Weak solutions of the incompressible Euler equations which are weak limits of vanishing viscosity Navier-Stokes solutions inherit, in two dimensions, conservation properties which are not available for general weak solutions. Research has focused on the behavior of energy, enstrophy and, more generally, the distribution function of vorticity, always in fluid domains with no boundary, with and without forcing. In this talk I will report on recent work in this direction.[-]