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Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions
Sergei B. Kuksin, Heriot-Watt University, Edinburgh, Scotland
A publication of the European Mathematical Society.
cover
Zurich Lectures in Advanced Mathematics
2006; 104 pp; softcover
Volume: 3
ISBN-10: 3-03719-021-3
ISBN-13: 978-3-03719-021-0
List Price: US$34
Member Price: US$27.20
Order Code: EMSZLEC/3
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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

Table of Contents

  • Function spaces
  • The deterministic 2D Navier-Stokes Equation
  • Random kick-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
  • Comments
  • Bibliography
  • Index
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