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

TL;DR

This paper explores higher-dimensional vortex filament equations, demonstrating finite-time collapses, and shows that the classical Hasimoto transform cannot be directly extended to higher dimensions, providing new evolution equations for membranes.

Contribution

It presents explicit higher-dimensional solutions collapsing in finite time and introduces generalized evolution equations for membranes, highlighting the limitations of the Hasimoto transform in higher dimensions.

Findings

Finite-time collapse solutions in higher dimensions

Counterexamples to straightforward higher-dimensional Hasimoto transform

Generalized evolution equations for membrane mean curvature and torsion

Abstract

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