On the very weak solution for the Oseen and Navier-Stokes equations
type: Item
instance of: scholarly article
title: On the very weak solution for the Oseen and Navier-Stokes equations
zbMATH Open document ID: 1198.35173
author: Chérif Amrouche
author: María Ángeles Rodríguez-Bellido
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2010-07-23T00:00:00Z
review text: The main objective of the paper consists in the development of a unified theory of very weak solutions of the Dirichlet problem for Stokes, Oseen and Navier-Stokes equations (and also for the Laplace equation). An important question in this context is the rigorous definition of traces of the vector functions that are belonging to subspaces of \({\mathbf L}^{p}(\Omega)\). The authors prove existence and regularity of very weak solutions \((\mathbf u, q) \in {\mathbf L}^{p}(\Omega) \times W^{-1,p}(\Omega)\) of Stokes and Oseen equations for any \(1 < p < \infty\) with arbitrary large data belonging to some Sobolev spaces with negative order. In the case of the Navier-Stokes equations the existence of very weak solutions is proved for arbitrary large external forces. Here a smallness condition is necessary for both the compressibility \(h\) and the boundary conditions \(\mathbf g\). If the data are small enough then the uniqueness of the very weak solution can also been proved.
reviewed by: Jürgen Socolowsky
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76D07
Mathematics Subject Classification ID: 35B30
zbMATH DE Number: 5760005
zbMATH Keywords: Stokes equations
zbMATH Keywords: Oseen equations
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: very weak solutions
zbMATH Keywords: Stationary solutions
MaRDI profile type: Publication
full work available at URL: dcdss.2010.3.159
OpenAlex ID: W1989465456
DOI: 10.3934/DCDSS.2010.3.159
description: scientific article; zbMATH DE number 5760005