The question of nite-time singularity formation for solutions to the gen-
eralized Constantin-Lax-Majda (gCLM) equation is considered. This equa-
tion was introduced by Constantin, Lax and Majda as a simpli ed model for
singularity formation in the 3D incompressible Euler equations. It was later
generalized by Okamoto, Sakajo and Wensch to include an advection term
with parameter a, which allows di erent relative weights for advection and
vortex stretching. There has been intense interest in the gCLM equation,
and it has served as a proving ground for the development of methods to
study singularity formation in the 3D Euler equations. Until recently little
has been known about singularity formation for general values of a in both
the dissipative and nondissipative equations. In this talk, we provide such
information via a combination of analysis, numerical computations, and ex-
act solutions focussing on the dissipative version of the equation. We nd
a signi cant di erence between the problems in the periodic and real-line
geometries.
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