  We study a 1D model for the 3D incompressible Euler equations in axisymmetric
geometries, which can be viewed as a local approximation to the Euler equations
near the solid boundary of a cylindrical domain. We prove the local
well-posedness of the model in spaces of zero-mean functions, and study the
potential formation of a finite-time singularity under certain convexity
conditions for the velocity field. It is hoped that the results obtained on the
1D model will be useful in the analysis of the full 3D problem, whose loss of
regularity in finite time has been observed in a recent numerical study (Luo
and Hou, 2013).
