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We study the long-time behavior of scale-invariant solutions of the 2d Eulerequation satisfying a discrete symmetry. We show that all scale-invariantsolutions with bounded variation on $\mathbb{S}^1$ relax to states that arepiece-wise constant with finitely many jumps.  All continuous scale-invariant solutions become singular and homogenize ininfinite time. On $\mathbb{R}^2$, this corresponds to generic infinite-timespiral and cusp formation. The main tool in our analysis is the discovery of amonotone quantity that measures the number of particles that are moving awayfrom the origin.  This monotonicity also applies locally to solutions of the 2d Euler equationthat are $m$-fold symmetric ($m\geq 4$) and have radial limits at the point ofsymmetry.  Our results are also applicable to the Euler equation on a large class ofsurfaces of revolution (like $\mathbb{S}^2$ and $\mathbb{T}^2$). Our analysisthen gives generic spiraling of trajectories and infinite-time loss ofregularity for globally smooth solutions on any such smooth surface, under adiscrete symmetry.

On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications