Vorticity and Incompressible FlowThis 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 comprise a modern applied mathematics graduate course on the weak solution theory for incompressible flow. |
Contents
An Introduction to Vortex Dynamics for Incompressible Fluid Flows | 1 |
11 The Euler and the NavierStokes Equations | 2 |
12 Symmetry Groups for the Euler and the NavierStokes Equations | 3 |
13 Particle Trajectories | 4 |
14 The Vorticity a Deformation Matrix and Some Elementary Exact Solutions | 6 |
15 Simple Exact Solutions with Convection Vortex Stretching and Diffusion | 13 |
16 Some Remarkable Properties of the Vorticity in Ideal Fluid Flows | 20 |
17 Conserved Quantities in Ideal and Viscous Fluid Flows | 24 |
66 The RandomVortex Method in Two Dimensions | 232 |
67 Appendix for Chapter 6 | 247 |
Notes for Chapter 6 | 253 |
References for Chapter 6 | 254 |
Simplified Asymptotic Equations for Slender Vortex Filaments | 256 |
71 The SelfInduction Approximation Hasimotos Transform and the Nonlinear Schrödinger Equation | 257 |
72 Simplified Asymptotic Equations with SelfStretch for a Single Vortex Filament | 262 |
73 Interacting Parallel Vortex Filaments Point Vortices in the Plane | 278 |
18 Lerays Formulation of Incompressible Flows and Hodges Decomposition of Vector Fields | 30 |
19 Appendix | 35 |
Notes for Chapter 1 | 41 |
References for Chapter 1 | 42 |
The VorticityStream Formulation of the Euler and the NavierStokes Equations | 43 |
21 The VorticityStream Formulation for 2D Flows | 44 |
22 A General Method for Constructing Exact Steady Solutions to the 2D Euler Equations | 46 |
23 Some Special 3D Flows with Nontrivial Vortex Dynamics | 54 |
24 The VorticityStream Formulation for 3D Flows | 70 |
25 Formulation of the Euler Equation as an Integrodifferential Equation for the Particle Trajectories | 81 |
Notes for Chapter 2 | 84 |
Energy Methods for the Euler and the NavierStokes Equations | 86 |
Elementary Concepts | 87 |
32 LocalinTime Existence of Solutions by means of Energy Methods | 96 |
33 Accumulation of Vorticity and the Existence of Smooth Solutions Globally in Time | 114 |
34 ViscousSplitting Algorithms for the NavierStokes Equation | 119 |
35 Appendix for Chapter 3 | 129 |
Notes for Chapter 3 | 133 |
References for Chapter 3 | 134 |
The ParticleTrajectory Method for Existence and Uniqueness of Solutions to the Euler Equation | 136 |
41 The LocalinTime Existence of Inviscid Solutions | 138 |
42 Link Between GlobalinTime Existence of Smooth Solutions and the Accumulation of Vorticity through Stretching | 146 |
43 Global Existence of 3D Axisymmetric Flows without Swirl | 152 |
44 Higher Regularity | 155 |
45 Appendixes for Chapter 4 | 158 |
Notes for Chapter 4 | 166 |
References for Chapter 4 | 167 |
The Search for Singular Solutions to the 3D Euler Equations | 168 |
51 The Interplay between Mathematical Theory and Numerical Computations in the Search for Singular Solutions | 170 |
52 A Simple ID Model for the 3D Vorticity Equation | 173 |
53 A 2D Model for Potential Singularity Formation in 3D Euler Equations | 177 |
54 Potential Singularities in 3D Axisymmetric Flows with Swirl | 185 |
55 Do the 3D Euler Solutions Become Singular in Finite Times? | 187 |
Notes for Chapter 5 | 188 |
Computational Vortex Methods | 190 |
61 The Random Vortex Method for Viscous Strained Shear Layers | 192 |
62 2D Inviscid Vortex Methods | 208 |
63 3D InviscidVortex Methods | 211 |
64 Convergence of InviscidVortex Methods | 216 |
65 Computational Performance of the 2D InviscidVortex Method on a Simple Model Problem | 227 |
74 Asymptotic Equations for the Interaction of Nearly Parallel Vortex Filaments | 281 |
75 Mathematical and Applied Mathematical Problems Regarding Asymptotic Vortex Filaments | 300 |
Notes for Chapter 7 | 301 |
Weak Solutions to the 2D Euler Equations with Initial Vorticity in L | 303 |
81 Elliptical Vorticies | 304 |
82 Weak L Solutions to the Vorticity Equation | 309 |
83 Vortex Patches | 329 |
84 Appendix for Chapter 8 | 354 |
Notes for Chapter 8 | 356 |
Introduction to Vortex Sheets Weak Solutions and ApproximateSolution Sequences for the Euler Equation | 359 |
91 Weak Formulation of the Euler Equation in Primitive Variable Form | 361 |
92 Classical Vortex Sheets and the BirkhoffRott Equation | 363 |
93 The KelvinHelmholtz Instability | 367 |
94 Computing Vortex Sheets | 370 |
95 The Development of Oscillations and Concentrations | 375 |
Notes for Chapter 9 | 380 |
Weak Solutions and Solution Sequences in Two Dimensions | 383 |
101 ApproximateSolution Sequences for the Euler and the NavierStokes Equations | 385 |
102 Convergence Results for 2D Sequences with L¹ and Lp Vorticity Control | 396 |
Notes for Chapter 10 | 403 |
The 2D Euler Equation Concentrations and Weak Solutions with VortexSheet Initial Data | 405 |
111 Weak and Reduced Defect Measures | 409 |
112 Examples with Concentration | 411 |
Decay Rates and Strong Convergence | 421 |
114 Existence of Weak Solutions with VortexSheet Initial Data of Distinguished Sign | 432 |
Notes for Chapter 11 | 448 |
Reduced Hausdorff Dimension Oscillations and MeasureValued Solutions of the Euler Equations in Two and Three Dimensions | 450 |
121 The Reduced Hausdorff Dimension | 452 |
122 Oscillations for ApproximateSolution Sequences without L¹ Vorticity Control | 472 |
123 Young Measures and MeasureValued Solutions of the Euler Equations | 479 |
124 MeasureValued Solutions with Oscillations and Concentrations | 492 |
| 496 | |
The VlasovPoisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions | 498 |
131 The Analogy between the 2D Euler Equations and the ID VlasovPoisson Equations | 502 |
132 The SingleComponent ID VlasovPoisson Equation | 511 |
133 The TwoComponent VlasovPoisson System | 524 |
Note for Chapter 13 | 541 |
| 543 | |
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Common terms and phrases
2D Euler equation algorithm approximate approximate-solution sequence asymptotic axis axisymmetric Banach space Biot-Savart law bounded chapter compute concentration condition conserved convergence curl defect measure defined denotes derivative dimensions discretization error dxdt energy error exact solutions example exists finite flows fluid formula global heat equation Hence Hölder Hölder continuous implies incompressible initial vorticity integral inviscid kernel Lemma limit linear Lipschitz continuous Majda Math mathematical matrix measure-valued solution Navier-Stokes equation nonlinear norm numerical particle trajectories perturbations Picard theorem potential theory Proof of Proposition prove regularization result satisfies scalar Section singularity smooth solutions Sobolev spaces stream function Subsection symmetric t)dy unique V-P system vector velocity field viscous vortex filaments vortex methods vortex patch vortex sheet vortex-sheet initial data vorticity control vorticity equation weak solution Young measure