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 L2(μ)L2(μ) with respect to the natural Gibbs measure μμ 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 NN defined on the space Cb,cyl1 of smooth cylinder bounded functions has a unique skew-adjoint mm-dissipative extension in the class of closed operators in L2(μ)×V′L2(μ)×V′ where V=D(N¯).