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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 $p\ge2$.

discrete and continuous dynamical systems. series b

Strong trajectory and global &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document} $\mathbf{W^{1,p}}$ \end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-attractors for the damped-driven Euler system in &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document} $\mathbb R^2$ \end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;