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Applied Analysis of the Navier-Stokes Equations

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ISBN-10: 052144568X

ISBN-13: 9780521445689

Edition: 1995

Authors: Charles R. Doering, J. D. Gibbon, M. J. Ablowitz, S. H. Davis, E. J. Hinch

List price: $62.00
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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…    
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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

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