Vanishing viscosity limit for global attractors for the damped Navier-Stokes system with stress free boundary conditions

We consider the damped and driven Navier-Stokes system with stress free boundary conditions and the damped Euler system in a bounded domain Ω⊂R2. We show that the damped Euler system has a (strong) global attractor in H1(Ω). We also show that in the vanishing viscosity limit the global attractors of the Navier-Stokes system converge in the non-symmetric Hausdorff distance in H1(Ω) to the strong global attractor of the limiting damped Euler system (whose solutions are not necessarily unique).