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Mathematics of Two-Dimensional Turbulence


Part of Cambridge Tracts in Mathematics

  • Date Published: September 2012
  • availability: Available
  • format: Hardback
  • isbn: 9781107022829

£ 69.99

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About the Authors
  • This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier–Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) – proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.

    • Essential mathematical tools for researching hydrodynamics statistics
    • Numerous exercises with hints and solutions
    • General overview of existing results
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    Product details

    • Date Published: September 2012
    • format: Hardback
    • isbn: 9781107022829
    • length: 336 pages
    • dimensions: 235 x 157 x 21 mm
    • weight: 0.6kg
    • contains: 7 b/w illus. 85 exercises
    • availability: Available
  • Table of Contents

    1. Preliminaries
    2. Two-dimensional Navier–Stokes equations
    3. Uniqueness of stationary measure and mixing
    4. Ergodicity and limiting theorems
    5. Inviscid limit
    6. Miscellanies
    7. Appendix
    8. Solutions to some exercises.

  • Authors

    Sergei Kuksin, Ecole Polytechnique, Palaiseau
    Sergei Kuksin is a Professor in the Centre Mathématiques Laurent Schwartz at École Polytechnique in Palaiseau, France.

    Armen Shirikyan, Université de Cergy-Pontoise
    Armen Shirikyan is a professor in the mathematics department at the University of Cergy-Pontoise (UCP), France, and served as the Head of Department from April 2008 to August 2012. He gained his PhD from Moscow State University in 1995 and his Habilitation thesis from the University of Paris-Sud in 2003. His current research interests are related to the ergodic theory for randomly forced equations of mathematical physics and controllability of nonlinear PDEs.

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