# Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions

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*Sergei B. Kuksin*

A publication of the European Mathematical Society

This 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.

#### Readership

Graduate students and research mathematicians interested in mathematical physics and differential equations