Kuksin S.B. Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions

European Mathematical Society, 2006. — 103 p. — (Zurich Lectures in Advanced Mathematics 03). — ISBN 9783037190210.Случайно вынужденные нелинейные уравнения в частных производных и статистическая гидродинамика в двух измерениях пространстваThis book gives an account of recent achievements in the mathematical theory of two-dimensional turbulence, described by the 2D Navier-Stokes equation, perturbed by a random force. The main results presented here were obtained during the last five to ten years and, up to now, have been available only in papers in the primary literature. Their summary and synthesis here, beginning with some preliminaries on partial differential equations and stochastics, make this book a self-contained account that will appeal to readers with a general background in analysis. After laying the groundwork, the author goes on to recent results on ergodicity of random dynamical systems, which the randomly forced Navier-Stokes equation defines in the function space of divergence-free vector fields, including a Central Limit Theorem. The physical meaning of these results is discussed as well as their relations with the theory of attractors. Next, the author studies the behaviour of solutions when the viscosity goes to zero. In the final section these dynamical methods are used to derive the so-called balance relations--the infinitely many algebraical relations satisfied by the solutions. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.Function spaces
The deterministic 2D Navier-Stokes Equation
Randomkick-forces
White-forced equations
Preliminaries from measure theory
Uniqueness of a stationary measure: kick-forces
Uniqueness of a stationary measure: white-forces
Ergodicity and the strong law of large numbers
The martingale approximation and CLT
The Eulerian limit
Balance relations for the white-forced NSE
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