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This work is a systematic and up-to-date investigation of the fundamental properties of the Navier-Stokes equations, including existence, uniqueness, and regularity of solutions in bounded as well as unbounded domains. When the region of flow is unbounded, asymptotic behavior is also investigated. The problems are treated within the framework of Lebesgue spaces, such a general approach allowing for a comprehensive and unified treatment.
The subject is divided into two volumes, which deal with steady flow (boundary value problem). Each volume is self-contained. In both volumes, the nonlinear analysis is preceded and supported by a suitable linear analysis; however, the nonlinear analysis is treated to a large extent due to independent interest.
This volume examines linearised approximations of the steady-state Navier-Stokes problem - specifically, the Stokes problem in bounded and exterior domains as well as in domains with noncompact boundaries. Moreover, Oseen equations and the associated boundary value problem are investigated in exterior domains.
This new edition has been revised and updated throughout to take into account the recent significant contributions to the field. A new chapter is dedicated to the motion of a rigid body in a fluid in the Stokes regime, which has found increased interest due to questions related to liquid-particle interactions.
The books are addressed to students (both graduate and advanced undergraduate) and to all mathematicians and applied mathematicians who wish to become acquainted with the subject.
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 thestyle 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)
The subject is divided into two volumes, which deal with steady flow (boundary value problem). Each volume is self-contained. In both volumes, the nonlinear analysis is preceded and supported by a suitable linear analysis; however, the nonlinear analysis is treated to a large extent due to independent interest.
This volume examines linearised approximations of the steady-state Navier-Stokes problem - specifically, the Stokes problem in bounded and exterior domains as well as in domains with noncompact boundaries. Moreover, Oseen equations and the associated boundary value problem are investigated in exterior domains.
This new edition has been revised and updated throughout to take into account the recent significant contributions to the field. A new chapter is dedicated to the motion of a rigid body in a fluid in the Stokes regime, which has found increased interest due to questions related to liquid-particle interactions.
The books are addressed to students (both graduate and advanced undergraduate) and to all mathematicians and applied mathematicians who wish to become acquainted with the subject.
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 thestyle 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)
From the reviews of the second edition:
"The book yields a comprehensive, detailed and self-contained study of the basic mathematical properties of different boundary-value problems related to the Navier-Stokes equations. These properties include existence, uniqueness and regularity of solutions. ... The book contains more than 400 carefully choosen exercises at different levels of difficulty that will help the young researcher. The comprehensive bibliography contains more than 500 items. Galdi's monograph can strongly be recommended to every mathematician (and theoretical physicist) interested in mathematical fluid mechanics or in PDEs." (Jürgen Socolowsky, Zentralblatt MATH, Vol. 1245, 2012)
"The book yields a comprehensive, detailed and self-contained study of the basic mathematical properties of different boundary-value problems related to the Navier-Stokes equations. These properties include existence, uniqueness and regularity of solutions. ... The book contains more than 400 carefully choosen exercises at different levels of difficulty that will help the young researcher. The comprehensive bibliography contains more than 500 items. Galdi's monograph can strongly be recommended to every mathematician (and theoretical physicist) interested in mathematical fluid mechanics or in PDEs." (Jürgen Socolowsky, Zentralblatt MATH, Vol. 1245, 2012)