Dense orbits for scale-invariant rotationally symmetric solutions of the 2D Euler equations
Abstract
Let m ≥ 4. We prove that there is a forward orbit of the 2D Euler system for scale-invariant m-fold symmetric vorticities, namely ∂tg + 2G∂θg = 0, 4G + ∂θθG = g, which is dense in S :={g ∈ Lm,odd∞(𝕊1): gsin(mθ) ≥ 0 and |g|L∞ ≤ 1} equipped with the topology of weak* convergence on L∞(𝕊1), where Lm,odd∞(𝕊1) denotes the space of odd and m-fold symmetric functions in L∞(𝕊1). 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.
Related
- On the non-uniqueness of solutions of the axi-symmetric swirl-free Navier-Stokes equations, I
- On Rotationally Symmetric Norms
- Seifert type magnetohydrostatic equilibria in rotationally symmetric toroidal domains
- Generic solutions to symmetric linear equations
- Global small data radial symmetric solutions of 3D semilinear Euler-Poisson-Darboux equations
BibTeX
@misc{suleiman2026dense,
title = {{Dense orbits for scale-invariant rotationally symmetric solutions of the 2D Euler equations}},
author = {Ibrahim Suleiman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20674},
eprint = {2609.20674},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20674},
}
RIS
TY - UNPB
TI - Dense orbits for scale-invariant rotationally symmetric solutions of the 2D Euler equations
AU - Ibrahim Suleiman
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20674
UR - https://arxiv.org/abs/2609.20674
AB - Let m ≥ 4. We prove that there is a forward orbit of the 2D Euler system for scale-invariant m-fold symmetric vorticities, namely ∂_tg + 2G∂_θg = 0, 4G + ∂_θθG = g, which is dense in S :={g ∈ L_m,odd^∞(𝕊^1): gsin(mθ) ≥ 0 and |g|_L^∞ ≤ 1} equipped with the topology of weak^* convergence on L^∞(𝕊^1), where L_m,odd^∞(𝕊^1) denotes the space of odd and m-fold symmetric functions in L^∞(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.
ER -