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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 \(x\).

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)
Full Text: DOI
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