The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in \(R^ 3\)
type: Item
instance of: scholarly article
title: The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in \(R^ 3\)
zbMATH Open document ID: 0641.35009
author: Keisuke Kikuchi
published in: Journal of the Mathematical Society of Japan
publication date: 1986-01-01T00:00:00Z
review text: We consider the motion of an ideal incompressible fluid past a finite number of isolated rigid bodies \(O_ 1,...,O_ m\) in \({\mathbb{R}}^ 3.\) The velocity \(v=(v\) 1(x,t),v 2(x,t),v 3(x,t)) and the (scalar) pressure \(p=p(x,t)\) of the fluid motion are governed by the Euler equation in \(\Omega ={\mathbb{R}}\) \(3\setminus (O_ 1\cup...\cup O_ m)\) \[ (1)\quad \partial v/\partial t+(v\cdot \nabla)v+\nabla p=f,\quad div v=0,\quad t\in [0,T] \] subject to the following conditions at infinity and on the boundary \(S=\partial \Omega\) \[ (2)\quad \lim_{| x| \to \infty}v(x,t)=v_{\infty},\quad v\cdot n|_ S=0,\quad t\in [0,T], \] satisfying the initial condition (3) \(v(x,0)=v_ 0(x)\), where \(f=f(x,t)\) is a given external force vector, \(v_ 0(x)\) is a given initial velocity and \(v_{\infty}\) is a given constant vector. The purpose of the present paper is to prove the existence of a solution \(\{\) v,p\(\}\) of (1), (2) and (3) which satisfies the asymptotic condition at infinity that v converges to \(v_{\infty}\) faster than \(| x|^{-(1+\delta)}\) for a certain \(\delta\geq 0\).
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 76B47
zbMATH DE Number: 4044308
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: ideal incompressible fluid
zbMATH Keywords: isolated rigid bodies
zbMATH Keywords: Euler equation
zbMATH Keywords: asymptotic condition at infinity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03840575
OpenAlex ID: W2054336934
DOI: 10.2969/JMSJ/03840575
description: scientific article; zbMATH DE number 4044308