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

We study the regularity of the De Gregorio (DG) model ωt + uωx = uxω on S for initial data ω0 with period π and in class X: ω0 is odd and ω0 ≤ 0 (or ω0 ≥ 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 R or the generalized ConstantinLax-Majda (gCLM) model on R or S 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 C initial data with any α ∈ (0, 1). On the other hand, for any α ∈ (0, 1), we construct a finite time blowup solution from a class of initial data with ω0 ∈ C∩C(S\{0})∩X. Our results imply that singularities developed in the DG model and the gCLM model on S can be prevented by stronger advection.