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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 central limit...

For the last 20-30 years, interest among mathematicians and physicists in infinite-dimensional Hamiltonian systems and Hamiltonian partial differential equations has been growing strongly, and many papers and a number of books have been written on integrable Hamiltonian PDEs. During the last decade though, the interest has shifted steadily towards non-integrable Hamiltonian PDEs. Here, not algebra but analysis and symplectic geometry are the appropriate analysing tools. The present book is the first one to use this approach to Hamiltonian PDEs and present a complete proof of the "KAM for...

Many problems of stability in the theory of dynamical systems face the difficulty of small divisors. This text illustrates developments of this theory both in finite and infinite dimension. A list of open problems (including some problems contributed by John Mather and Michel Herman) is included.