Vorticity and Incompressible Flow

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.

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; 1. An introduction to vortex dynamics for incompressible fluid flows; 2. The vorticity-stream formulation of the Euler and the Navier-Stokes equations; 3. Energy methods for the Euler and the Navier-Stokes equations; 4. The particle-trajectory method for existence and uniqueness of solutions to the Euler equation; 5. The search for singular solutions to the 3D Euler equations; 6. Computational vortex methods; 7. Simplified asympototic equations for slender vortex filaments; 8. Weak solutions to the 2D Euler equations with initial vorticity in L8; 9. Introduction to vortex sheets, weak solutions and approximate-solution sequences for the Euler equation; 10. Weak solutions and solution sequences in two dimensions; 11. The 2D Euler equation: concentrations and weak solutions with vortex-sheet initial data; 12. Reduced Hausdorff dimension, oscillations and measure-valued solutions of the Euler equations in two and three dimensions; 13. The Vlasov-Poisson equations as an analogy to the Euler equations for the study of weak solutions; Index.

This book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. 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.