涡和不可压缩流

Vorticity and Incompressible Flow is an essential read for mathematicians, scientists, and engineers with an interest in the mathematical theory of incompressible flows. The book begins with an introduction to vortex dynamics in incompressible fluid flows, including the Euler and Navier-Stokes Equations, symmetry groups, particle trajectories, vorticity, and exact solutions. Chapter 2 delves into the vorticity-stream formulation of the equations, and Chapter 3 explores energy methods for these equations, including local-in-time existence of solutions and the existence of smooth solutions globally in time. Chapter 4 covers the particle-trajectory method and its application to the local-in-time existence of inviscid solutions, while Chapter 5 discusses the search for singular solutions to the 3D Euler Equations, including a simple 1D model and potential singularities. Computational vortex methods are explored in Chapter 6, including the random-vortex method and 2D/3D inviscid-vortex methods. Simplified asymptotic equations for slender vortex filaments are introduced in Chapter 7, and Chapter 8 discusses weak solutions to the 2D Euler Equations with initial vorticity in L. Vortex sheets, weak solutions, and approximate-solution sequences for the Euler Equation are covered in Chapter 9, while Chapter 10 explores weak solutions and solution sequences in two dimensions, including convergence results. Chapter 11 focuses on the 2D Euler Equation, discussing concentrations and weak solutions with vortex-sheet initial data, reduced defect measures, and vortexicity maximal function decay rates. Chapter 12 covers the reduced Hausdorff dimension, oscillations, and measure-valued solutions of the Euler Equations in two and three dimensions, including Young measures and measure-valued solutions with oscillations and concentrations. Finally, Chapter 13 discusses the Vlasov-Poisson Equations as an analogy to the Euler Equations for the study of weak solutions.Overall, Vorticity and Incompressible Flow provides a thorough and comprehensive overview of the mathematical theory of vorticity and incompressible flow, with a focus on applied mathematics and a symbiotic interaction between theory, modeling, and physical phenomena. It is an essential reference for researchers and students in the fields of mathematics and fluid dynamics.