On the De Gregorio modification of the Constantin-Lax-Majda Model

We study a modification due to De Gregorio of the Constantin-Lax-Majda (CLM) model $\omega_t = \omega H\omega$ on the unit circle. The De Gregorio equation is $\omega_t+u \omega_x-u_x\omega =0, u_x = H\omega.$ In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when $t\to\pm\infty$, in a way resembling inviscid damping. We prove that such a behavior takes place near a manifold of equilibria.