The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution ω0\documentclass[12pt]{minimal}
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\begin{document}$$\omega _0$$\end{document}, predicting the long time behavior which can persist is an issue of fundamental importance. In the infinite time limit, some irreversible mixing of ω0\documentclass[12pt]{minimal}
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\begin{document}$$\omega _0$$\end{document} can occur. Since kinetic energy E\documentclass[12pt]{minimal}
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\begin{document}$${\textsf{E}}$$\end{document} is conserved, not all the mixed states are relevant and it is natural to consider only the ones with energy E0\documentclass[12pt]{minimal}
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\begin{document}$${\textsf{E}}_0$$\end{document} corresponding to ω0\documentclass[12pt]{minimal}
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\begin{document}$$\omega _0$$\end{document}. The set of said vorticity fields, denoted by Oω0¯∗∩{E=E0}\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {O}}_{\omega _0}}^*\cap \{ {{\textsf{E}}}= {\mathsf E}_0\}$$\end{document}, contains all the possible end states of the fluid motion. A. Shnirelman introduced the concept of maximally mixed states (any further mixing would necessarily change their energy), and proved they are perfect fluid equilibria. We offer a new perspective on this theory by showing that any minimizer of any strictly convex Casimir in Oω0¯∗∩{E=E0}\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {O}}_{\omega _0}}^*\cap \{ {{\textsf{E}}}= {{\textsf{E}}}_0\}$$\end{document} is maximally mixed, as well as discuss its relation to classical statistical hydrodynamics theories. Thus, (weak) convergence to equilibrium cannot be excluded solely on the grounds of vorticity transport and conservation of kinetic energy. On the other hand, on domains with symmetry (for example straight channel or annulus), we exploit all the conserved quantities and the characterizations of Oω0¯∗∩{E=E0}\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathcal {O}}_{\omega _0}}^*\cap \{ {{\textsf{E}}}= {\mathsf E}_0\}$$\end{document} to give examples of open sets of initial data which can be arbitrarily close to any shear or radial flow in L1\documentclass[12pt]{minimal}
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\begin{document}$$L^1$$\end{document} of vorticity but do not weakly converge to them in the long time limit.
