2D hydrodynamical systems: invariant measures of Gaussian type

Abstract

Gaussian measures $\mu^{\beta,\nu}$ are associated to some stochastic 2D hydrodynamical systems. They are of Gibbsian type and are constructed by means of some invariant quantities of the system depending on some parameter $\beta$ (related to the 2D nature of the fluid) and the viscosity $\nu$. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure $\mu^{\beta,\nu}$ is invariant for this flow and is unique. Finally, we prove that the deterministic inviscid equation has a $\mu^{\beta,\nu}$-stationary solution (for any $\nu>0$).