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In this paper, we will discuss the axisymetric, swirl-free Euler equation infour and higher dimensions. We will show that in four and higher dimensions theaxisymetric, swirl-free Euler equation has properties which could allowfinite-time singularity formation of a form that is excluded in threedimensions. We will also consider a model equation that is obtained by takingthe infinite-dimensional limit of the vorticity equation in this setup. Thismodel exhibits finite-time blowup of a Burgers shock type. The blowup resultfor the infinite dimensional model equation heavily suggests that smoothsolutions of the Euler equation exhibit finite-time blowup in sufficiently highdimensions.

On the regularity of axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions