Singularity Formation and Global Well-Posedness for the Generalized Constantin-Lax-Majda Equation with Dissipation

We study a generalization due to De Gregorio and Wunsch et.al. of the Constantin-Lax-Majda equation (gCLM) on the real line $${\omega }_t+au{\omega }_x=u_x\omega -\nu {\mathrm{\Lambda }}^{\gamma }\omega ,u_x=H\omega ,$$ where $H$ is the Hilbert transform and $\mathrm{\Lambda }=(-{\mathrm{\partial }}_{xx}{)}^{1\mathrm{/}2}$. 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 $a\le -1$, we prove global well-posedness for gCLM with critical and supercritical dissipation.