The book provides a comprehensive detailed and self-contained treatment of the fundamental
mathematical properties of boundary-value problems related to the Navier-Stokes equations.
These properties include existence uniqueness and regularity of solutions in bounded as well
as unbounded domains. Whenever the domain is unbounded the asymptotic behavior of solutions is
also investigated. This book is the new edition of the original two volume book under the same
title published in 1994. In this new edition the two volumes have merged into one and two
more chapters on steady generalized oseen flow in exterior domains and steady Navier-Stokes
flow in three-dimensional exterior domains have been added. Most of the proofs given in the
previous edition were also updated. An introductory first chapter describes all relevant
questions treated in the book and lists and motivates a number of significant and still open
questions. It is written in an expository style so as to be accessible also to non-specialists.
Each chapter is preceded by a substantial preliminary discussion of the problems treated
along with their motivation and the strategy used to solve them. Also each chapter ends with a
section dedicated to alternative approaches and procedures as well as historical notes. The
book contains more than 400 stimulating exercises at different levels of difficulty that will
help the junior researcher and the graduate student to gradually become accustomed with the
subject. Finally the book is endowed with a vast bibliography that includes more than 500
items. Each item brings a reference to the section of the book where it is cited. The book will
be useful to researchers and graduate students in mathematics in particular mathematical fluid
mechanics and differential equations. Review of First Edition First Volume: The emphasis of
this book is on an introduction to the mathematical theory of the stationary Navier-Stokes
equations. It is written in the style of a textbook and is essentially self-contained. The
problems are presented clearly and in an accessible manner. Every chapter begins with a good
introductory discussion of the problems considered and ends with interesting notes on
different approaches developed in the literature. Further stimulating exercises are proposed.
(Mathematical Reviews 1995)
