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## Über dieses Buch

The primary objective of this monograph is to develop an elementary and se- containedapproachtothemathematicaltheoryofaviscousincompressible?uid n in a domain ? of the Euclidean spaceR , described by the equations of Navier- Stokes. The book is mainly directed to students familiar with basic functional analytic tools in Hilbert and Banach spaces. However, for readers’ convenience, in the ?rst two chapters we collect, without proof some fundamental properties of Sobolev spaces, distributions, operators, etc. Another important objective is to formulate the theory for a completely general domain ?. In particular, the theory applies to arbitrary unbounded, non-smooth domains. For this reason, in the nonlinear case, we have to restrict ourselves to space dimensions n=2,3 that are also most signi?cant from the physical point of view. For mathematical generality, we will develop the l- earized theory for all n? 2. Although the functional-analytic approach developed here is, in principle, known to specialists, its systematic treatment is not available, and even the diverseaspectsavailablearespreadoutintheliterature.However,theliterature is very wide, and I did not even try to include a full list of related papers, also because this could be confusing for the student. In this regard, I would like to apologize for not quoting all the works that, directly or indirectly, have inspired this monograph.

## Inhaltsverzeichnis

### Chapter I. Introduction

Abstract
Throughout this book, $${\Omega \subseteq\mathbb{R}^{n}}$$ means a general domain, that is any open nonempty connected subset of the n-dimensional Euclidean space $${\mathbb{R}^{n}}$$. In the linearized theory we admit that $$n \geq 2$$, the nonlinear theory is restricted to n = 2 and n = 3 ; in the preliminaries, see Chapters I and II, we sometimes admit the case n = 1.
Hermann Sohr

### Chapter II. Preliminary Results

Abstract
Let $$L^{q}$$-norm of a function u with the $$L^{q}$$-norm of its gradient $$\nabla {u} = (D_{1}u,...,D_{n}u)$$.
Hermann Sohr

### Chapter III. The Stationary Navier-Stokes Equations

Abstract
Let $$\Omega \subseteq \mathbb{R}^{n},n \geq 2,$$ be any domain with boundary $$\partial \Omega$$. Our purpose is to investigate the Stokes system
$$\begin{array}{ccccccc}-\nu \Delta{u}+ \nabla{p} = \quad f \quad ,{\rm div} \; {u}=0, \\ u|\partial\Omega = 0.\end{array}$$
(1.1.1)
Hermann Sohr

### Chapter IV. The Linearized Nonstationary Theory

Abstract
$${\Omega \subseteq\mathbb{R}^{n}}$$ be an arbitrary domain with $$n \geq 2$$ and boundary $$\partial \Omega$$. In the linear time dependent theory we admit arbitrary dimensions $$n \geq 2$$.
Hermann Sohr

### Chapter V. The Full Nonlinear Navier-Stokes Equations

Abstract
In this chapter $${\Omega \subseteq\mathbb{R}^{n}}$$ means a domain with n = 2 or n = 3, and [0, T) is a fixed time interval with$$0 < T \leq \infty$$. The full nonlinear Navier-Stokes system in [0, T)$$\times \Omega$$, has the form
$$\begin{array}{rrrrrr} u_{t} - \nu \Delta u + u \cdot \nabla{u}+\nabla{p}\quad = \quad f \; , {\rm div} \; {u}=0,\\ {u|\partial \Omega \quad = \quad 0,\; u(0) = u_{0}}\end{array}$$
(1.1.1)
Hermann Sohr

### Backmatter

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