Higher-dimensional Euler fluids and Hasimoto transform: counterexamples and generalizations

The binormal (or vortex filament) equation provides the localized induction approximation of 3D incompressible Euler equation. We present explicit solutions in higher-dimensions that collapse finite time. local nature this phenomenon suggests a possibility singularity appearance nearby blob 5D and higher. Furthermore, Hasimoto transform takes to NLS barotropic fluid equations. show higher dimensions existence such would imply conservation Willmore energy skew-mean-curvature flows counterexamples for membranes based on products spheres. These (counter)examples there is no straightforward generalization 1D transform. derive its replacement, evolution equations mean curvature torsion form membranes, thus generalizing Da Rios