Inviscid limit for damped and driven incompressible Navier-Stokes   equations in ${{\mathbb R}^2}$
  We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in ${\mathbb R}^2$. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance. 
P. Constantin and F. Ramos