Heriot-Watt Mathematics Report Series
HWM04-19, 13 Sep 2004
A family of balance relations for the two-dimensional Navier-Stokes equations with random forcing
S B Kuksin and O Penrose
Abstract
For the 2D 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
vorticity field $\omega(t,\bx)$ satisfies $$
\E\big(g(\omega(t,\bx))|\nabla\omega(t,\bx)|^2\big){=}
\half M_1\E (g(\omega(t,\bx))),
$$ where $M_1$ is a number,
\sps independent of $g$, which measures
the strength of the random
forcing.
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