The Navier-Stokes Equations

The primary objective of this monograph is to develop an elementary and self contained approach to the mathematical theory of a viscous incompressible fluid in a domain 0 of the Euclidean space ]Rn, 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 first 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 O. 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 significant from the physical point of view. For mathematical generality, we will develop the lin earized theory for all n 2 2. Although the functional-analytic approach developed here is, in principle, known to specialists, its systematic treatment is not available, and even the diverse aspects available are spread out in the literature. However, the literature 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.

"The book is written in a well-arranged way, easy to survey and with utilization of the newest results..it is [written] for an extensive circle of readers, from students to expertsand specialists in the field."

--Zentralblatt Math

Contenu
I Introduction.- 1 Basic notations.- 2 Description of the functional analytic approach.- 3 Function spaces.- II Preliminary Results.- 1 Embedding properties and related facts.- 2 The operators ? and div.- 3 Elementary functional analytic properties.- III The Stationary Navier-Stokes Equations.- 1 Weak solutions of the Stokes equations.- 2 The Stokes operator A.- 3 The stationary Navier-Stokes equations.- IV The Linearized Nonstationary Theory.- 1 Preliminaries for the time dependent linear theory.- 2 Theory of weak solutions in the linearized case.- V The Full Nonlinear Navier-Stokes Equations.- 1 Weak solutions.- 2 Approximation of the Navier-Stokes equations.- 3 Existence of weak solutions of the Navier-Stokes system.- 4 Strong solutions of the Navier-Stokes system.