Invariant measures of the 2D Euler and Navier-Stokes equations

In any dynamical systems, the knowledge of invariant measures is extremely useful. For instance in a turbulence problem, it gives a solution to the hierarchy of equations that describes the moments of the velocity field. I will explain how it is possible to explicitly built sets of invariant measures that generalize statistical equilibrium measures for the two dimensional Euler and the Vlasov equations. I will describe new mathematical results about the asymptotic stability and explicit predictions of the large time asymptotics for the two dimensional Euler equation. These results are one of the few examples in statistical mechanics where the apparent paradox between microscopic reversibility and macroscopic irreversibility can be understood and analyzed thoroughly theoretically. Applications to model Ocean vortices and jets in the related Quasi-Geostrophic (Charney-Hasegawa-Mima) model will be briefly outlined. For the two-dimensional Navier-Stokes equations with weak stochastic forcing and dissipation, the existence of an invariant measure has been mathematically proved recently, together with mixing and ergodic properties. I will sketch how to use the invariant measures of the two dimensional Euler equations to describe self-consistently the invariant measures for the two dimensional Navier-Stokes equations. We predict for instance non-equilibrium phase transitions, and observe them in numerical experiments.