On the instability of De Gregorio excited states

Miguel M. G. Pascual-Caballo

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arXiv · 2610.03287

On the instability of De Gregorio excited states

Miguel M. G. Pascual-Caballo

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Abstract

We prove that all excited states $-\sin(nx)$, $n\geq2$, 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 $n\geq2$, the linearized operator has a real eigenvalue $λ_n\in(0.56,0.6)$. 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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Miguel M. G. Pascual-Caballo. 2026-10-02. On the instability of De Gregorio excited states. https://arxiv.org/abs/2610.03287

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