Remarks on the Balance Relations for the Two-Dimensional Navier–Stokes Equation with Random Forcing


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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 ζt,x) of a stationary solution has a finite exponential moment, and that for any a ∈ ℝ, t ≥ the expectation of the integral of | ∇xζ| over the level-set {x, |ζ(t, x) = a}, up to a constant factor equals the expectation of the integral of |∇xζ|−1 over the same set.

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