Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows -- 1.1. The Euler and the Navier-Stokes Equations -- 1.2. Symmetry Groups for the Euler and the Navier-Stokes Equations -- 1.3. Particle Trajectories -- 1.4. The Vorticity, a Deformation Matrix, and Some Elementary Exact Solutions -- 1.5. Simple Exact Solutions with Convection, Vortex Stretching, and Diffusion -- 1.6. Some Remarkable Properties of the Vorticity in Ideal Fluid Flows -- 1.7. Conserved Quantities in Ideal and Viscous Fluid Flows -- 1.8. Leray's Formulation of Incompressible Flows and Hodge's Decomposition of Vector Fields -- 1.9. Appendix -- 1.9.1. Elementary Properties of the Fourier Series and the Fourier Transform -- Fourier Series -- Fourier Transform -- 1.9.2. Elementary Properties of the Poisson Equation -- 1.9.3. Elementary Properties of the Heat Equation -- Notes for Chapter 1 -- References for Chapter 1 -- 2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations -- 2.1. The Vorticity-Stream Formulation for 2D Flows -- 2.2. A General Method for Constructing Exact Steady Solutions to the 2D Euler Equations -- 2.2.1. Radial Eddies for the Euler and the Navier-Stokes Equations -- 2.2.2. The Vorticity-Stream Formulation for Periodic Flows and More Exact Solutions -- 2.3. Some Special 3D Flows with Nontrivial Vortex Dynamics -- 2.3.1. Two-and-a-Half-Dimensional Flows -- 2.3.2. 3D Beltrami Flows -- 2.3.3. 3D Axisymmetric Flows -- Axisymmetric Flows without Swirl -- Axisymmetric Flows with a General Swirl -- 2.4. The Vorticity-Stream Formulation for 3D Flows -- 2.4.1. Hodge Decomposition, Potential Theory, and the Biot-Savart Law -- 2.4.2. Distribution Derivatives of Functions Homogeneous of Degree 1 - N and Singular Integrals.

2.4.3. Computation of the Velocity Gradients from the Vorticity by Means of Singular Integral Operators -- 2.4.4. The Vorticity-Stream Formulation in All of R -- 2.5. Formulation of the Euler Equation as an Integrodifferential Equation for the Particle Trajectories -- Notes for Chapter 2 -- References for Chapter 2 -- 3 Energy Methods for the Euler and the Navier-Stokes Equations -- 3.1. Energy Methods: Elementary Concepts -- 3.1.1. A Basic Energy Estimate -- 3.1.2. Approximation of Inviscid Flows by High Reynold's Number Viscous Flows -- 3.1.3. Kinetic Energy for 2D Fluids -- 3.2. Local-in-Time Existence of Solutions by means of Energy Methods -- 3.2.1. Calculus Inequalities for Sobolev Spaces and Properties of Mollifiers -- 3.2.2. Global Existence of Solutions to a Regularization of the Euler and the Navier-Stokes Equations -- 3.2.3. Local-In-Time Existence of Solutions to the Euler and the Navier-Stokes Equations -- Case 1: ν = 0 -- Case 2: ν > 0 -- 3.2.4. Existence of 2D Solutions with Locally Finite-Energy Decomposition -- 3.3. Accumulation of Vorticity and the Existence of Smooth Solutions Globally in Time -- 3.4. Viscous-Splitting Algorithms for the Navier-Stokes Equation -- 3.4.1. Viscous-Splitting Algorithm -- 3.4.2. Convergence of the Viscous-Splitting Algorithm -- 3.5. Appendix for Chapter 3 -- 3.5.1. Calculus Inequalities in the Sobolev Spaces -- 3.5.2. Properties of Mollifiers -- Notes for Chapter 3 -- References for Chapter 3 -- 4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation -- 4.1. The Local-in-Time Existence of Inviscid Solutions -- 4.1.1. The Choice of the Banach Space B and Elementary Calculus Inequalities for Hölder Continuous Functions -- 4.1.2. The Choice of the Open Subset O C B -- 4.1.3. Local Existence and Uniqueness of Solutions X(alpha,t) and Potential Theory Estimates.

4.2. Link Between Global-in-Time Existence of Smooth Solutions and the Accumulation of Vorticity through Stretching -- 4.3. Global Existence of 3D Axisymmetric Flows without Swirl -- 4.4. Higher Regularity -- 4.5. Appendixes for Chapter 4 -- 4.5.1. Calculus Inequalities in the Hölder Spaces -- 4.5.2. Potential Theory Estimates in the Hölder Spaces -- Notes for Chapter 4 -- References for Chapter 4 -- 5 The Search for Singular Solutions to the 3-D Euler Equations -- 5.1. The Interplay between Mathematical Theory and Numerical Computations in the Search for Singular Solutions -- 5.1.1. The Search for Potential Singularities Through Direct Numerical Simulation -- 5.2. A Simple 1D Model for the 3D Vorticity Equation -- 5.2.1. Integration of the Model Vorticity Equation and Explicit Breakdown of Solutions -- 5.3. A 2D Model for Potential Singularity Formation in 3D Euler Equations -- 5.3.1. Analogies between the 2D QG Active Scalar and the 3D Euler Equations -- The Geometric Analogy of Level Sets with Vortex Lines -- The Analytic Analogy with Vortex Stretching -- The Search for Potential Singularities through Direct Numerical Simulation -- 5.3.2. Rigorous Analytic and Geometric Constraints on Singular Solutions -- 5.4. Potential Singularities in 3D Axisymmetric Flows with Swirl -- 5.4.1. The 2D Boussinesq Equations as a Model for 3D Axisymmetric Swirling Flows -- 5.5. Do The 3D Euler Solutions Become Singular in Finite Times? -- Notes for Chapter 5 -- References for Chapter 5 -- 6 Computational Vortex Methods -- 6.1. The Random-Vortex Method for Viscous Strained Shear Layers -- 6.1.1. The Discrete Problem with Only Diffusion and Advection -- Case 1. No Diffusion -- Case 2. No Advection -- 6.1.2. Analysis of the Schemes with Both Diffusion and Advection -- 6.1.3. Computational Performance of the 2D Random-Vortex Method on Viscous Strained Shear Layers.

6.2. 2D Inviscid Vortex Methods -- 6.2.1. Smoothing of the Kernel K -- 6.2.2. Approximation of the Integral -- 6.2.3. Closing of the Equation by means of a Finite-Dimensional System of ODEs -- 6.3. 3D Inviscid-Vortex Methods -- 6.3.1. Smoothing of the Kernel K -- 6.3.2. Approximation of the Integral -- 6.3.3. Two Approximations of the Vorticity -- Approximation 1: Lagrangian Stretching -- Approximation 2: Eulerian Stretching -- 6.3.4. Closing of the Equations by means of a Finite-Dimensional System of ODEs -- 6.4. Convergence of Inviscid-Vortex Methods -- 6.4.1. Proof of Convergence of the 2D Vortex Method -- 6.4.2. The Consistency Error -- 6.4.3. The Stability Error -- 6.5. Computational Performance of the 2D Inviscid-Vortex Method on a Simple Model Problem -- 6.5.1. Numerical Results -- 6.6. The Random-Vortex Method in Two Dimensions -- 6.6.1. Motivation for the Need for a Stochastic Particle Method: The Failure of Core Spreading -- 6.6.2. Implementation of the Random-Vortex Method -- 6.6.3. Formulation and Main Theorem -- 6.6.4. Preliminary Estimates -- 6.6.5. Consistency Error for a Fixed Time -- 6.6.6. Stability Estimate for All Time -- 6.6.7. Convergence -- 6.7. Appendix for Chapter 6 -- Notes for Chapter 6 -- References for Chapter 6 -- 7 Simplified Asymptotic Equations for Slender Vortex Filaments -- 7.1. The Self-Induction Approximation, Hasimoto's Transform, and the Nonlinear Schrödinger Equation -- 7.1.1. Hasimoto's Transform and the Cubic Nonlinear Schrödinger Equation -- 7.2. Simplified Asymptotic Equations with Self-Stretch for a Single Vortex Filament -- 7.2.1. Sketch of the Asymptotic Derivation -- 7.2.2. The Mathematical Structure of the Asymptotic Equation -- Explicit Fourier Representation of the Nonlocal Operators -- Spectrum of the Linearized Operator and Singular Perturbation.

7.2.3. Asymptotic Equations for the Stretching of Vortex Filaments in a Background Flow Field -- 7.2.4. Analytical Properties of the Filament Equation with a Background Flow -- The Nonlinear Stability of Vortex Filaments in a Background Flow Field -- Linearized Theory for Planar-Strain Flows -- 7.2.5. The Creation of Kinks, Folds, and Hairpins in Vortex Tubes by Means of the Filament Equations -- The Finite Effects of Epsilon in the Asymptotic Theories -- 7.3. Interacting Parallel Vortex Filaments - Point Vortices in the Plane -- Conserved Quantities -- Elementary Exact Solutions -- 7.4. Asymptotic Equations for the Interaction of Nearly Parallel Vortex Filaments -- 7.4.1. Hamiltonian Structure and Conserved Quantities -- Other Conserved Quantities -- The Average Distance Functional -- 7.4.2. The Equations for Pairs of Interacting Filaments -- 7.4.3. Corotating Filament Pairs -- 7.4.4. Linearized Stability for the Filament Pair -- 7.4.5. Finite-Time Collapse and Wavelike Behavior for Nearly Parallel Pairs of Vortex Filaments -- 7.4.6. Finite-Time Collapse for Negative-Circulation Ratios -- 7.4.7. Wavelike Behavior Without Collapse for Positive Circulation Ratios -- 7.5. Mathematical and Applied Mathematical Problems Regarding Asymptotic Vortex Filaments -- 7.5.1. Applied Mathematics Issues -- 7.5.2. Mathematical Issues Regarding Properties of the Asymptotic Equations -- 7.5.3. Mathematical Issues Regarding Justification of the Asymptotic Approximation -- Notes for Chapter 7 -- References for Chapter 7 -- 8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L -- 8.1. Elliptical Vorticies -- 8.2. Weak L Solutions to the Vorticity Equation -- 8.2.1. A Weak Vorticity-Stream Formulation -- 8.2.2. The Existence of Weak Solutions -- 8.2.3. Potential Theory Estimates -- 8.2.4. The Uniqueness of Weak Solutions.

8.2.5. Propagation of Regularity.