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

Abstract

We study a generalization due to De Gregorio and Wunsch et.al. of the Constantin-Lax-Majda equation (gCLM) on the real line \[ ω_t + a u ω_x = u_x ω- νΛ^γ ω, \quad u_x = H ω, \] 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 $a\le -1$, we prove global well-posedness for gCLM with critical and supercritical dissipation.