The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary
type: Item
instance of: scholarly article
title: The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary
zbMATH Open document ID: 0915.76017
author: Nader Masmoudi
published in: Archive for Rational Mechanics and Analysis
publication date: 1999-06-23T00:00:00Z
review text: The author investigates the convergence of weak solutions of the Navier-Stokes equations when horizontal and vertical viscosities go to zero. Let \((v,p)\) satisfy the boundary value problem in \(Q=(0,T)\times \Omega\), \(\Omega=\omega\times (0,h)\) or \(\Omega=\omega\times (0,\infty)\), and \(\omega=\mathbb{T}^2\) or \(\omega=\mathbb{R}^2\), \(\frac{\partial v}{\partial t}+(v\cdot\nabla)v-\nu \frac{\partial^2 v}{\partial z^2 }-\eta\Delta' v+\nabla p=F\), \(\text{div} v=0\) in \(Q\), \(v| _{\partial Q}=0\), \(v| _{t=0}=v_0\), where \(\Delta' = \frac{\partial^2 }{\partial x^2 }+ \frac{\partial^2 }{\partial y^2 }\). Let further the Euler system in \(Q\) be written in the form \(\frac{\partial w}{\partial t}+(w\cdot\nabla)w +\nabla p=F^0\), \(\text{div} w=0\) in \(Q\), \(w\cdot n| _{\partial Q}=0\), \(w| _{t=0}=w_0\). It is proved that if \(v_0\to w_0,\;F\to F^0\) and \(\frac \nu\eta \to 0\) as \(\nu \to 0,\eta \to 0\), then there exist global weak solutions \(v\) of the first problem such that \(v\to w\) in certain spaces. A similar result is obtained for system describing rotating fluids.
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76U05
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 1218193
zbMATH Keywords: convergence of weak solutions
zbMATH Keywords: boundary value problem
zbMATH Keywords: global weak solutions
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: s002050050097
OpenAlex ID: W2091612438
DOI: 10.1007/S002050050097
description: scientific article; zbMATH DE number 1218193