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## Über dieses Buch

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)

## Inhaltsverzeichnis

### I Steady-State Solutions of the Navier–Stokes Equations: Statement of the Problem and Open Questions

Abstract
Let us consider a viscous fluid of constant density (in short: a viscous liquid) £ moving within a fixed region Ω of three-dimensional space ℝ3.
G. P. Galdi

### II Basic Function Spaces and Related Inequalities

Abstract
In this chapter we shall introduce some function spaces and enucleate certain properties of fundamental importance for further developments. Particular emphasis will be given to what are called homogeneous Sobolev spaces, which will play a fundamental role in the study of flow in exterior domains. We shall not attempt, however, to give an exhaustive treatment of the subject, since this is beyond the scope of the book. Therefore, the reader who wants more details is referred to the specialized literature quoted throughout. As a rule, we give proofs where they are elementary or relevant to the development of the subject, or also when the result is new or does not seem to be widely known.
G. P. Galdi

### III The Function Spaces of Hydrodynamics

Abstract
Several mathematical problems related to the motion of a viscous, incompressible fluid find their natural formulation in certain spaces of vector functions that can be considered as characteristic of those problems.
G. P. Galdi

### IV Steady Stokes Flow in Bounded Domains

Abstract
We now undertake the study of the mathematical properties of the motion of a viscous incompressible fluid. We shall begin with the simplest situation, namely, that of a steady, infinitely slow motion occurring in a bounded region Ω.
G. P. Galdi

### V Steady Stokes Flow in Exterior Domains

Abstract
In this chapter we shall analyze the Stokes problem in an exterior domain. Specifically, assuming that the region of flow Ω is a domain coinciding with the complement of a compact set (not necessarily connected) we wish to establish existence, uniqueness, and the validity of corresponding estimates for the velocity field v and the pressure field p of a steady flow in Ω governed by the Stokes approximation, i.e.
G. P. Galdi

### VI Steady Stokes Flow in Domains with Unbounded Boundaries

Abstract
So far, with the exception of the half-space, we have considered flows occurring in domains with a compact boundary. Nevertheless, from the point of view of the applications it is very important to consider flows in domains Ω having an unbounded boundary, such as channels or pipes of possibly varying cross section.
G. P. Galdi

### VII Steady Oseen Flow in Exterior Domains

Abstract
As we emphasized in the Introduction to Chapter V, the Stokes approximation may fail to describe the physical properties of a system constituted by an object ß moving by assigned rigid motion with “small” translational (v0) and angular (ω) velocities in a viscous liquid, at least at “large” distances from ß, where the viscous effects become less important.
G. P. Galdi

### VIII Steady Generalized Oseen Flow in Exterior Domains

Abstract
The Oseen approximation, which we analyzed to a large extent in the previous chapter, aims at describing the motion of a Navier–Stokes liquid around a rigid body, ß, that moves with a constant and “sufficiently small” purely translational velocity.
G. P. Galdi

### IX Steady Navier–Stokes Flow in Bounded Domains

Abstract
The objective of this and the following chapters is the study of steady motions of a viscous incompressible fluid described by the full nonlinear Navier–Stokes system. In the present chapter we shall focus on the case where the region of flow, Ω, is bounded. More specifically, we shall analyze the boundary value problem obtained by coupling the following system
G. P. Galdi

### X Steady Navier–Stokes Flow in Three-Dimensional Exterior Domains. Irrotational Case

Abstract
Objective of this and the next two chapters is to investigate the mathematical properties of steady flow of a viscous incompressible fluid that fills the entire space outside a finite number of “bodies ”, Ω1,…,Ωs, and whose motion is governed by the fully nonlinear Navier–Stokes equations.
G. P. Galdi

### XI Steady Navier–Stokes Flow in Three-Dimensional Exterior Domains. Rotational Case

Abstract
This chapter is devoted to the study of the mathematical properties of solutions to the exterior boundary–value problem (X.0.1)–(X.0.2), in the case $$\omega \neq 0,$$ so that in general, $$v_\infty = v_0 + \omega \times x.$$
G. P. Galdi

### XII Steady Navier–Stokes Flow in Two-Dimensional Exterior Domains

Abstract
In this chapter we shall study plane steady flow occurring in the complement of a compact region.
G. P. Galdi

### XIII Steady Navier–Stokes Flow in Domains with Unbounded Boundaries

Abstract
Let us consider a steady Navier–Stokes flow of a liquid filling a domain with two unbounded “outlets.”
G. P. Galdi

### Backmatter

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