On the Slightly Perturbed De Gregorio Model on $S^1$

Abstract

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) ${\omega }_t+au{\omega }_x=u_x\omega$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from smooth initial data for $a<1$. For the endpoint case where $a$ is close to and less than $1$, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection $a$ is slightly larger than $1$, we prove that the solution exists globally with $||\omega (t)|{|}_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.

19 pages. Added discussions in the introduction