Abstract
We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space H1. A similar result on the strong attraction holds in the spaces H1 ∩ {u : k curl ukLp < ∞} for p ≥ 2.