Periodic solutions of the Navier-Stokes equations in unbounded domains
type: Item
instance of: scholarly article
title: Periodic solutions of the Navier-Stokes equations in unbounded domains
zbMATH Open document ID: 0860.35095
author: Hideo Kozono
author: Mitsuhiro Nakao
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1996-07-18T00:00:00Z
review text: The authors consider the Navier-Stokes equations in an unbounded domain \(\Omega \subset \mathbb{R}^n\), i.e. \[ {\partial u \over \partial t} - \Delta u+(u \cdot \nabla)u + \nabla p=f, \quad \text{div} u=0\quad\text{ in } \Omega \times\mathbb{R}, \quad u=0 \text{ on } \partial \Omega \times \mathbb{R}. \] Here, \(\Omega\) is allowed to be the whole space \(\mathbb{R}^n\) or the half-space \(\mathbb{R}^n_+\) if \(n\geq 3\); in case \(n\geq 4\), \(\Omega\) may also be an exterior domain. The external force \(f\) is assumed to be periodic in time. Under suitable smallness conditions on \(f\) they prove the existence and uniqueness of a periodic strong solution. The idea is to rewrite the Navier-Stokes equations as an integral equation \[ u(t)= \int^t_{-\infty} e^{-(t-s)A} Pf(s)ds - \int^t_{-\infty} e^{-(t-s)A} P \bigl((u\cdot \nabla) u\bigr) (s)ds \] where \(A\) denotes the Stokes operator and \(e^{-tA}\) is the semigroup generated by \(A\). The main ingredients of the existence proof then are the \(L^p-L^r\) estimates for \(e^{-tA}\) and an iteration procedure.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 889574
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: unbounded domain
zbMATH Keywords: existence and uniqueness of a periodic strong solution
zbMATH Keywords: integral equation
zbMATH Keywords: Stokes operator
zbMATH Keywords: semigroup
reviewed by: Klaus Deckelnick
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225411
OpenAlex ID: W1977415811
cites work: \(L^ 2\) decay for the Navier-Stokes flow in halfspaces
cites work: Algebraic \(L^ 2\) decay for Navier-Stokes flows in exterior domains
cites work: On the Navier-Stokes initial value problem. I
cites work: Q4167065
cites work: Analyticity of the semigroup generated by the Stokes operator in \(L_r\) spaces
cites work: Domains of fractional powers of the Stokes operator in \(L_ r\) spaces
cites work: Solutions for semilinear parabolic equations in \(L^ p\) and regularity of weak solutions of the Navier-Stokes system
cites work: Solutions in \(L_ r\) of the Navier-Stokes initial value problem
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
cites work: A reproductive property of the Navier-Stokes equations
cites work: Strong \(L^ p\)-solutions of the Navier-Stokes equation in \(R^ m\), with applications to weak solutions
cites work: Q4024275
cites work: Q5576913
cites work: Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space
cites work: Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space
cites work: On nonstationary solutions of the Navier-Stokes equations in an exterior domain
cites work: Existence and periodicity of weak solutions of the Navier-Stokes equations in a time dependent domain
cites work: Q5673831
cites work: A note on the existence of periodic solutions of the Navier-Stokes equations
cites work: Q5723392
cites work: Q4273991
cites work: Q5592475
cites work: Q3207428
cites work: On the stability of periodic solutions of the Navier-Stokes equations in a noncylindrical domain
cites work: A solution formula for the Stokes equation in Rn+
cites work: Q3695717
DOI: 10.2748/TMJ/1178225411
description: scientific article; zbMATH DE number 889574