Decay properties of strong solutions for the Navier-Stokes equations in two-dimensional unbounded domains
type: Item
instance of: scholarly article
title: Decay properties of strong solutions for the Navier-Stokes equations in two-dimensional unbounded domains
zbMATH Open document ID: 0781.35053
DOI: 10.1007/BF01816552
author: Hideo Kozono
author: Takayoshi Ogawa
published in: Archive for Rational Mechanics and Analysis
publication date: 1993-08-29T00:00:00Z
review text: The initial-boundary value problem for the Navier-Stokes system is considered in an unbounded domain \(\Omega\subset \mathbb{R}^ 2\), \[ \begin{cases} u_ t- \Delta u+ u\cdot \nabla u+\nabla p=0,\quad\text{div }u=0,\quad x\in \Omega,\;t>0,\\ u=0\quad x\in\partial \Omega,\;t>0;\quad u|_{t=0}=a.\end{cases}\tag{1} \] Let \(\|\cdot\|_ q\) be the \(L^ q\)-norm on \(\Omega\) and \(A\) the Stokes operator in \(L^ 2\). The main result of the paper is the description of the behaviour of a strong solution of the problem (1) as \(t\to\infty\). If \(0\leq \mu<{1\over 2}\) and \(a\in\mathbb{R}(A^ \mu)\) then the strong solution \(u\) of (1) has the following decay properties \[ \| A^ \alpha u(t)\|_ 2=o(t^{- \mu-\alpha}),\;\| Au(t)\|_ 2=o(t^{-\mu-1}\sqrt{\log t}),\;\Bigl\| A^ \alpha {\partial\over\partial t} u(t)\Bigr\|_ 2= o(t^{-\mu-\alpha-1}) \] for \(0\leq \alpha<1\). Analogous estimates are obtained for \(L^ q\)-norms of the solution for \(q\geq 2\). New proofs for some results of \textit{W. Borchers} and \textit{T. Miyakawa} [Arch. Ration. Mech. Anal. 118, No. 3, 273-295 (1992; Zbl 0756.76018)] are adduced in the paper.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 399275
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: \(L^ 2\)-estimates
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: unbounded domain
zbMATH Keywords: strong solution
zbMATH Keywords: decay
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
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cites work: \(H^p\) spaces of several variables
cites work: On the Navier-Stokes initial value problem. I
cites work: Domains of fractional powers of the Stokes operator in \(L_ r\) spaces
cites work: \(L_ q-L_ r\) estimates for solutions of the nonstationary Stokes equation in an exterior domain and the Navier-Stokes initial value problems in \(L_ q\) spaces
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description: scientific article; zbMATH DE number 399275