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

TL;DR

This paper proves finite time blowup for a generalized Constantin-Lax-Majda model on a circle when parameter a is less than 1, and global existence with decay when a slightly exceeds 1, establishing a critical transition at a=1.

Contribution

It rigorously demonstrates the transition between finite time blowup and global existence for the gCLM model near the critical parameter a=1 on a circle.

Findings

Finite time asymptotically self-similar blowup for a<1.

Global existence with decay rate O(t^{-1}) for a>1.

Critical threshold at a=1 separating different behaviors.

Abstract

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) $\omega_t + a u\omega_x = u_x \omega$ 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.