Long-time loss of $L^2$ precompactness and compact-orbit generation

Establish the two long-time conjectured phenomena for the two-dimensional Euler equation: generic loss of $L^2$ precompactness and the generation of compact orbits by weak infinite-time limits.

Background

The paper situates its study within two broader long-time conjectures for two-dimensional Euler flows. These conjectures concern, respectively, the generic failure of $L^2$ precompactness along evolving solutions and the dynamical behavior of weak limits obtained along sequences of times tending to infinity, specifically whether such limits generate compact orbits. The paper addresses a particular scale-invariant, 3-fold-symmetric setting related to these conjectures, but the quoted passage states the broader conjectural phenomena themselves.