Uniqueness results for the generators of the two - dimensional Euler and Navier – Stokes flows . The case of Gaussian invariant measures

The Euler and Navier–Stokes equations for an incompressible fluid in two dimensions with periodic boundary conditions are considered. Concerning the Euler equation, previous works analyzed the associated (first order) Liouville operator L as a symmetric linear operator in a Hilbert space L2( ) with respect to a natural invariant Gaussian measure (given by the enstrophy), with domain the subspace of cylinder smooth bounded functions and have shown that there exist self-adjoint extensions of L. For the Navier–Stokes equation with a suitable white noise forcing term, the associated (second order) Kolmogorov operator K has been considered on the same domain as the sum of the Liouville operator L with the Ornstein–Uhlenbeck operator Q corresponding to the Stokes operator and the forcing term; existence of a C0 semigroup of contraction in L2( ) with generator extending the operator K has been proven. In this paper it is proven that both L and K are bounded by naturally associated positive Schrödinger-like operators, which are essentially self-adjoint on a dense subspace of cylinder functions. Other uniqueness results concerning L respectively K are also given.