L-Uniqueness of regularized 2D-Euler and stochastic Navier-Stokes equations

A regularization of the 2D-Euler equation with periodic boundary conditions is introduced, having the same infinitesimal invariants as the Euler equation. A flow of measure-preserving transformations is constructed on L¹-spaces induced by the Gaussian measure with covariance given by the inverse of the enstrophy and it is shown that this flow is the only measure-preserving flow inducing a strongly continuous semigroup on the corresponding L¹-space. We also prove similar uniqueness results for a corresponding class of regularized stochastic 2D-Navier-Stokes equations.