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 ω_t + auω_x = u_xω−νΛ^γω, u_x = Hω, where H is the Hilbert transform and Λ=(−∂_(xx))^(1/2) . We use the method in Chen J et al (2019 (arXiv:1905.06387)) to prove finite time self-similar blowup for a close to 1/2 and γ=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 ≤ -1, we prove global well-posedness for gCLM with critical and supercritical dissipation