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Applied Analysis of the Navier-Stokes Equations : Cambridge Texts in Applied Mathematics - Charles R. Doering

Applied Analysis of the Navier-Stokes Equations

Cambridge Texts in Applied Mathematics


Published: 28th August 1995
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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 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.

'The book is written for anyone who strives to understand the inherent difficulties of the nonlinear Navier-Strokes equations ... Hopefully, the book will see many editions.' P. Kahlig, Meteorology and Atmospheric Physics ' ... the authors have done a thoughtful job of expounding ideas that are not close to the hearts of many mathematicians yet are capable of providing new basic and computational understandings.' J. R. Ockendon, Journal of Fluid Mechanics 'The clear structuring of the scientific content is to be appreciated ... The exercises at the end of each chapter are well selected ... Hopefully the book will see many editions.' P. Kahlig, Meteorology and Atmospheric Physics

Prefacep. xi
The equations of motionp. 1
Introductionp. 1
Euler's equations for an incompressible fluidp. 1
Energy, body forces, vorticity, and enstrophyp. 7
Viscosity, the stress tensor, and the Navier-Stokes equationsp. 12
Thermal convection and the Boussinesq equationsp. 18
References and further readingp. 21
Exercisesp. 22
Dimensionless parameters and stabilityp. 23
Dimensionless parametersp. 23
Linear and nonlinear stability, differential inequalitiesp. 29
References and further readingp. 38
Exercisesp. 38
Turbulencep. 40
Introductionp. 40
Statistical turbulence theory and the closure problemp. 40
Spectra, Kolmogorov's scaling theory, and turbulent length scalesp. 49
References and further readingp. 59
Exercisesp. 60
Degrees of freedom, dynamical systems, and attractorsp. 61
Introductionp. 61
Dynamical systems, attractors, and their dimensionp. 62
The Lorenz systemp. 74
References and further readingp. 86
Exercisesp. 87
On the existence, uniqueness, and regularity of solutionsp. 88
Introductionp. 88
Existence and uniqueness for ODEsp. 89
Galerkin approximations and weak solutions of the Navier-Stokes equationsp. 96
Uniqueness and the regularity problemp. 104
References and further readingp. 113
Exercisesp. 113
Ladder results for the Navier-Stokes equationsp. 114
Introductionp. 114
The Navier-Stokes ladder theoremp. 117
A natural definition of a length scalep. 125
The dynamical wavenumbers k[subscript N,r]p. 127
Estimates for the Navier-Stokes equationsp. 128
Estimates for F[subscript 0]p. 129
Estimates for [left angle bracket]F[subscript 1 right angle bracket] and [left angle bracket]k[superscript 2 subscript 1,0 right angle bracket]p. 130
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]p. 131
A ladder for the thermal convection equationsp. 132
References and further readingp. 134
Exercisesp. 134
Regularity and length scales for the 2d and 3d Navier-Stokes equationsp. 137
Introductionp. 137
A global attractor and length scales in the 2d casep. 138
A global attractorp. 139
Length scales in the 2d Navier-Stokes equationsp. 139
3d Navier-Stokes regularity?p. 144
Problems with 3d Navier-Stokes regularityp. 144
A Bound on [left angle bracket]k[subscript N,1 right angle bracket] in 3dp. 146
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]p. 148
The Kolmogorov length and intermittencyp. 149
Singularities and the Euler equationsp. 152
References and further readingp. 155
Exercisesp. 155
Exponential decay of the Fourier power spectrump. 157
Introductionp. 157
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]p. 157
A bound on [double vertical line]e[superscript [alpha]t [down triangle, open] down triangle, open]u[double vertical line superscript 2 subscript 2]p. 163
Decay of the Fourier spectrump. 165
References and further readingp. 167
Exercisesp. 167
The attractor dimension for the Navier-Stokes equationsp. 169
Introductionp. 169
The 2d attractor dimension estimatep. 170
The 3d attractor dimension estimatep. 177
References and further readingp. 179
Exercisesp. 180
Energy dissipation rate estimates for boundary-driven flowsp. 181
Introductionp. 181
Boundary-driven shear flowp. 182
Thermal convection in a horizontal planep. 192
Discussionp. 197
References and further readingp. 203
Exercisesp. 204
Inequalitiesp. 205
Referencesp. 209
Indexp. 213
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521445689
ISBN-10: 052144568X
Series: Cambridge Texts in Applied Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 232
Published: 28th August 1995
Country of Publication: GB
Dimensions (cm): 23.5 x 15.88  x 1.27
Weight (kg): 0.32