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

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Abstract

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.

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References

  • S. B. Kuksin A. Shirikyan (2000) ArticleTitleStochastic dissipative PDE’s and Gibbs measures Comm. Math. Phys. 213 291–330

    Google Scholar 

  • S. B. Kuksin A. Shirikyan (2001) ArticleTitleA coupling approach to randomly forced nonlinear PDE’s. I Comm. Math. Phys. 221 351–366

    Google Scholar 

  • 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 

  • W. E. J. C. Mattingly Ya. G. Sinai (2001) ArticleTitleGibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation Comm. Math. Phys. 224 83–106

    Google Scholar 

  • 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.

  • S. B. Kuksin (2002) ArticleTitleErgodic theorems for 2D statistical hydrodynamics Rev. Math. Phys. 14 585–600

    Google Scholar 

  • S. B. Kuksin A. Shirikyan (2002) ArticleTitleCoupling approach to white-forced nonlinear PDE’s J. Math. Pures Appl. 81 567–602

    Google Scholar 

  • S. B. Kuksin A. Shirikyan (2003) ArticleTitleSome limiting properties of randomly forced 2D Navier–Stokes equations Proc. Roy. Soc. Edinb. 133 875–891

    Google Scholar 

  • S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Stat. Phys. 115:469–492 (2004).

    Google Scholar 

  • P. Constantin and C. Foias, Navier–Stokes Equations (University of Chicago Press, Chicago, 1988).

  • M. I. Vishik A. V. Fursikov (1988) Mathematical Problems in Statistical Hydromechanics Kluwer Dordrecht

    Google Scholar 

  • G. Da Prato J. Zabczyk (1992) Stochastic Equations in Infinite Dimensions Cambridge University Press Cambridge

    Google Scholar 

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Authors and Affiliations

  1. Sergei Kuksin, UK

    Sergei Kuksin & Oliver Penrose

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  1. Sergei Kuksin
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  2. Oliver Penrose
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Correspondence to Sergei Kuksin.

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Kuksin, S., Penrose, O. A Family of Balance Relations for the Two-Dimensional Navier--Stokes Equations with Random Forcing. J Stat Phys 118, 437–449 (2005). https://doi.org/10.1007/s10955-004-8816-2

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  • DOI: https://doi.org/10.1007/s10955-004-8816-2

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