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On the instability of De Gregorio excited states
Abstract
We prove that all excited states −sin(nx), n≥2, of the De Gregorio equation on the torus are nonlinearly unstable in the sense of Hadamard. More precisely, perturbations arbitrarily small in H2 leave a fixed H1 neighborhood of the state. For every n≥2, the linearized operator has a real eigenvalue λn∈(0.56,0.6). To prove this, we transform the eigenvalue problem into a Fuchsian equation on the complex disk and obtain a matching condition for the eigenvalue. This condition is verified rigorously using interval arithmetic. The nonlinear instability follows from a bootstrap argument in the spirit of Guo, Hallstrom, and Spirn.
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BibTeX
@misc{pascualcaballo2026instability,
title = {{On the instability of De Gregorio excited states}},
author = {Miguel M. G. Pascual-Caballo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03287},
eprint = {2610.03287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03287},
}
RIS
TY - UNPB
TI - On the instability of De Gregorio excited states
AU - Miguel M. G. Pascual-Caballo
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03287
UR - https://arxiv.org/abs/2610.03287
AB - We prove that all excited states −sin(nx), n≥2, of the De Gregorio equation on the torus are nonlinearly unstable in the sense of Hadamard. More precisely, perturbations arbitrarily small in H^2 leave a fixed H^1 neighborhood of the state. For every n≥2, the linearized operator has a real eigenvalue λ_n∈(0.56,0.6). To prove this, we transform the eigenvalue problem into a Fuchsian equation on the complex disk and obtain a matching condition for the eigenvalue. This condition is verified rigorously using interval arithmetic. The nonlinear instability follows from a bootstrap argument in the spirit of Guo, Hallstrom, and Spirn.
ER -