Vorticity and Incompressible Flow

, by Andrew J. Majda , Andrea L. Bertozzi

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ISBN: 9780521630573 | 0521630576
Cover: Hardcover
Copyright: 11/26/2001

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This book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. While the contents center on mathematical theory, many parts of the book showcase the interaction between rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. The first half forms an introductory graduate course on vorticity and incompressible flow. The second half comprises a modern applied mathematics graduate course on the weak solution theory for incompressible flow.

Preface

An Introduction to Vortex Dynamics for Incompressible Fluid Flows

(42)

The Euler and the Navier-Stokes Equations

(1)

Symmetry Groups for the Euler and the Navier-Stokes Equations

(1)

Particle Trajectories

(2)

The Vorticity, a Deformation Matrix, and Some Elementary Exact Solutions

(7)

Simple Exact Solutions with Convection, Vortex Stretching, and Diffusion

(7)

Some Remarkable Properties of the Vorticity in Ideal Fluid Flows

(4)

Conserved Quantities in Ideal and Viscous Fluid Flows

(6)

Leray's Formulation of Incompressible Flows and Hodge's Decomposition of Vector Fields

(5)

Appendix

(8)

Notes for Chapter 1

(1)

References for Chapter 1

(1)

The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations

(43)

The Vorticity-Stream Formulation for 2D Flows

(2)

A General Method for Constructing Exact Steady Solutions to the 2D Euler Equations

(8)

Some Special 3D Flows with Nontrivial Vortex Dynamics

(16)

The Vorticity-Stream Formulation for 3D Flows

(11)

Formulation of the Euler Equation as an Integrodifferential Equation for the Particle Trajectories

(5)

Notes for Chapter 2

(1)

References for Chapter 2

(2)

Energy Methods for the Euler and the Navier-Stokes Equations

(50)

Energy Methods: Elementary Concepts

(9)

Local-in-Time Existence of Solutions by Means of Energy Methods

(18)

Accumulation of Vorticity and the Existence of Smooth Solutions Globally in Time

114

(5)

Viscous-Splitting Algorithms for the Navier-Stokes Equation

119

(10)

Appendix for Chapter 3

129

(7)

Notes for Chapter 3

133

(1)

References for Chapter 3

134

(2)

The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation

136

(32)

The Local-in-Time Existence of Inviscid Solutions

138

(8)

Link between Global-in-Time Existence of Smooth Solutions and the Accumulation of Vorticity through Stretching

146

(6)

Global Existence of 3D Axisymmetric Flows without Swirl

152

(3)

Higher Regularity

155

(3)

Appendixes for Chapter 4

158

(10)

Notes for Chapter 4

166

(1)

References for Chapter 4

167

(1)

The Search for Singular Solutions to the 3D Euler Equations

168

(22)

The Interplay between Mathematical Theory and Numerical Computations in the Search for Singular Solutions

170

(3)

A Simple 1D Model for the 3D Vorticity Equation

173

(4)

A 2D Model for Potential Singularity Formation in 3D Euler Equations

177

(8)

Potential Singularities in 3D Axisymmetric Flows with Swirl

185

(2)

Do the 3D Euler Solutions Become Singular in Finite Times?

187

(3)

Notes for Chapter 5

188

(1)

References for Chapter 5

188

(2)

Computational Vortex Methods

190

(66)

The Random-Vortex Method for Viscous Strained Shear Layers

192

(16)

2D Inviscid Vortex Methods

208

(3)

3D Inviscid-Vortex Methods

211

(5)

Convergence of Inviscid-Vortex Methods

216

(11)

Computational Performance of the 2D Inviscid-Vortex Method on a Simple Model Problem

227

(5)

The Random-Vortex Method in Two Dimensions

232

(15)

Appendix for Chapter 6

247

(9)

Notes for Chapter 6

253

(1)

References for Chapter 6

254

(2)

Simplified Asymptotic Equations for Slender Vortex Filaments

256

(47)

The Self-Induction Approximation, Hasimoto's Transform, and the Nonlinear Schrodinger Equation

257

(5)

Simplified Asymptotic Equations with Self-Stretch for a Single Vortex Filament

262

(16)

Interacting Parallel Vortex Filaments-Point Vortices in the Plane

278

(3)

Asymptotic Equations for the Interaction of Nearly Parallel Vortex Filaments

281

(19)

Mathematical and Applied Mathematical Problems Regarding Asymptotic Vortex Filaments

300

(3)

Notes for Chapter 7

301

(1)

References for Chapter 7

301

(2)

Weak Solutions to the 2D Euler Equations with Initial Vorticity in L∞

303

(56)

Elliptical Vorticies

304

(5)

Weak L∞ Solutions to the Vorticity Equation

309

(20)

Vortex Patches

329

(25)

Appendix for Chapter 8

354

(5)

Notes for Chapter 8

356

(1)

References for Chapter 8

356

(3)

Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation

359

(24)

Weak Formulation of the Euler Equation in Primitive-Variable Form

361

(2)

Classical Vortex Sheets and the Birkhoff-Rott Equation

363

(4)

The Kelvin-Helmholtz Instability

367

(3)

Computing Vortex Sheets

370

(5)

The Development of Oscillations and Concentrations

375

(8)

Notes for Chapter 9

380

(1)

References for Chapter 9

380

(3)

Weak Solutions and Solution Sequences in Two Dimensions

383

(22)

Approximate-Solution Sequences for the Euler and the Navier-Stokes Equations

385

(11)

Convergence Results for 2D Sequences with L1 and Lp Vorticity Control

396

(9)

Notes for Chapter 10

403

(1)

References for Chapter 10

403

(2)

The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data

405

(45)

Weak-* and Reduced Defect Measures

409

(2)

Examples with Concentration

411

(10)

The Vorticity Maximal Function: Decay Rates and Strong Convergence

421

(11)

Existence of Weak Solutions with Vortex-Sheet Initial Data of Distinguished Sign

432

(18)

Notes for Chapter 11

448

(1)

References for Chapter 11

448

(2)

Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions

450

(48)

The Reduced Hausdorff Dimension

452

(20)

Oscillations for Approximate-Solution Sequences without L1 Vorticity Control

472

(7)

Young Measures and Measure-Valued Solutions of the Euler Equations

479

(13)

Measure-Valued Solutions with Oscillations and Concentrations

492

(6)

Notes for Chapter 12

496

(1)

References for Chapter 12

496

(2)

The Vlasov-Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions

498

(45)

The Analogy between the 2D Euler Equations and the 1D Vlasov-Poisson Equations

502

(9)

The Single-Component 1D Vlasov-Poisson Equation

511

(13)

The Two-Component Vlasov-Poisson System

524

(19)

Note for Chapter 13

541

(1)

References for Chapter 13

541

(2)

Index

543

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Vorticity and Incompressible Flow

Summary

Table of Contents

Supplemental Materials