  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 $H^1$. A similar result on the strong
attraction holds in the spaces $H^1\cap\{u:\ \|\mathrm{curl}
u\|_{L^p}<\infty\}$ for $p\ge2$.
