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
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.
General
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