On the instability of De Gregorio excited states

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