \infty} u=u_{\infty}, \;u|_{t=0}=u_0, \end{aligned} \] where the notations are the usual ones. The author studies the stability of the solution of the above system around the stationary solution \((w, \pi)\), i.e., the solution of the system \[ \begin{aligned} &-\nu \Delta w +(u_{\infty}, \nabla)w+(w,\nabla)w+\nabla \pi =\nabla \cdot F \;\text{ in } \Omega, \\ & \nabla \cdot w=0 \;\text{ in } \Omega, \\ & w|_{\partial \Omega}=-u_{\infty}, \;\lim_{|x|\rightarrow \infty} w=0, \end{aligned} \] which satisfies \(\int_{\Omega}|\nabla w(x)|^2dx < \infty\) (i.e., it is a Leray stationary solution). The main results are contained in Theorem 2.1, where \(\| w\|_{L^{3,\infty}(\Omega)}\) is assumed to be small.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 93D20
zbMATH DE Number: 6701594
zbMATH Keywords: Navier-Stokes in exterior domains
zbMATH Keywords: asymptotic stability
zbMATH Keywords: decay property
zbMATH Keywords: maximal \(L^p\) regularity
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
OpenAlex ID: W2576504309
DOI: 10.2969/JMSJ/06910373
description: scientific article; zbMATH DE number 6701594