Abstract
We use the balance relations for the stationary in time solutions of the randomly forced 2D Navier-Stokes equations, found in [10], to study these solutions further. We show that the vorticity ξ(t,x) of a stationary solution has a finite exponential moment, and that for any \(a\in{\mathbb R},\,t\geq 0\) the expectation of the integral of \(|\nabla_x\xi|\) over the level-set \(\{x\mid\xi(t,x)=a\}\), up to a constant factor equals the expectation of the integral of \(|\nabla_x\xi|^{-1}\) over the same set.
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References
J. Bricmont, A. Kupiainen and R. Lefevere, Exponential mixing for the 2D stochastic Navier–Stokes dynamics, Comm. Math. Phys.230 no. 1, 87–132 (2002).
P. Constantin and Ch. Doering, Heat transfer in convective turbulence, Nonlinearity9, 1049–1060 (1996).
P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago (1988).
P. Constantin, Navier–Stokes equations and area of interfaces, Comm. Math. Phys. 129, 241–266 (1990).
P. Constantin and I. Procaccia, The geometry of turbulent advection: sharp estimates for the dimension of level sets, Nonlinearity 7, 1045–1054 (1994).
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge (1992).
W. E, J. C. Mattingly and Ya. G. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation, Comm. Math. Phys. 224, 83–106 (2001).
F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEA 1, 403–423 (1994).
M. Hairer and J. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Preprint (2004).
S. B. Kuksin and O. Penrose, A family of balance relations for the two-dimensional Navier–stokes eqautions with random forcing, J. Stat. Physics 118, 437–449 (2005).
S. B. Kuksin and A. Shirikyan, Stochastic dissipative PDE's and Gibbs measures, Comm. Math. Phys. 213, 291–330 (2000).
S. B. Kuksin and A. Shirikyan, Coupling approach to white-forced nonlinear PDE's, J. Math. Pures Appl. 81, 567–602 (2002).
S. B. Kuksin and A. Shirikyan, Some limiting properties of randomly forced 2D Navier–Stokes equations, Proceedings A ofthe Royal Society of Edinburgh 133, 875–891 (2003).
S. B. Kuksin, Ergodic theorems for 2D statistical hydrodynamics, Rev. Math. Phys. 14, 585–600 (2002).
S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Stat. Physics 115, 469–492 (2004).
A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's, Ergodic Theory and Related Fields, to appear (2005).
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970).
M. I. Vishik and A. V. Fursikov, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht (1988).
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Kuksin, S.B. Remarks on the Balance Relations for the Two-Dimensional Navier–Stokes Equation with Random Forcing. J Stat Phys 122, 101–114 (2006). https://doi.org/10.1007/s10955-005-8084-9
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DOI: https://doi.org/10.1007/s10955-005-8084-9