On \(L^ 2\) decay of weak solutions of the Navier-Stokes equations in \(\mathbb R^ n\)
type: Item
instance of: scholarly article
title: On \(L^ 2\) decay of weak solutions of the Navier-Stokes equations in \(\mathbb R^ n\)
zbMATH Open document ID: 0607.35072
DOI: 10.1007/BF01162027
author: Ryuji Kajikiya
author: Tetsuro Miyakawa
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173722
review text: From the authors' Introduction: ''We consider the initial value problem for the Navier-Stokes equations in \({\mathbb{R}}^ n\), \[ \partial u/\partial t+u\cdot \nabla u-\Delta u+\nabla p=0\quad in\quad {\mathbb{R}}^ n\times (0,\infty), \] \[ \nabla \cdot u=0\quad in\quad {\mathbb{R}}^ n\times (0,\infty),\quad u(x,0)=a(x)\quad in\quad {\mathbb{R}}^ n, \] where \(n\geq 2\), \(u=(u^ i)^ n_{i=1}\) and p denote, respectively, unknown velocity and pressure, while \(a=(a^ i)^ n_{i=1}\) is a given initial velocity....Our purpose in this paper is to improve and generalize the result of \textit{M. E. Schonbek} [Arch. Ration. Mech. Anal. 88, 209-222 (1985; Zbl 0602.76039)] by the systematic use of a modified version of her method. In fact, for all \(n\geq 2\) and \(a\in L^ 2\) with \(\nabla \cdot a=0\), we shall show that the weak solution... tends to zero in \(L^ 2\) as \(t\to \infty\). Moreover, we show that if a is in \(L^ 2\cap L^ r\) for some \(1\leq r<2\), then the corresponding solution decays in \(L^ 2\) like \(t^{-(n/r-n/2)/2}\)''.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35D05
zbMATH DE Number: 3981695
zbMATH Keywords: weak solutions
zbMATH Keywords: initial value problem
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: decays
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3981695