  We study the regularity of the De Gregorio (DG) model $\omega_t + u\omega_x =
u_x \omega$ on $S^1$ for initial data $\omega_0$ with period $\pi$ and in class
$X$: $\omega_0$ is odd and $\omega_0 \leq 0 $ (or $\omega_0 \geq 0$) on
$[0,\pi/2]$. These sign and symmetry properties are the same as those of the
smooth initial data that lead to singularity formation of the De Gregorio model
on $\mathbb{R}$ or the generalized Constantin-Lax-Majda (gCLM) model on
$\mathbb{R}$ or $S^1$ with a positive parameter. Thus, to establish global
regularity of the DG model for general smooth initial data, which is a
conjecture on the DG model, an important step is to rule out potential finite
time blowup from smooth initial data in $X$. We accomplish this by establishing
a one-point blowup criterion and proving global well-posedness for initial data
$ \omega_0 \in H^1 \cap X$ with $\omega_0(x) x^{-1} \in L^{\infty}$. On the
other hand, for any $ \alpha \in (0,1)$, we construct a finite time blowup
solution from a class of initial data with $\omega_0 \in C^{\alpha} \cap
C^{\infty}(S^1 \backslash \{0\}) \cap X$. Our results imply that singularities
developed in the DG model and the gCLM model on $S^1$ can be prevented by
stronger advection.
