06-178 Sergei B. Kuksin

Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions (1221K, post-script) Jun 8, 06

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Abstract. The 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. Main results, presented here, were obtained during the last 5-10 years and can be found only in papers. The book starts with preliminaries on partial differential equations and on stochastic, which makes it self-contained and available for readers with general background in analysis. It goes on to recent results on ergodicity of the random dynamical system 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 behaviour of solutions when the viscosity goes to zero. The final section presents the balance relations - the infinitely-many explicit algebraical relations, satisfied by the solutions - and their discussion.

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