Applied Analysis of the Navier-Stokes Equations

Name: Applied Analysis of the Navier-Stokes Equations
Availability: InStock
ISBN: 9780521445689

ISBN-10: 052144568X

ISBN-13: 9780521445689

Edition: 1995

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

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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…

Book details

List price: $60.99
Copyright year: 1995
Publisher: Cambridge University Press
Publication date: 4/28/1995
Binding: Paperback
Pages: 232
Size: 5.94" wide x 8.94" long x 0.79" tall
Weight: 0.836
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