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Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions
Zurich Lectures in Advanced Mathematics

Sergei B. Kuksin (Heriot-Watt University, Edinburgh, UK)

Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions

ISBN print 978-3-03719-021-0, ISBN online 978-3-03719-521-5
DOI 10.4171/021
April 2006, 102 pages, softcover, 17.0 x 24.0 cm.
28.00 Euro

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. 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 the 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.

Further Information

Review in Bull. London Math. Soc. 39 (2007), 350–352

Review in Zentralblatt MATH 1099.35083

Review in MR 2225710 (2007h:60055)