On the Slightly Perturbed De Gregorio Model on S¹

Abstract

It is conjectured that the generalization of the Constantin–Lax–Majda model (gCLM) ω_t + auω_x = u_xω, 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)||_(H¹) 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.

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