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Description:

The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the fundamental dynamics of fluid motion. They are applied routinely to problems in engineering, geophysics, astrophysics, and atmospheric science. This book is an introductory physical and mathematical presentation of the Navier-Stokes equations, focusing on unresolved questions of the regularity of solutions in three spatial dimensions, and the relation of these issues to the physical phenomenon of turbulent fluid motion. The goal of the book is to present a mathematically rigorous investigation of the Navier-Stokes equations that is accessible to a broader audience than just the subfields of… More mathematics to which it has traditionally been restricted. Therefore, results and techniques from nonlinear functional analysis are introduced as needed with an eye toward communicating the essential ideas behind the rigorous analyses. This book is appropriate for graduate students in many areas of mathematics, physics, and engineering.Less

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Book details

List price: $62.00 Copyright year: 1995 Publisher: Cambridge University Press Publication date: 4/28/1995 Binding: Paperback Pages: 232 Size: 6.25" wide x 9.25" long x 0.50" tall Weight: 0.682 Language: English

AuthorTable of Contents

Preface

The equations of motion

Introduction

Euler's equations for an incompressible fluid

Energy, body forces, vorticity, and enstrophy

Viscosity, the stress tensor, and the Navier-Stokes equations

Thermal convection and the Boussinesq equations

References and further reading

Exercises

Dimensionless parameters and stability

Dimensionless parameters

Linear and nonlinear stability, differential inequalities

References and further reading

Exercises

Turbulence

Introduction

Statistical turbulence theory and the closure problem

Spectra, Kolmogorov's scaling theory, and turbulent length scales

References and further reading

Exercises

Degrees of freedom, dynamical systems, and attractors

Introduction

Dynamical systems, attractors, and their dimension

The Lorenz system

References and further reading

Exercises

On the existence, uniqueness, and regularity of solutions

Introduction

Existence and uniqueness for ODEs

Galerkin approximations and weak solutions of the Navier-Stokes equations

Uniqueness and the regularity problem

References and further reading

Exercises

Ladder results for the Navier-Stokes equations

Introduction

The Navier-Stokes ladder theorem

A natural definition of a length scale

The dynamical wavenumbers k[subscript N,r]

Estimates for the Navier-Stokes equations

Estimates for F[subscript 0]

Estimates for [left angle bracket]F[subscript 1 right angle bracket] and [left angle bracket]k[superscript 2 subscript 1,0 right angle bracket]

Estimates for lim[subscript t[right arrow infinity]F[subscript 1], [left angle bracket]F[subscript 2 right angle bracket], and [left angle bracket]k[superscript 2 subscript 2,1 right angle bracket]

A ladder for the thermal convection equations

References and further reading

Exercises

Regularity and length scales for the 2d and 3d Navier-Stokes equations

Introduction

A global attractor and length scales in the 2d case

A global attractor

Length scales in the 2d Navier-Stokes equations

3d Navier-Stokes regularity?

Problems with 3d Navier-Stokes regularity

A Bound on [left angle bracket]k[subscript N,1 right angle bracket] in 3d

Bounds on [left angle bracket double vertical line]u[double vertical line subscript [infinity right angle bracket] and [left angle bracket double vertical line]Du[double vertical line superscript 1/2 subscript [infinity right angle bracket]

The Kolmogorov length and intermittency

Singularities and the Euler equations

References and further reading

Exercises

Exponential decay of the Fourier power spectrum

Introduction

A differential inequality for [double vertical line]e[superscript [alpha]t[down triangle, open] down triangle, open]u[double vertical line superscript 2 subscript 2]

A bound on [double vertical line]e[superscript [alpha]t[down triangle, open] down triangle, open]u[double vertical line superscript 2 subscript 2]

Decay of the Fourier spectrum

References and further reading

Exercises

The attractor dimension for the Navier-Stokes equations

Introduction

The 2d attractor dimension estimate

The 3d attractor dimension estimate

References and further reading

Exercises

Energy dissipation rate estimates for boundary-driven flows

Introduction

Boundary-driven shear flow

Thermal convection in a horizontal plane

Discussion

References and further reading

Exercises

Inequalities

References

Index

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