  We consider the damped and driven Navier--Stokes system with stress free
boundary conditions and the damped Euler system in a bounded domain
$\Omega\subset\mathbf{R}^2$. We show that the damped Euler system has a
(strong) global attractor in~$H^1(\Omega)$. 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 $H^1(\Omega)$ to the the strong global
attractor of the limiting damped Euler system (whose solutions are not
necessarily unique).
