Mathematics of Two-Dimensional Turbulence

This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier-Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data.
They use this to study ergodic properties of u(t,x)- proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and the central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures.
The methods apply to other nonlinear PDEs perturbed by random forces.

This series is devoted to thorough, yet reasonably concise, treatments of topics in any branch of mathematics. Typically, a Tract takes up a single thread in a wide subject, and follows its ramifications, thus throwing light on its various aspects. Tracts are expected to be rigorous, definitive and of lasting value to mathematicians working in the relevant disciplines. Exercises can be included to illustrate techniques, summarise past work and enhance the book's value as a seminar text.
All volumes are properly edited and typeset, and are published, initially at least, in hardback.