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Abstract The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation $-{\mathrm{\Lambda }}^{\sigma }$, where $\widehat{{\mathrm{\Lambda }}^{\sigma }}=|k{|}^{\sigma }$, both for the problem on the circle $x\in [-\pi ,\pi ]$and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\sigma \text{⩾}1$and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when $a=1/2$, for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory. 

Numerically computed with high accuracy are periodic travelling waves at the free surface of a two-dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are the following. (i) There is a boundary layer where the angle rises sharply from $$0^\circ$$ at the crest to a local maximum, which converges to $$30.3787\ldots ^\circ$$ , independently of the vorticity, as the amplitude increases towards that of the extreme wave, which displays a corner at the crest with a $$30^\circ$$ angle. (ii) There is an outer region where the angle descends to $$0^\circ$$ at the trough for negative vorticity, while it rises to a maximum, greater than $$30^\circ$$ , and then falls sharply to $$0^\circ$$ at the trough for large positive vorticity. (iii) There is a transition region where the angle oscillates about $$30^\circ$$ , resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form.