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2013 | Buch

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Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models

verfasst von: Franck Boyer, Pierre Fabrie

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The objective of this self-contained book is two-fold. First, the reader is introduced to the modelling and mathematical analysis used in fluid mechanics, especially concerning the Navier-Stokes equations which is the basic model for the flow of incompressible viscous fluids. Authors introduce mathematical tools so that the reader is able to use them for studying many other kinds of partial differential equations, in particular nonlinear evolution problems.

The background needed are basic results in calculus, integration, and functional analysis. Some sections certainly contain more advanced topics than others. Nevertheless, the authors’ aim is that graduate or PhD students, as well as researchers who are not specialized in nonlinear analysis or in mathematical fluid mechanics, can find a detailed introduction to this subject.
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Inhaltsverzeichnis

Frontmatter

Chapter I. The equations of fluid mechanics

Abstract

This first chapter introduces the equations of fluid mechanics and the various models which we study later. Among the many books published on fluid mechanics, and more generally on the mechanics of continuous media, we recommend that readers consult, for example, references [90, 66, 102, 127, 126, 123] if they seek a more detailed description of the various concepts. Moreover, some of the thermodynamics concepts we use here are very briefly described in Appendix B.

Franck Boyer, Pierre Fabrie

Chapter II. Analysis tools

Abstract

The goal of this chapter is to describe the analysis tools that we use in later chapters. We have gathered together fundamental concepts required to study many linear or nonlinear evolution partial differential equations coming from many areas of physics and biology, for instance.

Franck Boyer, Pierre Fabrie

Chapter III. Sobolev spaces

Abstract

The first section of this chapter is dedicated to the basic definitions and properties of domains in ℝ^d. We particularly focus our attention on the case of Lipschitz domains for which we can easily define an integration theory on ∂Ω, the outward unit normal on ∂Ω, and finally prove the Stokes formula which is the keystone of the study of partial differential equations on domains.

Franck Boyer, Pierre Fabrie

Chapter IV. Steady Stokes equations

Abstract

The first section of this chapter is dedicated to the proof of the Necas inequality which says that, in the space L ²(Ω), the L ²-norm is equivalent to the sum of the H ^-1-norm of the function and of its gradient. Even if this seems to be a very natural property, the proof (given here in any Lipschitz domain with compact boundary) is far from being straightforward.

Franck Boyer, Pierre Fabrie

Chapter V. Navier–Stokes equations for homogeneous fluids

Abstract

The main matter of this chapter concerns existence, uniqueness, and regularity results in a bounded domain of ℝ^d, d = 2, 3 for solutions of the incompressible homogeneous Navier–Stokes equations. The study of steady solutions and their stability properties is also investigated.

Franck Boyer, Pierre Fabrie

Chapter VI. Nonhomogeneous fluids

Abstract

In this chapter we study the nonhomogeneous incompressible Navier–Stokes equations. The difference with the equations studied in the previous chapter is that the density is no longer a constant and has become a new unknown function.

Franck Boyer, Pierre Fabrie

Chapter VII. Boundary conditions modelling

Abstract

In this chapter, we consider two different problems related to boundary conditions that one may encounter when trying to compute numerical approximations of real flows. These problems arise when the computational domain is not exactly the original physical domain which is of interest.

Franck Boyer, Pierre Fabrie

Backmatter

Titel: Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models
verfasst von: Franck Boyer
Pierre Fabrie
Verlag: Springer New York
Electronic ISBN: 978-1-4614-5975-0
Print ISBN: 978-1-4614-5974-3
DOI: https://doi.org/10.1007/978-1-4614-5975-0

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