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

Abstract

We study the regularity of the De Gregorio (DG) model $ω_t + uω_x = u_x ω$ on $S^1$ for initial data $ω_0$ with period $π$ and in class $X$: $ω_0$ is odd and $ω_0 \leq 0 $ (or $ω_0 \geq 0$) on $[0,π/2]$. 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 $ ω_0 \in H^1 \cap X$ with $ω_0(x) x^{-1} \in L^{\infty}$. On the other hand, for any $ α\in (0,1)$, we construct a finite time blowup solution from a class of initial data with $ω_0 \in C^α \cap C^{\infty}(S^1 \backslash \{0\}) \cap X$. Our results imply that singularities developed in the DG model and the gCLM model on $S^1$ can be prevented by stronger advection.