On the regularity of the De Gregorio model for the 3D Euler equations

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.