Vorticity and Incompressible Flow

Vorticity and Incompressible Flow

by Andrew J. Majda, Andrea L. Bertozzi

ISBN-10: 0521630576

ISBN-13: 9780521630573

Pub. Date: 11/26/2001

Publisher: Cambridge University Press

This book introduces the mathematical theory of vorticity and incompressible flow, ranging from elementary material to current research topics. Although the contents center on mathematical theory, many parts of the book highlight interactions among rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. Early


This book introduces the mathematical theory of vorticity and incompressible flow, ranging from elementary material to current research topics. Although the contents center on mathematical theory, many parts of the book highlight interactions among rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. Early chapters can be used for an introductory graduate course on vorticity and incompressible flow. Later chapters comprise a modern applied mathematics graduate course on the weak solution theory for incompressible flow. Majda teaches at the Courant Institute of Mathematical Sciences of New York University. Bertozzi teaches mathematics and physics at Duke University. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Product Details

Cambridge University Press
Publication date:
Cambridge Texts in Applied Mathematics Series, #27
Product dimensions:
5.98(w) x 8.98(h) x 1.18(d)

Table of Contents

1An Introduction to Vortex Dynamics for Incompressible Fluid Flows1
1.1The Euler and the Navier-Stokes Equations2
1.2Symmetry Groups for the Euler and the Navier-Stokes Equations3
1.3Particle Trajectories4
1.4The Vorticity, a Deformation Matrix, and Some Elementary Exact Solutions6
1.5Simple Exact Solutions with Convection, Vortex Stretching, and Diffusion13
1.6Some Remarkable Properties of the Vorticity in Ideal Fluid Flows20
1.7Conserved Quantities in Ideal and Viscous Fluid Flows24
1.8Leray's Formulation of Incompressible Flows and Hodge's Decomposition of Vector Fields30
Notes for Chapter 141
References for Chapter 142
2The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations43
2.1The Vorticity-Stream Formulation for 2D Flows44
2.2A General Method for Constructing Exact Steady Solutions to the 2D Euler Equations46
2.3Some Special 3D Flows with Nontrivial Vortex Dynamics54
2.4The Vorticity-Stream Formulation for 3D Flows70
2.5Formulation of the Euler Equation as an Integrodifferential Equation for the Particle Trajectories81
Notes for Chapter 284
References for Chapter 284
3Energy Methods for the Euler and the Navier-Stokes Equations86
3.1Energy Methods: Elementary Concepts87
3.2Local-in-Time Existence of Solutions by Means of Energy Methods96
3.3Accumulation of Vorticity and the Existence of Smooth Solutions Globally in Time114
3.4Viscous-Splitting Algorithms for the Navier-Stokes Equation119
3.5Appendix for Chapter 3129
Notes for Chapter 3133
References for Chapter 3134
4The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation136
4.1The Local-in-Time Existence of Inviscid Solutions138
4.2Link between Global-in-Time Existence of Smooth Solutions and the Accumulation of Vorticity through Stretching146
4.3Global Existence of 3D Axisymmetric Flows without Swirl152
4.4Higher Regularity155
4.5Appendixes for Chapter 4158
Notes for Chapter 4166
References for Chapter 4167
5The Search for Singular Solutions to the 3D Euler Equations168
5.1The Interplay between Mathematical Theory and Numerical Computations in the Search for Singular Solutions170
5.2A Simple 1D Model for the 3D Vorticity Equation173
5.3A 2D Model for Potential Singularity Formation in 3D Euler Equations177
5.4Potential Singularities in 3D Axisymmetric Flows with Swirl185
5.5Do the 3D Euler Solutions Become Singular in Finite Times?187
Notes for Chapter 5188
References for Chapter 5188
6Computational Vortex Methods190
6.1The Random-Vortex Method for Viscous Strained Shear Layers192
6.22D Inviscid Vortex Methods208
6.33D Inviscid-Vortex Methods211
6.4Convergence of Inviscid-Vortex Methods216
6.5Computational Performance of the 2D Inviscid-Vortex Method on a Simple Model Problem227
6.6The Random-Vortex Method in Two Dimensions232
6.7Appendix for Chapter 6247
Notes for Chapter 6253
References for Chapter 6254
7Simplified Asymptotic Equations for Slender Vortex Filaments256
7.1The Self-Induction Approximation, Hasimoto's Transform, and the Nonlinear Schrodinger Equation257
7.2Simplified Asymptotic Equations with Self-Stretch for a Single Vortex Filament262
7.3Interacting Parallel Vortex Filaments--Point Vortices in the Plane278
7.4Asymptotic Equations for the Interaction of Nearly Parallel Vortex Filaments281
7.5Mathematical and Applied Mathematical Problems Regarding Asymptotic Vortex Filaments300
Notes for Chapter 7301
References for Chapter 7301
8Weak Solutions to the 2D Euler Equations with Initial Vorticity in L[superscript [infinity]303
8.1Elliptical Vorticies304
8.2Weak L[superscript [infinity] Solutions to the Vorticity Equation309
8.3Vortex Patches329
8.4Appendix for Chapter 8354
Notes for Chapter 8356
References for Chapter 8356
9Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation359
9.1Weak Formulation of the Euler Equation in Primitive-Variable Form361
9.2Classical Vortex Sheets and the Birkhoff-Rott Equation363
9.3The Kelvin-Helmholtz Instability367
9.4Computing Vortex Sheets370
9.5The Development of Oscillations and Concentrations375
Notes for Chapter 9380
References for Chapter 9380
10Weak Solutions and Solution Sequences in Two Dimensions383
10.1Approximate-Solution Sequences for the Euler and the Navier-Stokes Equations385
10.2Convergence Results for 2D Sequences with L[superscript 1] and L[superscript p] Vorticity Control396
Notes for Chapter 10403
References for Chapter 10403
11The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data405
11.1Weak- and Reduced Defect Measures409
11.2Examples with Concentration411
11.3The Vorticity Maximal Function: Decay Rates and Strong Convergence421
11.4Existence of Weak Solutions with Vortex-Sheet Initial Data of Distinguished Sign432
Notes for Chapter 11448
References for Chapter 11448
12Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions450
12.1The Reduced Hausdorff Dimension452
12.2Oscillations for Approximate-Solution Sequences without L[superscript 1] Vorticity Control472
12.3Young Measures and Measure-Valued Solutions of the Euler Equations479
12.4Measure-Valued Solutions with Oscillations and Concentrations492
Notes for Chapter 12496
References for Chapter 12496
13The Vlasov-Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions498
13.1The Analogy between the 2D Euler Equations and the 1D Vlasov-Poisson Equations502
13.2The Single-Component 1D Vlasov-Poisson Equation511
13.3The Two-Component Vlasov-Poisson System524
Note for Chapter 13541
References for Chapter 13541

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