  We study a generalization due to De Gregorio and Wunsch et.al. of the
Constantin-Lax-Majda equation (gCLM) on the real line \[ \omega_t + a u
\omega_x = u_x \omega - \nu \Lambda^{\gamma} \omega, \quad u_x = H \omega , \]
where $H$ is the Hilbert transform and $\Lambda = (-\partial_{xx})^{1/2}$. We
use the method in \cite{chen2019finite} to prove finite time self-similar
blowup for $a$ close to $\frac{1}{2}$ and $\gamma=2$ by establishing nonlinear
stability of an approximate self-similar profile. For $a>-1$, we discuss
several classes of initial data and establish global well-posedness and an
one-point blowup criterion for different initial data. For $a\leq-1$, we prove
global well-posedness for gCLM with critical and supercritical dissipation.
