Singularity formation and global well-posedness for the generalized Constantin-Lax-Majda equation with dissipation. (English) Zbl 1481.35320
Summary: We study a generalization due to S. De Gregorio [Math. Methods Appl. Sci. 19, No. 15, 1233–1255 (1996; Zbl 0860.35101)] and M. Wunsch [Commun. Math. Sci. 9, No. 3, 929–936 (2011; Zbl 1272.35049)] of the Constantin-Lax-Majda equation (gCLM) on the real line \(\omega_t+au\omega_x=u_x\omega-\nu\Lambda^\gamma\omega\), \(u_x=H\omega\) where \(H\) is the Hilbert transform and \(\Lambda=(-\partial_{xx})^{1/2}\). We use the method in [the author et al., Commun. Pure Appl. Math. 74, No. 6, 1282–1350 (2021; Zbl 1469.35053)] 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.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35C06 | Self-similar solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B44 | Blow-up in context of PDEs |
| 35B35 | Stability in context of PDEs |
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