Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence
type: Item
instance of: scholarly article
title: Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence
zbMATH Open document ID: 1236.35103
DOI: 10.4171/RSMUP/125-4
author: Reinhard Farwig
author: Hideo Kozono
author: Hermann Sohr
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2011-10-21T00:00:00Z
full work available at URL: https://eudml.org/doc/242112
full work available at URL: http://rendiconti.math.unipd.it/volumes/vol125.php
review text: Summary: Consider a smooth bounded domain \(\Omega\subseteq\mathbb{R}^3\) with boundary \(\partial\Omega\), a time interval \([0,T)\), \(0<T\leq\infty\), and the Navier-Stokes system in \([0,T)\times\Omega\), with initial value \(u_0\in L^2_\sigma(\Omega)\) and external force \(f=\text{div}\,F\), \(F \in L^2(0,T;L^2(\Omega))\). Our aim is to extend the well-known class of Leray-Hopf weak solutions \(u\) satisfying \(u|_{\partial\Omega}=0\), \(\text{div}\,u=0\) to the more general class of Leray-Hopf type weak solutions \(u\) with general data \(u|_{\partial\Omega}=g\), \(\text{div}\,u=k\) satisfying a certain energy inequality. Our method rests on a perturbation argument writing \(u\) in the form \(u=v+E\) with some vector field \(E\) in \([0,T)\times\Omega\) satisfying the (linear) Stokes system with \(f=0\) and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35D30
zbMATH DE Number: 5961388
zbMATH Keywords: global weak solution
zbMATH Keywords: Navier-Stokes equation
zbMATH Keywords: boundary data
zbMATH Keywords: divergence
zbMATH Keywords: perturbation term
zbMATH Keywords: existence
zbMATH Keywords: approximate system
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5961388