Skip to main content
Log in

On Distribution of Energy and Vorticity for Solutions of 2D Navier-Stokes Equation with Small Viscosity

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study distributions of some functionals of space-periodic solutions for the randomly perturbed 2D Navier-Stokes equation, and of their limits when the viscosity goes to zero. The results obtained give explicit information on distribution of the velocity field of space-periodic turbulent 2D flows.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold V.: Mathematical Methods in Classical Mechanics. 2nd ed., Springer-Verlag, Berlin (1989)

    Google Scholar 

  2. Freidlin M., Wentzell A.: Random Perturbations of Dynamical Systems. 2nd ed., Springer-Verlag, New York (1998)

    MATH  Google Scholar 

  3. Freidlin M.I., Wentzell A.D.: Averaging principle for stochastic perturbations of multifrequency systems. Stochastics and Dynamics 3, 393–408 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hairer M., Mattingly J.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164(3), 993–1032 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kuksin S.B., Piatnitski A.L.: Khasminskii - Whitham averaging for randomly perturbed KdV equation. J. Math. Pur. Appl. 89, 400–428 (2008)

    MATH  MathSciNet  Google Scholar 

  6. Krylov N.V.: Estimates of the maximum of the solution of a parabolic equation and estimates of the distribution of a semimartingale. Math. USSR Sbornik 58, 207–221 (1987)

    Article  MATH  Google Scholar 

  7. Kuksin S.B., Shirikyan A.: Randomly forced CGL equation: stationary measures and the inviscid limit. J. Phys. A: Math. Gen. 37, 1–18 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  8. Kuksin, S.B.: Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions. Zürich: Eur. Math. Soc. 2006, also available at http://www.ma.utexas.edu/ mp_arc 06-178

  9. Kuksin S.B.: Remarks on the balance relations for the two-dimensional Navier–Stokes equation with random forcing. J. Stat. Physics 122, 101–114 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Kuksin S.B.: Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model. Proc. Stelov Inst. Math. 259, 128–136 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sergei B. Kuksin.

Additional information

Communicated by G. Gallavotti

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuksin, S.B. On Distribution of Energy and Vorticity for Solutions of 2D Navier-Stokes Equation with Small Viscosity. Commun. Math. Phys. 284, 407–424 (2008). https://doi.org/10.1007/s00220-008-0577-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0577-3

Keywords

Navigation