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The question of global existence versus finite-time singularity formation isconsidered for the generalized Constantin-Lax-Majda equation with dissipation$-\Lambda^\sigma$, where $\widehat {{\Lambda}^\sigma}=|k|^\sigma$, both for theproblem on the circle $x \in [-\pi,\pi]$ and the real line. In the periodicgeometry, two complementary approaches are used to prove global-in-timeexistence of solutions for $\sigma \geq 1$ and all real values of an advectionparameter $a$ when the data is small. We also derive new analytical solutionsin both geometries when $a=0$, and on the real line when $a=1/2$, for variousvalues of $ \sigma$. These solutions exhibit self-similar finite-timesingularity formation, and the similarity exponents and conditions forsingularity formation are fully characterized. We revisit an analyticalsolution on the real line due to Schochet for $a=0$ and $\sigma=2$, andreinterpret it terms of self-similar finite-time collapse. The analyticalsolutions on the real line allow finite-time singularity formation forarbitrarily small data, even for values of $\sigma$ that are greater than orequal to one, thereby illustrating a critical difference between the problemson the real line and the circle. The analysis is complemented by accuratenumerical simulations, which are able to track the formation and motionsingularities in the complex plane. The computations validate and extend theanalytical theory.

Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains