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

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.