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([mu]) with respect to the natural Gibbs measure [mu] 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 of smooth cylinder bounded functions has a unique skew-adjoint m-dissipative extension in the class of closed operators in L2([mu])xV' where .