Limit search to available items
Book Cover
E-book
Author Foiaş, Ciprian.

Title Navier-Stokes Equations and Turbulence
Published Cambridge Cambridge University Press, 2001
©2004
Online access available from:
ProQuest Ebook Central Subscription Collection    View Resource Record  

Copies

Description 1 online resource (363 pages)
Series Encyclopedia of Mathematics and its Applications ; v.83
Encyclopedia of mathematics and its applications.
Contents Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- Acknowledgments -- I Introduction and Overview of Turbulence -- Introduction -- 1 Viscous Fluids. The Navier-Stokes Equations -- Nondimensional Form of the Navier-Stokes Equations -- 2 Turbulence: Where the Interests of Engineers and Mathematicians Overlap -- 3 Elements of the Theories of Turbulence of Kolmogorov and Kraichnan -- 4 Function Spaces, Functional Inequalities, and Dimensional Analysis -- The Fundamental Function Spaces -- Functional Inequalities -- More Inequalities -- Lebesgue Spaces -- Higher-Order Sobolev Spaces -- Sobolev Embeddings and Inequalities -- Compact Mappings, the Rellich Lemma, and Compact Sobolev Embeddings -- II Elements of the Mathematical Theory of the Navier-Stokes Equations -- Introduction -- 1 Energy and Enstrophy -- 2 Boundary Value Problems -- No-Slip Boundary Condition -- Space-Periodic Case -- Channel Flows -- Initial Condition -- Simplified Problems -- A Boundary Value Problem for the Pressure -- An Evolution Equation for the Velocity Field u -- 3 Helmholtz-Leray Decomposition of Vector Fields -- The Evolution Equation for the Velocity Field -- 4 Weak Formulation of the Navier-Stokes Equations -- Energy Equation -- 5 Function Spaces -- No-Slip Boundary Conditions -- Periodic Boundary Conditions -- Periodic Boundary Conditions with Zero Space Average -- Fourier Characterization of the Function Spaces for Periodic Flows -- Space-Time Function Spaces -- 6 The Stokes Operator -- The Stokes Operator in the No-Slip Case -- The Stokes Operator in the Space-Periodic Case with Vanishing Space Average -- The Stokes Operator in the General Periodic Case -- Alternative (Abstract) Definition of the Stokes Operator -- Asymptotic Behavior of the Eigenvalues of the Stokes Operator -- Galerkin (Spectral) Projectors
1.3 Definition and Properties of Generalized Limits -- 2 Invariant Measures and Stationary Statistical Solutions in Dimension 2 -- 2.1 Invariant Measures and Stationary Statistical Solutions Generated by Time Averages -- 2.2 Regularity of the Support of an Invariant Measure -- 3 Stationary Statistical Solutions in Dimension 3 -- 3.1 Stationary Statistical Solutions Generated by Time Averages -- 3.2 Regularity of the Support of Time-Average Measures -- 4 Attractors and Stationary Statistical Solutions -- 4.1 The 2-Dimensional Case: The Support of an Invariant Measure Is Included in the Global Attractor -- 4.2 The 3-Dimensional Case: The Support of a Time-Average Measure Is Included in the Weak Global Attractor -- 5 Average Transfer of Energy and the Cascades in Turbulent Flows -- 5.1 Energy Transfer and the Cascade in 3-Dimensional Turbulence -- Average Net Transfer of Kinetic Energy -- Direct Energy Cascade -- 5.2 Enstrophy Transfer and the Cascade in 2-Dimensional Turbulence -- Average Net Transfer of Enstrophy -- Direct Enstrophy Cascade -- Inverse Energy Cascade -- 5.3 Kraichnan's Cascade Mechanism -- The Effects of Walls on Kraichnan's Dissipation Wavenumber -- Appendix A New Concepts and Results Used in Chapter IV -- A.1 Background on Measure Theory -- Theorems in Measure Theory -- Theorems in Topology -- Theorems in Functional Analysis -- A.2 Banach Generalized Limits -- Appendix B Proofs of Technical Results in Chapter IV -- B.1 Equivalence between Invariant Measures and Stationary Statistical Solutions in Dimension 2 (Proof of Theorem 2.2) -- B.2 Time-Average Measures Are Stationary Statistical Solutions in Dimension 3 (Proof of Theorem 3.1) -- B.3 Proof of Proposition 3.2 -- B.4 Average Transfer of Energy -- Appendix C A Mathematical Complement: The Accretivity Property in Dimension 3 -- C.1 The 2-Dimensional Case: A Stronger Result
7 Existence and Uniqueness of Solutions: The Main Results -- Existence and Uniqueness in Dimension 3 -- Existence and Uniqueness in Dimension 2 -- Further Properties of the Solutions in Dimension 3 -- 8 Analyticity in Time -- Time Analyticity in the 3-Dimensional Case -- Global Analyticity in the 2-Dimensional Case -- Improvements in the 2-Dimensional Periodic Case -- 9 Gevrey Class Regularity and the Decay of the Fourier Coefficients -- Gevrey Spaces -- Estimates for the Nonlinear Term in the Periodic Case -- Analyticity in the 3-Dimensional Periodic Case -- Analyticity in the 2-Dimensional Periodic Case -- Exponential Decrease of the Fourier Coefficients -- 10 Function Spaces for the Whole-Space Case -- 11 The No-Slip Case with Moving Boundaries -- 12 Dissipation Rate of Flows -- Bounds on the Energy Dissipation for a 3-Dimensional Shear Flow -- Bounds on the Energy Dissipation for Periodic Flows -- 13 Nondimensional Estimates and the Grashof Number -- The 2-Dimensional Case -- The 3-Dimensional Case -- Appendix A Mathematical Complements -- A.1 Function Spaces -- Dual Spaces - Riesz Representation Theorem -- The Pair V, H -- Weak Topology in Normed Spaces -- Space-Time Function Spaces -- A.2 Weak and Strong Solutions of the NSE in Dimension 3 -- The Energy Inequality -- The Abstract Functional Equation and Nonlinear Inequalities -- A Priori Estimates -- A.3 Weak and Strong Solutions of the NSE in Dimension 2 -- A Priori Estimates Using the Energy Equation -- A Priori Estimates Using the Enstrophy Equation -- Improvements in the 2-Dimensional Periodic Case -- Appendix B Proofs of Technical Results in Chapter II -- B.1 Energy Equation and A Priori Estimates -- Proof of the Integral Form (A.19) of the Energy Inequality -- Proof of the A Priori Estimate (A.39) -- Proof of the A Priori Estimate (A.43) -- B.2 Time Analyticity
B.3 Bilinear Estimates in Gevrey Spaces -- B.4 Time Analyticity in Gevrey Spaces -- Analyticity in the 3-Dimensional Case -- Analyticity in the 2-Dimensional Periodic Case -- III Finite Dimensionality of Flows -- Introduction -- Elements of the Mathematical Theory of the NSE -- 1 Determining Modes -- Determining Modes in the No-Slip Case -- Determining Modes in the Space-Periodic Case -- 2 Determining Nodes -- Determining Nodes in the No-Slip Case -- Determining Nodes in the Space-Periodic Case -- 3 Attractors and Their Fractal Dimension -- 3.1 The Global Attractor for the 2-Dimensional Navier-Stokes Equations -- Attractor Dimension -- Improvements in the 2-Dimensional Space-Periodic Case -- An Example of Trivial Attractors for Arbitrarily Large Grashof Numbers in the Space-Periodic Case -- 3.2 The 3-Dimensional Navier-Stokes Equations -- The Hausdorff and Fractal Dimensions of Invariant Sets Bounded in V -- The Weak Global Attractor in Dimension 3 -- 4 Approximate Inertial Manifolds -- Appendix A Proofs of Technical Results in Chapter III -- A.1 Proof of the Generalized Gronwall Lemma 1.1 -- A.2 Proof of Lemma 2.1 -- A.3 Estimates for the Dimension of the Global Attractor -- A.4 Proof of the Triviality of the Attractor with Force in the First Mode -- A.5 Attraction and Compactness of the 3-Dimensional Weak Global Attractor -- Weak Attraction of the Weak Global Attractor -- Weak Compactness of the Weak Global Attractor -- A.6 Error Bounds for the FMT Approximate Inertial Manifold -- IV Stationary Statistical Solutions of the Navier-Stokes Equations, Time Averages, and Attractors -- Introduction -- 1 Mathematical Framework, Definition of Stationary Statistical Solutions, and Banach Generalized Limits -- 1.1 Weak and Strong Solutions of the Navier-Stokes Equations -- 1.2 Definition of Stationary Statistical Solution
C.2 The 3-Dimensional Case: Time-Average Measures Are Accretive -- C.3 Proofs of Lemma C.1 and Lemma C.2 -- V Time-Dependent Statistical Solutions of the Navier-Stokes Equations and Fully Developed Turbulence -- Introduction -- Elements of the Mathematical Theory of the Navier-Stokes Equations -- 1 Time-Dependent Statistical Solutions on Bounded Domains -- 1.1 Statistical Solutions in the Case of No-Slip Boundary Conditions -- Motivation for the Definition of Statistical Solutions -- Definition of Statistical Solutions -- Existence and Uniqueness Results -- 1.2 Statistical Solutions in the Space-Periodic Case -- Existence and Uniqueness Results -- 1.3 Hopf Equation on Bounded Domains -- 2 Homogeneous Statistical Solutions -- 2.1 Homogeneous Measures -- Nontrivial Homogeneous Measures -- Properties of Homogeneous Measures -- 2.2 Homogeneous Statistical Solutions on the Whole Space: The Periodic Case -- 2.3 Homogeneous Statistical Solutions on the Whole Space: The General Case -- 2.4 Hopf Equation on the Whole Space -- 3 Reynolds Equation for the Average Flow -- 4 Self-Similar Homogeneous Statistical Solutions -- 4.1 Rescaling Properties of Homogeneous Statistical Solutions -- Rescaling in the Deterministic Case -- Rescaling in the Statistical Case -- 4.2 Self-Similar Homogeneous Statistical Solutions -- Characterization of Self-Similar Statistical Solutions -- Evolution of the Kinetic Energy and the Enstrophy for Self-Similar Statistical Solutions -- Connection with a Perturbed Form of the Navier-Stokes Equations -- Remarks on the Transformation of the Energy Equation -- 5 Relation with and Application to the Conventional Theory of Turbulence -- 5.1 The Two-Point Correlation Function and the Energy Spectrum -- The Two-Point Correlation Function in the Homogeneous Case -- The Reynolds Number
The Two-Point Correlation Function in the Homogeneous and Isotropic Case
Summary This book presents the mathematical theory of turbulence to engineers and physicists, and the physical theory of turbulence to mathematicians
Notes Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2016. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Description based on publisher supplied metadata and other sources
Subject Turbulence.
Form Electronic book
Author Doran, Bruce J.
Flajolet, Philippe.
Ismail, M.
Lam, T. Y. (Tsit-Yuen), 1942-
Manley, O
Rosa, Rudolph, 1823?-1901.
Rota, Gian-Carlo, 1932-1999.
Temam, Roger.
Wutwak, E
MyiLibrary.
ISBN 9780511154225