Journal of Statistical Physics

, Volume 118, Issue 3–4, pp 437–449 | Cite as

A Family of Balance Relations for the Two-Dimensional Navier--Stokes Equations with Random Forcing



For the 2D Navier--Stokes equation perturbed by a random force of a suitable kind we show that, if g(F) is an arbitrary real continuous function with (at most) polynomial growth, then the stationary in time vorticity field ω(t,x) satisfies \(\mathbb {E} (g(\omega(t,x))|\nabla\omega(t,x)|^2)= \frac{1}{2} M_1\mathbb {E} (g(\omega(t,x)))\) where M_1 is a number, independent of g, which measures the strength of the random forcing. Another way of stating this result is that, in the unique stationary measure of this system, the random variables g(ω(t,x) and |ω(t,x)|2 are uncorrelated for each t and each x.


Two-dimensional Navier--Stokes equation stationary measures two-dimensional turbulence vorticity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Kuksin, S. B., Shirikyan, A. 2000Stochastic dissipative PDE’s and Gibbs measuresComm. Math. Phys.213291330Google Scholar
  2. Kuksin, S. B., Shirikyan, A. 2001A coupling approach to randomly forced nonlinear PDE’s.I Comm. Math. Phys.221351366Google Scholar
  3. J. Bricmont, A. Kupiainen and R. Lefevere, Exponential mixing for the 2D stochastic Navier–Stokes dynamics, Comm. Math. Phys. 230(1):87–132 (2002).Google Scholar
  4. Mattingly, W. E. J. C., Sinai, Ya. G. 2001Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equationComm. Math. Phys.22483106Google Scholar
  5. J. Bricmont, Ergodicity and mixing for stohastic partial differential equations, in Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. 1 (Higher Ed. Press, Beijing, 2002), pp. 567–585.Google Scholar
  6. Kuksin, S. B. 2002Ergodic theorems for 2D statistical hydrodynamicsRev. Math. Phys.14585600Google Scholar
  7. Kuksin, S. B., Shirikyan, A. 2002Coupling approach to white-forced nonlinear PDE’sJ. Math. Pures Appl.81567602Google Scholar
  8. Kuksin, S. B., Shirikyan, A. 2003Some limiting properties of randomly forced 2D Navier–Stokes equationsProc. Roy. Soc. Edinb.133875891Google Scholar
  9. S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Stat. Phys. 115:469–492 (2004).Google Scholar
  10. P. Constantin and C. Foias, Navier–Stokes Equations (University of Chicago Press, Chicago, 1988).Google Scholar
  11. Vishik, M. I., Fursikov, A. V. 1988Mathematical Problems in Statistical HydromechanicsKluwerDordrechtGoogle Scholar
  12. Prato, G. Da, Zabczyk, J. 1992Stochastic Equations in Infinite DimensionsCambridge University PressCambridgeGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Sergei KuksinUK

Personalised recommendations