The Navier-Stokes Equations

This book develops an elementary and self-contained approach to the mathematical theory of a viscous, incompressible fluid in a domain of the Euclidean space, described by the equations of Navier-Stokes.

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.

Elementary and selfcontained approach to the mathematical theory of the viscous incompressible Navier-Stokes equations Requires familiarity with the basic functional analytic tools in Hilbert and Banach spaces only Fills a gap by providing a systematic treatment of the subject? Includes supplementary material: sn.pub/extras

Auteur
Hermann Sohr is professor of mathematics, especially mathematical fluid mechanics, at the University Paderborn, Germany.

Texte du rabat

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 of the Euclidean space, described by the equations of Navier-Stokes.

Moreover, the theory is presented for completely general domains, in particular, for arbitrary unbounded, nonsmooth domains. Therefore, restriction was necessary to space dimensions two and three, which are also the most significant from a physical point of view. For mathematical generality, however, the linearized theory is expounded for general dimensions higher than one.

Although the functional analytic approach developed here is, in principle, known to specialists, the present book fills a gap in the literature providing a systematic treatment of a subject that has been documented until now only in fragments. The book is mainly directed to students familiar with basic tools in Hilbert and Banach spaces. However, for the readers'convenience, some fundamental properties of, for example, Sobolev spaces, distributions and operators are collected in the first two chapters.

*The book is written in a well arranged way, easy to survey and with utilization of the newest results. For its study, it is necessary to know only the basic functional analytic tools in Hilbert and Banach spaces. It is determined for an extensive circle of readers, from students up to experts in science and also for specialists in the field.

• (Zentralblatt MATH)

  *The author's purpose in this book is to develop an elementary and self-contained approach to the mathematical theory of the viscous incompressible Navier-Stokes equations from basic functional analytic tools. Another objective is to develop the results in reasonably full generality, in particular to allow for arbitrary nonsmooth, possibly unbounded, spatial domains.

• (Mathematical Reviews)

Contenu
Preliminary Results.- The Stationary Navier-Stokes Equations.- The Linearized Nonstationary Theory.- The Full Nonlinear Navier-Stokes Equations.