Vorticity and Incompressible Flow

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This comprehensive introduction to the mathematical theory of vorticity and incompressible flow begins with the elementary introductory material and leads into current research topics. While the book centers on mathematical theory, many parts also showcase the interaction among 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.

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Editorial Reviews

From the Publisher
"This book is destined to become a classic...This is the standard to which the rest of us need to aspire." Journal of Fluid Mechanics

"There are about 11 books currently available on the market covering incompressible flows. Majda and Bertozzi's book is unique in covering both the classical and weak solutions for the incompressible and inviscid flows and is excellently done." Mathematical Reviews

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Product Details

Table of Contents

1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows 1
1.1 The Euler and the Navier-Stokes Equations 2
1.2 Symmetry Groups for the Euler and the Navier-Stokes Equations 3
1.3 Particle Trajectories 4
1.4 The Vorticity, a Deformation Matrix, and Some Elementary Exact Solutions 6
1.5 Simple Exact Solutions with Convection, Vortex Stretching, and Diffusion 13
1.6 Some Remarkable Properties of the Vorticity in Ideal Fluid Flows 20
1.7 Conserved Quantities in Ideal and Viscous Fluid Flows 24
1.8 Leray's Formulation of Incompressible Flows and Hodge's Decomposition of Vector Fields 30
2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations 43
2.1 The Vorticity-Stream Formulation for 2D Flows 44
2.2 A General Method for Constructing Exact Steady Solutions to the 2D Euler Equations 46
2.3 Some Special 3D Flows with Nontrivial Vortex Dynamics 54
2.4 The Vorticity-Stream Formulation for 3D Flows 70
2.5 Formulation of the Euler Equation as an Integrodifferential Equation for the Particle Trajectories 81
3 Energy Methods for the Euler and the Navier-Stokes Equations 86
3.1 Energy Methods: Elementary Concepts 87
3.2 Local-in-Time Existence of Solutions by Means of Energy Methods 96
3.3 Accumulation of Vorticity and the Existence of Smooth Solutions Globally in Time 114
3.4 Viscous-Splitting Algorithms for the Navier-Stokes Equation 119
4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation 136
4.1 The Local-in-Time Existence of Inviscid Solutions 138
4.2 Link between Global-in-Time Existence of Smooth Solutions and the Accumulation of Vorticity through Stretching 146
4.3 Global Existence of 3D Axisymmetric Flows without Swirl 152
4.4 Higher Regularity 155
5 The Search for Singular Solutions to the 3D Euler Equations 168
5.1 The Interplay between Mathematical Theory and Numerical Computations in the Search for Singular Solutions 170
5.2 A Simple 1D Model for the 3D Vorticity Equation 173
5.3 A 2D Model for Potential Singularity Formation in 3D Euler Equations 177
5.4 Potential Singularities in 3D Axisymmetric Flows with Swirl 185
5.5 Do the 2D Euler Solutions Become Singular in Finite Times? 187
6 Computational Vortex Methods 190
6.1 The Random-Vortex Method for Viscous Strained Shear Layers 192
6.2 2D Inviscid Vortex Methods 208
6.3 3D Inviscid-Vortex Methods 211
6.4 Convergence of Inviscid-Vortex Methods 216
6.5 Computational Performance of the 2D Inviscid-Vortex Method on a Simple Model Problem 227
6.6 The Random-Vortex Method in Two Dimensions 232
7 Simplified Asymptotic Equations for Slender Vortex Filaments 256
7.1 The Self-Induction Approximation, Hasimoto's Transform, and the Nonlinear Schrodinger Equation 257
7.2 Simplified Asymptotic Equations with Self-Stretch for a Single Vortex Filament 262
7.3 Interacting Parallel Vortex Filaments - Point Vortices in the Plane 278
7.4 Asymptotic Equations for the Interaction of Nearly Parallel Vortex Filaments 281
7.5 Mathematical and Applied Mathematical Problems Regarding Asymptotic Vortex Filaments 300
8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L[superscript [infinity]] 303
8.1 Elliptical Vorticies 304
8.2 Weak L[superscript [infinity]] Solutions to the Vorticity Equation 309
8.3 Vortex Patches 329
9 Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation 359
9.1 Weak Formulation of the Euler Equation in Primitive-Variable Form 361
9.2 Classical Vortex Sheets and the Birkhoff-Rott Equation 363
9.3 The Kelvin-Helmholtz Instability 367
9.4 Computing Vortex Sheets 370
9.5 The Development of Oscillations and Concentrations 375
10 Weak Solutions and Solution Sequences in Two Dimensions 383
10.1 Approximate-Solution Sequences for the Euler and the Navier-Stokes Equations 385
10.2 Convergence Results for 2D Sequences with L[superscript l] and L[superscript p] Vorticity Control 396
11 The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data 405
11.1 Weak- and Reduced Defect Measures 409
11.2 Examples with Concentration 411
11.3 The Vorticity Maximal Function: Decay Rates and Strong Convergence 421
11.4 Existence of Weak Solutions with Vortex-Sheet Initial Data of Distinguished Sign 432
12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions 450
12.1 The Reduced Hausdorff Dimension 452
12.2 Oscillations for Approximate-Solution Sequences without L[superscript l] Vorticity Control 472
12.3 Young Measures and Measure-Valued Solutions of the Euler Equations 479
12.4 Measure-Valued Solutions with Oscillations and Concentrations 492
13 The Vlasov-Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions 498
13.1 The Analogy between the 2D Euler Equations and the 1D Vlasov-Poisson Equations 502
13.2 The Single-Component 1D Vlasov-Poisson Equation 511
13.3 The Two-Component 1D Vlasov-Poisson System 524
Index 543
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