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.

The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initally smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently, if the vorticity remains bounded, a smooth solution persists.

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Abstract.In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation.Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artificial boundaries.These boundary conditions not only guarantee stable difference approximations but also minimize the (unphysical) artificial reflections which occur at the boundaries.

The formation of smng and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied here through the symbiotic interaction of mathematical theory and numerical experiments. This active scalar represents the temperature evolving on the two dimensional boundary of a rapidly rotating half space with small Rosshy and Ekman numbers and canstant potential vorticity. The possibility of hntogenesis within this approximation is an important issue in the context of geophysical Rows. A striking mathematical and physical analogy is developed here between the structure and formation of singular solutions of this quasigeostrophic active scalar in two dimensions k d the potential formation of finite time singular solutions for the 3-D Euler equations. Detailed mathematical criteria are developed as diagnostics for self-consistent numerical calculations indicating strong front formation. These self-consistent numerical calculations demonstrate the necessity of nontrivial topology involving hyperbolic saddle points in the level sets of the active scalar in order to have singular behaviour; this numerical evidence is strongly supported by mathematical theorems which utilize the nonlinear structure of specific singular integrals in special geometric eon6gmtions to demonstrate the important role of nontrivial topology in the formation of singular solutions.

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Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.