The Navier-Stokes equations in the weak-\(L^n\) space with time-dependent external force
type: Item
instance of: scholarly article
title: The Navier-Stokes equations in the weak-\(L^n\) space with time-dependent external force
zbMATH Open document ID: 0965.35118
DOI: 10.1007/PL00004418
author: Masao Yamazaki
published in: Mathematische Annalen
publication date: 2001-07-19T00:00:00Z
review text: The boundary value problem and the initial-boundary value problem for the nonstationary Navier-Stokes system are investigated: \[ \begin{aligned} &\frac {\partial v}{\partial t}-\Delta v+ v\cdot\nabla v+\nabla p=f(x,t), \quad \nabla\cdot v=0 \quad \text{in \;} \{x\in\Omega,\;t\in \mathbb{R}\}\\ &v(x,t)=0 \quad \text{on \;}\{x\in\partial \Omega,\;t\in\mathbb{R}\}\tag{1}\\ &v(x,t)\to 0\;\text{as \;} |x|\to\infty.\end{aligned} \] \[ \begin{aligned} &\frac {\partial v}{\partial t}-\Delta v+ v\cdot\nabla v+\nabla p=f(x,t),\quad \nabla\cdot v=0 \quad \text{in \;} \{x\in\Omega,\;t\in (0,\infty)\}\\&v(x,t)=0 \quad \text{on \;}\{x\in\partial \Omega,\;t\in (0,\infty)\}\tag{2}\\ &v(x,0)=a(x), \;\;v(x,t)\to 0\;\text{as \;} |x|\to\infty.\end{aligned} \] Here \(\Omega=\mathbb{R}^n\) or \(\Omega=\{x\in \mathbb{R}^n,\;x_n>0\}\), or \(\Omega\) is an exterior domain in \(\mathbb{R}^n\) with smooth boundary \(\partial \Omega,\;n\geqslant 3\). The mild solutions of the problems (1) and (2) are studied in the paper. This solution is the function, bounded in an appropriate function space, which solves the integral equation associated with the Navier-Stokes equations in the sense of distributions. The author gives conditions on the function \(f\) sufficient for the unique existence of a small solution in the Lorentz-type space \(L_{\sigma}^{n,\infty}\times \mathbb{R}\). In particular, sufficient conditions for the unique existence and the stability of small time-periodic solutions are obtained. The regularity of the mild solutions is studied too.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35B15
zbMATH DE Number: 1515368
zbMATH Keywords: nonstationary Navier-Stokes equations
zbMATH Keywords: exterior problem
zbMATH Keywords: mild solution
zbMATH Keywords: stability
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: pl00004418
OpenAlex ID: W2100360505
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description: scientific article; zbMATH DE number 1515368