Energy conditional measures and 2D turbulence

We show that the invariant measures of point vortices, when conditioning the Hamiltonian to a finite interval, converge weakly to the enstrophy measure by conditioning the renormalized energy to the same interval. We also prove the existence of solutions to 2D Euler equations having the energy conditional measure as an invariant measure. Some heuristic discussions and numerical simulations are presented in Sec. VI.