Mathematical tools for the study of the incompressible Navier-Stokes equations and related models
type: Item
instance of: scholarly article
title: Mathematical tools for the study of the incompressible Navier-Stokes equations and related models
zbMATH Open document ID: 1286.76005
DOI: 10.1007/978-1-4614-5975-0
author: Franck Boyer
author: Pierre Fabrie
published in: Applied Mathematical Sciences
publication date: 2012-11-15T00:00:00Z
review text: In this interesting and very well-written book, the authors are concerned with the mathematical analysis of Navier-Stokes equations and related problems and models. The book is divided into seven chapters followed by two appendices, a list of references and an index. In the first chapter, the authors present the equations of fluid mechanics and related models that are studied in the forthcoming chapters. The second chapter reviews the main results in functional analysis that are used in the book, such as the weak and weak-\(*\) convergence in Banach spaces, the distribution theory, compactness results and some results from spectral theory. In the third chapter, the authors provide an introduction to Sobolev spaces in Lipschitz domains in \({\mathbb R}^d\). The fourth chapter examines the steady Stokes equations. The authors present a complete proof of Nečas inequality, which is used in the second section of this chapter for the study of some relations between gradient fields and divergence-free vector fields. Certain properties of divergence and curl operators are also presented. In the fifth section of Chapter four, the authors investigate the Stokes problem with Dirichlet boundary conditions, and show the well-posedness of the Stokes system. They also study the unbounded Stokes operator and present elliptic regularity for the Stokes problem. In the last three sections of Chapter four, the authors deal with Stokes equations with other boundary conditions, such as Neumann boundary conditions, interface conditions, and a boundary condition in terms of vorticity field. They prove existence, uniqueness and regularity results for these problems. The fifth chapter is devoted to steady and unsteady equations for homogeneous fluids. The authors prove the existence, uniqueness and regularity of solutions to the incompressible Navier-Stokes equations in bounded domains in \({\mathbb R}^d\), for \(d=2,3\). The first section provides an introduction to weak solutions, while th…
reviewed by: Mirela Kohr
Mathematics Subject Classification ID: 76-02
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 6106780
zbMATH Keywords: homogeneous fluid
zbMATH Keywords: nonhomogeneous fluid
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: weak solution
zbMATH Keywords: strong solution
zbMATH Keywords: Di Parma-Lions theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4614-5975-0
OpenAlex ID: W616225202
description: scientific article; zbMATH DE number 6106780