Vorticity and incompressible flow / Andrew J. Majda, Andrea L. Bertozzi

The book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flows, ranging from elementary introductory material to current research topics. In the first two chapters the Euler equations (for ideal fluids) and the Navier-Stokes equations (for viscous fluids) are formulated, and many their properties are described, including exact solutions with shear, vorticity, convection and diffusion. The vorticity equation is derived to show that, for inviscid flows, the vorticity is transported and stretched along particle trajectories for three-dimensional flows and is conserved along particle paths for two-dimensional flows. The vorticity equation allows for different reformulations of Euler and Navier-Stokes equations, including vorticity-stream formulation for two-dimensional flows and an integro-differential equation for particle trajectories.

In the next two chapters the existence, uniqueness, and continuation of local-in-time smooth solutions of Euler and Navier-Stokes equations are addressed. The energy method is shown to be general enough to be applied to both equations; the particle-trajectory method is performed for the Euler equations. ... (Zentralblatt)

Bibliogr. en fin de chapitres. Index

... This long-awaited book is a "comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. Although the contents center on mathematical theory, many parts of the book showcase the interactions among rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena'' (from the preface).
The first half of the book can be used for an introductory graduate course on vorticity and incompressible flow. It contains many excellent exact and explicit solutions. It sets up the equations in various modern forms for further research. In particular, the approach used in Chapter 5 to search for singular solutions is novel. The work in Chapter 7 on interacting asymptotic filaments is also quite novel in many respects.
The second half can be used for a graduate course on the theory of weak solutions for incompressible flow with an emphasis on modern applied mathematics. There are six chapters covering the following topics. The first is the elegant vortex patch theory in which Chemin's theorem (Theorems 8.7–8.8) is presented with a proof given by Bertozzi and Constantin. The next chapter covers the subtle theoretical and computational issues of vortex sheets, in particular Delort's theorem. The third chapter is on the concentration-cancellation phenomena in sequences of weak solutions. The fourth deals with the concept of measure-valued solutions. This chapter (Chapter 12) presents many open problems. The final chapter is on one-dimensional Vlasov-Poisson equations which serve as a simplified model for the two-dimensional Euler equations.
A key feature of the book is the natural and seamless incorporation of rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. This symbiotic interaction is apparent in 7 chapters out of 13.
This book is an outgrowth of several lecture courses by Majda, enriched by the authors' own research in the field and that of Majda's many students, collaborators and other researchers. Various students' lecture notes were in circulation and popular.
The chapters of the book are often supplemented with an appendix, as well as notes and references. The appendices are especially helpful for students. Interspersed reviews of topics such as singular integral operators are selected just right in presenting a complete and understandable exposition of the analysis. The book goes into deep mathematics without sacrificing much in the relative ease of understanding. ... (MathSciNet)