Dense orbits for scale-invariant rotationally symmetric solutions of the 2D Euler equations

Abstract

Let $m \ge 4$. We prove that there is a forward orbit of the 2D Euler system for scale-invariant $m$-fold symmetric vorticities, namely \begin{equation*} \partial_tg + 2G\partial_θg = 0, \quad 4G + \partial_{θθ}G = g, \end{equation*} which is dense in $S :=\{g \in L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1): g\sin(mθ) \ge 0 \text{ and }\|g\|_{L^\infty} \le 1\}$ equipped with the topology of weak$^*$ convergence on $L^\infty(\mathbb{S}^1)$, where $L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1)$ denotes the space of odd and $m$-fold symmetric functions in $L^\infty(\mathbb{S}^1)$. The system above was first derived by Elgindi and Jeong, who showed well-posedness in the $m$-fold symmetric class. In subsequent work by Elgindi, Murray and Said, it was shown that solutions with regulated vorticity relax to piece-wise constant steady states with finitely many jumps; our construction shows that this regularity assumption cannot be relaxed. The proof starts from a simple linear mechanism: by evolving prescribed step functions backwards from chosen times and superimposing the resulting data, one can arrange for a single linear orbit to approximate a countable dense family. We show that this mechanism persists for the nonlinear Euler coupling. The key stability estimates imply that the error introduced at each stage becomes arbitrarily small when the next approximation time is taken sufficiently far in the future, allowing the construction to be iterated indefinitely.