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 = ux omega - nu Lambda(gamma)omega, u(x) = H omega where H is the Hilbert transform and Lambda = (-partial derivative(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 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 <= -1, we prove global well-posedness for gCLM with critical and supercritical dissipation.