On the Slightly Perturbed De Gregorio Model on S^1

It is conjectured that the generalization of the Constantin–Lax–Majda model (gCLM) ω _t + a uω _x = u_x ω , 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 || ω (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.