  We present a novel method of analysis and prove finite time asymptotically
self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth
initial data on the real line with compact support. We also prove self-similar
blowup results for the generalized De Gregorio model \cite{OSW08} for the
entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous
initial data with compact support. Our strategy is to reformulate the problem
of proving finite time asymptotically self-similar singularity into the problem
of establishing the nonlinear stability of an approximate self-similar profile
with a small residual error using the dynamic rescaling equation. We use the
energy method with appropriate singular weight functions to extract the damping
effect from the linearized operator around the approximate self-similar profile
and take into account cancellation among various nonlocal terms to establish
stability analysis. We remark that our analysis does not rule out the
possibility that the original De Gregorio model is well posed for smooth
initial data on a circle. The method of analysis presented in this paper
provides a promising new framework to analyze finite time singularity of
nonlinear nonlocal systems of partial differential equations.
