Uniqueness of the generators of the 2D Euler and Navier–Stokes flows

A uniqueness result is proven for the infinitesimal generator associated with the 2D Euler flow with periodic boundary conditions in the space L 2 ( μ ) with respect to the natural Gibbs measure μ given by the enstrophy. This result remains true for the generator of the stochastic process associated with a 2D Navier–Stokes equation perturbed by a space–time Gaussian white noise force. The corresponding Liouville operator N defined on the space C b , cyl 1 of smooth cylinder bounded functions has a unique skew-adjoint m -dissipative extension in the class of closed operators in L 2 ( μ ) × V ′ where V = D ( N ¯ ) .