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

Abstract

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 ${\omega }_0\le 0$ (or ${\omega }_0\ge 0$) 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 ${\omega }_0(x)x^{-1}\in L^{\inf }$. On the other hand, for any $\alpha \in (0,1)$, we construct a finite time blowup solution from a class of initial data with ${\omega }_0\in C^{\alpha }\cap C^{\infty }(S^1\setminus \{0\})\cap X$. Our results imply that singularities developed in the DG model and the gCLM model on $S^1$ can be prevented by stronger advection.