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

Abstract

In this paper, we consider axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions. We show that in dimension $d\ge 4$, axisymmetric, swirl-free solutions of the Euler equation have properties which could allow finite-time singularity formation of a form that is excluded when $d=3$, and we prove a conditional blowup result for axisymmetric, swirl-free solutions of the Euler equation in dimension $d\ge 4$. The condition which must be imposed on the solution in order to imply blowup becomes weaker as $d\to +\infty$, suggesting the dynamics are becoming much more singular as the dimension increases.