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A family of balance relations for the two-dimensional Navier-Stokes equations with random forcing. (English) Zbl 1064.76027
Summary: For the two-dimensional Navier-Stokes equation perturbed by a random force of a suitable kind, we show that, if $$g(\cdot)$$ is an arbitrary real continuous function with (at most) polynomial growth, then the stationary in time vorticity field $$\omega(t,x)$$ satisfies $\mathbb E\left(g(\omega(t,x))| \nabla\omega(t,x)|^2\right) = \tfrac{1}{2} M_1 \mathbb E(g(\omega(t,x))),$ 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 $$g(\omega(t,x)$$ and $$| \omega(t,x)|^{2}$$ are uncorrelated for each $$t$$ and each $$x$$.

##### MSC:
 76D06 Statistical solutions of Navier-Stokes and related equations 76M35 Stochastic analysis applied to problems in fluid mechanics 76F55 Statistical turbulence modeling 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
##### Keywords:
two-dimensional turbulence; stationary measure
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##### References:
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