Navier-Stokes Equations: Theory and Numerical Analysis (AMS Chelsea Publishing)

Preface to the AMS Chelsea edition ix Preface to the third (revised) edition xi Foreword xiii Chapter 1. The Steady-State Stokes Equations viii CONTENTS 2. Application to the non-homogeneous steady-state Navier-Stokes equations Appendix II. Implementation of Non-conforming Linear Finite Elements (Approximation APX5-Two-dimensional Case) by F. Thomasset 0. Test problems 1. The triangulation Note to the reader : Two distinct bibliographies are available. Originally published in 1977, the book is devoted to the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluid. On the theoretical side, results related to the existence, the uniqueness, and, in some cases, the regularity of solutions are presented. On the numerical side, various approaches to the approximation of Navier-Stokes problems by discretization are considered, such as the finite dereference method, the finite element method, and the fractional steps method. The problems of stability and convergence for numerical methods are treated as completely as possible. The new material in the present book (as compared to the preceding 1984 edition) is an appendix reproducing a survey article written in 1998. This appendix touches upon a few aspects not addressed in the earlier editions, in particular a short derivation of the Navier-Stokes equations from the basic conservation principles in continuum mechanics, further historical perspectives, and indications on new developments in the area. The appendix also surveys some aspects of the related Euler equations and the compressible Navier-Stokes equations. The book is written in the style of a textbook and the author has attempted to make the treatment self-contained. It can be used as a textbook or a reference book for researchers. Prerequisites for reading the book include some familiarity with the Navier-Stokes equations and some knowledge of functional analysis and Sololev spaces. This book was originally published in 1977 and has since been reprinted four times (the last reprint was in 1984). The current volume is reprinted and fully retypeset by the AMS. It is very close in content to the 1984 edition. The book presents a systematic treatment of results on the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluids. Considered are the linearized stationary case, the nonlinear stationary case, and the full nonlinear time-dependent case. The relevant mathematical tools are introduced at each stage. The new material in this book is Appendix III, reproducing a survey article written in 1998. This appendix contains a few aspects not addressed in the earlier editions, in particular a short derivation of the Navier-Stokes equations from the basic conservation principles in continuum mechanics, further historical perspectives, and indications on new developments in the area. The appendix also surveys some aspects of the related Euler equations and the compressible Navier-Stokes equations. Readers are advised to peruse this appendix before reading the core of the book. This book presents basic results on the theory of Navier-Stokes equations and, as such, continues to serve as a comprehensive reference source on the topic. Chapter 1. The Steady-state Stokes Equations Chapter 2. Steady-state Navier-stokes Equations Chapter 3. The Evolution Navier-stokes Equation Appendix I. Properties Of The Curl Operator And Application To The Steady-state Navier-stokes Equations Appendix Ii. Implementation Of Non-conforming Linear Finite Elements (approximation Apx5 -- Two-dimensional Case) Appendix Iii. Some Developments On Navier-stokes Equations In The Second Half Of The 20th Century Bibliography To Appendix Iii By Roger Temam. Third (revised) Edition--p. Xi. Reprint. Originally Published: Amsterdam : North-holland, 1984. 3rd Rev. Ed. Includes Bibliographical References And Index. Cover Title page Dedication Contents Preface to the AMS-Chelsea edition Preface to the third (revised) edition Foreword The steady-state Stokes equations Steady-state Navier–Stokes equations The evolution Navier–Stokes equation Appendix I: Properties of the curl operator and application to the steady-state Navier–Stokes equations Appendix II: Implementation of non-conforming linear finite elements (Approximation APX5—Two-dimensional case) Appendix III: Some developments on Navier–Stokes equations in the second half of the 20th century Bibliography to Appendix III Comments Additional comments to the revised edition Bibliography Index Back Cover Presents a systematic treatment of results on the theory and numerical analysis of the Navier-Stokes equations for viscous incompressible fluids. Considered are the linearized stationary case, the nonlinear stationary case, and the full nonlinear time-dependent case. This title introduces the relevant mathematical tools at each stage. In this chapter we study the stationary Stokes equations; that is, the stationary linearized form of the Navier-Stokes equations.